Isotope shifts and relativistic shifts of Cr II for study of alpha-variation in quasar absorption spectra

Abstract

We use the combination of configuration interaction and many-body
perturbation theory method (CI+MBPT) to perform ab initio calculations the
low-energy spectra of Cr II with high accuracy. It is found that second-order
MBPT diagrams should be included in a consistent and complete way for the MBPT
to improve the accuracy of calculations in this five-valence-electron system.
This contrasts with previous ions with fewer valence electrons where it was
found that single-valence-electron diagrams dominate the corrections. Isotope
shifts and relativistic shifts (q-values) are calculated for use in
astronomical determination of the fine-structure constant in quasar absorption
spectra.

Full text

Page 1

arXiv:1110.2292v1 [physics.atom-ph] 11 Oct 2011
Isotope shifts and relativistic shifts of CrII for study of α-variation in quasar
absorption spectra
J. C. Berengut
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
(Dated: 11 October 2011)
We use the combination of configuration interaction and many-body perturbation theory method
(CI+MBPT) to perform ab initio calculations the low-energy spectra of CrII with high accuracy.
It is found that second-order MBPT diagrams should be included in a consistent and complete way
for the MBPT to improve the accuracy of calculations in this five-valence-electron system. This
contrasts with previous ions with fewer valence electrons where it was found that single-valence-
electron diagrams dominate the corrections.Isotope shifts and relativistic shifts (q-values) are
calculated for use in astronomical determination of the fine-structure constant in quasar absorption
spectra.
PACS numbers: 31.30.Gs, 31.15.am, 95.30.Ky, 06.20.Jr
I.INTRODUCTION
Quasar absorption systems provide a unique probe of
the value of fundamental constants throughout much of
the visible Universe. The many-multiplet (MM) method
enables the most complete analysis of optical spectra in
the search for space-time variation of the fine-structure
constant, α = e2/¯ hc [1, 2]. It makes use of all transi-
tions seen in all ions in a given quasar absorption system
to gain statistical significance and control systematics.
Early results using spectra taken from the Keck telescope
suggested that α may have been smaller in the past [3–
5], however when combined with new systems observed
with the Very Large Telescope (VLT) the data is more
consistent with a spatial variation in the fine-structure
constant [6]. The gradient in values of α reconciles all
existing measurements of α-variation [7]. In particular
the early Keck results that indicated a constant offset or
“monopole” model, are entirely consistent with the spa-
tial gradient “dipole” model since Keck mainly sees in
the northern hemisphere (the α-dipole axis is oriented
∼ 30◦from the equatorial axis). By contrast the VLT
data is taken mainly in the southern sky.
A spatial variation of α would manifest itself in a vari-
ety of terrestrial [8] and astrophysical [9] systems, which
could be used to confirm the dipole. It is also possible to
devise complementary tests using subsets of the quasar
absorption system data which may involve different sys-
tematics. One such test, currently underway, is a variant
of the many-multiplet method that only uses transitions
in CrII and ZnII [10]. The transitions have opposite
α-sensitivities and so a comparison of them is very sensi-
tive to α-variation: ZnII transitions are s − p and hence
their frequency increases if α increases, while the CrII
transitions are d − p so their frequency decreases with
increasing values of α. Furthermore the transitions are
very close in energy. This means that only a small part
of the optical spectrum is analysed, resulting in differ-
ent (perhaps smaller) systematics. Of particular concern
are “intra-order shifts”: velocity shifts of unknown ori-
gin within each echelle order in the spectrograph [11, 12].
This systematic may differently affect measurements of
α-variation when only CrII and ZnII lines are utilized,
compared to studies where a larger number and wider
variety of transitions are used.
One problem with using CrII and ZnII transitions ex-
clusively is that they are weak. Of course, this is the rea-
son why they don’t play a major role in the full MM anal-
ysis despite being included whenever available. However
there exist certain quasar absorption systems in which
ZnII lines are particularly strong [13], and from these
“metal strong” systems can be drawn a relatively large
sample with which to perform the CrII/ZnII analysis.
One potential systematic that has plagued all MM
analyses is isotope abundance [14–16]. Isotopic structure
cannot be resolved in the absorption spectra, so generally
terrestrial isotopic abundances are assumed for the ab-
sorber. Any deviation from terrestrial abundances would
shift the centroid of the line profile, and this might mimic
a change in α. Even quantifying the systematic can be
difficult because the isotopic structures themselves are
unknown for many of the UV transitions used in the MM
analysis. The systematic is lessened in the context of a
dipole result, since the isotope abundances would need
to vary according to direction in the sky, which in itself
would violate cosmological isotropy. Nevertheless, in or-
der to quantify possible systematics the isotope structure
should be known for all transitions used in the analysis,
hence considerable efforts by many groups to calculate
and measure them (see, e.g. [17–21]).
In this paper we calculate the isotope shifts and rela-
tivistic shifts of the CrII transitions seen in quasar ab-
sorption spectra. The corresponding parameters for ZnII
have been calculated previously [15, 22]. Our final results
are presented in Tables V and VI.
II.METHOD
The ab initio CI+MBPT method [23] is described in
full elsewhere [24]. Details of relevance for our CrII cal-
culation are presented below.

Page 2

2
A.Energy calculation
Any perturbative theory works best when the pertur-
bations are as small as possible.
electrons play an important role in shaping the atomic
core, and so they should be included in the initial ap-
proximation. As in previous works, our single-particle
wavefunctions are calculated using Dirac-Fock (relativis-
tic Hartree-Fock). We explore two Dirac-Fock configura-
tions: VN, which includes a half-filled 3d5subshell; and
VN−1, which includes 3d4. In both cases we simply scale
the potential due to the filled 3d subshell by the number
of electrons to provide a “configuration averaged” ini-
tial wavefunction. The choice of starting approximation
is essential to obtaining a good final spectrum for CrII,
but also leads to potentially large subtraction diagrams in
many-body perturbation theory, as will be demonstrated.
In CrII the d-wave
Once we have a Dirac-Fock potential for the core, we
diagonalize the Dirac-Fock Hamiltonian over a set of 40
B-splines [25] spanning 40 atomic units to obtain a large
set of valence and virtual orbitals from which we select
those with the lowest eigenvalues. A set of configura-
tions of valence electrons |I? are generated, from which
eigenfunctions of the complete Coulomb-potential Hamil-
tonian are calculated. We find that almost complete con-
vergence of the CI calculation can be obtained using the
basis 20spdf: that is we use s-wave states labeled 1 –
20, p-wave states labeled 2 – 20, etc. (For the lowest
eigenvalue states the label is just the principal quantum
number.) With 5 valence electrons it is not possible to
include all configurations and we must select those that
contribute most to the wavefunction. We include all con-
figurations that can be formed by one-particle excitations
from the leading configurations 3d5, 3d44s, and 3d44p, as
well as two-particle excitations from these same configu-
rations up to the 6sp9d6f orbitals. The effects of higher
orbitals and three-particle excitations were found to be
small and were not included.
Having achieved high saturation of the CI calcula-
tion, core-valence effects are included using second-order
MBPT by modifying matrix elements of the Hamiltonian:
HIJ→ HIJ+
?
M
?I|H |M??M|H |J?
E − EM
, (1)
where the states |M? include all Slater determinants that
have core excitations.The MBPT sum may be fur-
ther separated into one-, two-, and three-valence-electron
parts, denoted Σ(1), Σ(2)and Σ(3)in Refs. [24, 26]. Gold-
stone diagrams and analytical expressions for these are
presented in [24]. The states |M? include excitations from
all core states into virtual states up to 30spdfg. The ef-
fects on the energy calculation of including Σ(1), Σ(2),
and Σ(3)sequentially are shown in Tables I and II (for the
VN−1and VNstarting approximations, respectively).
TABLE I. Energy spectrum of CrII with orbitals calculated
in the VN−1approximation, relative to the experimentally
determined ground state, 3d5 6S5/2. Successive additions of
Σ(1), Σ(2), and Σ(3)are shown. (Energies in cm−1.)
Level
3d44s6D
J CI
8505 −7294
8682 −7128
+Σ(1)
+Σ(2)
6888
7095
40933
41028
41184
41401
41676
42008
42534
42686
42915
42725
43038
43880
43659
43741
43492
43959
44209
+Σ(3)
9491
9699
44533
44638
44798
45036
45320
45668
46074
46216
46420
46584
46907
47717
47460
47552
47330
47727
47994
Expt. [27]
12148
12304
46823
46905
47040
47227
47465
47752
48399
48491
48632
48749
49006
49706
49493
49565
49352
49646
49838
5/2
7/2
1/2 41720
3/2 41805
5/2 41945
7/2 42140
9/2 42387
11/2 42687
3/2 43214
5/2 43333
7/2 43515
1/2 43793
3/2 44057
5/2 44787
1/2 44594
3/2 44664
5/2 44451
7/2 44840
9/2 45054
3d44p6Fo
27542
27635
27788
28000
28271
28600
29890
29982
30132
30108
30328
30674
29189
29323
29510
29721
29940
3d44p6Po
3d44p4Po
3d44p6Do
TABLE II. Energy spectrum of CrII with orbitals calculated
in the VNapproximation, relative to the experimentally de-
termined ground state, 3d5 6S5/2. Successive additions of Σ(1),
Σ(2), and Σ(3)are shown. (Energies in cm−1.)
Level
3d44s6D
JCI
6688 −8662
6862 −8515
+Σ(1)
+Σ(2)
3953
4157
38794
38888
39044
39260
39534
39865
40888
41041
41265
41092
41320
42251
41749
41915
41591
41851
42108
+Σ(3)
6415
6617
41627
41726
41887
42114
42399
42747
43534
43674
43885
43995
44246
45175
44708
44861
44537
44815
45078
Expt. [27]
12148
12304
46823
46905
47040
47227
47465
47752
48399
48491
48632
48749
49006
49706
49493
49565
49352
49646
49838
5/2
7/2
1/2 41540
3/2 41626
5/2 41768
7/2 41966
9/2 42217
11/2 42521
3/2 43124
5/2 43242
7/2 43421
1/2 43816
3/2 44033
5/2 44812
1/2 44432
3/2 44552
5/2 44303
7/2 44543
9/2 44766
3d44p6Fo
26023
26110
26257
26461
26724
27047
27918
27987
28113
28030
28307
29073
28788
28878
28603
28849
29041
3d44p6Po
3d44p4Po
3d44p6Do
B.Isotope shift and relativistic shift
Isotope shifts in atomic transition frequencies come
from two sources: the finite size of the nucleus (field
shift), and the recoil of the nucleus (mass shift). This
mass shift is usually divided into the normal mass shift
(NMS), which is easily calculated from the transition fre-

Page 3

3
quency, and the specific mass shift (SMS). The mass shift
is more important for light elements, while for heavy el-
ements the field shift dominates. In the case of CrII,
the field shift is small; this paper is concerned with the
mass-shift contribution, which is more difficult to cal-
culate. The difference in the transition frequency, ω, be-
tween an isotope with mass number A′and an isotope A,
δωA′,A= ωA′− ωA, can be expressed as [15]
δωA′,A= (kNMS+ kSMS)
?1
A′−1
A
?
+ Fδ?r2?A′,A, (2)
where δ?r2?is the change in mean-square nuclear charge
radius. The normal mass shift constant can be expressed
(in atomic units ¯ h = e = me= 1)
kNMS=
1
2mu
?
i
p2
i= −ω
mu,
where mu = 1823 is the ratio of the atomic mass unit
to the electron mass, and the sum is over all electron
momenta, pi. The specific-mass-shift constant
kSMS=
1
mu
?
i<j
pi· pj
and field-shift constant F are more difficult to calculate.
We use the non-relativistic form of the mass-shift opera-
tor; relativistic corrections for optical transitions in light
atoms are on the order of few percent and can be ne-
glected [18].
To calculate kSMSwe use the all-order finite-field scal-
ing method. Here a rescaled two-body SMS operator is
added to the Coulomb potential everywhere that it ap-
pears in an energy calculation:
˜Q =
1
|r1− r2|+ λp1· p2. (3)
We recover the specific-mass-shift constant as
kSMS=dω
dλ
????
λ=0
. (4)
The operator˜Q has the same symmetry and structure as
the Coulomb operator (see Appendix A of Ref. [24]). We
have previously shown that good agreement with exper-
imental isotope shift can be obtained in many-valence-
electron atoms and ions by using this finite-field in a
CI+MBPT energy calculation [24, 26, 28, 29].
The relativistic shift of a transition may be calculated
in a similar fashion. We simply recalculate the transition
energies, ω, from the very beginning using different values
of α from the laboratory value α0. The sensitivity to
variation of the fine-structure constant is then extracted
using
q =
dω
dα2
????
α=α0
.(5)
TABLE III. Isotope shift constants kNMS and kSMS for transi-
tions to the ground state 3d5 6S5/2(GHz·amu).
LevelJkNMS
kSMS (CI)
VN
4520
4532
3964
3970
3980
3992
4009
4028
4072
4080
4093
3981
3983
4071
4016
4037
3982
3989
4003
kSMS (CI+Σ)
VN
4944
4950
4303
4314
4330
4346
4359
4362
4189
4143
4203
4227
4257
4271
4195
4267
4278
4280
4302
VN−1
4326
4337
3398
3403
3412
3423
3438
3456
3555
3563
3575
3495
3507
3493
3458
3467
3511
3451
3463
VN−1
5062
5069
4127
4140
4161
4178
4191
4188
4161
4090
4180
4068
4187
4188
4137
4156
4183
4129
4153
3d44s6D 5/2
7/2
1/2
3/2
5/2
7/2
9/2
11/2
3/2
5/2
7/2
1/2
3/2
5/2
1/2
3/2
5/2
7/2
9/2
−200
−202
−770
−771
−774
−777
−781
−785
−796
−797
−800
−802
−806
−817
−814
−815
−812
−816
−820
3d44p6Fo
3d44p6Po
3d44p4Po
3d44p6Do
III.RESULTS AND DISCUSSION
CrII has five valence electrons, and these have a sig-
nificant impact on the form of the basis orbitals. For the
CI+MBPT method to work well it is important to have
good initial orbitals, and so our Dirac-Fock and subse-
quent B-spline codes include the 3d4or 3d5configuration
in the core, as described in Section IIA. (Our calcula-
tions show that saturation of the CI can be met satis-
factorily in both VN−1and VNapproximations.) In the
CI+MBPT code, the 3d orbitals are then stripped from
the core and become valence orbitals for the purposes of
both the CI and MBPT components of the calculation.
In this way excitations from the 3d shell are treated non-
perturbatively.
To calculate the MBPT diagrams one must include
the change in effective core-potential, VN−5− VNor
VN−5− VN−1depending on the initial approximation.
MBPT diagrams that include this interaction are known
as subtraction diagrams, and in our calculation they are
huge. There are three subtraction diagrams in Σ(1)and
two in Σ(2)(these are shown in Figs. 2 and 4 of [24], re-
spectively). When the subtraction diagrams in Σ(1)are
included, they significantly and adversely affect the en-
ergies obtained, as can be seen in the CI+Σ(1)columns
(labelled “+Σ(1)”) of Tables I and II. However, it turns
out that these adverse effects are nearly completely com-
pensated when Σ(2)and Σ(3)are also included. When all
second-order diagrams are included consistently the en-
ergies and wavefunctions are improved by the addition.
One might consider what happens if the subtraction
diagrams are simply neglected from the calculation. In-
deed the CI+Σ(1)energies are improved. However when

Page 4

4
FIG. 1. Two large diagrams that affect the ground state 3d5
multiplet that partially cancel. The labels refer to their des-
ignation in [15].
3d 3d
α
3d
3d
3d
3d
3d
n
3 (a)
α
n
2 (a)
Σ(2)is added (either with or without the two-valence-
electron subtraction diagrams) the energy levels obtained
are again in very poor agreement with experiment. Thus
it is not only inconsistent to leave out the subtraction
diagrams, but it gives very poor results when all other
second-order MBPT terms are included.
The behaviour can be explained by examining the form
of the Σ(1)and Σ(2)diagrams. For example, consider
Fig. 2(a) and Fig. 3(a) from [24] with all external lines
representing 3d electrons (see Fig. 1). The subtraction
diagram 2(a) has opposite sign to the zero multipole
(k = 0) part of 3(a). This kind of cancellation is what
finally suppresses the large subtraction diagrams, and is
the reason why all second-order Σ diagrams must be in-
cluded consistently.
The energies obtained are slightly better in the VN−1
calculation. However, consider our calculations of the
isotope shift constant kSMS, presented in Table III. We
find that the pure CI calculations give different results
in the VNand VN−1approximations, yet when the
core-valence interactions are included self-consistently
the agreement is much improved: the disagreement is re-
duced from ∼ 13% to less than 4% and in the 3d44p6Po
transitions of astrophysical interest, more like ∼ 1%.
The calculation of q-values follow the same trend (Ta-
ble IV). In this case the results are far less sensitive to
details of the wavefunction: instead the relativistic ef-
fects are determined by the form of the wavefunctions
near the nucleus. We see very strong agreement between
our VNand VN−1results, especially after all Σ diagrams
are included consistently. In the transitions of astrophys-
ical interest, the different starting approximations leads
to disagreements of the order ∼ 25% in the pure CI case
but ∼ 5% when MBPT is included.
Despite the consistency with respect to starting ap-
proximation, our transition energies still differ from ex-
periment. The most likely explanation is that we haven’t
taken full account of the relaxation of the core 3p6elec-
trons. These have a strong effect on the 3d electrons
via the exchange potential, yet relaxation of these or-
bitals is only taken into account to second-order using
perturbation theory. Ideally one would include them as
valence electrons in the CI so that their relaxation could
TABLE IV. g-factors and relativistic shifts, q (cm−1), for
transitions to the ground state 3d5 6S5/2. Experimental g-
factors are taken from Ref. [27]; calculated values are for the
full CI+Σ method in the VN−1approximation.
LevelJgq (CI)
VN
q (CI+Σ)
VN
Expt.
1.669
1.578
Calc.
1.657 −2483 −2209 −2430 −2351
1.587 −2300 −2034 −2223 −2145
1/2 −0.689 −0.665 −2052 −1748 −1979 −1896
3/21.1241.067 −1959 −1661 −1875 −1792
5/2 1.3141.314 −1807 −1518 −1705 −1624
7/2 1.378 1.397 −1597 −1321 −1473 −1395
9/2 1.4161.434 −1333 −1073 −1180 −1106
11/21.454 −1016
3/2 2.3822.385 −1607 −1325 −1489 −1421
5/21.8751.880 −1479 −1209 −1340 −1280
7/21.7101.714 −1281 −1024 −1117 −1061
1/22.844 2.811 −2146 −1782 −2122 −2003
3/2 1.8021.786 −1913 −1517 −1847 −1704
5/21.6241.626 −1089 −1036
1/23.1553.186 −1512 −1316 −1390 −1357
3/21.8241.827 −1373 −1227 −1232 −1232
5/2 1.6281.634 −1651 −1204 −1547 −1396
7/21.577 1.585 −1417 −1171 −1281 −1220
9/21.570 1.552 −1200
VN−1
VN−1
3d44s6D 5/2
7/2
3d44p6Fo
−776
−830
−758
3d44p6Po
3d44p4Po
−916
−946
3d44p6Do
−977 −1036
−990
be treated non-perturbatively, however this is not pos-
sible because the CI Hamiltonian size grows too large.
In the VN−1approximation the 3p6core is more tightly
bound, lessening the magnitude of relaxation terms. This
likely explains the improved agreement with experiment.
The 3p electrons also pose a potential problem for the
isotope shift calculation. The scaled SMS operator that
appears in (3) manifests itself in the dipole part (k = 1)
of the multipole expansion of˜Q [15]. Therefore it may
be particularly affected if the 3p – 3d exchange terms are
not adequately described by the method. It is for this
reason that we conservatively use the difference between
the CI and the CI+Σ calculation as an estimate of the
uncertainty in kSMS, rather than the smaller difference
between the VNand VN−1calculations.
We present our final values of relativistic shifts, q, for
the6Potransitions of astronomical interest (that is, those
seen in quasar absorption spectra) in Table V. As ex-
plained in Section III we prefer our VN−1approximation
which gives better agreement with experimental transi-
tion energies, however the difference between the VNand
VN−1calculations is similar to the errors quoted. The
uncertainty is estimated as the difference between our
pure CI calculation and the full CI+MBPT calculation
including all Σ diagrams. Our calculated q-values are
seen to be in good agreement with the CI calculations
of Ref. [30]. In Table IV we see that the experimental
g-factors for these transitions are well reproduced by our
calculation. There are no close levels in this case, so the
methods of matching g-factors [30] are not required. The
4Po
5/2transitions are strongly mixed (as can
5/2and6Do

Page 5

5
TABLE V. Relativistic shifts, q, for transitions to ground
state 3d5 6S5/2.
LevelJωλq (cm−1)
(cm−1)
48399
48491
48632
(˚ A)
2066 −1421(70)
2062 −1280(70)
2056 −1061(70)
this workRef. [30]
−1360(150)
−1280(150)
−1110(150)
3d44p6Po
3/2
5/2
7/2
TABLE VI. Calculated velocity structure in wavelength space
of transitions to ground state 3d5 6S5/2in CrII.
Upper level Jλ (˚ A)(δλA,52/λ) · c (kms−1)
A = 50
0.252(45) 0.495(89)
5/2 2062 −0.522(84)
7/2 2056 −0.535(96)
A = 53A = 54
3d44p6Po
3/2 2066 −0.535(96)
0.246(39) 0.484(78)
0.252(45) 0.495(89)
be seen from the g-factors, which are well reproduced by
our calculation), so if these transitions are ever seen in
quasar absorption systems then a more careful analysis
may be required.
In a previous comparison between theory and exper-
iment in SrII it was found that the SMS was underes-
timated by theory at the ∼ 30% level [31]. This is a
single-valence-electron ion and so there is no CI; rather,
we added Σ(1)directly to our Dirac-Fock calculation.
In the case of SrII the addition of Σ(1)did not ade-
quately account for the effect of core-relaxation on the
SMS. On the other hand, using the same method good
agreement has been obtained between theory and exper-
iment for ZnII [15], which we also treated as a single-
valence-electron ion. Moreover, because the majority of
the isotope shift in CrII comes from the valence-valence
contributions which are treated to all orders using CI,
we have good reason to believe that our mass-shifts have
been calculated with reasonable accuracy.
We have also estimated the size of the field shift in
these transitions using a small CI basis to estimate F and
experimental values of δ?r2?taken from [32]. The field
shift is expected to be small for a light element like CrII.
Furthermore for the d−p transitions of astrophysical in-
terest the orbitals do not overlap the nucleus strongly
and so there is additional suppression. We find the field
shift is of order ∼ 0.010 kms−1or smaller, which is much
smaller than our uncertainty in kSMS. We neglect it en-
tirely.
In Table VI we present our isotope shift shift calcula-
tions for astronomically relevant transitions of CrII using
the VN−1results of Table III in (2). Again the uncer-
tainty quoted is the difference between the pure CI and
CI+MBPT calculations, i.e. the entire effect of Σ. This
is very much larger than the difference between our VN
and VN−1calculations. We quote the velocity structure
in wavelength space relative to the leading isotope52Cr.
This is the preferred form for use in astronomy: the ve-
locity shift is δv = λA,52/λ · c where λA,52= λA− λ52
and c is the speed of light in kms−1.
IV.CONCLUSION
We have shown that the CI+MBPT method can give
transition energies in good agreement with experiment
for low-lying transitions in CrII. Although the subtrac-
tion diagrams are very large when the orbitals are cal-
culated in the VN−1or VNapproximations, when all
second-order MBPT diagrams are taken into account
consistently the calculated energies are found to im-
prove. This may help to direct future efforts using the
CI+MBPT method in many-valence-electron ions such
as FeII, which is of importance to studies of α-variation
in quasar absorption systems.
The SMS in CrII is found to dominate the isotope
shift. For the 3d5 6S5/2→ 3d44p6Potransitions seen
in quasar spectra they are five times the magnitude
of the normal mass-shift and of opposite sign.
total mass-shift constant for these transitions, kNMS+
kSMS = 3365(112) GHz·amu (taking the J = 3/2 up-
per level), is consistent with an earlier CI estimate of
1900(1200)GHz·amu [16], although clearly at the limit of
the uncertainty. The CrII isotope shift is also quite large
in comparison to many of the other isotope shifts used in
the quasar analysis (although many are unknown). The
velocity shift between the even isotopes (∼ 500ms−1) is
comparable to the isotope shifts of the λλ2803 and 2796
lines of MgII (∼ 850ms−1). Fortunately in the case of
CrII there are stable isotopes on either side of the lead-
ing isotope, so one may hope that the total systematic
shift due to variation of isotope abundances is small.
The
ACKNOWLEDGMENTS
I thank Victor Flambaum and Michael Murphy for use-
ful discussions. This work was supported by an award un-
der the Merit Allocation Scheme on the NCI National Fa-
cility at the ANU and by the Australian Research Council
(DP110100866).
[1] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Phys.
Rev. Lett. 82, 888 (1999).
[2] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Phys.
Rev. A 59, 230 (1999).
[3] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J.
Drinkwater, and J. D. Barrow, Phys. Rev. Lett. 82, 884
(1999).