Structure factor amplitude Temperature parameter

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

Optical response in OPC Excess Rayleigh ratio Radius of gyration Siuface area Power spectrum Entropy function Structure factor Scattering amplitude function Absolute temperature Pulse amplitude Stokes parameter Velocity Volume... 

Figure 2.24 shows the temporal evolution of the compositional structure factor S<(, (a) and the orientational structure factor Ss (b) for the temperature quench into the lu region (T/Tn, = 0.6, (f)Q = 0.55) in Figure 2.22. The structure factor for concentration has a maximum at q, which corresponds to the peak wavenumber of coi(q). With time the corresponding mode grows exponentially and the peak position qm is invariant. Then the time evolution of the structure faaor S<, is lhe same as that of the Cahn-Hilliard theory for isotropic SD [102]. The amplitude of the peak at q = 0 decreases with time because s > 0 and another peak appears at q. In this quench, the concentration fluctuation initially induces the SD and the orientational ordering within the domain subsequently takes place due to the coupling between the two order parameters concentration-induced SD.

The C60 molecules were found to be executing large amplitude reorientations at room temperature, so that large anisotropic thermal displacement factors of the C60 carbon atoms were found. The thermal displacement parameters for some of the C60 carbon atoms at room temperature are, in fact, so large that the C60 atomic coordinates may well represent only an average over one or more disordered structures involving fractional atomic occupancy. On the other hand, the TDAE N and C atomic coordinates are well-defined already at room temperature. 

Rigid Molecule Group theory will be given in the main part of this paper. For example, synunetry adapted potential energy function for internal molecular large amplitude motions will be deduced. Symmetry eigenvectors which factorize the Hamiltonian matrix in boxes will be derived. In the last section, applications to problems of physical interest will be forwarded. For example, conformational dependencies of molecular parameters as a function of temperature will be determined. Selection rules, as wdl as, torsional far infrared spectrum band structure calculations will be predicted. Finally, the torsional band structures of electronic spectra of flexible molecules will be presented. 

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