Elastic, positronium formation and 2 p-excitation cross-sections
of e
1-Li scattering
S. Y. EL-BAKRY(1) and M. A. ABDEL-RAOUF(2)
(1) Department of Physics, Faculty of Science, Ain Shams University
Abbassia, Cairo, Egypt
(2) Physics Department, Faculty of Science, University of Qatar - Box 2713, Doha, Qatar
(ricevuto il 4 Settembre 1995; approvato il 10 Settembre 1996)
Summary. — Inelastic-collision processes, in which elastic, positronium formation
and 2p-excitation channels are open, have been investigated for the first time within the framework of the coupled-static and frozen-core approximations. Rayleigh-Ritz variational method has been employed for calculating the binding energies and wave functions of the 1s, 2s and 2p states of the lithium atom. The partial cross-sections corresponding to 7 values of the total angular momentum l ( 0GlG6) are determined for 9 values of the incident energy lying between 1.8 and 10 eV. Our results illustrate the stability of the iterative numerical technique employed. They demonstrate the role played by each partial wave in the total cross-sections and the importance of considering the positronium formation channel in all inelastic collisions of positrons by the atoms of alkali metals in the low-energy region. They also show that the 2p-excitation channel is very important in the same energy range.
PACS 34.50 – Scattering of atoms, molecules, and ions.
1. – Introduction
Recently, there has been an increasing interest among experimental and theoretical positron physicists in the study of positron-alkali atom collisions. The strongest justification of this interest is due to the possibility of positronium (Ps) formation even at zero incident energy. This means that we are facing from the beginning multichannel scattering processes in which at least two channels (the elastic and rearrangement ones) are open. This fact suggests that any serious theoretical treatment of these processes should consider, especially in the low-energy region (i.e. below 10 eV), the Ps formation channel. On the other hand, np excitation channels occur at very low energies (very few eV above the Ps formation channel) and emphasize the importance of their consideration. Lithium atom is the uppermost element of the alkali group and its inelastic collision with positrons can be treated theoretically with 637
reasonable accuracy. The first serious theoretical treatment of positron-lithium collisions was carried out by Guha and Ghosh [1] who applied the two-state close-coupling approximation considering only the elastic scattering and the ground-state positronium formation. Taking the effect of the adiabatic polarization potential into account in both the direct and rearrangement channels, their results for the differential, total and momentum-transfer cross-sections were computed at incident positron energies 0.5–10 eV. They found that the differential cross-section for both the elastic scattering and positronium formation shows a deep minimum at low energies of positron impact and at certain values of scattering angles. Wadehra [2] and Khare and Vijayshri [3] were mainly interested in the behaviour of the total collisional cross-sections of positron-lithium scattering at intermediate and high energies up to 1000 eV. They used Born and modified Glauber approximation (MGA), respectively, and obtained the total cross-sections which are in fair agreement with the one of electron-lithium scattering calculated by Guha and Ghosh [4]. Mazumdar and Ghosh [5] applied the distorted wave approximation (DWA) for determining positronium formation cross-sections of e1-Li scattering at few incident energies between 2 and 100 eV and found that the Ps formation cross-sections decrease with the increase of incident positron energies and at about 100 eV, Ps formation is found to be negligible. Gien [6] investigated the inelastic scattering of positrons by lithium atom using a modified Glauber (MG) and second Born (SB) approximation at energies ranging from 40 to 1000 eV. He employed Peach’s pseudo-potentials [7] for describing the interaction between the valence electron and the rest of the alkali atom. The author considered the inert-core and frozen-core assumption and used the Clementi wavefunctions [8] to represent the target electrons. His comparison between positron and electron cross-sections calculated by the same approach showed that those of positrons are somewhat smaller. Nahar and Wadehra [9] investigated the inelastic scattering of positrons by lithium atoms using first Born (FB) and distorted wave Born (DWB) approximations. They calculated the positronium differential and total cross-sections at very few intermediate energies using a Hellmann-type pseudo-potential [10] for representing the interaction between the valence electron and the core of the target. Abdel-Raouf [11] investigated the inelastic scattering of positrons by lithium using coupled-static and frozen-core approximations and employed the Clementi-Roetti [8] wavefunctions for describing the one-valence electron model of the target. He calculated the partial and total elastic and positronium formation cross-sections for eight values of the total angular momentum at 25 values of the incident energy ranging from 0.1 to 1000 eV. His results show that the total elastic cross-sections decrease monotonically, apart from a dip at incident energy 0.5 eV and a local minimum at 3 eV where regions of resonance could be expected. His results also show the oscillating behaviour and the monotonic decrease of the partial and total positronium formation cross-sections in the region 0.1–60 eV and the Ps formation has a larger cross-section than elastic scattering at 0.1 eV. Abdel-Raouf et al. [12] investigated the inelastic collisions of positrons with lithium atoms within the framework of the coupled-static and frozen-core approximations by employing Walters’ wavefunctions [13] for describing the target. They calculated the partial cross-sections for the first eight partial waves at 21 values of the incident energy lying between 0.1 and 1000 eV. Their results illustrate the fact that the value of the total Ps formation cross-section diminishes beyond 10 eV, a matter which supports the argument that the Ps formation does not play a fundamental role in the total collisional cross-sections of positron-lithium inelastic scattering in the intermediate and high-energy regions.
639
Ward et al. [14] performed five-state close-coupling calculations for e1-Li (2s-2p-3s-3p-3d) scattering in the energy range 0.5–50 eV. They used two different types of target wavefunctions, namely: numerical frozen-core Hartree-Fock wavefunctions and wavefunctions derived from the model potential of Peach [7]. They neglected the positronium formation channel and focused their attention to the 2 p, 3 d, etc. excitation channels. They calculated the elastic, excitation and total integrated cross-sections. The shape of their elastic differential cross-section changes appreciably at energies up to 20 eV but thereafter the basic shape is unaltered. Abdel-Raouf and Wood [15] investigated the possible appearance of resonances in the partial cross-sections of the inelastic collisions of positrons with lithium atoms at energies below 5 eV. They assumed that only elastic and rearrangement channels are open, while excitation channels are closed. Their work indicates the existence of resonance states in the s- and p-partial waves around 0.5 and 4.5 eV, respectively.
The most interesting point emphasized by all previous treatments of e1-Li inelastic scattering in which the Ps channel is ignored, is that the 2 p-excitation channel plays an important role in the very low-energy region. Therefore, in this paper we treat the collision of positrons with lithium atoms as a three-channel problem in which the elastic, rearrangement and 2 p-excitation channels are open.
Section 2 of this paper involves the details of the theoretical formalism starting with the representation of the target model and ending with the closed expressions of the total cross-sections. Section 3 deals with the discussion of our results and their quality compared to the results of the preceding authors. The analyses of the static potentials and the kernels of the coupled integro-differential (scattering) equations are given in appendix A, while the method of solving them iteratively is summarized in appendix B.
2. – Theoretical formalism
The quantum-mechanical treatment of any atomic scattering problem requires the solution of the Schrödinger equation
HNCb 4ENCb ,
(1)
where H and NCb are the exact total Hamiltonian and total wavefunction, respectively, corresponding to a given total energy E. Although in most atomic processes H and E are known, the evaluation of NCb is subjected to various methods of approximation. The first step in all these methods is to assume that the nucleus of the target is infinitely heavy and thus is regarded to be fixed at the origin of the configuration space. This assumption is usually referred to as Born-Oppenheimer’s adiabatic approximation. On the other hand, in the «frozen-core» treatment of the collisions of light particles (electrons, positrons, etc.) with alkali atoms, further assumtions are made, e.g., that the valence electrons are the only active electrons of the atoms; they move in the fields of their nuclei which are screened by fixed-core electrons. Thus, in the collision of positrons with lithium atoms, the latter is considered as a one-electron atom with its ( 2 s) valence electron moving in an effective potential produced by the nucleus and the two frozen 1 s electrons. In the present work, we concentrate ourselves on the inelastic
scattering of positrons by lithium atoms only when the first three channels (namely, the elastic, positronium (Ps) formation and 2 p-excitation) are open. Thus we have
(2) e1 1 Li ( 2 s) 6 K 7 e1 1 Li ( 2 s) Ps 1Li1 e1 1 Li ( 2 p) (elastic channel) , (positronium formation) , (2p-excitation channel) .
In the coupled-static approximation, it is assumed that the projections of the vector (H 2E)NCb onto the bound states of the three channels are zero. Thus, the following conditions aF2 sNH 2 ENCb 4 0 , (3) aF2 sNH 2 ENCb 4 0 (4) and aF2 pNH 2 ENCb 4 0 (5)
are satisfied. The total wavefunction NCb representing the three channels is defined by
NCb 4NF2 sb Nc1b 1NFPsb Nc2b 1NF2 pb Nc3b , (6)
Nc1b, Nc2b, Nc3b are the scattering wavefunctions of channels 1, 2 and 3, respectively. The bound-state wavefunctions can be expressed by
F1 s(ri) 4a1exp [2a1ri] , F2 s(r) 4 (a2exp [2a2r] 1a3r exp [2a3r] ) , (7a)
F2 p(r) 4 (a4r exp [2a4r] 1a5r2exp [2a5r] )(r× Q x×) , (7b)
FPs(r) 4
o
18 p exp [2rO2] , (7c)
where ri, r and x are the position vectors of the i-th core ( 1 s) electron, the valence
electron and the incident positron, respectively, with respect to the infinitely heavy nucleus (see fig. 1). (r× Q x×) is the cosine of the angle between the vectors r and x, and r is the internal distance of the Ps atom. The free parameters a1, a2, a3, a4and a5as well as the linear parameters a1, a2, a3, a4 and a5 are optimized variationally. The total
641
energy E is defined in the first, second and third channels, by
.
`
/
`
´
E 4E( 1 ) 4 E2 s1 K12, E 4E( 2 ) 4 EPs1 1 2 K 2 2 , E 4E( 3 ) 4 E2 p1 K32, (8) where K21 and K32are the kinetic energies (in Ry) of the positron in the first and third channels, respectively, and ( 1 O2) K2
2 is the kinetic energy of the centre-of-mass of the Ps atom relative to the infinitely heavy nucleus. E2 s, E2 p and EPsrefer to the binding energies of the 2 s, 2 p electrons and the ground state of the positronium, respectively. Rayleigh-Ritz [16] variational method has been employed for calculating the binding energies and wavefunctions of the 1 s, 2 s and 2 p states of the lithium atom. In complete agreement with Walters [13] we obtained a14 2 .5, a24 20 .42204505, a34 0 .1125214,
a14 2 .7, a24 2 .7, a34 0 .65 and E2 s4 20 .36392 Ry. The variational treatment of the Li in the 2 p state provides us with: a44 20 .1044689, a54 0 .0028277547, a44 0 .5, a54 0 .7 and E2 p4 20 .250576297 Ry (note that the experimental values [17] of E2 s and E2 p are given by E2 s4 20 .39645 Ry and E2 p4 20 .2605 Ry).
The one-valence-electron model of the lithium target is described (in Ry) by the Hamiltonian
HT4 2˜2r2
2
r 1 Vc(r) ,
(9)
where Vc(r) is an attractive core potential that expresses the net result of the rest of the Coulomb potential due to the nucleus and the potential due to the core electrons. It is defined by (10)
.
/
´
Vc(r)4!
i 41 2 8o
F 1 s(ri)N
2 Nr2riN 22 rN
F1 s(ri)p
2o
F2 s(ri)N
2 Nr2riNN
F1 s(ri)p
, Vc(r) 4VcD(r) 1Vcex(r) ,the dash over the first sum sign indicates that the term 22Or is repeated for each value of i. VcD(r) and Vcex(r) are the direct and exchange parts of the core potential, respectively. Substituting with NF1 sb and NF2 sb from eqs. (7a) into eq. (10), we obtain (11)
.
`
/
`
´
VcD(r) 42(4pa12Oa3)g
a11 1 rh
exp [22a1r] , Vex c (r) 48pa1{
a2 2 1 (a11 a2) r r (a11 a2) 3 exp [2(a11 a2) r] 1 1a3 6 14r(a11 a3) 1r2(a11 a3)2 r (a11 a3)4 exp [2(a11 a3) r]}
; the total Hamiltonians for the first, second and third channels are defined,respectively, by H( 1 )4 HT2 ˜2x1 Vint( 1 ), (12) H( 2 ) 4 HPs2 1 2 ˜ 2 s1 Vint( 2 ), (13) H( 3 ) 4 HT* 2˜2x 81 Vint( 3 ), (14) V( 1 )
int stands for the interaction potential between the incident positron and the one-valence-electron atom, i.e.
V( 1 ) int 44 2 x 2 2 r 1 V D c (x) , (15) where VD c (x) 42(4pa12Oa31)
g
a11 1 xh
exp [22a1x] . (16)HPs, eq. (13), is the Hamiltonian of the Ps atom, i.e.
HPs4 22 ˜2r2
2
r ,
(17) and V( 2 )
int is the interaction potential between the two particles of the positronium and the rest of the target, i.e.
Vint( 2 )4 2 x 2 2 r 1 Vc(r) 1V D c (x) . (18)
H *T is the Hamiltonian of the excited lithium atom which takes the form
HT* 42˜2r2
2
r 1 Vc* (r) ,
(19)
where Vc*(r) has the form (10) with F2 s replaced by F2 p and Vint( 3 ) is the interaction potential between the positron and the one-excited-electron atom which has the form of (15). The dash on the position vector x, eq. (14), is introduced in order to distinguish the arguments of the first and third channels.
Substituting from eqs. (12)-(19) and eqs. (6), (8) into eqs. (3), (4) and (5), we obtain (20) (˜2 x1 K12) Nc1b 4 4 Ust( 1 )(x) Nc1b 1 aF2 sNH( 2 )2 E( 2 )NFPsc2b 1 aF2 sNH( 3 )2 E( 3 )NF2 pc3b , (21) (˜2 s1 K22) Nc2b 4 4 Ust( 2 )(s) Nc2b 1 aFPsNH( 1 )2 E( 1 )NF2 sc1b 1 aFPsNH( 3 )2 E( 3 )NF2 pc3b ,
643
(22) (˜2
x 81 K32) Nc3b 4
4 Ust( 3 )(x 8)Nc3b 1 aF2 pNH( 1 )2 E( 1 )NF2 sc1b 1 aF2 pNH( 2 )2 E( 2 )NFPsc2b , where Schrödinger’s equations of the lithium (ground and excited) and Ps atoms are assumed to be verified. U( 1 )
st (x), Ust( 2 )(s) and Ust( 3 )(x 8) are the static potentials of channel 1, 2 and 3, respectively (see appendix A). Let us now consider the partial-wave expansions of the scattering wavefunctions Nc1b, Nc2b and Nc3b, i.e.
Nc1(x)b 4 1 x l 40
!
Q il ( 2 l 11) fl(x) Yl0(x×) , (23) Nc2(s)b 4 1 s l 40!
Q il ( 2 l 11) gl(s) Yl0(s×) , (24) Nc3(x 8)b 4 1 x 8 l 40!
Q il ( 2 l 11) hl(x 8) Yl0(x× 8) , (25)where fl(x), gl(s) and hl(x 8) are the radial partial wavefunctions corresponding to the
total angular momentum l of the first, second and third channels, respectively. Y0
l(x×),
Y0
l (s×) and Yl0(x× 8) are the related spherical harmonics, x×, s× and x× 8 are the angles
between the vectors x,s and x8 and the z-axis.
Substitution from eqs. (23)-(25) into eqs. (20)-(22), yields, for each value of l, the following coupled integro-differential equations:
(26)
g
d 2 dx2 2 l(l 11) x2 1 K 2 1h
fl(x) 4 4 Ust( 1 )(x) fl(x) 1 0 Q K12(x , s) gl(s) ds 1 0 Q K13(x , x 8) hl(x 8) dx 8 , (27)g
d 2 ds2 2 l(l 11) s2 1 K 2 2h
gl(s) 4 4 Ust( 2 )(s) gl(s) 1 0 Q K21(s , x) fl(x) dx 1 0 Q K23(s , x 8) hl(x 8) dx 8 , (28)g
d 2 dx 82 2 l(l 11) x 82 1 K 2 3h
hl(x 8) 4 4 Ust( 3 )(x 8) hl(x 8)1 0 Q K31(x 8, x) fl(x) dx 1 0 Q K32(x 8, s) gl(s) ds ,where the kernels K12, K13, K21, K23, K31and K32are expanded by (29) K12(x , s) 4 4 8(xs)
m
F2 sFPsg
2 1 2 ˜ 2 s2 1 2 K 2 2h
1 F2 sFPsVint( 2 )n
Yl0(x×) Yl0(s×) dx× ds×, (30) K13(x , x 8) 4 4 8(xx 8)]F2 sF2 p(2˜2x 82 K32) 1F2 sF2 pVint( 3 )( Yl0(x×) Yl0(x× 8) dx× dx× 8 , (31) K21(s , x) 42(8sx)]FPsF2 s(2˜2x2 K12) 1FPsF2 sVint( 1 )( Yl0(x×) Yl0(s×) dx× ds×, (32) K23(s , x 8) 4 4 2( 8 sx 8)]FPsF2 p(2˜2x 82 K32) 1FPsF2 pVint( 3 )( Yl0(x× 8) Yl0(s×) dx× 8 ds× , (33) K31(x 8, x) 4 4 8(x 8 x)]F2 pF2 s(2˜2x2 K12) 1F2 pF2 sVint( 1 )( Yl0(x×) Yl0(x× 8) dx× dx× 8 , (34) K32(x 8, s) 4 4 8(x 8 s)m
F2 pFPsg
2 1 2 ˜ 2 s2 1 2 K 2 2h
1 F2 pFPsVint( 2 )n
Yl0(x× 8) Yl0(s×) dx× 8 ds× .Because of the orthogonality of the wavefunctions F2 s(r) and F2 p(r) of the same target, K134 K314zero.
g
Note that the integrals on the right-hand sides of eqs. (29)-(34) are multiplied by 8 which represents the Jacobians of the transformations dr 48 ds and dr 48 dx .h
Let us now rewrite eqs. (26), (27) and (28) in theforms
g
d2 dx2 2 l(l 11) x2 1 K 2 1h
fl(x) 4Ust( 1 )(x) fl(x) 1Q1(x) , (35)g
d2 ds2 2 l(l 11) s2 1 K 2 2h
gl(s) 4Ust( 2 )(s) gl(s) 1Q2(s) 1Q4(s) , (36)g
d2 dx 82 2 l(l 11) x 82 1 K 2 3h
hl(x 8) 4Ust( 1 )(x 8) hl(x 8)1Q6(x 8) , (37) where Q1(x) 4 0 Q K12(x , s) gl(s) ds , (38)645 Q2(s) 4
0 Q K21(s , x) fl(x) dx , (39) Q4(s) 4 0 Q K23(s , x 8) hl(x 8) dx 8 , (40) Q6(x 8) 4 0 Q K32(x 8, s) gl(s) ds . (41)Equations (35)-(37) have the general form
(E 2H0) Nxb 4Nhb , (42)
where H0is an operator. Usually, H0is the main part of a Hamiltonian, which, as well as the total energy E and the wavefunction Nxb, describes a given quantum-mechanical system. It is well known that the solution of eq. (42) is given (formally) by the Lippman-Schwinger equation
Nxb 4Nx0b 1G0Nhb , (43)
where G0 is the Green’s operator (E 2H0)21 and Nx0b is the solution of the homo-geneous equation
(E 2H0) Nx0b 4N0b . (44)
Comparison between eqs. (35)-(37) and (42) enables us to set their solutions in the form (43). Also the partial wave expansions of the Green’s operators corresponding to the operators on the left-hand sides of eqs. (35)-(37) enable us to write their solutions in the following forms: (45) f(i) l (x) 4
{
di11 1 K1 0 Q gAl(K1x 8)[Ust( 1 )(x 8) fl( 1 )(x 8)1Q1(i)(x 8) ] dx 8}
fAl(K1x) 2 2 1 K1 gAl(K1x) 0 QfAl(K1x 8)[Ust( 1 )(x 8) fl(i)(x 8)1Q1(i)(x 8) ] dx 8 , i 41, 2, 3 ,
(46) g(i) l (s) 4
{
di21 1 K2 0 QgAl(K2s 8)[Ust( 2 )(s 8) gl(i)(s 8)1Q2(i)(s 8)1Q4(i)(s 8) ] ds8
}
Q Q fAl(K2s) 2 1 K2 gAl(K2s) 0 QfAl(K2s 8)[Ust( 2 )(s 8) gl(i)(s 8)1Q2(i)(s 8)1Q4( 1 )(s 8) ] ds8 ,
i 41, 2, 3 , (47) hl(i)(x 8) 4
{
di31 1 K3 0 QgAl(K3x 9)[Ust( 3 )(x 9) hl(i)(x 9)1Q6(i)(x 9) ] dx 9
}
fAl(K3x 8)2 2 1K3
gAl(K3x 8)
0Q
The delta-functions di j, i , j 41, 2, 3, specify two independent forms of solutions for
each of fl(x), gl(s) and hl(x 8) in channels i41, 2 and 3, according to the channel
considered. The functions fAl(m) and gAl(m) (m 4K1x K2s or K3x 8) are related to the spherical Bessel functions of the first and second kinds, jl(m) and yl(m), respectively, by
the relations fAl(m) 4mjl(m) and gAl(m) 42myl(m).
Equations (45), (46) and (47) can be solved iteratively and the iterative solutions are determined by (48) f(i , n) l (x) 4
{
di11 1 K1 0 X gAl(K1x 8)[Ust( 1 )(x 8) fl(i , n)(x 8)1Q1(i , n 21)(x 8) ] dx 8}
Q Q fAl(K1x) 2 1 K1 gAl(K1x) 0 X fAl(K1x 8)[Ust( 1 )(x 8) fl(i , n)(x 8)1Q1(i , n 21)(x 8) ] dx 8 , i 41, 2, 3; n F1 , (49) g(i , n) l (s)4{
di21 1 K2 0 S gAl(K2s 8)[Ust( 2 )(s 8) gl(i , n)(s 8)1Q2(i , n)(s 8)1Q4i , n)(s 8) ] ds8}
Q Q fAl(K2s) 2 1 K2 gAl(K2s 8) 0 S fAl(K2s 8)[Ust( 2 )(s 8) gl(i , n)(s 8)1Q2(i , n)(s 8)1Q4(i , n)(s 8) ] ds8 , i 41, 2, 3; n F0 , (50) hl(i , n)(x 8) 4{
di31 1 K3 0 X 8 gAl(K3x 9)[Ust( 3 )(x 9) hl(i , n)(x 9)1Q6(i , n)(x 9) ] dx 9}
Q Q fAl(K3x 8)2 1 K3 gAl(K3x 8) 0 X 8 fAl(K3x 9)[Ust( 3 )(x 9) hl(i , n)(x 9)1Q6(i , n)(x 9) ] dx 9 , i 41, 2, 3; n F0 ,where n is the order of iteration. X, X 8 and S represent the ranges of integrations over
x, x 8 and s, respectively. Physically X (4nh, where n is the number of mesh points and h is the Simpson step, or mesh size) and X 8 represent the distances at which we assume
the scattered positrons are not affected by the lithium atoms. S (4nh) is the distance at which the Ps atom and the rest of the target are totally separated.
In order to derive the partial cross-sections from eqs. (48), (49) and (50), we rewrite them in the forms
f(i , n) l (x) 4a1(i , n)fAl(K1x) 1b1(i , n)gAl(K1x) , (51) gl(i , n)(s) 4a2(i , n)fAl(K2s) 1b2(i , n)gAl(K2s) , (52) h(i , n) l (x 8) 4a3(i , n)fAl(K3x 8)1b3(i , n)gAl(K3x 8) , (53) where a(i , n)
647
b(i , n)
2 are the coefficients of fAl(K2s) and gAl(K2s) in eq. (49) and a3(i , n)and b3(i , n)are the coefficients of fAl(K3x 8) and gAl(K3x 8) in eq. (50).
Defining the two matrices
an4
.
`
´
k
( 1 OK1) a( 1 , n) 1k
( 2 OK2) a( 1 , n) 2k
( 1 OK3) a( 1 , n) 3k
( 1 OK1) a1( 2 , n)k
( 2 OK2) a( 2 , n) 2k
( 1 OK3) a( 2 , n) 3k
( 1 OK1) a( 3 , n) 1k
( 2OK2) a( 3 , n) 2k
( 1 OK3) a( 3 , n) 3ˆ
`
˜
, (54a) bn 4.
`
´
k
( 1 OK1) b( 1 , n) 1k
( 2 OK2) b( 1 , n) 2k
( 1 OK3) b( 1 , n) 3k
( 1 OK1) b1( 2 , n)k
( 2 OK2) b( 2 , n) 2k
( 1 OK3) b3( 2 , n)k
( 1 OK1) b( 3 , n) 1k
( 2 OK2) b( 3 , n) 2k
( 1 OK3) b( 3 , n) 3ˆ
`
˜
, (54b)we obtain the elements of the reactance matrix by applying the definitions
Rn
i j4 ]bn(an)21(i j.
(55)
The iterative transition matrix, Tn, is related to the reactance matrix Rnby
Tn
4 Rn(I 2iARn)21, (56)
where I is a 3 33 unit matrix and iA4k21 . The elements of the transition matrix can
be written as
Tn
i j4 ]Rn(I 2iARn)21(i j.
(57)
The iterative partial cross-sections corresponding to the total angular momentum l, i.e.
s(l , n)
i j ’s are determined (in a02units) by
s(l , n) i j 4 4 p( 2 l 11) K2 i NTi jnN2, i , j 41, 2, 3 . (58)
The analysis of eq. (57) provides us with the expressions (59a) NTn 11N24 ][R11( 1 2R22R331 R232) 1R12(R33R122 R23R13) 1 1R13(R22R132 R12R23) ]21 [R1221 R1322 R11(R221 R33) ]2( OD2, (59b) NTn 12N24 ][R12( 1 2R11R331 R132) 1R11(R33R122 R13R23) 1 1R13(R11R232 R12R13) ]21 [R11R121 R13R232 R12(R111 R33) ]2( OD2, (59c) NTn 13N24 ][R13( 1 2R11R221 R12) 1R112 (R13R222 R12R23) 1 1R12(R11R232 R12R13) ]21 [R11R131 R12R232 R13(R111 R22) ]2( OD2,
where (59d) D2
4 [ ( 1 1 R1221 R1321 R232) 2 (R11R221 R11R331 R22R33) ]21
1[ (R11R22R331 2 R12R23R13) 2 (R11R231 R222 R131 R332 R12) 2 (R111 R221 R332 ) ]2. Finally, the total cross-sections in the n-th iteration are calculated by
sn i j4
!
l 40 Q s(l , n) i j , i , j 41, 2, 3 . (60)3. – Results and discussion
The computation of the cross-sections was started by studying the behaviour of the static potentials of the first, second and third channels, U( 1 )
st (x), Ust( 2 )(s) and Ust( 3 )(x 8), respectively, to find the range at which these potentials die off. This provides us with the values of X, S and X 8 (see eqs. (48), (49) and (50)) to which the numerical calculations of the integrals should be extended. The static potential of the first channel is always repulsive and vanishes after 5.2 a0. The static potential of the second channel is also always repulsive and dies off after 3.5 a0. The static potential of the third channel has a repulsive part between 0 and 2.6875 a0and an attractive part in the region x 8 4 2.75 a0. The same potential has no effect after 15 a0.
Our computational process was proceeded by investigating the convergence of the elements of the reactance matrix, eq. (55), and the partial cross-sections, eq. (58), with the increase of the number of iterations (n) (see tables I and II) and the integration range (IR) (i.e. the values of X, S and X 8 at eqs. (48), (49) and (50)). It was found out that excellent convergence can be obtained with n 450 and this demonstrates the stability of our iterative technique. Therefore, we fixed this value for all further calculations. Table III contains the total cross-sections at different IRs. The value of TABLEI. – Variation of elements of matrix R with the number of iteration, at K124 3 eV and IR 4 15 a0, for the S-wave (Ri j4 Rji).
No. of iterations R11 R12 R13 R22 R23 R33 5 10 15 20 25 30 35 40 45 46 47 48 49 50 1.92983 21.93599 23.29292 23.89528 23.90353 23.90379 23.90386 23.90386 23.90386 23.90386 23.90386 23.90386 23.90386 23.90386 0.04654 21.39827 25.56501 26.38638 26.39784 26.39819 26.39827 26.39828 26.39828 26.39828 26.39828 26.39828 26.39828 26.39828 20.30339 1.28119 5.05720 5.79442 5.80477 5.80511 5.80518 5.80519 5.80519 5.80519 5.80519 5.80519 5.80519 5.80519 21.72143 22.24206 27.46172 28.50440 28.51922 28.51966 28.51976 28.51977 28.51977 28.51977 28.51977 28.51977 28.51977 28.51977 0.53776 2.72859 6.90787 7.72901 7.74076 7.74113 7.74121 7.74122 7.74122 7.74122 7.74122 7.74122 7.74122 7.74122 23.43051 24.61870 27.09574 27.55923 27.56590 27.56616 27.56620 27.56621 27.56621 27.56621 27.56621 27.56621 27.56621 27.56621
649
TABLE II. – Variation of the partial cross-sections with the number of iteration, at K2 14 3 eV
and IR 415a0, for the S-wave.
No. of iterations Elastic Ps formation 2p-excitation
5 14.32893 0.00017 0.02659 10 2.38928 2.18721 0.51575 15 4.72018 1.79964 0.90761 20 5.68899 1.48556 0.94176 25 5.70392 1.48129 0.94211 30 5.70433 1.48118 0.94213 35 5.70443 1.48115 0.94213 40 5.70443 1.48115 0.94213 45 5.70443 1.48115 0.94213 46 5.70443 1.48115 0.94213 47 5.70443 1.48115 0.94213 48 5.70443 1.48115 0.94213 49 5.70443 1.48115 0.94213 50 5.70443 1.48115 0.94213
TABLEIII. – Variation of the total elastic (s11), positronium formation (s12) and 2 p-excitation (s13) cross-sections (in pa02) of e1-Li scattering with the integration range IR (in a.u.).
K2 1(eV) s11 s12 s13 IR 48 15 8 15 8 15 2 31.19415 116.88652 18.21809 23.32164 4.84416 65.62015 3 32.93435 65.04433 15.59169 25.06456 2.97397 14.01416 4 34.56545 59.48263 12.53787 28.12687 2.20929 6.25953 5 36.03639 54.37752 10.18831 24.32729 1.95418 5.50042 7 36.34361 40.73203 6.77734 14.23922 3.01514 6.50287 10 26.06469 23.48527 4.47256 6.58602 5.61309 6.37207
IR was set equal to 15 a.u., which corresponds to 160 mesh points at mesh size h 43O32 and represents the maximum value allowed by our computing facilities. The final calculations were carried out for seven partial waves corresponding to 0 GlG6 at values of K12representing the low-energy region (1 . 8 GK12G 10 eV). Our energy scale starts slightly above the 2 p-excitation threshold (i.e. above 1.6 eV).
In table IV, we find the partial and total elastic cross-sections. From the table we notice that around the excitation threshold (at 1.8 eV) the G-wave has the very large contribution to the total cross-section and up to 2 eV the G-, P-, D- and S-waves, respectively, play the most important role in the total cross-sections. For higher energies the main contributions to the sn
11 are due to the P, D and F partial cross-sections. The total elastic cross-section falls off steadily with the increase of the incident energy.
Table V contains the partial and total positronium formation cross-section. The table illustrates the following points: 1) the P, F and G partial cross-sections produce the important parts of the total cross-sections and the G-wave becomes important after
TABLE IV. – Partial and total elastic cross-sections (in pa2
0) of e1-Li scattering calculated
by the coupled-static approximation. Integration range IR415a0. Number of iterations (n) 450. K2 1 (eV) l 40 1 2 3 4 5 6 Total 1.8 16.14728 38.93155 14.54834 4.92715 157.08947 0.00069 0.00000 231.64448 2 13.69020 36.68284 15.59304 4.12672 46.79161 0.00211 0.00000 116.88652 3 5.70443 29.75483 20.61611 6.15513 2.74560 0.06798 0.00025 65.04433 4 2.02914 24.07898 21.27847 9.61163 2.30019 0.18000 0.00422 59.48263 5 1.64313 20.59830 19.31347 10.37138 2.16958 0.26407 0.01759 54.37752 6 1.72052 17.69001 17.13737 8.26114 1.97481 0.32192 0.03630 47.14207 7 1.95697 14.67370 15.20890 6.71256 1.77374 0.35197 0.05419 40.73203 8 2.20877 11.82214 11.99536 5.55909 1.59745 0.35994 0.06765 33.60986 10 2.52796 7.69897 7.63499 3.87210 1.32125 0.34987 0.08013 23.48527
TABLEV. – Partial and total positronium formation cross-sections (in pa02) of e1-Li scattering
calculated by the coupled-static approximation. Integration range IR415a0. Number of
iterations (n) 450. K2 1 (eV) l 40 1 2 3 4 5 6 Total 1.8 2.00338 9.62470 2.02214 10.93786 2.72668 0.35993 0.01540 27.69009 2 1.84117 9.48458 1.55567 5.92080 3.87996 0.61165 0.02781 23.32164 3 1.48115 7.29854 1.24661 3.54020 7.92369 3.32957 0.24480 25.06456 4 2.30516 4.92932 1.09191 6.55215 8.45202 3.99068 0.80563 28.12687 5 1.95281 2.39572 0.81832 6.85906 7.35753 3.71801 1.22584 24.32729 6 1.62110 0.92259 0.66035 5.03174 5.56202 3.29925 1.35033 18.44774 7 1.33221 0.27314 0.84154 3.63905 4.08113 2.74469 1.32746 14.23922 8 1.07632 0.06517 0.68913 2.58606 3.04696 2.20576 1.20900 10.87840 10 0.69353 0.22744 0.35945 1.25563 1.71809 1.43547 0.89641 6.58602
2 eV. 2) sn12increases (after 1.8 eV) steadily until it reaches its maximum around 4 eV, and then falls off smoothly.
From table VI we remark that around the 2 p-excitation threshold (1.8 and 2 eV) the
G-wave also has the very large contribution to the total 2 p-excitation cross-sections
and for higher energies the F and G partial cross-sections become important. The total 2 p-excitation cross-section increases until its maximum value at 2 eV and then falls off steadily until 5 eV and increases again through the energy range 6-8 eV and after that falls off. This behaviour of the total 2 p-excitation cross-section of e1-Li scattering emphasizes the possible existence of resonant states.
Tables V and VI demonstrate that the 2 p-excitation cross-sections are leading in comparison with the corresponding Ps formation cross-sections around the 2 p-excitation threshold (1.8 and 2 eV) and the situation is reversed through the energy range 3–10 eV (see also fig. 2). This demonstrates the importance of considering the Ps rearrangement channel in all inelastic collisions of positrons by the atoms of alkali metals in the low-energy region and shows that the 2 p-excitation cross-section is also very important in the low-energy region.
651
TABLE VI. – Partial and total 2p-excitation cross-sections (in pa2
0) of e1-Li scattering
calculated by the coupled-static approximation. Integration range IR415a0. Number of
iterations (n) 450. K2 1 (eV) l 40 1 2 3 4 5 6 Total 1.8 1.22283 0.07157 3.20157 3.77656 44.46968 0.00005 0.00000 52.74226 2 1.23794 0.07053 2.26820 5.70076 56.34173 0.00099 0.00000 65.62015 3 0.94213 0.27233 1.70224 4.02853 6.80617 0.26142 0.00143 14.01416 4 0.67325 0.52397 1.02414 1.80527 1.73714 0.47082 0.02494 6.25953 5 0.61446 0.70570 0.60425 2.01961 1.12946 0.36180 0.06514 5.50042 6 0.59684 0.50443 0.56502 2.44003 1.35365 0.38476 0.08628 5.93101 7 0.55155 0.27733 0.99526 2.61500 1.50502 0.48858 0.11279 6.50287 8 0.48175 0.14859 1.18424 2.63160 1.55993 0.56316 0.15104 6.72031 10 0.31067 0.17702 1.20758 2.32301 1.53295 0.62097 0.19987 6.37207
The most interesting results are accumulated in table VII, where we find a comparison between total cross-sections (in pa02) determined by different authors using different approaches. In this table we present a comparison between the present total collisional cross-sections (elastic plus Ps formation plus 2 p-excitation) and the total collisional (elastic plus excitation) cross-sections determined by Khan, Dutta and Ghosh [18] and Ward et al. [14] using the five-state close-coupling approximation. The
Fig. 2. – Comparison between total elastic cross-sections (s11) (s), total positronium formation cross-sections (s12) (p) and total 2 p-excitation cross-sections (s13) (}) of e1-Li scattering.
TABLE VII. – Comparison between various total collisional cross-sections (in pa2
0) of e1-Li
scattering calculated by different authors. K2
1(eV) Present work (s111 s121 s13)
Ref. [14] Ref. [18] Ref. [15]
1.0 — 212.15 — 224.82 1.8 312.07683 — — — 2.0 205.82831 190.72 — 176.81 3.0 104.12305 179.16 — 141.15 4.0 93.86903 160.36 151.10 113.81 5.0 84.20523 145.11 135.29 92.00 6.0 71.52082 — — — 7.0 61.47412 121.04 112.80 63.70 8.0 51.20857 — — — 10.0 36.44336 97.27 94.70 34.75
last column of the table contains the total cross-sections determined by Abdel-Raouf and Wood [15] using Clementi-Roetti [8]–type wavefunction for the target and taking the Ps formation into account. The same authors have also included in their work the polarization effects in the elastic and Ps channels. From the table we conclude that our total collisional cross-sections around the 2 p-excitation threshold (1.8 and 2 eV) are higher than those obtained by the other authors. This is a sufficient proof for the importance of the rearrangement channel in the very low-energy region. It is also an indication for the valuable role played by the 2 p channel around its threshold. The last table illustrates also that the inclusion of higher excited states is necessary in order to reach the 5-state results obtained by Ward et al. [14].
* * *
Many thanks are due to Eng. N. A. Salem, Chairman of Egyptian Meteorological Authority (EMA) as well as Mr. A. Aamer and Mr. M. Abbas for allowing us to use the computer facilities of the EMA.
AP P E N D I X A
Analyses of the potentials and kernels
The first goal of this Appendix is to present the analytical form of the static potentials U( 1 )
st (x), Ust( 2 )(s) and Ust( 3 )(x 8). The second goal is to express the kernels
K12(x , s), K21(s , x), K23(s , x 8) and K32(x 8, s) as functions of x or x 8 and s. Let us define the static potential U( 1 )
st (x) as
Ust( 1 )(x) 4 aF2 s(r) NVint( 1 )NF2 s(r)b . (A.1)
653
therefore substitution into (A.1) and carrying out the integrations leave us with (A.2) Ust( 1 )(x) 42p
g
a22 a3 2h
g
a21 1 xh
exp [22a2x] 1 16 pg
a 2 3 a53h
g
1 x 1 3 a3 2 1 a 2 3x 1 1 3a 3 3x2h
exp [22a2x] 1 196 pg
a2a3 (a21 a3)4h
g
1 x 1 2 3(a21 a3) 1 1 6(a21 a3) 2 xh
exp [2(a21 a3) x] 1 14 pg
a 2 1 a3 1h
g
a11 1 xh
exp [22a1x] ,where the fact that aF2 s(r) NF2 s(r)b 41 is used. The static potential of the second channel is defined by (A.3) U( 2 ) st (s) 4 aFPsNVint( 2 )NFPsb 4 aFPs(r) N 2
N
s11 2rN
2 2N
s21 2 rN
NFPs(r)b 1 1aFPs(r) NVcexg
N
s2 1 2rN
h
NFPs(r)b . The first integral on the right-hand side of (A.3) vanishes because it consists of two parts which have the same magnitudes and different signs. The second integral, however, can be written asUst( 2 )(s) 4 1 2
0 Q exp [2r]{
21 11 Vcexg
N
s 2 1 2rN
h
dl}
r 2 dr , (A.4)where Vcex(Ns2 (1O2) rN) is obtained from eq. (11) after setting r4Ns2 (1O2) rN and
l 4cos u, u being the angle betweenrands. The integral in the bracket ] ( of eq. (A.4)
can be calculated using a Gauss quadrature and the integral from 0 to Q can be evaluated using a Gauss-Laguerre expansion.
The static potential of the third channel is defined by
U( 3 )
st (x 8) 4 aF2 p(r) NVint( 3 )NF2 p(r)b . (A.5)
Since F2 p(r)4
(
a4r exp [2a4r]1a5r2exp [2a5r])
(r× Q x×) and Vint( 3 )4 2 Ox 2 2 Or1VcD(x) therefore, substitution into (A.5) and carrying out the integrations leave us with (A.6) U( 3 ) st (x 8) 44p{
2x 823y
3 a2 4 2 a74 1 21 a 2 5 a95 1 2688 a4a5 a8z
}
1 14 p exp [22 a4x 8]{
x 823g
3 a2 4 2 a7 4h
1 x 822g
3 a 2 4 2 a6 4h
1 x 821g
7 a 2 4 2 a5 4h
11
g
33 a 2 4 12 a4 4h
1 x 8g
3 a 2 4 2 a3 4h
1 x 82g
a 2 4 2 a2 4h
}
1 14 p exp [22 a5x 8]{
x 823g
21 a2 5 a9 5h
1 x 822g
42 a 2 5 2 a8 5h
1 x 821g
15 a 2 5 4 a7 5 1 42 a 2 5 a7 5h
1 1147 a 2 5 4 a6 5 1 24 a 2 5 a5 5 x 81 66 a 2 5 5 a4 5 x 82 1 187 a 2 5 30 a3 5 x 83 1 77 a 2 5 30 a2 5 x 84 1 42 a 2 5 45 a5 x 85}
1 14 p exp [2ax 8]{
2688 a4a5 a8 x 8 23 1 2688 a4a5 a7 x 8 22 1 1504 a4a5 a6 x 8 21 1 640 a4a5 a5 1 1224 a4a5 a4 x 81 6944 a4a5 105 a3 x 8 2 1 84 a4a5 5 a2 x 8 3 1 168 a4a5 45 a x 8 4}
1 14 pa 2 1 a31g
a11 1 x 8h
exp [22a1x 8] , where a 4 (a41 a5).Let us now rewrite K12(x , s), eq. (29), in the form
(A.7) K12(x , s) 48sx
(a2exp [2a2r] 1a3r exp [2a3r] )g
1 8 ph
1 O2 exp [2rO2]Q Qm
21 2(˜ 2 s1 K22) 1g
2 x 2 2 N2 s 2 xNh
1 V ex c(
N2 s 2 xN)
n
Yl0(x×) Yl0(s×) dx× ds×.In order to exclude the operation with ˜2
s on the function of s× in eq. (A.7) we use
Green’s theorem which states that for any two functions h(r , r) and x(s) the following relation is valid:
h(r , s) ˜2
sx(s) 4
(
˜2sh(r , r))
x(s) .To operate with ˜2
son a function of r and r we have to derive a relation between ˜2sand
the derivatives with respect to r and r. This can be obtained by using the following relations:
.
`
/
`
´
r 42s2 x , 1 2 r4s2 x , ˇ ˇsi 4 2 ˇ ˇri 1 2 ˇ ˇri , i 41, 2, 3 . (A.8) Thus, we have ˜s24 4 ˜2r1 4 ˜r21 ˜rQ ˜r; (A.9)655
consequently, we can show that
(A.10) ˜s2](a2exp [2a2r] 1a3r exp [2a3r] ) exp [2rO2]( 4 4 exp [2rO2 ]
{
a2exp [2a2r]k
4 a2228 a2 r 1 1 2 4 r 1 4 a2(r× Q r ×)
l
1 1a3r exp [2a3r]y
4 a232 16 a3 r 1 8 r2 1 1 2 4 r 2 4(r× Q r×) r 1 4 a3(r× Q r ×)z
}
. Substitution from eq. (A.10) into eq. (A.7) yields(A.11) K12(x , s) 42 4 sx k8 p
exp [2rO2]{
a2exp [2a2r]y
( 2 E2 s1 2 K2 11 1 ) 1 14 a222 8 a2 r 1 1 2 4 r 1 4 a2(r× Q r ×) 2g
4 x 2 4 r 1 2 Vc(r) 12V D c (x)h
z
1 1a3r exp [2a3r]y
2 16 a3 r 1 4 a 2 31 8 r2 1 1 2 4 r 1 4 a3(r× Q r ×) 2 4 r(r× Q r ×) 1 1( 2 E2 s1 2 K121 1 ) 2g
4 x 2 4 r 1 2 Vc(r) 12V D c (x)h
z
}
Yl0(x×) Yl0(s×) dx× ds×.Thus, we reduced the integrand of (A.7) into functions of exp [2ajr], exp [2rO2], (r×Qr×)
and a polynomial in r and r. In order to evaluate the integrals over x× and s×appearing in (A.11) we have to use the transformation (A.8) and express (r× Q r×) as
(A.12) (r× Q r×) 4 (2s2
1 x22 3 sxm) O] 4(s21 x22 2 sxm)( 4 s21 x22 4 sxm)(1 O2,
where m 4 (x×Qs×). This enables us to reduce the integrals (A.11) into integrals over
m only, i.e. over the cosine of the angle between the vectors x and s, with limits from
m 421 to m411. In order to carry out these integrations, we assume the following
expansions of any function W
(
N2 s 2 xN , Ns 2 xN , h)
, where h 4 (r×Qr×):W
(
N2 s 2 xN , Ns 2 xN , h)
4 1sx l 40
!
Q
( 2 l 11) WA(s, x) Pl(m) ,
(A.13)
where WA(s, x) is a pure function of the variables s and x and can be evaluated by
WA(s, x) 4 1 2 sx
21 11 Pl(m) W(
N2 s 2 xN , Ns 2 xN , h)
dm . (A.14)The integral on the right-hand side of eq. (A.14) is calculated by means of a Gauss quadrature of order 10.
To express K12(s , x) in a closed form, we introduce a function Gl(x , s , J1, J2, m1, m2, n1, n2, R , n6), such that (A.15) Gl(x , s , j1, j2, m1, m2, n1, n2, R , n6) 4 1 6 sx
21 11 Pl(m) QQexp [2ajN2 s 2 xN] exp [2Ns 2 xN] N2 s 2 xNj1
(
2 Ns2xNj2)
QQ Fm1Fm2(r× Q x×) n1(r× Q s×)n2(r× Q x×)n3(r× Q s×)n4(r× Q r×)n5(x× Q s×)n6dm , where (r× Q x×) 4 (2sm2x)O
k
4 s2 1 x22 4 sxm , (A.16) (r× Q s×) 4 (s2mx)Ok
s21 x22 2 sxm , (A.17) (r× Q x×) 4 (sm2x)Ok
s21 x22 2 sxm , (A.18) (r× Q s×) 4 (2s2mx)Ok
4 s2 1 x22 4 sxm , (A.19) m 4 (x×Qs×) , (A.20) Fm14.
/
´
1 2 x 2 2 N2 s 2 xN 1 V ex c(
N2 s 2 xN)
for m14 1 , for m14 2 (A.21) and Fm24.
/
´
1 2 x 2 2 N2 s 2 xN for m24 1 , for m24 2 . (A.22)Let us now denote the function (A.15) as G(j)
l , where j is an integer corresponding to a
certain set of values of the integers j1, j2, m1, m2, n1, n2, n3, n4, n5and n6. Table VIII involves the set of j’s required for the final expressions of K12(x , s), K21(s , x),
K23(s , x 8) and K32(x 8, s). The function Gl(j) enables us to split K12(x , s) into the
form (A.23) K12(x , s) 4K( 1 ) 12 (x , s) 1K12( 2 )(x , s) 1K12( 3 )(x , s) , where (A.24) K( 1 ) 12 (x , s) 4 32 p k8 p ]a2[2(E2 s1 K 2 11 1 1 2 a22) Gl( 1 )2 2 a2Gl( 2 )1 Gl( 3 )] 1 1a3[ 8 a3Gl( 6 )2 (E2 s1 K121 1 1 2 a23) Gl( 7 )1 2 Gl( 8 )2 2 a3Gl( 9 )1 Gl( 10 )]( ,
657
TABLEVIII. – Arguments of G(j)
l at different values of j. j aj j1 j2 m1 m2 n1 n2 n3 n4 n5 n6 1 a2 0 0 1 1 0 0 0 0 0 0 2 a2 0 0 1 1 0 0 0 0 1 0 3 a2 0 0 2 1 0 0 0 0 0 0 4 a2 21 0 1 1 0 0 0 0 0 0 5 a2 0 21 1 1 0 0 0 0 0 0 6 a3 0 0 1 1 0 0 0 0 0 0 7 a3 1 0 1 1 0 0 0 0 0 0 8 a3 0 0 1 1 0 0 0 0 1 0 9 a3 1 0 1 1 0 0 0 0 1 0 10 a3 1 0 2 1 0 0 0 0 0 0 11 a3 21 0 1 1 0 0 0 0 0 0 12 a3 1 21 1 1 0 0 0 0 0 0 13 a2 0 0 1 2 0 0 0 0 0 0 14 a3 1 0 1 2 0 0 0 0 0 0 15 a4 1 0 1 1 1 0 0 0 0 0 16 a4 0 0 1 1 1 0 0 0 0 0 17 a4 0 0 1 1 1 0 0 0 1 0 18 a4 1 0 1 1 1 0 0 0 1 0 19 a4 1 0 1 2 1 0 0 0 0 0 20 a5 2 0 1 1 1 0 0 0 0 0 21 a5 1 0 1 1 1 0 0 0 0 0 22 a5 0 0 1 1 1 0 0 0 0 0 23 a5 2 0 1 1 1 0 0 0 1 0 24 a5 1 0 1 1 1 0 0 0 1 0 25 a5 2 0 1 2 1 0 0 0 0 0 26 a4 21 0 1 1 1 0 0 0 0 0 27 a4 1 21 1 1 1 0 0 0 0 0 28 a5 2 21 1 1 1 0 0 0 0 0 29 a4 1 0 2 1 1 0 0 0 0 0 30 a5 2 0 2 1 1 0 0 0 0 0 (A.25) K( 2 ) 12 (x , s) 4 32 p k8 p ] 4 a2a2G ( 4 ) l 2 4 a3Gl( 11 )( , (A.26) K12( 3 )(x , s) 4 32 p k8 p ] 2 a2G ( 5 ) l 1 2 a3Gl( 12 )( .
We notice from eq. (A.15) and table VIII that K( 1 )
12 (x , s) is a continuous function of x and s, while K( 2 )
12 (x , s) is singular at x 42s and K12( 3 )(x , s) is discontinuous at x 4s and
x 42s.
Let us now go to the treatment of the kernel K21(s , x), eq. (31), and rewrite it as (A.27) K21(s , x) 416sx
]a2exp [2a2r] 1a3r exp [2a3r](exp [2rO2] k8 p Q Q
m
2(˜2x1 K12) 1g
2 x 2 2 2 Ns2xNh
1 V D c (x)n
Yl0(x×) Yl0(s×) dx× ds×.From the transformations
.
`
/
`
´
r 42s2 x , 1 2 r4s2 x , ˇ ˇxi 4 2 ˇ ˇri 2 2 ˇ ˇri , i 41, 2, 3 , (A.28)we can show that
˜2x4 ˜2r1 4 ˜2r1 4 ˜rQ ˜r.
(A.29)
In order to avoid the operation with ˜2
x on the angles appearing in the integrand of
eq. (A.27) we use Green’s theorem mentioned previously and the transformation (A.29). This enables us to write K21(s , x) as
(A.30) K21(s , x) 4 16 sx
k8 p
exp [2rO2]{
a2exp [2a2r]y
K2 12g
2 x 2 2 r 1 V D c (x)h
1 1a222 2 a2 r 2 4 r 1 1 1 2 a2(r× Q r ×)z
1 a3r exp [2a 3r]y
K122g
2 x 2 2 r 1 V D c (x)h
2 24 a3 r 1 a 2 31 2 r2 2 4 r 1 1 1 2 a3(r× Q r ×) 2 2 r(r× Q r ×)z
}
Y0 l (x×) Yl0(s×) dx× ds×.Again, using eq. (A.15) and table VIII, we split K21(s , x) into the form
K21(s , x) 4K21( 1 )(s , x) 1K21( 2 )(s , x) 1K21( 3 )(s , x) , (A.31) where (A.32) K21( 1 )(s , x) 4 64 p k8 p]a2
[
2(
K 2 11 a221 1 2 VcD(x))
Gl( 1 )2 2 a2Gl( 2 )1 Gl( 13 )]
1 1a3[
4 a3Gl( 6 )2(
K121 a231 1 2 VcD(x))
Gl( 7 )1 2 Gl( 8 )2 2 a3Gl( 9 )1 Gl( 14 )]
( , (A.33) K21( 2 )(s , x) 4 64 p k8 p] 2 a2a 2 2Gl( 4 )2 2 a3Gl( 11 )( and (A.34) K( 3 ) 21 (s , x) 4 64 p k8 p] 4 a2G ( 5 ) l 1 4 a3Gl( 12 )( .Therefore, the kernel K21(s , x) is also singular at x 42s and x4s. Also, in the same manner and using table VIII and eq. (A.15) we split K23(s , x 8) into the form
K23(s , x 8) 4K23( 1 )(s , x 8)1K23( 2 )(s , x 8)1K23( 3 )(s , x 8) , (A.35)
659 where (A.36) K( 1 ) 23 (s , x 8) 4 64 p k8 p]a4
[
2(
E2 s1 K 2 12 E2 p1 a241 1 2 VcD(x 8))
Gl( 15 )1 14 a4Gl( 16 )1 2 Gl( 17 )2 2 a4Gl( 18 )]
1 a5[
2(
E2 s1 K122 E2 p1 a251 1 2 VcD(x 8))
Gl( 20 )1 16 a5Gl( 21 )2 6 Gl( 22 )2 2 a5Gl( 23 )1 4 Gl( 24 )1 Gl( 25 )]
( , (A.37) K( 1 ) 23 (s , x 8) 4 64 p k8 p]22 a4G ( 26 ) l ( and (A.38) K( 3 ) 23 (s , x 8) 4 64 p k8 p] 4 a4G ( 27 ) l 1 4 a5Gl( 28 )( .In complete analogy to K12, K21and K23we can use Gl(j)in order to express K32(x 8, s),
eq. (43), in the form
K32(x 8, s) 4K32( 1 )(x 8, s)1K32( 2 )(x 8, s)1K32( 3 )(x 8, s) , (A.39) where (A.40) K( 1 ) 32 (x 8, s) 4 32 p k8 p]a4[2(E2 s1 K 2 11 1 1 2 a24) Gl( 15 )1 1Gl( 29 )2 2 a4Gl( 18 )1 2 Gl( 17 )1 8 a4Gl( 16 )] 1 1a5[ 12 a5Gl( 21 )2 (E2 s1 K12111 2 a25) Gl( 20 )1 4 Gl( 24 )2 2 a5Gl( 23 )212 Gl( 22 )1 Gl( 30 )]( , (A.41) K( 2 ) 32 (x 8, s) 4 32 p k8 p]24 a4G ( 26 ) l ( and (A.42) K32( 3 )(x 8, s) 4 32 p k8 p] 2 a4G ( 27 ) l 1 2 a5Gl( 28 )( . AP P E N D I X B
The numerical iterative technique
This appendix contains a brief discussion of the iteration procedure used for calculating the elements of the reactance matrix
(
see eq. (55))
. This has been achieved by taking into consideration eqs. (48)-(50) and the iteration procedure starts byassuming that Q14 0 and the zeroth iteration of fl(i)(x) is calculated by (B.1) f(i , 0 ) l (x) 4
{
di11 1 K1 0 X gAl(K1x 8) Ust( 1 )(x 8) fl(i , 0 )(x 8) dx}
fAl(K1x) 2 2gAl(K1x) K1 0 X fAl(K1x 8) Ust( 1 )(x 8) fl(i , 0 )(x 8) dx 8 , i 41, 2, 3 .(Note that eq. (B.1) gives the static solution of the elastic scattering of positrons by lithium atoms.) The functions Q(i , n 21)
1 (x 8), Q2(i , n)(s 8), Q4(i , n)(s 8) and Q6(i , n)(x 9) which appeared in eqs. (48)-(50), are now defined by
Q1(i , n 21)(x 8) 4
0 S K12(x 8, s) g1(i , n 21)(s) ds , n F1 , (B.2a) Q2(i , n)(s 8) 4 0 X K21(s 8, x) f1(i , n)(x) dx , n F0 , (B.2b) Q4(i , n)(s 8) 4 0 X 8 K23(s 8, x 8) h1(i , n) (x 8) dx 8 n F0 (B.2c) and Q6(i , n)(x 9) 4 0 S K32(x 9, s) g1(i , n)(s) ds , n F0 . (B.2d)Thus, the iteration process starts by calculating fl(i , 0 )(x), i 41, 2, 3, using eq. (B.1) and
introducing its values into eq. (B.2b) in order to find Q2(i , 0 )(s 8) which can be used on the right-hand side of eq. (49) after setting Q44 0 for obtaining gl(i , 0 )(s). The values of the
last quantity can be introduced into eq. (B.2d) in order to find Q6(i , 0 )(x 9) which can be used on the right-hand side of eq. (50) for obtaining hl(i , 0 )(x 8). The values of gl(i , 0 )(s)
can be introduced into eq. (B.2a) in order to calculate Q1(i , 0 )(x 8) which may be employed in eq. (48) for determining f(i , 1 )
l (x). The values of hl(i , 0 )(x 8) can be introduced into
eq. (B.2c) in order to calculate Q(i , 0 )
4 (s 8) which can be employed with Q2(i , 0 )(s 8) in eq. (49) for determining g(i , 1 )
l (s). Q6(i , 0 )(x 9) can be employed in eq. (50) for determining
h(i , 1 )
l (x 8). This iteration process can be repeated as many times as we need and the
judgement of its quality is that the variation of the elements of the reactance matrix
Rn
ij, eq. (55), becomes negligible when n increases.
In order to calculate the integrals in eqs. (48)-(50), (B.1) and (B.2a)-(B.2d) we use Simpson’s rule. Thus, we expand Q1(i , n 21)(x), Q2(i , n)(s), Q4(i , n)(s) and Q6(i , n)(x 8) at the point q of the configuration space as follows:
Q1(i , n 21)(xq) 4
!
p 41 n ]Wp( 1 )K12( 1 )(xq, sp)1Wp( 2 )K12( 2 )(xq, sp)1Wp( 3 )K12( 3 )(xq, sp)(Q (B.3a) Q gl(i , n 21)(sp) , n F1 ,661 Q2(i , n)(sq) 4
!
p 41 n ]Wp( 1 )K21( 1 )(sq, xp)1Wp( 2 )K21( 2 )(sq, xp)1Wp( 3 )K21( 3 )(sq, xp)(Q (B.3b) Q fl(i , n)(xp) , n F0 , Q4(i , n)(sq) 4!
p 41 n ]Wp( 1 )K23( 1 )(sq, xp8 )1Wp( 2 )K23( 2 )(sq, xp8 )1Wp( 3 )K23( 3 )(sq, xp8 )( Q (B.3c) Q hl(i , n)(xp8 ) , n F0 and Q6(i , n)(xq8 ) 4!
p 41 n ]Wp( 1 )K32( 1 )(xq8 , sp)1Wp( 2 )K32( 2 )(xq8 , sp)1Wp( 3 )K32( 3 )(xq8 , sp)(Q (B.3d) Q gl(i , n)(sp) , n F0 where W( 1 )p ’s are the usual Simpson weights (hO3, 4hO3, 2hO3, R, 2hO3, 4hO3, hO3)
and Wp( 2 )’s, Wp( 3 )’s are modified weights used for avoiding the singularities at x 4s and
x 42s, respectively (see, e.g., Fraser [19] and Chan and Fraser [20]). The variables xp,
spand x 8p are chosen such that xp4 sp4 x 8p4 Ph, P 4 1 , 2 , R , n.
An essential point in the determination of the integrals in eqs. (48)-(50) is the calculation of the starting values of the functions f(i , n)
l (x), gl(i , n)(s) and hl(i , n)(x 8), i.e.
their values at xp4 h, sp4 h, x 8p4 h. (Note that fl(i , n)( 0 ) 4gl(i , n)( 0 ) 4hl(i , n)( 0 ) 40. ) In
order to find the starting value of f(i , 0 )
l (x) we consider the Taylor expansions of
Ust( 1 )(x), fAl(m) and gAl(m) around the origin. From eq. (A.2) we can prove that
U( 1 ) st (x) 42
g
Z x 1 C01 C1x 1Rh
, (B.4a) where Z 43 and C04 2pg
a 2 2 a2 2 1 16 a2a3 (a21 a3)3 1 3 a 2 3 ( 2 a43) 1 2 a2 1 a2 1h
. (B.4b)Also, it is known (see, e.g., ref. [21], p. 490), that
fAl(m) ` (m)l 11 ( 2 l 11)!!
{
1 2 m2 O2 1 ! ( 2 l 21) 1 (m2 O2 )2 2 ! ( 2 l 13)(2l15) 1 R}
(B.5) and gAl(m) ` ( 2 l 11)!! ( 2 l 11) ml{
1 1 m2O2 1 ! ( 2 l 21) 1 (m2O2 )2 2 ! ( 2 l 21)(3l22) 1 R}
, (B.6) where m 4K1x. Assuming that f(i , 0 )l (x) behaves close to the origin as
f(i , 0 )
l (x) 4C1x(l 11)1 C2x(l 12)1 C3x(l 13)1 R , (B.7)
therefore, substitution from eqs. (B.4a)-(B7) into eq. (B.1) yields C14 di1 K1(l 11) ( 2 l 11)!! , (B.8a) C24 C1 (l 11) (B.8b) and C34 C1 ( 2 l 13)
{
1 (l 11) 2g
1 1 1 2 K 2 1h
}
. (B.8c)Introducing the constants C1, C2 and C3 into eq. (B.7) and setting x 4h, we obtain a starting value for f(i , 0 )
l (x). (Note that the first three terms of eq. (B.7) are enough for
obtaining this value, especially when h is reasonably small.) Remark also that the main contribution to the starting values is due to the first two terms involved in U( 1 )
st (x). In order to find the starting values of f(i , n)
l (x), gl(i , n)(s) and hl(i , n)(x 8), we have to know
the behaviour of Q(i , n 21)
1 (x), Q2(i , n)(s), Q4(i , n)(s) and Q6(i , n)(x 8) close to the origin. Our investigation of this problem enables us to express Q(i , n 21)
1 (x) as
Q(i , n 21)
1 (x) 4C1xl 111 C2xl 121 R , (B.9)
where C1, C2, R depend on i and n. Again, assuming that fl(i , n)(x) is expanded by
f(i , n)
l (x) 4C1xl 111 C2xl 121 C3xl 131 R ,
(B.10)
one can show from eqs. (B.4a), (B.5), (B.6), (B.9) and (B.10) that C1and C2are identical to those given by eqs. (B.8a) and (B.8b), respectively, while C3 relates to C1 and C1 by C34 C1 ( 2 l 13)
{
1 (l 11) 2g
1 1 1 2 K 2 1h
}
1 C1 ( 2 l 13) . (B.11)The constant C1is roughly estimated by the relation
C1B
Q1(i , n 21)(h)
hl 11 2 C2h ,
(B.12a)
and C2by the expression
C2B 1 h
{
Q1(i , n 21)( 2 h) ( 2 h)l 11 2 Q1(i , n 21)(h) hl 11}
. (B.12b)Therefore, we can use eqs. (B.8a), (B.8b), (B.11) and (B.12a), (B.12b) in order to obtain a starting value for f(i , n)
l (x) at x 4h by employing eq. (B.10).
The above procedure can also be applied for calculating the starting values of
663
R E F E R E N C E S
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