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PROJECT TOPIC- FORTRAN 77 IMPLEMENTATION OF THE TRANSITION METAL MODEL POTENTIAL (TMMP) CODE: AN APPLICATION TO FCC-COBALT

PROJECT TOPIC- FORTRAN 77 IMPLEMENTATION OF THE TRANSITION METAL MODEL POTENTIAL (TMMP) CODE: AN APPLICATION TO FCC-COBALT

 

ABSTRACT

A Transition Metal Model Potential (TMMP) code originally written in FORTRAN IV and designed for mainframes was modified and upgraded to a FORTRAN 77 code and adapted to run on a Personal Computer (PC). After verification, the modified code which was found to be of high integrity was used to compute the phonon frequencies of a Face-centered cubic (FCC) cobalt crystal along lines of high symmetry in the crystal, namely, [001],[011] and [111] directions. The calculated phonon frequencies were plotted against the high symmetry directions to form phonon dispersion curves (PDCs), comparing experimental (measured) results with theory (the calculated). The phonon dispersion curves were without anomalies, and a high degree of agreement between theory and experiment was observed, corroborating earlier research findings on the nature of PDCs of transition metals with FCC structure. The highest value of calculated phonon frequency of 8.6040 THz was recorded at the L point of the Brillouin zone of the FCC crystal; while the lowest, 0.0000 THz, was recorded at the center.

CHAPTER ONE
GENERAL INTRODUCTION AND REVIEW

1.01 GENERAL INTRODUCTION:

All phases of matter are made up of atoms which are in constant motion even at absolute zero temperature (Kittel, 1996). In crystalline solids at finite temperatures, this atomic motion is reduced to vibrations of the atoms about their mean equilibrium lattice positions. The magnitude of these vibrations is dependent on temperature in the order of kBT (Born and Huang, 1954). Since crystalline solids have symmetry, these thermal vibrations become superimposed and can thus be analyzed in terms of the collective quantized modes of atomic motion (Kazanç and Özgen, 2008). These collective quantized modes of vibration are called phonons, each having an allowed wavelength and amplitude (Kittel, 1996). It is the study of these vibrations, which constitutes lattice dynamics (Kaxiras, 2003).
The theory of lattice dynamics which originated from the fundamental papers of Born (Born, 1912), von Karman (Karman, 1913) and Einstein and Debye (Brüesch, 1982; 1983), showed that the heat content of solids is dependent on the collective quantized modes of lattice vibrations associated with the small displacements of the basis ions from their equilibrium sites. Therefore, lattice dynamical studies generally seek to understand fully the nature of inter–atomic forces in solids, and further to apply the understanding to related properties of crystalline solids (Amar et al, 1999).

Some of these properties include specific heat capacity, electron-phonon interaction in transport theory and superconductivity, thermal conductivity in insulators, crystallographic phase transitions, etc (Animalu,1977; Guo et al, 2001). Hence, lattice dynamical studies form a major component of research work done to access the properties of the condensed state. In the theory of lattice dynamics, there are two normal modes of lattice vibration, namely, the longitudinal and the transverse modes, defined in terms of the direction of propagation of lattice waves with respect to the source of vibration (Ghatak and Kothari, 1972).

The determination of these normal modes presents a major problem in lattice dynamics. However, it is well known that the frequencies of these normal modes factor out along lines of high symmetry in crystals (Taura and Duwa, 2000). These phonon frequencies, w, are written as a function of wave-vectors, k (Srivastava, 1990). The relationship between w and k , (w = w( k )), is called phonon dispersion relation and it is often times presented graphically as a plot of w against k , known as phonon dispersion curves (PDCs).
The phonon dispersion relation is a powerful tool in lattice dynamical studies, since it reveals the nature of crystal vibrations. Phonon dispersion relation in crystals may be determined either experimentally or theoretically. Most experimental techniques determine the frequencies of lattice vibrations by inelastic neutron scattering experiments (Shapiro and Moss, 1977; Kazanç and Özgen, 2008), or by thermal diffuse x-ray scattering experiments (Holt et al, 1999).

Theoretical techniques focus on solving the Schrödinger wave equation (SWE) for a given lattice system and deriving the force constant (i.e. the second derivative of the total crystal potential energy with respect to atomic separation) of the system, from which the phonon frequencies are calculated by generating and using the corresponding dynamical matrix. The dynamical matrix is the Fourier transform of the force constant (Ruf, 2001; Bencherif et al, 2008) The theoretical approaches are divided into two broad groups, namely, the empirical (or phenomenological) and the ab initio (or first principles methods).

PROJECT TOPIC- FORTRAN 77 IMPLEMENTATION OF THE TRANSITION METAL MODEL POTENTIAL (TMMP) CODE: AN APPLICATION TO FCC-COBALT

 

The empirical methods make use of adjustable parameters to fit experimental data to constructed empirical models. On the other hand, ab initio methods have no need for empirical fitting parameters, but they generally employ a variational approach to calculate the ground-state energy of a many-body system from which the force constant and the associated dynamical matrix are obtained for use in computing the phonon frequencies (Ostanin and Trubitsin, 2000).

The major shortcomings of the empirical approaches are that they are not necessarily applicable to all types of systems of solids, and their parameters do not contain conceptual simplicity (Srivastava, 1990). On the other hand, the ab initio methods are only suitable for investigating systems in their ground-state configurations, thus unsuitable for investigating excited systems (Dreizler and Gross, 1990; Martin, 2004).
In this work, the pseudo-potential approach employing the Transition Metal Model Potential (TMMP) due to Animalu (1973b), was employed. This method has proved very efficient in the computation of phonon frequencies of transition metals (Animalu, 1973b; Oli and Animalu, 1976; Okoye and Pal, 1993).

The theoretical formulation of the TMMP method has been given in chapter two of this report. Similarly, for reasons of completeness and current update, a similar treatment has been given to the Density Functional Theory (DFT) approach, which is the state-of-the-art ab initio method employed in lattice dynamical studies.
The theoretical computation of phonon frequencies in 3-Dimensional crystals is a very tedious mathematical engagement. Considering this level of rigour, computer aided computations of phonon frequencies are increasingly frequent (Papanicolaou et al, 2000; Kazanç et al, 2006; Vora, 2008). Recently, Taura and Duwa (2000) modified a FORTRAN 66 code written by Keeler (1980) for mainframe computers into a FORTRAN 77 code for use with personal (micro) computers (PCs) in computing phonon frequencies in the simple cubic, body-centered cubic and face-centered cubic crystal structures. The current work is fashioned along this line, with three clear objectives aimed at:
(1) Modifying and adapting an existing TMMP code written in FORTRAN IV by Animalu (1973b) for the mainframe computers to run as a FORTRAN 77 program code on a personal (micro) computer (PC);
(2) Using the modified program code to calculate the phonon frequencies for FCC cobalt crystal along the high symmetry directions; and
(3) Comparing the calculated phonon frequencies with (measured) experimental results.
This work was motivated by some factors. First, the success recorded by researchers in the use of TMMP in the lattice dynamical studies of FCC transition metals; second, the availability of the TMMP-based code; third, the need to have the TMMP-based code in a format suitable for implementation on a portable machine; and lastly, the need to compare experiment with theory using the TMMP for cobalt, which hitherto was unavailable due to the problem of the martensitic transformation of Cobalt at high temperatures (Strauss et al, 1996).

1.02 REVIEW OF THEORETICAL DEVELOPMENTS:

The pioneer work in the theoreti

PROJECT TOPIC- FORTRAN 77 IMPLEMENTATION OF THE TRANSITION METAL MODEL POTENTIAL (TMMP) CODE: AN APPLICATION TO FCC-COBALT

 

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