theoretical side, William and Stelbovics (1997) and Msezane and Bessis (1997) ... We noticed that the Slater local-density exchange potential gave much larger.
J. Phys. B: At. Mol. Opt. Phys. 31 (1998) 4123–4135. Printed in the UK
PII: S0953-4075(98)90337-6
Relativistic structure description and relaxation effect on krypton 4p5 (2 P3/2,1/2 )5s excitation at small squared momentum transfer Qicun Shi†‡¶, Suimeng Zhang†, Hyuck Cho‡, Kezun Xu†, Jia-Ming Li†§ and Sabre Kaisk † Department of Modern Physics, University of Science and Technology of China, Hefei 230026, The People’s Republic of China ‡ Department of Physics, Chungnam National University, Taejon 305-764, Korea and Center for Molecular Science, Taejon 305-761, Korea § Institute of Physics, Chinese Academy of Science, Beijing 100080, The People’s Republic of China k Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA Received 6 January 1998, in final form 18 June 1998 Abstract. Relativistic generalized oscillator strengths (GOSs) at small squared momentum transfer (K 2 6 1 au) for krypton 4p5 (2 P1/2,3/2 )5s excitations are studied in detail under the first-order Born approximation based on the Dirac–Fock (DF) and Dirac–Fock–Slater (DS) theories. Comparisons are made with the existing experiment at 300 and 500 eV. The large GOS difference between the local-type and non-local-type exchange interactions in the target, and the large influence of the relativistic effects on the GOS are shown at K 2 less than 0.1. Particularly, the relaxation effect between the ground- and excited-state single configurations, and the Coulomb correlation effect by a model potential as a function of local density are investigated. Comparisons with the experiment show that the relaxation effect is more important than the correlation effect. The present DF calculation, although not involving the non-local correlation, yields both the GOS and the excitation energy closer to the experimental values than the DS calculation, due to the very small correlation effect in the outer-shell excitations. Results of krypton show that the average-configuration-state approximation in the DF theory is suitable to describe the excited-state single configuration including 4p1 4p4 5s1 and 4p2 4p3 5s1 subshells.
1. Introduction Electron–atom scattering at small angles, for example, 5◦ or less, is still a critical challenge to theorists and experimentalists even when the 0–180◦ relative experiment for the electron– helium excitation by Cubric et al (1997) is performed. One of difficulties on the experimental side is from the procedure of how to put relative measurement to an absolute scale through the zero momentum-transfer limit of generalized oscillator strengths (GOSs) measured either at intermediate and high incident energy such as 1500 eV by Xu et al (1996) or at low incident energy such as 15–100 eV by Panajotovi´c et al (1993). On the theoretical side, William and Stelbovics (1997) and Msezane and Bessis (1997) indicated the importance and necessity of dynamic effects at small angles which may have a large influence on differential cross sections (DCSs) and GOSs. ¶ Present address: Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA. c 1998 IOP Publishing Ltd 0953-4075/98/184123+13$19.50
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We pay attention to some small-angle experiments, for example, the 4p excitation of krypton by Takayanagi et al (1990), the 5p excitation of xenon by Suzuki et al (1996) and the 2p excitation of sodium by Bielschowsky et al (1991). A specific problem we are interested in stems from the relativistic static structure description on the averageconfiguration approximation (Grant 1970) for electron excitations of the 4p or 5p outer shell or 2p inner shell, where there is a closed shell involved before the excitation but an open shell left after the excitation. Another problem relates to the correlation effect based upon the static structure description of target. Of course, we have no reason to neglect the relaxation during the excitation. In the work of electron–helium elastic small-angle scattering at 1500 eV (Shi et al 1997) the DCSs are investigated using the well known eikonal-Born series approach, the Dirac– Fock (DF) static exchange approximation and the Dirac–Fock–Slater (DS) local-density approximation (the DS result and the results of inclusion of static polarization are not given in the paper). We noticed that the Slater local-density exchange potential gave much larger cross sections than the DF non-local exchange potential at 1500 eV at scattering angles of less than 15◦ , nearly five times larger than at 5◦ , although there was a satisfactory agreement over large scattering angles between the DF and DS calculations. A question is then proposed as to whether the DS theory is suitable to describe electron–atom excitations at small angles. Here the small angles approximately correspond to small squared momentum transfer (K 2 ) for an outer-shell excitation and a given incident energy, for example, larger than 200 eV. In this paper the DF and DS theories are, therefore, suggested to study the relativistic effects, the relaxation and correlation effects and the difference between different exchange potentials in a local or non-local type. Krypton is chosen to be the target, and the relativistic GOS result of 4p6 (1 S) → 4p5 (2 P3/2,1/2 )5s excitation is presented and compared with the non-relativistic calculation. For the 2 P3/2 transition the excitation energy given by Trajmar et al (1981) is near 10.0 eV which is 30 and 50 times smaller than 300 eV and 500 eV of the existing experimental incident energy, respectively, the first-order Born approximation is consequently thought to be correct, at least, to study the above effects accurately. Before the investigation of dynamic effects we make an a priori assumption that the dynamic influences are higher-order effects compared with the relaxation, correlation and relativistic effects. The 4p5 (2 P3/2 )5s and 4p5 (2 P1/2 )5s excitations of krypton were recently studied by Padma and Deshmukh (1992) using a relativistic local-density potential method. In their GOS calculation the single-electron wavefunction is approximately represented by the major component of the Dirac spinor which satisfies a second-order differential equation and the local Coulomb correlation effect is involved. Earlier theoretical calculations were published for lower incident energies than 100 eV by Meneses et al (1985) and Bartschat and Madison (1987). On the experimental side, the GOS was investigated only by Takayanagi et al (1990) using electron energy-loss spectroscopy at 300 eV and 500 eV at 1.5–10◦ with an experimental accuracy of −15 to +10%. They also provided the corresponding DCSs. Earlier works on DCS measurements were shown by Trajmar et al (1981) at 15–100 eV and references therein. In the following section, the main theoretical procedures to calculate the relativistic and non-relativistic GOS are given. More detailed GOS theory in the non-relativistic frame was given by Inokuti (1971) and Manson (1972). The results are then discussed in detail in section 3. Atomic units (au) are used throughout this paper unless otherwise specified.
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2. Theory 2.1. Generalized oscillator strengths In this calculation the relativistic contribution of the transverse electromagnetic field is neglected (Qu et al 1995) because we are particularly interested in an incident electron in the intermediate-energy region (of order 102 –103 eV). The relativistic GOS, f (w, K), in the Coulomb gauge is then determined under the first-order Born approximation and has the same form of non-relativistic equation (2.9) of Inokuti (1971), equation (2) of Manson (1972) and equation (9) of Padma and Deshmukh (1992) as 2w |I (K)|2 (1) K2 where w is the excitation energy and is approximated to be the energy eigenvalue difference between the ground and excited states corresponding to before and after the electron excitation. K is the length of the relativistic momentum-transfer vector K and is given by (w, K) =
K 2 = ki2 + kf2 − 2ki kf cos θ
(2)
in which ki and kf are incident and scattered electron momenta, respectively, and θ is the scattering angle. In this incident energy (Ei ) region, relativistic K may be approximated by the corresponding non-relativistic momentum transfer also represented by K without any confusion, s # " w w 2 −2 1− cos θ . (3) K = 2Ei 2 − Ei Ei As θ is chosen to be 180◦ and 0◦ , equation (2) gives Kmax = ki + kf ;
Kmin = |ki − kf |
(4)
where Kmin defines physical and non-physical regions in the K-axis for both the experiment and theory. According to the energy conservation and the experimental excitation energy w ≈ 0.369 (10.033 eV) of Trajmar et al (1981) for the 4p5 (2 P3/2 )5s excitation, a non2 relativistic estimation at the experimental incident energies 300 eV and 500 eV gives Kmin values 0.006 and 0.004, respectively, to define the approximate non-physical region 2 . 0 6 K 2 6 Kmin
(5)
Particularly, the region at K 2 6 1 corresponds to the scattering angle region, from equation (3), θ 6 12.3◦ at 300 eV and θ 6 9.5◦ at 500 eV, to which we shall pay much attention. In equation (1) I (K) is the transition matrix in the first-order Born approximation. Using the single-configuration approximation and the independent-particle approximation (IA), a simple single-electron excitation matrix reads, I (K) = hψf (r)|eiK·r |φi (r)i
(6)
in which φi (r) and ψf (r) express the initial (i) single-electron orbital |n0 k 0 m0 i and the final (f ) single-electron orbital |nkmi, respectively, on the self-consistent field (SCF) level. Quantum numbers n, n0 ; k, k 0 ; m, m0 as well as the related ones j, j 0 are all defined as those by Grant (1970). The different letters for the initial and final single-electron orbitals indicate that the initial and final states may be obtained on the different SCF potentials when a relaxation process is assumed to take place during the excitation. A consistent treatment of relaxation, according to the works of Theodosiou (1987) and Theodosiou and Fielder (1982) involves: (a) iteration of the different SCF potentials on the ground- and excited-state single
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configuration, respectively, for the initial and final single-electron orbitals, (b) calculation of overlap integrals from spectator electrons, and (c) implementation of orthogonalization between the initial- and final-state orbitals. Step (b) requires a modified equation (6) as I (K) = I0 hψf (r)|eiK·r |φi (r)i with I0 =
Y 0 hψi 0 |φi 0 in(i ) ,
(7) (8)
i0
where, for krypton i is for 4p or 4p0 and f is for 5s, n(i 0 ) is the electron number in each subshell of a non-excited core and i 0 goes through 1s, 2s, 2p, 2p0 , 3s, 3p, 3p0 , 3d, 3d0 , 4s, 4p, 4p0 subshells. These subshells are closed except 4p and 4p0 subshells with n(4p) = 1 for 2 P1/2 case and n(4p0 ) = 3 for 2 P3/2 case. Actually, equation (7) is also the result of the excitation matrix between initial and final single-configuration wavefunctions used instead of the two single-electron orbitals. Step (c) is realized if the excited configuration orbitals are Schmidt orthogonalized to the orbitals of the ground configuration. We will check the contribution of the three steps, two different SCF potentials, overlap integrals and orthogonalization, to the relaxation effect in the following section. Using the well known expansion for the Born scattering operator eiK·r and the Wigner– Eckart theorem, one obtains the matrix in equation (6) λ ∞ X X j λ j0 λ j −m I (K) = Dδ,0 (−1) (9) T1λ (e), 0 m δ m λ=0 δ=−λ λ in which Dδ,0 (usually called the D function) is related to the spherical harmonic function and expresses the transformation from the K space into the r space. T1λ (e) represents electric multipole from longitudinal electromagnetic field, and here is defined to be 1p j λ j0 T1λ (e) = i λ (2λ + 1)(−1)j − 2 (2j 0 + 1)(2j + 1) 1 0 − 12 2 Z ∞ (Pn0 k0 (r)Pnk (r) + Qn0 k0 Qnk )jλ (Kr) dr, (10) × 0
where P (r), Pnk (r) and Qn0 k0 (r), Qnk (r) are the relativistic major and minor components of radial wavefunctions, respectively. The procedures to average the initial substates and to sum the final substates should be made due to magnetic degenerate for a definite j or j 0 . The squared matrix then reads X |T λ (e)|2 1 1 |I (K)|2 = 0 δj 0 j λ . (11) 2j + 1 λ 2λ + 1 n0 k 0
Here δj 0 j λ represents the triangle rule of involving quantum numbers in the 3 − j symbol (Cowan 1980), and this rule limits the λ sum. Finally, the GOS is obtained by X |T λ (e)|2 2w 1 1 f (w, K) = 2 0 (12) δj 0 j λ . K 2j + 1 λ 2λ + 1 If one does a non-relativistic calculation for the optical-allowed transition such as the present 4p–5s transition, the procedure (equations (1)–(12)) to obtain the GOS expression in the IA is similar to that above without the two steps of the final-substate sum and the initial-substate average, λ1 0 2 2w X l λ l T1λ (e) (13) f (w, K) = 2 −m 0 m0 K λ=λ 0
Krypton 4p 5 (2 P3/2,1/2 )5s excitation in which T1λ (e)
=i
λ−λ0
p
(2λ + 1)
(2l 0
+ 1)(2l + 1)
l0 0
λ l 0 0
4127 Z
∞
Pn0 l 0 (r)jλ (Kr)Pnl (r) dr.
(14)
0
Here Pn0 l 0 (r) and Pnl are non-relativistic electron orbitals, λ1 = l + l 0 , λ0 = |l − l 0 |, λ = λ0 , λ0 + 2, . . . , λ1 and l, l 0 are orbital quantum numbers. Choosing the quantized axis along the K/K-axis in the K space for the optical-allowed transition np → (n+1)s or ns → (n+1)p similar to the work of Chen and Msezane (1994) when we use the non-relativistic function, we certainly reach the result m0 = 0 and m = 0. 2.2. Target structure and potentials Each single electron in the target is thought to move in an average static field of nuclei and other electrons and satisfies a coupling pair of radial equations as given by Herman and Skillman (1963). The potential of the average field is approximated to be central and may be represented by V (r) = Vs (r) + Ve + Vc (r)
(15)
where Vs (r) is the screened (s) Coulomb potential and Ve is the central exchange (e) potential which is of local type in the DS theory. Vc (r) is involved here to consider the Coulomb correlation (c) effect. The Slater exchange potential is obtained by, first, using the variational principle, then, making the statistical average over the total energy as a function of local charge density ρ(r) = ρ↑ (r) + ρ↓ (r), 3ρ(r) 1/3 Vex = −3α (16) 8π in which the coefficient α equals 1. If the two steps, the variation and the statistical average, are exchanged, the Kohn–Sham potential is reached in the same form as equation (16) but the coefficient α being 23 . If the coefficient equals 0 and there is no correlation potential in equation (15), the case in which there is only the static screened Coulomb potential is reached. The correlation potential should be introduced by calculating the Coulomb correlation energies as the same procedure as given by Vijayakumar et al (1989), using their universal value x = 34 for the ratio between the Coulomb- and Fermi-hole radii. The Coulomb correlation energies include two parts, one from spin-up electrons and the other from spindown electrons. For a spin-up electron the Coulomb correlation potential reads 4π (rc3 )↑ (rc3 )↓ ↑ Vc (r) = Vc (r) = − ρ↓ (r) + 27 1 + (rc )↑ 1 + (rc )↓ −1/3 1 3(r 2 )↓ + 2(rc3 )↓ 1 3 −1/3 + ρ↑ (r)ρ↓ (r) c (17) + π 2/3 81 n↑ 3 [1 + (rc )↓ ]2 where (rc )↑ is
−1/3 1 1 (rc )↑ = 0.75 π ρ↓ (r) + n↓ 3
(18)
and n↑ is the number of spin-up electrons and ρ↑ (r) is the charge density of spin-up electrons at r. For a spin-down electron in the present closed p shell (4p6 ) with a closed core the corresponding Coulomb correlation potential is the same function as equations (17) and (18)
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because n↑ equals n↓ and ρ↑ (r) equals ρ↓ (r). For a spin-down electron in the 5s shell, as in this case 4p5 5s, the potential function is approximately thought to be appropriate, based on the fact that 4p5 shell is near to a closed shell. This consideration will certainly introduce a few more errors. However, equation (17) is still employed to approximate the correlation potential of a spin-down electron in this estimation of the correlation influence on the GOS. The technique actually simplifies the SCF iteration. Finally, the total central potential V (r) by equation (15) is required in practice to smoothly match the Coulomb tail of Latter rule (1955) at large r in the DS calculation. If the DF theory is considered, the main difference in the target potential by equation (15) with respect to the DS theory is the exchange part Ve which now appears in the non-local exchange terms WP orQ (A, r) in a coupling pair of radial inhomogenous equations 1X κA 1 d BA QB (r)δκB κA + PA (r) − 2c + (A − V (r)) QA (r) = −WQ (A, r) − dr r c c B
κA d − dr r
1 1X BA PB (r)δκB κA QA (r) + [A − V (r)]PA (r) = +WP (A, r) + c c B
(19) (20)
where PAorB (r), QAorB (r), AorB , δκB κA and κAorB are defined as given by Grant (1970) and 1/c is the fine-structure constant related to the light speed c. Even though Ve is replaced by WP orQ (A, r) in the numerical iteration, there is little difficulty to solve the radial equations with the exception of a little more time. In this case the present local correlation potential is involved through V (r) = Vs (r) + Vc (r) before the non-local correlation potential is investigated. 3. Results and discussion In this section we first study three local-density exchange potentials on the DS theory, and the corresponding GOSs are compared with the theoretical result of Padma and Deshmukh (1992) and the experimental result of Takayanagi et al (1990). Furthermore, we consider relaxation in the DS calculation and continuously give the result calculated in the DF theory where the exchange potential is in the non-local type instead of the local type and the relaxation in the DF approach is then discussed. After the discussion of correlation in the DS calculation, we investigate relativistic effects. For the analyses of the physical effects which we are interested in, only the GOS of 4p6 (1 S)–4p5 (2 P3/2 )5s is shown in figures and tables in detail without repeat for the 4p6 (1 S)–4p5 (2 P1/2 )5s excitation unless the latter is involved. 3.1. Differences between exchange potentials The GOSs calculated through the Slater and Kohn–Sham potentials with α coefficients 1 and 23 , respectively, are shown by full and broken curves in figure 1. For comparison the experiment data of Takayanagi et al (1990) is also shown in the figure by triangles at 300 eV and diamonds at 500 eV. The experimental total accuracy is −15 to +10%. More clearly, the GOS at K 2 values of 0.01–0.3 is shown in figure 2 including the experimental error bars. The GOS in the case of Kohn–Sham potential at K 2 less than 0.2, particularly, the asymptotic plateau in the non-physical region defined by equation (5), is obviously closer to the experiment than results of the Slater potential. In our general codes if the exchange
Krypton 4p 5 (2 P3/2,1/2 )5s excitation
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Figure 1. The relativistic GOS calculations for the excitation of ground krypton to the 4p5 (2 P3/2 )5s state at the squared momentum transfer K 2 of 0.000 03–7.3. ——, Dirac–Fock– Slater (DS), α = 1; - - - -, DS, α = 23 ; · · · · · ·, DS, α = 0; — · —, DS on two self-consistentfield (SCF) potentials (TSCFP) for initial- and final-state orbitals, α = 1; , DS involving relaxation (RE) effect, α = 1; ×, Dirac–Fock (DF) on two SCF potentials (TSCFP); +, DF involving relaxation (RE) effect; – - –, DS (α = 1) involving correlation (C) effect; –•–, DS (α = 1) involving relaxation and correlation effects (RE+C); M, experiment (Exp.) of Takayanagi et al (1990) at 300 eV; ♦, experiment (Exp.) of Takayanagi et al (1990) at 500 eV; ; RC4 results of Padma and Deshmukh (1992).
◦
potential is cancelled out by setting α = 0, we reach the screened Coulomb potential approximation. In figure 1(a) a very large plateau indicated by a dotted curve in the nonphysical region appears and the GOS shows a steep change over 0.05 < K 2 < 0.3 which deviates distinctly from the experimental data either at 300 eV or 500 eV. In these calculations attention is also paid to the excitation energy w that is proportionally related to the GOS. For the three physical approximations at α = 1,
2 , 3
0,
(21)
the excitation energies obtained according to section 2.1 are listed in table 1, compared with the experimental value given by Trajmar et al (1981). It is noted that w produced by Slater potential (α = 1), different from the GOS plateau and first minimum, is closer to the experimental result than the value on the Kohn–Sham potentials (α = 23 ) although it is known that the Kohn–Sham approximation is a better one than the Slater approximation in the study of atomic structure. Furthermore, the parameters, GOS and w, by the three local-density exchange potentials as given in figures 1 and 2 and table 1 are compared with the relativistic data (small circles) of Padma and Deshmukh (1992) using the RC4 method. Their GOSs are too large. Tests to see if the exchange potential leads to the large deviation are made by choosing α from 0, 23 , 1 to 1.5, 2, 3, 4, 5, 6 (Zhang et al 1997). Although these α values except 0, 23 , 1
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Figure 2. As in figure 1 but K 2 is 0.01–0.3 and the experiment error of −15% to +10% is shown.
Table 1. The excitation energy (w) and the first minimum (K12 ) of the GOS calculated with different approximate approaches for 4p5 (3 P3/2 )5s excitation of krypton. DS: the Dirac–Fock– Slater theory; NR: the non-relativistic calculation; DF: the Dirac–Fock theory; RC4: the calculations of Padma and Deshmukh (1992) based on a developed relativistic local-densitypotential method of Vijayakumar et al (1989); Exp.: the experimental results of Takayanagi et al (1990); α: the Slater coefficient; RE: the relaxation effect; C: the correlation effect; RE + C: the mixture of the relaxation and correlation effects. DS α
1
2 3
w K12
0.337 1.25
0.225 1.12
a b
0 0.111 0.93
NR 1 RE 0.325 1.31
1 C 0.265 1.16
1 RE + C 0.255 1.22
DF
RC4
Exp.
RE 0.380 1.16
0.429a 1.21
0.369b
1 0.349 1.16
Including the Coulomb hole as given by Padma and Deshmukh (1992). The data from Trajmar et al (1981).
are unphysical, the SCF potential and SCF wavefunctions can be obtained mathematically. Final GOS results (not given in the figure 1 due to the unphysical reason) show that: (1) there is a lowest GOS plateau existing at α ≈ 1.5, and (2) at α > 1.5 the plateau rises as α increases, however, the fact that the plateau and GOS slope increase with α up to 6 is too slow to explain the large deviation between the experimental points and the data of Padma and Deshmukh. Here we choose the coefficient up to 6 because in their local-density potential the pure exchange part is near six times larger than the present Slater potential if the charge density of spin-up electrons is assumed to be that of spin-down electrons. Note that the Slater potential involves the electron self-interaction part and the pure exchange part. Therefore, the RC4 local-density potential method gives a less satisfactory description of
Krypton 4p 5 (2 P3/2,1/2 )5s excitation
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electron–atom excitation at small K 2 than its description of the corresponding target structure where the RC4 result approaches the DF calculation. 3.2. Relaxation effect 3.2.1. DS results. In the DS calculation one part of the relaxation effect or the influence under two SCF potentials (TSCFP) is estimated and shown by chain curve in figures 1 and 2 < K 2 < 1, the TSCFP consideration 2. As seen from the curve in the physical region Kmin gives a better GOS, with respect to the experiment, than the calculation without it (full curve), although the excitation energy is reduced and K12 is decreased as listed in table 1. 2 by nearly 30%. TSCFP raises the GOS at Kmin Furthermore, the overlap integral from the passive electrons in equation (7) and the orthogonalization between the initial- and final-state orbitals are introduced to reach a consistent consideration of the relaxation (RE). The corresponding GOS is shown with small squares in the figures, which is a little smaller than the experimental data but is still within the experimental error −15 to +10%. In comparison with the TSCFP consideration which raises the plateau, these two parts both decrease the GOS, the overlap integral does 2 . Additionally, the two by 1.7% at any K 2 and the orthogonalization does by 7% at Kmin parts give no influence on w in table 1 due to the present calculation on a SCF level. Hence, the TSCFP consideration gives the main contribution to the relaxation effect as expected through the matrix of equation (7). 3.2.2. DF results Based on our earlier works of the elastic scattering from helium (Shi et al 1997) as well as the structure calculation of the noble gases (Shi et al 1996), the nonlocal exchange terms are introduced and the initial single-electron DF orbital is obtained. Attention should be paid to the excited single-electron DF orbital for which the averageconfiguration-state approximation (Grant 1970) is utilized because the final 4p5 and 5s1 shells are open. The DF results (cross points) on two SCF potentials are somewhat larger than those of the corresponding DS calculation (chain curve). Contributions of the overlap integral and the orthogonalization to the relaxation effect are checked again and decrease the GOS by about 1.4% and 9%, respectively, which still supports the DS calculation. The final results (plus points) approach the experiment of Takayanagi et al (1990). From the GOS of 2 P3/2 and 2 P1/2 excitations, moreover, we can hardly distinguish which one of the DF and DS calculations is better due to the relatively large experimental error, particularly at 1.7◦ corresponding to K 2 ≈ 0.025 (at 300 eV). This is one of basic reasons why we choose the two separate approaches, the DS theory and the DF theory. However, we note that the excitation energy w produced by the DF calculation in table 1 is closer to the experimental value than that by the DS approach. Considering the exchange in a non-local type in the DF theory more exact than that of the local type in the DS theory, we think that the DF calculation is more suitable to describe the GOS at small K 2 (here < 1) than the DS calculation. 3.3. Correlation effect The local Coulomb correlation (C) has to be used to estimate the GOS influence utilizing the approximate local-density model as given in equation (17) before the non-local Coulomb correlation is investigated. The GOS results are shown by long–short-broken curve in figures 1 and 2, compared with the full curve of the initial DS calculation. It is interesting
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and important that the local correlation (full curve) mainly corrects the GOS a little at K 2 less than 0.2 and larger than 1, moves the first minimum (K12 in table 1) towards smaller K 2 a little and decreases w. It is noted that the 10% increment of the GOS plateau in the figures by the correlation supports the conclusion of Padma and Deshmukh (1992), and the change is smaller than that by the different local-density exchange potentials α = 1 and α = 23 as well as that by the relaxation effect. Thus the correlation influence, smaller than the experimental error of Takayanagi et al (1990), needs to be investigated with more accurate small-angle experiment of electron excitation and photon excitation experiment, both of which may verify zero-K 2 GOS limits. If both the relaxation and correlation effects (RE+C) are involved in the DS calculation, moreover, the dotted curve is shown in figures 1 and 2. This consideration, though producing small w (= 0.255), gives such a plateau closer to the experiment than the relaxation DF calculation (square points) that the non-local correlation reasonably improves the present DF results, increasing the plateau and decreasing w and K12 . Here we predict that theoretical K12 is about 1.1 based on data in table 1. 3.4. Relativistic effects Relativistic effects are investigated by comparing the relativistic GOS with the nonrelativistic GOS. The non-relativistic single-electron wavefunction is obtained by enlarging the light speed in the fine-structure constant 1c by a factor of 1000 in our relativistic codes. The approximate wavefunctions, as well as equations (13) and (14) for the non-relativistic GOS, are tested by results: (1) 0.051 the optical oscillator strength (OOS) for sodium 2p–3s, which agrees with the value 0.050 of Chen and Msezane (1994) using the Clementi and Roetti (1974) wavefunctions, and (2) the lithium-like GOS which is in agreement with the calculation by Qu et al (1995). In the relativistic study the α coefficient in equation (16) is chosen to be 23 in the Kohn–Sham local-density potential (there is no qualitative influence on this discussion if we choose 1 by Shi et al (1998)). The present non-relativistic GOS (full curve in figure 3) is then compared with the relativistic GOS sum (dotted curve) of two fine-structure transitions 2 P3/2 and 2 P1/2 . Neither the relativistic nor non-relativistic calculations involved the relaxation and correlation effects as estimated in sections 3.2 and 3.3, respectively. The experimental data of Takayanagi et al (1990) were measured almost at different K 2 values, so we cannot simply sum the points from their figures. The parabolic fitting is used for the 2 P3/2 data at 300 eV from their figures and an equation was obtained f3/2 = 0.161 611K 4 − 0.206 794K 2 + 0.069 2697
at 300 eV,
(22)
then, using fN R = f3/2 + f1/2 ,
(23) 2
2
(where NR denotes non-relativistic) the experimental data of P1/2 at a given K value and the corresponding fitted GOSs for 2 P3/2 are added to get the approximate value (open triangle) in figure 3. Similarly, the estimation at 500 eV is given by f3/2 = 0.106 177K 4 − 0.166 747K 2 + 0.062 7228
at 500 eV.
(24)
The sum values are represented by diamonds in figure 3. The figure shows that the sum data support the present relativistic calculation rather than the non-relativistic result. The relativistic effects do not increase the GOS but decrease the GOS. The conclusion about the influence of the relativistic effects on the GOS is furthermore verified by our calculations of the 2p excitation of sodium and agrees with the earlier results of Qu et al (1995). Therefore,
Krypton 4p 5 (2 P3/2,1/2 )5s excitation
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Figure 3. Comparison between the relativistic and non-relativistic GOS results. ——, nonrelativistic (NR) Kohn–Sham, α = 23 ; · · · · · ·, relativistic (R) Kohn–Sham, α = 23 ; M, from equations (22) and (23) on the experiment of Takayanagi et al (1990) at 300 eV; ♦, from equations (24) and (23) on the experiment of Takayanagi et al (1990) at 500 eV.
the relativistic effects cannot be thought of as one of the reasons to explain the too large GOS by the RC4 method in the small-angle region of K 2 6 0.8. Particularly, from equations (22) and (24) we know that the zero-K 2 limit of the experimental GOS for 2 P3/2 is within 0.063–0.069 in agreement with theoretical values 0.064 (squares), 0.069 (dotted curve) and 0.066 (plus symbols) in figures 1 and 2. 3.5. Conclusion We have studied the large influence of local and non-local exchange on the GOS results from the ground state 4p6 (1 S) to the excited state 4p5 (2 P3/2,1/2 )5s for krypton at small K 2 less than 1. The DS models give no consistent excitation energy and GOS result, although the DS calculation involving both the relaxation and correlation effects are quite close to the existing experiment. The DS and DF calculations clearly show the importance of the relaxation effects which raise the GOS at K 2 less than 0.01 about 20% and contributions of three parts, two different SCF potentials, overlap integrals and orthogonalization in the relaxation effect. The first part, giving the main contribution, increases the GOS over the present K 2 region while the second and third parts decrease the GOS. This estimation of the Coulomb correlation by the model potential shows that correlation influence up to 10% appears mainly at small K 2 less than 0.01. In this K 2 region the relaxation effect is stronger than the correlation effect, which may be verified with a smallangle electron-excitation experiment and a photon-excitation experiment. It is also thought that the relaxation and correlation effects may be very important to explain the 2p–3s inner-
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shell excitation of sodium (Bielschowsky et al 1991). A paper on this is in preparation. Based on all present physics considerations, the DF calculation involving the relaxation or more reasonably, the DF calculation involving both the relaxation and non-local correlation effect is recommended to obtain a consistent description of the GOS plateau, the first minimum and the excitation energy as well as the target structure. For the outer-shell excitation of the heavy and closed atom krypton, the relativistic effects decrease the GOS near 40% in the non-physical region. It may be explained mainly by the localization of the initial electron orbital which attempts to retain the closed atomic core in the electron–atom system. It is noted in this DF calculation that the average-configuration-state approximation for the excited electron orbital outside an open core like 4p2 4p3 5s1 or 4p1 4p4 5s1 is suitable to the GOS description. Further calculations for the 5p excitation of xenon support the conclusion. An exception appearing at 2s, 3s and 4s excitation of lithium, sodium and copper, respectively, as well as the application of the approximation to the 2p inner-shell electron excitation of sodium will be given later together with discussion in detail. Acknowledgments Financial support for this work was provided in part by the National Natural Science Foundation of China and the National Education Committee of China, The People’s Republic of China, and also by the Center for Molecular Science, Korea. QS would like to thank Professor Yang-Soo Chung. References Bartschat K and Madison D H 1987 J. Phys. B: At. Mol. Phys. 20 5839 Bielschowsky C E, Lucas C A and de Souza G G B 1991 Phys. Rev. A 43 5975 Chen Z and Msezane A Z 1994 Phys. Rev. A 50 3517 Clementi E and Roetti C 1974 At. Data Nucl. Data Tables 14 177 Cowan Robert D. 1980 The Theory of Atomic Structure and Spectra (Berkeley, CA: University of California Press) Cubric D, Mercer D, Channing J M, Read F H and King G C 1997 Proc. 20th Int. Conf. on the Physics of Electronic and Atomic Collisions (Vienna, Austria) ed F Mumayr et al Grant I P 1970 Adv. Phys. 19 747 Herman F and Skillman S 1963 Atomic Structure Calculations (Englewood Cliffs, NJ: Prentice-Hall) Inokuti M 1971 Rev. Mod. Phys. 43 297 Latter R 1955 Phys. Rev. 99 510 Manson S T 1972 Phys. Rev. A 5 668 Meneses G D, da Paix˜ao and Padial N T 1985 Phys. Rev. A 32 156 Msezane A Z and Bessis D 1997 J. Phys. B: At. Mol. Opt. Phys. 30 445 Padma R and Deshmukh P C 1992 Phys. Rev. A 46 2513 Panajotovi´c R, Pej´cev V, Konstantinovi´c M, Filipovi´c D, Boˇcvarski V and Marinkovi´c B 1993 J. Phys. B: At. Mol. Opt. Phys. 26 1005 Qu Yizi, Tong Xiao-Min and Li Jia-Ming 1995 Acta Phys. Sin. 44 1720 Shi Qicun 1996 Thesis (University of Science and Technology of China, Hefei) Shi Qicun, Feng Renfei, Ji Qing, Wu Sulan, Zhong Zhiping, Xu Kezun and Li Jia-Ming 1997 J. Phys. B: At. Mol. Opt. Phys. 30 5479 Shi Qi-Cun, Xu Ke-Zun, Chen Zhang-Jin, Cho Hyuck and Li Jia-Ming 1998 Phys. Rev. A 57 4980BR Suzuki T Y, Suzuki H, Currell F J and Ohtani S 1996 Phys. Rev. A 53 4138 Takayanagi T, Li G P, Wakiya K, Suzuki H, Ajiro T, Inaba T, Kano S S and Takuma H 1990 Phys. Rev. A 41 5948 Theodosiou C E 1987 Phys. Rev. A 36 3138 Theodosiou C E and Fielder W Jr 1982 J. Phys. B: At. Mol. Phys. 15 4113 Trajmar S, Srivastava S K, Tanaka H and Nishimura H 1981 Phys. Rev. A 23 2167
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