Hyperfine anomalies in Gd and Nd

The hyperfine anomalies in Gd and Nd have been extracted from the experimental hyperfine structure constants using a new method. In addition to the values of the hyperfine anomaly new improved values of the nuclear magnetic dipole moment ratios are derived.


Introduction
The study of hyperfine structure (hfs) in atoms has provided information of the electromagnetic moments of the nucleus as well as information of the properties of the electrons in the atom [1,2]. The magnetic hfs has in addition proved to give information on the distribution of magnetization in the nucleus through the so called Bohr-Weisskopf effect (BW-effect) [3,4,5]. Bohr and Weisskopf [3] showed that the magnetic dipole hyperfine interaction constant (a-constant) is smaller for an extended nucleus compared with a point nucleus. The extended charge distribution of the nucleus gives rise to the so-called Breit-Rosenthal effect (BR-effect) [6,7,8,9]. It was also shown that isotopic variations, in combination with the different contributions to the hfs from the orbital and spin parts of the magnetization in an extended nucleus, could yield large isotopic deviations from the point nucleus. The reason for this is that s-and p 1/2 -electrons have a probability of being inside the nucleus, thus probing the isotopic change in charge distribution as well as the distribution of magnetization. In this case, as in most cases, the differential BR-effect is negligible when two isotopes are compared. The BR-effect is therefore neglected in the following discussion. The differential hyperfine anomaly 1 ∆ 2 , the difference of the BW-effect between two isotopes, is normally defined as: where µ I is the nuclear magnetic dipole moment, and I the nuclear spin for the isotopes involved. The experimental a-constants should be corrected for second order hyperfine interaction. However, the value of the a-constants is fairly insensitive to this correction, so the only cases when this will have an effect is then the correction is large, that is when the experimental error is large, due to large errors in the fitting of the a-constant.
Using the effective operator formalism [10,11], the hyperfine interaction is divided into three parts, orbital, spin-dipole and contact (spin) interaction. The hyperfine interaction constants can then be expressed as a linear combination of effective radial parameters a ij l and angular coefficients k ij l , a(J) = k 01 l a 01 l + k 12 l a 12 l + k 10 l a 10 l + k 10 s a 10 s where the indices stand for the rank in the spin and orbital spaces, respectively. Of these, only the contact interactions (10) of s and p 1/2 electrons contribute to the hyperfine anomaly. We can thus rewrite the general magnetic dipole hyperfine interaction constant in a simpler form when dealing with hyperfine anomaly; a = a nc + a s + a p = a nc + k 10 s a 10 s + k 10 p a 10 p where a s and a p are the contributions due to the contact interaction of s and p 1/2 electrons, respectively, and a nc is the contribution due to noncontact interactions. The experimentally determined hyperfine anomaly, which is defined with the total magnetic dipole hyperfine interaction constant a, should then be rewritten to obtain the relative contributions to the hyperfine anomaly: where 1 ∆ 2 s and 1 ∆ 2 p are the hyperfine anomaly for an s-and p-electrons, respectively.
By performing an analysis of the hyperfine interaction it is possible to deduce the different contributions to the hyperfine interaction constants, and thus the hyperfine anomaly. That is the experimental hyperfine anomaly, which might show a J-dependence, can be used to extract the hyperfine anomaly for an s-(or p-) electron, 1 ∆ 2 s . It has been shown by Persson [12], that it is possible to extract the hyperfine anomaly without knowing the nuclear magnetic dipole moments, provided you know the contribution of the contact interaction to the hyperfine interaction constant in two atomic states; where A and B are two atomic states in the isotopes 1 and 2. The original use was for radioactive isotopes where the atomic factor, ( a A s a A − a B s a B ), were calibrated to a known hyperfine anomaly between two stable isotopes. However, it is possible to use this method on stable isotopes where the nuclear magnetic dipole moment is not known with high accuracy. I will show this by applying the method to Gd and Nd.

Hyperfine structure
Over the years many investigations of the hyperfine structure have been carried out in the rare-earth (4f-shell) region [13], and a vast amount of hyperfine interaction constants and isotope shift data has been obtained. One would expect that analysis of the hyperfine interaction in this region is difficult due to the large number of states. However, one finds that many states are very close to pure LS-coupling, especially the low-lying states. It is therefore relatively easy to perform an analysis. Even if the hyperfine structure has been well studied, the nuclear magnetic dipole moments are not always known to high accuracy. The nuclei in this region are often deformed leading to a large nuclear electric quadrupole moment and drastical changes in mean charge square radius [1]. It is therefore interesting to study the hyperfine anomaly, both in stable and unstable isotopes.

Hyperfine structure in Nd
The hyperfine structure of Nd has been studied with high accuracy by Childs and Goodman [14] and Childs [15], in the 4f 4 6s 2 5 I 4−8 and 4f 4 5d6s 7 L 5−11 , 7 K 4 states with the ABMR and LRDR methods, respectively. The studied states have also been found to be close to LS-coupling (98-100 % pure), making an analysis of the hyperfine interaction rather simple. The states show no sign of large second-order hyperfine interaction, since the experimental errors for the aand b-constants are small. The high accuracy of the hyperfine interaction constants and that an analysis has been performed [15] makes it possible to use equation 5. It is important that the atomic factor ( a A s a A − a B s a B ) attains a relatively large value, in order to avoid errors [12]. It is therefore important to choose the atomic states in the analysis with great care. From the the ratios of the a constants between the two stable isotopes 143 N d and 145 N d, we choose the 4f 4 5d6s 7 L 5 and 4f 4 5d6s 7 K 4 states for extraction of the hyperfine anomaly as states B in equation 5, and use 4f 4 5d6s 7 L 11 as state A. The experimental a-constants with ratios and contact contribution are given in table 1. The hyperfine anomaly for s-electrons are deduced and the result presented in table 2. The error of the hyperfine anomaly is only due to the experimental errors of the hyperfine interaction constants, as the errors in the contact contribution (e.i. the eigenfunctions) are not known. The error of the hyperfine anomaly is therefore too small and should be about two to four times larger when the uncertainties of the contact contributions are known. Once the hyperfine anomaly is determined it is possible to use this result as a way of obtaining the ratio of the nuclear magnetic dipole moments. The nuclear magnetic dipole moment has been measured by Smith and Unsworth [16] and the ratio is given with the calculated ratio in table 2. We also note that the a-constants ratio in 4f 4 6s 2 5 I 4−8 states, is the same as the "new" ratio, this is an indication that the hyperfine anomalies in these states are zero. This is not surprising, as the configuration does not contain an unpaired s-electron, thus having no contact interaction. This has also been found to be the case in other rare-earths. The difference between Smith and Unsworths experimental ratio and the calculated ratio could be due to systematic errors in the experiment.

Hyperfine anomaly in Gd
Precise studies of the hyperfine structure in Gd have been performed by Unsworth [17], who measured the 9 D term in the 4f 7 5d6s 2 configuration, and Childs [18], who studied the 4f 7 5d 2 6s 11 F term. Unsworth [17] found 1.626(12) 4f 4 6s 2 5 I(mean) [14] 1.60861(2) that the contact interaction was very small, indicating that no core-polarisation is present in the 4f 7 5d6s 2 configuration. The same have also been found in other rare-earths and this seem to be a general feature. In addition should the 4f 7 5d6s 2 9 D states exhibit no hyperfine anomaly, something that can also be seen from the lack of J-dependence of the a constant ratios for the two isotopes 155,157 Gd. The levels 4f 7 5d 2 6s 11 F term is reported to be 98-99% pure L-S coupled states [19], making it possible to use pure L-S coupling in the analysis of the hyperfine interaction [18]. In table 3 the a constants for the 11 F 2,3,8 states are given together with the s-electron contact contribution. Using equation 5 it is possible to derive the hyperfine anomaly, the result is given in table 4. The ratio of the nuclear magnetic dipole moments has also been extracted. The ratio obtained compares well with the ratio of the 4f 7 5d6s 2 9 D states, proving that the ratio is close to the actual ratio and that there is no hyperfine anomaly in these states. The ratio is also in agreement with the ENDOR measurement by Baker et al. [20], but the experimental error is one order of magnitude larger than the derived error. Baker et al. [20] were also able to deduce the hyperfine anomaly of Gd 3 + in   [20] 0.7633(45) 4f 7 5d6s 2 9 D(mean) [17] 0.76254(40)

Conclusions
The new method of deriving hyperfine anomaly [12] has been applied to 155,157 Gd and 143,145 Nd, giving the first precise values of the hyperfine anomaly for these isotopes. In addition have more precise values of the nuclear magnetic dipole moment ratios been obtained. These values are in agreement with the a-constant ratios for the ground terms 4f 7 5d6s 2 9 D(Gd) and 4f 4 6s 2 5 I(Nd), proving that the hyperfine anomalies in these states are negligible and that the electron core polarisation in the ground terms is very small. Using this method makes it possible to obtain values of the nuclear magnetic dipole moment ratio as well as a value of the hyperfine anomaly. This is of interest in the case of systematic studies over long chains of isotopes, to obtain information of nuclear structure.