Hyperfine structure and hyperfine anomaly in Pb

The hyperfine structure in the 6p2-configuration in lead has been analysed and the results is compared with calculations. The hyperfine anomaly and improved values of the nuclear magnetic moment for four lead isotopes is obtained, using the results from the analysis. The results open up for new measurements of the hyperfine structure in unstable lead isotopes, in order to extract information of the hyperfine anomaly and distribution of magnetisation in the nucleus.


Introduction
Hyperfine structures (hfs) and isotope shifts (IS) in PbI have been studied over the years using different techniques. The electronic ground configuration of lead is 6p 2 , which gives rise to five low-lying, even-parity, metastable states: 1 S 0 , 3 P 0,1,2 , 1 D 2 . The first odd-parity state (6p7s 3 P 0 ) has an energy of 34960 cm −1 , which places most transitions from the metastable states in the ultraviolet region. This has made high-resolution laser spectroscopy difficult until the advent of frequency doubled cw titanium-sapphire lasers. The high-lying metastable 1 D 2 state is accessible through transitions in the IR. The experimental results obtained by different authors is reviewed and an analysis based on the effective operator approach is performed in the 6p 2 configuration. The result of the analysis is used to investigate the statedependent hyperfine anomaly with application to unstable isotopes. The hyperfine anomaly of four lead isotopes and improved values of the nuclear magnetic moment has been obtained.

Experimental hyperfine structure constants
The hfs in Pb has been studied by different methods over the years, using optical spectroscopy as well as with the Atomic Beam Magnetic Resonance (ABMR) technique. With the advent of lasers, especially in the UV-region, more studies have been done. In table 1 an overview of the experimental hfs constants in 207 P b for a number of states of interest are given. The hfs constants from high accuracy measurements have been corrected with respect to the non-diagonal hyperfine interaction. 2602.060(1) [1] 2602.144(1) 2600.8 (9)[5] 6s 2 6p 2 1 D 2 609.818(8) [2] 609.818(8) 6s 2 6p7s 3 P 1 8802.0(1.6) [5] 8807.2(3.0) [7] There also exist studies of the hfs in unstable isotopes [3,6,7]. A compilation of the hfs constants obtained is given in table 2. These studies has mainly been concerned with the IS, i.e. the change in nuclear charge radii, hence the use of states without hfs.
As can be seen the hfs is known in only one state for most isotopes, with the exception of four isotopes. As we are interested in the hyperfine anomaly these isotopes will be studied in detail.
3 Analysis of hyperfine structure

Eigenvectors
Lead has a quite simple ground electronic configuration, but deviates from pure LS-coupling. In order to perform an analysis of the hfs, the break- down of LS-coupling must be taken into account and eigenvectors have to be obtained. The eigenvectors can be obtained by diagonalising the energy matrix of the spin-orbital and the electrostatic interactions or by an analysis of the experimental g J factors. The energy matrix has been derived by for example Condon and Shortley [8]. The agreement between the fitted and experimental energy levels using this energy matrix is not particular good. Landman and Lurio [2] included spin-spin, orbit-orbit and spin-other-orbit interactions but this did not improve the fit.
Instead of using the energy matrix, an analysis of the experimental g J factors will probably give a better description of the system. The experimental g J factors can be described as: where g LS J is the Lande g J factor for a pure LS-state corrected for the anomalous spin of the electron, α and β are the intermediate coupling coefficients.
The experimental g J factors must be corrected for diamagnetic and relativistic effects [9] [10]. In lead, these corrections are of the order 5 · 10 −4 , as can be seen when comparing the experimental g J factor for the 3 P 1 state with the corrected g J factor in table 3. A Hartree-Fock calculation of these diamagnetic and relativistic corrections has been done in [10] and the result is presented in table 3. In order to exclude coupling effects the sum of the g J factor for the J=2 states are given. The calculated corrections were not as large as expected, why configuration interaction effects should be important. It has been shown by Gil and Heldt [11] that there exists a configuration mixing between the 6p 2 and 6p7p configurations, by including configuration interactions in the energy matrix analysis. Even though their fit suffer from the same problems as in the ordinary matrix analysis, a calculation of the g J factors using their eigenvectors and including diamagnetic and relativistic corrections gave an excellent agreement in comparison with experimental data [10].
In this case we exclude the configuration interaction when analysing the g J factors, as a precaution, in order to obtain accurate eigenvectors, the estimated errors of the relativistic and diamagnetic corrections were enlarged.
All obtained eigenvectors are given in table 4. In case A the eigenvectors are obtained by analysing the energy levels according to the energy matrix of Condon and Shortley [8], in case B eigenvectors are derived by Landman and Lurio [2] and in case C the eigenvectors are obtained by analysing the experimental g J factors.

Hyperfine interaction
The analysis of the hyperfine interaction is based on an effective hyperfine hamiltonian, which for the magnetic dipole interaction is written as [12]: By determination of the angular parts, using the eigenvectors, the magnetic dipole interaction constants "A" can be expressed as a linear combination of the orbital (01), spin-dipole (12), and contact (10) effective radial parameters (a ij ).
The numbers in the parentheses correspond to the rank of the spherical tensor operators in the spin and orbital spaces. In this way can the effective radial parameters for the different eigenvectors be fitted to the corrected A factors.
The obtained effective radial parameters are presented in table 5. The errors in the effective radial parameters are mainly due to uncertainty of the eigenvectors, since the errors in the energy fit is quite large and hard to obtain, these errors are expected to be on the order of 10%. In the analysis of the experimental g J factors, the errors are possible to obtain from the fit.  (12) The effective radial parameters, proportional to the nuclear moment and the effective r −3 values can be expressed as [12]: Since the nuclear magnetic dipole moment has been determined independently (µ I = +0.592583(9) n.m.), it is possible to derive the effective r −3 values. These semi-empirical values are presented in table 6 together with calculated r −3 values using the Hartree-Fock (HF) and Optimized Hartree-Fock-Slater (OHFS) methods by Lindgren and Rosen [12].  (15) The calculated relativistic values of r −3 01 differ from the experimental value (case C) by 1.4% for the OHFS and 1.3%, for the HF method, while the corresponding difference between the calculated and experimental values of r −3 12 is 4.4% and 3.7%, respectively. The large difference between the experimental and calculated values of r −3 10 are mainly due to spin polarisation. Bouazza et al. [5] estimated the fraction of the spin polarisation to be 50.62%, as shown the isoelectronic Bi II [13], yielding a value of r −3 10 = −16.06a −3 0 , in reasonable agreement with the experimental values.

Hyperfine anomaly
In addition to the hyperfine interaction and nuclear magnetic dipole moment is it possible to obtain information on the distribution of magnetisation in the nucleus through the so called Bohr-Weisskopf effect (BW-effect) [14,15,16]. The first to consider the influence of the finite size of the nucleus on the hyperfine structure was Bohr and Weisskopf [14]. They calculated the hyperfine interaction of s 1/2 and p 1/2 electrons for an extended nucleus, and showed that the magnetic dipole hyperfine interaction constant (A) for an extended nucleus is generally smaller than for a point nucleus. The effect on the hyperfine interaction from the extended charge distribution of the nucleus gives rise to the so-called Breit-Rosenthal effect (BR-effect) [17,18,19,20]. In this case, as in most but not all cases, the differential BR-effect is negligible when two isotopes are compared. Inclusion of the BR-effect will not have any effect on the results, since the BW-and BR-effects show the same behaviour. The BR-effect is therefore neglected in the following discussion. Isotopic variations of magnetic moments became larger than those in the point dipole interaction since there are different contributions to the hfs from the orbital and spin parts of the magnetisation in the case of extended nuclei. The fractional difference between the point nucleus hfi constant (A point ) and the constant obtained for the extended nuclear magnetisation is commonly referred to as the Bohr-Weisskopf (BW) effect [16].The hfs constant A can therefore be written as where ǫ BW is the BW-effect, and A point is the A constant for a point nucleus. The BW-effect is dependent on both nuclear and atomic proper-ties, i.e. the electron density within the nucleus. The nuclear part, i.e. the distribution of nuclear magnetisation, can be calculated using different nuclear models [15,16]. Since electronic wavefunctions cannot be calculated with sufficient high accuracy in complex atoms, as they can be in hydrogenlike ions and muonic atoms, it is not possible to determine ǫ BW directly in atoms. However, it is possible to determine the difference of the BWeffect in two isotopes, the so-called (differential) hyperfine anomaly (hfa). Comparing the ratio of the measured hfs constants for two isotopes with the independently measured ratio of the nuclear magnetic dipole moments to extract the hfa, 1 ∆ 2 , for the isotopes 1 and 2, and a given atomic state, gives: where µ I is the nuclear magnetic dipole moment, and I the nuclear spin. In the case of electrons with a total angular momentum j>1/2 the anomalies may be disregarded as the corresponding wavefunctions vanish at the nucleus. The hfa can show a dependence of the atomic state, a state dependent hfa, where the values for different states can vary significantly. The reason for the state dependence is that the hyperfine interaction consists of three parts [21,22], orbital, spin-orbit and contact (spin) interaction, where only the contact interaction contributes to the hfa. Since the contribution of the different interactions differ between different atomic states, and it is only the spin interaction giving rise to the hfa, a state dependent hfa is the result. It is therefore suitable to rewrite the dipole hyperfine interaction constant as where A c is the contribution due to the contact interaction of s (and p 1/2 ) electrons and A nc is the contribution due to non-contact interactions. The experimental hfa, which is defined with the total magnetic dipole hyperfine constant A, should then be rewritten to obtain the relative contact contribution to the hfa: where 1 ∆ 2 c is the hfa due to the contact interaction, that is, for an s-or p 1/2 -electron.
From the discussion, one might come to the conclusion that one needs independent measurements of the nuclear magnetic moments and the A-constants in order to obtain the hfa, however, this is not true. As has been shown by Persson [23], it is possible to extract the anomaly solely from the A-constants of two different atomic states, provided the ratio Ac A differs for the different states. Comparing the A-constant ratio, for two isotopes, in two atomic states, gives: Where B and C denotes different atomic states and 1 and 2 denotes different isotopes. The ratio between the two A-constant ratios for the isotopes will therefore only depend on the difference in the contact contributions of the two atomic states and the hfa. It should also be noted that the ratio As A is isotope independent. Once determined for one isotopic pair, the ratio can be used for all pairs, which is very useful in the study of hfa in radioactive isotopes. It is possible to determine the ratio in two different ways; either by an analysis of the hyperfine interaction or by using a known hfa as a calibration. It should be noted that the atomic states used must differ significantly in the ratio As A , as a small difference will lead to an increased sensitivity to errors.
Since the hfa is normally very small (1% or less) it is often necessary to have high accuracy for the A-constants , preferably better than 10 −4 [16]. In stable isotopes there is no major problem to measure the nuclear magnetic moment with sufficient accuracy using NMR or ABMR, while for unstable isotopes it is more difficult. In most cases there does not exist any high precision measurements of the nuclear magnetic moment, in most cases the nuclear magnetic moment is deduced from the hfs while neglecting the effect of hfa. However, there might exist measurements of two A-constants, if the nuclear charge radius of the unstable isotopes has been measured by means of laser spectroscopy. In order to obtain the hfa one therefore needs to measure the A-constants with an accuracy better than 10 −4 , something that can be done by laser spectroscopy provided the A-constant is larger than about 1000 MHz, as being the case in Pb.

Hyperfine anomaly in unstable isotopes
From table 2 we see that the A constants are known for two states in four unstable isotopes, 191 P b, 193 P b, 195 P b and 197m P b. The complication is that one state has a small A constant and the other belongs to the 6p7s configuration. Still, it is possible to obtain a state dependent hyperfine anomaly using: with the A constants from 207 Pb, as reference nucleus (1). The state dependent hyperfine anomalies obtained are given in table 7. Note that the hfa contains contributions from both states involved. This makes the contact contribution of the hyperfine interaction is quite complicated with both s and p 1/2 electrons. However it is possible to examine the hyperfine interaction in the 6p7s 3 P 1 further. Bouazza et al [5] gives the eigenvector components for this state, and by assuming that the effective hyperfine interaction parametes for the p electrons are the same in the 6p 2 and 6p7s configurations, we can deduce a value of the s electron effective hyperfine interaction parameter.
Using the eigenvector we find that the A constant for the 6p7s 3 P 1 can be expressed in effective hyperfine interaction parameters as: Using the effective hyperfine interaction parameters for the p electrons in table 5, gives the s electron parameter a 10 s = 6884 M Hz , which is a reasonable value. It is now possible to calculate the contact contribution in equation 9, both for the 6p7s 3 P 1 and 6p 2 1 D 2 states.
Using this it is possible to obtain the state independent contact anomaly. It must be noted that the contact anomaly consists of both s-and p 1/2electron parts. If we assume that the contribution to the hyperfine anomaly is the same for s-and p-electrons we must correct the obtained state dependent hyperfine anomalies by a factor 1.13 ( Ac A (6p7s 3 P 1 ) − Ac A (6p 2 1 D 2 )), giving the state independent hyperfine anomaly in table 8. In order to check if the result is reasonable we can use the obtained hyperfine anomaly to calculate the nuclear magnetic dipole moment of the four isotopes using the measured A constants: rearranging gives Using this we can calculate the nuclear magnetic dipole moment using both atomic states, the results are given in table 9.
The agreement between the different states is much better for the corrected values, giving a better value for the nuclear magnetic moments. Table 9: Nuclear magnetic dipole moments from [3] and derived correcting for the hyperfine anomaly. The errors are only from experimental uncertainty.

The empirical Moskowitz-Lombardi formula.
The empirical Moskowitz-Lombardi (ML) formula was established in 1973 as a rule for the s-electron BW-effect in mercury isotopes [24].
where l is the orbital momentum for the odd neutron. It turned out that the empirical rule provided a better agreement with experimental hfa than the theoretical calculations performed by Fujita and Arima [15] using microscopic theory. The rule can be qualitatively explained by the microscopic theory used by Fujita and Arima [15], where the parameter α is more state independent than given by the theory. Further investigations gave an analogous expression for the odd-proton nuclei 191,193 Ir ,197,199 Au and 203,205 Tl, but also for the doubly-odd 196,198 Au nuclei. The results indicate that the spin operators g (i) s Σ (1) i are state independent for these nuclei. It is worth noting that all nuclei discussed lie close to the doubly closed shell nucleus 208 Pb, where one would expect the single particle model to provide a good description of the nucleus. In our case we would expect a fair agreement with the ML formula. However we have to keep in mind that the contact contribution is not a pure s anomaly, thus there will be an unknown numerical factor. We also have to take into account that the nuclear configuration of the unstable isotopes ( i 13/2 , I = l + 1 2 ) and the reference nucleus ( 207 P b, p 1/2 , I = l − 1 2 ) are different. The ML formula for calculating the hyperfine anomaly will therefore be: The calculated hyperfine anomalies are given in table 10 with the experimental hyperfine anomalies. From the results it seems that the ML formula still holds. The value of α seems to be too small, but one have to keep in mind that the ML formula uses s-electrons, the contact anomaly used in this case contains both s-and p-electron contributions, which must be evaluated further in order to make a more quantitative comparison.

Discussion
The hyperfine structure of 207 Pb has been analysed and the analysis has been used as the basis for determining the hyperfine anomaly in four unstable isotopes, 191 P b, 193 P b, 195 P b and 197m P b, using the method of Persson [23]. The derived hyperfine anomaly has then been used to obtain better values of the nuclear magnetic moment for the unstable isotopes. There exists measurements in other unstable isotopes, table 2, but only in one state that exhibits hyperfine structure, why it is not possible to derive the hyperfine anomaly in these isotopes. It would be possible if another atomic state is measured in these isotopes, preferably the 6p 2 1 D 2 . It is also possible to make the new measurements in the 6p 2 3 P 1 or 3 P 2 . The optimum would be to make measurements in all possible states in the 6p 2 configuration, thus giving in total four atomic states that enable a cross-check of the results. Due to constrains in the population of atomic states at accelerators, only the lower lying states would be feasible, thus excluding the relatively high energy state 6p 2 1 D 2 . The remaining states offer another complication, as the contact contribution to the hyperfine structure for the 6p 2 3 P 1 state ( As A = 0.452) is close to the contribution of the 6p7s 3 P 1 state ( As A = 0.422), which is not suitable for an analysis of the hyperfine anomaly [23]. The contact contribution to the hyperfine structure for the 6p 2 3 P 2 state ( As A = −0.247), is suitable for analysis with all other states. An experiment where the hyperfine structure of the 6p 2 3 P 2 state in unstable isotopes of lead are measured would give both better values of the nuclear magnetic moments as well as values of the hyperfine anomaly.
The empirical Moskowitz-Lombardi formula seems to be valid, from the results here. The mix of s-and p-electron contributions makes it difficult to make quantitative comparison, however, with more experimental data this should be possible.