Phonon Spectral Function Research Articles

Electron tunneling study of the superconducting proximity effect in Pb-Cd

Electron tunneling has been used to study the superconducting state in superimposed Pb-Cd thin films in which the Pb thickness varied from 950 to 330 A. The measured transition temperature T/sub c/ varied from 7.22 to 4.86 K in the samples, and the observed energy gaps varied from 1.38 to 0.43 meV. The tunneling density of states as recorded on either side of the sandwich resembles those predicted by the McMillan model, but there are some quantitative differences. The observed phonon structure in the density of states occurs at the same energy as that in bulk Pb, and the magnitude of the structure scales with T/sup 2//sub c/. The shape of the calculated phonon spectral function remains essentially unchanged throughout this series.

Electron-phonon interaction near a normal metal-superconductor boundary

Electron tunneling has been used to study the electron-phonon interaction in PbCd proximity effect samples by injecting electrons into Pb side. For Pb thicknesses ranging from 950 to 330 Å, the phonon structure occurs at the same energy as bulk Pb and the magnitude of the structure scales as T c 2 . The shape of the phonon spectral function, α 2 F( ω), remains essentially unchanged even though T c has been depressed by 35%.

Theory of ortho-para conversion in solid hydrogen at high densities

An expression is derived for the ortho-para conversion rate, as a function of density $\ensuremath{\rho}$, of solid ortho-${\mathrm{H}}_{2}$ in its rotationally disordered state. For $\ensuremath{\rho}$ greater than about 1.1 times the zero-pressure density ${\ensuremath{\rho}}_{0}$, conversion is shown to result from a one-phonon process. This rate is proportional to ${(\frac{\ensuremath{\rho}}{{\ensuremath{\rho}}_{0}})}^{\frac{10}{3\ensuremath{-}2\ensuremath{\gamma}}}$, where $\ensuremath{\gamma}$ is the Gr\"uneisen constant for solid ${\mathrm{H}}_{2}$, times a phonon spectral function which depends on the density as a parameter. Using the data of Pedroni et al. for the density dependence of the conversion rate for $1.0\ensuremath{\le}\frac{\ensuremath{\rho}}{{\ensuremath{\rho}}_{0}}\ensuremath{\le}1.7$, we extract the phonon spectral function and compare it to a theoretical spectrum calculated in a simple harmonic approximation.

Comparison of single-site approximations for the lattice dynamics of mass-disordered alloys

Calculations of phonon frequency distributions, spectral functions, and neutron-scattering cross sections have been carried out for the vibrations in monatomic fcc crystals with high concentrations of mass defects. The coherent-potential approximation (CPA), the average-$t$-matrix approximation (ATA), and a perturbation theory were used in the calculations. ATA predicts frequency distributions which can be markedly different from those for CPA; the spectral functions Im $〈{G}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})〉$ show an even greater difference. At all alloy concentrations and host-defect mass ratios, there are some ATA spectral functions which have $\ensuremath{\delta}$-function components, and thus an unphysical zero width. In some cases, ATA also predicts spectral functions near the top of the host band which have pronounced double peaks. These double peaks do not occur in CPA, and are inconsistent with the results of a neutron scattering experiment on ${\mathrm{Ni}}_{55}$${\mathrm{Pd}}_{45}$. However, for moderate concentrations ($c\ensuremath{\simeq}0.2$) of light defects, ATA may have some advantages over CPA.

Some effects of phonon dynamics on electron lifetime, mass renormalization, and superconducting transition temperature

Effects of the dynamical properties of phonons on the electronic properties of metals are treated through the use of the phonon spectral weight function. It is shown that the leading behavior of the electron quasiparticle lifetime is lowered to quadratic in the frequency or temperature from the cubic behavior which is well known to occur for infinitely sharp phonons. Explicit evaluations of $\frac{1}{\ensuremath{\tau}}$ are made using an extension of the Debye model which allows for a finite phonon lifetime. The mass-enhancement parameter $\ensuremath{\lambda}$ is discussed and is shown formally to be related to the static phonon propagator. Thus if there is a difference between the phonon relaxation at zero frequency and at the phonon frequency, it is possible for the phonon frequencies which determine $\ensuremath{\lambda}$ to differ significantly from the peak frequencies observed in inelastic neutron scattering. The implications of these results for superconducting transition temperatures are discussed with particular reference to recent neutron scattering experiments in ${\mathrm{Nb}}_{3}$Sn.

Phonon spectral function of a superconductor

The phonon spectral function of an acoustic phonon in a BCS superconductor is calculated at zero temperature and compared to the result for the normal metal. Shifts are found in the frequency and linewidth of the phonon depending on its wavevector. In addition to the phonon, the spectral function shows a structure around the gap energy of the superconductor, reflecting the increased polarizability of the superconducting electrons at this energy.

Computation of Phonon Spectral Functions and Ground-State Energy of Solid Helium. I. bcc Phase

Computations incorporating the three-phonon interaction process are presented of phonon spectral functions, yielding frequency and lifetime dispersion curves, for the bcc solid phase of both ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$. Comparison of the compressibilities obtained from ground-state energy and elastic constants as functions of molar volume quantify the effects of certain analytical approximations. A model calculation shows the sensitivity of elastic anisotropy and long-wavelength dispersions to the precise form of the three-phonon vertex.

Phonon Spectral Functions and Ground-State Energy of Quantum Crystals in Perturbation Theory with a Variationally Optimum Correlated Basis Set

A theory is presented of the damping and frequency shift of phonons and of the ground-state energy corrections due to interactions between phonons in quantum crystals with singular forces. The technique begins with the adoption of a trial ground-state wave function of the Jastrow form, together with trial excited-state wave functions constructed to represent one-, two-, and three-phonon excitations. The Hamiltonian matrix in this restricted basis is diagonalized, and the basis is optimized by minimizing the lowest eigenvalue with respect to variational phonon parameters. Using a lowest-order cluster expansion, the unambiguous prescription is obtained that a specific effective potential, softened by the Jastrow correlation function, replaces everywhere the true potential in the existing self-consistent theory of phonon damping applicable to nonsingular forces. Close analogies are drawn with the correlated basis function treatment, of superfluid liquid helium.

Calculation of the sound absorption in superfluid helium below 0.6 K

The sound absorption in superfluid helium is calculated using the renormalized perturbation expansion of the phonon self-energy. The second-orderωT4-result is derived without the assumption of a Lorentzian form of the phonon spectral function. The fourth-order contribution leads to better agreement with experimental data. These results are compared with simple perturbation theory. A discussion of the Boltzmann-equation approach is also given.

Self-Consistent Phonon Formulation of Anharmonic Lattice Dynamics

The self-consistent phonon theory of anharmonic lattice dynamics is derived via a stationary functional formulation. The crystal dynamics is approximated by a set of damped oscillators, and these are used to construct a trial action, analytically continued into the complex time-temperature plane. Using the action, a free-energy functional is required to be stationary with respect to the trial oscillators. The resulting phonon modes are undamped at the first order of approximation, whereas to second order the phonon spectral function is determined self-consistently. Expressions are obtained in first order for various thermodynamic derivatives, such as pressure, elastic constants, specific heats, and thermal expansion.

Theory of thermal motions in anharmonic crystals

A general theoretical treatment of transport properties of an anharmonic crystal is given, based on Green's function technique. Equations of motion for the phonon spectral function and for the generalized phonon number density at non-equilibrium are derived with no restriction on the frequency and wave vector of the thermal disturbance. The latter equation constitutes a generalized Boltzmann equation for the phonons. Based on these equations we are able to discuss in detail propagation and damping of “phonons” throughout the whole frequency range, stretching from the high frequency, essentially collisionless regime down to the low frequency, collision dominated, hydrodynamic regime. Particular attention is given to the interesting transition region between these two regimes. Here we have confined ourselves to the main lines. The general explicit solution of the generalized Boltzmann equation and a detailed discussion of propagation and damping of sound waves is given in a separate forthcoming paper.

Kinetic Equation for the Electron-Phonon Gas

The random phase approximation equations of motion for the electron-phonon system are solved to obtain long-time expressions for the basic operators that appear in the calculation of correlation functions. These operator expressions are used to obtain a kinetic equation for the electrons and an expression for the phonon spectral function. The kinetic equation has the same form as the Balescu-Lenard equation. Expressions for the density autocorrelation function and the two-particle correlation function are also obtained. The latter is shown to have the same form as the test-particle result for a classical plasma.