solid state physics - Reciprocal Lattices

Is there an easy way to understand and/or visualize the reciprocal lattice of a two or three dimensional solid-state lattice? What is the significance of the reciprocal lattice, and why do solid state physicists express things in terms of the reciprocal lattice rather than the real-space lattice?

Answer

Its questions like this one that keep me coming back to this site !

Your first question is:

Is there an easy way to understand and/or visualize the reciprocal lattice of a two or three dimensional solid-state lattice?

YES ! The reciprocal lattice is simply the dual of the original lattice. And the dual lattice has a simple visual algorithm.

1. Given a lattice $L$, for each unit cell of $L$ find the point corresponding to that cell's "center of mass" (see below).


2. Connect each such "center of mass" to its nearest neighbors.


3. The resulting lattice is the dual of $L$.

To find center of mass of unit cell (we consider 2d case, generalizes to arbitrary dimension):

1. Draw the perpendicular bisectors of the edges which bound the unit cell.


2. For regular lattices these lines should intersect at a single point in the interior of the cell. This point is the "center of mass" of the cell.

Performing these simple steps you find that the dual of a square lattice is also a square lattice, and that the triangular and hexagonal lattices are each others duals ! You can see a nice illustration of this fact here.

Your second question is:

What is the significance of the reciprocal lattice, and why do solid state physicists express things in terms of the reciprocal lattice rather than the real-space lattice?

As mentioned by others this has to do with fourier transforms. In solid-state physics we want to understand the excitations (waveforms) that a certain material, whose structure is given by some lattice $L$, can support. For a lattice only certain momenta are allowed due to its discrete structure. These allowed momenta correspond to the vertices of the dual lattice! For more see the wikipedia page or check out the first couple of chapters of little Kittel or Ashcroft and Mermin.

                                Cheers,

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Edit: This to clarify some doubts about my answer @wsc has expressed in the comments.

First of all, it is incorrect that reciprocal lattice vectors in 3D have dimensions $1/L^2$. Consider a 3D lattice with basis vectors $\{a_i\}$. The reciprocal lattice has basis vectors given by

$$ b_i = \frac{1}{2V} \epsilon_i{}^{jk} \, a_j a_k $$

in index notation, with summation convention. A more familiar way to write this is in vector notation:

$$ \mathbf{b}_i = 2\pi \frac{\mathbf{a}_j \times \mathbf{a}_k}{\mathbf{a}_i \cdot (\mathbf{a}_j \times \mathbf{a}_k)} $$

where $(i,j,k)$ are cyclic permutations of $(1,2,3)$. We can see that

$$ \dim[\mathbf{b}_i] = \frac{\dim[\mathbf{a}]^2}{\dim[\mathbf{a}]^3} = \frac{1}{L} $$

and in terms of the lattice spacing $a$, $\vert\mathbf{b}\vert \sim \frac{1}{a}$. In fact, this is a basic fact true in any dimension.

We can also understand the normalization of the reciprocal lattice vectors by the factor $\mathbf{a}_i \cdot (\mathbf{a}_j \times \mathbf{a}_k)$ as being nothing more than $V$ - the volume of the unit cell. Why? So that the transformation between the lattice and reciprocal lattice vector spaces is invertible and the methods of Fourier analysis can be put to use.

For all regular lattices AFAIK the "dual" and "reciprocal" lattices are identical. For irregular lattices - with defects and disorder - this correspondence would possibly break down.