Reciprocal Lattices

Question

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 1

To understand why reciprocal space is important it is perhaps useful to illustrate related concept of frequency space.

Anytime one is analyzing waves (be it sound waves, EM waves or any other kind) one can exploit the translational symmetry of the laws of motion. The dual quantity to position is momentum (for massless fields like EM waves this also corresponds to frequency) and because of the said symmetry everything becomes whole lot easier. Instead of working with general wave (which can be a pretty difficult beasts) one instead works with monochromatic (i.e. single frequency) waves. For this special kind of waves the differential equations simplify just to algebraic equations, so the problem becomes easily tractable.

Now, the method explained in the previous paragraph is known as Fourier analysis and Fourier transform and is very general. Put simply, anytime you have some nice symmetry you can use Fourier analysis to move to the dual space where the problem will simplify greatly. When applied to lattices (which have a lot of translational symmetry) one obtains the concept of a reciprocal lattice.

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Mathematical note:

From mathematical point of view one exploits that the system is essentially described by integrable functions on some locally compact abelian group $G$. By Peter-Weyl theorem we know that the space of such functions is parametrized by irreducible representations of $G$ which form a dual Pontryagin group $\hat{G}$.

For example when working with periodic functions one is actually working with the functions defined on the circle $S^1$. Now the irreps of $S^1$ are parametrized precisely by integers. So in this case we get the decompositions of a periodic function $$f(x) = \sum_{k \in \mathbb{Z}} f_k \rho_k(x)$$ where $f_k$ are the Fourier modes of $f(x)$ and $\rho_k(x) = \exp(-i k x)$ are irreps of $S^1$. In general the irreps of an LCA group are given by some exponential so this explains the ubiquity of exponentials in physics.

Answer 2

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.

Answer 3

The significance of the reciprocal lattice is tied with diffraction of waves on a crystal.

How do we determine the crystalstructure of some material? We usually do by bombarding a tiny piece of crystal with X-rays or neutrons or another type of wave of appropriate wavelength and properties. We then look at the diffraction pattern. Now, to make a long story short, the pattern that appears will essentially be a pattern in the reciprocal space.

http://en.wikipedia.org/wiki/X-ray_crystallography

Answer 4

One problem I had in understanding reciprocal space is the origin in a real space unit cell is at infinity in reciprocal space and vise-versa. How could any real construct be mapped out in reciprocal space in a finite cell that included the origin?

However it is important to remember that we tend to deal with waves in reciprocal space. So for example, when crystallographers use x-rays, neutron diffraction etc. to obtain a pattern in reciprocal space, simply applying the reciprocal conversion (i.e. $2\pi/a$) does not obtain the positions of the atoms, but rather the wavelengths of a set of waves that will denote the positions of the atoms. A wave with wavenumber zero will have infinite wavelength (i.e. is not a wave [is this correct?])

This is basically saying what the others have said about the reciprocal space being the Fourier transform of the real space - waves do not have position that is so easily defined, but they do have well defined wavelength!