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Interplay between lattice dynamics and the low-pressure phase of simple cubic polonium
Physics Letters A 375 (2011) 1695–1697
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Interplay between lattice dynamics and the low-pressure phase of simple cubic polonium A. Zaoui a,∗ , A. Belabbes b , R. Ahuja c , M. Ferhat b a
LGCgE (EA 4515) Lille Nord de France, Ecole Polytechnique de Lille, Université de Lille Sciences et Technologies de Lille, Cité Scientifique, Avenue Paul Langevin, 59655 Villeneuve D’Ascq Cedex, France b LEPM, Département de Physique, Université des Sciences et de la Technologie d’Oran, USTO, Oran, Algeria c Condensed Matter Theory Group, Department of Physics, Uppsala University, Uppsala, Sweden
a r t i c l e
i n f o
Article history: Received 29 December 2010 Received in revised form 3 March 2011 Accepted 4 March 2011 Communicated by P.R. Holland Keywords: Lattice dynamics High pressure Ab initio Polonium
a b s t r a c t Low-pressure structural properties of simple cubic polonium are explored through first-principles density-functional theory based relativistic total energy calculations using pseudopotentials and planewave basis set, as well as linear-response theory. We have found that Po undergoes structural phase transition at low pressure near 2 GPa, where the element transforms from simple cubic to a mixture of two trigonal phases namely, hR1 (α = 86◦ ) and hR2 (α = 97.9◦ ) structures. The lattice dynamics calculations provide strong support for the observed phase transition, and show the dynamical stability (instability) of the hR2 (hR1) phase. © 2011 Elsevier B.V. All rights reserved.
Most elements of the Periodic Table are characterized by their tendency to crystallise either in the face-centred cubic (fcc) (e.g. Cu, Ag, Au . . . ), body-centred cubic (bcc) (e.g. Li, Na, K, Rb . . . ), the hexagonal-closed-packed structure (hcp) (e.g. Re, Os, Ru . . . ), the diamond structure (cd) (e.g. C, Si, Ge, α -Sn), or other complex crystal structures. Exceptionally metallic polonium (Po) element with atomic number 84 is the only case, which crystallizes in the simple cubic structure (cs) at ambient conditions. Polonium was discovered by Marie and Pierre Curie in 1898 [1]. It is the first element in the Periodic Table for which all of its isotopes are radioactive. It is a very rare natural element with very unusual properties, such as its tendency to disperse itself with time. Another its property is a tendency to volatilize at surprisingly low temperature. Moreover, it also has some important biological applications. While polonium is radioactive element, it is a centre of academic interest due to its potential radiological health significance and application as a natural tracer of environmental processes. It can be also used in static eliminators, which are devices designed to eliminate static electricity in machinery. In addition, Po has been investigated as a heat source for thermoelectric power devices for space applications, and it has also been used as a light weight heat source to power thermoelectric cells.
*
Corresponding author. E-mail address: [email protected] (A. Zaoui).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.03.014
Crystal structure of polonium was discovered by Beamer and Maxwell in 1946 [2] during the investigation of the electrical resistively of this element. It exists in at least two allotropic modifications: a low temperature α -phase with a simple cubic lattice, and a high temperature β -phase with a simple rhombohedral (trigonal) structure. Experimental investigations of chemical and physical properties of Po are very difficult due to its strong radioactivity. Recent theoretical calculations [3–9] have addressed the electronic, structural stability and dynamical properties of the sc phase of polonium. To the best of our knowledge no experimental information on the pressure-induced phase transition of Po exists. On the theoretical side, recent calculations [6] predict transformation from a sc to a mixture of two trigonal phases at pressures between 1 and 3 GPa, although the structure of the new phase remains unknown. However, the study presented in Ref. [6] is far to be complete and raises deeper questions: (i) what is exactly the transition pressure, (ii) what is the rhombohedral angle corresponding the this transition, and more fundamentally, (iii) which one of the two obtained trigonal phases is more stable, since both have almost the same total energy. In order to clarify these points, in this Letter we address these issues in detail. All calculations reported here were carried out by means of the plane-wave pseudopotential method, as implemented in the PWscf code [10]. Electron–electron interactions are treated within the local density approximation (LDA) [11], as well as the Perdew– Burke–Ernzerhof (PBE) [12] form of the generalized gradient approximation (GGA). Scalar relativistic and full relativistic approx-
1696
A. Zaoui et al. / Physics Letters A 375 (2011) 1695–1697
Table 1 Structural parameters (lattice constant and bulk modulus) compared to firstprinciples calculations and experimental (Exp.) values.
This study: LDA This study: LDA + SO This study: GGA This study: GGA + SO Pseudopotentiala Flapwb : LDA Flapwb : LDA + SO Exp.c a b c
a (Å)
B (GPa)
3.28 3.32 3.34 3.41 3.28 3.27 3.33 3.345
52.6 42.8 45.6 35 56 57.1 42.3
Ref. [4]. Ref. [6]. Ref. [2].
Fig. 2. Total energy of the hR1 and hR2 phases.
Fig. 3. Phonon dispersions for the sc phase of Po at 0 GPa (a) and 2 GPa (b).
Fig. 1. Total energy along the rhombohedral transformation path.
imations with inclusion of spin–orbit interaction (SO) in ultrasoft pseudopotentials (US-PP) [13] were used. The electronic wave function was expanded with a plane-wave basis set with a kinetic energy cutoff of 60 Ry; the augmentation of charges was expanded up to 400 Ry and a Gaussian smearing of 0.02 Ry has been applied. The k-space integration on the Brillouin zone (BZ) for the self-consistent calculations was calculated with a 12 × 12 × 12 and 8 × 8 × 8 k-point mesh according to Monkhorst and Pack (MP) [14] for the simple cubic (trigonal) phase. The lattice-dynamical properties are calculated using the density functional perturbation theory (DFPT) [15]. In particular, 8 × 8 × 8 k-point mesh of MP was used; and obtained matrices were Fourier interpolated to obtain phonon dispersion curves. Table 1 shows the calculated equilibrium lattice constant, and the bulk modulus of sc Po, together with recent first-principles calculations and experimental data. It is clear that only the current LDA + SO and GGA results are in a good agreement with experimental data, and with the state-of-the-art full-potential linearized augmented plane-wave method (FLAPW) [6]. Further we will mainly discuss the LDA + SO results since SO splitting and relativistic effects are crucial for a correct description of the electronic and dynamical properties of heavy atom Po. The body-centred cubic, and face-centred cubic can be continuously connected with the simple cubic structure by means of
trigonal deformation path. The trigonal structures along this path can be described by a rhombohedral structure (Pearson symbol hR1 [16]), the sc structure can be viewed as a rhombohedron with angle α equal to 90◦ . While the primitive cells of bcc and fcc structures are rhombohedrons with angle α equal to 109.47◦ and 60◦ respectively. Thus, it is possible to generate all these cubic structures from a rhombohedron with the atomic position at (0, 0, 0) by varying the α angle. Fig. 1 shows our calculated total energy as a function of angle for Po at an equilibrium volume. Similar calculations were performed at several volumes covering low pressure of 2 GPa. For P = 0 GPa (V / V 0 = 1), the sc phase is the absolute minimum, while the fcc phase is the absolute maximum, and the bcc phase is the local maximum. This indicates that the sc phase exhibits the lowest total energy as expected and the unstable nature of the bcc and fcc phases for a given rhombohedral distortion. At P = 2 GPa (V / V 0 = 0.96), which corresponds relatively to a low pressure, the situation changes drastically. The sc phase is the absolute minimum, and two new local (absolute) minima appear at angles around 85◦ and 100◦ respectively. Po loses its stable sc structure, leading to a structural phase transition: sc → rhombohedral hR1 (α = 86◦ ) and/or hR2 (97.9◦ ) phases (see Fig. 2) respectively, what is in agreement with static total energy calculations of Legut et al. [6]. In order to estimate the dynamical response of polonium under compression and to check if the structural phase transition found previously is linked to dynamical instabilities of Po, we calculate phonon dispersions of sc phase at equilibrium volume and under compression. The calculated phonon dispersion curves of sc Po at equilibrium volume ( P = 0 GPa) along the high symmetry lines of simple cubic BZ are shown in Fig. 3. The calculated phonon dis-
A. Zaoui et al. / Physics Letters A 375 (2011) 1695–1697
Fig. 4. Phonon dispersion for the trigonal phases hR1 (a) and hR2 (b) of Po at 2 GPa.
persions at P = 0 compare well with those obtained by Verstraete [17]. The most striking features of the phonon dispersion of the sc Po are the phonon anomalies, spectacularly in all directions of the BZ, namely the transverse and longitudinal branch, which exhibit a sudden drop in frequency in the M–R, R–Γ , Γ –M, M– X , and X–Γ directions. The phonon dispersion curve of the sc Po is unique compared to other metals, and constitutes an additional type of anomalies among the class of elemental metal studied until now. The calculations strongly suggest that the observed phonon anomalies of sc Po are the Kohn effect [18] arising from a particular nesting feature of the Fermi surface of the sc Po [19]. The calculated phonon dispersion curves of sc Po at low pressure are shown in Fig. 3, where we notice that strong softening with decreasing volume occurs in [ξ ξ ξ ] and [ξ 00] directions, at volume V / V 0 = 0.96, corresponding to a pressure of 2 GPa. The phonon transverse frequencies become imaginary indicating structural phase instability. Thus, the simple cubic structure of Po is dynamically instable against trigonal distortion. These phonon anomalies are linked naturally to the structural phase transition sc → rhombohedral found previously from total energy calculations. However, we notice that the total energy difference between the trigonal hR1 and hR2 phases is extremely small, not exceeding 8 meV, which is at border of the accuracies of best to date ab initio methods. So in such conditions, where two rhombohedral minima have almost the same energy but very different angles, how can one predict which one is the real angle and which experiment can we follow [20]? This will make predictions at extreme conditions, where experiments cannot be performed, more difficult. Moreover the predicted phase transition can be unphysical, because the predicted phase could be dynamically unstable [21,22]. In order to check the dynamical stabilities of the mixture of the two trigonal structures for low pressure phase transition of sc Po found at 2 GPa, we calculate the phonon dispersions of the rhombohedral (α = 86◦ and α = 97.9◦ ) phases at the same pressure (2 GPa). The
1697
corresponding calculated phonon dispersion curves of the hR1 and hR2 phases along the high symmetry lines of the BZ, are shown in Fig. 4. The hR1 (α = 86◦ ) phase is found dynamically unstable (i.e. imaginary phonon modes), while the hR2 (α = 97.9◦ ) phase presents the absolute minimum and is found dynamically stable (only positive phonon modes). Consequently, the first low pressure phase transition of the sc Po forms the dynamical stable rhombohedral structure hR2 (α = 97.9◦ ). In conclusion, a low pressure phase transition of simple cubic → rhombohedral phase of polonium is fully explored by total energy and lattice dynamics calculations within the density-functional theory. It is predicated a bimodal mixture of two trigonal hR1 (86◦ ) and hR2 (97.9◦ ) phases of Po at 2 GPa. Our phonon calculations confirm this transition pressure. The fundamental question concerning the determination of the real minimum from the two trigonal phases having practically the same total energy is resolved by lattice dynamics calculations, where we have shown lattice dynamics as an efficient tool to determine true thermodynamic minimum. We demonstrate the dynamical instability (stability) of the hR1 (hR2) phases. Experimental works if possible on the dynamical and low-pressure behaviour of sc polonium are welcome, since they can shed a valuable light on the scenarios theoretically outlined in the present work. Acknowledgements Two of us (R.A. and M.F.) would like to thank VR-SIDA (MENA) for financial support. References [1] P. Curie, M. Curie, C. R. Acad. Sci. 127 (1898) 175. [2] W.-H. Beamer, C.-R. Maxwell, J. Chem. Phys. 14 (1946) 569. [3] M. Lach-hab, B. Akdim, D.-A. Papaconstantopoulos, M.-J. Mehl, N. Bernstein, J. Phys. Chem. Solids 65 (2004) 1837. [4] R.E. Kraig, D. Roundy, M.L. Cohen, Solid State Commun. 129 (2004) 411. [5] B.-I. Min, J.-H. Shim, M.-S. Park, K. Kim, S.-K. Kwon, S.-J. Youn, Phys. Rev. B 73 (2006) 132102. [6] D. Legut, M. Friák, M. Šob, Phys. Rev. Lett. 99 (2007) 016402. [7] K. Kim, H.-C. Choi, B.-I. Min, Phys. Rev. Lett. 102 (2009) 079701. [8] M. Šob, D. Legut, M. Friák, Phys. Rev. Lett. 102 (2009) 079702. [9] N.-A. Zabidi, H.-A. Kassim, K.-N. Shrivastava, AIP Conf. Proc. 1017 (2008) 255. [10] S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, http://www.pwscf.org. [11] J.-P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [12] J.-P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [13] A. Dal Corso, Phys. Rev. B 76 (2007) 054308. [14] H.-J. Monkhorst, J.-D. Pack, Phys. Rev. B 13 (1976) 5188. [15] S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys. 73 (2001) 515. [16] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, vol. 2, Pergamon, Oxford, 1967. [17] M.J. Verstraete, Phys. Rev. Lett. 104 (2010) 035501. [18] W. Kohn, Phys. Rev. Lett. 2 (1959) 393. [19] A. Belabbes, A. Zaoui, M. Ferhat, Solid State Commun. 150 (2010) 2337. [20] S.-I. Simak, U. Haussermann, R. Ahuja, S. Lidin, B. Johansson, Phys. Rev. Lett. 85 (2000) 142. [21] A.-S. Mikhaylushkin, U. Haussermann, B. Johansson, S.-I. Simak, Phys. Rev. Lett. 92 (2004) 195501. [22] V. Ozolins, A. Zunger, Phys. Rev. Lett. 82 (1999) 767.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Interplay between lattice dynamics and the low-pressure phase of simple cubic polonium A. Zaoui a,∗ , A. Belabbes b , R. Ahuja c , M. Ferhat b a
LGCgE (EA 4515) Lille Nord de France, Ecole Polytechnique de Lille, Université de Lille Sciences et Technologies de Lille, Cité Scientifique, Avenue Paul Langevin, 59655 Villeneuve D’Ascq Cedex, France b LEPM, Département de Physique, Université des Sciences et de la Technologie d’Oran, USTO, Oran, Algeria c Condensed Matter Theory Group, Department of Physics, Uppsala University, Uppsala, Sweden
a r t i c l e
i n f o
Article history: Received 29 December 2010 Received in revised form 3 March 2011 Accepted 4 March 2011 Communicated by P.R. Holland Keywords: Lattice dynamics High pressure Ab initio Polonium
a b s t r a c t Low-pressure structural properties of simple cubic polonium are explored through first-principles density-functional theory based relativistic total energy calculations using pseudopotentials and planewave basis set, as well as linear-response theory. We have found that Po undergoes structural phase transition at low pressure near 2 GPa, where the element transforms from simple cubic to a mixture of two trigonal phases namely, hR1 (α = 86◦ ) and hR2 (α = 97.9◦ ) structures. The lattice dynamics calculations provide strong support for the observed phase transition, and show the dynamical stability (instability) of the hR2 (hR1) phase. © 2011 Elsevier B.V. All rights reserved.
Most elements of the Periodic Table are characterized by their tendency to crystallise either in the face-centred cubic (fcc) (e.g. Cu, Ag, Au . . . ), body-centred cubic (bcc) (e.g. Li, Na, K, Rb . . . ), the hexagonal-closed-packed structure (hcp) (e.g. Re, Os, Ru . . . ), the diamond structure (cd) (e.g. C, Si, Ge, α -Sn), or other complex crystal structures. Exceptionally metallic polonium (Po) element with atomic number 84 is the only case, which crystallizes in the simple cubic structure (cs) at ambient conditions. Polonium was discovered by Marie and Pierre Curie in 1898 [1]. It is the first element in the Periodic Table for which all of its isotopes are radioactive. It is a very rare natural element with very unusual properties, such as its tendency to disperse itself with time. Another its property is a tendency to volatilize at surprisingly low temperature. Moreover, it also has some important biological applications. While polonium is radioactive element, it is a centre of academic interest due to its potential radiological health significance and application as a natural tracer of environmental processes. It can be also used in static eliminators, which are devices designed to eliminate static electricity in machinery. In addition, Po has been investigated as a heat source for thermoelectric power devices for space applications, and it has also been used as a light weight heat source to power thermoelectric cells.
*
Corresponding author. E-mail address: [email protected] (A. Zaoui).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.03.014
Crystal structure of polonium was discovered by Beamer and Maxwell in 1946 [2] during the investigation of the electrical resistively of this element. It exists in at least two allotropic modifications: a low temperature α -phase with a simple cubic lattice, and a high temperature β -phase with a simple rhombohedral (trigonal) structure. Experimental investigations of chemical and physical properties of Po are very difficult due to its strong radioactivity. Recent theoretical calculations [3–9] have addressed the electronic, structural stability and dynamical properties of the sc phase of polonium. To the best of our knowledge no experimental information on the pressure-induced phase transition of Po exists. On the theoretical side, recent calculations [6] predict transformation from a sc to a mixture of two trigonal phases at pressures between 1 and 3 GPa, although the structure of the new phase remains unknown. However, the study presented in Ref. [6] is far to be complete and raises deeper questions: (i) what is exactly the transition pressure, (ii) what is the rhombohedral angle corresponding the this transition, and more fundamentally, (iii) which one of the two obtained trigonal phases is more stable, since both have almost the same total energy. In order to clarify these points, in this Letter we address these issues in detail. All calculations reported here were carried out by means of the plane-wave pseudopotential method, as implemented in the PWscf code [10]. Electron–electron interactions are treated within the local density approximation (LDA) [11], as well as the Perdew– Burke–Ernzerhof (PBE) [12] form of the generalized gradient approximation (GGA). Scalar relativistic and full relativistic approx-
1696
A. Zaoui et al. / Physics Letters A 375 (2011) 1695–1697
Table 1 Structural parameters (lattice constant and bulk modulus) compared to firstprinciples calculations and experimental (Exp.) values.
This study: LDA This study: LDA + SO This study: GGA This study: GGA + SO Pseudopotentiala Flapwb : LDA Flapwb : LDA + SO Exp.c a b c
a (Å)
B (GPa)
3.28 3.32 3.34 3.41 3.28 3.27 3.33 3.345
52.6 42.8 45.6 35 56 57.1 42.3
Ref. [4]. Ref. [6]. Ref. [2].
Fig. 2. Total energy of the hR1 and hR2 phases.
Fig. 3. Phonon dispersions for the sc phase of Po at 0 GPa (a) and 2 GPa (b).
Fig. 1. Total energy along the rhombohedral transformation path.
imations with inclusion of spin–orbit interaction (SO) in ultrasoft pseudopotentials (US-PP) [13] were used. The electronic wave function was expanded with a plane-wave basis set with a kinetic energy cutoff of 60 Ry; the augmentation of charges was expanded up to 400 Ry and a Gaussian smearing of 0.02 Ry has been applied. The k-space integration on the Brillouin zone (BZ) for the self-consistent calculations was calculated with a 12 × 12 × 12 and 8 × 8 × 8 k-point mesh according to Monkhorst and Pack (MP) [14] for the simple cubic (trigonal) phase. The lattice-dynamical properties are calculated using the density functional perturbation theory (DFPT) [15]. In particular, 8 × 8 × 8 k-point mesh of MP was used; and obtained matrices were Fourier interpolated to obtain phonon dispersion curves. Table 1 shows the calculated equilibrium lattice constant, and the bulk modulus of sc Po, together with recent first-principles calculations and experimental data. It is clear that only the current LDA + SO and GGA results are in a good agreement with experimental data, and with the state-of-the-art full-potential linearized augmented plane-wave method (FLAPW) [6]. Further we will mainly discuss the LDA + SO results since SO splitting and relativistic effects are crucial for a correct description of the electronic and dynamical properties of heavy atom Po. The body-centred cubic, and face-centred cubic can be continuously connected with the simple cubic structure by means of
trigonal deformation path. The trigonal structures along this path can be described by a rhombohedral structure (Pearson symbol hR1 [16]), the sc structure can be viewed as a rhombohedron with angle α equal to 90◦ . While the primitive cells of bcc and fcc structures are rhombohedrons with angle α equal to 109.47◦ and 60◦ respectively. Thus, it is possible to generate all these cubic structures from a rhombohedron with the atomic position at (0, 0, 0) by varying the α angle. Fig. 1 shows our calculated total energy as a function of angle for Po at an equilibrium volume. Similar calculations were performed at several volumes covering low pressure of 2 GPa. For P = 0 GPa (V / V 0 = 1), the sc phase is the absolute minimum, while the fcc phase is the absolute maximum, and the bcc phase is the local maximum. This indicates that the sc phase exhibits the lowest total energy as expected and the unstable nature of the bcc and fcc phases for a given rhombohedral distortion. At P = 2 GPa (V / V 0 = 0.96), which corresponds relatively to a low pressure, the situation changes drastically. The sc phase is the absolute minimum, and two new local (absolute) minima appear at angles around 85◦ and 100◦ respectively. Po loses its stable sc structure, leading to a structural phase transition: sc → rhombohedral hR1 (α = 86◦ ) and/or hR2 (97.9◦ ) phases (see Fig. 2) respectively, what is in agreement with static total energy calculations of Legut et al. [6]. In order to estimate the dynamical response of polonium under compression and to check if the structural phase transition found previously is linked to dynamical instabilities of Po, we calculate phonon dispersions of sc phase at equilibrium volume and under compression. The calculated phonon dispersion curves of sc Po at equilibrium volume ( P = 0 GPa) along the high symmetry lines of simple cubic BZ are shown in Fig. 3. The calculated phonon dis-
A. Zaoui et al. / Physics Letters A 375 (2011) 1695–1697
Fig. 4. Phonon dispersion for the trigonal phases hR1 (a) and hR2 (b) of Po at 2 GPa.
persions at P = 0 compare well with those obtained by Verstraete [17]. The most striking features of the phonon dispersion of the sc Po are the phonon anomalies, spectacularly in all directions of the BZ, namely the transverse and longitudinal branch, which exhibit a sudden drop in frequency in the M–R, R–Γ , Γ –M, M– X , and X–Γ directions. The phonon dispersion curve of the sc Po is unique compared to other metals, and constitutes an additional type of anomalies among the class of elemental metal studied until now. The calculations strongly suggest that the observed phonon anomalies of sc Po are the Kohn effect [18] arising from a particular nesting feature of the Fermi surface of the sc Po [19]. The calculated phonon dispersion curves of sc Po at low pressure are shown in Fig. 3, where we notice that strong softening with decreasing volume occurs in [ξ ξ ξ ] and [ξ 00] directions, at volume V / V 0 = 0.96, corresponding to a pressure of 2 GPa. The phonon transverse frequencies become imaginary indicating structural phase instability. Thus, the simple cubic structure of Po is dynamically instable against trigonal distortion. These phonon anomalies are linked naturally to the structural phase transition sc → rhombohedral found previously from total energy calculations. However, we notice that the total energy difference between the trigonal hR1 and hR2 phases is extremely small, not exceeding 8 meV, which is at border of the accuracies of best to date ab initio methods. So in such conditions, where two rhombohedral minima have almost the same energy but very different angles, how can one predict which one is the real angle and which experiment can we follow [20]? This will make predictions at extreme conditions, where experiments cannot be performed, more difficult. Moreover the predicted phase transition can be unphysical, because the predicted phase could be dynamically unstable [21,22]. In order to check the dynamical stabilities of the mixture of the two trigonal structures for low pressure phase transition of sc Po found at 2 GPa, we calculate the phonon dispersions of the rhombohedral (α = 86◦ and α = 97.9◦ ) phases at the same pressure (2 GPa). The
1697
corresponding calculated phonon dispersion curves of the hR1 and hR2 phases along the high symmetry lines of the BZ, are shown in Fig. 4. The hR1 (α = 86◦ ) phase is found dynamically unstable (i.e. imaginary phonon modes), while the hR2 (α = 97.9◦ ) phase presents the absolute minimum and is found dynamically stable (only positive phonon modes). Consequently, the first low pressure phase transition of the sc Po forms the dynamical stable rhombohedral structure hR2 (α = 97.9◦ ). In conclusion, a low pressure phase transition of simple cubic → rhombohedral phase of polonium is fully explored by total energy and lattice dynamics calculations within the density-functional theory. It is predicated a bimodal mixture of two trigonal hR1 (86◦ ) and hR2 (97.9◦ ) phases of Po at 2 GPa. Our phonon calculations confirm this transition pressure. The fundamental question concerning the determination of the real minimum from the two trigonal phases having practically the same total energy is resolved by lattice dynamics calculations, where we have shown lattice dynamics as an efficient tool to determine true thermodynamic minimum. We demonstrate the dynamical instability (stability) of the hR1 (hR2) phases. Experimental works if possible on the dynamical and low-pressure behaviour of sc polonium are welcome, since they can shed a valuable light on the scenarios theoretically outlined in the present work. Acknowledgements Two of us (R.A. and M.F.) would like to thank VR-SIDA (MENA) for financial support. References [1] P. Curie, M. Curie, C. R. Acad. Sci. 127 (1898) 175. [2] W.-H. Beamer, C.-R. Maxwell, J. Chem. Phys. 14 (1946) 569. [3] M. Lach-hab, B. Akdim, D.-A. Papaconstantopoulos, M.-J. Mehl, N. Bernstein, J. Phys. Chem. Solids 65 (2004) 1837. [4] R.E. Kraig, D. Roundy, M.L. Cohen, Solid State Commun. 129 (2004) 411. [5] B.-I. Min, J.-H. Shim, M.-S. Park, K. Kim, S.-K. Kwon, S.-J. Youn, Phys. Rev. B 73 (2006) 132102. [6] D. Legut, M. Friák, M. Šob, Phys. Rev. Lett. 99 (2007) 016402. [7] K. Kim, H.-C. Choi, B.-I. Min, Phys. Rev. Lett. 102 (2009) 079701. [8] M. Šob, D. Legut, M. Friák, Phys. Rev. Lett. 102 (2009) 079702. [9] N.-A. Zabidi, H.-A. Kassim, K.-N. Shrivastava, AIP Conf. Proc. 1017 (2008) 255. [10] S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, http://www.pwscf.org. [11] J.-P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [12] J.-P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [13] A. Dal Corso, Phys. Rev. B 76 (2007) 054308. [14] H.-J. Monkhorst, J.-D. Pack, Phys. Rev. B 13 (1976) 5188. [15] S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Rev. Mod. Phys. 73 (2001) 515. [16] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, vol. 2, Pergamon, Oxford, 1967. [17] M.J. Verstraete, Phys. Rev. Lett. 104 (2010) 035501. [18] W. Kohn, Phys. Rev. Lett. 2 (1959) 393. [19] A. Belabbes, A. Zaoui, M. Ferhat, Solid State Commun. 150 (2010) 2337. [20] S.-I. Simak, U. Haussermann, R. Ahuja, S. Lidin, B. Johansson, Phys. Rev. Lett. 85 (2000) 142. [21] A.-S. Mikhaylushkin, U. Haussermann, B. Johansson, S.-I. Simak, Phys. Rev. Lett. 92 (2004) 195501. [22] V. Ozolins, A. Zunger, Phys. Rev. Lett. 82 (1999) 767.