Crystal thermal vibration

A) CRYSTAL MODELS WITH THERMAL VIBRATION INCLUSION... 

The practical importance of vacancies is that they are mobile and, at elevated temperatures, can move relatively easily through the crystal lattice. As illustrated in Fig. 20.21b, this is accompanied by movement of an atom in the opposite direction indeed, the existence of vacancies was originally postulated to explain solid-state diffusion in metals. In order to jump into a vacancy an adjacent atom must overcome an energy barrier. The energy required for this is supplied by thermal vibrations. Thus the diffusion rate in metals increases exponentially with temperature, not only because the vacancy concentration increases with temperature, but also because there is more thermal energy available to overcome the activation energy required for each jump in the diffusion process. 

Peng, L.-M. (1997) Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Cryst. A, 53, 663-672. 

Electrons of still lower energy have been called subvibrational (Mozumder and Magee, 1967). These electrons are hot (epithermal) and must still lose energy to become thermal with energy (3/2)kBT — 0.0375 eV at T = 300 K. Subvibrational electrons are characterized not by forbiddenness of intramolecular vibrational excitation, but by their low cross section. Three avenues of energy loss of subvibrational electrons have been considered (1) elastic collision, (2) excitation of rotation (free or hindered), and (3) excitation of inter-molecular vibration (including, in crystals, lattice vibrations). 

Figure 2. ORTEP drawing of the nonhydrogen atoms of one of the two crystal-lographically independent Th[(CHS)5C5]2[p-CO(CH2C(CHs)s)CO]Cl molecules in the unit cell of 5. The stereochemistry of the second molecule differs from this one primarily in the orientation of the t-butyl groups. All atoms are represented by thermal-vibration ellipsoids drawn to encompass 50% of the electron density...

The ionic atmosphere moves continually, so we consider its composition statistically. Crystallization of solutions would occur if the ionic charges were static, but association and subsequent dissociation occur all the time in a dynamic process, so even the ions in a dilute solution form a three-dimensional structure similar to that in a solid s repeat lattice. Thermal vibrations free the ions by shaking apart the momentary interactions. 

The ionic atmosphere is not a static structure, so its composition is best treated statistically. An aggregation of ionic charges, if static, would allow for crystallization if the solution was at ail concentrated. In dilute solutions, while the charges might instantaneously have a three-dimensional structure similar to that in an ionic repeat lattice, thermal vibrations soon cause such momentary interactions to break down (i.e. shake free) and reform. 

In solid state lasers the fluorescence lines are broadened 26) by statistical Stark fields of the thermal vibrating crystal lattice and furthermore by optical inhomogenities of the crystal. The corresponding laser lines are accordinglyjlarge at multimode operation 22)... 

The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward S = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. 

The ratio T between the actual intensity of a diffracted beam and the intensity which it would have if there were no thermal vibrations is e, where B is a constant for a particular crystal. B is related... 

Bradley (1935) points out that, since the effect of the absorption factor is opposite to that of thermal vibrations, the two may in some cases cancel each other approximately consequently it may be justifiable to ignore both factors. This naturally applies only to crystals of moderate or high absorption it does not apply to most organic substances, for which absorption is small and thermal vibrations large, so that the effect of the latter far outweighs the absorption effect. 

In setting out to discover the relative positions of the atoms in a crystal, it is best, when the unit cell dimensions have been determined and the intensities of the reflections measured, to calculate F for each reflection. (See Chapter VII.) Absolute values of F, derived from intensities in relation to that of the primary beam, form the ideal experimental materisi, though very many structures have been determined from a set of relative F s. The reliability of the set-of figures depends on the success with which the corrections for thermal vibrations, absorption, and extinction effects have been estimated. 

Molecular rotation In a normal crystal every atom occupies a precise mean position, about which it vibrates to a degree depending on the temperature molecules or polyatomic ions have precisely defined orientations as well as precise mean positions. When such a crystal is heated, the amplitude of the thermal vibrations of the atoms increases with the temperature until a point is reached at which the regular structure breaks down, that is, the crystal melts. But in a few types of crystal it appears that notation of molecules or polyatomic... 

The work of Cox, Cruickshank, and Smith (1958) on the crystal structure of benzene at — 3° C (a little below the melting-point) illustrates well this sort of application of the error synthesis. Fig. 215 shows the error synthesis (or difference synthesis) map in the plane of the benzene ring after a series of refinements in which only carbon atoms were included in the structure amplitude calculations, and thermal vibrations were assumed to be isotropic with a temperature factor B = 6-0 A2. 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

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