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Close coupling approach in optically allowed atomic transitions.

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Ann. Phys. (Leipzig) 16, No. 12, 783 – 790 (2007) / DOI 10.1002/andp.200610260
Close coupling approach in optically allowed atomic transitions
Smail Bougouffa∗
Department of Physics, Faculty of Science, Taibah University, P.O. Box 344, Madina, Saudi Arabia
Received 12 June 2006, revised 23 March 2007, accepted 25 June 2007 by G. Röpke
Published online 18 September 2007
Key words Electron-atom collisions, atomic transitions, close coupling approximation.
PACS 34.30, 34.80.Bm, 34.50.Fa, 34.80.Dp, 34.80.Nz
In general the calculations of the cross sections in atomic collisions theory need a treatment of a system
of coupled integro-differential equations. We perform a numerical technique for calculations of electrons
scattering with sodium atom. The cross sections are evaluated in the close coupling approach, where the
problem is formulated in three coupling channel approximation. It is found that the three-channel problem
results are typically in good agreement with experiment and two-channel calculations for intermediate energy range.The difference in the other range of energy can be assigned to the number of the used set of
coupled differential equations in the 32 S −→ 32 P transition of sodium.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
The problem of electron-sodium scattering has received considerable interest and some controversy over
the years. Measurements and calculations of cross sections for both elastic and inelastic scattering were
performed from 1972 to the present time [1–5]. During subsequent years, however, experimental studies
of electron-alkali metal atom scattering moved in new directions, leading to new demands on theory. The
most striking change has been an increased emphasis on collisions involving many effects [2]. Most of
the experimental work was performed on elastic scattering and excitation of the 32 P state and the existing
measurements are typically not in good accord with theory [6, 7]. From the theoretical viewpoint, several
methods have been reported [8, 9], and their results are typically in good agreement with measurements
near the first few excitation threshold or in the intermediate energy range. More recent calculations have
confirmed the validity of the non-relativistic approximation to the electron-sodium scattering problem by
comparing the results of a four state non-relativistic and seven state R-matrix (close coupling) calculations
[10]. Appreciable differences in the results were only found very close to the low-lying fine structure
excitation threshold, i.e., at energies too low for the present case of interest.
In previous papers [4, 5], we have reported calculations of the cross sections for the 32 S −→ 32 P
transition of Na by electron impact. Owing to the complexity of the treatment of the set of coupled differential equations, several simplifying approximations were used. The problem was formulated in the
two-channel approximation, i.e., l −→ l − 1. It has been found that the partial cross sections in the second case (Δl = +1) are very small for l > 2 and the effect of the third channel can be ignored. To
improve these calculations, we continue to use the Numerov numerical procedure [11–13] to solve the
coupled differential equations, and the third channel will be incorporated. The problem will be treated in
the three-channel approximation, the potentials are determined by taking the wave functions of the valence
electron of the sodium atom as hydrogen-like wave functions with an effective charge adjusted to fit the
experimental 3s −→ 3p line strength [4, 5].
This paper is organized as follows. In Sect. 2 we shall outline the key steps of the theory of electronatom excitation in the close-coupling scheme [3–5] which are essential in the ensuing discussion. In Sect. 3,
we shall focus on one class of electron-atom collisions which induce transitions of the type n , l, m −→
∗
E-mail: sbougouffa@hotmail.com, Phone: + 966 50 7670 489, Fax: + 966 484 61172
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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S. Bougouffa: Close coupling approach in atomic transitions
n, l ± 1, m. In this class, it is justified that the interaction potentials are calculated using the wave functions
of the one-electron orbital of the excited electron. In Sect. 4 we discuss the results obtained in this context,
and conclude in Sect. 5 with some comments on the possibility of extension of this work.
2 The coupled-channel formalism
The present work is focused on the coupled equations obtained by expanding the e-Na wave function in
terms of channel functions that is complete in the bound-electron coordinates, which we denote collectively
by re and the angular coordinates of the projectile, r̂. In this paper, we ignore the exchange of the incident
electron with the atomic electrons and the spin of electrons. Substituting this expansion into the (Ne + 1)electron Schrödinger equation and projecting out the channel functions yield the set of coupled differential
radial equations. The coupled system can be solved for the scattering matrix using the Numerov numerical
procedure and hence the cross sections are deduced.
The channel functions are eigenfunctions of the total orbital angular momentum L which is a constant
of the motion of the e-Na system. For a given channel γ = (α, l) corresponding to an atomic state α =
(Γ, la , mla ) for a particular electronic configuration Γ and projectile orbital angular momentum (i.e., partial
wave) l, where la and mla are the the quantum numbers of the Ne electron states of the atom target,
we build the channel functions ΦL
α l (re , r̂) from products of target wave functions φα (re ) and spherical
harmonics Ylml (r̂), where l and ml are the quantum numbers of the incident electron. these products can
be combined to construct the eigenfunctions of the constant of the motion. Therefore, we develop the
system wave function for a particular initial channel γ0 in this set, thus introducing the radial channel
L
(r) as
scattering functions Fγ,γ0
1 L
L
ΨL
Φαl (re , r̂)Fαl,α
(r).
(1)
γ0 ,E (re , r) =
0 l0
r
α,l
Not explicitly included in this expansion are (Ne + 1)-electron configurations constructed completely from
bound orbitals. Such configurations are sometimes used to represent resonances and bound-free correlation
effects [3, 14].
The number of Na target states required in expansion (1) to reach convergence depends in practice on
the scattering quantity being converged and on whatever independent physical variables governing this
quantity (e.g., the partial cross sections are functions of the orbital angular momentum and energy). On the
other hand, to render the solution of the coupled differential equations tractable some approximations must
be introduced and the number of terms included in the expansion of (1) should be limited to a manageable
size.
With a given selection of the target states in the close coupling expansion and for given values of the
total quantum numbers L and ML , we set up the scattering equations which are independent of ML
2
d
lγ (lγ + 1)
2
L
L
−
+
k
Uγγ
(2)
(r)Fγ γ (r).
γ Fγ,γ0 (r) =
0
2
2
dr
r
γ
The values of the angular momentum of the partial waves (lγ ) range from L+la to |L−la |. The nondiagonal
interaction potential Uγγ0 between a channel of even value of (la +l) and one with odd (la +l) vanishes [16].
Unlike Trail [3] and many others who have studied the e-atom interaction using the R-matrix method,
we have solved the coupled differential equations for e-Na atom scattering which induced transitions of
the type l −→ l ± 1, using a numerical technique based on the Numerov method [11–13] which has been
described formerly in some details [20, 21].
Far from the target, the radial functions satisfy the asymptotic conditions that identify the reactance
matrix R for scattering from initial channel γ0 to final channel γ, i.e.,
1
1
1
R
L
Fγ,γ0 ∼
sin(k0 r − l0 π)δγγ0 + cos(kγ r − lπ)Rγ,γ0 ,
(3)
2
2
kγ
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Leipzig) 16, No. 12 (2007)
785
where the final-state wave number kγ is related to the incident energy k02 , initial and final target energies,
and total system energy E by conservation of energy
E = ΓL +
1 2
1
k = Γ0 L0 + k02 .
2 γ
2
(4)
From the R-matrix we calculate the transition matrix
T=−
2iR
,
1 − iR
(5)
where T-matrix is diagonal with respect to the total quantum number L, and independent of the projection
of the total orbital angular momentum on the quantization axis. The partial cross sections between the
states γ, γ0 can be calculated from the T matrix.
σγL0 ,γ =
1
π
L
(2L + 1) | Tγ,γ
|2 ,
2
0
kγ0 (2lγ0 + 1)
(6)
l,l0
where L0 is the angular momentum of the target in the entrance channel γ0 .
3 Approximate forms of the interaction potential
L
The equations we must solve to obtain the R-matrix from the function Fαl,αl
(r) are coupled differential
0
in nature owing to the ignoring exchange electrons effect. The coupling terms are the potential matrix
elements that relate different channel functions and are given by
Ne
L
1
L
L Z
Φ Uγ,γ (r) = Φγ − +
| ri − r | γ
r
i=1
la lL
la l L
∗
=2
Cm
C
(r ) Vnla ma ,n la ma (r) Yl m (r) dr,
(7)
Ylm
ma m M
a mM
ma mma m
where the V’s functions are given by
Vnla ma ,n la ma (r) =
Φ∗nla ma (r1 , r2 , ...)
1
Z − +
r
|r − ri |
i
Φn la ma (r1 , r2 , ...) dr1 dr2 ... .
(8)
In this work we shall be concerned with the special category of electron-atom collision which requires:
i) strong coupling
ii) long-range interaction potential
iii) near resonance.
The optical allowed transitions 3s → 3p in sodium atom are good example of these assumptions.
Indeed, it has been shown by Seaton [15] that certain collision-induced transitions of the type (n , l, m −→
n, l ± 1, m) point up strong coupling with long-range interactions. In this case, we confine ourselves to
matrix elements
the single excited configuration like 1s2 2s2 2p6 nla for sodium. The
interaction potential
of Eq. 8 are to be evaluated between the φ1s ...φ2p φnla and φ1s ...φ2p φn la configurations. Though
in principle the core state functions in the two different configurations are dissimilar as denoted by φi
and φi (i = 1s, 2s, 2p), the numerical values [16] of the overlap integrals of the corresponding orbitals
< φi | φi > are very close to unity, and the integrals < φi | φj > with i = j are no greater than the order
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
786
S. Bougouffa: Close coupling approach in atomic transitions
of 10−2 [13]. Therefore in computing the nondiagonal elements of the interaction potential, we neglect
the difference between the set of φi and φi . On the other hand, since the core electrons extend only over
a small domain near the nucleus, we ignore the effect of the structure of the atomic core. Thus, the V’s
functions may be reduced to the following forms
1 1
φj (r1 ) ,
(9)
Vi,j (r) = − δij + φi (r1 ) r
|r − r1 | which are the expressions for one-electron atom.
On making multipole expansions of the potentials we have (in atomic units),
δ(γ, γ ) L
+
Uγγ (r) = 2 −
fλ (la l, la l ; L)yλ (Pnla , Pn la ) ,
r
(10)
λ
where la , la refer to the atomic electron and l, l to the scattered electron, Pnla are the bound radial functions
and the integrals yλ (A, B | r) are defined by:
r
∞
1
yλ (A, B | r) = λ+1
A∗ (r2 )B(r2 )r2λ dr2 + rλ
A∗ (r2 )B(r2 )r2−λ−1 dr2 .
(11)
r
0
r
The coefficients fλ are generalization of similar coefficients employed in atomic structure problems. The
explicit expansions have been tabulated in [17] for all transitions in which the atomic electron is initially
and finally in an s, p, d orbitals (i.e., la , la = 0, 1, 2). It should be noted that
fλ (la l; la l , L) = fλ (la l ; la l, L),
(12)
f0 (la l; la l , L) = δ(la l, la l ),
(13)
and that, due to conservation of parity, the atomic coefficients fλ are zero unless
(−1)la +l = (−1)la +l .
(14)
It can be seen from the parity conservation conditions on the angular momentum numbers that these occur
only for l = l ± 1 which is the condition for the transitions to be optical allowed. For this type of transition the system (2) may be reduced to three coupled differential equations corresponding to three channels
(ns, l), (np, l − 1) and (np, l + 1) which can be denoted by 1, 2, and 3 successively. The interaction potentials are calculated, using the hydrogen-like functions for the valence of sodium atom, with an effective
charge altered to fit the experimental 3s −→ 3p line strength. In Fig. 1, the graphical presentations of these
potentials are given for l = 4 and Z = 2.92.
We shall use these potentials as inputs in the coupled differential equations, which can be solved using
Numerov numerical procedure described previously [4, 5, 11–13, 18]. This proposed iteration code is not
restricted to this type of problem and can be extended to more complicated atomic transitions. On the other
hand, this technique does not required a considerable amount of computational work and converges rapidly
even though the strong coupling interactions are involved.
The solutions are initiated near the origin by a Frobenius kind expansion and developed out from the
origin using a Numerov method [11–13]. With a series of transformations, the solutions are matched to
the asymptotic form of Eq. (2) in order to obtain the R-matrix elements. Since some of the nondiagonal
members of Uγμ (r) have a long-range like r−2 , we have found it necessary in all cases to carry the numerical integration of the differential equations to a distance 250a0, where a0 is the Bohr radius, before
matching the solutions to the suitable asymptotic forms. When the number of channels is reduced, the time
consuming for computations decreases. In this case and to compare our results with previous work, we
consider the special transitions 32 S → 32 P in sodium where the problem is formulated in three-channel
states.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Leipzig) 16, No. 12 (2007)
787
Fig. 1 The potential interactions, in atomic units, for l = 4 calculated using hydrogen-like wave-functions with an
effective charge Z = 2.92. (a) The coupling term U12 . (b) The coupling term U23 . (c) The direct term U33 .
Fig. 2 Partial cross section (in units of πa20 ) versus the partial wave order for E = 10.52, 32.612,
57.7 eV in the 3-channel state.
4 Calculations and discussions
The problem of excitation of the 32 P state of sodium has been treated by several authors [2, 3, 19–21].
Calculations have been made for Na(3s, 3p), which is an excellent example of a near resonance and strong
coupling, with an energy separation ΔE = 2.104 eV and a transition line strength s2 = 19, as in [4].
The partial cross sections (in units of πa20 ) versus the partial wave order for E = 10.52, 32.612, 54.7
eV are presented in Fig. 2. As shown in this figure, the major contribution to the low energy is due to few
intermediate values of l, while for the intermediate and high energies the contribution is more uniformly
distributed among several different values [4, 5].
Because of the strong coupling between the 3s and 3p state produced by the interaction between the
incident electron and sodium atom, the use of the Born approximation results in a very serious overestimation of the excited cross sections. To correct for this overestimation, Seaton [15] has given modified
versions of the Born approximation (B’II) which preserve the unitary property of the S-matrix and satisfy
the conservation rule. Lane and Lin [19] calculated the 3s −→ 3p transition in Na by the resonance distortion method (RD), which consists of solving the limiting exact resonance problem as the zeroth-order
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
788
S. Bougouffa: Close coupling approach in atomic transitions
Fig. 3 Comparison of the partial cross sections (in units of πa20 ) in two- and three-channel calculations
by our technique (2-ch, 3-ch) with those of the resonance distortion approximation (RD), modified Bethe
method (B’II) and 3-state calculations by Kroff (3-st Kroff).
Table 1 Comparison of the partial cross sections (in units of πa20 ) in two- and three-channel calculations
by our technique (2-ch, 3-ch) with those of the resonance distortion (RD), modified Bethe (B’II) and 3-state
Kroff [16] calculations for two different values of energy, E = 10.52 eV and 13.144 eV.
l
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2-ch
0.00
0.02
4.46
7.11
6.65
5.94
5.15
4.35
3.62
2.97
2.43
1.97
1.59
1.28
1.23
0.83
E = 10.520 eV
3-ch Korff RD
1.10
0.65 0.93
1.16
0.33
2.58
0.76 4.31
6.99
4.01 7.58
8.33
5.46 8.25
7.91
4.75 7.44
6.78
3.85 6.29
5.50
3.26 5.18
4.70
2.84 4.21
3.90
2.47 3.39
2.82
2.10 2.72
2.24
1.75 2.18
1.82
1.46 1.74
1.46
1.19 1.39
1.12
0.98 1.11
0.891
0.89
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
B’II
2.62
6.41
8.37
8.52
7.73
6.61
5.47
4.46
3.59
2.88
2.30
1.83
1.46
1.16
0.92
2-ch
0.00
0.01
0.41
1.44
2.46
2.84
2.86
2.73
2.54
2.33
2.13
1.93
1.75
1.58
1.44
1.30
E = 23.144 eV
3-ch Korff RD
0.28 0.18
0.21 0.08
0.39 0.10 2.04
1.19 0.69 3.49
2.25 1.46 4.02
2.94 2.01 3.93
3.21 2.27 3.65
3.16 2.29 3.32
2.94 2.18 2.98
2.30 2.03 2.68
2.42 1.87 2.40
2.19 1.71 2.15
1.98 1.56 1.92
1.77 1.43 1.72
1.59 1.30 1.55
1.41 1.19 1.39
B’II
0.35
1.24
2.93
3.81
4.02
3.89
3.62
3.30
2.98
2.69
2.41
2.16
1.94
1.74
1.57
1.41
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Ann. Phys. (Leipzig) 16, No. 12 (2007)
789
approximation and using this solution to obtain the first-order solution by an iteration procedure. Barnes
et al. [21] performed two-state 3s → 3p close coupling calculations neglecting the exchange effects. In
the two-state close coupling calculations of Karule and Peterkop [22] which cover the energy region from
the threshold value to 5 eV, allowance was made for the exchange effect. Korff et al. [16] have calculated
the excited cross sections of the 32 P by taking three different sets of atomic states in the close-coupling
expansion, (i.e., 2-states, 3-states and 7-states). Bray [2] performed convergent close-coupling calculations
of electron scattering on atomic sodium. Trail et al [3] have applied the nonperturbative coupled-channel
R-matrix method in carefully converged calculations to generate a comprehensive data base of accurate
scattering quantities for studying some phenomena in e-Na collisions. Recently, we have reconsidered the
two-state close coupling problem with two channel coupling using a new theorem on the separation of
the coupled differential equations [23]. Application of this method has been made to schematic model
(2-channel problem) with an isotropic inverse-square interaction potential and to the problem of electronatom collisions with ns −→ np transition. To introduce the truth forms of the interaction potentials we
have performed an iteration techniques [4, 5], which were applied to the two-state close-coupling problem
in 2-channel case, i.e., l −→ l − 1 transition in Na. Compared to the results of the modified approximation,
all the more refined calculations cited above yield considerable small cross sections of the 32 P state. We
have recalculate the excitation cross sections of the 32 P state using the previous iteration method [4, 5] to
the two-state close coupling problem with 3-channels (i.e., l −→ l ± 1). The results of the partial-wave
cross sections (in units of πa20 ) of the 2- and 3-channel are summarized in Fig. 3 for two different values of
energy E = 7.364, and 16.832 eV and in Table 1 for two other values of energy E = 10.52, and 13.144 eV,
with those of the resonance distortion (RD), modified Bethe (B’II) and 3-state Kroff [16] calculations. Our
results in the 2- and 3-channel approximation are smaller then those obtained by RD and B’II methods over
the entire range of partial wave order l and they are about 35% greater than those obtained by Kroff [16] for
low partial wave order, while are about 15% for high values of l. The 32 P cross sections become greater
when the third channel is added (i.e., l −→ l + 1) transition to the manifold of the scattering equations. The
interaction potential U13 (r) which corresponds to the case (Δl = +1) is comparable in magnitude with the
U12 (r) which corresponds to the case (Δl = −1) as shown in Fig. 1, thus the transition terms (Δl = +1)
channeled the (l),(l+1) coupling state, resulting in an increase of the 32 S → 32 P excitation cross sections.
On the other hand, the 32 P cross sections become smaller when the 3d atomic state is added to the manifold of the scattering equations [16]. The indirect coupling via the 3d state in the Korff calculations and
the ignoring of the third channel in our previous calculations are seen to be effective in decreasing the 32 P
partial-wave cross sections over the entire range of partial wave order, while the 3-channel calculations are
smaller than those of the resonance distortion and modified Bethe results.
A comparison of our calculations with experiments and with other theoretical calculations is illustrated
in Fig. 4. Measurements of the excitation function of the 32 S −→ 32 P transition have been reported by Zapesochny (Z) [24], who subsequently derived an absolute excitation function of the 32 P state (with cascade
correlation). The cross sections given by Enemark and Gallagher (EG) [6] were obtained by normalizing
their optical measurements to the Born-approximation value at 1000 eV. Our results in the 2- and 3-channel
approximation are considerably higher than those of Zapesochnyi and Enemark and Gallagher. Our calculations (P2CH, P3CH) are in good agreement with the mearurments reported by Christoph (C) [25].
Further experimental work on absolute excitation cross sections should be particularly valuable for testing
the theory.
5 Conclusions
It has been demonstrated within close coupling calculations that the contribution of the 3-channel to the
excitation cross sections of the 32 P state in the low and intermediate energy range is quite significant.
In this energy range the electron exchange effect must be taken into account. On the other hand, in the
high energy range it is sufficient to consider just the initial and final states, and the coupling through the
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
790
S. Bougouffa: Close coupling approach in atomic transitions
Fig. 4 Excitation cross sections (in units of a20 ) as
function of the incident electron energy compered
with other calculations and experiment. B’II: Bethe
approximation, RD: resonance distortion approximation, K: Korff calculations, EG: Enemark and
Gallagher measurements, Z: Zapesochnyi measurements, C: Christopher measurements, and P2CH
and P3CH: our previous and present calculations.
intermediate states can be neglected [4]. In the intermediate as well as the high energy range, the method
of close coupling in the 2- or 3-channel approximation proves to be a useful tool for calculating excitation
cross sections of dipole-allowed states.
References
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[19] N. F. Lane and C. C. Lin, Phys. Rev. 133, A947 (1964).
[20] A. Z. Msezane, Phys. Rev. A 37, 1787 (1988).
[21] L. L. Barnes, N. F. Lane, and C. C. Lin, Phys. Rev. 137, A388 (1965).
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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