Single-particle vibrational motion

The aim of this chapter is to clarify the conditions for which chemical kinetics can be correctly applied to the description of solid state processes. Kinetics describes the evolution in time of a non-equilibrium many-particle system towards equilibrium (or steady state) in terms of macroscopic parameters. Dynamics, on the other hand, describes the local motion of the individual particles of this ensemble. This motion can be uncorrelated (single particle vibration, jump) or it can be correlated (e.g., through non-localized phonons). Local motions, as described by dynamics, are necessary prerequisites for the thermally activated jumps responsible for the movements over macroscopic distances which we ultimately categorize as transport and solid state reaction.. 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

In general, increasing the temperature within the stability range of a single crystal structure modification leads to a smooth change in all three parameters of vibration spectra frequency, half-width and intensity. The dependency of the frequency (wave number) on the temperature is usually related to variations in bond lengths and force constants [370] the half-width of the band represents parameters of the particles Brownian motion [371] and the intensity of the bands is related to characteristics of the chemical bonds [372]. 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

In quantum mechanics (18,19) the vibration of a diatomic molecule can be treated as a motion of a single particle having mass n whose potential energy is expressed by (1-21). The Schrodinger equation for such a system is written as... 

Through quantum mechanical considerations [2,7], the vibration of two nuclei in a diatomic molecule can be reduced to the motion of a single particle of mass p, whose displacement q from its equilibrium position is equal to the change of the intemuclear distance. The mass p is called the reduced mass and is represented by... 

The vibrational and rotational motions possible in real molecules lead to arrangements that a single atom can t have. A collection of real molecules therefore has a greater number of possible microstates than does the same number of ideal-gas particles. In general, the number of microstates possible for a system increases with an increase in volume, an increase in temperature, or an increase in the number of molecules because any of these changes increases the possible positions and kinetic energies of the molecules making up the system. We will also see that the number of microstates increases as the complexity of the molecule increases because there are more vibrational motions available. 

Conductivity can be deduced from vibrational spectra in IR spectroscopy, the absorption coefficient a(co) is related to tr(co) a(o)) = 4no(o))/nc, n being the refractive index and c the velocity of light. In Raman spectroscopy, the scattered intensity /(m) is related to conductivity by a(o ) oc o)I (o)/n(a)) + 1, n(co) being the Bose-Einstein population factor . Finally, the inelastic incoherent neutron scattering function P(o)) is proportional to the Fourier transform of the current correlation function of the mobile ions. P co) is homogeneous with a) /(cu) formalism. However, since P(co) reflects mainly single particle motions, its comparison with ff(co) could provide a method for the evaluation of correlation effects. (For further discussion, see also Chapter 9 and p. 333.)... 

Tsekov and Ruckenstein considered the dynamics of a mechanical subsystem interacting with crystalline and amorphous solids [39, 40]. Newton s equations of motion were transformed into a set of generalized Langevin equations governing the stochastic evolution of the atomic co-ordinates of the subsystem. They found an explicit expression for the memory function accounting for both the static subsystem—solid interaction and the dynamics of the thermal vibrations of the solid atoms. In the particular case of a subsystem consisting of a single particle, an expression for the fiiction tensor was derived in terms of the static interaction potential and Debye cut-off fi equency of the solid. 

With atoms and molecules taken to be single particles, earlier chapters have followed gas kinetic analysis of collisions, gas pressure, and transfer of energy as heat and work. However, the internal structure and mechanics of molecules— that they are not single point masses—can play a role in thermodynamic behavior and reaction energetics. This chapter focuses on the mechanics of vibration, an internal motion exhibited by all molecules. Though we start by using classical mechanics, it turns out to be an incomplete theory in that it fails to correctly describe very small, very low-mass particle systems. To go beyond classical pictures calls for us to invoke quantum mechanical ideas which are introduced here. The contrast and the correspondence between the classical and quantum pictures of the vibrational motion of molecules is a primary objective of this chapter. 

About half a century has passed after the publication of Hartree s and Fock s papers on the self-consistent field method and Bloch s and Jensen s > - on the collective sound-like vibrations of electrons in atoms. For more than fifty years our ideas have developed on single-particle and collective aspects of electron motion in atoms. 

The fraction of released energy that passes into translational and vibrational energies of product depends to a quite significant extent on the mass combination of reactant atoms. This mass effect is referred as kinematic effect. In order to understand mass effect, at least for a collinear reaction, we can transform the motion of the three particles on PES to that of a single... 

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