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High-pressure crystal structure studies of Fe, Ru and Os
Journal of Physics and Chemistry of Solids 65 (2004) 1565–1571 www.elsevier.com/locate/jpcs
High-pressure crystal structure studies of Fe, Ru and Os Anatoly B. Belonoshkoa,b,*, Sa Lic, Rajeev Ahujac, Bo¨rje Johanssona,c a
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Brinellva¨gen 23, Stockholm SE-100 44, Sweden b Condensed Matter Theory Group, Stockholm Center for Physics, Astronomy and Biotechnology, Department of Physics, The Royal Institute of Technology, Stockholm SE-10691, Sweden c Condensed Matter Theory Group, Department of Physics, University of Uppsala , Box 530, Uppsala S-751 21, Sweden Received 5 June 2003; revised 23 September 2003; accepted 12 November 2003
Abstract In order to reveal structural trends with increasing pressure in d transition metals, we performed full potential linear muffin-tin orbital calculations for Fe, Ru, and Os in the hexagonal close packed structure. The calculations cover a wide volume range and demonstrate that all these hexagonal close-packed metals have non-ideal c=a at low pressures which, however, increases with pressure and asymptotically approaches the ideal value at very high compressions. These results are in accordance with most recent experiment for Ru and Os. The experimental data for iron is not conclusive, but it is believed that the c=a ratio decreases weakly with increasing pressure at moderate compression. Since, the experimental and calculated equations of state for iron are in increasingly good agreement with increasing pressure, it is possible that either the negative c=a trend is valid only for a restricted pressure range, or related to the experimental difficulties (e.g. non-hydrostaticity). q 2004 Elsevier Ltd. All rights reserved. PACS: 61.66. 2 f; 64.30. þ t; 71.20.Be
1. Introduction A hexagonal close-packed (hcp) structure can be characterized by two structural parameters, corresponding to the size of a unit cell –c and a: If the hcp structure is ideal, then c=a ¼ 1:633; which is the so-called ideal ratio and atoms in such a structure are truly close packed. However, in most hcp metals the c=a ratio is slightly different from the ideal one. The present interest in the variation of the value of c=a with pressure in d transition metals can be easily understood, if we recollect that iron is believed to be the main constituent in the Earth’s core. It has been demonstrated that Fe at the pressure ðPÞ of the core and room temperature ðTÞ is stable in the hcp structure. It is also known that the Earth’s inner core possesses the seismic anisotropy, which might be related to the elastic anisotropy of an hcp structure, which, in * Corresponding author. Address: Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Brinellva¨gen 23, SE-100 44, Stockholm, Sweden. E-mail address: [email protected] (A.B. Belonoshko). 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2003.11.043
turn, is related to the c=a ratio. However, the experimental data on the hcp c=a ratio for iron, obtained from Xray diffraction, is not conclusive [1]. This is because experimental studies under extreme pressures are difficult. For example, the pressure becomes increasingly non-hydrostatic with increasing pressure [2], which might affect subtle changes in c=a: Recently, an experimental high-pressure study of d-metals (Ru and Os) was published [3]. It is interesting to note, that while the c=a value for Ru and Os increases with pressure, whereas for Fe c=a decreases. The data for Fe comes from several different groups, while the Ru and Os data is obtained using the same procedure. Since, one might expect similar structural trends with increasing pressure for d transition metals in the same row of the Periodic Table, it is of particular interest to calculate c=a for a wide pressure range. In Section 2, we report our results, which are further discussed and compared with existing theoretical estimates. We also discuss the possibility of applying the cell model [4] to calculate the c=a ratio for iron at high temperatures (Fig.1).
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Fig. 1. The calculated c=a ratio (shown by lines) as a function of pressure for Fe, Ru and Os in comparison with the experimental data. The Fe experimental data are denoted by diamonds [17] and inverted triangles [20], two points under high pressure are marked by a plus [1] and a star [19] separately. The Ru and Os experimental data at low pressure are denoted by circles and squares. Our calculated points for Fe, Ru and Os are marked by filled diamonds, circles and squares, correspondingly. The calculated data for Fe, Ru and Os are fitted using linear fit and shown by solid, dashed, and dotted lines, correspondingly.
2. Method In order to study the behavior of the hexagonal crystal structure of Fe, Ru and Os under high pressure, we have used the full potential linear muffin-tin orbital (FPLMTO) method [5]. All our results were obtained through the generalized gradient approximation (GGA) [6] for the parameterization of the exchange and correlation potential. Some results were compared with the results calculated by the local density approximation (LDA) [7]. The total energy as a function of volume as well as a function of internal parameters was obtained by means of first-principles self-consistent total-energy calculations with the non-local Perdew, Burke and Ernzerhof (PBE) [6] exchange correlation. For each volume ðVÞ; we calculated the total energy ðEÞ minimum regarding to the c=a ratio. The equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ and its first pressure derivative ðB00 Þ were estimated through a least-square fit of calculated V – E set to the integrated form of the third-order Birch – Murnaghan equation of state (EOS) [8,9] " 9 V3 V 7=3 B0 ð4 2 B00 Þ 02 2 ð14 2 3B00 Þ 04=3 EðVÞ ¼ 2 16 V V # V 5=3 þð16 2 3B00 Þ 02=3 þ E0 V
ð1Þ
Using the obtained B0 ; B00 and V0 ; the hydrostatic pressure P was determined from the volume derivative of the Eq. (1) " 5=3 # V0 7=3 V PðVÞ ¼ 1:5B0 2 0 V V " #) ( 3 0 V0 2=3 1 þ ðB0 2 4Þ 21 ð2Þ 4 V The bulk modulus is derived through the volume derivative of the Eq. (2) " # 7 V0 7=3 5 V0 5=3 BðVÞ ¼ 1:5B0 2 3 V 3 V " #) ( 3 0 V0 2=3 1 þ ðB0 2 4Þ 21 4 V " 5=3 # V0 7=3 V þ 1:5B0 2 0 V V " 2=3 # 1 0 V ðB0 2 4Þ 0 ð3Þ 2 V The FPLMTO calculations are all electron, fully relativistic, without a shape approximation for the charge density or potential. The crystal is divided into non-overlapping muffin-tin spheres and an interstitial region outside
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the spheres. Basis functions, electron densities, and potentials are expanded in combinations of spherical harmonic functions (with cutoff lmax ¼ 6) inside nonoverlapping spheres surrounding the atomic sites and in Fourier series in the interstitial region. The radial functions within the muffin-tin spheres are linear combinations of radial wave functions fn and their energy derivatives f_ n ; whereas outside the muffin-tin spheres the radial part of these functions are combinations of Neumann or Hankel functions depending on the sign of k2 : In these calculations, the muffin tins are kept as large as possible (without overlapping one another), so that the muffin tins fill about 80% of the total volume. The remaining 20% of the volume defines the interstitial region. To reduce the core leakage at the sphere boundary, a double basis with a total of four different k2 values is used. For Fe, 4s, 4p, 3d and 4f are taken as valence states, 3s and 3p as semi-core states. Semi-core states are those states, which belong to core states, however, to decrease the core leakage, we treated them as valence states in our calculation. For Ru, 5s, 5p, 4d and are valence states, 4s and 4p are semi-core state. For Os, 6s, 6p, 5d and 5f are treated as valence states, to reduce the core leakage at the sphere boundary, 5s, 5p and 4f are taken as semi-core states. The resulting basis forms a single, fully hybridizing basis set. This approach has previously proven to give a well-converged basis [5]. For the sampling of the irreducible wedge of the Brillouin zone, we use a special-k-point method [10]. The total energy convergence in respect to the number of k points was checked for all these metals. In our calculations, we used 65k points in the irreducible part of the hcp Brillouin zone. To speed up the convergence, a Gaussian broadening of width 20 mRy is associated with each calculated eigenvalue.
3. Results We have calculated the c=a ratio as a function of pressure for the d transition metals Fe, Ru and Os up to 400 GPa in the hcp crystal structure. At zero pressure, our GGA ˚ and c ¼ 3:8984 A ˚ are in reasoncalculated a ¼ 2:4611 A ˚ able agreement with the experimental data (a ¼ 2:5234 A
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˚ ) [1] for hcp Fe. At zero pressure the and c ¼ 4:0526 A experimental c=a ratio is around 1.606 [1] for Fe, 1.584 [3] for Ru and 1.580 [3] for Os. The experimental c=a ratio increases under pressure for Ru and Os. However, it decreases under pressure for Fe, although the decrease is very weak [12,13]. In our GGA calculation, we found that the c=a ratio are 1.584, 1.582, 1.580 for Fe, Ru and Os, respectively, at zero pressure, which is in good agreement with the result of Zheng et al. [14]. The comparison between theory and experiment for equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ; first pressure derivative of bulk modulus ðB00 Þ and c=a axial ratio for Fe, Ru and Os is shown in Table 1. From Table 1, we can see that LDA gives smaller equilibrium volume, but larger bulk modulus when comparing with GGA calculation. This is because LDA [7] always tends to overestimate the bonding. This, in turn, underestimates the volume, but overestimates the bulk modulus. The bulk modulus for Fe is too large in the LDA calculation. It is also found that GGA gives better agreement with EOS than LDA for hcp Fe, as shown in Fig. 2. The calculated EOS at low pressure poorly agrees with experiment and is in good agreement with previous calculations [18]. However, with increasing pressure, calculated and experimental Fe EOS are very close, except the constant volume shift (Fig. 2). This gives us confidence that the calculated pressure trend of c=a with increasing pressure is probably valid. B0 and B00 are highly correlated parameters in the Birch – Murnaghan EOS [15] and in Table 2 we show the comparison when B00 is taken to be the experimental value. GGA gives better agreement with experiment when B00 is fixed to the experimental data for Fe and Ru. Both GGA and LDA approximations give much larger bulk modulus for Os when B00 is fixed to the experimental value, namely 2.4. The bulk modulus for hcp Fe is plotted as a function of volume in Fig. 3 and the comparison between experiment and theory is shown. The calculated c=a ratio for all these metals is increased with increasing pressure as shown in Fig. 1. At the pressure 400 GPa, the c=a ratio for Fe, Ru and Os reaches the values 1.593, 1.606 and 1.598, respectively. Our result c=a ¼ 1:589 at a pressure of 185 GPa agrees well with the only point c=a ¼ 1:585 Dubrovinsky et al. [19] published for Fe at
Table 1 Comparison between experiment and theory as regards the equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ; first pressure derivative of bulk modulus ðB00 Þ and c=a axial ratio for Fe, Ru and Os GGA
Fe Ru Os a b c
LDA
Experiment
V0
B0
B00
c=a
V0
B0
B00
c=a
V0
B0
B00
c=a
10.224 13.845 14.244
303 331 442
4.1 4.1 4.0
1.584 1.582 1.580
9.599 13.239 13.751
349 371 471
4.3 4.3 4.1
1.583 1.583 1.581
11.175a 13.574c 13.978c
160.2 (2.1)a 348 (18)c 462 (12)c
5.82 (0.08)a 3.3 (0.80)c 2.4 (0.50)c
1.606 (0.002)b 1.584c 1.580c
Ref. [13]. Ref. [1]. Ref. [3].
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Fig. 2. Comparison of the experimental and theoretical EOS for hcp Fe using two different exchange and correlation functionals, LDA and GGA. It is shown that GGA gives better agreement than LDA. The experimental data was taken from Dubrovinsky et al. [19]. The starting point is the Fe bcc (a)-hcp(e ) transition pressure [38].
185 GPa and a temperature of 1115 K. Our results are also in good agreement with the calculations by So¨derlind et al. [11], (c=a ¼ 1:585 at P ¼ 13:4 GPa, which corre˚ 3/atom). Under extremely sponds to the volume of 9.8 A high pressure, we predict that an ideal packing becomes the most important factor for the c=a ratio. We calculated one point for Fe at pressure of 4000 GPa and found that c=a is near 1.633, i.e. the ideal c=a ratio, as shown in Fig. 4. Under very high pressure, the c=a ratio for all these three elements pffiffiffiffi approaches the ideal ratio of 8=3 ¼ 1:633:
4. Discussion The general trend of the c=a ratio is clear from our calculations. Being less than ideal at low pressures, it increases with increasing pressure and eventually pffiffiffiffi approaches the ideal ratio of 8=3 ¼ 1:633 at extreme pressures (Fig. 4). This seems, however, to be in contradiction with the experimental data [12,13] which shows a very weak decrease of the Fe c=a ratio with pressure. However, we have to remember that there is no systematic study of this issue for a really wide pressure range and in a hydrostatic environment. The so-called quasi-hydrostatic experiments become eventually substantially non-hydrostatic at high pressure [2]. Moreover, Fe samples tend to recrystallize during compression and become oriented with the c axes toward the anvil surface [16]. Therefore, the pressure tends to decrease the c size to a larger degree than the a size.
If this is indeed the real situation, then it might explain the contradiction between the subtle negative experimental and positive calculated trends. The experimental data on Ru and Os [3], is in good agreement with our calculations. The pressure derivative of the c=a ratio for Ru and Os (Fig. 1) is larger than that for Fe and is likely to be determined more reliably. Having the support from the comparison of our results with experimental data for Ru and Os and the experimental and calculated equations of state for iron at high pressure (Fig. 2), we suggest that the discovered theoretical trend for Fe is likely to be the correct one on a large pressure scale. This is probably true in a wide pressure range, since, the experimental and calculated equations of state for Fe shows close behavior at high pressure (Fig. 2). Still, we cannot exclude the possibility that at the pressure of several dozen GPa, our calculations do not reproduce the real situation, because in this pressure range our and experimental equations of states for iron differ most. In our Table 2 Comparison between experiment and theory for bulk modulus B at the experimental B00 GGA (B0 )
Fe Ru Os a b
260 371 606 Mao et al. (1990) [13]. Cynn et al. (2002) [3].
LDA (B0 )
374 400 609
Exp. B0
B00
160.2 (2.1)a 348 (18)b 462 (12)b
5.82 3.3 2.4
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Fig. 3. Comparison of calculated and experimental bulk modulus (B) of hcp Fe as a function of volume ðVÞ: The experimental B 2 V curve was derived by using the third order Birch– Murnaghan EOS (Eq. (3)) and parameters V0 ¼ 6:73 cm3/mol, K0 ¼ 160:2 GPa, K 00 ¼ 5:82 (according to Mao et al. [13]).
opinion, the behavior of the c=a ratio for iron in that pressure range is still an open question to be resolved by truly hydrostatic experiments with a careful control of recrystallization.
While the pressure trend is comparably easy to study, the effect of temperature at high pressure is a more difficult problem. Most recently, Steinle-Neumann et al. [21] calculated elastic constants of the hcp iron at pressures
˚ 3/atom (9266 GPa, Murnaghan EOS; 4373 GPa Birch–Murnaghan EOS; 3079 GPa Vinet Fig. 4. The total energy as a function of the c=a ratio for Fe at 3.1 A EOS). The total energy is shifted to be zero at the minimum ðc=a ¼ 1:6301Þ: The solid line was obtained by a ninth order polynomial fit to the calculated points.
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and temperatures relevant to the Earth’s inner core. Despite that they applied an ab initio method, we believe that their results may be erroneous. The effect of temperature has been calculated using the so-called cell model, also known as the Lennard – Jones – Devonshire (LJD) model [22]. In this model, only one atom is allowed to move in an otherwise frozen lattice. It was demonstrated [23,24] that the LJD model does not give an acceptable precision for the temperature dependence of the elastic constants unless one is satisfied with a precision of 500%. This error was calculated for a pair potential and cubic structure. It is possible that this error might be even larger when applying an ab initio model for hexagonal structure. The apparent failure of the LJD model when calculating elastic constants can be easily understood. This model does not take into account the long-wavelength vibration modes. This neglect does not lead to substantial errors in calculated thermal energies, but affects significantly the quality of calculations of the second derivative of the energy as a function of temperature [23]. The LJD method [22] has not been used for calculating elastic constants since it was demonstrated [23] that it is not suitable for that purpose. Steinle-Neumann et al. [21] obtained that the c=a ratio in hcp Fe increases with T and is very high (about 1.7) at the pressure in the center of the Earth and T ¼ 7000 K. However, this trend is not confirmed by the experimental data [1]. If we look at Fig. 4, we notice that the change in energy when the c=a ratio is changed is very small even at the extreme pressure of about 4000 GPa, which is roughly order of magnitude higher than in the center of the Earth. A variation of the c=a ratio of about 0.05 from the ideal (1.633) value leads to tiny energy changes of about 0.06 eV. This energy change is smaller at lower pressure. The cell model leads to errors of about 1 –2 percent in thermal energy [23]. According to Alfe et al. [25], the thermal energy of iron is about 2.5 ge V at 7000 K and high pressure. It is clear, that the error in the calculated thermal energy is quite comparable to the energy change when varying the c=a ratio. Therefore, the application of the cell model with an intrinsic error of 1 – 2 percent (0.025 –0.05 eV) might be meaningless for Fe. Recent ab initio molecular dynamics study demonstrated that c=a in Fe indeed approaches ideal close packing with increasing temperature [26]. We may note that c=a close to ideal packing under pressure exists not only in Fe, but also in some other hcp divalent metals. Novikov et al. [27] studied c=a for Cd and Zn as a function of pressure, and found that it decreases from , 1.9 at ambient pressure towards its ideal value with increasing pressure and it is effectively ideal in the 1 Mbar region. The same trend for Zn and Cd was also observed experimentally [28]. This conclusion was further corroborated by Godwal et al. for both Cd [29] and Zn [30]. There is still a debate on an anomaly in the value of c=a in the 10 GPa region which may be due to the electronic topological transition [31], but the main point is that with
increasing compression, metal becomes more and more free-electron-type and c=a approaches its ideal value. Velisavljevic et al. [32] measured c=a for Be up to 66 GPa and found essentially a constant value of , 1:57; which has a slight deviation from nearly free electron behavior. Errandonea et al. [33] measured c=a for Mg up to 20 GPa and also found a nearly constant value of ,1.62, very close to the ideal one. Two earlier measurements of Mg by Clendenen et al. [36] and Perez-Albuerne et al.[37] show different behavior of c=a above 20 GPa but with big uncertainties. Very recently, Verma et al. [34] calculated c=a for Re up to 1 TPa and found essentially a constant value of , 1.62. Haussermann and Simak [35] studied the effect of the electronic structure on the value of c=a: They have shown that the ideal hcp structure with c=a ¼ 1:633 is electronically unstable against c=a distortion. However, electrostatic energy, which favors the ideal close packing, increases with pressure faster relative to Eband energy and the nearly ideal closed packed structures can be obtained as high pressure modification.
5. Conclusions We have optimized the c=a ratio for the hcp d transition metals, namely, Fe, Ru, and Os as a function of pressure. The c=a ratio for all three metals is lower than the ideal one at low pressure and increases with increasing pressure. Eventually, it reaches the ideal value at extreme compression. The absolute values are slightly different from the experimental ones, indicating a possible imperfection of the applied first principles method. However, our calculations are in agreement with previous theoretical works. Since, the applied theoretical method in this work is physically sound, it is likely that the discovered trend is the correct one on a large pressure scale with the possibility that at the pressure 10– 30 GPa our method does not work for Fe for some unknown reason. The trend for the c=a for Fe is important from the geophysical point of view. While the pressure dependence of c=a ratio can be considered as established from our calculated data, its temperature dependence still represents a problem, which cannot be treated by such approximate methods as the cell model [4,21 –24], because of the small energy changes associated with the c=a ratio variations.
Acknowledgements We are grateful to J.M. Wills, Los Alamos, for letting us to use his FPLMTO code. Discussion with G. Shen was very informative. The study was supported by the Swedish Research Council (VR) and the Swedish Foundation for Strategic Research (SSF).
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High-pressure crystal structure studies of Fe, Ru and Os Anatoly B. Belonoshkoa,b,*, Sa Lic, Rajeev Ahujac, Bo¨rje Johanssona,c a
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Brinellva¨gen 23, Stockholm SE-100 44, Sweden b Condensed Matter Theory Group, Stockholm Center for Physics, Astronomy and Biotechnology, Department of Physics, The Royal Institute of Technology, Stockholm SE-10691, Sweden c Condensed Matter Theory Group, Department of Physics, University of Uppsala , Box 530, Uppsala S-751 21, Sweden Received 5 June 2003; revised 23 September 2003; accepted 12 November 2003
Abstract In order to reveal structural trends with increasing pressure in d transition metals, we performed full potential linear muffin-tin orbital calculations for Fe, Ru, and Os in the hexagonal close packed structure. The calculations cover a wide volume range and demonstrate that all these hexagonal close-packed metals have non-ideal c=a at low pressures which, however, increases with pressure and asymptotically approaches the ideal value at very high compressions. These results are in accordance with most recent experiment for Ru and Os. The experimental data for iron is not conclusive, but it is believed that the c=a ratio decreases weakly with increasing pressure at moderate compression. Since, the experimental and calculated equations of state for iron are in increasingly good agreement with increasing pressure, it is possible that either the negative c=a trend is valid only for a restricted pressure range, or related to the experimental difficulties (e.g. non-hydrostaticity). q 2004 Elsevier Ltd. All rights reserved. PACS: 61.66. 2 f; 64.30. þ t; 71.20.Be
1. Introduction A hexagonal close-packed (hcp) structure can be characterized by two structural parameters, corresponding to the size of a unit cell –c and a: If the hcp structure is ideal, then c=a ¼ 1:633; which is the so-called ideal ratio and atoms in such a structure are truly close packed. However, in most hcp metals the c=a ratio is slightly different from the ideal one. The present interest in the variation of the value of c=a with pressure in d transition metals can be easily understood, if we recollect that iron is believed to be the main constituent in the Earth’s core. It has been demonstrated that Fe at the pressure ðPÞ of the core and room temperature ðTÞ is stable in the hcp structure. It is also known that the Earth’s inner core possesses the seismic anisotropy, which might be related to the elastic anisotropy of an hcp structure, which, in * Corresponding author. Address: Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Brinellva¨gen 23, SE-100 44, Stockholm, Sweden. E-mail address: [email protected] (A.B. Belonoshko). 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2003.11.043
turn, is related to the c=a ratio. However, the experimental data on the hcp c=a ratio for iron, obtained from Xray diffraction, is not conclusive [1]. This is because experimental studies under extreme pressures are difficult. For example, the pressure becomes increasingly non-hydrostatic with increasing pressure [2], which might affect subtle changes in c=a: Recently, an experimental high-pressure study of d-metals (Ru and Os) was published [3]. It is interesting to note, that while the c=a value for Ru and Os increases with pressure, whereas for Fe c=a decreases. The data for Fe comes from several different groups, while the Ru and Os data is obtained using the same procedure. Since, one might expect similar structural trends with increasing pressure for d transition metals in the same row of the Periodic Table, it is of particular interest to calculate c=a for a wide pressure range. In Section 2, we report our results, which are further discussed and compared with existing theoretical estimates. We also discuss the possibility of applying the cell model [4] to calculate the c=a ratio for iron at high temperatures (Fig.1).
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Fig. 1. The calculated c=a ratio (shown by lines) as a function of pressure for Fe, Ru and Os in comparison with the experimental data. The Fe experimental data are denoted by diamonds [17] and inverted triangles [20], two points under high pressure are marked by a plus [1] and a star [19] separately. The Ru and Os experimental data at low pressure are denoted by circles and squares. Our calculated points for Fe, Ru and Os are marked by filled diamonds, circles and squares, correspondingly. The calculated data for Fe, Ru and Os are fitted using linear fit and shown by solid, dashed, and dotted lines, correspondingly.
2. Method In order to study the behavior of the hexagonal crystal structure of Fe, Ru and Os under high pressure, we have used the full potential linear muffin-tin orbital (FPLMTO) method [5]. All our results were obtained through the generalized gradient approximation (GGA) [6] for the parameterization of the exchange and correlation potential. Some results were compared with the results calculated by the local density approximation (LDA) [7]. The total energy as a function of volume as well as a function of internal parameters was obtained by means of first-principles self-consistent total-energy calculations with the non-local Perdew, Burke and Ernzerhof (PBE) [6] exchange correlation. For each volume ðVÞ; we calculated the total energy ðEÞ minimum regarding to the c=a ratio. The equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ and its first pressure derivative ðB00 Þ were estimated through a least-square fit of calculated V – E set to the integrated form of the third-order Birch – Murnaghan equation of state (EOS) [8,9] " 9 V3 V 7=3 B0 ð4 2 B00 Þ 02 2 ð14 2 3B00 Þ 04=3 EðVÞ ¼ 2 16 V V # V 5=3 þð16 2 3B00 Þ 02=3 þ E0 V
ð1Þ
Using the obtained B0 ; B00 and V0 ; the hydrostatic pressure P was determined from the volume derivative of the Eq. (1) " 5=3 # V0 7=3 V PðVÞ ¼ 1:5B0 2 0 V V " #) ( 3 0 V0 2=3 1 þ ðB0 2 4Þ 21 ð2Þ 4 V The bulk modulus is derived through the volume derivative of the Eq. (2) " # 7 V0 7=3 5 V0 5=3 BðVÞ ¼ 1:5B0 2 3 V 3 V " #) ( 3 0 V0 2=3 1 þ ðB0 2 4Þ 21 4 V " 5=3 # V0 7=3 V þ 1:5B0 2 0 V V " 2=3 # 1 0 V ðB0 2 4Þ 0 ð3Þ 2 V The FPLMTO calculations are all electron, fully relativistic, without a shape approximation for the charge density or potential. The crystal is divided into non-overlapping muffin-tin spheres and an interstitial region outside
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the spheres. Basis functions, electron densities, and potentials are expanded in combinations of spherical harmonic functions (with cutoff lmax ¼ 6) inside nonoverlapping spheres surrounding the atomic sites and in Fourier series in the interstitial region. The radial functions within the muffin-tin spheres are linear combinations of radial wave functions fn and their energy derivatives f_ n ; whereas outside the muffin-tin spheres the radial part of these functions are combinations of Neumann or Hankel functions depending on the sign of k2 : In these calculations, the muffin tins are kept as large as possible (without overlapping one another), so that the muffin tins fill about 80% of the total volume. The remaining 20% of the volume defines the interstitial region. To reduce the core leakage at the sphere boundary, a double basis with a total of four different k2 values is used. For Fe, 4s, 4p, 3d and 4f are taken as valence states, 3s and 3p as semi-core states. Semi-core states are those states, which belong to core states, however, to decrease the core leakage, we treated them as valence states in our calculation. For Ru, 5s, 5p, 4d and are valence states, 4s and 4p are semi-core state. For Os, 6s, 6p, 5d and 5f are treated as valence states, to reduce the core leakage at the sphere boundary, 5s, 5p and 4f are taken as semi-core states. The resulting basis forms a single, fully hybridizing basis set. This approach has previously proven to give a well-converged basis [5]. For the sampling of the irreducible wedge of the Brillouin zone, we use a special-k-point method [10]. The total energy convergence in respect to the number of k points was checked for all these metals. In our calculations, we used 65k points in the irreducible part of the hcp Brillouin zone. To speed up the convergence, a Gaussian broadening of width 20 mRy is associated with each calculated eigenvalue.
3. Results We have calculated the c=a ratio as a function of pressure for the d transition metals Fe, Ru and Os up to 400 GPa in the hcp crystal structure. At zero pressure, our GGA ˚ and c ¼ 3:8984 A ˚ are in reasoncalculated a ¼ 2:4611 A ˚ able agreement with the experimental data (a ¼ 2:5234 A
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˚ ) [1] for hcp Fe. At zero pressure the and c ¼ 4:0526 A experimental c=a ratio is around 1.606 [1] for Fe, 1.584 [3] for Ru and 1.580 [3] for Os. The experimental c=a ratio increases under pressure for Ru and Os. However, it decreases under pressure for Fe, although the decrease is very weak [12,13]. In our GGA calculation, we found that the c=a ratio are 1.584, 1.582, 1.580 for Fe, Ru and Os, respectively, at zero pressure, which is in good agreement with the result of Zheng et al. [14]. The comparison between theory and experiment for equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ; first pressure derivative of bulk modulus ðB00 Þ and c=a axial ratio for Fe, Ru and Os is shown in Table 1. From Table 1, we can see that LDA gives smaller equilibrium volume, but larger bulk modulus when comparing with GGA calculation. This is because LDA [7] always tends to overestimate the bonding. This, in turn, underestimates the volume, but overestimates the bulk modulus. The bulk modulus for Fe is too large in the LDA calculation. It is also found that GGA gives better agreement with EOS than LDA for hcp Fe, as shown in Fig. 2. The calculated EOS at low pressure poorly agrees with experiment and is in good agreement with previous calculations [18]. However, with increasing pressure, calculated and experimental Fe EOS are very close, except the constant volume shift (Fig. 2). This gives us confidence that the calculated pressure trend of c=a with increasing pressure is probably valid. B0 and B00 are highly correlated parameters in the Birch – Murnaghan EOS [15] and in Table 2 we show the comparison when B00 is taken to be the experimental value. GGA gives better agreement with experiment when B00 is fixed to the experimental data for Fe and Ru. Both GGA and LDA approximations give much larger bulk modulus for Os when B00 is fixed to the experimental value, namely 2.4. The bulk modulus for hcp Fe is plotted as a function of volume in Fig. 3 and the comparison between experiment and theory is shown. The calculated c=a ratio for all these metals is increased with increasing pressure as shown in Fig. 1. At the pressure 400 GPa, the c=a ratio for Fe, Ru and Os reaches the values 1.593, 1.606 and 1.598, respectively. Our result c=a ¼ 1:589 at a pressure of 185 GPa agrees well with the only point c=a ¼ 1:585 Dubrovinsky et al. [19] published for Fe at
Table 1 Comparison between experiment and theory as regards the equilibrium volume ðV0 Þ; bulk modulus at ambient pressure ðB0 Þ; first pressure derivative of bulk modulus ðB00 Þ and c=a axial ratio for Fe, Ru and Os GGA
Fe Ru Os a b c
LDA
Experiment
V0
B0
B00
c=a
V0
B0
B00
c=a
V0
B0
B00
c=a
10.224 13.845 14.244
303 331 442
4.1 4.1 4.0
1.584 1.582 1.580
9.599 13.239 13.751
349 371 471
4.3 4.3 4.1
1.583 1.583 1.581
11.175a 13.574c 13.978c
160.2 (2.1)a 348 (18)c 462 (12)c
5.82 (0.08)a 3.3 (0.80)c 2.4 (0.50)c
1.606 (0.002)b 1.584c 1.580c
Ref. [13]. Ref. [1]. Ref. [3].
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Fig. 2. Comparison of the experimental and theoretical EOS for hcp Fe using two different exchange and correlation functionals, LDA and GGA. It is shown that GGA gives better agreement than LDA. The experimental data was taken from Dubrovinsky et al. [19]. The starting point is the Fe bcc (a)-hcp(e ) transition pressure [38].
185 GPa and a temperature of 1115 K. Our results are also in good agreement with the calculations by So¨derlind et al. [11], (c=a ¼ 1:585 at P ¼ 13:4 GPa, which corre˚ 3/atom). Under extremely sponds to the volume of 9.8 A high pressure, we predict that an ideal packing becomes the most important factor for the c=a ratio. We calculated one point for Fe at pressure of 4000 GPa and found that c=a is near 1.633, i.e. the ideal c=a ratio, as shown in Fig. 4. Under very high pressure, the c=a ratio for all these three elements pffiffiffiffi approaches the ideal ratio of 8=3 ¼ 1:633:
4. Discussion The general trend of the c=a ratio is clear from our calculations. Being less than ideal at low pressures, it increases with increasing pressure and eventually pffiffiffiffi approaches the ideal ratio of 8=3 ¼ 1:633 at extreme pressures (Fig. 4). This seems, however, to be in contradiction with the experimental data [12,13] which shows a very weak decrease of the Fe c=a ratio with pressure. However, we have to remember that there is no systematic study of this issue for a really wide pressure range and in a hydrostatic environment. The so-called quasi-hydrostatic experiments become eventually substantially non-hydrostatic at high pressure [2]. Moreover, Fe samples tend to recrystallize during compression and become oriented with the c axes toward the anvil surface [16]. Therefore, the pressure tends to decrease the c size to a larger degree than the a size.
If this is indeed the real situation, then it might explain the contradiction between the subtle negative experimental and positive calculated trends. The experimental data on Ru and Os [3], is in good agreement with our calculations. The pressure derivative of the c=a ratio for Ru and Os (Fig. 1) is larger than that for Fe and is likely to be determined more reliably. Having the support from the comparison of our results with experimental data for Ru and Os and the experimental and calculated equations of state for iron at high pressure (Fig. 2), we suggest that the discovered theoretical trend for Fe is likely to be the correct one on a large pressure scale. This is probably true in a wide pressure range, since, the experimental and calculated equations of state for Fe shows close behavior at high pressure (Fig. 2). Still, we cannot exclude the possibility that at the pressure of several dozen GPa, our calculations do not reproduce the real situation, because in this pressure range our and experimental equations of states for iron differ most. In our Table 2 Comparison between experiment and theory for bulk modulus B at the experimental B00 GGA (B0 )
Fe Ru Os a b
260 371 606 Mao et al. (1990) [13]. Cynn et al. (2002) [3].
LDA (B0 )
374 400 609
Exp. B0
B00
160.2 (2.1)a 348 (18)b 462 (12)b
5.82 3.3 2.4
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Fig. 3. Comparison of calculated and experimental bulk modulus (B) of hcp Fe as a function of volume ðVÞ: The experimental B 2 V curve was derived by using the third order Birch– Murnaghan EOS (Eq. (3)) and parameters V0 ¼ 6:73 cm3/mol, K0 ¼ 160:2 GPa, K 00 ¼ 5:82 (according to Mao et al. [13]).
opinion, the behavior of the c=a ratio for iron in that pressure range is still an open question to be resolved by truly hydrostatic experiments with a careful control of recrystallization.
While the pressure trend is comparably easy to study, the effect of temperature at high pressure is a more difficult problem. Most recently, Steinle-Neumann et al. [21] calculated elastic constants of the hcp iron at pressures
˚ 3/atom (9266 GPa, Murnaghan EOS; 4373 GPa Birch–Murnaghan EOS; 3079 GPa Vinet Fig. 4. The total energy as a function of the c=a ratio for Fe at 3.1 A EOS). The total energy is shifted to be zero at the minimum ðc=a ¼ 1:6301Þ: The solid line was obtained by a ninth order polynomial fit to the calculated points.
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and temperatures relevant to the Earth’s inner core. Despite that they applied an ab initio method, we believe that their results may be erroneous. The effect of temperature has been calculated using the so-called cell model, also known as the Lennard – Jones – Devonshire (LJD) model [22]. In this model, only one atom is allowed to move in an otherwise frozen lattice. It was demonstrated [23,24] that the LJD model does not give an acceptable precision for the temperature dependence of the elastic constants unless one is satisfied with a precision of 500%. This error was calculated for a pair potential and cubic structure. It is possible that this error might be even larger when applying an ab initio model for hexagonal structure. The apparent failure of the LJD model when calculating elastic constants can be easily understood. This model does not take into account the long-wavelength vibration modes. This neglect does not lead to substantial errors in calculated thermal energies, but affects significantly the quality of calculations of the second derivative of the energy as a function of temperature [23]. The LJD method [22] has not been used for calculating elastic constants since it was demonstrated [23] that it is not suitable for that purpose. Steinle-Neumann et al. [21] obtained that the c=a ratio in hcp Fe increases with T and is very high (about 1.7) at the pressure in the center of the Earth and T ¼ 7000 K. However, this trend is not confirmed by the experimental data [1]. If we look at Fig. 4, we notice that the change in energy when the c=a ratio is changed is very small even at the extreme pressure of about 4000 GPa, which is roughly order of magnitude higher than in the center of the Earth. A variation of the c=a ratio of about 0.05 from the ideal (1.633) value leads to tiny energy changes of about 0.06 eV. This energy change is smaller at lower pressure. The cell model leads to errors of about 1 –2 percent in thermal energy [23]. According to Alfe et al. [25], the thermal energy of iron is about 2.5 ge V at 7000 K and high pressure. It is clear, that the error in the calculated thermal energy is quite comparable to the energy change when varying the c=a ratio. Therefore, the application of the cell model with an intrinsic error of 1 – 2 percent (0.025 –0.05 eV) might be meaningless for Fe. Recent ab initio molecular dynamics study demonstrated that c=a in Fe indeed approaches ideal close packing with increasing temperature [26]. We may note that c=a close to ideal packing under pressure exists not only in Fe, but also in some other hcp divalent metals. Novikov et al. [27] studied c=a for Cd and Zn as a function of pressure, and found that it decreases from , 1.9 at ambient pressure towards its ideal value with increasing pressure and it is effectively ideal in the 1 Mbar region. The same trend for Zn and Cd was also observed experimentally [28]. This conclusion was further corroborated by Godwal et al. for both Cd [29] and Zn [30]. There is still a debate on an anomaly in the value of c=a in the 10 GPa region which may be due to the electronic topological transition [31], but the main point is that with
increasing compression, metal becomes more and more free-electron-type and c=a approaches its ideal value. Velisavljevic et al. [32] measured c=a for Be up to 66 GPa and found essentially a constant value of , 1:57; which has a slight deviation from nearly free electron behavior. Errandonea et al. [33] measured c=a for Mg up to 20 GPa and also found a nearly constant value of ,1.62, very close to the ideal one. Two earlier measurements of Mg by Clendenen et al. [36] and Perez-Albuerne et al.[37] show different behavior of c=a above 20 GPa but with big uncertainties. Very recently, Verma et al. [34] calculated c=a for Re up to 1 TPa and found essentially a constant value of , 1.62. Haussermann and Simak [35] studied the effect of the electronic structure on the value of c=a: They have shown that the ideal hcp structure with c=a ¼ 1:633 is electronically unstable against c=a distortion. However, electrostatic energy, which favors the ideal close packing, increases with pressure faster relative to Eband energy and the nearly ideal closed packed structures can be obtained as high pressure modification.
5. Conclusions We have optimized the c=a ratio for the hcp d transition metals, namely, Fe, Ru, and Os as a function of pressure. The c=a ratio for all three metals is lower than the ideal one at low pressure and increases with increasing pressure. Eventually, it reaches the ideal value at extreme compression. The absolute values are slightly different from the experimental ones, indicating a possible imperfection of the applied first principles method. However, our calculations are in agreement with previous theoretical works. Since, the applied theoretical method in this work is physically sound, it is likely that the discovered trend is the correct one on a large pressure scale with the possibility that at the pressure 10– 30 GPa our method does not work for Fe for some unknown reason. The trend for the c=a for Fe is important from the geophysical point of view. While the pressure dependence of c=a ratio can be considered as established from our calculated data, its temperature dependence still represents a problem, which cannot be treated by such approximate methods as the cell model [4,21 –24], because of the small energy changes associated with the c=a ratio variations.
Acknowledgements We are grateful to J.M. Wills, Los Alamos, for letting us to use his FPLMTO code. Discussion with G. Shen was very informative. The study was supported by the Swedish Research Council (VR) and the Swedish Foundation for Strategic Research (SSF).
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