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Pressure induced phase transition in tin: Ab-initio calculations

First principles total energy calculations as a function of unit cell volume have been performed on α (diamond structure), β (double bct), bct, bcc and hcp structures of tin. The comparison of total energies of these structures at various volumes suggests high pressure structural transitions of α → β → bet → bct at pressures of ~ 0.15 GPa, 2.7 GPa and 28 GPa. Also, 300 K isotherm upto maximum pressures of ~ 400 GPa has been determined for this metal. Using theoretical isotherm, shock Hugoniot of tin has been derived. Further, theoretically evaluated volume dependent Gruneisen parameter in conjunction with Lindeman melting law has been used to determine the pressure dependent melting of tin. The intersection of the theoretical melting curve with theoretically determined temperature rise with shock pressure reveals that this metal will melt at shock pressure of ~ 38 GPa (temperature ~ 1428 K) in good agreement with experimental value of 39 GPa (1550+100 K).


Introduction
The group IV A element tin exists in double bct (β) structure at ambient conditions. In past, a number of investigations have been performed on this metal to examine its structural stability under pressure [1][2][3][4][5][6][7]. For example, in a high pressure x-ray diffraction measurements up to 10 GPa on tin Bernett et. al. [1] reported a β → bct transition at 9.8 GPa with the c/a ratio measured to be ~ 0.91. Subsequently, in high pressure energy dispersive x-ray diffraction study Olijnyk and Holzapfel [2] observed the β → bct → bcc structural transitions at 10 GPa and 35 GPa. In another high pressure x-ray diffraction study using diamond anvil cell (DAC) Liu and Liu [3] reported β → bct → bcc phase transitions at ~ 9.7 GPa and 40 GPa respectively. Later, Desgreniers et. al. [4] in a high pressure study up to 120 GPa reported the same structural sequence and demonstrated that the bcc phase remains stable up to 120 GPa. In a theoretical study, using ab-initio pseudopotential method Corkill et. al. [5] reported the structural transition sequence of α (diamond structure) → β → (bct, bcc) → hcp at pressures of 0.8 GPa, 23.5 GPa and 90 GPa, respectively. The β → bct transition pressure of 23.5 GPa reported in this study does not agree with experiments. Also, the bct → hcp transition predicted by these authors is not observed up to 120 GPa (the maximum pressure reached) in experimental study by Desgreniers et. al. [4]. In another study based on full potential linear muffin-tin orbital (FP-LMTO) calculations Christensien and Methfessel [6] reported α → β → bct →bcc transition sequence at pressures of 0.2 GPa, 10.4 GPa and 50 GPa, respectively. However, subsequently, Chun et. al. [7] using Cambridge Serial Total Energy Package (castep) which is again a total energy pesudopotential calculation method suggested the high pressure structural sequence of α → β → bct →bcc→ hcp in tin at pressures of 0.7 GPa, 12 GPa, 45 GPa and 160 GPa. Recently, using non local ultra soft pesudopotential method within generalised gradient approximations (GGA) Shouxin Cui et. al. [8] have determined the enthalpies of various high pressure phases of this metal and reported the α → β → bct →bcc transition pressures to be 1.9 GPa, 13 GPa and 32 GPa, respectively. Further, in contrast to Chun et.al. [7], these authors do not find bcc→ hcp up to maximum theoretical pressure of 200 GPa.
In addition to the pressure induced structural phase transitions, the room pressure melting and pressure effect on melting temperature of tin has also attracted the attention of researchers [9][10][11]. For example, Ravelo and Basakes [9] in their molecular dynamics calculations determined thermodynamic properties of tin at room pressure. On the basis of the calculated free energy differences between α and β, and between β and liquid phase, these authors reported the room pressure α → β and β to liquid phase transition at 283 K and 453 K, respectively. Similarly, through optimal potential method, which is a combination of both the first principle and classical molecular dynamics, Bernard and Maillet [10] determined the room pressure melting temperature of tin as 450 K as compared to experimental value of 510 K. Further, they determined the shock Hugoniot and pressure effect on melting up to 200 GPa and 50 GPa, respectively. The pressure dependence of melting temperatures determined by these authors is not in agreement with the melting line measured through DAC experiments [11]. In past experimental studies under shock compressions have also been reported on this metal [12][13][14][15]. Though a large body of Hugoniot data have been generated by these experiments, the bct to bcc phase transition have remained undetected as the volume change associated with this transition is very small ~ 0.8 % [3]. Recently, sound velocity measurements have been performed using velocity interferometer system for any reflector (VISAR) in plate impact experiments carried out in reverse impact geometry configuration [16]. In this study Hu et. al. [16] have reported shock induced bct to bcc and bcc to melting transitions as 342 GPa and 39 2 GPa, respectively.
In the present work, we have performed first principles total energy calculations as a function of hydrostatic compression on α, β, bct, bcc and hcp phases of tin and analysed its structural stability under pressure. Also, we have determined the 300 K isotherm up to ~ 400 GPa after adding the finite temperature contributions to the total energy calculated at 0 K. Further, the Hugoniot of tin has been derived from the theoretical isotherm up to ~380 GPa. Additionally, using theoretically evaluated volume dependent Gruneisen parameter in Lindeman melting law, the pressure dependent melting of tin has been determined. The shock induced melting of tin has been determined from the theoretical melting curve and temperature rise along Hugoniot.

Theoretical Procedure
The total energy calculations have been performed as a function of hydrostatic compression on α, β, bct, bcc and hcp phases of tin employing full potential linearised augmented plane wave method (FP-LAPW) method (WIEN 97 package) [17] within GGA [18] for exchange -correlation energy. The α phase is diamond structure and the β phase is tetragonal distortion of the α phase having double bct structure with c/a ratio 0.546. The calculation of α and β phase were performed in double bct cell having four atoms located at (000), (0 ½ ¾), (½ 0 ¼) and (½½ ½) [19] or equivalently at (0 ¾ 1/8), (½¾ 3/8) (½ ¼ 5/8) and (0 ¼ 7/8) (site 4a). It may be noted that for c/a =1.414, the cell becomes α structure. Similarly, the calculations of bct and bcc phases were carried out in bct cell. The bct phase has c/a ratio 0.91. The bct cell for c/a =1.0 corresponds to bcc structure. The muffin tin radius used for all the structures is 2.0 a.u. and the plane wave cut-off parameter RK max is fixed at 10. Before doing the calculations a k point convergence test is carried out at ambient volume. For this we varied the number of k points in the Brillouin zone from 2000 to 10000 and found that 5000 k points are sufficient to obtain energy convergence within 0.1 mRy/atom. Therefore, for further calculations a grid of 5000 k points is used for sampling of the full Brillouin zone. The structural stability analysis at 300 K is carried out by comparing the total free energy of α, β, bct, bcc and hcp phases under hydrostatic compression. The free energy (F) at T = 300 K is determined using the expression Where the entropy contribution 'TS(V)' and the total energy E(V,T) at T=300 is determined using the procedure followed by Walsh et. al. [20] and Godwal and Jeanloz [21]. The 300 K isotherm is determined using following expressions [21,22] ) , T) is thermal lattice contribution including zero point vibration energy and E e (V,T) = ½ T 2 is thermal electronic contribution to total energy. The various terms in these equations were determined using the procedures described by Godwal and Jeanloz [21] and Gupta et. al. [22].
The shock Hugoniot of tin is determined using the Eqs (2) and (3) together with the Rankine -Huginot equation. The pressure effect on melting is determined for bcc phase using the procedure followed by Soma et. al. [23]; according to which the melting temperature at any compression The variation in melting temperature as a function of compression is determined using theoretically determined (y). The melting temperature is plotted as a function of compression or corresponding pressure. Finally, temperature rise along the Hugoniot and melting temperature as a function of pressure are plotted together and crossover of the two curves yielded the shock pressure and corresponding temperature at which melting of Tin occurs under shock compression.   The free energy as a function of volume has been determined for α, β, bct, bcc and hcp phase of tin. The equilibrium lattice parameters determined for α, β and bct structures from the present calculations at ambient conditions have been compared with the available experimental data [1,3,24] in table 1. Therefore, based on this it is not possible to say whether the transformed structure will be bct or hcp. Since the pressure induced transformations are diffusionless in nature [25] it is possible to identify the path of various atoms involved in the transition. Energetically, the path taken by the atoms is the one which introduces the least activation barrier for the transition. This requires an extensive calculation in which the total energy surface is constructed in terms of all the independent variables describing the transformation. Qualitatively, however, it is expected that the path of the transition is the one which involves the least rearrangements or smallest displacements of atoms. To qualitatively examine which path of transition is more likely, we have estimated the atomic displacements required to form bct and hcp cells from β cell. As displayed in figure 2  632. This analysis suggests that the rearrangement of atoms required for formation of hcp directly from β is more than that for formation of bct from β indicating that formation of bct phase is more likely. Upon further compression, the bct phase transforms to bcc structure at V~ 20.55 (A 0 ) 3 /atom with corresponding hydrostatic pressure of ~ 28 GPa. In table 2, we have compared the theoretically determined transition pressures with the experimentally measured one.  bct→bcc 50 GPa [6] 45 GPa [7], 32 GPa [8] 28GPa 35 [2], 40 GPa [3] 45GPa [4] bcc→hcp 90 GPa [5] 160 GPa [7] -----  Figure 5. Theoretical melting line and temperature rise along Hugoniot. Also shown are DAC melting data [11] In figure 3 we have displayed the theoretically determined 300 K isotherm along with experimental data. The theoretical isotherm agrees well with experimental data [2][3][4]. Figure 4 shows the Hugoniot of tin derived from theoretical 0 K isotherm along with available experimental data from various sources [12][13][14][15]. The agreement of the Hugoniot with experimental data up to ~ 230 GPa is quite well, however, it starts deviating systematically from experimental data above this pressure. This systematic deviation of theoretical Hugoniot from experimental data above ~ 230 GPa could be due to the fact that the melting (or vaporization) of the bcc phase is not considered in our calculations.

Results and Discussions
Our theoretically determined pressure dependence of melting point along with the theoretical shock temperatures is shown in figure 5. Also shown in the figure is experimental melting data determined in high pressure high temperature diamond anvil cell (DAC) experiments [11]. The experimental data show a good agreement with the theoretical melting line. This indicates that the Debye model along with Lindeman law gives correct description of melting behaviour of tin at high pressure and high temperature. The room pressure melting temperature required for determination pressure dependence of melting is taken to be 505 K from literature [26]. The crossover of the melting and shock temperature curves suggests that tin will melt under shock compression of ~38 GPa, which is in good agreement with the experimental value of 39 2 GPa [16].