THE POSITRON ANNIHILATION TECHNIQUE IN FERMIOLOGY

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THE POSITRON ANNIHILATION TECHNIQUE IN FERMIOLOGY

S. Berko

To cite this version:

S. Berko. THE POSITRON ANNIHILATION TECHNIQUE IN FERMIOLOGY. Journal de Physique

Colloques, 1978, 39 (C6), pp.C6-1568-C6-1575. �10.1051/jphyscol:19786603�. �jpa-00218097�

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JOURNAL DE PHYSIQUE

Colloque C6, supplement au

no 8,

Tome

39,

aotit

1978,

page

C6-1568

S. Berko

Physics Department, Brandeis U n i v e r s i t y , Waltham, MA 0 2 1 5 4 , U.S.A.

RQsm6.- La correlation angulaire du rayonnement d'annihilation (ACAR) des positrons introduits dans un cristal orient6 donne la densitd d'impulsion des dlectrons vus par les positrons thermalisbs. De nouvelles techniques utilisant des ddtecteurs multiples produisent des surfaces ACAR bi-dimension- nelles correspondant a w projectionsde la densitd d'impulsion ; les discontinuites dans ces surfaces donnent la forme et la dimension de la surface de Fermi dans les mdtaux purs et les alliages.

Abstract. The angular correlation of annihilation radiation (ACAR) from positrons introduced in an oriented crystal gives the momentum density of electrons seen by the thermalized positron. Newmul- ti-detector techniques provide two-dimensional ACAR surfaces corresponding to projections of t h e m w mentum density ; discontinuities in these surfaces give the shape and size of the Fermi surface in pure metals and in alloys.

INTRODUCTION.- In this paper we discuss briefly the use of the positron (e+) annihilation technique for the study of electronic momentum densities in metals and alloys and, in particular, its relevance to Fermiology. The study of the behavior of slow e+-s in condensed matter has been the subject of intense experimental and theoretical investigation during the last two decades and the field has been reviewed thoroughly

111.

By studying the various properties of the annihilation quanta one obtains direct infor- mation about the' positron-electron system prior to annihilation. In particular, the 2y angular corre- lation of annihilation radiation (ACAR) reflects, by momentum conservation, the momentum density of

+ -

the annihilating e -e pair. Breaks in the density reflect directly the size and shape of the Fermi surface (FS) in pure metals and in alloys. If the behavior of the e+ is sufficiently well understood

+ -

and the e -e interaction sufficiently well accoun- ted for, the ACAR measurements can also lead to a test of the electronic wavefunctions of the system under investiga'tion.

Due to recent advances in multiple detector techniques, a renewed interest in high precision e+

annihilation experiments has developed, particular- ly in materials where the more standard FS solid state techniques such as the low temperature magne- to-oscillatory effects fail

-

for example in high concentration disordered alloys.

EXPERIMENTAL TECHNIQUES.?The usual e+ experiments proceed by injecting fast e+-s from a radioactive +Work supported by the National Science Foundation

source, such as 2 2 ~ a , 5 8 ~ o or 6 4 ~ u into a spatially oriented single crystal, usually kept at a low tem- perature. Since e+-s are known to be captured in low density defects such as vacancies and dislocations 121, well annealed samples are required. The e+-s slow down to thermal equilibrium with the sample prior to annihilation /I/. In 2y annihilation the two photons are non-collinear in order to conserve

+ -

the momentum p of the e -e pair. The deviationfrom - non-collinearity can be measured by counting coin- cidences between two detectors cl and c 2 as in figu- re I. Because of the large momenta of the photons ( = mc), the angles involved are only a few millira- dians ; thus the counters have to be positioned at a large distance from the samples (3-10 meters).The small angles 8 and

+

(figure 1) are directly rela- ted to the pZ (pZ = 8 mc) and py ( p y =

+

mc) compo- nents of the momentum

p

to be conserved. One milli- radian (mrad) corresponds to a momentum component of p = mc x = 0.137 a.u. (atomic units). The px component of

E

produces a Doppler shift (typically of a few keV) in the mc2 ( 5 1 1 keV) energy of each photon. Most experiments to date were performed with a "long-slit" geometry. corresponding to two long detectors with no energy resolution, measuring one component of

E

; given the momentum density D(F) of the annihilating pair, such a set-up measures N(p ) = // p(p)dpx dpy. The new multiple detector systems

"point-sli t geometry" and measure two compo- nents (no energy detection), thus yielding'N(pyxpz)=

/ p(p)dpx two dimensional (2D) surfaces. At Brandeis University we use two moveable sets of 32 NaI detec- tors each 131 at planes c l and c2 (figure 1, with

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786603

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L = 10 meters), measuring simultaneously 32 x 32 coincidences, with a present resolution of 0.5 x 1.5 mad2. Other laboratories have built or arebuilding 2D machines based on proportional wire chambers 141, on NaI ANGER cameras 151 or on multiple Ge detectors 161. The ultimate resolution of such 2D systems will be limited by source strength and by the thermal motion of the positron

-

^but

0.2 x 0.2 mrad2 resolutions seem feasible in the near future.

Fig. 1 : Geometry of the two-photon angular correlation experiments. Typically 8 and $ are a few milliradians

+ -

THEORY.- The importance of the e -e interaction, particularly in an electron gas, has been the subject of extensive many-body studies. The conclusion of such studies

-

born out by experiments

-

is that the e+-e- interaction does effect substantially the density of the electrons at the positron site,resul-

ting in a nearly ten-fold decrease of the positron lifetime from independent particle model (IPM) pre- dictions /I/ ; on the other hand the momentum distribution of the annihilating pair is not changed substantially from the IPM picture. The effect of

+ -

the e -e manybody interaction on the

shape

of the momentum density p(2) in an electron gas 171 is an increasing enhancement as E approaches the Fermimo- mentum, as plotted in figure 2. Plotted for compari-

son is the expected momentum density of an interacting electron gas, without the e+'s presence. The most important feature of figure 2 is that the dis-

continuity is not shifted from the FS due to the

+ -

e -e interaction. The predicted shape of

p(E)

^was

observed experimentally in oriented single crystals of Na by Donaghy and Stewart 181. Their long-slit measurement is shown in figure 3 ; the slight devia-

tion from a arabola (Planar cuts of a Fermi sphere) reflects the enhancement of figure 2 in a three di-

P

mensional isotropic distribution. The FS is found to be highly isotropic, as expected from band theory.

p in units of pF

Fig. 2 : The effect of many-body correlations on p(p). Curve (e -e ) , p(p) for a positron in an interacting electron gas ; curve (e -e ) , ~ ( p ) for an interacting electron gas without the positron ; e

,

non-interacting electron gas ; p(p)-s norma- llzes to equal volume in fre

p

space. Curves computed for electron density equivalent to that of the conduction band in Al.

angk between photons In m~lliradianr

Fig. 3 : Long-slit ACAR curves in Na, reference 181.

The experiment is compared to the many-body computation, reference 17) (-) and to a parabola (----) corresponding to a non-interacting electron gas ; a low intensity broad component has been sub- tracted from the raw data, corresponding to the co- re electron contribution

In the IPM theory, the p(~) measured by ACAR experiments is given at zero temperature, by

where u+(r) is the ground state

(k+

= 0) positron 0 -

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c6-1570 JOURNAL D E PHYSIQUE Bloch wave f u n c t i o n , $ k , n ( r ) = ~ ~ , ~ ( ~ ) e x p

-

( i k ~ ) i s

t h e e l e c t r o n wave f u n c z i o n w i t h c r y s t a l momentum&

( f i r s t zone) and band index n , and

l4

Z 1 ; t h e sum- mation i s over a l l occupied s t a t e s . One o b t a i n s

where t h e _Ds a r e t h e r e c i p r o c a l l a t t i c e v e c t o r s and AG(k,n) i s g i v e n by u , ( r ) u k n ( ~ )

+

=

A - ( & , n ) e ~ ~ ( i s * ~ ) . We n o t e t h z t i f one s e t s u + ( r )

1s 5

⁰-

c o n s t , e q u a t i o n (1) becomes i d e n t i c a l to t h e one governing t h e Compton p r o f i l e measurements / I / . To d a t e one p r e f e r s p o s i t r o n measurements f o r FS s t u - d i e s because of t h e p r e s e n t poor r e s o l u t i o n of y r a y Compton spectroscopy. A s d i s c u s s e d some timeago 191, e q u a t i o n s (1) and (2) show t h e e s s e n t i a l d i f - f e r e n c e between

k

and p space : an e l e c t r o n w i t h quantum number

k

c o n t r i b u t e s a t a l l ₌

k

+

2

i n p space, w i t h weights I ~ ~ ( k , n )

/ '.

Once t h e s-ation over occupied s t a t e s

iz

performed, t h e f o l l o w i n g c h a r a c t e r i s t i c s of p(p) a r e noted : 1) F i l l e d bands c o n t r i b u t e a continuous d i s t r i b u t i o n i n space ; 2) P a r t i a l l y f i l l e d bands appear a s d i s t r i b u t i o n s i n t h e r e p e a t e d zone scheme, bounded by d i s c o n t i - n u i t i e s a t t h e FS and modulated by t h e I ~ ~ ( k , n )

1'

amplitudes. The measurement of p (2) v i a

AEAR

experiments t h u s c o n t a i n s two s e t s of i n f o r m a t i o n : a) sharp b r e a k s i n p(p) r e f l e c t t h e topology and s i z e of t h e FS, whereas b) t h e a c t u a l shape of p(p) de- pends on t h e wave f u n c t i o n s v i a e q u a t i o n ( 1 ) . One problem i s t o s e p a r a t e t h e s e two e f f e c t s ; t h i s can b e achieved f o r simple ( n e a r l y - f r e e - e l e c t r o n ) bands, but f o r complicated FS t o p o l o g i e s , such as i n t r a n - s i t i o n m e t a l s , t h e o r y i s needed t o guide t h e expe- r i m e n t a l i n t e r p r e t a t i o n . Mijnarends has d i s c u s s e d t h e consequences of symmetry i n e q u a t i o n (1) / l o / . The c o n t r i b u t i o n t o p(2) of a band n o t belonging t o t h e t o t a l l y symmetric r e p r e s e n t a t i o n p o s s e s s e s nodes i n c e r t a i n symmetry d i r e c t i o n s . T h i s group theore- t i c a l s e l e c t i o n r u l e c a n l e a d t o momentum d e n s i t i e s t h a t a r e v e r y d i f f e r e n t from t h o s e obtained by t h e n a i v e assumptions t h a t ACAR ineasurements correspond

t o i n t e g r a l s through t h e FS i n

k

space.

I n t h e t h e o r e t i c a l IPM computations of p

(p)

one needs t o o b t a i n e+ wavefunctions, b e s i d e s t h e e l e c t r o n i c band wavefunctions. S i n c e t h e e a r l y Wigner-Seitz computation of u:(~) / 9 / , v a r i o u s

techniques have been used t o o b t a i n a r e a l i s t i c ; a n i s o t r o p i c e+ wavefunction, by u s i n g t h e s i g n in- v e r t e d p o t e n t i a l from band t h e o r y , w i t h o u t t h e ex- change t e - L l / , . I t i s t o be noted t h a t one s i n g l e

u:(~) i s needed i n e q u a t i o n (1) f o r any one material.

EXPERIMENTAL RESULTS.- Most of t h e experimental ACAR r e s u l t s have been d i s c u s s e d thoroughly i n t h e va- r i o u s r e v i e w s / l / . I n t h e f o l l o w i n g we s h a l l o n l y g i - v e a few i l l u s t r a t i v e examples t o demonstrate t h e power of t h e t e c h n i q u e , and t r y t o c o n c e n t r a t e on t h e most r e c e n t 2D ACAR measurements.

-

A l k a l i Metals.- The experiments on Na were a l - ready mentioned i n connection with t h e many-body problem. Although c o n v e n t i o n a l techniques such a s the de Haas-van Alphen e f f e c t can g i v e i n p r i n c i p l e much higher p r e c i s i o n FS v a l u e s , i n Na, and p a r t i c u -

l a r l y i n L i , t h e low temperature m a r t e n s i t i c t r a n s - formation p r e c l u d e s complete dHvA measurements I l l / . Long s l i t ACAR measurements i n s i n g l e c r y s t a l s of L i above t h e t r a n s i t i o n i n d i c a t e an a n i s o t r o p y ,

( 2 , 3-4 %) /8//12/ which i s s m a l l e r than p r e d i c t e d

by simple band theory. With t h e new 2D ACAR measurements i t w i l l b e p o s s i b l e t o o b t a i n a much more a c c u r a t e FS chape and s i z e f o r t h e s e m e t a l s .

-

Aluminum.

-

having a n e a r l y - f r e e - e l e c t r o n (NFE) conduction band h a s been used t o s t u d y many f e a t u - r e s of ei behavior i n m e t a l s . F i g u r e 4 shows t h e

Fig. 4 : 2D ACAR s u r f a c e from A l , r e f e r e n c e 1 3 1 . The o r i e n t a t i o n of t h e c r y s t a l i s i l l u s t r a t e d i n

terms of t h e B r i l l o u i n zone (BZ) i n t h e i n s e r t . Each c r o s s i n g of l i n e s i s a n independent measurement. Sample a t 100 K

r e c e n t 2D ACAR s u r f a c e o b t a i n e d a t Brandeis Univer- s i t y w i t h t h e NaI m u l t i c o u n t e r system 131. The d i s - t r i b u t i o n i s n e a r l y s p h e r i c a l i n t h e extended zone, r e f l e c t i n g t h e N F E f e a t u r e of t h e conduction band.

The i n f l u e n c e of t h e BZ on t h e FS and t h e wavefunc- t i o n s appears a s small d e v i a t i o n s from s p h e r i c i t y , p a r t i c u l a r l y around t h e r e c t a n g u l a r BZ f a c e . The d e t a i l e d shape of t h e ACAR s u r f a c e a g r e e s q u i t e w e l l w i t h an OPW computation, a s i n d i c a t e d i n f i g u r e 5 , f o r a s i n g l e curve (p : 0) of t h e f u l l 2D ACAR

Y

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Fig. 5 : A 1 ACAR c u r v e w i t h py = 0 , and t h e a b s o l u t e magnitude o f t h e f i r s t d e r i v a t i v e (second n e a r e s t n e i g h b o r s ) . Open c i r c l e s c o r r e s p o n d t o p, < 0 f o l d e d a b o u t p = 0 t o e x h i b i t t h e i n h e r e n t ' s p e t r y of t h e d a t a . F u i l c u r v e i s t h e t h e o r e t i c a l OPW p r e d i c t i o n ; Zashed c u r v e c o r r e s p o n d s t o a s i m p l e s p h e r e model. The i n s e t i n d i c a t e s t h e p r o j e c t i o n o f t h e BZ and of t h e ( 1 1 1 ) and (200) s p h e r e s used i n t h e s i m p l e model o n t o t h e (100) p l a n e

s u r f a c e 1131. At h i g h momenta t h e r a t h e r f l a t d i s - t r i b u t i o n o f f i g u r e 4 e x h i b i t s marked a n i s o t r o p i e s when m a g n i f i e d t e n - f o l d , due t o t h e h i g h momentum

components (HMC) of t h e c o n d u c t i o n e l e c t r o n s ( t h e s o - c a l l e d Umklapp a n n i h i l a t i o n s ) . A s t u d y of t h e s e HMC r e v e a l s t h a t one c a n d i f f e r e n t i a t e between a r e a l OPW and a pseudo-wavefunction, t h e e x p e r i m e n t p r e - f e r r i n g t h e f u l l OPW. R e c e n t l y a n u l t r a - h i g h p r e c i - s i o n l o n g - s l i t measurement w i t h 0.1-0.2 m a d r e s o l u - t i o n was performed i n A 1 1141, i n d i c a t i n g t h a t one c a n measure maximal FS o r b i t s w i t h a p r e c i s i o n of 0.25 % i n kF.

-

Copper.- Because o f t h e w e l l known FS o f c o p p e r many e + e x p e r i m e n t s have been performed t o t e s t va- r i o u s model c a l c u l a t i o n s of p ( ~ ) , a s w e l l a s t o s e r - v e a s a b a s e f o r a l l o y ' e x p e r i m e n t s . The w e l l known

< I l l > n e c k s c a n b e d i r e c t l y o b s e r v e d , even w i t h long- s l i t e x p e r i m e n t s 1151. A m o d i f i e d APW c o m p u t a t i o n 1161 p r e d i c t e d s u r p r i s i n g l y w e l l t h e observed a n i s o - t r o p i e s ( i . e . d i f f e r e n c e s i n ACAR c u r v e s a l o q g d i f - f e r e n t o r i e n t a t i o n ) . I n f i g u r e 6 we p l o t t h e 2D ACAR

< I l l > n e c k s a t p = pZ = 0 , a s o b t a i n e d i n r e f e r e n c e Y

/ 3 / . An a n a l y s i s of t h i s s u r f a c e , a s w e l l a s of p ( p ) o b t a i n e d from l o n g - s l i t measurements 1171 i n d i c a t e s r e a s o n a b l y good agreement w i t h t h e H a l s e CUES

/

181.

We n o t e t h a t i n a d d i t i o n t o t h e f i r s t zone c o n t r i - b u t i o n , one o b t a i n s i n copper a l a r g e c o n t r i b u t i o n i n ~ ( p ) c o r r e s p o n d i n g t o t h e f i l l e d 3d b a n d s , t h e c o r e e l e c t r o n s , a s w e l l a s HMC of t h e u n f i l l e d bands.

F i g . 6 : 2D ACAR s u r f a c e from Cu, r e f e r e n c e / 3 1

s u r f a c e i n Cu w i t h a n o r i e n t a t i o n showing o n e of t h e

-

T r a n s i t i o n m e t a l s and r a r e e a r t h s . - Most t r a n -

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C6-1572 JOURNAL DE PHYSIQUE

s i t i o n m e t a l work t o d a t a h a s used t h e l o n g - s l i t geometry. Because of t h e i r coinplex FS s t r u c t u r e , p ( p ) r e q u i r e s band c o m p u t a t i o n s f o r g u i d a n c e i n in-

-

t e r p r e t a t i o n . The r a r e e a r t h s a l s o e x h i b i t a r i c h a n i s o t r o p i c s t r u c t u r e , s i n c e t h e i r 5d-6s h y b r i d i z e d bands a r e s i m i l a r t o t h o s e of t r a n s i t i o n m e t a l s . From t h e many t r a n s i t i o n m e t a l e + e x p e r i m e n t s we s i n g l e o u t t h e r e c e n t work on V , C r , Mo and Nb by S h i o t a n i e t a l . / 1 9 / , and t h e t h e o r e t i c a l APW com- p u t a t i o n s by Wakoh e t a l . /20/.The t h e o r e t i c a l a n i - s o t r o p i e s a r e i n good agreement w i t h e x p e r i m e n t s , i n p a r t i c u l a r f o r Nb

-

see f i g u r e 7 f o r V and Nb. The a c t u a l i n d i v i d u a l ACAR s h a p e s a r e somewhat l e s s well p r e d i c t e d .

momentum I" a u momentum a u

Fig. 7 : The momentum a n i s o t r o p i e s

-

d i f f e r e n c e s between l o n g - s l i t ACAR c u r v e s a l o n g t h e d i r e c t i o x l s i n d i c a t e d

-

from V and Nb. The e x p e r i m e n t a l a n i s o - t r o p i e s (-, r e f e r e n c e 1191) a r e compared w i t h t h e t h e o r e t i c a l APW p r e d i c t i o n s (----, r e f e r e n c e / 2 0 / )

The r a r e e a r t h s have been s t u d i e d some t i m e ago w i t h l o n g - s l i t geometry /21./, and showed a com- p l e x FS v i a an a n i s o t r o p i c p ( p ) , t h e main f e a t u r e s

-

of which had been p r e d i c t e d by band t h e o r y 1221.

We n o t e i n c o n n e c t i o n w i t h t r a n s i t i o n m e t a l s and r a r e e a r t h s t h e development o.f t h e p o l a r i z e d e + a n n i h i l a t i o n t e c h n i q u e f o r t h e s t u d y of s p i n a l i g n e d moment* d e n s i t i e s i n f e r r o m a g n e t i c s u b s t a n c e s . The

t h e o r y , a s d e v e l o p e d by Berko / 2 3 / u s e s t h e s p i n po- l a r i z a t i o n o f e+ by p a r i t y n o n - c o n s e r v a t i o n i n b e t a d e c a y and t h e s e l e c t i o n r u l e f o r b i d d i n g 2y d e c a y from s p i n - t r i p l e t c o l l i s i o n s . E x p e r i m e n t s h a v e b e e n performed i n Fe 1241, i n N i / 2 5 / and i n Gd 1261.

M i j n a r e n d s compared t h e e x p e r i m e n t s i n Fe w i t h s p i n

p o l a r i z e d band c o m p u t a t i o n s o f p + ( p )

- o+(p)

/ l o / . B e s i d e s t h e above mentioned e x p e r i m e n t s , many o t h e r s have b e e n p e r f o r m e d i n a v a r i e t y of me-

t a l s and s e m i c o n d u c t o r s . P e r h a p s t h e most i m p o r t a n t u s e of t h e e + t e c h n i q u e i s i n t h e s t u d y of d i s o r d e - r e d n o n - d i l u t e s u b s t i t u t i o n a l a l l o y s , a s w e l l a s o t h e r a l l o y s y s t e m s and o r d e r e d m e t a l l i c compounds, n o t amenable t o s t a n d a r d FS t e c h n i q u e s / 2 7 / .

-

iso ordered alloys_.- Many ACAR measurements h a v e b e e n performed t o s t u d y t h e change i n s h a p e and s i z e of t h e FS upon a l l o y i n g . The a l l o y r e s u l t s have been reviewed up t o 1974 b y Berko and Mader 1281, and a more r e c e n t e + r e v i e w by M i j n a r e n d s / I / c o n t a i n s a n

up-to-date l i s t . As p r e d i c t e d by modern ATA and CPA band c o m p u t a t i o n s / 2 9 / , t h e FS r e m a i n s a v a l i d con- c e p t i n a l l o y s , a l t h o u g h t h e s m e a r i n g a t t h e FS i n - c r e a s e s w i t h e / a due t o s c a t t e r i n g . Most e x p e r i m e n t s have been performed on c o p p e r b a s e d a l l o y s . A compa- r i s o n w i t h t h e p u r e copper r e s u l t s i n d i c a t e s t h a t i n a-Cu Zn, a-Cu A l , a-Cu S i and a-Cu Ge t h e FS i n c r e a - s e s m o n o t o n i c a l l y w i t h e / a , f o l l o w i n g r e a s o n a b l y w e l l t h e r i g i d band p r e d i c t i o n s b a s e d on t h e Cu

bands. T h e r e i s however e v i d e n c e , p a r t i c u l a r l y i n t h e growth of t h e < I l l > neck d i m e n s i o n , t h a t n o t a l l t h e above a l l o y s y s t e m s s c a l e t h e i r FS on a u n i v e r s a l c u r v e a g a i n s t e / a . To i n d i c a t e t h e p r e c i s i o n o b t a i - n a b l e on

p(e)

i n d i s o r d e r e d a l l o y s we p l o t i n f i g u r e 8 t h e r e c e n t 2D ACAR s u r f a c e s o b t a i n e d i n a Cu-11.6%

Zn sample f o r two d i s t i n c t o r i e n t a t i o n s / 3 0 / . T h i s t i m e we p r e s e n t t h e 2D ACAR s u r f a c e a s a c o n t o u r map i n o r d e r t o e x h i b i t more q u a n t i t a t i v e l y t h e a n i - s o t r o p i e s observed i n ~ ( p ) . An a n a l y s i s of t h e s e r e - s u l t s i n d i c a t e t h a t i t i s p o s s i b l e t o measure a n i n c r e a s e i n t h e FS t o w i t h i n k.03 mrad, i . e . w i t h a 0.5 % a c c u r a c y . New measurements i n Cu-Pd and Ag-Pd w i t h l o n g - s l i t geometry 1311 i n d i c a t e d e v i a t i o n s from r i g i d band b e h a v i o r i n t h e s e systems, Suzuki e t a l . / 3 2 / have s t u d i e d t h e 5-Cu-Ge p h a s e s and s u g g e s t s t r o n g l y d i s t o r t e d FS s h a p e s .

With improved r e s o l u t i o n i t w i l l become f e a - s i b l e t o s t u d y t h e smearingsof t h e FS w i t h a l l o y i n g i n n o n - d i l u t e s y s t e m s , e q u i v a l e n t t o D i n g l e tempera- t u r e s t u d i e s by dHvA measurements a t low c o n c e n t r a - t i o n s . The t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e e + r e - s u l t s i n d i s o r d e r e d a l l o y s i s more c o m p l i c a t e d t h a n i n p u r e m e t a l s , i n view of t h e p o s s i b l e p r e f e r e n c e of e+-s f o r one o f t h e a l l o y c o n s t i t u e n t s / I / , a s w e l l a s due t o random s c a t t e r i n g of t h e e + i n t h e

a l l o y 1331. These e f f e c t s w i l l have a n i n f l u e n c e on

(7)

t h e s h a p e of p

(p),

b u t w i l l have l i t t l e a f f e c t o n n i c momentum d e n s i t i e s and Fermi s u r f a c e s i n p a r t i - t h e p o s i t i o n of t h e FS d i s c o n t i n u i t y . c u l a r . It i s c l e a r t h a t w i t h t h e i n c r e a s e d p r e c i s i o n

o f t h e 2 D ACAR measurements e x p e c t e d i n t h e n e a r fu- t u r e and w i t h a n improved t h e o r e t i c a l u n d e r s t a n d i n g

P, along [IOU in rncx10-~

+ 6 6 - 4 - 2 0 2 4 6 8 I0 of t h e i n f l u e n c e o f t h e e+-e- i n t e r a c t i o n s o n t h e

0 ,

7' ACAR r e s u l t s , ~ o s i t r o n a n n i h i l a t i o n t e c h n i q u e s w i l l

Q N

p l a y an i n c r e a s i n g r o l e i n t h e s t u d y o f m e t a l s , me-

.I 0

E .

t a l l i c compounds and d i s o r d e r e d a l l o y s .

--

E

o ACKNOWLEDGMENTS.- I wish t o t h a n k Dr. Frank S i n c l a i r

5

CU f o r a c a r e f u l r e a d i n g of t h i s m a n u s c r i p t .

I- IP U

e

-

0

E R e f e r e n c e s

111

West, R . N . , Adv. Phys.

2

(1973) 263 ; Berko, S., Compton S c a t t e r i n g , B. W i l l i a m s , e d . , C h a p t e r 9

(McGraw-Hill, London 1977) ; M i j n a r e n d s , P.E., P o s i t r o n s i n S o l i d s , H a u t o j a r v i , e d . ( S p r i n g e r -

P, along [log in ~ C X I O - ~

V e r l a g , t o b e ~ u b l i s h e d , 1978)

F i g . 8 : Contour g r a p h s c o r r e s p o n d i n g t o 2D ACAR s u r f a c e s f r o m a - b r a s s ( 1 1 . 6 a t . 7, Zn) f o r two o r i e n - t a t i o n s . The d a s h e d l i n e s c o r r e s p o n d t o t h e p r o j e c - t i o n of t h e BZ o n t o t h e (py,pZ) p l a n e . R e f e r e n c e 1301

-

M e t a l l i c compounds.- One of t h e most i n t e r e s t i n g m e t a l l i c compounds s t u d i e d by e+-s i s t h e h i g h tem- p e r a t u r e s u p e r c o n d u c t o r V3Si. L o n g - s l i t e x p e r i m e n t s 1341 h a v e been i n t e r p r e t e d a s e v i d e n c e f o r p l a n a r s t r u c t u r e s i n t h e FS a s e x p e c t e d by t h e l i n e a r - c h a i n model 1351. Two d i m e n s i o n a l ACAR measurements, howe- v e r ; e x h i b i t a v e r y complex s t r u c t u r e 1281. New 2D s u r f a c e measurements i n d i c a t e t h e a p p e a r a n c e of cy- l i n d r i c a l - t y p e h o l e s u r f a c e s w i t h a x e s a l o n g t h e cube e d g e s 1361, s i m i l a r t o t h e FS s u g g e s t e d f o r V3Ce from magnetothermal o s c i l l a t i o n s 1371 and f o r Nb Sn

3 from dHvA measurements 1381.

CONCLUSION.- We have b r i e f l y o u t l i n e d t h e u s e of t h e p o s i t r o n a n n i h i l a t i o n t e c h n i q u e i n s t u d y i n g e l e c t r o -

1 2 1 See f o r example t h e r e v i e w by S e e g e r , A., Appl.

Phys.

5

(1974) 183

/ 3 / Berko, S . , Haghgooie, M., and Mader, J . J . , Phys.

L e t t .

63A

(1977) 335

1 4 1 J e a v o n s , A . P . , Charpak, G . , S t u b b s , R . J . , Nucl.

I n s t r u m . Methods

124

(1975) 491 ; Manuel, A.A., Bongi, G.H., F i s c h e r ,

@.,

P e t e r , M . , Helv. Phys.

A c t a .

50

(1977) 166 ; Douglas, R . J . ,

S t e w a r t , A.T., HI4 IV I n t . Conf. on P o s i t r o n A n n i h i l a t i o n , ~ e l s i n g d r (1976)

151 West, R.N., p r i v a t e communication

1 6 1 T r i f t s h a u s e r , W . , l l n d I n t . Conf. on P o s i t r o n A n n i h i l a t i o n , K i n g s t o n (1971) 4.77-86

1 7 1 Kahana, S., Phys. Rev.

129

(1963) 1622 1 8 1 Donaghy, J . J . , S t e w a r t , A.T., Phys. Rev.

164

(1967) 396

/ 9 / B e r k o , S . , P l a s k e t t , J . S . , P h y s . Rev.

112

(1958) 1877

1101 M i j n a r e n d s , P.E., P h y s i c a

63

(1973) 235

/ I I / Lee, M . J . G . , CRC C r i t . Rev. S o l i d S t a t e S c i .

2

(1971) 8 5

1121 P a c i g a , J . J . , W i l l i a m s , D.L., Can. J . Phys.

2

(1971) 3227

1131 Mader, J . J . , Berko, S . , K r a k a u e r , H . , B a n s i l , A., Phys. Rev. L e t t .

37

(1976) 1232

1141 S t e w a r t , A.T., p r i v a t e communication

1151 Cushner, S . , E r s k i n e , J . C . , Berko, S . , Phys.

Rev.

81

(1970) 2852

1 1 6 1 B r o s s , H . , S t o h r , H . , Appl. Phys.

3

(1974) 307 1171 M i j n a r e n d s , P.E., Phys. Rev.

178

(1969) 622 I 1 8 1 H a l s e , M . R . , P h i l o s . T r a n s . R. Soc. London

A265 (1969) 507

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1191 S h i o t a n i , N . , Okada, T . , Mizoguchi, T . ,

Sekizawa, H . , J . Phys. Soc. J a p a n

2

(1975) 423 I 2 0 1 Wakoh, S . , Kubo, Y., Yamashita, J . , J. Phys.

Soc. J a p a n 2 (1975) 416.

1211 W i l l i a m s , R.W., Mackintosh, A.R., Phys. Rev.

168 (1968) 679

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C6-1574 ^JOURNAL^DE^PHYSIQUE

/22/ G u p t a , R.P., L o u c k s , T.L., P h y s . Rev.

176

(1968) 848

1 2 3 1 B e r k o , S . , P o s i t r o n A n n i h i l a t i o n , e d s .

A.T. S t e w a r t and L.O. R o e l l i g (Academic P r e s s , N.Y. 1967) pp. 61-79

/24/ a) M i j n a r e n d s , P.E., P h y s i c a

63

(1973) 248 ; b ) Berko, S . , M i l l s , A.P. J r . , J . P h y s i q u e

2

(1971) C1-287

/ 2 5 / S h i o t a n i , N., Okada, T . , S e k i z a w a , H.,

M i z o g u c h i , T., K a r a s a w a , T . , J. Phys. Soc.Japan 35 (1973) 456 ; see a l s o r e f e r e n c e / 2 4 b / -

/26/ Hohenemser, C., W e i n g a r t , J . M . , B e r k o , S . , Phys. L e t t .

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(1968) 41

/ 2 7 / S e e f o r e x a m p l e C o l e r i d g e , P.T., T e m p l e t o n , I.M., Can. ^{J .}Phys.

49

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sics,

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38 (1978) t o b e p u b l i s h e d

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40

(1978) 1590

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JOURNAL DE PHYSIQUE

Colloque C6, suppliment au no 8, Tome 39, aolir 1978, page

^C6-1575

POST DEADLINE PAPERS ACCEPTED FOR CONTRIBUTION

-

A new series of antiferromagnets

.

A.P. Filho, F. Palacio, C.J. Connor, R.L. Carlin

-

Inelastic neutron scattering study of solitons in C NiF3 in an external field

.

M. Steiner, J. Kjems 5

-

Effects of disorder on the thermoelectric power and electrical conductivity of TTT -I

P.M. Chaikin, S.K. Khanna, R.B. Somoano 2 3 '

-

p-state pairing and the ferromagnetism of ZrZn

C.P. Enz, B.T. Matthias ^{2 '}

-

Itinerant antiferromagnetism of TiBe 2 '

B.T. Matthias. A.L. Giorgi, V.D. Struebing, J.L. Smith

-

Some experiments on powdered PrNi5 in contact with liquid 3 ~ e

.

P.R. Roach, K.A. Webb, J.B. Ketterson

-

Crossover from long range to short range critical behaviour in the Ising ferromagnet Tb(0H)

A.T. Skjeltorp 3'

-

Magnetic ordering in Ce3A1,, compound

.

G. Chouteau, R. Tur, J. Flouquet, J.P. Keradec, J. Palleau, R. Tournier

-

Effect of antiferromagnetic ordering on electron kenetic phenomena in some metal-like compounds at low temperatures

.

V.I. Novikov, V.P. Zhuze, V.M. Sergeeva, S.S. Shalyt

-

Effect of hydrogen absorption on the superconducting and magnetic properties of CeRu2 and LaRu2

.

G.X. Tessema, J. Peyrard, A. Nemoz, J.P. Senateur, A. Rouault, R. Fruchart

-

Fermi liquid theory of the degenerate Anderson Model

.

Zawadowski

- AC susceptibility and ESR measurements on the Ag:Mn spin glass

.

E.D. Dahlberg, M. Hardiman, J. Souletie

- Temperature dependent spin-flip scattering in a spin glass

.

J. Schuller, R. Orbach, R.M. Chaikin

-

On spin-wave spectrum of systems without long-range magnetic order

.

J. Spatek, N. Rivier

-

Propagation of high-frequency phonons in thin SiO films at 60 mK

.

R.E. Horstman, J. Wolter, M.C.H.M. Wouters

-

Ultrasonic spectroscopy of the acceptor ground state in silicon

.

H. Zeile, 0. Mathuni, K. Lassmann

-

Critical behaviour of the thermal conductivity near the critical point of gadolinium

.

J.B. Sousa, R.P. Pinto, R.M. Amado, J.M. Moreira, M.E. Braga

-

Heat flow rectification in inhomogeneous copper

.

K. Balcer&

-

Inflammation of electron-hole radiation under the influence of ultrasound wave

.

A.S. Alekseev, T.I. Galkina

-

Further evidence for fractional charge on matter

.

G.S. Larue, W.M. Fairbank, J. Phillips