Surface relaxation and thermal expansion for the (001) face of α-Fe and Cu

The surface relaxation and the near-surface enhancement of thermal expansion have been calculated for the (001) face of a bcc crystal, α-Fe, and an fcc crystal, Cu. The calculations make use of the anharmonic perturbation formalism of Dobrzynski and Maradudin; the results for certain equal-time vibrational correlation functions which arise in this formalism are also presented. The crystal potential is described in terms of several kinds of short-range empirical interatomic potentials, such as have been used in studies of defects in bulk; in the near-surface region, the effects of surface redistribution of the electron distribution are modelled by the addition of a simple surface Madelung (SSM) force. The effect of the SSM force is to limit severely the usual outward relaxation driven by short-range interatomic potentials. For Fe(001), the five and one-half percent outward static relaxation driven by the short-range potentials acting alone is changed to a one percent inward static relaxation when the SSM force is incorporated; for Cu(001), the comparable change is from a one percent outward relaxation to a one-half percent outward relaxation. On the other hand, the SSM force makes only a small effect on the surface-enhanced thermal expansion coefficients (STEC) for interplanar spacings. The STEC for the outermost spacing is between 2.5. and 3.0 times of that for the bulk at the Debye temperature for both Fe(001) and Cu(001); for the second interplanar spacing, the STEC is smaller than 1.5 times of that of the bulk at the Debye temperature. The ratios of the near-surface mean-square amplitudes (MSA) to those of the bulk at high temperatures are, for Fe(001), about 1.75 for z-components (normal to surface) and 1.55 for x-components (parallel to surface) in the surface layer; for Cu(001), about 1.95 for z-components and 1.30 for x-components. The interplanar correlation functions, while smaller than the MSA on an absolute scale, do show considerable surface-enhancement, particularly for the zz-compoments. For example, the zz-correlation between an atom in the outermost layer and its nearest neighbor in the next layer is nearly twice the comparable bulk correlation above the Debye temperature for both Fe(001) and Cu(001).     

Studies of vibrational surface modes in ionic crystals: III. Vibrational contributions to the surface thermodynamic functions for the (001) face of LiF,MgO, NaF,NaCl, Nal,RbF, and RbCl

The temperature dependence of the vibrational contributions to surface specific heat, surface entropy, surface energy, and surface Helmholtz free energy have been calculated for the (001) face of seven crystals having the rocksalt structure. The calculations assume a perfect, unrelaxed surface and make use of shell models fitted to bulk phonon spectra determined from inelastic neutron scattering. In terms of the bulk zero-temperature Debye temperature θ₀, the surface specific heat C_v^s exhibits an effective power law behavior, T^α, from at least T = 0.02 θ₀ up to 0.05 θ₀ in most cases (and up to 0.07 θ₀ for NaF), with α ≈ 2.5 in most cases — in contrast with the result of α = 2 in a Debye-like model. (Below 0.02 θ₀, results derived for our 15-layer films depart significantly from intrinsic surface effects because of the finite thickness.) C_v^s attains a maximum at a temperature T(C_max^s) ranging from 0.14 θ₀ to 0.20 θ₀, in contrast with the result T(C_max^s) = 0.21 θ₀ for the Debye-like model. The peak value C_max^s ranges from 0.34 k_BA_SUC to 0.41 k_BA_SUC, where A_SUC is the area of the surface unit cell. The shap the peak in C_v^s differs characteristically between that class of crystals in which there is some overlap of the acoustical and optical bulk bands and that class in which there is an appreciable absolute gap between the acoustical and optical bulk bands; in the latter class the peak is flattened on the low side of the maximum, with the maximum pushed to somewhat higher temperature. On those points of comparison with the rather sparse existing data for surface-excess heat capacity in which the value of specific surface area is not required (e.g., the value of T(C^s_max)), the agreement ranges from encouraging to equivocal. On those comparisons which require the surface area of the experimental samples (e.g., the magnitude of C^s_max) the agreement ranges from only fair to bad. Further experimental work is needed, and great care in surface area determinations is necessary.     

Thermal properties of xenon adsorbed on graphite

We report theoretical results for dynamical and thermodynamic properties of the bare graphite (0001) surface, and with a physisorbed monolayer of xenon. The surface vibrations of graphite are less pronounced than those in more isotropic crystals. The surface-excess specific heat of the bare surface peaks at 45 K, that of the xenon at 13.5 K.     

Shear-horizontal surface waves on the (001) face of cubic crystals

A connection is made between (1) the result of elasticity theory that shear-horizontal (SH) surface waves cannot exist on a cubic (001) surface and (2) the result of a number of lattice studies of surface modes in thin films to the effect that SH surface waves do appear to exist along [110] for large, but finite wavelength λ. The SH surface wave frequency departs from the bulk band edge as λ^-3 and the localization depth is proportional to λ².     

Surface phonon frequencies for magnesium oxide

Surface phonon branches and the surface excess distribution of frequencies f^S(ω) have been calculated for the (001) face of MgO with a rigid shell model for a fifteen-layer film. Comparison with the f^S(ω) obtained by Rieder and Hörl from neutron scattering shows agreement with some features as well as the need for further experimental work.     

On the importance of sp-electrons in the surface relaxation of d-band metals

We present arguments to the effect that the self-consistent smoothing of the mobile sp-electron distribution gives rise to electrostatic forces which are a major contribution to the initial relaxing forces on the outermost ions at a d-band metal surface. Numerical estimates for the (001) face of α-Fe are given, showing that for the surface ions the surface Madelung force from the sp-electrons is inward and about $\text{3}\text{1}\text{2}$ times stronger than an outward short-range force simulating the effect of d-electrons.     

On the existence of long-wavelength shear-horizontal surface waves on the bcc (001) surface

Long-wavelength, low-frequency shear-horizontal surface waves are found to exist along 〈100〉, but not 〈110〉, directions of propagation on the (001) face for a simple but fairly realistic model of monatomic body-centered cubic crystals. This domain of existence contrasts with that found previously on the (001) face of monatomic simple cubic and both monatomic and rocksalt face-centered cubic structures, where such waves exist along 〈110〉, but not 〈100〉.