The quadrupole moments of Cd and Zn isotopes - an apology

Transcription

1 Hyperfine Interact (2016) 237:115 DOI /s The quadrupole moments of Cd and Zn isotopes - an apology H. Haas 1,4 M. B. Barbosa 2 J. G. Correia 3,4 Springer International Publishing Switzerland 2016 Abstract In 2010 we presented an update of the nuclear quadrupole moments (Q) for the Cd and Zn isotopes, based essentially on straightforward density functional (DF) calculations (H. Haas and J.G. Correia, Hyperfine Interact 198, (2010)). It has been apparent for some years that the standard DF procedure obviously fails, however, to reproduce the known electric-field gradient (EFG) for various systems, typical cases being Cu 2 O, As and Sb, and the solid halogens. Recently a cure for this deficiency has been found in the hybrid DF technique. This method is now applied to solid Cd and Zn, and the resultant quadrupole moments are about 15 % smaller than in our earlier report. Also nuclear systematics, using the recently revised values of Q for the long-lived 11/2 isomers in 111 Cd to 129 Cd, together with earlier PAD data for 107,109 Cd, leads to the same conclusion. In addition, EFG calculations for the cadmium dimethyl molecule further support the new values: Q( 111 Cd, 5/2 + ) =.683(20) b, Q( 67 Zn, gs) =.132(5) b. This implies, that the value for the atomic EFG in the 3 P 1 state of Zn must be revised, as it has been for Cd. Keywords Electric field gradients Hybrid density functional calculations Nuclear quadrupole moments Cd isotopes Zn isotopes This article is part of the Topical Collection on Proceedings of the International Conference on Hyperfine Interactions and their Applications (HYPERFINE 2016), Leuven, Belgium, 3-8 July 2016 H. Haas heinz.haas@cern.ch 1 Department of Physics and CICECO, University of Aveiro, Aveiro, Portugal 2 IFIMUP and IN - Instituto de Nanociencia e Tecnologia, Universidade do Porto, Rua do Campo Alegre, Porto, Portugal 3 C 2 TN, Instituto Superior Tecnico, Universidade de Lisboa, Lisboa, Portugal 4 EP Division, CERN, 1211 Geneve-23, Switzerland

2 115 Page 2 of 10 Hyperfine Interact (2016) 237:115 1 Motivation Precise nuclear quadrupole moments (Q) are essential not only for a detailed understanding of nuclei but also for any quantitative analysis of quadrupole interaction measurements in solids, gases, or atoms. As normally only the product of Q with the electric-field gradient (EFG, major component V zz ) created by the electronic environment, the nuclear quadrupole coupling constant, ν Q = eqv zz /h, is experimentally available, the EFG has to be calculated theoretically. Quantum mechanical methods have been generally quite successful in the interpretation of atomic and more recently also molecular data. With the advent of density functional (DF) methods a possibility to routinely calculate EFGs in solids has been opened up. For several test cases the results had been quite promising. This has lead us to apply standard DF methods to determine Q for the 5/2 +, 245 kev state of 111 Cd as well as the ground state of 67 Zn in an earlier work [1]. Since the numerical accuracy of such calculation is fairly high, the results seemed quite reliable, especially since they agreed in both cases with other methods of determining Q. The use of standard DF calculations to determine Q has in some cases resulted in values in full agreement with the most recently accepted ones [2], halogen molecules or solid In, e.g. [3, 4]. In others, however, halogen [5] or Sb[6] solids, e.g., the values found were clearly not correct [2]. It had actually been known for some time that these procedures badly fail in some cases. Typical examples are Cu 2 O[7], a semiconductor, As and Sb [8], semimetals, and the asymmetry parameter η for the solid halogens [9], insulators. In the last decade a more advanced extension of the standard DF methods, the admixture of some Hartree-Fock (HF) exchange to the DF [10], has been very successfully applied in chemical research. Such hybrid calculations for solids have also solved a long-standing problem of standard DF calculations, the failure to reproduce electronic gaps in semiconductors and insulators [11, 12]. It has been noted that this procedure also strongly changes the EFG values computed [13]. In the three problem cases mentioned above a substantial improvement of the agreement of the calculated EFG with the measured one has recently been achieved (see below). We have therefore now applied the hybrid DF technique in a redetermination of Q for Cd and Zn. 2 Input data The lattice constants used here for Zn/Cd were the ones from our previous report [1]: a = / Å, c = / Å. They had been found from early experimental work by extrapolation to T = 0. For Zn there exists a recent measurement at low temperatures [14] that agrees fairly well. We have not used these values, however, due to the unphysical linear T extrapolation employed in this work. For cadmium dimethyl no experimental structure is apparently available, so the theoretical one as discussed below had to be employed. For Zn and Cd the T=0 quadrupole coupling constants from our earlier work [1] were used. For comparison with the theoretical calculations these have to be increased slightly due the effect of zero-point vibrations as estimated in [15] for Cd. For Zn this effect has been scaled with the known Debye temperatures. The value of ν Q for Cd(CH 3 ) 2 is from [16]. The numbers are included in Table 1 together with the theoretical results.

3 Hyperfine Interact (2016) 237:115 Page 3 of Table 1 Experimental data, estimated zero-point vibration correction, and theoretical V zz values (in V/m 2 ) Zn Cd Cd(CH 3 ) 2 ν Q (MHz) 12.34(3) 137.2(1) 946(10) ZPC (%) V zz PBE V p zz PBE V d zz PBE δ p (%) δ d (%) V zz HYB Q (b) Computational details The computer code WIEN2k [17] has been shown to yield reliable results for the band structure of various solids with standard density functional methods. It allows to extract hyperfine properties at the nuclei and their local symmetry contributions. In most cases these are essentially the l-values of the local electrons. Also calculations for (virtually) isolated molecules can be performed with this code. To apply a solid state code to molecules is actually quite straightforward, though admittedly not very elegant. A simple sufficiently large unit cell with artificially separated molecules is being constructed. Obviously one has to check in the results that the separation chosen is adequate. 3.1 Standard density functional calculations The present calculations were performed with the PBE density functional including generalized-gradient corrections [18]. Checks with other functionals have shown that the results for the EFGs, of primary concern in the present work, only change to a very minor extent with different choices. They have allowed to obtain an idea about the theoretical errors, however. Standard technical parameters (R MT =2.3/2.7 bohr (Zn/Cd); l max =10; k max =9/R MT ;G max =12 bohr 1 ) were used. The calculations were made with an additional local d-orbital in order to increase basis set flexibility and included the semicore p-orbitals for zinc and cadmium. Calculations including spin-orbit interaction, presently not possible with hybrid functionals, were performed in order to include this (very small) contribution to the EFG. 3.2 Hybrid calculations In hybrid DF methods the exchange part of the standard functional is replaced by a fraction α with the exact exchange obtained in a Hartree-Fock procedure. The present calculations with such a hybrid functional used the YS-PBE0 method as described in [13]. The procedure is incorporated in the WIEN2k program package. It replaces only the short-range (SR) part of the PBE exchange with the corresponding HF term: E xc = E PBE xc + α ( Ex SR HF Ex SR PBE ). Since such calculations require about three orders of magnitude longer running times than the standard ones, however, extensive tests of the necessary input conditions were

4 115 Page 4 of 10 Hyperfine Interact (2016) 237:115 Fig. 1 Dependence of Zn EFG and its p- and d- components (in V/m 2 ) on number of k-points for PBE calculation performed. Initial calculations with just the minimum number of levels in the Hartree-Fock subroutine gave a first idea, but final results needed a cutoff energy at least 2 Rydberg above the Fermi level. The choice of the amount of HF exchange used (α =.25), the standard value in the PBE0 potential [19], is of utmost importance for the present work. Actually there exists a theoretical justification for the standard value in combination with the PBE functional [20]. One can, however, also consider another approach. For a large series of insulators it has been found that an optimum value for α is obtained with the relation α opt = /ε [21], with self-consistent calculation of the dielectric constant ε. For a metallic system one could consider replacing ε by the value computed from the atomic polarizability α at [22], using the classical relation ε = 1 + 4πα at /V at. In doing so, one obtains for Zn and Cd the α opt values.258 and.264, in full accord with the number chosen here. In a few cases also calculations with the unscreened HF potential were performed. It was found that with this procedure one can obtain similar results, with a smaller amount of HF admixture, however, typically α = Results of calculations 4.1 Standard calculation results Due to the complex Fermi surface of Zn and Cd the final results for the EFG have to be determined with an extremely dense k-mesh. We have used up to k-points in the full Brillouin zone. Final results (Table 1) were determined using extrapolation to 1/k 2 z = 0. Figure 1 shows this in a representative way for the PBE calculations of Zn. One can note here, that the final EFG is the sum of a positive p- and a negative d-contribution. Thus any error in the p-part would be amplified in the total value.

5 Hyperfine Interact (2016) 237:115 Page 5 of Fig. 2 Enhancement factor δ p as function of number of states above the Fermi level included in the Hartree- Fock procedure for Zn and Cd hybrid calculation The Cd-dimethyl molecule was placed into a rhombohedral cell, a = 16 bohr, c = 20 bohr. An eclipsed configuration of the two methyl groups was assumed. 6 k-points in the irreducible wedge of the Brillouin zone were found as highly sufficient to determine the structure and EFG. The obtained structure of the molecule is: d (Cd C) = Å, d (C H) = Å, θ (Cd C H) = o. The EFG results are included in Table Hybrid calculation results Hybrid calculations could unfortunately only be afforded for much fewer k-points. When comparing these with the PBE results for an identical k-mesh it was noted, however, that essentially only the p-contribution was changed appreciably. Thus enhancement factors 1+δ p for this, as well as 1+δ d for the d-contribution (of minor importance), were determined for various k-mesh choices and found not to be significantly different. These enhancement factors depend on the number of bands included in the HF subroutine and saturate rather slowly, as demonstrated in Fig. 2 for Zn and Cd. As final results the values at the highest number of k-points and number of excited bands included in the HF-procedure were adopted. Table 1 shows the theoretical EFG values from the PBE calculations and their corresponding p- and d-contributions as well as the δ factors and the final results for the quadrupole moments. 5 Error analysis and discussion In order to estimate the expected accuracy of the nuclear quadrupole moments determined a short discussion of possible errors is presented. Table 2 shows the various contributions considered:

6 115 Page 6 of 10 Hyperfine Interact (2016) 237:115 Table 2 Estimated error contributions (see text) to calculated quadrupole moments (in %) Zn Cd Cd(CH 3 ) 2 ν Q ZPC Struc DF α HFcut Q Fig. 3 Experimental ν Q for Cd and Cd(CH 3 ) 2 as function of theoretical value. Open symbols PBE calculation, filled symbols hybrid calculation Accuracy of the ν Q value (ν Q ), estimation of the zero point correction (ZPC), accuracy of the structure used (Struc), difference due to choice of density functional (DF), ambiguity due to choice of α (α), and error due to cutoff energy in HF (HFcut). As final values one obtains: Q( 111 Cd) =.683(20)bandQ( 67 Zn) =.132(5) b These numbers can now be compared with data from previous work. In Fig. 3 the results for the two treated Cd solids clearly show that our earlier result for 111 Cd has to be revised. In Fig. 4 the (updated) history of this value, see [1], is presented. Here the earlier experimental data have been shifted in accord with the recently revised values for the stable and long-lived isomeric Cd isotopes [23]. Since the ratios of Q for the 11/2 states in 107,109 Cd to the 5/2 + one in 111 Cd are experimentally quite well known from PAD measurements [24, 25], their values are obviously now also available. Figure 5 shows the systematics for 11/2

7 Hyperfine Interact (2016) 237:115 Page 7 of Fig. 4 History of quadrupole moment value for 111 Cd, 5/2 +, 245 kev state Fig. 5 New quadrupole moment values for 11/2 and 5/2 + states in light Cd isotopes from atomic (dots) and solid state (squares)data and 5/2 + states of the light Cd isotopes. It nicely confirms the early expectation that the decreasing trend for the 11/2 values continues, but with much reduced slope. In the case of 67 Zn there is some additional evidence suggesting that the only available reference value [26] (from 1969) could be wrong. Recent atomic calculations [27] have failed to reproduce the B-factor for the 3 P 1 state measured precisely a long time ago [28]. For normalization the authors have (apparently) employed the earlier value, Q( 67 Zn) =.150 b. With the new value of Q, however, a perfect agreement would have been obtained.

8 115 Page 8 of 10 Hyperfine Interact (2016) 237:115 Fig. 6 Theoretical V zz for PBE (dots) and hybrid (squares) calculations in 4sp and 5sp solids, normalized to the experimental values. Open symbols are preliminary results. The accuracy of the corresponding Q is shownbytheerrorbar 6 Conclusions and outlook The present results shed doubt on the precision of any DF calculation of electric-field gradients in solids. Obviously the influence of the partial inclusion of Hartree-Fock exchange would have to be studied in each case, and (presumably) would improve the results. That the effect, however, is markedly different in every case is exemplified in Fig. 6, where the results of (partially preliminary) calculations with and without HF exchange are compared for a series of 4sp and 5sp solids. In every case (accepting the new values of Q for Cd and Zn) it is noted that the hybrid calculation comes much closer to the experimental value. How could one then further test the reliability of the present results? Obviously a new calculation for the 3 P 1 state of Zn with another atomic code would be decisive. Also molecular calculations for ZnS, where the quadrupole coupling has been recently measured [29], could be of importance. For the 5/2 + state in 111 Cd the chances to obtain a still more precise value for Q are not as promising. Certainly a more sophisticated molecular calculation for cadmium dimethyl appears possible. Here, however, the experimental accuracy is the bottleneck. Only a measurement in the free molecule could solve this problem. Unfortunately, very early experiments at Berkeley (H. Rinneberg, 1972) and corresponding ones for CdCl 2 at Bonn (E. Bodenstedt, 1975) have failed. Perhaps these courageous efforts could be more successful with modern techniques of isotope production. Finally, one should consider to extend the nuclear theory calculations for the 11/2 states of Cd [30] to the cases of 107,109 Cd and compare with the present results.

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