The Dynamical Theory of X-Ray Diffraction

The Dynamical Theory of X-Ray Diffraction

R. W. JAMES Physics Department, University of Cape Town,South Africa

I. Introduction.. ...... . . . . . . . . . . . . . . . . 55 61 11. The Geometrical Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction ................................................. 2. The Transform Considered as a Distribution in Reciprocal Space. . . 3. The Essential Problem in Structure Determination.. . . . . . . . . . . . . . . 4. Periodic Structures.. .. .............................. 66 .............................. 68 5. The Reciprocal Lattice 68 6. The Transform of a Crystal Block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7. The Electron Density Expressed as a Fourier Series. . 8. Discussion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 9. Lattice Functions. ..................... 71 111. The Electromagnetic B ............................... 72 10. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 . . . . . . 73 11. The Effective Dielectric Constant of the Crystal.. . 12. The Dielectric Constant and its Fourier Coefficients. . . . . . . . . . . . . . . . . 77 13. The Wave Equation and its Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 14. The Magnetic Field.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 81 15. The Resonance Error.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Dispersion Surface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 16. The Definition of the Surface.. . . . . . . . . . . . 17. The Case of a Single Wave Train. . ............................ 84 85 18. The Solution for Two Effective Rec al-lattice Points (n = 2). . 19. The Nature of the Dispersion Surface for n = 2 . .................... 86 89 20. The Relative Amplitudes and Phases of the Waves when n = 2 . . . 21. The Transition from Two Waves to One.. . . . . . . . . . . . . . . . . . . . . . . 89 22. The Case of Three Waves.. ...... ......................... 90 23. Some Remarks on the Solution for ite Crystal.. . . . . . . . . . . . . . 92 V. The Field in a Bounded Crystal.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 24. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 94 25. The Cases of Transmission and Reflection.. . . . . . . . . . . . . . . . . . . . . . . . . . 26. The Determination of the Wave Points . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . 98 27. Some Necessary Geometrical Results. . 101 28. The Accommodation and the Resonance Errors.. . . . . . . . . . . . . . . . . . . . . 29. The Calculation of the Accommodation g . . . . . . . . . . . . . . . . 102 103 30. A Summary of Formulas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

54

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. W . JAMES

VI The Thick Crystal with Negligible Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 105 31 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 32 . Case I (Transmission): Some Geometrical Results . . . . . . . . . . . . . . . . . . . . 106 33 The Boundary Conditions and Wave Amplitudes . . . . . . . . . . . . . . . . . . . . 108 34 Discussion of the Fields in Case I ...................... 35 Reflection from a Thick Nonabsorbing Crystal . . . . . . . . . . 36 The Boundary Conditions... ............................. 114 37 The Reflection Coefficient ... ............................. 115 38 The Nature of the Wave Field when g is Complex.Primary Extinction . . 116 . . 118 39. The Magnitude of the Primary Extinction . . . . . . . . . . . . . . . . . . . . 40. The Range of Total Reflection and the Deviation from Bragg’s Law . . . . 118 41. The Integrated Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 42 Secondary Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 43 The Nature of the Wave Field in a Thick Nonabsorbing Crystal . . . . . . 124 126 44 The Field on the Atomic Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. The Nature of the Stationary Wave Field in the Crystal . . . . . . . . . . . . . . 129 130 46 The Oscillatory or Pendulum Solution in Case I . . . . . . . . . . . . . . . . . . . . . V I I . The Wave Field in Crystals with Finite Absorption . . . . . . . . . . . . . . . . . . . . . 132 47 Preliminary Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 48 Crystal without Symmetry Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 49. The Relation of the Absorption Coefficient to the Complex Amplitude . . 136 ....................... 137 50 The Absorption of the Wave Fields . . s...................... 138 51. The Calculation of the Absorption Co 52 Absorption in Case I (Transmission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 53 Absorption in Case I1 (Reflection) . . . . . . . . . . . . . . . . . . . . . 140 54 The Relation between the Complex Parameters 55. Discussion of the Absorption in Case I (Transmission). . . . . . . . . . . . . . . . 144 56 The Asymmetry of the Absorption in Case I . . . . . . . . . . . . . . . . . . . . . . . . 145 57. Discussion of. the Absorption in Case I1 (Reflection). . . . . . . . . . . . . . . . . . 148 V I I I The Reflection and Transmission Coefficients for Crystals of Finite Thickness. 153 153 58 The Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59. Case I . Transmission through an Absorbing Crystal . . . . . . . . . . . . . . . . 154 157 60 . Crystal Slice with Negligible Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 . The Fine Structure of R t ( p ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . 160 62 The Reflection Coefficient for a Very Thin Slice . . . 63. Numerical Discussion of t_heFormulas for Absorbing Crystals . . . . . . . . . 160 64. The Variation of Rr and T with the Thickness of the Slice . . . . . . . . . . . . 162 65. Reflection and Transmission Coefficients for a Slice of Finite Thickness 165 in Case I1 (Reflection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 66 Crystal Slice with Negligible Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 67. Discussion of the Formulas for the Nonabsorbing Crystal . . . . . . . . . . . . . 168 68 . Reflection Coefficient for a Thick Absorbing Crystal . . . . . . . . . . . . . . . . . . 171 TX . The Propagation of Energy in the Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . 175 ............................................. 175 69 Introduction . . urrent . . . . . . . . . . . . . . . . . . . . . . 71. Two Relevant Reciprocal-lattice Points . . . . . . . . . . . 72 . The Energy Current-density as a Function of the Angle of Incidence . . . 182 73 Crystals with Negligible Absorption . The Pendulum Solution.......... 183 74 . The Energy Current-densities in the Fields Associated with Individual .. 185 Wave Points . . . . . . . . . . . . . . . . . . . . . . . . . .

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T H E DYNAMICAL THEORY O F X-RAY DIFFRACTION

75. The Relation of the Direction of Energy Flow to the Dispersion Surface. . 76. The Resultant Energy Current-density and the Pendulum Solution. . . . . 77. The Lines of Energy Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. The Effect of Absorption on the Energy Current.. . . . . . . . . . . . . . . . . 79. The Nature of the Energy Flow-lines in an Absorbing Crystal. . . . . . . . . 80. The Periodic Component of the Energy Current in Absorbing Crystals. . 81. The Energy Current in Thick Absorbing Crystals.. . . . . . . . . . . . . . . . . . . X. Some Experimental Consequences of the Dynamical Theory. . . . . . . . . . . . . . 82. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. The Propagation of Beams of Limited Width. . . . . . . . . . . . . . . . . . . . 84. Limited Beams in Absorbing Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85. The Influence of Crystal Distortion on the Energy Propagation. . . . . . . . 86. Experimental Proof of Abnormal Transmission in Perfect Crystals . . . . 87. Transmission of Radiation through Wedge-shaped Crystals and the Experimental Demonstration of the Pendulum Effect. . . . . . . . . . . . . . . . .

55 189 192 194 199 200 202 205 207 207 207 209 211 212 213

1. Introduction

The first discussions of the diffraction of X-rays by crystals treated it as a problem of Fraunhofer diffraction in three dimensions. Radiation once scattered by an atom was supposed to pass o.ut of the crystal without further scattering, and in the crystal itself all waves, primary and scattered, were assumed to travel with the velocity of light in empty space. The resultant scattered amplitude was considered at distances from the crystal very large in comparison with its own dimensions. This procedure gives the directions of the diffracted beams, and sometimes too their intensities, with an accuracy that is adequate for many purposes, but the assumptions upon which it is based are in principle unjustifiable. The intensities of the diffracted beams, although usually considerably smaller than that of the primary beam, are by no means negligible. The rescattered beams must therefore be appreciable, and must combine with the primary beam and with one another, to give waves traveling in the crystal with phase velocities different from that of light in empty space. The simple procedure that neglects multiple scattering can give no real information about the wave field in the crystal itself, nor would one expect it to give correctly the intensities of the beams diffracted from it. The first attempt to give a more complete theory was made by C. G. Darwin1 in a remarkable paper published in 1914, about the time of the outbreak of World War I. Like W. L. Bragg, he considered diffraction by a cryatal as a matter of reflection from equally spaced parallel planes of atoms, but now made allowance for multiple reflections. He showed that, if exact regularity of spacing persists over a large number of successive 1C. G. Darwin,

Phil. Mag. [6] 27, 315, 675 (1914).

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R. W. JAMES

planes, the radiation, both primary and scattered, must traverse the crystal with a velocity different from that of light in empty space, and in general greater by a few parts in 100,000, so that the crystal may be considered to have a refractive index for X-rays slightly less than unity. He found too a slight divergence from Bragg's law of reflection, and that, when radiation falls on the surface of a crystal of negligible absorption, total reflection must occur over a small range of angles, within which the radiation can penetrate only to a very small depth in the crystal. This last effect Darwin called primary extinction. It is associated with the multiple scattering within the crystal, and would occur even if the ordinary photoelectric absorption were entirely negligible. Darwin showed too that the range of total reflection, and therefore the integrated reflection of the corresponding spectrum, is proportional to the amplitude scattered in the direction of the spectrum by a single crystal unit. The elementary theory, on the other hand, gives an integrated reflection proportional to the square of this amplitude, as in any case of Fraunhofer diffraction. Darwin himself saw that a quantitative test of this point was needed, and he and Moseley2 made some experimental estimates of the reflected intensities from rocksalt, which seemed rather to favor the elementary theory, a conclusion confirmed with the same crystal by A. H. Compton8 and by Bragg, James, and Bosanquet4 shortly after World War I. These apparent discrepancies between theory and experiment were later shown, as Darwin indeed suggested, to be related to the state of perfection of the crystal. Many crystals, most natural specimens of rocksalt among them, are so imperfect that a regular array of planes does not persist through a large enough volume for a wave field of the Darwin type to build up. Such crystals, generally called mosaic crystals, a term due to Ewald, can be considered as a collection of nearly parallel, but optically independent regions, to each of which individually the Fraunhofer treatment applies to a fair approximation; and the total intensity is therefore the sum of the independent intensities from these regions, and so proportional to the square of the amplitude scattered from a crystal unit. We may conveniently call the elementary way of treating the problem, which takes into account the positions of the atoms but neglects multiple scattering, the geometrical theory. The theory that is primarily based on the interaction of the scattered waves with each other, and with the crystal lattice, has been known since it was first treated by Ewald as the dynamical theory.

* H. G. J. Moseley and C. G. Darwin, Phil. Mug.[6] 26, 210 (1913).

aYA. @."Compton, Phys. Rev. 9, 29 (1917). 4iW. L. Bragg, R. W. James, and C. H. Bosanquet, (i) Phil. Mug.41, 309 (1921);(ii) 42, 1 (1921);(iii) 44, 433 (1922).

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

57

In determining crystal structures it is in fact usual to consider the mosaic crystal as the norm, and to use the much simpler geometrical theory, for as Professor von Laue once very happily put it, most crystals are so far from perfect that they do not deserve a better theory than the geometrical one. This procedure is on the whole justified by experience, although sometimes corrections for the state of perfection, which are very difficult to make with any certainty, have to be applied. Many crystals, however, do approach closely to perfection over considerable volumes, and their study, which shows features of great interest that verify in a striking way the essential correctness of the dynamical approach to the problem of diffraction, is becoming increasingly important. Because of the war, Darwin’s pioneer work on the dynamical theory did not become at all widely known for a time, and in the meanwhile the problem had been treated independently, and ‘more fundamentally, by Ewald6 in a series of classical papers that have formed the basis of all subsequent work. Ewald’s method of approach differs entirely from Darwin’s, and is based on the consideration of the type of stable field that can exist in a crystal supposed at the outset to be of infinite extent in all directions. The crystal is assumed to be b’uilt up of identical electric dipoles, situated at the points of a perfect space lattice and in a state of oscillation such that the electric moment of the dipole at any lattice point at any time is given by the displacement in a plane vector wave passing through the lattice. Ewald calls a wave of this kind a dipole wave, and it is important to be clear that it is simply a kinematical description of a steady state of oscillation of the dipoles themselves, and not an electromagnetic wave running through the medium in which the dipoles are situated. We may indeed suppose it to have been initiated by such a wave, but it now describes only the steady state of oscillation of the dipoles. Owing to the periodicity of the lattice, a state of oscillation describable in terms of one dipole wave can equally well be described in terms of any one of an infinite number. If KOis the wave vector of one such dipole wave, and r, the vector defining the position of a lattice point p relative to the origin, the state of vibration of the dipole at the point p is given by a function of the type Aexp ( 2 n i ( v t - Ko-rp)).

+

If now the wave vector becomes KO R,, where Rm is any vector of the reciprocal lattice, the corresponding function is

+ Rm).rp]];

A exp [2~i(vt- (KO

but since the scalar product of a crystal-lattice vector and a reciproP. P. Ewald, Ann. Physik 141 49, 1, 117 (1916); 64, 159 (1917).

58

R. W. JAMES

cal-lattice vector is always an integer, these two expressions are equal, for the exponents differ only by an integral number of times 2ri. As a mere matter of description of the state of oscillation of the dipoles, all waves of the set KO R, are equivalent, and may indeed be thought of as coexisting, once a state of oscillation has been chosen. The waves of the set differ of course not only in direction but also in velocity. Oscillating dipoles give rise to electromagnetic waves, which in their turn react on the dipoles. Ewald first investigated the type of wave field that will maintain itself and be consistent with the assumed state of oscillation of the dipoles in an infinite crystal. He showed that any dipole wave must be accompanied by an electromagnetic wave, traveling in the same direction and with the same velocity. We may call such a wave a concurrent wave (Ewald’s gleichlaufende Welle). The field in the crystal in the region between the dipoles may now be obtained by superposing the infinite set of concurrent waves that accompany the infinite set of dipole waves, in terms of any one of which the state of oscillation of the dipoles themselves may be described. If any one wave train, say that of wave vector KO,exists in the dynamically stable field, then in principle waves corresponding to every member of the set KO R, are also present. Ewald shows, however, that the amplitude of a wave is small unless its velocity, and thus that of the corresponding dipole wave, is very nearly equal to c, and so the corresponding value of 1 KO R, I to 1/X, c and X being the velocity and wavelength of a wave of frequency v in free space. When this is so, an effect analogous to resonance occurrj, and the amplitude of the corresponding wave becomes relatively large. Investigation shows that the velocity of a constituent wave of the field can never be exactly equal to c, for its amplitude would then become infinite; but the velocities of some of the waves under suitable conditions may approach c closely. In practice it is often enough to consider only the few strong waves whose velocities are very nearly c, and which therefore stand out from the weak general background of the wave field. These constitute the primary and diffracted waves in the crystal, but as long as we consider a crystal without boundaries there is no justification for thinking of any one of the waves as the primary wave. All are equally parts of a complete wave field, and when stability is attained, the existence of any one of them implies the existence of all the others. Each wave train has its own speed, differing very slightly from c if the amplitude is appreciable, so that we cannot speak of a definite single refractive index of the crystal for the frequency concerned. The directions of the waves of appreciable intensity are very nearly indeed those given by the geometrical theory. We may see this by using the familiar construction involving the Ewald sphere, drawn in reciprocal

+

+

+

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

59

FIGURE I

space. Let KObe the wave vector of one of the waves, which we may for convenience consider as the primary wave, drawn so as to terminate at the origin 0 of the reciprocal lattice. Its magnitude differs very little from 1/X, and it is represented by A 0 in Fig. 1. With center A , we describe the Ewald sphere of radius 1/X, which now passes nearly, but not exactly, through 0. Let M be a reciprocal-lattice point other than the origin, so that OM = R,. Then A M = K, = KO R,, and is the wave vector of a possible constituent wave of the wave field in the crystal; but that wave will have appreciable amplitude only if M lies very close to the sphere, so that the velocity of the corresponding wave is very nearly c. According to the geometrical theory, the condition for the occurrence of a diffracted wave is that the corresponding reciprocal-lattice point should lie exactly on the sphere. On the dynamical theory this is not possible, but, if waves of appreciable strength are to appear, the corresponding points must lie so nearly on the sphere that we may use the ordinary construction to determine which diffracted beams will occur. The main geometrical characteristics of the diffraction are therefore scarcely changed, but in principle the two viewpoints are entirely distinct. In the dynamical theory one considers the whole system, the field composed of the different wave trains and the oscillating dipoles, as a single electrodynamical entity; and if, for convenience, one disregards all but the two or three strongest waves, one must not forget that in principle all are present, as an actual, if virtually negligible, background. The situation is in fact even more complicated; for we shall find that if n waves are to be regarded as appreciable for any given angle of incidence, there are 2n positions of the point A consistent with the dynamical conditions, because of the two independent directions of polarization of the waves, and so 2n vectors A M corresponding to K,, differing slightly both in direction and magnitude. Clearly, too, by slightly altering the direction of AO, we shall not change the number of reciprocal-lattice points that lie near the sphere. As the direction of A 0 vanes, the 2n points A describe a

+

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R. W. JAMES

surface of 2n sheets, which Ewald called the dispersion surface. The first step in the dynamical solution is to determine this surface, for on it depend not only the directions but the possible relative intensities and phases of the waves that can coexist in the wave field. I n this preliminary survey of the theory we have considered an infinite crystal, and here the importance of the dynamical theory is at once apparent, for the geometrical theory cannot be applied to it at all. The important problem of the bounded crystal, in which wave trains enter the crystal at one surface and leave it again at the same surface or at another, requires separate consideration. This problem again was solved by Ewald by considering the conditions that must be satisfied at the surface of entry or exit by the electric and magnetic fields inside and outside the crystal. These conditions have the effect of limiting the possible positions of the point A on the dispersion surface, and in this way Ewald obtained Darwin’s results for the deviations from Bragg’s law and total reflection from nonabsorbing crystals. Neither Darwin nor Ewald considered the effect of finite absorption of the radiation in the crystal. I n 1930 Prinsa modified Darwin’s theory so as to take absorption into account in the case of reflection from perfect crystals, but it was not until 1949 that a detailed treatment of the dynamical theory of absorbing crystals was given by von Laue.’ Because of the difficulty of scientific communication during the war years 1914-1918, the significance of the dynamical theory was only slowly appreciated, and it was not until, in the years immediately following the war, quantitative measurements of the intensities of X-ray spectra began to be made that attention became focused on the importance in this connection of the state of perfection of the crystal. At this point it is desirable to indicate the scope of this article. It is intended primarily as an introduction to the dynamical theory for those wishing to get an idea of its general principles and its more important applications. The approach will be as far as possible physical, and the mathematical treatment will be that necessary for the development of the main lines of the theory. A treatment of this kind cannot be rigid a t every point, but where important simplifying assumptions have been made they have been indicated. So far as possible, the line of argument has been made self-contained. It is not my intention to deal exhaustively with all the recent applications of the theory, but to provide a foundation upon which a more detailed study of the various fields dealt with may be based. In particular, it has not been possible to deal with the dynamical theory of electron diffraction, and the treatment is confined to the diffraction of 6

7

J. A. Prins, Z. Physilc 6S, 477 (1930). M.von Laue, Ada Cryst. 2, 106 (1949).

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

61

X-rays. Many of the fundamental ideas are, however, common to both fields. Experimental work will be discussed only where it illustrates the principles under discussion. The treatment used is essentially that due to von Laue,* who in 1931 extended Ewald’s conception of a crystal composed of dipole oscillators to one in which the scattering matter is continuously distributed. After some consideration I have followed von Laue’s method of introducing the boundary conditions. It is a method familiar in classical electromagnetic theory, although, as von Laue himself points out, its use is not logically justifiable when the electric waves concerned have lengths comparable with the interatomic distances. In this respect Ewald’s method is logically superior. These points are, however, matters of detail. The dynamical theory must always be associated with the name of Ewald, and because of its beauty and ingenuity must be considered as one of the classical achievements of theoretical physics. It has shown itself capable of development and extension, and the work on absorption by perfect crystals carried out by Borrmann and his associates in Berlin has shown in a striking way its power and essential correctness. An excellent account of von Laue’s theory is to be found in his book, Rontgenstrahlinterferenzen,Bto which my own indebtedness will be very plain in what follows. I am grateful to Professor von Laue for his kindness in sending me copies of his most recent papers, a debt which, alas, it is no longer possible to acknowledge personally. To Professor G. Borrmann, Dr. A. R. Lang, and Dr. N. Kato my thanks are also due for reprints of papers not readily obtainable in Cape Town, where this article has been written. Mrs. Noel Taylor very kindly read most of the manuscript, and was instrumental in removing a number of obscurities of expression. I should like particularly to express my thanks to Dr. Aaron mug, who, during a visit to the Cape Town Laboratory, read all the manuscript except the last section in detail, and made a number of valuable suggestions for the clarification of the argument at certain points. II. The Geometrical Theory

1. INTRODUCTION

In order to introduce certain necessary ideas, and to make the treatment so far as possible self-contained, a brief account of the geometrical theory will first be given. We consider a distribution of scattering matter, not necessarily crystalline, confined within a region having linear dimen0

M. von Laue, Ergeb. Exact. Natunu. 10, 133 (1931). M. von Laue, “Rontgenstrahlinterferenzen.” Akad. Verlagsges. Leipzig, 1948.

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R. W. JAMES

sions small in comparison with the distance of any part of it from the point of observation. Through this region there sweeps a plane monochromatic wave of frequency v and wave vector ko = so& where sois the unit vector in the direction of the wave normal and X the wavelength of radiation of frequency v in free space. This primary wave we suppose to be scattered by the material of the distribution, but the secondary radiation so produced at any point is supposed to travel through the distribution and out to the point of observation without any further scattering, and with the same wavelength as the primary radiation. These are the essential assumptions of the geometrical theory.

FIGURE 2

In Fig. 2, let 0 be a suitable origin, chosen in the distribution, and P a point at which scattering occurs, at a vector distance r from 0. Q is a point of observation at a large distance D from 0, and the direction OQ of D is defined by the unit vector s. I D I is so large that at Q the waves scattered from any part of the distribution may be considered as plane, with wave normals all defined by the vector s and the wave vector k = s/X. The path difference on arrival at Q between waves scattered from P and 0 is equal to the difference between the projections of r onto the directions of s and so. If X is the wavelength, here assumed to be that of waves of frequency v in free space, the phase advance at Q of the waves from P over those from 0 is where

27r(s - so) .r/X = 2 ~ ( k -

R

=

ko).r/X

=

27rR.r

k - ko.

The wave vectors k and ko each have magnitude l / X velocity of light in empty space.

(1.1) (1.2)

=

v/c, c being the

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

63

The actual phase of a wave scattered in any direction is to be understood as referred to that of a wave scattered in the same direction, and under the same conditions, by a classical Thomson electron, supposed situated at the origin 0. The complex amplitude of the electric field due to such a wave at the point Q at the time t would be

- (ez//mc") (1/ 1 D I )E[k] exp {27riv(t - 1 D 1 / c ) j

(1.3)

exp (27rivt) being the component of the incident electric field perpendicular to the direction of scattering, and so to the wave vector k, at the time t. Let the density of the scattering matter a t P be p(r), so defined that an element of volume dr at P would scatter an amplitude p(r)dr times that scattered by a Thomson electron at the same point, under the same conditions. If the act of scattering is accompanied by a change of phase, p (r) itself will be complex. The element dr lies at a distance J D - r 1 from Q, but I r 1 is always small in comparison with I D , and, as far as the falling away of amplitude is concerned, we may suppose all elements of the distribution to lie at the same distance D I from Q, and may write for the resultant complex amplitude a t Q

Elk,

I

I

E

=

- ( e2/mZ)(11 I D I )E[kl

I

X exp { 2 k v ( t - D I / c )

1

1

p(r) exp (2mR.r) dr

(1.4)

where the integration extends over the whole distribution. As far as the distant point Q is concerned, the distribution therefore in effect scatters spherical waves radiating from 0; but their amplitude depends upon the nature and arrangement of the scattering matter within the distribution in a way that is determined by the integral in Eq. (1.4). Since the density p(r) is assumed to be zero everywhere outside the distribution itself, the value of the integral will not be changed by extending the limits to infinity. The integral so extended, which is a function of R, is known as the Fourier transform of the distribution p(r). We denote it by T(R), writing

T(R)=

0

p(r) exp 21Pi(R.r) dr.

The wave vectors koand k determine R, and so for a known distribution p (r) and given conditions of incidence and scattering the integral in Eq. (1.5) can in principle be evaluated, and the amplitude scattered by the distribution under the given conditions determined.

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R. W. JAMES

2. THETRANSFORM CONSIDERED AS

A

DISTRIBUTION IN RECIPROCAL SPACE

Let us suppose now that T(R) is known for all values of R, and that its value is plotted as a density distribution in the space of the vector R, in such a way that at the extremity of the vector R drawn from the origin the volume density in this space is T(R). In this density distribution we now draw the vector ko of the incident wave, starting from the point L and terminating at the origin 0. With L as center and radius LO = I ko [ = 1/X, let a sphere be described, the trace of which is shown in Fig. 3. Let P

FIQURE 3

be any point on this sphere. Then the radius LP is the wave vector k corresponding to the scattered wave in this direction. By Eq. (1.2), the appropriate value of R is k - ko, that is to say, R is the vector OP from the origin to the point P on the sphere. The value of T(R), the density of the distribution at .this point, gives the amplitude of the scattered wave. The position of the sphere is fixed by the direction of incidence, and by noting how it intersects the transform distribution we may at once get a general picture of the scattering in all directions for this particular direction of the primary beam. As this changes, we may suppose the sphere to swing about the origin as a fixed point. The corresponding variations of the scattering are given by the variations of T(R) at the intersection of the sphere with the transform. The use of this sphere was introduced by Ewaldlo for discussing problems of diffraction by crystal lattices, and in this connection it has often been called the sphere of reflection; but its value in all types of Fraunhofer diffraction problems, including those of ordinary optics, will be apparent, and it may more appropriately be called the sphere of diffraction, or the Ewald sphere, the term that will be used in this article. Formally, the transform T(R) may be complex, and equal say to A (R) i B ( R ) . I n such a case we might suppose two distributions to be drawn, one for A (R), the other for B(R). From the values of A(R) and

+

lop. P. Ewald, Physik. 2. 14, 465 (1913).

65

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

+

B(R) at the point P we could then calculate the intensity A2 B2 and the phase tan-l(B/A) of the radiation scattered in the direction LP. The integral (1.5) is a Fourier integral in real space. We may call the related space in which we have supposed the transform to be plotted reciprocal space, or Fourier space. Like the wave vectors, R, as defined by Eq. (1.2), has the dimensions of reciprocal length, and r.R, which, apart from a numerical factor, represents a phase, is a dimensionless number. In the figures, R and k are of course represented by lengths proportional to their magnitudes. 3. THEESSENTIAL PROBLEM IN STRUCTURE DETERMINATION In crystallographic applications of diffraction theory, we usually wish to determine the nature of the distribution p(r) from the diffraction patterns it produces, and from E q . (1.5) we can see the essential nature of this problem and to what extent it is in principle soluble. By Fourier’s integral theorem we obtain from Eq. (1.5) the corresponding relation p(r) =

0

T(R) exp ( - 2 ~ i r : R ) dR,

in which the integration is now to be taken over the whole of reciprocal space. To determine p(r) completely it is therefore in principle necessary to know T(R) at every point of reciprocal space. It is at once clear that we can never determine the whole transform experimentally. T (R) depends on observations of scattered amplitudes and phases, and with a given wavelength A, even if observations could be made at every possible angle of incidence, a sphere of radius 2/X about the origin contains the whole of reciprocal space accessible to observation. Only if the regions of transform space outside this sphere make no appreciable contribution to the integrand of E q . (3.1) can we determine with any approach to correctness the nature of the scattering distribution from observations of diffraction made with wavelength X. By decreasing the wavelength we may increase the radius of the limiting sphere, but there are obvious experimental limitations to this possibility. It is very easy to show from a simple example, say a Gaussian distribution of scattering matter, for which the necessary integration offers no difficulties, that detail on a scale much less than X can never be determined optically, a well-known result that is readily interpreted from the point of view we are considering. By using X-rays, for example, we may expect to resolve the individual atoms of a crystal optically, but not to learn much about their internal structure.

66

R. W. JAMES

Full knowledge of T(R) naturally involves knowledge of its phase; but since only I T ( R ) [ can be directly determined from observations of the intensities of X-ray spectra, it is clear that essential information necessary for evaluating the integral in Eq. (3.1) is lacking from our experimental data. The fundamental nature of the phase problem, the chief difficulty in the determination of crystal structures, is therefore evident, but further discussion of it lies beyond the scope of this article. 4. PERIODIC STRUCTURES We shall now apply the theory that has been outlined to scattering by three-dimensionally periodic structures, for here, although the phase difficulty remains, the nature of the transforms involved allows us to reduce integration over the whole of reciprocal space to a summation over a discrete set of points in it; it is this that makes possible the determination of crystal structures. Suppose the density of the scattering matter repeats itself exactly in each vector distance rp, where rp = p1a1

+ p2a2 + p3a3

=

C pjai

(4.1)

j

in which all a2, a3 are the primitive translation vectors of the space lattice upon which the structure is based, and p l , pz, p 3 are any three integers. The density will then have the same value at corresponding points of every unit cell, so that (4.2) P(r> = P(r

+ a.

Let F(R) be the transform of the scattering matter in one unit cell, say that associated with the origin, which is given by F(R)

=

/

exp (2hR.r) dr

p(r)

(4.3)

cell

and is the quantity usually known as the structure factor of the unit cell. On substituting r r, for r in Eq. (4.3), and using (4.1), and (4.2), we obtain for the transform of the unit cell associated with the lattice point (PI, Pz, P3)

+

F(R) exp (2rzRwhere

i

piai)

=

F(R) exp (271-i

p&)

(4.4)

j

R.ai

=

ti.

(4.5)

T H E DYNAMICAL THEORY OF X-RAY DIFFRACTION

67

The transform of a parallelepipedal block of the structure, having edges Nlal, N2a2, N3a3is obtained by summing the expression in Eq. (4.4) over all the cells in the block. It is equal to F(R) multiplied by three geometrical series of the type

The transform of the block may therefore be written

T(R)

=

T(5)

=

F(5)g(b)g(F2)g(f3)

(4.7)

=

F(f)G(t)

(4.8)

where 6 represents the triplet of numbers El, E2, t3. By Eq. (4.5), (j is constant over any plane in reciprocal space perpendicular to the vector ai in real space, that is to say to the ai axis of the crystal. The projection of R onto the direction of ai is R.ai/ I ai I = t i / I ai I , so that ( j measures the component of R in the direction of ai expressed as a multiple of 1/ ai 1 . The function g ( & ) of Eq. (4.6) is familiar from the theory of the diffraction grating. Consider for the moment g (1). It is periodic in El, with a period 11 I a1 I , and has main maxima of numerical magnitude N1 whenever t1 is a whole number hl, that is to say, over planes in reciprocal space perpendicular to the a1 axis of the crystal, lying at distances from the origin hl 1 a1 I , where hl is any integer positive, negative, or zero. Between each pair of main maxima there are N1 - 1 subsidiary maxima, which become negligible when N1 is as large as it usually is in crystal problems. The main maxima have a width 2/N1 in (], and so become extremely narrow for large N1. We may then consider the function g(&) to be represented by a distribution in reciprocal space confined virtually to a set of planes with constant spacing 11 I al I perpendicular to the al axis of the crystal. This distribution is of course the transform of an infinite row of points of spacing al. The functions g ((2) and g (t3)have, mutatis mutandis, the same properties. The function G(E) is the product of the three g functions, and therefore has appreciable values only at the common points of three sets of equally spaced parallel planes, perpendicular respectively to al, a2, a3, and having spacings 1/ I al , 1/ 1 a2 I , 1/ I a3 I . These points lie on a space lattice reciprocal to that upon which the structure is based. The point ( B , h2, h3) of this lattice is clearly the point of intersection of the planes hl, h2, h3 of the three sets.

I

I

68

R. W. JAMES

5. THERECIPROCAL LATTICE We denote the primitive translations of the reciprocal lattice by bl, bz, bs. Then b, lies in the line of intersection of planes perpendicular to a2 and a3,and so is itself perpendicular to the a2 and aa axes of the crystal. We thus have the relation b1-a4= bl.a3 = 0 (5.1) with similar ones for the other translations. The bl planes of the reciprocal lattice are those perpendicular to al, the spacing of which is 1/ I al I , and this spacing is therefore the projection of the reciprocal-lattice vector bl onto the direction of al. Thus or with corresponding relations for the other axes. These results may all be summed up in the relations

ai-bi = 0 if i # j = 1 if

i=j, (5.3) the usual defining equation for the reciprocal lattice, which here makes its appearance quite naturally as the transform of the space lattice upon which the structure is based. If the transform is to be a true space lattice, the crystal lattice itself must consist of an infinite array of scattering points. If the crystal units, while remaining points, are finite in number, G ( [ ) is still periodic, but its appreciable values axe no longer confined to the points of the reciprocal lattice, but extend through regions around them of dimensions inversely proportional to those of the finite scattering lattice; if the number of points in this lattice becomes very small, appreciable subsidiary maxima appear in the transform. 6. THETRANSFORM OF A CRYSTAL BLOCK

The transform of the whole crystal block is obtained by multiplying F(R), the structure factor of the unit cell, by the transform of the point lattice upon which the structure is founded. If the crystal block contains a large number of units, only the values of F(R) at the reciprocal-lattice points themselves are of importance, because of the nature of the function G ( [ ) . The transform becomes in effect the reciprocal lattice with each point m weighted by the corresponding value of F(Rm). The triplet of

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

69

numbers (hi, hz, h3) has here been replaced by the single symbol m, so defined that R, = hbi hzbz h3b3 = hkb. (6.1)

+

+

Ck

Since the transform now has appreciable values only at the reciprocallattice points, appreciable diffraction will occur only when a reciprocallattice point m lies on the Ewald sphere. The vector R, is perpendicular to the lattice planes (hl,hz, h3),and if 2e0is the angle between the incident and diffracted rays,

(&I

=

I k - kol = 2(sinBo)/A

(6.2)

which is equivalent to Bragg’s law. The structure factor F(R) becomes small for large values of I R I , although it does not fall away uniformly with increasing R I unless the structure is extremely simple. The ultimate falling away is determined by the transforms of the individual atoms of which the crystal is built up. These are the so-called atomic scattering factors, or f factors, given by

I

f(R)

=

0

pa(r>exp ( 2 m i . r ) dr

(6.3)

where pa(r) is the electron density in the atom itself, expressed as a function of the distance from the atomic center. Since this density extends through a finite volume, which is moreover effectively increased by the thermal motion of the atoms, the atomic transforms, and so the transform of the whole crystal, become small at large I R I ; and with X-rays of wavelength 10-8 cm most of the relevant part of the transform can be observed, as long as we are not concerned with the internal details of the electronic structure of the atom itself, but only with the arrangement of the atoms in the unit cell.

7. THE ELECTRON DENSITYEXPRESSED AS

A

FOURIER SERIES

In the integral of Eq. (3.1), p(r)

=

1T ( R )

exp (-22rrir.R) dR,

we now substitute the value of T(R) in terms of the parameters 5 from Eq. (4.8). The element of volume dR may be written dR

=

V*d[ld[& = (l/V) d t

70

R. W. JAMES

where V and V* are respectively the volumes of the unit cell of the crystal and of the reciprocal lattice. Then (7.1) This, as it stands, is an integral taken throughout Fourier space. If N , the number of crystal units in the block, is very large, the maxima of G(E) become very sharp, and have appreciable value only at the reciprocallattice points themselves. The function F(E), on the other hand, varies relatively slowly with 4, and by taking N large enough we may make G ( [ ) as sharp as we please in comparison with the period of the exponential function. In effect, there is then one integral for each reciprocal-lattice point extended through the volume V* surrounding it, and we can express Eq. (7.1) as the sum of these integrals over all the points. I n the integration for the mth point we may give R and F ( ( ) their values at the point itself, and so obtain, instead of Eq. (7.1), p(r) =

(1/V)

m

/,,, G(4) d4.

F(m) exp ( - 2 ~ i r . R ~ )

(7.2)

When G ( [ ) has the form calculated for the crystal block, it can be shown without difficulty that the integral in Eq. (7.2) approaches unity when N is very large; the same is true for a crystal of any reasonably regular form to a close approximation. Finally, therefore, the Fourier integral (7.1) becomes the Fourier series p(r ) =

(1/V)

m

F(m)exp (-2&r.R,),

(7.3)

where the sum is to be taken over all the reciprocal-lattice points. This is the standard series used to determine the density in terms of the structure factors, expressed in a compact vectorial form. The evaluation of the electron density for the periodic structure is practicable; for the relevant portions of the transform are limited to the reciprocal-lattice points, and its value is appreciable only at a finite number of them, most of which lie within the region of reciprocal space accessible to observation. 8. DISCUSSION

Only a truly periodic distribution of density can be represented at every point by a triple Fourier series, but even if the crystal is finite, its density at any point in a complete unit cell can of course be represented by the same series. The coefficients of the series are the values of the structure factors at the reciprocal-lattice points, supposed calculated by the ordinary

THE DYNAMICAL THEORY OF X-HAY DIFFHACTION

71

methods applicable to Fraunhofer difraction, without any consideration of multiple scattering or dynamical interaction, that is to say derived purely geometrically as we have used that term above. If the intensities of the spectra were given correctly by the geometrical theory, we could use it to calculate from the observed intensities the magnitudes of the structure factors I F ( m ) I for use in the series, although the difficulty of the indeterminate phase would still remain. Apart from the phase difficulty, however, we know that the geometrical theory does not in principle correctly describe the process of diffraction in a crystal, although it is true that in practice we may often employ it without great error. The problem is to obtain from the intensities of the spectra actually given by a crystal, which may be greatly affected by its state of perfection, the values of the geometrical structure factors 1 F ( m ) 1 . If the geometrical theory were applicable, the structure factor, apart from certain angular corrections, would be proportional to the square root of the intensity of the corresponding spectrum; if the crystal were perfect, and dynamical interaction were fully established, it would be directly proportional to the intensity. Usually the best result will be given by some compromise between these two extremes, which. may give very different figures for strong spectra. In practice it is customary to use the geometrical theory when determining crystal structures, and to apply corrections when necessary, if it is practicable to do so, for the state of perfection of the crystal specimens used. Most crystals are so imperfect in the strict sense in which perfection must be interpreted in the dynamical theory that this somewhat unsatisfactory compromise yields good results; but if we are interested not in structure determination, but in the state of the wave field inside the crystal itself, the geometrical theory can tell us nothing of value. The dynamical theory is then essential, and the remainder of this article will be devoted to its development. 9. LATTICEFUNCTIONS

Any function periodic in the identity periods of the space lattice we shall call a lattice function, a convenient term used by von Laue. Such a function may always be represented by a Fourier series of the same type as Eq. (7.3), if coefficients appropriate to the quantity concerned are used. It may be inferred that, when a stable dynamical field has been set up in an unbounded crystal, the amplitudes of the electric and magnetic fields will be lattice functions in this sense, and it will be shown in the next section that the wave equation valid in the crystal is in fact satisfied by fields of this type. It is easy to see that this leads at once to wave fields of the kind discussed by Ewald.

72

R. W. JAMES

Suppose the electric field in the crystal is represented by the wave function E = A(r) exp {2&(vt - K o - r ) ) . (9.1) We assume the amplitude of the wave at the point r to be a lattice function, so that we may write A(r)

=

Z L e x p {-2ni(r.R,)], m

(9.2)

the summation being over all reciprocal-lattice points, and the Fourier coefficients A, being independent of r or t. Substituting Eq. (9.2) in (9.1), we :get E(r) = Amexp [2&(vt - (KO R,) sr)]. (9.3)

+

Z m

Equation (9.3) represents a field consisting of an infinite number of plane waves, the wave vectors of which, K,, are related to the vector KO by the equation Km = KO Rm; (9.4)

+

that is to say, any vector obtained by adding to KO one of the reciprocallattice vectors K, is a possible vector of one of constituent plane wave trains of which we may consider the field to be composed. The relation of this to Ewald's approach, outlined in Part I, will at once be clear: it is of course only a slightly different approach to the same idea. What has been shown is that, if the wave field is such that its amplitude is a lattice function, it can always'be built up of a discrete set of plane waves, the directions and wavelengths of which are related by Eq. (9.4). Our next task is to seek solutions of this type consistent with the laws of electrodynamics and with the distribution of scattering matter in the crystal. 111. The Electromagnetic Basis of the Theory

10. INTRODUCTION

I n dealing with the scattering of X-rays, to which the discussion will be restricted, we may neglect the effect of the massive atomic nuclei, and consider the scattering to be due to the electrons alone. We shall suppose too that the frequency of the radiation is large in comparison with the critical absorption frequencies of any of the atoms in the crystal, and this will allow us to neglect the effects of anomalous scattering and dispersion. We assume the electron density applicable to the problem to be the Schrodinger charge-density

I I I #(r, t ) l2

dr, t) = - e

(10.1)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

73

where $(r, t ) is the electronic wave function, suitably normalized, and I e I is the numerical value of the electronic charge, expressed in electrostatic units. The electronic wave function can be expressed as a function of the coordinates in three-dimensional space only if the different electrons are independent, or if their interaction has been taken into account by some process such as that used by Hartree in his method of the self-consistent field. We shall assume this to have been done for the electrons of each atom, and shall further assume the total charge density at any point to be the sum a t that point of the densities of the individual atoms. The Schrodinger charge-density so calculated is continuous throughout the crystal, a function of the space coordinates and the time. The electric waves passing through the ciystal produce a periodic perturbation of the wave function, and from the corresponding periodically varying charge density we calculate the scattered waves by the methods of the electromagnetic theory. This is the method used by von Laue*in his modification of Ewald’s original treatment, in which the scatterers were assumed to be electric dipoles situated at the lattice points. It is not possible to give the necessary perturbation theory here, or to derive the appropriate expressions for the current density in terms of the wave functions. We must quote the relevant results, and refer the reader to the literature of the subject for details. A full account is given in von Laue’s book,B and a fairly detailed treatment, together with a number of references, will be found in the writer’s book,” Chapters 11, 111, and IV. 11. THEEFFECTIVE DIELECTRIC CONSTANT OF

THE

CRYSTAL

The electric properties of an insulating medium in which the dielectric constant varies from point to point can be represented as due to a suitable volume distribution of electric charge in vacuo, which can be calculated if the dielectric constant is known as a function of the coordinates. Here this process is to some extent reversed. From the perturbed Schrodinger charge and current densities we shall calculate the corresponding effective dielectric constant q . The formal theory upon which the procedure is based is as follows. Let D be the electric induction and E the electric field strength at a point where q is the dielectric constant, and let qo be the dielectric constant of empty space.llaThen D = qE (11.1) 11

R. W. James, “The Optical Principles of the Diffraction of X-Rays.” Bell, London,

1948. The medium, although crystalline, is assumed to be isotropic with respect to fields of the frequencies considered here. Maxwell units have been used, in which v o has the numerical value unity; but it has been retained explicitly in the equations.

11s

74

R. W. JAMES

and if free charge of density p is present divD

whence, using Eq. (11.1), we may write qo div E =

where

P’ = ( v o / ~ I ) P ,

P”

=

(11.2)

= 4rp,

47r(p1

+

p”)

- ( T O / T )C(E. grad 71)/4rl.

(11.3) (11.4)

The interpretation of Eq. (11.3) is that the field at any point in the medium may be calculated by assuming that throughout the volume occupied by it there is a real charge density p’, which is zero if there is no free charge in the medium, and a fictitious charge density p”, given by Eq. (11.4), which is zero if q is not a variable function of the coordinates. Let V be the potential at any point calculated from the charge densities defined by Eq. (1 1.4) and Vo the potential at the same point on the assumption that the actual free charge present were situated in vacuo, in the absence of matter. If V1 is the additional potential due to the presence of matter we may write

r

(11.5)

where T is the distance from the element of volume dv of the point at which the potential is to be calculated. By Eqs. (1 1.2) and (11.4) the last equation may be written

Vl

where

=

(1/4rqo)

=

- (1/q0) 4rP

1 1

div (TOE - D) dv/r

=

div P dv/r

(11.6)

D - qa.

(11.7)

The vector P is usually known as the dielectric polarization in the medium, and it is easy to show from Eq. (11.6) that it represents an electric moment per unit volume. Equation (11.6) shows that the effective charge density due to the presence of matter may be written pc =

- divP

(11.8)

while from Eqs. (11.7) and (11.1) the dielectric constant is 7 =

70

+ 4rP/E.

(11.9)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

75

Since we have assumed the medium to be isotropic for the fields involved,

E, D, and P are all in the same direction at any point.

Since the medium contains aa much positive as negative charge in the case with which we are concerned, and since the positive charge takes no part in the scattering, we may assume the latter to be so distributed that at every point it cancels the negative charge in the undisturbed state of the crystal. We may therefore put the real charge density equal to zero. The charge density with which we are concerned is pp the time-variable part of the Schrodinger density, which is the result of the perturbation of the atomic wave functions by the oscillating electric fields produced in the crystal by the X-ray waves. A variable charge density necessarily gives rise to a variable current density j, and the two are related by the equation of continuity div j &,/at = 0 (11.10) or, by Eq. (11.8), j = aP/at. (11.11)

+

The electric and magnetic fields in the medium obey Maxwell’s equations (11.12a) aB/at = - C curl E

qdE/at

+ 4xj

=

c curl H.

(11.12b)

The dielectric constant 110 is used here because the fields are supposed to be produced by an effective current density, in vmuo. All the quantities E,D, P, and j, vary periodically with a circular frequency w = 2xv. Let a typical one be represented by

F Then

=

Foexp (iwt).

aF/at

=

iu~.

( 11.13)

Using this result, and Eqs. (11.11) and (11.13)) we have

j,

iw(qoE

or

=

iwP,

+ 4xP} = c curl H iwD = ccurlH

and so since div curl H

divD = 0, 3

0.

(11.14)

(11.15)

76

R. W. JAMES

The charge density and current density are those of the Schrodinger theory. If A is the vector potential of the perturbing field, it can be showngJ1 that j,

=

(ih 1 e 1 /47rm) (9grad #* - #* grad #) - (e2/mc)A I #

and, by Eq. (3.1), P6 =

-le/

/#I2*

l2

(11.16) (11.17)

So long as dispersion effects can be neglected the first term of Eq. (11.16) makes no appreciable contribution to j,. The value of 1 # l2 in the second term may be taken with sufficient approximation to be that of the unperturbed wave function, so that e 1 # l2 can be put equal to the unperturbed electronic charge density pB. Thus j,

=

( I e 1 p6/mc)A.

(11.18)

The electric field and the vector potential are connected by the relation

E

=

- (l/c)aA/at - grad 4

(11.19)

4 being the scalar potential, which is zero for the wave field. Hence, by Eq. (11.13), E = -~wA/c (11.20) and by Eqs. (11.14), (11.18), and (11.20),

P/E = ( I e I /mu2)pa.

(11.21)

From Eq. (11.9), therefore, we see that for fields due to X-rays the crystal behaves as if it had a dielectric constant q‘, relative to that of a vacuum, given by (11.22) 9’ = d q o = 1 (47r 1 e I / m ~ 2 ~ o ) or q’ = 1 - (ez I # 12/7rrnv2vo) = 1 26, (11.23) where 6 = - (e2 I # 12/27rmv21)0). (11.24)

+

+

Here, v is the frequency of the radiation, and in the units used, q o is equal numerically to unity. This way of considering the matter in terms of an effective dielectric constant was first proposed by von Laue,s and justified in terms of wave mechanics by Kohler.12Its detailed working out is due to von Laue.13 M. Kohler, Sitzber. preuss. Akad. Wiss. Physik-Math. Kl. p. 334 (1935). M. von Laue, Ann. Physik [5] 23, 705 (1935).

77

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

CONSTANT AND ITSFOURIER COEFFICIENTS 12. THEDIELECTRIC or The value of 6, as given by Eq. (11.24), is of the order of and sometimes even less, for most crystals, and is always negative] so that the relative dielectric constant q’ is very slightly less than unity. It is a continuous function of the coordinates, varying from point to point in any unit cell, but the same at corresponding points in all unit cells, since this is true for I # 12. It is therefore a lattice function. We now define a quantity t$ by the relation 4?rP = t$D or, by Eqs. (11.7) and ( l l . l ) ,

4

=

1

- l/$.

(12.1) (12.2)

Using Eq. (11.23), remembering that 6 is very small in comparison with unity, we may write to a very close approximation =

26 = - (e2 I #

1*/?rmv2qo) =

- (e2//mc2) * ( 1 # I2X2/aqo)

(12.3)

where X is the wavelength of radiation of frequency v in free space. Since the quantity 4 is a lattice function it can be expressed as a Fourier series of the type (9.2). We write t$ = C $ m e x p (-2?ri(Rm.r)J.

(12.4)

m

The Fourier coefficientsare determined in the usual way, and +m

=

(1/V)

/,,, 9 exp { 2 k ( L * r )I dr

(12.5)

the integration being throughout the crystal lattice cell. Using Eq. (12.3), we obtain

or e2 X2 F ( m ) 4m -----mc2nqo

v

(12.7)

where P(m) is the structure factor of the unit cell for the spectrum m, which gives a physical significance to the Fourier coefficients.

78

R. W. JAMES

13. THEWAVEEQUATION AND ITSSOLUTION

The next step is to set up the equation that the wave field in the crystal must satisfy. From the Maxwell equation (11.12a) and Eq. (11.7), putting now, and in all that follows, po = q0 = 1, we obtain c curl (D - 47rP) = -8H/dt. Taking the curl of both sides of this equation, and using Eq. (ll.l2b), we get curl curl (D - 47rP)

-c-l(a/at) (curl H) = - (c2)---l(d2D/8t2),

=

+

or VD - grad div D 47r curl curl P = (1/c2)a2D/8t2,which, since div D = 0, by Eq. (11.5) we may write, using Eq. (12.1), in the form V2D - (l/c2)82D/dt2

=

- curl curl (+D).

(13.1)

This is the equation that must be satisfied by the electric induction D at every point in the crystal. We now consider the nature of the field that satisfies Eq. (13.1) in a crystal that extends to infinity in all directions, so that no boundary conditions need be assumed. We try a solution in the form of a plane wave of wave vector KO,having an amplitude that is a lattice function; that is to say that varies with the coordinates in such a way that it has the same value at corresponding points in every unit cell. The amplitude may therefore be expanded as a Fourier series in the way we have already discussed, and we may write for the electric induction

D

=

exp (27ri(v2 - Ko.r) }

D, exp { -2&(Rn.r) }

(13.2)

n

= n

D, exp {27ri(vt - K,.r)

1,

(13.3)

an expression representing an infinite set of plane wave trains with wave vectors related by the equation

Kn

=

+

(13.4)

KO Rn,

one for each reciprocal-lattice point. Our aim is to determine the Fourier coefficients D, for which a solution of this type will satisfy the wave equation (13.1). Since is also a lattice function we may write

+

4

=

C Q

+q

exp { -2m(Rq.r)

I

(13.5)

where q again denotes a reciprocal-lattice point, and the summation is

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

over all such points. By Eqs. (13.3) and (13.5), D+

=

C C+nDnexp [2&{vt n

=

- (Kn

q

79

+ Rq).r)l

c C &Dnexp {2&(vt - Kn+q-r)}.

(13.6)

n~

Each summation extends over all the reciprocal-lattice points, which are infinite in number. We may start either summation at any one of them, and it is convenient to take a new index number m = n q and to use the indices m and n as indices of summation, writing Eq. (13.6) in the form

+

D+

=

=

C m

n

drnFnDn exp (27ri(vt - K,*r) )

c (+D)mexp {2?ri(vt - K,.r) } m

where

(13.7) (13.8)

and is the Fourier coefficient m in the expansion of +D, which is also a lattice function. Each Fourier coefficient (+D), is itself a sum over all the reciprocal-lattice points. The next step is to substitute Eqs. (13.3) and (13.8) in the wave equation (13.1), which is most easily done by making use of the following results. If U(r, t ) = A exp {27ri(vt - Ker) 1 represents a plane vector wave, in which A and K are independent of r and t, it is a simple matter to show that divU

=

-2?rzlt(-U

(i)

v2u

=

-47r2k2U

(ii)

f32U/dt2

=

-4x2v2U

(iii)

curlU

=

-2m’(K x U)

(iv)

where the cross denotes the vector product, and curl curl U

=

47r2{K2U- K ( K - U ) ) .

(v)

The last expression has a simple geometrical interpretation. Let n be a unit vector in the direction of K. Then (v) may be written curl curl U

=

47r2K2{U- n(n-U)}.

80

R. W. JAMES

But n - U is the component of U in the direction of K, so that U - n(n. U) is its component perpendicular to K, which we denote by U[K]. Finally therefore curl curl U = 47r2K2U[KI. (4 We may apply these results directly to the solution of Eq. (13.1). Using Eqs. (13.6)-(13.8) we express each side of the wave equation (13.1) as a Fourier series; and equating corresponding Fourier coefficients on the two sides, at the same time using results (i) to (vi), we obtain at once (Km2

-

v2/C2)Dm

=

Km2

C b-nDn[m].

(13.9)

n

Here, Dnlrn, is the component of Dn perpendicular to K,. There is one equation of this type for each value of m, and (13.9) thus gives the amplitudes of an infinite set of plane wave trains that are related by Eq. (13.3), and which together, according to Eq. (13.4), constitute a wave field that satisfies the wave equation. If k = v/c = 1/X, we may write Eq. (13.9) in the form (Km2 - k2)Dm = K 2 C $m-nDn[m].

(13.10)

n

14. THEMAGNETIC FIELD

Before considering the significance and application of Eq. (13.10) we complete the preliminary discussion of the field in the crystal by determining the relation of D, to the corresponding magnetic quantity H,. The magnetic field is also a lattice function, and can be expressed as a Fourier series H = H,exp (27ri(vt - K,.r)). (14.1) Substituting this in the Maxwell equation D tions (iii) and (iv) of Section 13, we obtain

kD, and from the Maxwell equation same way kH,

=

fi =

-K, = -c

=

x H,,

c curl H, and using rela-

(14.2)

curl E we obtain in exactly the

K, x Em,

(14.3)

so that both D, and H m are perpendicular to K,. We now take the vector product of both sides of Eq. (14.2) with K,. Then k(K, x D,) = -K, x (K, x H,) = K2H,

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

81

since K m and Hmare mutually perpendicular. Thus

Hm

(k/Km2) ( K m

=

X

Dm).

(14.4)

We now form the vector product of both sides of Eq. (13.10) with k K m . Since D n [ m l is the component of D, perpendicular to Km,

Kn

x Dn[ml = K m x Dn,

and using Eqs. (14.4) and (14.2) we obtain (Km2 - k2)Hm

=

-

C

+m-nKm

x (Kn x Hn).

n

On expansion of the double vector product, this gives ( K m 2 - k2)Hm =

- Kn(Km*Hn)1,

C +m-n(Hn(Km*Kn) m

(14.5)

an equation for obtaining the Fourier components of H analogous to Eq. (13.10) for those of Dm. 15. THERESONANCE ERROR

Equation (13.9) shows that none of the waves that make up the field can travel in the crystal with exactly the velocity c, for in such a case I Km I = Ic, and the amplitude of the corresponding wave train becomes infinite. As we shall see, however, the speeds of one or more of the wave trains may approach c very closely indeed. It is therefore convenient to define a quantity ern by the equation

IKm I

=

k(1

+

(15.1)

em).

Ewald calls em the resonance error (Resonanzfehler). It is a convenient measure of the departure of the velocity of the corresponding wave from that of light in free space, and if it were possible for a single wave train to em would be the corresponding refractive index. travel in the crystal 1 In the examples we shall consider em is always very small in comparison with unity. When this is so, we may write to a close approximation

+

(Km2- k2)/Km2= 2em and Eq. (13.9) then becomes Dm

=

(%m)

-' C +m-nDn

Im],

(15.2)

n

an equation first derived by Ewald. For some of the waves, em may be of the order of or less, and for these the values of DmI given by Eq. (15.2) will be much greater than

I

82

R. W. JAMES

for any other possible waves. This has the important practical consequence that, although in principle Eq. (15.2) represents an infinite set of wave trains, the amplitudes of all of them are negligibly small except those for which em is very small, and so I K, I very nearly equal to k. The constituent waves of the set given by Eq. (15.2) are the concurrent waves of the Ewald theory, discussed in the introduction. As we saw there, the Ewald sphere of radius k may be used to determine which of the wave trains will have appreciable strength under any given conditions. They will be those for which the extremity of the wave vector K, lies very nearly on the sphere. Its distance from the sphere, upon which it can never lie exactly, is ke,. The point A of Fig. 1 from which the wave vectors K, radiate was called by Ewald the Ausbreitungspunkt. Here we shall call it the wave point. It lies at a distance -KO from the origin, and at a distance -K, from the reciprocal-lattice point m. Since the radius of the sphere is k, the distance of the reciprocal-lattice point from it is K, I - k = kern, so that m lies outside the sphere if em is positive and inside the sphere if it it is negative. For a point to lie near the sphere, as we are here to understand the term, or ern must be of the order of and this means that we may use the ordinary construction for the geometrical theory to determine which wave trains are to be considered as relevant. If it is decided that n points must be considered, we may carry out the subsequent calculations using only the corresponding 2n equations of (15.2), or n equations if we are able to deal with one independent state of polarization. This choice of a limited number of waves is, as we have seen, an approximation, but it is often a good one, and always a necessary one, since the general problem is mathematically quite intractable. It should be emphasized once more that the choice of any particular wave as the primary wave is arbitrary in the infinite crystal, for the existence of any one of the waves in principle implies the existence of all the others.

I

IV. The Dispersion Surface

16. THEDEFINITION OF THE SURFACE

By Eqs. (14.2) and (14.3), we see that all the waves of the field are transverse, and that D, is perpendicular to K,. For each wave direction K, there are thus two independent transverse components, corresponding to two independent directions of polarization at right angles to each other; so that if n reciprocal-lattice points correspond to appreciable wave intensities there are 2n equations of type (15.2) to be solved. The equations

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

83

are linear in Dn[,], and the condition that they may have a solution is 4 W h e determinant formed by the coefficients of D, and Dn[m] should

va?kh

the construction used to determine the possible wave trains by means of the Ewald sphere, it will be seen that once KO,and with it €0, is fixed, so are the position of the sphere and all the remaining values of em; for ke, is the distance of the reciprocal-lattice point m from the sphere. The quantities em are thus expressible in terms of €0, and, since the relevant values are all small, as linear functions of eo. The determinantal equation is therefore in effect an equation of degree 2n for eo, each solution of which fixes a possible position of the wave point A on KO. For any direction of KO for which n points lie near the Ewald sphere there are thus 2n possible wave points, All A2, * ., Ax.A slight deviation of the direction of KOwill not change the number or the identity of the relevant reciprocal-lattice points; and, as the direction varies, the 2 n points describe in reciprocal space a surface of 2n sheets, which Ewald calls the dispersion surface. Its determination by means of Eqs. (15.2) lies at the basis of the dynamical theory. If from any point on any of the sheets of the dispersion surface vectors are drawn'to the n relevant reciprocal-lattice points, these are the wave vectors of n wave trains that build up a field approximating very closely to a solution of the wave equation (13.1) ; although no finite number of wave trains can in principle give the true solution. Once a wave point is determined, so are the relative amplitudes and phases of the corresponding constituent wave trains. In the infinite crystal, any superposition, in any proportion, of the independent fields corresponding to different wave points is also a solution; but when the crystal is bounded the conditions that must be obeyed at its surface fix not only the positions of the possible wave points, but also the relative amplitudes and phases of the corresponding wave fields. For the moment, however, we consider only the unbounded crystal. The dispersion surface in the dynamical theory is the analog of the indicator surface in crystal optics. Because the interatomic distances are then much smaller than the wavelength of light, only one point, the origin itself, can ever lie on or near the Ewald sphere. In a doubly refracting crystal the indicator surface is a surface of two sheets, corresponding to the two states of polarization of waves in the crystal. If this analogy is borne in mind, the use of the dispersion surface in the dynamical theory will be more easily understood. The optical indicator surface is easily constructed if the Cauchy ellipsoid, or the optical indicatrix, is known. The construction is analogous to that used in deriving the biaxial wave surface from the Fresnel ellipsoid.

84 17. THECASEOF

R. W. JAMES A

SINGLEWAVETRAIN(n=1)

Suppose that no reciprocal-lattice point other than the origin, corresponding to the wave vector KO,lies near the sphere. Equation (15.2) then gives 2EODO = 4oDO (17.1) for the wave is transverse. In this particular case therefore €0

=

$40

(17.2)

=

- (e2/mc2)(X2/27r)F ( 0 )/ V

(17.3)

by Eq. (12.7), putting 70 = 1. This determines the magnitude of the wave and so the velocity with which the single wave train of frequency vector KO, v can be propagated in the crystal if no other appreciable wave is produced. While in principle the propagation of a single wave train is not possible, it is in practice closely realized when the direction of propagation is such that no appreciable diffracted waves are produced. In this case (17.4) and the quantity in the brackets is the refractive index of the crystal for a wave train of frequency v. This is the result obtained by Darwin, rather differently, in 1914. F ( 0 ) is in all practical cases a positive quantity, so that qjo is in genera] negative, and in discussing actual examples we shall often write it in the form - I 40 . It is, however, only when there is a single wave that we can properly speak of a refractive index. In this particular case it is easy to calculate the numerical value of e0. F ( 0 ) is equal to the number of electrons in the unit cell of the crystal. If there are n. atoms of type a, with atomic number Z,, in the cell

I

The quantity e2/mc2 is the so-called classical radius of the electron, and its value is 2.817 X lO-'3 cm. For rocksalt x n a Z a = 4(11 a

and For Cu Ka! radiation, X

V =

=

(E~639)~ X

+ 17),

cm3.

1.54 X 10" cm, we find €0

=

-6.64 X lo-',

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

85

a typical value as far as order of magnitude is concerned. This shows that 1 €0 will differ from unity at the most by a few parts in 1OO,OOO, and that reciprocal-lattice points that correspond to wave trains at all comparable in intensity with the primary wave must lie very close indeed to the Ewald sphere.

+

18. THESOLUTION FOR Two EFFECTIVE RECIPROCAGLATTICE POINTS (n = 2)

Much the most important case in practice is that in which two reciprocal-lattice points, say 0 and h, lie close to the sphere. The possible values of both m and n in Eq. (15.2) are now 0 and h, and we have therefore the equations ZED0

=

2eDh

= +hDo[hj

@O[O]

+ -k

@h[O] '#~ah[h].

(18.1)

The waves are transverse, and we distinguish between two independent states of polarization, state (a) in which the vectors Do and D h are perpendicular to the plane containing K Oand Kh, and so are parallel to one another, and state (b) in which Do and Dh lie in this plane, perpendicular to KOand K h respectively, but are not now parallel to one another. Denoting the amplitudes of Do and Dh, by Do and Dh, which may of course be complex, we see from Figs. 4 and 5 that in state (a) Eq. (18.1) becomes

+ (40

+ a 0

(18.2a) -

Kh Stole

(a)

C& and 4 upword

FIQURE 4

Stole (b) Ho and ff,, upword

FIQURE 5

86

R. W. JAMES

While in State (b) Do[q= Do,Dh[o]= Dh COS 28, Dh(h)= Dh, Dop] = Do COS 28, so that Eq. (18.1) becomes

(40 - 2Eo)Do 4 h COS 28Do

+6

COS

28Dh

+ (40 - 2Eh)Dh

=

0

=

0.

(18.2b)

Here, 28 is the angle between K Oand K h and 8 is very nearly equal to the Bragg angle 80 of the geometrical theory. Equations (15.2) and (18.1) are relations between vector components all of which are in the same direction in any one equation, so that Eqs. (18.2a) and (18.2b) are algebraic equations, giving relations between the amplitudes DOand Dh of the vector quantities Do and Dh. In state (a) of polarization DOand Dh are in the same direction; in state (b) they are not. The corresponding relations for the magnetic amplitudes are easily derived from Eq. (14.5) using Figs 4 and 5. Neglecting e0 and Q in comparison with unity, one finds that the magnetic equations may be obtained simply by replacing Do by Ho, and Dh by Hh, in Eqs. (18.2a) and (18.2b). Now, however, HOand Hh are parallel to each other, and perpendicular to the plane containing K Oand Kh,in state (b) . We shall return to the point when considering boundary conditions. Whether we use the electric or the magnetic equations, the determinantal condition in this simple case of two wave trains yields

(40 - 260) (40 - 2Eh)

=

(18.3)

c24hd'&

in which C = 1 in itate (a) of polarization, and C = cos 28 in state (b). In determining the nature of the dispersion surface we shall for the present assume the scattering density, and so 4 and the dielectric constant, to be real at every point, which is equivalent to assuming that the crystal does not absorb the radiation. F ( h ) and F ( h ) , and also 4h and &, are then conjugate complex quantities, if they are not actually real, as may be seen from Eq. (12.6). Eq. (18.3) now becomes

(40 - 2cO) (40 - 2 E h ) OF 19. THENATURE

THE

=

c2I $h 12.

DISPERSION SURFACE FOR n

(18.4) =

2

In Fig. 6 , the vectors K Oand Kh lie in the plane of the paper, and Q is +40)from each of the reciprocal-lattice a point in that plane distant k ( l points 0 and H . The distance of any practically relevant wave point A from Q will be extremely small in comparison with the distance of either from 0 and H - o n the scale of the figure, 0 and H would be a mile or more

+

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

a7

FIGURE6

away from Q-so that arcs of circles having centers at 0 and H and radii k(1 $40) can be represented within the region of the figure by two straight lines &TOand Q Th, perpendicular respectively to QO and QH. The radius of each circle is the wave vector of the wave train of frequency v that is propagated in the crystal when no other wave is excited. In actual examples 40 is negative, and the radius of the circles is less than k. The point L in Fig. 6 lies at a distance k( = l/X) from 0 and H and is the center of the Ewald sphere in the geometrical treatment of the problem when the Bragg condition is fulfilled. Following Ewald, we shall call L the Laue point. Let A be a position of the wave point consistent with Eq. (18.4). A 0 and AH are the wave vectors K Oand Kh, and their lengths are, therefore, t o ) and k ( l t h ) , respectively. It by the definition of t o and t h , k ( 1 should be noticed carefully that t ono longer has the value $40corresponding to a field with a single wave train. The perpendicular distances of A from &TOand QTn, ANo and AN,,, we denote by EO and t h , respectively. Then, since QO and QH are very nearly indeed parallel,

+

+

+

to

=

OA

-

ONo

=

k(1

+ to)

- k(1

+ $40)

=

$k(2ao - 40)

88

R. W. JAMES

and similarly, t h

=

$k (2€h - 40). Thus, to

= k(E0

[h

=

(19.1)

- $40)

k(Eh -

$40).

Equation (18.4) can now be written 50th

=

(k2/4)c2I 4h

12,

(19.2)

which is the equation of a hyperbola having the two lines Q Toand Q Th as asymptotes. Its diameter is found by putting f o = t h in Eq. (19.2) and is equal to V2Vl = Ck I 4 h I sec 8. (19.3) The diameter therefore depends on the state of polarization, since C = 1 in state (a) and cos 28 in state (b) . The hyperbolas corresponding to the two states of polarization have the same asymptotes, but the diameter for state (a) is greater than that for state (b) in the ratio 1:cos 28. It should be remarked here that the assumption that the surface is hyperboloidal in state (b) requires the angle e to be constant. This is not strictly true, since it varies with the position of A , but the variation within the relevant range of positions is so slight that we may take the value of 8 as constant, and equal to the Bragg angle e0 without sensible error, and this will always be assumed in what follows. Suppose now the diagram rotates about the line OH, that is to say about the reciprocal lattice vector Rh. In any plane containing 0 and H the argument that led to the hyperbolic loci of A remains valid for the components of D in and perpendicular to that plane, and the dispersion surface becomes two hyperboloids of revolution, one for each state of $40) cos 00 in polarization. The point Q describes a circle of radius k ( 1 a plane perpendicular to, and bisecting, OH. The angle of rotation need not be small, since the conditions remain valid as long as the sphere does not come near enough to other reciprocal-lattice points for other wave trains to be excited. This corresponds to the experimental result that the intensity of a spectrum is not altered by rotating a crystal face in its own plane, so long as the angle of incidence on the atomic planes remains constant. Here, again, we must except the cases, not in fact rare, in which other reflections may be excited. As long, therefore, as only two reciprocallattice points lie near the sphere, we may regard the dispersion surface as consisting of two hyperboloids of revolution, one for each of the states of polarization (a) and (b). This of course is only true as long as the wave point A is so close to Q that the circles &Toand QTh may be represented by their tangents at Q, a condition that is satisfied in all applications of the theory.

+

89

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

20. THERELATIVE AMPLITUDES AND PHASES OF THE WAVESWHENn

=

2

From Eq. (18.3) D h 260 =-=-

c'$i

Do

If '$ is real, so that

I '$h 1

=

'$0

c'$h

2eh

(20.1)

- '$0'

I '$ I ,i this gives for both states of 4

polarization (20.2)

Equation (20.1) can also be written, using ( M I ) , D h

DO

-

2fo

kC&

-

2fO'$h

(20.3)

kcl'$hl"

Since f o , k, and C are real quantities, the phase difference between D h and Do is governed by the phase of '$h. It is the same as the phase of '$h if lois positive, but differs from it by ?r if f o is negative. For a given state of polarization, f o and f h are positive over the sheet of the hyperbolic cylinder with the larger radius, and negative over that with the smaller radius. Over any one sheet of the dispersion surface the phase difference is thus constant, but differs by ?r for the two surfaces corresponding to one state of polarization. When 28 > u/2, C becomes negative, and the phase differences for states (a) and (b) then differ by T . It will be seen from Eq. (12.7) that the phase of '$h is opposite to that of the structure factor F ( h ) . 21. THETRANSITION FROM Two WAVES TO ONE

When the wave point lies at the apse of one of the generating hyperbolas = Do I . The two wave trains corresponding to the wave point then have equal amplitudes. Suppose A lies near the vertex Vz on the upper branch of the hyperbola in Fig. 6. As it moves from the vertex toward the asymptote &To,f o becomes smaller and f h greater. From Eq. (20.2) it will be seen that I D h I then becomes smaller in comparison with I DO1 until, when A lies very and the near the asymptote, there is effectively only one wave train KO, dispersion surface has become a sphere of radius k ( 1 &'$,I) with center 0, a state of affairs that is practically reached when the direction of KO differs quite slightly from that corresponding to A at the vertex. If A , while still on the same branch of the hyperbola, moves from the vertex towards the asymptote Q T h , the surface becomes more and more nearly a sphere of 3'$0) with its center at H ;and similar arguments apply to the radius k ( 1

lo = &,, and I Dh I

I

+

+

90

H. W. JAMES

FIGURE7

other branch of the hyperbola. The two surfaces of the hyperboloids for either state of polarization degenerate to spherical surfaces in the way indicated diagrammatically in Fig. 7, in which, however, the distance between the two surfaces of the hyperboloid is greatly exaggerated in comparison with the radii of the spheres. On the geometrical theory, two waves could occur only if A lay exactly on the circle described by the Laue point L as the diagram rotates about the line Oh. This is the circle of intersection of two spheres of radius k ( = l/A), which, since eo is negative, lie a little outside those corresponding to the dynamical theory.

22. THECASEOF THREE WAVES We can only consider briefly what happens when three reciprocallattice points, 0, h, and k lie near the Ewald sphere, so that three wave trains correspond to each wave point. Equations (15.2) now become

2&0

=

2~fiDfi= 2ekDk

=

+ dGDk[oj (~fiDo[fi]+ (~oDfi1fil+ @fi--kDk[fi] &Do[k] + +k-fiDfi[k]+ (boD~[k]. 4Dorol -k $ d h [ o ]

(22.1)

I t is difficult to proceed further with the general case because of the complexity of the polarization factors, but if the three wave vectors KO,Kfi, and Kk all lie in one plane the matter is comparatively simple, and as it illustrates certain principles that apply more generally it is worthwhile to consider it. We shall also assume that for all three wave trains the direction

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

91

of D is perpendicular to the plane that contains the three wave vectors. Equation (22.1) then becomes

+

(40 - 2~o)Do 4dh -k 9iDk

=

0

whence (22.3)

(22.4)

An interesting result follows at once. We notice from Eq. (22.3) that even if +h = 0, Dh does not vanish so long as neither +k nor 4 h - k is zero; and an analogous result holds for Dk and 4 k a That is to say that a spectrum, or more accurately, a component wave train in the field, can occur even if the structure factor ordinarily associated with it is zero, so that the spectrum is a “forbidden” one. This is the Renninger14 effect, the so-called Umweganregung. According to the simple theory of the effect, it may occur when the reciprocal-lattice points h and k both lie on the Ewald sphere. We can then think of K Oas giving rise to spectra in the directions Kh and Kk, where Kh = K O-k & But from these equations,

Kk Kh

=

KO-k &.

= Kk -k Rh-k

that is to say, we can think of Kk as giving rise to a spectrum in the direction Kh, associated with a reciprocal-lattice vector Rh-k, or, as we might say, reflected at the lattice planes ( h - k). This reflection will be in the direction Kh, but the structure factor F ( h ) is not involved, and can indeed be zero so long as F ( k ) and F ( h - k) are not zero. Even this simple way of looking at the matter goes beyond the strict geometrical theory, for it no longer neglects second reflections. Nothing is altered in principle if the three vectors are not coplanar. For further details reference may be made to von Laue’s book. 14

M. Ftenninger, 2.Physik 106, 141 (1937).

92

R. W. JAMES

23. SOMEREMARKS ON THE SOLUTION FOR

THE

INFINITE CRYSTAL

It must be clearly understood that up to this point the discussion applies only to an infinite crystal in which a wave field has been established. If conditions are such that two reciprocal-lattice points are relevant, any point on the dispersion surface of Fig. 6 is a possible wave point, giving rise to a wave field consisting of two infinite wave trains, with relative amplitudes and phases determined by the position of the wave point in the way we have discussed. Moreover, the superposition in any proportion of the wave fields corresponding to any number of different wave points is an acceptable solution. When the field is stimulated in a bounded crystal by radiation coming from outside, certain relations must hold between the field vectors at the boundary or boundaries of the crystal, and only certain wave points on the dispersion surface will give fields consistent with these conditions. This case, the important one in practice, will be taken up in the next section. One more point should, however, be mentioned here. According to the geometrical theory, a diffracted beam can occur in an infinite lattice only if the lattice points 0 and h lie exactly on the Ewald sphere. It is true that the geometrical theory cannot in any case apply to an infinite crystal; and if it is finite there is a certain tolerance. We may replace the reciprocallattice points by small regions of dimensions inversely proportional to those of the crystal, so that as the crystal gets larger the angular range over which diffraction can occur gets smaller. The spectrum has in fact a Fresnel width which becomes very small as the crystal gets very large. In the dynamical theory, on the contrary, the production of two wave trains of appreciable magnitude occurs over a finite range of angles even in an infinite crystal, and this range is not determined by the number of lattice units the crystal contains, but is nearly independent of it once the number is large enough for fairly complete dynamical interaction to be set up between the atoms and the wave field. V. The Field in a Bounded Crystal

24. INTRODUCTION

Up to this point the discussion has referred to a wave field already established in an infinite crystal, and the important case when the field lies partly inside and partly outside the crystal must now be considered. Inside the crystal the field must be of the type already dealt with. It must obey the wave equation (13.1) , and must be based on wave points situated on the dispersion surface; but the existence of boundary conditions at the crystal surface has the effect of limiting the permissible positions of the

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

93

wave points. By suitable addition of the fields corresponding to the permitted wave points, each of which satisfies the wave equation independently, the boundary conditions that must hold good at the surface can be fulfilled. A difEculty arises at the outset. In optical and electrical problems the wavelength of the radiation is usually very large in comparison with the atomic grain of the medium, and without serious error we may treat the latter as continuous and its boundary as a mathematical surface. Such a procedure is difficult to justify for X-rays, the wavelengths of which are less than the interatomic distances. Nevertheless, following von Laue, we shall use it here, in spite of this logical weakness, for it gives results in agreement with experiment, and being more familiar, is probably easier to follow than the method of boundary waves used by Ewald, which is logically more satisfactory. We accordingly assume the boundary between the electron-density distribution in the crystal and the empty space outside it to be a plane mathematical surface, upon which falls a plane wave of infinite lateral extent, giving rise to a wave field inside the crystal. A necessary condition at the surface is that of phase continuity. We assume that when a wave kooutside the crystal gives rise to a wave KOinside it the phase velocities of the two related waves parallel to the surface must be the same. That is to say, the components of the two wave vectors ko and KOparallel to the surface must be equal. This we shall suppose to be true for all pairs of directly related waves inside and outside the crystal. It is this condition that leads to Snell’s law of refraction in optics. The condition of phase continuity determines the directions of the waves. Their relative intensities have to be determined from the boundary conditions that must hold between the electric and magnetic field vectors. These are (i) that the normal component of the electric induction D, and (ii) that the tangential component of the electric field strength E must be continuous in passing from one medium to the other. In ordinary optical or electrical problems of this type D and E differ in magnitude, and to this is due the existence of a reflected wave at the surface of separation of the two media. If we except the particular case of total reflection at almost grazing incidence, the specular reflection of Xrays is negligibly small, because, as we have already seen, the dielectric constant of the crystal differs only very slightly from unity. For the purpose of setting up boundary conditions we shall therefore assume the tangential component of D, as well as its normal component, to be continuous, and shall neglect the very small specular reflections that must in principle always be present when wave trains impinge on boundaries between the two media, whether inside or outside the crystal.

94

R. W. JAMES

25. THECASESOF TRANSMISSION AND REFLECTION Consider now a slab of crystal with two parallel faces, upon one of which falls a plane wave train of frequency v and wave vector ko'. Its velocity outside the crystal is c, and

1k&I = k =

V/C = 1/X.

(25.1)

Suppose this wave train to give rise to a wave field inside the crystal consisting of two component wave trains, with wave vectors KOand Kh, both of which lie in the plane of incidence of koi. We suppose KOto be a direct continuation of koi, that is to say, it is the wave that would persist in the crystal if the direction of koi were altered continuously until the second wave train became negligible. We shall see in a moment that for either state of polarization there may be two separate wave fields, corresponding to two different wave points Al and Az, the positions of which are fixed by the boundary conditions. The existence of the wave Kh inside the crystal will in general imply the existence of waves khror khf outside it, and above it or below it respectively, both of which will have magnitude k; we must distinguish between two cams. In Case I, that of transmission, both KOand Kh inside the crystal are directed towards the lower surface. Ewald calls this the Laue problem. In Case 11, the reflection, or Bragg problem, Kh is directed towards the upper, or entrance surface of the slab. A particular example of Case 11, which will be considered in detail later, occurs when the crystal is very thick. If there is any. absorption at all, and there always is, no appreciable radiation will then reach the lower surface, and boundary conditions need be considered only at the surface of incidence. As we shall see, only one wave field then occurs in the crystal for any angle of incidence. In Case I, two fields will always occur for each state of polarization. The upper external wave khr is then absent, but both kotand kht occur at the lower surface. For either state of polarization there are then two wave fields inside the crystal, with wave vectors Kol,Khl and K02,Kh2, corresponding to two different wave points A1 and A2. This will also be true in Case I1 if the slice is thin enough for the radiation to penetrate to its lower surface. The two cases are illustrated in Figs. 8a and 8b; but it must be remembered that for each internal wave vector shown in the figure there may actually be two, differing very slightly in direction. The angle of incidence on the upper surface is denoted by lc.o and the angle of emergence of kh at either the upper or the lower surface by h . Each angle is measured in a clockwise direction from the inward-drawn normal.

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

95

(b)

FIG.8. (a) Case I (Transmission). (b) Case I1 (Reflection).

26. THEDETERMINATION OF THE WAVEPOINTS

In Fig. 9, let S S represent the trace of a plane parallel to the upper crystal surface. We consider wave trains inside the crystal related to the two reciprocal-lattice points 0 and h, represented in the figure by the points 0 and H . OH is then the vector Rh,which we suppose to lie in the plane of the diagram. The plane bisecting OH at right angles is parallel to the lattice planes h, by which, according to the geometrical theory, we could suppose the wave train I(,,to be produced by the reflection of KO.These planes make an angle x with S S . The point L lies at a distance k ( = 1/X) from both 0 and H , and is the Laue point, the center of the Ewald sphere for geometrical reflection at the Bragg angle. The angle OLH = 200, and LO would have been the direction of incidence necessary to produce the spectrum h on the geometrical theory. Let koi, the wave vector of the incident wave train, be represented by PoiO. This vector has magnitude k, and Poi lies on a circle of that radius, having 0 as center and passing through L, which it will be convenient to call the circle of incidence.

96

R. W. JAMES

FIQURE 9

We now apply the principle of phase continuity at the boundary. Since KOmust have the same component parallel to the surface as ko+,the corresponding wave point A must lie on the normal to the upper surface, PoCN, and since it must also lie on the dispersion surface DD its position is determined. Once the wave point is fixed in this way, so also are A 0 and A H , which are the wave vectors KOand Kh. The emergent wave vectors kof and kh' (or 41) are determined in a similar way. We consider the construction for the more general case of a wedge-shaped crystal, which is illustrated in Fig. 10. The position of A is determined by the conditions of incidence. Through it we draw a normal AN2 to the second surface. The point where this cuts the circle of incidence Po' in Fig. 10, determines the wave vector kd, which is represented by the line joining Pof to 0; for this vector has magnitude k, and the same component parallel to the surface of exit as KO. The transmitted emergent vector khf is determined by the point Pht where the normal to the second surface cuts the circle of center H and radius k that passes through L ; for the vector PhfH has the correct magnitude k and the same component parallel to the second surface as Kh. In Fig. 10,to avoid confusion, only one wave point A has been shown, and the trace of the dispersion surface upon which it lies has been omitted. If the two surfaces of the slab are parallel, the two normals through A coincide, Poi and Poi become one point, and P h f lies on the normal PoiN.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

97

s2

FIQURE 10

For a reflected wave khrit will always do so. The conditions for a parallelsided slab are shown in Fig. 9. In the cases we shall consider the dispersion surface consists of two hyperboloids of revolution about a common axis, one for each state of polarization, and PoCN will in general cut it in four different points. In some circumstances only one wave point may be effective for each state of polarization, and there may sometimes be no real point of intersection of the normal with the hyperbola, an important case that corresponds to Darwin’s range of total reflection. In general, however, four fields may coexist. The relative amplitudes and phases of the waves associated with any wave point are known, once its position is fixed. Each such pair of waves may be considered as an independent solution of the wave equation, and, as far as this is concerned, the wave fields associated with the different wave points may have any amplitude and phase; but if we take into account the conditions that must be satisfied at the surfaces by the field vectors we shall find that the relative magnitudes and phases of the waves associated with two wave points A1 and Az lying on the same normal for a given angle of incidence and given state of polarization are fixed, so that the whole field is determined. This question of the coherency of the field will be considered in more detail later.

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R. W. JAMES

27. SOME NECESSARY GEOMETRICAL RESULTS

The angle of incidence $'o on the upper surface varies as Poimoves along the circle of incidence in Fig. 9, and becomes #B, the angle corresponding to reflection a t the atomic planes ( h ) at the Bragg angle Bo, when Poi coincides with L. We write $0

= $'B

+

a$'O

(27.1)

where 6$0, shown on a greatly exaggerated scale in Fig. 9, is the angle between PoiO and LO, and is the angle of incidence referred to the Bragg angle as zero. The corresponding glancing angle of incidence on the crystal surface we denote by wo, which is equal to 7r/2 - $'a, and is also indicated in Fig. 9. The range of that need be considered in these problems is seldom more than a few seconds of arc, and the distances of 0 and H from L are some hundreds of thousands of times greater than the distances of Poi, Pht and All At from the same point. For most purposes therefore we may consider the pencils of rays from these points converging on 0 and H to form two parallel bundles, and in Fig. 11 they are so drawn. Within the limits of the diagram, the circle of incidence now becomes a straight line perpendicular to LO. I n discussing the geometrical magnitudes involved in the figure we may consider any ray in the pencil converging on 0 to make an angle $'B with the normal to the surface, and any ray in the pencil converging on H

FIQURE 11

99

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

+

an angle ( $ B 200). These angles will not differ by more than a few seconds from $0 and $h respectively. lye now put From Fig. 11,

cos$B

=

YO, p/2 -

so that yo =

sin ( X

+ Oo),

cos

x

($B

= $B

f 200)

(27.2)

= Yh.

+ eo

Yh =

sin ( X -

(27.3) (27.4)

00).

These formulas are true generally, for Case I1 as well as for Case I, if the following conventions are observed. (i) Let j be a unit vector parallel to the atomic planes, and bisecting the angle between KOand Kh. Then the figure is always to be viewed from that side of the plane of incidence toward which the vector product j x KOis directed. (ii) The angle x is to be measured in a clockwise sense from the direction of j to the crystal surface. (ic) The angles $0 and +h are to be measured from the direction of the inward-drawn normal to the crystal surface, positive in a clockwise direction, negative in an anticlockwise direction. I n Fig. 11, the two branches of the hyperbola for one state of polarization are shown. Q, the center of the hyperbola, lies a t a distance k (1 $40) from the reciprocal-lattice points 0 and H , and L, the center of the Ewald sphere, a t a distance k . I n the examples to be discussed @o is always negative, and the distance QO, or QH, becomes k (1 - 3 1 1 ) , which is always less than k. The point L will then lie on the side of the hyperbola remote from 0 and H . For a simple lattice composed of dipoles, such as that assumed by Ewald in his first papers, L would coincide with the apse Vt of the hyperbola for state (a) of polarization, although not for state (b) ; but for an actual crystal, owing to the decrease of the atomic scattering factor with increasing angle of scattering, L must always lie outside the hyperbola, and it has been so drawn in Fig. 11. By Figs. 9 and 11, me may write

+

=

LPo%/k= PoiM/k sin 200

where LM is perpendicular to PoiH. We now define a quantity equation PoiH = k ( 1 a h ) so that PO'L?~ = kCYh. Then S#o = ffh/sin 200.

+

(27.5) ffh

by the (27.6)

(27.7)

The parameter LYh, which is proportional to LPoi, and so to 6$0, is thus a convenient measure of the angle of incidence referred to the Bragg angle as zero. We count it as positive if Poi lies to the right of L in Figs. 9 or 11, so that Po*H> LH.

100

R. W. JAMES

0

F I G U R12~

For most purposes it is, however, more convenient to refer the angle of incidence to a zero at P,, the point at which the perpendicular to the surface of incidence through Q, the center of the hyperbola, meets the circle of incidence, In Fig. 12, a portion of Fig. 11 has been drawn on a larger scale, and, to avoid confusion, only one wave point A has been shown and the dispersion surface itself has been omitted. The lettering of the two figures corresponds. If &4, is the angle of incidence measured from P,, = PqPoi/k = S#o

6#q

+ P,L/k.

(27.8)

Let L F in Fig. 12 be perpendicular to P,Q. Then and since By Eq. (27.4) , 1

LP,

LQ

=

=

FL/COS#B = LQ COB X / Y O

ik I 4oI /COS eo

w, = wo- (40/Yo)

(COB X/COS

- Y ~ Y =O W Y O {sin ) ( X + eo) - sin ( X - e,)

e,).

1

(27.9) =

(2/r0) cos x sin eo

and so, using Eqs. (27.7) and (27.9),

(27.10)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

This we write in the form

p/(2 sin 280)

=

where

p

=

2ah

- $0[1 -

101 (27.11)

(Yh/?'O)]-

(27.12)

I n what follows, we shall always use 8, or some convenient multiple of it, as the incidence parameter. It is directly proportional to &bq, the angle of incidence measured from P,. The second term of Eq. (27.10) is in effect the deviation from Bragg's law introduced by the dynamical theory, for, as we shall see, the diffracted wave has its greatest intensity when /i? = 0. 28. THEACCOMMODATION AND THE RESONANCE ERRORS We have now to calculate the resonance errors, as defined by Eq. (15.1) , in terms of the positions of the wave points iixed by the boundary conditions. In Fig. 9, we denote the vector PodA,which is perpendicular to the crystal surface by PoiA = kgn (28.1)

n being a unit vector in the direction of the inward-drawn normal to the

surface of incidence. The quantity g was called by Ewald the accommodation (Anpassung),and is denoted in his papers by 6. It fixes the position of the corresponding wave point when the conditions of incidence are determined. In the case illustrated by Fig. 11, there are two values of the accommodation for any angle of incidence. Referring now to Fig. 12, we see that if A E is drawn perpendicular to Po'O, and A D to Po'H, But and so, since $0 and Eq. (27.2),

Also and and so

PoiE

=

I ko' I - I KOI

=

Po'E

=

PoiA cos $0

kg cos $0

$B

=

-k~.

are always very nearly equal, we may write, using €0

Po'D PoiD

= =

PoiH

- AH

kg cos eh

(28.2)

= -970.

=

($B ah

=

k(ah

+ 28,) - gYh.

=

- th) kgYh (28.3)

Equations (28.2) and (28.3) give the necessary relations between the resonance errors and the accommodation in terms of the angle of incidence and the position of the wave point.

102

R. W. JAMES

29. THECALCULATION OF THE ACCOMMODATION g

By Eq. (18.3), the equation of the dispersion surface for two waves may be written (29.1) (40 - 2EO) ($0 - 2eh) = C24h& and by Eq. (20.1), putting z = &/Do x = DdDo =

(260

- 40)/C$iio

(29.2)

In these equations, C = 1 if D lies perpendicular to the plane containing the wave vectors, and C = c 0 s 2 8 ~if D lies in this plane; that is to say, C = 1 in state (a) of polarization and cos 200 in state ( b ) . If cos 2e0 = I cos 2e0 I , 2e0 .c n/2, but if cos 2e0 = - I cos 2e0 1 ; 2e0 > a/2, and it is therefore convenient to write

c = I C I (-1)r

(29.3)

where r = 0 except when D lies in the plane of the wave vectors, and at the same time 280 > a/2, in which case 7 = 1. This case occurs not infrequently, and allowance must be made for it in the formulas. Using Eqs. (28.2) and (28.3), we write (29.1) in the form

and expressing this in terms of the parameter p, defined by Eq. (27.12), a t the same time putting (29.5) Y = 9 (do/2ro), obtain Y[y - (b/2Yh) 1 = 1 c I 2 ( 4 h 4 d 4 ~ o Y hi ) (29.6)

+

the solution of which gives

whence g can be calculated as a function of the incidence parameter p, and of the quantities 4h, $A,40 and YO, Yh, which depend on the nature of the crystal and the geometrical conditions of the problem. For any state of polarization there are two possible solutions for g, corresponding, when they are seal, to the two points of intersection, A1 and Az, of PO" with

THE DYNAMICAL THEOHY OF X-RAY DIFFItACTION

103

the hyperbola. The solution is not, however, always real, and the significance of this will be discussed in due course. We shall make the convention that A1 and A2 correspond to the positive and negative signs respectively for the square root in Eq. (29.7), and refer to the associated wave fields as field (1) and field (2).

30. A SUMMARY OF FORMULAS To complete this section, we summarize a number of formulas that will be much used in what follows. When the absorption of the radiation in the crystal is negligible, 4 is a real quantity; and and + while i, they may still be complex, are conjugate. Absorption may be taken into account by supposing the scattering charge density, and the related dielectric constant to be complex, in which case c#J~, 4h, &+it and p will all be complex. Absorption will be considered in some detail in a later section, and to avoid unnecessary repetition the relations that follow will be given in their more general forms. It is to be understood that some of the quantities occurring in them may become complex when absorbing crystals are discussed. By Eq. (29.7), the accommodation g is given. by g =

-(+o/~Yo)

+ ( 4 ~ h ) - ~ { P EDz + 4 I C l 2 4 n 4 i ( ~ h / ~ o ) ] ' j f

with = 2ffh

- 40[1 -

(yh/yO)].

(30.1) (30.2)

By Eqs. (29.2), (29.3), and (28.2), 2EO - 40

- - 2709 -k 40 I c I 4!i I c 14%.

x = -Dh = Do

-

(30.3)

We now introduce a new incidence parameter p , proportional to p, and having the same zero, defined by

P

=

2

I c I ( I ?'h I /Y0)'(4h9%)*p*

(30.4)

For a nonabsorbing crystal p is real, but we shall see later that even when the absorption is finite, and p consequently complex, its real part still measures the angle of incidence to a close approximation. In Case I (transmission) both yo and ~h are always positive, but in Case I1 (reflection) yois positive and Y h negative, since J / h > ~ / 2 so , that yh = - I Yh I . Using this result, Eq. (30.4), and the further abbreviations

104

R. W . JAMES

it is easy to express Eqs. (30.1) and (30.3) in the following forms: Case I

(Yh =

Case I1

(

1 YO1

-

~ = h

: g =

-4o/2ro

2 =

Dh/Do

+ aCP

f (p’

- G b f (p2

=

+ 1)*1

(30.6)

+ 1)”.

(30.7)

1 Yh 1 ): - a[p

g = -@0/2Yo 5 =

Dh/Do

=

(30.8)

- 1j *] ( p z - 1)”.

=t(p’

GC?, =t

(30.9)

For any value of p , there are in general two values both of g and x, and so two corresponding wave fields. The quantities referring to fields for which the square roots in the above expressions are taken as positive and negative respectively are denoted by the suffixes (1) and (2). While p is directly proportional to the angle of incidence, and so suitable for expressing actual results, in calculation it is often easier to use another parameter v , defined in terms of p , but somewhat differently in Cases I and 11. a. Case I P ut p

so that (p’

=

sinhv,

+ L)t + p

(p’

= eu,

+ l ) $= coshv (pz

+ I)* - p

(30.10) = e-u,

(30.11)

- ae-v

(30.12)

Then Eqs. (30.6) and (30.7) may be written gi

=

-4&0

+ ma,

x1 = -Gev,

g2 = -40/270 52

=

(30.13)

Ge-a.

+

CQ , and is zero when p The parameter v may vary from - CQ to that is to say, a t P , on the circle of incidence.

=

0,

b. Case 11

Assume for the time being that p is real. We have then to consider threeranges; (i) p < -1; (ii) -1 < p < 1; (iiij p > + l . I n ranges (i) and (iii) the square root in Eqs. (30.8) and (30.9) is real, and there are real solutions for g and x, but in range (ii) we must write the square root in the form i(1 - p z ) * , and there is no real solution for g. For Case 11, v is defined in the following ways:

+

In range (i), p

< - 1, we put

p = - coshv,

(pz - l)t

=

sinhv(v

> 0).

(30.14)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

Then and

+ ( p z - 1 ) t = -e-v,

p

91

= -4n/%o

+ w-',

-Ge-v,

XI =

In range (iii), p

p - (p2 - 1 ) t

>

82

=

-+o/270

52

=

-Gen

=

-en

+

105 (30.15) (30.16) (30.17)

+ I , we put p

=

coshv

(30.18)

and the corresponding expressions are = -4n/27o

gi

XI =

Inrange (ii), - 1

- aev,

Gev,

x2

=

2 =

and we now put

92 = -40/270 =

- ae-'

Ge-v.

(30.19) (30.20)

+l,

-40/2yo - a h f i(1 - p 2 ) q

(30.21)

GC?, f i(1 - p2)t-J

(30.22)

p

Then gi

= -40/2y0

XI

= Ge",

=

cosv(v

- aeio,

> O),

92 = -40/270 92

=

(30.23)

- ae-iv

Ge-a".

(30.24) (30.25)

In all three ranges v is positive. If p is complex, we may suppose the results still to apply formally, by making v complex. The formulas of this paragraph are the basis upon which most of the discussion that follows rests. VI. The Thick Crystal with Negligible Absorption

31. INTRODUCTION

I n this section we shall consider the wave field in a crystal bounded by an infinite plane surface, upon which falls a plane wave of infinite lateral extent. The absorption in the crystal we suppose to be so small that in calculating the wave field it may be treated as negligible, and yet to be finite, so that if the crystal is thick enough no energy will reach its lower surface, and we need consider the boundary conditions only at the surface of incidence.

106

R . W. JAMES

When absorption is negligible 9 is real. The Fourier coefficients (bh may still be complex if the crystal has no symmetry center; but it will be clear from Eq. (12.5) that if so, the coefficients are conjugate, so that we may write (bh

=

1 6 h I exp (ih) 7

$= i

1 $‘h 1 exp

( -ieh) ;

&& = 1 4 h

l2

(31.1)

where o h is a phase angle depending on the structure factor F ( h ) . By Eqs. (12.5) and (12.6), 90is always real and negative, and we shall therefore write 90= -190 * (31.2) Equations (30.5) now become

32. CASEI (TRANSMISSION) : SOMEGEOMETRICAL RESULTS

Both KOand & are in this case directed into the crystal, toward its lower surface. Equations (30.12) and (30.13), which apply here, now become (32.1) g1 = 140 I /2r0 ue’, 92 = 140 I / 2 ~ 0- ue-u

+

XI

=

-Gee,

22

=

Ge-”.

(32.2)

Equation (32.1) shows that g1 is positive for all values of v (or p ) , and that, for positive v, g1 > g2. Reference to Fig. 11, in which Poi is in a position corresponding to positive v, and PoiA1 and PoiAz are the corresponding values of g1 and g2, shows that field (1) must be derived from A1 on the lower branch of the hyperbola, and field (2) from A2 on the upper branch. The variation of the fields with changing angle of incidence can be expressed very simply in terms of the geometrical properties of the dispersion hyperbola. I n Fig. 13, the circle of incidence is represented by the straight line LI parallel to the asymptote &To, and lines V1$ and V& have also been drawn parallel to &TO through the vertices Vl and V 2 of the hyperbola. The distance between LI and &Tois equal to the difference in length between LO and QO, which is 3k 140I , and the ratio of QV2 to Q L is I C 1 I 6 h I / 140I ; so that if h is the distance between the line V 2 S z and the asymptote &TO,we find, using Eq. (31.3), h = 3k

I c I I $h I

= kU(7o

I Y h 1 )*.

(32.3)

The normal to the surface through Poi, which determines the wave points A1 and A2, cuts VIS1, V9S2, and &To in S1, Sz, and R respectively,

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

107

FIGURE 13

and since &To is inclined a t an angle #B to the surface, and cos #B

S2R

=

h / r ~= ka(

1 Y h I /YO)) = ka/ 1 G I

= yo,

(32.4)

by Eq. ( 3 1 . 3 ) , and

(32.5) PoiR = PQQ= +k I $0 I /yo, which is the limiting value of kgz for large positive v, in agreement with Eq. ( 3 2 . 1 ) -Thus, from Fig. 13, kgl = Po'Az = Po'R

- AZR

=

$k

140

I /yo - A2R

and on comparing this with Eq. (32.1) we find and similarly

AIR

=

kae-9

AIR

=

kaeo.

(32.6)

The distance between the two wave poiiits is therefore given by

A1A2 = 2ka cosh v

=

2ka(p2

+ 1)i.

(32.7)

When v = 0, PO'coincides with P,, and AlA2 becomes the diameter of the hyperbola that is normal to the crystal surface, and this gives a geometrical interpretation to the quantity a of Eq. ( 3 1 . 3 ) ,If QBp is the semidiameter perpendicular to the crystal surface, which is the value of ASR when v = 0, QBz = ka = 1 G 1 S2R (32.8) by Eqs. (32.4) and ( 3 2 . 3 ) .

108

R. W. JAMES

Let &, 5h2 be the perpendiculars A2No,A2Nhfrom A2 onto the asymp totes &TOand &Th, respectively. Then, by the equation of the hyperbola, (19.2), their product is constant, and since at V2 each is equal to h, E02Eh2 = h2. By Fig. 13 h/ro

and so

A2R/&R

=

=

E02h

SIR,

=

t 0 2 / ~ 0=

(502/th2)*

=

A2R

I Dia I / I Do2 I = 1x2 I

(32.9)

by Eq. (20.2). Thus, as Poi moves along the circle of incidence, and A2 along the hyperbola, the ratio A2R/S2Rgives the ratio of the amplitudes of the two wave trains of field (2) ; and in an exactly similar way it follows that the ratio of the corresponding amplitudes in field (1) is given by

AiR/SiR

=

AiR/S2R

=

I

I / I Do1 I = 121 1 .

(32.10)

33. THEBOUNDARY CONDITIONS AND WAVEAMPLITUDES

Equations (32.9) and (32.10), with the help of Fig. 13, enable one to see very easily how the relative amplitudes of the wave trains of either field vary with the angle of incidence. It is clear, for example, that when v is large and positive I DOZI is very much greater than I Dk2 I , but that when t ) is very nearly zero they are comparable; corresponding results follow for field ( 1 ) . In order now to determine the relative excitation of the waves belonging to the different fields (1) and (2), it is necessary to use the boundary conditions at the surface of incidence. Let Doi be the vector amplitude of the incident radiation. I n Case I there is no wave above the crystal associated with KA,and Dhr = 0. In state (a) of polarization all the induction vectors, inside and outside the crystal, are mutually parallel and are tangential to the surface, so that we may here apply the condition of the continuity of the tangential component of D directly; but we have seen in Section 24 that since the dielectric constant of the crystal is so nearly unity we may treat the resultant induction vector for associated wave trains as continuous on passing into the crystal for any state of polarization. It has already been shown that the wave vectors koi,Kol, and K02 differ in direction by angles of the order of seconds of arc only, and in state (b) of polarization, since the waves are transverse, the ~ DhZ. For same will be true for Doi, DO1,and Do,, and of course for D Aand either state of polarization the boundary conditions, allowing for the phase continuity of related fields at the surface, lead to the pair of equations

Doi Dhi

+ Do2 Doi +D A ~ 0 =

=

(33.1)

relating the amplitudes of the waves. From this it will be seen that

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

109

I Dh1 I = I Dh2 at all angles of incidence, but that the two waves are opposite in phase. The equations may also be written

I

+ Do2 ziDoi + ~2D02 Do1

which on solution give

Do1 = [ z 2 / ( ~ 2 zl)]Doi, or, by Eq. (30.13),

=

Doi

=

0

Do2

=

(33.2)

- [ ~ 1 / ( ~2 x1)]DOi (33.3)

(33.4) (33.5)

The geometrical interpretation of these results is simple. By Eqs. (32.4), (32.6), and (32.8), Eqs. (33.4)-(33.7) become, in terms of Fig. 13, Do1

=

(A&/ A1A2)Doi,

Do2

Dhi = -G(Q&/AiA2)Doi,

=

(A1R/A1A2)Doi

(33.8)

Dn2 = G(QB2/AiA2)Doi, (33.9)

which satisfies Eqs. (33.1). The following special cases are of importance. (i) When v = p = 0, A2 and A1 lie at B2 and B1, the intersections of the normal P,Q with the upper and lower branches of the hyperbola. Then and so

A1A2 = 2B2Q = 2A1R = 2A2R Do1

=

Do2

+Doi

=

-&

(33.10) ~GDo', results that also follow directly from Eqs. (33.4)-(33.7). (ii) When A2 lies at the vertex VZ,A2R = S2R, and by Eqs. (32.4), (32.6), and (32.8), eu = I G [ and p = sinhv = ( 1 G 12 - 1)/2 I G I , and also A I R = QB2 I G 1 , A2R = &B2/ I G I, so that Dn2 =

AJ.2 = Q&(

=

I G l2

+ I)/ I G 1.

(33.11)

110

R. W. JAMES

By Eqs. (33.8) and (33.9) therefore,

+

1 Do1 1 = I Dod I /( 1 G I2 1) I Doz I = I Do'[ I G la/( [ G l2 + 1) I Dhi [ = I Dhz 1 = I G 1' I Do'I / ( I G la -I- 1).

(33.12) (33.13)

The corresponding results when A1 lies at the vertex 'V1 are obtained by interchanging the suffixes (1) and (2) in Eqs. (33.12) and (33.13). 34. DISCUSSION OF THE FIELDS IN CASEI

From Fig. 13 it will be seen that when v is large and positive Az lies becomes negligibly close to the asymptote &TOso that the ratio AZRIAIAZ small, while at the same time the ratio A1R/A1A2tends to unity. By Eq. (33.8), therefore, I Do1 I becomes negligibly small for large positive v, while at the same time [ DOZI approaches I DO$[ . The values of I Dhl I and [ Dh2 I are equal, and considering Eq. (33.8) in conjunction with Fig. 13 we see they lie between those of I Do1 and 1 DOZ1 , but they are very small for large v, since &Bzremains constant while AIAsincreases rapidly as I v I increases. For large positive v the only wave train in the crystal is therefore KO),and its amplitude is Dod,that of the incident wave. For large negative v, on the other hand, AIR becomes small, while AIR approaches AIAz, and there is again only one appreciable wave train in the crystal, having the same direction and amplitude as the incident beam; but this time it is Kol. For large positive and negative values of v the wave trains inside the crystal are the ordinary refracted continuations of the incident beam that would occur if the medium were isotropic and had the same average electron density as the crystal. There is actually a very small deviation of the refracted beam, but the approximations used in setting up the boundary conditions in effect neglect this. As v becomes small, so that the angle of incidence on the atomic planes approaches 00, A1 and A2 move towards the vertices of the hyperbola. AZR grows and AIR becomes smaller, and each is equal to iA1AZ when v = 0, so that (34.1) Do1 = Do2 = +Do' while Drl = -3GDo', D u = 3GDo'. (34.2)

I

The interference field is now fully established, and all four waves are of comparable amplitude. I DOZI and I DM [ are equal when A2 lies at V,, and I Dol I and I Dhl I when A1 lies at VI.If the atomic planes are perpendicular to the surface I G I = 1, and all four waves have equal amplitude when v = 0.

111

T H E DYNAMICAL THEORY O F X-RAY DIFFHACTION

FIG.14. Values of 1 Do/Doi 1 and 1 Dh/Doi I as functions of p for x = No,J G

I

=

1.23.

As v becomes negative, the “reflected” waves Khl and Kh2 die away again, 1 Do2 1 rapidly diminishes, and 1 Do1 I rises to the limiting value I DOd1 for large negative u. The region of diffraction has iiow been passed over, and there is once more only a single diffra’cted wave, this time Kol. I n Fig. 14, the variation of the amplitudes of the four waves, expressed as fractions of the incident amplitude, are plotted as functions of p . I n order to show the actual range of angles involved, a scale of actual angles of incidence A+* is given in the figure. The relation between p and A+* is, by Eqs. (27.11) and (30.4), (34.3)

so that (34.4)

In drawing the figure, the value 1.23 was used for I G 1 [ = (yo/ I Y h I )+] which corresponds to the 200 spectrum from rocksalt Kith Cu Kcr radiation (A = 1.54 A), when the surface of the crystal is parallel to (111). The value of 1 4ia 1 for this spectrum is 2.12 X 10-6, and these figures give A+q = 3 . 2 seconds ~ of arc; but the example is of course rather artificial, since copper radiation is strongly absorbed by rocksalt, and the whole discussion in this section refers to a crystal with negligible absorption. The figures are intended only to give the general magnitude of the effect, and to emphasize once again that the range of angle involved is extremely small.

112

R. W. JAMES

The relation of the two wave points for large positive and negative values of v will be clear from Fig. 7, in which they are denoted by Az and AI. Each point lies effectively on the circle of radius k ( 1 - 3 I 40 I ) and center 0. I n the region of interference, in which the transition from one field to the other occurs, the angular range of which is enormously exaggerated in the figure, the field is a complicated one. It is made up of four wave trains, which are coherent, so that interference phenomena occur, the nature of which will be discussed in due course. I n any actual case, both states of polarization will as a rule occur, and to each of these will correspond a field of the kind we have discussed. If the incident radiation is unpolarized, states (a) and (b) may be treated aa independent and incoherent, and the intensities of the two fields may be added. If, however, the incident radiation were itself polarized in some state other than (a) or (b), it could be expressed in terms of these two states, but the resulting fields would be coherent, and the phenomena correspondingly involved. An optical analogy is the production of ellip tically polarized light by a crystal.

35. REFLECTION FROM

A

THICK NONABSORBING CRYSTAL

This is the problem discussed by Darwin in 1914 in terms of multiple reflections from the atomic planes. The appropriate dispersion-surface diagram, drawn for the case in which y o > I Y h 1 and I G 1 > 1, is shown in Fig. 15, in which the lettering corresponds to that in Fig. 13. When g is

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

113

real, we may write by Eqs. (30.8) and (30.9)

1 /2Yo - aCP f (P2- 1141

9 =

I40

x

Dh/Do

=

=

G[p f ( p 2 - l ) i ]

(35.1)

giving to u and G their values in Eq. (31.3). Let P , and P1' be the positions of Poi at which the perpendicular through it to the surface of the crystal becomes tangent to the hyperbola at C and C' respectively, a t which points p is equal to +1 and - 1. I n range (iii), Eq. (30.18), Poilies to the right of P t . The perpendicular POWthen makes two real intersections, with the same branch of the hyperbola, at the two wave points A 1 and A2. As v (or p ) diminishes, A1 and A2 draw together, and coincide at C, which marks the boundary between range (iii) and range (ii),when p = 1. When Poi lies between P t and Pt', and p between +1 and -1, the solution of Eq. (35.1) for g is complex, and the normal makes no real intersection with the hyperbola. When Poi lies to the left of P,' and p < - 1, there are again two real wave points, which both lie on the other branch of the hyperbola. Consider first range (iii), in which p > +l. By Eqs. (30.19) and (30.20), (35.2) g2 = I 4oI j2Y0 - ue-0 g1 = I $0 1 /2y0 - a e v ,

x1 with p

=

=

cosh v ( v

Gev,

22

=

Ge-"

(35.3)

> 0).

It will be clear on comparing Eq. (35.2) with Fig. 15 that field (2) corresponds to Az, the lower point of intersection with the hyperbola; for kgz is equal to PoiA2, and tends to the limit k I 40 1 /2yo for large v ag, A2 approaches the asymptote &To, while at the same time kg,, which is equal to PoiA1in magnitude, assumes large negative values.

Equations (32.9) and (32.10), which depend on the geometrical prop erties of the hyperbola considered in Section 32, still apply, and so

I XI 1 I x2 I 1 XI 1

=

AIR/&&

=

=

AzR/SzR

=

I Dhi I / I Do1 I 1 Dh2 1 / I Do2 I .

(35.4)

In range (iii), increases without limit as v increases while I x2 I decreases, which is also clear from Eq. (35.3) which is the geometrical expression of Eq. (35.4). I n range (i) the reverse is true. Here g1 =

140I /2ro

+ m-',

572

=

140

I /2yo

+ ueD

(35.5)

with p = -cosh v, ( v > 0). A1 is again the upper wave point, but it now approaches the asymptote &To as v becomes large, and p becomes large and negative. It should be noted that v is positive in all three ranges in Case 11.

114

R. W. JAMES

36. THEBOUNDARY CONDITIONS

There is now a reflected wave above the surface of the crystal, of amplitude Dhr, and the boundary conditions become

Do1

+ Do2 = Do'

XiDoi -k

=

x 9 0 2

(36.1)

Dhr.

I n range (iii), 1x1 1 increases without limit as v increases, and since the ratio 1 Dh' [ / 1 Doi 1 must be finite, and not greater than unity, we see that field (1) alone is not physically possible in this range. On the other hand, since all the necessary conditions can be fulfilled by field (2) alone, we may assume that in range (iii) it is the only field; while a similar argument shows that in range (i) field (1) alone exists. Using Eq. (36.1), we may therefore write

<

Inrange (i), p

-1:

Do, In range (iii), p

>

=

+ 1:

Doi, Do2

Do2 = Doi, Dm

Dhl =

Dh2

Dh2 =

Dhl

x ~ D o= ' Dhr

= =

=

0.

xpDoi

=

(36.2)

=

Dh'

0.

(36.3)

In both ranges ,(i) and (iii) only a single wave point is effective, range (i) and A2 in range (iii). In range (iii), by Eq. (35.4), 1x2

I

=

I Dh2 I / I Do2 I

=

I Dhr [ / I Doi [

=

A2R/S2R.

A1

in

(36.4)

I

By Eq. (36.3), the amplitude [ Do2 of the wave KO,is constant in this range, and equal to the amplitude of the incident wave. For large v, 1 Dh2 I is small, and effectively only one wave train is then present in the crystal, in the direction of the incident radiation, just as in Case I. It will be plain from Fig. 15 that I Dh2 I increases as v becomes smaller, and becomes equal to 1 Do2 1 when A2 coincides with the vertex V 2of the hyperbola. The field then consists of two wave trains, the direct and reflected trains. Their phase relationship depends on e h of Eq. (31.1), and this point will be considered for a particular case a little later. The wave point A 2 coincides with the vertex of the hyperbola when 1x2 \ = 1, or when (36.5) e-u = 1/ I G l = ( I Y h I / Y u ) ' .

Since v > 0, this is only possible when I G I > 1, the case illustrated in Fig. 15, when the inclination of the atomic planes to the surface is such that ?r/2 - #B > Bo. If ?r/2 - +B < Bo, so that I G 1 < 1, Eq. (36.5) cannot

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

115

be satisfied in range (iii). I n this case, the point A*, moving along the hyperbola as v diminishes, will meet the tangent point before arriving at the vertex VB,and I D M 1 is then always less than, I Do2 [ in range (iii) , The two possible cases are illustrated in Fig. 16. I n range (i) conditions are the same, except that everything that has just been said about field (2) there applies to field (1). When p is large and negative there is again a single wave in the crystal in the direction of the incident wave train, but it is now the wave KOI. 37. THEREFLECTION COEFFICIENT

We define the reflection coefficient R as the ratio of the energy reflected from any area of the crystal surface to the energy falling upon the same area from the incident beam. Since the latter makes an angle $0 with the normal to the surface and the reflected beam an angle #h, the cross sections of the beams incident on and reflected from a given area of the surface are, to our degree of approximation, in the ratio of 70to I Y h 1 , To the same approximation we may take the current density of the energy in each beam as proportional to the square of the electric induction, so that, using Eqs. (36.2) and 36.3), we may write for the reflection coefficients of the two fields

Since [ z1l2 = I G 12e-2p in range (i) and 1x2 I* has the same value in range (iii) , the reflection coefficient in each range is

R

=

eq-20,

(37.2)

116

R. W. JAMES

It decreases exponentially with increasing v, but in each case is equal to unity, that is to say the reflection is total, when v = 0 at the limits of range (ii). In range (ii) itself, in which -1 < p < +1, we may write x

so that for either field

=

R

GC1, f i(1 - p 2 ) > f ] , =

~ x I ~ / =( 1, G ~ ~

(37.3)

that is to say reflection is total throughout range (ii), between p = - 1 and p = $1, but outside this range falls away very rapidly with increasing I p I . This is the result obtained by Darwin, using an argument of a different type. 38. THENATUREOF THE WAVEFIELD WHEN g IS COMPLEX. PRIMARY EXTINCTION When Po'A is a real vector, perpendicular to the crystal surface, it is clear from Fig. 9 that KO= koi - kgn

Kh

=

+

RI, koi - kgn

(38.1)

where n is a unit vector in the direction of z, normal to the crystal surface and directed inwards. In range (ii) g is acomplex quantity, and so KOand K,,become complex wave vectors, if we suppose Eq. (38.1) still to apply formally. Suppose now that in the wave

u(x, t )

=

a exp (27ri(vt - K x ) 1

(38.2)

traveling in the direction x , the wave vector K is complex and equal to K , iKi.Substituting this in Eq. (38.2), we have

+

u(x, t ) where

=

a exp (27riKa) exp {27r(vt - K G ) }

= a exp ( - i u x ) u =

exp {2r(vt

-4nKi.

- KG) }

(38.3) (38.4)

This represents a plane wave of wave vector K , traveling in the direction of x , the amplitude of which varies exponentially with x , decreasing with z if u is positive, and so Ki negative, but increasing with x if Ki is positive. If Ki is negative, u is the linear absorption coefficient for the wave, calculated for intensity in the usual way.

THE DYNAMIC&

Putting g = gr

THEORY OF X-RAY DIFFRACTION

117

+ igi in Eq. (38.1),we find - ikgin Kh = (ko' + Rh - k g 4 ) - ikgin. KO= (ko' - kgrn)

(38.7)

The z components of the wave vectors are therefore complex, and since the imaginary part is negative, this represents an absorption of both fields with increasing depth in the crystal, with the same absorption coefficient ue, given by (38.8) ue = 4nkgi = 4 r k a ( l - p2)>" for each. The value of g used above is that for field ( 2 ) , Had we used the value for field (1)) Qr - igi, the complex part of the z component of the wave vector would have been positive, and this would have meant that both wave trains increased in amplitude exponentially with depth, a physically impossible state of affairs. We therefore conclude that in range (ii) also only a single field is possible-field ( 2 ) , The amplitudes of the two wave trains that constitute the field, KO, traveling downward, and Kh2 traveling upward, diminish with depth at the same rate. At the surface of the crystal this represents a deviation of the energy current of the incident beam into the direction of the reflected beam, or, in other words, total reflection. Inside the crystal there is an energy current parallel to the surface, diminishing very rapidly with depth. It must of course be borne in mind that both the crystal surface and the wave fronts are assumed to be of infinite extent, and a stationary condition, assumed to be already established, is being discussed. The existence of absorption of this type was shown by Danvin in his 1914 paper, and called by him extinction, and later primary extinction, to distinguish it from an effect of a different nature, to be considered briefly in Section 42. Primary extinction has nothing to do with the ordinary absorption of radiation in the crystal, which we have assumed to be negligible. I t involves no irreversible loss of radiation energy, but merely a redistribution, and is in fact an interference effect of a special type.

118

R. W. JAMES

39. THEMAQNITUDE OF THE PRIMARY EXTINCTION

By Eqs. (38.8) and (31.3), putting k = 1/X, we find for ue (39.1)

The extinction has its maximum value at the middle of the range of total reflection when p = 0, and is zero at either end of the range when p 2 = 1. From Eqs. (39.1) and (12.7), (39.2)

Extinction is thus most marked for strong spectra. To obtain an idea of the magnitude of the effect we assume the crystal planes to be parallel to the surface, so that 70 = 1 Y h I = sineo, and using the numbers given for rocksalt in Section 17, we find for state ( a ) of polarization, when I C I = 1, ur ( m a ) = @/sin 0), ( 1 F ( h )

1 / I F(0) I )

X 3.52 X lo8em-'

when X is expressed in A units. For a given spectrum from a given substance the extinction is independent of X, since 1 F ( h ) I is a function of (sin @/A. It will be greatest for strong spectra occurring at small glancing angles. For rocksalt, I F(200) I / F ( 0 ) I A 0.77, (sinBo)/X = 0.178, and so

I

ue ( m a ) =

1.52 X

lo4 cm-l.

This is nearly 1000 times the linear absorption coefficient of rocksalt for Mo K a radiation. 40. THERANGE OF TOTAL REFLECTION AND

BRAGQ'S LAW

THE

DEVIATION FROM

The range of total reflection extends from p = - 1 to p = +1, and so, by Eq. (34.4), over an angular range (40.1)

The middle of the range occurs when B ah

=

=

-$ 140 1 c1 - ( r h / Y o > ] = -3

0, arid so by Eq. (27.12), when 190

1 c1 + ( I Y h I /Yo)]

(40.2)

119

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

since TI, = the range,

- I T h I in Case 11. By Eq.

(27.10)therefore, at the middle of (40.3)

At the angle of incidence corresponding to reflection from the atomic = 0. The middle of the range planes at the Bragg angle, $0 = $B, and of total reflection thus occurs a t a value of $o smaller than $B, or at a value of the glancing angle of incidence wo larger than w g i by the same amount, where w g i = r/2 - $B. The middle of the range of total reflection thus occurs at a glancing angle wBi Awo, where

+

Aw0

=

-(I I $0 I 2 sin 2e0

+

Y),

(40.4)

By Eq. (40.4),AWOis always positive, and if we take the middle of the range of total reflection as the angle at which the spectrum is formed, the glancing angle predicted by Bragg's law is a little too small. If 1 - 6 is the refractive index of the crystal for the radiation when only one wave is appreciably excited, we may write Eq. (40.4)in the form AWO= 7 6 (I

sin 200

+

F),

(40.5)

and if the crystal planes are parallel to the surface this becomes ( A w o ) =~ A00 ~ ~ = 26/sin 200,

(40.6)

the result obtained by Darwin in 1914. Using Eq. (27.4) it is a simple matter to show that

+

3 190I { t a n & tan+B) = 3 140 I {tanOo+ cot w B i }

A ~= o

(40.7)

an expression first given by Ewald,ls which shows that Awo is abnormally large if w g i is small, that is to say, if the crystal is so cut that the incident radiation just grazes the surface when the spectrum is formed, a result that has been verified experimentally by Bergen Davis and von Nardroff.ls It is more usual to measure the glancing angle of emergence, and it is not difficult to calculate the deviation from Bragg's law in this case, which we denote by Awh. In Fig. 17 the various relevant angles are indicated. SS' is the crystal surface, PP' shows the direction of the crystal planes l6 P. P. Ewald, 2.Physik SO, 1 (1924). "Bergen Davis and R. von Nardroff, Proc. Natl. Acad. Sci. U.S. 8, 357 (1922).

120

R. W. JAMES A

1

N

FIGURE17

concerned, which is normal to Rh. AOA' indicates the actual path of the rays for the middle of the range of total reflection, and BOB' that given by Bragg's law; so that BOA = Aw0 and B'OA' = A w h . Because of the equality of the tangential components of corresponding wave vectors inside and outside the crystal, or

khi

- ko: = K h : - KO:= R h : k(sin $h - sin $0)

For related small deviations

A#h

=

=

constant

constant.

and A$o we may therefore write

COS h A h = COS $o&

or From Fig. 17, wo

=

- I YF I A $ h = YoA#o. ~ / 2- $0; w h = $h - ~ / 2 and , SO

(40.8)

According to the geometrical theory, the deviation of the radiation as a ~ may result of the reflection is 21%.The increase in the deviation, A ( 2 8 0 ) we therefore write A(200) = AwO

+

Awh

=

-I({

f

sin 21%

%)

f (1 f

6))

By Eq. (27.4), to our degree of approximation, YO = cos#o = sin

IYhI

=

1 cos $h 1

=

(00

+ x) = sin wo

sin (e, -

x)

=

sin W h

(40.10)

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

so that by substitution in Eq. (40.10) A (20,) = ( 6 cos x/cos 0,) { cosec w 0

+ cosec w,).

121

(40.11)

The first term in Eq. (40.11) becomes large if the crystal is cut so that the incident ray nearly grazes the surface, while the second term is large if the emergent ray does so. For symmetrical reflection, with the planes parallel to the surface, x = 0 and wo = wh. The right-hand side of Eq. (40.11) then becomes 26 sec B0 cosec e,, or twice Darwin's expression for the deviation from Bragg's law, which is correct.

FIQ. 18. The Darwin reflection curve.

The reflection coefficient R(w0) is shown in Fig. 18, plotted as a function of the incidence parameters p , wo,and A+*. The range of total reflection is proportional to I C#V, I , and so to the structure factor of the spectrum concerned. Even for the strongest spectrum, the range is only of the order of a few seconds of arc. It is greater for state (a) of polarization than for state (b) in the ratio 1 : I cos 2e0 1 , and becomes zero for state (b) when Bo = 45". This result is true, however, for both geometrical and dynamical theories, and depends on the nature of dipole scattering, upon which both are ultimately based. 41. THEINTEGRATED REFLECTION

The integrated reflection, which is the usual measure of the intensity of a diffracted beam, is defined by p =

J

R(w) dw

(41.1)

where the integral is taken over the whole range of appreciable reflection.

122

It. W. JAMES

In terms of the parameters p and

I),

we may write

= - (dw/dp)

/

R ( v ) (dp/dv) dv, (41.2)

since by Eq. (34.4) dw/dp, which is equal to - d ( A $ J / d p , is independent of p or v. Over the range of total reflection, which extends from p = - 1 to p = +1, R ( p ) = 1, and its contribution to p is therefore -2doldp. In range (iii), R ( v ) = e-2v, and p = cosh v, dp/dv = sinh v. The integration extends from v = 0 to v = rn , and the corresponding contribution to pis

- (dw/dp)

Am

e-2v sinh v dv = - (1/3) ( d w / d p )

Range (i) gives the same contribution as range (iii), and the total integrated reflection is therefore p =

-(2

+ $) dw/dp

(41.3) I /sin2@o)( I ^/h I /yo)* = % I cI I by Eq. (34.4). If the incident beam is unpolarized, consisting of equal and independent components in states (a) and (b) , we may use the average value of I C I for the two states, and, using Eq. (12.7), write

8 A* , I F ( h ) 1 e2 ( I Yh 3~ sin200 V n z ~ 2

p=-----

I ')OY/

1

+ I cos 2e0 I 2

(41.4)

This is the expression derived by Darwin by considering multiple reflections from a set of planes parallel to the crystal surface, in which case yo = 1 Yh 1 . The integrated reflection from a face, defined in this way, is a dimensionless number, and is proportional to the jirst power of the structure factor and not to its square, as it would have been according to the geometrical theory. If the crystal is regarded as a mosaic of very small, nearly parallel, but optically independent blocks, the expression corresponding to Eq. (41.4) for symmetrical reflection is

I n deriving it we suppose each block to be small enough to reflect geometrically, and allow for the absorption on the way to and from a block lying below the surface of the crystal by using the ordinary linear absorption coefficient p. I n deriving Eq. (41.4), p ww assumed to be.negligible

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

123

in comparison with the extinction over the reflecting range, which is nearly justifiable for a crystal such as aluminum, to take an example, with Mo Ka radiation. The difference between the results of the dynamical and geometrical theories is indeed well-illustrated by this example. James, Brindley, and Wood17 observed the absolute value of the integrated reflection 111 from a single crystal of aluminum to be 580 X lo-". The values calculated from the dynamical and mosaic formulas were respectively 19.6 X 10" and 818 X The spectrum 111 from aluminum is very strong, and for the weaker ones the discrepancy is less; it is still large, however, and it is quite clear that this crystal reflected much more nearly as a mosaic crystal, according to Eq. (41.5), than as a perfect one, according to Eq. (41.4). Analogous results are obtained with natural rocksalt, but RenningeP has shown that crystals of sodium chloride can be grown in the laboratory from a melt, and these crystals give integrated reflections in close agreement with the Darwin formula, if care is taken to choose very perfect specimens. 42. SECONDARY EXTINCTION

Brief mention must be made of an effect that is of importance for strong reflections from mosaic crystals. When a crystal is actually reflecting its effective linear absorption coefficient, to be used in Eq. (41.5), may be materially greater than the ordinary value when reflection is not taking place, even for a mosaic crystal. For our purpose, a typical mosaic may be supposed to consist of optically independent fragments, each so small that dynamical equilibrium cannot be set up within it. While any such fragment is reflecting, the radiation that passes through it is diminished, and less falls on any other fragment lying below it in the path of the beam. If the crystal planes of two such fragments are so nearly parallel that each reflects over the same angular range, the lower one will reflect less than it would have done had the upper one not been there and at the same time reflecting. The result is to produce an enhanced effective linear absorption coefficient, and is the more important the more nearly parallel the crystal fragments and the greater the reflection for the spectrum concerned. This effect is known as secondary extinction. Its existence was shown experimentally by Bragg, James, and Bosanquet4 (see ii) and it was discussed theoretically by Darwinlg in 1922. For a fuller account of the phenomenon and of the earlier experimental work, reference may be made to Optical Principles,ll Chapters I1 and VI. For more recent work a very interesting paper by Chandrasekhar20should be consulted. 17R. W. James, G . W. Brindley, and R. G. Wood, Proc. Roy. SOC.Al26, 401 (1929). M. Renninger, 2.Krist. 89, 344 (1934). lOC. G . Darwin, Phil. Mug. [6]43, 800 (1922). aa S. Chandrasekhar, AdYances in Phys. Phil. Mag. Suppl. 9, 363 (l9W).

124

R. W. JAMES

43. THENATUREOF THE WAVEFIELDIN CRYSTAL, CASEI1

A

THICKNONABSORBING

In Case 11, as we have seen, for any state of polarization only one wave field occurs in each of the three ranges of reflection. Field ( l ) ,associated with the wave point .A1, occurs in range (i), and field (2), associated with A*, occurs in ranges (ii) and (iii). The electric induction D at time t at a point defined by the vector r is therefore given by D =

Po+ Dhexp (-2airsRh)l

exp {2ai(vt - K o - r ) ) . (43.1)

This may be considered formally as a progressive plane wave, of wave vector KO,having an amplitude that is constant in time but is a function of the coordinates, repeating itself in the same way in each lattice cell. I n such a wave the propagation of energy will not as a rule be in the direction KO,and this point is discussed in more detail in Part X. The field may also be considered as the sum of the two ordinary plane wave trains KOand Kh;but neither of these alone is a solution of the wave equation, and their amplitudes are connected by relations that have already been derived. As before, we put Dh/Do = 2. I n the three ranges of incidence the relevant values of x are

(9 (ii) (iii)

<

-1

- coshv

-1,

x1

=

-Ge-P,

p =

$1,

22 =

Ge-IU,

p = cosu

z2 =

Ge-u,

p = coshv.

p >0+1,

(43.2)

In each range u is positive. I n ranges (i) and (iii) it extends from 0 to infinity, and in range (ii) from a to 0. The quantity in square brackets in Eq. (43.1) is a vector sum, but for the moment we consider state (a) of polarization, in which Doand Dh are parallel, and perpendicular to the plane containing KO and Kh.I n this case, if we put z = /x(ei6 (43.3) we obtain from Eq. (43.1)

I D/Do la

I 1 + z e x p { - 2 d ( r * R h ) } l2 = 1 + 1 z j2 + 2 I z 1 cos {2a(r-Rh)-

=

6).

(43.4)

Let us consider a particular case, that of a simple primitive lattice composed of atoms having absorption frequencies considerably lower than Y, the frequency of the incident radiation, so that dispersion effects can be neglected. We may then take the phase angle o h of Eq. (31.3) a8 equal to T , the change of phase of the radiation when it is scattered by atoms of

T H E DYNAMICAL THEORY O F X-RAY DIFFRACTION

125

FIGURE19

this nature. In state (a), 7

G

=

0. Then, by Eq. (31.3),

= ei"(yO/

1 Yh 1 ) '

=

-(YO/

1 Yh I If,

(43.5)

and by Eq. (43.2)) 6 = 0 for range (i), ir for range (iii), and ir - v for range (ii) . The relation between the phase 6 and the incidence parameters p and v is shown by Fig. 19. I n range ( i ) , field (1) alone is present, and, by Eq. (36.2)) 1 Do, 1 is equal to 1 Doi 1 , the amplitude of the incident radiation, and is independent of the depth in the crystal or of the angle of incidence; and in range (iii) the corresponding statements are true for field (2). In range (ii), only field ( 2 ) is present, but at a depth z below the surface of the crystal the intensity of the field is reduced by extinction to exp ( - u e z ) of its value a t the surface. It is convenient to express the ratio of the intensity of the field in the crystal to the intensity of the incident radiation as a function of the incidence parameter v and of position vector in the crystal in the form

1 DIDOil2

P(v, r)

(43.6)

1

+ 1 G 12e-20 + 2 I G I e-D cos (2n(roRh)]

(43.7)

1

+ 1 G j2e-2u - 2 I G 1 e-u

(43.8)

Then, in range (i),

P(v, r)

=

and in range (iii),

P ( v , r) while in range (ii)

P(v, r) = [l

=

+ I G l2

=

- 2 I G I cos {2ir(r'Rh)

COS

[ 2ir(r'Rh) }

+ v}] exp

(-uez)

(43.9)

where I G I = (YO/I Yh I 14. It will be seen from Eqs. (43.7)-(43.9) that apart from the effect of extinction in range (ii) which causes a diminution of the whole field with increasing depth, the intensity is constant over any plane parallel to the

126

R . W. JAMES

atomic planes, since over any such plane (raRh) is constant; and by Eq. (43.4) this is true whatever assumption may be made about Oh. In the particular case considered, in which Oh = T , and the state of poIarization is (a), the atomic planes themselves, for which r*Rh is integral, lie at the maxima of the field in range (i) and at the minima in range (iii). For a first-order spectrum, a minimum lies halfway between the planes in range (i) and a maximum in range (iii). For a spectrum of order n there will be n minima and n maxima between the planes in ranges (i) and (iii), respectively. For range (ii), the maxima lie on the planes when p = - 1, at the boundary between range (i) and range (ii), but as the angle of incidence passes through the range of total reflection the maxima pass away from the planes, and when v = 0 and p = + 1 a t the beginning of range (iii), the planes lie at minima. In ranges (i) and (iii) the maxima and minima of P ( v , r) are ( 1 =t 1 G I e-.) 2, and the difference between them is only marked when v is small and the wave Kh is appreciably excited. For large v, P ( v , r) approaches unity, and only Do1 is present in range (i) and DoSin range (iii) . Throughout range (ii) , the maxima and minima have their extreme values, (1 f I G 1 ) z , leaving the extinction factor out of account for the moment. These values become 4 and 0 when the planes are parallel to the surface of the crystal, and 1 G I = 1. The stationary maxima and minima considered here are not those of stationary waves in the usual sense, as the form of Eq. (43.1) shows. They are associated with a propagation of energy through the crystal, which in the range of total reflection must be parallel to the surface, and, when v is large, virtually in the direction of the incident beam. 44. THE FIELDON

THE

ATOMIC PLANES

On the atomic planes themselves, where r-Rhis integral, P ( v ) has the following values: (i)

P ( v ) = (1

(iii)

P(v)

(ii)

=

+ 1 G J e-u)2,

p

=

- coshv

(44.1)

+1,

p

=

coshv

(44.2)

- 2 p I G 1 ) exp ( - c e z ) , -1 < p < $1, p the value of I DIDOi J z

=

cosv.

(44.3)

(1 - 1 G 1 e-u)2,

P ( v , z ) = (1 = (1

< >

-1,

p

p

+ I G l2 - 2 1 G 1 cos v) exp(-uez)

+ I G 1'

In Eq. (44.3), P(v, z ) is at a depth z in a nonabsorbing crystal. In ranges (i) and (iii), P ( v ) is independent of z.

127

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION A 4

Field ( 2 )

I

-5

I

-4

I

-3

I

-2

Ronqe (i)

-I

I

0 Ranqe (ii)

I

I

I

2

I

3

I

4

I

5

Ranqe (iii)

FIGURE 20

In Fig. 20, P is plotted as a function of p . The full curve in range (ii) refers to a point just below the crystal surface, and here P ( p ) is a linear function of p in range (ii), having values ( 1 I G I ) 2 and (1 - I G 12) at p = - 1 and p = +1, respectively. At depths below the surface extinction comes into play. It is greatest at p = 0, and zero at p = f l . The dotted line represents its effect at a depth at which the maximum extinction factor is 0.05, still very near the surface. The curve in Fig. 20 is drawn for I G 1 < 1, corresponding to Fig. 16b. When p = +1, P ( p ) lies between 0 and 1, and when p = -1, P < 4. For symmetrical reflection, when the crystal planes are parallel to the surface, the corresponding values are 0 and 4. If I G I > 1, P ( p ) may become zero. By Eq. (44.2) this will occur when I G I = eu. In this case

+

P=3(1GI+1/lG/),

which is necessarily grea.ter than unity, so that the zero value will occur in range (iii) for a value of p a little greater than, and a value of the glancing angle of reflection a little less than, that corresponding to the limit of total reflection. P ( p ) approaches unity for large p and becomes less than unity as p diminishes in range (iii) . I n range (ii) it rises to a very sharp maximum, persisting over a very small range of angles near p = - 1. When p becomes less than - 1, the value of P ( p ) falls rapidly, approaching unity for large negative p , but always remaining above it. These results may be interpreted in terms of the wave trains existing in the crystal and the discussion in Sections 35 and 36. When p is large

128

R. W. JAMES

and positive Do2 alone exists. As p becomes smaller, I Do2 I remains constant, but I D h 2 1 meanwhile increases relative to it, as reference to Eq. (35.4) and Fig. 15 will show. In the particular case considered, 6 , the phase difference between the two waves of the field, is equal to s in range .(iii), and the resultant amplitude is I D02 I - 1 Dh2 I , which becomes smaller as 1 Dh2 1 grows and is zero when AS lies at the vertex of the hyperbola in Fig. 15. If I G I = 1 and the planes are parallel to the surface, this will occur when p = 1. When I G I > 1, the case illustrated in Fig. 15, it will occur when p > 1; but if I G I < 1, A2 cannot lie at the vertex of the hyperbola, and P cannot then become zero. Figure 20 refers to this case. In range (ii) , 6 changes progressively from s to 0 as p diminishes from +1 to -1, and the two waves come into phase. The resultant amplitude reaches a maximum when p = - 1. In range (i) , 6 = 0 and the two waves of the field are in phase. The resultant amplitude is I Dol 1 f 1 Dhl I , since now only field (1) is excited, and is a maximum when p = -1, and the two waves have nearly equal amplitude. As p becomes negative I Dhl 1 falls away rapidly and the resultant amplitude diminishes, approaching I DOl I , so that P ( p ) approaches unity from above for large negative p . The sudden change from field (2) to field (1) at the junction of ranges (ii) and (i) may at first sight appear paradoxical; but it must be remembered that in range (ii) there is no real wave point. The appropriate field is formally (2), because to get a physically acceptable solution in the range p2 < 1 we must take the negative sign for the square root in Eq. (37.3). When at p = -1 the wave points become real again; A1 and A z coincide, so that there is no &continuity involved. So far, only state (a) of polarization has been considered. In state ( b ) , DOand Dh lie in the plane of the wave vectors KOand Kh, with an angle 280 between them. Remembering that the amplitude in Eq. (43.1) is a vector sum, we obtain at once

I D/Do l2

=

1

+ I z l2 + 2 1 z 1 cos 2e0 cos (2s(r.Rh) - 6 ) .

(44.4)

The necessary modifications of the formulas follow very simply. For the symmetrical case, with 1 G I = 1, we now have

(i) (iii)

p p

< -1, > +1,

P(v)

=

P(v)

=

+ e-u)z - 4e-usin280 (1 - e-u)2 + 4e-"sin2BO.

(1

(44.5)

Then P = 4 cos2Bowhen p = -1, and 4 sin2Bowhen p = +1, so that P does not now vanish even in the symmetrical case at the lower limit of total reflection. This is to be expected, for the two field vectors DOand Dh are no longer parallel, and can never completely interfere, even when their amplitudes are equal and their phases opposite.

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

45. THENATUREOF

THE

STATIONARY WAVEFIELD IN

THE

129

CRYSTAL

The spacing of the nodes and loops of amplitude in the wave field is not determined by the wavelength, as in the more familiar types of stationary wave. The amplitude is a lattice function, and accommodates itself to the periodicity of the crystal. If an atom lies in a strong field or a weak field in one unit cell of the crystal, so does the corresponding atom in every unit cell; and this is a most important characteristic of the dynamical theory. We return for a moment to Case I, that of transmission, in which, as we have seen, fields corresponding to two wave points must always coexist. Once more we consider the atomic planes of the simple crystal discussed in the last paragraph, for which r-Rhis an integer. By Eq. (43.1) the amplitude on the atomic planes is Do Dh, a vector sum, in which each term may be complex. If we consider only state (a) of polarization, the addition is algebraic. By Eq. (30.13),

+

x1

=

-Ge',

xz = GerV

where now p = sinh v, and v may vary over the whole range from - a) to -l-03. Putting r = 0, and assuming 8 h to be T , as before, we find, using Eq. (31.3), 52 = - I G I e-". 2 1 = I G I eu, On the atomic planes, the two waves KOand Kh are therefore in phase for field (1) and out of phase for field (2) , and their amplitudes are I Do1 I 1 0 6 1 I and I Do2 - I D h 2 I . In the region of interference, therefore, when the two constituent waves of either field approach equality of amplitude, the field strength becomes large on the atomic planes for field (1) and small for field (2), and this has an important consequence. So far as field (2) is concerned, in the immediate neighborhood of an interference maximum all the atoms in the crystal lie in a field that may be much weaker than the average field due to an ordinary progressive wave of the same total energy. Since photoelectric absorption is due to the abstraction of energy from the wave field by the atoms, it must be reduced if the atoms lie only in regions of small intensity, and we should therefore expect the absorption coefficient of the crystal to be abnormally small for field (2) in the circumstances considered. On the other hand, so far as field (1) is concerned, the atoms lie in a field considerably more intense than the average field, and the absorption should be correspondingly enhanced. This expectation is confirmed by experiment, and the problem will be considered in detail in Part VII.

I

+

130

R. W. JAMES

46. THEOSCILLATORY OR PENDULUM SOLUTION IN CASEI

The resultant field in Case I has an interesting characteristic, first pointed out by Ewald. Suppose the angle of .incidence to be that corresponding to p = 0, so that the relevant wave points A2 and A1 lie on the normal to the surface that passes through Q, the center of the hyperbola and the maximum reflected wave train is excited. The geometrical circum-

F~GURE 21

stances are represented by Fig. 21. By the equations of (34.2), Section 34,

$Do'

Do1 = Do2

Dn2 = $GDoi.

Dhi = -iGDoi,

We consider state (a) of polarization in which all the induction vectors are parallel to one another and perpendicular to the plane of incidence. The resultant of the two wave trains KO,and KO,is

PO= +Do6{cos2?r(vt- Kol-r)

where

-

+ cos2a(vt - KO2-r)) - &-r))

= D O ~ C {O~S( K o P Kol)-r) cos {2?r(vt

g o

=

+

IKOI

=

k ( 1 - 3140I).

(46.1)

(46.2) $(Koi K02) and is equal to the vector from Q to the reciprocal-lattice point 0,so that (46.3)

Equation (46.1) represents a progressive wave of wave vector I(0 with planes of constant amplitude parallel to the surface of the crystal; for the first cosine factor is constant over planes perpendicular to the vector

131

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

KO,- Kol, which is equal to the vector A2AI. We may therefore put r(Ko2- Kol) or = a I KO2- Kol I z by Eq. (32.7), when p

=

=

rA2AI = 2akaz

0. Equation (46.1) thus becomes

PO= Doi cos (rz/Ao) cos [2a(vt - Roar)]

where

A0

=

1 / 2 h = (YOI Y h I )'/k

I+h

(46.4)

I 1 c 1.

(46.5)

The wave train POthus has loops, or planes of maximum amplitude, equal to I Doi 1 when z = ~ A o m, being an integer, and nodal planes, of zero amplitude, when z = ( m +)A,. We shall call A. the beat period in depth, or the beat thickness, when p = 0, since it is due to a beat phenomenon between two wave trains of very nearly equal direction and wavelength. A very large number of waves of wave-number goI lie a t any moment in one beat period. With Mo Ka! radiation (A = 0.7 X lo-* cm), and a rocksalt crystal with the (200) planes perpendicular to the surface, A. = 3.23 X cm in state (a) of polarization. The beat phenomenon is therefore on a scale that is large in comparison with the atomic spacing of the crystal or the wavelength of the radiation. The other two wave trains of the field, Khl and Kh2, directed towards the reciprocal-lattice point H , have the resultant

+

I

Ph =

since

@Do'[COS { 2 ~ (-~ Kh2.r)) t -

=

( 2 ~ (-~ Khl.r)]] t

GDoisin (az/Ao) sin 2r(vt - &-r)

Kh2 - Knl

Here, f(h

COS

=

i(Khi -b Kn2)

=

=

(46.6)

KO*- KO,= AnA1.

&H,

and

I

g h

I

=

Ig o I -

P h represents a wave traveling in the direction QH, which, like Po, has planes of maximum amplitude and nodal planes parallel to the crystal surface. Each wave train has the same beat period in depth, Ao, but the nodal planes of P h coincide with the planes of maximum amplitude of Po, and vice versa. At most depths both wave trains appear in appreciable strength, and since they are coherent, the resultant field is a matter of some complexity. At depths m a o , however, P h has zero amplitude, and Po alone occurs, traveling in the direction of the incident wave. If the crystal is a slice of thickness mAo, the boundary conditions on emergence can be satisfied by a single emergent wave train, traveling in the direction of the incident beam; and a t such thicknesses no reflected wave kh' would emerge from the crystal, even though the condition for maximum reflection were satis-

132

R. W. JAMES

+

fied. At depths ( m a)Ao, on the other hand, only the wave train Pi,, traveling in the direction Eh,will occur, and a slice of this thickness should transmit only the reflected beam kht,with no directly transmitted beam, whenp = 0. This periodic oscillation with the thickness of the direction of the emergent energy current was deduced by Ewald in his original papers, and it is now always known by the name he there gave to it, the “pendulum solution” (Pendellosung). We shall discuss it and its experimental verification more fully in Parts IX and X. In Fig. 22, the positions of the nodal n

Fr?. 22. The pendulum solution in Case I.

and loop planes of the two wave trains Po and P h are shown diagrammatically for symmetrical transmission, when the crystal planes are normal to the surface. VII. The W a v e Field in Crystals with Finite Absorption

47. PRELIMINARY CONSIDERATIONS

All real crystals absorb radiation, and we must now consider the effect of this on the dynamical wave field, the discussion of which has so far been confined to nonabsorbing crystals, which are purely ideal bodies. The intensities of Laue reflections in perfect crystals were discussed by Zachariasen*’ in 1945, and more fully by von Laue’ himself in 1949; the theory was extended to include Bragg reflection by Wagnerz2in 1956. The experi11

W. H. Zachariasen, “Theory of X-Ray Diffraction in Crystals.” Wiley, Yew York, 1945.

H.Wagner, 2.Physik 146, 127 (1956).

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

133

mental verification of the somewhat remarkable theoretical results by B ~ r r r n a n n and ~ ~ JothersZ6 ~ ~ ~ ~has done much to confirm the general correctness of the dynamical point of view and to suggest new problems. We shall treat the absorption of radiation in the usual way, by assuming formally that the dielectric constant of the medium through which it passes is complex. By Eq. (12.2), 9 and its Fourier coefficients t$h will then be complex, which in its turn implies complex atomic scattering factors. The physical interpretation of this is that the corresponding atoms scatter with a change of phase in such a way that the combination of the scattered with the primary radiation diminishes the resultant amplitude progressively as the waves pass through the crystal, an idea that is familiar enough in the treatment of optical absorption and dispersion. We therefore write (47.1) 9 = 9r i9i

+

+r and r#~i being by definition real quantities. The Fourier coefficients &, are given by (47.2) 4 h = 4hr '$hi with

+

+rr

= (I/v)

1 1

4,exp (27rir-~,,)dr,

call

9hi

=

(I/v)

oell

diexp

(27dr.Rh)

(47.3)

dr

by analogy with Eq. (12.5). Since R h = -&, while & and 9i are real, $hr, 9 i r and t$hi, dii are pairs of conjugate complex quantities. They are not, however, necessarily real unless the crystal cell has a symmetry center, which has also been chosen as origin of coordinates. The quantities 9or and &i, which by Eq. (47.3) are given by

(47.4) arb on the other hand always real. The complex nature of I#I~ depends upon the nature of the atoms that produce the absorption. If the atomic scattering factors themselves are complex, so is 90. Unlike and 40i, however, 157 (1941); (ii) 2.Physik 127,297 (1950); (iii) Z . Krist. 106, 109 (1954). G. Borrmann and W. Gerisch, 2. Krist. 106, 99 (1954). a G. Borrmann, G. Hddebrandt, and ;B.Wagner, Z . Physik 142, 406 (1955). *@Pa B. Hirsch, Acta Cryst. 6, 170 (1952).

G.Borrmann, (i) Physik. 2.42,

134

R. W. JAMES

4 h r and +hi depend not only on the nature of the atoms but also on their arrangement in the cell, and may be complex if the crystal has no symmetry center, even if the scattering factors themselves are real. Since, however, G h r , 4rr and +hi, +i;i are pairs of conjugate quantities, we may write 4hr '#'hi

I G h r I exp ( i d , = I &hi I exp ( i w h ) ,

I Ghr I exp ( - i q h ) @ i i = I G h i I eXp ( - i f d h ) .

=

Giir =

(47.5)

The quantity Gh#%, which occurs throughout the equations, now becomes by Eq. (47.2), h'#% =

= Phr

The quantities where

Phr

Phr

and

= IGhr

-

('#'h&r

+

Phi

+

Ghi4.i)

i('#'h&i

+

&+hi)

(47.6)

iPhi.

are real, and by Eqs. (47.5) and (47.6),

1' - I +hi 1'; Vh

phi

=

-

wh.

= 7]h

If the crystal has a symmetry center, cos Vh real, T h and Wh are either 0 or K.

2

IGhr

I I Ghj I cos Vh

(47.7) (47.8)

=

f l , since if

4hr

and

Ghi

are

48. CRYSTAL WITHOUT SYMMETRY CENTER

In the quantitative discussions that follow we shall in fact consider only crystals with symmetry centers, but in order to clarify the ideas of the last paragraph the more general case will be briefly considered. By Eq. (12.7), ' b A = -qF(h)i (48.1) where q = ( e z / m 2 )(X2/rVqo) and is constant for a given crystal and radiation, while F ( h ) is the structure factor of the unit cell for the spectrum h. I n terms of the atomic scattering factors of a set of discrete atoms, the integral for the structure factor F ( h ) becomes a sum over the atoms in the unit cell, in general complex, and may be written where

+ iB(h)

F(h)

=

A(h)

A (h)

=

x f k ( h ) COS 2?r(Rh*rk) k

B ( h ) = x f k ( h ) sin 2n(R~,.rk), k

(48.2) (48.3) (48.4)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTIOS

135

r k being the vector from the origin to the center of the atom k of the unit cell, the scattering factor of which isfk. B ( h ) is necessarily zero for a centrosymmetrical cell. We now suppose fk to be complex, writing

fk(h)

+

= fkr(h)

The structure factor now becomes where

(48.5)

ifki(h).

+ iFi(h)

F(h)

=

Fr(h)

Fr(h)

=

AT(h)

F,(h)

=

(48.6)

+ iBr(h) A % ( h )+ iB,(h).

(48.7)

The quantities A,, B, and A,, B , are formed by using successively the real and imaginary parts of f ~ ( hin) Eqs. (48.3) and (48.4). Since A ( - h ) = A ( h ) while B ( - h ) = - B ( h ) , it is plain that F,(h), F,(@ and F , ( h ) , F , (L) are pairs of conjugate complex quantities. By Eq. (48.1), 4hr =

-q{Ar(h) -I- i B r ( h ) ] ,

4hr

=

- ' J { A t ( h ) -I- i B , ( h ) } , (48.8)

from which it follows directly from Eqs. (47.5) and (47.6) that

+ B?),

1'

=

@(A?

tanTh

=

Br/Ar,

'20s Vh

=

Phr

=

$hr

Phi =

tanuh

1 4 h %1' =

=

+

'Jz(Az" Bz")

(48.9)

B,/A,

(48.10)

I

(48.11)

+ BrBt)/ 1 Fr(h) I F , ( h ) I q ' { ( A ? + B?) - ( A ? f Bz")]. 2'JZ(ArA,+ BrB,) (ArAt

(48.12) (48.13)

and is zero when the atomic scattering factors are real, whether or not the crystal has a symmetry center; and finally Phr/Phr

=

2(ArAt

+ BrBr)/(

~

Fr(h)

1' - 1

F t ( h ) l'}.

(48.14)

From these formulas the necessary quantities can be calculated if the geometrical structure factors and the real and imaginary parts of the atomic scattering factors are known. The effect of the asymmetry of the unit cell is given by Br and B,, which vanish if the cell is centro-symmetrical. The expressions then take the simpler forms COS Vh = ArAt/ I Ar I I A , 1, (48.13) which is either +1 or - 1, and Phl/Phr

=

2ArA,/(A,2 - A?)

=

2A,/Ar

(48.16)

136

R. W. JAMES

very nearly, if A i << A,, which is true unless the crystal contains atoms with absorption frequencies very near that of the radiation. By Eqs. (47.4)) (48.1), and (48.3),

dOr

= -q

dOi

xfkr(O), k

= -q

xfki(0).

(48.17)

k

It is necessary to take the thermal motion of the atoms into account in calculating the structure factors. The electron groups responsible for the absorption of radiation of a given frequency lie at distances from the atomic center small in comparison with the corresponding wave length. For these groups f,(h) would fall away very slowly with increasing h, so long as the atom is at rest; but the groups share the thermal motion of the atoms, and for this reasonf,(h) becomes smaller in comparison withfi(0) as h increases. 49. THERELATION OF AMPLITUDE

THE

ABSORPTION COEFFICIENT TO

THE

COMPLEX

Consider a train of waves KO passing through the crystal in a direction differing appreciably from that corresponding to the production of a spectrum. In these circumstances such a single wave, as we have seen, represents the wave field closely, and by Eqs. (15.1) and (17.2) we may write

Ko

=

k(1

+

&O)SO

(49.1)

so being a unit vector in the direction of the wave normal. If & is complex, the real and imaginary parts of the wave vector are

K, = k(1 Ki

=

+ 3dor)so

(49.2)

+kc$oiso.

We have already seen in Section 38 that the imaginary part of the wave vector, if negative, represents absorption. In all actual examples that will are negative, and we can write for the linear absorpconcern us dorand tion coefficient for intensity, just as in Eq. (38.4), p =

2rk I doi

I.

(49.3)

In practice, we use Eq. (49.3) to determine the value of I d0i I from the measured linear absorption coefficient. If the frequency of the radiation is not too near that of an atomic absorption edge we may calculate do, I fairly closely from the formula for the nonabsorbing crystal,

I

I $or

I = (e2/mc2)(X2/r> F (0)/V.

(49.4)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

137

The values of the linear absorption coefficients of rocksalt for Mo K a (X = 0.71 A) and Cu K a (A = 1.54 A) are, respectively, 16.4 cm-l and 160 cm-l, giving values 1.85 X lo-" and 3.92 X lo-' for 1 (boi I , The corresponding values of I (bar I , calculated from Eq. (49.4)) are 2.84 X and 1.33 X 10-s, so that the ratios I +oi/(bor I for these two wavelengths are 6.5 X lo-* and 2.9 X It is less easy to estimate the value of I (bhi I / (bar I when the field consists of more than one wave, but it may safely be assumed to be of the same order of magnitude, so that little error will be made in neglecting the squares of such ratios in comparison with unity. This approximation will be made in all that follows. Then, from Eq. (47.7), P h r > 0,P h i << P h r , and for most purposes we may put

I

Phr

=

I (bhr la

=

(49.5)

50. THEABSORPTION OF THE WAVEFIELDS

We now consider absorption in the composite field, when there are two appreciable wave trains for each wave point. We use Eq. (38.1),

KO= ko{- kgn Kh

= koi

+ Rh - kgn

(50.1)

where g is the accommodation, and n a unit vector normal to the surface of incidence and directed inward. In a nonabsorbing crystal g is real, except in the totally reflecting range in Case 11, but now, because there is finite absorption, both KOand K h are always complex vectors, and g, too, must be complex. We therefore write g = gr

+ igi.

(50.2)

Because the difference between K h and KOis Rh, a real vector, the complex parts of the two wave vectors are equal. We put Koi =

Khi

=

-kgin.

(50.3)

Each wave train associated with a given wave point will be equally absorbed for a given penetration below the surface, and if u is the absorption coefficient for intensity of the crystal, reckoned in terms of distance below the surface, u = 4~kgi. (50.4)

This represents absorption if

gi is positive, but an increase of intensity with depth if gi is negative. As we shall see, the values of u may be very different for fields (1) and (2).

138

R. TI'.

JAMES

51. THECALCULATION OF THE ABSORPTION COEFFICIENTS Some mathematical results must first be derived. Equation (30.1) may be written (51.1) g = -(+0/2YO) (4Yh)-l{P where W = (P2 f b2+h+l;) (31.2)

+

w}

*

in which the positive sign applies in Case I (transmission), and the negative sign in Case I1 (reflection) ; and (51.3)

b = 2 / C I ( / Y h I / Y o ) ' .

The cosine 70is always positive; Yh is positive in Case I but negative in Case 11. The quantities 40,0, and +h& are now complex, and we first have to calculate the real and imaginary parts of g in terms of them. Using Eq. (27.12), we put

P

=

Pr

+

$1

=

2CYh

+1

+or

1 C1 -

(Yh/YO)]

+i

I

140%

[1

-

(YV'YO)~.

(51.4)

It has here been assumed that both dot and +oz are negative, which is true for actual crystals. I n Eq. (30.4), a parameter p was introduced, defined by This may now be written pr

+ ip%

p =

=

( P / b ) (@h+l;)**

+

b-'(Pr

$0)

(phr

f

ipht)'

(51.5) on expanding, and neglecting terms of the second order of the small quantities ,B; and P h i . I n Eq. (51.5),

Q

= Phi/Phr.

(51.6)

To this order of accuracy, using Eq. (49.5), we may therefore write

Pr

=

b

I+hr

pr.

(51.7)

By Eqs. (51.4) and (51.7) we see that p r is a linear function of C Y ~and so of the angle of incidence, and may be used as the incidence parameter. Equation (51.2) may now be written

v'

=

b

1 +hr I

((pr

+ iB)*f ( 1 + iQ)Ii*

(51.8)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

Here

B = pi/b =

1 @hr 1

1 @Oi/@hr 1 [(YO - Y h ) / ( Y O 1 Y h 1 )*](2 [ c I )-I

139 (51.9) (51.10)

by Eqs. (51.3) and (51.4). By Eq. (47.7), where

Q

=

Phi/Phr

= 2

1 @ h i / @ h r I COS Vh

A =

= 2A

I @Oi/@hr 1,

1 @hi/@Oi i cos vh

(51.11) (51.12)

and is always less than, but of the order of, unity. Neglecting B2 in comparison with unity, we may now write Case I :

W

Case 11:

w

1 = b I @hr I = b

j@hr

{Pi2

+1

{p?

-

1 f

i(2Bpr

Q ) I*

(51.13)

i(2Bpr

- Q)]+.

(51.14)

In each equation, B is given by (51,10), but in Case 11, YA = - 1 Y h I . By Eqs. ( j l . 1 ) and (51q4),the real and imaginary parts of g may be written g r = ( 1@Or 1 /270) ( 4 7 h ) - ' { / % f wr] (51.15)

+

gi = (

l@Oi

I /270)(YO-'

+

7h-l)

f

(Wi/47h).

(51.16)

The symbols Wr and W i denote respectively the real and imaginary parts of W . I n Eqs. (51.15) and (51.16), the positive sign applies to field (1) and the negative to field ( 2 ) . By Eqs. (49.3), (50.4), and (51.16), the corresponding absorption coefficients are given by or (51.17)

If ' u l and u2 are the absorption coefficients for fields (1) and ( 2 ) , respectively, i(U1 'JZ) = ac((YO-' 7h-l) = b (51.18)

+

i('J1

- 'J2)

+

= PWi/27h

I @Oi I

= K

(51.19)

and the absorption coefficients can be written 41,2

= d

f K,

(51-20)

a form that is often very convenient, since d does not depend on p r and is the same for both fields. The difference between u1 and u2 is clearly shown by this method of expression.

140

R. W. JAMES

We may also write 2rk(grl

rkWd/'yh

=

K

and by Eq. (51.15)

(51.21)

- Q ~ P ) = rkWr/Yh,

(51.22)

a result that will be needed later. 52. ABSORPTION IN CASEI (TRANSMISSION)

+

+

In Eq. (51.13), (2Bpr Q) is always small in comparison with pr2 1, since B and Q are each proportional to I dod/$hr I . We may therefore expand the square root by the binomial theorem, and neglecting terms of the second and higher orders, obtain wr =

b

=

2

I4hr

IcI

I (p?

+ 1)'

(Th/TO)'

I4hr

I (p?

+ 1)'

(52.1) (52.2)

expressions that are valid over the whole relevant range of pr. Putting in the values of B and Q from Eqs. (51.10) and (51,11), we get (7h-l

K = i(u1

- u2)

=

5 =

+

= +c((yO-'

$(Q1

U2)

The positive sign gives (52.1) and (51.22),

+c(

UI

- YO-') pr

+

(p," Yh-l),

and the negative

+

2A I C 114

I /(rod'

(52.3) (52.4)

U1,2

= 5 f K.

UP.

In this case, by Eqs.

53. ABSORPTION IN CASEI1 (REFLECTION) '. In deriving the corresponding results for Case I1 it is necessary to distinguish once more between ranges (i) and (iii) in which p,2 > 1, and range '(ii) in which pr2 < 1. We consider first the ranges (i) and (iii). In Eq. (51.14), put x = p12 - 1, y = 2Bpr - Q so that W = b 1 $hr I (Z iy)' and (53.1) wr = b I 4 h r I ($[(z2 y2)' XI)'

+

wi = b

+ +

I { a [ ( X 2 -k

8')'

- XI)'.

(53.2)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

141

These formulas are valid over the whole of ranges (i) and (iii), and are easy to evaluate if the numerical values of z and y are known. By Eqs. (51.18) and (51.19), remembering that Y h = - I Yh 1 , we have

Also

In these expressions

When p , = f l , z = 0 and W = b I @hr I (iy)i. By Eq. (53.6), since 1 C I and I A 1 are both less than unity, while 70 + I Y h 1 2. 2 ( 7 0 I Yh I )*, y

is positive when p r 1 1 and negative when p , I -1. Using the result that the imaginary part of ( f i I y 1 ) * is equal to f (3 1 y I )*, we find for the value of K when p r = f l where

K(fl)

= F [ H

I c I /(YO I Yh I I*] I $hr/'$Oi I (iI y I 1'

(53.7)

If y2 << x2, we may expand Eq. (53.3) by the binomial theorem, obtaining K =

-

142

R. W. JAMES

Here, the upper sign of the ambiguity refers to field (1) and the lower to field ( 2 ) . This expression is in fact valid over most of ranges (i) and (iii) , except quite near p , = f l , where it clearly cannot apply; and to the same approximation

I n range (i),p,2

W Putting now x

=

<

=

1, and Eq. (51.14) becomes

ib

I $'hr 1

1 - pr2,y

(1 =

pr2

+ i(&

which gives

-11

-b

1

(53.12)

Q - 2Bpr, we obtain, just as before,

1 +hr i { + ( V ' Z ~ wi = b 1 4 h r 1 {a(Wr =

- 2Bpr)}'.

+

~2

+

-

Z)

Z)}f,

(I I

In that part of the range where y2 << x2 we may expand as before, and put

_-

2ak d

'

Y

3

I

::$

(l - p 2 )'(1 + -

(53.14)

and (53.13)

It will be noticed that the magnitude of extinction coefficient ue of Section 39. For range (ii)

or in the range where expansionis valid,

K

is virtually th a t of the primary

143

THE DYNAMICAL THEORY OF X-ltAY DIFFRACTION

54. THERELATION BETWEEN THE COMPLEX PARAMETERS 8, p , AND v By Eq. (51.7),

8r

=

b

I $hr 1

(54.1)

and from Eqs. (51.5), (51.10), and (51.12),

an expression that holds for both Case I and Case I1 if the sign of taken into account. The relation of p to v is easily derived. In Case I, p,

+ i p i = sinh (v, + i v i ) .

7 h

is

(54.3)

On expanding the right-hand side of this equation, using the relations sinh (iv) = i sin v,

cosh (iv) = cos v,

and equating real and imaginary parts on the two sides of the resulting relation, we obtain p,

=

sinh v,.cos vi

(54.4)

pi

=

coshv,sinv;

(54.5)

I sinh v 12

=

sinh2 vr

(54.6)

+ sin2v i .

In the examples we shall consider, v i is usually so small that we can put cosvi

sinvi =

=

1,

=

sinh vr

vi,

so that the approximations pr

P' = v.,coshv,

or

+

(54.7) vi = pi/(pr2 1)' are generally adequate. In Case I1 we must distinguish between ranges (i) and (iii) in which p 2 > 1 and range (ii) in which p 2 < 1. In range (iii), which leads to

p,

+ ipi p,

=

cash (v,

+ i~i),

(54.8)

= COS~V~COSV~

p i = sinhv, sin v;

1 cosh v l2

=

cosh2v, - sin2vi,

(54.9)

144

R. W. JAMES

with the approximations

pr

=

COShvr

(54.10)

vi = pi/sinhvr = p i / ( p , 2 - l)*, the last of which is not valid if I p , I is too near unity. ipi = -cosh (v, ivi), so that In range (i) p ,

+

(54.11)

+

p,

=

vi =

In range (ii) ,

p,

- Cosh~, - p i / ( p ? - 1)t.

+ ipi

=

cos (v,

(54.12) (54.13)

+ iVi)

and this in a similar way gives (54.14)

=

cos V , cash ~i - sin vr sinh vi

=

cos2vr

(54.16)

p,

=

pi

I cos v l2

(54.15)

+ sinh2

vi,

with the approximations p,

=

cosv,

v i = -pi/(l OF 55. DISCUSSION

THE

(54.17)

- pr2)'.

(54.18)

ABSORPTION IN CASEI (TRANSMISSION)

The absorption coefficient u gives the diminution of the field intensity with depth below the surface of the crystal surface, and 'J1.2

= $(a1

+ a?,)

f K = $P(YO-'

+

7h-l)

fK

(55.1)

where K is given by Eq. (52.3). The first term in Eq. (55.1) is constant, and is equal to the ordinary linear absorption coefficient p multiplied by the average of the secants 1/70 and 1/Yh, a factor that appears because p refers to the actual path traversed by the radiation in the crystal and a to the depth to which it has penetrated below the surface, which is shorter, so that is greater than p. Both ul and u2 are positive in Case I. To show this, we make the substitution p , = sinhv, from Eq. (54.7) in the expression for K given by (52.3). Equation (55.1) becomes

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

145

Here, the upper sign of each ambiguity applies to field (1) and the lower sign to field (2). By Eq. (51.12), 1 A 1 < 1, and 1 C 1 is either unity or 1 cos 200I , and therefore, since aez be-= 5 2(ab)t for all values of z if a and b are positive, it follows that the expression in brackets in (55.2) is always positive, whether the sign of its last term is positive or negative. The intensities of both fields therefore decrease as the depth below the surface of incidence increases. If A is positive, as it is for the simple type of crystal considered in Section 43, u1 > u2 in the range where interference occurs, and the difference between the two coefficients may be large. To get an idea of the magnitude of the effect, we suppose the crystal planes h to be perpendicular to the surface, so that yo = yh. I n this case Eq. (55-2)becomes

+

(55.3) For large values of 1 pr 1 , that is to say for angles of incidence well on either side of the region of interference, both u1 and u2 tend to p l y o , the coefficient to be expected from ordinary absorption. If A is positive, u1 > p / y o > us. When p , = 0, (55.4) ~ 1 , 2= (c(/Yo) (1 C 1 A I ).

*

Sometimes A may be nearly unity. For N'aC1200 its estimated value at room temperature is about 0.95, so that for state (a) of polarization, when 1 C 1 = 1, u1 could have nearly double its mean value a t pr = 0, and u2 at the same time only between 5 and 10% of it. The value of the second term in Eq. (55.3)decreases rapidly as I pr i increases, so that the range of anomalous transmission is very small. Over this range, however, the crystal should show a quite abnormal transparency to the radiation. This result is fully confirmed by the experimental work of Borrmann and his collaborators in Berlin. 56. THEASYMMETRY OF THE ABSORPTION IN CASEI Equation (55.2)may be written 41,2

where

=

(56.1)

(56.2) (56.3)

146

E. W. JAMES

r

FIQ.23. Absorption coefficients ul and 02 as functions of pr for NaCl200 when crystalplanes are parallel t o (111) (x = 54’ 42’) for Cu Ka radiation. Case I (Transmission).

is symmetrical in p,, but u’ changes sign with p,, tending to - l/&) for p, = f c o , and becoming zero when p , = 0. The unsymmetrical component vanishes for all values of p , when the transmission is symmetrical and yo = Yh. In Fig. 23, ul, UZ, and the asymmetrical component of the absorption a‘ are shown, plotted as functions of p , for rocksalt 200, when the surface of the crystal is parallel to (111). The angle made by the planes (200) with the surface is then 54’42’. The figure is drawn for Cu K a radiation (A = 1.54 A, p = 160 cm-l). Here, Bo = 15’50’, #o = 19’30’, and I,bh = 51’10’. These figures give l/rh - 1 / 7 0 = 0.534, l/yl, l/yo = 2.6.55, l/(?’O I Yh 1 ) * = 1.300. The value of I @hi/$Oi I has been taken as 0.95. The numbers used here, as in other numerical examples involving rocksalt in the article, have been taken from a paper by Renninger.18 The value of I $oi I is calculated from the value of p, using (49.3), and the difference between I @hi I and 1 @oi I is assumed to be due to thermal motion, for which the results of Waller and James2’ have been used. The exact values are not important in discussing the general magnitude of the effect;although more recent values have been The term

u”

= t ( p / 2 ) (l/rh

+

e7

I. Waller and R. W. James, Proc. Roy. SOC.A117, 214 (1927).

THE DYNAMICAL T HE OR Y O F X-RAY DIFFRACTION

147

obtainedlZ8it is convenient to use these figures, the derivation of which is discussed in detail in Chapter VI, page 325, of Optical Principles.11 I n the case discussed cosVh = 1, and so A = 0.95. For state (a) of polarization,

and from these figures the curves were drawn. The asymmetry due to the component u' is clearly shown. I n the example considered, 7 0> yh, and x < go", so that the maxima and minima of u occur when pr has a small positive value. If x > go", ut changes sign, and the asymmetry is reversed. The curves A and B, for u1 and u2, are the mirror images of each other in the line D, which gives the constant value of 3 (u1 4. For large positive values of p,, u1 and u2 approach the limiting values p / y h and p / y O , respectively, and for large negative values of p , these limits are reversed. These results are to be expected, since outside the angular range within which interference occurs there will be only one wave train in the crystal, the continuation of the incident beam, for which the appropriate absorption coefficient in depth is p / y O . As we have seen, this wave train is carried by field (1) alone if p , is large and negative, and by field (2) alone if p , is large and positive. It should be remembered that the whole range of p , shown in Fig. 23 corresponds to a n angle of 12 or 15 seconds of arc on either side of p , = 0, so that the angular width of the region of abnormal absorption is very small. It is interesting a t this point to consider a few actual figures. Suppose radiation passed through a slice of rocksalt 0.1 cm in thickness. The average absorption would reduce the intensity of the field to exp (-21.3), or 5.6 X of its surface value in the thickness of the slice. The maximum value of u1 is 415 cm-l, which corresponds to a reduction factor of about 10-l8; but the minimum value of u2 is only about 12 cm-l, which gives a reduction factor of 0.30. Over a small range of angles, therefore, the slice is relatively transparent for field ( 2 ) , but virtually opaque for field ( I ) . For state (b) of polarization the absnormal transmission is less marked. In the example considered here, I C I A is about 0.81 for state ( b ), and the numerical coefficient in u" is 168 instead of 198 as in state ( a ) . This leads to a minimum value of u2 equal to about 40 cm-1, and while this is much smaller than the normal coefficient, it is considerably greater than the minimum value of u2 in state ( a ) , If the crystal slice is more than a millimeter or two in thickness the transmitted radiation in the range of abnormal transparency will consist of field ( 2 ) , almost entirely in state (a) of polarization.

+

rsM.Renninger, Aciu Cryst. 6, 711 (1952); 8, 597 (1955).

148

R . W. JAMES

57. DISCUSSION OF THE ABSORPTION IN CASEI1 (REFLECTION) By Eq. (53.4), the absorption coefficients may now be written (57.1)

where

(57.2)

and K is given by Eq. (53.3) for ranges (i) and (iii), and by (53.13) for range (ii) . In symmetrical reflection, when the crystal planes are parallel to the surface of incidence, d is zero, and u1 and uz are then equal respectively to &K. When the planes are inclined to the surface d will no longer vanish, and its numerical value may be considerable when either the incident or the emergent beam makes a small angle with the surface; but it will still in general be smaller than the numerical value of K . The signs of the absorption coefficients for the two fields will thus still be opposite, and the same as for symmetrical reflection. Unless p , is very nearly equal to & l , Eq. (53.10) is valid in ranges (i) and (iii) ; and making the substitutions p , = - cosh vr in range (i) and p , = cosh vF in range (iii) we obtain the following expressions for the absorption coefficients in the ranges when the reflection is unsymmetrical. In range (i) exp 2 sinh v,

(*Ur)

YO

+ expI

(TU,) Yh

j

+

(70

''

I Yh

A ~

}

)#

(57.3)

and in range (iii)

Here the upper and lower signs in each ambiguity apply respectively to fields (1) and (2). As we have already seen, the expressions within the brackets are always positive, so that in range (i) u1 is positive and u2 negative, and in range (iii) u1 is negative and u2 positive; the same will be true for unsymmetrical reflection. It is probably easiest to discuss the implications of these results in the light of an actual example; and for this purpose we shall consider reflection from a rocksalt crystal in which the (200) planes are inclined a t an angle of 10' to the surface, the geometrical conditions being those illustrated by Fig. 16a. For Cu Kcu radiation, 00 = 15'50' for 200, and l/yo = 2.295, 1/ 1 Y h 1 = 9.839.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

149

In numerical work it is convenient to express the absorption coefficients

u as multiples of the quantity

I Yh 1, p / ~ oand p / I Y h ] , the

(57.5) absorption coeffiwhich is the geometric mean of cients' in depth of waves traveling in the crystal in the directions of incidence and reflection, respectively, and subject only to the ordinary linear absorption coefficient p. For the example chosen, 5, = 4.75~~ and 5 of Eq. (57.2)is equal to -0.8005,. The absorption coefficients in depth are greater than the linear absorption coefficients because the wave trains concerned travel in directions inclined at relatively small angles to the surface, and in penetrating to a depth d travel the greater actual distances d/yo and d / I Yh I in the crystal. For state (a) of polarization, ] C I = 1, and Eq. (53.6) gives drn = P / d %

Y

=

2 I +oi/+or

I (1.276pr - A ) .

Taking A = 0.95 and I +hr/$Oi I = 25 as appropriate values for rocksalt 200,it is now easy to calculate K in terms of b, using Eq. (53.3)in ranges (i) and (iii) and (53.13)in range (ii). In practice, the much simpler expressions (53.9) and (53.14) are adequate over a surprisingly large range of the variable pr. The values of I u I /d, so obtained, plotted as functions of p,, are shown in Fig. 24 in which the dotted curves correspond to negative values of the absorption coefficients.

1 IUl/qm

(ii)

-2 -I

(iii)

0

FIQ.24. Relative absorption coefficients in Case I1 (Reflection).

150

R. W. JAMES

In a thick crystal, of the type discussed in Part VI, a field with a negative value of the absorption coefficient is not physically possible; for if its intensity were finite near the surface it would increase without limit as the depth in the crystal increased. I n such a crystal, therefore, field (1) alone can exist in range (i) and field (2) in range (iii), in agreement with the conclusions already reached in Section 38. For large values of I p , I there will be only one effective wave train in the crystal, in the direction of the incident radiation, and this will belong to field (1) in range (i) and to field (2) in range (iii) . I n either case the appropriate absorption coefficient in depth will be p / y o , and it will be seen that both positive branches of the curves in Fig. 24 tend to this limit. In the range where interference occurs, and the wave train Kh becomes appreciable, abnormal absorption effects will occur. When A is positive, the absorption is always greater than ply0 in range (i), and becomes very much greater as p , approaches -1. In range (ii), which will be considered in more detail below, the absorption is largely governed by the primary extinction; but when p , = +1, the absorption is still greater than the normal absorption, but falls very rapidly to a minimum considerably less than 1 / y 0 for a value of p , a little greater than +1, and thereafter slon~lyincreases as p , increases, tending to p / y o from below for large values. If the crystal is in the form of a slice of finite thickness the argument used above against the possibility of negative absorption coefficients is no longer valid. It may then happen that, as a result of the boundary conditions a t the lower surface, a wave train in the direction Kh will be set up, traveling toward the upper surface and diminishing in intensity as it does so. To such a wave train, and to the wave train KO to which it must necessarily give rise in the region of interference, the negative absorption coefficients will apply. Clearly, such wave trains must belong to field (2) in range (i) and to field (1) in range (iii), and to them the dotted branches of the curves in Fig. 24 apply. For large values of I p , 1 these give limiting values of 1 u I equal to p/ 1 yh I , appropriate to a single wave train traveling in the crystal in the direction of the reflected radiation. This is to be expected, for in such fields the wave train Kh plays a part analogous to that of the wave train KO in the direct beam. We have taken an example in which p/ I Y h I > p/yo. Had we taken the case illustrated by Fig. lGb, with the atomic planes still making an angle of 10" with the crystal surface but with the glancing angle of incidence smaller than the glancing angle of reflection, the full and dotted curves of Fig. 24 would have been interchanged. For symmetrical reflection the two sets of curves coincide, and we shall consider this case in a little more numerical detail, now including range (ii) in the discussion.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

For symmetrical reflection, yo = I yh I = sin Bo, and (53.6) becomes y = 2 1 '#'Oi/d'hr 1 ( p r - 1 c I A )

151

+ = 0, while Eq. (57.6)

and this may be used over the whole of ranges (i) and (iii) in Eq. (53.3) and over the whole of range (ii) in (53.13). I n the example considered, however, when p , < - 1.75 or p , > +1.04 the simpler expression (53.10), which now becomes

1 I /an = 1 p , - A 1

C

I 1 / ( p T 2-

1);

(57.7)

is accurate enough. Here a, = p/sin

Bo

(57.8)

and is the value of crnfor symmetrical reflection. When p , = f l , Eq. (53.7) now gives

I a ( A 1 ) 1 /a!<= I c l'{ 1 +h,/+Oi I I 1 =F I c I A I 14,

(57.9)

where the smaller value applies to p , = + 1 when -4is positive. For state (a) of polarization and the values of A and I &/+Oi 1 used above, Eq. (57.9) gives I a 1 /a, = 6.98 and 1.12 for p , = +1 and -1, respectively. At p , = 1.05 the ratio has fallen to a minimum value of 0.31, and thereafter rises with increasing p,, tending to unity for large values. In range (i), the ratio falls away, at first rapidly and then more slowly from a value of about 7 at p , = -1, tending again to unity, but this time from above, for large negative values. In range (ii), which corresponds closely to Darwin's range of total reflection, the values of I a 1 /a, can be calculated for the example considered with sufficient accuracy from Eq. (53.14), which now becomes

I 1 /an 0-

=

I

+h,/+"i

I

(1

- P,")*,

(57.10)

if p , lies between -0.7 and +0.95. Over the rest of the range Eq. (53.13) must be used. The curve shown in Fig. 25 was calculated from the latter formula. In that part of range (ii) in which Eq. (57.10) is valid the absorption differs very little from the primary extinction; but extinction becomes zero at both ends of the range, and here there is a definite absorption effect much greater when p , is near - 1 than when it is near +l. The reason for this may be understood in general terms from the discussion in Section 45, in which the nature of the stationary distribution of intensity in a nonabsorbing crystal was considered. It was there shown that the intensity of

152

R. W. JAMES

FIG.25. I u [ / I un I for symmetrical Bragg reflection from rochalt 200, Cu K a radiation; I.( = 160 cm-1, p/sin Oo = 586 cm-1.

the field on the atomic planes was a minimum for p = 1, and that it rose as p passed through the range of total reflection, reaching a maximum a t p = -1 (see Fig. 20). True absorption can occur only if the atoms that produce it lie in a field of appreciable strength, so that we should expect the greatest absorption to occur near p , = - 1. The unsymmetrical shape of Prins's reflection cirves, which are discussed in Section 68 (Fig. 32), are due to the same cause. As p , passes through the range of large extinction, the field penetrates so slightly into the crystal that the atoms have little opportunity to cause absorption, and the reduction of field strength with depth depends virtually on extinction alone, whether or not the crystal is absorbing. At the edges of the range the extinction becomes small and the radiation penetrates to depths that are governed by true absorption. When reflection is not symmetrical, there will be two separate curves in range (ii), continuations of those shown in Fig. 24, each similar to the curve shown in Fig. 25, but separated from each other by a constant difference in ordinates. One of these curves, that corresponding to field ( 2 ) , gives positive values of u in range (ii), and is continuous with the positive branches of the curves shown in Fig. 24 for ranges (i) and (iii) . The other, corresponding to field (1) and to negative values of the absorption, is continuous with the negative or dotted curves in Fig. 24.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

153

VIII. The Reflection and Transmission Coefficients for Crystals of Finite Thickness

58. THEBOUNDARY CONDITIONS

For each state of polarization, when the crystal has finite thickness, there will in general be two effective wave points A1 and As, and so two different wave fields active simultaneously, both for transmission and reflection. Each field will satisfy the wave equation independently, but their relative strengths and phases are not arbitrary and are determined by the boundary conditions that must apply at the two surfaces. The assumptions made in setting up the boundary conditions have already been discussed in Sections 24 and 33. If DO' and Dhe are the amplitudes of the external waves corresponding respectively to the internal waves KOand Kh, the boundary conditions given for a special case in Eq. (33.1) take the following more general form, applicable at either surface, CoiDoi Chi&

+ +

cozDoz = coDo' ChnDhz

=

ChDh'.

(58.1)

Each coefficient cmj is a phase factor of the form cmj

and Co

=

exp { - 2 ~ i ( K , p r ) )

= eXp { - 2 & ( k o i - r ) ) ,

Ch

= eXp { -2Ti(kh"r) ]

(58.2) (58.3)

in which kh' is the external wave vector corresponding to K h . It becomes kh' for reflection, and kht for transmission. K m j is the wave vector from the wave point j to the reciprocal-lattice point m.I n the example considered, j may be either (1) or ( 2 ) and m either 0 or A. The vector r is drawn from an origin chosen in the upper surface to the point at which the field is considered. If, as before, x1 = Dhl/DO1, x2 = Dht/Doz, Eqs. (58.1) become

+ COZDOZ = CODO' +

CoiDoi XiChiDoi

X&hzDoz = ChDo',

(58.4)

which apply to either state of polarization if the appropriate values of x1 and xp are used.28a **a

The boundary conditions could have been handled rather differently. In state (a) of polarisation all the induction vectors D are parallel to each other and to the surface, and so Eq. (68.1) follows directly. In state (b) the vectors H are parallel to the surface, and a corresponding relation holds between them. To the numerical approximation to which we are working, however, we may put Dm = H,, and this lee& at once to the relation (58.1) between the amplitudes.

154

R. W. JAMES

The vectors concerned obey the relation

+

(58.5) K,j = koi R, - kgjn where n is a unit vector normal to the surface, directed inward. For a point in the upper surface of the crystal nor = 0, so that, by Eqs. (58.2), (58.3),and (58.5), c01 = c02 = CO. The components of Khl and K ~ parallel z to the surface are each equal to the tangential component of khe,so that, since r lies in the surface, K h j - r = khe-r, and Chl = ch2 = c h . At the upper boundary Eqs. (58.1) therefore become

+ Do2 = Doe +d o n =

Do1

(58.6) At the lower surface nor = d , the thickness of the slice, and using Eqs. (58.1) and (58.5) we obtain for the boundary conditions XiDoi

Dhe.

exp (iald)Dol

+

+ exp (iad)Do2 = Doe

{ 51 exp (iaid)Do1 52 exp (%hid) Do*} = = exp { -2ai(koi -k Rh) or) Ch'

where ch' and for brevity we have written

ChDh',

(58.7)

2agk = a. (58.8) These equations are of general application when there are two active reciprocal-lattice points 0 and h, and are valid for an absorbing crystal if g is replaced by the appropriate complex quantity. We shall deal first with this more general case, from which the particular results applicable to a crystal slice with negligible absorption may easily be deduced. 59. CASEI. TRANSMISSION THROUGH

AN

ABSORBING CRYSTAL

The wave trains kof and kht now both emerge from the lower surface of the slice, as in Fig. 8a. There is no reflected wave above the upper surface, and Dho = 0 in Eq. (58.6). Above the upper surface we put Doe = Doi, and below the surface of emergence Doeand Dhe are put equal to Dot and Dht, respectively, t denoting transmission. Equations (58.6) then become Do1 Don = Do'

as in Eq. (33.2) ; whence

XiDoi

+ + xD02 = 0

(59.1)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

155

which, substituted in Eq. (58.7), gives Dot ChDht

= (52 =

- z~)-’[zzexp ( i d ) - zlexp (iazd)]Do‘

Ch’XlzZ [exp ( i a d ) - exp (iad)]Do’. 2 2 - 21

(59.3) (59.4)

Assuming g, and therefore a, to be complex, we write

+ ia”

u = a‘

and so

ia

=

+ igi)

(59.5)

- a”,

ia’

=

2nk(g,

(59.6)

With this substitution, Eqs. (59.3) and (59.4) become

Do’ =

(ZZ-

ChDh’

=

s)-’[zZexp (ial’

- al”)d

-[exp

- al”)d

Ch’ZIZt

x2

- 21

(ial’

- a e x p (ia’- az”)d]Do’ (59.7) - exp (iaz’ - az”)d]Doi. (59.8)

Since v is now complex, we write Eq. (30.13) in the form z1 = -Gexp (v,

whence 121

lz

=

z1z** =

I x1z21 2 = I G 1’ =

+ ivi),

zt

=

Gexp ( - v r

- ivi),

(89.9)

I la = 1 G I2exp (-2v,) z~*zZ = - I G 12 exp (-2ivi)

I G Iz exp (2v,), - I G 12exp (2ivi), I G 14, ( Y O / Y h ) I #h/4fi

1x2

- 1’ 21

=

4 1 G 1’

I cash v 1’

(59,ga)

*

We now define the direct and reflected transmission coefficients, T(v) and R d v ) , by T ( v ) = 1 Dot/Doi 1 2 ,

Rt(v) = Yh I Dh‘ 1 2 / ~ o I Do’

I*.

(59.10)

Substituting Eq. (59.9) in (59.7) and (59.8), and taking the squares of the moduli of both sides of the resulting equations, one obtains exp ( - dd) [cash ( 2 v r ( ~ 1 ” - &”)dl T(v) = 2 I coshv I2 -k COS ( (UI’ - az’)d 2 ~ , ) ] (59.11)

+

-

where

(59.12) b =

a1”

+ a”.

156

R. W. JAMES

By Eqs. (50.4), (58.8), and (59.5), -

and so 5 = &(ui

and by Eq. (56.1)

+

ai” - az”

UZ),

K

=

(59.13)

u2” = +u2

TU1, 1

u’

= +(.I

- uz) =

+ g”

K

(59.14) (59.15)

By Eq. ( 5 2 . 5 ) )

Equation (59.16) may be written in the physically significant form ~ 1 ’

where

UZ’ =

A

and

=

(59.17)

2a/A,

&/(p?

+ I)*

YO 1 Y h I

(59.18)

c1

i,

(59.19) which is the beat thickness in the crystal when p , = 0 for the Ewald pendulum solution, which was defined in Section 46 for the special case of a nonabsorbing crystal. By Eqs. (54.7) and (54.9), we may put to a close enough approximation AO =

pr = siiihv,,

1 coshv , 2

)’/

=

~

cosh2v, = pr2

+1

(59.20)

in Eqs. (59.11) and (59.12), and using (59.17) obtain for the reflection coefficients and the direct transmission coefficients expressed as functions of p , exp (-d) [cosh2 (v, i ~ d) sin2 { i r ( d / A ) - v i ) ] (59.21) T(pr) = pp“ 1

+

+

In these expressions, v, = sinh-’ pr, and K and v, may be expressed as functions of p , by Eq. (59.15) in conjunction with (56.2) and (56.3), and by (54.7). The factor 1 I$h/I#Ji 1 in Eq. (59.22) is unity unless the crystal has no center of symmetry and is also absorbing, in which case I #% 1 and I +,j 1 may differ, that is to say, the intensities of reflection from the opposite sides of the set of planes may not be equal. This was demonstrated experimentally by Coster, Knol, and PrinsZ9in the case of zincblende as early as 1930, but many examples are now known. 29

D. Coster, K. S. Knol, and J. A. Prins, 2.Physik 63, 345 (1930).

T H E DYNAMICAL THEORY O F X-RAY DIFFRACTION

157

60. CRYSTAL SLICEWITH NEGLIGIBLE ABSORPTION

Before discussing the general formulas (59.21) and (59.22) numerically, we shall consider the form they take when the absorption of the crystal is negligible. The appropriate formulas are obtained simply by putting 5, K , and v i equal to zero, and 1 +h/&, 1 = 1. Then, since p is now real, Eqs. (59.21) and (59.22) become (60.1)

T ( p ) = 1 - Rt(p).

(60.2)

The sum of the two coefficients R t ( p ) and T ( p ) is necessarily equal to unity if there is no absorption in the crystal involving irreversible loss of energy, but must be less than unity if the absorption is finite. By Eq. (60.1), whenever the thickness d of the slice is an integral multiple mA of A, R t ( p ) is zero, whatever the value of p . T ( p ) is then unity, and the transmitted radiation on emergence from the crystal is entirely in the direction of incidence. The maxima of R t ( p ) occur when a’ = ( m + ) A , and their value is l,/(p2 1). T ( p ) is then equal to p*/(p2 l ) , and is small for small p , so that most of the transmitted energy is in the direction of reflection. We have thus again arrived a t the Ewald pendulum solution. Kh en p is small, and the interference effects in the crystal nearly a t their maximum, the direction of the transmitted radiation alternates between those of incidence and reflection as the thickness changes, with a period of alternation in depth equal to A. This is the beat thickness, and we see by Eq. (59.18) that it depends on p . Only when p = 0, as in the case discussed in Section 46, does complete interchange take place between the two directions of emergence as the thickness changes. The reflection coefficient is always zero when d = mA, and for these thicknesses the whole emergent radiation is in the direction of incidence; but the intermediate maxima of R t ( p ) become smaller as p increases, so that T ( p ) no longer falls to zero. The maxima and minima in T ( p ) become less marked, and when p is large and there is no longer an appreciable interference field in the crystal, the direct beam carries virtually the whole of the transmitted energy.

+

+

+

G1. THEFIKESTRVCTURE OF R t ( p )

We now suppose the thickness of the slice to remain constant, but allow the direction of incidence, assumed as in the whole of this discussion to be sharply defined, to vary. By Eq. (59.18), A itself is a function of p , and so R t ( p ) will vary with the angle of the incidence while d remains

158

R. W. JAMES

-

1.8"-

FIQ. 26. Fine structure of transmitted reflection coefficient R&).

constant. Equation (60.1) may be written RdP)

=

sin2 { ~d (pz p2

+ 1) * / A o )

+1

(61.1)

In Fig. 26, Rt(p) is shown plotted as a function of p for rocksalt 200 with Mo Ka radiation, when the reflecting planes are perpendicular to the surfaces of the slice, the thickness of which is 10A0. As we saw in Section 46, Ao, the beat thickness when p = 0, is in this case 3.23 X cm, so that the slice considered has a thickness of about $ mm. Rocksalt is not of course a nonabsorbing crystal; but a slice of this thickness would transmit about 0.6 of the radiation incident upon it in so far as the ordinary absorption of the material for Mo Ka radiation is concerned, and the example will serve to give the order of magnitude of the effect. It will be seen from Eq. (61.1) that R t ( p ) becomes zero whenever ( d / N (pZ

+ 114

= n,

an integer. In the simple example we have chosen, d = 10A0, and it is an easy matter to calculate the values of p for which Rt(p) is zero. The curve consists of a series of maxima, separated by zero minima, and bounded by 1). In the figure, a zero value occurs at p = 0, but the curve of l/(p2 this is because d has been taken as a multiple of Ao. According to the thickness, R t ( 0 )may have any value between 0 and 1.

+

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

159

The maxima are not evenly spaced, but become more nearly so as p increases. The figure shows clearly the wider spacing for the smaller values of p . Four maxima lie between p = 0 and p = 1, and eight between p = 1 and p = 2 . The thicker the slice the more cron-ded together are the maxima, and doubling the thickness approximately doubles the number within a given range of p . I n the case considered, a change of unity in p corresponds to a change of angle of incidence of about 1.8 seconds of arc; and the angular width of the peak a t half-value is about 3.6 seconds. It is clear that a very small divergence of the incident beam, such as can hardly be avoided in practice, is likely to cause complete smearing out of the maxima and minima, even if the thickness of the slice is uniform. The phenomena due to the pendulum effect are therefore very difficult to observe directly, but a brief account of some of the experimental work in this field will be given in Part X. Unless very special precautions are taken to ensure sharply defined beam directions, the practically relevant curve for the reflection coefficient is that obtained by averaging Eq. (61.1), and this will be closely equal to

+

(62.1) R d p ) = 1 / 2 ( p 2 1). In Fig. 27 the average reflection and transmiss’ion curves 8,and 5? are shown for a nonabsorbing crystal. They apply for any thickness. To get Rt and T we superpose on each average curve a “ripple” the amplitude of which is given by the I?, curve in Fig. 26. The ripples are closer together the thicker the crystal slice, and the maxima in R t ( p ) always coincide in p with minima in T ( p ) .

FIG.27. Mean values of direct and reflected transmission coefficient9 p ( p ) and &(PI.

160

R. W. JAMES

62. THEREFLECTION COEFFICIENT FOR

A

VERYTHINSLICE

Using Eq. (59.18), we may write (60.1) in the form (62.1) If the thickness of the slice is much less than the beat thickness A the second factor on the right-hand side of Eq. (62.1) is of the order of unity, and R t ( p ) is proportional to the square of the thickness of the crystal, and to 1 +A 1 2 , as would be expected from the geometrical theory, for AOis inversely proportional to I +I, I . From the figures given for AOwe may conclude that with rocksalt and Mo KCY radiation the slice would have to be of the order of or cm in thickness for this to be true, which is in agreement with other estimates that have been made of the limiting size of the fragments of crystal in perfect array if the mosaic formula is to be used for strong spectra. For a discussion of the transition from the dynamical to the geometrical theory, reference may be made to von Laue's book. Space does not permit a discussion of it here.

63. NUMERICAL DISCUSSION OF THE FORMULAS FOR ABSORBING CRYSTALS We shall now take absorption into account, and apply Eqs. (59.21) and (59.22) to the case of a rocksalt crystal with Cu K a radiation, for which p = 160 cm-I. I n each of these equations, the square bracket contains a nonperiodic ,term, together with a periodic one representing a ripple on the first. We have already discussed this for zero absorption, when the amplitude of the ripple on the reflection coefficient Rt is equal to l&. As the absorption increases, the periodic term in Eq. (59.22) becomes less important in comparison with the other term, especially as the crystal becomes thicker, so that the oscillatory character of the solution gets less and less marked. The reason for this is clear enough. The periodic term is due to a n interference effect between the t v o fields ( I ) and (2) that correspond to the two different wave points; and we have already seen that in the neighborhood of p , = 0 the absorption coefficient is very much greater for one field than for the other. As the crystal becomes thicker, or the absorption coefficient greater, one field will become smaller and smaller relative to the other and the interference effects more and more negligible. We shall calculate the average coefficients F ( p , ) and B t ( p , ) , obtained by putting the sine-squared and cosine-squared terms in Eqs. (59.21) and (59.22) equal to 1/2. On these curves we must imagine a ripple to be superposed, the amplitude of which becomes relatively less and less important the greater the value of Kd.

T H E DYNAMICAL THEORY OF X RAY DIFFRACTION

161

For comparison, we take first the case for which the absorption curves of Fig. 23 were drawn, a slice of rocksalt, with its surface parallel to the planes ( l l l ) ,the reflecting planes being (200). Then, as in Section 56, 5 =

2

1

+

UZ) = $p(yo-'

+ 1 I-') Yh

=

213 cm-'

FIG. 28. R r ( p ) and F ( p T ) for NaCl 200. Planes (111) parallel to crystal surface. Cu Ka radiation, p = 160 cm-1. d = 0.01 cm. Curve A, F ( p t . ) ;Curve B, Rt(pr):Curve C, amplitude of cosine ripple.

The curves of Fig. 28 were drawn, using these figures, for a crystal slice of thickness 0.02 cm. Curve A shows the average transmission coefficient p and curve B the average reflection coefficient €?,,while curve C shows the amplitude of the ripple to be superposed on each curve, in this case still very considerable as the crystal is relatively thin. Because 0% is not now zero, although it is still small, the ripples on T and Rt are no longer exactly opposite in phase. Curves of this type were first given by von Laue.' Their most striking feature is the abnormally large transmission coe5cient when p , is a little greater than zero, that is to say, for glancing angles of incidence on the atomic planes a little less than that corresponding to maximum reflection. This corresponds to the abnormally low values of uz over a small range of angles of incidence, shown in Fig. 23, and discussed in Section 56. It may

162

R. W. JAMES

be inferred that the transmitted beam in this range consists almost entirely of field ( 2 ) . For small negative values of p,, the transmission coefficients are abnormally low, and the fall from the maximum is extremely sharp. I n Fig. 28, a change of unity in p , corresponds to an angular change of about 4 seconds of arc. For both large positive and large negative values of p , , p tends to the same limit,2Qaexp (-&/yo) , corresponding to the reduction of intensity with depth of a wave that travels through the crystal in the direction of the incident radiation and is subject to the ordinary linear absorption of the material of the crystal. For large positive p r the curve tends to this limit from above, and for large negative values from below. The line N N in Fig. 28 indicates this normal transmission coefficient, which is considerably exceeded by the maximum transmission. As we shall see, the relative excess is even greater for thicker slices. The reflection coefficient I?, also rises above the normal transmission. Its maximum occurs at a small positive value of p,, and the curve itself is not symmetrical in p,, since by Eq. (63.1) K is not. When the reflecting planes are perpendicular to the surface the maximum occurs a t pr = 0, and the curve is then symmetrical; but, quite generally, I?,(O) = p(O), which follows directly from Eqs. (59.21) and (59.22). 64. THEVARIATION

OF

R , AND T'

WITH THE

THICKNESS OF THE SLICE

We close the discussion of Case I by considering how 8, and vary as the thickness of theaslicechanges, and for this purpose we shall take the case of a slice of rocksalt in which the planes (200) are normal to the surface, so that y o = Th = cos B0. With Cu Ka: radiation we then have K

= 158/(p,2

+ l)i,

d = ~ / y o=

166.4 cm-l.

The curves A to E in Figs. 29 and 30 were drawn for the following five cm; (b) 5 X lo-* cm; (c) thicknesses (a) cm; (d) 2 X low2cm; and (e) 4 X cm. The transmission coefficients for the direct beam are *ga

If in Eq. (59.11) the average of the cosine term is put equal to zero and the hyperbolic cosine is expanded, with sinh u, = p , and al" - a;' = K the result

follows a t once. For very large 1 p , 1, since pr and sinh sign, we obtain Lt F ( p , ) = exp 1- (d - K ' ) d ) K'

5

being the value of

-

K'

P/yO.

K

for very large p,, which is

fM(Yh-'

K

d always have the same

- yo-*) by Eq. ( 5 2 . 3 ) .Thus

"HE DYNAMICILL THEORY OF X-RAY DIFFRACTION

163

-

0.6

-

0.5 -

NB

0.2

-

-Nc

-

ND

' - 6

-4

-2

0

2

4

6

P t NE

FIG. 29. Average tranamhion coefficients !f'(pr) for slices of rocksalt of different thicknesses, d. Surface of slice parallel to (010).Spectrum 200. Cu Ka radiation p = 160 cm-1; Curves A, B, C, D, and E are for d = 10-8, 5 X 10-8, 10-9, 2 X 10-4 and 4 x lo-* cm, respectively. N A etc. indicate the normal transmission factors.

shown in Fig. 29 and for the reflected beam in Fig. 30. The normal transmission coefficients are indicated by the arrows marked N , with the appropriate su&. cm, differs only slightly from that for Curve A of Fig. 29, that for a crystal with negligible absorption, but a slight excess of transmission above the normal occurs for p r greater than about 4. There is a minimum of transmission, corresponding roughly but not exactly with the maximum of reflection. For this one case, the curves for transmission and reflection are plotted together in Fig. 30 for comparison with the corresponding curves for the nonabsorbing crystal shown in Fig. 27. As the thickness of the slice increases, the asymmetry of the transmission curve becomes more marked. A maximum develops for small positive values of p , and, as the thickness grows, this rapidly becomes higher relative to its surroundings, although absolutely lower, and the value of p , at which it occurs becomes smaller. For greater thicknesses the transmission becomes very small, except in the region of the peak, particularly for negative p,, but the peak values both of 5? and 8,remain surprisingly large.

164

R. W. JAMES 0.1

0.E 0.7 0.E

0.5

0.4

-NB

0.3 0.2 0.I C

-

FIQ.30. Average reflection coefficients l?t(pr) for slices of rocksalt of different thicknesses, d. Surface of slice parallel to (010). Spectrum 200. Cu Ka radiation, p 180 cm-1; Curves A, B, C, D, and E are for d = 10-8, 5 X 10-8, lo-*, 2 X 10-2 and 4 x 10- cm, respectively.. NA,etc. indicate the normal transmission factors.

In Table I the values of the normal transmission factor exp (- Bd) for the symmetrical case, and the maximum values of F and R , are shown for a number of thicknesses of the slice. The last column gives the corresponding ratios of the maximum transmission factor to the normal transmission factor. The greatest thickness for which figures are given is 40 times the smallest, an increase that reduces the normal transmission factor from 0.847 to 0.0013; but the maximum transmission is reduced only from 0.857 to 0.234, and the maximum reflection factor from 0.429 to 0,181. The figures in the last column of Table I are very significant, for they show how a beam transmitted in the optimum direction would stand out above the general background. At the greatest thickness for which figures are given, 0.4 mm, F (max) is still as large as 0.234, and is decreasing quite slowly with increasing thickness. Provided that they stood out sufficiently from the background, very much weaker transmitted beams

165

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

TABLE I d X 108 cm

exp ( - 3 d )

1 5 10 15 20 25 30 40

0.847 0.435 0.189 0.0824 0.0358 0.0156 0.0068 0.0013

-

-

T (max)

R c (max)

0.857 0.581 0.434 0.368 0.321 0.291 0.269 0.234

0.429 0.288 0.240 0,224 0.212 0.205 0.197 0.181

-

Tmaxexp (3d ) 1.012 1.310 2.293 4.46 8.97 18.64 39.60 183

could easily be detected. Considering the numbers in columns 3 and 4 of Table I in comparison with those in the last column, we might expect detectable abnormal transmission of this kind to occur with quite thick crystals, a conclusion fully confirmed by the work of Borrmannt3 (see ii) .28b The calculations in this paragraph have been made for rocksalt and Cu Ka radiation, but a glance a t Eqs. (59.21) and (59.22) will show that . the results would apply to a and Rt are functions of the product ~ d All crystal twice as thick with a value of K half as large. Now K is proportional to p, but it also depends on y o and yh, and so on the geometry of the reflection, and we cannot simply say that the formula for a given value of K d , evaluated for a thickness d, and an absorption coefficient p , would also apply for the same spectrum for a thickness nd and a wavelength for which the absorption coefficient is p / n , As a rough guide, however, this approximation can be safely used. For example, Table I would give the general magnitude of the transmission coefficients of perfect slices of rocksalt, with thicknesses ten times those tabulated, for PI10 K a radiation, for which I.( in rocksalt is about 16.4 cm-I. COEFFICIENTS FOR 65. R E ~ E C T I OAND N TRANSMISSION FINITETHICKKESS IN CASEI1 (REFLECTION)

A

SLICEOF

There is now no transmitted beam Dhe leaving the lower surface, and we denote the reflected beam from the upper surface by Dhr. Equations (58.6) for the upper surface now become

+ Doz XIDOI+ ~2Doz Doi

*gb

=

Dod

(65.1)

=

Dhr

(65.2)

See also A. Authier, Actu Cryst. 14, 287 (1961).

166

R . W.

and (58.7), for the lower surface, exp (iald)Do1 XI

exp (iald)Do1

JAMES

+ exp (iazd)DOZ

=

DO‘

(65.3)

=

0.

(65.4)

+ xz exp (iuzd)DOZ

Using Eqs. (65.1) and (65.4)) we first find expressions for Dol and Dozin terms of Doi, and substituting these in (65.2) and (65.3), obtain Dhr

=

XI22 [exp 26

(iazd)

- exp (iuld)]Doi

(65.5) (65.6)

where

u = xz exp (iad) - x1 exp (iald).

(65.7)

+

If the crystal is absorbing we must put a = a’ id’, as before. In Case I1 we have always to distinguish between the three ranges:

- cosh u,

Range (i) :

p =

Range (ii) :

p = cos u,

Range (iii) : p

+

=

cosh u,

XI =

-Ge-u,

52

x1

GelY,

x2 = Ge+

=

z1 = GeU,

=

-Gev

x2 = Ge-v,

and u = Vr ivi, if the crystal is absorbing. Proceeding just as in the discussion of Case I in Sec,tion 59, we may obtain without difficulty from these results for range (i) ,

(65.9)

Here, as before, K =:

all’ - az” = Ul’

- u2’

- 02) = 2?rk(gr, - grz). 1 2 (01

(65.10) (65.11)

The correct values of these quantities are those given in Eqs. (53.3) and (53.5) of Section 53, but the approximations (53.9) and (53.11) are valid except in the near neighborhood of p r = f l . In range (iii) Eqs. (65.8) and (66.9) remain valid if the signs of the terms Zv, and 2vi are interchanged. In range (ii) , pr2 < 1. The results for this range are easily derived from those of range (i) if it is noticed that u in the formulas for range (i) becomes - i v in those for range @), so that vy iui becomes vi - iUr. Thus

+

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

we need only replace v by -iv, v, by (65.9), which gives at once

R(v) =

vi

and

vi

167

by -v, in Eqs. (65.8) and

cash Kd - cos ( ~ 1 ' - az')d cash { Kd 2vi) - COS { ( ~ 1 ' - a;)d

+

+ 2vr] I 4 d 4 ~I

(65.12)

I sin 21 l2 (65.13) cash { Kd + 2vi) - cos { (ax' - a$)d + 2 ~ , ). The values of K and a2 - a21 are now those for range (ii) , and are given by T(v) =

2 exp (-ad)

Eqs. (53.13) and (53.16) of Section 53. I n that part of the range in which the approximation (53.14) is valid, which in practice is a considerable part of it, K is numerically very nearly equal to the primary extinction coefficient. For symmetrical reflection and state (a) of polarization, the maximum value of 1 K 1 is, from Eq. (53.14), given by (65.14) which indicates how the extinction coefficient at p , ordinary linear absorption coefficient.

=

0 is related to the

66. CRYSTAL SLICEWITH NEGLIGIBLE ABSORPTION The evaluation of the formulas just deduced for an absorbing crystal slice will not be undertaken here, although reflection from a very thick slice will be dealt with in some detail in Section 68. It seems worth while at this point, however, to discuss reflection from a slice of nonabsorbing crystal of finite thickness. The necessary formulas are easily deduced as special cases of those for the absorbing crystal. In ranges (i) and (iii), 5 and K become zero by Eqs. (53.3) and (53.4) , when absorption is negligible, while v, = v and vi = 0. Since 4"iis now zero, y of Eq. (53.6) is zero, and by (53.5) and (58.8) (66.1)

where

(66.2) AObeing the beat thickness, as defined in Eq. (46.5). By Eq. (65.8) therefore, 1 - cos (2xd/A)

cosh2v - cos (2rd/A) sin2 (sd/A)

(66.3) (66.4)

168

R. W. JAMES

and, for a nonabsorbing crystal,

T(v) = 1 - R(v)

(66.5)

of necessity; and this is easily verified by making the appropriate substitutions in Eq. (65.9). Since cosh2v = p 2 , Eq. (66.4) may be written, after some obvious transformations, and using (66.2) , in the form

R(p)

=

{p'

+ ( p 2 - 1) cot2 [ d ( p 2 - l)*/Ao]}-',

(66.6)

which applies in ranges (i) and (iii). I n range (ii), ul' - u2' vanishes for a nonabsorbing crystal, by Eq. (53.16), since y is now zero; while, by (49.3) and (53.13), K

=

- 2 ~ ( 1- P'))/Ao.

In this range, p = cos v, and by processes similar to those used in deriving Eq. (66.6) we obtain from (65.12)

R(p)

=

(p'

+ (1 - p 2 ) Coth'[Td(l

- p2)*/Ao])-'.

(66.7)

This result can also be deduced directly from Eq. (66.6) by replacing (pz - 1)t in Eq. (66.2) by i ( 1 - p 2 ) t , and using the identity cot (ix) = -i coth (2). 67. DISC~SSION OF THE FORMULAS FOR THE NONABSORBING CRYSTALS Equation (66.6) applies when p 2 > 1, outside the range of total reflection for a thick crystal. The solution is here of a n oscillatory type, and R ( p ) varies between zero, when ( p 2 - 1)*= nAo/d,

n being a n integer, and l/p2, when (p2 - 1)t

=

(n

+ &)Ao/d.

(67.1) (67.2)

When R ( p ) is zero the transmission coefficient T (p ) is unity, and all the energy is in the transmitted beam. For a given value of the incidence parameter p , the reflection is zero and the energy wholly transmitted when d

=

nAo/(p2 - 1)t

(67.3)

so that the beat thickness corresponding to the angle of incidence represented by p is Ao/(p2 - l ) f ,and the reflection coefficient varies between zero and l / p 2 as the thickness varies by one-half this amount. If the thickness d is constant, and equal to mA0, R ( p ) is zero and T ( p ) unity whenever (67.4) p2 = 1 (n/m)2

+

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

169

n being integral, but greater than zero. The right-hand side of Eq. (66.6) is indeterminate when p = & l l and this case must be investigated separately. The solution in ranges (i) and (iii) is t’husof the same general type as that investigated for Case I in Sections 60 and 61. I n range (ii) Eq. (66.7) applies. The periodic cotangent is now replaced by the nonperiodic hyperbolic function, and the solution is no longer oscillatory. At p = 0 the reflection coefficient is R ( 0 ) = tanh2 (ad/Ao)

(67.5)

and this is the maximum value of R ( p ), which is symmetrical about p = 0. R(0) is extremely nearly unity if d is more than a very few times Ao. The value of R ( p ) a t the limits of range (ii), p = =tl, are easily calculated from Eq. (66.7) if we remember that (sinh X ) / X and cosh X both tend to unity as X tends to zero. As I p [ tends to unity, the second term ) ~ the first in the denominator of Eq. (66.7) therefore tends to ( A ~ / n dand to unity; and Eq. (66.6)) similarly treated, yields the same result. Thus

R(&1)

= [l

+ (Ao/~d)~]-’.

(67.6)

This approaches unity closely if d is much greater than Ao. For such thicknesses reflection is very nearly total over the whole of range (ii), in agreement with the Darwin formula derived in Section 37. A “thick” crystal, in the sense in which the term was used in Part VI, is thus a crystal whose thickness is relatively large in comparison with Ao, As an example, we now consider reflection from a parallel-sided slice of rocksalt, with the (200) planes parallel to the surface. In order to illustrate the theory, we shall make the assumption that ordinary absorption may be neglected in slices of the thickness considered. For symmetrical reflection, yo = I Y h I = sin Oo, and in state (a) of polarization Eq. (59.19) becomes A sin Bo Ao=

sin Oo 1‘ amc2 = --I $% 1 A I F ( h ) I e2

by Eq. (12.7), and is independent of the wavelength since 1 F ( h ) I is a function of (sin B O ) / X . On comparison with Eq. (39.2), we see that

PO = 27~/~max

(67.7)

where ulnsxis the maximum value of the primary extinction coefjicient for the same reflection. Using the values given in Section 39, we obtain for

170

R. W. JAMES

FIG. 31. Reflection coefficients R ( p ) from slices of non-absorbing crystal for thicknesses *A0 (A), A0 (B) and 2A0 (C), A. being the beat thickness. is the Darwin curve for a thick crystal.

rocksalt 200, and symmetrical reflection A. = 4.13 X l F 4 c m . This is the actual value of the beat thickness when p = 6. In Fig. 31, curves A , B, and C show the variation of R ( p ) with p for slices of thickness :Ao, Ao, and 2A0, respectively, which, in the example considered, are approximately 2, 4, and 8 times cm. The zero values of R ( p ) in ranges (i) and (iii), where oscillation takes place, occur at values of p given by p2. = 1 4n2, p2 = 1 n2,and p 2 = 1 n2/4, respectively where ?t takes successively the values 1, 2, 3, * * It will be seen from curve C that when the thickness is 240, less than 1/100 mm, reflection is nearly total over the whole df range (ii). The figures suggest again that for the mosaic reflection formula to be applicable in this instance, the optically independent perfect regions can have a maximum thickness not greater than between 10-6 and 10-4 cm. If they were as thick as the slices considered here, the crystal would show strong primary extinction. In ranges (i) and (iii), the rate of oscillation with varying p becomes rapidly greater as the thickness increases. In Fig. 31, A , the curve for the thinnest slice, has been drawn in both ranges, but to avoid confusion curves B and C have been drawn in one of the oscillatory ranges only, B on the right, C on the left. One could hardly hope to observe the oscillatory effect directly. Unavoidable small variations in thickness and lack of complete parallelism of the incident beam will produce an averaged curve which in an actual case will probably lie very close to the Darwin curve D in most of ranges (i) and (iii).

+

+

+

0 .

171

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

68. REFLECTION COEFFICIENT FOR

A

THICK ABSORBINQ CRYSTAL

When the crystal slice is so thick that no appreciable field reaches its lower surface, only positive values of Q are possible, and as we saw in Section 57, this means that field (1) alone can occur in range (i) and field (2) in ranges (ii) and (iii). I n range (i), Eqs. (65.1) and (65.2) then give 21

=

Dhr/Doi

(68.1)

and the reflection coefficient is equal to

R

=

(

I Y h I /YO) I Dhr/DOi /* = ( I

I /YO) 1x1

(68.2)

a result already given in Section 87 for a thick nonabsorbing crystal. By Eq. (30.17), R ( v ) = ( I 7% 1 /YO) I @ l2 I e-" 12, (68.3) but v is now complex, and equal to v, write this

R(vr)

=

+ ivi. Using Eq. (30.5), we may

I h/$% I exp (-2vr)

(68.4)

which may be compared with Eq. (37.2). A similar argument shows that the same formula applies in range (iii) . In range (ii) field (2) is the appropriate one, but here, by Eq. (80.25), xg = Ge--IU1and so (68.5) R = I6h/$L I exp (2vi). Equations (68.4) and (68.5) are very easy to apply in those ranges of incidence in which the approximations for vr and v i given in Sections 53 and 54 are valid, that is to say unless p , is near f l . For the moment we assume the approximations to hold, and, using Eqs. (!54.10), (54.12), and (54.18), find for ranges (i) and (iii), (IprI

> 1)

R(pr) = I h / d L I exp

with vr =

and for range (ii) , using (54.2),

where, since now

Yh =

I COSh-Ipr I

(68.6) ;

- 1 Yh I ,

This is an essentially positive quantity, for in range (ii), I p , and A , is less than unity, and YO I Y h I > 2(r0 1 Y h 1 ) ?

+

I , like I C I

172

R. W. JAMES

0.2 -

-3

-2

-I

0

I

2

3 Pr

FIG.32. Reflection coefficient R ( p , ) for a thick crystal of rocksalt. Spectrum 200, Cu K a radiation ( p = 160 cm-1). The curves for states (a) and (b) of polarization are shown in relation to the Darwin curve D. To obtain the relative anges of reflection the abscissae of curve (b) must be reduced in the ratio I cos 26 I:1, or 0.85:l.

It will be clear that in the part of range (ii) in which Eq. (68.7) applies the numerical value of the exponent is smaller for a positive value of pr than for the corresponding negative value, so that the curve giving R as a function of p , has a positive slope (see Fig. 3 2 ) . When p , = 0, R (0)

=

1 $ h / & I exp

( -2B)

For symmetrical reflection, when the crystal planes are parallel to the surface, YO = 1 Y~ I , and Eqs. (68.7) and (68.9) become

(68.11) (68.12) The reflection coefficient is thus smaller the greater p and the smaller I $ h r 1 , that is to say, the smaller the structure factor of the spectrum; and for a given spectrum it is also smaller for state (b) of polarization than for state ( a).

173

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

It is perhaps of interest to note that R ( 0 ) for symmetrical reflection can be expressed in terms of the normal absorption coefficient in depth, a, ( = p / y o ) , and ue, the primary extinction coefficient for the crystal if it is assumed to be nonabsorbing, as defined in Eq. (39.1), in the form (68.13)

The quantity I I$h/C#Jhr I differs little from unity,290and Eq. (68.13) shows that the reflection coefficient is larger the smaller the absorption and the greater the primary extinction. An accurate formula, valid over the whole range, was given by Prins in 1930.6To obtain it, we use the value of x given by Eq. (30.3),

(68.14)

We may express this in terms of by i multiplying the nuinstead of +, merator and denominator of Eq. (68.14) by ,f3 F W , which gives (68.15)

Then, just as in deriving Eq. (51.8), and using (68.2), remembering that b = 2 I C I ( I y h I / y o ) * , weobtain

(68.16)

Using Eqs. (47.5), (47.7), and (51.11) it may be shown that

I neglecting terms of order

R ( P ~=)

4hnr I 2 / p h r Q2,

=

so that 1

I p,

1

+ Q tan

+ Q tan

vh

Vh

+ iB =I= [ ( p , + iB)2- (1 + iQ)]+

l2

(68.17)

From Eqs. (47.2), (47.5), and (47.6) it is easy to show that for a centrosymmetrical I +j,i/+j,, I* which for rocksalt 200 differs from unity by crystal [ + h / + b I = 1 leas than 1 part in 1,000; but the correction may be rather greater for higher orders.

+

174

R. W. JAMES

which, apart from a difference of notation, is Prins's formula. The upper sign of the ambiguity refers to field (1). Putting (1

z=p2-

+ B'),

y = 2Bpr

-Q

in Eq. (68.17) we obtain (68.18) Ranges (i) and (iii) are now those in which x > 0, and range (ii) that in which x < 0. The negative sign of the ambiguity applies in range (i), and the positive sign in ranges (ii) and (iii). If in range (ii) we now put z = - I z 1 in Eq. (68.18), we obtain

(68.19) To evaluate these expressions numerically, we use the result (121

(68.20)

z t i j y l ) ' = ufiw

where u =

{+[(x'

1L' =

(+[(x'

+ y")" +

+ y')f

15

- Iz

I I}+

1 I]*.

Since B and Q are both positive, and B > Q, y is negative in range (i) and positive in range (iii), whence, using Eqs. (68.18) and (68.20), the expression

+ Q tan ( 1 pr I + u)' + ( B + 1

R(pr)

=

Yh

4 2 '

(68.21)

valid in both ranges, follows at once. The corresponding formula in range (ii) is (68.22) Here it must be noticed that p , is the algebraic value, and not the numerical value as in Eq. (68.21). The positive sign applies if 1~ is positive, and the negative if y is negative. I n practice, y will be negative in range (ii) , except very near to p , = + l , and the sign of w will in practice usually be negative. We shall apply the formulas to the example already considered, that of rocksalt 200, n-ith the crystal planes parallel to the surface, using the figures previously employed for the same crystal. The calculations are made for Cu K a radiation ( p = 160 cm-l) , I n this simple case 70= I Th 1 ,

THX DYNAMICAL THEORY OF X-RAY DIFFRACTION

and tan

Vh

= 0,

175

because the crystal is centro-symmetrical; and

B

=

I C I-' I +Oi/+hr

I

Q

1

=

I $hi/+hr I

by Eqs. (51.10) and (51.11). For state (a) of polarization, when the induction is perpendicular to the plane of incidence, we find for this examplem

B

= 0.0407,

y = 2Bp,

and for state (b)

-Q

Q

= 0.0776

=

O.O814p,

= 0.0956pV

- 0.0776

- 0.0776.

We shall neglect B2in comparison with unity, putting I x 1 = I p,2 - 1 1 . The reflection curves for rocksalt 200 for the two states of polarization, plotted as functions of p,, are shown in Fig. 32. It is simple enough, if a little tedious, to evaluate Eqs. (68.21) and (68.22) ; but the approximate formulas (68.6) and (68.7) apply quite closely over a considerable fraction of the ranges, and with the values of the reflection coefficient at p , = f l , which are very easily obtained from the accurate formulas, should determine the course of the curve for mist purposes. For the example under discussion and

R(-1)

=

0.575,

R(+1) = 0.912 forstate (a),

R(-1)

=

0.440,

R(+1) = 0.821 for state (b).

IX. The Propagation of Energy in the Wave Field

69. INTRODUCTION

We shall consider the flow of energy in the crystal in Case I, that of transmission, when an indefinitely broad beam, sharply defined in direction, is incident on its surface. The boundary conditions on incidence limit the revelant wave points to a discrete set, lying at the intersections of the dispersion surface with a line perpendicular to the surface and passing *Od

For the source of these figures reference may be made to the paper by Renninger"

already quoted, or to the Optical Principles", Chapter VI, page 326. The symbols used in this article are related to those of the earlier work in the following way: ++a A1

&h

BIA

=8

+ iB

+

f iB1,

ht~fi AZ ~ B z

+Qi/+hri

B/A

'$hi/+hr.

176

R. W. JAMES

through the point Poion the circle of incidence. Each wave point A , of the set is associated with a group of wave trains with wave vectors Kma, directed from A , to the reciprocal-lattice points m. Once the position of the wave point is known, the relative amplitudes and phases of these waves are determined; but nothing less than the complete set of wave trains associated with a wave point is a solution of Eq. (13.1) that governs wave propagation in the crystal, and for any such set there is, at each point in the crystal and at any instant, a completely determined direction of energy flow, that of Poynting's vector S for the energy current-density, given by (69.1) S = (c/47r)E x H where E and H are the resultant electric and magnetic vectors for the whole field associated with the wave point. The cross here denotes the vector product. For a definite state of polarization of the incident radiation the fields associated with the different wave points of the discrete set are mutually coherent. Their relative amplitudes and phases are determined by the boundary conditions at the surface of incidence, and in a proper discussion of the energy transmission we must consider the complete coherent field, including any interference effects that may take place between the fields associated with the different wave points. To different states of polarization there correspond different sets of wave points. If the incident radiation is unpolarized, the fields corresponding to two mutually perpendicular directions of polariza'tion are independent and incoherent, and this is a common case in practice; but if the incident radiation is itself polarized the fields associated with all the wave points are coherent. For the time being, we shall assume this to be so. 70. THEMEANENERGY CURRENT

To obtain an observable quantity, we first take the time-average of the energy current over a time very long in comparison with the period of the waves, which is given by

8' =

(c/Sa)R(E x H*)

(70.1)

where the symbol R denotes that the real part of the expression following it is to be taken.se The field-equations deduced in the earlier sections con-

+

If we write E = EOcos wt, H = HOcos (wt p ) , then S = ( 4 4 s ) (EoX Ho) X (cose wt COB p - 3 sin 2wt sin p ) , the time average of which is (48s)(Eo X Ho) cos p, and this is the real part of ( c / S r ) (E X H*) when E and H are written in complex form.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

177

tain D and not E; but because the effective dielectric constant for X-ray frequencies differs so slightly from unity a t every point, no appreciable error either in magnitude or direction will be made in calculating the energy flow by writing D instead of E in Eq. (70.1). Accordingly, we put

St

=

( c / 8 r ) R ( D x H*)

(70.2)

noticing, however, that when the dielectric constant is not explicitly given the equation is not dimensionally correct. By Eqs. (13.3) and (14.1)) D

=

x

exp {2?ri(vt- Koa.r)}

z

D,, exp { -27riRm*r}

m

u

H = x e x p {27ri(ut - KoB-r)}x H n b e x p {-27riRn-r}. B

(70.3)

n

Here, a and /3 denote wave points, and m and n reciprocal-lattice points, and in each case the sum is to be taken over all the wave points belonging to a coherent set of waves, and over all reciprocal-lattice points. From Eq. (70.3)

D x H*

= u

z e x p {27ri(Kop* - Ko,).r] B

m

z D m , x H,p* n

X exp {2ai(Rn - R,) ar].

(70.4)

We take the absorption of the crystal into account by using Eqs. (50.1), (50.2)) and (50.4), from which KOB*- KO, = - (gB* = =

where

=

-gdkn

+ i ( g , i + gpi) }kn (gar g s . ) k n + (i/2?r)a,~n $ ( a , + a@), { (gar - gpr) -

(70.5) (70.6)

u. and US being the absorption coefficients in depth associated with fields (a)and ( p ) , respectively. On substituting these results in Eq. (70.4) we

obtain

D x H* =

zz

exp (-aaaz) exp (27rik(g,, - g s r ) z )

a

@

X

xz m

n

D,,

x H,B* exp (27ri(Rn - Rm) or}

(70.7)

in which z = n-r, and is the depth of the point under consideration below the surface of the crystal.

178

H. W. JAMEB

In the summation of Eq. (70.7) , the absorption factors change very little in the depth of a single cell of the crystal, even if the absorption coefficient is relatively large. The first periodic factor also changes very little in the same depth; for k ( = 1/X) is of the same order of magnitude as the reciprocal of the cell depth, while gar - gsr expresses the distance between the wave points a and p as a fraction of k , and is very small in comparison with unity in the relevant ranges of incidence. Within any given cell we may therefore with little error assign to each of these factors its value a t the cell center. The exponential factor under the double summation varies within a lattice cell, but in the same way in all cells, and to obtain a physically relevant quantity we shall replace it by its mean value taken over a single cell. R, - R, is itself a reciprocal-lattice vector, which we may denote by ( 70.8) j

where b j is one of the three primitive translation vectors of the reciprocal cell, and q, is a corresponding integer. The vector r from the origin, chosen at some convenient point in the surface, to a point in the pth unit cell of the crystal lattice may be written (70.9)

where ak is a primitive vector of the lattice cell, pk is an integer, and uk is one of the three fractional coordinates of the point in the unit cell, so that 0 < uk < 1. Then exp (271+R,.r) = exp ( 2 r i C C qjbj. ( p k j

k

+uk)ak)

(70.10)

since p k q j is a whole number and, by the properties of the reciprocal lattices, bj-ak = 1 if J' = k , but vanishes if J' # k. The mean value of the right-hand side of Eq. (70.10) over a unit cell, which we denote by M , is given by

M

= (1/v)

/

0011

exp (2ni

C q k u k ) dV k

and, since dV = Vdvl dv, d v 3 , this is the product of three integrals of the type Mk =

I'

eXp ( 2 i T i q k u k )

dUk

179

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

and this vanishes unless q k = 0, in which case its value is unity. M itself therefore vanishes unless all three values of q k are zero, that is to say unless m = n, when it is equal to unity. On averaging over the unit cell, the double sum over m and n thus reduces to a single summation, say that over n. The mean value of D x H* over a cell at depth z is therefore

D

X

H*

=

C C e x p (2?riz/A(cu@)} exp (-a,@) C (Dna a

8

X

Hnb*).

n

(70.11) Here we have written

A(a@> = l/k(ga, - g d ,

(70.12)

which is the period in depth of an interference effect between the coherent fields associated with the wave points CY and p. This period is generally large in comparison with the depth of a single cell, so that in spite of the averaging processes used in obtaining it we can, in effect, treat Eq. (70.11) as a continuously varying function of the depth z below the surface of the crystal. By Eq. (14.4), HnB = (k/Kn&Knp x DnB = (k/Kn8)sn@

X

DnB

(70.13)

where snp is a unit vector in the direction of Kn8, that is to say of the line joining the wave point @ to the reciprocal-lattice point n. The magnitudes of k and K,,b differ by not more than a few parts in a hundred thousand, and in what follows we shall write

(70.14)

The waves are all transverse, and the directions of the wave vectors from the different wave points to a given reciprocal-lattice point n differ in the relevant range of incidence by angles of the order of seconds of arc only. In state (a) of polarization the vector s , ~is perpendicular to Dna.In state (b) it is perpendicular to D,, when CY = 8, and very nearly indeed so when a # @. We may therefore neglect the second term on the righthand side of Eq. (70.15).To the same degree of approximation, we may also replace snp by a unit vector s, in the direction of the radius of the

180

R. W. JAMES

Ewald sphere from L to the reciprocal-lattice point n. Equation (70.11) can then be written

D x H*=

Q

xexp B

and

(70.16)

where en(a,9) is the angle between the directions of the vectors Dn, and Dnb, and Sn(aO) is the phase difference between them. Using Eq. (70.2), substituting (70.17)in (70.16),and taking the real part, we obtain for the time average of the mean energy current over a unit cell at depth z the expression

since to the approximation we have used cos en (a@)may be taken as unity. The mean energy current-density is thus a vector sum, having components in each of the directions Sn from the center of the Ewald sphere to the reciprocal-lattice points n. That 9 can be expressed in this way was first shown by von Laue,ao who gave an expression equivalent to Eq. (70.18)except for the presence of absorption factors. The expression within the square brackets in Eq. (70.18)gives the magnitude of the component of 9 in the direction 6,. A typical term in the sum over a gives the contribution to this component from the wave train K,,, supposing it alone to have been present. The vector sum of all such contributions over all wave points and all reciprocal-lattice points would give the total energy current-density if the fields associated with the different wave points were incoherent; but since they are in fact coherent, there are interference effects, and these are taken into account by the second summation, which extends over all pairs of diferent wave points, for each value of n. The appropriate absorption coefficient for any term associated with two wave points a and @ is u , ~ , the mean of uQ and up which correspond to the fields associated with these wave points. M. von Laue, Acfa Cryet. 8, 619 (1952).

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

181

RECIPROCAL-LATTICE POINTS 71. Two RELEVANT When a number of reciprocal-lattice points lie near the Ewald sphere and the state of polarization is arbitrary, the discussion of Eq. (70.18) is a matter of some complexity. We shall therefore confine the detailed discussion to the two-beam problem, when only two points lie near enough to the sphere to give rise to appreciable wave fields. In this case n can take the values 0 and h. As before, we consider two independent perpendicular directions of polarization, (a) and (b). The dispersion surface consists of two hyperboloids, one for each of these states of polarization. We denote where the the corresponding induction vectors by Dnaa,Dnpa,Dnab,D,,B~, upper index denotes the state of polarization, and CY and /3 may be either 1 or 2. The index 2 refers to a wave point on the upper branch of the hyperbola corresponding to the state of polarization concerned, the index 1 to a point on the lower branch. These points will of course differ for the two states of polarization. Whatever the values of CY and 8, Dnaaand Dnpbare mutually perpendicular, so that the corresponding scalar products vanish. As one would expect, therefore, fields belonging to two mutually perpeqdicular states of polarization make no contribution to the interference terms in Eqs. (70.16) and (70.18). We therefore separate the terms in the summation belonging to the states of polarization (a) and (b), and treat them independently. Using Eqs. (70.2) and (70.16), and omitting for compactness the polarization indices (a) and (b) in the symbols for the induction, we find for either state of polarization

s = (cPR) (exp (-mz) ( I DO,I2so + I Dhl

I2sh)

+ exp (-a#) ( I Do2 I2so+ I Dh2 12%) + exp ( -62) cos ( 2 ~ 2 / A ) X [(Doi*Do2*+ Dol**Doz)~o + (Dhi.Dm* + Dhi*.Dh2)Sh]). (71.1)

Here, 5

= $(ul

+

u2),

A =

as in Part VII, while 1

k(&l - g72)

-

A(Y0

I Yh 1)'

I c I I @Ar I ( p 2 +

(71.2)

and is the beat period in depth, as defined in Eq. (59.18)-the period of the interference term between the two fields (1) and (2). The value of A is greater for state (b) of polarization than for state (a), and also varies with the angle of incidence; and, as we have seen already, the two absorptions coefficients u1 and u2 vary markedly with the angle of incidence, and over a very small range of angles in the region of interference may differ very greatly.

182

R. W . JAMES

72. THEENERGY CURRENT-DENSITY AS A FUNCTION OF

INCIDENCE

THE

ANGLEOF

In Eq. (71.1) we now use the relations between the field vectors in the crystal and in the incident radiation imposed by the boundary conditions on incidence. By Eq. (59.2) Dai/Do'

= 52/(~2-

Doz/Doi

=

-xi/

(52

xi),

-

XI)

,

Dhi/Doi

= ~ i ~ 2 / (~ 2XI)

Dhz/Do'

=

-XlX2/(22

- XI)

from which, by Eqs. (59.9) and (59.9a), it follows a t once that

(72.2) The quantities in the periodic term of Eq. (71.1) are necessarily real; and since all the induction vectors are parallel in state (a) of polarization, while the pairs of vectors Dol, Do2 and Dhl, Dh2 are very nearly parallel in state (b), we may write the scalar products as ordinary products, and, taking due regard of signs, obtain

Doi.Doz*

+ Doi**Do2

=

cos 2vi

2 j coshv

l2

j Do' l2

(72.3)

(72.4)

If

sois the mean energy current density in the incident beam, 1 So

1

=

( ~ / 8 nI )Doi l2

(72..?)

and using Eqs. (72.1)-(72.6) we may write Eq. (71.1) in the form

Here, v

=

v,

+ iui, and to a close approximation we may assume

1 coshv l2

=

cosh2v,

=

p,2

+ 1,

u, = sinh-l p,.

(72.7)

183

T H E DYNAMICAL THEORY OF X-RAY DIFFRACTION

1 G l2 = (yO/yh) I +h/& I , which becomes yo/yh if the crystal is centrosymmetrical, as we shall assume in what follows. We now write U l = b + K ,

U 2 = d -

K,

as in Eq. (52.4), and, taking out the common factor exp Eq. (72.6), collect the coefficients of SO and sh. This leads to

8=

I 80 I exp (-52) 2 cosh2vr

[{cash (20,

(-52)

from

+ + cos ( ~ T z / A )cos 2 ~ ; )

SO

KZ)

+ 1 G 12{cosh

KZ

- cos (27rz/A)}sh].

(72.8)

From Eq. (72.8) the energy current-density in the crystal can be calculated a8 a function of the angle of incidence, By Eqs. (51.18) and (52.3), 5 = $P(rO-'

f7h-l)

(72.9)

where I.( is the ordinary linear absorption coefficient of the crystal. The parameter p , is proportional to the angle of incidence of the radiation on the surface, measured from the direction a t which the normal to the surface through the point Po'on the circle of incidence passes through Q, the center of the hyperbola. If this angle is A+gl (72.11)

[see Eq. (34.3)]. The actual angles concerned are always small, and a change of unity in p , will correspond to a change of only a few seconds in A#*.

73. CRYSTALS WITH NEGLIGIBLE ABSORPTION.THEPENDULUM SOLUTION

When the absorption of radiation in the crystal is negligible, 01

=

02

= d = 0;

vi

=

0, vT = v = sinh-lp

and Eq. (72.6) becomes

g=-

I so I

4 cosh2v

[{e-20so

+[G

['Sh}

+ [e*"so + 1 G +2

COB

/%h}

(2a~/A){So

- I G I2Sh)].

(73.1)

The right-hand side of this equation is the sum of three groups of terms, of which the first two represent 8, and 82, the energy current-densities

184

R. W. JAMES

due to the fields associated with the wave points A , and Az, respectively, either of which can be propagated independently. The two fields are, however, coherent, and the third term takes account of the interference between them when both fields are present and their relative amplitudes and phases are fixed by the boundary conditions on incidence. A simple transformation of Eq. (73.1) gives

-

s=-

i s 0 1

coshZv

=-

[(cosh2 v

- sin2 (?rz/A) ) S O f I G l2 sin2 (?rz/A)sh] (73.2)

I so I

p2

+ 1 [{ p z + cos2 (?rz/A)} + I G l2 sin2 (?rz/A) SO

sh]

(73.3)

from which it is evident that the direction of the energy current varies with the depth below the surface with a period A. Whenever z = ma, m being an integer, the component of in the direction s h vanishes. The resultant energy current is then entirely in the direction SO, and is equal to I 5 0 I so, the mean energy current density in the incident radiation. This is true whatever the angle of incidence, but it must be remembered that A depends on the angle of incidence in the way given by Eq. (71.2). When p (or v ) vanishes, and the diffracted beam is a maximum, the energy current is entirely in the direction of s h when the depth below the + ) A . At intermediate values its direction lies somewhere surface is ( m between these limits. As the depth varies, the direction of the energy current swings backward and forward between so and Sh, the directions of incidence and of reflection from the crystal planes. This is Ewald's pendulum solution, which we have already discussed in Sections 46, 60, and 61, considered from the point of view of energy transmission. When p is not zero, the so component never vanishes entirely. Equation (73.3) shows that the SO and s h components are respectively of magnitude I I and zero when z = mA,and p2 1 I / ( p 2 1) and I G l2 I SOI / ( p z 1) when z = ( m + ) A : A itself varies inversely as ( p 2 1)'. The main features of the pendulum solution follow simply and directly from Eq. (73.3), and we shall discuss it in a little more detail in Section 76. Whether we consider the total energy current-density or the individual current densities and s z , we have to deal with two components, one in the direction of SO, the other in the direction of Sh, and these we derepresents the incident note by Pso and Qshl respectively. In Fig. 33 energy current-density, which is always very nearly in the direction so since the total range of angles of incidence considered extends over a range of only a few seconds on either side of the direction corresponding to a diffraction maximum.

s

+

so

so

+

+

+

+

s,

s,

so

185

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

FIGURE 33

Let j be a unit vector in the direction of the line bisecting the angle between SO and Sh, and let e be the angle between the direction of the resultant energy current and j, taken as positive if S lies between so and j. Then, since SO’% = cos 2e0 = 2 coszBo - 1

s = PSo + QSh

1 l2 tan

(P -

=

e =

&)2

p-Q ~

+

+ 4PQ cos2

tan 00,

80

(73.4) (73.5) (73.6)

formulas that are easy to apply to the different cases that arise.

CURRENT-DENSITIES IN THE FIELDS ASSOCIATED WITH 74. THEENERGY INDIVIDUAL WAVEPOINTS In the more detailed discussion we shall consider first the energy currents associated with the individual wave points, leaving until later the question of their interaction. By Eq. (71.1), when the absorption is

186

R. W. JAMES I. 0

,,A, -5

-15"

-4

-3 -10"

-2

-I

-5"

0

0

2

I

5"

3 10"

4

5 p 15" A$

FIQ. 34. Values of I Do/Doi la and I Dh/Doi /* as functions of p for fields (1) and (2) tor rocksalt 200 with Cu Ka radiation. Planes (111) parallel to surface. x = 54" 42', I G 1 (ro/rdt = 1.23.

We may now use the results obtained in Section 33, summarized algebraically in Eqs. (33.3)-(33.7) , and graphically in Fig. 14. Figure 34 shows corresponding curves, modified by plotting the squares of the quantities concerned, so that the ordinates give directly the values of P1, PI, Q1,QI for unit incident energy current-density. As a consequence of the boundary conditions on incidence, Q1 = Q2 for all angles of incidence. The abscissas give the values of the incidence parameter p , and, for comparison, a scale of actual angulhr values for the particular case illustrated is included, For large positive values of p , and from Fig. 34 it will be evident that in this connection I p 1 = 10 is already a large value, only one of the four components is appreciable, PI, the SO component of % Virtually the whole of the energy current is carried by field (2), and to a very close approximation & = 1 so 1 so. For large positive values of p , both PI and Q1are very small, and so consequently is the energy current density 5,; but Q1>> P1,and so the vestigial current carried by field (1) is in the direction s h . For large negative p , on the other hand, conditions are reversed; is now vestigial and in the direction s h , and virtually the whole of the energy current is carried by field (1) in the direction so. The discussion in this paragraph should be read in conjunction with that given in Section 34. The relevant dispersion hyperbola is shown in Fig. 13. We now consider SI in more detail. Since only PZis appreciable for large positive p , 81 is approximately equal to 00; but as p becomes smaller PI falls and QZ rises, so that the direction of 6'1 swings away from that of so toward that of j, which it reaches when the curves for P2 and QZ in Fig. 34 intersect and the two components are equal. The energy current in

sI

THE DYNAMICAL THEORY O F

X-RAY DIFFRACTION

187

field (2) is then parallel to the crystal planes, and diffraction is well established. The wave point A z in Fig. 13 then coincides with the vertex V s of the hyperbola. For the case illustrated in the figure, in which J G 1 > 1, this occurs for a positive value of p equal to & ( I G lz - 1)/ 1 G 1 . The angle €2 is now zero. As p becomes still smaller, Qs becomes greater than Pt, and €2 becomes negative. As p itself becomes increasingly negative, S2 swings rapidly towards Sh, a t the same time falling in magnitude, and for large negat'ive p it is very small indeed and virtually in the direction sA. When the two components are equal and the energy current is parallel to the atomic planes,

S2 = 2 cos eo 1 D ~ ~ / D (,2+I So1 j

(74.3) by Eqs. (73.4) and (33.11).

Fro 35. The radii from 0 t o the curve, for different values of p , give the corresponding values of I & 1/1 SOI, which approach unity for large positive p .

188

R. W. JAMES

The curve in Fig. 35 shows the locus of the extremity of the vector 9, as p varies, and some of the corresponding values of p are indicated. As the angle of incidence varies over a range of a minute of arc or less, the direction of the energy current swings from so to s h , a range of 32' in the example considered, and it will be seen that most of the change in direction occurs over a range of f 2 in p on either side of zero. When the diffraction spectra are strongly excited the direction of energy flow is always nearly parallel to the atomic planes concerned. It is not easy to test these results directly, for any ordinary X-ray beam is likely to have a divergence greater than the whole effective range of incidence here considered, and will give a fan of energy-current vectors, filling the range between SO and sh. From Eq. (73.1))and from the symmetry of Fig. 34, it is clear that Sl(P) = S2(-p) (74.4) so that the curve of Fig. 35 is also the locus of the extremity of the vector &; but it will be described in opposite directions by the two vectors as p varies. When p is large and positive S,is in the direction soand has magnitude I SOI , while Slis in the direction s h but is very small. As p decreases, the two vectors swing together, one decreasing, the other increasing. They coincide both in direction and magnitude when p = 0 and the interference field in the crystal is fully excited. In this case,

Q1 = &2 = I G I SOI /4 by Eq. (73.1), and tanel,2 = tanOO(l- I G lz)/(l I G ) ,1 by Eq. (73.6)) so that el,, is negative if I G I > 1, and in the example considered is about -8'30'. As p diminishes still further the two vectors cross over, Sl grows, and its direction is parallel to the atomic planes when A , lies at the lower vertex of the hyperbola in Fig. 13. The directions of both S1 and & are shown in Fig. 35 for p = 0.5. In the special case in which the atomic planes concerned are normal to the crystal surface both energy currents travel normal to the surface and parallel to the atomic planes when p = 0. All four components are then equal, I G l2 = 1 and, by Eq. (74.3))

PI = P2

=

190I /4

and

+

S,

=

S, = + I So I cos eon

where n is a unit vector perpendicular to the crystal surface. The sum of the two current densities is then equal to the normal component of the energy current-density in the incident beam, and from this we may infer that the periodic term in Eq. (73.1) represents a component parallel to the crystal surface. The form of the term shows this to be generally true.

189

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

The magnitudes of s1 and s2 for any angle of incidence may be calculated from Eq. (73.1). The second group of terms may be written (74.5) PzSo

=

+

&z%.

By Eq. (73.5), therefore,

and this is also equal to I %(v) By Eq. (73.6), tanez(v)

=

I.

tanel(--)

=

eu eu

- e-u

+ e-u

I G l2 I G l2

tan 00.

(74.7)

These formulas can be expressed as functions of the incidence parameter p by using the results cosh2v = p 2

+ 1,

eu = ( p z

+ 1)i + p,

e-'

=

(P2

+ I)* - p.

If the atomic planes are perpendicular to the surface, 1 G j2 in this special case

tanEZ(p) = tanel(-p)

= =

(P2

+ 1)' tan80

tanhv t a n &

=

1, and

(74.9) (74.10)

75. THERELATION OF THE DIRECTION OF ENERGY FLOW TO THE DISPERSION SURFACE Ewald3l and Katoa2 have shown generally that the direction of the associated with a wave point A , is that of the energy current-density normal to the dispersion surface a t A,. This result enables one to visualize rather simply the nature of the energy flow for different conditions of incidence, and is very easy to prove for the special case of the hyperboloidal dispersion surface.

s,

*1P.P. Ewald, Acta Cryst. 11, 887 (1958). a N. Kato, Acta Cryst. 11, 885 (1958).

190

R. W. JAMES Y

Figure 36 shows the hyperbolic trace of the dispersion surface for a single state of polarization. The equation of the hyperbola is €O[h =

I c 18 ' COB2 00

(75.1)

where d is the semidiameter, and toand [ h are the perpendicular distances of a point on the curve from the two asymptotes. It should be noted that €0 and t h are parallel respectively to SO and s h . We refer the hyperbola to rectangular axes x and y, bisecting the angle between the asymptotes and having their origin a t the center, Q, of the hyperbola. The positive direction of the y axis$ -j. Then €0

= y cos Oo

.$I,

= y cos Oo

- x sin Oo

+ x sin flo

and the equation of the hyperbola becomes

u(x,y) = y2

- x2tan200 - 1 C 12d2 = 0.

The slope of the noimal to the hyperbola a t the point is equal to

(2,

(75.2) y), or (50, &),

(75.3) Suppose now the point (to, &) to lie on the upper branch of the hyperbola, so that the wave point considered is A'. By Eq. (71.1),when absorption is negligible,

Sz

(C/fh)

( I Doz

l28o

-k 1 Dhz I28h)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

and by Eq. (20.2) whence

I Dh2 lz/ 1 Doz 1' S,

= ( ~ / 8 ~ 5 h )

=

+

191

fo/h

t0Sh)

I Do2 1 2 -

(75.4)

The components of Sz parallel to so and sh are therefore proportional respectively to ti, and to,and its components parallel to 2 arid y to ( l h - 50) sin Bo and - (& to)cos eu. is therefore parallel to a line the slope of which referred to the axes z and y is

+

s2

+

- t h ) cot 0 0 the same as the slope of the normal to the hyperbola.82' The sign of the y component of Szshows that the energy current is directed downward. This result is illustrated by Fig. 37, in which the hyperbolic trace of the dispersion surface is shown for a hypothetical example, in which 0 0 = 30" and I r$h/+O I = 0.5. The distance from the center of the hyperbola to the center of the Ewald sphere is then equal to the diameter of the hyperbola. The geometrical conditions correspond fairly closely to those for rocksalt 400 when the crystal planes are inclined a t about 75" to the surface. The center of the circle of incidence, which is also the origin of the reciprocal lattice, would then lie at a distance from L equal to about 2.5 X lo6times the semidiameter of the hyperbola, a distance of the order of a mile on the scale of the figure, so that it is fully justifiable to represent the circle of incidence by a straight line. A change of unity in p corresponds to a distance cosec BO(yh/yO)4 times the semidiameter of the hyperbola, measured along the circle of incidence, and in this case to a change of angle of incidence equal to about 1.3". The whole range of angles included in the figure is therefore less than 15". The directions and relative magnitudes of s1 and s ~ calculated , from Eq. (74.8), are shown for a series of values of p . The energy current is (€0

th)/(EO

m The corresponding result in the theory of doubly refracting crystals is well-known.

In the optical region there is only one active reciprocal-lattice point, the origin iteelf; for the radius Lif the Ewald sphere is then very much less than the reciprocallattice spacing. The dispersion surface becomes the optical indicator surface, a double surface derived from the Cauchy ellipsoid in the same way that the biaxial wave surface is derived from the Fresnel ellipsoid. The circle of incidence is a great circle of a sphere of radius k lying entirely inside the indicator surface. A vector from any point on it to the origin determines a direction of incidence on the crystal surface, and a normal to the crystal surface through the point on the sphere determines two wave points by its intersections with the indicator surface. The vectors from the two wave points to the origin give the wave vectors of the two planepolarLed waves that can be propagated in the crystal; and the corresponding ray directions, or the directions of energy flow, are given by the normals to the indicator surface a t the corresponding wave points.

192

R. W. JAMES

FIG.37. The relations of the directions of energy flow to the dispersion surface for different angles of incidence.

always normal to the hyperbola at the corresponding wave point. The dotted lines show the loci of the extremities of the vectors and s2.From this figure it is easy to follow out most of the phenomena discussed in Section 74. It will be Been that the two fields (1) and (2) occur together in comparable strength only over the small range within which the curvature of the hyperbola is considerable. The fields are coherent, and the actual direction of the total energy flow is their vector sum, in obtaining which, however, the periodic term due to their interference must also be taken into account.

s1

76. THERESULTANT ENERGY CURRENT-DENSITY AND

SOLUTION

THE

PENDULUM

We now suppose both fields to be present in the proportions determined by the boundary conditions on incidence. Rearranging the terms in Eq. (73.2), and putting cosh2v = p2 1, we obtain for the resultant energy current, expressed as a function of p ,

+

(76.1)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

193

FIGURE 38

In Fig. 38, AN is the unit vector n, perpendicular to the crystal surface. N B is parallel to the surface, and AC and A B &re in the directions of SO and s h , respectively. Then N C = tan rlo,N B = tan #h. From the figure, 60 = T d C , s h = Y h A B ; SO 1 G l*Sh = ( Y O / Y h ) S h = r a .The Coefficient Of the periodic term in Eq. (76.1) therefore becomes SO

- I G i2Sh = ro(AC - AB)

=

-roCB,

and the equation can be written in the form (76.2)

=

I so I {AC +

sin* ( x z / A ) CB}

(76.3)

( p 2 + 1)

The energy current-density therefore has two components, one constant and in the direction of the incident radiation SO,the other parallel to the crystal surface and varying periodically with the depth below it. Whenever the depth z is an integral multiple of the beat thickness A the periodic component parallel to the surface vanishes, and the mean energy current-density is i soI so in the direction of incidence AC. The periodic term makes its maximum contribution when z = ( m + ) A , and the direction of s is then along A T , where CT = C B / ( p 2 1). As the depth varies the direction of mean energy flow swings between A C and AT, completing a period of alternation in each successive depth A. This

+ +

194

R. W. JAMES

way of regarding the problem shows very clearly the nature of the variation of direction of energy flow with depth that is characteristic of the Ewald pendulum solution. The maximum swing of direction, between so and s h , takes place when p = 0, and this case is discussed in Sections 46 and 73. The fields (1) and (2) are then equally excited in both direct and reflected wave trains, and interference effects will be a maximum. As 1 p 1 increases, one field, in the direction of incidence, predominates increasingly, and the range of oscillation in direction becomes rapidly smaller. T in Fig. 38 approaches C, and the resultant direction of energy flow deviates less and less from AC. It is always AC when z = mA, but it should be remembered that A itself decreases quite rapidly with 1 p 1 , as Eq. (71.2) shows. When z # mA, there is always a component of the mean energy current in the direction s h as well as in the direction so, and to satisfy the boundary conditions on emergence from a slice of such thickness transmitted beams in both directions are necessary. This point has been discussed more fully in Sections 60 and 61. 77. THELINESOF ENERGY FLOW When the reflecting planes are perpendicular to the crystal surface YO = Y h = cos Bo, and 1 G l2 = 1. The formulas then become very simple; but all the essential principles can be studied from this example, and to avoid merely algebraic complication we shall confine ourselves to it in what follows. The extension to the general case offers no special difficulties. The form of Fig. 38 relevant to this case is shown in Fig. 39, from which it is clear that tanBot AC = n

+

CB

=

-2 tanOot

t being a unit vector parallel to the crystal surface, so directed that so lies between n and t. Substituting these values of AC and CB in Eq. (76.3),

we obtain

[

1

+ p 2 + pcos2 +( 21r z / A ) tanBot . (77.1) If + is the angle made by the direction of s with the normal, taken as posiS ( p ) = ISo/cose0n

tive when S lies between n and t, tan+

=

p2

+ cos (27rz/A) p2

+1

tan Bo.

(77.2)

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

195

FIQURE39

We now refer the lines of energy flow to axes of x and z, parallel respectively to t and n. By Eq. (77.2), the slope of a curve of energy flow at the point ( x , z ) is dx/dz = A B cos (27rz/A) (77.3) where B = tan O o / ( p z 1). (77.4) A = p 2 tan Oo/(p2 l ) ,

+

+

+

The equation of a line of energy flow that passes through the origin of coordinates, obtained by integrating Eq. (77.3) with respect to z between the limits 0 and z, is (77.5) z = Az (BA/27r) sin ( 2 r z / A ) ,

+

It is the equation of a sinuous line that swings from side to side of the straight line 5 = Az = Cp2/((p2 l ) ] tanBoz, (77.6)

+

+

intersecting it when z = m A or ( m $ ) A . When z = m A the slope of the curve is tan00 and the energy flow is in the direction of so;when z = ( m 3) A the slope is ( p 2 - 1) tan Oo/(p2 1). Between the points of intersection the curve swings from side to side of the straight line with an amplitude equal to (77.7) a ( p > = (A/%) tan Oo/(p2 1)

+

+

+

196

R. W. JAMES

parallel to the surface. The maximum elongations occur when 2

=

(m f 1 / 4 ) A ,

and the energy flow is then parallel to the straight line z = Az of Eq. (77.6), which it will be convenient to call the mean line of energy flow in what follows. It will be clear that the mean line coincides in direction with n when p = 0, and very nearly with so for large values of p , positive or negative. For all values of p other than p = 0 it lies between so and n. By Eq. (71.2), A ( p ) , the beat thickness for an angle of incidence corresponding to p , is given by

+ I)#.

(77.8)

A(P) = A(0)/(PZ

The amplitude of the swing of the flow curve parallel to the surface is therefore equal to ~ ( p tan ) eo - A ( 0 ) tan 00 (77.9) a(p) = 2n(p2 1)*’2 ’ 2n(p9 1) which decreases rapidly as p increases, both absolutely and as a fraction of the beat thickness itself. I n Fig. 40, a set of mean lines drawn for the spectrum 200 of rocksalt on the assumption that the crystal is nonabsorbing is shown. The figure includes a depth of lOA(O), which for Cu K a radiation in state (a) of cm. Along each radiating line depths polarization is about 1.48 X below the surface corresponding to multiples m of A ( p ) are marked off, and a set of curves ha8 been drawn, each connecting points for which m is constant. In the diagram, the curve for m A ( p ) is denoted by d ( m ) . It intersects the mean flow line p at a depth m A ( p ) below the surface, and a t any such point of intersection the whole energy flow is in the direction of so. The figure shows plainly the rapid decrease of the beat thickness with increasing p . No d(m)lines are drawn for 1 p I > 2, where they lie too close together to be shown separately in the diagramaaZb In Fig. 40,the amplitude of the line of energy flow for p = 0 is shown to the correct scale. The mean flow line is here perpendicular to the surface, and is cut by the true flow line when z = m A ( 0 ) and z = ( m $ ) A ( O ) ,

+

+

+

)*b

It should be borne in mind that throughout this discussion only directions of incidence very near to SO, the direction of incidence corresponding t o reflection from the atomic planes a t the Bragg angle, are considered. For geometrical, as distinct from interference purposes we can in effect treat the direction of the incident beam, or of the wave train in the crystal derived directly from it, as being SO. Our argument applies only to the range of incidence within which this assumption can be made. If the direction of incidence departs more widely from SO, there is effectively a single wave train in the crystal which, apart from a very small deviation due to refraction whose effects may generally be neglected, will lie in the same direction.

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

197

FIQURE 40'

where the energy flow is parallel respectively to so and sh. I n Fig. 41 some of the curves are shown again on a larger scale for a thickness of only 2A(O), and the oscillations for p = 1 are shown. They are already much less marked than for p = 0. At the points of intersection of the mean lines with the d(m)lines the energy current is always parallel to so, but a t the intermediate nodes it is in the direction s h only when p = 0. A line drawn parallel to the crystal surface in Fig. 40, a t a constant depth in the crystal, will intersect successive d ( m ) lines a t increasing values of p I . Moving along this line corresponds physically to varying the angle of incidence of a beam of radiation sharply defined in direction, while the crystal thickness remains constant. Whenever p is such that the line z = const intersects the d(m) line, the flow of energy a t that depth is entirely in the direction so, and no reflected beam, but only a directly transmitted beam, would emerge from the lower surface of a slice of this

I

198

R. W. JAMES

Z/A(o)

FIGURE41

thickness. For intermediate values of p the energy current has a component in the direction Sh, and some of the emergent radiation is in the direction of the reflected beam. This is of course precisely the phenomenon already discussed in Sections 60 and 61, and illustrated by Fig. 26. The numerical values given in Part VIII refer to MoKa radiation; but the formulas of the present section show that for a constant multiple of the beat thickness the curve of Fig. 26, expressed as a function of p , is independent of the wavelength and of eo. Figure 26 corresponds to the intersection of the curves of Fig. 40,by the line z = 1OA (0). The actual values of the beat thickness and of the angles concerned naturally depend on the crystal and the wavelength used. The values of p for which R t ( p ) is zero are very easily calculated. Let the thickness of the crystal be mA (0). Then the line z = mA ( 0 ) intersects the curves d(n) when mA(0) = nA(0)/(p2 l)+,or when p 2 = (n2/m2)- 1,

+

THE DYNAMICAL THEORY O F X-RAY DIFFRACTION

199

n being an integer greater than m. Using this result, it is easy to construct the curves of Fig. 40,and to determine the zero values of R t ( p ) in Fig. 26. O F ABSORPTION ON THE ENERGY CURRENT320 78. THEEFFECT

We must now return to the more general expressions (72.6) and (72.8) in order to see what modifications are introduced by the absorption in the crystal, and for this purpose we shall again limit the discussion to the case in which the crystal planes are normal to the surface, so that I G l2 = 1, yo = ' y h = cos 00. Equations (72.9) and (72.10) then become 0 = p K =

pA

sec 0,)

(78.1)

1 C I sec eo -

(P,"

+ 1)'

Figure 39 still applies, and, putting I G

M

(P,"

Iz

=

+ 1)'

1,

so sec O0 = n

+ tan Oat,

n

- tanOot

sh

sec Oo

=

in Eq. (72.8), we obtain

(78.2) *

- 1 So I exp ( -52) cos eo [(cash ( v , + KZ)cosh vr

S=

cosh2v,

- cos (27rz/A) sin2vi)n

+ {sinh (v, + KZ)sinh v,

-I- cos (27rz/A) cos2v i ) tan &t]. (78.3) The periodic component of the energy current is no longer strictly parallel to the surface of the crystal; but in actual examples, since ~li is a number of the order of or less, sin2vi, the coefficient of the term in the normal component involving cos (27rz/A), is very small, and in what follows we shall neglect it and also put cos2vi = 1. If is the angle made by the direction of the energy current with the downward drawn normal at depth z,

+

where

m See

tan+

=

(A

+ B ) tan&

+ KZ)tanhv, B = cos (2rz/A)/cosh (v, +

A

=

(78.4)

tanh (v,

also N. Kato, Acta Cryst. 13, 349 (1960).

(78.5) KZ)

cosh v,.

(78.6)

200

R. W. JAMES

Neglecting for a moment the periodic term in Eq. (78.4), we calculate from it a mean line of energy flow, which, however, will no longer be a straight line, The actual line of flow oscillates about this mean line, but now the oscillation not only decreases as I p I increases, but, as the form of Eq. (78.6) shows, also for all values of p I dies out rapidly as the depth in the crystal increases. The slope of the mean flow line is

I

dx/dz

=

tan eo tanh V , tanh

(vr

+

(78.7)

KZ)

and the equation of the line passing through the origin is accordingly 2 =

lz

(dx/dz) dz

=

K-~

tanh v,(ln cosh

(0,

+

KZ)

or, using Eq. (78.2), and putting tanh Vr = pr/(pr2

x

=

( p , / M ) tan Bo(lncosh

(or

+

KZ)

- In cosh v,] tan eo

+ l)f

- In cosh ZJ~].

(78.8) (78.9)

The family of curves obtained from Eq. (78.9) by giving different values to p , replaces the set of straight mean flow lines applicable when the absorption of the crystal is negligible. 79. THENATUREOF THE ENERGY FLOW-LINES IN AN ABSORBING CRYSTAL Equation (78.7) shows that when z is very small the slope of the mean flow line approximates closely to

(d~/dz),,~= tanh2v, t a n &

=

+ 1)

pr2taneo/(pr2

(79.1)

which is the slope of the mean flow line for a crystal with negligible absorption, and is independent of the sign of p,. This is to be expected; for very near the surface very little absorption has taken place and so both fields (1) and (2) are active, and still have much the same relative values as in the nonabsorbing crystal. When, however, z is large, the factor tanh (v, K Z ) approaches unity, and

+

(dz/dx) z:-03

=

tanh Ur tan 00

=

p , tan Bo/(p,Z

+ l)*.

(79.2)

Comparing this with Eq. (74.10), we see that it is the direction of energy flow when only one field is present; and this again is to be expected. We saw in Part VII that when p , is not very different from zero, so that interference is taking place, there is a very strong preferential absorption of one of the fields, so that at any considerable depth in the crystal only one field, associated with a single wave point, is present in appreciable strength.

201

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

Here therefore we should expect the energy transmission to be of the singlefield type. In these circumstances, as p , varies from large positive to large swings across from so to s h , and negative values, the direction of tan e ( p ) = - tan e ( - p ) ; whereas when both fields are present, as p , becomes negative the direction of swings back again towards so, and ) tan e ( - p ) . The next step is to consider how one type of field tan ~ ( p = changes to the other as the depth increases. When p , is positive, the slope of the mean curve is always positive, but, for a given value of p,, increases from the value given by Eq. (79.1) to that given by (79.2) as a limit as z increases from zero to large values. When p , is negative, on the other hand, the slope of a mean flow curve is positive, and the same as that for the corresponding positive value when z = 0, but changes sign at a certain depth, and becomes

s s

for large z. For negative p , and z =

dx/dz

- I p , I tanW(p,2

= -

tanh I v, I tanh

- ( I p , I / M ) tan Bo( In cosh

(KX

+ I)*

(KZ

- I v , I ) tanBo

- I v, I )

-

In cosh I v,

(79.3)

I1

The mean flow line is therefore normal to the surface when dx/dz when z = I U, 1 / K = [(pl" l)*/M] 1 sinh-I pr I .

+

(79.4) =

0, or

(79.5)

The maximum value of x occurs at this depth, and is equal to xmax= ( I p , 1 / M ) In cosh v, tan eo = +( I pr I / M ) In (pl"

+ 1) tan eo. (79.6)

The curve crosses the z axis when x 20

=

=

2

0, and so, by Eq. (79.4), when

1 V, I / K .

(79.7)

The slope of the mean flow line at this point is equal in magnitude, but opposite in sign, to its slope at the surface. Some curves of mean flow for NaCl 200 are given in Fig. 42. In this case A(0) = 1.48 X lo-* cm and M = 158 cm-' for Cu Ka radiation. To show the nature of the curves more clearly, the scale of the z axis has been magnified fourfold. It will be seen that zo, the depth at which the mean flow line crosses the axis when p , is negative, increases rapidly as I p , I increases. The crossing of the axis is an indication that the flow is approaching the single-fieldtype, for which tan E ( - p ) = -tan e ( ~ .) A glance at Fig. 23 will show that as I p , 1 increases the difference between ul and

202

R. W. JAMES x Io-"cm

- 0.1

C

ai

0.2

0.3

2.0

FIG.42. Mean flow-curves for absorbing crystal near the surface for positive and negative pr. Horizontal scale four times vertical.

becomes smaller, so that the double-field type of energy flow will persist to greater depths. For small pr I , when one absorption coefficient is very much greater than normal and the other very much less, one field very rapidly becomes inappreciable in comparison with the other, and zo becomes very small. In the case illustrated in the figure, when pr = -1, zo is 1.58 X cm, or rather more than lOA(0). The depth A(0) contains about 2500 repetitions of the crystal pattern, and the depth illustrated in the figure about 40,000, which will give some idea of the grain of the fine structure of the crystal relative to the phenomena considered. US

I

80. THEPERIODIC COMPONENT OF THE ENERGY CURRENT IN ABSORBINQ CRYSTALS

The true energy flow curve oscillates from side to side of the mean flow curve considered in the last few paragraphs. To the value of 2 calculated

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

203

from Eq. (78.9) must be added a component xp, periodic in depth, which by Eqs. (78.4) and (78.6) will be equal to

x,

=

tan Oo

/

-

cOshVr o

cos (2nz/A) sech (v,

+

KZ)

(80.1)

dz.

Although the integral cannot be evaluated directly, it is not difficult to estimate the magnitude of xp a t a given depth. To do this, we express z as a multiple of A , putting so that, by Eq. (78.21,

z = fA

=

KfA = fMA(O)/(p?

Then Tcp =

Q8r

A tan (P,z 111

+

/ol

(80.2)

fA(0)/(p2 4-l ) + ,

(80.3)

4- 1).

cos (2nf) sech (v,

+ K ~ Adf.)

cos 2rrf sech (u,+fKA)

f

-0.81

.

f

p, =-I

(b)

FIGURE 43

(80.4)

204

R. W. JAMES

In Fig. 43, the integrand of Eq. (80.4) is shown, plotted as a function off. It is necessary to distinguish between positive and negative values of pr, and curves are shown in the figure for p, = +l and - 1. The value of ill for NaCl200, with Cu Ka radiation, 158 cm-I, was used. The correspond1). ing value of A(0) is 1.48 X cm, and KA = 0.234/(p,2 The value of z p for any value off is the area beneath the sinuous curve from the origin, multiplied by the factor before the integral in Eq. (80.4). It is zero when f = 0, and reaches its first positive maximum when f = 1/4, the area of the curve being then that of the first positive half-loop. It becomes zero again when f is slightly greater or slightly less than 1/2, according to whether p, is positive or negative. I n either case, the first negative maximum is reached when f = 3/4. Continuing thus, we see that maxima and minima occur a t f = m =t 1/4, m being an integer, while zero 3. values occur very nearly, but not in general exactly, a t f = m It will be clear from the figure that we can obtain a very close estimate of the positive amplitude of oscillation of the flow curve in the mth beat period by supposing f in the second factor of the integrand of Eq. (80.4) to retain the constant value m while integrating the cosine factor from f = m to f = m 1/4. For positive p,, the amplitude in the mth period is given closely by

+

+

+

The amplitude thus dies away with increasing depth in the crystal for a given value of p,, and with increasing p , for a given depth. For negative pr

which increases with increasing depth until z = I u, dies away. The maximum amplitude reached is about A(0) tan&/2n(pr2

I /K,

and thereafter

+ 11,

and the mean flow-curve is then normal to the surface, by Eq. (79.5). The physical significance of this at first sight paradoxical result may be understood by referring to Fig. 37, which shows the vectors contributing to the energy flow from the two fields. The maximum oscillation of the flow curve may be expected to occur when the contributions from the two fields are nearly equal; and with a nonabsorbing crystal, for which Fig. 37 was drawn, this is when p = 0. I n the absorbing crystal, however, the absorption is different for the two fields. In the case considered it is greater for

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

205

TABLEII. VALUESOF TBE AMPLITUDESOF OSCILLATIOX OF THE ESERGY-FLOW CURVES, AS FRACTIONS OF THE BEATTHICKNESS A(p,) EXPRESSED m. BEATPERIODS

FOR

DIFFERENT

The depth in the crystal is given nearly enough by mA(p,). The figures refer to NaCl200 with Cu K a radiation (A = 1.54 A) in state (a) of polarization. cm. A(0) = 1.48 X 10- cm, A(1) = 1.046 X

m 1 2 4 6 8 10 15 20

a( m )/A ( 0 ) pr = 0

a ( m )/A (1 p, = 1

0.044 0.041 0.030 0.021 0.013 0.0086 0.0027 0,0008

0.021 0.019 0.016 0.013 0,010 0.0081 0.0045 0.0026

a(m)/A ( - 1) p , = -1 0.025 0.026 0.029 0.031 0.032 0.031 0.023 0.014

field (1) than for field ( 2 ) . For positive p , the' absorption increases the disparity of the two fields and so decreases the oscillation of the flow curves; but as p , becomes negative the two fields are brought into equality, and at a greater depth the larger p,, so that until this happens we may expect an increase in the degree of oscillation as the depth increases. It must be remembered, of course, that we are dealing here only with the direction of the flow curves. The actual magnitude of the energy current in a n absorbing crystal becomes rapidly very small as p , increases in magnitude, so that the effect would not be easy to observe. In Table I1 some values of the amplitude, expressed as fractions of the relevant beat thickness, are given for the example considered.

81. THEENERGY CURRENT IN THICK ABSORBING CRYSTALS Except for very thin slices of crystal, transmission in the region of interference will be governed almost entirely by the smaller of the two absorption coefficients, in the example we have considered, by uz. I n Eq. (72.6) we may then neglect all terms on the right-hand side except the second, and write

where u2 =

a-

K.

206

R. W. JAMES

This equation differs in form from (74.5) only in being multiplied by the absorption factor exp ( -azz). For the symmetrical arrangement, when I G i2 = 1, we have therefore

which gives the mean energy current-density a t depth z as a function of the incidence parameter p,.

'h

-a5 -03

-

FIG. 44. Variation of energy current with pr for rocksalt 200, Cu K a radiation; A a t a depth of 1 m, B a t a depth of 2 mm.

I n Fig. 44, the magnitude of the energy current is shown for a series of directions corresponding to different values of p , for slices of rocksalt 1 and 2 mm in thickness, for the spectrum 200 with Cu Kcr radiation, for which (TZ = 166.4 - 158/(pr2 1)f.

+

T H E DYXAMICAL T H E O R Y O F X-RAY DIFFRACTION

207

Figure 44 is in effect Fig. 35 in which the energy-current vectors have been multiplied by the absorption coefficients appropriate to the different directions. The radius in any direction gives the corresponding value of I I / 1 $0 1 . It will be seen that the angular range of effective transmission gets narrower as the thickness of the crystal increases, but that over this range the transmission coefficient remains quite appreciable for thicknesses that would be virtually impenetrable if the ordinary linear absorption coefficient of the crystal were operative. Indeed in this case it would scarcely be possible to shorn the transmission coefficients on the scale of the diagram. Even in the symmetrical case the transmission is not symmetrical in p,. This is clear from Eq. (81.2), but Fig. 44 should also be compared with Fig. 29, which is also drawn for symmetrical transmission.

s

X. Some Experimental Consequences of the Dynamical Theory

82. INTRODUCTION Any detailed discussion of the recent experimental work that has been based on the dynamical theory lies beyond the scope of the present article, which has been planned as an introduction to the main principles of the theory, upon which further reading in special fields may be based. In this final section, however, a few of the more interesting experimental consequences of the ideas discussed in the last three sections will be very briefly considered, and references to some of the more important papers will be given. Two very good general accounts of much of this work are to be found in a lecture delivered before the German Academy of Sciences by Professor von Laue33 in 1958, and in a article written by Professor Borrmann34for the occasion of von Laue’s eightieth birthday in 1959.

83. THEPROPAGATION

OF

BE;.khrs

OF

LIMITED\ v I D T H

The whole of the discussion in the previous sections has assumed the beam incident upon the crystal to be of infinite width and sharply defined in direction, neither of which conditions can ever be really fulfilled in practice. We shall now, while still supposing the direction of the beam to be sharply defined, assume its width to be limited. The full discussion of the M. von Laue, “Rontgenwellenfelder in Kristallen,” Sitzber. Deut Akad. Wiss., No. 1. Berlin, 1959. M G .Borrmann, Rontgenwellenfelder, in “Beitrage zur Physik und Chemie des 20 Jahrhunderts,” p. 262. 1959. I*

208

R. W. JAMES

propagation of a limited beam is a matter of considerable difficulty, We shall assume, however, that as long as the width of the beam is large in comparison with the wavelength of the radiation, and this will be true in the examples considered, we may consider the propagation of energy within it to be not very different from that in a wide beam, provided only that those parts of the beam lying within a few wavelengths of its lateral boundaries are neglected. This is in fact the assumption made, usually tacitly, in most experimental work on optics in which diffraction effects due to the finite width of the beam are not specifically discussed, We suppose a beam of finite width to be incident on a parallel-sided slice of crystal of negligible absorption, and, for simplicity, further assume the atomic planes concerned to be normal to the surfaces of the slice. If the direction of incidence corresponds to a diffraction maximum, so that p = 0, the mean direction of energy flow, as was shown in Section 77, will be parallel to the atomic planes and normal to the crystal surface for both s, and &; and, within the beam, interference will take place, giving rise to the pendulum effect. If, however, the direction of incidence differs will take different slightly from that corresponding to p = 0, s, and is an independent solution of the paths through the crystal; for each field wave equation and can be propagated independently. It is only in the region, quite near the surface of incidence, where the two beams overlap, that an interference field can be set up. The two beams belonging to one state of polarization travel through the crystal in defined directions, which in the simple case considered are equally inclined to the crystal planes, and it is only on emergence from the crystal a t the lower surface of the slice that separation into primary and reflected beams in the directions so and s h takes place. The beams corresponding to the two fields are not of equal strength, and in this connection reference should be made to Section 74 and to Fig. 35. Corresponding to either state of polarization (a) or (b) there will be four emergent beams a t the lower surface, one parallel to so and one to s h at each of the two regions of exit. By Eq. (74.9), however, the angles el and e2 made with the atomic planes by the two beams within the crystal depend on the state of polarization, and are larger for state (b) than for state ( a ) . If the incident radiation contains both states of polarization there will therefore be 2 close pairs of beams &, s,, and sZa,S B b in the crystal, and so 8 beams in all on emergence. There is in fact a species of multiple refraction produced by the crystal slice, and this is illustrated diagrammatically by Fig. 45. When p = 0.5 for state (a) of polarization, the values of e corresponding to states (a) and (b) are respectively 7'14' and 8'11' if the figures for rocksalt 200 with Cu K a radiation are used.

s2

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

209

.

FIG.45. Energy paths in the crystal for different fields and states of polarization, and separation into direct and reflected beams on emergence.

84. LIMITEDBEAMSIN ABSORBING CRYSTALS

In a perfect crystal with negligible absorption the effects discussed in the last paragraph would be largely masked by the fanning out of the beams & and $& as a result of the almost unavoidable small divergence of any actual incident beam, an effect discussed in Section 74, and illustrated by Fig. 35. Some of the results can, however, be confirmed by making use of the anomalous absorption effects that occur in actual perfect crystals. It was shown in Part VII that when radiation passes through the crystal in directions near that corresponding to a diffraction maximum the absorption coefficient for one of the two fields is abnormally small, while that for the other is abnormally large. If, as in the examples discussed numerically in Part VII, field (2) has the small absorption coefficient, it will be clear from the discussion in Section 56 that only field (2), in state (a) of polarization, will penetrate effectively to the lower surface, if the slice is thick enough. Over a very small angular range, however, as Figs. 23 and 44 show, the transparency of the crystal to the favored field is quite abnormal; this in effect selects a slightly divergent beam, in the examples considered &, for transmission.

210

R. W. JAMES

(b)

FIQTJR~: 46

Using this effect, Borrmann2aiiwas able to show that the separation of the transmitted energy current into primary and reflected beams does in fact take place only on emergence from the crystal, as the theory indicates that it should. The principle of the experiment can be understood from Fig. 46a, in which the atomic planes are again considered to be normal to the crystal surface. The incident beam so gives rise to an average energy within the crystal, propagated parallel to the atomic current-density planes when p = 0, and consisting, when the depth in the slice is great enough, virtually only of a single field. On emergence, it gives rise to the direct and reflected beams T and R, in the directions of so and s h respectively, which fall upon a photographic film. The incident beam is not entirely monochromatic, and some of the radiation of shorter wavelength, which takes no part in the interference phenomena, passes directly through the slice to 0, producing a convenient reference mark on the film. When the distance of the film from the slice is altered, the separation between T and 0 remains unchanged, but that between R and T varies in a way that shows the point of divergence of the rays to lie in the exit surface of the slice. This behavior should be compared with that to be expected from the geometrical theory, which is illustrated in Fig. 46b. Here, the direct beam passes straight through the slice, giving rise to reflected contributions from the whole of its length, and this in fact does normally occur when crystals having an ordinary

s

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

21 1

degree of imperfection are used.348 For some similar results with Bragg reflections from calcite reference should be made to a paper by Borrmann, Hildebrandt, and Wagner.25 85. THEINFLUENCE OF CRYSTAL DISTORTION ON THE ENERGY

PROPAGATION

In order to demonstrate the effects considered in the last two paragraphs it is necessary to use very perfect crystals. In some experiments with a calcite crystal 3.2 mm in thickness, Borrmann and Hildebrandtss found that a temperature difference of less than a degree centigrade between the opposite faces produced enough distortion of the lattice to destroy the dynamical character of the reflection. More recent work by Borrmann and Hildebrandt3s-37 indicates that in crystals very slightly distorted by establishing a temperature gradient between their faces the path of the energy current is curved. Relatively simple considerations suggest the possibility of such an effect. By Eq. (71.1), it is clear that the mean direction of th,e energy current 5 2 , for example, the density of which is given by 52

= ( c / ~ T )exp

(-g22)

( I Doz /2so

+1

Dhz

I2sh)

depends on the ratio I Dh21 2 / I Dozl2 = 1 z2 1 2 . If I x2 remains constant, as it does in a perfect crystal, so does the direction of 5,. If, however, the temperature of the crystal varies with the depth the ratio I zz l2 will change also, and with it, one may suppose, the direction of the energy current. It is evident that an argument of this kind is little more than an indication that curved energy trajectories may perhaps be expected in a crystal within which there is a temperature gradient; it is equally evident that any proper theoretical discussion of the effect would be a matter of very considerable complexity. The effect is certainly a very interesting one, and may well be of much theoretical importance, but space does not permit of more than this very brief reference. In the light of these results it is necessary to reconsider the genesis of double Laue spots from perfect crystals, examples of which are shown in Fig. 107, p. 298, of Optical Principles.11 They could perhaps be explained by supposing that while the exit surface is the normal source of the reflected beam from a perfect crystal, slight distortions which may well occur near the surface of incidence may produce there a local mosaic, giving rise to a contribution to the reflection from that surface also. G. Borrmann and G. Hildebrandt, Z.Nuturforsch. l l a , 585 (1956). 84 G. Borrmann and G. Hildebrandt, Z . Physik 166, 189 (1959). *'G. Hddebrandt, Z.Krist. 112, 312, 340 (1959).

Ma

212

R. W. JAMES

86. EXPERIMENTAL PROOF OF ABNORMAL TRANSMISSION IN PERFECT CRYSTALS

The most direct proof of the existence of abnormally low absorption in perfect crystals when diffraction maxima are produced is given by the beautiful photographs obtained by BorrmannZ3(see ii) 24,34 and his fellow workers with calcite and germanium. The principle of his method is very simple, and is illustrated by Fig. 47. A is the anticathode spot of an X-ray tube, S a screen with a circular hole, and EF the diameter of the circle on the photographic film P that is irradiated by A through the hole in S. C is the slice of crystal, which may be as much as 2 or 3 mm in thickness, even when copper radiation is used. The voltage applied to the tube is so adjusted that a good intensity of the characteristic radiation, but relatively little short-wave radiation, is produced. So far as the ordinary absorption is concerned, only quite weak radiation can pass through the crystal to irradiate the region lying within the circle EF. When the crystal is perfect enough, however, beautifully sharp traces of the Kossel cones produced by diffraction at the crystal lattice make their appearance. Those due to the directly transmitted beam T lie within the circle EF, those due to the reflected radiation R outside it. The lines are very much more intense than the weak background, and are exceedingly sharp. Some very striking examples of such photographs are reproduced as illustrations to the lecture by von Lauea3referred to above, which show in the clearest way the strong abnormal transmission and the very narrow angular range over which it occurs.

D

FIGURE47

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

213

87. TRANSMISSION OF RADIATION THROUGH WEDGE-SHAPED CRYSTALS AND THE EXPERIMENTAL DEMONSTRATION OF THE PENDULUM EFFECT Results depending directly on the existence of the pendulum effect have been obtained by Kato and Lang,38 who studied the transmission of X-rays through wedge-shaped slices of crystal. 'The boundary conditions for phase continuity in such cases have already been briefly discussed in Section 26, and are illustrated by Fig. 10, of which Fig. 48 is an amplified version. //

..............................

////////////////

FIQURE 48

We consider a beam of radiation, sharply defined in direction, but of unlimited lateral extent, incident upon the upper surface of a wedgeshaped slice of crystal of angle a, which we assume to have negligible absorption. The plane of incidence, which is the plane of the paper in Fig. 48,is normal to both surfaces of the wedge, and to the crystal planes conKato and A. R. Lang, Ada Crysl. 12, 787 (1959).

214

R. W. JAMES

cerned in the diffraction, which make an angle x with the surface of incidence. LO and LH are the radii of the Ewald sphere from the Laue point to the reciprocal-lattice points 0 and H . LI is the circle of incidence, of center 0 and radius k, and LR the circle of the same radius that has H as center. Poi is the point on the circle of incidence corresponding to an angle of incidence $0, and A2 and A1 are the wave points determined by this angle of incidence, lying respectively on the upper and lower branches of the dispersion hyperbola for the state of polarization considered. Let AIMl and A2M2be the normals through A1 and A2 to the lower or exit surface of the wedge, which intersect the circles LI and LR in the pairs of points Poll, Pod and Phi', P d , respectively. Then Polt is the emergent wave vector kolt,corresponding to the wave vector KOIin the crystal, for it has the same component parallel to the exit surface as A10( = Kol); and, similarly, POZ' is the emergent wave vector corresponding to KOZ. In the same way, Phl'H and P d H are the emergent wave vectors khlt and k d , corresponding to Kh1 and I ( h 2 in the crystal. Had the crystal slice been parallel-sided, the points POI',Pod, and Poi would all have been coincident, and both Kol and K02 would have given rise to an emergent wave train with a wave vector kol, the same as that of the incident beam. Phi' and Ph2' would also have been coincident on the normal Poi", and again there would have been a single wave vector of emergence, equal to koi,in the transmitted reflected beam. When, however, the crystal is wedge-shaped both direct and reflected beams on emergence consist of two superposed wave trains, of the same wavelength but differing slightly in direction. The crystal has in effect different refractive indices for fields (1) and (2). In each beam the two wave trains are coherent, and interference will occur between them. From Fig. 48, in which the angles between the vectors corresponding to a given wave point are enormously exaggerated, it will be seen that the angle between the emergent wave vectors kolt and koztin the directly ~ ' ) that / ~ , the difference transmitted beam is equal to (arc P O ~ ~ P Oand O Zsimilar ~ ; results are between kO2'and koltis the vector distance P O ~ ~ Pand true for the reflected beam. We may therefore write

1 ko2' - koi' I I kh2' - khl' I

=

Po2'P0it

=

Ph2'Phl'.

(87.1)

To calculate the geometrical relations and magnitudes of these vector quantities we refer to Fig. 49 in which, with the usual justifiable approximation, the circles LI and LR are shown as straight lines, perpendicular to LO and LH, respectively. The lettering in Figs. 48 and 49 corresponds.

T H E DYNAMICAL THEORY O F X-RAY DIFFRACTION

215

I

\

-

FIGURE 49

The angles made by LI and LR with the lower surface of the wedge are respectively a - +o and a - Go - 200, and these are equal to the angles made by LO and L H with the normal to the lower surface; that is to say they are equal within a few seconds of arc to the angles of emergence of the direct and reflected beams, which we denote by +ot and $ h t respectively. Then, from Fig. 49, where

M I M Z= PoltP0ztcos $ u t $ot

= a

?;OW AlA2 =

=

- +",

PhltPhZtcos +ht +ht

=

a -

+o

=

-

AlA2 sin LY

(87.2)

2eo.

k 1 g1 - g2 1 (87.3)

where A is the beat thickness for the angle of incidence concerned, as defined by Eqs. (59.18) and (59.19). Using this result, and Eqs. (87.1) and (87.2), we obtain

1 kOzf- kolt! 1 khat - khlt 1

=

( l / A ) sin a/cos Got

(87.4)

=

( l / A ) sin a/cos

(87.5)

#ht

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R. W. JAMES

which are the significant forms of the equations for our purpose. If x is the angle made by the crystal planes with the upper surface of the wedge,

+

r/2 - (x eo>, and using this we may write Eqs. (87.4) and (87.5) in the form +o =

I k d - koltI I khd - khlt I

(x + a + 00)

= (l/A)

sin &in

= (l/A)

sina/sin ( x

+ a - eo).

(87.6) (87.7)

It will be seen from Eqs. (87.4) and (87.5) that the component parallel to the lower surface of the difference of the vectors of the two constituent wave trains is the same for both direct and reflected beams, and is equal to ( l / A ) sin a,which is the reciprocal of the component of the beat thickness parallel to the lower surface. 88. THENATUREOF THE EMERQENT WAVEFIELDAND ITSRELATION TO THE PENDULUM EFFECT

We consider the nature of the emergent radiation when the angle of incidence is such that the reflected beam is at its maximum and p = 0. In this case all the induction vectors are parallel, and, by Eq. (33.lo), Do2 = Do1 = *Do';

Dh2 = -Dhl = $GDoi.

(88.1)

In the region below the surface of exit both emergent beams consist of a pair of wave trains of equal amplitude, with wave vectors of equal magnitude that differ slightly in direction. The resultant displacement due to the two wave trains'that constitute the directly transmitted beam (0) is

Ao(r, 1) where

- kolt*r) + cos 2r(vt - kozt-r)} = Doi cos {n(kozt - kolt)or} cos 2r(vt - h t - r )

=

+DO~{COS 2n(vt

Eo'

=

3(koit

+ koz')

(88.2) (88.3)

and is a vector in a direction bisecting the angle between the vectors kol' and kod. Equation (88.2) represents a progressive plane wave train, with a wave vector gotand an amplitude that is a function of r, but is constant over planes perpendicular to the vector k02' - kolt,that is to say perpendicular to LI in Fig. 49, and parallel to Eat. The amplitude is zero when

(k02' - kolt)- r = m

+3

(88.4)

m being an integer, and so over a set of equally spaced parallel planes in the region below the wedge, which are perpendicular to LI and at a distance 1/ IkoZ' - kol' I , that is to say l/PoztPol', apart.

21 7

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

Midway between these nodal planes, the amplitude reaches its maximum value 1 Doi 1 , and along any line perpendicular to them the phases of the successive maxima differ by T . Between the nodal planes, waves travel outward from the crystal surface in the direction Eat. The resultant directly transmitted beam is in fact a set of interference fringes. The reflected beam consists of a similar set of interference fringes. In this case, using Eq. (88.1),we may write instead of'(88.2) with

&(r, t )

=

GDoisin { n ( k d - khlt)' r ) sin 2 n ( v t

- ht-r)

(88,5)

k,'= &(khz'+ khl').

The nodal planes are now defined by (khz'

- khl') nr

=

m

(88.6)

and are perpendicular to the direction of the vector khot - khlt, and so to LR in Fig. 49, and the spacing of the fringes is 1/ 1 khz' - khl' 1 , or 1 / P h 2 ' P h l t , By Eqs. (87.4)and (87.5), do and d h , the spacings of the interference fringes in the direct and reflected beams perpendicular to their directions of emergence are given by do = A COseC CY

COS $o',

dh = A COseC CY COS $h'

(88.7)

and since # 0 6 and $h' are equal within a few seconds of arc to the angles made by the wave vectors gotand t h ' with the normal to the surface of exit, both sets of interference fringes intersect this surface a t equal intervals A cosec a, the maxima of one set coinciding with the minima of the other on emergence. It should, however, be remembered that all the wave trains are coherent, so that with a broad beam, the actual field below the crystal surface is of considerable complexity, although it is convenient to think of it as made up of two sets of interference fringes. These results are very closely related to those discussed in Sections 77 and 46, where it was shown that in a semi-infinite crystal the direction of energy propagation varies with the depth below the surface of incidence, and is constant over planes parallel to the surface. In particular, when p = 0, the total energy current lies in the direction SOof the incident radiation when z = mA, and in the direction s h of the reflected beam when z = ( m 4- + ) A , Consider now a wedge-shaped crystal, of angle a. The planes z = m A intersect its lower surface at intervals A cosec a, and along these lines of intersection the whole energy current is in the direction SO and can give rise to an emergent beam only in this direction. At depths ( m + ) A , on

+

218

R. W. JAMES

the other hand, the energy current in the direction so is zero, and there is no emergent beam in this direction, We may picture the formation of the fringe system that constitutes the direct emergent beam as an injection of energy at depths mA in the correct direction, along the maxima of the interference fringes. At depths ( m +)A, the energy is injected into the maxima of the interference system constituting the reflected beam, and at depths mA there is no energy current in this direction.

+

FIQ.50. The relation between the pendulum effect and interference effecta obtained with wedge-shaped crystals.

These conditions are represented diagrammatically in Fig. 50, in which the dispersion surface, the wave points, and the direction of the energy current-density s in the crystal are also indicated. The close relation between the formation of the interference fringes and the pendulum effect will be clear. The argument given here is intended merely as a comparatively simple outline of the physical principles involved. The careful consideration of the boundary conditions that would be necessary to base the result securely is beyond the scope of an article of this kind, and the reader is referred to the work of Katoas and others for more detailed discussions. Kato and Langas have observed interference fringes of the type just considered by a method indicated diagrammatically in Fig. 51. A beam of radiation, collimated by the slits S1and SZ,falls on the surface of a crystal wedge in the direction to form an interference maximum. The radiation *9

N. Kato, Acta Cryst. 14, 526, 627 (1961).

THE DYNAMICAL THEORY OF X-RAY DIFFRACTION

219

travels through the crystal in the general direction of the lattice planes concerned in the reflection, and divides into direct and reflected beams only on emergence from the second surface. The direct beam is intercepted by a screen S3, pierced with a hole that transmits the reflected beam and allows it to fall onto a photographic film. As we have already seen, a very small divergence of the incident beam is enough to cause a wide variation of the mean direction of energy transmission through the crystal; but the and S3 in effect limit the direction positions of the three apertures in Sl, SZ, of incidence to a very small angle, insofar as the radiation that can fall on the film is concerned. The screens are fixed in position, but the film holder and the crystal wedge are rigidly connected, and are caused to travel backward and forward in a direction accurately parallel to the plane of the film. Each position in the traverse corresponds to a certain thickness of the wedge, and in this way the fringe system corresponding to the reflected beam may be recorded. The fringes are relatively narrow, since their spacing is of the general order of magnitude of the beat thickness, although this can be magnified to some extent by suitable geometrical arrangements. Kato and Lang observed fringes in a number of cases with spacings of the order of 0.1 mm. They found fair agreement with theory, although the observed fringe width tended to be a little less than the calculated one. For experimental details and reproductions of the fringes, and for a fuller account of the theory of such fringes reference should be made to the original paper. As Kato and Larig have pointed out, the fringe width in an interference system of this kind-a geometrical quantity, directly measurable with a scale-is inversely proportional to the diameter of the dispersion hyperbola

220

R. W. JAMES

through Q and perpendicular to the surface of incidence. This diameter, for any given spectrum and radiation is a known multiple of the structure factor 1 F ( h ) I , I n principle, therefore, we have here a means of determining the structure factor absolutely, without the necessity of measuring X-ray intensities. I n its present state of development the method would certainly not be a n easy one to apply, but that such a method in principle exists is a point of some importance. As early as 1940, M a ~ G i l l a v r ydrew ~ ~ attention to the same possibility in a paper on periodic phenomena in electron diffraction in crystals. In some respects indeed conditions would seem to be more favorable for observing the pendulum effect with electrons than with X-rays, and the first observations of it were so made. It has, however, been necessary to limit this article to the discussion of diffraction of X-rays, and it is possible here only to mention the work of Kato," and of Whelan and H i r s ~ h who ,~~ observed the effect of the oscillation of the wave field in thin metallic foils with the electron microscope. C . H. LIacGillavry, Physica 7, 329 (1940). N. Kato, J . Phys. SOC.Japan 7, 397, 496 (1953); 8, 350 (1954). 4 1 M. J. Whelan and P. B. Hirsch, Phil. Mag. 2, 1121, 1303 (1957). 40

41