Thermal vibrations of atoms

To = a constant characterizing the thermal vibrations of atoms inside the molecules ... 

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 ... 

All these errors can be properly corrected by appropriate corrections and experimental techniques (92), (e.g., low-temperature measurement and measurement of symmetry-related reflections). In particular, low-temperature measurements are essential to reduce the thermal smearing of the deformation density due to the thermal vibration of atoms. 

Thermal vibrations of atoms which have a frequency of about 10 second" are slow compared to the X-ray frequency, which is about 10 second". Consequently, the atoms appear to be stationary to the X-rays and the diffraction pattern represents a time average of many instantaneous states. If the motion of the atoms is harmonic so that the restoring force is proportional to the distance of the atom from its rest position and if the motion is isotropic so that the mean square displacements of the atom in all directions are the same, then is related to the temperature factor, B, by... 

The thermal vibrations of atoms in molecules lead to absorption bands in the infrared (IR) region (Bellamy, 1964 Colthup et al., 1964 Hesse et al., 1984). IR bands are most intense if a dipole is induced by the vibration (OH, NH, CH, C=0, C=N). The mass of the interacting atoms Ml and M2 and the bond strengths defined by a force constant f determine the wave number n or energy of an infrared absorption band v = K(f/M ) , where the reduced mass is M = MjM2/Mj+M2 and K is a constant conversion factor. The frequency n for the CH-stretch vibration is around 2900 cm for C=0 close to 1700 cm. Hydrogen bonds lead to a broadening and low-frequency shift of OH and NH vibrations (3400 cm —> 3200 cm ). 

The channeling effect can be seen most clearly in tungsten, which is characterized by small-ampUtude thermal vibrations of atoms. Consequently, dechanneUng of particles wiU be weak. In this case, the three sections corresponding to the three types of trajectory shown in Fig. 8.5 are quite pronounced on the concentration distribution curve, as shown in Fig. 8.5. Particles scattered by angles on... 

Allowance for the Thermal Vibrations of Atoms. Up to now, in studies of secondary electron scattering we have assumed that atom positions in... 

According to Hosemann [59,60], the lattice distortions, in addition to those due to thermal vibrations of atoms, may be classified as of a) first kind if the long-range periodicity is preserved with respect to the average positions over all the lattice points, and of b) second kind if the position of each lattice... 

The difficulty of experimental determination of atomic radii comes from several reasons, mainly the blurring effect of thermal vibrations of atoms and the extreme complexity of theoretical interpretation of the experimental data [204]. Johnson [205, 206] noted that electron density of an atom in a metallic structure shows a... 

Obviously, thermal vibrations of atoms in a solid are strongest on the verge of melting. Sutherland was the first (1891) to suggest that melting occurs when the amplitude of vibrations reaches a certain fraction (equal for all the elements) of the atomic size [13]. In 1910, Lindemann [14] developed this idea and related the critical amplitude to the temperature of melting (Tm) and atomic oscillation frequency v proportional to the characteristic Debye temperature ( ). In its modern form [15] the Lindemann s rule states that a material melts at the temperature at which the amplitude of thermal vibration exceeds a certain critical fraction of the interatomic distance, and this fraction depends somewhat on the crystal structure, position in the Periodic Table, and perhaps other unspecified physical quantities. These works initiated numerous... 

Surface phonon bands along symmetry lines of the SBZ are given for fee metals in Figs. 5.2-49-5.2-55 and in Table 5.2-20. In all figures the horizontal axis is the reduced wave vector, expressed as the ratio to its value at the zone boundary. Table 5.2-21 gives the surface Debye temperatures for some fee and bcc metals, as well as the amplitudes of thermal vibrations of atoms in the first layer p as compared with those of the bulk pb-In the harmonic approximation, the root mean square displacement of the atoms is proportional to the inverse of the Debye temperature. 

Thermal noise Electric noise power of a device or component that is due to thermal vibration of atoms, molecules, and electrons. It is proportional to a product of the frequency bandwidth of interest and the absolute temperature of the device or component. The proportionality constant is Boltzmann s... 

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. 

Temperature can also influence the magnetic characteristics of materials. Recall that raising the temperature of a solid increases the magnitude of the thermal vibrations of atoms. The atomic magnetic moments are free to rotate hence, with rising temperature, the increased thermal motion of the atoms tends to randomize the directions of any moments that may be aligned. 

ROOT-MEAN-SQUARE AMPLITUDE OF THERMAL VIBRATIONS OF ATOM COMPLEXES... 

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