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On the sagittal focusing of synchrotron radiation with a double crystal monochromator
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
Nuclear Instruments and Methods in Physics Research A328 (1993) 588-591 North-Holland
Section A
On the sagittal focusing of synchrotron radiation with a double crystal monochromator V.I. Kushriir
t, J.P. Quintana and P. Georgopoulos DUNU Synchrotron Research Center, Robert R . McCormick School of Engineering and Applied Science, Northwestern University, 1033 University Place, Suite 140, Evanston, IL 60201-3138, USA
Received 27 October 1992 A method to avoid the anticlastic bending of the second crystal in a two-crystal monochromator for synchrotron radiation is proposed . It is analytically shown that the anticlastic curvature is zero at the center of the crystal for a simply supported isotropic crystal loaded with a constant moment provided that the crystal's aspect ratio is equal to a "golden value" dependent on the Poisson coefficient v. For v = 0.262 (equal to v in the Si(111) plane) this ratio is 2.360. Finite element results are presented on the case of the clamped crystal and show that there is a similar "golden value" approximately equal to 1 .42 for v = 0.262 . 1. Introduction Horizontal focusing of a synchrotron beam with the second crystal in a double crystal monochromator (so called sagittal focusing) is of great importance to the synchrotron radiation user [1-7]. By focusing the relatively large horizontally divergent beam, the optics increase the beam intensity at the sample position . While sagittal focusing using total external reflection from mirrors is limited to a few milliradians due to the small critical angles for X-rays (especially for energies larger than 10 keV), a bent perfect crystal focuses a much larger divergence . However, due to the nonzero Poisson ratio, anticlastic bending causes deviations of the second crystal's diffraction planes from the Bragg position and reduces the throughput [1]. Various solutions to the anticlastic bending problem have been proposed . Sparks et al . [1,2] advocate using crystals with stiffening ribs in the anticlastic direction. Knapp et al . [7] describe the most recent experiment with this design . Batterman and Berman [3], Mills et al . [4] and Matsushita et al . [5] propose a variant of the ribbed crystal which is a "slotted crystal" with periodic parallel slots. However, while successfully eliminating anticlastic bending, ribbed or slotted crystals behave more like polygonal segmented mirrors than cylinders, thereby limiting the achievable focus size to twice the width of the stiff ribs . * Present address : Advanced Photon Source - XFD/362, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439-4815, USA.
In this work, we show that the crystal curvature in the anticlastic direction is a function of the crystal aspect ratio (length divided by width) and becomes zero at an aspect ratio dependent on the Poisson coefficient . 2. Problem formulation We consider a rectangular thin crystal of dimensions 2 X X 2Y which is simply supported at two opposite edges and free at the remaining edges with a constant bending moment applied to the supported edges. A typical shape for the resulting crystal is shown z
Fig. 1. Typical shape for an isotropic crystal of dimensions 2X X 2Y simply supported on the y = ± Y edge with a constant bending moment applied along the supported edges. The crystal is shown translated up from the x-y plane for clarity.
0168-9002/93/$06.00 C 1993 - Elsevier Science Publishers B.V . All rights reserved
V.L Kushner et al. / Sagittal focusing of synchrotron radiation in fig. 1. We employ the following approximations : 1) The crystal thickness is small compared to the length and width (thin crystal approximation) . 2) The maximum deformation is less than the crystal thickness. 3) While Si crystals are orthotropic we nevertheless describe the crystal properties with a single Young's modulus and Poisson ratio and note that the Poisson ratio in the (111) plane is isotropic in the plane [6]. With these approximations, the crystal shape satisfies the biharmonic equation [8] a4 a7
+2
a4 +-)z(x, y)=0 . ay 4 ax 2 ay 2 a
4
(2a) a2 a2 Mx = ( - + v ) z = constant . ax2 y
(2b)
On the free edges x = ±X, and moment and force are equal to zero [8]: (a2 M .= +v- z=0 (2c) y aye axe a2) Fy
a3
a3
f ay 3 +(2-v)axay 2 )
Z=0
(2d)
The solution of eq. (1) under the constraints given by eq . (2) is : z(x, y) =
K
2Y(Y2-y2)
+ KY T_
n=1
(An
cosh(knx)
+xk,B, sinh(k n x)) cos(k n y),
(3)
where kn = (-rr/Y)(n - 1/2) and Cn k 2Y2 n
An : X
(1+v) sinh(k,X)-k n X(1-v) cosh(k,X) ( k n X(1- v) 2- sinh(k n X)(v +3)(1- v) cosh(k,X) ) (4)
Cn sinh(k n X) k,2 Y 2 ( sinh(k,X)(v+3)cosh(k,X)-k n X(1-v) )
C n =2vl (
-1 )n+1
40 30 20 10
IE
0
~H
-10 -20 -30 -40'
On the supported edges y = ± Y and the displacement and moment are given by :
~ .
kn Y
K is a normalization constant dependent on the loading moment and bending stiffness. It will be chosen later to achieve an appropriate value of Ry , the sagittal bending radius . The series in eq. (3) converges in less than ten terms since the coefficients A n and Bn effectively decrease exponentially with n.
589
2
Fig. 2. The anticlastic curvature Kx =(-1/R X)=(a 2z/ax e ) as a function of the crystal aspect ratio y = Xl Y for v = 0.262 . The asymptotic behavior of this function as y exponentially approaches zero.
The crystal shape depends on two parameters : the Poisson coefficient v and aspect ratio y =X/Y. For y -* 0 this solution gives the ratio of the two main curvatures equal to BX/R Y = -1/v in agreement with Sparks et al . [1]. The crystal curvature in the anticlastic direction (i .e . 1/Rx = a2 z/ax e) in the middle of the crystal (x =y = 0) as a function of the aspect ratio is shown in fig. 2 for v = 0.262. This value of v is equal to the in-plane Poisson coefficient for the Si(111) plane. For y - -, this curvature decreases exponentially . In order to obtain a small curvature at the crystal center point (x =y = 0), the crystal's aspect ratio must be large (y > 7) or the aspect ratio must be chosen equal to some "golden value", yo . For v = 0.262 this value is 2.360. The value of yo as a function of v is presented in fig. 3. This curve was obtained by numerically solving for the crossover point seen in fig. 2 over the range of v given in fig. 3. Fig. 4 shows the crystal shape in the central zone along the line y = 0 with a crystal aspect ratio greater than, equal to, and smaller than yo along with the results of a finite element analysis of the crystal shape using the ANSYS [9] program. Here, the crystal dimensions are Y = 4 cm, X = yY, v = 0.262 and the crystal curvature is set to Ry = 1 m. For y < yo , the anticlastic curvature is positive, and for y > yo it is negative . Both cases cause a deviation of the crystal reflecting planes from the exact Bragg position . In practice, it is difficult to provide a "simply supported edge" boundary condition and load it with a constant moment . The more realistic case is of a clamped crystal with the given slope az/ay = "constant" at the two clamped edges y = ±Y. However, this case is difficult to solve analytically . Ferrer et al . [10] attempt to analyze this problem by minimizing the crys-
V.1. Kushnir et al. / Sagittal focusing of synchrotron radiation 10 .0 90 80 7 .0 6 .0 5 .0 4 .0 3 .0 20 10 00 -1 .0 0.0
0 1
0 .2
0 .3
04
05
0 6
0 7
08
x [cm]
Fig. 3. Golden ratio y o = X/ Y as a function of Poisson's coefficient v for an anticlastic bending curvature of 0 at the center of an isotropic rectangle which is simply supported at the edges y = ± Y and free at the edges x = ± X.
Fig. 4. Crystal shape determined analytically and numerically for a simply supported crystal along the central line y = 0 for various aspect ratios y. Crystal width and length are 8 X 8y cm with v = 0.262 . For y = 2.36, the anticlastic curvature is 0 at x = 0. [a] denotes the solution was obtained analytically, [n] denotes the solution was obtained numerically. 0 .40 0 .35 030 025
tal's elastic energy assuming that the shape is a simple fourth order polynomial . However, they do not show that this shape is the proper solution to the differential equation . We have performed finite element calculations using ANSYS [91 for the clamped crystal boundary condition (i .e . z(x, y) = 0 and az/ay ="constant" at y = f Y) . The results of these calculations on the crystal centerline are shown in fig. 5. For y = 1 .40 the anticlastic curvature is negative at the crystal center while at y = 1 .44, the anticlastic curvature is positive. The anticlastic curvature is near zero for y = 1 .42 in the central part of the crystal . The polynomial used by Ferrer et al . [10] does not predict this change in curvature . The centerline slope in the central zone of the clamped and simply supported crystals at or near y 0 is shown in fig. 6. The maximum angular deviation inside a ±7 mm zone for the simply supported crystal and ±4 mm for the clamped crystal is smaller than ±1 grad, which is much better than the 8 .8 wrad Si(111) Darwin width at 30 keV. 3 . Conclusion
0 .20 0 .15 0 10 005 0 -0 .05
Fig. 6. The slope error along the centerline of a simply supported and clamped crystal of width 8 cm and length 8y cm bent to a nominal radius of 1 m. [a] denotes the solution was obtained analytically, [n] denotes the solution was obtained numerically.
0
0 .2
04
0 .6
0 .8 x [cm]
1 .0
1 .2
1 4
Fig. 5. The crystal shape for a clamped crystal of width 8 cm and length 8y cm, v = 0.262 and a sagittal bend of 1 m. Shape is along the central line y = 0 for various aspect ratios y. [n] denotes the solution was obtained numerically.
The analytical case of a cylindrically bent rectangular crystal is considered with certain approximations (the crystal is isotropic, thin, simply supported and loaded with a uniform moment along the supported edges) . At a particular value of the aspect ratio which is dependent on the Poisson coefficient v, the anticlastic curvature in the center of the crystal is equal to zero resulting in an extended central zone with slope errors less than the Darwin width for Si(111) at 30 keV at a 1 m bending radius . A similar aspect ratio exists for the clamped case which is smaller than the aspect ratio for
V.I. Kushmr et al. / Sagittal focusing of synchrotron radiation the simply supported case . These results suggest that the anticlastic bending can be minimized without the use of ribs or slots by using a crystal with the proper aspect ratio. The method to avoid the anticlastic bending of a crystal proposed in this work employs a flat rectangular crystal. This method may be useful to construct X-ray monochromators for existing and future synchrotrons . We are currently building a monochromator based on these principles and will report the results in a future publication . Acknowledgements This work was supported by the State of Illinois through Technology Challenge Grant no . 92-82139 and Du Pont Corporation through their support of the DUNU Synchrotron Research Project for the Advanced Photon Source .
591
References [1] C.J . Sparks, Jr ., B.S . Boric and J.B. Hastings, Nucl . Instr. and Meth . 172, (1980) 237. [2] C.J . Sparks, Jr ., G.E . Ice, J. Wong and B.W . Batterman, Nucl . Instr. and Meth . 195 (1982) 73 . [31 B.W. Batterman and L. Berman, Nucl . Instr . and Meth . 208 (1983) 327. [4] D.M . Mills, C. Henderson and B.W . Batterman, Nucl . Instr. and Meth . A246 (1986) 356 . [5] T. Matsushita, T. Ishikawa and H. Oyanagi, Nucl . Instr. and Meth . A246 (1986) 377. [6] J.J. Wortman and R.A. Evans, J. Appl . Phys ., 36(1) (1965) 153. [7] G.S . Knapp, M. Ramanathan, H.L. Nian, A.T . Macrander and D.M . Mills . Rev. Sci. Insu . 63(1) (1992) 465 . [8] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959) p. 82 . [9] Swanson Analysis Systems, Inc., Houston, PA, USA. [10] S. Ferrer, M. Krisch, F. de Bergevin and F. Zontone, Nucl . Instr and Meth . A311 (1992) 444.
Nuclear Instruments and Methods in Physics Research A328 (1993) 588-591 North-Holland
Section A
On the sagittal focusing of synchrotron radiation with a double crystal monochromator V.I. Kushriir
t, J.P. Quintana and P. Georgopoulos DUNU Synchrotron Research Center, Robert R . McCormick School of Engineering and Applied Science, Northwestern University, 1033 University Place, Suite 140, Evanston, IL 60201-3138, USA
Received 27 October 1992 A method to avoid the anticlastic bending of the second crystal in a two-crystal monochromator for synchrotron radiation is proposed . It is analytically shown that the anticlastic curvature is zero at the center of the crystal for a simply supported isotropic crystal loaded with a constant moment provided that the crystal's aspect ratio is equal to a "golden value" dependent on the Poisson coefficient v. For v = 0.262 (equal to v in the Si(111) plane) this ratio is 2.360. Finite element results are presented on the case of the clamped crystal and show that there is a similar "golden value" approximately equal to 1 .42 for v = 0.262 . 1. Introduction Horizontal focusing of a synchrotron beam with the second crystal in a double crystal monochromator (so called sagittal focusing) is of great importance to the synchrotron radiation user [1-7]. By focusing the relatively large horizontally divergent beam, the optics increase the beam intensity at the sample position . While sagittal focusing using total external reflection from mirrors is limited to a few milliradians due to the small critical angles for X-rays (especially for energies larger than 10 keV), a bent perfect crystal focuses a much larger divergence . However, due to the nonzero Poisson ratio, anticlastic bending causes deviations of the second crystal's diffraction planes from the Bragg position and reduces the throughput [1]. Various solutions to the anticlastic bending problem have been proposed . Sparks et al . [1,2] advocate using crystals with stiffening ribs in the anticlastic direction. Knapp et al . [7] describe the most recent experiment with this design . Batterman and Berman [3], Mills et al . [4] and Matsushita et al . [5] propose a variant of the ribbed crystal which is a "slotted crystal" with periodic parallel slots. However, while successfully eliminating anticlastic bending, ribbed or slotted crystals behave more like polygonal segmented mirrors than cylinders, thereby limiting the achievable focus size to twice the width of the stiff ribs . * Present address : Advanced Photon Source - XFD/362, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439-4815, USA.
In this work, we show that the crystal curvature in the anticlastic direction is a function of the crystal aspect ratio (length divided by width) and becomes zero at an aspect ratio dependent on the Poisson coefficient . 2. Problem formulation We consider a rectangular thin crystal of dimensions 2 X X 2Y which is simply supported at two opposite edges and free at the remaining edges with a constant bending moment applied to the supported edges. A typical shape for the resulting crystal is shown z
Fig. 1. Typical shape for an isotropic crystal of dimensions 2X X 2Y simply supported on the y = ± Y edge with a constant bending moment applied along the supported edges. The crystal is shown translated up from the x-y plane for clarity.
0168-9002/93/$06.00 C 1993 - Elsevier Science Publishers B.V . All rights reserved
V.L Kushner et al. / Sagittal focusing of synchrotron radiation in fig. 1. We employ the following approximations : 1) The crystal thickness is small compared to the length and width (thin crystal approximation) . 2) The maximum deformation is less than the crystal thickness. 3) While Si crystals are orthotropic we nevertheless describe the crystal properties with a single Young's modulus and Poisson ratio and note that the Poisson ratio in the (111) plane is isotropic in the plane [6]. With these approximations, the crystal shape satisfies the biharmonic equation [8] a4 a7
+2
a4 +-)z(x, y)=0 . ay 4 ax 2 ay 2 a
4
(2a) a2 a2 Mx = ( - + v ) z = constant . ax2 y
(2b)
On the free edges x = ±X, and moment and force are equal to zero [8]: (a2 M .= +v- z=0 (2c) y aye axe a2) Fy
a3
a3
f ay 3 +(2-v)axay 2 )
Z=0
(2d)
The solution of eq. (1) under the constraints given by eq . (2) is : z(x, y) =
K
2Y(Y2-y2)
+ KY T_
n=1
(An
cosh(knx)
+xk,B, sinh(k n x)) cos(k n y),
(3)
where kn = (-rr/Y)(n - 1/2) and Cn k 2Y2 n
An : X
(1+v) sinh(k,X)-k n X(1-v) cosh(k,X) ( k n X(1- v) 2- sinh(k n X)(v +3)(1- v) cosh(k,X) ) (4)
Cn sinh(k n X) k,2 Y 2 ( sinh(k,X)(v+3)cosh(k,X)-k n X(1-v) )
C n =2vl (
-1 )n+1
40 30 20 10
IE
0
~H
-10 -20 -30 -40'
On the supported edges y = ± Y and the displacement and moment are given by :
~ .
kn Y
K is a normalization constant dependent on the loading moment and bending stiffness. It will be chosen later to achieve an appropriate value of Ry , the sagittal bending radius . The series in eq. (3) converges in less than ten terms since the coefficients A n and Bn effectively decrease exponentially with n.
589
2
Fig. 2. The anticlastic curvature Kx =(-1/R X)=(a 2z/ax e ) as a function of the crystal aspect ratio y = Xl Y for v = 0.262 . The asymptotic behavior of this function as y exponentially approaches zero.
The crystal shape depends on two parameters : the Poisson coefficient v and aspect ratio y =X/Y. For y -* 0 this solution gives the ratio of the two main curvatures equal to BX/R Y = -1/v in agreement with Sparks et al . [1]. The crystal curvature in the anticlastic direction (i .e . 1/Rx = a2 z/ax e) in the middle of the crystal (x =y = 0) as a function of the aspect ratio is shown in fig. 2 for v = 0.262. This value of v is equal to the in-plane Poisson coefficient for the Si(111) plane. For y - -, this curvature decreases exponentially . In order to obtain a small curvature at the crystal center point (x =y = 0), the crystal's aspect ratio must be large (y > 7) or the aspect ratio must be chosen equal to some "golden value", yo . For v = 0.262 this value is 2.360. The value of yo as a function of v is presented in fig. 3. This curve was obtained by numerically solving for the crossover point seen in fig. 2 over the range of v given in fig. 3. Fig. 4 shows the crystal shape in the central zone along the line y = 0 with a crystal aspect ratio greater than, equal to, and smaller than yo along with the results of a finite element analysis of the crystal shape using the ANSYS [9] program. Here, the crystal dimensions are Y = 4 cm, X = yY, v = 0.262 and the crystal curvature is set to Ry = 1 m. For y < yo , the anticlastic curvature is positive, and for y > yo it is negative . Both cases cause a deviation of the crystal reflecting planes from the exact Bragg position . In practice, it is difficult to provide a "simply supported edge" boundary condition and load it with a constant moment . The more realistic case is of a clamped crystal with the given slope az/ay = "constant" at the two clamped edges y = ±Y. However, this case is difficult to solve analytically . Ferrer et al . [10] attempt to analyze this problem by minimizing the crys-
V.1. Kushnir et al. / Sagittal focusing of synchrotron radiation 10 .0 90 80 7 .0 6 .0 5 .0 4 .0 3 .0 20 10 00 -1 .0 0.0
0 1
0 .2
0 .3
04
05
0 6
0 7
08
x [cm]
Fig. 3. Golden ratio y o = X/ Y as a function of Poisson's coefficient v for an anticlastic bending curvature of 0 at the center of an isotropic rectangle which is simply supported at the edges y = ± Y and free at the edges x = ± X.
Fig. 4. Crystal shape determined analytically and numerically for a simply supported crystal along the central line y = 0 for various aspect ratios y. Crystal width and length are 8 X 8y cm with v = 0.262 . For y = 2.36, the anticlastic curvature is 0 at x = 0. [a] denotes the solution was obtained analytically, [n] denotes the solution was obtained numerically. 0 .40 0 .35 030 025
tal's elastic energy assuming that the shape is a simple fourth order polynomial . However, they do not show that this shape is the proper solution to the differential equation . We have performed finite element calculations using ANSYS [91 for the clamped crystal boundary condition (i .e . z(x, y) = 0 and az/ay ="constant" at y = f Y) . The results of these calculations on the crystal centerline are shown in fig. 5. For y = 1 .40 the anticlastic curvature is negative at the crystal center while at y = 1 .44, the anticlastic curvature is positive. The anticlastic curvature is near zero for y = 1 .42 in the central part of the crystal . The polynomial used by Ferrer et al . [10] does not predict this change in curvature . The centerline slope in the central zone of the clamped and simply supported crystals at or near y 0 is shown in fig. 6. The maximum angular deviation inside a ±7 mm zone for the simply supported crystal and ±4 mm for the clamped crystal is smaller than ±1 grad, which is much better than the 8 .8 wrad Si(111) Darwin width at 30 keV. 3 . Conclusion
0 .20 0 .15 0 10 005 0 -0 .05
Fig. 6. The slope error along the centerline of a simply supported and clamped crystal of width 8 cm and length 8y cm bent to a nominal radius of 1 m. [a] denotes the solution was obtained analytically, [n] denotes the solution was obtained numerically.
0
0 .2
04
0 .6
0 .8 x [cm]
1 .0
1 .2
1 4
Fig. 5. The crystal shape for a clamped crystal of width 8 cm and length 8y cm, v = 0.262 and a sagittal bend of 1 m. Shape is along the central line y = 0 for various aspect ratios y. [n] denotes the solution was obtained numerically.
The analytical case of a cylindrically bent rectangular crystal is considered with certain approximations (the crystal is isotropic, thin, simply supported and loaded with a uniform moment along the supported edges) . At a particular value of the aspect ratio which is dependent on the Poisson coefficient v, the anticlastic curvature in the center of the crystal is equal to zero resulting in an extended central zone with slope errors less than the Darwin width for Si(111) at 30 keV at a 1 m bending radius . A similar aspect ratio exists for the clamped case which is smaller than the aspect ratio for
V.I. Kushmr et al. / Sagittal focusing of synchrotron radiation the simply supported case . These results suggest that the anticlastic bending can be minimized without the use of ribs or slots by using a crystal with the proper aspect ratio. The method to avoid the anticlastic bending of a crystal proposed in this work employs a flat rectangular crystal. This method may be useful to construct X-ray monochromators for existing and future synchrotrons . We are currently building a monochromator based on these principles and will report the results in a future publication . Acknowledgements This work was supported by the State of Illinois through Technology Challenge Grant no . 92-82139 and Du Pont Corporation through their support of the DUNU Synchrotron Research Project for the Advanced Photon Source .
591
References [1] C.J . Sparks, Jr ., B.S . Boric and J.B. Hastings, Nucl . Instr. and Meth . 172, (1980) 237. [2] C.J . Sparks, Jr ., G.E . Ice, J. Wong and B.W . Batterman, Nucl . Instr. and Meth . 195 (1982) 73 . [31 B.W. Batterman and L. Berman, Nucl . Instr . and Meth . 208 (1983) 327. [4] D.M . Mills, C. Henderson and B.W . Batterman, Nucl . Instr. and Meth . A246 (1986) 356 . [5] T. Matsushita, T. Ishikawa and H. Oyanagi, Nucl . Instr. and Meth . A246 (1986) 377. [6] J.J. Wortman and R.A. Evans, J. Appl . Phys ., 36(1) (1965) 153. [7] G.S . Knapp, M. Ramanathan, H.L. Nian, A.T . Macrander and D.M . Mills . Rev. Sci. Insu . 63(1) (1992) 465 . [8] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959) p. 82 . [9] Swanson Analysis Systems, Inc., Houston, PA, USA. [10] S. Ferrer, M. Krisch, F. de Bergevin and F. Zontone, Nucl . Instr and Meth . A311 (1992) 444.