The Atomic Nucleus

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The Atomic Nucleus
A.-M. Mårtensson-Pendrill and M.G.H. Gustavsson
Volume 1, Part 6, Chapter 30, pp 477–484
in
Handbook of Molecular Physics and Quantum Chemistry
(ISBN 0 471 62374 1)
Edited by
Stephen Wilson
 John Wiley & Sons, Ltd, Chichester, 2003
Chapter 30
The Atomic Nucleus
A.-M. Mårtensson-Pendrill1 and M.G.H. Gustavsson1,2
1
2
Göteborg University and Chalmers University of Technology, Göteborg, Sweden
Ericsson Microwave Systems AB, Sensors and Information Networks, Mölndal, Sweden
1 Introduction
2 Nuclear Charge Distributions
3 Finite Nuclear Charge Distributions
4 Physics Near the Nucleus
5 Concluding Remarks
References
1
1
4
6
7
7
1 INTRODUCTION
disadvantages from a computational point of view? How
should calculated results be presented in order to make
possible meaningful comparisons between results obtained
using different approaches? How much is known experimentally and how is that information best included in the
calculations? These are a few of the questions addressed in
this chapter.
This chapter focuses mainly on the distribution of
charge, which, however, also influences other properties.
In Section 4, we give a brief discussion of physical effects
close to the nucleus.
As molecules and heavy atoms are invoked for studies of
fundamental interactions and properties, the demand for
more accurate descriptions of the wave function within
and close to the nucleus grows more stringent. With the
generally increasing precision of quantum chemistry calculations, the simple point nucleus approximation resulting
in the well-known Z/r potential is no longer completely
adequate. The inadequacy becomes more and more evident
as quantum chemistry takes on the challenge of molecules
including heavy, or even super-heavy, atoms: For large Z,
the behaviour close to the nucleus becomes too singular
for a point-nuclear charge, and for the – still hypothetical – case of Z > 137, the Dirac equation breaks down.
For finite nuclear distributions, the possible range extends
much further.
Many properties depend on details of the nuclear structure. What are the alternatives for the nuclear distributions used in calculations? What are their advantages and
2 NUCLEAR CHARGE DISTRIBUTIONS
Handbook of Molecular Physics and Quantum Chemistry,
Edited by Stephen Wilson. Volume 1: Fundamentals.  2003
John Wiley & Sons, Ltd. ISBN: 0-471-62374-1.
Experimental information about charge distributions is
derived from many sources, including electron scattering. The experimental data indicate that R0 1.2 fm
The early theorists, without access to computers, had strong
reasons to use analytical descriptions of charge distributions
and potentials, which enabled series expansions of analytical solutions of the wave functions within and close to the
nucleus. A common choice was the homogeneous charge
distribution inside a radius R = R0 A1/3 , where A is the
mass number of the nucleus. The most important parameter for many properties is the expectation value r 2 , which
has the value 3R 2 /5 for the homogeneous nucleus. This
simple distribution gives the correct analytical behaviour of
the electronic wave functions at r = 0 and has been used
in many early analyses and is discussed in more detail in
Section 3.2. These expansions are also useful for a general
understanding of the effects involved.
2.1 Experimental information
2 Approximate separation of electronic and nuclear motion
gives a reasonable approximation for the homogeneous
distribution. Clearly, the tail of a real nucleus is less sharp
than indicated by the homogeneous distributions. It is often
described in terms of a ‘skin thickness’ t, defined as the
distance in which the charge density falls from 90% of its
central value to 10%. Experiments indicate that t is about
2.3 fm for most nuclei.
The primary data from electron-scattering experiments
are expressed in terms of a ‘Fourier–Bessel expansion’.
Figure 1 shows the resulting charge distribution for several
different nuclei. It is possible to use these data directly using
a numerical approach, as shown in Section 3.1. Tabulations
often give values relating to additional parameterizations, in
particular, the two and three-parameter Fermi distributions
as well as the Gaussian expansions,(1,2) that are discussed
in more detail in the following text.
Another source are muonic X-ray energies. These
probe somewhat different moments, the ‘Barrett moments’
r k e−αr , of the nuclear distribution. Nevertheless, the
results are quoted also in terms of r 2 .(3)
Optical isotope shifts provide an important source of
complementary information, particularly for chains of
radioactive isotopes. It is found that the changes in charge
radius along an isotope chain are, in general, smaller than
those indicated by the aforementioned textbook formula.
The isotope shifts also reveal an ‘odd–even staggering’
of the r 2 values, providing evidence of nuclear shell
structure.(4,5) A spectacular recent application is the precision determination of the ‘deuteron structure radius’ from
the hydrogen-deuterium isotope shift of the 1s–2s twophoton resonance.(6)
2.2 Nuclear distributions and electronic wave
functions
The electrostatic potential energy, V (r), for an electron in
a spherically symmetric charge distribution, ρ(r), is given
by the well-known expression
4π r
e
2
ρ(r )r dr V (r) = −
4π0 r 0
∞
1
2
+ 4π
ρ(r )r dr
r
r
Outside the nucleus, this expression reduces to the same
potential, −Ze2 /4π0 r, as from a point nucleus. For a
0.10
0.09
0.08
0.07
t
Pb
ρ(r )
0.06
0.05
t
Ni
0.04
58Ni
0.03
12C
208Pb
0.02
t
O
0.01
16O
0
0
1
2
3
4
5
6
7
8
9
10
r (fm)
Figure 1. The radial charge distribution for several nuclei determined by electron scattering. The skin thickness t is shown for O, Ni,
and Pb; its value is roughly constant at 2.3 fm. These distributions were obtained from Fourier–Bessel expansion data tabulated by
de Vries, H., de Jager, C.W., and de Vries, C. (1987).(1)
The atomic nucleus 3
homogeneous distribution, where ρ(r) = 3Ze/4πR 3 within
a nuclear radius R, the electrostatic potential for r ≤ R is
V (r) = −
1 Ze2
4π0 r
3r
r3
−
2R
2R 3
G is valid for the ‘ls-convention’ for the coupling of spin
and orbital angular momentum.)
In the following text, we use atomic units, where e =
me = 4π0 = h̄ = 1 and c = 1/α ≈ 137. (A word of caution: Relativistic atomic calculations form a meeting ground
between traditional atomic and molecular calculations and
field theoretical approaches, where the equations are instead
expressed in ‘natural units’, where c = me = 4π0 = h̄ = 1
and e2 = α ≈ 1/137).
For a point nucleus, the leading term is r γ , where γ =
√ 2
[κ − (Zα)2 ] for both F (r) and G(r). The solution r −γ is
singular for r = 0 and must be rejected for point nuclei. For
an extended nucleus, however, this term enters with a small
coefficient in the matching of outer and inner solutions.
Inside an extended nucleus, the leading term in the
upper component is r l . For j = l + 1/2 orbitals, the lower
component carries an extra factor r, whereas for j =
l − 1/2 orbitals, the lower component has a small term r l−1
that dominates for very small r.
We note that atomic orbitals with j = 1/2 (i.e., s and
p1/2 ) have an appreciable probability within the nucleus,
where their density is essentially constant. The effect
(1)
We note that the potential is essentially constant at V0 =
−(1/4π0 ) × (3Ze2 /2R) very close to the origin. This
determines the behaviour of the orbitals for very small r,
which is obtained by a series expansion of the coupled
radial equations from the Dirac equation:
κ
d
−
G(r) = [ε − V (r)] F (r)
−h̄c
dr
r
d
κ
F (r) = ε + 2me c2 − V (r) G(r) (2)
h̄c
+
dr
r
where κ = ∓(j + 1/2) for j = l ± 1/2, ε is the binding
energy, and F (r) and G(r) are the radial parts of the ‘large’
and ‘small’ components, respectively, of a relativistic wave
function. (The sign relation in equation (2) between F and
−0.30
−0.35
〈r 2〉1/2 = 5.523
−0.40
E + 104318 eV
〈r 2〉1/2 = 5.521
−0.45
〈r 2〉1/2 = 5.519
−0.50
〈r 2〉1/2 = 5.517
−0.55
〈r 2〉1/2 = 5.515
−0.60
1155
1160
1165
1170
〈r 4〉
1175
1180
1185
1190
(fm4)
Figure 2. The dependence of the ground state energy of hydrogen-like bismuth on the nuclear charge distribution. The lines show
results for constant values of the nuclear root-mean-square (rms) radius and were obtained by fitting a linear expansion in r 4 to the
results of explicit numerical calculations (marked by crosses) for different values of the nuclear parameters.
4 Approximate separation of electronic and nuclear motion
of the nuclear distribution on atomic properties then
becomes
proportional
to the expectation value, r 2 =
2
r ρ(r) dV / ρ(r) dV , of the nuclear distribution. This
observation is well known from studies of optical isotope
shifts, which are changes in transition frequency between
different isotopes of an element. The ‘field’ or ‘volume’ isotope shift is the part due to the different charge
distributions and is expressed as δν = F δr 2 , where F =
(2π/3)(Ze2 /4π0 )|(0)|2 .
For heavy nuclei, the next terms in the expansion of
the wave functions also become significant. Thus, higher
moments δr 4 , δr 6 , . . . of the nuclear distribution also
contribute. As an illustration, Figure 2, shows the 1s energy
for hydrogen-like Bi (Z = 83) for Fermi distributions with
various values of r 2 and r 4 .
The electron density then varies within the nucleus,
giving contributions from higher moments. These ‘Seltzer
corrections’(7) can be accounted for by including a factor κ,
which is, for example, about 0.92 for Bi.(8) These factors are
of course not needed if the distribution is included directly
in the calculation.
In the following text, we consider in more detail the
expression for potentials and orbitals for homogeneous,
Fermi, and Gaussian models of the nuclear distributions.
Several other distributions have been considered in detail by
Andrae,(9) who provides expressions for wave functions and
energy corrections for a wide range of nuclear distributions
as well as a comparison of first-order energy corrections
for one-electron systems with exact ones for nonrelativistic
and relativistic treatments.
3 FINITE NUCLEAR CHARGE
DISTRIBUTIONS
The accuracy is limited by the finite range of q values.
The Fourier–Bessel analysis, introduced by Dreher and
co-workers,(10) assumes a cut-off radius for the nucleus and
uses an analytical approximation for the form factor F (q)
for q values beyond the maximum available value. The
charge distribution is then written as
νπr aν j0
for r ≤ R
ρ(r) =
R
ν
0
for r > R
where j0 (x) = sin(x)/x is the zeroth-order spherical Bessel
function. The first few coefficients are deduced from experiments and are given in tabulations.(1,2)
The expressions for the total charge Q and the potential
energy V (r) are
R 3
νπ(−1)ν+1
νπ
ν

R 2 νπr 


−4π
aν
j0
− (−1)ν


νπ
R
ν
V (r) =
for r ≤ R



Z

−
for r > R
r
Q = 4π
aν
This potential has been applied in the case of the
ground state hyperfine structure (hfs) of hydrogen-like
Bi.(11)
The second moment, r 2 , of the charge distribution
can also be expressed in terms of the Fourier–Bessel
parameters:
r 2 =
5
4π R
aν
νπ(−1)ν 6 − (νπ)2
Q ν
νπ
In this section, we give a brief presentation of models
commonly used to account for the experimentally obtained
information about the nuclear distribution. We also present
the resulting potential and discuss the related initial conditions and orbital behaviour for small r values.
and is usually quoted in tabulations as well as values
for the Fermi and Gaussian parametrizations, which are
discussed later.
3.1 Electron scattering and the Fourier–Bessel
expansion
The uniform distribution of charge within a radius R
was discussed briefly in Section 2.2. Obviously, there is a
discontinuity in the charge distribution, ρ(r), at the nuclear
boundary, leading to a discontinuity in the second derivative
of the potential at r = R. Care must thus be taken in
numerical integration schemes to avoid spurious effects
related to the nuclear boundary. In numerical approaches,
the discontinuity also leads to slower convergence with the
grid spacing.(12)
To find a detailed description for the orbital behaviour
for very small r, a power series Ansatz for the orbitals
The data from electron scattering are analysed using the
Plane Wave Born Approximation, and the charge distribution is obtained as the Fourier transform of the form
factors F (q), giving for a spherically symmetric charge
distribution
sin(qr) 2
1
F (q)
q dq
ρ(r) =
2
2π
qr
3.2 Uniform distribution and orbital expansions
The atomic nucleus 5
is inserted together with the expression for the potential
equation (1) in the radial Dirac equation (2).
To facilitate the notation in the orbital expansion, we
write the potential in the form V (r, R) = V0 + V2 r 2 , where
V0 = −3Z/2R and V2 = −V0 /3R 2 for the homogeneous
distribution. The other potentials discussed in the following
text lead, in addition, to terms of the type V3 r 3 and higher,
which affect the higher-order terms in the series expansion
of the orbitals. Using these parameters and the binding
energy and keeping the lowest-order terms gives for
κ<0
(ε − V0 )(ε + 2c2 + V0 ) |κ|+2
F (r) = N r |κ| −
·
·
·
r
2c2 (1 + 2|κ|)
ε − V0
(ε − V0 )2 (ε + 2c2 + V0 )
r |κ|+1 +
G(r) = N −
c(1 + 2|κ|)
2c3 (1 + 2|κ|)(3 + 2|κ|)
V2
r |κ|+3 · · ·
+
c(3 + 2|κ|)
and for κ > 0
(ε − V0 )(ε + 2c2 − V0 )
F (r) = N r κ+1 −
2c2 (3 + 2κ)
V2 (1 + 2κ)
κ+3
+
r
···
(ε + 2c2 − V0 )(3 + 2κ)
c(1 + 2κ) κ ε − V0 κ+2
G(r) = N
r
−
·
·
·
r
ε + 2c2 − V0
2c
where N is a normalizing constant.
The energy dependence of these expressions is very
weak, since the binding energy ε is much smaller than the
potential energy V0 = −3Z/2R at r = 0, particularly for
low Z.
short-range character of the strong interaction, which holds
the nucleus together.
The shape of an arbitrary nuclear distribution can often
be adequately described by the moments r 2n of the
distribution. Varying the parameter in the Fermi distribution
makes it possible to study the influence of the higher nuclear
moments, r 2n , given to a good approximation,(11) by the
relations
3 2
c +
5
3
r 4 ≈ c4 +
7
3
r 6 ≈ c6 +
9
r 2 ≈
7 2 2
πa
5
18 2 2 2
πa c +
7
11 2 2 4
πa c +
3
31 4 4
πa
7
239 4 4 2 127 6 6
πa c +
πa
15
5
The electrostatic potential cannot be obtained analytically
for the Fermi distribution equation (3). However, convenient expressions for the evaluation of the potential have
been derived(15) and are used in the procedure adopted in
the general atomic structure package of computer programs
GRASP2 ,(16,17) and are summarized briefly in the following
text.
The radial integrals can be written in terms of infinite
series, which converge to machine precision with just a
few terms included. The potential energy for r < c can be
written as
rπ2 a 2
3ra 2
3r
r −c
r3
−
S
− 3+
2c 2c
2c3
c3 2
a
3
3
6a
6a
c
r −c
+ 3 S3
− 3 S3 −
c
a
c
a
Vr<c (r) = −
N
r
and for the case r > c, we have
3.3 Fermi distribution
The two-parameter Fermi model gives a realistic description
of the nuclear distribution,(13,14) and at the same time
provides considerable flexibility in the analysis:
ρ(r) =
ρ0
1 + e(r−c)/a
(3)
where c is the half-density radius and a is related to the
skin thickness t by t = (4 ln 3)a, giving values of a that is
roughly constant at 0.524 fm. For vanishing skin thickness,
the Fermi form reduces to the homogeneous distribution
with a radius R = c.
The shape of the Fermi distribution is identical to
that of the Woods–Saxon potential. The analogy between
the nuclear distribution and potential is related to the
N
π2 a 2
3ra 2
r −c
1 + 2 + 3 S2 −
Vr>c (r) = −
r
c
c
a
3
3
6a
r −c
6a
c
+ 3 S3 −
− 3 S3 −
c
a
c
a
where the Sk functions are given by
Sk (x) =
∞
n=1
(−1)n
exn
nk
and the normalization condition gives
N=
Z
π2 a 2
6a 3 c 1 + 2 − 3 S3 −
c
c
a
6 Approximate separation of electronic and nuclear motion
In the expression for the potential within the nucleus, we
recognize the constant term V0 = −3Z/2c from the uniform distribution for the case of vanishing skin thickness.
Gaussian tail. The charge distribution is then expressed
as
2
2
Ai e−[(r−Ri )/γ] + e−[(r+Ri )/γ]
ρ(r) =
i
3.3.1 Deformed nuclei
Generalizations of the Fermi model to describe deformed
nuclei have been applied, for example, in the study of
energies for highly charged uranium.(18,19) The nuclear
radius parameter c in the Fermi distribution equation (3)
is then replaced by
R(r) = c 1 + β2 Y20 (r) + β4 Y40 (r)
(4)
which includes an explicit dependence on the angular
coordinates.
3.3.2 The three-parameter Fermi model
The three-parameter Fermi distribution makes it possible to
reproduce the small dip found in the charge distribution for
small r, as seen in Figure 1.
3.4 Gaussian expansions
Many quantum chemistry programs expand the electronic
wave functions in terms of Gaussian-type functions. Using
a Gaussian charge distribution then has the advantage of
making it possible to evaluate electron–nucleus interactions
using the same integral routines as the electron–electron
interactions. Visscher and Dyall(12) in their comparison of
different approaches use a one-parameter Gaussian distribution ρ(r) = Z (ξ/π)3/2 exp(−ξr 2 ) where the width is
determined from the experimental rms radius ξ = 3/2r 2 ,
giving a potential related to the error function V (r) =
√
−(Z/r) × erf[( ξ)r]. Visscher and Dyall(12) found that
this one-parameter Gaussian potential reproduced about
99% of the correction to the energies for high Z, whereas
the uniform distribution gave about 100.1% of the correction, indicating the importance of the higher r 2n moments.
In order to obtain a model-independent fit of the charge
distribution, a parametrization of the experimental data
into a sum of Gaussians was suggested by Sick.(20) The
width γ of the Gaussians is chosen to be equal to the
smallest width of the peaks in the nuclear radial wave
functions calculated by the Hartree–Fock method. Only
positive values of the amplitudes of the Gaussians are
allowed so that no structures narrower than γ can be created
through interference. An advantage of the use of Gaussians
is that values of ρ(r) at different values of r are decoupled
to a large extent because of the rapid decrease of the
where the coefficients Ai are given by
Ai =
2π3/2 γ3
ZeQi
1 + 2Ri2 /γ2
In this definition, the values of Qi indicate the fraction
of the charge contained in the ith Gaussian, normalized
such that
Qi = 1
i
The second term for each parameter, centred around a
nonphysical negative Ri value, is included to ensure that the
slope of the distribution is indentically zero at the origin. An
additional advantage over the Fourier–Bessel expansion is
that the sum-over Gaussians by construction can give only
nonnegative values for ρ(r).
The relevant parameters can be found tabulated, as in
the case of the Fourier–Bessel expansion.(1,2) For highspin nuclei, it might be valuable to include an angular
dependence in analogy with equation (4). In general, they
give a better description of the inner part of the nucleus
better than the Fourier–Bessel expansion, which, on the
other hand, gives a better description in the outer part
where the Gaussian tail decays too rapidly. Since an
electronic wave function is more sensitive to the outer part
of the nuclear distribution, the Fourier–Bessel expansion is
preferable. On the other hand, the main part of the energy
correction (which in itself is only 0.12% for Z = 100)
is accounted for already with the one-parameter Gaussian
potential, so the Gaussian expansion should clearly be
adequate for most applications.
4 PHYSICS NEAR THE NUCLEUS
The energy of a system is only weakly influenced by details
in the nuclear charge distribution. Even for Z = 100, using
an extended nuclear charge gives only an 0.1% reduction
of the binding energy. Differences between different distributions are less than a percent of the correction itself.
However, several atomic properties depend directly on the
wave function close to the nucleus. Using a point charge,
with the resulting unphysical behaviour of the wave function at r = 0 then leads to overestimates by several percent
for high Z.
An important observation in the study of nuclear
properties is that the orbital behaviour within the nucleus
The atomic nucleus 7
depends on the angular momentum but is essentially
independent of the binding energy. For very heavy nuclei,
the binding energies are a larger fraction of the nuclear
potential energy, giving a small n-dependence in the orbital
behaviour close to the nucleus, as demonstrated, for example, by Shabaev(21) for hfs corrections for different values
of Z.
For a many-electron system, all electrons are affected but
mainly through their interaction with the modified j = 1/2
orbitals. It is then the sensitivity of the interaction of the
j = 1/2 orbitals that needs to be studied in detail. A striking
illustration is the comparison by Visscher and Dyall(12) of
the relation between the corrections for the 1s orbital energy
and the total energy. The ratio between the corrections
to the point nuclear value for Fermi, homogeneous, and
Gaussian distribution were plotted for both cases, giving
two figures differing only in their captions. In this way,
nuclear size effects obtained for one charge distribution
could be rescaled to another distribution by applying
correction factors determined from the effects on the
s orbitals. (The difference between the higher moment
contributions for s and p1/2 is relatively insignificant.)
As discussed in previous sections, the shape can be
parametrized in terms of the moment r 2 , r 4 , and r 6 of the nuclear charge. To facilitate comparisons between
different calculations, we therefore suggest that whatever
nuclear distribution is used, these moments should be
given (or sufficient information provided to make their
extraction easy).
4.1 Hyperfine structure
4.2 Parity nonconserving operators
The study of parity nonconserving weak interaction has been a fruitful branch of atomic physics. Recent experiments
for Cs are accurate at the percent level, providing a serious
challenge to atomic theory. The electro-weak interaction
involves the neutron distribution, and James and Sandars(23)
have analysed in detail the sensitivity to details in the distribution. Experimental information can possibly be obtained
from hyperfine anomalies, as in the recent experiments for
Fr isotopes.(24)
The near-energy independence of the orbital shape close
to the nucleus forms the basis for derivations of relations
between different interactions close to the nucleus by using
the series expansion of the wave function. The ratios are,
however, modified by many-body corrections if the operators have different angular structure, as discussed, for
example, in Reference 25. This has long been used to estimate field isotope shifts from observed hfs(26,4) and more
recently for various P and T violating operators.(27,28,29)
The ‘Schiff moments’, for example, involve the difference
between the charge and electric dipole distributions.
5 CONCLUDING REMARKS
The best-known example of nuclear-structure effects in
atomic spectra is the magnetic hfs, which arises from a
coupling between the magnetic moments of the nucleus and
the electron, given by
hhfs
p = r̂·
arises from differences in the charge distribution (the
so-called Breit–Rosenthal effect). Another contribution to
the hyperfine anomaly, the ‘Bohr–Weisskopf effect’ arises
from the distribution of nuclear magnetic moment. In this
way, the hyperfine anomaly can give information about
magnetic moment distribution.
µ×α
r2
for a point magnetic dipole.
The resulting energy splitting is often expressed as AI·J,
where I and J, are the angular momenta of the nucleus
and the electron, respectively. The hfs was an early source
of information about nuclear spin, magnetic moments.
In general, magnetic moments are known better through
other sources, such as nuclear magnetic resonance (NMR).
Hyperfine structure can thus be used to test atomic wave
functions.(22)
The ratio between the magnetic moment and the ‘Afactor’ is, however, found to vary slightly between various
isotopes of an element. Part of this ‘hyperfine anomaly’
As quantum chemists approach systems with increasingly
heavy atoms, the use of an extended charge distribution
becomes essential, and the charge distribution used must be
clearly specified to make possible comparison of different
results.
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8 Approximate separation of electronic and nuclear motion
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