Next: Computer programs Up: POINT DEFECTS, LATTICE STRUCTURE Previous: Calculation of the shear

# Isothermal-isotension ensemble

An extended ensemble, which was implemented in this project, is the isothermal-isotension ensemble with constant number of atoms (NtT). We study the behavior of vanadium containing self-interstitials and vacancies, at temperatures close to the melting point. In this case, one expects changes of the volume and shape of the sample due to the thermal expansion and point defects. Therefore, in order to describe the system accordingly we have to implement an algorithm, which allows for fluctuation of the shape and volume of the computational cell, and at the same time controls the pressure of the system.

Parinello and Rahman in 1980 [66] invented a new method, in which the shape and volume of the computational cell are dynamical variables. This technique is very helpful in study of the phase transitions. The statistical ensemble designed by Parrinello and Rahman, combined with Nose-Hoover thermostat is identified as an isothermal-isotension (NtT) ensemble.

The realization of this method is in the following way. First of all, we introduce scaled coordinates in addition to the usual ones :

 (11.1)

where the Greek indices are the coordinate indices , the Latin indices count the atoms, and is a transformation matrix.

The volume of the computational box is given by:

 (11.2)

One can introduce a metric tensor by using the matrix:
 (11.3)

or
 (11.4)

where is stands for transpose. A distance between any two atoms are and is simply given by:
 (11.5)

where and are the coordinate indices.

The Lagrangian of the system has to be extended from 3N coordinate degrees of freedom to 3N+9, including the new 9 degrees of freedom of the real matrix :

 (11.6)

where the extra 'kinetic' energy and the corresponding 'potential' energy are added to the original Lagrangian to account for the new decrees of freedom.
The 'kinetic' energy term is given by:
 (11.7)

where is a parameter, its physical meaning will be explained latter.
The 'potential' energy term is:
 (11.8)

here is volume of the system and its pressure, which can be obtained according to the virial theorem:
 (11.9)

where is the force acting on atom due to the atom , and is the vector which connects the atoms and , and stands for ensemble averaging.

The modified Lagrangian is written in the terms of new variables:

 (11.10)

here, for the sake of simplicity we omit for a while the terms corresponding to the Noose-Hoover thermostat. The equations of motion are obtained in the usual way:
 (11.11)

 (11.12)

Therefore we obtain:
 (11.13)

 (11.14)

where and ,
the is the stress tensor which describes anisotropy of the solid.

The stress tensor is defined in the following way [67]:

 (11.15)

here is th component of the atom velocity,
and is th component of the vector connecting atoms and .
The hydrostatic pressure components are the diagonal elements of the stress tensor:
 (11.16)

The stress tensor can be rewritten in the terms of the scaled variables :
 (11.17)

here is the th component of the force acting on the atom due to the atom .

The system is driven by dynamic imbalance between the applied external stress and the internally generated stress. The parameter corresponds to the relaxation time of recovery from the imbalance between the external and the internal stress. It can be shown that the value of does not influence the ensemble average [67].

The equations of motion generated by the Lagrangian can be effectively solved numerically. One has take into account that the Parinello-Rahman model is represented by a system of coupled ordinary differential equations for scaled coordinates and for the elements of the matrix. These equations have to be solved simultaneously.

This can be done using the predictor-corrector (PG) method, which includes the following steps:
1.) New scaled positions and velocities are predicted on the basis of the old ones,
including the accelerations :

 (11.18)

 (11.19)

here are the (PG) prediction coefficients.
2.) The force acting on the atom is calculated using the predicted coordinates :
 (11.20)

3.) The temperature of the system and the stress tensor are evaluated at :
 (11.21)

 (11.22)

4.) The values of the elements of the matrices are predicted:
 (11.23)

 (11.24)

5.) The values of the matrix is evaluated at using the predicted values of the matrix according to:
 (11.25)

6.) The acceleration of of the atom is calculated:
 (11.26)

if we take into account the Noose-Hoover thermostat:
 (11.27)

 (11.28)

7.) Now the values of and are corrected:
 (11.29)

 (11.30)

here are the (PG) correction coefficients.
8.) Finally, the cycle is completed by correction of positions and velocities of all atoms:
 (11.31)

 (11.32)

9.) Go to the step 1.
The structural phase transitions can be studied using this algorithm. We used this algorithm to find the volume and the shape of the sample under the zero external pressure, close to the melting point .

Next: Computer programs Up: POINT DEFECTS, LATTICE STRUCTURE Previous: Calculation of the shear
2003-01-15