Ramesh C. Shukla

Anharmonic contribution to the Debye-Waller factor for copper, silver, and lead.

Using high intensity ({similar_to}70 Ci) {sup 183}Ta Moessbauer sources, we have measured the elastic scattering fraction values, {ital scrF}, and the relative integrated scattering intensities for the (200), (400), (600), and (220) Bragg planes of copper and silver single crystals; and for the (200), (400), and (600) reflections of a lead single crystal. The experiments were done as a function of temperature from 82 K to a high temperature of 1086, 1211, and 507 K for Cu, Ag, and Pb, respectively. The {ital scrF} values were found by Moessbauer line-shape studies, and were used to correct the measured integrated intensities for thermal-diffuse scattering, so that accurate Debye-Waller factors (DWF`s) could be evaluated. The measured DWF`s for Cu, Ag, and Pb each have a significant anharmonic contribution at about 50% of the melting temperature ({ital T}{sub {ital m}}). Contrary to what Martin and O`Connor have reported for copper, we observed no {ital Q}{sup 4} contribution to the DWF within our experimental errors, which we estimate to be smaller than those reported by Martin and O`Connor.

Derivation of the self-consistent phonon theory from Zubarev type Green's function

We have presented a new method for the derivation of the Helmholtz free energy (F) of an anharmonic crystal from the Zubarev type Greens function. The Hamiltonian (H) employed in the derivation contains the contributions from all the even terms of the Taylors expansion of the crystal potential encrgy. In the language of perturbation theory (PT) these are essentially all the first order PT contributions summed to infinity to the free energy, the self-energy of the Greens function, and the renormalized phonon frequencies. The self-consistency condition arises because in evaluating the correlation functions from the Zubarev type Greens functions the full Hamiltonian is required instead of the usual harmonic Hamiltonian. The final equations which determine F and the self-consistent phonon frequencies are shown to be identical to those of the first order self-consistent phonon (SC1) theory.

Higher-order perturbation theory for the thermodynamic properties of a solid with a truncated potential energy expansion

A perturbation theory (PT) is developed in the classical limit which is based on an infinite series of diagrams composed of the loops and bubbles arising from the first- and second-order matrix elements of the PT, respectively. This theory leads to a closed form expression for the free energy, which on expansion gives an infinite power series in the temperature. Results from this theory are obtained for a Hamiltonian in which the Taylor expansion of the potential energy is truncated at the quartic term. These results are compared with results of finite summation versions of the theory up to O(λ8), with results of standard PT of O(λ2) and O(λ4), and with results of molecular dynamics (MD) simulations carried out for the same potential energy surface (i.e., the potential energy expansion truncated at the quartic term). The results show that the theory which includes all powers of temperature gives better agreement with the MD results throughout a wide temperature range than does the standard PT of O(λ2) and...

Debye-Waller factor in Cu : a Green's function approach

Abstract We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Greens function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...

Molecular dynamics simulations of the effects of truncation of the Taylor expansion of the potential energy on the thermodynamic properties of a crystal

Molecular dynamics simulations are carried out on a Lennard‐Jones crystal, for the potential energy surfaces generated from the full Hamiltonian and the Taylor expansion of the potential energy truncated at the quartic term, to determine the accuracy of the quartic truncation with regard to the thermodynamic properties of a crystal. The results show that the errors arising from the quartic truncation become significant only for temperatures T≳0.2Tm, and are only on the order of 5% at T=0.8Tm, where Tm is the melting temperature. The quartic truncation represents a significant improvement over the quadratic (harmonic) truncation, and the errors associated with the quadratic truncation are decreased by 75%. The sources of error in the λ2 perturbation theory are investigated; the errors are found to arise from the truncation of the potential energy expansion at low temperatures, and primarily from the truncation of the perturbation expansion at high temperatures.

Green's function and atomic mean-square displacement: Phonon shift and width contributions

Abstract We present a derivation of the finite-temperature expressions for the lowest-order anharmonic (λ2) contributions to the atomic mean-square displacement (MSD) from the Greens function (GF) method and show that there is no contribution to MSD from the polarization mixing in the self-energy of the GF. These contributions arise from the cubic- and quartic-phonon shifts and the cubic-phonon width for a phonon mode qj where q is the wave-vector and j is the polarization index. Since the present MSD expressions are valid for all temperatures they can be used in numerical evaluation of the λ2 contributions for Ne and other quantum crystals where the high-temperature expressions are not valid. In the high-temperature limit the total MSD from the cubic shift and width agrees with the cubic contribution to MSD derived in the classical limit by Maradudin and Flinn (MF) and the cubic MSD derived in the ω → 0 limit of the self-energy of the GF by Shukla and Hubschle (SH). The quartic contribution to MSD in th...

Atomic mean-square displacement in fcc metals: Repulsive potentials

We present a method for the calculation of the atomic mean-square displacement (u2) of an anharmonic crystal from potentials involving repulsive interactions. The quasiharmonic and the lowest-order cubic and quartic anharmonic contributions to (u2) are evaluated from the knowledge of seven Brillouin zone (BZ) sums which are tabulated in the interval -0.1 a1 0.0. The parameter a1 characterizes the volume dependence of the BZ sums and is negative for repulsive potentials. All the BZ sums are evaluated in the limit L, where L is the step length from the origin to the boundary of the BZ. The method is applicable to a nearest-neighbour central force model of the fcc lattice, and it can be extended to a bcc lattice. We present two applications of the method. One is the calculation of (u2) from the r-1 2 repulsive potential, and our results are in good agreement with those obtained by Monte Carlo methods. The other is to the nearest-neighbour Born-Mayer potential with a volume-dependent e ective coe cient alpha....

Atomic mean-square displacements in f.c.c. metals

Abstract In two recent letters, one by Zoli in 1991, and the other by Schober in 1992, the evaluation of the quasiharmonic and anharmonic contributions to the atomic mean-square displacement (MSD) for f.c.c. metals has been discussed. In Zolis work, the difference in the two contributions is found to be 91%. Schober, on the other hand, has not evaluated the explicit anharmonic contribution to MSD. The huge difference in Zolis work is shown here to be due to an inaccurate evaluation of the explicit anharmonic contribution to MSD. A proper self-contained method as presented here, which employs the same model in the quasiharmonic and anharmonic calculations of MSD or Debye-Waller factor, indeed shows that the two contributions differ from each other by 10–15% depending on the temperature. Larger differences exist at higher temperatures. Some numerical results are given for a model of the f.c.c. lattice, namely a nearest-neighbour central force model employing a Lennard-Jones potential (applicable to rare-g...

Higher-order anharmonic contributions to the Debye-Waller factor

Abstract We have presented a detailed analysis of the higher-order anharmonic contributions to the average square atomie displacement (u 2) or Debye-Waller factor by the diagrammatic method and the Zubarev type Greens function method. The Hamiltonian employed in the detailed analysis contains the anharmonic terms up to O(λ4)that is the cubic, quartic, quintic and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to (u 2) of O(λ2) and O(λ4), presented in the diagrammatic language, arise from the first-, second-, third- and the fourth-order perturbation theory. This analysis establishing interrelationships between the Helmholtz free energy (F), self-energy and (u 2) reveals that a total of 16 diagrams contribute to (u 2) for a Bravais lattice or a lattice with a basis where every atom is at a site of inversion symmetry. A simple prescription is given for the derivation of (u 2) diagrams from the F diagrams. Out of 16 diagrams, two are of O(λ2) and 14 of O(λ4), where λ is a perturbation expansion parameter. It is also shown that at least up to O(λ4) the contributions to (u 2) from nine out of 16 diagrams are included in the Greens function iterative solution if the latter is generated from the anharmonic Hamiltonian containing the cubic and auartic terms. Exact and approximate numerical magnitudes for all the diagrams are obtained for a nearest-neighbour 6-12 Lennard-Jones fec solid by extrapolating the Brillouin zone (BZ) sums to the q → 0 limit. A disproportionate contribution to the BZ sums comes from this region of q space in the calculation of (u 2) In the context of (u 2) the importance of the various self-energy diagrams involving the same type of anharmonic vertices is established. It is shown that there exists a heavy cancellation among these 14 diagrams; nevertheless their total contribution to (u 2) is sufficient to bring it in almost exact agreement with the classical Monte Carlo results, which are also obtained for the same model.