The Inverted Statistical Chopper Facility for Elastic and Inelastic Neutron Scattering Experiments EUR 5047

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THE INVERTED STATISTICAL CHOPPER FACILITY

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-FOR ELASTIC AND INELASTIC NEUTRON

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SCATTERING EXPERIMENTS

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by

W. KLEY and W. MATTHES

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1974

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LEGAL NOTICE

document was prepared under the sponsorship of the Commission European Communities.

oi tne european communities.

Neither the Commission of the European Communities, its contractors nor any person acting on their behalf:

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to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method r process disclosed in this document may not infringe privately owned rights; or Η * 5 Ι « ί ( | % ^ ί ^ ^ |

EUR 5047 e

T H E INVERTED STATISTICAL CHOPPER FACILITY FOR ELASTIC A N D INELASTIC N E U T R O N SCATTERING EXPERIMENTS

by W. KLEY and W. MATTHES Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Luxembourg, January 1974 — 36 Pages — 16 Figures — B.Fr. 50.—

In this paper it is shown that the inverted time of flight spectrometer (inverted statistical chopper) has a large accessibility in the K, ω-plane and that the energy resolution function and the intensity is rather constant over a large energy range. The inverted geometry is therefore very favourable to perform precision measure­ ments of line width and line shape of local modes, high energy phonons and magnons. In principle a 47iEe-MgO window detector can be built for high intensity low background experiments. The role of the statistical chopper is primarily to reduce the continuous elastic and inelastic background.

EUR 5047 e

THE INVERTED STATISTICAL CHOPPER FACILITY FOR ELASTIC A N D INELASTIC N E U T R O N SCATTERING EXPERIMENTS

by W. KLEY and W. MATTHES

Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Luxembourg, January 1974 — 36 Pages — 16 Figures — B.F.r 50.—

In this paper it is shown that the inverted time of flight spectrometer (inverted statistical chopper) has a large accessibility in the K, ω-plane and that the energy resolution function and the intensity is rather constant over a large energy range. The inverted geometry is therefore very favourable to perform precision measure­ ments of line width and line shape of local modes, high energy phonons and magnons. In principle a 4rcEe-MgO window detector can be built for high intensity low background experiments. The role of the statistical chopper is primarily to reduce the continuous elastic and inelastic background.

EUR 5047 e

T H E INVERTED STATISTICAL CHOPPER FACILITY FOR ELASTIC A N D INELASTIC N E U T R O N SCATTERING EXPERIMENTS

by W. KLEY and W. MATTHES

Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Luxembourg, January 1974 — 36 Pages — 16 Figures — B.Fr. 50.—

All the results concerning the merits or advantages of a conventional statistical chopper compared to a common time of flight spectrometer are as valid for the inverted spectrometer.

An inverted statistical chopper facility is a useful and interesting spectrometer for high energy transfer measurements as well as for high resolution powder diffraction work.

All the results concerning the merits or advantages of a conventional statistical chopper compared to a common time of flight spectrometer are as valid for the inverted spectrometer.

An inverted statistical chopper facility is a useful and interesting spectrometer for high energy transfer measurements as well as for high resolution powder diffraction work.

All the results concerning the merits or advantages of a conventional statistical chopper compared to a common time of flight spectrometer are as valid for the inverted spectrometer.

EUR 5047 e

COMMISSION OF THE EUROPEAN COMMUNITIES

THE INVERTED STATISTICAL CHOPPER FACILITY

FOR ELASTIC AND INELASTIC NEUTRON

SCATTERING EXPERIMENTS

by

W. KLEY and W. MATTHES

1974

ABSTRACT

In this paper it is shown that the inverted time of flight spectrometer (inverted statistical chopper) has a large accessibility in the K, ω-plane and that the energy resolution function and the intensity is rather constant over a large energy range. The inverted geometry is therefore very favourable to perform precision measure­ ments of line width and line shape of local modes, high energy phonons and magnons. In principle a 47rBe-MgO window detector can be built for high intensity low background experiments. The role of the statistical chopper is primarily to reduce the continuous elastic and inelastic background.

All the results concerning the merits or advantages of a conventional statistical chopper compared to a common time of flight spectrometer are as valid for the inverted spectrometer.

An inverted statistical chopper facility is a useful and interesting spectrometer for high energy transfer measurements as well as for high resolution powder diffraction work.

KEYWORDS

TIME-OF-FLIGHT SPECTROMETERS ENERGY RANGE

ENERGY RESOLUTION PHONONS

MAGNONS RESOLUTION

MEASURING INSTRUMENTS PULSED NEUTRON TECHNIQUES SCATTERING

CONTENTS

pa g e

I. Introduction 5

Π. The C h a r a c t e r i s t i c P a r a m e t e r s of a Conventional and

Inverted Statistical Chopper Facility 7

II. 1 - General Considerations Concerning the Intensity and Resolution of a Time of Flight Scattering Ex­

p e r i m e n t Using a White Neutron Beam at a Hot

Neutron Source 7 II. 2 - Comparison of a Conventional and an Inverted Sta­

tistical Chopper Facility 13

II. 2. 1 - The K,u) - r a n g e of the Two Instruments 13

II. 2. 2 - The Resolution Functions of the Inverted and Conventional Statistical Chopper F a c i ­

lity 1 4

II. 2. 3 - Comparison of the Intensity of the Two I n s t r u ­

ments 15

III. Fields of Application 2 0

IV. The Theory of the Inverted Statistical Chopper Facility 21

V. Conclusions 26

I. Introduction

In I968 the c o r r e l a t i o n technique h a s been i n t r o d u c e d by a n u m b e r of

f "7

a u t h o r s for t i m e of flight s p e c t r o m e t e r s at s t a t i o n a r y r e a c t o r s . The

application of c o r r e l a t i o n techniques for pulsed n e u t r o n s y s t e m s h a s b e e n / 7 ­ I 4 7

d i s c u s s e d a l s o in the following y e a r s . With the p r o g r e s s of the u n d e r ­

standing of the c o r r e l a t i o n technique, the l i m i t a t i o n s of this m e t h o d for s t a ­

t i o n a r y r e a c t o r s a p p e a r e d c l e a r l y . Weak l i n e s cannot be m e a s u r e d in the

p r e s e n c e of i n t e n s e lines as is v e r y often the c a s e for c o h e r e n t e l a s t i c and

i n e l a s t i c , and in p a r t i c u l a r , for i n c o h e r e n t e l a s t i c , q u a s i ­ e l a s t i c and i n e l a s ­

tic s c a t t e r i n g e x p e r i m e n t s . Also weak l i n e s in the p r e s e n c e of l a r g e a v e r a g e

i n t e n s i t i e s cannot be o b s e r v e d as obtained often in i n c o h e r e n t i n e l a s t i c s c a t ­

t e r i n g e x p e r i m e n t s . T h e s e negative a s p e c t s do not e n t e r so m u c h in c o n s i d e r ­

ation for i n c o h e r e n t q u a s i ­ e l a s t i c s c a t t e r i n g e x p e r i m e n t s , w h e r e the q u a s i ­

e l a s t i c line is dominant within a l a r g e K-range and of i n t e r e s t a l o n e . The

A /

v a r i o u s t h e o r e t i c a l a n a l y s e s and a l s o an e x p e r i m e n t by Sköld have d e ­

m o n s t r a t e d that the m a i n advantage of the pseudo ­ s t a t i s t i c a l chopper a p p e a r s

in i t s application for e x p e r i m e n t s that have to be c a r r i e d out with a l a r g e and

u n c o r r e l a t e d b a c k g r o u n d o r i g i n a t e d from the n e u t r o n b e a m impinging on the

s a m p l e and from the r o o m background of insufficiently s h i e l d e d d e t e c t o r s .

C e r t a i n p r o b l e m s in p h y s i c s (diffraction, q u a s i ­ e l a s t i c s c a t t e r i n g , high e n e r ­

gy phonons, local m o d e s , i n t e r n a l m o l e c u l a r m o d e s , m a g n e t i c s c a t t e r i n g ,

e t c . ) r e q u e s t i n f o r m a t i o n of the s c a t t e r i n g law S(/<,u>) in a r a t h e r l a r g e range

of K and u; w h e r e

« = k. -

κ

1 f

stands for the m o m e n t u m ­ and

2

2m v"i

■t ^ ti , 2 2N

tiu;= E. _ E , = (k. ­ k,)

for the e n e r g y t r a n s f e r in the s c a t t e r i n g p r o c e s s , k., k . and E . . E denote the

1 f 1 f

— 6 —

This type of experiments r e q u i r e s high energy neutrons up to 1 eV and sometimes even higher. F o r a number of these experiments it is indicated to use a white neutron beam in o r d e r to cover the l a r g e s t possible range in

the h,UJ -plane. A white neutron b e a m , from a hot neutron s o u r c e , is the

most adequate one. F o r the energy definition of the neutrons of a white neutron beam, the time of flight method is the m o s t appropriate one. The application of a statistical chopper in the white neutron beam that modulates neutrons with an energy below 1 eV, but not above 1 eV, is m o s t useful, since a high duty cycle (& 50%) can be used in the time of flight s p e c t r o m e t r y and the continuous fast neutron background of the p r i m a r y beam can be s u p p r e s s e d by the correlation technique as well as the room background.

II. The Characteristic Parameters of a Conventional and Inverted Statistical

Chopper Facility

II. 1 General Considerations Concerning the Intensity and Resolution of a

Time of Flight Scattering Experiment Using a White Neutron Beam at

a Hot Neutron Source

According to the notations of Fig. 1, the intensity of a scattering expe­

riment at a pulsed neutron source is given by

(1) J^ =/ ¿0(E.,&.,t)cLE..dA.dt.V.f.F_.N. d.

v ' Detector J 4T Γ ι ι ι ι ι Τ

2

(Ε.,Λ., Εί }i l ). dn dEf. f . ε (E J / n / s e c j

where

* díLdE v i ' i ' "

<f> ( Ε . , Λ . . , t ) i s t h e flux p e r u n i t e n e r g y a n d p e r u n i t s o l i d a n g l e a t t h e s u r ­

f a c e of t h e n e u t r o n s o u r c e a t t i m e t

d E . i s t h e e n e r g y s p r e a d of t h e n e u t r o n s w i t h e n e r g y E . a t t h e t a r g e t

dSL. i s t h e s o l i d a n g l e e x p a n d e d b y a n a r e a e l e m e n t d F . of t h e n e u ­

t r o n s o u r c e a r e a F . (dQ­. =dF./'C w h e r e *-. i s t h e d i s t a n c e

s i l i ι

s oure e­target)

dt the time integration differential

t the time variable

V* the neutron source pulse frequency

f. is a factor taking into account the intensity loss in the appara­

tus between the neutron source and the target

F is the effective target area (= neutron beam cross section)

Ν is the atom density of the target material

¿F(

E

i'

n

i'

E

A)= ^ r - Γ ·

3

^

ia

i

i

i s t h e d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n

k , k . a r e t h e i n i t i a l a n d f i n a l n e u t r o n m o m e n t a ι f

^ = k . ­ k i s t h e m o m e n t u m t r a n s f e r of t h e n e u t r o n on t h e t a r g e t i n t h e i f

s c a t t e r i n g p r o c e s s

hcO= E . ­ E i s t h e e n e r g y t r a n s f e r of t h e n e u t r o n on t h e t a r g e t i n t h e s c a t ­

t e r i n g p r o c e s s

d­ü­. i s t h e s o l i d a n g l e e l e m e n t e x p a n d e d by a n a r e a e l e m e n t d F of

t h e n e u t r o n d e t e c t o r a r e a F (diX = d F J Ζ , w h e r e / i s t h e

d i s t a n c e t a r g e t ­ d e t e c t o r )

d E i s t h e e n e r g y s p r e a d of t h e n e u t r o n s w i t h e n e r g y E a t t h e d e ­

t e c t o r

£ (E ) i s t h e n e u t r o n d e t e c t o r e f f i c i e n c y

f i s a f a c t o r t a k i n g i n t o a c c o u n t t h e i n t e n s i t y l o s s i n t h e a p p a r a ­

t u s b e t w e e n t h e t a r g e t a n d t h e d e t e c t o r

If n e u t r o n s a r e s c a t t e r e d c o h e r e n t l y from, a s i n g l e c r y s t a l , t h e e l a s t i c a s

w e l l a s t h e i n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n i s g i v e n i n t e r m s of o ­ f u c t i o n s

a n d t h e i n t e g r a t i o n / e q u a t i o n ( l ) / h a s t o be c a r r i e d o u t ( i n t e g r a t e d i n t e n s i t y ) .

F o r a l l i n c o h e r e n t e x p e r i m e n t s t h e c r o s s s e c t i o n i s a r a t h e r s m o o t h f u n c t i o n

of I H I a n d CO a n d t h e r e f o r e e q u a t i o n ( l ) c a n be w r i t t e n a c c o r d i n g to M a i e r ­ / I 57

L e i b n i t z i n a m o r e s i m p l i f i e d m a n n e r . A s s u m i n g f u r t h e r m o r e a c o n t i ­

n u o u s n e u t r o n s o u r c e , t h e i n t e n s i t y of n e u t r o n s c a t t e r i n g e x p e r i m e n t s u s i n g

t i m e of flight ( T . O. F . ) ­ t e c h n i q u e s a n d c o n v e n t i o n a l c r y s t a l s p e c t r o m e t r y

a s o u t l i n e d i n F i g . 2, c a n be c a l c u l a t e d b y

(2) J „ , = 7 ­ 9 ( E . , i l . ) . A E . . £ Q . . . A t . V . f . . Fr r. N . d.

D e t e c t o r 4 / Γ7 ι ι ι ι ι Τ

_ ¿ X _ _ . Δ ΰ , , Δ Ε f e ( E ) / n / s e c 7 .

— 9 —

A c c o r d i n g to F i g . 3

F A k( x ). A k( y )

(3)

*V~f

=

l

r

1

1 ¿Z k2

ι ι

(4) tfl, = v<;.

2

ΔΕ Ak Δ Ε , A k ,

1 1 I I

(6)

ΛΕ. = f - k . . A k

( z )

; ΔΕ = f - . k . Δ ^

ζ )

ι 2m i i f 2m f f

(7)

AV. ^ k f

x )

. A k

( y )

. A k f

Z

U

3

1 1 1 1

(8)

AV = A k !

X

W

y )

. A l > U

3

f f f f

and since for a s c a l a r flux

­ E . / E E. Δ Ε .

(9)

f a ^ ^ t

l

.

F

\ ^

and f u r t h e r m o r e

^ 2 ^ 2 Λ2 2

10 E = k „ . T = —­.kT ; E. = ­ — . k . ; E = — ­ . k

Τ Β 2m Τ ι 2m ι f 2m f

■ρ

(11) Κ = k. ­ k . ; Tico = E. ­ E = — (k. ­ k J

i f i f 2 m i f

we can r e w r i t e equation (2)

(12) J n t = A ( T ) . f . . F N . d . (Γ . S ( K , ^ ) . f Δ ν , . Δ ν L(E )

— 10 —

2 / 1 . 2

(13) A(T) =

­kí/k

total e

Φ

2 +Α

87Γ xah k

Τ AT ΤΩ.

/»(Τ)

a n d

(14) /> (Τ) = Δη <£ total m e

-Φ4

AV.

ι

2—

_Α

4

'

Ρ (Τ) stands for the n e u t r o n density in m o m e n t u m s p a c e . The i n t e n s i t y

at the t a r g e t and the d e t e c t o r is t h e r e f o r e p r o p o r t i o n a l to

-k

2

/¿

ι 1

( 1 5 ) J

T a r g e t * ^ T 4 ­

k

T

Av.

<K

(16) J ^ feS (%uj).

D e t e c t o r 4

. Δ ν . . ÄV .

In a T. O. F . e x p e r i m e n t as i l l u s t r a t e d in F i g . 2, w h e r e the collimation

o r Δ&. is the s a m e for all k., the m o m e n t u m s p a c e volume Δ ν . is p r o p o r ­

tional to k. since

*P

At

, λ " ι Δι

( 1 , ) k. ­ t ­

1

At

1

y.-1

"At

Ζ '

L 1

*

m . k. = const, k i i

a n d

(18) Δ Λ . = AkW . A k( y )/ k2

and t h e r e f o r e

(19) û V . * M . . k .

1 1 1

— 11 —

­ki/ kT 4 ­ E . / ET E , 2

(20) I ­ ­ a ­ T ­ ­ . k S . * T . S 1 = e "X. x2

T a r g e t 4 i \ E /

kT l

In F i g . 4 the r e l a t i v e i n t e n s i t y at the position of the t a r g e t is plotted as

a function of χ = E . / E It can be s e e n that the i n t e n s i t y is a slowly v a r y i n g

function in a r a t h e r l a r g e e n e r g y r a n g e . Since in any n e u t r o n s c a t t e r i n g ex­

p e r i m e n t the r e s o l u t i o n h a s to be adjusted to the n e u t r o n s o u r c e s t r e n g t h ,

the T. O. F . ­technique i s an a p p r o p r i a t e m e t h o d for diffraction and n e u t r o n

d o w n ­ s c a t t e r i n g e x p e r i m e n t s . The i n v e r t e d s t a t i s t i c a l chopper facility as

d i s c u s s e d in this context is a T. O. F . ­ s p e c t r o m e t e r with s p e c i a l f e a t u r e s .

As it can be s e e n from F i g . 2, the d i s t a n c e T a r g e t ­ C r y s t a l ­ D e t e c t o r can

be changed at will and t h e r e f o r e the r e s o l u t i o n and i n t e n s i t y , d e t e r m i n e d

by AV adjusted a p p r o p r i a t e l y . F r o m equation (16) it follows that the i n ­

t e n s i t y d i s t r i b u t i o n at the d e t e c t o r is d e t e r m i n e d by the s t a t i c and dynamic

s t r u c t u r e of the t a r g e t m a t e r i a l . F o r d o w n ­ s c a t t e r i n g e x p e r i m e n t s with the

t a r g e t m a t e r i a l at a t e m p e r a t u r e Τ for which k . Τ 4í neo, the intensity at

the d e t e c t o r is d e t e r m i n e d by the double differential c r o s s s e c t i o n o r the

s c a t t e r i n g law S(f\,<¿>) which is given in the c a s e of the h a r m o n i c o s c i l l a t o r

by

d2

m=0 f f

and for m = 1

2 2

2 k <u >

<

2 2

> Α

=

<Λ

ΪΓ·

6

"

2

.(«

2

<4.¿(

Ef

-vH>

f f i

a n d

2 . 2 ,

(23) S(H,u>) = e 2 . ( Λ υ Ζ» . ί ( Ε E. + ^ W )

— 12 —

The scattering law has an optimum at

(24) K2

opt . 2.

where u stands for the oscillator amplitude. For the isotropic harmonic

oscillator this amplitude can be calculated easily from

(25) mco<u> = [n(co )+~] · t)U>

o o ¿ o

2 Á2 1

<u > =

2m * hijo

o

where n(¿J ) stands for the Bose occupation number. The maximum intensity

is therefore obtained in experiments with

(26) K2 = ~ ( 2 * ω ) .

opt ±¿ o

. 2

Fig. 5 shows the Κ ­dependence of the scattering law of the isotropic har­

monic oscillator. In a white neutron beam, with the intensity distribution

as shown in Fig. 4, the optimum intensity condition can be obtained easily

for any vibrational state and frequency distribution function. In a real expe­

riment where multi­phonon processes occur, it is necessary to measure 2

the energy transfer and the natural line shape as a function of Κ in the 2 2

range Οζ Κ £ Η in order to eliminate the multiphonon contributions opt

that lead to ambiguous results. Therefore it is desirable to collect data in

a large space of the K,(^> ­plane. For measurements concerning the density

of state distribution function, where a large range of the possible energy

transfer nU) ought to be measured at small Κ ­values, the white neutron

beam experiment is most suitable. It may be mentioned that the energy ca­

libration of the crystal spectrometer and the determination of the reflecti­

vity of the crystals can be done by using a Vanadium target and the time of

13

efficiency h a s to be d e t e r m i n e d only for one n e u t r o n e n e r g y . No m e c h a n i ­

cal d r i v e s y s t e m s a r e needed for the c r y s t a l s p e c t r o m e t e r .

II. 2 C o m p a r i s o n of a Conventional and a n I n v e r t e d S t a t i s t i c a l Chopper

F a c i l i t y

II. 2. 1 ­ The K,<­J­range of the two i n s t r u m e n t s

In F i g . 6 and F i g . 7 the two c a s e s i n question a r e i l l u s t r a t e d in r e a l ­

and m o m e n t u m s p a c e r e p r e s e n t a t i o n . The flexibility of the two i n s t r u m e n t s

can be c h a r a c t e r i z e d by the c o r r e s p o n d i n g HfO ­ r a n g e , t h e r e s o l u t i o n func­

tion and the i n t e n s i t y in this r e g i o n . F r o m Fig. 1 and the c o n s e r v a t i o n of

e n e r g y

(27)

^ = £ ( k

2

-

k f 2

)

follows for the i n v e r t e d s t a t i s t i c a l chopper

fZ

(28) ¿OJ = f - [2(/?,k.)-*Z] = E. - E =

2 m ι i f

2 m 2k. cosïKf \\Κ1/ -k sinf $ - 2k^ s i n V +Λ:' f

and for the conventional s t a t i s t i c a l chopper

, λ . t,2 Γ 2 2 , r 2 2 - , 2 , , 2 y 21 l / Z ^ 1

(29 £ω = τ— 2k. .sin J + 2k. /k. .sin ¿/(sin V-l)+ cos ¿ΛΚ J -K \

2m ι - ι L ι J J

Using t h e s e e q u a t i o n s , the IK ,t*>-range can be c a l c u l a t e d and i s shown in

F i g . 8, 9 and 10. F o r the i n v e r t e d c a s e the Ζ**,¿^­region i s given for a p y r o ­

lytic g r a p h i t e , a B e ­ s i n g l e c r y s t a l and a Be­MgO window filter i n the a n a l y s e r

s p e c t r o m e t e r . The i n v e r t e d s p e c t r o m e t e r h a s an "open" region t o w a r d s l a r g e

/C,6J­values and c o l l e c t s m o s t of the data in a region with low y<­values. This

14 —

has been chosen). It is i m p o r t a n t to avoid l a r g e data collection in regions

with high multiphonon background.

II. 2 . 2 ­ The Resolution­functions of the i n v e r t e d and conventional s t a t i s t i c a l

chopper facility

The r e l a t i v e e r r o r of an energy t r a n s f e r nlo ­ m e a s u r e m e n t is given by

(30)

A f 2k..Ak.+ 2k . A kr

Lhuc ι ι f f

küO _{v2 v2}

ki ­kf

Since for an i n v e r t e d t i m e of flight s p e c t r o m e t e r

/ s àt h 2

31 Ak. = y ­ . — . k.

ι / . m ι

ι

the r e s o l u t i o n function of the i n v e r t e d facility is given by

(32) Δ* CO

tìOO

iL·.

ÎL

IL

¿. ' m 2m

ι­ ι +

­— . 2k.. Δ k .

2m 1 f

co

Ρίω

where

(33) A t = ^ tc + ^ +

Δ ^ 2 1/2

s t . c h . M .

k

' m f

)}

and for the conventional case by

3/2

(34)

Δ/δω

ICO

àt _{S t . C h}

_{A A}

₂

£

'

m " 2m

— . 2k..Ak. 2m ι ι

i t o

In F i g . 11, 12, 13, 14 and 15 the r e s o l u t i o n functions a r e given for v a

-T (~* X-, (~* (~* "U

rious s e t s of k.. Ak., kr. Akr and At with Κ . ' = 15 m and £ ' ' = 4 m .

1 1 f f 1 f

— 15 —

on ti(jo since s m a l l Tiu;-values a r e m e a s u r e d as the difference of two large n u m b e r s , demonstrating clearly the disadvantage of the conventional faci­ lity. Limiting in this case the a c c e s s i b l e region in the H,Ό -plane by

cho-sing s m a l l k . - v a l u e s , of course any value fii-C can be m e a s u r e d with the r e ­ quested energy resolution compatible with the available source strength. However, this p r o c e d u r e costs time and money.

II. 2 . 3 - Comparison of the intensity of the two i n s t r u m e n t s

In Fig. 8, 9 and 10 it is d e m o n s t r a t e d that the accessibility in the /c,CO plane is quite different for the two i n s t r u m e n t s and in F i g . 11-15 it is shown

that the energy resolution functions a r e again very different for a given range of fiui. Therefore the performance c h a r a c t e r i s t i c s of the two instruments

a r e quite different for a c e r t a i n setting of k., A k . and k _, Δ k -values. Conse-1 ι f f

quently, a c o m p a r i s o n of the intensity in a c e r t a i n /f,6<> -range is not possible. Hence we limit our considerations to two c a s e s :

1. h<-0 = 150 meV + 5.4% and K = 4. 88 Å'1 ± 3. 1 %

and

2. hco = 150 meV + 4. 5% and Hi _, = 8.75 Â" + 23%

— I. Ch —

and (Kr -,, = 8. 7 5 X"1 + 6. 14%

C. Ch —

F o r the second c a s e with l a r g e momentum t r a n s f e r it is a s s u m e d that in the conventional s p e c t r o m e t e r neutrons a r e detected between 0 ^ V 4 90 and in the inverted s p e c t r o m e t e r in a 47T-Be-MgO window filter detector. In this case experiments a r e c o m p a r e d that have the same energy resolution but dif-ferent and possible t\ - r e s o l u t i o n s .

16

of 4 m , defining a Δ,β = ~— = 0. 3 5 « 20 . This is about the l i m i t for 'f 4

s t r a i g h t d e t e c t o r s for which

A k (z) '

(35) s i n ^ ) ^

The d e t e c t o r s of the i n v e r t e d s p e c t r o m e t e r have a length of 0. 7 m and a r e

positioned 2 m (-^J) from the t a r g e t . F o r r e a s o n s c o n c e r n i n g the r e s o l u t i o n

the distance t a r g e t ­ d e t e c t o r ( ¿ J o r Δ(?> m u s t fulfil the following condition

k

1 ­

A

Kf (z)

(36) c o s ( ­ û | îf) 4

A k( z )

1 + — . tan <x

k.

where cV. denotes the Bragg a n g l e .

F o r o p t i m u m intensity and a c c u r a c y one chooses for this p a r t i c u l a r com­ p a r i s o n for both c a s e s k . . Ä k . = k ..Ak,. In g e n e r a l the intensity ratio of the

r ι ι f f ö }

i n v e r t e d and conventional s p e c t r o m e t e r s is t a k e n as the figure of m e r i t :

(37) M{6u),K)

! 2h 2

-k A

R ( kf) . £ ( kf) . FT. e .AV..AVf_

I. Ch.

' 2 ­k 2 / kT ~

R ( k . ) . 6 ( kf) . FT. e . A V . . A Vi C i C h

where R(k) stands for the reflectivity of the c o r r e s p o n d i n g c r y s t a l s and neu­

t r o n energy and £ (k ) for the d e t e c t o r efficiency.

F o r the conventional s p e c t r o m e t e r a v a r i a b l e k. is m a n d a t o r y and con­

sequently a double c r y s t a l s p e c t r o m e t e r m u s t be i n s t a l l e d since the l a r g e

d e t e c t o r i n s t a l l a t i o n cannot be m o v e d . The figure of m e r i t m u s t be c a l c u ­

(38)

a n d

í'-.flCO

ito (tf, co)

C . C h .

17 —

Ahoo

L

i

co

(K¿*>)

I. Ch.

(39) {K, co)

C . C h .

Δ Κ

(Ar,c-)

I. Ch.

A s s u m i n g no l i m i t a t i o n s in s a m p l e size (a r e a s o n a b l e a s s u m p t i o n for in­

c o h e r e n t down­s c atte ring e x p e r i m e n t s ) and equal d e t e c t o r efficiency e q u a ­

tion (37) r e d u c e s for t h e s e p a r t i c u l a r c a s e s to

(40) M ( / u i , A ) =

R ( kf) . FT. /Av . . ¿ vf J L C h >

\ * \ ) .

F

Δν,.Δν

I l i C . C h .

The figure of m e r i t is d e t e r m i n e d by c a l c u l a t i n g t h e p o s s i b l e m o m e n t u m

space e l e m e n t s i m p o s e d by the g e o m e t r i c a l conditions of the two s p e c t r o ­

m e t e r s and equations (35), (36), (38) and (39). In spite of t h e s e boundary con­

d i t i o n s , the volume e l e m e n t s and t h e i r product m a y be quite different.

In Table 1 and 2 the c h a r a c t e r i s t i c p e r f o r m a n c e data for the two c a s e s

a r e s u m m a r i z e d .

F o r the f i r s t c a s e we obtain:

M = 1. 12

and for the second c a s e :

M = 14. 4 .

F r o m the r e s u l t s , the l a r g e a c c e s s i b i l i t y in the Κ,<-¿ ­ p l a n e , the wide range

of the flat r e s o l u t i o n function and the p o s s i b i l i t y of a 47T­detector s y s t e m , it is

concluded that the i n v e r t e d s p e c t r o m e t e r is v e r y suitable for d o w n ­ s c a t t e r i n g

Parameters Dimensions Inverted Spectrometer Conventional Spectrometer Tico meV 150 150 ΔΤιω Tico % 5,41 5,41 Κ

o J A 4,88 4,88 ΔΚ Κ % 3,1 3,1 k¡ °-1 A 10 10

iti

"i

7o

1,17 1,09 "f

o J A 5,29 5,29 kf % 3,06 3,06

Ι

m 15 0 'f m 2 4 At St.Ch. Li sec. 20 36,5 At

u sec.

27 A°<¡ Rad IO3 0,6667 2,15

aß\

Rad I O3

3,333

20

A A ¡

Sterad IO6

2,22

43

AV¡

A -J

106

25,9

469

1 ■ ι

Parameters Dimensions Inverted Spectrometer Conventional Spectrometer A<Xf Rad X103 11,1 30

A£

f Rad ΧΙΟ3 700 350 Afîf Sterad X106 7777 10500 AVf

° 3 A -J xlO6 35200 48500 FT r 2 Cm 15x3 5x1 RBe % 0,5 0,4

3-Degree 10,2 10,2

* B e

Degree

20

1041

Be

Rad x l O3

11,1 2,15 AVf A V¡ 1360 103

A A f

Ailj

3500

244

M(1ico,k) = 1.12

Parameters Dimensions Inverted Spectrometer Conventional Spectrometer "hoj meV 150 150 Anco "h co % 4,5 4,5 K

A "1

8,75 8,75 AK K

ν.

23 6,14 k¡

Å

J 8,62 8,62 Ak, k¡ % 2,01 19 kf

X -

1

1,54 1,54 kf

7o

6,14 6,14 li m 15 0 If m 0,6 4 At St.Ch. ^j sec. 40 250 At

u sec.

55 250 A<*j Rad x103 0,666 7 4,1

A f t

Rad x103 3,3333 20 AA¡ Sterad x106 2,22 82 AV¡

A "3 x106

29

1060

L .. . I

Parameters Dimensions Inverted Spectrometer Conventional Spectrometer Ao<f Rad 2 R 1,57 A/3f Rad 2rc 0,35 AAf Sterad 4Π1 0,55 AVf

° 3 A -J

2,81

8x102

FT

C m2

15x3

5

RBe

%

0,4

^ * B e

Degree Degree

o°_

360°

0 ° _ 90°

-12,1

* * B e

Rad

x103

-1,51

T rB e

7.

0,8 AVf AV¡" 9,69 x10A 75,4 AHf Ail," 5,66

x l O6

6700

M (Tico) = 14,4

Tab. 2 THE CHARACTERISTIC PERFORMANCE DATA OF THE TWO SPECTROMETERS

20

III. Fields of Application

A white neutron beam experiment as discussed in this context has a wide range of application. In diffraction experiments m o s t of the lines appear in backscattering directions well accessible in the geometry of an inverted s p e c t r o m e t e r . F o r the analysis of complex s t r u c t u r e s large numbers of lines have to be m e a s u r e d and a wide K - r a n g e is requested. As illustrated in Fig. 17, this can be done at the inverted s p e c t r o m e t e r . F o r certain expe-riments concerning phase t r a n s i t i o n s , it is of i n t e r e s t to m e a s u r e simulta-neously a.3 a function of t e m p e r a t u r e , p r e s s u r e , e l e c t r i c and magnetic fields the static s t r u c t u r e , phonons and frequency density distributions.

21

IV. The T h e o r y of the I n v e r t e d S t a t i s t i c a l C h o p p e r F a c i l i t y

In t h i s c h a p t e r we d e s c r i b e :

1) how to p e r f o r m the c r o s s c o r r e l a t i o n b e t w e e n the ( p s e u d o s t a t i s t i c a l ) o n

-a n d o f f - p -a t t e r n of the s t -a t i s t i c -a l c h o p p e r -a n d the n u m b e r of c o u n t s in

the d i f f e r e n t a n a l y s e r c h a n n e l s ;

2) how to e x t r a c t i n f o r m a t i o n a b o u t the s c a t t e r i n g function out of the m e a

-s u r e d c o r r e l a t i o n function a n d c o n -s i d e r

3) the i n f l u e n c e of the s t a t i s t i c a l e r r o r of the c o u n t s in the a n a l y s e r c h a n n e l s

on the s t a t i s t i c a l a c c u r a c y of the ( m e a s u r e d ) s c a t t e r i n g function.

The a r g u m e n t a t i o n is b a s e d on the m e t h o d and r e s u l t s p r e s e n t e d in the

p a p e r s ¿9j a n d / l 6 y .

F o r a n a n a l y t i c a l d e s c r i p t i o n of the c h o p p e r s a m p l e d e t e c t o r c o n f i g u r a

-t i o n we m e a s u r e -t h e -t i m e a -t -the c h o p p e r and d e -t e c -t o r p o s i -t i o n in i n -t e r v a l s of

e q u a l l e n g t h Q a n d i n t r o d u c e the q u a n t i t i e s :

Cii

1 if t h e c h o p p e r w a s o p e n i n t i m e i n t e r v a l i

« 0 if t h e c h o p p e r w a s c l o s e d in t i m e i n t e r v a l i

Q(i) n u m b e r of d e t e c t o r c o u n t s in t i m e i n t e r v a l i if the c h o p p e r w a s open

in t i m e i n t e r v a l " 0 " only

C(k) t o t a l n u m b e r of d e t e c t o r c o u n t s in t i m e i n t e r v a l k ( d e t e c t o r c h a n n e l k)

b(k) t o t a l n u m b e r of b a c k g r o u n d c o u n t s in d e t e c t o r c h a n n e l k ( a s s u m e d to

be i n d e p e n d e n t of k: b(k) = b)

T h e n we h a v e :

22

As the s t a t i s t i c a l chopper works p e r i o d i c a l l y in t i m e with p e r i o d N o ,

we have

(42) a. = a. + ¿ N ( · / = 1,2, . . . )

K ' ι ι

F u r t h e r m o r e the sequence a. is chosen to have the p s e u d o s t a t i s t i c a l pro­

p e r t y :

(43)

2N

a

rr¿

= m(l

­

c)

Ld(N)

+ mC

where m is the n u m b e r of t i m e s for which a. = 1 within one p e r i o d and ι

c = ( m ­ l ) / ( N ­ l ) .

After m e a s u r i n g o v e r K p e r i o d s we p e r f o r m the c o r r e l a t i o n o p e r a t i o n :

KN N K­1

(44) ^ / _ a C(i+k) = / \ a. / _ . C(i+¿N+k)

i=l * i=l x ¿=0

Denoting the sum of the d e t e c t o r counts in c o r r e s p o n d i n g channels o v e r

K p e r i o d s again by C(i+k) we obtain, u n d e r the a s s u m p t i o n that

/ ¿ 0 for H i ^ N

(45) Q(i) \

L= 0 for N < i

the r e s u l t :

N

(46) Y ' a.C(i+k) = K m ( l ­ c ) Q ( k ) + KmcQ + Kbm

L a

i=l

w h e r e

(47) Q = T Q(i)

i ­ ­ o o

On the other hand we h a v e :

N

— 23 —

Combining this expression with (47) gives us finally

N

t

49

'

Ω

« = iwïTT)

_{Km(l ­c) 4—»}

L

K­c).c(i

_i +

k)

_m b

As the C(i) are statistically independent quantities with a Poisson distri­

bution, we obtain for the variance of Q(k)

with

(51) K ( k ) =N ( Q ( k H b )

Q­2Q(k)­­ m

The first factor in (50) is just the variance of Q(k) if the measurement would

have been performed in the conventional way using K pulses of width θ .

This leads us to write:

<

52

>

c2(k)

K(k)} . . / f o w l

. ,

v J statistical v J conventional

chopper chopper

where

(53)

G 2

(

k

) = Ü ^ L ·

-Ν +c<* (k)

can be interpreted as a gain factor, indicating that the statistical chopper is 2

superior to the conventional chopper when G > 1.

A detailed parameter study of G was made by Scherm , who showed 2

that basically G (k) will be larger than one, if (X(k) is larger than one. The 2

discussion of the gain factor G (k) reduces therefore to a discussion of C<(k),

especially to the determination of those parts of Q(k) in which oc (k) is larger

— 24·

b / m and Q(k) compared to Q.

Then we obtain:

, , , . Q[k) + b Q N

(55) Q =¿ 2 ,

Q ( Í )

i=l

This is the well known result, that for stationary neutron beams the s t a t i s -tical chopper is superior to the conventional one for those p a r t s of Q(k) for which oc (k) is l a r g e r than one and that this part is determined by the mean value Q of the whole spectrum Q(k).

Now we turn to the discussion of the function Q(k) for two special time of flight experiments:

a) input energy fixed (normal chopper, Fig. 17 a), b) output energy fixed (inverted chopper, Fig. 17 b).

±

With the normal (statistical) chopper neutrons of fixed energy E enter the c h o p p e r - s a m p l e - d e t e c t o r system during time interval 0 at the position of the (statistical) chopper.

The target is just behind the chopper.

The s c a t t e r e d neutrons after the t a r g e t have different energies and a r r i v e in different t i m e - i n t e r v a l s k at the detector.

In this case we have:

(56) Q(k) = M(E*) . S(E* k)

where

— 25

S ( E , k) is the probability that a neutron (out of the input pulse) of

energy E generates a count in the detector channel k .

With the inverted (statistical) chopper neutrons out of a white spectrum

enter the s y s t e m during time interval " 0 " at the position of the (statistical) chopper.

The t a r g e t is just before the detector.

The incoming neutrons have different energies and a r r i v e at different t i m e -intervals k at the t a r g e t .

In this case we have

(57) Q+(k) = M(k).S(k,E*)

w h e r e :

M(k) is the total number of neutrons a r r i v i n g at the t a r g e t in time i n t e r ­ val k

S(k, E ) is the probability that a neutron arriving in time interval k at the

t a r g e t will generate a count in the detector after scattering into the

energy (-interval / \ E around) E .

If we denote by

Ι(Ε)ΔΕ the intensity of the incoming neutrons with energy ΐ η Δ Ε around

E, by

S(E, Ε ')ΔΕ ' the scattering probability for a neutron from energy E into the

interval Δ Ε ' around E ' ( f o r the s c a t t e r i n g - g e o m e t r y chosen), and by

E('r) the energy n e c e s s a r y for a neutron to cover the flight path from

the chopper to the detector in time t

then we have for both c a s e s :

— 26

(59) Q+(k) = 1(E). S(E, E*) | y á E * d2

If a Maxwellian s p e c t r u m is used for the e x p e r i m e n t s , the principle of de-tailed balance:

(60) K E ^ S Í E * E) = I(E)S(E, E*)

says that both experiments give us the s a m e information and from the point of view of the statistical e r r o r they a r e equivalent.

Note that in the definition of the quantities entering equations (58) and (59) we put the resolution-functions for the incoming and outgoing neutrons equal to one for both versions of the statistical chopper. This is not c o r r e c t , but the relative m e r i t s of the resolution functions for both types of statistical choppers is discussed in the other chapters of this paper.

V. Conclusions

It has been shown that the inverted time of flight s p e c t r o m e t e r has a large accessibility in the K,u> -plane and that the energy resolution function and the intensity is r a t h e r constant over a large energy range. The i n v e r ted geometry is therefore very favourable to perform precision m e a s u r e -ments of line width and line shape of local m o d e s , high energy phonons and magnons. In principle a 4/T-Be-filter-detector can be built for high intensity low background experiments. The role of the statistical chopper is p r i m a r i -ly to reduce the continuous elastic and inelastic background. All the r e s u l t s concerning the m e r i t s or advantages of a conventional statistical chopper compared to a common time of flight s p e c t r o m e t e r a r e of course as valid for the inverted s p e c t r o m e t e r .

27 —

VI. L i t e r a t u r e

¿ l / BECKURTS, K. H . , K e r n f o r s c h u n g s z e n t r u m K a r l s r u h e , KFK, I n ­ t e r n a l R e p o r t (1968)

[Ζ] SKØLD, Κ . , Nucí. I n s t r u m . M e t h . , 63, 1 1 4 ( 1 9 6 8 ) , Nucí. I n s t r u m . M e t h . , 6,3, 347 (1968)

[ï] P A L , L . , KROÓ, N. , PELLIONISZ, P . , SZLARIK, F . , VIZI, I . , Neutron I n e l a s t i c S c a t t e r i n g Meeting, I A E A ­ P r o c e e d i n g s S y m p o s i u m , Copenhagen, Vol. 2, 407 (1968)

¿ 4 / GOMPF, F . , REICHARDT, W. , GLAESSER, W. , BECKURTS, Κ. Η. , Neutron I n e a l s t i c S c a t t e r i n g Meeting, I A E A ­ P r o c e e d i n g s S y m p o s i u m , Copenhagen, Vol. 2. 417 (196 8)

¿ 5 / HOSSFELD, F . , AMADORI, R. , SCHERM, R. , I A E A ­ P r o c e e d i n g s of the P a n e l Conference on I n s t r u m e n t a t i o n for Neutron In­ e l a s t i c S c a t t e r i n g R e s e a r c h , Vienna, (1970)

¿ 6 / HOSSFELD, F . , AMADORI, R. , K F A ­ J ü l i c h ­ R e p o r t 6 8 4 ­ F F , (.1970)

¿IJ KROO, N. , I A E A ­ P r o c e e d i n g s of the P a n e l Conference on I n s t r u m e n ­ tation for N e u t r o n I n e l a s t i c S c a t t e r i n g R e s e a r c h , Vienna (1970)

¿ 8 / KLEY, W. , P r o c e e d i n g s of the Joint E u r a t o m ­ J a p a n e s e Information Meeting on the Design Status and the P r o j e c t e d Use in Science and Technology of the E u r a t o m P r o j e c t SORA and the J a p a n e s e L I N A C ­ B o o s t e r P r o j e c t , held at I s p r a on Sept. 1 7 ­ 1 8 , (1971), EUR­4954 e

¿9_/ MATTHES, W . , I A E A ­ P r o c e e d i n g s S y m p o s i u m , G r e n o b l e , N e u t r o n In­ e l a s t i c S c a t t e r i n g , M a r c h 6­10 (1972)

¿Ì0_/ AMADORI, R. , HOSSFELD, F . , S y m p o s i u m G r e n o b l e , N e u t r o n I n e l a s ­ tic S c a t t e r i n g , M a r c h 6­10 (1972) 747

¿U_/ KROO, N . , PELLIONISZ, P . , VIZI, I . , ZSIGMOND, G. , ZHUKOV, G., NAGY, G. , I n e l a s t i c S c a t t e r i n g , M a r c h 6­10 (1972) 763

[\Z] PELLIONISZ, P . , KROo', N. , MAZEI, F . , I n e l a s t i c S c a t t e r i n g , M a r c h 6­10 (1972) 787

¿ 1 3 / KLEY, W. , E U R ­ R e p o r t in p r i n t

¿ 1 4 / KLEY, W. , E U R ­ R e p o r t in p r i n t

¿ 1 5 / M A I E R ­ L E I B N I T Z , Η . , Nukleonik, 8 ( 1 9 6 6 ) 6 1

— 28

F-,· ι T a r g e t - A r e a

FD : N e u t r o n - D e t e c t o r - A r e a

F i g . 1 G E O M E T R I C A L REPRESENTATION OF A NEUTRON SCATTERING EXPERIMENT

U ;

Be-Filter

Target

_MgP_

Detector

d Mobile-Detectors

Statistical Chopper Mobile - Single - Crystals

— 29 —

Detector

Fiq.3. A NEUTRON SCATTERING EXPERIMENT WITH MONOCHROMATIC NEUTRON BEAMS

IN REAL. AND MOMENTUM.SPACE

Relative Intensity at the Target

10' .

10

io-1 J

10

Fig.A THE RELATIVE INTENSITY AT THE TARGET AS A FUNCTION

OF THE RELATIVE NEUTRON ENERGY x'= E' /E 1

30

/2^...2

fC<ir>

Fi q. 5 The Scattering Law of an isotropic harmonic oscillator

Be_ Crystal

Vi Length of Detector

U

A pf= - ^ = 0,35^20°

— 31 —

Detectors

Kt

FIG. 7 - THE INVERTED STATISTICAL CHOPPER IN REAL-AND MOMENTUM - SPACE

I 2 3 t 5 6 7 8 9 10 I , , , , *-κ

1 12 13 H 15 16 17 "

A tìUj(meV)

i 00

sno

9.10-Kf=IO,«X"' ƒ Ef.227,3meV

Kf„ 1,813 A

Ef= 6,9 meV

Reactor

0 1 2 3 4 6 7 8 9 IO 11 12 13 Η 15 16 _Κ Fig.9 The K.uu range of the inverted statistical chopper

tiuu(meV) 4

electors

K.= 1 0 Â " ' ; E,:208,SmeV ■ 3 = 1 0 , 3 5 *

K(Å-')

1 2 3 U 5 6 7 8 9 10 11 12 13 U 15 16

Rq.10

The Κ,υυ range of the conventional

s t a t i s t i c a l chopper f a c i l i t y

— 33

14 13 12 11 10

9 8

7 ■■

6 5 ·

U

3 2

1

ΔΤιω

M

Bg.Mg O-Filter <;

ω

©

©

©

* J l . o

kf

kf

kf

i i L = 6.K.10-2

kf

AL. _ .. — 2

A ,s t . c h =2 0-1 0"6 s e c ..

..

·■

R

St.Ch."

F i g J l THE RESOLUTION FUNCTION FOR THE INVERTED FACILITY FOR kf = 1,54 Å '1 AND lf = 0.6m

= 20.10 sec

— 34 —

15 f 14 13 12 11 10 9 8 t 7 6 5 4

3 ■

2

1 ■

Tiu)

[v.]

ω

^ i l , „,_ =

20.10-6sec. St. Ch.

,-2

® AJíí - io"

© Akf_ . 2.10" kf

[me V]

50 100 150 200 250 300 350 400 450 500

Î -1

FigJ3 THE RESOLUTION FUNCTION OF THE INVERTED FACILITY FOR kf = 5A ~ AND lf = 2m

15 -14 . 13 12 -11 10

-9 8 7

-e

5 4 3 2 -1

Δΐιω

"huj

[%]

© Ail _- o _kf

© i i i = io"2

kf

At =20.10" sec St. Ch.

"hui [m.VJ

.100 100 200 300 400 500

Î -1

— 36

I h . k . l )

( h , k , l )

Mosaic Spread Fiq.16 THE TAILORED MOMENTUM SPACE ELEMENTS

I 0,0,0 )

Fixed input energy E * TARGET

ST. CH.

Fig. 1 7 α : Normal (STATISTICAL) CHOPPER

— · - · DETECTOR

White Input Beam TARGET

) · - « DETECTOR

ST.CH. Fixed output energy E

■m

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