Calculation of francium hyperfine anomaly

The Dirac-Hartree-Fock plus many-body perturbation theory (DHF+MBPT) method has been used to calculate hyperfine structure constants for Fr. Calculated hyperfine structure anomaly for hydrogen-like ion has been shown to be in good agreement with analytical expressions. It has been shown that the ratio of the anomalies for $s$ and $p_{1/2}$ states is weakly dependent on the principal quantum number. Finally, we estimate Bohr--Weisskopf corrections for several Fr isotopes. Our results may be used to improve experimental accuracy for the nuclear $g$ factors of short-lived isotopes.


I. INTRODUCTION
In recent years, the precision achieved in laser spectroscopy experiments coupled with advances in atomic theory has enabled new atomic physics based tests of nuclear models. The hyperfine structure constants and isotope shifts are highly sensitive to the changes of charge and magnetization distributions inside the nucleus because they depend on the behavior of the electron wave function in this region. The hyperfine structure (HFS) measurements can serve as very useful tool for understanding of shape coexistence phenomena in atomic nuclei [1].
The ratio of magnetic hyperfine constants A for different isotopes is usually assumed to be equal to the ratio of their nuclear g factors g I = µ/I, where µ and I are magnetic moment and spin of the nucleus. However, this is true only for the point-like nucleus. For the finite nucleus one should take into account (i) distribution of the magnetization inside the nucleus and (ii) dependence of the electron wave function on the nuclear charge radius. Former correction is called magnetic, or Bohr-Weisskopf (BW) correction [2] and the latter one is called charge, or Breit-Rosenthal (BR) correction [3,4]). These corrections break proportionality between magnetic hyperfine constants and nuclear g factors. This phenomenon is called hyperfine anomaly (HFA) [2]. Below we discuss how to calculate HFA for many-electron atoms with available atomic package [6], which is based on the original Dirac-Hartree-Fock code [7]. This package was often used to calculate different atomic properties including HFS constants of Tl [8,9], Yb [10], Mg [11], and Pb [12].

II. THEORY AND METHODS
It is generally accepted that the observed HFS constant A can be written in the following form: Here g I is a nuclear g factor, g I A 0 is a HFS constant for the point-like nucleus, δ and are the nuclear charge distribution (BR) and magnetization distribution (BW) corrections respectively. A 0 is independent on the nuclear g factor. In the case of hydrogen-like ions the expression for A 0 was obtained in the analytical form by Shabaev [5]: Here α is the fine-structure constant, Z is the nuclear charge, m and m p are electron and proton masses, j is the total electron angular momentum, κ is a relativistic quantum number, N = n 2 r + 2n r γ + κ 2 , n r is a radial quantum number, γ = κ 2 − (αZ) 2 . We use the model of the homogeneously charged ball of the radius R = 5 3 r 2 1/2 . The extended nuclear magnetization leads to a modification of the hyperfine interaction. It was shown in Refs. [26,27] that the corresponding contribution to the HFS constant may be factorized by "atomic" and "nuclear" factors. Following Refs. [26,27] the corresponding factor d nuc depending on the nuclear spin and configuration, was introduced. Then corrections δ and for a given Z and electron state can be written as [28]: where b N and b M are factors, which are independent of the nuclear radius and structure. It follows from Eqs. (1) and (3), that if we calculate HFS constant numerically for different R and d nuc , we should get following dependence on the radius in the first order in δ and : Within the point-like magnetic dipole approximation (d nuc = 0) the Bohr-Weisskopf correction is equal to zero, and the HFS constant can be fitted by the function: On the other hand, for d nuc = 1 one obtains: Let us compare HFS constants for two isotopes with nuclear g factors g (1) I and g (2) I , slightly different nuclear radii R (1,2) = R ± r, and nuclear factors d I , 0, R)/∂R A(g (2) I , 0, R) .
Then the part of the HFS anomaly related to the change of the nuclear charge distribution 1 ∆ 2 BR (R) is: Nuclear radii of heavy isotopes are typically very close, then r R, and anomaly (7) is therefore small. For isotopes with the same nuclear factors d nuc similar dependence on the nuclear radii holds for the magnetic part of the HFS anomaly 1 ∆ 2 BW . However, the nuclear factors may significantly vary from one isotope to another. In this case we can neglect the radial dependence of the magnetic part of the HFS anomaly and write it as nuc , d nuc ). Thus, the HFS anomaly can be divided into two terms related to the nuclear charge and magnetization distributions: In this work we calculate the magnetic hyperfine constants and HFS anomalies for low-lying states of Fr atom within the Dirac-Hartree-Fock (DHF) approximation and the DHF plus many-body perturbation theory (DHF+MBPT) method. The effects of the Breit corrections and spin-polarization of the core are also considered.

A. HFS anomaly for H-like francium ion
Here we calculate HFS constants of the 1s, 2s, and 2p 1/2 states of Fr 86+ for the different nuclear radii R and compare our results with analytical expressions from Ref. [5]. Figure 1 shows the dependence of the hyperfine     The ground configuration of the neutral francium atom is [Rn]7s. If we treat francium as an one-electron system with the frozen core, we can do calculation using Dirac-Hartree-Fock (DHF) method. In this case the dependence of the HFS constants on the nuclear radius is similar to the one-electron ion.

R (fm)
In the DHF approximation the HFS constant A(7p 3/2 ) = 0.56 GHz is very small and practically does not depend on R (see Table II). At the same time, the HFS constants A(7s) and A(7p 1/2 ) are well described by Eqs. (5)  Situation changes when we include spin-polarization of the core via random phase approximation (RPA) corrections. These corrections lead to effective mixing of different partial waves, thus A(7p 3/2 ) constant acquires contributions from the s and p 1/2 waves. Due to the RPA corrections the value of the constant A(7p 3/2 ) is significantly changed. At the same time this constant becomes sensitive to the distributions inside the nucleus. To account for that, we can use Eq. (3) with the same γ as for s and p 1/2 states. The RPA corrections for the 7s and 7p 1/2 states are smaller than for 7p 3/2 , but they are also significant. Due to the RPA corrections the ratios of the parameters b N and b M for 7s and 7p 1/2 states change by ∼ 15%: b N (7s) b N (7p 1/2 ) = 3.153 and b M (7s) b M (7p 1/2 ) = 3.073. Core-valence and core-core electron correlations were taken into consederation within DHF+MBPT method [6]. Electron correlation corrections significantly change A 0 values. The parameters b N and b M also change, but ratios of these parameters for the 7s and 7p 1/2 states remain stable. Without RPA correc- According to Mårtensson-Pendrill [21] the ratio of b N parameters obtained by scaling the Breit-Rosenthal corrections for Tl is equal to 3.2 in good agreement with our result. For the ratio of the b M parameters the value of 3.0 was used in [21] also in agreement with our results. Information about parameters b N and b M can be extracted from the experimentally measured ratio of HFS constants ρ = A(7s)/A(7p 1/2 ). This ratio can be written as a function of nuclear radius and nuclear factor: where ρ 0 = A 0 (7s)/A 0 (7p 1/2 ). Several experimentally measured values of ρ for odd-odd and even-odd isotopes [13] and corresponding fits by Eq. (10) are presented in Fig. 2. Even-odd Fr isotopes with neutron number N ≤ 126 (A ≤ 213) have spin I = 9/2 and nearly constant magnetic moments. When going from A = 213 to A = 207, the magnetic moment µ(A, 9/2) changes only by 3%. Their ground states are regarded as a pure shell-model h 9/2 states, therefore one can assume that d nuc factor is also constant within 3% limits. Factor d nuc was calculated by the simple shell-model formula [21]: d nuc = 0.3 for A = 207 − 213. Then the one-parameter fit with ρ 0 as free parameter gives us the following relation: ρ = 8.456 (1 − 0.033 R 2γ−1 ), where we used our final results for b N and b M from Table II, or ρ = 8.404 (1 − 0.031 R 2γ−1 ) within two-parameter fit. Comparing these two results we can estimate the error bars for fitting parameters to be: ρ 0 = 8.43 (3) and b N + d nuc b M = 0.032 (1). Note that the theoretical value of ρ 0 obtained within DHF+MBPT+Br+RPA method is equal to 8.85, which is 5% larger. Taking into account the FIG. 2: Experimentally measured ratios ρ = A(7s)/A(7p 1/2 ) for even-odd and odd-odd Fr isotopes [13]. The nuclear radii R are taken from Ref. [25]. Lines are the one-parameter fits by Eq. (10), dashed line corresponds to two-parameter fit. For even-odd isotopes we use dnuc = 0.3 [21] and parameters ρ0 (one-parameter fit), or ρ0 and (bN + dnuc bM ) (two-parameter fit). Then for odd-odd isotopes we fix the obtained by the one-parameter fit ρ0 value and fit nuclear factor dnuc, with the result dnuc = 0.49. For 221 Fr the fit gives dnuc = 0.05. possible change of the d nuc factor ( 3%) and its possible deviation from the shell-model value, the correspondence between fitted and calculated ρ values should be regarded as satisfactorily.
We used formulas from Ref. [26] to calculate d nuc factor for the odd-odd Fr isotopes. Spins and configurations for odd-odd Fr isotopes with A = 206 − 212 are different (I = 5, 6, 7, 3). Correspondingly, d nuc factor is different for different isotopes. However, it can be shown that for all these cases d nuc = 0.5 (1). To check the general applicability of our approach we used the same nuclear factor for the all considered odd-odd Fr isotopes. We fix ρ 0 obtained for even-odd Fr isotopes and fit nuclear factor for odd-odd ones which gives us d nuc = 0.49 in agreement with the shell-model estimation. The deviation of the experimental ρ values for 206m Fr and 206g Fr from the fit line (see Fig. 2) is obviously connected with the structural changes in these nuclei resulting in the changes of the d nuc factor (see discussion in Ref. [13]). For 221 Fr the fit gives d nuc = 0.05. This result can be of a particular interest for nuclear physics and more detailed analysis will be presented in the forthcoming paper.
The accuracy reached in our calculations of HFS constants for neutral Fr can be estimated in comparison with available experimental and theoretical data presented in Table III. Due to Bohr-Weisskopf correction calculated

IV. CONCLUSIONS
In this work we use the method developed in Ref. [28] to calculate the hyperfine anomaly by the analysis of the HFS constants of Fr as functions of nuclear radius. The HFA in this method can be parametrized by coefficients b N and b M . We tested our method by calculating HFS constants of H-like francium ion and obtained fairly good agreement with analytical expression from Ref. [5]. Then we made calculations for neutral Fr, described as a system with one valence electron. We show that the ratios of b N (7s)/b N (7p 1/2 ) and b M (7s)/b M (7p 1/2 ) are practically the same, as in H-like ion and rather stable within the DHF and DHF+MBPT approximations. However when we include spin-polarization of the core by means of RPA corrections, these ratios change by 10 -15%.
The corrections caused by redistribution of the magnetization inside the nucleus were estimated using experimentally measured ratio of the HFS constants A(7s)/A(7p 1/2 ). Estimated Bohr-Weisskopf corrections for odd-odd francium isotopes 206, 208, 210, and 212 were found to be 1.62 times larger, then for even-odd isotopes 207, 209, 211, and 213. The Bohr-Weisskopf correction for 221 Fr is significantly smaller, than for other even-odd isotopes. This information can be used to ob-tain more accurate values for the nuclear g factors of the Fr isotopes from the ratios of the HFS constants. The reliability of the applied method enables one to determine the nuclear factor d nuc which gives important nuclearstructure information and may be compared with the theoretical predictions.