Monochromators for x-ray synchrotron radiation

MONOCHROMATORS FOR X-RAY SYNCHROTRON RADIATION

R. CACIUFFO, S. MELONE, F. RUSTICHELLI Università di Ancona, Ancona, Italy and A. BOEUF Commission of the European Communities, J. R, C. Ispra, Ispra, Italy

I

NORTH-HOLLAND

-

AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 152. No. I (1987) 1—71. North-Holland. Amsterdam

MONOCHROMATORS FOR X-RAY SYNCHROTRON RADIATION R. CACIUFFO*, S. MELONE. F. RUSTICHELLI (!niri’r.si(a di Ancona, Ancona, ku/v

and A. BOEUF (oni,nvssion of the Luropean ( o,nrnuflities. I. R. ( .

Ispru. I.ipra . ku/v

Received February 1987

( Ofltt’flt,s 1. Introduction 2. Properties of synchrotron radiation 2.1. Introduction 2.2. Fundamental equations 3. Reflection of X-rays by optical surfaces 3.!. Reflectivity at small grazing angles 3.2. Mirror optics 3.3. Characteristics of synchrotron radiation mirrors 4. Curved crystal monochromators optics 5. Graphic methods for synchrotron radiation X-rays optics 5.1. DuMond diagram 5.2. Phase-space analysis 6. X-ray optical systems adopted in some laboratories 6.1, Introduction 6.2. Monoehromators for diffraction experiments 6.3. Monoehromators for small angle X-ray scattering cxperiments

3 4 4 4 7 7

10 12 13 16 16 19 24 21 26

6.4. Monochromators or EXAFS and XANES measurements 6.5. Monoctiromators for other experimental techniques 6.6. Crystal monochromators for harmonic suppression 7. Dynamical theory of X-ray diffraction 7.1. Introduction 7.2. Fundamental equations of the Dynamical Theory 8 ay diffractton by deformed crystals 8.1. Introduction 8.2. Fundamentals of Taupin theor~ 8.3. Reflectivity of elastically bent perfect crystals in Laue geometrs 8.4. A model for the dynamical X.rav diffraction by deformed crystals in Bragg geometry Appendix. On the polarization correction References

3(1 32 36

II 4! 42 SI 52 35

68

28

Present address: European Institute for Transuranium Elements. Postfaeh 2340. D-7501) Karlsruhe I. F.R.C

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R. Caciuffo et a!., Monochromators for X.ray synchrotron radiation

3

Abstract: A review on the characteristics of several X-ray monochromators used at different synchrotron radiation sources is presented and the theoretical instruments necessary to calculate the diffraction properties of both perfect and curved crystals are discussed. After a description of the source properties, the reflection of X-rays by optical surfaces and the optics of curved crystals monochromators are considered. Then, the graphic methods used for synchrotron radiation X-ray optics and the optical systems adopted in different experimental apparati are reviewed. Afterwards, the basic concepts of dynamical theory of X-ray diffraction by perfect crystals and the modifications induced in the diffraction properties by bending the crystals are discussed. Finally, a physical model useful to predict the main diffraction characteristics of bent monochromators is described and used for the interpretation of some experimental results,

1. Introduction Among the experimental techniques which make use of Synchrotron Radiation (SR) X-ray sources, only a few require a polychromatic beam. In general, a monochromator device is needed to extract from the primary beam the component with the suitable wavelength and handwith. In the experiments so far performed with SR X-rays, wavelengths ranging from about 0.2—0.3 A to about 15—20 A have been employed, whereas the required bandwidth z~A/Avaries over a range of six orders of magnitude, from about 10_i for small angle X-ray scattering and fluorescence analysis to about i07 for inelastic X-ray scattering spectroscopy. Different solutions must then be adopted to match the requirements of the various experimental techniques. The easiest way to obtain a monochromatic beam in the X-ray region consists in using Bragg diffraction from perfect or mosaic crystals. This system has been widely used for conventional X-ray sources with both organic and inorganic crystals (see table 1). However, the peculiar characteristics of SR beams (high power, collimation in the vertical plane, correlation between position in the beam cross section and divergence, polarization, continuous energy distribution,. .), must be carefully considered in the design of the monochromator apparatus and, in some cases, particular problems not connected with the conventional sources may arise. For instance, the suppression of higher order harmonics in the monochromated beam can be much more a stringent exigence than in the case when the characteristic emission line of a conventional source is selected. The aim of this article is to review the characteristics of the monochromators existing at the different radiation sources, to discuss in a critical way their performances and to provide the simplest theoretical instruments to design future monochromators, aiming to an optimization not only of the geometrical properties but also of the energy resolution and reflectivity. At first a description of the source characteristics and of the general requirements for mono.

Table 1 Crystals used for X-ray monochromators Crystals

Crystal system

Space group

a

b

c

a

13

~

Silicon Germanium LiF CaF 2 Cu Al NaCI InSb a-quartz SiO. Al2O~,3-alumina Graphite Calcite CaCO~

cubic cubic cubic

Fd3m Fd3m Fm3m

5.431 5.657 4.017

id id id

id id id

900 90° 90°

id id id

id id id

cubic cubic cubic cubic cubic hexagonal hexagonal hexagonal rhombohedral

Fm3m Fm3 Fm3 Frn3m F43m P3,2 P6m2 C6mc R3c

5.463 3.615 4.049 5.63 6.478 4.913 4.763 2.456 5.361

id id id id id

id id id id id 5.405 13.003 6.696

90° 90°

id id id id id id 90° 90°

id id Id id id id 120° 120°

9(30

90° 90° 120° 90° 90° 46°6’

4

R. C’aciuffo et a!. - Monochrornator,c Jir X-ray synchrotron radiation

chromators (based both on diffraction by perfect crystals and on total mirror reflection) will be presented together with a synthetic description of the practical solutions adopted until now in this field. Then the basic concepts of dynamical theory of X-ray diffraction by perfect crystals which are necessary to understand the phenomena involved in monochromatization, will be presented. Afterwards, the modifications induced in the diffraction properties by bending the crystals will be considered according to the rigorous calculations based on dynamical theory. Then it will be shown that a simple model developed by the authors is able to predict the main diffraction characteristics of bent monochromators, with a satisfactory accuracy. as demonstrated by comparison with the above mentioned rigorous theories. This model will be used to interpret some experimental results obtained on bent monochromators in several laboratories.

2. Properties of synchrotron radiation 2.1. Introduction Synchrotron radiation (SR) is the electromagnetic radiation emitted by charged particles undergoing curved motion with highly relativistic velocities. The theory of SR for circular particle accelerators was first developed by Ivanenko and Pomeranchuk [1] and by Schwinger 12, 3]. An exhaustive treatment of this subject can be found in refs. [4, 5]. Considered originally just as a parasitic by-product of particle accelerators, SR is now recognised as a powerful research tool in many different fields such as physics, materials science, biology, biophysics. chemistry, metallurgy and medicine. This fact is due to the peculiar properties of SR: the continuous spectrum of the emitted radiation ranges from the infrared to the X-ray region; the produced pulsed beam is highly collimated in the instantaneous direction of motion of the emitting particles and linearly polarized in the plane of the orbit, with its electric vector in the direction of acceleration; the pulse duration can be as short as iOops, allowing time-resolved experiments to be performed. For these reasons, numerous high-energy electron or positron storage rings are being designed as “dedicated” SR sources with parameters optimized for the production of stable and long lifetime beams. A list of operated or planned SR sources is given in table 2. Extensive reviews of SR research can be found in refs. [6, 111. 2.2. Fundamental equations The energy radiated per unit time, per unit azimutal angle, per unit wavelength by a single electron moving on a circular orbit with energy E, is given by [3] 1(~,A)

=

~

~

(~)~i

+

(y~)2I2[K~(~)+ 1 +(y~)2K~.~)j

(2.1)

where y = EImc2 2’trR ~=-~-~--y

(2.2) ‘(l+(y~i))’.

(2.3)

R. Caciuffo et al., Monochromators for X-ray synchrotron radiation

5

Table 2 Dedicated synchrotron radiation sources. E = particle energy; I = initial current; R = magnet radius; = characteristic photon energy. Table adapted from refs. [10, 11] Location

Name

E [GeVI

China Europe France

HESYRL ESRF ACO SuperACO DCI

0.8 6.0 0.54 0.8 1.72

Germany

BESSY DORIS COSY BONN I BONN II

Italy

R [m]

r~[keV]

Notes

300 100 150 500 300

2.22 134 1.11 1.75 3.82

0.51 19.2 0.31 0.65 2.95

C P 0 C 0, PD

0.78 3.7 0.56 2.5 0.5

250 70 300 30 30

1.78 12.12 0.38 76 1.7

0.59 9.27 1.03 4.56 0.16

0 0, PD P 0 0

ADONE

1.2

80

5

0.77

0, PD

Japan

PF-PING SOR UVSOR TERAS SUPERSOR INS-ES

3.0 0.4 0.6 0.6 1.0 1.3

250 300 500 100 500 80

8.66 1.1 2.2 2.0 3.66 4

6.92 0.13 0.22 0.24 0.61 1.22

0 0 C 0 P 0

Sweden

MAX LUSY

0.55 1.2

40

1.2 3.6

0.31 1.06

C 0

UK

SRS

2.0

250

5.56

3.19

0

USA

SURFII SPEAR SXRL TANTALUS ALADDIN NSLSI NSLSII ALS DUAL PING

0.28 4.0 1.0 0.24 1.0 0.75 2.5 1.3 2

50 100 100 200 500 200 500 400 200

0.84 12.7 2.0 0.64 2.08 1.9 6.88

0.041 11.18 1.11 0.06 1.07 0.49 5.04

N-100 UEPP-2M UEPP-3 S-60 MOSCOW

0.10 0.67 2.25 0.68 2

25 100 100 150 1000

0.5 1.22 6.15 2 5

USSR

I [mA]

19.1

0.93

0 0, PD C, PD 0 C 0 0 P P

0.004 0.54 4.1 0.35 3.55

0 0, PD 0, PD 0 P

C = In construction. O = Operated. PD = Partially dedicated. P = Proposed.

w0 denotes the orbital frequency, R the radius of the curvature, K~13and K213 the modified Bessel functions of the second kind and i~(ithe angle between the line of observation and its projection on the orbital plane (see fig. 1). 3 or the corresponding An important parameter is the “characteristic wavelength” A~ 4i~R / 3y critical energy Ec = hcIA~which is defined in such a way that one half of the total power is radiated below the critical energy. In practical units E~(eV)= 2218 E3(GeV)IR(m)

(2.4)

6

R. Caciuffo et a!., Monochromator,s for X-ray sv~ichrotronradiation

/4, z~

/

Fig. I. Geometry of Synchrotron Radiation emission. From ref. [7].

A C (A)

=

5.59 E3(GeV) R(m)

=

B(kG)E~(GeV) 186.4

(2.5)

where E is the electron beam energy and B the bending magnetic field. Generally, the intensity is useful out to about four times the critical energy. Equation (2.1) can be integrated over all angles ~1iand over all wavelengths yielding the total power radiated by a single electron. If the total current is j, the total power radiated for 2~rad is Q(keV)

=

88.5 E4(GeV) j(mA)IR(m).

(2.6)

Equation (2.6) allows the calculation of the thermal load on optical elements. The intensity integrated over all angles ~i, expressed in photons/s mrad eV. is given as a function of the photon energy by (for £ ~ E~)

1= 4.5 x 10t2R13(m) ~23(eV) j(mA).

(2.7)

The photon intensity for several SR sources is reported as a function of the photon energy e in fig. 2. The radiation is emitted into a small forward angular cone having an opening angle of the order of y = mc2I E; for an electron of 5 GeV this corresponds to about 0.1 mrad. As a consequence. the emitted radiation is highly collimated in the plane perpendicular to the orbit, while in the orbital plane a fan of radiation several degrees wide is produced, due to the continuous electron beam curvature provided by the bending magnets. This fact allows the installation of several facilities, each using a different part of the fan. Another parameter which characterizes the performance of a SR source is the electron beam “emittance”, i.e. the product of transverse beam size and divergence. The convolution of the electron emittance with the distribution of emitted photons yields the emittance of the SR source. A small emittance corresponds to a high source brightness (ph/s cm2 sterad eV). The formulae quoted above refer to SR emitted by electrons moving on circular orbits induced by bending magnets. More sophisticated magnetic devices (periodic wigglers and undulators) can be employed to enhance the brightness, to extend the spectral range or to change the plane of polarization.

R. Caciuffo et al., Monochromators for X-ray synchrotron radiation

7

-c

12I~

-~

C

I

i02

Wavelength (A) 12.4 1.24

0.12

I

1 102 Photon energy (keV)

Fig. 2. Photon intensity for several Synchrotron Radiation sources as a function of the photon energy. From ref. [11].

Wigglers and undulators cause the electrons to deflect periodically in a straight section of the storage ring and thus to radiate. If the angular deflection 3 of the electron beam is much greater than the natural opening angle ‘y ~,no interference effects are allowed and the magnetic device is called a wiggler. A simple three pole wiggler, reducing the radius of curvature R, produces an extension of the spectrum to shorter wavelength (eq. (2.5)); adding more poles, a flux “multiplication” can be obtained. In an undulator the angular excursion 3 is less than or comparable to y ~,so that in a multipole device constructive interference effects occur between the radiation emitted at the different source points. This results in a very narrow cone of radiation with a discrete spectrum peaked at wavelength given by -

A=

2w)’

(1 +

1(3)2

+

(2.8)

202)

where A~is the spatial period of the sinusoidal magnetic field, n is the harmonic number and 0 is the angle between the average beam direction and the observation direction. The intensity at the beam position is proportional to N2, N being the number of magnetic poles. The linewidth z~A/Ais proportional to (nN)1 and assumes values of the order of 102_10t~

If y3 1, all the emitted power is essentially concentrated on the fundamental harmonic (n resulting in a very high spectral brightness. °~

=

1),

3. Reflection of X-rays by optical surfaces 3.1. Reflectivity at small grazing angles The interaction of X-rays with a medium can be described in terms of a complex dielectric function, given by e= 1 26 21$. The quantities 3 and /3 are wavelength dependent and account for the —

—

K

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

dispersive properties of the material and for any absorption effect: r 2~

(3.1)

(3.2) where r1 = 2.818 X l0~ m is the classical electron radius, A is the X-rays wavelength, ~sis the linear absorption coefficient and N is the number of electrons per unit volume of the reflecting material. The refractive index is given by (3.3) In the absence of absorption (/3 = 0), the refractive index is a real quantity with a value less than unity. As a consequence, for any given wavelength, total external reflection occurs for grazing angle of incidence 0 smaller than a critical value 0~.By using 5 Snell’s law, one obtains cos 0~= (1 —6) and, since the values of 6 are typically of the order of i0~ =

=

2.34 x i0~

A

(3.4)

where Z, p and A are the atomic number, the mass density expressed in CGS units and the atomic mass of the reflecting material, respectively. The reflectivity is equal to 1 for 0 < 0~and 0 for 0> 0~. Table 3 reports the 0~values at A = 1.54 A for some mirror materials. Due to the absorption, the sharp cut-off is broadened; the complex reflection coefficients are given, as a function of the grazing angle, by the Fresnel equations [12, 131: r,~=

~sin0—VE—cos20 r sin 0 + cos 0

(3.5a)

.,

—

r,,.=

sin0_V~_cos2O

(3.5b)

2

sinO+Vc—cos 0 where the suffixes r and ii refer to the states of polarization with the E vector parallel to the plane of incidence and perpendicular to it, respectively. The corresponding reflectivities are R~=F~=~Fj2

~

Table 3 Critical angles of total reflection at A = 1.54 A calculated by eq. (3.4) for some mirror materials Material

Quartz

Crown glass

Ni

Au

Pt

O~(mrad)

5.1

4.1 (from ref. [16])

7.42

10

10.5

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

9

At small grazing angle, the polarization dependence can be neglected and a single expression for the reflectivity of the X-rays is obtained [14] R_h(0I0V2~1) —

(36

h+(0/0~)\/2(h—1)

where h = (0I0~)2+ \/[(0/0~)2 — 112 +

(3.7)

(/3/6)2.

Equation (3.6) is valid for a perfectly smooth homogeneous mirror surface and can be used for wavelengths smaller than about 10 A; for longer wavelength the polarization dependence should be taken into account [15]. Figure 3 shows the reflectivity curves R(0) at A = 1.54 A for quartz and Au-coated quartz mirrors. By coating the quartz surface with a heavy element the critical angle increases (O~ \/~) but the greater absorption lead to a less sharp cut-off and to a smaller reflectivity near O~. As pointed out by Witz [16] the total flux reflected by the mirror can be estimated by the product O~R(0~)for a cylindrical mirror and by 0~R(O~)for a toroidal mirror. For a quartz mirror 0~R(0~) 2.55 mrad whereas for the gold-coated surface O~R(O~)—4 mrad; an increase of the reflected intensity of about 1.5 can then be obtained. In fig. 4 the reflectivities for quartz and gold mirrors are shown as a function of the wavelength for a given glancing angle. 0. Only X-rays with wavelengths down to a critical value of A~are reflected efficiently, whereas more energetic X-rays are not reflected. The critical wavelength is given by (3.8)

A~=O\/~7~~N.

A mirror can then be used as a high-frequency filter valuable for harmonic rejection when used with a crystal monochromator. For these applications a sharp cut-off and, hence, materials with low absorption for X-rays are preferable, even though this implies smaller values for O~. From eq. (3.8) it is evident that a material with a high electron density reflects X-rays with shorter wavelengths than does a low electron density material set at the same glancing angle. This is the reason why the contamination by carbonaceous materials of the surface exposed to intense synchrotron radiation beams degrade the

jO.5~~1Omr;d

Glancing angle (mrad) Fig. 3. Reflectivities of uncoated and gold-coated quartz mirrors calculated as a function of the glancing angle 0 for A = 1.54 A. The corresponding critical angle is 9~(eq. (3.4)). From ref. [201.

10

R. (aciuffo ci a!.. Monochrornators Jdr X-ray synchrotron radiation

~

)<.ray wavelength

(A]

Fig. 4. Reflectivities of uncoated and gold-coated quartz mirrors calculated as a function of the X-ray wavelength. The grazing angle is set at 6 = 3.8 mrad; A~is the critical wavelength (eq. (3.8)). From ref. 20].

reflectivity at the short wavelength limit. This effect can be reduced by putting the mirror after a monochromator [17] or in a ultra-high vacuum container. Any appreciable change in reflectivity for a Pt-coated mirror operating at 10~~Torr and exposed to a white SR beam has been observed after more than a year of continuous irradiation [18]. It must be emphasized that all the expressions quoted above are correct only for perfectly smooth homogeneous mirror surfaces. A review of the theories which take account of the surface roughness and of the thick coatings has been done by Bilderback [19]. The surface roughness causes a steeper descent of the reflectivity as the glancing angle approaches the critical value 0~., while an oscillatory behaviour of the reflectivity is provided by the thick coating. Moreover, a reduction of the reflectivity is produced by the non-specular scattering of X-rays from the surface irregularities, whose amplitude should be less than 100 A for a mirror to be used with A = 1 .5 A X-rays [201.The parasitic scattering due to the surface defects is enhanced by the radiation damage caused by the incident beam. A granular structure of 2—3 ~i.mperiod and depressions of 10—50 ~imdiameter have been observed on irradiated surfaces of silica mirrors by Franks et al. [21]. All these effects make very difficult the prediction of the reflectivity for a surface in a practical case. 3.2.

Mirror optics

Curved mirrors at grazing incidence are widely used as focusing elements both for conventional and for SR sources. The principles of curved mirrors optics are the same as those of symmetrically cut curved crystal monochromators. The formulae developed for the latter case can simply be adapted by putting a (the asymmetry angle) equal to zero, i.e., the glancing angle of incidence must be equal to the glancing angle of reflection. Hence, the focusing condition is given by the “lens equation”: 2 R

sin0 p

—=-—----+--——

sin0 q

(3.9)

where R is the radius of curvature of the mirror surface, p and q are the source—mirror and the mirror—focus distances, respectively. For unit magnification p = q and R = p/sin 0. The radiation emitted from a divergent point source S can be focussed into a line I if the mirror surface has the shape of an ellipsoidal cylinder with foci in S and I and semi-axis a and b given by [22]

34

R. (‘aciuffo ci a!.. Monochromators for X-ray synchrotron radiation

~BA°

/ /Rowland circle Fig. 29. Schematic drawing of an X-ray spectrometer for low energy resolution inelastic scattering: M = double crystal Si(400) nsonochromator: S = sample: B = beam stopper: A = spherically bent analyser Si(333) crystal: D = detector. From ref. [99].

The intensity delivered to the detector could be increased by an order of magnitude by using a Ge(400) double crystal monochromator and a spherically bent LiF(420) analyser. An inelastic X-ray scattering spectrometer for very high energy resolution experiments (~E< 10meV) has been proposed by Dorner and Peisl in ref. [1001. These authors suggest the use of spherically bent Si crystals in almost backscattering geometry (see fig. 30). A monochromator with a radius of curvature of 20 m produces an image of 1/3 of the source at the sample position, while a 1: 1 imaging is produced by the analyser to collect a large solid angle. The energy scans could be performed at constant momentum transfer by changing the temperature difference between monochromator and analyser. Finally, we report the characteristics of a channel-cut crystal monochromator used at CHESS 11011 for X-ray standing-wave excited fluorescence [102, 103] in the wavelength range 1.0< A <2.6 A. Standing-wave experiments require high angular resolutions which can be obtained by asymmetrically cut monochromators. However, in this case, the angle between the incident beam and the crystal surface should usually be less than 10, a condition putting an abrupt upper limit at the accessible energy. A narrow angular transmission can also be obtained by multiple Bragg reflections [57] and a collimation method was proposed by Bollman et al. [104] based on successive reflection from separate

~

~nalyser detector

source

L

2

‘

d~

ample

monochromator ~mm

Fig. 30. Proposed instrument for high energy resolution inelastic X-ray scattering. Spherically bent Si crystals in haekscattering geometry are used as monochromator and analyser. From ref. [101)].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

d+~d2-~(f1-f2)2(f1+f2)2

2

a

=

b2

=

where

2

a2

—

(f~+f

11

(3.lOa)

214

(3.lOb)

2)

f~and f

2 are

the distances defined in fig. 5 and /

4~f \2

1/:2L:2\.i.~i(~ U2~fl~J2)i

Jtf2 c)

~_

ft

Il

J2

For different shapes of the mirror surface the image will be affected by spherical aberration. For instance, for a cylindrical mirror (R = R0) of length L, the distance from the focus of the rays coming from the mirror extremities is given by [23] SAcyt.

=

[

2 ~3 [sin2O 2 0 q 0

qL

]

(3.12)

sin 22O~1 p0

—

where the suffix 0 refers to the quantities measured in correspondence of the mirror extremity closer to the source point. For a logarithmic spiral surface (0 = 00 measured along radial lines emanating from S), one has [23] SA10g=qL 213 1~sin 200 2 LO

q0

—

7o sin 20~] 2 ] 0

(3.13)

p0

However, for practical applications, the radii of curvature are, in general, so large that the detailed surface figure is not particularly important. Mirrors with a double-curved surface can be used to obtain a point focus. The ideal figure is, in this case, an ellipsoid of revolution or toroid, with its foci at the source and at the image points. Hollow toroidal mirrors in gold-coated epoxy resin are produced for conventional source camerae [24—25].For

-E

~

—~

Fig. 5. Geometry of an X-ray focusing mirror.

12

R. Caciuffo ci a!.. Monochromators for X-ray synchrotron radiation

SR applications, due to the lengths needed, hollow toroidal mirrors are very hard to manufacture. It is much easier to obtain an approximated toroidal figure by bending to a radius R1 a mirror whose substrate is ground to a fixed radius curvature The focusing conditions satisfied 2 0 of [261.A doubleR7. curvature fused quartz mirror are 0.6 m in lengthif R1 = 2pq/(p + q) sin 0 and R7 = R~ sin has been used to point focus a soft X-ray SR source [261in a device which proved the feasibility of a scanning X-ray microscope [27]. Another 0.6 m long Pt-coated fused quartz one to one focusing toroidal mirror with an angular acceptance of 6 x 0.2 mrad2 has been used in a four circle diffractometer at the SPEAR storage ring [28, 29, 301. In this case, because of its high energy cut-off, the mirror also provides good harmonics rejections. 3.3. Characteristics of SR mirrors The performances of a focusing mirror for X-ray SR are affected by several factors depending both on the machining and on the physical properties of the mirror material. High figure accuracy and surface finish must be achieved, with micro-roughness not exceeding 50—100 A; the mirror material must have a good resistance to radiation damage and such thermal and mechanical properties as to provide high dimensional and shape stability under irradiation. Float glass has been used as a mirror material at SPEAR [31, 32, 33] and at CHESS [34] but the most widely employed material for SR mirrors has been fused quartz (spectrosil). Silicon carbide has been recently proposed [25] as an alternative material for mirror to be used with powerful multiple wiggler sources. Platinum- and gold-coated quartz mirrors have also been employed [18, 27, 28, 29] and the use of rhodium with an Al substrate has been proposed at NSLS [36]. For technical reasons, the optical elements in a SR beam cannot be located too close to the source point; typical distances are of the order of 10—20m. In order to collect the full vertical divergence of the SR beam (0.2—0.4mrad), long mirrors are needed. For instance, a spectrosil mirror 1.5 m in length is required to collect 0.3 mrad at 15 m from the source and at a glancing angle of 3 mrad. By coating the quartz surface with Pt or Au the critical angle 0~increases by more than a factor of two and the mirror length can be reduced by the same factor. Long mirrors are expensive and, moreover, under strain they are subject to flow. For this reason, polylithic mirror systems consisting of several precisely aligned short segments have been widely employed [22, 37, 41]. On the other hand, segmented mirrors require more precise mechanical holders. The focusing is achieved by adjusting the heights of the segments until the reflected beams overlap at the focus position. The radii of curvature of mirrors for SR applications are usually of the order of 4—5 km. The sagitta equation gives the relation between the radius of curvature R and the deflection 8 at the centre of a mirror of length L [11]: 6=L2/8R.

(3.14)

For L = 1 m and R = 4 km, eq. (3.14) gives 6 p~m.For a thin mirror, this bending can be obtained under gravity [27], but the presence of strains leads to flow. The traditional technique for elastically bending a mirror consists of applying forces on pairs of steel pins sitting some cm aside on the front arid rear surface at each end of the mirror [38]. 25

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

13

4. Curved crystal monochromators optics Even with highly collimated synchrotron radiation, focusing of the incident beam is required if the intensity at the sample position must be maximized. In fact, due to experimental constraints, the sample is often at several meters from the source point, where the beam cross-section is usually larger than the specimen size. Focusing can be obtained either by curved mirrors or by curved crystal monochromators which, whatever the bending radius is, behave as mirrors when used with a white source. The first focusing spectrometers using bent crystals were built by Johann [42] and Cauchois [43, 45], while the exact focusing conditions were firstly obtained by Johansson [46] and de Wolff [47]. A review on focusing devices used with conventional X-ray sources has been given by Witz in ref. [16]. In this section, the geometrical problems associated with curved crystals are considered. The focusing geometry is shown in fig. 6; the plane of the figure is perpendicular to the crystal OA, of length 1, lying on the curved surface MOM’. The rays emitted by the point source S are focused, after Bragg reflection, to the image point I. If the crystal surface is cut at an angle a to the Bragg planes, the glancing angle of incidence and emergence will be 0B + a and 0B a, respectively. The distance between the source S and a general point L of the crystal surface is indicated by p, while q is the distance between L and the image point I. The radius of curvature is given by —

2 R

sin(0 + a) p

+

sin(0 q

(4.1)

—

If the reflecting lattice planes are parallel to the crystal surface (a = 0) eq. (4.1), together with the additional constraint p = q required by the Bragg law, gives the so-called Johann focusing conditions [42] p=q=RsinO.

(4.2)

In a SR instrument, the source to monochromator distance must be large and so would be the focal

S

I

R 0

qB~ L

B

Fig. 6. Focusing geometry; the monochromator MOM’ (curved to a radius R) focuses to I the X-rays coming from S. From ref. [23].

14

R. (.acwjto

Ct a!.. Monochro,nutors for X-ray synchrotron radiation

length, making difficult the scan recording. Shorter focal length can he obtained with a +0. In this case the Guinier focusing condition holds [481: p/q

=

sin(O

+

a)/sin(0

—

(4.3)

a)

and the object and image distances are given by p

=

R sin(0

+

a) (4.4)

q

=

R sin(O

—

a)

In the case of monochromators with the shape of circular cylinders, the Guinier condition corresponds to achromatic focusing. In fact, the variation of the angle of incidence along the crystal surface =

L [sin(On_ a)

—

sin(00+ a)

(4.5) 0B

to

~

isthezero if the Guinier is satisfied, and so the corresponding contribution ~A/A =is cot reflected radiationcondition bandwidth. In general, theis spectral width of the reflected beam given by ~A/A = cot

0B

\/w~+

h2/p +~0~

(4.6)

where co 5 is the angular width of acceptance of a perfect crystal (eq. (7.34)) and h is the horizontal extension of the source. The width of the line focus depends on several factors, such as the angular width of emergence w~of the diffracted beam, the source demagnification hq/p and the aberration of the optical system. An achromatic focusing can also be obtained if the form of the monochromator is a logarithmic0Bspiral + a polar equation exp(a~)and theinhomogeneity. source S as origin [47,this 49].geometry In this case, fact, can the angle iswith constant and therer is= no wavelength With a linein focus be obtained from an extended source sitting on a logarithmic spiral caustic having the same origin as the crystal profile. The geometries of various curved crystal monochromators are shown in fig. 7. In the Johann geometry (fig. 7a), the crystal lies on a circle of radius R centred in 0, and it is tangent at its central point C to the focal circle (Rowland circle) of radius R/2. The source and the focal caustics are both circles with centre in 0 and radius R cos(0 0 + a) and R cos(00 a), respectively. This arrangement is easy to realize but suffers from geometrical aberrations. Figure 7b shows the same design used in Laue geometry to obtain a convergent beam from an extended source [43]. The focusing problems can be treated in the same way as in the reflection case by introducing an optically virtual source with a caustic circle of radius R sin(0~+ a). A monochromator free of geometrical aberrations can be obtained if the so-called Johansson geometry is used [46, 50]. A crystal, whose surface is ground to a radius R12, is bent so that the Bragg planes lie on a circular cylinder of radius R (fig. 7c). In this way, the crystal surface is tangent to the Rowland circle and a line focus F is obtained from a point source S. Ideal point-to-point focusing could be obtained by a double-bent crystal having the form of an ellipsoid of revolution with its foci at the source and at the image points. Such a figure is difficult to he --

R. Caciuffo et a!., Monochromatorsfor X-ray synchrotron radiation

15

F F

/ ~Sj~ i’0,Io()

o

RsJne6~)

Fig. 7. Geometries of different curved crystal monochromators: (a) Johann geometry in Bragg reflection; (b) Johann geometry in Laue reflection; (c) Johansson geometry.

manufactured and only approximate solutions have been so far adopted (section of sphere [51], bent cylinders [52, 56]). Focusing in the scattering plane is referred to as in-plane or meridian focusing, while focusing out of the plane of scatter is called sagittal focusing. The geometry for sagittal focusing was first discussed by Gouy [168] and von Hamos [169, 170] and, more recently, by Sparks et al. [171]. The focusing conditions for a doubly curved optical element are given by [172] R5 = 2pq sin 0/(p Rm

=

0

+

—

q)

(4.7)

R~Ipq

(4.8)

where R~and Rm are the sagittal and meridional radii of curvature and 0 is the angle of incidence of the central ray. For a cylindrically curved crystal, aberrations are reduced if R~is set such that the magnification M = qIp ~ [171]. Sagittally bent crystals for SR focusing were successfully tested at SPEAR, CHESS and NSLS [96, 172, 173, 174]. The devices used by Sparks et al. [172] and by

16

R. Caciuffo

Ct

a!., Monochro,nators for X-ray synchrotron radiation

Batterman and Berman [173] consisted of a non-dispersive double-crystal monochromator with a flat Si(111) crystal followed by a Si(111) crystal bent to a radius R~allowing to collect about 3mrad of horizontal divergence. With this configuration, the anticlastic effect (appearance of a transverse curvature in a plate subjected to pure bending in a plane of symmetry) can destroy the parallel double-crystal relationship between crystal 1 and crystal 2, resulting in a considerable loss of intensity. In fact, meridian rays divergent from the central ray deviate from the Bragg condition and are not diffracted from crystal 2. To prevent anticlastic bending a Si crystal cut with stiffening ribs transverse to the bending curvature [1721or triangular-shaped Si crystals with slots cut to provide a polygonal approximation to a cylindrical surface [173] can be used.

5. Graphic methods for SR X-ray optics 5.1. DuMond diagram The behaviour of an X-ray optical system involving single or multiple diffraction can be qualitatively determined by a graphic representation in the A—0 plane introduced by DuMond [57]. The Bragg diffraction for a set of planes with interplanar spacing d is described in the DuMond diagram by a sinusoidal strip centred on the curve A = 2d sin 0 and having a thickness along the 0-axis equal to the crystal’s rocking curve width /3(A). The angular width of acceptance for any given wavelength A55 is then represented by the length of the segment defined by the intersection of a horizontal straight line A = A5 with the sinusoidal strip (fig. 8). In the plane normal to the electron orbital plane, the white SR source has a divergence defined by the opening angle ~, As a consequence it can be represented in the DuMond diagram by the region delimited by two vertical straight lines separated by a distance L~OSR= 2’)’ A collimating slit can also be described by a linear vertical strip SS’, whose width defines the range of incidence angles ~ (fig. 9). The acceptance of the optical system is then defined by the parallelogram abed in fig. 9. The accepted photon flux is proportional to the sampled area, while the wavelength spread is given by ~A A55 cot 0~(~0~ + /3). In the case of symmetric reflection from a perfect crystal, /3 is equal to the Darwin width co,, given by eq. (7.24), and the acceptance region coincides with the transmitted one. The situation is different if asymmetric diffraction is used: the angular width of acceptance w55 = w~/\[/i is larger than the angular - ~.

~.

Fig. 8. The DuMond diagram for symmetric single crystal diffraction.

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

17

>‘ ~Sr

Fig. 9. The acceptance region of an optical system composed by a crystal and a collimating slit. The accepted photon flux is proportional to the area of the parallelogram abed; ~ is the angular range of acceptance defined by the slit SS’ and &k is the wavelength spread.

width of emergence Wh = w~’/~ if the asymmetry factor b = sin(OB a)Isin(OB + a) is smaller than one, vice versa if b > 1. Moreover, the reflection curve H is shifted from the acceptance curve 0 by an angle [58, 59] —

(5.1)

where ~ and ~°H (given by eqs. (7.36) and (7.37)) are the deviations from the Bragg law for the 0 and the H beam, respectively. This situation is illustrated in fig. 10, where the refraction displacement ~ is exaggerated for the sake of clarity. The window of emergence efgh is the projection along the 0 axis of the acceptance window abed on the reflection curve H. The angular interval z~O11represents the divergence of the delivered beam. Figure 10 also allows to qualitatively determine the variations in the reflected intensity, when the asymmetry angle a is changed, by measuring the sampled area abed. The DuMond diagram is particularly useful when multiple diffraction devices are involved [20, 60]. In fig. 11 the DuMond diagrams for various symmetric double crystal arrangements are shown, as an example. The reflected radiation is represented by the intersection region ABCD of the two DuMond curves C1 and C2. The horizontal and the vertical extensions of the parallelogram ABCD define the angular and the wavelength spread of the emerging beam, respectively. In the so-called (+H1, +H2) dispersive arrangement, the X-ray beam is reflected by two nonidentical sets of crystallographic planes with reciprocal lattice vectors H1 and H2, as shown in fig. 1 la. In the DuMond diagram, the 0-axes of the corresponding A—0 curves have opposite directions and the origins of C1 and C2 are separated by an angle ~1s given by the sum ~ + °B2 of the Bragg angles in the two crystals. Only a small fraction of the available SR source is used in this case, because the angular acceptance of the apparatus is of the order of w5. On the other hand, a highly monochromatic beam is obtained. In fact, from fig. ha it is possible to estimate that

18

R. (aciuffo

Ct a!., Monochromators for X-ray synchrotron radiation

0/ e~

-

—

e Fig. 10. DuMond diagram for asymmetric diffraction with 6< I; abed is the acceptance window while the reflected radiation is represented by the parallelogram cfgh.

st

0132 +

tan 0~ w, tan °Bl + tan 0132

(5.2) (5.3)

cost + ~

=

A

tan 0 tan

tan 013 +

tanOBI+tanOB2

For instance, if H1 and H. correspond to the (111) reflection of Si and Ge respectively, the values 6 x 10~’rad and LXA/A S x i0’~are obtained for A = 1.5 A; the dispersive arrangement operates in a high resolution mode. In the low dispersive (+ H~,—H2) configuration shown in fig. lib, lattice planes with nearly antiparallel reciprocal space vectors H1 and H2 are0Btused. ThefIg. DuMond curves C1 and From lib one obtains C2 have equioriented 0-axes and their origin is displaced by 013~ co~ tan0~+co~tan0 = s H (5.4) tan °B~ tan 0 13j -

‘~

—

L~A =

A

—

W5~ +

(i)2

(5.5)

tanOBS—tanOBj

In this case, for the (lii) reflection of Si and Ge at A = 1.5 A, one has ~ 3 x 10 ~rad and ~A/A=2.5 x 10 2 Finally, the non-dispersive setting (+H, —H) is obtained by using the same reflection in two identical antiparallel crystals. In the DuMond diagram the two curves would be coincident and the divergence of the accepted radiation would be equal to the angular divergence ~°SR of the source. Hence =

z~0~

~A/A = cot O~(~0sR+ w5).

(5.6) (5.7)

R. Caciuffo et a!,, Monochromators for X-ray synchrotron radiation

19

A CA~ 3~

\~t

-~

Ci

xc ~

a) dispersive -0 (+H~~ H~)

- -

8SR ~

A

A

/

-0 dispersive b)low (+Ht -Hi) -0

a’

Ci

OiPt -.

A k~51~52

Hi

~~Tet c)non dispersive ~ (+H,-H)

0

-

--

__

/

Fig. 11. DuMond diagrams for different double crystal arrangements: (a) dispersive: (b) low dispersive; (c) non-dispersive.

The DuMond diagram clearly shows that a double crystal non-dispersive arrangement must be very stable, due to the small values of w~.Moreover, a careful relative alignment of the two crystals must be provided, and this is more easily achieved by using a monolithic grooved single crystal. Many multi-crystals configurations have been realized or proposed for SR applications and reviews can be found in refs. [58, 59, 61, 62, 63]. 5.2. Phase-space analysis

The direction of emission of a SR source is correlated to the position of the radiating electron in the

20

R. Caciuffo et a!., Monochromator.s for X-ray synchrotron radiation

source plane. Consequently, the optical properties of the source depend also on the electron paths [29, 33, 64, 65]. This problem can be conveniently treated in the frame of a method first developed for the description of the transport of charged particle beams [66, 671 and subsequently extended to X-ray SR optics problems [68—71].Constant luminosity contours are plotted in an optical phase-space whose axes represent the angular divergence of the emitted radiation and the displacement of the radiating electron from the centre of the bunch, respectively. This corresponds to the Hamiltonian dynamics phase-space representation, where the generalized momenta p, and p5 are replaced by the angular divergences x’ and y’. The Cartesian system of reference here utilized is shown in fig. 12; the z-axis is tangent to the electrons orbit and it is oriented along the direction of emission, while the y-axis points vertically upwards. With this choice, the phase-space representation in the vertical (y’, y) and in the horizontal (x’, x) planes can be treated separately. In the vertical plane the electron beam is represented by an ellipse which gives, for any value of y, the FWHM of the angular distribution of the electron beam along y’ [67].The phase-space diagram of the SR is obtained by2IE folding ellipse with a Gaussian function having FWHM equal so to the natural of thethis radiation emitted by a single electron (fig. a13). The curve obtained is divergence y of= polar mc equations [18]: still an ellipse

y= y

0 cos ~ a

(5.8)

y =-~-sin~+~y55cos~ with 6 given by 6=

(5.9)

,

Vi

+

where ~ is the polar angle, 2y55 is the spatial extension of the photon beam in the vertical at 2 is thedirection maximum zelectron = 0, a and /3 are instrumental parameters of the storage ring and y[~ = (y5516)V1 + a beam divergence value. At the origin point (z = 0), the radiation_divergence reaches its maximum value Y~ax= (y 2 in correspondence to YM(°)= y 2. + a z from a point A to a point B, 55a/\/ 1 + a by a distance L, the When the X-ray beam5516)V1 moves along separated electron

~~eIectron

trajectory

bunch

Fig. 12. The Cartesian system of reference used to analyse the optics of X-ray synchrotron radiation in the phase-space.

R. Caciuffo et al., Monochromators for X-ray synchrotron radiation

—

21

“I,

ELECTRON ELLIPSE

PHASE

PHASE ELLIPSE INCLUDING PHOTON OPENING ANGLE y

VMAX ~

Fig. 13. Vertical phase-space diagram for a SR light source. From ref. [29].

phase-space ellipse is deformed according to y~=y~,

y~y~+Ly~.

(5.10)

As a consequence, at the point Zv = —aôI(h + a2) the Yr~iaxvalue is obtained at YM(Zv) = 0 and the principal axes of the ellipse are parallel to they- and y’-axis (fig. 14). This position can be considered as the virtual source of the system, as conventional geometrical optics can be applied to the beam coming from this point where the luminosity of the source can be factorized in the product of a function of y only and a function of y’ only. The fact that the virtual source is not at z = 0 must be considered in the design of focusing optical devices. For instance, the distance from the focusing elements and the virtual source should appear in the lens equation [72, 73]. In fig. 14 a slit of width 2y~positioned at z = L is represented by two tilted straight lines both at the ytm rad) y’(mrad) REAL SOURCE

~ Z~-.~-2J8m

=

~~m)

ORIGIN

(z=O)

Fig. 14. Representation at z 0 and at the virtual source point of a slit positioned at z represent the slit in the z L phase-space plot. From ref. [68].

=

SLITS

= L and having a width 2y,. The dash-dotted vertical lines

22

R. Caciuffo et a!.. Monochromator.s for X-ray synchrotron radiation

X’(mrad)

x=9~_x0

/

I

2.x x~_

0

X~m)

\ Fig. 15. Horizontal phase-space diagram for a SR light source. From ret. [29J.

origin z = 0 and at the virtual source point. The region between these two lines defines the portion of the source which is accepted by the aperture. The dash-dotted vertical lines represent the same slit as will appear in the z = L phase-space plot. In order to study the interaction between the various slits in the optical system it is convenient to transform all the apertures back to the same point. The phase-space representation on the horizontal plane is complicated by the finite curvature of the electron orbit and by the spatial broadening of the beam due to the spread of the electron energy.

~

XO

(b)

or

X

____E2 _____ F

$

~_

I

~[_ E3~

I

~,Ai I

i ~

IA4

XoorXk

E4

Fig. 16. Representation in the real space (a) and in the phase-space (b) of a flat perfect crystal in asymmetric diffraction. The accepted beam A is transformed in E. From ref. [20].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

23

x~0tx~~ (b)

Ez

~

w~

Fig. 17. Representation in the real space (a) and in the phase-space (b) of a curved crystal in the Johansson geometry. The accepted beam is represented by region A, while regions E and F represent the beam immediately after diffraction and at the focus, respectively. From ref. [201.

Moreover, the source is not bounded in the X’-direction. It can be shown [29] 2 ±that X the source representation in the plane of the synchrotron is given by two parabolae X = ~RX’ 0, where R is the local radius of curvature of the electron orbit and 2X0 is the width of the electron beam at the tangent point (fig. 15). The optical phase-space description of a number of crystal monochromators has been given in refs. [74—76],both in reflection and in transmission geometry. A detailed discussion on these problems can be found in the review of Matsushita and Hashizume [20]. As an example, the phase-space description of a flat perfect crystal and of a focusing monochromator in the Johansson geometry are shown in figs. 16 and 17 respectively. For the flat perfect crystal, the extension of the acceptance window is given by the angular width of reflection w0 (eq. (7.34)) beam. The 0h =and bw by the lateral size 10 of the incident 1h = 10 lb. emergence window covers an angular range ~ 0 and it has a spatial extension In fig. 18 the focusing effect obtained with a mirror-monochromator camera at SSRP [29], is shown both in the horizontal (a) and in the vertical (b) planes. A 10: 1 demagnification of the source is obtained at the expenses of the angular resolution. Other graphic approaches to X-ray SR optics have been developed to take into account both wavelength-angle and wavelength-position correlations [74—77,78]. A review on these topics is given in ref. [20].

24

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

x’(mrad)

SOURCE

y’ (mrad)

(a)

2.0

16

1.6 1.2

---FOCUSED ~z BEAM H 8

~ox(mm)

(b)

I

6-0.4’

li-B ‘—12

it

-1.6 —16 -2.0 Fig. 18. Phase-space diagrams in the horizontal (a) and in the vertical (b) planes. showing 10:1 demagnification. From ret. [29].

6. X-ray optical systems adopted in some laboratories 6.1. Introduction One of the most important properties of a material to be used as monochromator for a SR beam is the resistance to the radiation damage. For this reason, organic crystals are not a suitable choice. Moreover, a good monochromator crystal should provide a high reflected intensity over a wide range of wavelength with the appropriate energy resolution. High mechanical and thermal stabilities are also needed, especially for focusing systems. Hence, materials with high thermal conductivity and small expansion coefficients should be employed. At present, the most widely used monochromator crystals have been germanium, silicon, quartz and graphite. The first three are perfect crystals with narrow angular acceptance. For instance, in the case of the Si(11h) reflexion the intrinsic Bragg reflection width is w, = 7.395 seconds of arc, much smaller than the angular divergence of a typical SR beam. This leads to a considerable loss of intensity. Higher integrated reflectivities can be obtained with Ge(w (111) = 16.338 sec of arc) or by using asymmetrically cut crystals (see eq. (7.41)). Moreover, the Ge(222) reflexion is virtually forbidden and then, by using Ge(hhh) crystals, a monochromatic beam not contaminated by the second order harmonic A/2 can be obtained. Considering that the percentage of A13 photons in the primary beam is of the order of 10% for A = A~l2,no higher harmonics will be practically present for A < 1.5 A and A~= 3—4 A. However, it must be noted that the reflectivity of germanium drastically decreases near A = 1.12 A where an absorption edge is located. Some considerations on the design of high resolution optical systems for SR X-rays are discussed in ref. [58]. In table 4 the reflection width w5, the energy resolution z~.E/E= co, cot 0B and the integrated intensity I for some reflections of Si, Ge and a-Si0 2 at A = 1.5 A are reported. Wider bandwidths can be obtained by means of mosaic crystals but the divergence of the monochromatic beam is increased. High Oriented Pyrolytic Graphite (HOPG) crystals with a mosaic spread of about 0.5 A have been used for photographing the fiber diffraction diagram of DNA [79]. Theoretical reflectivities of some mosaic

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

25

Table 4 Bragg reflection widths w,, energy resolutions LsEIE, and integral reflecting powers I of some reflections of Si, Ge and a-quartz perfect crystals at A 1.54 A. From ref. [201.

=

Crystal Silicon

Germanium

a-quartz

~E/E 4) (X10 14.1 6.04 2.90 2.53 1.44 1.47

39,9 29.7 16.5 19.3 11.8 15.5

1.989 2.675 1.907

0.88 0.96 0.60

9.9 14.0 9.3

111 220

16.338 12.449

32.64 14.46

85.9 67.4

311

7.230

6.92

37.1

400 331 422 333 (511) 440 531

7.951 5.076 6.178

5.94 3.34 3.34

42.3 25.4 32.4

4.127 5.339 3.719

2.00 2.14 1.33

20.2 27.5 17.7

100 101 110 102 200 112 202 212 203 301

3.798 7.453 2.512 2.488 2.252 2.927 2.072 2.042 2.430 2.368

10.00 15.26 3.69 3.36 2.81 3.03 1.93 1.47 1.74 1.69

18.8 40.9 12.2 12.9 11.5 15.5 10.6 10.7 12.9 12.6

hkl 111 220 311 400 331 422 333 (511) 440 531

(second of arc) 7.395 5.459 3.192 3.603 2.336 2.925

I (X106)

crystals in symmetrical Bragg geometry have been calculated by Freund in ref. [175]. It appears that substantial gains in reflectivity could be obtained by using mosaic crystals instead of perfect ones. In particular, beryllium seems to be much promising and recent progress in growing single crystals of this material [176] suggests the possibility of its use in the near future. The performance of a number of monochromator crystals suitable for works in the energy region 550—5000 eV (A = 2.5—22 A) has been tested at SSRL and the results have been reported in ref. [80].In particular, the resolution, the intensity and the harmonics contamination have been studied as a function of the incident photon energy. In fig. 19 the full width at half maximum (FWHM) of the measured double-crystal rocking curves are reported for some of the investigated samples. It appeared that good results could be obtained with beryl (1010) in the wavelength range 6—15.5 A (0.8—2 keV) where an energy resolution i~ElEof the order of 5 x 10~has been measured. The energy interval 1.68 < E <4.3 keV could be covered by InSb (111), while the (2000) reflection of 13-alumina could be used for low resolution experiments in the

26

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

ft -Atumin~(2000) 8 2dr22.49A 2 BerylOofo); 2do 15.96.A 1 1.0 . ~_~r~-quartz(i~o). 0.5 . /~77~ ~ 0500 I 1500 2000 /InSb Photo,

B 6 ~ .

4

energy(e’~/ ~

I

2000

Ge(tl)~

—--—-~

2

I I 0 3000 4000 5000 Photon energy (eV)

Fig. 19. Full width at half maximum of double-crystal rocking curves measured for different materials as a function of the incident photon energy. From ref. [80].

energy range 0.55< E < 1.5 keV (8 ~ A ~ 22 A). A high vulnerability to radiation damage has been observed for a-quartz in the wavelength interval S ~ A ~ 8 A. Some examples of monochromator systems adopted in different laboratories are reported below. 6.2.

Monochromators for diffraction experiments

The high brightness of SR allows structural studies to be performed on very small crystals in a short time. This is particularly interesting for crystallographic studies of organic materials and a number of facilities for protein crystallography have been built on SR sources [11]. The monochromator apparatus of a diffraction instrument must provide a high brightness and wavelength tunability in order to allow anomalous dispersion studies to be performed. The diffractometer usually operates in the vertical plane because of the polarization of SR. One of the first SR tunable diffractometers was installed at the SPEAR Laboratory at Stanford [31, 32, 33]. The monochromator was a logarithmic spiral type curved Si crystal with the surface cut at 8.5°with respect to the (111) Bragg planes. The monochromator crystal was placed at 14.4 m from the source while the crystal to focus distance can be varied between 0.6 to 3 m. Vertical focusing was achieved with a single sheet of float glass 1.2 m in length bent to an asymmetric elliptical profile and positioned immediately before the monochromator (see fig. 20). The angular acceptance of the optical system was 2 x 0.35 mrad2 (horizontal x vertical) and the focal spot size 0.5 x 0.5 mm2. With SPEAR operating at 3.7 GeV, 20 mA an intensity of 6 x i08 ph/s was measured at A= 1.74A and A four circle diffractometer at SPEAR [28, 29, 30] is equipped with a channel-cut Si (220) crystal which can be substituted by a double crystal Ge (111) monochromator. With the latter arrangement a small A /2 contribution from the forbidden (222) reflection is obtained. A 0.6 m long platinum-coated fused quartz toroidal mirror at 10 m from the source provides one-to-one focusing. A focal spot size of 4.2 x 2.4 mm2 (h x v) is obtained at 20 m from the source point with an angular acceptance of 6 x 0.2 mrad2. An advantage of this system is that the wavelength can be changed without disturbing the focus. ~A/A—104.

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

!~

27

1640cm

(RANGE 60 Be WINDOW 10300cm) ON 26o2~° 26hi BEAM PIPE ~CRYSTAL~

L-7~m~

~

SPECIMEN 8cm FOCUS

HOLDER

BEAMSTOP HOLDER

Fig. 20. Schematic representation of a double focusing mirror-crystal optical system used in a diffraction camera for biological samples at SPEAR. From ref. [33].

An 12ph/s intensity of SPEAR 3.2 X lO~~ was and measured SPEAR~A/A~103—104). operating at 3.7 GeV and 20 mA and with atph/s 3.7GeV 4OmAwith (A=1.7A; lx At 10the DCI laboratory in Paris, a focusing monochromator rotation camera for protein crystallography and low angle diffraction has been operated [81, 82]. The instrument is equipped with a cylindrically bent Ge (111) crystal, cut at 10°to the (111) Bragg planes and mounted at 15.5 m from the source. A line focus (0.56 x 4.5 mm2) is obtained at 1.7 m from the monochromator. The angular acceptance of the apparatus is 1.7 X 0.6 mrad2 and the wavelength can be tuned between 0.7 A and 3.5 A. The intensity measured at A = 1.54 A is 6 x 1010 ph/s (DCI at 1.72 GeV, 120 mA) while an intensity of 1.5 x ~çJtl ph/s has been recorded at A = 1.4 A and DCI operating at 1.72 GeV, 300 mA. The cylindrical curvature was obtained in an original way by displacing the free tip of a triangularly shaped germanium crystal with the basis secured to a steel plate. A constant curvature is obtained because the linear increase of the bending moment is compensated by the linear increase of the crystal section. If the force applied to the tip is W, the induced radius of curvature R is [22] R=Ebt3ll2hW

(6.1)

where E is the modulus of elasticity of the material, h is the height of the triangle, b its base and t is the thickness. This kind of curved crystal monochromator, whose principle is shown in fig. 21, is now widely used [18, 22, 23, 83]. It must be noted that some differences in the diffraction geometry exist between the singly bent triangular perfect crystal monochromator and other conventional experimental arrangements. In particular, in the former case the beam has a spectral bandwidth variable over a wide magnitude and an asymmetric crossfire. Moreover, a correlation exists between the direction of the incident ray and its energy. These characteristics have been discussed in refs. [78, 85] where the necessary modifications in the data elaboration procedures in oscillation camera works were derived. At the film data collection diffractometer SRS-1 of SRS, Daresbury, a 20 cm long singly bent triangular Ge( 111) monochromator is used; the crystal surface is cut at 10.4° with respect to (111) planes [18]. The optical system is completed by a bent Pt-coated fused quartz single segment mirror providing 1:1 vertical focusing. A focus 1.1 x 0.3 mm2 large is obtained at 2.52 m from the monochromator and 23.42 m from the source. The intensity measured at A = 1.49 A was 6.6 x 1010 ph/s with SRS at 2 GeV and 250 mA. The Ge monochromator can be substituted with a set of oblique cut Si(111)

28

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation source

Ge(I11)~ ~t0f

~

differential

screw

0 (crvsta[ rotation)

28 (frame rotation) Fig. 21. A cylindrical curvature is obtained by displacing the free tip of a triangularly shaped crystal with its basis secured to a steel plate. From ref. [81].

crystals (asymmetry angle from 6.75° to 14.75°) in order to obtain better wavelength resolutions (~A/A—3x 10~). At the Stanford SR Laboratory a high resolution X-ray scattering camera [86] has been installed at an 8 poles electromagnetic wiggler providing 2.3 X lOto ph/s mrad at A = 1.75 A (SPEAR at 3 GeV, 50 mA). The optical system consists of a toroidal mirror and of a double crystal Si(111) or Ge(111) monochromator operating in He atmosphere and oriented in such a way to reflect photons with A = 1.75 A wavelength. The first crystal has a thickness t~= 3.18cm and is water-cooled to prevent thermal distortion whereas no cooling is needed for the second thinner crystal (t2 = 0.64 cm). The toroidal mirror figure is obtained by bending to a radius R1 = 1.6 km a 58cm long Pt-coated (.-.-.1000 A) fused quartz slab ground to a fixed radius of curvature R, = 10cm (grazing angle 0 = 7.7 mrad; p = q =2 13dimensions m). The optical system has a horizontal angular acceptance and a focal is obtained. With SPEAR operating at 3 GeV, of 50 4.6 mAmrad an intensity of spot of 2 x 4 mm 10t3 ph/s (Ge(111)). The monochromator, observed energy resolution is L~E/E 3.2 x i0~and = 4 was for measured the Ge(111) and Si(111) respectively. At Q == 1.4498At the ~EIE momen1.3 x i0 tum resolution of the spectrometer is L~QlQ= 0.7 x iü~.A schematic representation of the instrument is shown in fig. 22. In ref. [87] the performances of monochromator and analysers used at CHESS for SR X-ray powder diffraction are discussed. A perfect Si(220) double crystal monochromator is used at a wavelength ranging from 1.07 A to 1.54 A. The intensity delivered at A = 1.54 A with the synchrotron operating at 5 GeV and 10 mA is of about 2 x i09 ph/s. Different analysers (Si, Ge, LiF, A1 203) are employed as “receiving slit” and to suppress the specimen fluorescence. The highest resolution is achieved with a Si(111) analyser but an integrated intensity 3.7 times bigger is obtained with LiF(200). 6.3. Monochromators for small angle X-ray scattering experiments The high brightness of SR is of great advantage also for small angle scattering (SAXS) experiments and a large number of SAXS instruments has been installed at the SR source around the world. A typical configuration for a SR SAXS camera consists of a mirror and a monochromator operating in orthogonal planes in order to obtain a double focusing of the primary beam. A camera of this kind

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

29

DETECTOR S5 SPECTROMETER SI. MONITOR

/53 MONOCHROMATOR 52

MIRROR 51

—.~

I I

aLniuuuuuu

_ II

I

26m

WIGGLER SOURCE

24rn

13m

Om

Fig. 22. High resolution X-ray scattering camera at the Stanford laboratory. The optical system consists of a toroidal mirror and of a double crystal monochromator. From ref. [86].

was installed at the DESY Laboratory in Hamburg [37, 88]. The monochromator was a cylindrically bent quartz (lOll) crystal with 7°oblique cut (or a curved Ge(l11) crystal with an asymmetry an,gle of 70) mounted at 30.6 m from the source point. The mirror was realized from two 20 X 2.6 x 0.5 cm3 pieces of fused quartz polished to a microroughness of 20 A, providing an angular acceptance of 0.85 X 0.05 mrad2 (horizontal X vertical) [20]. A focus with dimensions 0.25 x 1.0 mm2 was obtained at a distance from the monochromator ranging from 1.0 to 3.0 m. The intensity measured at A = 1.5 A was 7 X 108 ph/s with DESY at 7.2 GeV and 10 mA [3]. Of similar design were the SAXS camerae of the DORIS storage ring in Hamburg [22, 23]. In the instrument described in ref. [23], the monochromator is a 18 cm long cylindrically bent Ge( ill) perfect crystal with its external surface at 7°with respect to the Bragg plane. The monochromator-source distance is p = 22 m. The mirror consists of 8 pieces of 20 X 5 x 2 cm3 quartz segments having a reflectivity of about 50%. The focus dimensions are 0.5 x 0.3 mm2 and the focal distance q can be varied between 1.5 and 2.5 m. An intensity of about 1010 ph/s was measured at A = 1.25 A and DORIS at 3.7 GeV, 10 mA [20]. A second double focusing SAXS camera operating at DORIS [22]utilizes an identical optical system, but the monochromator is here triangularly shaped (symmetric Ge(111) or 7°cut Ge(lll)). The optical bench has been constructed to work at a fixed wavelength, namely A = 1.5 A. The vertical aperture is 0.3 mrad and the focus size is 1.1 x 0.7 mm2 (horizontal x vertical). With a wavelength resolution ~AlA— ~ the intensity is 5 x 1011 ph/s (DORIS at 4.6GeV, 20 mA). A focusing mirror-monochromator camera for time resolved small angle X-rays diffraction on biological substances has also been realized at the Photon Factory, Japan [83]. A cylindrically bent triangular shaped Si single crystal (5 cm base, 17 cm height, 0.1 cm thickness) selects out of the white beam a wavelength band ~A IA -= i0~centred at a wavelength A 0 tunable between 1 A and 2.2 A. The crystal surface is cut at 7.8°with respect to the (111) Bragg planes. Focusing in the direction is obtained means 7 cylindrically fused quartzthe segments 3) vertical positioned at a distance of by 13 m fromofthe source andbent of 4.5 m from mono(20 x 6 x 1.5 cm A focus of 2.63 x 1.55 mm2 dimension is obtained at 4.6 m from the monochromator. chromator crystal. The angular aperture of the optical system at A = 1.5 A is 1.12 x 0.36 mrad2 (h x v) and the expected intensity is of the order of 10t1~1012 ph/s, if the PF storage ring works at 2.5 GeV and 0.5 A. The spectrometer arrangement is shown in fig. 23. A different principle is adopted for the SAXS instrument of the NBS materials science beam lines at NSLS [89], where single diffraction from a matched pair of asymmetrically cut Si crystals is used. The

31)

R. (‘aciiiff0 ct a!., Monochrornators for X-ray synchrotron radiation

SOURCE

BENT CRYSTAL

t.~6~RROR.

Fig. 23. Focusing mirror-inonochroniator camera br time-resolved small angle X-ray diffraction at the Photon Fact~rv.From ret. [83].

sample is positioned between the two crystals and the SAXS scan is provided by rocking the second crystal. The advantages of using asymmetrically cut perfect crystals in this kind of spectrometer are described in refs. [90,91]. By changing the asymmetry angle a the intensity can be increased at the expense of the resolution. Moreover, by increasing a the signal to noise ratio rSN can he improved. For a = 0.5°. r~= 5 x i01 is obtained at NSLS. An intensity increase of a factor 15 could he obtained by using Ge crystals rather than silicon. A double monochromator system consisting of two equal single crystals, vertically displaced by l.22m. is employed at the X-15 slits camera at DORIS [84] for resonant SAXS experiments in the wavelength range 0.6 A < A <3.25 A. The first crystal is 24 m from the source, while the second one can he moved along a 3 m long optical bench. This allows to change the diffraction angle 20M from 18°to 60°.keeping constant the exit beam position. Four pairs of crystals are available, namely Ge( 111) and Si(220) perfect crystals with asymmetry angle a = 7°,Ge( 111) with a 5’ mosaic spread and a = 11°and graphite crystals with mosaic width 0.50. Each of the crystals can be put in the Bragg position by rotating a tetragonal prism on which they are mounted. The monochromator has an angular acceptance of 1 mrad X 0.2 mrad (h X v) and a resolution ~A/A of the order of 10~can be obtained. The intensity observed with DORIS at 4.6 GeV, 20 mA ranges between 109~10b0ph/s. 6.4. Monochromators for EXAFS and XANES measurements In an EXAFS (extended X-ray absorption fine structure) experiment the X-ray absorption coefficient of a material is measured as a function of the photons energy around the absorption edges of one of the elements contained in the sample. The energy interval investigated is usually of the order of 1 keV. An EXAFS monochromator must, then. offer the possibility to rapidly change the photons wavelength and it must provide a beam with an energy band width narrower than the lifetime broadening of the absorption edge. Energy resolutions of the order of 10 ~are usually requested for energy ranging from about 4 keV to about 30 keV (A from —3 A to —0.4 A). Moreover, in an EXAFS camera the total intensity at the sample position is of importance, and the whole available beam should be concentrated to a focus of a few mm2. The optical system of a conventional EXAFS camera consists of a double focusing mirror followed by a channel-cut perfect crystal monochromator (see fig. 24). The wavelength is tuned by rotating the grooved monochromator above its first wall. In this arrangement the incident and the diffracted beams are parallel, but their

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

31

ION CPIAt-IBER I ION CHbJIBER a

SAMPLE

STORED BEAM

_____________________________________________

I

HELIUM ATMOSPHERE

DOUBLE CRYSTAL MONOCHROMATOR (FLAT)

Ito V PREAMP

TOROIDAL. MIRROR

SPEAR STORAGE RING

2CHANNEL COUNTER

COMPUTER

Sfl PPIN6 MOTOR DRIVER

Fig. 24. Schematic representation of a point focusing optical system for EXAFS experiments. From ref. [29].

spatial separation h is wavelength dependent. In fact, if I is the distance between the two walls of the groove, one has h = 21 cos In the EXAFS spectrometer of SSRL [29],two flat perfect Ge(l11) crystals, rotated as a unit in the (n, n) parallel setting, were used instead of the monolithic channel-cut monochromator. In this way it was possible to detune the primary wavelength and reduce the harmonics. One-to-one focusing was provided by a toroidal Pt-coated mirror located midway between the source and the focal spot; the focus position is independent on the selected wavelength (this is not the case when focusing is achieved by means of a fixed curved crystal). The focussed beam bandwidth at 7 keV was found to be 8—12 eV [20] and an intensity of 1012 ph/s was measured with SPEAR operating at 3.7 GeV and 40 mA of current. The optical apparatus described above is the same as those employed at the SPEAR four-circle diffractometer described earlier. A focusing monochromator for EXAFS studies to be installed at NSLS has been described in ref. [92].Four Si crystals are arranged in two stages which can be rotated in opposite directions (see fig. 25), with the 2nd and 3rd crystals in dispersive arrangements. Focusing in the horizontal plane is achieved by bending the fourth crystal. The radius of curvature or —

Fig. 25. Focusing monochromator for EXAFS studies at NSLS. Focusing is achieved by bending the fourth crystal. From ref. [92].

32

R. Caciuffo et a!., Monochrornators for X-ray synchrotron radiation

the position of the curved crystal can be changed in order to maintain a fixed focal position as the wavelength is tuned. The energy resolution of this system is about i0~. A separated two perfect Si(111) crystals monochromator is used at the CHESS Laboratory for EXAFS and XANES measurements between about A = 5.7 A and A = 0.95 A [931. In order to obtain a wavelength-independent parallel displacement between the incident and the diffracted beams, the two crystals are rotated about the point of intersection between the plane of the second crystal and the normal to the first one (point A in fig. 26)~in this way, the beams are separated by h = 2d~,where d( is defined in fig. 26. The crystal B can be moved along AC but it is constrained to lie along the incident path, whereas the crystal D is allowed to move along AE. The same result can be obtained with a monolithic channel-cut monochromator with the groove walls suitably curved. In fact, at the Photon Factory Spieker et al. [94]succeeded in the realization of a fixed exit beam position monolithic Si monochromator. Their arrangement requires a very careful spatial positioning but it can simplify the EXAFS apparatus. By using (111) and (333) Bragg reflections, the energy scan can be performed between 5 keV and 76 keV. Recently a different approach to EXAFS measurements was proposed [951 and tested at SPEAR. A quasi-parallel SR beam is focused by a curved crystal to the sample position; as the Bragg angle varies across the crystal surface, a relation exists between the direction of the diffracted rays and their energy. By means of a position sensitive detector it is then possible to observe simultaneously the whole EXAFS spectrum (see fig. 27). 6.5. Monochromators for other experimental techniques Sparks et a!. [96] have used the SR of SPEAR for the study of fluorescence excitation of superheavy elements. Their spectrometer consisted of a cylindrically bent pyrolithic graphite monochromator with a curvature of 10 m’ and positioned at 17 m from the source. A focal spot 1 x 1 mm2 was obtained at 1.0 m from the monochromator. The angular acceptance of the spectrometer was 2 x 0.05 mrad2. With SPEAR operating at 3.5 GeV electron energy and 30 mA current, the observed intensity at A = 0.34 A was 8 X 10°~ ph/s. At the spectroscopy station of the NBS materials science beamlines at NSLS high brightness is achieved by focusing 6 mrad of radiation into a focal spot less than 0.3 mm wide [89]. The monochromator is an asymmetrically cut, cylindrically bent crystal providing focusing in the horizontal plane with a demagnification ratio between 5: 1 and 3: 1. For experiments between 5 and 10 keV (1.2 A ~ A ~ 2.5 A) a flat mirror is used after the monochromator crystal for harmonics suppression. The mirror can be substituted by a flat double crystal monochromator for operations in the high resolution mode.

Y

d 0

Fig. 26. A separated two crystals monochromator with fixed exit beam position. From ref. 93].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

33

Curved ctystal

~

Fig. 27. Schematic representation of an energy dispersive EXAFS apparatus tested at SPEAR. From ref. [95].

A Laue—Bragg monolithic monochromator with fixed exit beam height has been described in ref. [97]. It consists of an L-shaped Si crystal (see fig. 28) which can be rotated about an axis passing through the centre of the Laue crystal and lying in the plane of the Bragg surface. A constant displacement between the incident and the exit beam is so obtained. Using the (400) Bragg reflection, a wavelength as short as 0.4 A can be selected. A review of other monolithic devices used in SR optics can be found in ref. [98]. In ref. [991the characteristics of an X-ray spectrometer, suitable for low energy resolution (z~E= 1 eV) inelastic scattering experiments, are discussed. This instrument is installed at HASYLAB and a schematic drawing of it is shown in fig. 29. A double-crystal Si(400) monochromator selects photons with A0 = 2.08 A (E0 = 5.948 keV) out of the white SR beam. The energy distribution of the photons scattered by the sample at an angle 0~is analysed by a double focusing, spherically bent Si(333) crystal prepared by gluing a 0.38 mm thick silicon disc into a glass lens of bending radius R = 0.85 m. The analyser is oriented in nearly backscattering geometry (OBA = 85.81°to fit the incident energy E0) in order to match the monochromator energy resolution. Moreover, the double curvature produces an increase of the solid angle M? of scattered radiation which is sent by the analyser to the detector, thus approaching the optimization conditions requiring a ~Q value fitted to the momentum transfer resolution. INCIDENT 0

EXIT BEAM Fig. 28. Laue—Bragg monolithic monochromator with fixed exit beam position. From ref. [97].

58

R. Caciuffo Ct a!., Monochromators for X-ray synchrotron radiation

Y

1 ko-2k

~

~:::

7

_____

REGION

ytu~~2~~~~.\REGION

~B~-II

4 III

—

‘

—

(a)

.—

‘~uIf —

— (b)

Fig. 49. Schematic representation of the model for a curved crystaL (a) shows the different regions in the actual crystal and (h) shows how the model consisting of perfect crystals is built up.

=

—{y(O)

=

—{y(O)+ 1}/c.

—

i}/c

(8.37) (8.38)

The thickness of region II is given by A 1~=2/c.

(8.39)

As the condition —1
=

r~P11,

T~ =

t~P11

(8.40)

.

The second layer will act on the power T~as the first layer on the power P11, so that the diffracted power P2 and the transmitted power T2 are given by P.

=

r2T1

where r2 and

=

t2

r2t~P11 ,

T.

=

t2T~= t2t~P1

(8.41)

represent the reflectivity and the ratio of the transmitted to the incident intensity,

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

35

WEA PIEZOELECTRIC TRANSDUCER

1 cm

Fig. 31. Channel cut Si(111) monochromator used at CHESS for X-ray standing-wave excited fluorescence. The angular offset between the two linked grooves can be changed by elastically bending the thin link. From ref. [101].

crystals. The device presented in ref. [101] is a four-bounce channel cut Si(l11) monochromator consisting of two linked grooved crystals. The multiple reflection is here used to reduce the tails of the Darwin—Prins curve, as suggested in ref. [106]. An angular offset a of a few seconds of arc between the two grooves can be introduced by elastically bending the thin link (fig. 31). In this way, a transmission curve with an angular width We = a is obtained, where w~is the width of the Darwin—Prins curve for a two-bounce channel-cut crystal (fig. 32). As shown in ref. [101],for a given angular resolution the multiple reflection monochromator delivers the same intensity as an asymmetrically cut crystal. However, the channel-cut device offers the possibility to vary the angular collimation by changing the offset angle a, and it can be of smaller dimensions since the angle of incidence is usually large. In fig. 33 the theoretical transmission functions of the monochromator shown in fig. 31 are reported together with the rocking curve measured at CHESS using a single Si(lll) sample crystal in the parallel configuration and A = 2 A photons. It can be seen that for a = 0 the rocking curve is given by the convolution of two Darwin—Prins curves, while with increasing a, its shape becomes more and more similar to that of a single crystal Darwin—Prins curve. A similar device has been described by Hart Ct a!. [107]as a variable resolution monochromator and by Bonse et al. [108] as a monochromator for harmonic rejection (see paragraph 6.6). —

1.0

I

I

I

j,_i

I

I

~i

I I

0.8-

-

>I.-

I

0 i

i/

I

2 II

oc -10

•

~

I

-5

...~(

i

I

/r>~..

0 88B

-

I ‘r-.~i

5

I

10

(ARC-SEC)

Fig. 32. Theoretical reflectivity from the two-bounce channel cuts of the monochromator shown in fig. 31. The net transmission is represented by the cross-hatched region. From ref. [101].

36

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

~O2

~

0.8

LIlIlI

(b)

-

0,~-

/J

(d)

-

--

-

-

--

-

e-o~(ARC-SEC)

10

III~II

eo

0

(,ARC-SEC)

Fig. 33. Theoretical reflectivity (a) of the monochromator shown in fig. 31 for different values of the offset angle a (a = 0, 0.55w,, 0.8w,) compared with the single crystal Darwin—Prins curve (dashed line); (b)—(d) measured rocking curves for a Si(lll) sample in the parallel configuration for a = 1). 0.55w, and 0.80w,, respectively. From ref. [101).

6.6. Crystal monochromators for harmonic suppression Higher order and Laue spots often contaminate the monochromatic beam obtained from SR sources by crystal Bragg diffraction [109—112].The former occurs because the reflection conditions, for a given family of lattice planes, are simultaneously satisfied for both the fundamental wavelength A1 and for the harmonics A~= 2(d/n) sin ~ where d is the interplanar spacing and n is an integer number representing the harmonic order. The Laue spots are due to oblique lattice planes coming into the reflecting position and contributing to the signal with unwanted photons. Their effect can be reduced, in some cases, by means of diaphragms or by rotating the second crystal of a non-dispersive (+ H, H) arrangement about the normal to the principal lattice plane [108]. The integrated intensity delivered by a centrosymmetric weakly absorbing perfect crystal in the Bragg geometry (eq. (7.41)) is proportional to 2. (6.2) I, F~ e~A~/n (n) -DW(n) Both the real part of the crystal structure factor Fizr and the Debye—Waller factor e become smaller with increasing n, and hence the integrated intensity of the higher-order harmonics decrease more rapidly than n2 (see table 5). Due to this fact and to the spectral distribution of the SR beams, only the first two or three harmonics are usually of appreciable intensity. In some cases, a practically harmonic-free monochromatic beam can be obtained from Ge or Si(11l) reflection by choosing a fundamental wavelength somewhat smaller than the source critical wavelength A~.In fact, the second-order harmonic structure factor is virtually zero, in this case, and only a few percent of A/3 photons could be present in the direct beam. However, particular solutions must be in general adopted in order to suppress the harmonics contamination, especially in excitation and irradiation experiments or when photographic techniques are used. One possible choice is the use of a totally reflecting mirror as a high frequency filter, as was discussed in section 3. Moreover, several monochromators have been specially designed in order to provide a beam of the lowest harmonic contamination but still retaining —

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

37

Table 5 Integrated intensity I~for harmonic reflection in perfect Si crystal in symmetric Bragg geometry for if polarized incident beam; A 1 = 1.6 A. Although the highest intensity is delivered by the (111) reflection, the (331) or the (511) reflections are preferable when a harmonic free monochromatic beam must be obtained (from ref. [114]) h 1 3 4 5

k 1 3 4 5

I 1 3 4 5

n 1 3 4 5

l~II~0 1.0 0.0672 0.0384 0.0119

1.0 0.0672 0.0384 0.0119

2 4 6 8

2 4 6 8

0 0 0 0

1 2 3 4

1.0 0.1633 0.0405 0.0121

0.7769 0.1268 0.0314 0.0094

3 9 12

1 3 4

1 3 4

1 3 4

1.0 0.0314 0.0126

0.4379 0.0138 0.0055

4 12

0 0 0

0 0 0

1 2 3

1.0 0.1193 0.0219

0.4190 0.0611 0.0112

3 9 12

3 9 12

1 3 4

1 3 4

1.0 0.0184 0.0065

0.3196 0.0058 0.0021

4 8 12

2 4 6

2 4 6

1 2 3

1.0 0.0904 0.0143

0.4190 0.0379 0.0060

5 15

1 3 4

1 3 4

1 3 4

1.0 0.0122 0.0039

0.2806 0.0034 0.0011

8

20

jI)/j

high fundamental intensities [29, 83, 108, 109, 114, 122]. A review of the order-sorting methods based on crystal diffraction is given in ref. [108]. The working principles of these devices rest on the differences between the diffraction patterns of the fundamental and those of the harmonic on the dependence ofcc n2 the (eq. 2components, (7.24)) and inof particular the deviation fromorder Bragg’s law reflection range ~ IF~le’~’~In (eq. (7.38)). As an example, the acceptance R~”~(O 0) and the reflection R~(OH)curves for asymmetric Si(2n, 2n, 0) reflections calculated at fundamental wavelength A1 = 1.6ofA incidence for an asymmetry factor 0H are the angles and emergence, brespectively, = 0.4 [114] a re shown in fig. 34. Here, 00 and related by (0~ = b(0 0 °~). The acceptance angular range has a width w~= ~~°~i’/~ and it is centred at °B + 30~,where [113, 123]

aor

—

0~)

—

/

2

\

1\

(K3

~,.

bi~

The width of the reflection curve is w

=

w ~‘~‘/~ and its centre falls at

On + 6O~,with

38

R. Cactuffo et a!., Monochromators for X-ray synchrotron radiation

0

4.0

8.0

12.0

16.0

20.0

24.0

20.0

24.0

-8~ (SEC OFARC)

0

4.0

12.0 16.0 8e (SEC OFARC)

8.0 -

Fig. 34. Acceptance R~(~,)and reflection R~(O,,)curves for asymmetric Si(2n. 2n, 0) reflections, calculated at a fundamental wavelength A 1 = 1.6 A and for an asymmetry factor 6 = 0.4 (a polarization). From ref. [114].

~O~~I)

(6.4)

If two identical crystals are used in the nondispersive (+, —) Bragg—Bragg arrangement, order-sorting can be achieved by detuning one crystal with respect to the other [109, 115—117, 124]. The intensity of the harmonic components decreases more rapidly than that of the fundamental when the angle y between the two crystals is increased. In particular, if wW/2> y > the overlap between the diffraction patterns of the two crystals is removed for all but the fundamental component and the harmonic content of the monochromatic beam is minimized [108]. For instance, an order-sorting monochromator routinely running at SRS [121] makes use of two Si (220) 11~ crystals and it operates at a = 17.4” and w~= 3.0”). fundamental wavelength A1 = of 3.1about A (for8” Si(220) at A~content = 3.1 A is one has w while the intensity of the With a relative angular misfit the harmonic minimized, fundamental is about half that delivered with y = 0. Better efficiency can be obtained by using multiple reflection from grooved crystals in order to reduce the tails of the intrinsic diffraction curves [106]. This fact has been exploited by Hart and Rodrigues [117] who designed and tested a channel-cut Si monochromator in which variable misfit angles between the walls of the groove can be produced. Harmonic rejection can also be obtained with Laue—Bragg (+, arrangements [63, 124]. In Laue geometry, in fact, M~”~ = 0 for all n and both the fundamental and all the harmonics are reflected at the same angle. As a consequence, only for the fundamental component will the diffraction patterns of the two crystals overlap if a misfit angle y = ~ is introduced. An advantage of the Laue—Bragg configuration is that it is possible to suppress the fundamental wavelength whereas the diffracted intensities of higher-order harmonics are maximized. This fact can be useful when it is necessary to work with short wavelengths [97]. Another method to suppress higher-order radiation was proposed by Bonse et al. [63] at the DESY synchrotron. A Si(220) and a Ge(220) were used in the slightly dispersive (+H, —H’) setting, as it is W(2)

—)

R. Caciuffo et al, Monochromators for X-ray synchrotron radiation

39

shown in fig. 35. It can be demonstrated [108]that harmonics rejection can be achieved in this case only if the divergence of the incident beam is smaller than the quantity ~(d1 sin A1 — d2 sin A2 \ 1 (2) (2) o-0tan ~dtcosA1—d2cosA2) ~ +W2 )

(6.5)

with _~O~1) 2~ —sin~(A

A~=~O~

1/2d1)

(6.6)

where i = (1, 2) refers to the first and to the second crystal, respectively, and d. is the lattice spacing of the reflecting planes. The results obtained by Bonse et al. [63] are also shown in fig. 35; with a convenient orientation of the Ge crystal, the reflected intensity for the fundamental frequency (A1 2.5 A) is about 300 times bigger than that of the second harmonic. A double grooved monochromator with tunable harmonic rejection was recently presented by Bonse et al. [108]. This monochromator was fashioned from a monolithic silicon single crystal and it is composed of two grooves connected by a thin link (fig. 36). The primary beam is reflected three times from the first groove and twice from the second one. The harmonic suppression is obtained by elastically bending the link in order to change the relative orientation between the two grooves. The main improvement with respect to the monochromator of Hart and Rodrigues [117]is that the angle of incidence in each groove is the same for all the reflections and this allows optimization of the value of the angular offset ‘y between the two channel-cut crystals. The monochromator was tested at wavelengths ranging from 2 A to 2.9 A and a good monochromaticy was obtained with a transmission T= i”~(y)IP’~(0) for the fundamental frequency of the order of 60%. The device built by Hashizume et a!. [114, 122] is based on a slightly different principle. A

Ge-220 io2

101

..~ I 100

~n ,

10

I

20

30 ENERGY

I

40

(keV)

.

•

• I

50

—~

Fig. 35. Slightly dispersive arrangement for higher-order radiation suppression. The harmonics are suppressed by a small angular rotation of the Ge(220) crystal, as shown by the spectrum measured by the solid state detector (SSD). From ref. [63].

40

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation 0.8mm

Fig. 36. Double grooved monochromator with tunable harmonic rejection. The relative Orientation between the two grooves can hc changed by elastically bending the link L. in order to obtain the harmonic suppression. The beam is reflected three times from the groove a and twice from the groove b. From ref. [108].

channel-cut monochromator is fashioned from a perfect Si crystal, with the groove walls cut at different angles a1 and a11 to the reflecting Bragg plane (fig. 37). If the corresponding asymmetry factors b1 and b11 are such that the condition ~

DW(n) <~1 h~F~r~

1

-

<

CIF(t~ h~r~

DW(t

(6.7)

is fulfilled, the acceptance curve of crystal II overlaps the reflection curves of crystal I only on the tails for all the harmonics but the fundamental, for which a partial peak-to-peak overlap is still preserved [114, 118—120]. This fact is illustrated in fig. 38, which shows the diffraction curves calculated by Hashizume [114] at A = 1.6A for Si(111) reflections with cut angles a1 = 0°and ait = 8.1°. Experimental tests were performed on a grooved Si(111) monochromator with asymmetry angles a1 = 0°and a~= 7°;a harmonic contamination of about 3.3% was obtained at fundamental wavelengths ranging from 1.2 A to 1.6 A, with a transmission for the fundamental frequency of about 50% (ratio between the intensity delivered by the non-parallel groove and that reflected by a standard channel-cut monochromator) [114]. Harmonic contaminations down to 0.1% could be reached by using Si(331) or Si(511) reflections. An inconvenience presented by this device is that harmonic rejection can be obtained only for a restricted range of fundamental wavelength values. In fact, the central term in eq. (6.7) depends on the wavelength through the Bragg angle O~,while the other terms in the inequality are almost A independent. As a consequence, the conditions expressed by eq. (6.7) cannot longer be fulfilled if the fundamental wavelength is tuned over a wide interval.

CRYSTAL I

X-RAY CRYSTAL II Fig. 37. Monolithic Si crystal monochromator for harmonic suppression. The groove walls are cut at different angles a and a, to the reflecting Bragg planes. From ref. [114].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

~

0

0

0

4.0

4.0

8.0 8H1

-

0B

6.0 0011 ~0B

4.0

0

I 8.0

0H11

12.0 16.0 (SEC OFARC)

20.0

24.0

12.0

20.0

24.0

I 20.0

I 24.0

16.0

(SECOFARC)

I 12.0 -

41

I

I

I 16.0 I

0B (SECOFARC)

Fig. 38, Si(111) harmonic reflection curves for an asymmetrically grooved crystal with cut angles a~ = 00 and a 0 = 8.10 (A = 1.6 A; ir polarization). R~is the reflectivity of crystal I (fig. 37), R~is the acceptance of crystal II and R~,is the reflectivity of the whole system. From ref. [1141.

7. Dynamical theory of X-ray diffraction 7.1. Introduction The conventional kinematical theory of X-ray diffraction in single crystals only gives reliable predictions if the interaction between the incident wave and the wavefields excited inside the crystal is very weak. This condition is readily met in the so-called mosaic crystals, i.e. crystals composed of small perfect blocks slightly misoriented around a mean direction and diffracting the incident wave independently [125]. If the spatial dimensions of the mosaic blocks are small with respect to the extinction length (a parameter depending on both the wavelength and the crystallographic structure factor) the extinction of the primary beam can be neglected and the amplitude of the incident wave assumes the same value on each scattering centre. This is the basic assumption of the kinematical theory of X-ray diffraction, and this theory can be used to evaluate the intensity distribution of the waves diffracted by heavily distorted crystals. For instance, in the case of an idea! mosaic crystal the integral reflecting power will be given by 3 ~C’2 Fh12 e~~’ (7.1) r~A 21LV2 sin 2O~~ where r 0 = 2.818

X

10

15

m is the classical electron radius, A is the wavelength of the incident radiation,

42

R. Caciuffo el a!.. Monochrornators for X-ray synchrotron radiation

is the linear absorption coefficient, V is the volume of the unit cell, C is the polarization factor (C = or C = cos2OB~),On is the Bragg angle, F, is the crystal structure factor and et~ is the Debye—Waller factor. The situation is different when the diffraction takes place in a large perfect crystal where multiple reflections occur on the lattice planes. Thus, the interaction between the incident and the reflected waves must be considered in order to account for the observed diffraction phenomena. This problem is faced by the so-called Dynamical Theory of X-ray diffraction, originally proposed by Darwin [126], Ewald [127—130]and Laue [131], The theory was later extended by Zachariasen [132].James [133, 134] and Kato [135] and numerous review papers have been published on this subject [136—143].Here, only the fundamental principles of the Dynamical Theory are reviewed together with some useful formulae for understanding the behaviour of perfect crystal monochromators. j.t

7.2. Fundamental equations of the Dynamical Theory The interaction between an electromagnetic wave and a medium characterized by a triply periodic electron density p(r) can be described in terms of the electric susceptibility, which is given by r A2

x=— —~--—p(r).

(7.2)

Thus, the electric susceptibility is also a triply periodic function of the position vector r and it can be expanded in Fourier series

x

=

Xh

(7.3)

exp(2irih . r)

with rA2 =

— —~~-F,,

rA2 =

—

~

—~-~~-

,f~exp(2irih .

r,)

(7.4)

where h is a reciprocal lattice vector, Fh is the crystallographic structure factor and f 1 is the form factor of the jth atom localized by the position vector r1. In general, the electric susceptibility is a complex quantity x,, = Xhr + ix,,1 whose imaginary part is related to the linear absorption coefficient in the forward direction through the relation

Xoj

=

/20!2ITK.

(7.5)

In the case of X-rays, the electric susceptibility assumes very small values and the index of refraction is thus given by n=1

+~I2.

(7.6)

If D = e0( 1 + ~)E is the electric displacement field of a monochromatic electromagnetic wave of frequency i.’ and wavenumber K, the propagation equation has the form 2K2D = 0. (7.7) z~D+ curl curl ~D + 4ir

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

43

Its general solution can be expressed as a linear combination of Bloch waves D1 (7.8)

D—>JA1D1 where

(7.9)

DJ=~DhJexp(—21riKhJ~r) and KOJ=KhJ—h.

(7.10)

By inserting eqs. (7.3) and (7.9) in eq. (7.7) an infinite set of linear homogeneous equations is obtained: K~. DhJ

—

2

KhJ-K

2 ~

Xh-h”~h’hf=

0

(7.11)

h’

where DhhJ is the projection of DhI on a plane normal to KhJ. A given wavefield thus contains an infinite number of waves DhJ. However, most often only the displacements D01 and D~1have significant amplitude (two waves approximation). Hence, a given wavefield D1 contains a wave propagating along K01 and a wave propagating along K~1= K01 + H, that is Bragg’s diffraction occurs: D1

=

D01 exp(—2iriK01 r) + D~1exp{—2lTi(K01 + H). r}.

(7.12)

With this approximation, eq. (7.11) reduces to a system of linear homogeneous equations. After projection on the plane K01, KHJ and on the normal to this plane, one obtains 2(1 + ~o)]Do 2x~CD I [K~1— K —1K 111 = 0 2XHCDOI + [K~ 2(1 + x L.—K 1— K 0)] DHJ = 0

(7.13)

where C = 1 (C = cos 20) for a polarization normal (parallel) to the plane K0, KH. Equations (7.13) have a non-trivial solution only if the secular equation is satisfied: 2(1 + xo)] [K~ 2(1 + x 4C2XHX~. [K~1— K 1— K 0)] = K Thus, for any given value K of the wavevector outside the crystal, there are two solutions D

(7.14)

1 and D2 of the propagation equation (7.7), corresponding to the two solutions K01, KHI and K02, KH2 of eq. (7.14). The ensemble of wavevectors that can propagate inside the crystal defines a surface of the reciprocal space (the dispersion surface) whose equation is given by eq. (7.14). In the case of ~ = XH = 0, the dispersion surface is represented by a couple of spheres of radius R = K and centred at the points 0 and H of the reciprocal space; these are the Ewald spheres of the kinematical theory. The intersection between the dispersion surface and the plane of incidence (K0, KH), is represented in fig. 39. The

44

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

~

Fig. 39. Intersection between the dispersion surface and the plane of incidence in the two waves approximation. 0 and L are the Lorentz point and the Laue point defined in the text.

intersection of two spheres having a radius R = nK = (1 + ~,,/2)K and centre in 0 and H, respectively, defines the Lorentz point Q. The intersection between the Ewald spheres is the Laue point L. Any given point A of the dispersion surface (tie-point) is associated with a wavefield represented by the wavevectors K0 = OA (refracted wave) and K~= HA (reflected wave). It should be noted that the scale of fig. 39 is not correct; the points 0 and H should be at several km from 0. The region of the dispersion surface close to the point Q is shown on an enlarged scale in fig. 40 S

\\

\ \ \ \ \ \ \ \ \\\

,~

SUR

S

Sfi

n

-.

AL. .....

I

/

~~.Q2e8

..

\~

///I\

/1

S0

/

/~ /,~

~

\~ \~

0 Fig. 40. The dispersion surface close to the Laue point L. The circles of radius nK have been approximated by their tangents T and are the tangents in L to the circles of radius K centred in 0 and H, respectively. SS’ represents the crystal surface.

TH.

S,, and SH

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

45

where the circles of radius nK have been approximated by their tangents T0 and TH. It is possible to show that, for tie-points A close to 0, the dispersion equation (7.14) can be written in the form [134]: 122 ~0~H

=

~K C

(7.15)

XHXH

where ~ and ~H are the distances of a tie-point A from T0 and TH, respectively. Where eq. (7.15) holds, the dispersion surface is thus represented by a hyperbola having T0 and TH as asymptotes. The dispersion surface is particularly useful in describing the propagation of the waves inside the crystal. For instance, it is possible to determine the excited wavefields by imposing the continuity condition to the tangential component of the wavevectors at the entrance surface (SS’in fig. 40). In fact, let I be the point defined on the Ewald sphere S0 by the incident wavevector K, pointing towards 0. The excited wavefields are represented by the points of intersection between the two branches of the hyperbola and the normal to the entrance surface passing through I. If the entrance surface is oriented in such a way that no intersection points are obtained, there is no real solution of the propagation equation and total reflection occurs (fig. 41). The direction of propagation of a wavefield is, moreover, determined by the normal to the dispersion surface passing through the excited tie-point. As a consequence, the wavefields corresponding to MM’ in fig. 40 propagate in the same direction. They are coherent but their wavenumber is slightly different. In this situation an interference phenomenon occurs, characterized by a periodic exchange of energy (Pendellosung). The ratio between the amplitudes of the two waves that constitute the two wavefields is given by

DHIIDOJ

=

V~0jIS~HJ.

(7.16)

In order to obtain the absolute value of the two amplitudes it is necessary to impose at the crystal surfaces the boundary conditions expressing the continuity of the tangential component of the electric field and of the normal component of the electric displacement. Two cases must be distinguished; in the ,w,,,,,,

L

/

/

k

k

(.4)5

O Fig. 41. Total reflection in Bragg geometry. From ref.

[1441.

46

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

Bragg (or reflection) geometry the diffracted wave emerges from the same face illuminated by the incident beam (fig. 42a). In the Laue (or transmission) geometry, the diffracted wave leaves the crystal from the rear surface (fig. 42b). Moreover, the crystal is called asymmetric if the angle a between the diffracting plane and the entrance surface is different from zero, otherwise the term “symmetric reflection” is used. It must be noted that in the Bragg case the two excited tie-points lie on the same branch of the hyperbola and that the condition corresponding to total reflection can be verified for a finite ensemble of incident wavevectors. The angular width of total reflection is thus related to the diameter of the hyperbola (fig. 41). A detailed discussion on the boundary conditions in both the Laue and the Bragg case can be found, e.g., in refs. [132, 137]. Once the amplitudes are obtained, the intensity distributions can be calculated. For a plane parallel non-absorbing crystal plate, in the Laue geometry, the reflecting powers are thus given by [132]: R — sin(irtlfl,,) ~i + y2 1+y2 R,,

=

1

—

(7.17)

R, 1

where

t

is the crystal thickness, A1, is the “extinction length” defined as

A,= ~

(7.18)

y1) and y1~being the direction cosines of incident and diffracted beams, with respect to the unit inward crystal surface normal. The deviation parameter y is a function of the difference ~0 between the angle given by the Bragg law of the geometrical theory O~and the angle of incidence 0: 20B —

(~I’YH)

+ 2X

~0 sin

0(’ — Yo’YH)

(7.19) The diffraction pattern for the non-absorbing Laue case is shown in fig. 43. The rapid oscillations with respect to y (Pendellösung) are characteristics of the diffraction from perfect crystals. However, the (b)

,//~± \,,~I/IIBragg planes

(a)

~

~

/~ bO

b’0

Fig. 42. Real space representation of diffraction in Bragg geometry (a) and in Laue geometry (b). From ref. ]145].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

RH

47

J.8

0.6

Fig. 43. Diffraction pattern for a non-absorbing Laue case. From ref. [145].

experimental resolution is in general too poor to allow the observation of such an angular dependence and only the average intensity RH—

~I(1+y2)

(7.20)

can be measured. Equation (7.20) shows that the average diffracted intensity in the Laue case is represented by a Lorentzian curve having a full width at half maximum Ay =2 and the maximum at y = 0 corresponding to 0

—O +7

max

B

( 721 . )

~11xo~7o~i) 2sin2On

Hence, only in the symmetric case (Yo = YH = cos On) the maximum of the rocking curve RH(y) coincides with the Bragg angle On. The deviation from the Bragg law is positive or negative depending on the sign of the ratio YH’Yo~ In the Bragg geometry, the diffraction pattern for a non-absorbing crystal is given by 1 y +(y —1)cot [(irtIA0)~y 2

=

2

2

1

2

y>

—1] ;

(7.22)

y~1.

Also in this case the diffraction pattern is symmetrical with respect to y 0. Moreover, as the ratio 70’YH in Bragg geometry is always negative, 2 the mean onediffraction has Omax>pattern On. If eq.is obtained: (7.22) is integrated with respect to i,-t1A0 over a range ~(irtIA0)> IT!

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

48

R~(y)°°° I

;

y~1 (7.23) y~>1.

The diffraction pattern for a non-absorbing perfect crystal in Bragg geometry (Darwin curve), is shown in fig. 44. In the angular scale, the width of the range of total reflection in the symmetric case is given by 2 rA2 C Fhre_DW. (1)5 = sin 2O~ (7.24) The integrated reflecting power in the y-scale is obtained as I~=

J

RH(y) dy

=

~ tanh(ITtIA

0).

(7.25)

If the condition tIA0 ~ 1 is satisfied, the integrated intensity tends towards a constant value ‘H = ir. In the opposite case of tin0 1, ‘H is proportional to the crystal thickness [1321: 2tIA = ~r 0= QtIy0 (7.26) °~

where Q is the crystallographic factor of the kinematical theory. Equation (7.26) shows that in the limiting case of tiA0 1 the Dynamical Theory yields the same results as the geometrical one. The deviation from the proportionality to the crystal thickness is due to the “primary extinction”, a consequence of the intensity reduction of the incident wave along its path inside the crystal. In general, the absorption cannot be neglected and both x and the parameter y must be considered as complex quantities. In the following, only some results valid in Bragg geometry are reported, because this is the most interesting case when dealing with X-ray monochromators. °~

I 0.8

_~_

Fig. 44. Diffraction pattern for a non-absorbing crystal in Bragg geometry (Darwin curve). From ref. [145).

R. Caciuffo et a!., Monochrotnators for X-ray synchrotron radiation

49

The formulae quoted below refer to the more general case of asymmetric diffraction. The geometry in the real space is shown in fig. 45. The diffracting planes and the entrance crystal surface form an angle a, which is considered positive when the angle between the incidence direction and the crystal surface is smaller than the Bragg angle On. An asymmetry factor is defined by b = sin(OB

—

a) isin(On + a).

(7.27)

The diffraction curve is given by [132, 146] RH(y)=L—VL2—1

(7.28)

where L= 2X~r2 XHr+XHi

(7.29)

[y2±g2+2_g2_~+~)2±4(gy_~)2]

XHr

XHr

and (see eq. (7.19)) 1—b ~b(O0—On)sin208 2CVj~J XHr C\1jNXHr —

x0~

—

(7.30) 731

From eq. (7.30) one obtains the angle O~between the direction of incidence and the diffracting planes: 0—

+~~+~

X~r

2sin2On

+

XHr

20B ~ 0H between the diffracted beam and the Bragg planes can also be obtained by replacing b by The lib angle in eq. (7.30) 0 —

~

bi

\/~

sin

Fig. 45. Real space representation of asymmetric diffraction in Bragg geometry from a perfect crystal (b < 1). From ref. 120].

50

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

(7.33) From eqs. (7.32) and (7.33) it is possible to see that in the asymmetric case (b distribution R~(00)is different from R~(0~) (fig. 46).

1) the intensity

In fact, the diffracted intensity has an appreciable value in the range —1
(7.34)

=

and to an angular range of emergence (7.35) where w~is the width of the Darwin curve, given by eq. (7.24). Moreover, the centre of the diffraction curve (y = 0) is displaced from the Bragg angle by ~0,,= ~(1+1ib)~05

(7.36)

in the case of the acceptance curve and by =

1(1

+

b) SO.,

(7.37) R(0~)

0~i:__~0;530

80- O~

R(e4)

0.4

J~_t.IiH—l\

\~

0.2

________ -5

A9~

0

5

5\\

~

L

10

15

20

25

__~

30

9N~9B

(sec of arc) Fig. 46. Reflection curves for Si(ill) at 1.6 A. R(O,,) is the reflectivity as a function of the incidence angle 9~while R(051) represents the intensity reflected at a reflection angle 9~for a plane wave incident at 8~.(Solid lines for b = 0.4; broken lines for b = 1 where R(O,) = R(9,,).) From ref. [20].

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

in the case of the emergence curve. Here, ~ symmetrical diffraction

=

sin2On

=

sin 2On

51

is the deviation from the Bragg law in the case of

~W ‘~0r~

(7.38)

From eqs. (7.34) and (7.35) one can see that by asymmetric diffraction from perfect crystals it is possible to change the angular divergence of a beam [113, 147]; in fact it is =

bw0.

(7.39)

Moreover, if S0 and SH are the spatial cross-section of the incident and diffracted beams, the Liouville’s theorem requires that S0 ~ = SH WH and hence SH

=

S0!b.

(7.40)

As a consequence, if b < 1 the range of total reflection for the emergent beam is b times smaller than that of the incident beam, while its spatial cross-section is 1/b times greater (Fankuchen effect). Finally, we report an expression that can be used to evaluate the integral reflecting power for a weakly absorbing crystal in Bragg reflection [134, 146] IHj —

R

‘O’dO HkO)

— tI~

1

8 r0ITV C Fhre DW 3sin2On

741

8. X-ray diffraction by deformed crystals 8.1. Introduction As discussed in previous paragraphs, curved crystals are commonly used as focusing X-ray monochromators in several laboratories. In spite of this, only the problems related to the focusing geometry are usually theoretically treated while no attempts are made to evaluate the monochromator diffraction characteristics and, hence, the intensity of the diffracted beam via the reflectivity and the associated wavelength bandwidth. This is probably due to the complexity of the rigorous theoretical treatments [148—152]that are quite cumbersome to handle in practical cases. On the other hand, simple physical models have been elaborated that are more easily applied in evaluating the performances of curved crystal monochromators, both in Laue [153—155]and in Bragg geometry [156]. In this section, the fundamental results of the dynamical diffraction theory of deformed crystals developed by Taupin [150] are resumed. Then, some models useful to predict the reflectivity of elastically bent perfect crystals in Laue geometry are discussed and a model developed by the authors to evaluate the X-ray diffraction characteristics of curved monochromators in Bragg geometry is described. Finally, the use of this model for the interpretation of experimental data and in evaluating the performances of bent crystals used in real cases is illustrated by some examples.

52

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

8.2. Fundamentals of Taupin theory The physical basis of the theory developed by Taupin to describe X-ray diffraction by deformed crystals will be considered, together with the more important approximations and final results. The adopted hypothesis on the type of deformation of the crystal leads to a complete freedom in the shape of the deformation itself, but poses some quantitative conditions as to the magnitude of the deformation at each point and as to the corresponding spatial variation. If w is the vector describing the elastic displacement of the crystal atoms, one supposes that the following conditions are satisfied: ,9w

(8.1)

which implies that the deformations are much smaller than one, and d2w

(8.2)

A2~A~,

ax 1 ax1

ax,

which implies that the variations of the deformations from one atom to the other are small as compared to the deformations themselves. The form given in the vacuum to the incident wave, supposed quite arbitrary, is 2IT t~,,(r))] (8.3) D(r) = D0(r) exp[i(w0t — where r is a point in the actual space. ~ 0(r)and D,,(r) are supposed to be real but space dependent, whereas the frequency w,~remains constant also inside the crystal. For a plane wave one has (8.4) with K constant. Actually it is supposed to impinge on the crystal a wave which is almost plane, i.e. with a curvature radius R much bigger than the X-ray wavelength A in the vacuum. Inside the crystal the same expression (8.3) is given to the wave by maintaining the same function 40(r) for the phase and by introducing in D11(r) the perturbations originated in the wave by the crystalline medium. As a consequence D,1(r) becomes in general a complex quantity. For the reflected wave an expression analogous to that of Zachariasen [132] is used (see eq. (7.9)), i.e. D(r)

=

~ D~(r)exp[i(w0t

—

21r4H(r))]

(8.5)

where 4H(r)=40(r)+nH(r)

(8.6)

with nH(r) an integer index associated with the lattice surfaces forming the plane family H = (h, k, 1),

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

53

where h, k, 1 are the Miller indices. The value of nH(r) increases in the direction of the reciprocal lattice vector H, defined for a crystal supposed not deformed, and changes its value of one unit by passing from a plane to the next one. As a consequence one obtains (8.7)

H=gradn~(r).

Although the deformed crystal is no longer periodic with respect to r, it can be supposed to be so with respect to ~H in each family of the lattice planes. One takes the following form for the dielectric constant e inside the crystal: r~0(1+ x)

=

(8.8)

where ~ is the dielectric constant of the vacuum and

x

=

XH

exp(—2lrinH)

(8.9)

where, as in the theory for perfect crystals (eq. (7.4)),

=

— ~

(8.10)

F0

xu=—~~-FH.

(8.11)

By inserting eq. (8.5) in the propagation equation (7.7), an infinite set of linear homogeneous equations is obtained. However, for the case when the crystal is supposed to be near the Bragg conditions for a reciprocal lattice point, so that only the incident wave and one reflected wave exist, and assuming that the crystal has a centre of symmetry (XH x11) a system of only two equations can be considered: 2 i— XODH+XHDOaHDH~~DHL14H AÔD iA A aD 1

~-~-

2 0 =xoD(,+xHDH—1. i— A

(8.12) D 04q~

where XH and X0 are defined by rr~X0S0+XHSH. Here, S0 and S~are unit vectors in the incident and reflected directions, a11 is defined by 2(K2 aH = A 11 + 2K~H) = 2 sin 2On(On — O~)

(8.13)

(8.14)

and it represents a quantity measuring the deviation from the Bragg conditions at each point of the crystal.

54

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

Taupin showed that the two Laplacians in eqs. (8.12) are usually negligible as compared to the other terms if the curvature radius of the incident wave is larger than the absorption distance, a situation which is always obtained in the case of X-rays. A particular case is now considered, i.e. a cylindrical curvature of the crystal with the plane of incidence perpendicular to the cylinder generatrix. Moreover, the polarization factor is assumed to be C = I and the impinging wave is of the type called by Taupin “onde propre”, i.e. a wave such as a1~ depends only on a spatial variable, the depth below the surface of the crystal. Before writing the form assumed by the system (8.12) for this particular case, some definitions are introduced. If n is a unit vector perpendicular to the surface of the crystal, one writes: ~AA11i1T;~’~~

(8.15)

=

—AA,/7rX~1

(8.16)

=

XHr + ‘XHi

X11

Xi-i

=

=

XHr(’

+

iK*)

(8.18)

.

=

(8.17)

(8.19) b

(8.20)

=

yA A=—

H

+yA H

1)

°—depth

Xo~ 2V~~ —

g=

i—i~— absorptton

1—b X11r y - 2V~ ~r

b

Q(A) -

Q,(A)

aFI

-

1

-

\‘T~J

(8.21) (8.22) 23 (8.)

D(A) D0(A)’

By using these definitions the system (8.12), where the Laplacians are neglected, is reduced, by subtraction, to 2(1+iK*)_2X(y+ig)+(l+iK*). (8.25) i~ X It must be noted that X is in general complex and that it is related to the reflectivity of the crystal by RH

=

X(0)2

(8.26)

and y to A by y = y(0) + cA

(8.27)

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

55

with y(O) being the value of y when A = 0 and c defined by c=dy/dA

(8.28)

is a dimensionless quantity related to the radius of curvature R~. For instance, in the case of a cylindrically curved crystal it is 2(b 1) [1+ by~(1+ K*)]. c = IT7A3FHbRC 0V

(8.29)

The quantity A is related to the thickness t of the crystal by

ITCXur

A.

(8.30)

The calculation of the diffraction pattern for a curved crystal consists in evaluating the reflectivity R as a function of y(O). Some examples will be given below in connection with a check of validity of the model elaborated for the Bragg reflection case. 8.3. Reflectivity of elastically bentperfect crystals in Laue geometry The intensity of X-rays reflected in Laue geometry by perfect crystals can be increased upon bending [157]provided that the induced curvatures are not too small. This effect is quantitatively explained by a detailed dynamical diffraction theory in the case of weak bending [141], while several models have been elaborated in order to account for the experimental results in the medium and strong bending regions [153—155]. The relation between the integrated reflectivity I for a given family of crystallographic planes and the curvature p provided by the dynamical diffraction theory shows that, after an initial interval (p 1) in which I is almost constant, an increase of reflectivity with increasing p takes place, with a slope S = dI/dp tending towards a constant value ‘~

—

— exp(—p~ 0t0!y0) /L0(1—b)

exp(—tO/yH)

a

8 31

where t0 is the thickness of the crystal slab and p~is the linear absorption coefficient. Moreover, if the reflection occurs from the concave side of the plane (positive curvature) a higher intensity is obtained. The general expression for the integrated reflectivity of a cylindrically bent perfect crystal for the wavefield j = 1, 2, is given by [141, 153]

=f ~

jJ.c

JJ~C

Bj~cA1dx

=z~o0v~zexp{_!±PLci(I 2 Yo

—

711

56

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

xli11

2(1—~)+1/z

-I ~(l — 2)(1 x exp~± ii~~ t, 1 log

—

~2

p~

+ z~)

p

Z\/~~i~

Z

t 7H)

1~t)~(Y)

2

1

(8.32)

where I~’is the reflected intensity of a single ray with polarization mode c (r or IT) passing through the position parameter x at the exit surface of the crystal plate, J~,is the intensity of the incident beam, Bjc and A describe the anomalous and the normal absorption respectively, p~is the linear anomalous absorption coeffIcient, Z~O,,= cXH/sin 2O~,z = p t,, sin 2O0icx11/~3~i1 and ~ = xiz. The sign + or — in eq. (8.32) corresponds to] = 1 or] = 2, respectively. It must be noted that the parameter p appearing in eq. (8.32) is the effective curvature combining the geometrical curvature with the effects of the interplanar spacing changes along the beam path [155]. The experimental observations support the results of eq. (8.32) only for small curvature values. Following the suggestion of Balibar et al. [154], Kalman et a!. [1551 proposed an expression for the integrated reflectivity valid in the intermediate and strong bending case =

I

—

exp(—GIp) p

i

A

(8.33)

where G is determined under the hypothesis that the ideal mosaic crystal reflectivity is exponentially approached as p —~ ~. Another model which takes into account the effects due to the anomalous transmission and to the elastic anisotropy of the crystal has been elaborated by Kalman and Weissmann [153]. This model provides a reflectivity versus curvature relation which reproduces the results of eq. (8.32) for weak curvatures and shows an asymptotic behaviour towards the reflectivity Mc of an ideal mosaic crystal for strong degrees of bending. For any given effective curvature p. two values p1 and p2 close to p are chosen so that p.

JI.C(~)

+

J

1);

(i

=

1,2).

(8.34)

Hence, the parameters Sc(p) = [JC(p1) — JC(p2)]/(p1 — p,) and Fc(p) and the integrated reflectivity is computed according to I~(p)= Fc(p)

+

{i

-

exp(-

i

M~_Fc(p)

~)}

p Sc(p)

=

JC(p)

—

p, Sc(p) are calculated

(8.35)

2/Sa A sin 20B is assumed for the reflectivity of an ideal mosaic where the expression Mr =provided (ITXHIC)by eq. (8.35) for the Si(422) Laue reflection at A = 0.71 A are crystal [105]. The results compared in fig. 47 with the previsions of eq. (8.32) for both positive and negative curvature. The departure between the two theories is already appreciable for radius of curvature of the order of 10 m. Figure 48 shows some of the experimental results reported in ref. [153]. The measurements were performed on a Si perfect crystal with a thickness t 0 = 0.8 mm using MoK~radiation. The reflectivity values evaluated by eq. (8.35) are represented in fig. 48 by solid lines and appear to be in fairly good agreement with the experimental results in the entire curvature range investigated.

R. Caciuffo et al., Monochromatorsfor X-ray synchrotron radiation

-60

57

a~

b

>~ > I-

-

U

w -J

U-

Ui

1—

I

.05

I

I

.1 .15 CURVATURE

.2

—

.25 rn~

Fig. 47. Integrated reflectivity for Si(422) curved crystal in Laue geometry calculated as a function of the effective curvature p; A = 0.71 A. Curves are obtained from eq. (8.32), curves b~are provided by eq. (8.35). The signs ± refer to positive and negative curvature. From ref. [153].

8.4. A model for the dynamical X-ray diffraction by deformed crystals in Bragg geometry 8.4.1. Calculation of the reflectivity We consider a curved crystal oriented with respect to a monochromatic incident beam in such a way that the deviation from the Bragg law at the crystal surface is y(O). The curved crystal will be divided into three regions [156],as shown in fig. 49. Region I is defined in such a way that y(O) < y < —1, region II corresponds to that part of the crystal for which —1 1. At the centre of region lithe Bragg law is verified and y is equal to zero. According to eq. (8.27), this is obtained at a depth =

—y(0)/c.

(8.36)

On the other hand, the upper limit ~A11and the lower limit e~4iiof region II are given by

~

~104~~

IMPRESSED CURVATURE—~ m1

IMPRESSEDCIJRVAT(JRE—.

(a) Fig. 48. Experimental reflectivity of a Si perfect crystal with a thickness

t 0=

From ref. [153].

m1

(b) 0.8mm at A = 0.71

A. The solid curves are the predictions of eq. (8.35).

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

\~\~//

59

/3//V

Fig. 50. The energy balance considered to evaluate the reflectivity of a curved crystal.

respectively, of layer 2. As the orientation of layer 2 is different from that of layer 1, r2 and t2 are different from r1 and t1. It is also assumed that the power P2 on the way out cannot be diffracted back by layer 1, as would be the case in diffraction by a perfect crystal. However, the power P2 will be partially absorbed inside layer 1. By calling P21 the fraction of power P2 emerging from layer 1 (see fig. 50), and ~ the X-ray absorption coefficient, one obtains =

P2 e~

=

r2t1P0 eS1

(8.42)

where S~is the X-ray path inside layer 1 which is given by S~= t0/sin 0 where

t0

(8.43)

is the thickness of each layer. The reflectivity of the two layers is defined as =

(8.44)

‘~H~~’~O

where P11 is the power diffracted by the two layers P11

P11

=

+

(8.45)

P21.

From eqs. (8.40), (8.42), (8.44) and (8.45) one obtains 2~ = r R~ 1 + r2t1 e~~1 .

(8.46)

Extending eq. (8.46) to the case of n layers, one obtains: ~

=

~

r1

tk

e~).

(8.47)

Once the orientation of the curved crystal, i.e. y(O), is fixed, the orientation, y. of the ith layer is known through eq. (8.27). For a given orientation y~and a given thickness A of the ith layer, the reflectivity r1 is calculated by using eq. (7.28) or a more general expression derived by Zachanasen (eq. (3.138) in ref. [132]) for absorbing crystal plates of arbitrary thickness:

60

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

r

=

Q

+

.,

U sinhaw

—

V sinavC+

2av

sinh2aw}

x11LU {sin q~sinh~2aw~ ~\/ + ~\/V +

—

(8.48)

.,

~, —

q~sin~2av~

where Q=~q+z2~

(8.49)

U=Q+~z~2

(8.50)

V=Q—~z~.

(8.51)

For the definition of the quantities appearing in eqs. (8.48—8.51) one should refer to ref. [132]. The ratio t 1 of the transmitted to the incident intensity of the ith layer can be obtained in a similar way from eq. (3.137) of ref. [132]: —

—

Q + U sinh

2aw

—

Qexp(2A(I+b)g/(l—b)) + ~\/U2 q~2sinh~2aw~ +

V sin2av

—

______ —

~—

q~2sin~2av~ -

(852

Equations (8.48) and (8.52) are functions of the quantities y and A, which in the present case take the values y~and A 1 respectively. It must be emphasized that (8.48) and (8.52) take into account the X-ray absorption inside the crystal layer i. By inserting eqs. (8.48) and (8.52) into (8.47) one obtains the X-ray diffraction pattern R = R(y(0)) for the curved crystal. 8.4.2. Check of the model by comparison with the Taupin dynamical theory In order to check the validity of the model, a comparison of its predictions was made [156]with the diffraction patterns obtained by Taupin [150]with its theory. Figures 51(a) and (b) show the comparison for the curvature corresponding to c = 0.1, i.e. for a slight curvature, for g = —0.1 and K* = 0 and K* = 0.1 respectively. Figures 51(c) and (d) present the diffraction patterns corresponding to c = I, the other physical quantities remaining the same (for a physical discussion of the curvature corresponding to c = 1, see ref. [159]). Figures 51(e) and (f) report the comparison for c = 10, i.e. for a large curvature. It appears that the agreement, although not excellent, is satisfactory, in particular as far as the integral reflecting power is concerned. 8.4.3. Interpretation of some experimental data concerning bent monochromators In this section some experimental results are interpreted on the basis of the model described above. Kohra et al. [58] reported measurements of integrated intensity from elastically bent crystals of germanium and a-quartz as a function of the bending radius at two X-ray wavelengths. These data show that a gain G in the integrated intensity of about 30 can be obtained by inducing in a quartz crystal a curvature of about 0.5 m. The experimental set-up used [160]is presented in fig. 52. A monochromatic beam is obtained by a perfect crystal of the same material as the curved crystal whose diffraction pattern is then recorded. Figures 52(a) and (b) show the integrated reflectivity of Ge(111) and

R. Caciuffo et a!., Monochromatorsfor X-ray synchrotron radiation D

c=0.1

c=0.1

g=-0.i

61

-~

g=-0.1

0.959

k— 0.1

0.819

4i~4~7L

kI~/~

0000

__________________________ ________________________ Y(ot 2 1 0 —1 —2

oop 1

Y~t 2

a)

R~.

__________________________

0

b)

2

1

0870 _____________________________________________

0000’0

_____________________________________________________ Y(o) 2 0 -2 4 6 8

,

o~0

4Y1o1 2

0

-2

0

0

4

-6

-8

d)

C) c=1

gr-Q.J k~0 10Y~e~5

R~*

jbo~.a..o~0.1

0

-5

e)

-10

—15

-20

-25

V(.) 10 5 ~0~00~l00

-5

0,0

-10

f)

-15

-20 -25

Fig. 51. Comparison of the model predictions (circles) with the dynamical theory prediction [150]for X-ray diffraction by curved crystals of different bending radii.

a-Si0

2(101) bent crystals as a function of the bending radius for CuK~radiation (A = 1.54 A) and for MoK~(A = 0.71 A) radiation, respectively. The experimental data are represented by dots while the broken lines represent the prediction of the model. The agreement is quite satisfactory. Figure 53 shows the diffraction patterns predicted by the model for the extreme curvatures used in the experiment with MoK~radiation. The most striking fact appearing from these results is that, whereas for larger bending radii the a-Si02(101) crystal has a lower reflectivity than Ge(111) at both wavelengths, for lower bending radii and MoK~radiation the reflectivity of the quartz increases (b) ~~

(a)

z

-

z

Ui

w

Ia

Ui lx 0

>

I-

100

~

(1)

UJ

____

(2)

I—

~~~0.

z

10 2~102

I

3

i0

IO~ I

_____

~ a 100-

4

lx

0 Ui

.-----.-

Bending Radius

(mm) I

______

102 ~ iU~ ________________________ (mm) Bending Radius

Fig. 52. Integrated intensities from bent single crystals of Ge (curve 1) and ct-quartz (curve 2) as a function of the bending radius (a) for CuK, radiation, (b) for MoK, radiation. The dots represent the predictions of the model.

62

R. Caciuffo ci a!.. Monochromators for X-ray synchrotron radiation

1.0ROT

1.0Ge~MI~

R=~ o-sIo

0000Mo

0 ~5~=o.tos.iô~ Rc5i0~mm

0

RT

~I~5

Rr~

aSi02(101)MoK~

05 -

—

e•

Rc5~10’rflm C =10.52

S6sec

~

Fig. 53. Diffraction patterns predicted by the model for the extreme curvatures used in the experiment for MoK, radiation.

strongly, becoming much larger than the corresponding reflectivity of Ge(111). On the other hand, the integrated reflectivity of Ge( 111) remains practically constant at both wavelengths and for all the bending radii. In order to explain this behaviour one can roughly assume that, because of absorption the curved crystal is equivalent to a crystal of finite thickness ta given by:

t~=!(I+I)

/L

Yo

YH

(8.53)

where p is the linear absorption coefficient for the considered radiation. Equation (8.53) is easily obtained by imposing the condition that the maximum path of the diffracted X-rays inside the crystal is equal to 1/p. It is well known that if one progressively deforms a perfect crystal, the integrated reflecting power increases to a saturation value which is the kinematic reflecting power. Therefore we expect that the saturation value for a given curved monochromator should be roughly equal to the kinematical integrated reflecting power 111(kin) corresponding to a crystal thickness ta• As shown in section 7, 111(kin) is given by (eq. (7.26))

111(kin)=J R11(y)dy= ITA

(8.54)

where A is the thickness of the crystal in proper units, defined in eq. (8.30). Then, by calling A0 the value corresponding to the thickness ~a’ one obtains a saturation value of the integrated reflectivity given by

R. Caciuffo et al., Monochromators for X-ray synchrotron radiation

Values of

c

63

Table 6 and A, with bending radius of a Ge(111) and an ct-Si0

2(101)

crystal Crystal

A (A)

Ge(111)

1.54(CuK,) 0.71

Bending radius (m)

(M0K,)

1.54 (CuK,)

ct-SiO,(101)

0.71 (M0K,)

IH(sat)

=

ITAa

=

IT

c

A,

1 0.5

0.078 0.16

4.71 4.71

1

0.32

0.5

0.64

2.60 2.60

1 0.5

1.00 2.00

6.40 6.40

1 0.5

5.26 10.5

28.37 28.37

CXHr

2

(8.55)

ta~ AV~YO7H~

0~I2pand For symmetrical reflections,

sin

ta =

I~(sat)= ITdX (8.56)

11r~

Thus, the increase of the integrated reflectivity, that can be obtained by increasing the curvature c of a bent crystal, is limited by the absorption phenomena. The maximum gain in reflectivity with respect to a perfect crystal (111(dyn) = ir) is reached as c —~ and it is given by A a The value of c and A a calculated for the monochromators used by Kohra et al. [58] are reported in table 6. One can see that for the Ge(111) bent crystal for both wavelengths, the values of c and Aa remain relatively small for all the bending radii considered; as a consequence, the integrated intensity cannot increase strongly. Si(lll) OC=8°30’

1.0 0.8

0.6~

~s:l.54A 5m

1a50m -

¶caüOO8

‘ii

\C-0.08

4’=im

f~aa4

-HH~-~~ I

H

:1:1111 LiiiJ~I~1 I

/ I

A9=6.4.lO5rad

0

—

Angle ot incident 8 (rad) Fig. 54. Diffraction patterns calculated at A = (1/pSOm, Sm, im).

1.54 A

for Si(111) asymmetric curved crystal (a

= 8°30’) for

different values of the bending radius lip

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

64

Si(111) cU~~30

tIi.o-~

__

Xzl.54A .~

5rad

0.8 -~

—-

AGo 6./.lO c=1.7 —

::I912~EI1r2i:i Angle of incidence

9 (rad)

Fig. 55. Diffraction patterns calculated at A = 1.54 A for Si( 111) asymmetric curved crystal; bending radius I/p = 0.25 in, 0.1 m.

The same conclusion is reached with Si0 2 a-quartz for CuKa radiation, because of the low values of c and Aa~On the other hand, the reflectivity of the a-quartz should increase by a factor G 30 when MoK~radiation is used and c becomes large, as it was experimentally observed. 8.4.4. Evaluation of the diffraction characteristics of some bent crystal monochromators for X-ray synchrotron radiation The above described model was used in ref. [156] to study the characteristics of curved monochromators used at that time in the laboratories of Hamburg, Orsay and Stanford. Several diffraction patterns were obtained corresponding to different monochromator materials (Ge, Si) for different a=8°30’

Si(111)

—l

—I

0.8

-

—-—

z5Om Cr0003

~

—

—

—i

~ =5m — — — — ~=1m c=0.03

—-

c=0.18

P~L~ Angle of incidence

0 (rad)

Fig. 56. Diffraction patterns calculated at A = 2.29 A for a Si(Ill) asymmetric curved crystal; radius of curvature lip

=

50 m, S m, 1 m.

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

R-~Po

1

1.0

Si (111) -

~k,=830’

Ar2.29A

—

c =

65

—

—

—

-

0.71

0.6~-—--

::________________ Angle of incidence e(rad) Fig. 57. Diffraction patterns calculated at A = 2.29

A

for a Si(l1l) asymmetric curved crystal; radius of curvature lip = 0.25 m, 0.1 m.

reflecting planes (111), (220), asymmetry angles, X-ray wavelengths and curvature radii. Here, only the results concerning one of the studied monochromators are reported as an example, namely the X-ray focusing crystal used at SPEAR [31—33] and described in section 6.2. This monochromator was a logarithmic spiral type curved Si crystal with the surface cut at 8.5°to the (111) Bragg planes and it allowed a selection of wavelengths between 0.5 A and 3 A. The diffraction patterns were calculated for a cylindrically curved Si crystal having the same Si (111)

Rmax

a:8°30

Rmax— -

0.8

—

-

w

\

-

810~

__2•~

~(meter)H Fig. 58. Peak reflectivity Rm,, and full width at halfmaximum was a function of the curvature p for A diffraction by a curved asymmetric Si(lll) crystal (a

= 8~30’).

= 1.54 A

(CuK,) and A = 2.29 A (CrK0) in the

66

R. Caciuffo ci a!., Monochromators for X-ray synchrotron radiation

S i(111)

-~ ~

~

i=1.54A

8’30

~.29~30__

I (meterYt Fig. 59. Integral reflecting power l~of a Si(l1l) asymmetric (a 2.29 A.

=

8°30’)curved crystal as a function of the curvature p for A = 1.54

A

and

characteristics for different values of the wavelength A and of the bending radius lip. The polarization was assumed to correspond to the electric field vibrating in the horizontal plane (C = cos 20B) and the crystal thickness was assumed to be 1 mm or more, i.e. much longer than the penetration depth due to absorption. Figure 54 shows the diffraction patterns predicted for the CuK radiation wavelength for radii of curvature, p =50m, and fig. 55 those for p =25 and 10 cm. Figures 56 and 57 show the corresponding diffraction patterns for the CrKa radiation wavelength. Figure 58 reports the peak reflectivity and the full width at half maximum for A = 1.54 A and 2.29 A as a function of the radius of curvature. Figure 59 reports the integral reflecting power of the monochromator as a function of the radius of curvature. These results are discussed in ref. [156]by using simple physical considerations based on the concepts exposed in section 8.4.3. For more details the reader should consult ref. [1561.

Appendix. On the polarization correction In a diffraction experiment the observed intensity is related to the Bragg intensity I by the relation (Al) where L and P are the kinematical and polarization factors, respectively. In a SR apparatus the polarization factor P is determined by the polarization properties of the source and by the degree of polarization introduced by the monochromator. This must be known in order to correctly relate the measured intensities to the corresponding structure factors, especially when radiations of wavelength greater than 1 A are employed [109]. An expression for the polarization factor of a typical crystalmonochromated SR diffraction camera has been derived in ref. [82]following the arguments developed in ref. [1091.

R. Caciuffo et a!., Monochromators for X-ray synchrotron radiation

67

The polarization of the beam incident on the monochromator is described by the ratio A=(Iff—I~)II0

(A2)

where I~= E~,I~= E~and I~= I~,+ I,~is the intensity of the SR beam; the suffixes if and ir refer to the polarization states with the E vector in and orthogonal to the plane of incidence, respectively (see fig. 60). The intensity reflected by the monochromator is given by I’ = I~’+ I~,where [109] 2I, I,’, = aH 1, 2I~ I~= H and the polarization of the monochromated beam is then described by A’—

—

I’

— —

(A3a) (A3b)

a(1+A)—(1—A) a(1+A)+(1—A)~

(A4)

The quantity H in eqs. (A3) is a polarization-independent proportional factor while a is known as the “polarization ratio” of the monochromator. The intensities which appear in eqs. (A3) should correspond to the integrated intensities if the divergence of the incident beam is much greater than the monochromator rocking curve width, or to the peak reflectivities in the opposite case. With these definitions, it is possible to show that the polarization factor P for a beam diffracted by a

x

E~ E

‘

\ MONOCH.

~E~c

~

~

~j

I

~ sample

29 / /

/ /

/ ~etecto plane

/~/

r

I 1/

// fl\/ /~ /

VI

/ /

I

Fig. 60. Geometry for evaluating the polarization correction in a diffraction experiment. From ref. [82].

R. Caciuffo et a!.. Monochromators for X-ray synchrotron radiation

68

crystalline sample is given by [82] 1+cos22o 2

A’ cos2psin2O

AS

2

the angle 0 and p being defined in fig. 60. An error in A’ corresponds to an error in the Bragg intensity given by —

IP

~P

—

cos2p sin2 20 M’ 2P

A6

The quantity A’ depends on the degree of polarization A of the white SR beam and on the polarization ratio a of the monochromator crystal. The former quantity can be calculated from the parameters of the X-ray source even though, as pointed out in ref. [82], vertical displacements of the X-ray beam with respect to the sample crystal during the data collection time produce a change in A. The polarization ratio a can be expressed as [161] a=cosm(20M)

(A7)

where °Mis the Bragg angle for the monochromator crystal and m is a number (ranging from zero to two) related to the extinction coefficient [161—163].For a mosaic monochromator obeying the kinematical diffraction theory it is m = 2, while for a perfect crystal with no absorption the dynamical theory of diffraction yields a = cos2OMI. For graphite monochromators in Bragg geometry at A= 1.54 A, the theoretical values of m range from 0.8 to 1.2. An increase of the secondary extinction leads to a smaller y value and to a polarization ratio closer to unity. Moreover, a decrease of m with the rocking curve width has been observed [164—167]. Accurate values of A’ can be obtained by standard measurements of the polarization ratio a [163].A tabulation of some 40 a values measured in different apparati can be found in ref. [161].

Acknowledgement It is a pleasure to thank Dr. AK. Freund of ILL and ESRF, Grenoble, for helpful discussion during the preparation of the manuscript.

References [1]D.D. Ivanenko and I. Pomeranchuk, Phys. Rev. 65 (1944) 343. [2] J. Schwinger, Phys. Rev. 70 (1946) 798. [3] J. Schwinger, Phys. Rev. 75(1949)1912. [4] A.A. Sokolov and I.M. Ternov, Synchrotron Radiation (Pergamon. New York, 1968). [5] J.D. Jackson, Classical Electrodynamics (Wiley, New York. 1975). [6] E. Koch, Handbook on Synchrotron Radiation (North-Holland, Amsterdam, 1983). [7] C. Kunz, ed., Topics in Current Physics, Synchrotron Radiation, Techniques and Application (Springer Verlag. Berlin, 1979). [8] H. Winick and S. Doniach, eds., Synchrotron Radiation Research (Plenum, New York, 1980). [9] Physics Today, June 1983, in a special issue on synchrotron radiation.

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