HAL Id: jpa-00225821
https://hal.archives-ouvertes.fr/jpa-00225821
Submitted on 1 Jan 1986
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
INTEGRATION OF SINGLE CRYSTAL NEUTRON DATA USING A P.S.D : A CASE OF LARGE, WEAK
AND OFTEN OVERLAPPING REFLECTIONS
M. Roth, A. Lewit-Bentley
To cite this version:
M. Roth, A. Lewit-Bentley. INTEGRATION OF SINGLE CRYSTAL NEUTRON DATA USING A P.S.D : A CASE OF LARGE, WEAK AND OFTEN OVERLAPPING REFLECTIONS. Journal de Physique Colloques, 1986, 47 (C5), pp.C5-27-C5-34. �10.1051/jphyscol:1986504�. �jpa-00225821�
JOURNAL DE P H Y S I Q U E
C o l l o q u e C5, s u p p l 6 m e n t a u n o 8 , Tome 47, a o O t 1 9 8 6
INTEGRATION OF SINGLE CRYSTAL NEUTRON DATA USING A P.S.D : A CASE OF LARGE, WEAK AND OFTEN OVERLAPPING REFLECTIONS
M. ROTH a n d A . LEWIT-BENTLEY
Institut Laue-Langevin and E M B L , F-38042 Grenoble, France
~ 6 s u m 6 . S o n t p r 6 s e n t k s l e s d k v e l o p p e m e n t s a p p o r t k s 2 n o s programmes d v i n t 6 g r a t i o n d e s r 6 f l e x l o n s , u t i l i s g s d e p u i s p l u s i e u r s a n n k e s e n c r i s t a l l o g r a p h i e n e u t r o n i q u e , e n v u e d l u n e u t i l i s a t i o n d a n s d e s c o n d i t i o n s m o i n s r e s t r i c t i v e s . L e p r i n c i p e d e b a s e e s t r e s t 6 l e mgme, 2 s a v o i r l e c a l c u l d e m a s q u e s
e l l i p s o i d a u x p o u r c h a q u e & f l e x i o n 2 p a r t i r d e s p a r a m ' e t r e s d e 1 1 e x p 6 r i e n c e d e d i f f r a c t i o n . C e t t e mgthode d v i n t 6 g r a t i o n e s t t r 2 s r a p i d e c a r b e a u c o u p d e v a r i a b l e s p e u v e n t s t r e c a l c u l 6 e s p a r a v a n c e . Le programme d v i n t 6 g r a t i o n e s t p r 6 v u p o u r f o n c t i o n n e r e n l i g n e p e n d a n t l l a c q u i s i t i o n d e s d o n n g e s . L e s i n d i c e s d e s
r g f l e x i o n s a p p a r a i s s a n t 2 c h a q u e b t a p e d e l ' e x p 6 r i e n c e s o n t c a l c u l 6 s ?m e s u r e . Une n o u v e l l e m g t h o d e d e t r a i t e m e n t d u b r u i t i d e f o n d , f a i s a n t a p p e l 5 l v i n t e r p o l a t i o n l i n g a i r e
b i d i m e n s i o n n e l l e e t 2 d e s c a l c u l s d e moyenne m o b i l e , a 6 t 6 d g v e l o p p g e p o u r t e n i r m i e u x c o m p t e du f a i t q u e l e b r u i t d e f o n d e s t l e p l u s s o u v e n t non u n i f o r m e e t non s t a t i o n n a i r e .
A b s t r a c t . We h a v e e x t e n d e d o u r d a t a t r e a t m e n t s e r i e s o f p r o g r a m m e s , u s e d r o u t i n e l y f o r s e v e r a l y e a r s now f o r o f f - l i n e i n t e g r a t i o n o f l o w - r e s o l u t i o n n e u t r o n d i f f r a c t i o n d a t a , t o more g e n e r a l c o n d i t i o n s o f u s e . The b a s i c p r i n c i p l e h a s n o t c h a n g e d : i t c o n s i s t s o f t h e a p r i o r i c a l c u l a t i o n o f e l l i p s o i d a l m a s k s f o r e v e r y r e f l e c t i o n . The r e s o l u t i o n f u n c t i o n a l g o r i t h m i s r a p i d , a s m o s t p a r a m e t e r s c a n b e c a l c u l a t e d b e f o r e h a n d . The d a t a t r e a t m e n t i s o n - l i n e , t h e r e f l e c t i o n s a p p e a r i n g on t h e d e t e c t o r a t a n y r o t a t i o n a n g l e a r e p r e d i c t e d p r o g r e s s i v e l y . A c c u r a t e b a c k g r o u n d c o r r e c t i o n i n t h e c a s e o f n o n - u n i f o r m a n d n o n - s t a t i o n a r y
b a c k g r o u n d h a s b e e n d e v e l o p p e d u s i n g l i n e a r t w o - d i m e n s i o n a l s p l i n e f u n c t i o n i n t e r p o l a t i o n a n d moving a v e r a g e t e c h n i q u e s .
I - INTRODUCTION:
The work p r e s e n t e d h e r e a f t e r i s a n e x t e n s i o n a n d a m o d i f i c a t i o n o f a s y s t e m o f d a t a c o l l e c t i o n a l r e a d y p r e s e n t e d i n 1982 / I / , and u s e d e x t e n s i v e l y s i n c e t h e n f o r d a t a c o l l e c t i o n i n n e u t r o n d i f f r a c t i o n e x p e r i m e n t s w i t h a 2 - d i m e n s i o n a l P o s i t i o n S e n s i t i v e D e t e c t o r . The e x p e r i m e n t s w e r e c a r r i e d o u t on t h e s m a l l a n g l e s c a t t e r i n g
s p e c t r o m e t e r Dl7 a t t h e I n s t i t u t Laue - L a n g e v i n and c o n c e r n e d p r o t e i n c r y s t a l l o g r a p h y a t l o w r e s o l u t i o n ( i . e . f o r B r a g g d s p a c i n g s > 15 !?I.
The c o r r e s p o n d i n g d i f f r a c t i o n a n g l e r a n g e i s 28 < 45 d e g . , u s i n g n e u t r o n s o f a b o u t 10 1 w a v e l e n g t h .
A new i n s t r u m e n t f o r l o w r e s o l u t i o n c r y s t a l l o g r a p h y , D B 2 1 , h a s r e c e n t l y b e e n c o n s t r u c t e d a t t h e ILL i n c o o p e r a t i o n w i t h t h e EMSL, G r e n o b l e O u t s t a t i o n , a n d a new s y s t e m o f d a t a c o l l e c t i o n h a s b e e n
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986504
C5-28 JOURNAL DE PHYSIQUE
developed for it. The work is not yet completed, as some parts have still to be tested and integrated into the system.
Both instrument2 have a flat square 2-dimensional PSD, 64x64 cm2 on Dl7 with a 5x5 mm resolution, 20x20 cm2 on DB21 with a 1.5x1.5 mm2 resolution. In both cases the size of a spectrum is 128x128 pixels. A typical size for a diffraction reflection observed on these detectors is 7x7 pixels.
I1 - METHOD OF INTEGRATION:
At low resolution it is possible to use a rather broad beam, with a large wavelength spread (monochromatisation by velocity selector on Dl71 or a large divergence (intercalated pyrolytic graphite
monochromator on DB21). As a consequence (see Fig.1):
1 ) the omega range of diffraction (in normal beam geometry) for a
given reflection is large (>2.0 deg) and variable from one reflection to another.
2) the angle of diffraction of a reflection (i.e. its position on the detector) changes markedly during the scan.
In addition, the reflections are often close to each other and even overlap, because of the large unit cells of the crystals and because the reflections remain excited over a large rotation angle range.
Given this rather complex situation, we have chosen the following method of integration:
- the intensity of a reflection in a given spectrum is integrated by adding the neutron counts inside an appropriate mask whose size and position are calculated a priori from the parameters of the beam and of the crystal.
- the background is estimated from the data outside the masks.
I11 - A PRIOR1 CALCULATION OF POSITION AND SIZE OF MASKS ON THE PSD.
A - CALCULATION OF THE DIFFRACTED INTENSITY DISTRIBUTION.
The following assumptions are made:
1) point diffraction geometry, i.e. sample smaller than the PSD spatial resolution.
2) the incident beam intensity distribution is a 3-dimensional
Gaussian function with respect to the incident wave vector components:
3 2
i(k) = exp(
-t(
( g - h . ) . e i ) I 9 2 (1 1i t 4
& = any incident wave vector, 3, = the average incident wave vector,
9 ; = the 3 principal orthonormal axes (i = 1,2,3) of the distribution
o; = the widths of the distribution along these axes (i = 1,2,3).
In the version of this calculation for Dl7 /I/, the intensity
distribution is assumed symmetrical about the mean beam direction, an hypothesis hardly compatible with a beam monochromated by Bragg diffraction from a single crystal monochromator.
3) the crystal mosaicity is assumed to be isotropic and uniform in reciprocal space with a 2-dimensional Gaussian angular distribution.
An approximate analytical solution of the problem can be found in the form of a Gaussian distribution of the intensity in the diffracted beam, in the case where the angular width of mosaicity is smaller than the beam divergence. It reads:
I(X,Y,w) = exp -
"04 w = rotation angle
= rotation angle corresponding to maximum intensity angles ( spherical coordinates )
= corresponding reflection coordinates on detector
0, ,+',= diffraction angles of maximum intensity at a given w setting.
These parameters and the corresponding variances and covariance appearing in relation (2) are functions of the crystal orientation, the reciprocal lattice vector /H,K,L/ under consideration, the primary beam intensity distribution parameters, the crystal mosaicity and the diffractometer geometry. Their rather lengthy expressions will not be given here.
B - SIZE OF THE MASKS.
To calculate the size of a mask using the expression of I(X,Y,o) it is necessary to define a cut-off limit to the Gaussian function in relation (2) :
i f Definition of the cut-off c : one considers a pixel of coordinates (X,Y) as belonging to the mask M(c) of the reflection under
consideration, for a given angle setting of the crystal, if the value of I(X,Y,w) corresponding to it satisfies the inequality:
This is analogous to the definition of a statistical test. M(c) represents the domain of validity of the hypothesis: "pixel (X,Y) belongs to the reflectionn, and I(X,Y,w), if properly normalized, represents the corresponding probability distribution function.
The threshold of the test is given by:
J,,
I(X,Y,w) sin 8 de d@ dws = i -
5..
I(X,Y,w) sin 0 de d@ d ws is the relative cut-off error due to integration in the limited mask M(c).
ii) To give a value to s, we use the following criterion: The absolute cut-off error on the integrated intensity Ic, corrected for
background, has to be small compared to the standard deviation of Ic,
where: B = average background per pixel
n = number of pixels used to integrate the reflection
(?5-30 JOURNAL DE PHYSIQUE
I n ( 5 1 , i t i s a s s u m e d t h a t t h e e r r o r on B c a n b e n e g l e c t e d , a n d t h a t t h e u n c o r r e c t e d i n t e g r a t e d i n t e n s i t y ( I c + n . B) i s a P o i s s o n random v a r i a b l e . I n p r a c t i c e we u s e t h e r e l a t i o n :
T h i s l e a d s t o t h e d e f i n i t i o n of a mask whose s i z e v a r i e s a s a f u n c t i o n o f t h e r e a l i n t e n s i t y o f t h e r e f l e c t i o n c o n s i d e r e d : t h e s i z e o f t h e m a s k s v a r i e s b o t h a s a f u n c t i o n o f w - w o , a n d f r o m o n e r e f l e c t i o n t o a n o t h e r .
The v a l u e o f n c a n b e c a l c u l a t e d a p r i o r i f r o m t h e v a l u e o f t h e e x p e r i m e n t p a r a m e t e r s a n d t h e v a l u e o f c . To make u s e o f ( 6 1 , we n e e d t o know t h e s c a l e f a c t o r b e t w e e n t h e r e a l i n t e n s i t y o f a g i v e n
r e f l e c t i o n i n a g i v e n r o t a t i o n s e t t i n g a n d i t s t h e o r e t i c a l c o u n t e r p a r t ( 2 ) . T h i s f a c t o r i s c a l c u l a t e d by i n t e g r a t i n g i n X a n d Y , w i t h
b a c k g r o u n d c o r r e c t i o n , t h e d i f f r a c t e d i n t e n s i t y i n a s i n g l e s p e c t r u m m e a s u r e d a t w n e a r t o w, f o r t h e r e f l e c t i o n u n d e r c o n s i d e r a t i o n , a n d d i v i d i n g t h e r e s u l t by t h e c o r r e s p o n d i n g i n t e g r a t e d v a l u e o f
I ( X , Y , w ) .
The c a l c u l a t i o n o f m a s k s must b e m o d i f i e d i n two c a s e s :
- f o r s t r o n g r e f l e c t i o n s , t h e m a s k s o b t a i n e d a r e e n l a r g e d by a d d i n g a r i n g o f o n e p i x e l w i d t h . T h i s h a s t o b e d o n e t o t a k e i n t o a c c o u n t t h e s c a t t e r i n g o f n e u t r o n s o f t h e d i f f r a c t e d beam by t h e m a t e r i a l o f t h e e n t r a n c e window o f t h e d e t e c t o r .
- f o r v e r y weak r e f l e c t i o n s , t h e r e l a t i o n ( 6 ) i s n o t a p p l i c a b l e ( s would b e > 1 ) . The c u t - o f f c i s t h e n g i v e n a v a l u e s u c h t h a t t h e number o f p i x e l s i n c l u d e d i n t h e mask a t w = wo ( i . e . a t maximum i n t e n s i t y ) i s e q u a l t o 2 .
I V - DETERMINATION CRYSTAL MOSAICITY.
The w i d t h o f t h e G a u s s i a n d i s t r i b u t i o n o f t h e m o s a i c i t y i s d e t e r m i n e d by a l e a s t - s q u a r e s a d j u s t m e n t o f t h e t h e o r e t i c a l a n d
o b s e r v e d v a r i a t i o n s o f t h e v a l u e s o f Xm and Ym, t h e c o o r d i n a t e s o f t h e i n t e n s i t y maximum o f a g i v e n r e f l e c t i o n i n t h e s p e c t r a w h e r e i t i s a c t i v e , a n d o f i t s i n t e n s i t y i n t e g r a t e d i n t h e s e s p e c t r a , a s a f u n c t i o n o f t h e r o t a t i o n a n g l e . T h i s d e t e r m i n a t i o n h a s t o b e made on a s m a l l number o f r e f l e c t i o n s b e f o r e t h e s t a r t o f t h e f u l l d a t a r e d u c t i o n .
V - FLOW D I A G R A M OF ON-LINE DATA TREATMENT.
The d a t a t r e a t m e n t h a s t o be made i n two s t a g e s : S i n c e a c o r r e c t b a c k g r o u n d c a l c u l a t i o n c a n o n l y be made on a s p e c t r u m w h e r e a l l masks h a v e b e e n c a l c u l a t e d , a n d b e c a u s e t h e s i z e of t h e m a s k s d e p e n d s o n t h e i n t e n s i t y o f t h e r e f l e c t i o n s , i t i s n e c e s s a r y t o make a f i r s t
e v a l u a t i o n o f t h e i n t e n s i t y o f t h e p r e d i c t e d r e f l e c t i o n s ( t h i s i s made, f o r e v e r y r e f l e c t i o n , on t h e s p e c t r u m w h e r e t h e a v e r a g e B r a g g c o n d i t i o n s a r e s a t i s f i e d ) . They a r e t r e a t e d e f f e c t i v e l y l a t e r o n , w i t h a c o n s t a n t l a g NL s u f f i c i e n t l y l a r g e t o b e s u r e t h a t a l l r e f l e c t i o n s a c t i v e a t N-NL h a v e a l r e a d y h a d t h e i r maximum a t N.
58.0 60.5 w 63.0 deg
F i g u r e 1: E x a m p l e o f v a r i a t i o n s , as a f u n c t i o n o f r o t a t i o n a n g l e w , o f : I , t h e i n t e g r a t e d i n t e n s i t y p e r s p e c t r u m ; B , t h e a v e r a g e
b a c k g r o u n d p e r p i x e l i n a box a r o u n d t h e r e f l e c t i o n ; a n d Xm a n d y m , t h e c o o r d i n a t e s o f t h e p o s i t i o n o f t h e i n t e n s i t y maximum, I n t h e c a s e o f r e f l e c t i o n / I l l / o f a N u c l e o s o m e C o r e P a r t i c l e C r y s t a l i n 100% D20.
C5-32 JOURNAL DE PHYSIQUE
V I - ON-LINE (STEP BY STEP) H,K,L PREDICTION.
+ S p e c t r u m
The method u s e d t o p r e d i c t t h e r e f l e c t i o n s HKL whose i n t e n s i t y maximum ( a v e r a g e Bragg c o n d i t i o n ) o c c u r s i n t h e omega r a n g e :
number N+l N
N-NL
where 6 w i s t h e omega s t e p , i s t h e f o l l o w i n g :
1 ) C a l c u l a t i o n o f HI, H2, K1, K2 , L 1 , L2 d e f i n i n g t h e s e t E l o f HKL1s:
H1 < H < H2, K1 < K < K2, L1 < L < L2
where HI, H2, K1, K2, L1, L2 a r e t h e n a r r o w e s t l i m i t s t o be u s e d f o r g e n e r a t i n g t h e HKL1s t o b e s e l e c t e d . T h e s e l i m i t s a r e c a l c u l a t e d as t h e e x t r e m a o f H , K , and L w i t h t h e c o n s t r a i n t s :
a ) Bragg c o n d i t i o n f o r t h a t omega s e t t i n g ,
b ) d i f f r a c t e d beam i n s i d e a c o n e s u r r o u n d i n g t h e PSD, u s i n g a L a g r a n g e m u l t i p l i e r method p r o p o s e d by T a u p i n / 2 / .
Measurement i n p r o g r e s s
-
F i r s t e v a l u a t i o n o f r e f l e c t i o n p a r a m e t e r s on r e f l e c t i o n s which a r e a t t h e i r maximum on t h i s s p e c t r u m :
1 ) p r e d i c t t h e v a l u e s o f HKL o f t h e s e r e f l e c t i o n s . 2 ) c a l c u l a t e a f i r s t e s t i m a t e o f t h e i r
i n t e n s i t y and d e t e r m i n e t h e i r i n t e g r a t i o n c u t o f f c . 3 ) c a l c u l a t e t h e i r omega l i m i t s (w)min a n d ( w ) m a x .
4) s t o r e H K L , ( w ) m i n , ( w ) m a x , c and o t h e r p a r a m e t e r s i n a n i n d e x e d s c r a t c h f i l e w i t h (w)min a s key ( V A X - s p e c i f i c ) . F u l l i n t e g r a t i o n o f t h e r e f l e c t i o n s , s p e c t r u m by s p e c t r u m :
r e f l e c t i o n s t h a t a r e b e i n g i n t e g r a t e d a r e k e p t i n a t e m p o r a r y a r r a y which i s u p d a t e d a t e a c h s t e p .
1 ) r e c a l l from t h e s c r a t c h f i l e t h e HKL1s which a r e b e g i n n i n g t o be a c t i v e on t h i s s p e c t r u m ( s e a r c h w i t h key ( w ) m i n ) and p u t them i n t o t h e t e m p o r a r y a r r a y .
2 ) c a l c u l a t e t h e masks f o r a l l r e f l e c t i o n s o c c u r i n g on t h i s s p e c t r u m .
3 ) c a l c u l a t e b a c k g r o u n d .
4 ) i n t e g r a t e i n t e n s i t i e s w i t h i n masks and t r e a t o v e r l a p s . 5 ) f o r a n y r e f l e c t i o n s f i n i s h e d ((w)max r e a c h e d ) :
- o u t p u t t h e t o t a l i n t e g r a t e d i n t e n s i t y , c o r r e c t e d f o r background and d e t e c t o r e f f i c i e n c y ,
- c a l c u l a t e t h e s t a n d a r d d e v i a t i o n a n d L o r e n t z c o r r e c t i o n
- e r a s e from t h e t e m p o r a r y a r r a y .
2) S e l e c t i o n i n t h e s e t El o f t h e s u b s e t E2 v e r i f y i n g t h e c o n d i t i o n - -
( 7 , I
3 ) S e l e c t i o n from s u b s e t E2 o f t h e s u b s e t E 3 of t h e r e f l e c t i o n s f a l l i n g e f f e c t i v e l y on t h e d e t e c t o r .
T h i s c a l c u l a t i o n i s v e r y f a s t f o r low r e s o l u t i o n r e f l e c t i o n s .
VIf
-
BACKGROUND CALCULATION AND CORRECTION.METHOD 1 : t h e method u s e d up t o now was t o c a l c u l a t e a n a v e r a g e v a l u e o f t h e b a c k g r o u n d a r o u n d a r e f l e c t i o n a n d t o c o r r e c t t h e
i n t e g r a t e d i n t e n s i t y by s u b t r a c t i n g t h i s v a l u e m u l t i p l i e d by t h e number o f p i x e l s u s e d i n t h e i n t e g r a t i o n . The a v e r a g e was c a l c u l a t e d by s a m p l i n g t h e b a c k g r o u n d c o u n t s f o u n d i n t h e p i x e l s o u t s i d e t h e masks i n t h e v i c i n i t y o f t h e c o n s i d e r e d r e f l e c t i o n , o v e r a l l s p e c t r a w h e r e t h i s r e f l e c t i o n i s a c t i v e .
T h i s method i s s i m p l e a n d f a s t , and a l s o p r e c i s e b e c a u s e a h i g h number o f p i x e l s i s s a m p l e d . N e v e r t h e l e s s i t i s n o t s a f e i f t h e b a c k g r o u n d i s n o t u n i f o r m , b e c a u s e t h e r e i s no c o n t r o l o n t h e b a c k g r o u n d s a m p l i n g .
N o n - u n i f o r m i t y o f b a c k g r o u n d i s i n d e e d o b s e r v e d f r e q u e n t l y , a s w e l l a s v a r i a t i o n s o f b a c k g r o u n d w i t h t h e r o t a t i o n o f t h e c r y s t a l . F o r i n s t a n c e , o n e o b s e r v e s a n a p p a r e n t i n c r e a s e o f b a c k g r o u n d n e a r i n t e n s e r e f l e c t i o n s d u e t o s c a t t e r i n g o f t h e d i f f r a c t e d beam by t h e d e t e c t o r , o r d u e t o t h e r m a l d i f f u s e s c a t t e r i n g ( s e e B i n F i g . 1 ) .
I n t h e method o f Xuong e t a 1 . / 3 / , a n a v e r a g e b a c k g r o u n d i s c a l c u l a t e d f o r e a c h p i x e l i n d e p e n d e n t l y , u s i n g a method o f
a u t o r e g r e s s i v e f i l t e r i n g o f t h e b a c k g r o u n d m e a s u r e d i n t h e same p i x e l b e f o r e t h e a p p e a r a n c e o f t h e r e f l e c t i o n . T h i s s o l v e s t h e p r o b l e m o f n o n - u n i f o r m i t y o f b a c k g r o u n d r a t h e r w e l l b u t i t c a n n o t b e u s e d i n o u r c a s e a s a g i v e n p i x e l c a n r e m a i n i n s i d e masks o v e r a l a r g e p a r t o f t h e c r y s t a l r o t a t i o n r a n g e .
METHOD 2: t h i s i s t h e method w e a r e u s i n g a t p r e s e n t . I t i s a c o m b i n a t i o n o f :
a ) a n i n t e r p o l a t i o n o f t h e b a c k g r o u n d i n e a c h s p e c t r u m by
t w o - d i m e n s i o n a l l i n e a r s p l i n e f u n c t i o n s u s i n g t h e unmasked p i x e l s , and b ) a moving a v e r a g e o f t h e s p l i n e p a r a m e t r e s o v e r t h e M most r e c e n t s p e c t r a .
The method o f i n t e r p o l a t i o n w i t h l i n e a r s p l i n e s h a s b e e n a p p l i e d by S c h w a r t z / 4 / . It i s b a s e d on a d i s t r i b u t i o n o f n o d e s on t h e
d e t e c t o r a r e a , e a c h n o d e b e i n g t h e v e r t e x o f a t r i a n g l e . A l i n e a r v a r i a t i o n o f t h e b a c k g r o u n d i s assumed i n s i d e t h e s e t r i a n g l e s . The v a l u e o f b a c k g r o u n d on t h e n o d e s i s d e t e r m i n e 8 by l e a s t - s q u a r e s . The s y s t e m o f l i n e a r e q u a t i o n s ( t y p i c a l s i z e 100 * 1 0 0 ) i s s o l v e d i n o u r c a s e by a c o n j u g a t e g r a d i e n t method / 5 / .
The c a l c u l a t i o n o f t h e moving a v e r a g e i s a p p l i e d t o t h e v a l u e o f b a c k g r o u n d a t t h e n o d e s and i s made e i t h e r by a s s u m i n g s t a t i o n a r y b o n d i t i o n s o r a l l o w i n g f o r a l i n e a r t i m e e v o l u t i o n o v e r t h e M s p e c t r a , o r w i t h some i n t e r m e d i a t e a p p r o x i m a t i o n .
The a d v a n t a g e o f t h i s method i s t h a t t h e s a m p l i n g o f t h e b a c k g r o u n d i s l e s s l o c a l a n d c a n b e c o n t r o l l e d , p e r m i t t i n g t h e t r e a t m e n t o f n o n - u n i f o r m and n o n - s t a t i o n a r y b a c k g r o u n d c o n d i t i o n s
I t i s , h o w e v e r , r a t h e r s l o w ( 4 s e c / s p e c t r u m on a V A X 7 5 0 ) a n d may i n v o l v e l a r g e a r r a y s .
The b a c k g r o u n d c o r r e c t i o n i s made by a d d i n g t h e i n t e r p o l a t e d a n d a v e r a g e d v a l u e s o f b a c k g r o u n d c a l c u l a t e d i n t h e m a s k s , a t t h e same t i m e a s i t i s made f o r t h e i n t e n s i t y i n t e g r a t i o n a n d t h e t o t a l b a c k g r o u n d i s s u b t r a c t e d a t t h e e n d . T e s t s h a v e shown t h a t t h e p r e c i s i o n i s t h e same a s f o r t h e f i r s t method i . e . i t g i v e s t h e same o r d e r o f m a g n i t u d e f o r t h e s t a n d a r d d e v i a t i o n s o f t h e c o r r e c t e d i n t e g r a t e d i n t e n s i t i e s .
C5-34 JOURNAL DE PHYSIQUE
REMARK ON DETECTOR NORMALISATION.
The m e a s u r e d l o c a l v a r i a t i o n s o f c o u n t i n g i n t h e c a s e o f u n i f o r m f l u x a r e d u e m o s t l y t o a n i m b a l a n c e b e t w e e n n e i g h b o u r i n g p i x e l s ( a n e f f e c t c o m p a r a b l e t o a n o n - u n i f o r m i t y o f t h e p i x e l a r e a s ) a n d n o t t o a l o c a l v a r i a t i o n i n c o u n t i n g e f f i c i e n c y ( s e e f o r i n s t a n c e / 6 / ) . T h i s e f f e c t s h o u l d n o r m a l l y b e i n c l u d e d i n t h e b a c k g r o u n d c a l c u l a t i o n b u t b e c a u s e o f t h e l a r g e number o f p i x e l s i n v o l v e d i n r e f l e c t i o n
i n t e g r a t i o n , i t i s p r e s u m a b l y a v e r a g e d o u t a n d t h e r e l e v a n t c o r r e c t i o n c a n b e d i s c a r d e d . T h i s i s g o i n g t o b e t e s t e d .
V I I I - DISCUSSION.
The p o s i t i v e f e a t u r e s o f t h e method o f a p r i o r i c a l c u l a t e d m a s k s a r e :
- f a s t c a l c u l a t i o n
- a good c o n t r o l o f t h e c o l l e c t e d d a t a ( i . e . c o n t r o l o f t h e q u a l i t y o f t h e c r y s t a l a n d o f t h e d i f f r a c t i o n c o n d i t i o n s )
- a n e a s y t r e a t m e n t o f r e f l e c t i o n s v e r y c l o s e t o e a c h o t h e r o r o v e r l a p p i n g .
The m e t h o d a l s o p r o v i d e s a n e a s y way o f c a l c u l a t i n g a n a p p r o x i m a t e v a l u e o f t h e i n t e g r a t e d i n t e n s i t y f r o m p a r t i a l d a t a .
I t s d i f f i c u l t i e s a n d l i m i t a t i o n s a r e :
- i t r e q u i r e s a good k n o w l e d g e o f t h e o r i e n t a t i o n o f t h e c r y s t a l
- t h e r e f l e c t i o n p r e d i c t i o n a n d mask c a l c u l a t i o n h a v e t o be e x h a u s t i v e ( n o u n p r e d i c t e d r e f l e c t i o n s )
- t h e i n c i d e n t beam a n d t h e c r y s t a l h a v e t o b e " r e g u l a r w ( c l e a n c o l l i m a t i o n a n d no c r y s t a l o r w a v e l e n g t h s p l i t t i n g ) .
Our e x p e r i e n c e s e e m s t o show t h a t t h e G a u s s i a n a p p r o x i m a t i o n b a s e d on t h e r e a l p a r a m e t e r s o f t h e d i f f r a c t i o n e x p e r i m e n t i s good e n o u g h f o r p r e d i c t i n g m a s k s , b u t i t i s n o t a c c u r a t e e n o u g h f o r a p r o f i l e f i t t i n g . N e v e r t h e l e s s , i t may p r o v i d e a s t a r t i n g p o i n t f o r a p r o f i l e a n a l y s i s .
REFERENCES.
/ I / ROTH, M. a n d LEWIT-BENTLEY A . , A c t a C r y s t . A38 ( 1 9 8 2 ) 670-679.
/ 2 / TAUPIN, D . , J . A p p l . C r y s t . 18 ( 1 9 8 5 ) 2 5 3 - 2 5 7 7
/ 3 / XUONG, N . , H . , FREER, S . , T . , HAMLIN, R . , NIELSEN, C . a n d VERNON, W . , A c t a C r y s t . A34 ( 1 9 7 8 ) 289-296.
/ 4 / SCHWARTZ, S . , C . , X p o r t s i n P a t t e r n A n a l y s i s No. 1 3 2 , D i v . o f Appl. M a t h . , Brown U n i v . , P r o v i d e n c e , Rhode I s l a n d , USA ( 1 9 8 3 ) . / 5 / BARKER, V . , A . , NIELSEN, H.,B. a n d THOMSEN, P . , G . " F o r m u l a e a n d
m e t h o d s i n E x p e r i m e n t a l D a t a E v a l u a t i o n n , v o 1 . 3 , ~ h . ~ , L i n e a r A l g e b r a f f , p u b l i s h e d by E u r o p e a n P h y s i c a l S o c i e t y ( 1 9 8 4 1 , CERN, G e n e v a , S w i t z e r l a n d .
/ 6 / HAMLIN, R . , C O R K , C . , HOWARD, A , , NIELSEN, C . , VERNON, W . , MATTHEWS, D . a n d X U O N G , Ng., H . , J . A p p l . C r y s t . 1)1 ( 1 9 8 1 ) 85-93.
