INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 1707–1725
PII: S0953-4075(02)31668-7
The Li+ –H2 system in a rigid-rotor approximation:
potential energy surface and transport coefficients
I Røeggen1 , H R Skullerud2 , T H Løvaas2 and D K Dysthe3
1
Department of Physics, University of Tromsø, 9037 Tromsø, Norway
Department of Physics, Norwegian University of Science and Technology,
7491 Trondheim, Norway
3 Department of Physics, University of Oslo, 0316 Oslo, Norway
2
E-mail: helge.skullerud@phys.ntnu.no
Received 6 December 2001, in final form 25 February 2002
Published 26 March 2002
Online at stacks.iop.org/JPhysB/35/1707
Abstract
An accurate potential energy surface for the Li+ –H2 system has been calculated
ab initio at an H–H distance of 1.449 a0 , using an extended geminal model.
The potential has been used to find elastic and inelastic total and transport
cross sections, which have subsequently been used in moment-type transport
calculations with a Kramers–Moyal-type expansion of the collision integral.
The ion mobilities have also been measured at 295 K and E/n0 values from
10 to 450 Td, which corresponds to mean centre-of-mass energies from 0.04
to 7 eV. The agreement between experimental and calculated values is excellent
below 220 Td. At higher E/n0 values, the agreement deteriorates gradually,
possibly because of the neglect of vibrational excitation.
1. Introduction
A knowledge of the transport coefficients for ions moving in electrostatic fields in gases
is necessary for the modelling of many technically important gas discharge and plasma
phenomena, as well as for the analysis of a range of ionized gas phenomena occurring in
Nature. Besides these applications, there has been since the beginning of ion transport studies
in gases about a century ago a profound interest in trying to understand the phenomena in depth,
and to establish a firm connection between a ‘microscopic description’ and the observable
‘macroscopic’ transport coefficients. Thus, one would like to start with the Schrödinger
equation for the ion-neutral system, find the transport coefficients ab initio, and compare
the results with experimentally determined values.
For closed-shell atomic ions in atomic gases, a quantitative agreement between experimentally measured mobilities and diffusion coefficients and values calculated ab initio has now
0953-4075/02/071707+19$30.00
© 2002 IOP Publishing Ltd
Printed in the UK
1707
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I Røeggen et al
been obtained for several systems—after revisions of both experimental and theoretical methods. For open-shell atomic systems the interaction potential calculations have yet not been
sufficiently accurate to achieve the same degree of agreement.
The theoretical calculation of transport properties for closed-shell atomic ions in molecular
gases is considerably more complicated than similar calculations for atomic gases, even for
the relatively simple case of homonuclear diatomic molecules. The latter should however be
possible with the introduction of current techniques and suitable approximations. The gases
of most immediate interest would be hydrogen, nitrogen, and oxygen.
Viehland et al reported in 1992 a theoretical investigation of Li+ ions in N2 gas, which
was in good agreement with the experimental mobilities of Selnæs et al (1990), in the range
of ratios of electric field to density where the calculations converged. We will report here the
study of another and seemingly simpler ion–diatom system, Li+ ions in H2 gas.
A major difference between the Li+ –N2 system and the Li+ –H2 system lies in the
spacings of the molecular rotational levels in the two gases. In nitrogen, the spacings
are small compared to room temperature thermal energies 3kB T /2 ∼ 38 meV. This
permits the assumption of a continuous rotational spectrum and the use of a classical
scattering approximation, as was done by Viehland et al. In hydrogen, on the other hand,
the spacings range from 44 meV upwards, and full quantum cross section calculations
should be performed. In both cases, one may probably neglect vibrational excitations
except at very strong fields, and a rigid-rotor approximation would presumably be
reasonable.
The transport theory used by Viehland et al starts out from the Wang–Chang–Uhlenbeck
extension of the Boltzmann equation, which is based on arguments about ‘inverse collisions’
and detailed balance. The validity of these arguments for rotating systems has sometimes been
questioned. We have circumvented the problem by starting out from the conceptually simpler
Maxwell-type ‘equations of balance’ for various velocity averages.
The transport theory of Viehland et al also presumes a priori that the ion velocity
distribution function may be expanded in polynomials around a Gaussian weight function,
and this assumption is what gave rise to convergence problems in their calculations.
We have used a quite different way of treating the so-called ‘collision integral’, and
generalized a ‘Kramers–Moyal expansion method’ introduced by Kumar et al (1980) to
incorporate inelastic collisions. This permits the use of more flexible velocity-space
basis sets.
The Li+ –H2 system was studied extensively earlier, and elastic, rotational, and vibrational
excitation cross sections have been calculated by a variety of methods. The comparison with
experiment has however only been with beam data, which are inherently less accurate than
swarm data at low centre-of-mass energies. There has therefore apparently not been much
incentive to find a potential energy surface more accurate than the combined SCF–asymptotictheory surface calculated by Lester (1971). To make a comparison with experimental swarm
data meaningful, we have calculated a new and more accurate potential energy surface, to be
used with a rigid-rotor model and an H–H distance approximately equal to the mean separation
in the rotational ground state. Also, we have measured the mobilities for Li+ ions in hydrogen
at room temperature with a higher accuracy than in experiments reported earlier.
We present in the following first the calculation of the new potential surface, then
the transport theory and the scattering calculation, and finally a comparison between
experimental mobility values and values calculated from the new potential surface, and—to
show the improvement—also values calculated from the old SCF–asymptotic-theory surface
of Lester (1971).
The Li+ –H2 system
1709
2. The potential energy surface
An extended geminal model (Røeggen 1999) was adopted for the calculation of the potential
energy surface. For a system comprising only two electron pairs, the electronic energy is given
by
E EXG = E RH F + 1 + 2 + 12 .
(1)
RH F
Here E
denotes the restricted Hartree–Fock energy for the dimer, 1 and 2 are singlepair correlation terms, and 12 is the double-pair correlation term. Numerically, a single-pair
correlation term is calculated as a full configuration interaction (FCI) term in a truncated virtual
orbital space plus a second-order correction to the FCI term based on the complementary virtual
space; i.e. a basis set extension (BSE) effect:
k = ˜KF CI + KBSE .
(2)
Similarly, we have for the double-pair correlation term
F CI
BSE
KL = ˜KL
+ KL
.
(3)
The basis set expansion correction is in this case calculated at the MP2 level, i.e.
BSE
MP 2
MP 2
KL
= KL
− ˜KL
.
MP 2
The terms KL
MP 2
and ˜KL
(4)
are dispersion-type MP2 corrections calculated by using respectively
the complete virtual space and the truncated virtual space. The latter is utilized for the FCI
F CI
calculation which generates ˜KL
.
In this work we use the counterpoise correction scheme advocated by Boys and Bernardi
(1970) for eliminating the basis set superposition error. This implies that the supermolecule
basis is adopted for the calculation on the subsystems, i.e. we perform subsystem calculations
for each chosen supersystem geometry.
The one-electron basis adopted is an uncontracted Gaussian-type-family basis: (13s, 7p,
5d, 3f) for hydrogen and (14s, 6p, 5d, 4f, 3g) for the lithium ion. To avoid linear dependency
we had to use slightly different basis sets for the hydrogen atoms. The s-type functions are
an even-tempered set where the exponents are of the form ηi = αβ i ; i = 1, . . . , 13. For both
atoms we have β = 2.5, and α = 0.004 9603 for one of the atoms with α = 0.003 6548 for the
other. The exponents for the polarization functions are drawn from the set of s-type exponents.
The p-type sets start with a lowest exponent 0.031 0020 and 0.022 8425, respectively; the
d-type sets start with a lowest exponent 0.077 5050 and 0.057 1062, respectively; the f-type
sets start with a lowest exponent 0.193 7625 and 0.142 7656, respectively. The chosen basis
set is so large that the slightly unsymmetric character in the hydrogen basis has a negligible
effect on the calculated surface. The s-type functions for Li+ are an even-tempered set where
the exponents are of the form ηi = αβ i ; i = 1, . . . , 14; α = 0.006 7322 and β = 2.5. As
in the hydrogen case, the exponent for the polarization functions are drawn from the set of
s-type exponents. The lowest exponent for the p-, d-, f-, and g-type functions is, respectively,
0.105 1904, 0.262 9760, 0.262 9760, and 0.657 4400. The polarizations for both hydrogen and
the ion are defined with spherical harmonics.
In all calculations we use the Beebe–Linderberg two-electron integral approximation
(Beebe and Linderberg 1977). The errors generated by this approximation are estimated
to be less than 10−9 au.
The dimensions for the truncated virtual orbital space are 65 and 60, respectively, for
the single- and double-pair corrections. Errors generated by adopting these approximations
instead of utilizing the complete virtual space in the FCI calculations are estimated to be less
than 10−6 au.
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I Røeggen et al
Figure 1. The H2 –Li+ coordinate system.
We consider the interaction between a Li+ ion and a hydrogen molecule, with a fixed
distance R between the two protons, as shown in figure 1. From the Kolos and Wolniewicz
(1964) H–H potential, we find the mean distances between the protons in the vibrational ground
state and the two lowest rotational states j = 0 (parahydrogen) and j = 1 (orthohydrogen) to
be Rj =0 = 1.4483 a0 and Rj =1 = 1.4505 a0 . We will wish to model both normal hydrogen (a
3:1 mixture of orthohydrogen and parahydrogen) and pure parahydrogen at low temperatures,
and have chosen R = 1.449 a0 as a reasonable compromise.
The potential energy was calculated as a function of the Li+ –H2 distance r, for seven
values of the angle θ between r and R (see figure 1): θ = 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ , and 90◦ .
For use in scattering calculations, the potential is conveniently expanded in Legendre
polynomials in cos θ:
V (R̂, r) =
Vλ (r)Pλ (cos θ).
(5)
λ=even
The Legendre components, with λ = 0, 2, 4, and 6, are listed in table 1.
3. Elements of ion transport theory
The elements needed to establish a practical calculational method for finding the transport
coefficients for ions in a molecular gas can be found in a comprehensive article by Kumar
et al (1980), and we have largely followed procedures suggested there. It may however
not be generally clear how this is done and what are the underlying assumptions. We will
therefore sketch our approach from first principles, and also give enough details to enable our
calculational schemes to be reproduced by others.
3.1. Formulation of the problem
We consider a ‘swarm of ions’ moving in an electrostatic field E and a homogeneous,
unbounded gas with temperature T and randomly oriented molecules. The ions may interact
both elastically and inelastically with the gas molecules, but we will for the sake of simplicity
assume that ion–molecule reactions do not occur, i.e. that the number of ions does not change
with time.
To avoid misconceptions, we will remind the reader that ‘a swarm experiment with N
ions’ is not one experiment where N ions are released simultaneously, but ideally a succession
of N one-ion experiments performed under ‘macroscopically identical conditions’. Averages
are thus formed as averages over ensembles of repeated experiments, typically
h(r , v, t) = lim N −1
N →∞
N
k=1
h(rk , vk , t)
(6)
The Li+ –H2 system
1711
Table 1. The Legendre components Vλ (r) of the intermolecular potential for H2 Li+ at RH−H =
1.449 a0 , in mHartree.
r
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.50
9.00
9.50
10.0
11.0
12.0
13.0
15.0
20.0
30.0
50.0
λ=0
527.773 50
312.329 63
180.910 17
101.081 41
53.213 93
25.033 41
8.887 69
0.023 89
−4.495 23
−6.476 25
−7.023 53
−6.804 93
−6.217 09
−5.488 92
−4.746 49
−4.053 72
−3.438 30
−2.907 59
−2.458 10
−2.081 57
−1.768 08
−1.507 72
−1.291 50
−1.111 59
−0.961 45
−0.835 65
−0.729 79
−0.564 21
−0.443 47
−0.353 73
−0.285 84
−0.192 97
−0.135 18
−0.097 60
−0.054 69
−0.017 21
−0.003 40
−0.000 45
λ=2
λ=4
λ=6
399.890 87
233.629 64
141.905 78
87.870 21
55.294 62
35.433 83
23.206 75
15.601 87
10.817 79
7.768 62
5.794 31
4.490 74
3.608 49
2.992 69
2.546 94
2.211 56
1.949 17
1.736 56
1.559 27
1.408 00
1.276 79
1.161 64
1.059 73
0.968 99
0.887 89
0.815 16
0.749 78
0.637 78
0.546 23
0.470 90
0.408 53
0.312 90
0.244 59
0.194 67
0.128 99
0.055 89
0.016 98
0.003 73
101.434 87
44.280 13
22.317 12
12.058 72
6.746 34
3.870 75
2.266 44
1.351 88
0.821 48
0.508 71
0.322 02
0.208 41
0.138 72
0.095 33
0.067 88
0.050 01
0.003 83
0.030 14
0.024 37
0.020 15
0.016 88
0.014 26
0.012 11
0.010 37
0.008 98
0.007 83
0.006 81
0.005 36
0.004 22
0.003 32
0.002 68
0.001 73
0.001 19
0.000 84
0.000 45
0.000 15
0.000 04
0.000 01
26.677 78
7.829 96
2.934 68
1.270 54
0.613 89
0.308 96
0.161 28
0.086 68
0.047 23
0.026 35
0.014 97
0.008 51
0.004 82
0.002 69
0.001 60
0.001 00
0.000 52
0.000 28
0.000 05
−0.000 00
−0.000 00
−0.000 11
−0.000 15
−0.000 15
−0.000 12
−0.000 08
−0.000 02
−0.000 09
−0.000 00
−0.000 05
−0.000 07
−0.000 06
−0.000 05
−0.000 04
−0.000 04
−0.000 02
−0.000 01
0.000 00
where rk and vk are the position and velocity of the ion in the kth experiment. As there is in
principle only one ion present at a time, the concept of ion number density becomes physically
meaningless. We therefore do not define our ‘transport coefficients’ as belonging to some
gradient expansion of an equation for the number density, as is often done, but define them
from the behaviour of position vector moments. The drift velocity vdr and the diffusion tensor
⇒
D are thus written as
d
N =const
v
r → lim t→∞ dt
t→∞
⇒
d 1 ∗ ∗ N =const
r r
→ lim r ∗ v ∗ D = lim
t→∞ dt 2
t→∞
vdr = lim
(7)
(8)
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I Røeggen et al
where r ∗ = r − r and v ∗ = v − v . As r ∗ ≡ 0 by definition, the expression for the
diffusion tensor may also be written as
⇒
D = lim r ∗ v
(9)
t→∞
which is a slightly more convenient starting point for our calculational scheme.
In practice, the ‘one-particle-at-a-time’ condition for the swarm experiment can be relaxed
somewhat as long as the number of ions present at the same time is kept sufficiently low that ion–
ion interactions can be neglected. A typical experiment for measuring ion transport coefficients
is thus usually a succession of ‘few-ion experiments’.
3.2. Maxwell-type balance equations
Our primary interest is in finding the steady-state values of the ensemble averages v and
r ∗ v. More generally, we would seek quantities of type ψ(
v ) and r ∗ ψ(
v ), with ψ(
v)
being some polynomial in v. We thus start out by establishing equations which describe the
evolution of such quantities with time, and afterwards seek the steady-state limit.
A quantity ψ(
v ) changes continuously with time because of the acceleration in the electric
field a = q E/m:
v ψ = a · ∇
v ψ
∂ta ψ(
vk ) = ∂ta vk · ∇
and discontinuously because of collisions, vk → vk and ψ(
vk ) → ψ(
vk ):
∂tc ψ(
v ) ≡ C ψ(
v) =
νj (
v )[ψ(
v ) − ψ(
v )]
(10)
j
where νj is the collision frequency for a collision of type j , and C is the ‘collision operator’.
v ) change also because of the free motion of the ions between
The quantities r ∗ ψ(
collisions, ∂t rk∗ = vk∗ , giving an extra term
∂tv (r ∗ ψ) = v ∗ ψ.
Summing together the contributions, we have
v ψ + C ψ
∂t ψ(
v ) = a · ∇
v (r ∗ ψ) + C (r ∗ ψ) + ∂t r ∗ ψ = a · ∇
v ∗ ψ.
Taking the steady-state limit of these equations, we have the ‘balance equations’ which we
will use as a starting point for finding the transport coefficients;
v ψ + C ψ = 0
a · ∇
(11)
v (r ∗ ψ) + C (r ∗ ψ) = −
a · ∇
v ∗ ψ.
(12)
In the expression (10) for the collision term, ‘a collision of type j ’ can be taken as a collision
between an ion and a molecule with velocity in dV around V before impact and ending up
with a relative velocity in dg around g after impact. The frequency of such collisions would
be
νj = [No density of molecules] × [relative velocity] × [cross section]
→ f0 (V ) dV × g × σ1 (g, χ ) dĝ where f0 (V ) is the Maxwellian distribution function of the molecules, normalized to the gas
number density n0 , σ1 the angular cross section, χ the polar scattering angle, and dĝ the
solid-angle element. Replacing the sum over j with an integral over dV and dĝ gives
C ψ = dV f0 (V )[ψ(
v ) − ψ(
v )]gσ1 (g, χ ) dĝ .
(13)
The Li+ –H2 system
1713
In the derivation of this expression, no assumption was made about the type of collision. The
expression is thus equally valid for elastic and inelastic collisions. The specific collision type
is taken into account in the calculation of v (
v , V , ĝ ).
If the molecules have a range of initial states i with relative densities xi and a range of
final states f , one must sum over these;
Cψ →
xi
Cif ψ.
(14)
i
f
3.3. Kramers–Moyal expansion of the collision operator
The expression C ψ in equation (13) can be evaluated conveniently if one inserts for ψ(
v) a
Taylor series expansion around v. The collision operator C is then transformed to a differential
operator on ψ(
v ), and the integrals over dV and dĝ can be performed independently of
the function ψ(
v ). Also, if ψ is a polynomial in v, the resulting expression will give an
exact representation of C ψ, if terms are kept in the differential expansion to the order of the
polynomial. We thus insert in equation (13)
v ψ + · · · .
v ) + (
v − v) · ∇
ψ(
v ) = ψ(
Because of the conservation of momentum in a collision we can write
g − g) ≡ µ0 h
v − v = (m0 /(m + m0 ))(
where h is the change of relative velocity in a collision. This gives
∞
(N )
N
ψ(
v)
C ψ(
v) =
µN
T
(
v
)
(
∇
)
v
0
N =1
T (N ) (
v ) = (1/N!) dV f0 (V )hN gσ1 (g, χ ) dĝ (15)
(16)
which is a Cartesian tensor representation of our ‘Kramers–Moyal expansion’.
The tensor T (N ) (
v ) in equation (16) has 3N components. However, the outcome of a
scattering between an ion with velocity v and a collection of molecules with a perfectly random
orientation of absolutely all properties must of necessity have a cylindrical symmetry around
v, and the components of T (N ) can therefore not all be linearly independent. In a cylindrical
v )N can only contain components constructed from combinations
symmetry, the operator (∇
v and the spherically symmetric operator (∇ 2 ). C ψ may hence be rewritten in the form
of v̂ · ∇
v
[N/2]
∞
N −2L
2 L
∇
ψ(
v ).
(17)
C ψ(
v) =
µN
T
(v)(
v̂
·
)
(∇
)
NL
v
0
v
N =1
L=0
That is, they may rewritten in a form containing only ([N/2] + 1) linearly independent
‘cylindrical Kramers–Moyal coefficients’ for each N . Our scheme for calculating these
coefficients numerically is presented in appendix A.
3.4. Solving the balance equations
If all collision frequencies are constant, the balance equations (11) and (12) can be solved
directly and exactly, as shown by Maxwell. In general, this is however not possible, and we
have in our work resorted to a ‘method of least residues’, as sketched below.
The swarm can be assigned a probability distribution function f (r , v, t) in the sixdimensional (r , v) space, such that the probability of finding the ion in an arbitrary selected
1714
I Røeggen et al
one-ion experiment in a volume element 5r 5
v at time t is 5P = f (r , v, t) 5r 5
v . The
ensemble averages ψ(
v ) and r ∗ ψ(
v ) can then be expressed as
v , t)ψ d
v
ψ(
v ) = dP ψ =
f dr ψ d
v ≡ f (0) (
(18)
v ) = dP r ∗ ψ =
r ∗ f dr ψ d
v ≡ f(1) (
v , t)ψ d
v
r ∗ ψ(
v ) is the velocity distribution function of the ions regardless of their position, and
where f0 (
f(1) (
v ) is a vector function describing the correlation between the velocity v and the mean
random displacement r ∗ (
v ). These functions are by definition normalized to f (0) d
v =1
(1)
∗
and f d
v = r ≡ 0, and must in a steady state be independent of time t.
To find the drift velocity, the velocity distribution function f (0) is expanded in some
suitable and finite basis set of linearly independent functions {φ(
v )}:
v ) :=
f (0) (
imax
ξi φi (
v)
(19)
i=1
and this expansion is inserted in the balance equations (11) for a set of linearly independent
polynomials {ψ(
v )}. Combined with the normalization condition, this gives a linear system
of equations for the expansion coefficients ξi :
ξi φi d
v=1
i
(20)
v + C )ψj d
ξi φi (
a·∇
v = 0.
i
When the matrix elements of these equations have been calculated, the equations are solved
and give values for the expansion coefficients, and the quantities ψj can then be calculated.
With a reasonable choice of the functions φ and ψ, the value of the drift velocity v , for
example, will, one hopes, converge towards the correct value as the size of the basis {φ} is
increased, although this is in no way guaranteed.
To find the diffusion tensor from equations (12), we proceed similarly, the only difference
being that the equations contain inhomogeneous terms which first must be found from the
solution of the drift equations.
Our choice of velocity-space moment and basis functions, and some details about the
calculation of the matrix elements of the collision operator, are given in appendix B.
4. The cross section calculations
The cross sections needed to perform transport calculations can be found from the Li+ –H2
potential surface by methods which are well described in the literature. We have essentially
followed Child (1974) in the formulation of the theoretical framework, and have used routines
from the MOLSCAT program package of Hutson and Green (1994) for the actual computations.
The scattering problem was formulated in a space-fixed coordinate system oriented along
the incident relative velocity, as described by Arthurs and Dalgarno (1960), and the resulting
close-coupled equations were solved using the modified log-derivative method of Manopolous
(1986), to give the corresponding T -matrices. We departed slightly from the strict rigidrotor model, and used ‘the energy-corrected rigid-rotor approximation’ of Lester and Schaefer
(1973) with tabulated rotational levels Ej (Dabrowski 1984) instead of the rigid-rotor values
Ej ∝ j (j + 1).
The Li+ –H2 system
1715
Following Child, we then obtained the T -matrix elements TJ (j m, j m ) describing the
transition from a molecular state (j, m), with chirality m referred to the incident direction, to a
state (j , m ), with chirality m referred to the outgoing direction, from the space-fixed matrix
elements (j l|T J |j l ) as
j
J
l
j
J
l l−l TJ (j m, j m ) =
(2l + 1)(2l + 1)
i (j l|T J |j l )
m −m 0
m −m 0
ll (21)
where the (:::) are 3j -symbols. The angular substate-to-substate cross sections are given by
the corresponding T -matrices as
∞
2
2 −1 J
σ1 (j m, j m ; χ ) = (4kj ) (2J + 1)TJ (j m, j m )dmm (χ )
(22)
J =0
(J )
Dmm
(0, χ , 0),
J
dmm
(χ )
(J )
Dmm
(λ)
=
being a Wigner D-function.
where
The Legendre components σ (χ ) of the cross sections, defined in equation (A.5), are
(λ)
(α, β, γ ) and the integral condition (see
readily found by using the identity Pλ (cos α) = D00
e.g. Edmonds (1957))
A B C
A B C
(A) (B) (C)
3
Daa
(23)
Dbb Dcc dα d cos β dγ = 8π
a b c
a b c
as
σ (λ) (j m, j m ; χ ) = (π/kj2 )
(2J + 1)
(−1)m−m (2J + 1)
×
J
J
m
λ
0
J
−m
J
J
−m
J
m
λ
0
(|TJ |2 − 21 |TJ − TJ |2 ).
For λ = 0, this reduces to usual expression for the total cross section:
σ (0) = (π/kj2 )
(2J + 1)|TJ |2 .
(24)
(25)
J
In transport calculations we can instead of the partial cross sections σ (λ) use ‘transport cross
sections’ σλ (equation (A.8)), which for elastic and weakly inelastic collisions will converge
much faster than the σ (λ) when summing over the total angular momentum index J . The σλ
can be obtained from the T -matrices as
2
σλ (j m, j m ; χ ) = (π/kj )
(2J + 1) fJ (J, m, m , λ)|TJ |2
J
+
1
2
J +λ
gJ (J, J , m, m , λ)|TJ − TJ |2
(26)
J =J −λ
where
gJ (J, J , m, m , λ) = (2J + 1)(−1)m+m
J
m
J
−m
λ
0
J
m
J
−m
λ
0
(27)
and
fJ = 1 −
J +λ
gJ .
J =J −λ
For m = m , the coefficient fJ ≡ 0.
(28)
1716
I Røeggen et al
σ(0) (nm)2
0.100
2
4
6
0.010
8
10
12
14
0.001
0.01
0.10
1.00
10.00
W (eV)
Figure 2. Total inelastic cross sections σ (0) (nm2 ) for collisions between Li+ ions and hydrogen
molecules in initial state j = 0, as functions of the centre-of-mass kinetic energy W (eV). The
final states are indicated on the curves.
σ(0) (nm)2
0.100
3
5
7
0.010
9
11
13
15
0.001
0.01
0.10
1.00
10.00
W (eV)
Figure 3. Total inelastic cross sections σ (0) (nm2 ) for collisions between Li+ ions and hydrogen
molecules in initial state j = 1, as functions of the centre-of-mass kinetic energy W (eV). The
final states are indicated on the curves.
As input for the transport calculations, we determined substate-averaged total cross
sections and transport cross sections:
(σ (0) , σλ )(W ; j, j ) =
1 (0)
(σ , σλ )(W ; j m, j m )
2j + 1 m,m
(29)
for j = (0, 2, 4) and j = (0, 2, . . . , 16) (parahydrogen), and for j = (1, 3) and j =
(1, 3, . . . , 17) (orthohydrogen), and initial centre-of-mass kinetic energies W in the range
[10−4 , 50] eV. Ideally, we should have included all channels, including closed ones, to at least
two j -values above the last open channels (or up to the dissociation limit), but this would have
unduly increased the computational effort. Because of the restricted j -range, the results are
not reliable for W above 3–4 eV.
As an illustration, we show in figures 2–4 total inelastic cross sections for scattering from
the three lowest rotational states j = 0, 1, and 2, in the kinetic energy range W ∈ [0.01, 10] eV.
For j = 2, the superelastic 2 → 0 cross section is included also.
The Li+ –H2 system
1717
σ(0) (nm)2
0.100
0
0.010
4
6
8
10
12
14
0.001
0.01
0.10
1.00
10.00
W (eV)
Figure 4. Total inelastic cross sections σ (0) (nm2 ) for collisions between Li+ ions and hydrogen
molecules in initial state j = 2, as functions of the centre-of-mass kinetic energy W (eV). The
final states are indicated on the curves.
5. Experimental and calculated mobilities
5.1. The basic results
The mobility—that is the ratio between drift velocity and electric field—of Li+ in hydrogen
has been measured by various groups, the most extensive measurements being those of the
Georgia Tech. group (Ellis et al 1976), performed at a gas temperature of 300 K and with
mass spectrometric identification of the ions. The experimental scatter in these experiment
was, however, somewhat larger than we would be comfortable with, and there also seems to
be some systematic error at high fields.
We have made new mobility measurements, at a temperature of 295 K, using the Tyndall–
Powell technique and a thermostated, static-gas, variable-length drift tube. The apparatus has
been described in detail earlier (Løvaas et al 1987). The measured mobilities are estimated to
have an uncertainty of the order of ±0.5% at field-to-density ratios4 E/n0 100 Td, rising
to ±1% in the range 100–200 Td. Above 200 Td, the end effects in the experiment become
very marked, until above 450 Td the drift gap essentially becomes all boundary regions and the
measurements cannot be used to find a mobility value. We consider the measured mobilities
above 200 Td to be uncertain by approximately 2%, but would not be too surprised if there
were unnoticed systematic errors larger than this at the highest E/n0 values.
We have also calculated the mobilities, as well as the ratios between the longitudinal and
transverse diffusion coefficients and the mobility, DL /µ and DT /µ, for the same temperature
and E/n0 range as in the experiments, assuming a thermal equilibrium composition of the
hydrogen gas. Final rotational states up to jfmax = 17 were included in the calculations.
Typically, the cross sections were calculated at 100 incident kinetic energy values per decade
in the low-energy range where quantum oscillations are prominent, and at 10–40 values per
decade at higher energies.
The measured and calculated mobilities and diffusion coefficients are listed in table 2,
with the mobility given as the reduced value
K0 = NL−1 vdr /(E/n0 )
(30)
where NL = 2.6868×1025 m−3 is Loschmidt’s number. The table also shows calculated values
4
1 Td = 10−21 V m2 .
1718
I Røeggen et al
Table 2. Experimental and calculated mobilities, calculated D/µ ratios and mean laboratory and
centre-of-mass energies Wlab and Wcm , for 7 Li+ ions in normal hydrogen at 295 K.
E/n0
(Td)
2
5
10
20
30
40
50
60
70
80
90
100
110
120
130
140
160
180
200
220
240
260
280
300
350
400
450
Theory, jfmax = 13
Experiment,
K0
(cm2 V−1 s−1 )
K0
(cm2 V−1 s−1 )
DL /µ
(V)
DT /µ
(V)
Wlab
(eV)
Wcm
(eV)
12.26
12.27
12.27
12.35
12.54
12.88
13.52
14.60
16.00
17.68
19.24
20.50
21.49
22.22
22.96
23.14
23.08
23.01
22.65
22.33
22.20
21.88
21.37
20.88
20.59
12.29
12.29
12.29
12.30
12.33
12.42
12.61
12.98
13.63
14.68
16.12
17.78
19.37
20.69
21.69
22.38
23.11
23.30
23.26
23.10
22.89
22.68
22.48
22.25
21.90
21.68
21.62
0.0255
0.0257
0.0267
0.0308
0.0379
0.0497
0.0703
0.1125
0.206
0.388
0.649
0.904
1.072
1.1409
1.15
1.14
1.14
1.19
1.29
1.45
1.63
1.84
2.10
2.41
3.31
4.58
6.39
0.0254
0.0255
0.0259
0.0273
0.0297
0.0333
0.0386
0.0468
0.0607
0.0840
0.1190
0.1641
0.215
0.268
0.322
0.376
0.484
0.595
0.712
0.839
0.975
1.121
1.279
1.450
1.950
2.57
3.36
0.038
0.039
0.043
0.058
0.084
0.122
0.174
0.25
0.36
0.54
0.81
1.20
1.67
2.2
2.8
3.4
4.6
5.8
7.1
8.4
9.8
11
13
15
19
25
31
0.038
0.039
0.039
0.043
0.049
0.057
0.068
0.085
0.110
0.150
0.21
0.30
0.40
0.52
0.65
0.78
1.05
1.33
1.61
1.91
2.2
2.6
2.9
3.3
4.3
5.5
7.0
of the mean ion energy Wlab = mv 2 /2 and an estimated ‘mean centre-of-mass energy’, taken
to be
m0
m
Wcm ∼
Wlab +
(3kB T /2).
m + m0
m + m0
5.2. Comparison between experimental and theoretical mobilities
Figure 5 shows the mobility as a function of E/n0 . Also shown are values calculated using
jfmax = 9 instead of jfmax = 17 (dashed curve), and values calculated using the SCF potential
surface of Lester (1971).
There is a good agreement between experimental and calculated values up to around
220 Td, while the SCF potential as expected gives rather inaccurate predictions of the mobilities.
To exaggerate the differences between the mobilities obtained in various ways, we have
plotted (figure 6) the mobility values relative to values calculated using jfmax = 13. In the
range from 10 to 220 Td, which corresponds to a Wcm -range from thermal to about 2 eV,
the measured mobilities are of the order of 0.5% lower than the theoretical ones, which is
comparable to the experimental accuracy in a single measurement, but more than we expect of
The Li+ –H2 system
1719
24.0
22.0
K0 (cm2/Vs)
20.0
18.0
16.0
14.0
12.0
0
100
200
300
400
E/n0 (Td)
Figure 5. The reduced mobility K0 (cm2 V s−1 ) for Li+ ions in normal hydrogen at 295 K as a
function of E/n0 (Td). Filled circles: experimental values. Full curve: calculated values, with
jfmax = 17. Dashed curve: calculated values, with jfmax = 9. Dash–dot: calculated from the SCF
potential surface of Lester (1971), with jfmax = 9.
1.06
1.04
jfmax =9
K0rel
1.02
jfmax =13
1.00
jfmax =17
0.98
0.96
0
100
200
300
400
E/n0 (Td)
Figure 6. Mobilities for Li+ ions in normal hydrogen at 295 K relative to the values calculated using
rotational states up to jfmax = 13, K0rel . Filled circles: present experimental values. Squares: Ellis
et al (1976). Dashed curve: calculated using jfmax = 9. Dash–dot: calculated using jfmax = 17.
a systematic error. The difference is most probably due to the use of the rigid-rotor model with
a constant H–H distance instead of a more complicated ‘vibrating-rotor’ model. It does not
stem from the transport calculations: as a check on those we also ran Monte Carlo simulations,
using integrated cross sections:
χ
σI (χ ) = 2π
σ1 (χ ) d cos χ ξ =−π
as input, for all E/n0 values up to 120 Td, and the mobility values thus obtained never differed
by more than 0.14% from the kinetic-theory values.
Above 220 Td, the disagreement between experimental and theoretical mobilities increases
rapidly with increasing E/n0 . This is apparently not due to the size of the rotational basis:
1720
I Røeggen et al
26
B
24
K0 (cm2/Vs)
22
A
20
18
16
14
12
0
100
200
300
E/n0 (Td)
Figure 7. The reduced mobility K0 for Li+ ions in normal hydrogen at 295 K, as a function of
E/n0 . Curve A shows values calculated using an isotropic interaction potential V = V0 (r), i.e.
neglecting higher-order Legendre components. Curve B was calculated using a full interaction
potential but neglecting inelastic energy losses in the transport calculations. The full curve and the
filled circles give our best calculated and experimental values, respectively.
a comparison between the data calculated with jfmax = 9, 13, and 17 would indicate that
the calculations have converged with respect to the jfmax -value up to around 350 Td, and the
remaining difference between 220 and 350 Td must have some other cause. The collisions
become rather violent in this energy range, and we would expect that our model with a
constant H–H distance would no longer be very good, and possibly vibrational excitation
will also become important. However, in the high-E/n0 range the mean kinetic energy in
the laboratory system, Wlab , is becoming comparable to the total available energy in the drift
tube—the voltage over the drift gap was not increased above 150 V. It may thus also be that the
mean velocity in the swarm did not quite reach the steady-state value vdr at the highest E/n0
values, although apparently the mean arrival time increased proportionally to the increase in
drift length for the longest drift lengths used.
5.3. The importance of anisotropy and inelasticity
The anisotropy of the interaction potential influences the angular distribution in the scattering,
and is also the cause of inelastic transitions.
To get an insight into how anisotropy and inelasticity influence the transport properties,
we have performed two different model calculations: (A) using only the isotropic part V0 of
the interaction potential in the scattering calculations, resulting in curve A in figure 7; and
(B) using the full anisotropy in the scattering calculations, but neglecting the inelastic losses in
the transport calculations, i.e. setting g = g in equation (A.7), resulting in curve B in figure 7.
The anisotropies clearly have a significant influence on the angular distribution in the
scattering even at low energies, and the use of only the isotropic part of the potential is thus
never a very good approximation. For the scattering calculations, this has the implication that
closed channels must also be included to find the low-energy elastic cross sections.
The inelastic losses are unimportant at very low E/n0 values, as is well known: even with
inelastic channels open, detailed balancing will remove the effect on the transport properties
when the ion distribution function is close to thermal. With increasing E/n0 values and
centre-of-mass energies, the inelastic losses do, on the other hand, reduce the mobilities by up
The Li+ –H2 system
1721
24
295K
K0 (cm2/Vs)
22
20
18
16
78K
14
12
0
100
200
300
E/n0 (Td)
Figure 8. The calculated reduced mobility K0 as a function of E/n0 for Li+ ions in normal
hydrogen and in parahydrogen, at 78 and 295 K. Full curves: normal hydrogen. Broken curves:
parahydrogen.
to about 10%, and the mobility measurements are thus quite sensitive to the magnitudes of the
corresponding cross sections.
5.4. Mobilities in normal hydrogen and parahydrogen
Hydrogen gas can be regarded as a mixture of two components, parahydrogen with nuclear
spin S = 0 and orthohydrogen with nuclear spin S = 1. In parahydrogen only even rotational
states (j = 0, 2, . . .) are populated, and in orthohydrogen only odd states (j = 1, 3, . . .).
The transition between the two spin states will under normal experimental conditions only
occur in the presence of some catalyst, most prominently some paramagnetic substance. The
mixing ratio between the two components will therefore often be ‘frozen’, and not vary if the
temperature is changed (see e.g. Farkas (1935)).
At high temperatures, the equilibrium mixing ratio is northo :npara = 0.75:0.25, and the
equilibrium ratio at room temperature is practically the same (0.7494:0.2506 at 295 K). A
hydrogen gas with this composition is usually called ‘normal hydrogen’.
Pure parahydrogen can be produced by condensing hydrogen on a catalyst at liquid
hydrogen temperature, and then slowly evaporating it. The gas can then be used in experiments
at higher temperature.
If measurements are made in parahydrogen at liquid nitrogen temperature (78 K), 99.5%
of the gas will be in the rotational ground state j = 0, and it will be much easier to analyse
the measurements theoretically than if more initial states were present. To follow this up and
see whether such experiments really would give results different to those from experiments in
normal hydrogen, we have calculated mobilities in both normal hydrogen and parahydrogen,
at room temperature 295 K as well as liquid nitrogen temperature 78 K. The results are shown
in figure 8.
At 78 K the predicted mobilities in parahydrogen are up to 7% lower than in normal
hydrogen. This is qualitatively as expected, and occurs because parahydrogen has a lower
inelastic threshold (0.0439 eV) than orthohydrogen (0.0728 eV) and also because of the
relatively large value of the j = 0 → 2 cross section (see figure 2).
At 295 K, parahydrogen is a nearly 1:1 mixture of the rotational states j = 0 and 2.
The mean energy losses in this parahydrogen mixture are not far from the energy losses in
1722
I Røeggen et al
orthohydrogen in the ground state and thus at 295 K the mobilities are almost the same in
parahydrogen and normal hydrogen.
6. Conclusions
The mobility of Li+ ions in H2 at room temperature has been calculated ab initio, i.e. with the
non-relativistic Schrödinger equation for the system as the basic input.
At E/n0 values below 220 Td the resulting mobility values are systematically about 0.5%
higher than our experimental values. This difference is comparable with the experimental
uncertainty. It was in fact quite surprising to find that the rigid-rotor model used in the
scattering calculations could give such good predictions of the transport properties.
The use of a more accurate ‘vibrating-rotor’ model might possibly remove the discrepancy,
but that would necessitate the calculation of interaction potential surfaces at more than one H–H
distance, and would also substantially increase the computing requirements in the scattering
calculations.
Above 220 Td, there is an increasing discrepancy between the calculated and measured
mobilities. This is presumably at least partly due to the breakdown of the rigid-rotor
approximation for ‘violent collisions’.
Calculations of the mobility for normal hydrogen and parahydrogen at liquid nitrogen
temperature, show a difference of up to 7%. A classical calculation would not have shown
any difference, and would thus clearly have been inadequate for this case. Experimental
measurements at this temperature have never been performed, but would have given an
interesting and stricter test of the theory than the room temperature measurements.
Acknowledgments
We are most grateful to Professor J M Hutson for providing us with the source code for the
MOLSCAT program package, and to Dr M Syvertsen for valuable assistance in implementing
the scattering program. We would also like to thank Dr M T Elford and Dr K Kumar for many
helpful discussions and suggestions.
Appendix A. The cylindrical Kramers–Moyal coefficients
We present here our scheme for the calculation of the cylindrical Kramers–Moyal coefficients
TN L (v) in equation (17). For a comprehensive background, we refer the reader to the papers
by Kumar et al (1980) and Larsen et al (1988). In these papers, the collisions were assumed to
be elastic. We have removed this restriction—the modifications needed were not very drastic,
and the somewhat heavy algebra needed to go from a Cartesian to a cylindrical representation
is the same whatever the collision type might be.
We start at the outermost level in the expression for TN L , and work inwards. TN L may be
written as
∞
TN L (v) =
tN(n)L ξnl (v)
(A.1)
n=0
where
tN(n)L
= (−1)
l−r
2n
2l + 1
n! [2(n + l) + 1]!!
l
2(l − r)
r
l
(A.2)
The Li+ –H2 system
1723
with l = N − 2n and r = L − n. The function ξnl (v) is given by
ξnl (v) = dV f0 (V )Pl (ĝ · v̂)g N +1 σnl (g)
∞
= 4πn0 (m0 /2kB T )3/2
exp[−(m0 /2kB T )(v − g)2 ]
0
× g 2n+l+1 σnl (g)[(π/2z)1/2 Il+1/2 (z) exp(−z)]g dg.
(A.3)
Here, Pl is a Legendre polynomial, Il+1/2 a modified Bessel function, z = m0 vg/kB T , and kB
Boltzmann’s constant.
The function σnl has the dimension of a cross section, and can be written as
n+l
λ (λ)
anl
σ (g)
(A.4)
σnl (g) =
λ=0
where
σ
(λ)
= 2π
Pλ (cos χ )σ1 (g, χ ) d cos χ
(A.5)
and
2n+l
h
2λ + 1
Pl (cos κ)Pλ (cos χ ) d cos χ .
(A.6)
2
2g
Here, g, g , and h are the sides of a triangle with χ = (g, g ) and κ = (g, h), and one thus
has
h = g 2 + g 2 − 2gg cos χ
(A.7)
cos κ = (g − g cos χ )/ h.
In a collision
with an energy difference between the initial and final state of the molecule,
√
g/g = (1−/W ), where W = mr g 2 /2 is the centre-of-mass kinetic energy before collision.
λ
The integrand in the expression for the coefficients anl
(equation (A.6)) is a polynomial
both in g /g and in cos χ , and the integral can written out in full. In practice, we have found it
most convenient to evaluate it using a Gauss–Legendre quadrature, which gives a ‘numerically
exact’ value.
The quantities σnl (equation (A.4)) can alternatively be expressed in terms of the total
cross section σ (0) and ‘transport cross sections’ σλ , defined by
λ
=
anl
σλ = σ (0) − σ (λ)
(A.8)
as
σnl (g) = (1 − g /g)2n+l σ (0) (g) −
n+l
λ=1
λ
anl
σλ (g)
(A.9)
where the first term disappears for elastic collisions.
When there are a range of initial molecular states i with relative densities xi and a range
of final states f , the expression for σnl to be used to find the function ξnl from equation (A.3)
must be evaluated as
if
xi
σnl .
(A.10)
σnl (g) =
i
f
The final integral in equation (A.3) must be evaluated numerically, and we have used an
adaptive trapezoidal rule—there may be rapid quantum oscillations in the cross sections, and
more sophisticated quadrature formulae do not then work particularly well.
With the ξnl found, the Kramers–Moyal coefficients TN L are finally found by insertion in
equation (A.1).
1724
I Røeggen et al
Appendix B. Moment and basis functions
For the moment functions {ψ}, we have chosen polynomials of form
v ) → ψlr (
v ) = v l+2r Ylm (v̂)
ψj (
with
Ylm (v̂) →
Pl (â · v̂)
Pl1 (â · v̂)(â × v̂)
(B.1)
drift and longitudinal diffusion
transverse diffusion
where Pl1 is an associated Legendre polynomial. From equation (17) we then get
C ψlr (
v ) = −νlr (v)ψlr (
v)
(B.2)
with ‘collision frequencies’ νlr given by
νlr =
l+2r
µN
0
N =1
[N/2]
L=0
βNlrL v −N TN L (v)
(B.3)
where
r!
(l + 2r − 2L)! (2l + 2r + 1)!!
.
(r − L)! (l + 2r − N )! (2l + 2r + 1 − 2L)!!
βNlrL = 2L
(B.4)
For the basis functions {φ} we have chosen functions of form
v ) → φlr (
v ) = x l+2r exp(−x 2 )Ylm (v̂)
φi (
with
x (v) =
2
v
ṽ dṽ/C(ṽ)
(B.5)
(B.6)
0
and C(v) a speed-dependent ‘temperature function’.
To estimate a reasonable form for C(v), we first determine an effective centre-of-mass
collision frequency for momentum transfer from the momentum balance equation:
νeff (v) = (1 + m/m0 )ν10 (v) = T10 (v)/v
(B.7)
and a ‘speed-dependent drift velocity’ as
vd (v) = a/ν10 (v) = (1 + m/m0 )a/νeff (v).
(B.8)
C should conform with the ion temperature estimated by Wannier (1953) in the bulk of the
distribution, while in the high-energy snout of the distribution, it should fall off approximately
as exp(−νeff a/v). In line with this, we have used a temperature function
C(v) = kB T + (h21 + h22 )/(h1 + h2 )
with
h1 = va/ν10 = (1 + m/m0 )vvd (v)
h2 = (1/3)(m + m0 )vd2 (v).
(B.9)
v )C ψj (
v ) d
v , were subsequently
The matrix elements of the collision operator, Cij = φi (
found by numerical integration, transforming to x as an integration variable and using a 32point Gauss–Laguerre quadrature.
Some additional flexibility in the basis set was introduced by varying the effective collision
frequency, νeff (v) → kνeff (v), with the factor k chosen to give an optimum numerical
convergence in the solution of the drift and diffusion equations. Typically, we would have
0.8 < k < 1.4.
and
The Li+ –H2 system
1725
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