IOP PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 3015–3023
doi:10.1088/0953-4075/40/15/003
Scattering of low-energy electrons by isomers of C4H10
M H F Bettega1, M A P Lima2 and L G Ferreira3
1
Departamento de Fı́sica, Universidade Federal do Paraná, Caixa Postal 19044,
81531-990 Curitiba, Paraná, Brazil
2 Instituto de Fı́sica ‘Gleb Wataghin’, Universidade Estadual de Campinas, 13083-970 Campinas,
São Paulo, Brazil
3 Instituto de Fı́sica, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo,
São Paulo, Brazil
Received 24 May 2007, in final form 19 June 2007
Published 13 July 2007
Online at stacks.iop.org/JPhysB/40/3015
Abstract
We report elastic integral, differential and momentum transfer cross sections
for low-energy electron collisions with the two isomers of C4H10, butane
and isobutane. We employ the Schwinger multichannel method with
pseudopotentials at the static-exchange-polarization approximation, and cover
energies from 1 eV to 20 eV. We investigate the influence of polarization on the
isomer effect comparing the cross sections of these isomers and concluded that
polarization has a small influence in that effect. We also compare our computed
elastic cross sections with available total cross sections. Total ionization cross
sections by electron impact are computed using the binary-encounter-Bethe
(BEB) model and the results compared with available experimental data. The
ionization cross sections for butane and isobutane are similar, showing that they
also have a small isomer effect in the ionization process.
1. Introduction
The electron collisions with isomers of small hydrocarbons have been investigated by
experimental [1–4] and theoretical [5–8] groups over the last couple of years. The main
goal of these studies was to investigate the isomer effect. This effect is responsible for
differences in the cross sections of hydrocarbons isomers and allows one to identify each
isomer of a molecule by its cross section. If the cross sections are similar it means that in this
case the isomer effect is small.
In a previous study, Lopes et al [7] investigated the isomer effect for the two isomers of
C4H10, butane and isobutane, in calculations carried out with the Schwinger multichannel
method with pseudopotentials in the static-exchange (SE) approximation. The authors
compared the cross sections of butane and isobutane and found small differences between
the integral and momentum transfer cross sections of these isomers in the energy range
considered (from 5 eV to 50 eV). The differential cross sections of these isomers presented
0953-4075/07/153015+09$30.00 © 2007 IOP Publishing Ltd Printed in the UK
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M H F Bettega et al
a different oscillatory behaviour for energies below ∼20 eV. The authors concluded then that
the isomers of C4H10 presented a small isomer effect, since their cross sections were similar
in the energy range considered.
There are few studies regarding electron interactions with C4H10 molecules. Floeder
et al [9] measured total cross sections for several hydrocarbons including butane and
isobutane. The measurements were carried out in a transmission experiment for impact energies
up to 400 eV. Their results for butane and isobutane present a broad shape resonance around
10 eV for each molecule. Calculations of total (elastic+absorption) cross sections for a series
of hydrocarbons including C4H10 were performed by Raj [10] for energies from 100 to
500 eV. These calculations employed the independent centre approximation, and the cross
sections were computed by the optical theorem. More recently, Jiao et al [11] measured
absolute cross section for ionization of butane by electron impact for energies from 10 to
200 eV.
In this paper, we report integral, differential and momentum transfer cross sections for
butane and isobutane obtained in the static-exchange-polarization (SEP) approximation for
energies from 1 up to 20 eV. Our goal in this study is to investigate the influence of polarization
effects on the isomer effect, i.e., if the inclusion of polarization in the present calculations will
make the isomer effect more evident than in the previous calculations. We also compare the
elastic cross sections with the total cross sections of Floeder et al.
We also computed total ionization cross sections for butane and isobutane using the
binary-encounter-Bethe (BEB) model [12], which has been successfully employed in
calculations of total ionization cross sections of molecules by electron impact [13].
In the next sections, we present a brief discussion of the theoretical method and the
computational procedures used in our calculations. We present our results and discussions
and close with a brief summary of our results and conclusions.
2. Theory
The Schwinger multichannel (SMC) method [14–16] and the implementation with
pseudopotentials [17] were described in detail in previous publications. Here we are interested
in discussing the points which are relevant to the present calculations.
The SMC method is a variational method for the scattering amplitude. The resulting
expression to the scattering amplitude is
1 Skf V |χµ (d −1 )µν χν |V Ski
(1)
f (kf , ki ) = −
2π µ,ν
where
dµν = χµ |A(+) |χν (2)
and
(Ĥ P + P Ĥ ) (V P + P V )
Ĥ
−
+
− V G(+)
(3)
P V.
N +1
2
2
In the above equations, Ski,f is a solution of H0 , the unperturbed Hamiltonian, and is written as
a product of a target state and a plane wave, V is the interaction potential between the incident
electron and the electrons and nuclei of the target, {|χµ } is a set of (N + 1)-electron Slater
determinants (configuration state functions (CSFs)) used in the expansion of the trial scattering
wavefunction, Ĥ = E − H is the total energy of the collision minus the full Hamiltonian of
A(+) =
Scattering of low-energy electrons by isomers of C4H10
3017
the system, with H = H0 + V , P is a projection operator onto the open-channel space defined
by the target eigenfunctions, and G(+)
P is the free-particle Green’s function projected on the
P-space. The direct configuration space is constructed as
{|χµ } = {A(|1 ⊗ |ϕµ )}
(4)
where |1 is the target ground-state wavefunction, described at the Hartree–Fock level of
approximation, |ϕµ is a one-electron function represented by a virtual (unoccupied) orbital,
and A is the antisymmetrizer.
Polarization effects are included enlarging the configuration space by including CSFs of
the type:
{|χν } = {A(|j ⊗ |ϕµ )}
(5)
where |j are virtual states (closed channels) of the target obtained from the ground state by
single excitations and |ϕµ , as before, is a one-particle function. To construct the |j states,
we considered only single excitations from the (valence) occupied orbitals to a set of polarized
orbitals {|ϕi,k } [18] defined as
ϕj |rk |ϕi |ϕi,k =
|ϕj (6)
Ej − Ei
j ∈virtuals
where |ϕi is an occupied orbital, rk (k = 1, 2, 3) is a component of the dipole moment operator
r, and the summation runs over the Hartree–Fock virtual orbitals. To construct an orthonormal
set of orbitals (from the polarized and virtual orbitals) the Schmidt orthogonalization procedure
is used. All unoccupied (polarized and virtual) orbitals are used as scattering orbitals.
The ionization cross sections were calculated using the binary-encounter-Bethe model
[12]. The expression for the ionization cross section per molecular orbital is given by
ln t
S
1
1
ln t
1− 2 +1− −
σBEB (T ) =
t +u+1 2
t
t
t +1
where T is the incident electron energy, t and u are normalized energies, t = T /B and
u = U/B; B is the binding energy and U is the electron kinetic energy of the molecular
orbital, S = 4π a02 N R 2 /B 2 where N is the orbital occupation number, a0 = 0.5292 Å and
R = 13.61 eV. The total ionization cross section is obtained by summing σBEB (T ) over the
molecular orbitals that satisfy T > B.
3. Computational procedures
The cross sections calculations were carried out in the static-exchange-polarization (SEP)
approximation at the ground-state equilibrium geometry [19]. We used the pseudopotentials
of Bachelet, Hamann and Schlüter [22] in order to replace the 1s core electrons of the carbon
atoms. In our bound state and scattering calculations the basis set used for the carbon atoms
are the same used in [7] and was generated according to [23]. For the hydrogen atom the basis
set is from [24]. We have not included in our calculations the 3s symmetric combination of
d-type functions, namely [(x 2 + y 2 + z2 ) exp(−αr 2 )], in order to eliminate any possible linear
dependence in the basis set.
Isobutane is a polar molecule. The calculated value of the dipole moment is 0.114D,
which is in very good agreement with the experimental value of 0.132D [19]. Since this value
is too small, we have not included in our calculations the Born closure of the long-range dipole
potential in order to compute the higher partial waves. In these calculations the scattering
amplitude is expanded in partial waves up to = 10. The final conclusions of this study will
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M H F Bettega et al
Figure 1. Geometrical structures of butane and isobutane.
not be affected by using a truncated scattering amplitude. To quantify the changes caused by
the dipole potential, we carried out a calculation at 4 eV including this long-range interaction.
At this energy the changes in the differential cross section caused by the dipole potential occur
for angles below 5◦ and the integral cross section increases by 0.4%.
The calculated polarizabilities are 8.16 ×10−24 cm3 , for butane and 8.23 ×10−24 cm3 ,
for isobutane. These values agree well with the experimental values of 8.12 ×10−24 cm3 for
butane and 8.14 ×10−24 cm3 for isobutane [20].
In figure 1, we present the geometrical structures for butane, which belongs to C2h group
and for isobutane, which belongs to the C3v group. For symmetry reasons the calculations for
both molecules were performed in the Cs group. The number of CSFs per symmetry used for
butane was 7673 for A and 7399 for A with a total of 15 072 CSFs, and for isobutane was
7277 for A and 7245 for A with a total of 14 522 CSFs. It is important to have a balanced
description of polarization for both molecules (the same amount of CSFs for each molecule)
in order to compare results obtained at the same level.
The values of the binding energies B and the kinetic energies U needed in the calculation
of σBEB (T ) for these isomers were computed using the package GAMESS [21] in a RHF
calculation with a 6-311++G(2d) basis set.
4. Results and discussion
In figure 2, we present a comparison between the integral and momentum cross sections for
butane obtained in the SE and SEP approximations. The SE results are from [7]. This figure
shows that polarization effects lower the cross sections for energies below ∼6 eV and move
down the resonance position from 12 eV (obtained in a SE calculation from [7]) to ∼10 eV.
Figure 3 shows the comparison between the cross sections of isobutane obtained in the SE
(also from [7]) and SEP approximations. The same behaviour observed for butane is observed
for isobutane: the polarization lowers the cross sections for energies below ∼5 eV and moves
down the resonance position from 12 eV to 11 eV.
In figure 4, we compare the integral and momentum transfer cross sections for butane and
isobutane obtained in the SEP approximation. The cross sections for both molecules have the
same shape and magnitude maintaining the behaviour observed in the SE calculations of [7].
We conclude then that polarization does not affect the isomer effect since the cross sections
remain similar. In this figure, we also show the total cross section of Floeder et al for butane
and isobutane. We shifted the energy scale of the experimental data in order to place the
shape resonance at the same location as ours (the experimental position is about 1.5 eV below
ours for both molecules). Our calculated elastic cross sections show the same shape of the
measured total cross sections.
Scattering of low-energy electrons by isomers of C4H10
3019
60
50
40
2
10
cross section (10
20
-16
cm )
30
0
SEP
SE
0
4
0
4
8
12
16
20
8 12 16
energy (eV)
20
40
30
20
10
0
Figure 2. Integral (upper panel) and momentum transfer (lower panel) cross sections for butane
in the SE and SEP approximations.
60
50
40
2
10
cross section (10
20
-16
cm )
30
0
SE
SEP
0
4
0
4
8
12
16
20
8 12 16
energy (eV)
20
40
30
20
10
0
Figure 3. Integral (upper panel) and momentum transfer (lower panel) cross sections for isobutane
in the SE and SEP approximations.
To investigate the origin of the small differences seen in the integral cross sections of
butane and isobutane we show in figure 5 the symmetry decomposition of the integral cross
section of these molecules according to the Cs group. The cross sections of these molecules
for the A symmetry are very similar in shape, the cross section for butane lying above the
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M H F Bettega et al
60
butane (teo)
2
cm )
40
isobutane (teo)
20
cross section (10
-16
butane (expt)
isobutane (expt)
0
0
5
15
20
0
5
15
10
energy (eV)
20
10
40
30
20
10
0
Figure 4. Integral (upper panel) and momentum transfer (lower panel) cross sections for butane
and isobutane in the SEP approximation. Total cross sections of [9] are also shown for comparison.
The energy scale of experimental is shifted (see text for discussion).
36
30
24
A´
2
6
cross section (10
12
-16
cm )
18
0
butane
isobutane
0
5
10
15
20
24
18
12
A´´
6
0
0
5
15
10
energy (eV)
20
Figure 5. Symmetry decompositions of the integral cross sections for butane and isobutane in the
SEP approximation.
cross section for isobutane. For the A symmetry the cross sections for both molecules are
quite similar for energies above ∼11 eV. The A cross sections for both molecules present a
peak, the peak of butane being located at a lower energy of about 2.5 eV than that of isobutane.
This figure also shows that the broad structure seen in both integral cross sections presented
Scattering of low-energy electrons by isomers of C4H10
3021
10
4 eV
7.5 eV
10
cross section (10
-16
2
cm /sr)
butane
isobutane
1
0
60
120
180
1
0
60
10 eV
10
1
0
120
180
15 eV
10
60
1
120
180 0
60
scattering angle (degrees)
120
180
Figure 6. Differential cross sections for butane and isobutane in the SEP approximation.
in figure 4 is in fact a superposition of two resonances, one belonging to A and the other
to A .
In figure 6, we present a comparison of the differential cross sections (DCSs) of butane
and isobutane at 4, 7.5, 10 and 15 eV. There are differences in the oscillatory pattern shown by
the DCSs of these isomers at 4, 7.5 and 10 eV. The oscillations are more evident in the DCSs
of isobutane than in the DCSs of butane. These oscillations are due to the different angular
momentum coupling. Although not shown here, the same differences were observed in the
DCSs obtained in the SE calculations from [7].
The most significant isomer effect is seen in the oscillatory behaviour of the DCSs. As
discussed in [8], the most symmetric molecule tends to present more oscillations in the DCSs.
The explanation for the similarities presented by the ICS of butane and isobutane may be
explained by the ‘scaling’ obtained using a method based on geometrical optics [25], which
will not be discussed here. The results obtained using this method show that the cross sections
of these isomers follow a scaling law (which explains the similarities between the isomeric
integral cross sections) and that the carbon atoms are more important in the scattering process
than the hydrogens.
Figure 7 shows a comparison of the total ionization cross sections for butane and isobutane.
We also show the experimental data from [11] for butane. The ionization cross sections of
these isomers are very similar which indicates that the isomer effect is also small with respect
to the ionization process.
5. Summary
We presented elastic integral, differential and momentum transfer cross sections for butane
and isobutane, which are isomers of C4H10. We compared the cross sections of both molecules
obtained in calculations which included polarization effects. We concluded that the isomer
effect was not affected by the inclusion of polarization since the cross sections remain similar.
A method based on geometrical optics provides a scaling law for the cross sections of these
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M H F Bettega et al
12
cross section (10
-16
2
cm )
9
6
experiment (butane)
theory (butane)
theory (isobutane)
3
0
0
50
100
energy (eV)
150
200
Figure 7. Total ionization cross sections for butane and isobutane. The experimental data of [11]
are also shown for comparison.
isomers, which explains their similarities. This method also indicates that the carbons are
better scatterers that the hydrogens. We also computed total ionization cross sections for
these isomers. The cross sections are similar indicating that the isomer effect is also small in
the ionization process.
Acknowledgments
The authors acknowledge support from Brazilian agency Conselho Nacional de
Desenvolvimento Cientı́fico e Tecnológico (CNPq). MHFB also acknowledges support from
the Paraná state agency Fundação Araucária and from FINEP (under project CT-Infra 1)
and computational support from Professor Carlos M de Carvalho at DF-UFPR and from
CENAPAD–SP.
References
[1] Szmytkowski Cz and Kwitnewski S 2002 J. Phys. B: At. Mol. Opt. Phys. 35 3781
[2] Nakano Y, Hoshino M, Kitajima M, Tanaka H and Kimura M 2002 Phys. Rev. A 66 032714
[3] Makochekanwa C, Kawate H, Sueoka O, Kimura M, Kitajima M, Hoshino M and Tanaka H 2003 Chem. Phys.
Lett. 368 82
[4] Szmytkowski Cz and Kwitnewski S 2003 J. Phys. B: At. Mol. Opt. Phys. 36 2129
[5] Lopes A R and Bettega M H F 2003 Phys. Rev. A 67 032711
[6] Lopes A R, Lima M A P, Ferreira L G and Bettega M H F 2004 Phys. Rev. A 69 014702
[7] Lopes A R, Bettega M F H, Lima M A P and Ferreira L G 2004 J. Phys. B: At. Mol. Opt. Phys. 37 997
[8] Makochekanwa C, Kato H, Hoshino M, Tanaka H, Kubo H, Bettega M H F, Lopes A R, Lima M A P and
Ferreira L G 2006 J. Chem. Phys. 124 024323
[9] Floeder K, Fromme D, Raith W, Schwab A and Sinapius G 1985 J. Phys. B: At. Mol. Opt. Phys. 18 3347
[10] Raj D 1994 Indian J. Pure Appl. Phys. 32 867
[11] Jiao C Q, DeJoseph C A and Garscadden A 2007 J. Phys. D: Appl. Phys. 40 409
[12] Kim Y-K and Rudd M E 1994 Phys. Rev. A 50 3954
[13] Kim Y-K, Hwang W, Weinberger N M, Ali M A and Rudd M E 1997 J. Chem. Phys. 106 1026
Ali M A, Kim Y-K, Hwang W, Weinberger N M and Rudd M E 1997 J. Chem. Phys. 106 9602
Nishimura H, Huo W M, Ali M A and Kim Y-K 1999 J. Chem. Phys. 110 3811
Scattering of low-energy electrons by isomers of C4H10
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
Takatsuka K and McKoy V 1981 Phys. Rev. A 24 2473
Takatsuka K and McKoy V 1984 Phys. Rev. A 30 1734
Lima M A P, Brescansin L M, da Silva A J R, Winstead C and McKoy V 1990 Phys. Rev. A 41 327
Bettega M H F, Ferreira L G and Lima M A P 1993 Phys. Rev. A 47 1111
Lengsfield B H III, Rescigno T N and McCurdy C W 1991 Phys. Rev. A 44 4296
Lide D R (ed) 1998 CRC Handbook of Chemistry and Physics 79th edn (Boca Raton, FL: CRC Press)
Miller K J 1990 J. Am. Chem. Soc. 112 8533
Schmidt M W et al 1993 J. Comput. Chem. 14 1347
Bachelet G B, Hamann D R and Schlüter M 1982 Phys. Rev. B 26 4199
Bettega M H F, Natalense A P P, Lima M A P and Ferreira L G 1996 Int. J. Quantum Chem. 60 821
Dunning T H Jr 1970 J. Chem. Phys. 53 2823
Ferreira L G, Lopes A R, Lima M A P and Bettega M H F 2006 J. Phys. B: At. Mol. Opt. Phys. 39 1045
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