Temperature factor distribution functions

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

These simple values for the first few coefficients are the result of our choice of number density and temperature as multiplicative factors in the zero-order distribution function, and of defining the expansion variable in terms of the mean velocity and temperature. 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

The studies of recombination luminescence for samples irradiated at one temperature, 71, but then stored at another, T2, allow one, in some cases, to determine the activation energy for tunneling recombination. An instantaneous temperature jump is expected to produce no change in the distribution function over the distances between the reacting particles. When the temperature jump results only in a change of the frequency factor v but not of the parameter a in the expression for W(i ), then the ratio of the luminescence intensities for any fixed moment of time is proportional to the ratio of the corresponding v values, i.e. 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

As mentioned at the beginning of this section, the temperature factor absorbs other unaccounted or incorrectly accounted effects. The most critical are absorption, porosity and other instrumental or sample effects (see Chapter 3), which in a systematic way modify the diffracted intensity as a function of the Bragg angle. As a result, the B parameters of all atoms may become negative. If this is the case, then the absorption correction should be re-evaluated and re-refined or the experiment should be repeated to minimize the deleterious instrumental influence on the distribution of intensities of Bragg peaks. 

Pre-exponential factor independence of the coverage of heterogeneous surface has been postulated at the above calculation of distribution functions on chemisorption activation energies on the basis of kinetic isotherms measured at different temperatures. However, it appeared to be in contradiction with some experimental data and theoretical conclusions [128]. Compensation relationship is commonly observed for the catalytic processes proceeding on the heterogeneous surface [129]... 

The best way to correct determination of chemisorption activation parameters on heterogeneous oxides surface includes the calculation of its both distribution functions on the activation energy and on logarithms of the pre-exponential factor and their central moments from the kinetic isotherms measured at different temperatures. Obtained from such calculations the isokinetic temperature and logarithm of rate constant at this temperature reflect the gaseous organic compound reactivity toward active sites of the heterogeneous oxides surface. 

Figure 2.9 Electron relaxation dynamics for GaAs (100). (a) Compares the hot electron lifetimes as a function of excess energy (above the valence band) of a pristine surface prepared using MBE methods with device-grade GaAs under the same conditions. The higher surface defect density of the device-grade material increases the relaxation rate by a factor of 4 to 5. (b) The electron distribution as a function of excess energy for various time delays between the two-pulse correlation for MBE GaAs. The dotted lines indicate a statistical distribution corresponding to an elevated electronic temperature. The distribution does not correspond to a Fermi-Dirac distribution until approximately 400 fs. The deviation from a statistical distribution is shown in (c) where the size of the error bars on the effective electron temperature quantifies this deviation.

The inadequacy of the Debye approximation in describing the details of the frequency distribution function in a real solid is well known. This results in noticeable disparities between Debye temperatures derived from the results of different experimental techniques used to elucidate this parameter on the same solid, or over different temperature ranges. Substantial discrepancies may be expected in solids containing two (or more) different atoms in the unit cell. This has been demonstrated by the Debye-Waller factors recorded for the two different Mdssbauer nuclei in the case of Snl4,7 or when the Debye-Waller factor has been compared with the thermal shift results for the same Mdssbauer nucleus in the iron cyanides.8 The possible contribution due to an intrinsic thermal change of the isomer shift may be obscured by an improper assignment of an effective Debye temperature. 

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