Design relations for synchrotron radiation curved two-crystal monochromators

Nuclear Instruments and Methods in Physics Research A319 (1992) 141-148 North-Holland

NUCLEAR INSTRUMENTS IN PHYSICS

RESEARCH Section A

Design relations for synchrotron radiation curved two-crystal monochromators M. Popovici and W.B. Yelon

Research Reactor, University of Missouri-Columbia, Columbia, MO 65211, USA Formulae are given for the necessary radii of curvature in double bent-crystal monochromators for synchrotron radiation . It is shown that it is possible to overcome the heat load problem by suitably curving the second crystal : in this way, the reflection efficiency is fully restored, and may be enhanced due to the reflectivity increase on curving. By controlling the curvatures of the two crystals one can focus the monochromatic beam at the sample without the use of curved mirrors. Results of computations made with an adapted neutron optics program are presented. 1. Introduction Double flat-crystal monochromators are standard equipment on synchrotron radiation beamlines with tunable wavelength . Curved mirrors are commonly used in conjunction with them to keep the beam focused spatially onto the sample . Bent crystal focusing, which requires curvatures that depend on the Bragg angle, is used mainly in single monochromator fixed wavelength instruments. However, even in fixed wavelength instruments, a curved double monochromator may be desirable. Moreover, in any flat-crystal double monochromator the first crystal may be curved by the incident beam heating. Thus the optics of synchrotron radiation double reflection by curved crystals should be of interest; however, it has received little attention . It has been occasionally considered [1-3] on the basis of the wellknown formulae for characteristic X-rays, i .e. for the case of monochromatic focusing. The requirement of monochromatic focusing is excessive, if only because flat-crystal double monochromators do not give monochromatic beams in the sense of characteristic X-ray optics. An independent development of curved crystal optics has occurred for neutrons with results relevant to the synchrotron radiation case: an equivalent of the flat-crystal parallel position was found to exist for bent crystals [4] ; formulae for bent two-crystal neutron monochromator design were derived and experimentally checked [5] . The optical design of curved two-crystal monochromators for synchrotron radiation was considered in a recent paper of ours [6] starting from neutron results. Both for neutrons and synchrotron radiation, one can ensure that all the radiation diffracted by the first crystal automatically has the right conditions to be

diffracted by the second one. For this to happen, one has to choose the right curvature for the second crystal in the synchrotron radiation case, and the right curvature and the right reticular spacing in the neutron case. If curved, the two crystals must have different reticular spacings in a neutron monochromator, but may have the same spacing in a synchrotron radiation monochromator. The difference is due to the fact that in the meridional plane a synchrotron radiation source is practically pointlike while a neutron source is usually rather wide. These considerations can be applied to overcoming the heat load problem in synchrotron radiation double monochromators by suitably curving the second crystal to accommodate the curvature of the thermally distorted first crystal [6] . In focusing the beam onto the sample it may be convenient to use two curved crystals instead of one . The formulae for the necessary radii of curvature are discussed in this paper for two situations: (1) a thermally distorted first crystal, and (2) the source image required at the sample position.. Results of computations with a neutron optics program adapted to the synchrotron radiation conditions are presented .

2. Theoretical background An analytical treatment of the curved crystal neutron optics has been developed in refs. [5,7-9] . It is similar to the one presently used for synchrotron radiation [1,2] in that it considers the problem in the phase space and uses matrix language. The neutron case is slightly more complicated because one has to deal with thick crystals and large sources and samples . The crystals may have large reflectivity widths and different

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IV(a). CHARACTERISTICS

142

M. Popoi ici, W.B. Yelon / Curred two-crystal nronochrornators

types of lattice spacing gradients [10-13]. This increases the number of variables in the problem and requires the use of matrix computational techniques throughout . Programs for numerically performing these computations have been developed [8,9]. The programs compute the covariance matrices of the radiation intensity distribution with respect to the phase space coordinates after reflection on the monochromator, and then the covariance matrices of the resolution functions for elastic scattering [9] or inelastic scattering [8] . The programs are written in Fortran 77 and have been tested on CDC Qyber, Vax and PC/AT computers . The double crystal monochromator option is available only in the elastic scattering program [9], which is presently being put in a user-friendly form for release. In these programs the intensity distributions and the resolution functions are approximated by multi-dimensional Gaussians, so that line shape details are lost. From this point of view the synchrotron radiation ray-tracing programs [14-17] are superior. However, the matrix computations have the advantage of allowing resolution-intensity optimizations to be conveniently done. Analytical formulae for phase space focusing conditions have been derived [5.7-9] for use as starting rots in numerical optimizations . The real-space focusing conditions are the usual ones but for neutron instruments they are not so important because of the large source areas and crystal thicknesses . For point sources, thin crystals and monochromatic focusing, the neutron formulae reduce to the known X-ray ones. In this work, only the meridional (vertical plane) optics will be considered . In the linear approximation it is independent of the horizontal plane optics for which the usual saggital focusing conditions should apply . The term crystal may mean a multilayer mirror as well. Multilayers with laterally variable reticular spacing are particularly promising for curved crystal applications [18,19] but here the spacing is assumed to be constant. The extension of the formulae to the case of a variable reticular spacing is straightforward . The meridional optics formulae are most conveniently expressed in terms of the two focal lengths of thin curved crystals [7] : f=(R12) sin(0 + X)sgn(0 +X),

(la)

f* = (R12) sin(0 -,y)sgn(0 +X),

(1b)

where R is the meridional radius of curvature (positive if the radiation strikes the concave side of the crystal), 0 is the Bragg angle and X the cutting angle (X = 0 .means symmetric reflection), both measured in the same trigonometric sense (so they have signs). The relation f * =f1b holds, b being the usual asymmetry parameter b = sin(O + X)/sin(O - X) ( I b I > I for a

Fankuchcn cut). The first focal length f always has the same sign as R while f * has the sign of R in the reflection range ( I X I < 10 1, b > 0) and the opposite sign in the transmission range ( Ix I > 10 1, b < 0). The above notation unifies the formulae for the reflection range and the less usual transmission range [13]. For a perfect crystal, the following correlation exists between the phase space coordinates y, y', Ak1k of the reflected radiation at a distance 1* behind the crystal : AkIk = cot

Of

-yl(2f *) - y'[ 1 - 1*/(2f*)] + b4), (2a)

where Aklk is the relative deviation from the mean wavevector k and ~ is the deviation from the local Bragg angle within the Darwin curve width. A similar expression can be written for Ak/k in terms of the variables before reflection, at a distance I before the perfect crystal [7] : Ak/k=cotOl-y/(2f)+y'[I-//(2f)]+~) . (2b) In the following the quantities 0, X, b, f and f * will bear the subscripts I and 2. For a point source situated at a distance L, before crystal I the image is formed after reflection at a distance L i given by the relation fIlL,

+fl*lL i = 1 .

(3a)

At the image position, Ulk will be given by eq. (2a) with y = 0 and I* = L ; . A second crystal, placed at a distance L from the first, can reflect radiation having Aklk given by eq. (2b) with y = 0 and I = L - L i. For the double reflection to be efficient the two expressions for Ak/k should give the same result. For ~ = 0 this happens if the following condition is satisfied : aLi lf l*

- (L - Li )lf2 = 2(a -

1),

(4)

where a is the usual dispersion parameter, a = -tan 02/tan 9r. Any number of curved crystals can be put in subsequent reflection, provided that for each subsequent pair of crystals, conditions like eq. (4) are satisfied . In ref. [6], the condition (4) was derived from a general result obtained for neutrons in ref. [5] . For two crystals of identical reticular spacings, a equals I in the parallel position ( 0 2 = -8r ) and -I in the antiparallel ( 0 2 = Od, so that for flat crystals in parallel position eq. (4) is fulfilled automatically . For bent crystals eq. (4) is a condition imposed on the necessary meridional radii of curvature that can be satisfied in both the parallel and the antiparallei ( +, + ) positions. After reflection by the second crystal the beam will be focused at a distance L 2 given by the relation f2l( L - Li) +f2*IL2 = 1 (3b) Formally eqs. (3a) and (3b) are conventional optics

M. Poporici, W.ß. Yelon / Curved two-crystal monochromators relations for lenses with two different focal lengths (normal lenses placed between media of different refraction indices). If the focusing is required to be monochromatic, then the usual conditions 2f, = L, and 2f2* = L, should be fulfilled to ensure that the source and its images are placed on proper Rowland circles. Then eq. (4) is automatically fulfilled too . However, for the usual symmetric reflection geometry, the resulting configuration is of limited interest because large distances between the two crystals arc needed. Interesting configurations become possible when the requirement of monochromatic focusing is dropped. This is discussed in section 4. In the next sections, the above formulae are illustrated by numerical computations made with one of the mentioned neutron optics programs. The adaptation to the synchrotron radiation case consisted merely of working with appropriate perfect crystal reflectivity parameters and with effective crystal thicknesses given by the absorption depths. An automatic generation of those parameters will have to be implemented to make the program suitable for general use in synchrotron radiation computations . 3. Application to the heat load problem In a flat double monochromator, the first crystal is not actually flat because of the heating by the incident beam. There is a loss of intensity as the flat second crystal no longer efficiently diffracts the beam given by the thermally distorted first crystal . The "rocking curve" (the intensity profile measured by rotating one of the crystals) broadens and may correspondingly have a lower peak intensity. This heat load problem has been the object of much work over the last decade (see e.g. ref. [20]). As some thermal distortion is unavoidable at the high heat loads from the beams of the next generation synchrotron facilities, no matter how intense the cooling or how thin the first crystal [21,221, it should be of interest to compensate for this distortion by suitably curving the second crystal . Such a compensation should be possible if the thermal distortion is made uniform enough so that a definite (negative) meridional radius of curvature could be assigned to the crystal surface . Then the necessary radius of curvature of the second crystal can be computed through eq. (4). For identical crystals in symmetric reflection and in parallel position this relation simplifies to

R2 :" -R,(1 - L/Li) .

If the monochromator sees the source directly then the quantity L i here is given by eq. (3a). It will be negative so that the second crystal should be concave with

143

0 r N 0.8

v c

ô

j

®.dO4 0.06 o.0os ®.0'o 0.012 Second crystal curvature 1/132 (1/m) Fig . 1 . Computed dependence of the rocking curve widths and peak intensities on the second crystal curvature 132' for the instrument geometry of the CHESS A2 beamline [231. The radii of curvature of the first crystal are thought to correspond respectively to the gallium-cooled ( -233 m) and water-cooled ( -113 m) Si (1 11) double monochromators set to 20 keV at a power load of about 0.4 W/mm2 [21,241. 0.0

0.002

R 2 >- I R,1. If a curved minor stands between the source

and the monochromator, then Li is defined by the joint effect of the mirror and the first cr,,stal. Some results of numerical computations are shown in figs . 1 and 2. Tine computations refer to the monochromator on the A2 beamline at the Cornell High Energy Synchrotron Source [231 for which the heat load effects are well documented [21,241. The monochromator sees the source directly and uses Si(1 11) crystals at high energies for a 7.3 cm beam offset . Fig . 1 shows the computed rocking curve width and peak intensity vs the second crystal curvature 132' for two radii of curvature of the first crystal . The values of these two radii (-113 and - 233 m) were inferred from the slope errors of the reported model computations [241 for the water-cooled and gallium-cooled monochromators, respectively [211, set for an energy of 20 keV at a power density of about 0.4 W/mm2. It is seen that by curving the second crystal to the right radius, a practically complete restoration of the rocking IV(a). CHARACTERISTICS

M. Poporici, W.B. Yelon / Curved two-crystal monochromators

144

first crystal is meridionally curved to a fixed radius giving a high energy resolution (2f, = L,) in the middle of the scanned energy range . The second crystal is convex with the same (negative) radius, in concordance with eq . (5) for L << Li. A subsequent ellipsoidal mirror is used to focus the beam onto the sample . The beam can be focused, though, at the sample by the two bent crystals with no mirror necessary. The radii of curvature needed to achieve this are obtained by eliminating the parameter Li from eqs. (3a), (3b) and (4):

2-crystal Si (111), E = 20 keV R, = -11 .3 m

A

R1/Rmono=2[L - (1 - a)L 2 b 2 ]

0.0 30.

/[L +L1/b, - (1 - 2a)L,62 ], R,/R2ono=2[L - (1 - 1/a)L1/bl]

(6a)

/[ L + L,b, - (1 - 2 /a)L1/b1 ], (6b) Rmo"o and Rmono are the radii for monochrofocusing given by 2f, = L, and 2f2* = L 2, re-

where matic spectively. In the

(a= 1), one has L 1 /b, + L2b2). This corresponds to the configurations discussed in ref. [3]. More generally, one has both R, = Rmono and R, = Rmono for any a, provided that L = L1 /b, + L2b2: for a monochromatic beam it does not matter whether the second crystal is in the parallel or in the antiparallel position and it may have any reticular spacing . For a = -1 one has L,/b, = L2 b2 as a special case: then both f, = L, and = L2 for any L. The value of L does not matter because the beam is planeparallel between the crystals . Generally, this happens when L,/b, = -aL 2b2, i .e . either for a < 0 with both crystals in reflection or both in transmission, or for a > 0 with one of the crystals in transmission and another in reflection . The configuration could be of interest for double crystal small-angle scattering instruments . For the image magnification M = (L i/L 1 )[ L 2/( L - Li )] one obtains parallel

position

case

R 1 /Rmono = R2/R2 ono = 2L/(L + 2-crystal Si (111), E = 20 keV ®.O

&610

0.020 ®MO 0-040 0.050 - 0.00 0.670 Second crystal curvature 1/R2 (11m)

0.680

Fig. 2. The same as in fig. 1, for a power load 10 times higher and correspondingly 10 times smaller curvature radii . curve width to its flat-crystal value can be achieved . The peak intensities are slightly higher than for two flat crystals due to the reflectivity increase of curving . The computations shown in fig . 2 refer to a heat load 10 times higher than that in fig. 1 . Correspondingly, the values of R, were assumed to be 10 times smaller . The lattice spacing gradient no longer allows the rocking curve widths to be restored to their flatcrystal values, but the residual broadening is accompanied by a marked intensity increase . This effect is related to the large absorption depth of silicon (about 100 FLm at 20 keV) and for germanium it is virtually absent . At high energies and small enough radii of curvature, silicon double monochromators should give more intense beams than germanium ones [6]. It is seen from fig . 2 that, at high heat loads, the setting of the second crystal curvature has to be quite precise . As the power density depends on the Bragg angle, for tunable monochromators R, will vary and R2 will have to be controllable .

4. Application to beam focusing We know of only one synchrotron radiation monochromator with two curved crystals [25,26] . The

f2

M= (b,b2 ) - 1 [ L - (1 - a)L,b2] /[aL + (1 - a)L 1 /b, ] . This is essentially a conventional optics formula that does not account for the phase space picture of the reflection on two curved crystals . It is correct as long as the source meridional width is small enough for the point source approximation used in deriving eqs . (6a) and (6b) to be justified . This will be discussed below. For a = 1 one has M = 1 /(b I b 2): with two identical crystals in symmetric reflection (b, = b2 = 1), no demagnification can be achieved in the parallel ( +, - ) position . By contrast, one curved crystal in symmetric reflection can give large image demagnifications [2729] .

M. Popovici, W.B. Yelon / Cun-ed two-crystal monochromators Similar large demagnifications can, however, be obtained in the (+, - ) configuration by using two asymmetrically cut crystals with b,, 62 >> 1 (both in Fankuchen cuts) . The Darwin curve broadening for bent crystals will ensure an enhanced reflection efficiency. For flat crystals such an asymmetric configuration is not particularly efficient because of the misfit between the angular emergence and acceptance windows of the two crystals. Another possibility of obtaining image demagnifications in the equivalent of the parallel position is to work at a 0 1 by using crystals with different reticular spacings. For a = -1, eq. (7) gives M = - (L2/Li)[ 1 -L/(2L,b,)] /[1 -L/(2L,/b,) ] .

Therefore, with identical crystals in antiparallel symmetric reflection, large demagnifications can be obtained just by correctly choosing the distance L between the two crystals. Asymmetric reflection with the second crystal at b2 << 1 (inverse Fankuchen cut) is useful if small L values are desired. In fixed-energy instruments the particular value 0 = 45° could be used to extract the beam in the horizontal plane. The a = -1 configuration is also of interest for heat loaded double monochromators because it allows focusing to be obtained with convex first crystals. For the parallel position (a = 1) this is possible only with

100.

150.

14 5

crystals in transmission : eq. (6a) gives R, < 0 only if L + L,/b, + L2b2 < 0 and this can happen only if b I < 0. A thin first crystal in transmission could also help in reducing the heat loading effects. Curved crystals in symmetric transmission are useful when no Darwin curve broadening is desired . However, they can focus in the meridional plane only. To illustrate the peculiarities of the antiparallel (+, + ) configuration with two curved crystals some computational results are shown in figs. 3-6. The starting configuration is the same as in the preceding figures, corresponding to the water-cooled monochromator with an assumed radius of curvature R, = -113 m, placed at L, = 14.5 m from a source 1.5 mm wide meridionally . Fig . 3 shows the computed beam width dependence on the distance L, from the second crystal for the parallel ( + , - ) and antiparallel ( + , + ) positions. The case of a germanium second crystal is also shown, as it gives a deeper minimum at the focus. This is due to the smaller absorption depth of germanium with a corre= spondingly lower aberration due to toeam penetration . Fig . 4 details the beam intensity and width at the focus dependences on the second crystal curvature. For silicon the shape of the intensity curve is asymmetric around the optimal value R 2 = 27.2 m due to the reflectivity increase with increasing curvature. For germanium there is no such asymmetry . The beam intensities are calibrated to that computed for flat silicon

200.

250.

300.

Distanc e L2 from monochromator (cm)

Fig. 3. Computed beam width vs distance from monochromator for the same instrument geometry as in figs. 1 and 2, water-cooled first crystal (R, _ -113 m assumed), optimally curved Si or Ge (111) second crystal in parallel and respectively antiparallel positions . IV(a). CHARACTERISTICS

M. Popovici, W.B. Yelon / Curved two-crystal monochromators

146 400.

0 3 V

(+,+),

3 E

E = 20 keV

First Si, R, =1 .83 m Second Ge

1 .25

o

(+,+), E = 20 keV First Si, R, = 1 .83 m Second Ge

0 .75

0.®Q

Q®10 0-015 0.

0.

0.030 0.035 0.040 M045 0.050 0.055

Secorid crystal curvature 1IR2 (1/m)

Fig. 4. Beam intensity and width at the foci of fig. 3. as functions of the second crystal radius of curvature.

0 .45 0 .46

0.47

0.48

0 .49

0.50

0.51

0.52

0.53

Second crystal curvature 1/R2 (1/m)

0.54

0 .!~5

Fig. 6 . Computed beam intensity and width at focus for the

configuration of fig. 5.

va c 0 L U

E È

3 v rr

3 E m

m

12 .5

13.0 13 .5 14 .0 14 .5 15.0 15 .5 Distance L2 from monochromator (cm)

16.0

16.5

17.0

Fig. 5. The same as in fig. 3, for a possible configuration with strong image demagnification using a first Si and a second Ge (111) curved crystals of optimal curvatures .

M. Popovici, W.B. Yelon / Curved two-crystal monochromators

crystals . It is seen that, in contrast to the situation for flat crystals in the (+ + ) r_onfiguratinn the intensity is not markedly reduced . The beam widths at the foci seen in fig . 4 are smaller than those computed directly through eq. (7). As the assumed source width is reduced the results predicted by eq. (7) come closer to the widths at foci computed with the program. This is due to the fact that a narrow band from the meridional source width is actually selected on reflection by two curved crystals when their radii of curvature are chosen according to the point-source eqs . (6a) and (6b). The width of the selected band can be increased by correctly choosing, with the neutron case formulas [5], the second crystal reticular spacing too . Figs. 5 and 6 refer to a possible optimal configuration for microfocusing on the same beamline. The focus is very sharp, as fig . 5 shows. The setting of the crystal curvatures should be very precise (fig. 6), which is not surprising [28,29] . The full beam intensity is expected to be higher than for two flat silicon crystals in parallel position. This is because the neutron case conditions are satisfied in the chosen configuration and the full available source width is seen. The beam flux (number of quanta per unit area) at focus is expected ïo be larger than in the present CHESS A, configuration (with the water-cooled monochromator) by a factor of about 1000. A further enhancement is possible through saggital focusing . 5. Conclusions The crystals in curved double monochromators may have different reticular spacings. The curvatures and the reticular spacings are additional degrees of freedom that can be exploited to improve the monochromator performances. Curving the second crystal in double monochromators, as a way to overcome the heat load problem, can work at very high power densities and can turn a loss of intensity into a net gain due to the reflectivity increase on curving . At high energies, this gain makes silicon superior to germanium . Large image demagnifications for microfocus applications can be obtained with two curved crystals either in parallel asymmetric or in antiparallel positions. For curved crystals, the distinction between parallel (+ , - ) and antiparallel ( +, + ) configurations is no longer relevant from the intensity point of view: both give beam intensities comparable to or better than that for flat crystals in parallel ( +, - ) position . An essential difference, however, is that the full beam intensity can be easily condensed into a narrow focus in the ( +, + ) configuration, while in the ( + , - ) configuration this can be obtained only if both crystals are set in strongly

147

asymmetric reflection. Here germanium is superior to silicon as it gives smaller aberrations associated with beam penetration in curved crystals .

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M. Popovici, W.B. Yelon / Curved two-crystal monochromators

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[29] F. van Langevelde, D.K. Bowen, G.H.J . Tros, R.D . Vis, A. Huizing and D.K.G. de Boer, Nucl . Instr. and Meth . A292 (1990) 719.