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.

Maya: 3 days ago

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

$$S=\sum_{\vec k}|\Delta_{\vec k}|^4.$$

Now you can do the switch to an integral without the need to worry about Dirac delta's dimensionality.