DATA REDUCTION FROM AREA DETECTORS USED WITH CONTINUOUS WAVELENGTH NEUTRON SOURCES

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Submitted on 1 Jan 1986

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DATA REDUCTION FROM AREA DETECTORS USED WITH CONTINUOUS WAVELENGTH

NEUTRON SOURCES

C. Wilkinson

To cite this version:

C. Wilkinson. DATA REDUCTION FROM AREA DETECTORS USED WITH CONTINUOUS

WAVELENGTH NEUTRON SOURCES. Journal de Physique Colloques, 1986, 47 (C5), pp.C5-35-

C5-42. �10.1051/jphyscol:1986505�. �jpa-00225822�

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JOURNAL DE PHYSIQUE

Colloque C5, supplement au n o 8, Tome 47, aoOt 1986

DATA REDUCTION FROM AREA DETECTORS USED WITH CONTINUOUS WAVELENGTH NEUTRON SOURCES

C

.

^WILKINSON

King

's

College London (KQC) , Strand, GB-London WCPR 2LS.

Great-Britain

~6sum6

-

Une technique pour 1 ' i n t e g r a t i o n de 1 'i n t e n s i t e des r e f l e x i o n s de monocristaux a 6 t 6 d6veloppGe pour l e s m u l t i d e t e c t e u r s bidimensionnels avec analyse de temps de v o l . El l e e s t t r e s proche de l a technique a ( I ) / I de reconnaissance de forme pr6cedemment d e c r i t e p a r N i l kinson & Khamis / I / pour l e s faisceaux monochromatiques. Son e f f i c a c i t e a

e t e

mise en evidence p a r l ' a n a l y s e de donnees obtenues

a

Argonne s u r l e d i f f r a c t o m e t r e IPNS pour mono- c r i s t a u x .

A b s t r a c t

-

^Atechnique has been developed f o r t h e i n t e g r a t i o n o f area d e t e c t o r s i n g l e c r y s t a l r e f l e c t i o n s observed w i t h t i m e o f f l i g h t a n a l y s i s . I t i s c l o s e l y r e l a t e d t o t h e p a t t e r n r e c o g n i t i o n o ( I ) / I technique f o r use w i t h monochromatic beams p r e v i o u s l y described by Glilkinson and Khamis / I / . The e f f e c t i v e n e s s o f t h e technique has been observed by a n a l y s i s o f data gathered on t h e s i n g l e c r y s t a l d i f f r a c t o m e t e r a t IPNS, Argonne.

I

-

INTRODUCTION

I n a previous paper

/ I /

a method has been described f o r t h e optimal i n t e g r a t i o n o f d i f f r a c t i o n peaks from a c r y s t a l w i t h an area d e t e c t o r and a monochromatic beam.

I t s implementation i n t h e form o f t h e PEKIrlT program and i t s use i n t h e a n a l y s i s o f data c o l l e c t e d on t h e D l 9 "banana" d e t e c t o r a t ILLl,is described i n t h i s volume /2/.

For t h a t case t h e t h r e e dimensional data a r r a y i n d e t e c t o r space" has t h e two s p a t i a l dimensions o f t h e detector, w i t h t h e t h i r d dimension being steps i n c r y s t a l angle r o t a t i o n . I n t h e case o f a continuous wavelength pulsed beam and an area d e t e c t o r t h e normal method f o r r e c o r d i n g a s i n g l e c r y s t a l r e f l e c t i o n i s t o leave t h e c r y s t a l s t a t i o n a r y and t o analyse t h e t i m e o f a r r i v a l o f d i f f r a c t e d neutrons o f d i f f e r e n t wavelengths, thereby p r o v i d i n g t h e t h i r d dimension i n t h e data a r r a y . The data f o r r e f l e c t i o n s has a s i m i l a r s u p e r f i c i a l resemblance t o t h a t f o r a monochromatic beam, b u t two s i g n i f i c a n t d i f f e r e n c e s should be noted.

The f i r s t i s t h a t t h e w h i t e beam data i s recorded simultaneously i n a l l elements o f t h e data a r r a y and i n f o r m a t i o n i s a v a i l a b l e a t a l l times on r e f l e c t i o n s which fa17 w i t h i n t h e s p a t i a l grasp o f t h e s t a t i o n a r y detector. The experimental s t a t i s t i c s a r e o f course improved as a f u n c t i o n o f t h e measurement time i n t h a t p o s i t i o n . I n t h e monochromatic beam case t h e data i s produced i n a d i f f e r e n t sequence, w i t h r e f l e c t i o n s being completed a t d i f f e r e n t times corresponding t o d i f f e r e n t stages o f c r y s t a l r o t a t i o n . Since t h e b a s i c philosophy of t h e i n t e g r a t i o n method o f PEKINT i s t h a t t h e parameters d e s c r i b i n g t h e shapes o f s t r o n g peaks a r e used t o improve t h e accuracy w i t h which t h e i n t e n s i t i e s o f t h e weak peaks can be measured, t h i s d i f f e r e n c e presents an improved o p p o r t u n i t y f o r data processing w h i l e measurement i s i n progress as t h e s t r o n g peaks are always a v a i l a b l e f o r a c h a r a c t e r i s a t i o n o f t h e i r p r o p e r t i e s .

The second i s t h a t w h i l e i n t h e monochromatic beam case t h e shapes o f t h e peaks i n t h e t h r e e dimensional a r r a y u s u a l l y approximate t o Gaussian e l l i p s o i d s i n v a r i o u s o r i e n t a t i o n s r e l a t i v e t o t h e s p a t i a l and r o t a t i o n dimensions, i n t h e pulsed w h i t e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986505

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C5-36 JOURNAL

DE

PHYSIQUE

beam case t h e s p a t i a l shape i s approximately Gaussian b u t t h e time shape i s i n h e r e n t l y asymmetric due t o t h e d i f f e r e n t r i s e and decay times o f t h e neutron pulses a r r i v i n g a t t h e specimen. The consequence o f t h i s f o r t h e i n t e s r a t i o n method, which p a r t i t i o n s t h e peak according t o t h e volumes o f p a r t i c u l a r i n t e n s i t y contours i s t h a t these volumes a r e no l o n g e r c o n c e n t r i c i n t h e pulsed beam case.

This i s n o t a fundamental problem b u t means t h a t i f t h e peaks a r e t o be p a r t i t i o n e d i n t h i s way t h a t t h e p o s i t i o n s of centres o f t h e volumes o f p a r t i c u l a r i n t e n s i t y contours r e l a t i v e t o t h e centres o f g r a v i t y o f t h e s t r o n g peaks should be recorded.

I 1

-

COElPUTATION OF PEAK SHAPES FOR PULSED BEAM BRAGG REFLECTIONS

Using r e c i p r o c a l s p a c e / r e f l e c t i n g sphere geometry the i n t e n s i t y d i s t r i b u t i o n i n s i n g l e c r y s t a l r e f l e c t i o n s has been c a l c u l a t e d from t h e beam divergence, t h e c r y s t a l 1 it e spread o f t h e specimen, t h e wavelength ( t i m e ) p r o f i l e o f t h e p u l s e a r r i v i n g a t t h e specimen from t h e moderator and t h e p h y s i c a l s i z e o f t h e specimen.

The combined e f f e c t s o f beam divergence and c r y s t a l l i t e spread on t h e peak a t d i f f r a c t i o n angles

r ,

v places i n t e n s i t y a t p o i n t s d r , dv, dT from t h e peak c e n t r e i n a plane i n r

,

v

,

T space s a t i s f y i n g t h e equation

dk =

3

⁼ ( c o s r s i n v dv + s i n r c o s v d r )

- A T 2(1

-

c o s r c o s v

1 ...

^{( 1 )}

(This equation may be d e r i v e d by d i f f e r e n t i a t i o n o f Bragg's law w r i t t e n i n t h e form cos y cos v = 1

-

k2/2d2, keeping t h e i n t e r p l anar spacing d constant. )

Each p o i n t i n t h a t plane i s then convoluted w i t h t h e time d i s t r i b u t i o n o f t h e p u l s e a r r i v i n g a t t h e specimen. F i n a l l y t h e d i s t r i b u t i o n must be convoluted w i t h t h e shape o f t h e specimen. The form o f h a l f h e i g h t peak contours f o r d i f f e r e n t d i f f r a - c t i o n angles

r ,

v i s shown i n f i g . 1 f o r a model i n which t h e primary beam d i v e r g - ence has been made i s o t r o p i c a l l y Gaussian w i t h 0.667' f u l l w i d t h a t h a l f h e i g h t (fwhh) and t h e c r y s t a l l i t e spread i s o t r o p i c a l l y Gaussian w i t h 0.25" fwhh. The i n t e n s i t y d i s t r i b u t i o n has been p r o j e c t e d down t h e time a x i s . No account has been taken i n t h i s case o f f i n i t e specimen size.

v

= 30'

Fig. 1

-

V a r i a t i o n o f t h e shape o f t h e d i f f r a c t e d beam w i t h d i f f r a c t i o n angles

r , v .

I s o t r o p i c beam divergence o f 0.667" fwhh and c r y s t a l 1 i t e spread o f 0.25" fwhh have been assumed. The dashed o u t l i n e i n d i c a t e s t h e area covered by t h e SCD

d e t e c t o r a t Argonne Laboratory.

v= -30°

% - --e-@-@

r=

^30' so" 90" 120"

r=18@

These c a l c u l a t i o n s have been compared w i t h measurements made by Nelmes e t a1 /3/ on a c r y s t a l o f P r o u s t i t e (Ag3AsS3) w i t h the SCD s i n g l e c r y s t a l d i f f r a c t o m e t e r /4/ a t Argonne National Laboratory. The d e t e c t o r was s e t w i t h i t s c e n t r e a t

r

= 90"

v = 0' and spanned t h e range i n

r ,

shown i n d o t t e d o u t l i n e i n f i g u r e 1. i t

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can be seen t h a t i n t h i s r e g i o n t h e p r i n c i p a l axes o f t h e peak shape e l l i p s o i d s should be almost p a r a l l e l t o t h e X, Y d i r e c t i o n s and t h i s was i n f a c t found t o be t h e case. Equation ( 1 ) i n d i c a t e s t h a t i f measured a t t i m e i n t e r v a l s dT which a r e p r o p o r t i o n a l t o T a l l peaks should be t h e same shape a t a p a r t i c u l a r

r,v.

T h i s i s i l l u s t r a t e d i n f i g . 2 where the t i m e v a r i a t i o n o f t h r e e r e f l e c t i o n s a r r i v i n g a t w i d e l y d i f f e r e n t times (and i n p o s i t i o n s marked P,S,Q i n f i g . 1 ) i s seen t o be very s i m i l a r when p l o t t e d as a f u n c t i o n o f dT/T.

Fig. 2

-

^Timedependance o f p u l s e f o r t h r e e

d i f f e r e n t r e f l e c t i o n s from P r o u s t i t e c r y s t a l as a f u n c t i o n o f dT/T.

The r e f l e c t i o n shape i n t h e X,Y d i r e c t i o n s was found t o vary l e s s across t h e

d e t e c t o r area t h a n i n d i c a t e d i n f i g u r e 1, presumably due t o specimen s i z e broadening e f f e c t s . An i s o m e t r i c p r o j e c t i o n o f t h e i n t e n s i t y i n t h e c e n t r a l X,T s e c t i o n o f t h e s t r o n g peak S i s shown i n f i g u r e 3. The maximum count i s 10765 and t h e average background l e v e l i s 17.

10765 F i g . 3

-

Central X,T s e c t i o n through s t r o n g peak S. Maximum count 10765, average background 17.

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JOURNAL DE PHYSIQUE

The assymetric v a r i a t i o n w i t h t i m (sharp r i s e , slow f a l l ) which can be seen i n f i g . 2 and a l s o i n f i g . 3 means t h a t i f t h r e e dimensional constant i n t e n s i t y contours a r e drawn f o r a peak they a r e n o t concentric, b u t t h a t t h e i r c e n t r e s o f g r a v i t y are displaced along t h e t i m e axis. T h i s i s f u r t h e r discussed below.

111

-

THE a ( I ) / I PEAK INTEGRATION METHOD

The o ( I ) / I technique was f i r s t described by Lehmann and Larsen /5/ and has s i n c e been considerably extended

/ I /

t o deal w i t h PSD data.

Suppose t h a t a s t r o n g r e f l e c t i o n i s c h a r a c t e r i s e d by measuring t h e i n t e n s i t y I ( p ) contained i n p peak p o i n t s which l i e w i t h i n a s e r i e s o f shapes s e t up t o approximate t o i n t e n s i t y contours i n t h e peak. L e t I o ( p o ) be t h e " t o t a l " i n t e n s i t y o f t h e peak contained w i t h i n po p o i n t s and l e t x ( p ) be t h e r a t i o I ( p ) / I ( p o ) . Suppose a l s o t h a t t h e r e i s a weak peak o f t o t a l i n t e n s i t y WO(po) which has t h e same shape as t h e s t r o n g peak and s i t s on t h e same background B. I t can then be shown /6/ t h a t t h e q u a n t i t y o(W)/W and a l s o t h e q u a n t i t y o(W/x) [

-

o(Wo)] i s minimised when t h e eauation

i s s a t i s f i e d and t h e r e f o r e corresponds t o t h e minimum e r r o r i n measurement o f W o when t h e peak i s d i v i d e d up i n t h i s way. ( c i s a constant t h e value o f which depends on t h e r a t i o o f t h e number o f background p o i n t s t o t h e number o f peak p o i n t s a t which a(W)/W i s a minimum. I t i s 1 when t h e number o f p o i n t s over which t h e value o f B i s measured i s much g r e a t e r than t h e number o f peak p o i n t s and 2 when t h e two a r e c o n s t r a i n e d t o be equal .)

Equation 2 can be solved f o r t h e s t a t i s t i c a l l y optimum number o f p o i n t s f o r i n t e - g r a t i o n o f t h e weak peak when t h e q u a n t i t i e s IodP/dI and 2p/x have been obtained from nearby s t r o n g peaks. F i g u r e 4 shows these q u a n t i t i e s d e r i v e d from t h e s t r o n g peak S whose c e n t r a l s e c t i o n i s i l l u s t r a t e d i n f i g u r e 3.

" '?

Y

^lo

. -Byy

^I

1

^a!

a i F i g . 4

-

V a r i a t i o n o f x, I d ~ / d I _{, .}- and 2p/x f o r - -

&! numbers o f peak p o i n t s

i n c l u d e d w i t h i n i n t e g r a - O? t i o n contour e l l i p s o i d s

strono Peak S

g = 139287 o.I f o r s t r o n g peak S. The

1; 17 lowest contour e l 1 ip s o i d

a. contained 520 p o i n t s .

0.3

The i n t e g r a t i o n volumes were those e l l i p s o i d s w i t h t h e same moment o f i n e r t i a tensors and same centres as those corresponding t o t h e chosen peak contours, a t t a c h i n g u n i t weight t o each peak p o i n t . T h e i r n a j o r axes a r e so c l o s e t o t h e d e t e c t o r and time axes t h a t f o r s i m p l i c i t y they have been assumed t o l i e e x a c t l y i n

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t h e s e d i r e c t i o n s . Figure

5

shows t h e s e e l l i p s o i d s f o r peak S

i n

X,T s e c t i o n s through t h e i r c e n t r e s .

^A

displacement o f up t o 0.6 elements i n T

is

apparent between high and low contours.

Fig.

5

- X,T s e c t i o n s through i n t e g r a t i o n volumes f o r peak S.

E l l i p s o i d a l i n t e g r a t i o n volumes can be used t o p a r t i t i o n t h e s t r o n g peak I o ( p ) and generate t h e curves of Figure 4 even though t h e t r u e i n t e n s i t y contours a r e n o t s t r i c t l y e l l i p s o i d a l . There is no consequent l o s s of accuracy in estimating

W o

provided t h a t t h e i n t e g r a t i o n volumes a r e c o r r e c t l y positioned r e l a t i v e t o t h e c e n t r e of t h e weak peak.

IV - TEST OF THE METHOD FOR INTEGRATION OF WEAK PEAKS

In o r d e r t o t e s t t h e r e l i a b i l i t y of i n t e g r a t i o n of t h e peak within a a ( I ) / I minimum volume f o r weak peaks, an a r t i f i c i a l s e r i e s o f t h e s e with i n t e n s i t i e s 0.1, 0.01 and 0.001 of and t h e same shape a s t h e s t r o n g peak S were generated.

Including t h e model peak t h e s e represented s i g n a l t o noise r a t i o s (over t h e whole e x t e n t o f t h e peak) o f 16,1.6,0.16 and 0.016. An i s o m e t r i c p r o j e c t i o n through t h e c e n t r a l X,T s e c t i o n o f t h e weakest peak i s shown i n f i g u r e

6.

I t i s viewed from t h e same p o s i t i o n a s f i g u r e

3 .

A peak point (average value 10.8) can just be detected r i s i n g above t h e background (average value 17).

Fig. 6 - Isometric

p r o j e c t i o n o f c e n t r a l X,T

s e c t i o n i n t e n s i t y o f a

peak which has t h e same

shape a s

S,

but is 1000

times weaker and s i t s on

an average background of

17.

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C5-40 JOURNAL DE PHYSIQUE

The q u a n t i t y o(W/x) [ 5 o(Wo)

1

has been c a l c u l a t e d f o r i n c r e a s i n g numbers o f peak p o i n t s a t these s i g n a l t o n o i s e r a t i o s and i s shown as a f u n c t i o n o f t h e normalised v a r i a b l e p/po i n f i g u r e 7. The p r e d i c t e d p o s i t i o n s o f t h e u ( I ) / I minima from equation 1 a r e i n d i c a t e d . (Since t h e s t r o n g e s t peak can n o t be taken as a

"bootstrap" model f o r i t s e l f , t h e minimum value o f u(Wo) occurs a t t h e f u l l l i m i t o f t h e peak, w h i l e i t s o ( I ) / I value l i e s much c l o s e r t o t h e peak c e n t r e . ) T h i s i s a consequence o f t h e very "sharp" nature o f t h e time r i s e o f t h e peaks compared w i t h t h e slower s p a t i a l v a r i a t i o n , and u ( I ) / I p o i n t s can be seen t o g e t i n - c r e a s i n g l y c l o s e t o t h e peak c e n t r e as t h e s i g n a l gets weaker.

F i g . 7

-

o(Wo) c a l c u l a t e d f o r peaks o f t h e same shape as S, ( b u t w i t h v a r y i n g s i g n a l t o n o i s e r a t i o s ) as a f u n c t i o n o f i n t e g r a t i o n volume.

I t can be seen t h a t gajns i n accuracy below 80 peak p o i n t s a r e o n l y s l i g h t and thought t o be t o o r i s k y due t o " s t a t i s t i c a l " mis-centering o f t h e i n t e g r a t i o n volume t o be worthwhile. S e t t i n g a minimum i n t e g r a t i o n volume o f 80 p o i n t s , ( r o u g h l y t h e 1% h e i g h t contour, c o n t a i n i n g 94% o f t h e o v e r a l l i n t e n s i t y ) i s s a t i s f a c t o r y , i n t h a t mis-centerings o f up t o 1 peak element i n any d i r e c t i o n produce o n l y a small e r r o r i n I. T h i s has been e m p i r i c a l l y estimated by d i s p l a c e - ment o f t h e i n t e g r a t i o n volume w i t h i n peak S i n X,Y and T d i r e c t i o n s . The

v a r i a t i o n i s shown i n f i g u r e 8.

Fig. 8

-

V a r i a t i o n i n t h e i n t e n s i t y i n c l u d e d w i t h i n t h e 1% contour as a f u n c t i o n o f t h e c e n t r e of t h e i n t e g r a t i o n volume i n X,Y and T d i r e c t i o n s .

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The o v e r a l l gains i n accuracy o f e s t i m a t i n g 5 (W,) by choosing an i n t e g r a t i o n volume o f 80 r a t h e r than 520 p o i n t s a r e shown i n Table 1. I t can be seen t h a t t h e g a i n i s more than 300% f o r t h e weakest peak considered.

Also shown i n t h e f i n a l column o f Table 1 i s t h e "computationally simulated" value o f u (Wo) which was obtained by 100 repeated generations of peaks o f those

s t r e n g t h s w i t h random n o i s e applied. These values agree w e l l w i t h those p r e d i c t e d by equation 2.

Table 1

-

o(W0) as a f u n c t i o n o f i n t e g r a t i o n volume. Also shown i s t h e

"computationally simulated" value o f o(W0) obtained by repeated peak generation.

V

-

DISCUSSION AND CONCLUSIONS

o(W,) by repeated peak generation

(80 p o i n t s )

- -

127 52 38 SignalINoise

R a t i o

16 1.6 0.16 0.016

The a (1111 method o f peak i n t e g r a t i o n has been adapted and found t o work s a t i s - f a c t o r i l y f o r white beam pulsed source neutron d i f f r a c t i o n .

When measured as a f u n c t i o n o f X,Y p o s i t i o n elements on t h e d e t e c t o r and constant dT/T i n t h e t i m e dimension t h e d i f f r a c t i o n peaks have been found t o vary o n l y s l i g h t l y i n shape over t h e whole volume o f r e c i p r o c a l space sampled. T h i s means t h a t t h i s method o f i n t e g r a t i o n , which r e l i e s on i n t e r p o l a t i o n o f s t r o n g peak parameters a t t h e p o s i t i o n o f weak r e f l e c t i o n s i s even more s u i t a b l e f o r t h i s case than the monochromated beam/crystal r o t a t i o n method, where i t . h a s p r e v i o u s l y been demonstrated t o work. Indeed, because o f the "sharp" n a t u r e o f t h e observed d i f f r a c t i o n peaks the method can be s i m p l i f i e d s t i l l f u r t h e r and t h e "minimum"

i n t e g r a t i o n volume adopted f o r a l l peaks which are n o t considered "strong" ( d e f i n e d t o have s i g n a l t o n o i s e r a t i o s measured over t h e

whole

range o f t h e peak of

g r e a t e r than 1).

o(W0) estimated from 520 p o i n t s 60 p o i n t s

400 400

185 130

148 58

143 45

T h i s has p a r t i c u l a r advantages when t h e peaks a r e c l o s e t o g e t h e r in t h e o v e r a l l data array, as e r r o r s due t o o v e r l a p o f t h e o u t e r regions o f peaks can be avoided.

It a l s o means t h a t on7y p a r t i a l l y recorded ("edge") peaks can be d e a l t w i t h s a t i s f a c t o r i l y , thereby i n c r e a s i n g t h e o v e r a l l number o f r e f l e c t i o n s which can be e x t r a c t e d from a data array. T h i s f e a t u r e i s o f course even more valuable when' u s i n g a "banana" d e t e c t o r .

w

Q

139287 13929 1392 139

By i t s nature, t h e method i s p a r t i c u l a r l y s u i t a b l e f o r t h e time o f f l i g h t method, as o n - l i n e processing f o r t h e establishment o f a p p r o p r i a t e peak i n t e g r a t i o n volumes can proceed as t h e data s t a t i s t i c s b u i l d up and a l i s t o f i n t e g r a t e d i n t e n s i t i e s be r a p i d l y e s t a b l i s h e d as soon as measurement has f i n i s h e d . The o ( I ) / I procedure w i 11 be implemented f o r t h e s i n g l e c r y s t a l d i f f r a c t o m e t e r

(SXD) on t h e I S I S s p a l l a t i o n source a t t h e K u t h e r t o r d Appleton Laboratory.

V l

-

ACKNOWLEDGEMENTS The author i s g r a t e f u l t o Dr. W.I.F. David f o r t h e p r o - v i s i o n o f experimental data on Ag,AsS,, t o Dr. J.B. F o r s y t h and Dr. R. S t a n s f i e l d f o r u s e f u l discussions and Miss V. Leavers f o r assistance i n p r e p a r i n g some o f t h e diagrams.

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C5-42 JOURNAL DE PHYSIQUE

REFERENCES

/1/ Wilkinson, C. and Khamis, H.W. i n P o s i t i o n S e n s i t i v e D e t e c t i o n o f Thermal Neutrons, Academic Press (1983) 358.

/ 2 / S t a n s f i e l d , R.F.D. T h i s volume.

/3/ Nelmes, R.J., Howard, C.J., Ryan, T.W., David, W.I.F., Schultz, A.J. and Leung, P.C.W., J. Phys. C. 17 (1984) L861.

/4/ Schultz, A.J., Srinivasan, S(T, T e l l e r , R.G., Williams, J.M. and Lukeheart, C.M.

3. Am. Chem. Soc. 106 (1984) 999.

/5/ Lehmann, P1.S. and E s e n , F.K., Acta Cryst. A30 (1974) 580.

/6/ Wilkinson, C. and Khamis, H.W., A.I.P. Conference Proceedings No 89 (1981) 111.