Inclusion of Higher Order Anharmonic Contributions in Self-Consistent Phonon Theory

The ansatz method of infinite summation of higher order diagrams given in Shukla and Cowley, Phys. Rev. 58, 2596, (1998), is extended to the self-consistent phonon theory. We demonstrate the high accuracy of this approach with respect to the first order self-consistent (SC1) and improved self-consistent (ISC) phonon theories, by comparing the results from the ansatz method with their exact counterparts. The ISC theory is then extended to include the remaining diagrams of $O(\lambda^4)$, which could not be included in its earlier formulation. This makes the ISC theory consistent, at least to $O(\lambda^4)$. The new ISC theory offers a substantial improvement over the current ISC theory. The results of the equation of state for a face centered cubic (fcc) nearest neighbor interaction Lennard-Jones solid from the new ISC theory are shown to be in excellent agreement with the results of the classical Monte Carlo method also obtained for the same model.

Phys. Rev. B, 1, August, 2000

Higher-order anharmonic contributions to the Debye-Waller factor

We have presented a detailed analysis of the higher order anharmonic contributions to the average square atomic displacement $\langle u^2 \rangle$ or Debye Waller factor by the diagrammatic method and the Zubarev type Green's function method. The Hamiltonian employed in the detailed analysis contains the anharmonic terms up to $O(\lambda^4)$, i.e. the cubic, quartic, quintic, and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to $\left\langle u^2 \right\rangle$of $O(\lambda^2)$ and $O(\lambda^4)$, 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 $\langle u^2 \rangle$ reveals that a total of 16 diagrams contribute to $\langle u^2 \rangle$ 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 $\langle u^2 \rangle$ diagrams from the F diagrams. Out of 16 diagrams 2 are of $O(\lambda^2)$ and 14 of $O(\lambda^4)$ where $\lambda$ is a perturbation expansion parameter. It is also shown that at least up to $O(\lambda^4)$ the contributions to $\langle u^2 \rangle$ from 9 out of 16 diagrams are included in the Green's function iterative solution if the latter is generated from the anharmonic Hamiltonian containing the cubic and quartic terms. Exact and approximate numerical magnitudes for all the diagrams are obtained for a nearest-neighbour 6-12 Lennard-Jones fcc solid by extrapolating the Brillouin Zone (BZ) sums to the ${\bf q} \rightarrow 0$ limit. A disproportionate contribution to the BZ sums comes from this region of ${\bf q}$ space in the calculation of $\langle u^2 \rangle$. In the context of $\langle u^2 \rangle$ 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 $\left\langle u^2 \right\rangle$ is sufficient to bring it in almost exact agreement with the classical Monte Carlo (MC) results which are also obtained for the same model.

Phil. Mag. B, August, 2000

Successful theory of anharmonicity in the classical limit

We present a critical assessment of the equation of state results for a fcc Lennard-Jones (LJ) solid calculated from two entirely different summation procedures for an infinite set of free energy diagrams. The first is the recent procedure given by Shukla and Cowley (R.C. Shukla and E.R. Cowley, Phys. Rev. B 58, 2596 (1998)) where the diagrams of the same order of magnitude generated from the Van Hove ordering scheme, but arising in different orders of perturbation theory (PT), are summed to infinity. The second procedure is the self-consistent phonon theory (SC) which has been in use for some time. In the first order version of this theory (SC1) only the first order PT diagrams are summed and in the improved self-consistent (ISC) theory the first important contribution (cubic term) arising from the second order PT, omitted in SC1, is included as a correction to the SC1 free energy. We have calculated the equation of state results from the ISC theory by averaging the cubic tensor force constant and also without averaging this constant (ISCU). This brings out the effect of averaging which is a necessary requirement in the SC1 theory but not in ISC. The results from SC1 and ISCU are poor. The results from ISC and Shukla-Cowley summation procedures agree with each other at low temperature. At high temperatures, the ISC results are in poor agreement with the classical Monte Carlo (MC) results whereas the Shukla-Cowley procedure yields results in excellent agreement with MC.

Phys. Rev. B, 60, 14 500-2, 1999.

Summation of free-energy diagrams of an anharmonic crystal and equation of state for a Lennard-Jones solid

We present a method of calculating the Helmholtz free energy of an anharmonic crystal. The exact expression for F, obtained by summing an infinite series of contributions, from all the loops and bubbles (quartic and cubic contributions to the self-energy of the Green's function), is evaluated numerically and the equation of state results for a Lennard-Jones solid are compared with the lambda-squared perturbation theory (PT) which contains only the lowest-order cubic and quartic contributions. It is shown that the infinite sum results are considerably improved over the lambda-squared PT results for higher temperatures. Next, we have presented a powerful ansatz approach of evaluating the same sum. The numerical results from this method are shown to be identical to the exact sum except at near T_m where they are very slightly different. The ansatz method is then extended to the higher-order lambda-to-the-fourth diagrams and here too the numerical results are found to be improved over the results from the lambda-to-the-fourth PT. The ansatz procedure is then extended to the propagator renormalization and the numerical results obtained seem to have the best agreement with the results of classical Monte Carlo simulations.

Phys. Rev. B, 58, 2596-602, 1998.

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 Green's function. The hamiltonian (H) employed in the derivation contains the contributions from all the even terms of the Taylor's expansion of the crystal potential energy. 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 Green's function, and the renormalized phonon frequencies. The self-consistency condition arises because in evaluating the correlation functions from the Zubarev type Green's 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.

phys. stat. sol. (b) 205, 481-92, 1998.

Quantum corrections to the simulated properties of solids

It is shown that a practical procedure for including both anharmonic and quantum effects in the calculation of the properties of solids is to combine classical molecular-dynamics simulations with quantum corrections obtained with the quasiharmonic approximation. The procedure is simple to implement and possesses an ordered set of anharmonic quantum corrections. It is tested by calculations on a Lennard-Jones model for solid Ar with nearest-neighbor interactions. The results obtained are competitive with the predictions of effective-potential Monte Carlo (EPMC) and are in very good agreement with path-integral Monte Carlo results, which were obtained with a constant-pressure algorithm that includes higher-order corrections to the Trotter expansion. The lowest-order perturbative correction to EPMC is shown to be the same as the cubic part of the anharmonic quantum correction.

Phys. Rev. B, 57, 833-8, 1998.

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(lambda-to-the-eighth), with results of standard PT of O(lambda-squared) and O(lambda-to-the-fourth), 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(lambda-squared) and O(lambda-to-the-fourth).

J. Chem. Phys., 107, 7409-17, 1997.

Erratum: Anharmonic contributions to the Debye-Waller factor for copper, silver and lead

Phys. Rev. B, 54, 15548-97, 1997.

Molecular dynamic 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 > 0.2 Tm, 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 lambda-squared 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.

J. Chem. Phys., 105, 4185-90, 1996.

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

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 (lambda-squared) anharmonic perturbation theory and a Green's function (GF) method which sums an infinite series of the lambda-squared-type anharmonic terms. The static approximation in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hübschle is further justified by showing that in the high-temperature limit the exact results for the cubic and quartic anharmonic contributions 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 lambda-squared perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mössbauer and X-ray data in the entire temperature range, 300 K < T <1200 K. The GF and lambda-squared PT results for the Morse potential agree very well with each other but are lower than the experimental values in the high-temperature range, T > 600 K.

Phil. Mag. B, 74, 1-11, 1996.

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

We present a derivation of the finite-temperature expressions for the lowest-order anharmonic (lambda^2) contributions to the atomic mean-square displacement (MSD) from the Green's 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. Since the present MSD expressions are valid for all temperatures they can be used in numerical evaluation of the lambda-squared 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 limit of the self-energy of the GF by Shukla and Hübschle (SH). The quartic contribution to MSD in the high-temperature limit similarly agrees with those derived by MF and SH. Thus it is shown that the static approximation limit employed by SH leads to the exact result for MSD in the high-temperature limit.

Phil. Mag. B, 74, 13-23, 1996.

Average square atomic displacement: A comparison of the Monte-Carlo and Green's function results

Highly accurate results for the mean-square displacement (MSD) of an atom for a nearest-neighbor Lennard-Jones model of a fcc solid are presented. These results are obtained from the Monte-Carlo (MC) method and the Green's function method. The dependence of MSD on the sample size (N) in the MC method and similarly the dependence of MSD on the number of wave vectors used in the calculation by the Green's function method is discussed in detail. The results are presented by both methods for the infinite sample size limit as well as for the finite sizes, i.e., 32-, 108-, and 256- atom sample sizes and the same number of corresponding wave vectors. It is shown that an analytical method like the Green's function method reproduces almost exactly the results of the MC method (a completely numerical procedure) for all temperatures except at the melting point (T_m). The computing time required in the former method is only a fraction of the latter one.

phys. stat. sol. (b) 195, 73-83, 1996.

Molecular dynamics and higher-order perturbation-theory results for the anharmonic free energy and equation of state of a Lennard-Jones solid

The anharmonic contribution to the Helmholtz free energy (F_A) is calculated as a function of temperature (T) and volume (V) for a fcc crystal of atoms interacting via a nearest-neighbor 6-12 Lennard-Jones potential, by the molecular-dynamics (MD) method and by perturbation theory (PT) to O (lambda-squared) and O (lambda-to-the-fourth). The volume-dependent coefficients for the T^2 and T^3 terms of (F_A) terms have been extracted from the MD results, and compared with the corresponding coefficients of lambda-squared and lambda-to-the-fourth PT. The agreement between the results of the MD and PT methods is excellent. The coefficient of the T^4 term of F_A is also extracted from the MD results, to give an approximation to the corresponding term in O (lambda-to-the-sixth) PT. Various thermodynamic properties are calculated from this approximation and are compared to exact MD results to provide a picture of the convergence properties of PT. The results suggest that lambda-to-the-sixth PT would be accurate up to approximately 60% of the melting temperature.

Phys. Rev. B, 54, 3266-72, 1996.

Atomic mean-square displacement in fcc metals: repulsive potentials

We present a method for the calculation of the atomic mean-square displacement of an anharmonic crystal from potentials involving repulsive interactions. The quasiharmonic and the lowest-order cubic and quartic anharmonic contributions are evaluated from the knowledge of seven Brillouin zone (BZ) sums which are tabulated in the interval-0.1 < a_1 <0.0. the parameter a_1 characterizes the volume dependence of the BZ sums and is negative for repulsive potential. All the BZ sums are evaluated in the limit L to infinity, where L is the step length from the origin to the boundary of the BZ. The method is applicable to a nearest-neighbor 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 ....from .... repulsive potential, and our results are in good agreements with those obtained by Monte Carlo methods. The other is to the nearest-neighbor Born-Mayer potential with a volume-dependent effective coefficient alpha. In the case of Cu, the agreement with other theoretical calculations as well as the experimental values from X-ray data at 300 and 400 K is excellent.

Phil. Mag. Lett., 73, 79-84, 1996.

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

Phys. Rev. B, 52, 168-76, 1995.

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

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 fcc metals has been discussed. In Zoli's 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 Zoli's 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 fcc lattice, namely a nearest-neighbor central force model employing a Lennard-Jones potential (applicable to rare-gas solids) and the Morse potential as a model for Cu.

Phil. Mag. Lett., 70, 255-9, 1994.

Uncorrelated-factors approximation and a comparison of theories for predicting thermal properties: A Lennard-Jones solid

The equations for determining the free energy of a solid with two-body interactions in the uncorrelated-factors approximation (UFA) are derived from the correlated-factors theorem. A self-consistent choice of the parameters in the harmonic Hamiltonian causes the approximation to be accurate through second order. The specific heat, thermal expansion, and bulk modulus of an fcc Lennard-Jones solid with nearest-neighbor interactions only are calculated in the UFA and the results are compared with the predictions of lowest-order and improved self-consistent phonon theory (SC1 and ISC), perturbation theory through fourth order, and other approximations. The predictions of the UFA are in very good agreement with new classical Monte Carlo estimates and with recent effective potential Monte Carlo results. The calculational effort required in the UFA is similar to that in SC1, while the accuracy of the predictions is similar to that of ISC.

Phys. Rev. B, 49, 8732-7, 1994.

Mean-square displacement from Mössbauer and x-ray data for solid krypton: A comparison of theory and experiment

The Mössbauer recoilless fraction is calculated from the lowest-order anharmonic-perturbation theory, the Green's function method, which sums the lowest-order anharmonic contributions to all orders, and the Monte Carlo method. In all cases we have employed a nearest-neighbor-interaction Lennard-Jones and Morse potential. Excellent agreement is shown to exist between the theory and the Mössbauer and x-ray experimental results.

Phys. Rev. B, 49, 9966-8, 1994.

Comparison of Monte Carlo and anharmonic-lattice-dynamics results for the thermodynamic properties and atomic mean-square displacement of Xe using the Morse potential

Monte Carlo (MC) and anharmonic-lattice-dynamics (the lambda-squared and lambda-to-the-fourth) perturbation-theory (PT) calculations of the thermodynamic properties of Xe are presented for the temperature range 60-160 K using a nearest-neighbor central-force (NNCF) model of the fcc crystal with atoms interacting via a Morse potential. In particular, we calculate the equilibrium lattice parameter at zero pressure and the corresponding specific heats at constant volume and at constant pressure, volume expansivity, adiabatic and isothermal bulk moduli, and Grüneisen parameter. We also calculate the atomic mean-square displacement (MSD) from the MC method and the lowest-order (lambda-squared) PT for the same NNCF model and the Morse potential. For the thermodynamic properties, the MC results are found to agree more closely with the lambda-squared PT and the lambda-to-the-fourth PT results. Similarly, the MSD results from the MC method agree quite well with those from the lambda-squared theory. This may be due to the fact that the exact solution of the Schrödinger equation for the vibrational states of the Morse potential for a one-dimensional or an isotropic three-dimensional model agrees exactly with the lambda-squared PT. We show that this is indeed true by evaluating the lambda-squared and lambda-to-the-fourth contributions to vibrational energy for the above model of the Morse potential and showing that all the lambda-to-the-fourth contributions add up to zero and that the total lambda-squared contribution is in agreement with the solutions of the Schrödinger equation.

Phys. Rev. B, 45, 12812-20, 1992.

Debye-Waller factor of sodium: A comparison of theory and experiment

We have compared the Debye-Waller factor of sodium calculated by three different theoretical methods with the recent measured values in the temperature range 80-295 K, using the Mössbauer gamma-ray-scattering technique. The Mössbauer results are also compared with the two sets of earlier x-ray measurements, one of which extends to 365 K. The three theoretical methods are the following: the lowest-order anharmonic perturbation theory, a Green's-function method that includes anharmonic contributions of the lowest-order perturbation theory summed to infinity, and the molecular-dynamics method, which includes the anharmonic contributions to all orders. In all three methods the Ashcroft pseudopotential with the Vashihta-Singwi screening function is employed to generate the real-space two-body potential function whose range is cut off at the sixth-neighbor distance. Excellent agreement is found between the results of these three methods and the Mössbauer experimental results. The x-ray results are also in very good agreement with the Mössbauer data where the temperatures overlap in the measurements.

Phys. Rev. B, 45, 10765-8, 1992.