Force derived from lattice modes

As pointed out in Section IIB, it is possible to approach the lattice dynamics problem of a molecular crystal by choosing the cartesian displacement coordinates of the atoms as dynamical variables (Pawley, 1967). In this case, all vibrational degrees of freedom of the system are included, i.e., translational and librational lattice modes (external modes) as well as intramolecular vibrations perturbed by the solid (internal modes). It is then obviously necessary to include all intermolecular and intramolecular interactions in the potential function O. For the intramolecular part a force field derived from a molecular normal coordinate analysis is used. The force constants in such a case are calculated from the measured vibrational frequencies, The intermolecular part of O is usually expressed as a sum of terms, each representing the interaction between a pair of atoms on different molecules, as discussed in Section IIA. 

From the frequency of the transverse optical mode in a simple AB lattice with k = Q a force constant can be derived which is a measure of the restoring forces experienced by the atoms as they are distorted from the equilibrium position. This force constant, Fflattice), is a linear combination of internal force constants, since in a lattice a linear combination of equilibrium distances and angles yields a coordinate of this vibration. Based on this assumption, the GF method (Wilson et al, 1955) can be applied. For diamond (or zinc blende), the following relation is obtained ... 

The largest uncertainty in the calculation of partition function ratios for minerals is the magnitude of frequency shifts on isotope substitution. Because direct spectroscopic measurements of minerals made with the less abundant isotope are not widely available, these shifts must usually be calculated or estimated in some other way. The detailed approach to calculating frequency shifts employs the methods of lattice dynamics to determine force constants for each vibrational mode, from which the vibrational frequencies for a mineral and its isotopic derivative can be predicted (e.g. Bottinga 1968 ... 

The reaction scheme of Bode [11] was derived by comparison of the X-ray diffraction patterns of the active materials with those for the model compounds. How the 8-Ni(OH)2 in battery electrodes differs from the model compound is discussed in Section 5.3.I.3. In recent years, the arsenal of in situ techniques for electrode characterization has greatly increased. Most of the results confirm Bode s reaction scheme and essentially all the features of the proposed a/y cycle. For instance, recent atomic force microscopy (AFM) of o -Ni(OH)2 shows results consistent with a contraction of the interlayer distance fiom 8.05 to 7.2 A on charge [61-63]. These are the respective interlayer dimensions for the model a-Ni(OH)2 and y-NiOOH compounds. Electrochemical quartz crystal microbalance (ECQM) measurements also confirm the ingress of alkali metal cations into the lattice upon the conversion of a-Ni(OH)2 to y-NiOOH [45,64,65]. However, in situ Raman and surface-enhanced Raman spectroscopy (SERS) results on electrostretching modes that are consistent with a weakening of the O-H bond when compared with results for the model a- and 8-Ni(OH)2 compounds [66]. This has been ascribed to the delocalization of protons by intercalated water and Na ions. Similar effects have been seen in passive films on nickel in borate buffer electrolytes [67]. 

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