Resolution in experiments of angular correlation of annihilation photons

NUCLEAR

INSTRUMENTS

AND METHODS

IO 9

(1973) 557-56o;

7~) N O R T H - H O L L A N D

PUBLISHING

CO.

R E S O L U T I O N IN E X P E R I M E N T S OF ANGULAR C O R R E L A T I O N OF ANNIHILATION PHOTONS

G. POLETTI*

CISE, Segrate P.B. 3986, Milan, Italy F. ROSSITTO

CESNEF, Politecnico di Milano, Via Ponzio 34/3, Milan, Italy Received 16 October 1972 and in revised form 26 February 1973 The overall resolution function for angular correlation devices is evaluated, removing the point source approximations. The non-uniform intensity distribution in the source is taken into

account. The impact of the resolution on the features of the recorded angular correlation curves is discussed, with reference to the electron positron system annihilation.

1. Introduction

on the direction of annihilation photons, but also on the position of any individual annihilation event in the source. It is on this basis that the overall resolution function for a linear-slit device, designed to measure angular correlation curves of annihilation photons from the electron-positron system, is derived in the following. First, the slit resolution function (R~) is calculated in the light of the angular acceptance limits imposed by the slits as a function of the position P in the sample, in which any annihilation event takes place. Next the overall or "'optical" resolution function, (Rop,), which also takes into account the intensity

In angular correlation measurements (e.g. of annihilation photons from the electron-positron system) it is customarily assumed that the geometrical resolution function of the experimental set-up, due to the finite slit opening at the detectors, can be conveniently described by a triangular function~-3), depending only on the position of the mobile detector. This assumption is, in fact, not supported by geometry: a more correct approach would take into account the dependence of the slit resolution function not only * G . N . S . M . Researcher.

z

sf

~' " " '""

ocs

Fig. 1. Sketch of linear-slit device for angular correlation measurements.

557

558

G. P O L E T T I

A N D F. R O S S I T T O

distribution of positrons in the sample, is derived and a simple but illustrative case is discussed.

2. Slit resolution function A sketch of the considered (linear-slit) experimental set-up is shown in fig. I. The slit resolution function R~ gives the acceptance by the slits as a function of 0 - 0o, where 0o is the angular setting of the mobile slit with respect to the fixed slit, and 0 is the angle between the directions of the correlated annihilation photons. In general, the angle subtended to the source by the slits in the ),-direction greatly exceeds the angular range of the correlation curve, so that the measured photon intensity is the integral over the y-coordinate. Therefore the study of R~ can be restricted to the (x, z) plane. For a given angular setting 0 0` to any annihilation event taking place at a point P(x, z) in the sample (see fig. 2), there is a corresponding set of limiting angles for acceptance (¢p~,tp2 for slit l, and ~ ; , ¢¢2 for slit 2). The following relations define these angles in terms of the geometrical parameters of the device: --Z--~S

(3) R~ = 0,

(4)

6o = x20°--xLO°-2zL L 2 __x 2

is the shift of the centre of R~ with respect to the angular setting 0o, and 2sL

A =6M--~m--

(5)

/ 2 __X2

is the width of the R, function. Eqs. (2), (4) and (5), when applied to the usual geometry of the experimental devices, can be simplified as follows: Z

fir. = - 2 - - : e , L

tP2 -- - - ,

L-x

otherwise.

Therefore R, is rectangular, as shown in fig. 3, where

--2+~S

~Pl - - - ,

Z

tim = - 2 - + c t , L

L-x

t/J1 = L O ° - ½ s - : , L+x

~2 - LO,,+ ~ s - z L+x

x'- Oo - xLOo - 2 z L L 2 __X 2

sL

X 200-- XLOo-- 2 z L + s L 12 __X 2

S

Fig. 2. G e o m e t r y requirements

..................

5o = - 2 - ,

(I)

Correspondingly the angle 0 between the directions of any two correlated annihilation photons accepted by the slits ranges from 0,,~. = ~p~ +~bt to 0 ...... = ~P2 +~b2, that iS0m~.~<0~<0 .... . By using 0 - 0o = 6, t h i s c a n b e written 6m <<-6 ~ 6s~, where:

6M =

The acceptance is unity in this range and zero everywhere outside; therefore the required slit resolution function R, = R~(x, z, 6) is:

Z

A

= 2:t,

L

where :t -- s/L. In the same approximation, which is actually quite valid, R~ depends only on the z-coordinate of the point P in which the annihilation event occurs, so that & = R~(,~, z).

3. Optical resolution function

(2)

The overall resolution function is also affected by the penetration of positrons into the sample, which gives rise to an intensity distribution dependent on the physical properties and the size of the sample.

/

[ ..,.

for the acceptance by the slits o f the ;,-rays from an annihilation event occurring at the (x.z) point of the sample.

ANGULAR

CORRELATION

OF A N N I H I L A T I O N

559

PHOTONS

! Rs(~Iz) ~°

.- 0 - 0 o Fig. 3. Slit resolution function for the linear-slit apparatus.

Following the reasoning given above, the positron intensity distribution within the sample is only z-coordinate dependent and may be described by a suitable function P(z), which has to be defined on the basis of the geometry of the sample-source system and of positron penetration into the sample. The optical resolution function, Rop., is then defined

where 1~ is the positron absorption coefficient of the sample. Since, for positrons in condensed matter l~ >> I cm - 1 the semi-infinite sample approximation is quite good. Rop, can then be evaluated by observing that:

as

Rop,(6) = j

Rs(6, z) P ( z ) d z .

Zm = ~ ( - - L f - - s ) ,

z.~, = ~ ( - L f i + . s ) ,

z o = - ~L6,

A. = s.

By straightforward calculation we get:

)[sample]

Therefore, the measured setting 0o is given by:

intensity at the angular

Root(6 ) ---

eL

M(Oo) =

N(O) Rom(O-Oo) dO,

where N(O) is the true distribution of the intensity of the annihilation photons, retaining, that is, all the effects due to the positron motion and to the temperature of the sample, as could be recorded if Rot , were a delta function. As an application of the outlined approach, let us calculate Rop, for the device sketched in fig. 1, supposing an external source irradiates positrons on a semiinfinite sample whose surface is located at z = 0 , aligned with the centrelines of the slits at 0o = 0. [n this case p ( : ) = /0,

[ eu:,

for z > 0, for : ~< 0,

2 sinh(½pL:0 exp(-}j~L~i),

6 > ~.

4. D i s c u s s i o n

The optical resolution function just derived was compared with the generally accepted one based on a triangular slit function"5), using the geometrical and physical parameters of the arrangement used by Stewart e t a [ . 6) to measure the angular correlation curves of positrons annihilating in alkali metals. First the slit resolution function Rs was calculated on the grounds of the given values of the involved parameters of the device, and then, taking into account the source and sample features relative to the experiment

560

G. POLETTI AND F. ROSSITTO

/'\

"l

I

I

I

1

5. Conclusions

I

T h e o u t l i n e d a p p r o a c h a l l o w s the e v a l u a t i o n o f general, optical r e s o l u t i o n functions, w h e n e v e r the g e o m e t r y o f the e x p e r i m e n t a l set-up and o f the s a m p l e are k n o w n , for a p o s i t r o n s o u r c e b o t h externttl to the s a m p l e a n d d i s t r i b u t e d within the sample. O u r investig a t i o n brings o u t clearly a r e m a r k a b l e influence of" the i n s t r u m e n t a l p a r a m e t e r s on the a n g u l a r c o r r e l a t i o n m e a s u r e m e n t s o f the a n n i h i l a t i o n p h o t o n s o f the e l e c t r o n - p o s i t r o n s y s t e m : this sensitivity o f the results to i n s t r u m e n t a l effects c a n n o t be d i s r e g a r d e d p a r t i c u larly in high r e s o l u t i o n e x p e r i m e n t s . W e c o n c l u d e that results based on small effects a p p a r e n t in the r e c o r d e d c u r v e s s . 8 - ~ ) s h o u l d be critically reviewed.

Zr a.,qular gilt ft.ncticn F4ectancJu:ar sl.t functlo:l The %llt-fJnCilOnS are r s ~ m a '.'zed to eqLjal area

_

__

- -

12

/

/,

I

[

~o

i

0.9 f P

}

3.8

g ~ °.,,

]27mr

1

~.'3mr

References O~

/

\

/

°'i/j 3 -:2

1 -0~

I 0

[ O'

I O.2

I C.S -e-(%

0 1

0.5

r~

7

(mr)

Fig. 4. Comparison of the optical resolution functions evaluated for triangular and rectangular slit functions. on N a 7), the optical r e s o l u t i o n f u n c t i o n Ropt was c a l c u l a t e d . In fig. 4 this o p t i c a l r e s o l u t i o n f u n c t i o n is c o m p a r e d with t h a t used by S t e w a r t et al.: a s t r i k i n g difference b o t h in s h a p e and f w h m is e v i d e n t .

I) O. E. Mogensen, LTF II no. I (Technical University of Denmark, Lyngby, 1968). e) p. E. Mijnarends, Phys. Re','. 178A (1969) 622. z) p. U. Arifov, V. 1. Goldanskii and V. U. S. Sayasov, Soviet Phys. Solid State 6, no. 10(1965) 2484. 4) S. M. Kim, P h . D . Thesis (University of North Carolina, 1900) unpublished. 5) K. Fujiwara and O. Sueoka, J. Phys. Soc. Japan 21, no. 10 (1966) 1947. ~i) A. T. Stewart, in Positron attnihilation (eds. A. T. Stewart and L. O. Rocllig; Academic Press, New York, 1967). 7) A. T. Stewart, J. B. Shand and S. M. Kim, Proc. Phys. Soc. 88 (1966) 1001. s) S. M. Kim, A. T. Stewart and J. P. Carbotte, Phys. Rev. Letters 18, no. 11 (1967) 385. ") J. J. Donaghy and A. T. Stewart, Phys. Rev. 164, no. 2 (1967) 396. 1o) A. Perkins and J. P. Carbotte, Phys. Re','. B i, no. I (1970) 101. tl) j. j. Donaghy and A. T. Stewart, Phys. Re,,'. 164, no. 2, (1967) 391.