DECAY OF THE POSITRONIUM MOLECULE INT0 A POSITRONIUM

ATOM AND PHOTONS

Yu. A. Tisenko UDC 538.13:530,145

Decays of the positronium '.molecule Ps 2 into para - or orthopositronium Ps in the ground state and photons are investigated. The differential

probabilities of the decays are determined. The total probabilities (psi) o~ Ps 2 annihilation with the production of two and three ~'t~ 's~) a n d w ~

p h o t o n s and p o s i t r o n i u m a r e c a l c u l a t e d t o be

w(Ps~) _ 1,6. I0 I0 sea -I , ,,.(Ps=) = 0.43. I0 ~ sea -I . 2T - - ~3T

The curve of the angular correlation of the y rays on the decay of PS 2 into

two photons and parapositronium is studied. The width of this curve is

AIP I = 0.128 a.u. (IPI is the total photon momentum), which corresponds to a deviation of the emission angle of the T rays from ~: 0 < 0.934 mrad. The maximmn in the distribution of the photons with respect to

the momenta P in the center of mass of the annihilating pair is

attained at P = 0.175 a.u. The calculations were made on an M-222 computer, and their accuracy is determined by the choice of the wave function of the positronium molecule and the accuracy in the computer calculation of the integrals.

In this paper, we investigate the decay of the positronium molecule Ps 2 into

para- or orthopositronium Ps in the ground state and photons. The positronium molecule

is classified by Gol'danskii [11 as a "light" electron--positron system. It is interest-

ing, in particular, in that in it two-electron and two-positron exchange bonds of the

ordinary covalent type can be realized simultaneously. The properties of such systems depend strongly on the magnitude of the exchange interaction. The part played by the

Pauli principle in many-particle systems was studied in [2-41. The formation of complex "light" electron--positron systems in gases was considered in [5, 6S. In [7], the

reasons for the difficulties in the experimental detection of such systems in matter are

explained, and [8] notes the possible existence of bound states of a positron and electrons in defect ionic crystals (F~ and F~" centers).

The positronium molecule was investigated theoretically for the first time in [9, 10].

In [iI], the molecule was shown to be stable against decay into two positronium atoms, the

Ps 2 wave function was established, and the binding energy was calculated and found to be

0 504 a u. In the present paper, we determine the probabilities -~ (P%) and w~ Psi) of an- nihilation of the positronium molecule with the production of two and three photons and a

positronium atom in the ground state, and we also study the correlation curve and distribu- tion of the y rays with respect to their total momentum when Ps 2 decays into paraposi-

tronium and two photons. Unless stated otherwise, relativistic units are employed: h = c = I.

I. The vectors of the initial and final states in the reaction Ps 2 § Ps + 27 can be written as

+ + + [r = ~ T~ (P,, P=, P3, P,) Z,,=o~,,,ao, (p,) b~ (p=) a~, (p~) b,, (p,)JO>, (1) <r <,,',.':~ ~p,l (p;, p~) Zo,,('s's") .<Ola,(p;)b,(p~)c~,(~Oc~=(~D"

Here, IO) is the vacuum state; ~, ~(ps) and Z, Z (s'sz) are the wave and spin functions of

Ukraine and Russian Reunion Tercentenary State University, Dnepropetrovsk~ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No~ 2, pp. 3-6, February, 1981. Original article submitted July 31, 1979; revision submitted January 8, 1980.

0 0 3 8 - 5 6 9 7 / 8 1 / 2 4 0 2 - 0 0 9 9 5 0 7 . 5 0 �9 1981 Plenum P u b l i s h i n g C o r p o r a t i o n 99

t h e Ps 2 and Ps ; S and S z a r e t h e q u a n t u m n u m b e r s o f t h e p o s i t r o n i u m ~tom;

L(o,c) .... ~ 2 -1:~ (~:~ .+2 - - ~ . ,2~. : ) ;

t h e b r a c k e t s {z,p}, { ~ ' , p ' } mean t h a t t h e s u m m a t i o n i s o v e r t h e f o u r u n p r i m e d a n d two p r i m e d s p i n s and momenta , ~t, %, Pt, P3 and ~ , ~ , P2, P4 b e i n g t h e s p i n s a nd momenta o f t h e e l e c t r o n s and p o s i t r o n s i n t h e p o s i t r o n i u m m o l e c u l e ; t h e n o t a t i o n ~ , p~ and ~ , p~ r e f e r s t o t h e e l e c t r o n and p o s i t r o n w h i c h fo rm t h e p o s i t r o n i u m a tom a f t e r t h e a n n i h i l a t i o n ; a + and a , b + and b , and c + and c a r e t h e o p e r a t o r s and a n n i h i l a t i o n o f e l e c t r o n s , p o s i t r o n s , and p h o t o n s , r e s p e c t i v e l y ; ~ 1 , ~2 and ~ , ~2 a r e t h e p h o t o n p o l a r i z a t i o n s and momenta . The v e c t o r s I+i > and (@fl a r e o r t h o g o n a l , a nd t h e n o r m a l i z a t i o n c o n d i t i o n < ~ i [ ~ i > = < ~ / [ ~ / > = 1 m u s t be t a k e n i n t o a c c o u n t by m u l t i p l y i n g t h e d e c a y p r o b a b i l i t y by a f a c t o r d e t e r m i n e d by t h e n u m b e r o f i d e n t i c a l p a r t i c l e s i n t h e i n i t i a l a nd f i n a l s t a t e s . I t i s a s sume d t h a t t h e Ps 2 and Ps wave f u n c t i o n s i n t h e momentum r e p r e s e n t a t i o n , ~ and ~ ( P s ) , a r e n o r m a l i z e d by t h e c o n d i t i o n

The s p i n f u n c t i o n X i s d e t e r m i n e d by

The c a l c u l a t i o n s a r e made i n t h e c e n t e r - o f - m a s s s y s t e m o f t h e p o s i t r o n i u m m o l e c u l e . The m a t r i x e l e m e n t i s

<%,1 s I m,> = ( 2x ) 'iB(4) (P/-- Pt ) T (psO 2-f

, . , ~ , , . , (s,s,~ _ 2 ' S a + ' - " --4 ~ ~ ~(Pt, P,,P,,P=)"~(Ps)(pz,P=)Z=:,,+.'=Z=,%<. T [ =,tP,)b+(P2)IO>, l =,,.,.p,,p,} l ~',p' }

( 3 )

P~ = pt + p~ + p~ + p'.., P, = O, P: = ~ + ~ + p ] + p ; .

The f a c t o r 4 h a s a p p e a r e d a f t e r s u b s t i t u t i o n o f t h e e x p r e s s i o n s f o r (~f[ a nd [@i > and t h e a p p r o p r i a t e c o m m u t a t i o n s . We h a v e t a k e n i n t o a c c o u n t t h e f a c t t h a t ~ i s s y m m e t r i c and • a n t i s y m m e t r i c u n d e r t h e p e r m u t a t i o n o f two e l e c t r o n s o r p o s i t r o n s , w h i c h i s a c o n s e q u e n c e o f t h e i d e n t i t y o f t h e e l e c t r o n s a n d p o s i t r o n s i n t h e Psg.. The a m p l i t u d e (@f iS I@ i ) c a n be e x p r e s s e d i n t e r m s o f t h e m a t r i x e l e m e n t o f t w o - p h o t o n a n n i h i l a t i o n o f an e l e c t r o n and a p o s i t r o n ( b o t h f r e e a n d n o n r e l a t i v i s t i c ) :

<271Sa+(p,)b+(P2)[O> = (2~)' iB(4) ( P , - - D/) '~(~176 �9 ( 4 ) ,,p(=:=)

I n ~2~ , we h a v e r e t a i n e d o n l y t h e z e r o t h a p p r o x i m a t i o n i n P l / m and P2/m (m i s t h e e l e c t r o n m a s s ) . I t f o l l o w s f r o m t h e a n a l y s i s o f t h e s p i n f u n c t i o n s t h a t i n t h e c o n s i d e r e d c a s e p o s i t r o n i u m i s f o r m e d o n l y i n t h e p a r a s t a t e w i t h S = 0 , S z = 0 . C o m p a r i n g (3) w i t h ( 4 ) , we o b t a i n

T(PS,) = , - , , , ( s , s z ) T ( ~ : ~ ) 2T --4 "~ ~ ~,(P,,P2, P,,P'=)~?(P')(P,, P ; ) Z ..... ,%Z=':; 2~ . "[ =,,'%,P,,P=} l n ' , p ' }

T (~:~) i n t h e z e r o t h a p p r o x i m a t i o n i n t h e momenta P l and P2 We t a k e t h e e x p r e s s i o n f o r 2~ T h e r e f o r e , d o e s n o t d e p e n d on P l a n d P 2 . U s i n g t h i s , we c a n r e l a t e T ~ s")" t o t h e

~(Ps) m a t r i x e l e m e n t 12T o f a n n i h i l a t i o n o f t h e p a r a p o s i t r o n i u m g r o u n d s t a t e . F o r t h i s , we

~(Ps) r e p r e s e n t ~2T i n t h e fo rm

,r,(Ps) (0,0) T (~:~)

{ ~I,==,P.P . . }

w h e r e T~ '~) i s c a l c u l a t e d i n t h e z e r o t h a p p r o x i m a t i o n i n P l and P2" I t f o l l o w s t h a t

( 5 )

I n a d d i t i o n

(o,o),..(~:,) - - T~ Ps) ~ t % 1 2 7 - -

':..,'. (p,, p,) p i "~ P l P= Z

( 6 )

,,(s'sz)TP:') = O, S 1. L~t% ZT

(7)

i00

12 (I~I) l ~ ( a . u)

Fig. 1

l~i (a.u) I

O'- iR~- I f ( I~ I ) I ~ (a.u)

~" t .~t~t(a.u )

F i g . 2

Expanding ~-~ ~ff_~X~,~/~'X(7'~ z) in (5) with respect to the functions ~(s~s~,~ , using (6) and 47) {~,~ I ~ ale ~

and s q u a r i n g ( 5 ) , we o b t a i n a f t e r some s i m p l e m a n i p u l a t i o n s

V I ~(Ps) 12 ~2x IF., <8) ] r ~ -'> I ~ = 1 6 . 1 / ( P ) I -~ 4:1 r (o )

f<P)= ~ ~(P,,P..,P:,P'~)?(Ps)(P:,P;L +(P~)(O)---V'P ~ ~(P~)(p,,p,.,). 49) {p,,p,,p],p~} ~p,,p,>

I n ( 9 ) , P i s t h e p o s i t r o n i u m momentum; i t o c c u r s i n ~<Ps) a s a p a r a m e t e r . I n 48) , we have summed o v e r t h e p o l a r i z a t i o n s o f t h e two p h o t o n s i n t h e f i n a l s t a t e .

S e p a r a t i o n o f t h e n o r m a l i z a t i o n v o l u m e s V f rom t h e wave f u n c t i o n s a n d a l l o w a n c e ~or the presence of identical particles before annihilation leads to the differential proba- bility

d3~ld3tr d~aT~,)= If(P)I 2 d " P . "ps" d ~ s) ( 2 ~ ) , ~ ( 4 ) ( p / _ p , ) 2~ ~ I r (0 ) I ' �9 a ~ , , = r ~P~ ]~ . ( l O )

After integration in (I0) over the momenta of the photons with allowance for their identity, we obtain ,aj(Ps) d3p

d~Ps2 ) = -2~ . ~. ~ ltps>(0)j~ [/(P)l (2~y' (11)

where ~s) is the parapositronium annihilation probability. The function f(P) depends only on IP] and is determined by the relation 49); an expression for the Fourier transform

(P,, Pi, P],P~) is given in Ref. 11.

In the calculation of /(IP]), the expression on the rlght-hand side of (9) was re: duced by lengthy transformations using Feynman's parametric method to a single integral over the parameter x 40 < x < I), which was calculated by means of an M-222 computer. We took 1000 mesh points in the range of integration of the variable x. Figure 1 shows the

the function Jf(JPl)l~0 which is proportional to the photon angular correlation curve. The maximal value of ~(IPI)I 2 is 0.594-103 a.u. and is attained at [PI = O, and the width

of the curve is 0.128 a.u., which corresponds to a deformation of the emission angle of

the 7 rays from ~: 0 < 0.934 mrad. The function 4~.IPII.I/(IP[)I ~ is proportional to the distribution of the photons with respect to their total momentum IPI and is shown in Fig. 2. The maximum of this dependence is attained at IF[ = 0.175 a.u. and is 0.68"102 a.u. Bearing in mind that [~(Ps)(0)[~ = (8~) -I a.u. for the ground state of Ps in accordance with [12], we obtain, integrating (11) over P,

w~ s') = 2.006.w~ p~).T = 1.6- 10 '~ s e c - 1 ,

The a c c u r a c y o f o u r r e s u l t s i s d e t e r m i n e d by t h e c h o i c e o f t h e wave f u n c t i o n o f t h e p o s i t r o n i u m m o l e c u l e a n d t h e a c c u r a c y o f t h e c o m p u t e r c a l c u l a t i o n o f t h e i n t e g r a l s .

2 . S i m i l a r c a l c u l a t i o n s f o r t h e r e a c t i o n P s 2 - ~ P s ~ - 3 7 l e a d t o

u,(Ps,) = v.--aT j 2 -- = 0.43- 10 ~ sec-1,

!01

where ~0,<Ps) is the probability of orthopositronium annihilation. The decay ratio ~V(Pso)/r0, psi-) =372 is equal to the ratio of the probabilities of 27 and 37 annihila-

2 T - /--3}' t i o n o f u n p o l a r i z e d e l e c t r o n s and p o s i t r o n s i n a f r e e c o l l i s i o n .

I t h a n k V. S. V a n y a s h i n , Yu. F. S m i r n o v , V. P. S h a n t a r o v i c h , a nd P. U. A r i f o v f o r h e l p f u l d i s c u s s i o n s and i n t e r e s t i n t h e work .

LITERATURE CITED

1. V. I. Gol'danskii, Physical Chemistry of the Positron and Positronium [in Russian], Nauka, Moscow (1968).

2. V. I. Kukulin, V. G. Neudachin, Yu. F. Smirnov, and R. M. El'-Khovari, Izv. Akad. Nauk SSSR, Ser. Fiz., 3_88, 2123 (1974).

3. Yu. F. Smirnov and R. M. El'-Khovari, Izv. Akad. Nauk SSSR, Set. Fiz., 39, 2084 (1975). 4. V. I. Kukulin, V. G. Neudachin, and Yu. F. Smirnov, Fiz. Elem. Chastits At. Yadra, I__O0,

1236 (1979). 5. G. Ferrante and R. Geracitano, Phys. Rev. 182, 215 (1969). 6. S. C. Mukherjee and D. P. Sural, Phys. Rev. A 6, 125 (1972). 7. U. A. Arifov and P. U. Arigov, Physics of Slow Positrons [in Russian], FAN UzSSR,

Tashkent (1971). 8. E. P. Prokop'ev, in: Quantum Properties of Atoms and Ions and Positron Diagnostics

[in Russian], FAN UzSSR, Tashkent (1975), Ch. 4. 9. J. A. Wheeler, Ann. New York Acad. Sci., 48, 219 (1946).

10. A. Ore, Phys. Rev., 70, 90 (1946). 11. E. A. Hylleraas and A. Ore, Phys. Rev., 7_! , 493 (1947). 12. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory,

Oxford (1971).

CHARACTERISTICS OF GROWTH OF COMPLEX FERROELECTRIC OXIDE FILMS

BY PLASMA-ION SPUTTERING

V. M. Mukhortov, Yu. I. Golovko, V1. M. Mukhortov, and V. P. Dudkevich

UDC 537.226.33

An e x p e r i m e n t a l i n v e s t i g a t i o n was made o f t h e p r o c e s s o f g r o w t h o f a c omp l e x o x i d e f i l m , s u c h a s BaTiO 3 o r (Ba, S r ) T i 0 3 , by p l a s m a - i o n s p u t t e r i n g . I t was f o u n d t h a t i o n b o m b a r d m e n t o f a c e r a m i c t a r g e t k n o c k e d o u t n e u t r a l e x c i t e d a t o m s . T h e s e a toms l o s t e n e r g y away f rom t h e t a r g e t by c o l l i s i o n s a nd a t a c e r t a i n c r i t i c a l d i s t a n c e h c r t h e y w e r e c a p a b l e o f o x i d a t i o n t o p r o d u c e HaO, TiO, T i O 2, a n d SrO. T h e r e f o r e , d e p e n d i n g on t h e d i s t a n c e b e t w e e n t h e c a t h o d e and t h e s u b s t r a t e , t h e " c o n s t r u c t i o n " m a t e r i a l a r r i v e d i n t h e fo rm o f a toms o r m o l e c u l e s o f s i m p l e o x i d e s . T h e s e two ( a t o m i c a nd m o l e c u l a r ) d e p o s i t i o n m e c h a n i s m s c o r r e s p o n d e d t o two m e c h a n i s m s o f s y n t h e s i s a nd c r y s t a l l i z a t i o n d i f f e r i n g i n r e s p e c t o f t h e d e p e n d e n c e s o f t h e g r o w t h r a t e , u n i t c e l l p a r a m - e t e r s , a n d i ~ Q t h e r s t r u c t u r a l p r o p e r t i e s on t h e d e p o s i t i o n t e m p e r a t u r e . The r o l e o f r e - e v a p o r a t i o n and o f o x i d a t i o n - r e d u c t i o n p r o c e s s e s was a n a l y z e d .

F u n d a m e n t a l i n v e s t i g a t i o n s and p r a c t i c a l a p p l i c a t i o n s o f f e r r o e l e c t r i c f i l m s i n m i c r o e l e c t r o n i c s a r e b e i n g h i n d e r e d by t h e a b s e n c e o f a r e l i a b l e m e t h o d f o r g r o w i n g c o m p l e x f i l m s and b y t h e l a c k o f s u f f i c i e n t d a t a on t h e p h y s i c o c h e m i c a l g r o w t h p r o c e s s e s . We s h a l l r e p o r t e x p e r i m e n t a l d a t a on t h e c h a r a c t e r i s t i c s o f t h e s y n t h e s i s a nd c r y s t a l l i z a - t i o n o f c o m p l e x o x i d e s ( s p e c i f i c a l l y , BaTiO 3 and B a T i O 3 - S r T i O 3 s o l i d s o l u t i o n s ) f rom t h e p r o d u c t s o f d i s s o c i a t i o n c a u s e d by r e a c t i v e h f c a t h o d e s p u t t e r i n g . The p r o c e s s o f f i l m growth was divided into the following nominal stages: sputtering of a complex oxide by ion bombardment, transport of the dissociation products across an hf-discharge plasma to the substrate, surface processes on the substrate, and oxidation-reduction processes.

State University, Rostov-on-Don. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizlka, No. 2, pp. 7-11, February, 1981. Original article submitted August 6, 1979; r e v i s i o n s u b m i t t e d J a n u a r y 7 , 1980 .

102 0038-5697/81/2402-0102507.50 �9 1981 Plenum Publishing Corporation