35, NO. 2. APRIL 1997. Resonances in Three-Body Atomic Systems Involving ... theoretical methods to investigate electron atom interactions can, .... The transformation of Eq. (1) has a great computational advantage for systems with ... where 1 and 2 denote the electron and positron, respectively, and p denotes the proton.
CHINESE JOURNAL OF PHYSICS
VOL. 35, NO. 2
APRIL 1997
Resonances in Three-Body Atomic Systems Involving Positrons Y. K. Ho Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 300, R.O.C.
(Received November 18, 1996) We review the theoretical studies of atomic resonances in three-body systems involving positrons. These works employ a method of complex-coordinate rotation. Investigations on resonances in positron-hydrogen scattering below various hydrogen and positronium thresholds are discussed, as well as resonances in e+-He+ scattering. Furthermore, resonances in electron-positronium scattering will also be reviewed. The theoretical results for doubly-excited states of Ps- reveal the supermultiplete structures of such an ion, and its tri-atomic molecular characters in analogy to an XYX molecule. PACS. 31.15.Ar - Ab initio calculations. PACS. 32.80.D~ - Autoionization.
I. Introduction In this paper, we provide an overview of the recent theoretical investigations of atomic resonance phenomena involving positrons. Recently, there has been increasing interest to investigate the interactions of positrons and positronium with atoms. From the viewpoint of non-relativistic quantum physics, a positron is just a positively charged electron. All theoretical methods to investigate electron atom interactions can, in principle, be applied to positron atom scattering. One major difference between the electron and positron atom scattering is that a static potential in the positron case is repulsive, whereas it is attractive in the electron counterpart. Polarization potentials are attractive for both electrons and positrons. In the positron case, the static potential and polarization potential cancel each other. Such a cancellation effect would provide a stringent test for various atomic theories. Furthermore, there are some unique processes that appear only in positron atom interactions. These are positronium (a bound positron-electron pair) formation and positron annihilation (a positron annihilates with an electron to produce gamma-rays). Examinations of such processes would provide additional insight of atomic system involving positrons. From the experimental front, great advances have recently been made in the area of atomic scattering involving positrons [l]. As intense mono-energetic positron beams have become available in laboratories around the globe, rich sets of positron scattering data with improving accuracy now exist in the literature [2]. For example, in addition to the total cross section measurements with gases, and the recent measurements with alkali atoms [3], differential cross sections for some atomic systems have also been measured [4], 97
@ 1997 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
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and so do positronium formation cross sections [5]. Ail these experimental activities call for theoretical investigations. Studies of positron-atom interactions would also help to interpret some astrophysical findings. These include the observations of positron annihilation in solar flares [6], in the center of our galaxy, the Milky Way [7], and in gamma ray bursts (81. One of the areas in positron atom interactions which has attracted considerable theoretical interests is atomic resonances involving positrons. To my knowledge, none of the atomic resonances involving positrons that I am going to describe to you have been observed experimentally. But the recent intensive theoretical activities would certainly provide guideline for future experiments. Let me first give an outline of my talk. I will discuss in detail the resonances in scattering of positrons by atomic hydrogen, as well as the new resonances in e+-He+ scattering. Recent studies on resonance states of positronium ions will also be discussed. II. Method of complex-coordinate rotation In this talk, I will discuss mostly the theoretical work performed by using the method of complex-coordinate rotation [g-11]. Let me first briefly outline the essence of the method. Let us consider the scattering of a positron (or electron) by a target atom consisting of N particles. Figure 1 shows the energy spectra for the (N+l) particle system before and after the complex transformation. In the figure there are three distinct parts of the spectrum, which are relevant to the present discussion: (1) Bound state poles (if they exist) of the (N-l-l) particle system are located on the negative side of the real axis in the complex E plane.
-i-
-RESONANCES
HIDDEN
BOUND STATES I RESONANCES EXPOSED
/ ROTATED CUTS -’
FIG. 1.
Energy spectrum of an atomic Hamiltonian before and after the complex-coordinate rotation.
Y. K. HO
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99
(2) A series of branch cuts extend from the elastic and all inelastic thresholds (for the N, N-l, . . ., ets. particle systems) to the right-hand side infinity (+oo) of the energy plane. (3) Resonant poles are located near the cuts and are hidden in higher sheets of the Riemann surface. To locate resonant poles, analytic continuation from the physical scattering regions to the unphysical Riemann surfaces can be used. The method of complex-coordinate rotation can be viewed as one method of the analytic continuation. In such a method, all the inter-particle radial coordinates r;j are transformed into Tij+Tije
i0
,
where 8 is real and positive. Under the transformation of Eq. (l), the spectrum of the (now so-called rotated) Hamiltonian is transformed to the following (see Fig. 1): (1) The bound state poles remain unchanged under the transformation. (2) The cuts are now rotated downward, making an angle of 28 with the real axis. (3) The resonant poles are “exposed” by the cuts once the “rotational angle” 8 is greater than l/2 arg (EreS), where E,,, is the complex resonance energy, i.e. E,,, = E, - iI’/2 = IEle-@,
(2)
with p the phase factor of the complex energy, and E, and I? the usual resonance position and width, respectively. The transformation of Eq. (1) has a great computational advantage for systems with Coulomb interactions. The kinetic part of the Hamiltonian would scale as exp(-2iO), and the potential part of the Hamiltonian would simply scale as exp(-8). Under such a transformati’on, one just calculates the kinetic and potential matrix elements separately, and then scales them according to the above scheme. A complex eigenvalue problem will then be solved, and the eigenvalues are obtained by diagonalizing the analytical.ly continued Hamiltonian E =
/ < iplis >,
(3)
wherein the eigenvalues and eigenvectors are complex, while the basis functions 9 are real. Discrete resonance complex eigenvalues have the form of
E = E, - X/2,
(4)
where E, gives the resonance position and I? the resonance width. When we diagonalize the complex Hamiltonian of Eq. (3), the whole spectrum of complex eigenvalues would come out. The complex eigenvalues that represent various parts of the energy spectrum of the (rotated) Hamiltonian are then examined. Some of the eigenvalues represent the bound and
100
RESONANCES IN THREE-BODY ATOMIC SYSTEMS
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resonance states, and some represent the cuts. The complex-eigenvalues that represent the rotated cuts have a strong dependence of 8, and those for resonances are stationary with respected to the changes of 19, once 19 is sufficiently large to discover such resonances. We have E ae M 0 as a condition for a resonance. We should point out that there are no variational bounds, like those obtained for bound states in variational methods, on the eigenvalues obtained in this method described above. However, the viral theorem 2 < Q]T]QP >= - < Q]V]@ > is’satisfied when 8E/d0
(5)
= 0.
III. (a) Resonances in positron-hydrogen scattering
Investigations of the existence of resonances in e+-H scattering have a long history. It is well known that resonances below the H (N=2) threshold in e--H scattering are the result of the 2s-2p degeneracy of the target atom [12] . Mittleman [13] suggested that the attractive dipole potential that behaves like rm2 asymptotically in e--H scattering would be the same as that in e+-H scattering. In the latter case, there exists a lower-lying positronium formation channel (see Fig. 2). It was argued 1141, however, that the effect of opening such a channel is of short range and would not affect the existence of such dipole resonances (the positions and widths may change). However, since no numerical results were reported in Ref. [13], the investigation of resonances in e+-H scattering continued (see Refs. [9,14-161).
c-0.25 H(N=2)
- 1.0
I
I r-l.0
H(N=I
FIG. 2. Energy levels for the e+-H system.
I
I
-0.256
I
I
-0.252
I
I
-0.248
ReE
FIG. 3. S-wave resonances in e+-H scattering below the H (Nk2) threshold.
Y. K. HO
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101
The first definitive calculation to establish that S-wave resonances do exist in e+H scattering below the H (N=2) threshold was carried out by Doolen et al. [17] using the method of complex-coordinate rotation. They used the Pekeris functions in which the coordinates for positronium atoms were explicitly included. They determined the resonance parameters for the lowest S-wave resonance by assuming the nucleus as infinitely heavy, and obtained E, = -0.257374 f 1 x 10m6 Ry and I/2 = 0.0000667 f 1 x 10m6 Ry. The nuclearmass effect on such a resonance was examined when the finite mass for the nucleus (mass of the proton=1836 m,, the mass of the electron) was used in a complex-coordinate calculation
P81.
In
Ref. [18], the Hamiltonian for the ef+H system is given by
H=T+V,
(6)
v=$+&$7
(8)
where 1 and 2 denote the electron and positron, respectively, and p denotes the proton. Hylleraas-type wave functions were used in Ref. [18]:
with k + m + n 5 w, and w, k, m and n all being positive integers or zero. Table I shows the convergence behavior for the lowest S-wave resonance when different expansion lengths for the basis sets are used. With the use of finite nucleus mass, the resonance energy and width for the lowest S-wave resonance were determined as E, = -0.257245 Ry and I ’ / 2 = 0.0000666 Ry. It indicates that the nuclear-mass effect on this resonance is about AE = 1.3 x lo-* Ry. But the nuclear-mass effect on its width is negligible. In addition to the complex-coordinate calculations. Other theoretical methods have also been used to investigate resonances in e+-H scattering. These methods include the use of hyperspherical functions [19-201. T heoretical justifications for the existence of resonances below the H (N=2) threshold were provided by Choo et al. [21] and by Treml [22]. Resonances in e+-H scattering associated with the Ps (N=2) threshold have been investigated by using different theoretical approaches, including the use of complex-coordinate rotations [23-251. Other S-wave resonances associated with higher H and Ps thresholds have also appeared in the literature [25]. Table II shows S-wave resonances below the H (N=3) threshold, as well as the resonances below the H (N=4 and 5) and Ps (N=3 and 4) thresholds [25]. All the S-wave resonances are shown in Figs. 3, 4, 5, and 6. Resonance positions (from Re E) and widths (from 2 Im E) for individual resonances are plotted in the figures. Fig. 2 shows S-wave resonances below the H (N=2) threshold. Also in the figure, resonances that are believed to belong to the same dipole series are connected by dashed lines. According to Gailitis and Damburg [12], the energy and width ratios R for neighbouring resonances due to the attractive rV2 dipole potential would be
102
RESONANCESIN THREE-BODYATOMICSYSTEMS ...
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TABLE I. Convergence behaviours for the lowest S-wave resonance in e+-H scattering when different expansion lengths (M) of Hylleraas functions were used. The finite nuclear mass was used for calculation [ 181.
W
10 11 12 13 14 15
M
E@Y)
286 364 455 560 680 816
-0.2572454 -0.2572443 -0.2572455 -0.2572450 -0.2572452 -0.2572453
JJ/2(RY > 0.0000675 0.0000673 0.0000669 0.0000665 0.0000668 0.0000666
x
Re
FIG. 4
E
Re E(Ry)
FIG. 5. S-wave resonances in e+-H scattering associated with the H (N=4) and Ps (N=3) thrersholds.
S-wave resonances in e+-H scattering below the Ps (N=2) and H (N=3) thresholds. H CN=5)
-5
x10-
,
RlN.4) + I.
. I
,
0’
I’
/‘I
,
I
,
,
’
d
,
-4 -
,=
.I’
,’
,‘I ,’ ’ :
1’
_A_ /CC
A _B_ o/r
**
cc
_
:
A’
= a w
:
I
c_
:’
I’ I :
/
d -121
’ -CUM6
’
’ t -a42
’ v -0.036
* ’ -CD34
d -w30
ReEU?y)
FIG. 6. S-wave resonances in e+-H scattering associated with the H (N=5) and Ps (N=4) thresholds.
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103
TABLE II. S-wave resonances in e+-H scattering associated with various hydrogen and positronium thresholds. The number inside the prenthesis represents the power of ten by which the preceding number is to be multiplied.
Complex-rotation [18,24,25]
Hyperspherical coordinate [28]
WRY)
WRY)
%(RY)
~/WY)
Between the Ps (N=l) and H (N=2) thresholds -0.257312 -0.250302
-0.2572453 -0.25017
6.64(-5) 3.91(-6)
6.66(-5)
EGY)
~/WY >
Below H (N=2) threshold -0.254132
8.152(-6)
-0.254112
8.1(-6)
-0.148181 -0.13073 -0.12733
2.97(-4) 1.67(-4) 7.3(-5)
Below Ps (n=2) threshold -0.148071 -0.130666 -0.127343
2.936(-4) 1.480(-4) 6.73(-5)
shows convergence behaviour for the lowest P-wave resonance below the Ps (N=2) threshold. Table IV summarizes all the P-wave resonances obtained by using the method of complexcoordinate rotation [27]. Table IV also shows results obtained by using hyperspherical coordinates [28]. Other earlier investigation of P-wave resonances below the H (N=2) threshold was a three-state (ls-2s-2p) close-coupling calculation [29]. However, in Ref. [29] the lower-lying positronium formation channel was not included. Very recently, several methods have been used to calculate resonances in e+-H scattering [29-311. The complexcoordinate results [18,24,25,27] can be used as assessment of merits for these calculations. III.(b) Resonances in e+-He+
scattering
Recently, two S-wave resonances in e+-He+ scattering were calculated by Bhatia and Drachman [32] using the stabilisation method. Fig. 7 shows the energy levels for the e+He+ system. Two resonances were found lying at E, = -0.73 Ry and E, = -0.39 R y , respectively [32]. One possible explanation for their existence is that these resonances are the results of the He++ ion attaching to the degenerate 2s-2p excited states (E, = -0.125 Ry) of the Ps atom. Since the He++ ion is doubly charged, the attraction between the He++ ion and the exited Ps atom is larger than that for the Ps-H+ counterpart, and it is reasonable for the PsHe2f resonances to lie lower than those for the PsH+ system. The location of the first resonance at E, = -0.73 Ry is quite interesting since it lies far away from the proposed parent threshold. In fact, it lies even lower than the ground state of the Ps atom (ET = -0.5 R y ) . The interaction between the He++ ion and the ground state positronium atom may play a role for the existence of the lowest resonance. The resonances in e+-He+ scattering were also investigated using the method of complex-coordinate rotation [33].
-_.
106
RESONANCES IN THREE-BODY ATOMIC SYSTEMS .‘.
00
VOL. 35
r __o,,25 ps~N;2j~ ------‘+0.X
- 0.2
H:+(N-5)
+-A Hc+(Nz4) A
- 0.4
+-A Hei(N-3)
37 u tn 9,
-0.5 Ps(N-I)--+ ------- 0.6
ii -1 2
- 0.8
2 2
‘I
- 1.0
4---1.0
H&N:Z)
t-4.0
H&N111
c
- 4.0T
FIG. 7. Energy levels for the e+-He+ system The kinetic and potential energy operators of the Hamiltonian for e+-He+ system are given by
T = -0; - 0;
(12)
where 1 and 2 denote the electron and positron, respectively, and 2 is the charge of the positively charged particle. Also, in Eq. (12) th e nucleus is assumed to be infinitively heavy. Using the elaborate Hylleraas-type wave functions (Eqs. (9) and (ll)), resonance parameters (energy and total width) for two S- and two P-wave resonances were determined. The resonance parameters for these resonances are shown here in Table V. The complexcoordinate results for the S-wave resonance positions of -0.74099 Ry and -0.3712 Ry are consistent with those of -0.73 Ry and -0.39 Ry obtained by using the stabilisation method [32]. No widths were however reported in Ref. [32], and the complex-coordinate calculation was the first work to report widths for such resonances. In order to shed light of the “mechanism” of the resonances, the charge of the infinitively heavy nucleus was changed from 2=2 to Z=l for the systems (e+, e-, Z), where 2 is the charge ofthe positively charged heavy particle (see Eq. (13)). By changing Z from Z=2 to Z=l, the systems from e +-He+ to e+-H were investigated. Fig. 8 and Fig. 9 show the results from Ref. [33] for S- wave and P-wave resonances, respectively. It is seen that the S-wave resonance at E, = -0.3712 Ry and the P-wave resonance at E, = -0.36956 Ry are
Y. K. HO
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107
TABLE V. S- and P-wave resonances in e+-He+ scattering [33].
%(RY)
V-WY) S-wave
-0.74099f0.00005
0.12943fO.C0005
-0.3712f0.00005
0.0393f0.0005 P-wave
-0.7086910.00004
-0.36956f0.00010
0.17752f0.00004 0.04317f0.00010
the results of the He2+ ion attaching to the excited Ps (N=2) threshold. We can also rule out that the Ps (N=2) state as the parent state for the S-wave resonance at E, = -0.74099 Ry and the P-wave resonance at E, = -0.70869 Ry. These two lower-lying states probably are the result of a He++ ion attaching to the ground Ps (N=l) state. Also they are located above the parent N=2 state. The autoionization route to e++He+ (N=2) is now possible. The opening of such a route may explain the huge resonance widths for these two lower-lying states. Their total widths of 3.52 eV and 4.82 eV for the S-wave and P-wave resonances, respectively, raise the possibility that such resonances might be observed in e+-He+ scattering experiments. III.(c) .Resonances in electron-positronium scattering
Theoretical calculations of the ground state energy of Ps- have a long history ever since the early work of Wheeler [34]. Activities have been intensified recently due to, in part, the discovery of such a system by Mills [35] and his measurements of its annihilation rate [36,37]. Th e recent investigations include variational calculations using Hylleraas functions [38-401, exponential valiational functions [41,42], and using a numerical finite-element method [43]. A non-variational calculation using hyperspherical harmonic functions was also used to calculate the ground state energy of Ps- [44]. A calculation of one-photon annihilation rate has appeared in the literature [45]. In a continuing effort to calculate resonance parameters in Ps-, we have in recent years reported calculations of ‘t3Se [46] states, r13Po [47], 3Pe [48], 1*3De [49] and ‘13Do [50] doubly-excited resonances using the method of complex-coordinate rotation. In the following, we will describe the Hamiltonian for Ps - and the wave functions used for various angular momentum states. The kinetic and potential energy operators, respectively, are
RESONANCES INTHREE-BODYATOMICSYSTEMS ...
108
0.0
I
VOL. 3s
I
I
I
Ps (N-2)
s
+s FL w
E. Ill
-0.6
-0.6 -
-0.8
1.0
1.2
t e+- f-i
1.4
1.6
Z
1.8
2.0 t
e+- He+
FIG. 8. Thresholds and S-wave resonances for systems from e+-H to e+-He+ as Z is changed.
-1
.o
I
1.0 t
e+- H
1.2
I 1.4
Z
I
I
1.6
1.8
2.0 t
e+- He'
FIG. 9. Thresholds and P-wave resonances for systems from ef-H to et‘-He+ as Z is changed.
T = --$V; - -+ - -+,
(14)
v=-e-$+-$,
(15)
where 1, 2 and p, denote the electrons 1, 2, and the positron, respectively. The mass for particle i is m;; and rij represents the distance between particles i and j. For lv3Se states, wave functions of Hylleraas-type were used with the form (16) kin
with k + 1+ n 5 w, where w, k, 1, n are positive integers or zero. In Eq. (16), the upper sign is for the singlet-spin states, and the lower sign for the triplet-spin states. Also, for the ‘Se
109
Y. K. HO
VOL. 35
states, the summation indexes in Eq. (16) are k >_ 1 > 0 and n 2 0. For 3Se states, the indexes are k > 1 > 0 and R, 2 0. Calculations can be simplified by expressing the kinetic operator (Eq. (14)) in terms of distance coordinates, and Eq. (14) becomes
-(~+~)(~+~&J -(~+~)(~+~&-)
(17)
a2 - 2 - cos(kr?J) dr12arlp ml d2 -$
c442,2p)
dT12aT2p
a2 2 _cos(k2,) &@+ m?J with 4hp,zp)
For
the
=
rfp + T&, - 32 2r@‘zp
7
etc.
3P” states, Hylleraas type wave functions were used, with
@(Tl,Tz) = (Sin
6$2)TlT2 CClmn[T:T~e-(ar1+pr2) lmn
+ (1 H 2)]Ty2@+,
where the ‘D functions involve the symmetric Euler angles describing the orientation in space of the coordinate vectors [51,52]. In Eq. (19), we have I -I- m -I- n < w, where W, I, m, and n are positive integers or zero. The T and V operators of the Ps- Hamiltonian are given by T =
(20)
-2V; - 27722,
v=-2vr.v2 - 217.1 - 2/T2+2/T12, where
TV
(21)
and ~2 are the coordinates of electrons with respect to the positron, and ~12 =
IFi--1.
The most general two-electron wave function for ‘13P states of odd parity is qq, T-2) = - cos
- sin
(F)
(fk jp:+
$
(fTf)#_
(
>
(22)
-_ -. _
RESONANCES IN THREE-BODYATOMICSYSTEMS ...
110
VOL. 35
where 2) are the rotational harmonics, depending on the symmetric Euler angles 8, c$, 9. The trial function f is of the Hylleraas type and is given by f(Tl,T2,T12)
= e
--ar~-pT~
Tl
c
&mT:T~T;2.
(23)
/,77I,ll>O
It is understood that f(Tr,
(24)
T2,T12) = f(T27T~T12).
In Eq. (23), 1 + m f n 5 w, where w, 1, m, and n are positive integers or zero. For ly3De wave functions, we also use Hylleraas type @ = (f f f>[-@+(B, $4 Q) t
~(cos~l2)~;+(4 $7 qE‘)
t(f+f)~sinel22)~-(e,~,Q)
(25)
t(9 G)[- c0se12~~+(e,~,IE.)t~'D~+(e,~,rk.)],
where the 2) are the rotational harmonics, depending on the symmetric Euler angles 8, c$, Q. The trial radial functions f and g are given by
c
f(Tl, T2, Tg) = f?lT1--O1'zT;
c~~,T~T~T~2,
(26)
k,m,n>O
and g(T1, T2, T3) = 6--azr1--P2r2TlT2
c
~(2) k m n km’1 T2 T12.
(27)
k,m,n>O
It is implied that f(Q, rz,rs) = f(r2, rrJr2)
(28)
j(T1,T2,T12) = !3(Tz,Tl,T12),
(29)
and
with k f m + n < w, where k, m,n, and w are positive integers or zero. In Eq. (25) the upper signs represent the singlet-spin states, and the lower signs the triplet-spin states. The most general D-state wave function of odd parity for two electrons is @ = sin Q12{ [( j+j)
c0s( $I12)D~+ + (f i f)
+[(5c0s e12 - I)(Gj) c0s(+e12p;+
t(5 c0se12 t l)(g kg)
sin($912)2):-]}
sin( +8r2)~~-]
- ..-__
Y. K. HO
VOL. 35
111
where the 2, are the rotational harmonics, depending on the symmetric Euler angles 0,4, $J. These functions are eigenfunctions of exchange and satisfy the following property:
The trial wave function is of the Hylleraas type and the radial functions f and g are given by f(Tr, Tz,Tr2) = e-(~lr1+61+;T~ c c c c,‘~!Mr;z I>_0 m>o n>o g(rr,rz,rrz) =
(2) 1 e--(Y2’1+62TZ)~f~~ C C C Clmn~1~y~;L2 IlO m>o n>o
(32) (33)
with I+mfn< w, and 1, m, n, and w being positive integers or zero. Also in Eq. (30), we have f = f (7’2, Tl, 7.12)
j = 9 (T2,Tl, 7’12).
(34)
The upper signs in Eq. (30) correspond to the singlet states and the lower signs to the triplet states. The first term in Eq. (30) corresponds to (pd) configuration and the second term the (df) configuration with total angular momentum L=2. From the available theoretical results, it is apparent that this (e-e+e-) three-body system behaves very much like a triatomic XYX molecule, as illustrated in Fig. 10. Due to the vibrational character of such a molecule, the 3P” and ‘PO states that have the same “q uantum numbers” would be nearly degenerate [47,48]. Similarly, the ‘D” and 3Do pair with the same quantum numbers would be nearly degenerate. In Tables VI and VII we show all the calculated doubly excited intra-shell (the two electrons occupying the same shell) states associated with the N=2 and N=3 I’s thresholds, respectively. The rotational character of Ps- implies that states such as ’ Se, 3Po, and lDe that have the same quantum numbers would belong to the same rotor series. In Tables VI and VII, each state is classified by a set of quantum numbers (K, T, N, n, L, S, R), where L, S, N, n, and K have the usual spectroscopic meanings. The quantum numbers K and T are “approximately good” quantum numbers, and can be described briefly as follows: K is related to < - cos 012 >, where Or2 represents the angle between the two electron vectors. The more positive the value of K, the closer the value of < - cos 012 > to unity. The two electrons in this situation are located near the opposite sides of the positron. The quantum number T describes the orientations between the orbitals of the two electrons. For example, a state with T=O implies that the two electrons are moving on the same plane. The quantum numbers K and T hence describe the angular correlations between the two doubly excited electrons. For more discussions for the quantum numbers K and T readers are referred to Ref. [53,54]. In Tables VI and VII, states having the same KT quantum numbers, hence belonging to the same rotor series, are grouped together. For a given set of K, T, and N, the allowed L values for the doubly excited states are
-I‘.
E!!!
RESONANCES IN THREE-BODY ATOMIC SYSTEMS
112
VOL. 35
L = T, T + 1, ... , K + N - 1.
(35)
The highest L value for a given [K, T] rotor series is therefore governed by the relationship L( maz)=K+N-l. For example, when K=2 and N=3 (the [2, 0] series in Table VII) the highest L states is an L=4 state (lGe state in this case). Similarly, when K=l and N=3 the rotors are cut off after L=3 (F states). As for the K=O series, the rotors stop at L=2. In Fig. 11, 12, and 13, we show the rotational characters of the “molecule” associated with the positronium N=2, N=3, and N=4 thresholds, respectively. In Fig. 14, 15, and 16, we use the results for the doubly excited intrashell states associated with the positronium N=2, N=3, and N=4 thresholds, respectively, to construct the I-supermultipiet structures [55]. The quantum number I is defined as I=L-T,
(36)
BENDING VIBRATION
FIG. 10.
The tri-atomic X-Y-X character for the e-e+e- system.
TABLE VI. Doubly excited intra-shell states of Ps- associated with the N=2 Ps threshold (threshold energy=-0.125 Ry).
State
KTNn
lSe 3P0 lD” lPO 3Pe
Resonance
Ref.
0.000043034 0.0001278 2.6 x 1O-6
Feshbach Feshbach Feshbach
[461 P71 [491
0.00045 0.00027
shape shape
WRY)
P/2(Ry)
1022 1022 1022
-0.152060883 -0.1466547 -0.1358255
0122 0122
-0.12434 -0.12440
1471 [481
113
Y. K. II0
VOL. 35
TABLE VII. Doubly-excited intra-shell states of Ps- associated with the N=3 Ps threshold (threshold energy=-0.05555 Ry).
T/2(RY >
Resonance
Ref.
-0.0706837708 -0.0697428 -0.0678289
7.4657x 1O-5 6.06x 1o-5 2.56x 1O-5
Feshbach Feshbach Feshbach
Ml WI WI
-0.05545 -0.05450 -0.05236
0.000042 0.00046 0.00203
shape shape shape
[461
3P” lDe
0033 0033 0033
lPO 3De
1133 1133
-0.0632447 -0.0608225
0.0002206 0.0001322
Feshbach Feshbach
WI WI
1133 1133
-0.0632613 -0.0608825277
0.0001791 7.19537x 1o-5
Feshback Feshbach
WI WI
0233 0233
-0.0548 -0.05497
0.00032 0.0003
shape shape
PI WV
State
KTNn
lSe
2033 2033 2033
3P0 lDe
lSe
WRY 1
-0.05
-0.11 -
z-0.14-0.15-0.16-
t
Ps (N=2) threshold
- 0 IZ‘P ’ ‘P ” _ _ _ _ _ _ _ _ _ _ _ _I _ _ _ _ _ - - - _ _ - - _- _ _ - --,-0.13CK
WI Ml
‘0 ’
;: ‘.$ - 0 0 6 w
‘P ’ -
‘S’
ID”_ _
--‘D
yJ.___
--‘p
I Ps(N=3) threshold (E=-0.0555SSRy)
ID’-0.07
’ p’-.‘S FL01
[ KT 1 FIG. 11. The rotational character of the doubly-excited Ps- associated with the Ps (N=2) threshold.
LOA
11.11
rJ.01
W.Tl
FIG. 12. The rotational character of the doubly-excited Ps- associated with the Ps (N=3) threshold.
c!!!
RESONANCES IN THREE-BODY ATOMIC SYSTEMS
114
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‘Q .__ --‘D
-0 030 -
p- -,p ___-____-------------
___---
Ps (N=2) threshold
f Ps(N=4) threshold (E=-0.03125) ‘D’- -
;:
--‘D
I -0.12
‘D’- .:gI=
3 -0.035 -
_o.ai 1 g ‘p- -‘p ‘D -
-0.15 -0.16
[l,Ol
PA
IO,11
‘D ’
I~.
;:
%_-0
w
06 t
ap.
‘p ”
‘P B
t
I
-0.025
‘D ”
IWll IVY
A
'S'
Il.01
L1 .Ol I=0
I=1
I=2
I
?Gi
.ropl. ‘D ” ‘D ’ riiiliij
FIG. 15. The vibrational character of the doubly-excited Ps- with the Ps (N=3) threshold.
0
1
0
0
Ps (N=4) threshold t
‘D ’ >P’
1
FIG, 14. The vibrational character of the doubly-excited Ps- associated with the Ps (N=2) threshold.
Ps (N=3) tieshold
W.,~~.[o;zt.
xi
1
I
FIG. 13. The rotational character of the doubly-excited Ps- associated with the Ps (N=4) threshold.
-0 05
‘P ”
‘D ’
w’- 0.14
-m
[Zll
i3.01
z-o.13 II:
‘P ’
F [L ‘c w
I
-0.030 c
-0 035 I
FIG. 16. The vibrational character of the doubly-excited Ps- with the Ps (N=4) threshold.
and has the same meaning as the ro-vibrational quantum number R used in molecular physics [55]. For example, states with I=0 are the ground states of various rotor series. From these figures, the vibrational characters of the “molecule” is evident. At present, only doubly-excited states with total angular momentum of L52 have been calculated. It would be highly desirable to investigate doubly-excited angular momentum states with LL3.
Y. K. HO
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115
TABLE VIII. lSe resonances in Ps-. The number inside the prenthesis represents the power 01 ten by which the preceding number is to be multiplied.
Ps threshold
Complex-rotation[46] WRY)
VWY >
N=2 (1) (2) (3)
-0.152060883 -0.127298353 -0.1252026
4.3034(-5) 8.68(-6) 1.2(-s)
N=3 (1) (2) (3) shape
-0.0706837708 -0.05969229 -0.056582 -0.05545
7.4657(-5) 5.271(-5) 1.23(-5) 4.2(-5)
N=4 (1) (2) (3) (4)
-0.04042783 -0.0350208 -0.03462187 -0.0326359
1.3009(-4) 1.315(-4) 1.5782(-4) 4.78(-5)
N=5 (1) (2) (3) (4) (5) shape
-0.02606188 -0.02344563 -0.0230391 -0.0213983 -0.0209959 -0.019875
1.0512(-4) 4.344(-5) 1.163(-4) 5.97(-5) 2.11(-5) 1.9(-5)
N=6 (1)
-0.018176 -0.016828 -0.01633 -0.01518 -0.01516 -0.014877 -0.01456 -0.01443
7.6(-5) 7.2(-6) l.O(-4) 5.0(-5) 7.1(-6) 1.015(-4)
‘,s (4) (5) it; (8)
Molecular Approximation [56]
2.5(-5)
-0.05490
-0.026068 -0.023370 -0.023642 -0.021192 -0.020796 -0.01962
116
RESONANCES IN THREE-BODY ATOMIC SYSTEMS ."
VOL. 3.5
For completeness, in Table VIII we list resonance parameters for doubly-excited ‘S” states of Ps- up to the Ps (N=6) threshold. One interesting aspect of these results is the existence of the ‘S” shape resonances associated with the N=3 and N=5 positronium thresholds. Their counterparts in H- are however located below the respective thresholds and become Feshbach-type resonances [57-591. III.(d) Resonances from e+-H to e+-Ps
Next, we will discuss the resonances from e+-H to e+-Ps when the proton mass mp (see Eq. (7)) is changed from infinte to mp=l. Fig. 17 shows the changes for thresholds and S-wave resonances from l/m,=0 to l/m,=l. Here we show changes for the lowest S-wave below the parent N=2 threshold, as well as for the resonance below the parent N=3 threshold. The changes in the widths are quite interesting. Fig. 18 shows the change in the total width for the lowest S-wave resonance below the N=2 parent threshold. In this case, with the exception for the e+-Ps system [60], all the resonances for mp # 1 have two autoionization routes. Each of the resonances would decay to either e+ + H (N=l) or to Ps (N=l)+g+, with H representing the hydrogenic atom with mass mp. Our results in Fig.18 show the total width decreases from l/m, = 0, to about l/m, = 0.45. When I/m, reaches 0.45 approximately, the total width starts to increase and stay increasing to I/m, = 1 (e+-Ps case). The possible explanation of our findings are the following: Near the ef-H end, the autoionization process is dominated by the decay route to Ps (N=l)+n+. The dominance and hence the partial width for such decay route decrease all the way from l/m, = 0 (e+-H) to l/m, = 1 (e+-Ps). Meanwhile, the partial width for the decay route to ef + $? (N=l) starts to increase from l/m, = 0 for increasing l/m,. Such increase starts quite slowly until l/m, reaches 0.45 approximately. After that, the partial width for decaying to the e+ + a (N=l) route would increase rapidly and dominate the total width, and stays increasing to l/m, = 1. Of course, in our investigation we have not calculated the partial widths directly. It is hoped, nevertheless, that our findings would stimulate other investigations for such interesting phenomemon. Fig. 19 shows the l/m, dependence on the width for the S-wave resonance below the H(N=3) threshold. The total width is seen decreasing for the whole l/m, range from I/m, = 0 to l/m, = 1. In this energy region, the resonance has four decay routes to which it can autoionize. They are (Ps(N=2)+g+), (Ps(N=l)t R+), (E(N=2)+e+), and (H(N=l)+e+). N ear the e+-H end (l/ mp = O),the major autoionization process would be (Ps(N=2)+Hf). The partial width (so does for the Ps(N=l)+E+ route) decreases rapidly for increasing l/m,. Meanwhile, the partial width for the (T?(N=2)te+) route (as well as for the H(N=l)+ e+ route), increase for increasing l/m,. All these four combinations would lead to the overall l/m, behaviour as shown in Fig. 19 for the total width. Apparently the partial width increments for the H(N=l) and H(N=2) routes are not pronounced enough to have the total width increasing at the l/m, = 1 end of the mp spectrum.
Y. K. HO
VOL.35
- -- resonances
-0.30 1 0.0
I 0.2
/
I
,
I
0.4
0.6
0.8
1.0 t
t
e’-H
’ i 5,
e ’- Ps
FIG. 17. Thresholds and S-wave resonances for systems from e+-H to e+-Ps as I/m, (m,=being proton charge) is changed from l/m,=0 to l/m,=l.
e*-H
e+-Ps
FIG. 18. The resonance width for the lowest S-wave resonance below the IV=2 parent threshold for system from e+-H to e+-Ps as l/m, (mp b e i n g the proton charge) is changed from 0 to 1.
IV. Summary and concluding remarks
I have given you an overview of atomic resonances in three-body atomic systems involving positrons. These systems are e+-H, e+-He++, and eS-Ps (or its conjugate equal e--Ps). We have also discussed the changes of the charge or the mass of one of its particles. As such, resonance characters from one system to the other can be examined. The method we have been using is the method of complex-coordinate rotation, a rigorous theoretical method and a powerful computational technique for atomic resonance calculations. In closing, let me go back to the resonances in e+-H scattering. Up to now, the resonances in ef-H have not been observed experimentally. The technical difficulty is partly due to the un-avaibility of a high-intensity high-resolusion positron beam. Now that the anti-hydrogen atoms have been successfully produced in the laboratory, cold anti-hydrogen atoms may also be produced in the near future [61,62]. Anti-hydrogen atoms may be stored and accumulated with abundances by using the atomic trapping technique. The resonances in
118
RESONANCES
0.0 t e'-H
INTHREE-BODYATOMICSYSTEMS
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
l/m,
VOL.
35
t e*-Ps
FIG. 19. Resonance width for the S-wave resonance below the N=3 parent threshold for systems from e+-H to e+-Ps as l/m, (mp being the proton charge) is changed from 0 to 1. positron-hydrogen scattering could also be observed in electron-anti-hydrogen scattering, since the energy levels for the (e+, e-, P) system are the same as those for its conjugate (e-, e+, F) counterpart, where P and p represent the proton and anti-proton, respectively. Of course, here we only need to consider the atomic Coulomb potential and ignor all the high order effects. Since high-intensity high-resolusion electron beams are available in nowadays laboratories, the resonances in e+-H scattering could well be first observed through the route of electron-anti-hydrogen atom scattering experiments. Acknowledgments The works described in this review were carried out over a period of many years. The earlier works were supported by the United States National Science Foundation and the recent works by National Science Council of Republic of China with Grants No. NSC 85-2112-M-001-016 and No. NSC 86-2112-M-001-012. I also thank Dr. A. K. Bhatia for our collaborative works on Ps-.
VOL. 35
Y. K. HO
119
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RESONANCES IN THREE-BODY ATOMIC SYSTEMS
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[38] [39] [40] [41]
