Monoatomic Crystal Research Articles

Defect modes due to substitutional impurities in FCC lattice: Green's function approach

Lattice dynamical study of monoatomic fcc crystals containing substitutional impurities has been made by the Green's function technique, using group theory. The impurity and its 12 nearest neighbors constitute an XY 12 impurity space having O h symmetry. The phonon Green function matrix is analyzed according to the irreducible representations of the point group pertaining to the substitutional impurity in the fcc lattice. The effects due to change in mass at the impurity site and the change in nearest neighbor force constants for the impurity-host atom interactions are taken into account. Analytical expressions for the various modes of vibrations pertaining to the defect space have been obtained. Local mode frequencies due to various substitutional impurities, corresponding to F 1u mode (defect atom moving) have been computed. A special model is chosen for the defecthost force and it is assumed that there are no distortions of the lattice structure due to the defect. A Kihara hardcore potential with parameters fitted to neutron data has been used to compute lattice dynamics and Green functions of the host lattices. Our theoretical results have been compared with available experimental and theoretical results. Our results show reasonably good agreement with experimental results.

Effect of vacancies on the thermodynamics of crystal evaporation

An investigation is made of the evaporation heat and the equilibrium vapor pressure over the crystal containing vacancies. It is shown that in the case of a monoatomic crystal the vtacancies reduce both of these characteristics; the same effect takes place in a dilute intersitial alloy. In the case of dilute substitutional alloys the vacancies reduce both the vapour pressure and the evaporation heat of the host component and increase the impurity vapour pressure. It is found that the coefficient of the isotope separation changes considerably due to the vacancies. [Russian Text Ignored].

Determination of self-diffusion mechanisms from high-field nuclear-spin-relaxation experiments

The theory of high-field nuclear-spin relaxation due to random-walk diffusion in monoatomic crystals is extended to correlated diffusion mechanisms. It is predicted that the effect of the diffusion mechanism on the relaxation times ${T}_{1}$, ${T}_{2}$, and ${T}_{1\ensuremath{\rho}}$ can be observed in three different ways: (i) In single crystals the relaxation times are shown to depend on the crystallographic orientation of the magnetic field. Whereas on the low-temperature side of the ${T}_{1}$ minimum these anisotropies should characterize the diffusion mechanism, they are expected to disappear on the high-temperature side. (ii) Shape and width of the ${T}_{1}$ and ${T}_{1\ensuremath{\rho}}$ minimum as a function of temperature are found to depend on the diffusion mechanism---in both single crystals and polycrystalline samples. (iii) Assuming that in a single crystal the decay of the transverse magnetization can be described by a single relaxation time ${T}_{2}$ (usually true at temperatures above the ${T}_{1}$ minimum), deviations from the exponential decay, which should be characteristic for a given diffusion mechanism, are predicted in polycrystalline samples. The numerical results obtained for diffusion via monovacancies in bcc and fcc crystals are compared with those obtained for the limiting case of random-walk diffusion. It is found that for these two mechanisms the orientation dependences of the relaxation times are similar, but that considerable differences in the shapes of the relaxation rates versus temperature should exist.

Surface Phonons and Gas Surface Interactions

Modifications to the band frequencies of a crystal occur with gas adsorption on the surface. If the adsorbed species are small mass or if they are strongly bonded, theoretical calculations predict the appearance of new frequencies which lie above the band frequencies of the crystal (surface phonons) and of a modification of the Rayleigh waves. Results of calculations with different models of monoatomic crystals and one adsorbed layer are presented. Two main consequences are discussed: the entropy variation due to adsorption, which yields a modification of the pre-exponential factor of the desorption rate equation; the amount of energy exchange between a colliding particle and a crystal, which decreases as the surface phonon-frequencies increases. Experimentally the existence of surface phonons correlated to adsorption can be supported by data from energy loss of low energy electrons and by infrared absorption measurements. Indirect support is also provided by experimental results dealing with the pre-exponential factor of desorption rates, and with reflection of light atom beams at a crystal surface.

Green's function approach to the quantum many-body theory of the solid state

TheGreen's function approach is used to develop a quantum many-body theory of the solid state which should work at low temperatures as well as in the neighbourhood of phase transition points. The theory is applicable also in those cases where the traditional expansion of the potential in powers of the atomic displacements is entirely inadequate (crystalline helium). The starting point of our approach is the concept of broken symmetry since the invariance of the equilibrium ensemble under the continuous group of infinitesimal translations is reduced in a crystalline solid to the invariance under finite translations through a lattice vector. A homogeneous integral equation is derived which has nontrivial solutions in the crystalline state. By this equation it is shown that the umklapp phonons are the symmetry restoring collective modes expected due to a general theorem ofGoldstone. The single particle excitations and the structure of the Dyson mass operator in the crystalline state are discussed. It is further shown that the homogeneous Bethe-Salpeter equation for the linear response to an external disturbance possesses symmetry breaking solutions which are connected to the lattice dynamics of the solid state. These collective excitations (phonons) are exhibited in RPA and tight-binding approximation for monoatomic cubic crystals with a Bravais lattice in order to demonstrate how the present theory reproduces well-known results.

Dynamics of a crystal lattice containing isotopes

Synopsis By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δ μ p > A υ = ∑ i c i Δ μ i p = ∑ i c i ( μ i - μ > A υ ) p , where ci is the concentration of the i-th isotope and Δμi the deviation of its inverse atomic mass μi from the average value 〈μ〉 = Σi ciμi. The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμi are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass 〈μ〉Av−1. The next two terms are proportional to 〈Δμ2〉Av and 〈Δμ3〉Av; the fourth term brings contributions in 〈Δμ4〉Av and in 〈Δμ2〉2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained. Each term but the first one brings contributions of the form ∫ δ(λ − λ′) K(λ′) dλ′, where λ is the square of the angular frequency. These functionals may not be simply put equal to K′(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ′ function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial.