Mean-square displacement amplitudes

Mean-square displacement amplitude The average, in a given direction, of the square of the deviation of the instantaneous position of an atom from its average position. It is generally represented by or v )- The latter is used here. 

The components of the mean-square displacement amplitude matrix (MSDA)... 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

The temperature dependence of the mean-square-displacements of Au adatom in the normal to the surface direction is shown in Figure 4 for the three low-index faces of Cu. We note that up to 500"K the MSD s on the three different faces are almost equal, while at higher temperatures the vibrational amplitudes of Au on Cu(llO) present enhanced anharmonicity and become much larger than on the other faces. These results denote that... 

The mean-square displacements of each of the atoms in the crystal, which affect the the X-ray scattering amplitudes, are obtained by summation over the displacements due to all normal modes, each of which is a function of ea(j Icq), as further discussed in section 2.3. The eigenvalues of D are the frequencies of the normal modes. 

Thus, in the high-temperature limit, the mean-square displacement of the harmonic oscillator, and therefore the temperature factor B, is proportional to the temperature, and inversely proportional to the frequency of the oscillator, in agreement with Eq. (2.43). At very low temperatures, the second term in Eq. (2.51a) becomes negligible. The mean-square amplitude of vibrations is then a constant, as required by quantum-mechanical theory, and evident in Fig. 2.5. 

From the plot of ln(I/I0) versus Q2 the values for the temperatures below Tc were derived. At higher temperatures deviations from linear behavior occur at large Q. Figures 29a and 29b show the values of the mean square vibration amplitude, , as a function of temperature for both copolymers. At low temperature the mean square displacement follows a nearly linear temperature dependence as expected for harmonic vibrations. A stronger and quasiexponential temperature dependence sets in around T = 250 K for the 60/40 copolymer and T = 230 K for the 80/20 copolymer. It should be noted that the temperatures where a deviation from the harmonic behavior occurs corresponds to the glass transition in the rase of both copolymers [6]. We can attribute this behaviour to the appearance of a new degree of freedom in this region. Similar... 

Finally, there is one further source of information on the harmonic force field that has been used occasionally, namely mean square amplitudes of vibration in the various intemuclear distances, as observed by gas-phase, electron-diffraction techniques. These can be measured experimentally from the widths of the peaks observed in the radial distribution function obtained from the Fourier transform of the observed diffraction pattern. They are related to the harmonic force field as follows.23 If < n > denotes the mean square displacement in the distance between atoms m and /t, then the mean amplitudes <2 > are given as the diagonal elements of a matrix 2, where... 

Similarly, comparison of first shell coordination numbers and mean-square displacements along the metal-ligand bond can be obtained by comparing the amplitude functions Aj and Aj of the two systems (Eq. (1))... 

The diffusivity D and [by using Eq. (31)] also the mean square displacement then may be determined straightforwardly from the slope of a logarithmic plot of the spin echo amplitude versus the squared gradient intensity. 

The mean-square displacement of surface atoms should be sensitive to changes in the number and type of neighboring atoms. The adsorption on the clean surface of gases that chemically interact with the surface atoms (e.g., oxygen on nickel or tungsten) strongly affects the vibrational amplitude of surface atoms. 

The Einstein model (Kittel 1968) assumes that the solid is composed of a large number of independent linear oscillators each vibrating at the same angular frequency (o. (Each atom is represented by three oscillators, one for each dimension.) As the temperature increases, the oscillators keep vibrating with the same frequency, however, the amplitude of vibration - and so the mean square displacement of the atoms - becomes larger. The Einstein fi equency coe is associated with the Einstein temperature 6 ... 

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