Question:

# Constrained Brilloiun zone sum to integral

Valentina: 3 days ago

In order to evaluate the two-phonon Raman scattering cross section I'm trying to compute the following double sum in $$k$$-space (See Phys. Rev. B 17, 4951 (1978), or the book Light Scattering in Solids II, by M. Cardona, page 149):

$$S=\sum_{\vec{k}_1,\vec{k}_2}{|\Delta_{\vec{k}_1}|^2|\Delta_{\vec{k}_2}|^2\delta(\vec{k}_1+\vec{k}_2)}$$

where $$|\Delta_\vec{k}|$$ is the phonon displacement, and $$\delta(\vec{k}_1+\vec{k}_2)=1$$ when $$\vec{k}_1+\vec{k}_2=0$$, and otherwise $$\delta(\vec{k}_1+\vec{k}_2)=0$$.

This means that only the pairs of vectors $$\vec{k}$$ and $$-\vec{k}$$ will contribute to the sum.

Note that $$\Delta_\vec{k}=\Delta_{-\vec{k}}$$, so when $$\delta(\vec{k}_1+\vec{k}_2)=0$$ we have $$|\Delta_{\vec{k}_1}|^2|\Delta_{\vec{k}_2}|^2=|\Delta_{\vec{k}_1}|^4$$

To evaluate $$S$$, I first change the summation over the brillouin zone into an integral:

$$\sum_\vec{k}{\Delta_\vec{k}}=\frac{V}{(2\pi)^3}\int_\vec{k}{\Delta_\vec{k}d^3\vec{k}}$$

and obtain:

$$S=\frac{V^2}{(2\pi)^6}\int_{\vec{k}_1}{|\Delta_{\vec{k}_1}|^2d^3\vec{k}_1}\int_{\vec{k}_2}{|\Delta_{\vec{k}_2}|^2d^3\vec{k}_2}$$

At this point one must define the limits of the summation over the spherical coordinates in order to ensure that the two integrals only take the pairs $$\vec{k}_1=\vec{k}$$ and $$\vec{k}_2=-\vec{k}$$. My only idea on how to do this is to integrate $$|\Delta_{\vec{k}_1}|^2|\Delta_{\vec{k}_2}|^2$$ together over the entire Brillouin zone instead of separately, such that:

$$S\sim\int|\Delta_{\vec{k}}|^2|\Delta_{-\vec{k}}|^2d^3k$$

However this is wrong, as it would change the dimensionality of the result, since the factor $$\frac{V}{(2\pi)^3}$$ would only appear once in front of the integral. Otherwise I can separate the integrals first so that $$\frac{V}{(2\pi)^3}$$ does come out squared, but then by using the delta function to reduce the second integral to simply $$|\Delta_{\vec{k}_1}|^2$$ again changes the dimensionality, since I'm not integrating over a volume any more.

I feel like I am missing something trivial. If anyone could explain what that is, or provide any resources (books, papers) which might help, I would appreciate it.

Why do you have to begin with a switch to an integral? You could simply expand your initial sum using the properties of $$\delta$$ as
$$S=\sum_{\vec{k}_1,\vec{k}_2}{|\Delta_{\vec{k}_1}|^2|\Delta_{\vec{k}_2}|^2\delta(\vec{k}_1+\vec{k}_2)}= \sum_{\vec{k}}{|\Delta_{\vec{k}}|^2|\Delta_{-\vec{k}}|^2}.$$
Then, using your note that $$\Delta_\vec{k}=\Delta_{-\vec{k}}$$, we can further develop this into
$$S=\sum_{\vec k}|\Delta_{\vec k}|^4.$$