Harmonic crystal

We may now discuss the imphcations of the results foimd for the self-motion of hydrogens in the a-relaxation regime by neutron scattering. It is well known that for some simple cases - free nuclei in a gas, harmonic crystals. 

These features are exploited in an experiment which is taking place in our laboratory. A schematic diagram of this experiment is shown in figure 4. The system consists of an all-lines violet mode-locked Kr+ ion laser operating at a repetition rate of about 250 MHz Which synchronously pumps a C102 dye laser. The dye laser typically produces about 300 mW of average power and pulse durations of about 3 psec. This is frequency doubled to 243 nm in a crystal of p-barium borate to produce in excess of 2 mW average power. The output from the second harmonic crystal is then mode-matched into an ultra-violet enhancement cavity. The free... 

In a harmonic crystal the DOS is the real part of the Fourier transform of VACF t), the mass weighted power spectrum Z(co), which is almost identical to the DOS calculated by LD and measured by INS, the differences will be discussed latter. 

In the perfectly harmonic crystal the total vibrational energy equals the sum of the energies of the simple harmonic oscillators or normal modes which have infinitesimal amplitude and ignore each other. The introduction of anharmonic coupling between oscillators leads initially to small shifts in... 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

Periodicity is an important attribute of crystals with significant implications for their properties. Another important property of these systems is the fact that the amplitudes of atomic motions about their equilibrium positions are small enough to allow a harmonic approximation of the interatomic potential. The resulting theory of atomic motion in harmonic crystals constitutes the simplest example for many-body dynamics, which is discussed in this section. 

For a harmonic crystal the Hamiltonian is (a, y, rj run here over the cartesian indices x, > )... 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is ... 

The model of non-interacting harmonic oscillators has a broad range of applicability. Besides vibrational motion of molecules, it is appropriate for phonons in harmonic crystals and photons in a cavity (black-body radiation). 

It is possible to calculate derivatives of the free energy directly in a simulation, and thereby determine free energy differences by thermod5mamic integration over a range of state points between the state of interest and one for which we know A exactly (the ideal gas, or harmonic crystal for example) ... 

In a perfectly harmonic crystal the elastic constants would be strictly independent of temperature. However, due to the existence of third- and fourth-order anharmonic terms in the crystal potential there is a coupling between the homogeneous strains and the phonon coordinates. This will lead to a background temperature dependence of the elastic constants. It can be described within a quasiharmonic approximation (Ludwig 1967), in which the anharmonic contributions to the crystal potential are implicitly included by assuming a strain dependence of the phonon frequencies which can be characterized by the... 

According to the classical Debye theory for a harmonic crystal... 

In harmonic crystals, the vibrations are phonons, i.e. plane waves involving the motion of all atoms in the sample. Phonons are described by a frequency illuminated volume seen by the detector, and fields irradiated by aU polarizable units interfere. As a consequence, one phonon scattering processes (first-order Raman scattering) are subjected to the selection rules ... 

For a harmonic crystal the phonon lifetime is infinite and there is no scattering of thermal phonons.To understand the mechanism on how the guest-host interactions lead to the anomalous temperature dependence of the thermal conductivity, the lifetimes were calculated for phonon-phonon scatterings as a result of the anharmonic terms in the xenon-water potential of xenon hydrate in the small and large cage. The inverse relaxation time (lifetime), of a lattice vibration with frequency C0j q) (/ is the branch index and q is the direction of the momentum transfer) is related to the transition rate, W, of the lattice wave scattered from state qj q f by a defect according to, ... 

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