D. V. Fursa

English

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Electron scattering from the ground state of
mercury
D. V. Fursa∗ and I. Bray†
∗School of Chemistry, Physics and Earth Sciences, Flinders University, GPO Box 2100, Adelaide
5001, Australia
†Centre for Atomic, Molecular, and Surface Physics, Murdoch University, Perth 6150 Australia
Abstract. We give a short review of e-Hg scattering and present some results of our recent close-
coupling calculations. We look at the challenges facing theorists in the calculation of elastic scatter-
ing and excitation of the 6s6p 1,3P1 levels.
INTRODUCTION
In the field of electron-atom scattering the e-Hg collision systems is one of more
interesting and important cases. Mercury is an important constituent of various industrial
plasmas where electron collisions play crucial role in many technologically important
processes, as, for example, in fluorescent and high intensity discharge lamps [1, 2].
Additionally, as a heavy atom, mercury is a useful target in studies of relativistic effects
in electron-atom scattering (see Andersen et al. [3] for a recent review).
In this report we would like to give a short review of the present status of e-Hg
scattering and present some of our recent results for this scattering system concentrating
on elastic scattering and excitation of the 6s6p 1,3P1 levels.
THEORETICAL METHODS
Electron scattering form mercury presents a serious challenge to theorists. Comparing
to light atoms, where good agreement between theory and experiment was established,
there are a number of processes which make the theoretical modelling substantially
more difficult. These difficulties originate from the fact that mercury is a heavy atom
of nuclear charge Z = 80. For such an atomic system it is important to to take into
account relativistic effects, as well as to model electron-electron correlations between
two active outer electrons and the rest of the electrons.
Relativistic effects
There are two different aspects to relativistic effects, the description of the target states
and the scattered electron. While both are important and observed in experiments, an ad-
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equate description of the mercury wave functions seems to be the most important. The
two most prominent relativistic effects relevant to the accurate description of elastic scat-
tering and excitation of the 6s6p 1,3P1 levels of mercury are the relativistic contraction
of the orbitals which leads to an increase in the ionization energy (experimental value
is 10.43 eV) by about 1.5 eV, and the singlet-triplet mixing between the nonrelativistic
6s6p 1P1 and 6s6p
3P1 configurations (the mixing coefficient is 0.171).
The most consistent way to account for the relativistic nature of mercury is to use
methods based on the Dirac equation. This approach was employed for the study of
elastic scattering by Walker [4], Sin Fai Lam [5], Haberland and Fritsche [6], McEachran
and Stauffer [7], Sienkiewicz [8], Sienkiewicz [9], and McEachran and Elford [10].
Excitation of the 6s6p levels has been studied by Srivastava et al. [11] using distorted
wave approximation. Wijesundera et al. [12] have applied a five-state fully relativistic
Dirac R-matrix method to study low energy e-Hg scattering and have provided elastic
and excitation (6s6p 3P0,1,2) cross sections.
Breit-Pauli approximation has been used extensively in e-Hg scattering calculations.
Bartschat and Madison [13] have studied excitations of the 6s6p 1,3P1 levels using
a distorted-wave Born approximation (DWA) and Scott et al. [14], Bartschat et al.
[15] have used five-state R-matrix method (RM(5)) to calculate elastic scattering and
excitation. In all these calculations only a one-body spin-orbit term was used to model
relativistic effects.
The present calculations use the CCC method [16] to study e-Hg scattering. The
relativistic contraction of the Hg orbitals has been modelled by means of a short-ranged
potential. Singlet-triplet mixing in the 6s6p 1,3P1 manifold has been accounted for in the
semi-relativistic approximation by combining nonrelativistic amplitudes for the 6s6p 1P1
and 6s6p 3P1 states with appropriate mixing coefficients [17].
Electron correlations
In most of the calculations the mercury atom is modelled as two active electrons above
a frozen inert Hg++ core of [Xe]4 f 145d10. Within this approximation it is important to
include configurations with the “inner” electron being described by a combination of
the 6s, 6p and 6d orbitals. The “outer” electron is expanded using sufficiently many
Laguerre based orbitals. Such an approach leads to a good ground state and excited
levels based on the [Xe]4 f 145d10 core. However, the mercury discrete spectrum contains
a number of states corresponding to the excitation out of 5d10 shell. The importance of
such core excitations can be appreciated by noting that about one-third of the mercury
ground state static dipole polarizability (αd = 34.4 a.u. [18]) comes from the 5d
96s2nl
manifold. Opening of the 5d10 shell also leads to a reduction of the optical oscillator
strength f for the 6s6p 1P1 level by nearly a factor of two [12]. An error in the value of
α d or f can substantially affect scattering calculations, as will be discussed later.
Only the Dirac R-matrix method of Wijesundera et al. [12] attempted to take into
account valence-core correlations directly by allowing excitation out of the 5d10 shell.
Most of the theoretical methods applied to e-Hg scattering take no account of core
excitations. Our calculations use two- and one-electron polarisation potentials to model
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valence-core correlations, which leads to a good agreement for the 6s6p 1P1 level optical
oscillator strength, but the error in the value of α d remains.
Channel coupling
There are very few calculations of e-Hg scattering which take into account channel-
coupling. These are the Dirac R-matrix method of Wijesundera et al. [12] and the Breit-
Pauli R-matrix method of Scott et al. [14], Bartschat et al. [15]. Both calculations have
included only five low lying states of Hg. Such calculations are expected to be accurate
only at low energies where states not included in the calculations are closed. The present
CCC calculations include 54 states (nine 1S, eight 3S, 1,3De, 1,3Po, two 3Pe and one
1Pe, 1,3Do) comprising both the discrete spectrum states and the positive-energy states
modelling the target continuum.
RDD
Kaussen et al.
S
scattering angle θ (deg)
1501209060300
1
0.5
0
-0.5
-1
RDD
CCC
Holtkamp et al.
Panajotovic et al.
25 eV
scattering angle θ (deg)
C
ro
ss
se
ct
io
n
(a
.u
.)
1501209060300
100
10
1
0.1
0.01
FIGURE 1. Differential cross section and Sherman function S for elastic electron scattering on the
ground state of mercury at 25 eV. The present CCC calculations are described in the text. The RDD
calculations are due to McEachran and Elford [10]. Measurements (DCS) are due to Panajotović et al.
[19] and Holtkamp et al. [20], and measurements of Sherman functions S are due to Kaussen et al. [21].
Elastic scattering
In Fig. 1 we present comparison between theory and experiment for e-Hg elastic
scattering at 25 eV. Differential cross section (DCS) measurements of Panajotović et al.
[19] and Holtkamp et al. [20] are found to be in good agreement with present calculations
and relativistic dynamic distortion (RDD) calculations of McEachran and Elford [10] at
all scattering angles except for forward scattering. The discrepancy at forward scattering
angles is related to an underestimation of the static dipole polarizability α d in the present
CCC calculations as well as in the RDD calculations.
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Elastic e-Hg scattering can be used to test the relativistic effects associated with
scattered electrons. Spin polarisation (Sherman function S) of the unpolarised electrons
scattered elastically from unpolarised mercury has been studied by Kaussen et al. [21]
and presented in Fig. 1 together with the results of the RDD calculation. The agreement
between the experiment and RDD is very good. Note that this is a pure relativistic effect,
a nonrelativistic calculation, such as CCC, yields exactly zero spin polarisation in elastic
e-Hg scattering.
Excitation of the 6s6p 1,3P1 levels
Excitation of the 6s6p 1,3P1 levels in mercury are strongly affected by electron corre-
lations and relativistic effects. In Fig. 2 we present the DCS for excitation of the 6s6p 1P1
and 6s6p 3P1 levels at 15 eV. For the 6s6p
1P1 DCS there is good agreement between the
experimental data of Zubek et al. [22], Panajotović et al. [19] and Peitzmann and Kessler
[23]. Present CCC calculations are in good agreement with the experimental data, how-
ever results of RDWA [11] and DWA [13] calculations are substantially larger. This is
apparently related to the larger value of the 6s6p 1P1 level optical oscillator strength in
the RDWA and DWA calculations.
DWA
RDWA
CCC
6s6p3P1
scattering angle θ (deg)
1501209060300
Peitzmann and Kessler
Panajotovic et al.
Zubek et al.
6s6p1P1
scattering angle θ (deg)
C
ro
ss
se
ct
io
n
(a
.u
.)
1501209060300
100
10
1
0.1
0.01
FIGURE 2. Differential cross sections for electron-impact excitation of the 6s6p 1P1 and 6s6p
3P1
states from the ground state of mercury. The present CCC calculations are described in the text, RDWA
calculations are due to Srivastava et al. [11], DWA calculations are due to Bartschat and Madison [13].
Measurements are due to Zubek et al. [22], Panajotović et al. [19] and Peitzmann and Kessler [23].
Singlet-triplet mixing in the 6s6p 1,3P1 manifold is important for an accurate descrip-
tion of the excitation of the 6s6p 3P1 level. In our calculations this is achieved by com-
bining the nonrelativistic amplitudes for the 6s6p 3P1 and 6s6p
1P1 levels with mixing
coefficients 0.985 and -0.171, respectively as given by Lurio [24]. Note that for calcu-
lations of the DCS this is equivalent to combining corresponding nonrelativistic DCS
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multiplied by the square of the mixing coefficients, singlet-triplet interference terms dis-
appear. Comparing present CCC results with the experimental data of Zubek et al. [22]
we find mixed agreement that appears worst at the forward scattering angles. This is
puzzling because in this region the 6s6p 1P1 DCS dominates, and good agreement with
experiment for the 6s6p 1P1 DCS suggests a wrong choice of the mixing coefficients or
importance of relativistic effects for the continuum electron which are not accounted for
in our method and lead to a nonzero interference term between singlet and triplet ampli-
tudes resulting in a larger cross section. The RDWA and DWA results are substantially
above the experiment at most scattering angles leaving our understanding of this DCS
incomplete.
Finally, in Fig. 3 we look at recent spin-resolved and spin-averaged measurements of
orientation parameter J+
⊥
by Herting et al. [25], and compare them with the present CCC
results, RM(5) and RDWA calculations [25]. In the nonrelativistic theory the spin-up
and spin-down values for J+
⊥
are the same. The experimental results clearly demonstrate
substantial differences between observed spin-up and spin-down J+
⊥
values. Good agree-
ment between experiment and our semi-relativistic results indicate that singlet-triplet
mixing is the major relativistic effect for this transition. Note, that in this case the in-
terference between the nonrelativistic singlet and triplet amplitudes is crucial to achieve
agreement with experimental data.
spin down
scattering angle θ (deg)
60300
RDWA
RM(5)
spin up
scattering angle θ (deg)
60300
CCC
spin averaged
scattering angle θ (deg)
60300
1
0.5
0
-0.5
-1
FIGURE 3. Spin-averaged and spin-resolved orientation parameter J+
⊥
for electron-impact excitation of
the 6s6p 3P1 state from the ground state of mercury at 15 eV. The present CCC calculations are described
in the text. The RM(5), RDWA and the experiment are due to Herting et al. [25].
CONCLUSIONS
Though many e-Hg scattering processes have been adequately described by available
theoretical methods, there are still many processes where theoretical methods are yet
to achieve quantitative agreement with experiment. Our ultimate goal is to develop a
fully relativistic multi-channel close-coupling approach that will not be subject to the
semi-relativistic approximations necessary when applying the present CCC method.
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ACKNOWLEDGMENTS
The support of the Australian Research Council and the Merit Allocation Scheme on the
National Facility of the Australian Partnership for Advanced Computing are gratefully
acknowledged.
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Electron scattering from the ground state of mercury

http://researchrepository.murdoch.edu.au/id/eprint/12684/1/Electron_scattering_from_the_ground_state_of_2003.pdf

Electron scattering from the ground state of mercury

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