A computer technique for the solution of laue back reflection patterns of cubic crystals. Part II
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If the s o u g h t - f o r r e f l e c t i o n is not a l r e a d y in the e x p e r i m e n t a l set, then the " m o s t p r o b a b l e " location (x*, y*) of a r e f l e c t i o n with d e s i r e d i n d i c e s (hlel) m a y be d e t e r m i n e d by solving the t w o - d i m e n s i o n a l minimization problem:
z* = z(x*, y*) = Min z(x, y),
[1]
x ,y
w h e r e the s u m of s q u a r e d e r r o r s , n
zCx, y) = ~ [Cos ~i(x, Y) - Cos ~i] 2
[2]
i=l
Following the notation p r e v i o u s l y used, the actual i n t e r p l a n a r a n g l e s ~ i a r e given by the t r a d i t i o n a l e x pression:
~h ยง ~k + td Cos ~i
[3]
(h~ + ~ + l~) 1'2 (k s + ~ + Z2)"~
The calculated i n t e r p l a n a r angle &i a s a function of the a s s u m e d r e f l e c t i o n p o s i t i o n i s given by:
c o s ~i = [~(1 + ~ ) ] ' ~
[4]
where: 28. R. L. Fullman:GeneralElectricResearchand DevelopmentCenter,Schenectady, NewYork,privatecommunication,circa 1955.
ui =
D2 + x i x + YiY bi b
[5]
bi = (x~ + y2 + D2)Uz
[6]
b = ( ~ + y2 + Dz)I/2
A Computer Technique for the Solution of Laue Back Reflection Patterns of Cubic Crystals. Part II J . H. CHRISTENSEN, W. H. HUANG, AND R. J . BLOCK
and D is the f i l m - t o - s p e c i m e n d i s t a n c e . T h i s e x p r e s sion has b e e n d e r i v e d f r o m the angle f o r m u l a p r e v i o u s l y u s e d , ~ and involves significantly l e s s c o m p u t~ttion. The highly efficient method of F l e t c h e r and R e e v e s 2 i s u s e d to solve the m i n i m i z a t i o n p r o b l e m . T h i s method r e q u i r e s the evaluation of the p a r t i a l d e r i v a t i v e s (Oz/ax) and (az/Oy); t h e s e can be shown to be:
I N a r e c e n t p a p e r ~ the p r e s e n t a u t h o r s d e s c r i b e d a c o m p u t e r p r o g r a m which had b e e n developed to index Laue b a c k r e f l e c t i o n p a t t e r n s of cubic c r y s t a l s . The p r o g r a m a c c e p t s a s i n p u t the C a r t e s i a n c o o r d i n a t e s of the r e f l e c t i o n s to be indexed obviating the need for any of the u s u a l n e t s o r c h a r t s . The capability of the p r o g r a m has now b e e n extended to p r o v i d e d i r e c t l y the a n g l e s b e t w e e n a set of axes e s t a b l i s h e d on the s p e c i m e n such a s the c r y s t a l f a c e s , and any d e s i r e d set of c r y s t a l l o g r a p h i c d i r e c t i o n s . The m o d i f i c a t i o n of the p r o g r a m was a c c o m p l i s h e d a c c o r d i n g to the following s c h e m e . n e x p e r i m e n t a l r e f l e c t i o n s located at (xi, Yi) a r e a s s i g n e d the M i l l e r i n d i c e s (hikil i) w h e r e i = 1, 2 . . . . n, by the s e a r c h technique p r e v i o u s l y outlined. The s u m (h~ + k~ + l~) p r o v i d e s a r a p i d m e a n s of d e t e r m i n i n g w h e t h e r a p e r m u t a t i o n of any d e s i r e d set (hkl) i s a l r e a d y among the e x p e r i m e n t a l points, s i n c e this sum i s unique for r e f l e c t i o n s having i
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