Oscillations, harmonic solid lattice

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Equation (1) Is the first term of a Taylor expansion valid for (6/T) 2rr. For the case of the non-existence of zero point energy, one predicts an isotope effect for the Pb/ ° Pb vapor pressure ratio two orders of magnitude larger than the prediction from Eq. (1) at 600 K. In addition the no zero point energy case predicts the Pb to have the larger vapor pressure at 600 K. 

Stem s estimate of the difference in vapor pressures of Ne and Ne at 24.6 K througih Eq. (1) led to the first separation of isotopes on a macro scale by Keesom and van Dijk ( ). The same theory, without the approximation (6/T) 1, was used by Urey, Brickwedde, and Murphy ( ) to design a Raleigh distillation concentration procedure to enrich HD in H2 five fold above the natural abundance level, which was adequate to demonstrate the existence of a heavy isotope of hydrogen of mass 2. 

It is important to look into the implications of Eq. (1) since the development of the quantum-statistical mechanical theory of Isotope chemistry from 1915 until 1973 centers about the generalization of this equation and the physical interpretation of the various terms in the generalized equations. According to Eq. (1) the difference in vapor pressures of Isotopes is a purely quantum mechanical phenomenon. The vapor pressure ratio approaches the classical limit, high temperature, as t . The mass dependence of the Isotope effect is 6M/M where 6M = M - M. Thus for a unit mass difference in atomic weights of Isotopes of an element, the vapor pressure isotope effect at the same reduced temperature (0/T) falls off as M 2. Interestingly the temperature dependence of In P /P is T 2 not 6X0/T where 6X.0 is the heat of vaporization of the heavy Isotope minus that of the light Isotope at absolute zero. In fact, it is the difference between 6, the difference in heats of vaporization at the temperature T from ( that leads to the T law. 

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For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

This linear problem is thus exactly soluble. On the practical level, however, one cannot carry out the diagonalization (4.11) for macroscopic systems without additional considerations, for example, by invoking the lattice periodicity as shown below. The important physical message at this point is that atomic motions in solids can be described, in the harmonic approximation, as motion of independent harmonic oscillators. It is important to note that even though we used a classical mechanics language above, what was actually done is to replace the interatomic potential by its expansion to quadratic order. Therefore, an identical independent harmonic oscillator picture holds also in the quantum regime. 

The last reference system we discuss is the lattice of interacting harmonic oscillators. In this system each atom is connected to its neighbors by a Hookean spring. By diagonalizing the quadratic form of the Hamiltonian, the system may be transformed into a collection of independent harmonic oscillators, for which the free energy is easily obtained. This reference system is the basis for lattice-dynamics treatments of the solid phase [67]. If D is the dynamical matrix for the harmonic system (such that element Dy- describes the force constant for atoms i and j), then the free energy is... 

The details of lattice vibrations will not be discussed here. But for the sake of discussion, the main results of one of the simpler models, namely, the Einstein solid, are given below without proof. By assuming the solid to consist of Avogadro s number jVav of independent harmonic oscillators, all oscillating with the same frequency Einstein showed that the thermal entropy per mole is given by... 

A number of theoretical studies have investigated multiphonon relaxation in solids. Nitzan and Jortner considered a harmonic oscillator coupled to a harmonic lattice the coupling potential was taken to be linear in the vibrational coordinate and of high order in phonon displacements. 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

The ab initio potentials used in solid nitrogen are from Refs. [31] and [32]. They have been respresented by a spherical expansion, Eq. (3), with coefficients up to = 6 and Lg = 6 inclusive, which describe the anisotropic short-range repulsion, the multipole-multipole interactions and the anisotropic dispersion interactions. They have also been fitted by a site-site model potential, Eq. (5), with force centers shifted away from the atoms, optimized for each interaction contribution. In the most advanced lattice dynamics model used, the TDH or RPA model, the libra-tions are expanded in spherical harmonics up to / = 12 and the translational vibrations in harmonic oscillator functions up to = 4, inclusive. 

The model of harmonic oscillations of the Mossbauer atom as described above, is quite satisfactory for solids at low temperatures. However, at higher temperatures the smaU amplitude motion of the atom around its equilibrium site may become unstable such that the atom has a flnite probability of making a sudden jump to an adjacent lattice site. Although this type of motion can in principle occur even at very low temperatures via the mechanism of quantum tunnelling, there is little evidence for this phenomenon with Mossbauer isotopes. The diffusive motion can therefore be considered in terms of thermally activated jumps... 

To this date, no stable simulation methods are known which are successful at obtaining quantum dynamical properties of arbitrary many-particle systems over long times. However, significant progress has been made recently in the special case where a low-dimensional nonlinear system is coupled to a dissipative bath of harmonic oscillators. The system-bath model can often provide a realistic description of the effects of common condensed phase environments on the observable dynamics of the microscopic system of interest. A typical example is that of an impurity in a crystalline solid, where the harmonic bath arises naturally from the small-amplitude lattice vibrations. The harmonic picture is often relevant even in situations where the motion of individual solvent atoms is very anhaimonic in such cases validity of the linear response approximation can lead to Gaussian behavior of appropriate effective modes by virtue of the central limit theorem. ... 

The theory of lattice specific heat was basically solved by Einstein, who introduced the idea of quantized oscillation of the atoms. He pointed out that, because of the quantization of energy, the law of equipartition must break down at low temperatures. Improvements have since been made on this model, but all still include the quantization of energy. Einstein treated the solid as a system of simple harmonic oscillators of the same frequency. He assumed each oscillator to be independent. This is not really the case, but the results, even with this assumption, were remarkably good. All the atoms are assumed to vibrate, owing to their thermal motions, with a frequency v, and according to the quantum theory each of the three degrees of freedom has an associated energy of which replaces the kT as postulated by... 

The separation works characterize the energy state of the atoms in the crystal lattice and are closely related to the corresponding chemical potentials In the frameworks of the Einstein s solid state model considering the crystal s atoms as independent linear harmonic oscillators [1.5,1.93], the chemical potential jUi,o of the ith atom of a given crystallographic face is presented as [1.4,1.11,1.94] ... 

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