Department ofPhysics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 400-1. A. K. Bhatia. Laboratory for Astronomy and Solar ...
PHYSICAL REVIEW A
VOLUME 47, NUMBER 2
FEBRUARY 1993
BRIEF REPORTS Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Rapid Communications; howeuer, the same standards of scientific quality apply. (Addenda are included in Brief Reports )A . Brief Report may be no longer than 4 printed pages and must be accompanied by an abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors
P-wave shape resonances in positronium
ions
Y. K. Ho Department
of Physics and
Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803 400-1
A. K. Bhatia Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771 (Received 12 October 1992)
A 'P' shape resonance in Ps lying above the n =2 Ps threshold is calculated using the method of complex-coordinate rotation with Hylleraas-type wave functions. Comparison is made with a calculation using adiabatic potential curves. We also report new results for P' shape resonances lying above the n =3 and 7 thresholds.
PACS number(s): 36. 10.Dr
In this work we present a calculation of P-wave shape resonances in Ps . These resonances include a 'P' shape resonance lying above the n =2 Ps threshold, and P' shape resonances lying above the n =3 and n =7 Ps thresholds, respectively. The results are obtained by using the method of complex-coordinate rotation (see Ref. [1] and references therein) together with employing Hylleraas-type wave functions. Studies of Ps are due to the observations of such species in the laboratory by Mills [2], and the subsequent measurement of its annihilation rate [3]. Theoretical studies of various properties of Ps have also continuously appeared in the literature (see Ref. [4] and references therein). These studies include calculations of its ground-state energy and annihilation rate [5 —8], and the nonexistence of a bound P'
TABLE I. Convergence behavior for the onance in Ps (a =P= 0. 25).
N
0.6 0.7 0.8
E„(Ry) =1140
n
= 2 'P'
state [9, 10]. Resonance phenomena in Ps has also attracted considerable interest (see Ref. [4] and references therein). Several methods have been used to investigate resonance phenomena in Ps . These methods include rotation [11,12], the closethe complex-coordinate coupling scattering approximation [13], and using adiabatic potential curves [14—16]. In the earlier complexcoordinate calculations, we have reported results for P' resonances [17,18] and for P' resonances [19]. P The most general two-electron wave function for states of odd parity is
'
'
4(r, , r2) = —cos
—sin
816 969 1140 1330
~i2
(f + f )2)I
where 2) are the rotational harmonics, depending on the symmetric Euler angles 8, $, $ (see Refs. [20,21]). The trial function is of the Hylleraas type and is given by
I /2 (Ry)
—0. 124 3400 —0. 124 340 5 —0. 124 344 7
0. 000426 3 0. 000451 1 0.000 469 7
E„(Ry)
I /2 (Ry)
—0. 124 326 —0. 124 331 —0. 124 341 —0. 124 349
(f+f )2)I)+
shape res-
f
TABLE II. Comparison above the n
=2 Ps threshold.
of the 'P' shape reonance lying
Present calculation, complex-coordinate rotation
0=0.7
15 16 17 18
~i2 2
0. 000 460 0. 000 455 0. 000 451 0. 000 448
E„(Ry) I"
— (Ry) 2
47
1497
Hyperspherical coordinate [14]
—0. 124 34+0.000 03
—0. 1242
0. 000 45+0.000 03
0.0004
1993
The American Physical Society
BRIEF REPORTS
1498
TABLE III. Convergence behavior for the (a = P = 0. 165 ).
=3 P'
n
r, and r2 are the coordinate of electrons with respect to the positron and r, 2=~r, — r2~. Energy units are in Rydbergs. In the complex-rotational method, the radial coordinates are rotated through an angle 0:
shape
where
resonance in Ps
E„(Ry)
I /2 (Ry)
X =1140
0.4 0.5 0.6 0.7
—0.054 580 —0.054 519 —0.054 509 —0. 054 504
0.000 495 0. 000 460 0. 000 456 0. 000 455
r~r H
I /2 (Ry)
—0. 054488 5 —0. 054 5070 —0.054 503 8 —0.054 502 6
f(r, , r2, r, ~)=e
—ar1 —pr~
r)
g ~o CI
n
1
E = (NH4)
.
It is understood that )
=
2V]
2V~
2V ' V
2/T ]
2/r~
/(@4),
The theoretical aspects of the complex rotation method have been discussed in previous publications [l]. Here, we only briefly describe the computational procedures. First, we use the stabilization method to obtain optimized
+ 2/r ]2 = T + V (4)
TABLE IV. Shape resonances in Ps
E„ r" 2
n
—0. 124 34+ 3 X 10 n
Ry)
0. 000 46+1 x 10-' n
—0. 030 975+1 x 10-"
=4 (E = —0. 03125
Ry)
—Q. Q3Q 972+5 X 1Q
0. 000030+1x10 '
0. 000032+5 X 10
I
=5 (E = —0.02
Ry)
—0.019 708+5 X 10 0.000033+5 x10
2
2
—0. 124 40+ 3 X 10 0.00027+3 x 10-'
—0. 054 50+1 X 10
n
E„ r"
Ry)
=3 (E = —0.05555
2
2
=2 (E = —0. 125
3pe
0. 00045+3 X 10
E„ r"
E„ r"
(units in Ry) associated with different Ps thresholds. 3po
1PO
—0.013 750+ 5 X 10
n
=6 (E = —0.0138888
'
Ry)
—0. 013 761+1 X 10
0.000 026+5 X 10
0. 000020+1 X 10 n
r" 2
'Present calculation. Reference [19]. 'Reference [18].
the ex-
E„ i I /2
+res
and l +m +n ~co, where ~ is a positive integer or zero. The Hamiltonian is given by
H
are calculated by diagonalizing
wherein the wave function is real. Since the rotated Hamiltonian is complex, complex eigenvalues are obtained. The resonance parameters are determined by finding a root which is stable with respect to the variation of the nonlinear parameters a, P, and the angle 0, provided that it is greater than arg(E„, )/2. The complex resonance energy is given by
l, m, n
f (r ), r2, r)2) =f (r~, r, , r(2
.
pression
0. 000 469 5 0. 000458 3 0. 000455 1 0. 000 455 2
„r,r~m r)q
can be written as
= T exp( —2i0)+ V exp( —i0)
The eigenvalues
0=0. 7 816 969 1140 1330
exp(io),
and the Hamiltonian
E„(Ry) 15 16 17 18
47
=7 (E = —0.010204
Ry)
—0.010 08+1 x 10-" 0. 000020+1x10
'
BRIEF REPORTS
47
calculawave functions with which complex-coordinate tions will then be carried out. Once the stabilized wave functions for a particular resonance are obtained, a straightforward complex rotation method is applied, and the so-called "rotational paths" are examined after the complex transformation exp(ig) is made. We then determine the optimized nonlinear parameters by examining the resonance complex eigenvalue when it exhibits the most stabilized characters, i.e. , t)~E~/t)9=min. This is usually done by employing smaller basis expansion sets. Once the optimized a, P, and 9 are obtained, we can examine the convergence behaviors for the resonance parameters for different expansion lengths N. Table I shows the convergence behaviors for the n =2 'P' shape resonance. The optimized nonlinear parameters for this state are determined as a=P=0. 25 and 0=0. 7. Different expansion lengths are then used to determine the resonance parameters. We conclude that 0. 12434 and I /2=0. 00045 Ry, the results are E, = — with uncertainty of +0.00003 Ry assigned to both the real and imaginary parts. In Table II, we show a comparison with a calculation using adiabatic potential curves [14]. It is seen that their resonance position lies outside
r~r
our estimated error. In addition to the n =2 'P' shape resonance, we also report new results for two P' shape resonances, one each lying above the n =3 and n =7 Ps thresholds, respectively. A P' shape resonance lying above the n =5 Ps threshold was reported in an earlier publication [18].
[1] Y. K. Ho, Phys. Rep. 99, 1 (1983). [2] A. P. Miills, Jr. , Phys. Rev. Lett. 46, 717 (1981). [3] A. P. Mills, Jr. , Phys. Rev. Lett. 50, 671 (1983). [4) Y. K. Ho, in Positron Annihilation in Gases and Galaxies, NASA Conference Publication 3058, edited by Richard J. Drachman (NASA, Greenbelt, MD, 1989), p. 243. [5] Y. K. Ho, J. Phys. B 16, 1503 (1983). [6] Y. K. Ho, Phys. Lett. A 144, 237 (1990). [7] A. K. Bhatia and R. J. Drachman, Phys. Rev. A 42, 5117
(1990). [8] A. M. Frolov and A. Yu Yeremin, J. Phys. B 22, 1263 (1989). [9] A. P. Mills, Jr. , Phys. Rev. A 24, 3242 (1981). [10) A. K. Bhatia and R. J. Drachman, Phys. Rev. A 28, 2523 (1983).
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Table III shows the convergence behaviors for the P' shape resonance lying above the n =3 Ps threshold. The nonlinear are found as optimized parameters a=P=O. 165 and 8=0. 7. Resonance parameters are deduced by using different expansion lengths in the wave functions. We determine the resonance parameters as I /2=0. 00046 E„=—0. 054 50+1 X 10 and Ry + 1 X 10 Ry. Recalling that the threshold energy for the Ps (n =3) state is E = — 0. 055 55 Ry, our results inP' shape resonance would lie dicate that the about 0. 001055 Ry above the n =3 Ps threshold. A shape resonance lying above the n = 7 Ps threshThe old is determined. resonance similarly are E, = — 0. 010 08+1 X 10 parameters Ry and I /2=0. 000020+1 X 10 Ry. This resonance lies about 0. 000 124 Ry above the n =7 Ps threshold (threshold en0. 010204 Ry). A P' shape resonance lying ergy E = — threshold above the n =5 was at reported E„=—0. 019 708+5 X 10 Ry with half width of I /2=0. 000033+5X10 Ry [18]. In summary, we have reported a calculation for three shape resonances in Ps . Results are obtained using the rotation with Hylleraas method of complex-coordinate functions. For completeness, we also list in Table IV the other shape resonances that we have calculated for Ps [18,19]. We now conclude that for 'P' and P' states shape resonances exist above the positronium n =2, 4, and 6 thresholds. For P states shape resonances exist above the positronium n = 3, 5, and 7 thresholds.
'
[11]Y. K. Ho, Phys. Rev. A 19, 2347 (1979). [12] Y. K. Ho, Phys. Lett. A 102, 348 (1984). [13] S. J. Ward, J. W. Humberston, and M. R. C. McDowell, J. Phys. B 20, 127 (1987). [14] J. Botero and C. H. Greene, Phys. Rev. Lett. 56, 1366 (1986). [15] J. Botero, Phys. Rev. A 35, 36 (1987). [16] J. Botero, Z. Phys. D 8, 177 (1988). [17] A. K. Bhatia and Y. K. Ho, Phys. Rev. A 42, 1119 (1990). [18] Y'. K. Ho and A. K. Bhatia, Phys. Rev. A 44, 2890 (1991). [19] Y. K. Ho and A. K. Bhatia, Phys. Rev. A 45, 6268 (1992). [20] A. K. Bhatia and A. Temkin, Rev. Mod. Phys. 36, 1050 (1964). [21] A. K. Bhatia and A. Temkin, Rev. Mod. Phys. 182, 15 (1969).
