Hyperfine Interact DOI 10.1007/s10751-010-0211-6 The quadrupole moments of Zn and Cd isotopes an update H. Haas J. G. Correia Springer Science+Business Media B.V. 2010 Abstract The nuclear quadrupole moments of 111 *Cd (245 kev state) and 67 Zn were determined from the quadrupole coupling constants and lattice parameters of the metals at low temperature. The required electric field gradients were obtained with density functional calculations employing the WIEN2k code. The resultant numbers, 0.765(15) and 0.151(4) b, are in line with the previously used values but considerably more precise. Calculations for various other solids confirm the results, with less accuracy, however. Keywords Nuclear quadrupole moments Density functional calculations Zn isotopes Cd isotopes 1 Introduction Precise nuclear quadrupole moments (Q) are a prerequisite for any quantitative analysis of measured quadrupole coupling constants. While historically good precision of calculated electric field gradient (EFG) values for normalization could only be achieved for atoms, the progress of calculations for molecules and solids in recent times has opened up a new possibility. Thus in many cases now values for Q have been obtained by this route [1]. Here we have used the full-potential linearized augmented plane waves (FLAPW) method to calculate the EFG in some solids in order to derive updated values of Q for the Zn and Cd isotopes by comparison with the experimental quadrupole coupling H. Haas J. G. Correia Instituto Tecnológico e Nuclear, Estrada Nacional 10, 2685 Sacavém, Portugal H. Haas (B) J. G. Correia CERN/PH-IS, 1211 Geneve-23, Switzerland e-mail: heinz.haas@cern.ch
Table 1 Lattice parameters and quadrupole coupling constants for the systems treated H. Haas, J.G. Correia a (Å) c (Å) u ν Q (MHz) η Zn 2.65489 4.85335 12.25 Cd 2.96313 5.51890 137.23 ZnF 2 4.7048 3.1338 0.3024 7.93 0.29 8.2 0.15 CdCl 2 3.85 17.46 0.2524 48.7 CdSiP 2 5.678 10.431 0.2057 112 CdGeP 2 5.740 10.775 0.2194 90 CdSnP 2 5.901 11.513 0.2452 35 constants ν Q. Particular emphasis was given to the treatment of the pure metals, as the best data sets are available for these. 2 Input data The most reliable calculations are possible for the hexagonal metals, since no internal structure parameters have to be considered. Accurate lattice constants at 4 K may be determined by extrapolation using the well known room temperature values [2] and the published thermal expansion data [3]. The resultant lattice parameters are included in Table 1. It should be noted that the number of primary importance for the EFG in the hcp metals, the c/a ratio, deviates considerably from the room temperature values, due to the anisotropic thermal expansion, for Zn from 1.8563 to 1.8281 and for Cd from 1.8856 to 1.8625. These might seem like small changes, but since essentially only the difference between the real c/a and the ideal value 1.633 is responsible for the EFG, any use of the room temperature values would lead to about 10% too large results. Only room temperature structural data were available for the other compounds treated here. Furthermore precise values for the coupling constants for the metallic systems 111 *CdinCdand 67 Zn in Zn as function of temperature are known from the literature [4], permitting extrapolation to T = 0. All input data are summarized in Table 1. 3 Calculations All calculations were performed with the full-potential linearized augmented plane waves (FLAPW) code WIEN2k [5]. For error analysis numerous calculations were made using different density functionals and computational parameters. In order to check the appropriateness of various density functionals for the present case, the theoretical unit cell volume was calculated with the local density approximation (LDA) [6] and 2 versions with generalized gradients, here approximated WC [7] and PBE [8]. The experimental volume for Zn is best reproduced with the PBE functional, while for Cd the WC functional is closer. A particular problem for the hcp metals is the complicated Fermi surface structure. This required a very dense mesh of K-points in the Brillouin zone. The final results were obtained by extrapolation to an infinite number of K-points and averaged according to the atomic volume results.
The quadrupole moments of Zn and Cd isotopes an update Fig. 1 Experimental EFG at Cd (in 10 21 V/m 2 ) versus calculated one for various solids using the present Q 8 7 6 5 EFG-exp Q =.765 b CdGeP2 CdSiP2 Cd 4 3 CdCl2 2 CdSnP2 1 EFG-the 0 0 1 2 3 4 5 6 7 8 4Results Various check calculations make us confident that the present EFG results for the metals are considerably more accurate than earlier ones, allowing an error margin of only 2%. The final results (absolute values) obtained are: Q ( 67 Zn, g.st. ) = 0.151 (4) b Q ( 111 Cd, 5/2+ ) = 0.765 (15) b Calculations of the EFG in various other compounds give results in agreements with these values of Q, of less precision, however. These calculations generally involve the determination of one internal crystal structure parameter u in addition to the lattice constants. This was theoretically determined by minimizing the internal forces computed. For the chalcopyrite [9] compounds CdSiP 2,CdGeP 2, and CdSnP 2, calculated with the WC functional, the dependence of the EFG on u is quite small, but for CdCl 2 [10] this is limiting the result. A comparison of the experimental EFG values (using the new value of Q) with the theoretical ones is shown in Fig. 1. A particular situation exists for ZnF 2. While the EFG is reproduced within 3% with the gradient corrected functionals, well within the experimental uncertainty, there is apparently no way to reproduce the large η value of the second experiment. The calculations therefore strongly suggest that η from the first experiment is the correct one. 5 Discussion For 67 Zn there is a complete agreement of our value for Q with the result from optical spectroscopy. The situation for Cd is more complex. In Fig. 2 the various values published to date are summarized. Our result agrees with the direct measurements coming from the two newer nuclear orientation experiments [11 13], though these unfortunately have quite large error bars. Most values obtained by indirect methods
H. Haas, J.G. Correia 2 1.6 1.2 0.8 0.4 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Fig. 2 Published values of Q (in b) for 111 *Cd as function of time. Grey markers represent data from estimated EFG values only Table 2 Nuclear quadrupole moments of Cd isotopes Isotope 105 107 109 111* 107* 109* 111 m 113 m 115 m E (kev) 0 0 0 245 846 463 396 264 173 Spin 5/2 + 5/2 + 5/2 + 5/2 + 11/2 11/2 11/2 11/2 11/2 Q (b) +0.452 +0.713 +0.721 +0.765 0.973 0.956 0.883 0.738 0.570 Err (b) 0.025 0.038 0.038 0.015 0.034 0.029 0.044 0.039 0.053 may generally be neglected [14, 15, and others]. The most reliable earlier number apparently comes from a nuclear physics analysis. The trends of coupling constants for the 11/2 states from optical spectroscopy and perturbed angular distribution measurements has been considered for this purpose [16, 17]. Using the experimentally determined ratios of Q from the literature [18], mostly measured with high accuracy, the moments of the other Zn and Cd isotopes can be obtained, generally with higher accuracy than previously known. While for Zn this procedure is straightforward, in the case of Cd the nuclear systematics analysis mentioned above has to be considered. We are assuming for the ratio of Q between the 11/2 states in 109 Cd and 111 Cd a value of 92/85, as argued by Sprouse et al. [17] and assign an error of 2% to it. The resultant numbers are summarized in Table 2. It may be noticed that also for the long lived states measured by optical methods the errors are typically reduced by a factor of 2 from the earlier ones [18], while the changes induced are somewhat less than this. Note added in proof S. Cottenier has pointed out to us that the experimental coupling constants should include effects of zero-point motion. This correction (D. Torumba et al., Phys. Rev. B 74, 144304 (2006) will increase the quadrupole moments quoted for Cd (and Zn) by approximately 1.6%. References 1. Pyykkö, P.: Year-2008 nuclear quadrupole moments. Mol. Phys. 106, 1965 (2008, and references therein)
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