Chapter 3

Elastic Constant Relations

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In 1914 I had published a paper on diamond. I assumed two kinds of forces, a radial force between two nearest neighbours and an angular force involving three neighbours. Therefore, I had two independent atomic constants. But the (cubic) crystal has three elastic constants; therefore the theory provided one relation, namely 4 C₁₁ (C₁₁ - C₄₄)/(C₁₁ + C₁₂)^2 = 1. At the time no measurements of the elastic constants of diamond existed. I had to wait 31 years. Then, in 1945 in Edinburgh, I learned about new supersonic methods to measure elastic constants. Remembering the old formula, I wrote to my friend Franz Simon... and suggested to him to put one of his pupils on to this problem. Before I had finished this letter, the postman brought me my mail which included a paper by the Indian physicist Bhagavantam. It contained just these three measurements. Inserting his figures into the formula I obtained instead of 1 on the right hand side, the value 1.1 -- quite a satisfactory confirmation.... This paper started off a series of investigations... establishing relations between macroscopic constants by making simple, natural approximations about the lattice forces.

-- Max Born

A useful theoretical approach to guide the development of potentials is to predict elastic properties implied by generic functional forms and compare with experimental or ab initio data. Recently, this method has been pursued by only a few authors , but apparently they did not search for elastic constant elastic constant relations implied by simple functional forms, which is our approach. This tool for understanding interatomic forces dates back to the 19th century, when St. Venant showed that the assumption of central pairwise forces supported by Cauchy and Poisson implies a reduction in the number of independent elastic constants from 21 to 15. The corresponding six dependencies, given by the single equation C₁₂ = C₄₄ if atoms are at centers of cubic symmetry, are commonly called the Cauchy relations. They provide a simple test for selecting which materials can be described by a pair potential. Once it was realized that the Cauchy relations are not satisfied by the experimental data for semiconductors, a number of authors in this century, led by Born, derived generalized Cauchy relations for noncentral forces in the diamond structure.

Born's ingenious idea was to consider an underdetermined model (with fewer degrees of freedom than the number of independent elastic constants) and derive the implied elastic constant relations. If these relations are not satisfied, then the functional form cannot reproduce the data, no matter how it is fit. If they are satisfied, then we have compelling evidence that the functional form is correct. This kind of information is rare in the field of interatomic potentials; usually the validity of a functional form can only be assessed by fitting experience, which is time-consuming and inconclusive.

In this chapter we analyze the elastic properties of several general classes of many-body potentials in the diamond and graphitic crystal structures in order to gain insight into the mechanical behavior of sp^2 and sp^3 hybrid covalent bonds, respectively. These high symmetry atomic configurations must be accurately described by any realistic model of interatomic forces in a tetravalent solid. We only discuss results for three-body cluster potentials, ignoring bond order cluster functionals in the interest of simplicity. In this work the so-called Valence Force Field (VFF) models, which can only describe small distortions of the diamond lattice, are also not considered. The goal in the VFF approach is to reproduce lattice dynamics as accurately as possible, paying little or no attention to broader transferability (with some exceptions. Over forty such potentials have been produced for Si alone, with the most recent displaying superb agreement with experiment for elastic constants and phonon frequencies. However, as described earlier, unifying themes of this thesis are simplicity and transferability, and hence our motivation is quite different from VFF.

1. Taylor Expansion of the Cohesive Energy
2. Diamond sp^3 Hybrid Covalent Bonds
   1. First Neighbor Interactions
   2. Quantum-Mechanical Interpretation
   3. Second Neighbor Interactions
3. Graphitic sp^2 Hybrid Covalent Bonds
   1. Ab Initio Elastic Constants for Graphitic Silicon
   2. First Neighbor Interactions
4. Comparison of sp^2 and sp^3 Bonds
5. Conclusion

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© 1997 Martin Zdenek Bazant.