One molecule in the unit cell

In this section we discuss the solutions of eqn (10.10) for the case of a crystal with one molecule in the unit cell. The results of this section will also be helpful when we consider crystals with two molecules in the unit cell. Let %2 = 0. Then one of the solutions of eqn (10.10) is J1 = c2Q2jec = cu2av(q). It is just the dispersion relation of the bare cavity photon (see eqn 10.2). From the first of eqns (10.7) it follows for this solution that the fields ) and Et are related by Ei = Et tan y . The total electric field Ec in this mode and in our coordinate system reads  

The second solution of eqn (10.10) for the region of the Rabi splitting where u wcav(q) can be found from the equation 

Thus the cavity polariton dispersion has a simple interpretation. Let us calculate the electric fields from eqn (10.7) for the modes (10.16). Neglecting small terms of the order of q2 /n2, the fields Ei and Et are related in these modes by Ei = —Et cot (p. Then the y-component of the fields Eu,l is equal to zero. In other words, with accuracy up to small terms (of the order of q2 /k2) the total in-plane electric field in the polaritonic modes is parallel to the dipole moment Pi for any direction of the wavevector q, and the value of the Rabi splitting energy thus does not depend on the wavevector direction. 

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In a crystal of hexamethylbenzene (triclinic, with one molecule in the unit cell) the vibration direction for the lowest refractive index is almost exactly normal to the 001 plane (Bhagavantam, 1930), indicating that... 

The two molecules in the unit cell are then related by symmetry, consequently only two packing parameters (y and w) defining the orientation and height of one molecule in the unit cell are needed. 

The absorption spectrum, in the 2800 A region, of single oriented crystals of hexamethylbenzene has been studied at room temperature (Nelson and Simpson, 1955) and at 20°K (Schnepp and McClure, 1957). Single crystals of hexamethylbenzene are particularly favourable for such study because of their simplified geometry. The crystals are triclinic with one molecule in the unit cell, so that all molecules in a single crystal are identically orientated (Fig. 5). 

Copper(I) nitride, Cu3N, crystals are also cubic, OPm3m, a0 = 3.814 A, with one molecule in the unit cell. Each N has an octahedral arrangement of six Cu atoms. Cu atoms have linear bonds to two N atoms and eight nearest Cu neighbors (see Figure 5.24). These are two sets of four planar Cu atoms bonded to the N atoms. The N atoms occupy sites for a simple cubic cell. To accommodate Cu and N in the P system, Cu atoms occupy P sites in an ABC sequence with N atoms in O sites. The P sites are three-quarters occupied and O sites are one-quarter occupied to give the notation 3 2P3/4Oi/4. 

Fig. 5. Orientational probability distribution of the molecular axes for the delocalized (a) and localized (b) mean field states in /3-nitrogen, (a) applies to both molecules in the unit cell, (b) is drawn for one molecule in the unit cell the other molecule in the cell is rotated over = 180°. The distribution does not change qualitatively up to (at least) T = 70 K it just becomes slightly wider with increasing temperature. Reading of the contour plot as in Fig. 2.

To demonstrate the mechanism of Bethe splitting in molecular crystals we assume that a unit cell contains only one molecule but the term / under consideration is ro-fold degenerate. In the case being considered, quite analogous to that of the equation system (2.19), we obtain a system of ro equations for coefficients u r The solution of the secular equation shows that in crystals with one molecule in the unit cell ro excitonic bands appear which correspond to an ro-fold degenerate molecular term. 

In his work on this topic Frenkel considered crystals with one molecule in the unit cell. Crystals with molecular cells containing a few molecules, in the context of excitonic properties, were first investigated by Davydov (11). 

To demonstrate this effect we first consider an exciton in a crystal with one molecule in the unit cell having a nondegenerate molecular term /. In this case the exciton energy, in the Heitler-London approximation, is given by the relation (2.12). 

An analysis of 25,322 organic molecules with just one molecule in the unit cell from the CSD showed that there is a low degree of correlation (r < 0.42) between 2D and 3D shape measures, even for molecules with no conformational flexibility the graph-theoretical (2D) and geometrical (3D) shape descriptors / were defined as... 

Fig. 25. Restriction plots calculated from the experimental spectral data associated with one molecule in the unit cell of NAV (A) N overtone shift, (B) N/ H dipolar splitting, (C) H chemical shift, (D) H/ N dipolar splittings. (E) the intersection of all four types of experimental data, (F) the intersection of the data obtained with a N separated local field experiment alone, (G) the intersection of the data with the N overtone shift and H chemical shift alone, and (H) the intersection of the data obtained with N overtone shift and H/ N dipolar splitting alone. (Reproduced ftrom McNamara et with permission.)...

A common complication in ESR and ENDOR single crystal studies is the occurrence of different spectra due to identical paramagnetic species that are differently oriented due to crystal symmetry. The species are said to be located in different sites, and the overlap of spectra due to different orientations is referred to as site splitting. In the triclinic crystal system, e.g. for the malonic acid discussed in Chapter 2, there is only one molecule in the unit cell and no site-splitting occurs. Two symmetry-related sites with coordinates (x, y, z) and (-x, y, -z) give identical ESR spectra when the crystal is rotated about the y axis. The result is plausible since the magnetic field... 

Splitting, which is commonly observed in the spectra of crystals with more than one molecule in the unit cell. Obviously, this phenomenological characteristic of weak coupling is restricted to systems with distinct vibrational structure. A detailed consideration leads to the important result that weak coupling exists only in these systems. ... 

The presence of more than one molecule in the unit cell oriented in a different way leads to splitting of the IR band according the Bethe theory and the submaxima possess different polarization. In these cases the eUmination is obtained to an inflex point as shown in Figure 2.26, where the consequent elimination of the submaxima at 1738 cm- in 4 -cyanophenyl-4-n-heptylbenzoate is obtained. 

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