Hyperfine Quenching of the $4s4p ^{3}P_{0}$ Level in Zn-like Ions
EPJ manuscript No.
(will be inserted by the editor)
Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
J. P. Marques1, F. Parente2, and P. Indelicato3
1 Centro de F́ısica Atómica e Departamento F́ısica, Faculdade de Ciências, Universidade de Lisboa,
Campo Grande, Ed. C8, 1749-016 Lisboa, Portugal, e-mail: jmmarques@fc.ul.pt
2 Centro de F́ısica Atómica da Universidade de Lisboa e Departamento F́ısica da Faculdade de Ciências e Tecnologia da
Universidade Nova de Lisboa, Monte da Caparica, 2825-114 Caparica, Portugal, e-mail: facp@fct.unl.pt
3 Laboratoire Kastler Brossel, École Normale Supérieure; CNRS; Université P. et M. Curie - Paris 6
Case 74; 4, place Jussieu, 75252 Paris CEDEX 05, France, e-mail: paul.indelicato@spectro.jussieu.fr
Received: October 24, 2018/ Revised version: date
Abstract. In this paper, we used the multiconfiguration Dirac-Fock method to compute with high precision
the influence of the hyperfine interaction on the [Ar]3d104s4p 3P0 level lifetime in Zn-like ions for stable
and some quasi-stable isotopes of nonzero nuclear spin between Z = 30 and Z = 92. The influence of this
interaction on the [Ar]3d104s4p 3P1 − [Ar]3d
104s4p 3P0 separation energy is also calculated for the same
ions.
PACS. 31.30.Gs – 31.30.Jv – 32.70.Cs
1 Introduction
It has been found before that the hyperfine interaction
plays a fundamental role in the lifetimes and energy sep-
arations of the 3P0 and
3P1 levels of the configurations
1s2p in He-like [1,2,3,4,5,6,7,8,9], [He]2s2p in Be-like [10,
11,12], and [Ne]3s3p in Mg-like [11,13] ions, and also in
3d4J = 4 level in the Ti-like ions [14].
In the He-like ions, in the region Z ≈ 45, these two lev-
els undergo a level crossing [15] and are nearly degenerate
due to the electron-electron magnetic interaction, which
leads to a strong influence of the hyperfine interaction on
the energy splitting and on the 3P0 lifetime for isotopes
with nonzero nuclear spin. In Be-like, Mg-like and Zn-like
ions a level crossing of the 3P0 and
3P1 levels has not been
found [16], but hyperfine interaction still has strong influ-
ence on the lifetime of the 3P0 metastable level and on the
energy splitting, for isotopes with nonzero nuclear spin.
Until recently, laboratorymeasurements of atomic mes-
tastable states hyperfine quenching have been performed
only for He-like systems, for Z = 28 to Z = 79 [3,4,5,8,
17,18,19,20]. Hyperfine-induced transition lines in Be-like
systems have been found in the planetary nebula NGC3918 [12].
Measured values of transition probabilities were found to
agree with computed values [11,21].
Divalent atoms are being investigated, both theoreti-
cally and experimentally, in order to investigate the possi-
ble use of the hyperfine quenched 3P0 metastable state for
ultraprecise optical clocks and trapping experiments [22].
Recently, dielectronic recombination rate coefficients
were measured for three isotopes of Zn-like Pt48+ in the
Send offprint requests to: J. P. Marques
Heidelberg heavy-ion storage ring TSR [23]. It was sug-
gested that hyperfine quenching of the 4s4p 3P0 in isotopes
with non-zero nuclear spin could explain the differences
detected in the observed spectra.
In this paper we extend our previous calculations [2,
10,13] to the influence of the hyperfine interaction on the
1s22s22p63s23p63d104s4p levels in Zn-like ions.
In Fig. 1 we show the energy level scheme, not to scale,
of these ions. The 3P0 level is a metastable level; one-
photon transitions from this level to the ground state are
forbidden, and multiphoton transitions have been found to
be negligible in similar systems, so the same behavior can
be expected for Zn-like ions. Therefore, in first approxima-
tion, we will consider the lifetime of this level as infinite.
The energy separation between the 4 3P0 and 4
3P1 lev-
els is small for Z values around the neutral Zn atom and
increases very rapidly with Z. Hyperfine interaction is ex-
pected to have a strong influence on the energy splitting
and on the 4 3P0 lifetime for isotopes with nonzero nuclear
spin.
The different steps of this calculation are described in
Refs. [2,24]. Here we will emphasize only the fundamental
topics of the theory and the characteristic features of Zn-
like systems.
In this work we used the multi-configuration Dirac-
Fock code of Desclaux and Indelicato [25,26,27] to evalu-
ate, completely ab initio the 4s4p fine structure energies
and transition probabilities.
Several terms, such as the nonrelativistic (J indepen-
dent) contribution to the correlation, are the same for all
levels. In this calculation Breit interaction is included, as
well as radiative corrections, using the method described,
http://arxiv.org/abs/0704.1946v1
2 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
Fig. 1. Energy level and transition scheme for Zn-like ions (not to scale).
for instance, in Ref. [28] and references therein. The MCDF
method, in principle, allows for precise calculations be-
cause it can include most of the correlation relatively eas-
ily, i. e., with a small number of configurations. Here, cor-
relation is important in the determination of transition
energies to the ground state, which are used in the calcu-
lation of transition probabilities. We found that the largest
effect is obtained by using [Ar]3d104s2 and 4p2 as the con-
figuration set for the ground state. For the excited states
we included all configuration state functions (CSF) origi-
nated from [Ar]3d104s4p, 4p4d, and 4d4f (which is usually
defined as intrashell correlation), because in second-order
perturbation theory the dominant energy difference de-
nominators correspond to these configurations. Correla-
tion originating from interaction with the Ar-like core has
been neglected. In particular we included, in a test cal-
culation, spin-polarization from s subshells and found a
negligible influence in both energies and transition proba-
bilities. We note that the fine structure energy separation,
∆E0;fs = E3P1 −E3P0 , which is the important parameter
in the calculation of the hyperfine quenching, is not very
sensitive to correlation, similarly to what we found for sys-
tems with smaller number of electrons [10,13]. The same
set of CSF have been used for energy, transition proba-
bilities, and for the calculation of the hyperfine matrix
elements.
All energy calculations are done in the Coulomb gauge
for the retarded part of the electron-electron interaction,
to avoid spurious contributions (see for example Refs. [29,
31]). The lifetime calculations are all done using exact
relativistic formulas. The length gauge has been used for
all transition probabilities.
2 Relativistic calculation of hyperfine
contribution to fine structure splitting and to
transition probabilities
In the case of a nucleus with nonzero spin, the hyperfine
interaction between the nucleus and the electrons must be
taken into account. The correspondent Hamiltonian can
be written as
Hhfs =
(k)·T(k), (1)
where M(k)and T(k) are spherical tensors of rank k, rep-
resenting, respectively, the nuclear and the atomic parts
of the interaction. As in the case of He-like, Be-like and
Mg-like ions, the only sizable contribution from Eq. (1)
is the magnetic dipole term (k = 1). The hyperfine inter-
action mixes states with the same F = J + I values. In
our case, we are interested in the 3P0 level, so J = 0 and
F = I. The contribution of this interaction for the total
energy has been evaluated through the diagonalization of
the matrix:
Htot =
iΓ0 +W0,0 W0,1 W0,2
W1,0 E1 +
iΓ1 +W1,1 W1,2
W2,0 W2,1 E2 +
iΓ2 +W2,2
Here, Ef is the unperturbed level energy and Γf is
the radiative width of the unperturbed level (f = 0, 1, 2
stands, respectively, for 3P0,
3 P1,
1 P1). In reality there is
a fourth level, 3P2, which we included in all calculations
but was found unnecessary, because the influence of this
level in the 3P0 lifetime is negligible. This can be ex-
plained by the large energy separation between the 3P2
and 3P0 levels and also because the probability of the
allowed M2 transition 4s4p 3P2 → 4s
2 1S0 is many or-
ders of magnitude smaller than those of the E1 transitions
4s4p 3P1 → 4s
2 1S0 and 4s4p
1P1 → 4s
2 1S0. Also, the
magnetic dipole hyperfine matrix element between the 3P2
and 3P0 levels is very small. This also leads to a negligi-
ble influence of the 3P2 level on the
3P1 −
3 P0 separation
energy. This is consistent with the results of Plante e John-
son [32], who found that the magnetic quadrupole term of
the hyperfine interaction affects the 3P2 level only at high
Z. The influence of the 1P1 level, however, must be taken
into account, specially for light nuclei, because the large
energy separation between 1P1 and
3P0 levels is compen-
sated by the much shorter lifetime of the 1P1 level. The
J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 3
hyperfine matrix element
Wf,f ′ = Wf ′,f = 〈[Ar] 3d
104s4p f |Hhfs|[Ar] 3d
104s4p f ′〉
may be written as
WJ1,J2 = 〈I, J1, F,MF |M
(1)·T(1)|I, J2, F,MF 〉, (3)
where I is the nuclear spin and F the total angular mo-
mentum of the atom, and may be put in the form:
WJ1,J2 = (−1)
I+J1+F
I J1 F
J2 I 1
× 〈I||M(1)||I〉〈J1||T
(1)||J2〉. (4)
The 6j symbol leads to W0,0 = 0. Also the nuclear mag-
netic moment µI in units of the nuclear magneton µN may
be defined by
µIµN = 〈I||M
(1)||I〉
I 1 I
−I 0 I
, (5)
with µN = eh/2πmpc.
The electronic matrix elements were evaluated on the
basis set |3P0〉, |
3P1〉, |
1P1〉 with all intrashell correlation
included.
The final result is then obtained by a diagonalization
of the 3× 3 matrix in Eq. (2), the real part of each eigen-
value being the energy of the correspondent level and the
imaginary part its lifetime.
3 Results and discussion
The MCDF method has been used to evaluate the influ-
ence of the hyperfine interaction on the [Ar]3d104s4p 3P0,
3P1 and
1P1 levels for all Z values between 30 and 92 and
for all stable and some quasi-stable isotopes of nonzero
nuclear spin. Contributions from other levels have been
found to be negligible. A detailed list of the contributions
to the the theoretical 3P0,
3P1 and
1P1 level energies is
presented in Table 1, for Z = 36, 54, and 82.
In Table 2 we present, for all possible values of the
nuclear spin I, Z, and the mass number A, the diagonal
and off-diagonal hyperfine matrix elements Wi,j , and the
nuclear magnetic moment, µI [37], in nuclear magneton
units. The indexes 0, 1, and 2 in the hyperfine matrix
elements stand for 3P0,
3P1, and
1P1, respectively.
In Table 3 are presented, for all possible values of I, Z,
and A, the unperturbed separation energies∆E0 = E3P1−
E3P0 , the hyperfine-affected separation energy ∆Ehf, the
4s4p 3P1 and 4s4p
1P1 levels lifetime values, τ1 and τ2,
respectively, which are not affected, within the precision
shown, by the hyperfine interaction, and finally the per-
turbed lifetime τ0 of the 4s4p
3P0 level. As we pointed
out before, the unperturbed value of τ0 is assumed to be
infinite.
In Fig. 2 is plotted the difference E = ∆Ehf −∆E0 as
a function of Z for the different possible values of nuclear
spin. The influence of the hyperfine interaction on this
energy is shown to increase slowly with Z, the increase
becoming more rapid for Z ' 60.
In Fig. 3 is plotted the perturbed 4s4p 3P0 level life-
time τ0, for the different nuclear spin values I, as a func-
tion of Z. One can easily conclude that the opening of
a new channel for the decay of the 4s4p 3P0 level has a
dramatic effect on its lifetime.
After submitting this paper, it was brought to our at-
tention the work by Liu et al [38], which contains inde-
pendent calculations of the hyperfine quenched lifetime of
the 3P0 level in several Zn-like ions. Our results for this
lifetime are three times higher than the values found by
Liu et al for Z = 30 and 1.5 higher at Z = 47. To look for
the origin of this discrepancy we used the 3P1 →
1 S0 and
1P1 →
1 S0 transition energies and probabilities of Liu et
al to diagonalize the matrix in eq. 2 and obtained results
very close to our own calculations. As we do not know the
values of the hyperfine matrix elements calculated by Liu
et al, the reasons for this discrepancy remain unknown.
One of the most interesting practical implication of
these calculations comes from the relationship between
the 4s4p 3P0 − 4s4p
3P1 levels energy separation and the
4s4p 3P0 level lifetime (Eq. 2). As referred in [13] this
energy separation can be estimated from a measurement
of the hyperfine-quenched 4s4p 3P0 lifetime of Zn-like ions
with nuclear spin I 6= 0. This method has been demon-
strated for heliumlike Ni26+ [4], Ag45+ [3], Gd62+ [17] and
Au77+ [8]. In the Zn-like ions case, as in the correspond-
ing Be-like and Mg-like ions, the situations is different, be-
cause, even for the highest Z values, the lifetimes involved
are much longer than in heavy heliumlike ions. However,
measurements of Be-like hyperfine quenched transition rates
have been performed from astrophysical sources [12] and
the hyperfine quenching of the Zn-like 195Pt48+ ion was
observed in the TSR heavy-ion storage-ring [23].
The continuous progress in storage rings, ion sources
and ion traps leads us to believe that lifetimes between
0.1 s and 10 µs could be measured, with some accuracy,
by directly looking at the light emitted by the ions as a
function of time, after the trap has been loaded. Very good
vacuum inside the trap is needed, of course, if long life-
times are to be measured. It remains to be demonstrated
that this method is experimentally feasible. Such experi-
ments would be able to provide, for different isotopes, the
unperturbed energy separation, because nuclear magnetic
moments are well known. This would be an interesting
way to test our relativistic calculations.
Acknowledgments
This research was partially supported by the FCT projects
POCTI/FAT/50356/2002 and POCTI/0303/2003 (Portu-
gal), financed by the European Community Fund FEDER,
and by the French-Portuguese collaboration (PESSOAPro-
gram, Contract n◦ 10721NF). Laboratoire Kastler Brossel
is Unité Mixte de Recherche du CNRS n◦ C8552.
4 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
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J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 5
Table 1. Contribution to the energy of the 4s4p3P0,
3 P1 and
1P1 levels (in eV).
Z = 36
Coulomb† −75616.613 −75616.265 −75609.264
Magnetic† 42.785 42.779 42.765
Retardation (order ω2)† −4.095 −4.095 −4.094
Retardation (> ω2) −0.197 −0.197 −0.197
Self-energy (SE) 31.358 31.363 31.366
Self-energy screening −2.735 −2.740 −2.743
VP [ α(Zα)]correction to e-e interaction 0.033 0.033 0.033
Vacuum Polarization α(Zα)3 + α2(Zα) 0.010 0.010 0.010
2nd order (SE-SE + SE-VP + S-VP-E)‡ −0.031 −0.031 −0.031
Recoil −0.003 −0.003 −0.003
Relativistic Recoil♯ 0.009 0.009 0.009
Total energy −75549.478 −75549.137 −75542.148
Z = 54
Coulomb† −195348.063 −195344.974 −195317.667
Magnetic† 171.169 171.142 170.943
Retardation (order ω2)† −17.418 −17.418 −17.417
Retardation (> ω2) −1.711 −1.713 −1.731
Self-energy (SE) 127.610 127.615 127.669
Self-energy screening −9.166 −9.170 −9.205
VP [ α(Zα)]correction to e-el interaction 0.124 0.124 0.124
Vacuum Polarization α(Zα)3 + α2(Zα) 0.242 0.242 0.242
2nd order (SE-SE + SE-VP + S-VP-E)‡ −0.253 −0.253 −0.253
Recoil −0.012 −0.012 −0.012
Relativistic Recoil♯ 0.042 0.042 0.042
Total energy −195077.437 −195074.376 −195047.265
Z = 82
Coulomb† −513070.812 −513061.827 −512876.005
Magnetic† 710.586 710.511 708.897
Retardation (order ω2)† −72.655 −72.654 −72.655
Retardation (> ω2) −14.666 −14.668 −14.965
Self-energy (SE) 600.177 600.173 600.249
Self-energy screening −38.198 −38.203 −38.200
VP [ α(Zα)]correction to e-e interaction 0.572 0.572 0.570
Vacuum Polarization α(Zα)3 + α2(Zα) 4.192 4.192 4.189
2nd order (SE-SE + SE-VP + S-VP-E)‡ −2.975 −2.975 −2.974
Recoil −0.051 −0.051 −0.051
Relativistic Recoil♯ 0.191 0.191 0.191
Total energy −511883.638 −511874.738 −511690.755
† Contains the Uheling potential contribution to all order and all order Breit interaction.
‡ Calculated using the results of Ref. [33,34,35].
♯ The formulas and definitions used to evaluate this term are Appendix A of Ref. [36]
6 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
-0.005
0.005
0.015
0.025
25 35 45 55 65 75 85 95
I=1/2
I=3/2
I=5/2
I=7/2
I=9/2
Fig. 2. Influence of the hyperfine interaction on the 4s4p 3P1 −
3P0 energy separation, as a function of the nuclear spin I
and the atomic number Z. The quantity E = ∆Ehf −∆E0 is the contribution of the hyperfine interaction to the fine structure
splitting ∆E0. The symbols represent values for the differente nuclear spins; some elements have several isotopes with identical
spins but different µI values.
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
25 35 45 55 65 75 85 95
I=1/2
I=3/2
I=5/2
I=7/2
I=9/2
Fig. 3. Influence of the hyperfine interaction on the lifetime of the 4s4p 3P0 level, as a function of the nuclear spin I and the
atomic number Z. The unperturbed lifetime is, to a very good approximation, infinite.
J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 7
Table 2. Hyperfine Matrix elements Wi,j in eV. The indexes 0, 1, and 2 stand for
3P1, and
1P1 respectively. I is the
nuclear spin and µ the nuclear magnetic moment in nuclear magneton units.
I Z A W1,1 W2,2 W0,1 W0,2 W1,2 µI
1/2 34 77 −3.312×10−05 −3.245×10−06 −2.382×10−05 1.708×10−05 3.268×10−05 0.535042
39 89 2.697×10−05 4.139×10−07 1.670×10−05 −1.055×10−05 −2.199×10−05 −0.137415
45 103 4.648×10−05 −4.491×10−06 2.564×10−05 −1.321×10−05 −2.923×10−05 −0.0884
47 107 7.878×10−05 −1.013×10−05 4.218×10−05 −2.011×10−05 −4.507×10−05 −0.11368
47 109 9.057×10−05 −1.164×10−05 4.849×10−05 −2.311×10−05 −5.182×10−05 −0.130691
48 111 4.696×10−04 −6.718×10−05 2.480×10−04 −1.136×10−04 −2.560×10−04 −0.594886
48 113 4.912×10−04 −7.027×10−05 2.594×10−04 −1.188×10−04 −2.678×10−04 −0.622301
50 115 9.289×10−04 −1.573×10−04 4.788×10−04 −2.018×10−04 −4.591×10−04 −0.91883
50 117 1.012×10−03 −1.714×10−04 5.216×10−04 −2.199×10−04 −5.002×10−04 −1.00104
50 119 1.059×10−03 −1.793×10−04 5.457×10−04 −2.301×10−04 −5.233×10−04 −1.04728
52 123 9.385×10−04 −1.801×10−04 4.734×10−04 −1.834×10−04 −4.199×10−04 −0.736948
52 125 1.132×10−03 −2.172×10−04 5.708×10−04 −2.212×10−04 −5.063×10−04 −0.888505
54 129 1.232×10−03 −2.602×10−04 6.102×10−04 −2.170×10−04 −4.984×10−04 −0.777976
69 169 1.391×10−03 −3.983×10−04 6.364×10−04 −1.217×10−04 −2.710×10−04 −0.231
70 171 −3.210×10−03 9.272×10−04 −1.463×10−03 2.694×10−04 5.974×10−04 0.49367
74 183 −1.035×10−03 3.081×10−04 −4.659×10−04 7.391×10−05 1.610×10−04 0.117785
76 187 −6.581×10−04 1.982×10−04 −2.942×10−04 4.344×10−05 9.378×10−05 0.064652
78 195 −7.176×10−03 2.186×10−03 −3.189×10−03 4.387×10−04 9.383×10−04 0.60952
80 199 −6.867×10−03 2.112×10−03 −3.030×10−03 3.892×10−04 8.250×10−04 0.505885
81 203 −2.364×10−02 7.305×10−03 −1.040×10−02 1.290×10−03 2.724×10−03 1.622258
81 205 −2.387×10−02 7.377×10−03 −1.050×10−02 1.303×10−03 2.751×10−03 1.638215
82 207 −9.267×10−03 2.876×10−03 −4.060×10−03 4.873×10−04 1.024×10−03 0.592583
3/2 31 69 −1.363×10−05 −1.123×10−06 −2.652×10−05 2.024×10−05 1.535×10−05 2.01659
31 71 −1.732×10−05 −1.427×10−06 −3.370×10−05 2.572×10−05 1.950×10−05 2.56227
33 75 −2.179×10−05 −2.286×10−06 −3.670×10−05 2.687×10−05 2.235×10−05 1.43948
35 79 −5.715×10−05 −4.904×10−06 −8.849×10−05 6.206×10−05 5.431×10−05 2.1064
35 81 −6.161×10−05 −5.286×10−06 −9.538×10−05 6.690×10−05 5.855×10−05 2.270562
37 87 −1.198×10−04 −6.370×10−06 −1.745×10−04 1.165×10−04 1.056×10−04 2.75131
54 131 −3.649×10−04 7.708×10−05 −4.042×10−04 1.438×10−04 1.477×10−04 0.6915
56 135 −5.429×10−04 1.235×10−04 −5.920×10−04 1.932×10−04 1.987×10−04 0.837953
56 137 −6.073×10−04 1.382×10−04 −6.622×10−04 2.161×10−04 2.222×10−04 0.937365
64 155 3.449×10−04 −9.328×10−05 3.595×10−04 −8.381×10−05 −8.504×10−05 −0.2572
64 157 4.556×10−04 −1.232×10−04 4.750×10−04 −1.107×10−04 −1.123×10−04 −0.3398
65 159 −2.936×10−03 8.052×10−04 −3.049×10−03 6.825×10−04 6.901×10−04 2.014
76 189 −2.239×10−03 6.745×10−04 −2.238×10−03 3.305×10−04 3.191×10−04 0.659933
77 191 −5.500×10−04 1.666×10−04 −5.481×10−04 7.811×10−05 7.506×10−05 0.1507
77 193 −5.974×10−04 1.810×10−04 −5.954×10−04 8.484×10−05 8.154×10−05 0.1637
79 197 −6.243×10−04 1.911×10−04 −6.182×10−04 8.215×10−05 7.824×10−05 0.148158
80 201 2.535×10−03 −7.798×10−04 2.501×10−03 −3.212×10−04 −3.045×10−04 −0.560226
91 231 −1.960×10−02 6.265×10−03 −1.845×10−02 1.656×10−03 1.506×10−03 2.01
5/2 30 67 −1.975×10−06 −3.436×10−08 −6.734×10−06 5.115×10−06 2.303×10−06 0.875479
37 85 −3.537×10−05 −1.880×10−06 −7.865×10−05 5.252×10−05 3.118×10−05 1.353352
40 91 6.158×10−05 −2.554×10−07 1.273×10−04 −7.804×10−05 −4.821×10−05 −1.30362
42 95 6.081×10−05 −2.597×10−06 1.207×10−04 −6.931×10−05 −4.375×10−05 −0.9142
42 97 6.209×10−05 −2.652×10−06 1.233×10−04 −7.078×10−05 −4.467×10−05 −0.9335
44 99 5.826×10−05 −4.623×10−06 1.116×10−04 −5.968×10−05 −3.835×10−05 −0.6413
44 101 6.530×10−05 −5.182×10−06 1.251×10−04 −6.689×10−05 −4.298×10−05 −0.7188
46 105 7.773×10−05 −8.788×10−06 1.442×10−04 −7.151×10−05 −4.664×10−05 −0.642
51 121 −7.648×10−04 1.385×10−04 −1.332×10−03 5.382×10−04 3.597×10−04 3.3634
53 127 −8.004×10−04 1.617×10−04 −1.366×10−03 5.072×10−04 3.406×10−04 2.813273
59 141 −2.218×10−03 5.482×10−04 −3.622×10−03 1.040×10−03 6.985×10−04 4.2754
63 151 −2.567×10−03 6.842×10−04 −4.106×10−03 9.970×10−04 6.642×10−04 3.4717
63 153 −1.133×10−03 3.020×10−04 −1.812×10−03 4.401×10−04 2.932×10−04 1.5324
66 161 4.556×10−04 −1.264×10−04 7.196×10−04 −1.548×10−04 −1.021×10−04 −0.4803
66 163 −6.384×10−04 1.771×10−04 −1.008×10−03 2.169×10−04 1.431×10−04 0.673
70 173 8.428×10−04 −2.434×10−04 1.312×10−03 −2.416×10−04 −1.568×10−04 −0.648
75 185 −6.034×10−03 1.807×10−03 −9.246×10−03 1.415×10−03 8.981×10−04 3.1871
75 187 −6.096×10−03 1.826×10−03 −9.341×10−03 1.429×10−03 9.073×10−04 3.2197
82 205 −2.226×10−03 6.908×10−04 −3.331×10−03 3.998×10−04 2.460×10−04 0.7117
92 233 1.702×10−03 −5.451×10−04 3.263×10−03 −2.839×10−04 −1.256×10−04 0.59
8 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
Table 2. Continued
I Z A W1,1 W2,2 W0,1 W0,2 W1,2 µI
7/2 34 79 9.003×10−06 8.821×10−07 2.967×10−05 −2.128×10−05 −8.883×10−06 −1.018
51 123 −4.141×10−04 7.500×10−05 −9.673×10−04 3.910×10−04 1.948×10−04 2.5498
53 129 −5.326×10−04 1.076×10−04 −1.220×10−03 4.528×10−04 2.267×10−04 2.621
55 133 −6.480×10−04 1.424×10−04 −1.459×10−03 4.970×10−04 2.494×10−04 2.582025
55 135 −6.857×10−04 1.507×10−04 −1.544×10−03 5.260×10−04 2.639×10−04 2.7324
57 139 −8.527×10−04 2.001×10−04 −1.892×10−03 5.915×10−04 2.968×10−04 2.783046
60 143 4.324×10−04 −1.093×10−04 9.424×10−04 −2.593×10−04 −1.296×10−04 −1.065
60 145 2.664×10−04 −6.731×10−05 5.805×10−04 −1.597×10−04 −7.984×10−05 −0.656
62 147 3.935×10−04 −1.032×10−04 8.483×10−04 −2.147×10−04 −1.069×10−04 −0.812
62 149 3.235×10−04 −8.486×10−05 6.975×10−04 −1.765×10−04 −8.790×10−05 −0.6677
67 163 −3.107×10−03 8.721×10−04 −6.561×10−03 1.356×10−03 6.644×10−04 4.23
67 165 −3.063×10−03 8.598×10−04 −6.467×10−03 1.337×10−03 6.550×10−04 4.17
68 167 4.484×10−04 −1.271×10−04 9.431×10−04 −1.875×10−04 −9.150×10−05 −0.56385
71 175 −2.238×10−03 6.514×10−04 −4.657×10−03 8.257×10−04 3.979×10−04 2.2323
72 177 −8.583×10−04 2.519×10−04 −1.781×10−03 3.041×10−04 1.459×10−04 0.7935
73 181 −2.765×10−03 8.176×10−04 −5.722×10−03 9.414×10−04 4.495×10−04 2.3705
92 235 −3.699×10−03 1.185×10−03 −5.287×10−03 4.599×10−04 2.731×10−04 −0.38
9/2 32 73 3.089×10−06 3.176×10−07 1.419×10−05 −1.060×10−05 −3.306×10−06 −0.879468
36 83 1.124×10−05 7.920×10−07 4.326×10−05 −2.963×10−05 −1.029×10−05 −0.970669
38 87 1.959×10−05 6.773×10−07 7.138×10−05 −4.640×10−05 −1.662×10−05 −1.093603
41 93 −1.930×10−04 4.552×10−06 −6.573×10−04 3.901×10−04 1.449×10−04 6.1705
43 99 −2.464×10−04 1.512×10−05 −8.076×10−04 4.478×10−04 1.696×10−04 5.6847
49 113 −5.500×10−04 8.617×10−05 −1.648×10−03 7.241×10−04 2.856×10−04 5.5289
49 115 −5.512×10−04 8.636×10−05 −1.651×10−03 7.257×10−04 2.862×10−04 5.5408
72 179 5.392×10−04 −1.582×10−04 1.403×10−03 −2.395×10−04 −9.166×10−05 −0.6409
83 209 −7.663×10−03 2.388×10−03 −1.922×10−02 2.230×10−03 8.126×10−04 4.1103
5 57 138 −7.965×10−04 1.869×10−04 −2.439×10−03 7.625×10−04 2.772×10−04 3.713646
83 208 −7.773×10−03 2.422×10−03 −2.146×10−02 2.491×10−03 8.243×10−04 4.633
7 67 166 −1.322×10−03 3.711×10−04 −5.264×10−03 1.088×10−03 2.827×10−04 3.6
71 176 −1.588×10−03 4.624×10−04 −6.233×10−03 1.105×10−03 2.824×10−04 3.169
9 73 180 −2.189×10−03 6.472×10−04 −1.083×10−02 1.781×10−03 3.558×10−04 4.825
83 210 −2.545×10−03 7.929×10−04 −1.217×10−02 1.412×10−03 2.699×10−04 2.73
J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 9
Table 3. Influence of the hyperfine interaction on the 4s4p 3P1 − 4s4p
3P0 energy separation and on the lifetime of the
level, as a function of the nuclear spin I and the atomic number Z. ∆E0 is the unperturbed energy separation (in eV), and
∆Ehf is the perturbed energy (in eV) when the hyperfine interaction is taken into account (the 5 digits do not necessarily
represent the accuracy of the calculation - they are intended to show the effect at low Z). τ0, τ1 and τ2 represent the perturbed
lifetimes (in s) of 4s4p 3P0,
P1 and
P1 levels respectively.
I Z A ∆E0 ∆Ehf τ1 τ2 τ0
1/2 34 77 0.20461 0.20458 2.139×10−07 1.340×10−10 7.847×10+00
39 89 0.61806 0.61809 1.393×10−08 5.025×10−11 1.317×10+01
45 103 1.41748 1.41753 1.976×10−09 2.375×10−11 5.111×10+00
47 107 1.74706 1.74714 1.221×10−09 1.923×10−11 1.847×10+00
47 109 1.74706 1.74715 1.221×10−09 1.923×10−11 1.397×10+00
48 111 1.92132 1.92179 9.804×10−10 1.739×10−11 5.286×10−02
48 113 1.92132 1.92181 9.804×10−10 1.739×10−11 4.831×10−02
50 115 2.28581 2.28674 6.623×10−10 1.433×10−11 1.391×10−02
50 117 2.28581 2.28682 6.623×10−10 1.433×10−11 1.172×10−02
50 119 2.28581 2.28687 6.623×10−10 1.433×10−11 1.071×10−02
52 123 2.66778 2.66872 4.673×10−10 1.185×10−11 1.397×10−02
52 125 2.66778 2.66891 4.673×10−10 1.185×10−11 9.609×10−03
54 129 3.06295 3.06418 3.425×10−10 9.804×10−12 8.274×10−03
69 169 6.17717 6.17856 7.752×10−11 2.398×10−12 7.234×10−03
70 171 6.38647 6.38326 7.246×10−11 2.174×10−12 1.367×10−03
74 183 7.22323 7.22219 5.650×10−11 1.464×10−12 1.353×10−02
76 187 7.64172 7.64106 5.051×10−11 1.196×10−12 3.400×10−02
78 195 8.06074 8.05357 4.545×10−11 9.804×10−13 2.895×10−04
80 199 8.48051 8.47364 4.115×10−11 7.937×10−13 3.215×10−04
81 203 8.69084 8.66722 3.922×10−11 7.194×10−13 2.725×10−05
81 205 8.69084 8.66699 3.922×10−11 7.194×10−13 2.672×10−05
82 207 8.90152 8.89226 3.745×10−11 6.494×10−13 1.794×10−04
3/2 31 69 0.06865 0.06864 4.446×10−06 4.528×10−10 8.711×10+00
31 71 0.06865 0.06863 4.446×10−06 4.528×10−10 5.395×10+00
33 75 0.14992 0.14990 4.831×10−07 1.808×10−10 3.524×10+00
35 79 0.26849 0.26843 1.071×10−07 1.044×10−10 5.357×10−01
35 81 0.26849 0.26843 1.071×10−07 1.044×10−10 4.610×10−01
37 87 0.42429 0.42417 3.448×10−08 6.944×10−11 1.270×10−01
54 131 3.06295 3.06259 3.425×10−10 9.804×10−12 1.883×10−02
56 135 3.46760 3.46706 2.611×10−10 8.197×10−12 8.660×10−03
56 137 3.46760 3.46699 2.611×10−10 8.197×10−12 6.920×10−03
64 155 5.12988 5.13022 1.130×10−10 3.876×10−12 2.277×10−02
64 157 5.12988 5.13034 1.130×10−10 3.876×10−12 1.304×10−02
65 159 5.33938 5.33645 1.041×10−10 3.534×10−12 3.155×10−04
76 189 7.64172 7.63948 5.051×10−11 1.196×10−12 5.872×10−04
77 191 7.85115 7.85060 4.785×10−11 1.081×10−12 9.806×10−03
77 193 7.85115 7.85055 4.785×10−11 1.081×10−12 8.310×10−03
79 197 8.27049 8.26987 4.329×10−11 8.850×10−13 7.729×10−03
80 201 8.48051 8.48305 4.115×10−11 7.937×10−13 4.730×10−04
91 231 10.82070 10.80116 2.591×10−11 2.538×10−13 8.872×10−06
5/2 30 67 0.04546 0.04546 1.834×10−05 1.371×10−09 1.771×10+02
37 85 0.42429 0.42425 3.448×10−08 6.944×10−11 6.251×10−01
40 91 0.72914 0.72920 9.434×10−09 4.348×10−11 2.221×10−01
42 95 0.97889 0.97895 4.695×10−09 3.344×10−11 2.395×10−01
42 97 0.97889 0.97895 4.695×10−09 3.344×10−11 2.297×10−01
44 99 1.26339 1.26345 2.584×10−09 2.646×10−11 2.732×10−01
44 101 1.26339 1.26346 2.584×10−09 2.646×10−11 2.175×10−01
46 105 1.57892 1.57900 1.541×10−09 2.132×10−11 1.597×10−01
51 121 2.47489 2.47413 5.525×10−10 1.302×10−11 1.779×10−03
53 127 2.86396 2.86316 3.984×10−10 1.079×10−11 1.661×10−03
59 141 4.08585 4.08364 1.828×10−10 6.211×10−12 2.274×10−04
63 151 4.92049 4.91793 1.233×10−10 4.274×10−12 1.747×10−04
63 153 4.92049 4.91936 1.233×10−10 4.274×10−12 8.977×10−04
66 161 5.54887 5.54933 9.615×10−11 3.205×10−12 5.667×10−03
66 163 5.54887 5.54823 9.615×10−11 3.205×10−12 2.885×10−03
70 173 6.38647 6.38731 7.246×10−11 2.174×10−12 1.702×10−03
10 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions
Table 3. Continued
I Z A ∆E0 ∆Ehf τ1 τ2 τ0
5/2 75 185 7.43246 7.42645 5.348×10−11 1.325×10−12 3.433×10−05
75 187 7.43246 7.42639 5.348×10−11 1.325×10−12 3.364×10−05
82 205 8.90152 8.89930 3.745×10−11 6.494×10−13 2.670×10−04
92 233 11.03725 11.03356 2.506×10−11 2.288×10−13 1.090×10−04
7/2 34 79 0.20461 0.20462 2.139×10−07 1.340×10−10 5.060×10+00
51 123 2.47489 2.47448 5.525×10−10 1.302×10−11 3.371×10−03
53 129 2.86396 2.86343 3.984×10−10 1.079×10−11 2.084×10−03
55 133 3.26430 3.26365 2.976×10−10 8.929×10−12 1.435×10−03
55 135 3.26430 3.26362 2.976×10−10 8.929×10−12 1.281×10−03
57 139 3.67249 3.67164 2.304×10−10 7.463×10−12 8.424×10−04
60 143 4.29381 4.29424 1.645×10−10 5.650×10−12 3.350×10−03
60 145 4.29381 4.29408 1.645×10−10 5.650×10−12 8.830×10−03
62 147 4.71127 4.71166 1.351×10−10 4.695×10−12 4.108×10−03
62 149 4.71127 4.71159 1.351×10−10 4.695×10−12 6.075×10−03
67 163 5.75836 5.75527 8.929×10−11 2.915×10−12 6.802×10−05
67 165 5.75836 5.75531 8.929×10−11 2.915×10−12 6.999×10−05
68 167 5.96779 5.96824 8.264×10−11 2.646×10−12 3.294×10−03
71 175 6.59569 6.59346 6.757×10−11 1.972×10−12 1.350×10−04
72 177 6.80490 6.80404 6.369×10−11 1.786×10−12 9.241×10−04
73 181 7.01409 7.01133 5.988×10−11 1.618×10−12 8.953×10−05
92 235 11.03725 11.03895 2.506×10−11 2.288×10−13 2.864×10−04
9/2 32 73 0.10442 0.10442 1.298×10−06 2.644×10−10 2.655×10+01
36 83 0.34167 0.34168 5.882×10−08 8.403×10−11 2.142×10+00
38 87 0.51641 0.51643 2.146×10−08 5.882×10−11 7.376×10−01
41 93 0.84950 0.84931 6.536×10−09 3.802×10−11 8.194×10−03
43 99 1.11697 1.11672 3.448×10−09 2.967×10−11 5.277×10−03
49 113 2.10110 2.10055 8.000×10−10 1.577×10−11 1.184×10−03
49 115 2.10110 2.10055 8.000×10−10 1.577×10−11 1.179×10−03
72 179 6.80490 6.80544 6.369×10−11 1.786×10−12 1.491×10−03
83 209 9.11260 9.10502 3.584×10−11 5.848×10−13 8.025×10−06
5 57 138 3.67249 3.67170 2.304×10−10 7.463×10−12 5.069×10−04
83 208 9.11260 9.10493 3.584×10−11 5.848×10−13 6.433×10−06
7 67 166 5.75836 5.75705 8.929×10−11 2.915×10−12 1.057×10−04
71 176 6.59569 6.59411 6.757×10−11 1.972×10−12 7.540×10−05
9 73 180 7.01409 7.01193 5.988×10−11 1.618×10−12 2.501×10−05
83 210 9.11260 9.11009 3.584×10−11 5.848×10−13 2.003×10−05
	Introduction
	Relativistic calculation of hyperfine contribution to fine structure splitting and to transition probabilities
	Results and discussion
