Angular force

The same effect can be seen in the zig-zag chain of Fig.. 3-11. It is remarkable that we can compute the angular force constant in that model exactly, as well as in the Bond Orbital Approximation (see Problems 8-1 and 8-2). The results turn out to be identical for the homopolar semiconductors, but for polar semiconductors, the exact solution has a, replaced by . Sokel has shown that the result is not so simple for the tetrahedral solid, but turns out quantitatively to be very close to an dependence. We will also find an ot dependence when we treat tetrahedral solids in terms of the chemical grip in Section I9-F. This suggests the approximation to the full calculation,... 

The angular forces arc considerably weaker. However, they are strong compared to angular forces in other types of solids and they represent the rigidity... 

The change in the second is equal and opposite. There are equal changes in two of the angles centered at each atom of the other type, and in our model we have used the same angular force constant for each so that the total angular energy ehange per atom pair is... 

From an expansion of this energy for small deviations from tc/2 (or from the second derivative of the energy with respect to ip), obtain the angular force constant C1, defined in analogy with Eq, (8-18), such that the total energy change for the /V atoms is equal to... 

The large difference between the angular force constants Ci determined in the two ways appears to be the principal defect in application of the valence force field theory to the treatment of covalent solids. The defect is not readily repaired by the... 

It is not clear whether the discrepancies here come from errors in the parameters used or from the approximations used. The physical origins of the bending and the rigidity are presumably those given here. In Section 11-F we shall add an angular force at the silicon, and we shall discuss the vibrational spectra. 

It is possible to make a valence force field analysis on this basis, by introducing radial forces between nearest neighbors and angular forces at the silicon and at the oxygen. For this, the model described earlier by Kleinman and Spitzer (1962) is appropriate. The three force constants are fitted by matching the highest three observed infrared-active modes with the frequencies and Then, these... 

We first show that the second-order term in Eq. (19-20) does not lead to angular forces. We focus on a particular titanium ion. For each of the d states, y, we must sum over the ncigliboring oxygen ions, but these are independent sums. We may have initially constructed our d states as listed in Table 19-2, with respect to some laboratory coordinate system, but we can reexpress them in terms oft/ states defined with respect to a coordinate system with z-axis in the direction of the oxygen ion being considered, The details are not crucial here but will be when we go to fourth order, so the point should be staled carefully. (More detail is given by Rose, 1957, or Weissbluth, 1978.)... 

To test whether the diamond structure for C, when randomized and annealed according to the WWW algorithm, would yield ta-C and not just recrystallize because of the strong angular forces for sp -bonding in diamond, a 216-atom model [3] was constructed with the force constants appropriate to diamond a = 1.293 x 10, /3 — 0.8476 x 10 dyn/cm. Even increasing by a factor of 10 did not result in recrystallization. 

The vAl-O-Al IR band was used to study the effects of the caustic ratio on the structure of sodium aluminate solutions.118 Variable-temperature Raman spectra of the spinel MgAl204 showed a band at 727 cm" 1 for samples annealed above 800°C. This was ascribed to cation disorder in these samples.119 The angular force constant method was used to analyse vibrational modes for MA1204, where M = Mg or Zn.120 Ab initio calculations have been made of vibrational wavenumbers involving the OH group in topaz, Al2Si04(0H)2.121... 

As will be shown in coming sections, wide classes of total energy descriptions fall under this general heading including traditional pair potential descriptions, pair functionals, angular force schemes and cluster functionals. 

As we have noted above and will see in more detail later, the pair potential description is littered with difficulties. There are a few directions which can be taken that will be seen to cure some of them. We begin with a qualitative discussion of these issues, and turn in what follows to the development of both angular force schemes and effective medium theories that patch some of these troubles, primarily through the device of incorporating more of what we know about the nature of bonding in solids. 

Fig. 4.11. Schematic of the triangular lattice used in computing the total energy using angular forces.

While these potentials are notationally complex, they are conceptually simple in terms of the basic physics they incorporate. In particular, potentials of this type invoke both of the new physical ideas presented in this section environmental dependence and angular forces. In particular, the quantity denoted bij guarantees that the interaction between two given atoms will depend upon both the number and disposition of neighboring atoms. For our purposes, the key advantage of the Tersoff potential is its attempt to capture effects which are outside of the purview of more traditional potentials. 

As a preliminary to our discussion of the emergence of both pair functionals and angular forces from tight-binding theory, we first discuss the moments of the density of states which serve as a useful basis for these developments. We begin with the recognition that every distribution has its moments. In particular, the n moment of the density of states is defined by... 

Moments and Angular Forces. Our aim in the present setting is to exploit the geometry of the densities of states in order to construct effective potentials for describing the bonding in solids which go beyond the pairwise forms considered thus far. We follow Carlsson (1991) in formulating that description. First, we reiterate that our aim is not the construction of the most up-to-date potentials that... 

Fig. 6.9. Energy of various competing structures in Si as computed using a variety of different total energy methods (a) density functional results of Yin and Cohen (1982), (b) pair potential results of Ackland (1989), (c) angular force results of Stillinger and Weber (1985), (d) empirical results of Tersoff (1988), (e) tight-binding results of Mercer and Chou (1993) and (f) density functional results of Needs and Mujica (1995).

Vacancy Formation Energy using Angular Forces... 

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