Atom mean-square amplitude

The intensity of light scattering, 7, for an isolated atom or molecule is proportional to the mean squared amplitude... 

The isotropic equivalent thermal parameters are on the whole larger than in the PbTX-1 dimethyl acetal structure or the structure of the natural product. The B values for atoms on the fused ring skeleton range from 4.7 to 12.6 A (mean square amplitudes of 0.059 and 0.16 A ). Curiously, the largest values are associated with C17-C20 of the 9-membered E ring—the ring that adopts two conformations in crystalline PbTX-1. The acyclic atoms do not have appreciably higher thermal parameters, with the exception of hydroxyl 013, which has a B of 22.4 A 2. 

With h 6) - 1/sin 0)5(0 — Oq), one obtains the same result as given by (4.58), which implies that the anisotropy of the/factor cannot be derived from the intensity ratio of the two hyperfine components in the case of a single crystal. It can, however, be evaluated from the absolute/value of each hyperfine component. However, for a poly-crystalline absorber (0(0) = 1), (4.66) leads to an asymmetry in the quadrupole split Mossbauer spectrum. The ratio of l-Jh, as a function of the difference of the mean square amplitudes of the atomic vibration parallel and perpendicular to the y-ray propagation, is given in Fig. 4.19. 

The frequency vD at the edge of the Brillouin zone is thus equal to vsqD/2n. The Debye temperature 0D is defined as hvD/(kB). As shown below, 0D is an inverse measure for the vibrational mean-square amplitudes of the atoms in a crystal at a given temperature. 

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. 

Uncorrelated motion is most likely to be found when the bonds are weak, for example, when the cation is a large alkali metal. In this case the mean square amplitude, A, is given by the sum of the components of the atomic displacement parameters, U, of the two atoms along the bond direction, that is. 

Fig. 9.3. Rate of increase of the mean square amplitude of thermal vibration with temperature plotted as a function of bond valence. The circles represent the sum of uncorrelated atomic displacements along the bond direction, the lines represent the expected bond vibrational amplitudes calculated from eqn (9.9).

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... 

The mean amplitude of molecular vibration can be calculated from the vibrational frequency [74] and vice versa. For the frustrated rotation of an upright diatomic molecule adsorbed on heavy substrate atoms and a vibrational mode which is doubly degenerate, the mean square amplitude at equilibrium temperature Ts is given by [68]... 

The coefficients eja governing the mathematical transformation from normal coordinates Qa to atomic Cartesian coordinates 7j provide a transparent description of mode character. The vector ej parallels the motion of atom j in normal mode a, while the sqnared magnitude describes its relative mean squared amplitude. Normalization, according to = then ensures that the mode... 

In the mid 1950s Durwaxd W. J. Cruickshank " noted that atoms in, for example, a rotating molecule, are displaced towards the rotation axis. Rotational oscillations of molecules, such as found in crystalline benzene near its melting point, will cause an apparent displacement of atomic positions from their true positions because the best fit to the electron density should be curvilinear but, with the limitations of present-day techniques, is generally linear (Figure 13.15). If the root-mean-square amplitude of libration about an axis is o> (in radians), then the apparent (but not real) shortening of the bond, d, is ... 

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 Mn-O bond vector is closely aligned with the crystal X axis and the anisotropic thermal parameters Bjs and B12 are close to zero. The mean-square amplitude of the oxygen atom along the Mn-O bond vector is then given directly by the parameter Bn, without the need for a co-ordinate transformation. 

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... 

It is generally agreed that thermally induced vibrations of atoms in solids play a major role in melting [2.144]. The simple vibrational model of Linde-mann predicts a lattice instability when the root-mean-square amplitude of the thermal vibrations reaches a certain fraction / of the next neighbor distances. However, the Lindemann constant/varies considerably for different substances because lattice anharmonicity and soft modes are not considered, thus limiting the predictive power of such a law. Furthermore, Born proposed the collapse of the crystal lattice to occur when one of the effective elastic shear moduli vanishes [2.138], Experimentally, it is found instead that the shear modulus as a function of dilatation is not reduced to zero at Tm and would vanish at temperatures far above Tm for a wide range of different substances [2.145]... 

Crystal structure determination requires the development of a suitable model for periodic electron density distribution in the crystal that is related by Fourier transform to the structure factors that can be derived from the experimentally measured diffraction intensities. The model, in practice, is atomistic and consists of coordinates and atom types for all symmetry independent atoms as well as parameters that are used to describe the (mean square amplitude of) displacement of the atoms about their mean positions, which arises due to molecular motions and small variations in the mean position across the collection of unit cells that comprise the crystal. 

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... 

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. 

Here/(ro) is the normalised number of phonon modes of frequency co summed over all phonon modes, s, and wave vectors, q, in the Brillouin zone of the crystal, Cp (ro) is the displacement of the pth hydrogen atom, of mass Mp, in the th mode summed over all the phonons of frequency co, and < Mp > is the mean square amplitude of displacement of the pth hydrogen averaged over all modes. 

---