Amplitude of vibration, mean-square

Reduction by hydrogen completely alters the chemical reactivity and its variation with size. At the same time, the Pt particle size is reduced. The mean-square amplitude of vibration follows this reactivity. 

Mean square amplitudes of vibration were calculated by Baran (14) based on the frequencies and estimated geometry reported by Smith et al. (270). The UV absorption spectrum of FCIO2 was studied by Sicre and Schumacher (264) and Pilipovich et al. (228). From a mass spectro-... 

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. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

In the classical approximation, which is sufficiently accurate at room temperature and above, the mean square amplitude of vibration of a bond, A ), at... 

A ) cannot be measured directly, but the mean square amplitudes of vibration of the individual atoms, U, can be found using X-ray or neutron diffraction. Ai ) can be determined from U only if we know both the amplitudes of the atomic vibrations along the direction of the bond and how they are correlated. While the amplitudes are readily measured, the correlations between their motions are unknown. There are, however, two limiting cases, the first when the bond connecting the two atoms is strong and rigid so the atoms always move in phase, and the second when bond is weak so that the atomic motions are uncorrelated. 

Mossbauer spectroscopy should also be mentioned here as a very promising method for combining the structural and dynamic studies of biomolecular systems. The asymmetry of Mossbauer spectra caused by the anisotropy of vibrations of Mossbauer atoms allowed—for example, to find that the mean square amplitude of vibrations of Fe atoms normal to the plane of porphyrin ring (which are responsible for many important biological functions of hemoproteins) is about five times larger than in the... 

T12S04, Cs2W04, Cs2S04 and Cs2Mo04 all385 possess a structure which is presumably close to the D2(i structure (86). The extraordinary large root-mean-square amplitudes of vibration observed for these four compounds have been inter-... 

Long, D. A., and E. A. Seibold Root-Mean-Square Amplitudes of Vibration in Some Group 4 Tetrahalides. Trans. Faraday Soc. 56, 1105-1109 (1960). 

The mean square amplitude of vibrations have been calculated for these ions 550), and Bastiansen-Morino shrinkage effects reported 767, 768). 

Essentially the characteristic temperature is a measure of the temperature at which the atomic heat capacity is changing from zero to 6 cal deg for silver (0 = 215 K) this occurs around 100 K, but for diamond (0 = 1860 K) with a much more rigid structure, the atomic heat capacity does not reach 5 cal deg i until 900 K. Those elements that resist compression and that have high melting points have high characteristic temperatures. Equations have been derived relating y/ u ) to the characteristic temperature 0. At room temperature diamond, with a characteristic temperature of 1860 K, has a root-mean-square amplitude of vibration, / u ) of 0.02 A, while copper and lead, with characteristic temperatures of 320 and 88 K, respectively, have values of 0.14 and 0.28 A for (u ). - Similar types of values are obtained for crystals with mixed atom (or ion) types. For example, average values of / u ) for Na+ and Cl in sodium chloride (0 = 281 K) are 0.14 A at 86 K and 0.23 A at 290 K. ° ... 

The loss of intensity with increasing temperature in the quadrupolar d—d transitions of the tetrahedral halide complexes [C0X4 ] is accommodated by a calculation of the mean-square amplitudes of vibration in the electronic ground state as a function of temperature for the four tetrahedral modes ). The observed temperature dependence of the d—d transitions to the Ti (F) and Ti (P) excited state from the... 

Results on a large number of linear-chain complexes of platinum are summarised in Table 11. Harmonic wavenumbers and anharmonicity constants have been determined in all cases. The normal coordinate seems to be related to the halogen movements involved in the proposed hopping process for the conductivity of these linear-chain mixed-valence complexes (95). The chain halogen atoms would need to move, on average, 0.54,0.38 and 0.22 A for chlorides, bromides and iodides, respectively, in order to reach the point midway between the two platinum atoms, i.e. to the situation of a platinum (III) chain. These values only differ by a factor of about two from the root-mean-square amplitudes of vibration of Vi in the Vj = 16 states these are calculated (91) to be 0.22 A for X = Cl (wi = 319.5 cm-i) and 0.20 A for X = Br (cji = 179.6 cm ). These distance changes are related to the shift in the equilibrium... 

Let us also note that the mean square amplitude of vibration ((2s 2) ), as given in eq. (5.A43) and which can be easily calculated in the case of a harmonic vibration governed by the Hamiltonian Hq(Q ) of eq. (5.A44), increases with temperature T. Adding corrective quadratic terms in g, to the development of co in powers of g that is writing... 

Mean square amplitudes of vibration determined from gas-phase electron diffraction studies. 

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... 

L is the usual matrix relating R to Q, R = LQ, and A is a diagonal matrix of the thermally averaged mean square amplitudes of vibration in the normal modes ... 

When Up is diagonalized, one has the three principal components of the mean square amplitude of vibration for nucleus p. 

A valence density map of uracil is shown in Figure 3. Notice that contrast between O and N is clearly evident, but the density about the protons and C nuclei does not appear as a distinct maximum. For data that extend to v 1.0 A-1 in sin0/X and for root-mean-square amplitudes of vibration larger than 0.14 A, these valence density maps should be free from series termination error.15... 

In Eq. (1), k is the photoelectron wave vector relative to Eq (k = 0) N is the the number of neighboring atoms of the same kind at a distance r., of is the mean-square relative displacement (MSRD) of the absorber-scatterer atom pair from their equilibrium inter-atomic distance or in molecular spectroscopy terminology, the mean-square amplitude of vibration other terms have their usual meaning Using standard Fourier transform and curve fitting procedures, we can derive the coordination number, bond length and local dynamics (MSRD) from EXAFS. 

T = exp [-27i (UiihV +U22k b +U33l c +2Ui2hka b cosY +2Ui3hla c cosp +2U23klb c cosa )] where Uy values are the thermal parameters denoted in terms of mean-squared amplitudes of vibration. 

The basic tool for the measurement of the mean-square amplitude of vibrations is X-ray technique. The energy of the diffracted X-rays is reduced due to independent heat atom vibrations. 

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