Crystal lattices thermal motion effects

Another aspect of high chain symmetry is the possibility of molecular motion within the crystal lattice contributing to higher T. For example, polyethylene and polytetrafluoroethylene are both sufficiently symmetrical to be considered as smooth, stiff cylindrical rods. In the crystal, these rods tend to roll over each other and change position when thermally agitated. This motion within the crystal lattice, called premelting, effectively stabilizes the lattice. Consequently, more thermal energy is required to break down... 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

As described above, the electrons in a semiconductor can be described classically with an effective mass, which is usually less than the free electron mass. When no gradients in temperature, potential, concentration, and so on are present, the conduction electrons will move in random directions in the crystal. The average time that an electron travels between scattering events is the mean free time, Tm. Carrier scattering can arise from the collisions with the crystal lattice, impurities, or other electrons. However, during this random walk, the thermal motion is completely random, and these scattering processes will therefore produce no net motion of charge carriers on a macroscopic scale. 

Although the effects of thermal motion might be considered undesirable in crystallography, they provide, in fact, important information about the properties of the molecules under study (see, for example, Finzel et al., 1984). In terms of refining a crystal structure, the stmcture factors calculated assuming the atoms to be rigidly fixed on the lattice will be too large compared to the observed structure factors. The thermal factors for each atom reduce the sizes of the calculated structure factors by an amount appropriate for the thermal motion present. 

Bearing in mind that the nuclear spin system is an ensranble perturbed by the rf-field, we have to consider the possibility that energy from the rf-field is transferred to the nuclear spins and dissipated further to the crystal lattice. These effects can be described by relaxation times that characterize the rates with which the system returns to thermal equilibrium after the perturbation has been switched off. There are the longitudinal or spin-lattice relaxation time Tj and the transverse or spin-spin relaxation time Tj. Including the relaxation effects the equations of motion in the rotating frame (cf. Eq. (19)) are... 

Finally, there is the thermal motion problan in X-ray crystallography." Molecules vibrate internally, and molecules in a aystal lattice also vibrate with what is referred to as the rigid-body motion. That is simply the whole molecule moving back and forth and undergoing rotational types of oscillation in the crystal lattice. (These are motions that correspond to translations and rotations in the isolated molecule that are converted into vibrations by the constraints imposed by the crystal lattice.) The molecule as a whole is quite heavy, so these vibrational frequencies tend to be quite low, and the vibrational amplitudes are large. The most serious part of this problem comes not from the translational motions but rather from the rotational oscillations of the molecule in the crystal lattice. The effect of these is to shorten the apparent bond lengths and compact the apparent size of the molecule. 

The thermal vibration of atoms in the crystal lattice is strongly temperature dependent and is less effective in assisting dislocation motion at low temperatures. The interaction of dislocations with thermal vibrations is complicated, but it is nonetheless satisfying to find that ductility usually decreases somewhat with a decrease in temperature. 

In an ideal ionic crystal, all ions are held rigidly in the lattice sites, where they perform only thermal vibratory motion. Transfer of an ion between sites under the effect of electrostatic fields (migration) or concentration gradients (diffusion) is not possible in such a crystal. Initially, therefore, the phenomenon of ionic conduction in solid ionic crystals was not understood. 

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

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