Two body harmonic oscillator in 3D, and 2D diamond lattice vibrations
Posted by peeterjoot on January 7, 2014
Abridged harmonic oscillator notes
Motivation and summary of harmonic oscillator background
After having had some trouble on a non-1D harmonic oscillator lattice problem on the exam, I attempted such a problem with enough time available to consider it properly. I found it helpful to consider first just two masses interacting harmonically in 3D, each displaced from an equilibrium position.
The Lagrangian that described this most naturally was found to be
This was solved in absolute and displacement coordinates, and then I moved on to consider a linear expansion of the harmonic potential about the equilibrium point, a problem closer to the exam problem (albeit still considering only two masses). The equilibrium points were described with vectors as in fig. 2.1, where .
Using such coordinates, and generalizing, it was found that the complete Lagrangian, to second order about the equilibrium positions, is
Evaluating the Euler-Lagrange equations, the equations of motion for the displacements were found to be
Observe that on the RHS above we have a projection operator, so we could also write
We see that the equations of motion for the displacements of a system of harmonic oscillators has a rather pleasant expression in terms of projection operators, where we have projections onto the unit vectors between each pair of equilibrium position.
A number of harmonically coupled masses
Now let’s consider masses at lattice points indexed by a lattice vector , as illustrated in fig. 2.2.
With a coupling constant of between lattice points indexed and (located at and respectively), and direction cosines for the equilibrium direction vector between those points given by
the Lagrangian is
Evaluating the Euler-Lagrange equations for the mass at index we have
where . Equating both, we have in vector form
This is an intuitively pleasing result. We have displacement and the direction of the lattice separations in the mix, but not the magnitude of the lattice separation itself.
Two atom basis, 2D diamond lattice
As a concrete application of the previously calculated equilibrium harmonic oscillator result, let’s consider a two atom basis diamond lattice where the horizontal length is and vertical height is .
Indexing for the primitive unit cells is illustrated in fig. 2.3.
For mass assume a trial solution of the form
The equations of motion for the two particles are
Insertion of the trial solution gives
Regrouping, and using the matrix form for the projection operators, this is
As a single matrix equation, this is
Observe that this is an eigenvalue problem for matrix
and eigenvalues .
To be explicit lets put the and functions in explicit matrix form. The orthogonal projectors have a simple form
For the and projection operators, we can use half angle formulations
After some manipulation, and the following helper functions
the block matrices of eq. 22.214.171.124 take the form
A final bit of simplification for possible, noting that , and , so
It isn’t particularly illuminating to expand out the determinant for such a system, even though it can be done symbolically without too much programming. However, what is easy after formulating the matrix for this system, is actually solving it. This is done, and animated, in twoAtomBasisRectangularLatticeDispersionRelation.cdf