Potential Born-Mayer

As mentioned previously, some texts prefer to consider only nearest interactions for the repulsive term and write it in the form Ur =B exp(—/3r). 

Taking derivatives to obtain the equilibrium energy and configuration as before, we find that 

Comparison of Bom-Mayer potential (dashed line) with the Born potential (1/r ) for the repulsive term (Equation 3.6) (solid line) with ro/3 = n. Only the region around r=ro is shown because this is where the difference between the two is the largest. 

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Equation 5 is often used to decribe the interaction between the incoming ion and the target atoms. The interaction between two target atoms generally occurs at low energy where the Thomas-Fermi potential overestimates the interaction. Under this situation a Born-Mayer potential is more appropriate , i.e. ... 

As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipole-and EE-based models of molecular systems often use the Lennard-Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born-Mayer potentials, especially when ionic systems are being modeled. 

The first term is the classical interaction between point charges and the second term is a repulsive Born-Mayer potential. 

The form that (p p, the harmonic force constant, takes depends on the nature of the interaction between atoms k and k in the crystal. Although the interactions are, in fact, quite complex, the assumption of effective two-body interactions such as a Born-Mayer potential... 

Dolan et al. [ 176] and Rinzler et al. [ 142] presented a semi-empirical approach based on a single configuration coordinate model (see Sect. 3.1) for separating the effects of local compressibihty and crystal field strength on pressure-induced changes in the T2 energy of Cr + in several fluoride elpasohte systems. Their approach is based on an empirical form, motivated by the Born-Mayer potential, for the local compressibility k(P) ... 

Our study was extended to the binary system silica-calcia. The parameters of the Born-Mayer potential (Equation 3.4.2) were taken from Ref. [15]. The interatomic potential was calculated from the potential used for pure silica performed by TTAM [3] and derived from the ab initio Hartree-Fock self-consistent field calculations for model clusters of silica. The parameters used in Equation 3.4.2 are summarized in Table 3.4.2. 

Table 3.4.1 Born-Mayer potential used in Equation 3.4.2 for pure silica liquid...

For both Morse and Born-Mayer potentials —Vq z) = 2Wi z). Hence for the Bom-Mayer potential... 

Third-order elastic constants of LaSe at 0 K were calculated using the Born-Mayer potential model. The repulsive interaction was considered up to the second nearest neighbors. The interatomic distance To = 3.030A leads to the values in lO N/m ( lO dyn/cm ) Cin = -21.439, c°i2 = Cii6=-1.860, C123 = cJse = C144 = 0.743. The temperature dependence for is given by = + where are (in lO N-m K ) am =6.837, an2 = 3.601,... 

In Problem 5.6.4, the Griineisen parameter for the q = 0 optical modes of crystals with NaCl structure is evaluated on the basis of the Born-Mayer potential. 

Using (4.56,62,65,66,72), show that in terms of the volume independent parameters b and p of the Born-Mayer potential (4.19), the TO and LO-mode frequencies for q = 0 of the NaCl structure are given by... 

As the C-H acid molecules approach some base, their equilibrium distance is determined by the sum of the corresponding van der Waals radii. In this case, the distance between two equilibrium positions of the proton is equal to about 1.6 A. For such a tunneling distance, the exponential term in (4.14) turns out to be extremely small, i.e. 10 . Clearly, the proton transfer reaction can take place only at much shorter distances for which the tunneling probability sharply increases in accordance with (4.14). However, the molecules are hindered from coming close to each other by the repulsive forces which sharply increase with decreasing distance. These forces may be described by the Born-Mayer potential[446] ... 

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