(2020) focuses on the importance of having more hyperfine splitting
information for higher excited levels to better understand not only the Fr
atom, but the properties of the different nuclear isotopes.

Figure 4: $A(n^{*})^{3}$ scaling, with $A$ in MHz, vs. $n$ for the 210, 211
and 212 Fr isotopes. Experimental data: red open squares for 210Fr, black
diamond for the 211Fr single value, and blue open triangles for 212Fr, joined
by a dashed line. Theoretical predictions with colored dots joined by a
continuous line for 210Fr by Sahoo  _et al._ (2015), for 211Fr by Grunefeld,
Roberts, and Ginges (2019), and for 212Fr by Lou  _et al._ (2019). Error bars
for experiment and theory are too small to be visible. The effective quantum
numbers are derived from the quantum defects of Simsarian  _et al._ (1999).
The 211Fr theoretical data follow closely a horizontal line of the
$1/(n^{*})^{3}$ scaling law.

###  VI.2 Anomalies

The hyperfine anomalies, first introduced by Bohr and Weisskopf (1950), are
defined as the $A$ deviations from Eq. (7) produced by the finite size of the
nucleus. As a measure of the finite structure influence on the dipole
constants of isotopes 1 and 2, following Persson (2013) the expression for the
${}^{1}\Delta^{2}$ hyperfine anomaly is:

${}^{1}\Delta^{2}=\frac{A^{1}}{A^{2}}\frac{g_{I}^{2}}{g_{I}^{1}}-1,$ (11)

where $(A^{i},g^{i}_{I})$ are the hyperfine magnetic dipole constant and the
nuclear gyromagnetic ratio, respectively, of the $i=(1,2)$ isotopes. For a
point-like nucleus, the hyperfine anomaly is null. The article of Persson
(2013) represents the most recent review of the atomic anomalies. Those of the
$6^{2}S_{1/2}$ Fr states are examined in Zhang  _et al._ (2015). Recent
theoretical studies of Fr anomalies can be found in Konovalova  _et al._
(2018); Konovalova, Demidov, and Kozlov (2020); Roberts and Ginges (2020).
Anomalies can be derived from measured $A$ constants and accurate values of
the nuclear $g$-ratio. This is the case for light alkali atoms with their
precise values in the gas phase determined in the sixties and seventies, as
discussed in Arimondo, Inguscio, and Violino (1977). Anomalies for those atoms
are reported in Table 9, derived from the weighted mean values and variances
of the dipole constants reported in the Tables of the previous Section. Only
anomalies with a value significantly different from zero are presented in
Table 9. For the $5^{2}P_{1/2}$ state of the Rb isotopes, where a very large
discrepancy between the measured values was noted in subsection V, the value
is missed because none reasonable anomaly is associated to the different set
of data. The Rb values of Table 9 are in good agreement with those derived by
Pérez Galván  _et al._ (2007); Pérez Galván, Zhao, and Orozco (2008); Wang
_et al._ (2014b) for ${}^{2}S$ and ${}^{2}D$ states. For the Rydberg $S$
states, enormous anomalies are obtained, $\approx-50(5)$ percent, a quite
surprising result, because the interaction of a Rydberg electron with the
nucleus should be comparable to that of low orbitals. The fairly constant
anomaly for all the ${}^{2}S$ states of the Rb isotopes was pointed out by
Pérez Galván  _et al._ (2007). For Rb ${}^{2}P$ states the situation is not
well defined, with the the $5^{2}P_{1/2}$ value reflecting the discording
results associated to this state, and for the ${}^{2}P_{3/2}$ states the
quadrupole interaction playing an important role. The constant value of the
$4^{2}P_{1/2}$ anomaly applies also to the K isotopes.

Table 9: Light alkali hyperfine anomalies ${}^{1}\Delta^{2}$ with states listed in order of increasing $L$, then of increasing $n$ and finally of increasing $J$ Element | Isotope 1 | Isotope 2 | State | ${}^{1}\Delta^{2}(\%)$
---|---|---|---|---
Li | 6 | 7 | 2$S_{1/2}$ | 0.0068067(8)
|  |  | 2$P_{1/2}$ | -0.1734(2)
|  |  | 2$P_{3/2}$ | -0.155(8)
K | 39 | 40 | 4$S_{1/2}$ | 0.467(2)
|  |  | 4$P_{1/2}$ | 3.90(13)
K | 39 | 41 | 4$S_{1/2}$ | -0.22937(13)
|  |  | 4$P_{1/2}$ | 3.9(3)
Rb | 85 | 87 | 5$S_{1/2}$ | 0.35141(2)
|  |  | 6$S_{1/2}$ | 0.361(19)
|  |  | 7$S_{1/2}$ | 0.342(3)
|  |  | 5$P_{1/2}$ | 0.55(8)
|  |  | 5$P_{3/2}$ | 0.168(5)
|  |  | 6$P_{1/2}$ | 0.31(7)
|  |  | 6$P_{3/2}$ | 0.46(5)
|  |  | 4$D_{5/2}$ | 0.60(15)
|  |  | 5$D_{3/2}$ | 0.279(6)
|  |  | 5$D_{5/2}$ | 0.44(5)

The francium case is different, because the list of isotope data is quite long
and therefore, interesting information about the nuclear structures could be
derived from the anomaly determinations. However, for francium an important
element is missing in Eq. (11) because a direct measurement of the nuclear
gyromagnetic ratio is available only for 211Fr in Stone (2005). For the
remaining isotopes, the $g_{I}$ ratios are derived by assuming a zero anomaly:
see Raghavan (1989). In the recent theoretical studies of Fr anomalies by
Konovalova  _et al._ (2018); Roberts and Ginges (2020); Konovalova, Demidov,
and Kozlov (2020), the information on the nuclear structure is replaced by
derivations of the radial nuclear structure and of the nuclear radius. In
order to obtain the anomalies without relying on such theoretical analyses,
Grossman  _et al._ (1999), following Persson (1998), concentrated their
attention on the $7^{2}S_{1/2}$ and $7^{2}P_{1/2}$ states. The ${}^{2}P_{1/2}$
electron probes the nucleus with a more uniform radial dependence of the
interaction than does the ${}^{2}S_{1/2}$ electron. They introduced the
following $R(S/P)$ ratio of their hyperfine constants:

$R(S/P)=\frac{A(S_{1/2})}{A(P_{1/2})}$ (12)

as a probe of the nuclear magnetization distribution. Since both states are
spin-1/2, these ratios are independent of quadrupole effects that complicate
the extraction of the nuclear structure information. For several francium
isotopes, the ratio is presented in Fig 5. Notice the staggered isotope
behaviour of $R(S/P)$ vs. the even/odd isotope number. Such behavior evidences
the different radial distributions of the nuclear magnetization for the
even/odd number of neutrons. The ratios of the hyperfine constants of the
$7^{2}S_{1/2}$ and $7^{2}P_{3/2}$ states, as well as those of the
$7^{2}P_{1/2}$ and $7^{2}P_{3/2}$ ones, do not exhibit such a clear staggered
dependence as the $R(S/P)$ ratio plotted in the figure. We have tested the
presence of a $R(S/P)$ staggered dependence also for the Rb isotopes using the
85 and 87 data in Table LABEL:Table:85Rb,LABEL:Table:87Rb, and the 82 isotope
ones from Zhao  _et al._ (1999). This limited data set appears to confirm the
$R(S/P)$ staggered behavior.
The ratio of the hyperfine constants of the ${}^{2}P_{1/2}$ and
${}^{2}P_{3/2}$ states has been proposed in Konovalova, Demidov, and Kozlov
(2020) as an additional test of the nuclear structure, even if the nuclear
quadrupole coupling is important for the latter state. This ratio calculated
from the francium Table LABEL:Table:Fr data does not exhibit a clear
dependence on the isotope number.

Figure 5: $R(S/P_{1/2})$ ratio for the $7^{2}S_{1/2}$ and $7^{2}P_{1/2}$
states of francium isotopes vs the isotope number. Mean values and error bars
are derived from the francium Table LABEL:Table:Fr. Notice the staggered
dependence of $R(S/P_{1/2})$ on the even/odd isotope number. The 221Fr isotope
value with a large error bar agrees with that dependence.

##  VII Conclusions

The previously published experimental values for the stable isotopes of the
light alkali atoms and for all the nuclear-ground-configuration francium
isotopes have been compiled. The Tables report the most accurate data obtained
before 1977 and all those published after that time. For each measured
hyperfine constant we present a recommended value. A critical examination of
the most interesting cases, or of the most discording ones, is presented. For
the discording cases we have calculated a weighted enhanced error following
the procedure of the Particle Data Group in Zyla  _et al._ (2020). For those
cases a comparison of our data analysis to the cluster maximum likelihood
estimator introduced by Rukhin (2009, 2019), is presented in the supplementary
information (SI) of this review Allegrini, Arimondo, and Orozco (2022). We
encourage future reviewers of larges sets of data to look into these more
recent methods, that are currently gaining adepts in the community, but are
not followed by those in charge of the recent redefinition of the fundamental
constants of physics and chemistry Tiesinga  _et al._ (2021).
The experimental interest in measuring hyperfine constants is renewed in the
recent years. Today, the laser sources required to excite energy levels not
easily assessed at the optical pumping time are available on the market. Those
sources and the associated atomic species can be important for quantum
simulation and computational investigations. As in the past, the quest for
higher precision spectroscopic measurement could represent an additional step
for refining the experimental tools required in those areas.
Instead of concentrating the attention on a specific atomic state, the recent
global theoretical analyses cover all the atomic states of a single species,
possibly of all the isotopes. For this global approach, precise experimental
data are required for a large set of states, stimulating the hyperfine data
search in several directions.
Spectroscopic investigations of alkali atoms are also associated with the
theoretical progress on the electron-nucleus hyperfine coupling. The hyperfine
coupling being an important probe of the nuclear structure, the Grossman  _et
al._ (1999) francium research has introduced the ratio of the $S$ and $P$
state hyperfine couplings as a new tool for studying nuclear properties. This
has stimulated a large theoretical effort, but it has pointed out the need for
more precise experimental data. The application of this ratio to lighter
alkalis could be an interesting exploration direction.
The electron-nucleus interaction contains a term characterized by the nuclear
magnetic octupole moment. In recent years three different measures of that
moment are reported for 133Cs $6^{2}P_{3/2}$ by Gerginov, Derevianko, and
Tanner (2003), for 87Rb $5^{2}P_{3/2}$ in the experimental data of Ye  _et
al._ (1996) reexamined by Gerginov, Tanner, and Johnson (2009), and more
recently for the 133Cs $6^{2}D_{3/2}$ state by Chen  _et al._ (2018). The
parity violation search in alkali atoms may also find new life. Up to now, the
most accurate data on the atomic parity-non-conserving interaction was derived
from the $6^{2}S_{1/2}\to 7^{2}S_{1/2}$ transition in cesium by Wood  _et al._
(1997). To obtain a more accurate value of the nuclear weak charge producing
the parity-violating Hamiltonian, it would be desirable to consider other
candidates. Gwinner and Orozco (2022) with their collaborators are pursuing
its measurement in the $7^{2}S_{1/2}\to 8^{2}S_{1/2}$ transition in a variety
of Fr isotopes, while Aoki  _et al._ (2017) have proposed to search for that
violation operating on the ${}^{7}S_{1/2}\to 6^{2}D_{3/2}$ electric quadrupole
transition in 210Fr. It is expected that advances in the understanding of
hyperfine interactions will continue to illuminate atomic parity nonconversion
and vice versa.
As a final expectation, in the near future, a few totally new entries will be
added to the above Tables, counterbalanced by several refined entries.

##  VIII Acknowledgments

The authors are grateful to Marianna Safronova for a guide through the recent
progress in the theoretical analyses of the hyperfine structures and a careful
reading of the manuscript, and to Roberto Calabrese for hints on the francium
spectroscopy. The authors are grateful and indebted to the referee for ample
comments that have made this a much better review and for providing the
algorithm of the CMLE approach. Hassan Jawahery, Peter J. Mohr and William D.
Phillips have helped elucidate the evaluation of the errors to report
recommended values. The authors acknowledge Wang-Yau Cheng, Pierre Dubé,
Johannes Deiglmayr, Randy J. Knize, Mark Lindsay, Frederic Merkt, Dieter
Meschede, Priyanka Rupasinghe and Sun Svanberg for communicating data of
various atomic states. The very kind help of the librarians Armelle Michetti
and Odile Richaud in Grenoble, and Massimiliano Bertelli in Pisa is also
acknowledged.

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