- Email: [email protected]
Ab-initio calculation of electronic structure and electric field gradients in HfAl2 and ZrAl2 Laves phases
Available online at www.sciencedirect.com
Computational Materials Science 41 (2007) 164–167 www.elsevier.com/locate/commatsci
Ab-initio calculation of electronic structure and electric field gradients in HfAl2 and ZrAl2 Laves phases ˇ avor *, V. Koteski, B. Cekic´, A. Umic´evic´ J. Belosˇevic´-C Institute of Nuclear Sciences Vinca, P.O. Box 522, 11001 Belgrade, Serbia Received 4 December 2006; received in revised form 16 March 2007; accepted 25 March 2007 Available online 17 May 2007
Abstract A detailed theoretical study of the structure, electronic properties and electric field gradients (EFG) of the HfAl2 and ZrAl2 Laves phases is presented. Using all-electron augmented plane waves plus local orbitals (APW + lo) formalism, the equilibrium volumes, bulk moduli and EFGs for the two compounds are calculated. The obtained results are compared with the available experimental and theoretical data. Better agreement with the experimental data is found by employing supercell calculations with Ta and Cd impurities. 2007 Elsevier B.V. All rights reserved. PACS: 71.15.Ap; 71.20.Lp; 71.70.Jp Keywords: HfAl2; ZrAl2; Electric field gradient; Augmented plane wave
1. Introduction Zirconium–aluminium and hafnium–aluminium alloys are potential structure materials in thermal nuclear reactors due to good mechanical properties at high temperature combined with low absorption cross-sections for thermal neutrons [1–3]. Extensive work has been reported on the Zr–Al alloys regarding amorphisation [4], nano-phase formation [5], formation of several metastable phases [6] and making zirconium based pressure tubes with aluminium linings [7]. Also, stability [8], getter potentialities [9], elastic moduli [10] and combustion synthesis [11] of Zr–Al intermetallic compounds have been investigated. However, there is a lack of electronic structure calculations for these alloys. The same is true for the Hf–Al alloys. Our purpose in this work is to investigate the structural, electronic and hyperfine interaction properties of ZrAl2 and HfAl2, which are Laves phases with MgZn2-type structure (C14). Studying hyperfine structure of nuclei is a pow-
*
Corresponding author. Tel.: +381 11 2453 681; fax: +381 11 3440 100. ˇ avor). E-mail address: [email protected] (J. Belosˇevic´-C
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.03.009
erful tool for investigation of interactions of atomic nucleus with local electric and magnetic fields. These interactions cause shifts and splitting of nuclear energy levels and enable us to obtain information about the symmetry of the charge distribution around the nucleus, about the electronic configurations of atoms and ions as well as about the peculiarity of the atomic structure of solids.
2. Computational details The calculations have been performed using the augmented plane waves plus local orbitals (APW + lo) method as implemented in the WIEN 2k code [12], within the framework of the density functional theory (DFT) [13]. In this method, the space is divided into two regions. Near the atoms all quantities of interest are expanded in spherical harmonics and in the interstitial region they are expanded in plane waves. The first type of expansion is defined within a so-called muffin-tin sphere of radius Rmt around each nucleus. In our calculations the muffin-tin radii for Hf, Zr and Al were 2.3, 2.2 and 2.15 a.u., respectively. The Hf 5s, 5p, 6s, 5d, 4f, the Zr 4s, 4p, 5s, 4d and the
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167
Al 3s, 2p, 3p states were put in the valence panel. The standard basis set was extended with local orbitals: 5s and 6p for Hf, 4s and 5p for Zr and 2p for Al. The exchangecorrelation potential was calculated by the generalized gradient approximation, using the scheme of Perdew– Burke–Ernzerhof [14]. The core states were treated fully relativistically, while the valence states were treated within the scalar relativistic approximation. The Brillouin zone integrations within the self-consistency cycles were performed via a tetrahedron method [15], using 165 k points in the irreducible wedge of the Brillouin zone. The cut-off parameter RmtKmax for limiting the number of plane waves was set to 8.5, where Rmt is the smallest value of all atomic sphere radii and Kmax is the largest reciprocal lattice vector used in the plane wave expansion. In our calculations the self-consistency was achieved by demanding the convergence of the integrated charge difference between last two iterations to be smaller than 105 electron, since it ensures better stability of the calculated values than the corresponding energy criterion. As the first step of structural relaxation, the atomic positions were relaxed according to Hellmann–Feynman forces calculated at the end of each self-consistent cycle. The force minimization criterion was 1 mRy/a.u. During this relaxation the cell volume and the c/a value were fixed to their experimental values. Then the theoretical equilibrium volume was determined by fixing the atomic positions to their optimized values and further keeping the c/a ratio fixed. A series of calculations was carried out, changing the volume within ±5% of its experimental value and calculating the total energy as its function. In the last step the c/a ratio was optimized by changing it within ±2% of the experimental value while keeping the optimized volume fixed. 3. Results and discussion The theoretically determined cell and structure parameters for the two compounds, along with the experimental values obtained from X-ray diffraction measurements [16,17] are given in Table 1. In both cases the theoretical volume overestimates the experimental one by 3.5%. Table 1 The parameters of the HfAl2 and ZrAl2 structure
HfAl2 u v a c ZrAl2 u v a c
Experimental results from X-ray diffraction [16,17]
WIEN 2k
0.062 0.83 5.241 8.673
0.064 0.829 5.256 8.688
0.0653 (3) 0.828 (1) 5.281 (1) 8.742 (1) ˚ The distances are in A.
0.064 0.829 5.300 8.756
165
The bulk moduli B0 obtained by fitting the data to the Murnaghan’s equation of state [18] are 114 GPa and 113 GPa for HfAl2 and ZrAl2, respectively. To the best of our knowledge, up to date, there is no experimental information regarding the HfAl2 bulk modulus, while in the case of ZrAl2 our calculated value is somewhat smaller than both the measured and earlier calculated ones [10,19]. The first coordination shell around Al 2a atoms in both compounds consists of 6 Al 6h and 6 Hf i.e. Zr atoms, whereas around Hf(Zr) atom it breaks into five Al 6h, two Al 2a and three Hf(Zr) subshells. The interatomic distances in ZrAl2 are slightly larger than the corresponding ones in HfAl2, which is expected because of the larger ZrAl2 unit cell. After determining the self-consistent charge density we obtained the EFG tensor Vij using the method developed in Ref. [20]. The usual convention is to designate the largest component of the EFG tensor as Vzz. The asymmetry parameter g is then given by g = (Vyy Vxx)/Vzz, where jVzz j P jVyyj P jVxxj. In Table 2, the decomposition of the EFG at all three non-equivalent lattice sites in the fully relaxed HfAl2 and ZrAl2 structure is presented. We also predict the signs of the EFGs, which have not been determined from the experiment. The EFGs at the two Al sites have opposite sign when compared with the EFG at Hf(Zr) but, as it is often the case in transition metal compounds, the main contribution to the EFG at all three sites comes from the p electrons. The contribution from the d electrons in the case of Hf(Zr) is about 50% smaller than the p contribution, and for Al it is almost negligible in both compounds at both lattice sites. The use of local orbitals enabled us to estimate the semicore p–p contributions (the s–s and s–d semicore contributions are negligible) to the calculated EFG at all lattice sites. It turned out that the semicore p–p contributions are quite small at the transition metal sites (on the order of 3% in HfAl2 and 4% in ZrAl2), but rather large at the Al site (on the order of 43% for HfAl2 and 34% for ZrAl2). If we compare the on-site contribution to the EFG (Vzz in Table 2) with the lattice contribution (EFG-Vzz), it is clear that the latter, which in the implementation of the APW method stems from the charge outside the MT spheres, is virtually zero at the two transition metal lattice sites in both compounds. In contrast, the lattice contribution at both Al Table 2 Decomposition of the calculated Vzz values in units of 1021 V/m2 for the fully relaxed HfAl2 and ZrAl2 structures p–p
d–d
Others
Vzz
EFG
g
HfAl2 Al 2a Al 6h Hf 4f
1.35 0.22 1.67
0.02 0.01 0.77
0.03 0.02 0.02
1.30 0.19 2.46
1.47 0.32 2.46
0 0.731 0
ZrAl2 Al 2a Al 6h Zr 4f
1.53 0.40 0.82
0.02 0.01 0.37
0.03 0.06 0.01
1.48 0.33 1.20
1.63 0.49 1.20
0 0.518 0
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167
166
2a and 6h sites is considerable and especially large at the Al 6h site in HfAl2. This is no surprise, and can be expected for lighter elements with much extended wave functions. As far as the direction of the EFG is concerned, the situation is clear for both Al 2a and Hf(Zr) sites, where we have an axially symmetric electric field gradient (g = 0) and the direction of the principal component of the EFG is along the z axes. For the Al 6h position in ZrAl2 we found that Vzz points along the [1,2,0] axes, but for the same lattice site in HfAl2, due to the competition between Vzz and Vyy, the direction shifts toward the [2,1,0] axes and the EFG changes sign. However, it has to be noted that the magnitude of the calculated EFG at Al 6h is very small, and consequently it is difficult to make definite conclusions, since small changes in the structure can alter the charge asymmetry and therefore the calculated hyperfine interactions parameters. The EFGs in HfAl2 and ZrAl2 were investigated experimentally by employing the time differential perturbed angular correlation (TDPAC) method [21], in which the measurement actually takes place on the 181Ta i.e. 111Cd probe ion instead on the Hf or Al nucleus. In order to compare the calculated results with the measured ones and to estimate the Ta(Cd) impurity effect, we constructed 2 · 2 · 2 supercells, starting from the pure HfAl2 and ZrAl2 unit cell. The EFGs at the impurities substituting on all lattice sites in the C14 structure were calculated. The point group symmetry around the impurity atom is still the same as around the original atom, but the number of non-equivalent positions is larger (20–39) and the complexity of the calculations increases. The calculated EFGs and asymmetry parameters in the investigated compounds are given in Table 3. In the case of Cd probe substituting the Al site (both 2a and 6h), the magnitude of the calculated EFG in the two compounds is practically identical (7.2 vs. 7.1 · 1021 V/m2 at Al 2a, and 3.7 vs. 3.9 · 1021 V/m2 at Al 6h site). This reproduces the experimental result, where the relatively similar values of the EFG and g indicated similar charge distribution around the probe atom in both compounds. The only difference is that our calculations deliver somewhat larger values for the EFGs and smaller values for the asymmetry parameters. The value of the EFG at Al 2a site is about two times larger than the one at Al 6h site, Table 3 Comparison between the calculated and measured EFG values in units of 1021 V/m2 Phase
HfAl2
Probe
111
Cd
181
Ta
ZrAl2
111
Cd
181
Ta
Lattice site
EFG measured [21]
g measured
EFG calculated
g calculated
4f 2a 6h 4f
±7.8 ±5.5 ±2.7 ±1.7
(1) (1) (1) (1)
0 0 0.41 (2) 0.17 (2)
1.7 7.2 3.7 2.3
0 0 0.35 0
2a 6h 4f
±6.0 (1) ±2.9 (1) ±1.6 (1)
0 0.39 (5) 0.15 (1)
7.1 3.9 3.2
0 0.29 0
which is also in agreement with the experimental results. The values of EFGs at Ta probe substituting for Hf and Zr are similar, again in agreement with the experimental results. There is no large difference in the absolute values of the calculated and measured EFGs, especially having in mind the temperature dependency of the EFG, i.e. the T 3/2 power law [21], and the fact that the calculated values correspond to T = 0 K, whereas the measurements have been performed at 773 K. The obtained values for the asymmetry parameters, which should not depend strongly on temperature, are in better agreement with the experimental results. In the case of the Ta probe at transition metal sites, the comparison of the results given in Tables 2 and 3 indicates that the inclusion of the Ta impurity changes the EFG not as much at Hf site, but more pronounced at Zr site, which is also similar to the findings in many transition intermetallic compounds [22]. For the Al sites in both compounds, the inclusion of Cd impurities in the calculations is essential, as it improves significantly the agreement with the experimental EFG values. However, the large discrepancy found for the Cd probe at 4f lattice site in HfAl2 is symptomatic, especially compared to the relatively good agreement between our calculated EFGs and the experimental results at all the other lattice sites. Therefore, the assignment of the measured EFG to the 4f lattice site in HfAl2 in this case is strongly questionable. 4. Summary In summary, we have presented ab-initio calculations of the fully relaxed structures of HfAl2 and ZrAl2 and provided further insights into the electronic and structure properties, as well as electric field gradients of these intermetallic compounds. We have shown that the main contribution to the EFGs at all three non-equivalent sites (2a, 6h and 4f) in both compounds comes from the p electrons. The directions of the EFG, and the semicore and lattice contributions to the EFG have also been determined. The results of the EFG calculated at Ta and Cd impurities in both compounds have shown fair agreement with the experimental data, and provided additional information for interpreting the experimental results. With exception of the 4f lattice site in HfAl2, our calculations have successfully reproduced the measured hyperfine interaction parameters at all lattice sites in the investigated intermetallics. Acknowledgement This work has been supported by the Grant No. 141022 G from the Serbian Ministry of Science and Environmental Protection. References [1] E.M. Schulson, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds, vol. 2, John Wiley and Sons Ltd., 1994, pp. 133– 146.
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
E.F. Ibrahim, B.A. Cheadle, Can. Metall. Q. 24 (1985) 273–281. A. Laik, K. Bhanumurthy, G.B. Kale, Intermetallics 12 (2004) 69–74. H.W. Sheng, K. Lu, E. Ma, J. Appl. Phys. 85 (1999) 6400–6407. Z. Fu, W.L. Jonson, Nanostruct. Mater. 3 (1993) 175–180. S. Gialanella, A.R. Yavari, R.W. Cahn, Scripta Metallurgica et Materialia 26 (1992) 1233–1238. E.M. Schulson, M.J. Stewart, Metall. Trans. 7B (1976) 363–368. M. Alatalo, M. Wienert, R.E. Watson, Phys. Rev. B 57 (1998) R2009–R2012. M. Ferrini, D. Gozzi, J. Alloys Compd. 317–318 (2001) 583–589. F. Willis, R.G. Leisure, I. Jacob, Phys. Rev. B 50 (1994) 13792–13794. U. Anselmi-Tamburini, G. Spinolo, G. Flor, Z.A. Munir, J. Alloys Compd. 247 (1997) 190–194. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN 2k an Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, Vienna University of Technology, Vienna, Austria, 2001. A.R. Williams, J. Kubler, C.D. Gelatt Jr., Phys. Rev. B 19 (1979) 6094–6118.
167
[14] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865– 3868. [15] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223– 16233. [16] A. Israel, I. Jacob, J.L. Soubegoroux, D. Fruchart, H. Pinto, M. Melamud, J. Alloys Compd. 253–254 (1997) 265–267. [17] A.E. Dwight, J.W. Downey, R.A. Conner Jr., Acta Crystallographica 14 (1961) 75–76. [18] F.D. Murnaghan, Proceedings of the National Academy of Sciences USA 30 (1944) 244–247. [19] H. Anton, P.C. Schmidt, Intermetallics 5 (1997) 449–465. [20] P. Blaha, K. Schwarz, Phys. Rev. Lett. 54 (1985) 1192–1195. [21] P. Wodniecki, B. Wodniecka, A. Kulinska, M. Uhrmacher, K.P. Lieb, J. Alloys Compd. 335 (2002) 20–25. [22] L.A. Terrazos, H.M. Petrilli, M. Marszalek, H. Saitovich, P.R.J. Silva, P. Blaha, K. Schwarz, Solid State Commun. 121 (2002) 525– 529.
Computational Materials Science 41 (2007) 164–167 www.elsevier.com/locate/commatsci
Ab-initio calculation of electronic structure and electric field gradients in HfAl2 and ZrAl2 Laves phases ˇ avor *, V. Koteski, B. Cekic´, A. Umic´evic´ J. Belosˇevic´-C Institute of Nuclear Sciences Vinca, P.O. Box 522, 11001 Belgrade, Serbia Received 4 December 2006; received in revised form 16 March 2007; accepted 25 March 2007 Available online 17 May 2007
Abstract A detailed theoretical study of the structure, electronic properties and electric field gradients (EFG) of the HfAl2 and ZrAl2 Laves phases is presented. Using all-electron augmented plane waves plus local orbitals (APW + lo) formalism, the equilibrium volumes, bulk moduli and EFGs for the two compounds are calculated. The obtained results are compared with the available experimental and theoretical data. Better agreement with the experimental data is found by employing supercell calculations with Ta and Cd impurities. 2007 Elsevier B.V. All rights reserved. PACS: 71.15.Ap; 71.20.Lp; 71.70.Jp Keywords: HfAl2; ZrAl2; Electric field gradient; Augmented plane wave
1. Introduction Zirconium–aluminium and hafnium–aluminium alloys are potential structure materials in thermal nuclear reactors due to good mechanical properties at high temperature combined with low absorption cross-sections for thermal neutrons [1–3]. Extensive work has been reported on the Zr–Al alloys regarding amorphisation [4], nano-phase formation [5], formation of several metastable phases [6] and making zirconium based pressure tubes with aluminium linings [7]. Also, stability [8], getter potentialities [9], elastic moduli [10] and combustion synthesis [11] of Zr–Al intermetallic compounds have been investigated. However, there is a lack of electronic structure calculations for these alloys. The same is true for the Hf–Al alloys. Our purpose in this work is to investigate the structural, electronic and hyperfine interaction properties of ZrAl2 and HfAl2, which are Laves phases with MgZn2-type structure (C14). Studying hyperfine structure of nuclei is a pow-
*
Corresponding author. Tel.: +381 11 2453 681; fax: +381 11 3440 100. ˇ avor). E-mail address: [email protected] (J. Belosˇevic´-C
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.03.009
erful tool for investigation of interactions of atomic nucleus with local electric and magnetic fields. These interactions cause shifts and splitting of nuclear energy levels and enable us to obtain information about the symmetry of the charge distribution around the nucleus, about the electronic configurations of atoms and ions as well as about the peculiarity of the atomic structure of solids.
2. Computational details The calculations have been performed using the augmented plane waves plus local orbitals (APW + lo) method as implemented in the WIEN 2k code [12], within the framework of the density functional theory (DFT) [13]. In this method, the space is divided into two regions. Near the atoms all quantities of interest are expanded in spherical harmonics and in the interstitial region they are expanded in plane waves. The first type of expansion is defined within a so-called muffin-tin sphere of radius Rmt around each nucleus. In our calculations the muffin-tin radii for Hf, Zr and Al were 2.3, 2.2 and 2.15 a.u., respectively. The Hf 5s, 5p, 6s, 5d, 4f, the Zr 4s, 4p, 5s, 4d and the
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167
Al 3s, 2p, 3p states were put in the valence panel. The standard basis set was extended with local orbitals: 5s and 6p for Hf, 4s and 5p for Zr and 2p for Al. The exchangecorrelation potential was calculated by the generalized gradient approximation, using the scheme of Perdew– Burke–Ernzerhof [14]. The core states were treated fully relativistically, while the valence states were treated within the scalar relativistic approximation. The Brillouin zone integrations within the self-consistency cycles were performed via a tetrahedron method [15], using 165 k points in the irreducible wedge of the Brillouin zone. The cut-off parameter RmtKmax for limiting the number of plane waves was set to 8.5, where Rmt is the smallest value of all atomic sphere radii and Kmax is the largest reciprocal lattice vector used in the plane wave expansion. In our calculations the self-consistency was achieved by demanding the convergence of the integrated charge difference between last two iterations to be smaller than 105 electron, since it ensures better stability of the calculated values than the corresponding energy criterion. As the first step of structural relaxation, the atomic positions were relaxed according to Hellmann–Feynman forces calculated at the end of each self-consistent cycle. The force minimization criterion was 1 mRy/a.u. During this relaxation the cell volume and the c/a value were fixed to their experimental values. Then the theoretical equilibrium volume was determined by fixing the atomic positions to their optimized values and further keeping the c/a ratio fixed. A series of calculations was carried out, changing the volume within ±5% of its experimental value and calculating the total energy as its function. In the last step the c/a ratio was optimized by changing it within ±2% of the experimental value while keeping the optimized volume fixed. 3. Results and discussion The theoretically determined cell and structure parameters for the two compounds, along with the experimental values obtained from X-ray diffraction measurements [16,17] are given in Table 1. In both cases the theoretical volume overestimates the experimental one by 3.5%. Table 1 The parameters of the HfAl2 and ZrAl2 structure
HfAl2 u v a c ZrAl2 u v a c
Experimental results from X-ray diffraction [16,17]
WIEN 2k
0.062 0.83 5.241 8.673
0.064 0.829 5.256 8.688
0.0653 (3) 0.828 (1) 5.281 (1) 8.742 (1) ˚ The distances are in A.
0.064 0.829 5.300 8.756
165
The bulk moduli B0 obtained by fitting the data to the Murnaghan’s equation of state [18] are 114 GPa and 113 GPa for HfAl2 and ZrAl2, respectively. To the best of our knowledge, up to date, there is no experimental information regarding the HfAl2 bulk modulus, while in the case of ZrAl2 our calculated value is somewhat smaller than both the measured and earlier calculated ones [10,19]. The first coordination shell around Al 2a atoms in both compounds consists of 6 Al 6h and 6 Hf i.e. Zr atoms, whereas around Hf(Zr) atom it breaks into five Al 6h, two Al 2a and three Hf(Zr) subshells. The interatomic distances in ZrAl2 are slightly larger than the corresponding ones in HfAl2, which is expected because of the larger ZrAl2 unit cell. After determining the self-consistent charge density we obtained the EFG tensor Vij using the method developed in Ref. [20]. The usual convention is to designate the largest component of the EFG tensor as Vzz. The asymmetry parameter g is then given by g = (Vyy Vxx)/Vzz, where jVzz j P jVyyj P jVxxj. In Table 2, the decomposition of the EFG at all three non-equivalent lattice sites in the fully relaxed HfAl2 and ZrAl2 structure is presented. We also predict the signs of the EFGs, which have not been determined from the experiment. The EFGs at the two Al sites have opposite sign when compared with the EFG at Hf(Zr) but, as it is often the case in transition metal compounds, the main contribution to the EFG at all three sites comes from the p electrons. The contribution from the d electrons in the case of Hf(Zr) is about 50% smaller than the p contribution, and for Al it is almost negligible in both compounds at both lattice sites. The use of local orbitals enabled us to estimate the semicore p–p contributions (the s–s and s–d semicore contributions are negligible) to the calculated EFG at all lattice sites. It turned out that the semicore p–p contributions are quite small at the transition metal sites (on the order of 3% in HfAl2 and 4% in ZrAl2), but rather large at the Al site (on the order of 43% for HfAl2 and 34% for ZrAl2). If we compare the on-site contribution to the EFG (Vzz in Table 2) with the lattice contribution (EFG-Vzz), it is clear that the latter, which in the implementation of the APW method stems from the charge outside the MT spheres, is virtually zero at the two transition metal lattice sites in both compounds. In contrast, the lattice contribution at both Al Table 2 Decomposition of the calculated Vzz values in units of 1021 V/m2 for the fully relaxed HfAl2 and ZrAl2 structures p–p
d–d
Others
Vzz
EFG
g
HfAl2 Al 2a Al 6h Hf 4f
1.35 0.22 1.67
0.02 0.01 0.77
0.03 0.02 0.02
1.30 0.19 2.46
1.47 0.32 2.46
0 0.731 0
ZrAl2 Al 2a Al 6h Zr 4f
1.53 0.40 0.82
0.02 0.01 0.37
0.03 0.06 0.01
1.48 0.33 1.20
1.63 0.49 1.20
0 0.518 0
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167
166
2a and 6h sites is considerable and especially large at the Al 6h site in HfAl2. This is no surprise, and can be expected for lighter elements with much extended wave functions. As far as the direction of the EFG is concerned, the situation is clear for both Al 2a and Hf(Zr) sites, where we have an axially symmetric electric field gradient (g = 0) and the direction of the principal component of the EFG is along the z axes. For the Al 6h position in ZrAl2 we found that Vzz points along the [1,2,0] axes, but for the same lattice site in HfAl2, due to the competition between Vzz and Vyy, the direction shifts toward the [2,1,0] axes and the EFG changes sign. However, it has to be noted that the magnitude of the calculated EFG at Al 6h is very small, and consequently it is difficult to make definite conclusions, since small changes in the structure can alter the charge asymmetry and therefore the calculated hyperfine interactions parameters. The EFGs in HfAl2 and ZrAl2 were investigated experimentally by employing the time differential perturbed angular correlation (TDPAC) method [21], in which the measurement actually takes place on the 181Ta i.e. 111Cd probe ion instead on the Hf or Al nucleus. In order to compare the calculated results with the measured ones and to estimate the Ta(Cd) impurity effect, we constructed 2 · 2 · 2 supercells, starting from the pure HfAl2 and ZrAl2 unit cell. The EFGs at the impurities substituting on all lattice sites in the C14 structure were calculated. The point group symmetry around the impurity atom is still the same as around the original atom, but the number of non-equivalent positions is larger (20–39) and the complexity of the calculations increases. The calculated EFGs and asymmetry parameters in the investigated compounds are given in Table 3. In the case of Cd probe substituting the Al site (both 2a and 6h), the magnitude of the calculated EFG in the two compounds is practically identical (7.2 vs. 7.1 · 1021 V/m2 at Al 2a, and 3.7 vs. 3.9 · 1021 V/m2 at Al 6h site). This reproduces the experimental result, where the relatively similar values of the EFG and g indicated similar charge distribution around the probe atom in both compounds. The only difference is that our calculations deliver somewhat larger values for the EFGs and smaller values for the asymmetry parameters. The value of the EFG at Al 2a site is about two times larger than the one at Al 6h site, Table 3 Comparison between the calculated and measured EFG values in units of 1021 V/m2 Phase
HfAl2
Probe
111
Cd
181
Ta
ZrAl2
111
Cd
181
Ta
Lattice site
EFG measured [21]
g measured
EFG calculated
g calculated
4f 2a 6h 4f
±7.8 ±5.5 ±2.7 ±1.7
(1) (1) (1) (1)
0 0 0.41 (2) 0.17 (2)
1.7 7.2 3.7 2.3
0 0 0.35 0
2a 6h 4f
±6.0 (1) ±2.9 (1) ±1.6 (1)
0 0.39 (5) 0.15 (1)
7.1 3.9 3.2
0 0.29 0
which is also in agreement with the experimental results. The values of EFGs at Ta probe substituting for Hf and Zr are similar, again in agreement with the experimental results. There is no large difference in the absolute values of the calculated and measured EFGs, especially having in mind the temperature dependency of the EFG, i.e. the T 3/2 power law [21], and the fact that the calculated values correspond to T = 0 K, whereas the measurements have been performed at 773 K. The obtained values for the asymmetry parameters, which should not depend strongly on temperature, are in better agreement with the experimental results. In the case of the Ta probe at transition metal sites, the comparison of the results given in Tables 2 and 3 indicates that the inclusion of the Ta impurity changes the EFG not as much at Hf site, but more pronounced at Zr site, which is also similar to the findings in many transition intermetallic compounds [22]. For the Al sites in both compounds, the inclusion of Cd impurities in the calculations is essential, as it improves significantly the agreement with the experimental EFG values. However, the large discrepancy found for the Cd probe at 4f lattice site in HfAl2 is symptomatic, especially compared to the relatively good agreement between our calculated EFGs and the experimental results at all the other lattice sites. Therefore, the assignment of the measured EFG to the 4f lattice site in HfAl2 in this case is strongly questionable. 4. Summary In summary, we have presented ab-initio calculations of the fully relaxed structures of HfAl2 and ZrAl2 and provided further insights into the electronic and structure properties, as well as electric field gradients of these intermetallic compounds. We have shown that the main contribution to the EFGs at all three non-equivalent sites (2a, 6h and 4f) in both compounds comes from the p electrons. The directions of the EFG, and the semicore and lattice contributions to the EFG have also been determined. The results of the EFG calculated at Ta and Cd impurities in both compounds have shown fair agreement with the experimental data, and provided additional information for interpreting the experimental results. With exception of the 4f lattice site in HfAl2, our calculations have successfully reproduced the measured hyperfine interaction parameters at all lattice sites in the investigated intermetallics. Acknowledgement This work has been supported by the Grant No. 141022 G from the Serbian Ministry of Science and Environmental Protection. References [1] E.M. Schulson, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds, vol. 2, John Wiley and Sons Ltd., 1994, pp. 133– 146.
J. Belosˇevic´-Cˇavor et al. / Computational Materials Science 41 (2007) 164–167 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
E.F. Ibrahim, B.A. Cheadle, Can. Metall. Q. 24 (1985) 273–281. A. Laik, K. Bhanumurthy, G.B. Kale, Intermetallics 12 (2004) 69–74. H.W. Sheng, K. Lu, E. Ma, J. Appl. Phys. 85 (1999) 6400–6407. Z. Fu, W.L. Jonson, Nanostruct. Mater. 3 (1993) 175–180. S. Gialanella, A.R. Yavari, R.W. Cahn, Scripta Metallurgica et Materialia 26 (1992) 1233–1238. E.M. Schulson, M.J. Stewart, Metall. Trans. 7B (1976) 363–368. M. Alatalo, M. Wienert, R.E. Watson, Phys. Rev. B 57 (1998) R2009–R2012. M. Ferrini, D. Gozzi, J. Alloys Compd. 317–318 (2001) 583–589. F. Willis, R.G. Leisure, I. Jacob, Phys. Rev. B 50 (1994) 13792–13794. U. Anselmi-Tamburini, G. Spinolo, G. Flor, Z.A. Munir, J. Alloys Compd. 247 (1997) 190–194. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN 2k an Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, Vienna University of Technology, Vienna, Austria, 2001. A.R. Williams, J. Kubler, C.D. Gelatt Jr., Phys. Rev. B 19 (1979) 6094–6118.
167
[14] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865– 3868. [15] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223– 16233. [16] A. Israel, I. Jacob, J.L. Soubegoroux, D. Fruchart, H. Pinto, M. Melamud, J. Alloys Compd. 253–254 (1997) 265–267. [17] A.E. Dwight, J.W. Downey, R.A. Conner Jr., Acta Crystallographica 14 (1961) 75–76. [18] F.D. Murnaghan, Proceedings of the National Academy of Sciences USA 30 (1944) 244–247. [19] H. Anton, P.C. Schmidt, Intermetallics 5 (1997) 449–465. [20] P. Blaha, K. Schwarz, Phys. Rev. Lett. 54 (1985) 1192–1195. [21] P. Wodniecki, B. Wodniecka, A. Kulinska, M. Uhrmacher, K.P. Lieb, J. Alloys Compd. 335 (2002) 20–25. [22] L.A. Terrazos, H.M. Petrilli, M. Marszalek, H. Saitovich, P.R.J. Silva, P. Blaha, K. Schwarz, Solid State Commun. 121 (2002) 525– 529.