A Model for the One-dimensional Crystal

The first (bonding) and the last (antibonding) energy levels of the linear polyene chain are reported with respect to Nin Table 3.1 in units of (3. In the last column of the table is the difference in energy between two successive levels. The asymptotic approach of x towards 2 and of % towards -2 is apparent from the numbers given in the table, as well as is the decreasing distance between two successive levels, which tends to zero for N — oo. 

These results can be easily established in general as follows. Using formula (3.14) for the orbital energy of the kth MO of the N-atom linear polyene chain  

ForN — oo, therefore, the polyene chain becomes the model for the one-dimensional crystal. We have a bonding band with energy 

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So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The interaction parameters z, z, and Ji are defined in the usual way, and t) = /S"//8, where /3" is the resonance integral between nearest neighbors in the adsorbed layer. If rj = 1, the eigenvalue condition. Equation (19), is the same as for the one-dimensional model. The only change is that the discrete localized states (CP and 91) of the one-dimensional model now appear as bands of surface states (CP or 91 bands) associated with the adsorbed layer and the crystal surface. At most, two such bands may be formed, and each band contains levels. This is the number of atoms in the adsorbed layer. Depending on the values of the interaction parameters z and z, these bands may or may not overlap the normal band of crystal states. All this was to be expected, and Fig. 2 gives the occurrence of (P and 91 surface bands when = 1. It is when tj 7 1 (and this will be the usual situation) that a new feature arises. In this case, the second term in the second bracket in Equation (19) does not vanish, and the eigenvalue condition is not the same as in the one-dimensional model. In fact we have z - - 2(1 — jj )(cos 02 - - cos 03) in place of z, and this varies between z - - 4(1 — ij ) and z — 4(1 — tj ). We can still use Fig. 2 if we remember that z varies between these two limits. Then if, for example, this variation... 

It is also shown that the electron-phonon interaction is operative in the polymerization process of the one-dimensional conjugated polymeric crystals a simple dynamical model for the polymerisation in polydiacetylenes is presented that accounts for the existing observations. 

The strict generalization of the one-dimensional model treated in Sec. III,A leads to a crystal with its surface completely covered by adsorbed atoms. For this system, the tight-binding approximation gives a difference equation and boundary conditions which can be solved directly. The results show an important new feature. For a simple cubic lattice, the energy levels are given by the usual equation... 

To establish the fundamental laws of the process of the kinetics of particle accumulation in solids, initially the quasicontinuum model of a crystal was studied [107]. A one-dimensional crystal was represented in the form of a segment containing L cells with periodic boundary conditions (the ends of the segment are closed). The simulation was conducted for different dimensions L of the crystals and magnitudes l of the recombination region. The results of the simulation are given in Table 7.2. 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

Cholesteric liquid crystals are similar to smectic liquid crystals in that mesogenic molecules form layers. However, in the latter case molecules lie in two-dimensional layers with the long axes parallel to one another and perpendicular or at a uniform tilt angle to the plane of the layer. In the former molecules lie in a layer with one-dimensional nematic order and the direction of orientation of the molecules rotates by a small constant angle from one layer to the next. The displacement occurs about an axis of torsion, Z, which is normal to the planes. The distance between the two layers with molecular orientation differing by 360° is called the cholesteric pitch or simply the pitch. This model for the supermolecular structure in cholesteric liquid crystals was proposed by de Vries in 1951 long after cholesteric liquid crystals had been discovered. All of the optical features of the cholesteric liquid crystals can be explained with the structure proposed by de Vries and are described below. 

Particle-in-a-box states for an electron in a 20 Pd atom linear chain assembled on a single crystal NIAI surface. The left set of curves shows the predictions of the one-dimensional parti-cle-in-a-box model, the center set of images is the 2D probability density distribution, and the right of curves is a line scan of the probability density distribution taken along the center of the chain. The chain was assembled and the probability densities measured using a scanning tunneling microscope. 

FIGURE 1 A one-dimensional crystal structure model for a tiny perfect crystallite, illustrating, on the left, the scattering density function for the crystal, the shape function, the infinite structure model, the lattice, and the contents of a single unit cell the Fourier transforms of these functions are given on the right. 

Also worthy of further development is the theory of surface biphonons. The conditions required for the formation of these states are different from those of the formation of surface states for the spectral region of the fundamental vibrations. It was demonstrated on the model of a one-dimensional crystal (26) that situations may exist, in general, in which the surface state of the phonon is not formed and the spectrum of surface states begins only in the frequency region of the overtones or combination tones of the vibrations. 

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