Crystals with one molecule per unit cell

We will compute the eigenvalues of AH for the cases of a crystal with one molecule per unit cell and in the general case separately. 

In crystals with one molecule per unit cell the indices n, m become the crystal latttice vectors n and m. Diagonalization of the excitation energy operator 

In consequence, the operators B (k) appear as Bose operators of creation of states with quantum numbers / and wavevectors k. The operators Bf(k) are annihilation operators of those states. 

The operator (3.39) is diagonal with respect to the operators Nf = B (k)B/(k) being the occupation numbers of excitonic states k/. Therefore the eigenfunction of (3.39) has the form 

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To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

In the special case of crystals with one molecule per unit cell eqns (3.97) have the simpler form... 

As a second example of the application of the above described theory we consider crystals with one molecule per unit cell. We wish to consider polaritonic states with frequencies near to the nondegenerate electronic excitation of the molecule, where the transition from the ground to the excited state is dipole allowed. 

As was shown by Born and Huang (4) in cubic crystals with one molecule per unit cell the tensor Q"f is reduced to the scalar QtJ = (An/ yv)dlJ where v is the volume of the unit cell, if we ignore spatial dispersion. Moreover, a - = aSij so that eqn (5.3) yields the tensor Ai3 = Adij where A = [1 — (47ra/3v). Substituting this expression into eqn (5.6) we obtain c,3 = eSl3 where... 

In this approximation (the additive refraction or mean polarizability approximation) eqn (5.42) with the addition of eqn (5.37) fully determines the dependence of the dielectric constant tensor on the impurity concentration c. The optical properties of mixed crystals with large impurity concentrations are discussed in Section 5.6. As we did above for crystals with one molecule per unit cell, we shall discuss here the case of small values of c when we can ignore terms of the order of c2, c3, etc. in the expansion of the tensor (5.42) in powers of c. Then we obtain ... 

Crystals with one molecule per unit cell are most easily analyzed. Here eqn (6.24) is of the form... 

For the discussion of the excitonic spectrum in a one-dimensional molecular crystal (with one molecule per unit cell) we use the following Hamiltonian ... 

The crystal structure of Pb2Li7 is hexagonal with one molecule per unit cell, a = 4.751 and c = 8.589 A. Figure 9.40 shows a projection of the unit cell. The heights along c are ... 

The crystal structure of Ni2Al3 is hexagonal, with one molecule per unit cell, D3d, C 3m, a0 = 4.036, and c0 = 4.901 A. The structure is the same as La203 (Section 5.5.10) with different spacings. A projection of the cell is shown in Figure 9.41a. Aluminum atoms alone are in A positions, with Ni and A1 in B and C positions. These layers are filled. The heights along c0 are ... 

In discussing infrared absorption spectra, we refer, first of all, to the papers by Dows and Schettino (54) and of Schettino and Salvi (55). Dows and Schettino (54) investigated the CO2 crystal spectrum in the frequency region corresponding to the combination tone of the intramolecular vibrations v and 1/3 u /y3 Ri 3720 cm-1). Schettino and Salvi (55) measured the infrared (IR) spectra of N2O and OCS crystals. The CO2 and N2O molecules are linear, have no permanent dipole moments, and form a simple cubic lattice upon crystallization. This lattice has four molecules per unit cell, which are oriented along the axes of a tetrahedron. The OCS molecule is also linear, but it forms a crystal of the trigonal system with one molecule per unit cell. 

Assume a crystal with four molecules per unit cell is prepared with a molar purity of 99.999%. If the lattice constants are all 5.68 A, what is the number of impurity molecules in one cm sample ... 

A crystal structure usually is described by the unit cell dimensions, space group and coordinates of the atoms (or orientation and position of the molecules) in the asymmetric unit. This, in fact, is the order in which the information is obtained when a crystal structure is determined by X-ray or neutron diffraction experiments. However, an equivalent way to describe a structure is to place the center of a molecule at the origin of an orthogonal coordinate system and to specify its molecular surroundings. This alternative is especially powerful in crystals with one molecule per asymmetric unit because the orientations of the surrounding molecules are related to the central molecule by crystallographic symmetry. The coordination sphere or environment of the structure then is defined as those surrounding molecules which are in van der Waals contact, or nearly in contact, with the central molecule. 

A crystal with only one molecule per unit cell does not possess translational vibrations. 

In order to exemplify of the formulas (3.126) and (3.127) we consider crystals of anthracene type with two molecules per unit cell. In such crystals one usually considers a mixing of the first excited state (1) with the second excited state (2). Since the oscillator strength of the first excited state is much smaller than those of the second excited state, the role of mixing is more relevant for the lower electronic transition (for details, see Section 3.10 of this chapter). 

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