static allows the user to select from a pre-defined set of lattice types, or to define an arbitary lattice. This choice is made in the subroutine setlat.f.

In many cases, however, you may want to start with a basic lattice type, say a face-centered cubic (fcc) lattice, and then distort the lattice. For example, you might want to calculate the elastic constants of a crystal based on the tight-binding parametrization.

setlat.f allows you to generate an arbitrary strain, or to select one of several defined strain types. What we mean by strain is defined in many solid-state theory textbooks, including Kittel and Ashcroft & Mermin. In general, a strain is represented by a strain tensor, which transforms a set of primitive lattice vectors.

To see this, let's start with an arbitrary primitive lattice:

where X, Y, and Z are unit vectors pointing along the Cartesian axis and the parenthesis surround matrices.

A general strain will bend and perhaps change the length of the unit vectors. Since a rotation of the coordinate axis does not change the total energy of the system, we can restrict ourselves to non-rotating strains. These strains can be represented by a symmetric strain matrix, which has the effect of changing the relationship between X, Y, and Z. This matrix is usually expressed in terms of e_i, the strain coefficients:

The strain matrix changes the unit vectors in (1) to the primed vectors of (2). Thus the new lattice can be described as:

By Hook's law, if the strains e_i are small, the change in the energy of the lattice is proportional to the square of the strains:

+ O[e3]     .      (4)

The linear terms vanish when we are at equilibrium, or if the strain causes no change in the volume of the crystal. Otherwise, the "deltas" are related to the strain on the crystal, and the C_ij are the elastic constants. Since the strain matrix (2) is symmetric, we can assume the C_ij tensor to be symmetric, too:

C_ij = C_ji   .     (5)

There are thus 21 independent elastic constants. This number can be reduced by symmetry. For example, in a cubic crystal, such as copper, niobium, or diamond, symmetry demands that, e.g. C₁₁ = C₂₂. In fact, only three independent elastic constants remain, so that (4) reduces to

E = E₀ + ½ C₁₁ [ e₁¹ + e₂² + e₃³] + C₁₂ [ e₁e₂ + e₂e₃ + e₃e₁] + ½ C₄₄ [ e₄⁴ + e₅⁵ + e₆⁶] + O[e³]     .     (6)

(Equation (6) assumes that we start from equilibrium, i.e., there are no stress terms.)

It is obvious the several linear combinations of elastic constants and strains will occur frequently. They occur so frequently, in fact, that the static program has several pre-defined strains included in the code. These strains are defined in the setlat.f subroutine. The table below describes the strains already defined. Any user can of course add additional strains by editing setlat.f. In addition, as can be seen in option zero, a general strain can be entered.

In each entry of the table, only the listed e_i are non-zero. The column labeled "Parameters Entered" indicates the actual input expected by the code, in the order it should be given. The remaining column labels should be self-explanatory, except for column 5. There, "QE/V" means "the quadratic part of the energy (4) divided by the volume."

Suggestions on how to use these strains can be found in the examples section of this manual.

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