A new formalism, using standard-basis matrix operators, is presented for the study of the collective excitations and the thermodynamic properties of an ensemble of identical interacting quantum-mechanical systems subsequently called "ions, " each ion having discrete energy levels. A model Hamiltonian is formulated in terms of these operators. The Hamiltonian contains terms which express the interaction of the ion with the crystal field and the external fields as well as terms which arise from the mutual interaction of ions. Using the doubletime temperature-dependent Green's-function technique, the equations of motion of the Green's functions of standard-basis operators are developed in the random-phase decoupling approximation.It is demonstrated that the temperature-dependent correlation functions of standard-basis operators, which are obtained from associated spectral Green's functions, lead to a set of equations that can be solved for the occupation probabilities of the singleion energy levels. Hence, one can calculate the thermal-average expectation value of any quantum-mechanical operator representing a microscopic observable of a single ion, or a pair of ions (correlations). An important feature of the standard-basis matrix-operator formalism is that the single-ion terms, such as crystal field, molecular field, or external fields, are always treated exactly in any Green's-function decoupling scheme. In contrast, Green's-function methods which use angular momentum operators often necessarily treat single-ion terms in a decoupling approximation.As an example, the general standard-basis operator theory is applied to the Heisenberg ferromagnet in the presence of uniaxial singleion crystal-field anisotropy, which has received extensive theoretical attention previously, with widely varying results.
ELASTIC SURFACE WAVES IN CRYSTALS 951the order of a wavelength, then we findwhere we have neglected higher-order terms in u/(wdv/du>).Here v' is a characteristic velocity defined asWhen \a 2 \ is small compared to |ai|, the energy density in the surface wave is u>^ -la) 2 dv/dQ)}f?Z\ w= n 0 , which shows thatNumerical calculations given by Lim 17 have shown that the leaky waves very often have an attenuation per period which is only 10 -4 -10~3. In these cases the approximate results given above should be quite good.The low-temperature thermal transport properties of an insulating crystal with an arbitrary concentration of randomly located phonon scattering centers are found for slab geometry. The phonon distribution function, the distribution of temperature, the Kapitza resistance at the boundaries, and the heat flux are evaluated by analytically and numerically solving the complete Boltzmann equation with energy-dependent scattering cross section.
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