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Let’s tackle a problem like the 2D problem of the final exam, but first more generally. Instead of a square lattice consider the lattice with the geometry illustrated in fig. 1.1.

Here, and are the vector differences between the equilibrium positions separating the masses along the and interaction directions respectively. The equilibrium spacing for the cross coupling harmonic forces are

Based on previous calculations, we can write the equations of motion by inspection

Inserting the trial solution

and using the matrix form for the projection operators, we have

This fully specifies our eigenvalue problem. Writing

we wish to solve

Neglecting the specifics of the matrix at hand, consider a generic two by two matrix

for which the characteristic equation is

So our angular frequencies are given by

The square root can be simplified slightly

so that, finally, the dispersion relation is

Our eigenvectors will be given by

or

So, our eigenvectors, the vectoral components of our atomic displacements, are

or

**Square lattice**

There is not too much to gain by expanding out the projection operators explicitly in general. However, let’s do this for the specific case of a square lattice (as on the exam problem). In that case, our projection operators are

Our matrix is

where, specifically, the squared sines for this geometry are

Using eq. 1.0.6, the dispersion relation and eigenvectors are

This calculation is confirmed in oneAtomBasisPhononSquareLatticeEigensystem.nb. Mathematica calculates an alternate form (equivalent to using a zero dot product for the second row), of

Either way, we see that leads to only horizontal or vertical motion.

**With the exam criteria**

In the specific case that we had on the exam where and , these are

For horizontal and vertical motion we need , or for a difference in the absolute values of the sine arguments

That is, one of

In the first BZ, that is one of or .

**System in rotated coordinates**

On the exam, where we were asked to solve for motion along the cross directions explicitly, there was a strong hint to consider a rotated (by ) coordinate system.

The rotated the lattice basis vectors are , and the projection matrices. Writing and , where , or . In the basis the projection matrices are

The dot products that show up in the squared sines are

So that in this basis

With the rotated projection operators eq. 1.0.5.5 takes the form

This clearly differs from eq. 1.0.16d, and results in a different expression for the eigenvectors, but the same as eq. 1.0.20.20 for the angular frequencies.

or, equivalently

For the and case of the exam, this is

Similar to the horizontal coordinate system, we see that we have motion along the diagonals when

or one of

**Stability?**

The exam asked why the cross coupling is required for stability. Clearly we have more complex interaction. The constant surfaces will also be more complex. However, I still don’t have a good intuition what exactly was sought after for that part of the question.

**Numerical computations**

A Manipulate allowing for choice of the spring constants and lattice orientation, as shown in fig. 1.2, is available in phy487/oneAtomBasisPhonon.nb. This interface also provides a numerical calculation of the distribution relation as shown in fig. 1.3, and provides an animation of the normal modes for any given selection of and (not shown).