Differential cross sections for low-energy electron elastic scattering by lanthanide atoms:
La, Ce, Pr, Nd, Eu, Gd, Dy, and Tm
Z. Felfli,
1
A. Z. Msezane,
1
and D. Sokolovski
2
1
Department of Physics and Centre for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia 30314, USA
2
School of Mathematics and Physics, Queen’s University of Belfast, Belfast BT7 1NN, United Kingdom
Received 4 March 2009; published 12 June 2009
Elastic differential cross sections DCSs in angle of electron scattering by the representative lanthanide
atoms La, Ce, Pr, Nd, Eu, Gd, Dy, and Tm have been calculated in the electron-impact energy range 0 E
1 eV. Additionally, the DCSs in electron-impact energy are also presented at scattering angles =0°, 90°,
and 180° for unambiguous identification of the binding energies BEs of the negative ions formed during the
collisions as resonances. The shape resonances and the DCSs critical minima are identified as well. A Thomas-
Fermi-type potential incorporating the vital core-polarization interaction is used for the calculations. Dramati-
cally sharp resonances are found to characterize the near-threshold electron elastic DCSs, whose energy
positions are identified with the BEs of the resultant negative ions. A procedure is suggested for measuring
reliably the BEs of tenuously bound BE 0.1 eV, weakly bound BE 1 eV, and complicated open d- and
f -subshell negative ions through the elastic DCSs both in scattering angle and electron-impact energy.
DOI: 10.1103/PhysRevA.79.062709 PACS numbers: 34.80.Bm
I. INTRODUCTION
In a recent paper 1 the near-threshold electron attach-
ment mechanism in electron-lanthanide atom scattering,
manifesting itself as Regge resonances, was investigated us-
ing the recently developed Regge-pole analysis through the
calculation of the electron elastic total cross sections TCSs
and the Mulholland partial cross sections 2. Generally, the
TCSs were found to be characterized by dramatically sharp
resonance structures whose energy positions were identified
with the binding energies BEs of the resultant negative ions
formed during the collisions as Regge resonances. This is
consistent with the conclusion 3 that the existence of a
large peak in the electron-atom scattering TCS at low energy
represents the signature of the ground state of the negative
ion, with the proviso that the second or even the third empty
orbital be at low energy and has orbital angular momentum,
l 0. Indeed, the careful scrutiny of the imaginary part of the
complex angular momentum, L ,Im L was used 1 to distin-
guish between the bound states of the negative ions and the
shape resonances. For the latter Im L was found to be several
orders-of-magnitude greater than that corresponding to the
former. Ramsauer-Townsend minima, shape resonances, and
the Wigner threshold law were also determined. The BEs
extracted from the resonances were compared with those
from recent measurements and calculations. In particular, the
negative ions whose binding energies agreed very well with
the most recently measured and/or calculated values are
among those selected for use in the present investigation of
the electron elastic differential cross sections DCSs.
Here we have selected typical lanthanides, determined
through their formation of tenuously bound BE 0.1 eV,
weakly bound BE 1 eV, and complicated open d- and
f -subshell negative ions in the near-threshold electron elastic
scattering, to investigate the structure of the DCSs in angle in
the electron-impact energy range 0 E 1 eV. The DCSs in
energy at the scattering angles =0°, 90°, and 180° are cal-
culated as well; these readily yield the BEs of the negative
ions formed during the collisions 4. Also determined are
the so-called DCS critical minima 5 which correspond to
the DCS minima in the plane of scattering angle and projec-
tile energy. The accurate experimental determination of the
depth of the critical minimum is virtually impossible due to
the finite resolution of an experimental apparatus 6. The
recent experimental investigations of the DCSs and their
critical minima in the elastic electron-Kr scattering at inter-
mediate incident electron energies 100–260 eV and scatter-
ing angles 30 ° – 110° 5, as well as in the elastic
electron-Yb scattering at 10, 40, and 80 eV in the angular
range 10° 160° 7 demonstrate the experimental diffi-
culties and the need for theoretical guidance, particularly as
E → 0 eV and → 0°.
It is therefore appropriate to explore the near-threshold
electron DCSs for the lanthanides to guide future experimen-
tal and theoretical investigations. The DCSs provide strin-
gent test of theoretical calculations when the results are com-
pared with those of reliable measurements. However, for the
lanthanides the only available detailed near-threshold
electron-scattering data, to our knowledge, are the TCSs and
partial cross sections 1. Also, the published electron DCSs,
including the positions of the critical minima, for the lan-
thanide atoms are only those for Yb by Predojevic et al. 7
and Kelemen et al. 8 but the former cover the electron
energy range E 10 eV while for the latter the energy range
is 2 eV–2 keV; these are much higher than the energy range
of interest here. There are some theoretical and experimental
results for e-Yb scattering 9 but neither the theory nor the
experiment explored the energy region near threshold, i.e.,
below 0.1 eV. For the energy range E 2 eV, comparison
between the measured and calculated 9 DCSs for e-Yb
scattering have been discussed 10. We note that Remeta et
al. 11 calculated low-energy, below 2 eV, electron elastic
forward and backward scatterings by some atoms, including
Ca and Yb, using their optical potential approach. They ob-
tained the total elastic and differential cross sections. How-
ever, the adjustable parameter used in the calculation of the
PHYSICAL REVIEW A 79, 062709 2009
1050-2947/2009/796/06270910 ©2009 The American Physical Society 062709-1