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A PROPOSED MULTI-BEAM X-RAY
MONOCHROMATOR FOR SYNCHROTRON RADIATION : THEORETICAL CONSIDERATIONS
C.-M. Lo, S.-L. Chang
To cite this version:
C.-M. Lo, S.-L. Chang. A PROPOSED MULTI-BEAM X-RAY MONOCHROMATOR FOR SYN-
CHROTRON RADIATION : THEORETICAL CONSIDERATIONS. Journal de Physique Colloques,
1987, 48 (C9), pp.C9-75-C9-78. �10.1051/jphyscol:1987909�. �jpa-00227204�
A PROPOSED MULTI-BEAM X-RAY MONOCHROMATOR FOR SYNCHROTRON RADIATION : THEORETICAL CONSIDERATIONS
C.-M. LO
and
S.-L. CHANGSynchrotron Radiation Research Center and Department of Physics National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Abstract
-
An X-ray monochromator u t i l i z i n g multi-beam simultaneous r e f l e c t i o n i s proposed f o r synchrotron r a d i a t i o n . The monochroator i s an asymmetrically c u t p e r f e c t s i n g l e c r y s t a l , which i s s e t f o r a m u l t i p l e d i f f r a c t i o n . The s p a t i a l width, t h e angular divergence and t h e wavelength dependence of t h e m u l t i p l y d i f f r a c t e d beam i n r e l a t o n t o t h e i n c i d e n t beam parameters a r e considered t h e o r e t i c a l l y , based on a two-beam approximation of t h e dynamical theory of X-ray m u l t i p l e d i f f r a c t i o n . The transformation m a t r i c e s between t h e parameters of t h e i n c i d e n t and t h e d i f f r a c t e d beams a r e derived i n t h e position-angle-wavelength scheme. I t i s found t h a t a tunable angular acceptance f o r t h e i n c i d e n t beam and a reduction i n t h e d i f f r a c t i o n beam-divergence can be achieved by s e l e c t i n g a proper m u l t i p l e d i f f r a c t i o n . .I . I n t r o d u c t i o n
-
Monochromatic X-radiation i s u s u a l l y d e s i r e d t o have f o r most X-ray d i f f r a c t i o n experiments, with e i t h e r c l a s s i c a l Bremsstrahlung o r synchrotron r a d i a t i o n . In t h e c a s e of synchrotron r a d i a t i o n , s i n g l e c r y s t a l monochromators, curved, channel and asymmetrically c u t , a r e f r e q u e n t l y used i n t h e hardx-ray s p e c t r a l range. Since two-beam dynamical d i f f r a c t i o n ( t h e Bragg d i f f r a c t i o n ) from a s i n g l e c r y s t a l i s t h e main mechanism r e s p o n s i b l e f o r t h e monochromatization of an i n c i d e n t r a d i a t i o n , t h e dynamical theory of X-ray d i f f r a c t i o n [ I ] i s t h e fundamental f o r designing an X-ray monochromator. I n a d d i t i o n , t h e X-ray o p t i c s needed f o r synchro- t r o n r a d i a t i o n should t a k e i n t o account t h e phase diagram o f t h e emitted photons t h e e l e c t r o n s t o r a g e r i n g . The position~angle-wavelength djagram [2] i s a h e l p f u l guide f o r designing two-beam d i f f r a c t i o n monochromator f o r synchrotron r a d i a t i o n .k l t i - b e a m d i f f r a c t i o n , u n l i k e t h e two-beam d i f f r a c t i o n , h a s not been s e r i o u s l y considered a s a means f o r monochromatizing r a d i a t i o n s , though t h e coherent i n t e r - a c t i o n among t h e d i f f r a c t e d beams may provide a beam with an extremely small angular divergence [3,4]. In t h i s a r t i c l e , we provide a fundamental study o n t h e p o s s i b i l i t y of using m u l t i p l e d i f f r a c t i o n f o r X-ray monochromatization. The dynamical e f f e c t and t h e position-angle-wavelength r e l a t i o n f o r multi-beam d i f f r a c t i o n i n an asymmetrically c u t c r y s t a l a r e considered.
11. Theoretical Considerations
(a) Geometry of m u l t i p l e d i f f r a c t i o n : To g e n e r a t e a m u l t i p l e d i f f r a c t i o n , t h e c r y s t a l i s f i r s t a l i g n e d f o r a given G - r e f l e c t i o n , t h e primary r e f l e c t i o n , a n d i s t h e n r o t a t e d around t h e r e c i p r o c a l l a t t i c e v e c t o r
2
o f t h e G - r e f l e c t i o n t o b r i n g a d d i t i o n a l(secondary) s e t s of atomic planes H t o s a t i s f y t h e Bragg c o n d i t i o n . The m u l t i p l e d i f f r a c t i o n (G,H) t h e r e f o r e invoive t h e a n g l e o f j n c i d e n c e 0 f o r t h e G - r e f l e c t i o n and t h e azimuthal a n g l e $ o f r o t a r i o n around t h e g. The condition f o r m u l t i p l e d i f f r a c t i o n , r e f e r r i n g t o [ S ]
,
i scos ,8 = CoX/sinOG
2 +
where Co= (h -h,,g)/Zh~
.
h i s t h e r e c i p r o c a l l a t t i c e v e c t o r of t h e H-reflection.h,, and h, a r e t h e components of
g
p a r a l l e l and perpendicular t o3 .
OG i s t h e Bragg angle f o r G. Bis t h e angle between hL and t h e plane o f incidence of t h e G-reflec-Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987909
JOURNAL DE PHYSIQUE
t i o n . X i s t h e wavelength used. The Lorentz f a c t o r [6] t a k e s t h e form
LF(3-beam) = 1/ (Ah cos
oG
sin$) (2)where $ i s t h e angle between t h e G and H r e f l e c t i o n d i r e c t i o n s p r o j e c t e d on t h e p e r p e n d i c u l a r t o t h e p l a n e o f i n c i d e n c e o f t h e G - r e f l e c t i o n . Note t h a t LF(2-beam) =
l / s i n 20G.
@) Beam parameters: The beam parameters, such a s t h e a n g u l a r p o s i t i o n s and widths of t h e i n c i d e n t and d i f f r a c t e d beams, of a 3-beam (G,H) c a s e , can be d e r i v e d a n a l y t i c a l l y , a s i n t h e two-beam c a s e [Z], provided t h a t t h e two-beam approximation [9] i s employed, t r e a t i n g t h e 3-beam c a s e a s a p e r t u r b e d two-beam case.
Adapt t h e n o t a t i o n of [2] and r e w r i t e t h e parameter W, derived from t h e two- beam approximation f o r a 3-beam (G,H) d i f f r a c t i o n [6] :
w
= -b [ ( l - l / b ) x o / 2 + (Oo-O~) s i n 2 0 ~-
~ A X G X H x H - G l 4 + aoxH%/4b1 / (-1 XG, HI ) (3) W r e p r e s e n t s t h e d e v i a t i o n o f t h e a n g l e of i n c i d e n c e from B G . b i s t h e asymmetry parameter. S i m i l a r l y , t h e parameter r e p r e s e n t i n g t h e d e v i a t i o n of t h e r e f l e c t i o n a n g l e Og from BG can be o b t a i n e d by r e p l a c i n g b by l / b [Z]. x ~ , ~ / ~ T i s t h e e f f e c - t i v e e l e c t r i c s u s c e p t i b i l i t y d e f i n e d a sXG,H = PXG - XHXG-H/Z where
a = -P3X(Tsin$cosOG t a n @ ) , XG = -TFG exp(-M) and
r
= reXL/(.rrv).FG and M a r e t h e s t r u c t u r e f a c t o r and t h e Deby-Waller f a c t o r . r e and V a r e t h e c l a s s i c r a d i u s o f t h e e l e c t r o n and t h e volume o f t h e c r y s t a l u n i t c e l 1 , r e s p e c t i v e l y . P and Pg a r e t h e two-beam and 3-beam p o l a r i z a t i o n f a c t o r s . Equations (3) and (4) a r e n o t v a l i d a t t h e e x a c t 3-beam d i f f r a c t i o n p o s i t i o n , @=0. n - e v a r i a b l e s a a d a ' can b e t r e a t e d a s c o n s t a n t s . From eq. ( 3 ) , t h e peak p o s i t i o n s AOo=Oo-O~ a%d A0 g- -O
O -OG, measured from 0 ~ can be derived by s e t t i n g W=O. The r e l a t i o n between AOoand
~8~
0 i swhere
C = -b R1/R2 with
R1 =
1 xo 1
+ aoXH-GXG-H-
a ~ X H X B +44v I
x G , H I W andR~ = 2 (1-b)
Ixo l
+ aGHXB-
b a b ~ ~+ - ~ ~I X
G , H ~I
-w
~The corresponding a n g u l a r widths, wo and w o f acqeptance and of r e f l e c t i o n a r e g'
wo= 2 I x ~ , ~ I / ( ~ s i n 2 ~ ~ 1 , w g = 2 4 % ~
I x ~ , ~ ~
/ s i n 2OG (7) The s p a t i a l widths So and Sg o f t h e i n c i d e n t and t h e primary r e f l e c t e d beams s a t i s f y t h e r e l a t i o nSg = s 0 / l b l (8)
Equations ( 3 ) - ( 7 ) can be reduced t o t h e i r two-beam forms by l e t t i n g
x ~ - ~
and XHequal t o z e r o .
(c) D e r i v a t i o n o f t h e t r a n s f e r m a t r i x .
Following t h e n o t a t i o n s of [Z], l e t <=A@, ,xP=A0 and %=So, xl=aOg and %=
g g
So, x =S I n t h e 3-beam c a s e , t h e energy dispersion comes i n t o pla$ through eq.
(1). ~ F ' d i f f e r e n t i a t i n g eq. (1) with r e s p e c t t o A , we o b t a i n
-
(A0) t a n 8-
(AB) tan6 = AX/X. (91
This i m p l i e s t h a t 8 and
B
i n c r e a s e a s X d e c r e a s e s . Let t h e a n g u l a r increments be AX;=AO and x ' = A8. The t o t a l a n g u l a r v a r i a t i o n s of t h e i n c i d e n t and r e f l e c t e d beams must s a t f s f y t h e c o n d i t i o n , eq. (S), namely,X I
-
x t = C(X;-
Ax;)g g (10)
with
Ax1 = = A8 =
-
[(AB)/tanB + (AX/X)]coteg (11)
[ z~ 1 = 1 o c
(c-1) tanBtanoG (c-l)coteG][;i
ax/?,
0 0 0 1ax/x
(d) Determination of t h e e f f e c t i v e X G f a c t o r .
The a n g l e s , wo and wg, of a c c e p t & c e and of r e f l e c t i o n depend mainly on t h e e f f e c t i v e X G , ~ f a c t o r f o r a given b v a l u e . While t h e X G , ~ , d e r i v e d fromthetwo-beam approximation, i s an approximate q u a n t i t y . Since t h e approximation i s v a l i d i n t h e a n g u l a r range about 5 t o 10 seconds of a r e o f f t h e exact 3-beam p o s i t i o n [ 7 ] , t h e v a l u e of XG H can be determined by matching t h e azimuthal a n g u l a r width of t h e d i f f r a ~ t i o n ' i n t e n s i t ~ p r o f i l e with t h e a n g u l a r width d e r i v e d from t h e Lorentz f a c t o r :
The i n t e g r a t e d i n t e n s i t y I G of t h e G - r e f l e c t i o n n e a r $=0 i s p r o p o r t i o n a l t o t h e r e a l p a r t of t h e product X G , H K M , ~ , Re[xG HX For centrosymmetric c r y s t a l s [ 8 ] , t h e minimum and i n f l e c t i o n of Ic; occur l h t%i! t a i l of t h e i n t e n s i t y p r o f i l e . The a n g u l a r p o s i t i o n s , Qmin and 4 i n f , of t h i s minimum and t h e p o i n t s of i n f l e c t i o n ( s e e F i g , 1) can be found by e q u a t i n g t o z e r o t h e f i r s t and second d e r i v a t i v e s of
Re[XG,HXH with r e s p e c t t o @. T h i s l e a d s t o @min
'
P3 F H F ~ - ~ / ( F ~ - P s ~ ~ + c o s O ~ ) and$inf = - ( 3 / ~ ) @ , ~ ~ . The i n t e r v a l between
amin
and $ i n f can b e used a s a r e f e r e n c e s c a l e f o r t h e peak width. Numerical c a l c u l a t i o n (Fig. 1 ) [fj] shows t h a t @min=(2_/_3)4inf
and t h e peak width W+ i s - a f r a c t i o n f of Qinf, with f"5%, f o r Ge (000) (222)(111) and (000) (222) (113) d i f f r a c t i o n s f o r CuKcil. From t h i s d i s c u s s i o n , t h e peak width i s expressed a s
W = f P ~ X F ~ F ~ - ~ / ( F - P sin$cosOG)
4
C- (13)A l t e r n a t i v e l y , from t h e Lorentz f a c t o r , W can b e w r i t t e n a s
@ W = 2 1 ~ ILF(S-beam).
$ G ,H (14)
By combining eqs. ( 2 ) , (13), and (14), t h e e f f e c t i v e X G , H i s determined:
2 .
~ x ~ , ~ I
= ~ ( F G FH-G/Fc) (P3 h ~ x slnB)/(2P s i n $ ) . (15)A O(Seconds of arc )
F i g 4 1 C a l c u l a t e d i n t e n s i t y p r o f i l e s : ( a ) Ge 2 2 2 / i i 1 and (b) Ge 222/113
f o r CuKa,
.
Fig. 2 Position-Angle-Wavelengthdiagram f o r 3-bem d i f f r a c t i o n . (e) Position-angle-wavelength scheme
The c h a r a c t e r i s t i c s o f synchrotron r a d i a t i o n a r e governed by t h e t r a j e c t o r i e s o f e l e c t r o n s and photons. They a r e e l l i p s e s i n t h e y-y' space ( v e r t i c a l ) and hyper- b o l a e i n t h e x-XI space ( h o r i z o n t a l ) [9]. By c o n s i d e r i n g t h e e l l i p t i c a l t r a j e c t o r y and t h e t r a n s f e r m a t r i x , eq. (12), t h e p o s i t i o n - a n g l e - w a v e l e n g t h diagram i n t h e y-y'
-A
space can be drawn (Fig. 2 ) . ys i n Fig. 2 i s t h e width of t h e s l i t . The u s e f u l r e g i o n i s t h e volume c o n f i n e d by Bragg's law, eq. ( I ) , t h e geometrical f a c t o r f o r a 3-beam d i f f r a c t i o n and t h e e l l i p s o i d due t o t h e SR source [the shaded p a r t i n Fig.2).111. P r a c t i c a l c o n s i d e r a t i o n
The following requirements, among o t h e r s , should be s e r i o u s l y considered f o r
JOURNAL
DE
PHYSIQUEp r a c t i c a l use: (1) l a r g e wo and small wg and (2) h i g h peak-to-background r a t i o . Based on eqs. ( 7 ) , (15) and Fig. 2 , t o f u l f i l l t h e f i r s t requirement, one should choose -l<<b<O. To s a t i s f y t h e second requirement, a weak r e f l e c t i o n G and two s t r o n g r e f l e c t i o n s , H and G-H should b e chosen. This i s because when F H F K . ~ > F ~
,
a Umweg peak [ i O j i s obtained, and consequently t h e peak-to-background r a t i o i s enhanced. Equation (15) a l s o i n d i c a t e s t h a t when choosing a proper m u l t i p l e d i f f r a - c t i o n , with a p p r o p r i a t e B and $ a n g l e s , FHFH-~/FC r a t i o and f f a c t o r , one could t u n e w, and wg.
A few arrangements a r e proposed f o r p r a c t i c a l u s e w i t h SR. They a r e shown i n Fig. 3. T h e i r c h a r a c t e r i s t i c s a r e l i s t e d i n Table 1. The arguments I and I 1 i n Table 1 mean t h e f i r s t and second c r y s t a l s . T s t a n d s f o r ' t u n a b l e ' .
IV. Conclusions
We have d i s c u s s e d t h e fundamental f e a t u r e s of t h e multi-beam monochromator and have proposed a few arrangements f o r p r a c t i c a l use. Experimental v e r i f i c a t i o n o f t h e s e f e a t u r e s and arrangements i s needed f o r f u r t h e r understanding of t h i s type o f monochromators.
Table 1
C h a r a c t e r i s t i c s of t h e proposed arrangements of monochromators
(L: Large, S:Small, T:Tunable)
Fig. 3 P o s s i b l e arrangements of multi-beam monochromator.
References:
[ I ] Laue, M. von, Exakten Naturwiss.
2
(1931) 133[Z] Matsushita, T. and Hashizume, H., i n ''Handbook on Synchrotron Radiation", Vol.
1, e d i t e d by E. E. Koch (North-Holland, Amsterdam) 1983, Chap. 4.
[3] Kshevetsky, S. A. and Milkhailyuk, I . P., Sov. K r i s t a l l o g r . (1976) 381 [4] Chang, S. L .
,
Agpl. Phys. L e t t .40
(1982) 793[5] Cole, H . , Chambers, F. W. and Dunn, H. M., Acta C r y s t .
25
(1962) 138 [6] Chang, S. L . , "Multiple D i f f r a c t i o n of X-rays i n C r y s t a l s " ( S p r i n g e r ,Heidelberg) 1984
[7] J u r e t s c h k e , H. J . , Phys. Rev. L e t t . 48 (1982) 1487 [8] Chang, S. L . , Phys. Rev. L e t t . 48 (1982) 163 [9] Green, G. K . , BNL Report No. 5 0 E 2 (1976) [ l o ] Renninger, M . , Z . Phys.
