- Email: [email protected]
Momentum transfer cross sections for slow electron elastic scattering on Ca, Sr and Ba atoms
PhysicsLettersA 164 (1992) 73—82 North-Holland
PHYSICS LETTERS A
Momentum transfer cross sections for slow electron elastic scattering on Ca, Sr and Ba atoms C.F. Cribakin, B.V. Gul’tsev, V.K. Ivanov MI. Kalinin Polytechnical Institute, 195291 Saint Petersburg, Russia
M.Yu. Kuchiev A.F. loffe Physico-Technical Institute of the Academy of Sciences of Russia, 195021 SaintPetersburg, Russia
and A.R. Tan~ié Boris Kjdrië Institute for Nuclear Sciences, Institute for Theoretical Physics, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 24 September 1991; revised manuscript received 27 January 1992; accepted for publication 4 February 1992 Communicated by B. Fricke
The momentum transfer cross section for the elastic scattering ofslow electrons on calcium, strontium and barium is calculated for energies belowthe ionization threshold. We present the results for the s, p, d and fphase shifts and corresponding momentum transfer cross sections which are initially evaluated in the frozen-core Hartree—Fock (HF) approximation. The correlation corrections are found by solving the Dyson equation for the reducible self-energy part of the one-particle Green function.
1. Introduction The theoretical analysis of slow-electron scattering on atomic and molecular systems represents a very difficult task for the scattering theory, requiring an exact treatment of the exchange effects and polarization of the target by the Coulomb field ofthe incident electron as the most important contributions. There exist various approximation methods which have been developed for the treatment ofthese processes [1—3]but the majority of these methods fail to include these effects accurately, or have computational difficulties. A method, based on the many-body theory, to obtain the optical potential is proposed in this paper. Basically, the solution of the slow-electron elastic scattering problem reduces to finding the matrix elements of the selfenergy part (i.e. the optical potential) of the one-particle Green function which is directly related to the scattering phase shifts. In a previous paper [4], the slow-electron phase shifts were calculated in the first order perturbation theory in the optical potential. A remarkable improvement was made by Amusia et al. [5] by iterating the irreducible self-energy part I. We are extending this treatment to the elastic electron scattering on the alkaline-earth atoms (Ca, Sr and Ba). The most interesting peculiarity of this problem, and corresponding difficulties, is connected with the exact treatment of the large value of their polarizabilities. That results in a strong polarization potential attracting the electron to the atom, and then the picture of the lowenergy electron atom scattering changes noticeably. In order to solve this problem it is convenient to use the Dyson equation method which was successfully used in our previous paper [6] for solving the negative ion problem, the p wave scattering and the photodetachment. Now, we present here our calculations of the s, p, d and f electron phase shifts and corresponding 0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
73
Volume 164, number 1
PHYSICS LETTERS A
momentum transfer cross sections gases),
6 April 1992
aM
(which are important when considering diffusion of electrons through
2(ô,—ô/+I),
(1)
t7M=—~-—~(1+l)Sifl
where k is the electron momentum, and ô 1 is the phase shift. Throughout this paper atomic units are used, except for the numerical values of the energy, which are given in Ry.
2. Theory The wave function
çoE(r)
describing our problem satisfies the equation
H~~~(r)+JEEr,r’~Er’dr’=E~Er),
(2)
where HHF is the static Hartree—Fock (HF) Hamiltonian of the atom, and ‘E ( r, r’) is the energy-dependent nonlocal potential which completely includes the correlational interaction of the “extra” electron with the atom. At large distances E~( r, r’) in (2) turns into the well-known polarization potential ~ a~/r~ ad electron being thewith di2>0 describes an pole static polarizability theatom. atom.Just Theassolution of eq. in (2)our with E= e =papers ~k momentum k scattered byofan demonstrated previous [4—61we have chosen a set of eigenfunctions of the atomic single-particle HF (nonrelativistic) Hamiltonian, as a zeroth order approximation. For an atom with N electrons the first N lowest eigenfunctions are the ground HF state orbitals. The remaining eigenfunctions represent the excited states of an electron moving in the frozen core of the atomic HF ground state, corresponding thus to the total system of N+ 1 electrons. The phase shifts ô 1( e) are determined by the asymptotic expression for the radial part ofthe ~E( r) [5]. They may be presented as a sum of ô~”~ (e), which is the phase shift in the (frozen core) HF approximation (obtained by the asymptotic form of the excited continuum-state radial functions P~’), and the additional phase shift due to the polarization potential . (r, r’): —
(3) If we know the self-energy matrix (where v = ni, or v= el, for the discrete or continuum states, respectively), the phase shift &5~(e) may be obtained directly from the calculation of the wave function [5]: (4)
where =
$
+~ E—~”
(5)
is the matrix element ofthe reducible self-energy operator. All notations used in the present paper follow those in ref. [6]. The problem then reduces to the calculation of the HF values of the phase shifts and their correlation corrections, i.e., to the calculation of the irreducible self-energy part and finding the solution of eq. (5) for the reducible self-energy part £. We shall use the random phase with exchange (RPAE) method in order to find 1 [71.That approximation corresponds to the summation of sequences of diagrams in fig. 1. Since the HF approximation is of the zeroth order, the first order diagrams are included in the chosen HF basic functions ~ The second order diagrams shown in figs. la—id contribute to I in the lowest order of the Coulomb interaction and may be expressed analytically as 74
Volume 164, number 1
PHYSICS LETTERS A
~
=
6 April 1992
+
:‘1
(b)
(a)
_
£l>
+ (c)
~IIIIIII1~~
÷(d) ________
el
+
eI
+
l’~
+ d,~
+
II~~1
e’I +
gi
Fig. 1. Lowest order Feynman—Goldstone diagrams contributing to the irreducible self-energy part.
(2L+l)(21+i)
~3l3~F
~313>F
6
1S,C’l>
‘\F — ~I1I,~2~F)
31311 VLIIC212,Cl1I E—e >
‘
( )
where the summation v ~ F is performed over occupied (hole) states and v> F means the summation over free (particle) states. Reducedmatrix elements of the multipole L components of the Coulomb interaction are determined in accordance with ref. [7].
75
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
It has been shown earlier that the main contribution to the RPAE comes from the diagrams forward in time and diagonal in the hole states [4,7]. Such an additional approximation can be accomplishedby retaining them by the previously mentioned diagrams using a new set of excited state functions P~/~(denoted in fig. 2 with double lines) instead of the p~F1 for particle line v2. Such a new set of excited state functions P~/~1, (i = 2, 4 in fig. 1) is found in the frozen core field of a singly ionized atom with a hole in the state v3. This is equivalent to the summation of an infinite series of RPAE diagrams (forward in time) in which the hole states are unchangeable [4,7]. The states el, P1, Cl are described by the wave functions P~thisis our basic approximation.
3. Results The momentum transfer cross sections of the low-energy elastic scattering of electrons has been calculated
~l
I I
L/”i ~l
=
+
+
+
(b)
(a)
=
V.,
+
(c)
+
+
.,
(d)
1~~V
V
+
+
Fig. 2. The basic approximation which is used to calculate
76
Volume 164, number I
13 4
-—
__-,
O
0
0. I
0.
0. 2
0.3 K(au)
ond order diagrams, only; (III) with the contribution of some third order diagrams).
0.2
0.3
0.4
0.5
K(a~)
0.2
0.3
0.4
0.5
K(o.i)
____ 0
0.1
6 April 1992
2P channel) in Fig. 3. p phase shift for electron scattering by Ba ( the HF approximation (dashed curves) and with the polarization potential taken into account (solid curves: (II) with the see-
//
-
IT
PHYSICS LETTERS A
0
0.1
0.2
0.3
0.4
0.5
0.2
D.3
0.4
fo.r
K(~j)
I’’
0.1
K(aj)
Fig. 4. Calculated phase shifts ô, for s, p, d and f waves (1=0, 1, 2, 3) on Ca in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position ofthe momentum which corresponds to the opening of the inelastic channel (4s—4p).
for Ca, Sr and Ba atoms. The contributions from the s, p, d, and f partial waves only are taken into account because for the given low-energy range the higher partial wave contributions are negligible. The ground state orbitals for the given atoms are computed by the atomic self-consistent HF program, while all other discrete and continuum state radial functions are computed by the frozen core HF program [8]. The continuum functions, describing the scattering electron in zeroth order (functions ~~‘) are evaluated in the frozen HF field. Their states are denoted in fig. 2 with single lines. Each of the partial waves is presented by a set of 15—17 wave functions. 77
Volume 164, number 1
0
10
0.1
PHYSICS LETTERS A
0.2
-—
0.3
0.4
I
0.1
0.2
0.3
0.5
K
0
6 April 1992
0.1
0.2
0.3
0.4
—
I
0.4
0.5
K
0
0.1
0.2
0.3
0.4
0.5
KIQ.j)
0.5
K(~j)
—
Fig. 5. Calculated phase shifts ö, for s, p. d and f waves (1=0, 1, 2, 3) on Sr in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position of the momentum which corresponds to the opening of the inelastic channel (5s—4d).
The double lines in fig. 2 present the second type of the excited states. They have been evaluated in the frozen HF field with a hole in one of the outer shell orbitals. Our estimations point out that the contribution of the next deeper subshell is 10—20 times smaller. The set of such states includes usually the three lowest discrete states and 13 states of the continuum (with a fixed step size in momentum). The monopole, dipole, quadrupole and octupole excitations of the given subshell were calculated. For example: in the case of the p wave scattering the main contribution to I comes from the dipole p—d excitation; the dipole p—s excitation is 5—7% of the p— d transition; the monopole and quadrupole p—ftransitions are 41—52% of the p—d transition; the octupole transitions are 5—6% of the p—dtransition, only. This indicates that the magnitudes of the diagrams rapidly decrease with the growth of the orbital momentum, transferred from the electron to the atom through the Coulomb interaction. The results of different calculations of the electron affinity values for one and the same atom differ very much. Our results are generally larger than the results of the other authors [61. Within our approach there may be several origins of possible corrections to the EA values; the inclusion of relativistic effects [9] and the contributions of inner shell excitations to. Besides that, the problem of higher order diagrams in the self-energy matrix is very complicated. It is known that the HF calculations tend to increase polarizabilities. Because ofthat fact it would be necessaryto consider higher order corrections. We carefully estimated the possible third order corrections to the diagrams (fig. 2) which dominate in I for alkaline-earthatoms (the dipole and quadrupole excitations). The corresponding diagrams (which involve the doubly excited states) and correlation coefficients to the second order diagrams are shown in our previous paper [61. With these corrections the magnitude ofthe self-energy matrix reduces about 78
Volume 164, number 1
PHYSICSLETTERS A
6 April 1992
iT
0
0.1
0.2
0
0.1
0.2
t0.3
0.3 I
0.4
K1~~1
0
0.1
0.2
f 0.3
0.4
K(oj)
0.4
KIm)
0
0.1 III
0.2
f 0.3
0.4
K(a.j)
Fig. 6. Calculated phase shifts J, for s, p. d and f waves (1=0, 1, 2, 3) on Ba in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position ofthe momentum which corresponds to the opening of the inelastic channel (6s—5d).
20—30%, and the values of a~come much closer to their exact values [6]. Concerning the phase shift calculations, the sensitivity to the contribution of given third order corrections is much smaller. For example, we show in fig. 3, the electron p wave scattering on the Ba atoms. The results of the calculations are presented in figs. 4—9. The phase shifts of the p wave scattering by Ca, Sr and Ba atoms were presented earlier [6]. We repeat shortly some of the conclusions: the HF p wave shifts show the resonant behaviour. These resonances correspond to the quasistationary ns2np states of the negative ions. The addition ofthe polarization potential changes the picture of the HF p wave phase shifts dramatically: the electron resonance (HF approximation) is shifted to the negative energy region, turning into a real bound state. The s phase shifts ö~(figs. 4—6) in the HF approximation for all three elements decrease monotonically from the maximum value (at k= 0). After taking into account the polarization potential the s phase shift behaviour is qualitatively different; one can notice the increase up to the maximum values: c5~’(k=0.05 au)=0.21; o~r(k= 0.06 au) = 0.18 and ö~(k= 0.04 au) = 0.18. After the maximum, the values of the phase shifts decrease and after the saddle point become parallel to the HF phases. The behaviour of the phase shifts of the s waves suggests the existence of the Ramsauer minimum in the momentum transfer cross sections. The d phase shifts in the HF approximation are presented in figs. 4—6. Comparing the d phases for each element, one can notice that the momentum at which ô 2 starts to increase shifts towards lower values, and the rise becomessteeper with the increase ofthe charge of the nucleus. Therefore, the d phase shift shows resonance behaviour already in the HF approximation. Taking into account the polarization potential all phases d2 start to increase at lower k values than ö~!F The phase shifts change for ~x (with respect to their values at k= 0) 79
Volume 164, number I
/
II
/
/
000
I
-
PHYSICS LETTERS A
—~ \
6 April 1992
11
\
1500
\
Ca
/
,1
000
~
/
-
Sn \
500
K(au)
1000
I 0.1
I 0.2
I 0.3
0.4
K(au) Fig. 7. Calculated momentum transfer cross section (Ca atom) in the HF approximation (dashed curve), and with the polarization potential taken into account (solid curve).
3261
1 500
-
1000
-,
500—
286)
/ t\i~
2000-
\
J
Ba
\
I
I
I
1000
/ 1
‘N
01 2
.3 KIau)
80
Fig. 8. Calculated momentum transfer cross section (Sr atom) in the HF approximation (dashed curve), and with the polarization potential taken into account (solid curve).
Fig.the 9. HF Calculated momentum transfer crossand section atom) in approximation (dashed curve), with (Ba the polarization potential taken into account (solid curve).
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
at the following momenta: k = 0.355 au, ksr = 0.27 au and kBa = 0.12 au. Therefore we have the resonance behaviour of the d wave cross sections for the Sr and Ba atoms. Figs. 4—6 show that the f phase shifts in the HF approximation are very small; for example: t5~(k= 0.5 au)=0.065; t5~f(k=0.5 au)=0.088 and ö~(k=0.5au)=0.165. The phase ö3 becomes important already at k=0.2 au due to the polarization corrections. The values of the polarization corrections for f scattering are not larger than 0.32 units. At momenta which correspond to the opening of the inelastic channels, the cross sections, corresponding to the f scattering, become comparable to the other cross sections. In fig. 7 the momentum transfer cross section for the Ca atom is given. The Ramsauer minimum is at k= 0.04 au. At small values of the momentum, the aM is predominantly determined by the s wave contributions. After the Ramsauer minimum the contribution of the p waves becomes dominant. In the vicinity of the inelastic channel the f and d wave scattering become influential. In fig. 8 the momentum transfer cross section of the elastic electron scattering on the Sr atom is given. The increased minimum in comparison with the scattering on the Ca atom is mostly induced by the p scattering contributions. In fig. 9 the momentum transfer cross section for the Ba atom is given. The sudden change of the d phase shift in the“Ba appearance a large maximum in ö~.This suggests thetransfer existence of the metastable 2D inresults the system atom +d of electron”. The minimum of the momentum cross section is more state pronounced than in both above mentioned elements (mostly due to the partial cross sections of the s and d waves). Unfortunately we are not able to compare directly our results with the results of other authors. In ref. [10] the interaction of electrons with Ca atoms is studied on the basis of the Dyson equation (method). The basic result is the existence of the resonance in p wave scattering. By using semi-empirical methods, in ref. [11] the total cross section was calculated with the first three partial waves, and their results qualitatively agree with ours. Experimental data for the scattering of the low-energy electrons on Ca, Sr and Ba atoms (for the zeroth angle) are given in ref. [12]. By indirect comparison to the previously mentioned experimental results the best agreement with our results is in the case of the Ca atom. In conclusion, we noted that the excitations of the outer ns2 subshell of the alkaline-earth atom play the main role in the polarization interaction of a slow electron with the atom. We neglect the excitations of the deeper subshells because their polarizabilities are much smaller. The largest contributions to the matrix element are due to the diagrams with dipole atomic excitations, just as has been expected. We also concluded that the Dyson equation method is a very sensitive theoretical method for investigating the low-energy scattering of electrons on the heavier atoms. Using the Dyson equation method one should examine carefully the possible contributions of higher order diagrams to the I. The use of the presented method shows that in comparison with the HF approximation, there are some qualitative changes in the phase shift behaviour and consequently in the momentum transfer cross sections.
Acknowledgement The authors are grateful to Professor M.Ya. Amusia for numerous and helpful discussions.
References [I ] N.F. Mott and H.S.W. Massey, The theory ofatomic collisions, 3rd Ed. (Clarendon, Oxford, 1965). [211.1. Fabrikant, Atomic processes (Zinatne, Riga, 1974) p. 80 [in Russian]. [3]H.A. Kurtz and K.D. Jordan,J. Phys. B 14(1981)4361. [4JM.Ya. Amusia, N.A. Cherepkov, L.V. Chernysheva, 5G. Shapiro andA.R. Tanëió, Zh. Eksp. Teor. Fiz. 68 (1975) 2023 [Soy. Phys. JETP41 (1976) 1012]. [5] M.Ya. Amusia, N.A. Cherepkov, L.V. Chernysheva, D.M. Davidoviá and V. Radojevi~,Phys. Rev. A 25 (1982) 219.
81
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
[6] G.F. Gribakin, B.V. GuI’tsev, V.K. Ivanov and M.Yu. Kuchiev, J. Phys. B 23 (1990) 4505. [7] M.Ya. Amusia and N.A. Cherepkov, Case Stud. At. Phys. 5 (1975) 47. [8] M.Ya. Amusia and LV. Chernysheva, Automated system for atomic structure investigation (Nauka, Moscow, 1983) [in Russian]. [9] C. Froese Fischer, Phys. Rev. A 39 (1989) 963. [10] M.Ya. Amusia and V.A. Sosnivker, Zh. Eksp. Teor. Fiz. 59 (1989) 28. [l1]J.YuanandZ.Zhang,J.Phy.B22(1989)2751. [121 NI. Roman’uk, 0. Shpenik and I. Zapesochny, Pis’ma Zh. Eksp. Teor. Fiz. 32 (1980) 472.
82
PHYSICS LETTERS A
Momentum transfer cross sections for slow electron elastic scattering on Ca, Sr and Ba atoms C.F. Cribakin, B.V. Gul’tsev, V.K. Ivanov MI. Kalinin Polytechnical Institute, 195291 Saint Petersburg, Russia
M.Yu. Kuchiev A.F. loffe Physico-Technical Institute of the Academy of Sciences of Russia, 195021 SaintPetersburg, Russia
and A.R. Tan~ié Boris Kjdrië Institute for Nuclear Sciences, Institute for Theoretical Physics, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 24 September 1991; revised manuscript received 27 January 1992; accepted for publication 4 February 1992 Communicated by B. Fricke
The momentum transfer cross section for the elastic scattering ofslow electrons on calcium, strontium and barium is calculated for energies belowthe ionization threshold. We present the results for the s, p, d and fphase shifts and corresponding momentum transfer cross sections which are initially evaluated in the frozen-core Hartree—Fock (HF) approximation. The correlation corrections are found by solving the Dyson equation for the reducible self-energy part of the one-particle Green function.
1. Introduction The theoretical analysis of slow-electron scattering on atomic and molecular systems represents a very difficult task for the scattering theory, requiring an exact treatment of the exchange effects and polarization of the target by the Coulomb field ofthe incident electron as the most important contributions. There exist various approximation methods which have been developed for the treatment ofthese processes [1—3]but the majority of these methods fail to include these effects accurately, or have computational difficulties. A method, based on the many-body theory, to obtain the optical potential is proposed in this paper. Basically, the solution of the slow-electron elastic scattering problem reduces to finding the matrix elements of the selfenergy part (i.e. the optical potential) of the one-particle Green function which is directly related to the scattering phase shifts. In a previous paper [4], the slow-electron phase shifts were calculated in the first order perturbation theory in the optical potential. A remarkable improvement was made by Amusia et al. [5] by iterating the irreducible self-energy part I. We are extending this treatment to the elastic electron scattering on the alkaline-earth atoms (Ca, Sr and Ba). The most interesting peculiarity of this problem, and corresponding difficulties, is connected with the exact treatment of the large value of their polarizabilities. That results in a strong polarization potential attracting the electron to the atom, and then the picture of the lowenergy electron atom scattering changes noticeably. In order to solve this problem it is convenient to use the Dyson equation method which was successfully used in our previous paper [6] for solving the negative ion problem, the p wave scattering and the photodetachment. Now, we present here our calculations of the s, p, d and f electron phase shifts and corresponding 0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
73
Volume 164, number 1
PHYSICS LETTERS A
momentum transfer cross sections gases),
6 April 1992
aM
(which are important when considering diffusion of electrons through
2(ô,—ô/+I),
(1)
t7M=—~-—~(1+l)Sifl
where k is the electron momentum, and ô 1 is the phase shift. Throughout this paper atomic units are used, except for the numerical values of the energy, which are given in Ry.
2. Theory The wave function
çoE(r)
describing our problem satisfies the equation
H~~~(r)+JEEr,r’~Er’dr’=E~Er),
(2)
where HHF is the static Hartree—Fock (HF) Hamiltonian of the atom, and ‘E ( r, r’) is the energy-dependent nonlocal potential which completely includes the correlational interaction of the “extra” electron with the atom. At large distances E~( r, r’) in (2) turns into the well-known polarization potential ~ a~/r~ ad electron being thewith di2>0 describes an pole static polarizability theatom. atom.Just Theassolution of eq. in (2)our with E= e =papers ~k momentum k scattered byofan demonstrated previous [4—61we have chosen a set of eigenfunctions of the atomic single-particle HF (nonrelativistic) Hamiltonian, as a zeroth order approximation. For an atom with N electrons the first N lowest eigenfunctions are the ground HF state orbitals. The remaining eigenfunctions represent the excited states of an electron moving in the frozen core of the atomic HF ground state, corresponding thus to the total system of N+ 1 electrons. The phase shifts ô 1( e) are determined by the asymptotic expression for the radial part ofthe ~E( r) [5]. They may be presented as a sum of ô~”~ (e), which is the phase shift in the (frozen core) HF approximation (obtained by the asymptotic form of the excited continuum-state radial functions P~’), and the additional phase shift due to the polarization potential . (r, r’): —
(3) If we know the self-energy matrix
where =
$
(5)
is the matrix element ofthe reducible self-energy operator. All notations used in the present paper follow those in ref. [6]. The problem then reduces to the calculation of the HF values of the phase shifts and their correlation corrections, i.e., to the calculation of the irreducible self-energy part and finding the solution of eq. (5) for the reducible self-energy part £. We shall use the random phase with exchange (RPAE) method in order to find 1 [71.That approximation corresponds to the summation of sequences of diagrams in fig. 1. Since the HF approximation is of the zeroth order, the first order diagrams are included in the chosen HF basic functions ~ The second order diagrams shown in figs. la—id contribute to I in the lowest order of the Coulomb interaction and may be expressed analytically as 74
Volume 164, number 1
PHYSICS LETTERS A
~
=
6 April 1992
+
:‘1
(b)
(a)
_
£l>
+ (c)
~IIIIIII1~~
÷(d) ________
el
+
eI
+
l’~
+ d,~
+
II~~1
e’I +
gi
Fig. 1. Lowest order Feynman—Goldstone diagrams contributing to the irreducible self-energy part.
(2L+l)(21+i)
~3l3~F
~313>F
6
1S,C’l>
‘\
31311 VLIIC212,Cl1I E—e >
‘
( )
where the summation v ~ F is performed over occupied (hole) states and v> F means the summation over free (particle) states. Reducedmatrix elements of the multipole L components of the Coulomb interaction are determined in accordance with ref. [7].
75
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
It has been shown earlier that the main contribution to the RPAE comes from the diagrams forward in time and diagonal in the hole states [4,7]. Such an additional approximation can be accomplishedby retaining them by the previously mentioned diagrams using a new set of excited state functions P~/~(denoted in fig. 2 with double lines) instead of the p~F1 for particle line v2. Such a new set of excited state functions P~/~1, (i = 2, 4 in fig. 1) is found in the frozen core field of a singly ionized atom with a hole in the state v3. This is equivalent to the summation of an infinite series of RPAE diagrams (forward in time) in which the hole states are unchangeable [4,7]. The states el, P1, Cl are described by the wave functions P~thisis our basic approximation.
3. Results The momentum transfer cross sections of the low-energy elastic scattering of electrons has been calculated
~l
I I
L/”i ~l
=
+
+
+
(b)
(a)
=
V.,
+
(c)
+
+
.,
(d)
1~~V
V
+
+
Fig. 2. The basic approximation which is used to calculate
76
Volume 164, number I
13 4
-—
__-,
O
0
0. I
0.
0. 2
0.3 K(au)
ond order diagrams, only; (III) with the contribution of some third order diagrams).
0.2
0.3
0.4
0.5
K(a~)
0.2
0.3
0.4
0.5
K(o.i)
____ 0
0.1
6 April 1992
2P channel) in Fig. 3. p phase shift for electron scattering by Ba ( the HF approximation (dashed curves) and with the polarization potential taken into account (solid curves: (II) with the see-
//
-
IT
PHYSICS LETTERS A
0
0.1
0.2
0.3
0.4
0.5
0.2
D.3
0.4
fo.r
K(~j)
I’’
0.1
K(aj)
Fig. 4. Calculated phase shifts ô, for s, p, d and f waves (1=0, 1, 2, 3) on Ca in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position ofthe momentum which corresponds to the opening of the inelastic channel (4s—4p).
for Ca, Sr and Ba atoms. The contributions from the s, p, d, and f partial waves only are taken into account because for the given low-energy range the higher partial wave contributions are negligible. The ground state orbitals for the given atoms are computed by the atomic self-consistent HF program, while all other discrete and continuum state radial functions are computed by the frozen core HF program [8]. The continuum functions, describing the scattering electron in zeroth order (functions ~~‘) are evaluated in the frozen HF field. Their states are denoted in fig. 2 with single lines. Each of the partial waves is presented by a set of 15—17 wave functions. 77
Volume 164, number 1
0
10
0.1
PHYSICS LETTERS A
0.2
-—
0.3
0.4
I
0.1
0.2
0.3
0.5
K
0
6 April 1992
0.1
0.2
0.3
0.4
—
I
0.4
0.5
K
0
0.1
0.2
0.3
0.4
0.5
KIQ.j)
0.5
K(~j)
—
Fig. 5. Calculated phase shifts ö, for s, p. d and f waves (1=0, 1, 2, 3) on Sr in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position of the momentum which corresponds to the opening of the inelastic channel (5s—4d).
The double lines in fig. 2 present the second type of the excited states. They have been evaluated in the frozen HF field with a hole in one of the outer shell orbitals. Our estimations point out that the contribution of the next deeper subshell is 10—20 times smaller. The set of such states includes usually the three lowest discrete states and 13 states of the continuum (with a fixed step size in momentum). The monopole, dipole, quadrupole and octupole excitations of the given subshell were calculated. For example: in the case of the p wave scattering the main contribution to I comes from the dipole p—d excitation; the dipole p—s excitation is 5—7% of the p— d transition; the monopole and quadrupole p—ftransitions are 41—52% of the p—d transition; the octupole transitions are 5—6% of the p—dtransition, only. This indicates that the magnitudes of the diagrams rapidly decrease with the growth of the orbital momentum, transferred from the electron to the atom through the Coulomb interaction. The results of different calculations of the electron affinity values for one and the same atom differ very much. Our results are generally larger than the results of the other authors [61. Within our approach there may be several origins of possible corrections to the EA values; the inclusion of relativistic effects [9] and the contributions of inner shell excitations to
Volume 164, number 1
PHYSICSLETTERS A
6 April 1992
iT
0
0.1
0.2
0
0.1
0.2
t0.3
0.3 I
0.4
K1~~1
0
0.1
0.2
f 0.3
0.4
K(oj)
0.4
KIm)
0
0.1 III
0.2
f 0.3
0.4
K(a.j)
Fig. 6. Calculated phase shifts J, for s, p. d and f waves (1=0, 1, 2, 3) on Ba in the HF approximation (dashed curves), and with the polarization potential taken into account (solid curves). Arrows indicate the position ofthe momentum which corresponds to the opening of the inelastic channel (6s—5d).
20—30%, and the values of a~come much closer to their exact values [6]. Concerning the phase shift calculations, the sensitivity to the contribution of given third order corrections is much smaller. For example, we show in fig. 3, the electron p wave scattering on the Ba atoms. The results of the calculations are presented in figs. 4—9. The phase shifts of the p wave scattering by Ca, Sr and Ba atoms were presented earlier [6]. We repeat shortly some of the conclusions: the HF p wave shifts show the resonant behaviour. These resonances correspond to the quasistationary ns2np states of the negative ions. The addition ofthe polarization potential changes the picture of the HF p wave phase shifts dramatically: the electron resonance (HF approximation) is shifted to the negative energy region, turning into a real bound state. The s phase shifts ö~(figs. 4—6) in the HF approximation for all three elements decrease monotonically from the maximum value (at k= 0). After taking into account the polarization potential the s phase shift behaviour is qualitatively different; one can notice the increase up to the maximum values: c5~’(k=0.05 au)=0.21; o~r(k= 0.06 au) = 0.18 and ö~(k= 0.04 au) = 0.18. After the maximum, the values of the phase shifts decrease and after the saddle point become parallel to the HF phases. The behaviour of the phase shifts of the s waves suggests the existence of the Ramsauer minimum in the momentum transfer cross sections. The d phase shifts in the HF approximation are presented in figs. 4—6. Comparing the d phases for each element, one can notice that the momentum at which ô 2 starts to increase shifts towards lower values, and the rise becomessteeper with the increase ofthe charge of the nucleus. Therefore, the d phase shift shows resonance behaviour already in the HF approximation. Taking into account the polarization potential all phases d2 start to increase at lower k values than ö~!F The phase shifts change for ~x (with respect to their values at k= 0) 79
Volume 164, number I
/
II
/
/
000
I
-
PHYSICS LETTERS A
—~ \
6 April 1992
11
\
1500
\
Ca
/
,1
000
~
/
-
Sn \
500
K(au)
1000
I 0.1
I 0.2
I 0.3
0.4
K(au) Fig. 7. Calculated momentum transfer cross section (Ca atom) in the HF approximation (dashed curve), and with the polarization potential taken into account (solid curve).
3261
1 500
-
1000
-,
500—
286)
/ t\i~
2000-
\
J
Ba
\
I
I
I
1000
/ 1
‘N
01 2
.3 KIau)
80
Fig. 8. Calculated momentum transfer cross section (Sr atom) in the HF approximation (dashed curve), and with the polarization potential taken into account (solid curve).
Fig.the 9. HF Calculated momentum transfer crossand section atom) in approximation (dashed curve), with (Ba the polarization potential taken into account (solid curve).
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
at the following momenta: k = 0.355 au, ksr = 0.27 au and kBa = 0.12 au. Therefore we have the resonance behaviour of the d wave cross sections for the Sr and Ba atoms. Figs. 4—6 show that the f phase shifts in the HF approximation are very small; for example: t5~(k= 0.5 au)=0.065; t5~f(k=0.5 au)=0.088 and ö~(k=0.5au)=0.165. The phase ö3 becomes important already at k=0.2 au due to the polarization corrections. The values of the polarization corrections for f scattering are not larger than 0.32 units. At momenta which correspond to the opening of the inelastic channels, the cross sections, corresponding to the f scattering, become comparable to the other cross sections. In fig. 7 the momentum transfer cross section for the Ca atom is given. The Ramsauer minimum is at k= 0.04 au. At small values of the momentum, the aM is predominantly determined by the s wave contributions. After the Ramsauer minimum the contribution of the p waves becomes dominant. In the vicinity of the inelastic channel the f and d wave scattering become influential. In fig. 8 the momentum transfer cross section of the elastic electron scattering on the Sr atom is given. The increased minimum in comparison with the scattering on the Ca atom is mostly induced by the p scattering contributions. In fig. 9 the momentum transfer cross section for the Ba atom is given. The sudden change of the d phase shift in the“Ba appearance a large maximum in ö~.This suggests thetransfer existence of the metastable 2D inresults the system atom +d of electron”. The minimum of the momentum cross section is more state pronounced than in both above mentioned elements (mostly due to the partial cross sections of the s and d waves). Unfortunately we are not able to compare directly our results with the results of other authors. In ref. [10] the interaction of electrons with Ca atoms is studied on the basis of the Dyson equation (method). The basic result is the existence of the resonance in p wave scattering. By using semi-empirical methods, in ref. [11] the total cross section was calculated with the first three partial waves, and their results qualitatively agree with ours. Experimental data for the scattering of the low-energy electrons on Ca, Sr and Ba atoms (for the zeroth angle) are given in ref. [12]. By indirect comparison to the previously mentioned experimental results the best agreement with our results is in the case of the Ca atom. In conclusion, we noted that the excitations of the outer ns2 subshell of the alkaline-earth atom play the main role in the polarization interaction of a slow electron with the atom. We neglect the excitations of the deeper subshells because their polarizabilities are much smaller. The largest contributions to the matrix element
Acknowledgement The authors are grateful to Professor M.Ya. Amusia for numerous and helpful discussions.
References [I ] N.F. Mott and H.S.W. Massey, The theory ofatomic collisions, 3rd Ed. (Clarendon, Oxford, 1965). [211.1. Fabrikant, Atomic processes (Zinatne, Riga, 1974) p. 80 [in Russian]. [3]H.A. Kurtz and K.D. Jordan,J. Phys. B 14(1981)4361. [4JM.Ya. Amusia, N.A. Cherepkov, L.V. Chernysheva, 5G. Shapiro andA.R. Tanëió, Zh. Eksp. Teor. Fiz. 68 (1975) 2023 [Soy. Phys. JETP41 (1976) 1012]. [5] M.Ya. Amusia, N.A. Cherepkov, L.V. Chernysheva, D.M. Davidoviá and V. Radojevi~,Phys. Rev. A 25 (1982) 219.
81
Volume 164, number 1
PHYSICS LETTERS A
6 April 1992
[6] G.F. Gribakin, B.V. GuI’tsev, V.K. Ivanov and M.Yu. Kuchiev, J. Phys. B 23 (1990) 4505. [7] M.Ya. Amusia and N.A. Cherepkov, Case Stud. At. Phys. 5 (1975) 47. [8] M.Ya. Amusia and LV. Chernysheva, Automated system for atomic structure investigation (Nauka, Moscow, 1983) [in Russian]. [9] C. Froese Fischer, Phys. Rev. A 39 (1989) 963. [10] M.Ya. Amusia and V.A. Sosnivker, Zh. Eksp. Teor. Fiz. 59 (1989) 28. [l1]J.YuanandZ.Zhang,J.Phy.B22(1989)2751. [121 NI. Roman’uk, 0. Shpenik and I. Zapesochny, Pis’ma Zh. Eksp. Teor. Fiz. 32 (1980) 472.
82