Positron annihilation in positronium negative ions

Volume 144, number 4,5

PHYSICS LETTERSA

5 March 1990

P O S I T R O N A N N I H I L A T I O N IN P O S I T R O N I U M NEGATIVE I O N S Y.K. HO Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received 15 November 1989; revised manuscript received 15 January 1990; acceptedfor publication 16 January 1990 Communicatedby B. Fricke

The positron annihilation rate of positronium negativeions was calculated as 2.08613 ns- ~by using a 946 term Hylleraas-type wave function.

A positronium negative ion ( P s - ) is a bound threeparticle system consisting of two electrons and one positron which interact via Coulomb forces. The calculation of the binding energy of this system has a long history that can be traced back to the early work of Wheeler [ 1 ]. This system was observed for the first time in the laboratory by Mills [2 ]. Later he also measured [ 3 ] the positron annihilation rate for P s - . The production of these positronium negative ions has stimulated intense theoretical investigations. The role that the positronium negative ions would play in astrophysics and space physics has been suggested by Sivaram and Krishan [ 4 ]. In the last decade, several progress reports and review articles on P S - have appeared in the literature #1 [ 5-8 ]. In many aspects the positronium negative ions have properties similar to those of hydrogen negative ions, H - , a system which has been intensively studied by both theorists and experimentalists. Recently, the muonium ions, M u - , have also been observed [ 9 ]. These three systems differ in the mass of the positively charged particles. There are, however, many properties which are unique in P s - and have no counterparts in H and M u - . These properties involve the annihilation of the positrons in the positronium negative ions. There have been several calculations [ 10,11 ] for the positron annihilation rate of P s - . The agreements between the theoretical values and the earlier experimental result [ 3] are generally good. A recent improved experimental measurement for the posi#~ For referencesup to 1982, see ref. [ 5 ].

tron annihilation rate is now in progress [ 12 ]. It is therefore of interest to extend the earlier calculations. This work presents such a calculation. The Hamiltonian of the positronium negative ion is (1)

H=T+V,

where T and V are the kinetic energy and potential energy operators, respectively, and T=-~I ml V=-

V~-~l V~----1V~, m2 mp

2

2

rip

r2p

2

+--,

(2) (3)

r12

where 1, 2 and p, denote the electrons 1, 2, and the positron, respectively. The mass for particle i is mi; and rij represents the distance between particles i and j. Atomic units are used in this work, with energy expressed in Rydbergs. (In energy units, 1 a.u. = 2 Ry. ) Hylleraas-type wavefunctions of the form ~=

2

Cklm exp[ -- ot (rip +r2p) ]

l>~ m >~O k>~O

k l m X rl2(rlpr2p + r~'prt2o)

(4)

are employed to represent the system, where k + l + m < ~ e ) , where co is a positive integer or zero. The calculation can be simplified by expressing the kinetic operator in terms of distance coordinates, and eq. (2) becomes

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237

Volume 144, number 4,5

T=

PHYSICS LETTERS A

5 March 1990

( m~1 + m2J\Or~2 1 ~( 02 + --0~12) r122 0 (1

!"](02

- ~

(

-- ~

1

20

)

×

+ rnpJ\Or21p + rl---~pOTlp

2o)

= 100.617<61p ) .

1"~( 02 --~r2p + mp} k.OrZp + r2p

2 - rn~ cos(O12,1p)

Ori2Orlp

2 - m~ cos(012,2p)

Or120r2,

2 - __m--cos(01p,Zv)

Or, pOrzp '

The lifetime o f P s - against annihilation is

02

1

z=?.

(7)

02

02

(5)

with 2 2 r22 COS(01p,2p ) -- r i p + r 2 p -, 2rlpr2p

etc.

One o f the experimentally interesting p a r a m e t e r s is the annihilation rate, F, given in units o f n s - l, Table 1 Convergence of ground state energy for Ps-. ~o

N

a

- E (Ry)

16 17 18 19 20

525 615 715 825 946

0.433 0.453 0.465 0.478 0.510

0.52401013898 0.52401013964 0.52401014901 0.52401014019 0.52401014041

In eq. ( 6 ) , the correction term p r o p o r t i o n a l to a is due to the triplet lifetime [ 13 ] a n d the leading radiative correction to the singlet lifetime [14]. Absent in eq. ( 6 ) are bound-state and relativistic effects, which have not yet been calculated. In this work we use wave functions up to 946 terms ( ~ o = 2 0 ) . The ground state energy is a p p r o x i m a t e l y optimized for each expansion length N. Table 1 shows the energy together with the non-linear parameters used. We show results ranging from ~o= 16 ( N = 525) to o~=20 ( N = 946). It is seen that the convergence rate for functions with m a x i m u m even values o f k (powers of rl2 ) differs slightly from that for functions with m a x i m u m o d d powers. In this work, no a t t e m p t is m a d e to extrapolate the ground state energy to infinite co. F o r calculation o f positron annihilation rates, we use the 946 term wave function. In table 2 we compare our ground state energy result with other calculations. The most accurate ground state energy of P s - in the literature is that o f Frolov and Yeremin [ 17 ] who used a 700 term ex-

Table 2 Ground state energy of Ps- (in Ry). Ho (1983),ref. [10] ( 125 term Hylleraas function, one non-linear parameter) Bhatia and Drachman (1983), ref. [ 11 ] (220 term Hylleraas function, two non-linear parameters) Frolov ( 1987 ), ref. [ 15 ] ( 325 exponential variational expansion ) Petelenz and Smith ( 1988 ), ref. [ 16 ] ( 150 term exponential variational expansion) Frolov and Yeremin (1989), ref. [ 17] (700 term exponential variational expansion) Ho ( 1989 ), present calculation (946 term Hylleraas function)

238

(6)

-0.524009790 -0.5240101300 -0.5240101404 -0.524010140 -0.5240101404656 ± 1× 10-12 -0.52401014041

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5March 1990

Table 3 Positron annihilation rate in Ps- calculated by using eq. (6).

Bhatia and Drachman ( 1983 ) (220 term Hylleraas function ) Ho (1983) ( 125 term HyUeraas function) Haftel and Mandelzweig (1989) ( 169 hyperspherical function) Ho ( 1989 ) (946 term Hylleraas function) Mills ( 1983 ) experiment

F (ns- 1)

Pip

Ref.

2.0861

-0.50000

[ 11 ]

2.0850

-0.4991

[ 10] a)

2.08582

-0.5

[20]

2.08613

- 0.50002

present work

2.09 + 0.09

[3]

~) In ref. [ 10 ], the two-photon annihilation rate without the radiative correction was reported as F = 2.0908 ns-1. This was based on F = 100.94< J~p> ns -1. ponential variational expansion wave function. No p o s i t r o n a n n i h i l a t i o n rate was h o w e v e r c a l c u l a t e d in ref. [17 ]. O n c e the e n e r g y - m i n i m i z e d w a v e functions are o b t a i n e d , t h e y can be u s e d to calculate F by the use o f eq. ( 3 ) . S o m e r e c e n t results are s h o w n in table 3. It is seen t h a t t h e y c o m p a r e q u i t e well w i t h the experimental measurement of F= 2 . 0 9 + 0 . 0 9 ns -1 [ 3 ] . F u r t h e r m o r e , an i m p r o v e d m e a s u r e m e n t o f t h e a n n i h i l a t i o n rate is in progress

[12]. This would stimulate further theoretical studies. T h e q u a l i t i e s o f the w a v e f u n c t i o n s can be tested by calculating the e l e c t r o n - e l e c t r o n a n d e l e c t r o n p o s i t r o n cusp values. F o r a system i n t e r a c t i n g t h r o u g h C o u l o m b forces, the a v e r a g e v a l u e o f the cusp c o n d i t i o n b e t w e e n particles i a n d j is g i v e n by [181

Table 4 Expectation values of various functions of interparticle distances for Ps-. Re£


Pip







[10] a) [11] b) [16] c) [19] a) [20] e) present work f) exact

0.020713 0.020733

-0.4991 -0.50000

5.4891

48.3936

0.3398

5.4896333

0.33982102

0.0207303 0.0207333023

-0.5 -0.50002 -0.5

5.488352 5.48963188

48.418936 48.4152 48.379317 48.4188427

0.3398313 0.33982103

0.09094923 0.09093535

Ref.



v12

(rl2>



(ri51 >

< (r12rlp)-l>

8.5476

93.1283

0.1556

8.5485808

93.178633 93.1714 93.1006970 93.1784456

0.15563190

[10] a) [11] b) [16] c) [19] a) [20] e) present work r) exact

0.0001715

0.0001801517 0.0001710105

0.4971 0.49508

0.49968 0.5

8.54611129 8.54857794

0.1556543 0.15563191

< (rlp~p)-l>

0.06070779 0.06069769

a) Ref. [ 10], Ho (1983), 125 term Hylleraas function. b) Ref. [ 11 ], Bhatia and Drachman (1983), 220 term Hylleraas function. c) Ref. [ 16 ], Petelenz and Smith ( 1987 ), 150 variable exponential function. d) Ref. [ 19 ], Bhatia and Drachman ( 1985 ). e) Ref. [ 20 ], Haftel and Mandelzweig (1989), 169 hyperspherical function. f) Ho, present calculation, 946 term Hylleraas function. 239

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References

~J= < ~lS(rij)OlOrijl ~> ( ~ l d ( r e j ) I ~>

'

(8)

a n d the exact v a l u e for vii is

v~j = q~qjll~i ,

(9)

where q~ is the charge for the particle i a n d / z 0 is the reduced mass for particles i a n d j. T h e exact values for e l e c t r o n - e l e c t r o n a n d e l e c t r o n - p o s i t r o n c o n d i tions are hence + 0 . 5 a n d - 0 . 5 respectively. T h e p o s i t r o n - e l e c t r o n cusp values calculated i n the recent calculations are also s h o w n here in table 3. It is seen that the cusp results are very close to the exact one. It also indicates that the p o s i t r o n - e l e c t r o n cusp values would give i n d i c a t i o n s o f the accuracy for ann i h i l a t i o n rate calculations. I n this work we also use the 946 t e r m wave function to calculate the e x p e c t a t i o n values o f v a r i o u s f u n c t i o n s o f interparticle distances. T h e results are s h o w n in table 4, c o m p a r e d with p r e v i o u s calculations. T h e p r e s e n t results are o f accurate calculations, a n d c a n be used as useful references for other investigations. In s u m m a r y , we have carried out a n extensive calc u l a t i o n for the p o s i t r o n a n n i h i l a t i o n rate in posit r o n i u m negative ions. T h e result i n c l u d e s contrib u t i o n s from the triplet lifetime a n d the radiative correction for the singlet lifetime. W i t h the expected i m p r o v e m e n t of the e x p e r i m e n t a l a n n i h i l a t i o n rate, it w o u l d allow us to test these calculations a n d w o u l d s t i m u l a t e further i n v e s t i g a t i o n o f the corrections d u e to the b o u n d - s t a t e a n d relativistic effects.

240

5 March 1990

[ 1] J.A. Wheeler, Ann. N.Y. Acad. Sci. 48 (1946) 219. [ 2 ] A.P. Mills Jr., Phys. Rev. Lett. 46 ( 1981 ) 717. [3] A.P. Mills Jr., Phys. Rev. Lett. 50 (1983) 671. [4] C. Sivaram and V. Krishan, Astron. Space Sci. 85 (1946) 31. [ 5 ] D.M. Schrader, in: Positron annihilation,eds. P.G. Coleman, S.C. Sharma and L.M. Diana (North-Holland, Amsterdam, 1982) p. 71. [6] M.R.C. McDowell, in: Positron (electron)-gas scattering, eds. W.E. Kauppila, T.S. Stein and J.M. Wadehra (World Scientific, Singapore, 1985 ) p. 11. [7] J.W. Humberston, Adv. At. Mol. Phys. 22 (1986) 1. [8] R.J. Drachman, in: Atomic physics with positrons, eds. J.W. Humberston and E.A.G. Armour (Plenum, New York, 1987) p. 203. [9 ] Y. Kuang et al., Phys. Rev. A 35 ( 1987 ) 3172. [10] Y.K. Ho, J. Phys. B 16 (1983) 1530. [ 11 ] A.K. Bhatia and R.J. Drachman, Plays. Rev. A 28 (1983) 2523. [ 12 ] A.P. Mills Jr., Workshop on Positron annihilation in gases and galaxies, held in NASA/Goddard Space Flight Center, Greenbelt, MD, July 1989. [13] A. Ore and J i . Powell, Phys. Rev. 57 (1949) 1696. [ 14 ] I. Harris and L.M. Brown, Phys. Rev. 105 ( 1957 ) 1656. [ 15 ] A.M. Frolov, Zh. Eksp. Teor. Fiz. 92 (1987) 1959. [16] P. Petelentz and V.H. Smith Jr., Phys. Rev. A 36 (1987) 5125. [ 17 ] A.M. Frolov and A.Yu. Yeremin, J. Phys. B 22 ( 1989 ) 1263. [ 18 ] T. Kato, Commun. Pure Appl. Math. 10 (1957) 151; D.P. Chong and D.M. Schrader, Mol. Phys. 16 ( 1969 ) 137. [ 19 ] A.K. Bbatia and D.J. Drachman, Phys. Rev. A 32 ( 1985 ) 3745. [20] M.I. Haftel and V.B. Mandelzweig, Phys. Rev. A 39 (1989) 2813.