Theoretical approach to predict the phase transition of NaH under pressure

This paper is dealing with the high pressure behavior of alkali halide (NaH) and a structural phase transition from NaCl to CsCl phase is predicted under high pressure. We have adopted an effective interionic interaction potential, which includes the long range Coulomb, van der Waals (vdW) interaction and the short-range repulsive interaction upto second-neighbor ions within the Hafemeister and Flygare approach. The thermodynamic requirement, for the equality of free energies, at a phase boundary, suggests that the transition pressure of NaH occur at 37 GPa. The transition involves the coordination number increasing from 6 for the NaCl-type structure to 8 for the CsCl-type structure.


Introduction
The alkali-metal hydrides (CsH, RbH, KH, NaH, and LiH) represent a series of partly ionic materials whose high-pressure behavior is expected to parallel that of the alkali-metal halides. The alkali metal hydrides are ionic compounds which can be treated as analogues to alkali metal halides. In alkali matal hydrides hydrogen considered as the first member of the halogen group of elements and formally it behaves as an H -ion. Support for this view can be obtained from the observed crystal structures of these compounds which are typical for ionic systems (NaCl and CsCl). But at normal condition alkali metal hydrides have NaCl or B1-type crystal structure, while the alkali metal halides can have both NaCl or B1-type structure or CsCl or B2 type crystal structure [1]. Pressure is most important thermodynamical quantity to bring out the mechanical properties of solids and alloys. The phase change in solids at high pressure usually involves a change in crystal structure. Co ordination number is important in phase transition due to high pressure. This is because thermodynamically the molar volume of high pressure phase must be smaller than low pressure phase. Hochheimer et al. [2] has reported the transition pressures for CsH, RbH, and KH, as well as the equation of state of NaH to 28 GPa. The change in volume can be estimated using known equation of state (EOS) [3]. Bashkin et al. [4] have pointed out, using a hard-sphere ion model, that the B1 to B2 transition is thermodynamically possible in NaH, KH, RbH, and CsH compounds. The Duclous et al determined the specific volumes of each phase by summing the metal and hydride ion radii, which depend on the structure and coordination number, to determine the B2 lattice constant [5].
In particular, NaH show a structural phase transition from NaCl to CsCl. For example, Ahuja et. al. [6] calculated transition pressure for NaH to be 37 GPa. In search of B1-B2 transition Duclous et al. [5] submitted sodium hydrides to pressure as high as 54 GPa which is in good agreement with the experiment value of around 51 GPa. To estimate the possibility of the NaCl to CsCl transition in NaH under pressure, we consider the relationship of the Gibb's free energy of both phases. The phase transition pressures and the volume collapses were obtained from the effective interionic potential model (EIOP), which incorporates the long range Coulomb, the short-range vdW attraction and the overlap repulsive interactions operative up to the second neighbor ions within the Hafemeister and Flygare framework. The vander Waals coefficient are calculated by the Slater Kirkwood variation method as a first step. Then, we evaluate volume collapse.

Theory and method of computation
The effective interionic potential is expressed as Here, long-range Coulomb is represented by first term, second term correspond to Hafemeister and Flygare [7] form of short-range repulsive energies and van der Waals multipole are represented by third and fourth terms respectively. The symbols cij and dij represents the van der Waals coefficient and ij being the pauling coefficients, respectively. Zm is the modified ionic charge and parametrically includes the effect of Coulomb screening effect, b and  are short-range parameters. Thus, the effective interionic potential contains only three free parameters (Zm, b and ) which can be determined from the crystal properties [8]. Further, an isolated phase is stable only when its free energy is minimum for the specified thermodynamic conditions. As the pressure or any other variable acting on the systems is changed, the free energy of the system changes smoothly and continuously. A phase transition is said to occur when the changes in structural details of the phase are caused by such variations of free energy. The test materials transform from their initial B1 to B2 structure under pressure. The stability of a particular structure is decided by the minima of Gibbs's free energy, G = U +PV -TS. Here U is internal energy, which at 0 K corresponds to the cohesive energy, S is the vibrational entropy at absolute temperature T, pressure P and volume V. The Gibbs's free energies GB1(r) = UB1(r) + 2Pr 3 for NaCl (B1) phase and GB2(r') = UB2(r') + [8/3√3]Pr' 3 for CsCl (B2) phase become equal at the phase-transition pressure P and at zero temperature i. e., G (= GB1 -GB2). Here, UB2 and UB1 represent cohesive energies for B1 and B2 phases, and are ), ).
(3) Here r and r' are nearest-neighbor (nn) separations corresponding to NaCl and CsCl phases, respectively.
Here, cij and dij being the vdW coefficients due to dipole-dipole (d-d) and dipole-quadrupole (d-q) interactions. We follow SKV method [9] to derive cij and dij as  Here, me being the electron mass. αi is the electronic polarizability and Ni denotes the effective number of electrons of the ith ion, respectively. The values of the overall vdW coefficients (C and D) are obtained following Tosi approach [10] in terms of appropriate lattice sums [S6(0), S6(r), S8(0) and S8(r)].  Table 2. Crystal data and model parameters for NaH compound.  Table 3. Calculated (reported) transition pressures and volume collapses in NaH compound.

Results and discussions
The effective interionic potential is constructed in a hierarchical and easy generalizable manner. We have undertaken such structural and elastic properties in an ordered way. For such purposes we have then three free parameters, namely, modified ionic charge, range and hardness parameters (Zm,  and b). To estimate the free parameters, we first deduce the vdW coefficients from the Slater-Kirkwood variational method [11] and are listed in table 1. It is perhaps worth to remark that we have deduced the values of free parameters modified ionic charge (Zm), range () and hardness (b) from the knowledge of equilibrium distance and the bulk modulus following the equlibrium conditions [12]. The input data along with their relevant references and the model parameters for NaH compounds are given in table 2.  In an attempt to reveal the structural phase transition of the test materials, we minimise the Gibbs's free energies GB1(r) and GB2(r') for the equilibrium interatomic spacing (r) and (r'). The Gibbs's free energy difference G [= GB2(r) -GB1(r ' )] have been plotted as functions of pressure (P) in figure 1 using the interionic potential discussed above for NaH.
Let us now estimate the values of relative volumes associated with various compressions following Murnaghan equation of state.
V0 being the cell volume at ambient conditions. The estimated value of pressure dependent radius for both structures, the curve of volume collapse with pressure to describe the phase diagram is exemplified in figure 2 for NaH. It is noticed from the graph that our access has predicted correctly the relative stability of competitive crystal structures, as the values of G are positive. The magnitude of the discontinuity in volume at the transition pressure is obtained from the phase diagram and their values are tabulated in table 3, which is in fair good agreement with those revealed from experiments [4,6,13] and other theoretical works [14,15].

Conclusions
A high pressure structural phase transition from NaCl (B1) to CsCl (B2) in alkali metal hydride (NaH) was predicted following the developed EIOP model. It has been found that the B1 type structure of NaH is unstable at after pressure 37 GPa. This was established by calculating the Gibb's free energies in both the structures under consideration. Further, the associated volume collapses at the phase transitions have been found to be 4.3% for sodium hydride.