- Email: [email protected]
Crystal Diffraction Methods for Investigation the Structure of Matter
Chapter 22
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter Outline 22.1 22.2 22.3 22.4
A Short History of the Crystal Diffraction Method Fundamentals of Crystal Diffraction Method Diffraction of Laue Diffraction of Bragg
313 313 314 316
22.5 Diffractometry of Polycrystals 22.6 Decoding of the Proteins and DNA Structure References
317 318 319
22.1 A SHORT HISTORY OF THE CRYSTAL DIFFRACTION METHOD In 1912 the German scientist Max von Laue with his coworkers first discovered diffraction of X-ray radiation, which finally proved its wave nature. This achievement was in 1914 marked by the Nobel Prize in Physics. A little later, William Henry Bragg together with his father started to study diffraction of the X-ray radiation. W.H. Bragg, the Senior, was born and educated in England, but in 1885 he moved to Australia. Here, in Adelaide in 1904e08, he performed his famous studies with alpha particles, which showed that charged particles lose most of their energy and produce the greatest ionization effect at the end of the range. The curves illustrating this phenomenon are called the Bragg curves (Section 5.7.4). At the same time, Bragg formulated the rule for calculating the path and energy loss in a complex matter, known as the Bragg rule (Section 5.2.8). According to this rule the energy losses of a charged particle in a complex matter are composed of losses on individual elements with a weight equal to the relative content of the elements. In 1908, Bragg’s family returned to the United Kingdom. His son, William Lawrence Bragg Jr., having completed his education in Cambridge, started his research of the problem of X-ray diffraction and in 1913 he derived the famous formula known as the BraggeWulff rule. The derivation of this formula is given in Section 22.4. Georgiy Viktorovich Wulff, the Russian physicist, regardless of Bragg formulated the condition for diffraction of X-rays on crystals. Based on the theory formulated by his son W.L. Bragg constructed a diffractometer and together they carried out a study of diffraction on many crystal types. For these studies, they both were awarded the Nobel Prize in Physics in 1915. Bragg Jr. appears to be the youngest Nobel laureate for all time of the Prize existence; that year he was 25 years old. X-ray diffraction on crystals quickly became a powerful method both for measuring the parameters of X-ray radiation and for studying the structure of matter, not only in the crystalline state but also in liquid and amorphous. After the death of Ernest Rutherford, Bragg Jr. became the director of the famous Cavendish Laboratory in Cambridge. He promoted the use of the diffraction method for deciphering the structure of proteins. It was during his directorship of the laboratory that F. Crick and J. Watson, using a diffraction method, discovered the DNA structure, the famous double helix.
22.2 FUNDAMENTALS OF CRYSTAL DIFFRACTION METHOD X-ray radiation is an electromagnetic wave process and as any wave process it manifests the basic wave properties, i.e., interference, diffraction, and polarization. Interference and diffraction reflect one of the most important features of wave processesdsuperposition of waves, the superposition of waves without mutual perturbation. Particles exchange energy during their collision, whereas waves do not exchange energy and are not distorted when crossing. At the intersection, the waves are superimposed on each other in accordance with the phases and then diverge, without changing their shape, as if
Radiation. https://doi.org/10.1016/B978-0-444-63979-0.00022-7 Copyright © 2019 Elsevier B.V. All rights reserved.
313
314 PART | II Applications
they had not met. Mathematically, the superposition principle is expressed as follows. If one wave is expressed by the function j1 and the other by the functionj2, then by overlay the following state is produced: j ¼ s 1 j1 þ s 2 j2 ;
(22.1)
i.e., linear combination of initial waves, wheres1, s2 are constants. It is because of the superposition that interference, diffraction, or, e.g., standing waves occur. We are going to illustrate the property of superposition by the simplest example. Let us consider a superposition of two waves with the same frequency and wave number. Generally, the amplitudes of these waves can differ, and the phases may (also) differ, but the phase difference must remain constant, at least for a rather long time. Obviously, under the accepted conditions, the phases can differ only because the initial phases are different. We are going to solve the problem of superposition of two such waves, representing waves in a complex form j1 ¼ A1 exp½iðut kx þ a1 Þ;
(22.2a)
j2 ¼ A2 exp½iðut kx þ a2 Þ.
(22.2b)
Let us sum up these waves j ¼ A1 exp½iðut kx þ a1 Þ þ A2 exp½iðut kx þ a2 Þ ¼ ½A1 expðia1 Þ þ A2 expðia2 Þexpðut kxÞ.
(22.3)
Thus, the amplitude of the total wave is equal A ¼ A1 expðia1 Þ þ A2 expðia2 Þ.
(22.4)
The intensity of the total wave is equal to the square of the amplitude, which for complex numbers is calculated as the product of the number itself on its conjugate A2 ¼ AA ¼ ½A1 expðia1 Þ þ A2 expðia2 Þ½A1 expðia1 Þ þ A2 expðia2 Þ ¼ A21 þ A22 þ A1 A2 ½exp iða1 a2 Þ þ exp iða2 a1 Þ.
(22.5)
As expðiaÞ þ expðiaÞ ¼ cosa þ i sina þ cosa i sina ¼ 2cosa;
(22.6)
A2 ¼ A21 þ A22 þ 2A1 A2 cosða2 a1 Þ.
(22.7)
then
The obtained formula (22.7) shows that the total intensity is made up of the intensities of each of the waves and of an additional, so-called the interference term. The magnitude of the interference term is determined by the cosine value, i.e., phase difference. It is evident that it can be both positive and negative. The maximum positive cosine value is equal to unity. In this case, when the amplitudes of the original waves are equal, the total intensity is equal to the quadruple intensity of the individual wave. If the phase difference of two waves is such that the cosine equals minus one, then when the amplitudes of the original waves are equal, the total intensity is zero, i.e., waves suppress each other. The above relations are theoretical justification of such pure wave phenomena as interference and diffraction. At present, several different methods for obtaining a diffraction pattern are known and used. All of them are combined by the term X-ray diffractometry (XRD) [1e3].
22.3 DIFFRACTION OF LAUE In optics, the phenomenon of diffraction is demonstrated as a result of the passage of light through various barriers: a circular aperture, a circular disk, the edge of a half-plane, a slit, and, finally, a grating, which is the set of a large number of identical slots located at the same distance from each other. A convenient diffraction grating for X-ray radiation can be crystals, as the order of the interatomic distances in crystals is the same as the wavelength of the X-ray radiation with an energy of w 5e25 keV. A crystal is a set of particles (atoms, ions, molecules) that are naturally distributed in three-dimensional space. For determination of the diffraction conditions on such a three-dimensional structure and for manifestation of the diffraction pattern, let us consider sequentially one-dimensional, two-dimensional, and three-dimensional crystal lattices.
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
FIGURE 22.1
315
Diffraction on one-dimensional grating.
A one-dimensional lattice is a series of atoms located on the straight line with equal intervals a between them, as shown in Fig. 22.1. In fact, scattering centers for X-rays are electrons, but in an idealized representation let us assume that all the electrons of the atom are concentrated at one point. Let a plane wave fall on a linear chain of atoms at an angle a0. By definition the angle of incidence is measured between the beam and the perpendicular to the surface at the point of incidence. So, the angle a0 is an angle of slip. Each of the scattering centers is the source of new spherical waves diverging in all directions. Let us choose one arbitrary direction, characterized by the angle a. The difference in the path of the rays scattered by two neighboring atoms is aðcosa cosa0 Þ.
(22.8)
The diffraction maximum in the direction of a is obtained when the path difference is equal to an integer number of wavelengths aðcosa cosa0 Þ ¼ hl;
(22.9)
where h is an integer, called the order of interference. It can be seen that for different wavelengths the values of the directing cosine for the diffraction maximum differ. The directions corresponding to the maximum of the interference lie on the surface of the cone with the angle at the vertex a. If there existed such a linear lattice, then on a photographic plate or a fluorescent screen at some distance from the lattice, a family of lines of the second order would appear, i.e., traces of intersection of diffraction cones with the screen plane. The shape of these tracks depends on the location of the screen. If the screen is parallel to a linear grid, a family of hyperbolas is produced; if it is perpendicular, then a family of circles appears; and if not completely perpendicular, then ellipses occur. Passing from the linear lattice to the planar lattice and then to the bulk lattice, the analogous conditions for the appearance of diffraction maxima can be obtained: aðcosa cosa0 Þ ¼ h1 l;
(22.10a)
aðcosb cosb0 Þ ¼ h2 l;
(22.10b)
aðcosg cosg0 Þ ¼ h3 l;
(22.10c)
where a, b, g are angles in three mutually perpendicular directions, h1, h2, h3 are integers. Eq. (22.10) are called the Laue equations. Each of the conditions (22.10) would lead to the appearance of curves on the screen of the system, and in combination the conditions (22.10) are satisfied only for certain points lying at the intersection of these curves. Thus, the diffraction pattern should look like a system of points or, given a certain scatter of angles, as a system of spots often called reflexes. By the position of the reflexes and by their intensity, one can judge the structure of matter. In addition to the conditions (22.10), for any direction, the following conditions must be met: cos2 a0 þ cos2 b0 þ cos2 g0 ¼ 1; cos2 a þ cos2 b þ cos2 g ¼ 1:
(22.11)
From the joint solution of Eqs. (22.10) and (22.11), one finds 2aðh1 cosa0 þ h2 cosb0 þ h3 cosg0 Þ . l ¼ 2 h1 þ h22 þ h23
(22.12)
316 PART | II Applications
FIGURE 22.2 The mutual arrangement of the source of X-ray radiation, the sample, and the film when the diffraction patterns are recorded by the Laue method.
Thus, the diffraction maximum of a certain order (h1, h2, h3) is obtained only for a certain wavelength, satisfying Eq. (22.12). And this means that for the Laue diffraction it is necessary for the X-rays to have a continuous spectrum. The detection of diffraction patterns by the Laue method is usually carried out with the help of film. The mutual arrangement of the X-ray source, the sample, and the film under investigation are shown in Fig. 22.2. The central section of the film opposite the source is usually covered from a straight beam or cut out. Depending on the type of specimen, these or other images are obtained on the film. Diffuse rings around the diffuse central reflex are characteristic of amorphous samples. A sharp symmetrical pattern of spot reflexes is given by monocrystals; polycrystalline samples manifest themselves as a system of concentric rings. The crystal under study is at point A, and the XYZ coordinate system is associated with it. For a simple cubic lattice, it is convenient that the coordinate axes coincide with the crystallographic axes of the lattice. The screen is perpendicular to the beam at a distance L. The diffracted beam makes a spot on the screen at point B with the coordinates Xʹ and Yʹ on the screen. The angles between the diffraction beam AB and the axes are shown in the figure. In the above example, the Laue conditions (22.10) are performed as follows: acosa ¼ h1 l;
(22.13a)
acosb ¼ h2 l;
(22.13b)
aðcosg þ 1Þ ¼ h3 l.
(22.13c)
h1 : h2 : h3 ¼ cosa: cosb: ðcosg þ 1Þ.
(22.14)
wherein It is seen from the graph, that cosa ¼ Xʹ/AB; cosb ¼ Yʹ/AB; cosgʹ ¼ L/AB. The distance from the crystal to the diffraction spot is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (22.15) AB ¼ X 02 þ Y 02 þ L2 . Then h1 : h2 : h3 ¼ X 0 : Y 0 : ðAB LÞ.
(22.16)
22.4 DIFFRACTION OF BRAGG In the abovementioned Laue method, diffraction of X-rays passing through the crystal is observed. W.H. Bragg and W.L. Bragg, and also G.V. Wulff suggested to consider the rays reflected from the crystal. In Fig. 22.3 the reflection scheme is shown. A plane wave is incident on a series of parallel crystallographic planes spaced apart by a distance d, so that the
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
317
FIGURE 22.3 To the derivation of the WolfeBragg equation.
FIGURE 22.4 The scheme of the diffractometer for obtaining diffractograms by the Bragg method. 1dX-ray tube, 2, 3dmonochromator, 4dcrystal, 5ddetector, 6dgoniometer.
angle between the plane and the direction of the wave equalsq (q is an additional angle to the angle of incidence). The rays reflected from the crystallographic planes interfere and, depending on the difference in the path traveled, quench or amplify each other. The beam M1A1N1 passes a shorter distance than that of the ray M2A2N2. The path difference is B2A2 þ A2C2. From simple geometric considerations it is clear that B2A2 ¼ A2C2 ¼ d$sinq. Hence the BraggeWulff condition is 2d sinq ¼ nl;
(22.17)
where n ¼ 1, 2, 3 . It can be seen from Eq. (22.17) that diffraction are observed only for d > l/2, i.e., under the condition l 2d, the diffraction peaks are absent. To carry out the measurements, the test sample is placed in a special installation, called a diffractometer. The scheme of the diffractometer for obtaining the diffractograms by the Bragg method is shown in Fig. 22.4. The crystal is located on a so-called goniometer, a device that allows to rotate both the crystal and detector, accurately counting the angles and distances. The goniometer ensures a simultaneous rotation of the sample by an angle q and rotation of the detector by an angle of 2q. A typical diffractogram is shown in Fig. 22.5.
22.5 DIFFRACTOMETRY OF POLYCRYSTALS XRD makes it possible to work not only with single crystals but also with polycrystalline samples, powders, metal plates, etc. In polycrystalline samples, the crystals can be in different orientations. When monochromatic radiation hits the sample at a certain angle, among the crystals of different orientations, those can be found for which the diffraction condition is met. When incidence angle changes, different sets of crystallographic planes get into the reflecting position. Thus, to obtain a diffraction pattern, the sample must be rotated and the irradiation has to be carried out by monochromatic radiation. To obtain the roentgenograms, the sample is placed in a goniometer. Measuring methods have been developed both in the reflected beam mode and in the transmission mode. The scattered radiation is detected either by a film or by a detector. By the X-ray patterns a set of reflection intensities and corresponding interplanar distances are calculated, and then the necessary parameters are determined, comparing the attained results with the known ones. At present, information on powder diffractograms of crystalline substances is stored at the International Center for Diffraction Data and at the Powder
318 PART | II Applications
FIGURE 22.5 Typical diffractogram of NaCl, radiation of Cu-Ka. The Miller indices are indicated above the peaks.
Diffraction File, which contain about 50,000 experimental and more than 500,000 calculated powder diffraction patterns for all classes of compounds. Even with a known chemical composition, the determination of the parameters of the crystal lattice can cause difficulties, but if the composition is unknown, then the problem most likely does not have an unambiguous solution, as similar X-ray diffraction patterns can occur in different isostructural substances. Because decoding the radiographs is a rather complicated task, methods have been developed to facilitate it. For example, it is the isotopic substitution method. When analyzing the diffractograms of long polymer molecules, the sample stretching is used, as well as many other methods.
22.6 DECODING OF THE PROTEINS AND DNA STRUCTURE Particularly complicated is the problem of decoding X-rays by studying such complex systems as proteins and DNA. A detailed description of the methods for determining the structure of such complex molecular systems can be found in the literature [4e6]. The interpretation of X-ray diffraction patterns by long molecules, such as polymer molecules, proteins, or DNA, is based on the fact that the location and intensity of the diffraction spots are directly related to the peculiarities of the fiber structure. The vertical line passing through the center of the roentgenogram is called the meridian, and the horizontal line passing through this point is called the equator. Reflexes on the meridian give information on the order of reflection along the fiber axis, equatorial reflexes reveal ordering in the perpendicular direction. Decoding the structure of proteins and DNA is one of the greatest achievements of X-ray structural analysis. The overwhelming majority of the most important functions in a living cell are performed by proteins. In the dry weight cells of the famous Escherichia coli proteins make up about 50%, whereas DNA makes only 3%. All enzymes that catalyze the basic processes in the cell consist of proteins. They catalyze splitting of complex molecules and their synthesis, DNA replication and repair processes and matrix RNA synthesis; they regulate transcription, translation, splicing, and other processes. From the proteins, structural elements of the cystoskeleton are constructed: collagen and elastin are the main components of the intercellular substance of connective tissue (for example, cartilage); hair, nails, feathers of birds, and some shells consist of other structural proteins, i.e., keratin. Proteins carry out motor functionsdcontraction of muscles, movement of cells, movement of flagella, and cilia. On the basis of proteins, receptors are constructed that perceive light, sound, mechanical action, smell, and taste. And hormones that carry out signaling functions are also proteins. Proteins are long linear polymers of amino acids. The sequence of amino acids in the protein significantly affects the properties of the protein and the functions that it performs in the cell and the body. It is the primary structure of the protein. But apart from the sequence of amino acids, the properties of the protein are determined by the fact that it is folded into a unique three-dimensional structure. In many proteins, the polypeptide chain is curved and coiled. Lynus Pauling was the first to detect the specific shape of this spiral, that famous alpha helix, as well as the laws of its formation, by using X-ray analysis, for which he was awarded
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
319
the Nobel Prize in Chemistry in 1954. About this discovery, J. Watson noted: “The alpha helix was discovered not by simple contemplation of radiographs; the main trick was to ask yourself the question: which atoms prefer to be neighbors with each other? The main working tools were not paper and pencil, but a set of molecular models, at first glance, reminiscent of children’s toys.” [7]. The X-ray diffraction analysis method was perfected by M. Perutz, who applied this method to the analysis of hemoglobin structure. M. Perutz performed crystallization in the presence of salts of heavy metals. Also in Cambridge, in the laboratory, which was headed by W.L. Bragg, J. Kendrew, with the use of the same X-ray diffraction method, determined the spatial arrangement of polypeptide chains in the myoglobin protein molecule and established its detailed structure, confirming the presence of alpha helices in it. For these achievements M. Perutz and J. Kendrew in 1962 were awarded the Nobel Prize in Chemistry. And another one more great success of X-ray analysis was the decoding of the DNA structure by J. Watson and F. Crick. They themselves received X-ray photographs before, but the most valuable material was contained in the X-ray diffraction patterns of M. Wilkins and R. Franklin, mainly because they got better samples of DNA crystals. Because of the complexity of decoding, Watson and Crick also solved this by the method of molecular models. On the basis of chemical conceptions, they compiled models and determined what kind of diffraction pattern can such a model present, compared it to X-ray diffraction patterns, made corrections, constructed new models, and finally produced a famous double helix.
REFERENCES [1] [2] [3] [4] [5] [6] [7]
B. Rupp, Biomolecular Crystallography: Principles, Practice and Application to Structural Biology, Garland Science, New York, 2009. W. Massa, Crystal Structure Determination, Springer, Berlin, 2004. W. Clegg, Crystal Structure Determination (Oxford Chemistry Primer), Oxford University Press, Oxford, 1998. A. McPherson, Introduction to Macromolecular Crystallography, John Wiley & Sons, 2003. D. Blow, Outline of Crystallography for Biologists, Oxford University Press, Oxford, 2002. D.E. McRee, Practical Protein Crystallography, Academic Press, San Diego, 1993. J. Watson, The Double Helix. A Personal Account of the Discovery of the Structure of DNA, A Signet Book, 1968.
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter Outline 22.1 22.2 22.3 22.4
A Short History of the Crystal Diffraction Method Fundamentals of Crystal Diffraction Method Diffraction of Laue Diffraction of Bragg
313 313 314 316
22.5 Diffractometry of Polycrystals 22.6 Decoding of the Proteins and DNA Structure References
317 318 319
22.1 A SHORT HISTORY OF THE CRYSTAL DIFFRACTION METHOD In 1912 the German scientist Max von Laue with his coworkers first discovered diffraction of X-ray radiation, which finally proved its wave nature. This achievement was in 1914 marked by the Nobel Prize in Physics. A little later, William Henry Bragg together with his father started to study diffraction of the X-ray radiation. W.H. Bragg, the Senior, was born and educated in England, but in 1885 he moved to Australia. Here, in Adelaide in 1904e08, he performed his famous studies with alpha particles, which showed that charged particles lose most of their energy and produce the greatest ionization effect at the end of the range. The curves illustrating this phenomenon are called the Bragg curves (Section 5.7.4). At the same time, Bragg formulated the rule for calculating the path and energy loss in a complex matter, known as the Bragg rule (Section 5.2.8). According to this rule the energy losses of a charged particle in a complex matter are composed of losses on individual elements with a weight equal to the relative content of the elements. In 1908, Bragg’s family returned to the United Kingdom. His son, William Lawrence Bragg Jr., having completed his education in Cambridge, started his research of the problem of X-ray diffraction and in 1913 he derived the famous formula known as the BraggeWulff rule. The derivation of this formula is given in Section 22.4. Georgiy Viktorovich Wulff, the Russian physicist, regardless of Bragg formulated the condition for diffraction of X-rays on crystals. Based on the theory formulated by his son W.L. Bragg constructed a diffractometer and together they carried out a study of diffraction on many crystal types. For these studies, they both were awarded the Nobel Prize in Physics in 1915. Bragg Jr. appears to be the youngest Nobel laureate for all time of the Prize existence; that year he was 25 years old. X-ray diffraction on crystals quickly became a powerful method both for measuring the parameters of X-ray radiation and for studying the structure of matter, not only in the crystalline state but also in liquid and amorphous. After the death of Ernest Rutherford, Bragg Jr. became the director of the famous Cavendish Laboratory in Cambridge. He promoted the use of the diffraction method for deciphering the structure of proteins. It was during his directorship of the laboratory that F. Crick and J. Watson, using a diffraction method, discovered the DNA structure, the famous double helix.
22.2 FUNDAMENTALS OF CRYSTAL DIFFRACTION METHOD X-ray radiation is an electromagnetic wave process and as any wave process it manifests the basic wave properties, i.e., interference, diffraction, and polarization. Interference and diffraction reflect one of the most important features of wave processesdsuperposition of waves, the superposition of waves without mutual perturbation. Particles exchange energy during their collision, whereas waves do not exchange energy and are not distorted when crossing. At the intersection, the waves are superimposed on each other in accordance with the phases and then diverge, without changing their shape, as if
Radiation. https://doi.org/10.1016/B978-0-444-63979-0.00022-7 Copyright © 2019 Elsevier B.V. All rights reserved.
313
314 PART | II Applications
they had not met. Mathematically, the superposition principle is expressed as follows. If one wave is expressed by the function j1 and the other by the functionj2, then by overlay the following state is produced: j ¼ s 1 j1 þ s 2 j2 ;
(22.1)
i.e., linear combination of initial waves, wheres1, s2 are constants. It is because of the superposition that interference, diffraction, or, e.g., standing waves occur. We are going to illustrate the property of superposition by the simplest example. Let us consider a superposition of two waves with the same frequency and wave number. Generally, the amplitudes of these waves can differ, and the phases may (also) differ, but the phase difference must remain constant, at least for a rather long time. Obviously, under the accepted conditions, the phases can differ only because the initial phases are different. We are going to solve the problem of superposition of two such waves, representing waves in a complex form j1 ¼ A1 exp½iðut kx þ a1 Þ;
(22.2a)
j2 ¼ A2 exp½iðut kx þ a2 Þ.
(22.2b)
Let us sum up these waves j ¼ A1 exp½iðut kx þ a1 Þ þ A2 exp½iðut kx þ a2 Þ ¼ ½A1 expðia1 Þ þ A2 expðia2 Þexpðut kxÞ.
(22.3)
Thus, the amplitude of the total wave is equal A ¼ A1 expðia1 Þ þ A2 expðia2 Þ.
(22.4)
The intensity of the total wave is equal to the square of the amplitude, which for complex numbers is calculated as the product of the number itself on its conjugate A2 ¼ AA ¼ ½A1 expðia1 Þ þ A2 expðia2 Þ½A1 expðia1 Þ þ A2 expðia2 Þ ¼ A21 þ A22 þ A1 A2 ½exp iða1 a2 Þ þ exp iða2 a1 Þ.
(22.5)
As expðiaÞ þ expðiaÞ ¼ cosa þ i sina þ cosa i sina ¼ 2cosa;
(22.6)
A2 ¼ A21 þ A22 þ 2A1 A2 cosða2 a1 Þ.
(22.7)
then
The obtained formula (22.7) shows that the total intensity is made up of the intensities of each of the waves and of an additional, so-called the interference term. The magnitude of the interference term is determined by the cosine value, i.e., phase difference. It is evident that it can be both positive and negative. The maximum positive cosine value is equal to unity. In this case, when the amplitudes of the original waves are equal, the total intensity is equal to the quadruple intensity of the individual wave. If the phase difference of two waves is such that the cosine equals minus one, then when the amplitudes of the original waves are equal, the total intensity is zero, i.e., waves suppress each other. The above relations are theoretical justification of such pure wave phenomena as interference and diffraction. At present, several different methods for obtaining a diffraction pattern are known and used. All of them are combined by the term X-ray diffractometry (XRD) [1e3].
22.3 DIFFRACTION OF LAUE In optics, the phenomenon of diffraction is demonstrated as a result of the passage of light through various barriers: a circular aperture, a circular disk, the edge of a half-plane, a slit, and, finally, a grating, which is the set of a large number of identical slots located at the same distance from each other. A convenient diffraction grating for X-ray radiation can be crystals, as the order of the interatomic distances in crystals is the same as the wavelength of the X-ray radiation with an energy of w 5e25 keV. A crystal is a set of particles (atoms, ions, molecules) that are naturally distributed in three-dimensional space. For determination of the diffraction conditions on such a three-dimensional structure and for manifestation of the diffraction pattern, let us consider sequentially one-dimensional, two-dimensional, and three-dimensional crystal lattices.
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
FIGURE 22.1
315
Diffraction on one-dimensional grating.
A one-dimensional lattice is a series of atoms located on the straight line with equal intervals a between them, as shown in Fig. 22.1. In fact, scattering centers for X-rays are electrons, but in an idealized representation let us assume that all the electrons of the atom are concentrated at one point. Let a plane wave fall on a linear chain of atoms at an angle a0. By definition the angle of incidence is measured between the beam and the perpendicular to the surface at the point of incidence. So, the angle a0 is an angle of slip. Each of the scattering centers is the source of new spherical waves diverging in all directions. Let us choose one arbitrary direction, characterized by the angle a. The difference in the path of the rays scattered by two neighboring atoms is aðcosa cosa0 Þ.
(22.8)
The diffraction maximum in the direction of a is obtained when the path difference is equal to an integer number of wavelengths aðcosa cosa0 Þ ¼ hl;
(22.9)
where h is an integer, called the order of interference. It can be seen that for different wavelengths the values of the directing cosine for the diffraction maximum differ. The directions corresponding to the maximum of the interference lie on the surface of the cone with the angle at the vertex a. If there existed such a linear lattice, then on a photographic plate or a fluorescent screen at some distance from the lattice, a family of lines of the second order would appear, i.e., traces of intersection of diffraction cones with the screen plane. The shape of these tracks depends on the location of the screen. If the screen is parallel to a linear grid, a family of hyperbolas is produced; if it is perpendicular, then a family of circles appears; and if not completely perpendicular, then ellipses occur. Passing from the linear lattice to the planar lattice and then to the bulk lattice, the analogous conditions for the appearance of diffraction maxima can be obtained: aðcosa cosa0 Þ ¼ h1 l;
(22.10a)
aðcosb cosb0 Þ ¼ h2 l;
(22.10b)
aðcosg cosg0 Þ ¼ h3 l;
(22.10c)
where a, b, g are angles in three mutually perpendicular directions, h1, h2, h3 are integers. Eq. (22.10) are called the Laue equations. Each of the conditions (22.10) would lead to the appearance of curves on the screen of the system, and in combination the conditions (22.10) are satisfied only for certain points lying at the intersection of these curves. Thus, the diffraction pattern should look like a system of points or, given a certain scatter of angles, as a system of spots often called reflexes. By the position of the reflexes and by their intensity, one can judge the structure of matter. In addition to the conditions (22.10), for any direction, the following conditions must be met: cos2 a0 þ cos2 b0 þ cos2 g0 ¼ 1; cos2 a þ cos2 b þ cos2 g ¼ 1:
(22.11)
From the joint solution of Eqs. (22.10) and (22.11), one finds 2aðh1 cosa0 þ h2 cosb0 þ h3 cosg0 Þ . l ¼ 2 h1 þ h22 þ h23
(22.12)
316 PART | II Applications
FIGURE 22.2 The mutual arrangement of the source of X-ray radiation, the sample, and the film when the diffraction patterns are recorded by the Laue method.
Thus, the diffraction maximum of a certain order (h1, h2, h3) is obtained only for a certain wavelength, satisfying Eq. (22.12). And this means that for the Laue diffraction it is necessary for the X-rays to have a continuous spectrum. The detection of diffraction patterns by the Laue method is usually carried out with the help of film. The mutual arrangement of the X-ray source, the sample, and the film under investigation are shown in Fig. 22.2. The central section of the film opposite the source is usually covered from a straight beam or cut out. Depending on the type of specimen, these or other images are obtained on the film. Diffuse rings around the diffuse central reflex are characteristic of amorphous samples. A sharp symmetrical pattern of spot reflexes is given by monocrystals; polycrystalline samples manifest themselves as a system of concentric rings. The crystal under study is at point A, and the XYZ coordinate system is associated with it. For a simple cubic lattice, it is convenient that the coordinate axes coincide with the crystallographic axes of the lattice. The screen is perpendicular to the beam at a distance L. The diffracted beam makes a spot on the screen at point B with the coordinates Xʹ and Yʹ on the screen. The angles between the diffraction beam AB and the axes are shown in the figure. In the above example, the Laue conditions (22.10) are performed as follows: acosa ¼ h1 l;
(22.13a)
acosb ¼ h2 l;
(22.13b)
aðcosg þ 1Þ ¼ h3 l.
(22.13c)
h1 : h2 : h3 ¼ cosa: cosb: ðcosg þ 1Þ.
(22.14)
wherein It is seen from the graph, that cosa ¼ Xʹ/AB; cosb ¼ Yʹ/AB; cosgʹ ¼ L/AB. The distance from the crystal to the diffraction spot is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (22.15) AB ¼ X 02 þ Y 02 þ L2 . Then h1 : h2 : h3 ¼ X 0 : Y 0 : ðAB LÞ.
(22.16)
22.4 DIFFRACTION OF BRAGG In the abovementioned Laue method, diffraction of X-rays passing through the crystal is observed. W.H. Bragg and W.L. Bragg, and also G.V. Wulff suggested to consider the rays reflected from the crystal. In Fig. 22.3 the reflection scheme is shown. A plane wave is incident on a series of parallel crystallographic planes spaced apart by a distance d, so that the
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
317
FIGURE 22.3 To the derivation of the WolfeBragg equation.
FIGURE 22.4 The scheme of the diffractometer for obtaining diffractograms by the Bragg method. 1dX-ray tube, 2, 3dmonochromator, 4dcrystal, 5ddetector, 6dgoniometer.
angle between the plane and the direction of the wave equalsq (q is an additional angle to the angle of incidence). The rays reflected from the crystallographic planes interfere and, depending on the difference in the path traveled, quench or amplify each other. The beam M1A1N1 passes a shorter distance than that of the ray M2A2N2. The path difference is B2A2 þ A2C2. From simple geometric considerations it is clear that B2A2 ¼ A2C2 ¼ d$sinq. Hence the BraggeWulff condition is 2d sinq ¼ nl;
(22.17)
where n ¼ 1, 2, 3 . It can be seen from Eq. (22.17) that diffraction are observed only for d > l/2, i.e., under the condition l 2d, the diffraction peaks are absent. To carry out the measurements, the test sample is placed in a special installation, called a diffractometer. The scheme of the diffractometer for obtaining the diffractograms by the Bragg method is shown in Fig. 22.4. The crystal is located on a so-called goniometer, a device that allows to rotate both the crystal and detector, accurately counting the angles and distances. The goniometer ensures a simultaneous rotation of the sample by an angle q and rotation of the detector by an angle of 2q. A typical diffractogram is shown in Fig. 22.5.
22.5 DIFFRACTOMETRY OF POLYCRYSTALS XRD makes it possible to work not only with single crystals but also with polycrystalline samples, powders, metal plates, etc. In polycrystalline samples, the crystals can be in different orientations. When monochromatic radiation hits the sample at a certain angle, among the crystals of different orientations, those can be found for which the diffraction condition is met. When incidence angle changes, different sets of crystallographic planes get into the reflecting position. Thus, to obtain a diffraction pattern, the sample must be rotated and the irradiation has to be carried out by monochromatic radiation. To obtain the roentgenograms, the sample is placed in a goniometer. Measuring methods have been developed both in the reflected beam mode and in the transmission mode. The scattered radiation is detected either by a film or by a detector. By the X-ray patterns a set of reflection intensities and corresponding interplanar distances are calculated, and then the necessary parameters are determined, comparing the attained results with the known ones. At present, information on powder diffractograms of crystalline substances is stored at the International Center for Diffraction Data and at the Powder
318 PART | II Applications
FIGURE 22.5 Typical diffractogram of NaCl, radiation of Cu-Ka. The Miller indices are indicated above the peaks.
Diffraction File, which contain about 50,000 experimental and more than 500,000 calculated powder diffraction patterns for all classes of compounds. Even with a known chemical composition, the determination of the parameters of the crystal lattice can cause difficulties, but if the composition is unknown, then the problem most likely does not have an unambiguous solution, as similar X-ray diffraction patterns can occur in different isostructural substances. Because decoding the radiographs is a rather complicated task, methods have been developed to facilitate it. For example, it is the isotopic substitution method. When analyzing the diffractograms of long polymer molecules, the sample stretching is used, as well as many other methods.
22.6 DECODING OF THE PROTEINS AND DNA STRUCTURE Particularly complicated is the problem of decoding X-rays by studying such complex systems as proteins and DNA. A detailed description of the methods for determining the structure of such complex molecular systems can be found in the literature [4e6]. The interpretation of X-ray diffraction patterns by long molecules, such as polymer molecules, proteins, or DNA, is based on the fact that the location and intensity of the diffraction spots are directly related to the peculiarities of the fiber structure. The vertical line passing through the center of the roentgenogram is called the meridian, and the horizontal line passing through this point is called the equator. Reflexes on the meridian give information on the order of reflection along the fiber axis, equatorial reflexes reveal ordering in the perpendicular direction. Decoding the structure of proteins and DNA is one of the greatest achievements of X-ray structural analysis. The overwhelming majority of the most important functions in a living cell are performed by proteins. In the dry weight cells of the famous Escherichia coli proteins make up about 50%, whereas DNA makes only 3%. All enzymes that catalyze the basic processes in the cell consist of proteins. They catalyze splitting of complex molecules and their synthesis, DNA replication and repair processes and matrix RNA synthesis; they regulate transcription, translation, splicing, and other processes. From the proteins, structural elements of the cystoskeleton are constructed: collagen and elastin are the main components of the intercellular substance of connective tissue (for example, cartilage); hair, nails, feathers of birds, and some shells consist of other structural proteins, i.e., keratin. Proteins carry out motor functionsdcontraction of muscles, movement of cells, movement of flagella, and cilia. On the basis of proteins, receptors are constructed that perceive light, sound, mechanical action, smell, and taste. And hormones that carry out signaling functions are also proteins. Proteins are long linear polymers of amino acids. The sequence of amino acids in the protein significantly affects the properties of the protein and the functions that it performs in the cell and the body. It is the primary structure of the protein. But apart from the sequence of amino acids, the properties of the protein are determined by the fact that it is folded into a unique three-dimensional structure. In many proteins, the polypeptide chain is curved and coiled. Lynus Pauling was the first to detect the specific shape of this spiral, that famous alpha helix, as well as the laws of its formation, by using X-ray analysis, for which he was awarded
Crystal Diffraction Methods for Investigation the Structure of Matter Chapter | 22
319
the Nobel Prize in Chemistry in 1954. About this discovery, J. Watson noted: “The alpha helix was discovered not by simple contemplation of radiographs; the main trick was to ask yourself the question: which atoms prefer to be neighbors with each other? The main working tools were not paper and pencil, but a set of molecular models, at first glance, reminiscent of children’s toys.” [7]. The X-ray diffraction analysis method was perfected by M. Perutz, who applied this method to the analysis of hemoglobin structure. M. Perutz performed crystallization in the presence of salts of heavy metals. Also in Cambridge, in the laboratory, which was headed by W.L. Bragg, J. Kendrew, with the use of the same X-ray diffraction method, determined the spatial arrangement of polypeptide chains in the myoglobin protein molecule and established its detailed structure, confirming the presence of alpha helices in it. For these achievements M. Perutz and J. Kendrew in 1962 were awarded the Nobel Prize in Chemistry. And another one more great success of X-ray analysis was the decoding of the DNA structure by J. Watson and F. Crick. They themselves received X-ray photographs before, but the most valuable material was contained in the X-ray diffraction patterns of M. Wilkins and R. Franklin, mainly because they got better samples of DNA crystals. Because of the complexity of decoding, Watson and Crick also solved this by the method of molecular models. On the basis of chemical conceptions, they compiled models and determined what kind of diffraction pattern can such a model present, compared it to X-ray diffraction patterns, made corrections, constructed new models, and finally produced a famous double helix.
REFERENCES [1] [2] [3] [4] [5] [6] [7]
B. Rupp, Biomolecular Crystallography: Principles, Practice and Application to Structural Biology, Garland Science, New York, 2009. W. Massa, Crystal Structure Determination, Springer, Berlin, 2004. W. Clegg, Crystal Structure Determination (Oxford Chemistry Primer), Oxford University Press, Oxford, 1998. A. McPherson, Introduction to Macromolecular Crystallography, John Wiley & Sons, 2003. D. Blow, Outline of Crystallography for Biologists, Oxford University Press, Oxford, 2002. D.E. McRee, Practical Protein Crystallography, Academic Press, San Diego, 1993. J. Watson, The Double Helix. A Personal Account of the Discovery of the Structure of DNA, A Signet Book, 1968.