MATHEMATICAL SIMULATION OF X-RAY OPTICAL SYSTEMS USING MONTE CARLO METHOD

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MATHEMATICAL SIMULATION OF X-RAY OPTICAL SYSTEMS USING MONTE CARLO

METHOD

V. Nikolaev, R. Plotnikov, O. Marenkov, T. Singarieva

To cite this version:

V. Nikolaev, R. Plotnikov, O. Marenkov, T. Singarieva. MATHEMATICAL SIMULATION OF X- RAY OPTICAL SYSTEMS USING MONTE CARLO METHOD. Journal de Physique Colloques, 1984, 45 (C2), pp.C2-161-C2-164. �10.1051/jphyscol:1984236�. �jpa-00223949�

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Colloque C2, supplément au n°2, Tome 45, février 1984 page C2-161

MATHEMATICAL SIMULATION OF X-RAY OPTICAL SYSTEMS USING MONTE CARLO METHOD

V.P. Nikolaev, R.I. Plotnikov, O.S. Marenkov and T.V. Singarieva VNIINauehprybop, Leningrad, U.S.S.B.

Résumé - On a effectué une simulation mathématique des systèmes d'optique des rayons X à cristaux à double courbure. On a mon- tré que les courbes de rendement de réflexion des cristaux à double courbure pour le cas d'une source ponctuelle possèdent deux maxima : R2/Ri = -1 (sphère) et R./R. = sin26Q (surface en forme de tonneau à focalisation ponctuelle). Des cristaux sphériques sont recommandés pour dés montages de rayons X à fentes (diaphragmes).

Abstract - There is performed mathematical simulation of X-ray optical systems with double-bent crystals. It is shown that curves of reflection efficiency for double-bent crystals have two maxima at Ro/R* = 1 (sphere) and R2/Ri = s^n e°

(barrel-shaped surface with point focusing) when using a point radiation source. For X-ray optical systems with slit sources (diaphragms) it is advisable to use spherical crystals.

To compute parameters of complex X-ray optical systems (efficiency, resolution, aberrations, tolerances for manufacture and arrangement of system components and so on) we applied a technique of mathemati- cal simulation by Monte Carlo method /1,2/. The developed technique was used for computation of X-ray optical systems with double-bent crystals which are of high interest in development of X-ray spect- rometers of high efficiency /3,4/. In the present work there was studied distribution of reflecting areas and reflection efficiency for crystals bent along surfaces created by rotation of an arc of radius R.. lying in a focal plane around an axis lying in the same plane. In this case a central point of the arc has rotation radius R2. It was assumed that reflecting planes of the crystal are paral- lel to its surface.

When trajectory of a photon is assigned as a straight line passing via the point radiation source and current point on the crystal sur- face, a glancing angle 6 complementary to the angle between photon trajectory and normal to the crystal surface in the current point may be found. For those points on the surface under study which are located at random or as a net there was computed angle 6 and deter- mined difference between this angle and Bragg angle © . As a result of computation there was obtained a net listing (topogram) with sym- bols of three types of points corresponding to the following cases:

where S is a crystal mosaic. The second case corresponds to ref-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984236

C2-162 JOURNAL DE PHYSIQUE

l e c t i o n . Then t h e r e w a s computed a value of s o l i d angle S2 a t which a r e f l e c t i n g a r e a i s seen from t h e source ( r e f l e c t i o n e f f i c i e n c y ) . I n Fig. 1 t h e r e i s shown succession of topograms f o r various values of 'f = R2/R1

.

those corresponding t o t h e t h i r d c a s e a r e marked by +

Fig. 1 - Topograms of c r y s t a l s u r f a c e ( h e i g h t h/R1 1 0 4 , width b/R1 = 0.2 ) f o r d i f f e r e n t values of 'f = R2/R1 with 00=450, 8 = 0.00058, Source i s l o c a t e d i n focal. plane,

A s i t i s seen from t h e f i g u r e f o r l a r g e values of f j t h e topograms correspond t o I o h m n focusing. A s t h e surface shape approximates t o a sphere a c r o s s i s transformed i n a s l i g h t l y bent v e r t i c d b e l t , Then w i t h decrease of f t h i s b e l t i s compressed in an e l l i p t i c a l spot which i s again extended along t h e v e r t i c d changing a t = 0.5 sin24!j0 ate a s t r a i g h t closed v e r t i c a l b e l t corresponding t o t h e condition of p o i n t focusing a t t h e barrel-shaped s u r f a c e /4/, With t h e f u r t h e r decrease of 7 a c r o s s similar t o t h e i n i t i a l one appears again. It may be shown a n a l y t i c a l l y that t h e h e i g h t of r e f - l e c t i n g a r e a i n t h e c e n t e r of c r y s t a l has two m a x i m a ( a t 7 ^{= 1 and}

7 ³ S i n 00) corresponding t o t h e c r y s t a l bending a l o n t h e sphere 2 and barrel-shaped s u r f a c e with p o i n t focusing (Fig. 27.

When t h e r e f l e c t i n g s u r f a c e i s o b t a h e d by r o t a t i o n of 1ogarithm;lc s p i r a l a r c o r t h e c r y s t a l i s made according t o Iohansson t h e above- mentioned dependence i s a c h a r a c t e r i s t i c of r e f l e c t i n g a r e a height of t h e whole c r y s t a l , i.e. of i t s e f f i c i e n c y . R e f l e c t i o n e f f i c i e n c y as a f u n c t i o n of "f f o r t h e case shown in Fig. 1 i s represented in Fig, 3. A s i t i s seen from Fig. 3 t h e r e a r e 2 t y p i c a l maxima of r e f - l e c t i o n l o c a t e d n e a r t o ? =I and 7 =0,5. The l a t t e r maximum i s nar- row enough and so t h e r e a r e high requirements t o t h e accuracy of R2.

Requirements t o t h e accuracy of R2 when bending along t h e sphere a r e considerably lower. S i m i l a r computations f o r o t h e r values of 80 show t h a t t h e g e n e r a l behavlour of r e f l e c t i o n e f f i c i e n c y curves i s reserved ( maxima a r e n e a r t o =I and 'f = ~ i n 2 8 , ) , t h e s e maxima a r e sharpened with t h e decrease of 8,.

I n Fig. 4 t h e r e a r e shown c r y s t a l s u r f a c e topograms i n c a s e of sour- ce displacement along a no- t o t h e f o c a l plane f o r v a l u e s = 0.5 and 7 ⁼ 1. Corresponding curves of r e f l e c t i o n e f f i c i e n c y a r e shown

F i g * 2 - R e f l e c t i n g a r e a h e i g h t ZR .;Z/R (in d i r e c t i o n o f normal t o f o c a l p l a n e ) as a f u n c t i o n o f f o r c o n d i t f o n s shown i n Fig. 1.

Fig. 3 - R e f l e c t i o n e f f i c i e n c y R as a f u n c t i o n o f 7 f o r c o n d i t i - ons shown in Fig. 1 .

i n Fig. 5. A s i t i s seen from Pigs.4 and 5 arrangements with c r y s t a l s b e n t a l o n g t h e barrel-shaped s u r f a c e w i t h p o i n t f o c u s i n g a r e e x t r e - mely c r i t i c a l t o displacement o f t h e p o i n t s o u r c e a l o n g t h e normal t o t h e f o c a l p l a n e t h a t hampers t h e u s e of slit sources. Unlike t h i s c a s e f o r t h e systems w i t h s p h e r i c a l l y b e n t c r y s t a l s i t i s p o s s i b l e and a d v i s a b l e t o u s e s l i t sources. Computations f o r o t h e r v a l u e s of 8, confirm t h e s e conclusions.

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Fig. 4 - Surface topograms o f t h e same c r y s t a l f o r v a r i o u s d i s p l a c e - ments o f t h e source (ZSR =ZS/R,) a l o n g normal t o f o c a l p l a n e with

5 = 0 . 5 (A) and y = 1 (B).

Fig. 5 - R e f l e c t i o n e f f i c i e n c y o f t h e same c r y s t a l as a f u n c t i o n of ZSR With 'f = 1 ( 1 ) and 'f ⁼ 0.5 ( 2 ) .

References

1. PLOTNIKOV R.I. e t al., Apparatura i metody rentgenovskogo a n a l i z a 23 (1980) 27.

2. PLOTNIKOV R.I. e t a l , , Apparatura i metody rentgenovskogo a n a l i z a 28 (1982) 42.

3 . - ^EGGS J.U., Z. angew. Phys. 20 No.2 (1965) 118.

4. BEMIEMAIJ D.W. e t a l . , Rev. S c i . I n s t r . 3 (1951) 1219.