Solid state physics is a fascinating subject. Learning it however, is amazingly difficult. After reading through the first half of the classic intro by Kittel twice (the book where all important details are in the figure captions) 1, complementing with one round through half of Ashcroft & Mermin (the book where all important details are in the foot notes) 2, I can only claim that I “kind of get it”. Nevertheless, there are some interesting interdisciplinary connections between the subject and the theory of signals, that I found insightful and want to share with others.

Primary target audience is physics and signal people interested in the other field. I aim to write this post such that it is understandable for either category.

Contents

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Introduction

In physics, traveling plane waves are usually described using an amplitude $A$, a wave vector $\vec k$ (a.k.a. directional spatial frequency) and temporal frequency $\omega$, and an optional phase offset $\varphi$

$$ \psi(\vec r, t) = A \cos(i(\vec k^T \vec r - \omega t + \varphi)) = \mathfrak{Re} (C \exp(i(\vec k^T \vec r - \omega t))) $$

where the phase $\varphi$ is baked in to the (complex) amplitude $C = A\exp(i\varphi)$. For all practical purposes the complex signal can be manipulated directly, keeping in mind that there is a corresponding implicit conjugate signal present at all times.

The temporal frequency is related to the “normal” (i.e. revolutionary) frequency/time $f = 1 / T$ as $\omega = 2 \pi f = 2 \pi / T$, and the wave vector to the wave length $\lambda$ as $||\vec k|| = 2 \pi / \lambda$. When studying static conditions such as standing waves in matter (i.e. refraction and reflection), the wave vector $\vec k$ will be the only parameter of interest, and for all practical purposes in this post we can refer to it as simply “the frequency”, keeping in mind that there are actually three components, one in each dimension, and not just one.

Solid state physics builds a theory based on crystals, which exhibit regular periodic structure that enables analytical computations on matter to be performed. The periodicity can classified into exactly 14 different Bravais lattices, which are the equivalence classes of symmetry groups acting on a periodic set of points. The 14 lattice classes can be further grouped into 7 lattice systems; cubic, hexagonal, etc. The smallest non-repetitive volume in a lattice is called a unit cell (strictly speaking that would be a primitive unit cell, while a slightly larger but perhaps mathematically more simple is known as a convenient unit cell). Equipping a unit cell with translation vectors $\vec a_1, \vec a_2, \vec a_3$ originating from one corner and extending along three of the edges, then any point $\vec p$ can be assigned coordinates $c_1, c_2, c_3$ w.r.t. this basis such that $\vec p = \sum c_k \vec a_k$. It is worth mentioning that the word basis is (unfortunately) ambiguous in solid state physics terminology, because that is also (confusingly) the name given to the set of atoms present in a primitive unit cell. The interesting thing about crystals is that, for any property $\xi(\vec p)$ that vary through space, it will exhibit translational symmetry such that $\xi(\vec p) = \xi(\vec p + \vec T)$ for any lattice point $\vec T$. In other words, only periodic functions are allowed in crystals. Note that the lattice points themselves have integral coordinates w.r.t. the translation vectors $\vec a_k$.

The first Brillouin zone and the Nyquist frequency

When studying crystalline materials, the mathematical formalism leads to the reciprocal lattice construction, which is nothing more than a three dimensional discrete Fourier space. It is acknowledged that wave vectors, just like any other property, exhibit translational symmetry as well, such that any vector large enough is equivalent to smaller ones, if they can be translated between each other through a reciprocal lattice vector. The set of “small” frequencies constitute the so called first Brillouin zone.

The underlying physics behind this is easy to understand, and is known as aliasing in the signal formalism. Essentially, due to the discrete nature of atoms and the finite spacing in between those, waves with sufficiently high frequency can not be represented through displacements of the atoms, but will be “folded back” onto the lower part of the frequency spectrum. The correspondence here is that atoms are discrete samples of a continuous signal, that is, a digital signal. The maximal frequency before the aliasing phenomena kicks in is known as the Nyquist frequency.

If you have ever watched the wheels of a car (de-)accelerating, then you have most likely noticed that at a certain speed it looks like the wheels stop turning, and eventually start to revolute in the wrong direction! Surely, this is an optical illusion, but have you ever tried to explain it? My explanation of it is that our vision can not process arbitrary high spatio-temporal frequencies, and there is a threshold due to some discrete process in either our eyes, our brain or somewhere in the nervous system in between. If you think about it, neurons operate by firing discrete pulse trains, which is a form of digital signal. I wouldn’t quote myself on it though, I did quickly check it with a friend holding a masters degree in cognition science, but probably a good idea to double check this with a neuroscientist if you want to be sure. But it does make sense, and it is “common knowledge” that buying a computer screen with higher than 120 Hz is a waste of money.

Back to solid state physics; this aliasing phenomena is the underlying mechanism behind so called umklapp scattering where phonons (quantized atomic lattice vibrations) scatter in counterintuitive ways. Phonons, being waves would normally just superimpose upon collision. However, when the sum of the spatial frequencies $\vec k = \vec k_1 + \vec k_2$ (which can also be identified as a crystal momentum) exceed the Brillouin zone they will wrap back such that $||\vec k|| < ||\vec k_1|| + ||\vec k_2||$. Sometimes $1 + 1 = -1$ is actually valid math.

To summarize, atoms are samples and the first Brillouin zone is a three dimensional Nyquist frequency.

The structure factor is a band pass filter

Light can interact with matter in many ways, including

  1. Absorption; the photon energy is converted to thermal energy
  2. Excitation; the photon energy energy is transferred to an electron which jump up to a higher shell, or even eject from the atom
  3. Stimulation; an excited electron is “encouraged” to jump down again (a.k.a. LASER)
  4. Scattering; an electron will quickly absorb and re-emit the photon in a random direction

The latter effect is of uttermost interest when studying X-ray reflections in crystalline materials, for which matter is highly transparent, which leads to interesting interference patterns between the scattered photons from different unit cells, and between the atoms within each unit cell, in the crystal. Picture an incoming ray $\vec k$ that scatters at position $\vec r$ into $\vec k^\prime$, and assume elastic scattering such that $||\vec k || = || \vec k^\prime ||$. The difference in phase $\Delta \varphi$ between the incoming and outgoing rays can shown to be $\Delta \varphi(\vec k, \vec r) = (\vec k - \vec k^\prime)^T \vec r = \Delta \vec k^T \vec r$. Let $n(\vec r)$ denote the electron distribution within a unit cell, then we can define a scattering factor $F$ as an integral

$$ F(\vec k, \vec k^\prime) = \int n(\vec r) \exp(i\Delta \varphi(\vec k, \vec r)) d\vec r $$

which encode how the material responds to different wave vectors. This is essentially a linear filter defined in frequency space! See the correspondence to signal theory? It is a bit difficult to spot since it depends on both input and output, but remember that we assume elastic scattering such that the wave length does not change (if it did it would not be linear), but since we are dealing with 3D frequencies now there is a directional dependence as well. To make this clear, take an input signal $X(\vec k)$ (the frequency spectrum of a general polychromatic X-ray beam), the intensity of the output signal $Y$ (the reflected beam) in direction $\vec k^\prime$ depends on $F$ as

$$ Y(\vec k^\prime) = \int F(\vec k^\prime, \vec k)X(\vec k)d \vec k $$

Hopefully this should be convincing enough that the relationship is linear. This expression is on the same form as a matrix/vector product, but with summation replaced with an integral, and row/column indices replaced with continuous vectors. To prove it, fix the input and output directions $(\theta, \phi)$ and $(\theta^\prime,\phi^\prime)$, and take the derivative of the output $Y$ w.r.t. the input $X$ at specific input and output scalar frequencies $k = ||\vec k||$ and $k^\prime = ||\vec k^\prime||$, such that

$$ \frac{dY(k^\prime ; \theta^\prime, \phi^\prime)}{dX(k ; \theta, \phi)} = F(k,k^\prime ; \theta, \theta^\prime, \phi, \phi^\prime) \delta (k, k^\prime) $$

where I have used a semi-colon to separate variables from constants, and the Dirac delta comes from the assumption on elastic scattering, proving that it is linear.

It can be shown that $F$ become vanishing small unless $\Delta \vec k = \vec G$ for some reciprocal lattice vector $\vec G$. What does this imply? It turns out there is a one to one correspondence between reciprocal lattice vectors and crystal planes. For many lattice types they correspond to the normals of the crystal planes. For these crystals, there will only exist reflections due to constructive interference for scattering vectors aligned with a crystal plane normals. This result is intuitive, just imagine the crystal planes as mirrors; only reflections through these makes sense.

Sometimes this is visualized through the Ewald sphere. For elastic scattering, the incident and reflected beams will lie on a sphere, since they have the same norm. The diffraction conditions is satisfied when it is possible to translate the sphere such that both beams point at reciprocal lattice points (because then they are separated by a reciprocal lattice translation). Personally I do not find the construct too helpful when learning about X-ray diffraction, but it is widely used and it is good to know about it.

Let us continue to analyze the scattering factor evaluated at the reciprocal lattice vectors. The following calculations are taken more or less directly from Kittel 1. Note that $F$ only depends on the frequency shift $\Delta \vec k$, and we define $S(\vec G) = F(\vec k, \vec k + \vec G)$ for arbitrary $\vec k$, which can be calculated to be

$$ S(\vec G) = \int n(\vec r) \exp(i\Delta \varphi) d\vec r = \int n(\vec r) \exp(-i \vec G^T \vec r) d \vec r $$

In general, a unit cell will have a polyatomic basis. We can make an approximation and try to decompose the electron density $n(\vec r)$ as a superposition of the density contributed by each atom

$$ n(\vec r) = \sum_j n_j (\vec r - \vec r_j) $$

where $\vec r_j$ are the positions of the atoms in the unit cell. Substituting back into $S$ gives

$$\begin{align} S(\vec G) &= \sum_j \int n_j(\vec r - \vec r_j) \exp(-i \vec G^T \vec r) d \vec r \newline &= \sum_j \exp(-i\vec G^T \vec r_j) \int n_j(\vec r - \vec r_j) \exp(-i \vec G^T (\vec r - \vec r_j)) d \vec r \newline &= \sum_j \exp(-i\vec G^T \vec r_j) \int n_j(\vec r) \exp(-i \vec G^T \vec r) d \vec r \newline &= \sum_j f_j \exp(-i\vec G^T \vec r_j) \end{align} $$

where we have utilized the fact that any property has to be periodic, including the electron densities $n_j$, such that taking an integral over a unit cell is necessarily translation invariant. I have introduced the so called atomic form factor $f_j = \int n_j(\vec r) \exp(-i \vec G^T \vec r) d \vec r$ for atom $j$, which is usually looked up in a table created either through numerical pre-computation or physical experimentation.

In the simple case for monoatomic crystals, the atomic form factor can be taken outside the summation as a constant, and evaluating $S$ for different discrete patterns of $\vec G$ will reveal that certain wave vectors are completely canceled out through destructive interference, known as forbidden reflections. This shows that materials exhibit band pass filter properties! For example, monoatomic crystals arranged in face centered cubic (FCC) lattices will cancel out any frequency with scattering vector coordinates of unequal parity. It is worth pointing out that the reciprocal lattice vector coordinates (a.k.a. Miller indices) depends on the choice of unit cell, and currently we consider the simple cubic convenient unit cell (for monoatomic FCC, that contains 4 atoms). To be concrete, enumerating the reciprocal lattice vectors (uniquely up to permutation), we can mark which are allowed and not

Reciprocal lattice vector ($\vec G$) Allowed?
(1,0,0) No
(1,1,0) No
(1,1,1) Yes
(2,0,0) Yes
(2,1,0) No
(2,2,0) Yes
(2,2,1) No
… etc.

This is what makes analyzing X-ray reflections interesting; we can get information about a material by looking at the diffractogram for different angles of reflection, which will contain distinct peaks unique to that material, which are allowed to pass through “the material band pass filter”. The so called “material phase” is identified by comparing this “fingerprint” against a database of known diffraction spectra.

Reciprocal lattice and dual spaces

I would like to round off with some insights into the origin of the terminology of the reciprocal lattice, and connect to some more generic mathematical constructs. The reciprocal lattice is the set of points generated by discrete translations of the reciprocal axis vectors $\vec b_1, \vec b_2, \vec b_3$, defined such that $\vec a_i^T \vec b_j = 2 \pi \delta_{ij}$. At first glance it might look like $\vec b_k = 2\pi \vec a_k$, but remember that the translation vectors does not necessarily constitute an orthonormal basis, so except from the cubic lattice system they will point in quite different directions. Mathematically, the reciprocal axis vectors $\vec b_1, \vec b_2, \vec b_3$ constitute a scaled dual basis of the primal basis $\vec a_1, \vec a_2, \vec a_3$. I call it is scaled because a proper dual basis should not have the $2 \pi$ term in the inner product above. The dual basis is a basis in the so called dual space. In linear algebra, any vector space have a corresponding dual space, and the dual of the dual is the original (a.k.a. primal) vector space. The dual space can be identified as the set of linear functionals operating on the vectors in the primal.

One interesting thing is that coordinates in these two spaces have a reciprocal relationship: given a basis in the primal consisting of vectors of large norm, the coordinates of a vector will be small. Since the coordinate of a vector in the primal is given by the inner product with a vector from the dual, the corresponding dual vector must necessarily be small as well. Conversely, if the basis vectors are small in the primal, the corresponding dual basis vectors and primal coordinates are large. I believe that this is the origin of the name of the reciprocal lattice.

What this means is that given any “primal quantity” $\xi$ in with some associated uncertainty $\sigma$, the corresponding “dual quantity” $\tilde \xi$ will have a reciprocal uncertainty $\tilde \sigma \propto 1/\sigma$. This is the uncertainty principle; in quantum mechanics it appears in the form of position/momenta pairs, in signal theory it comes in the form of time/frequency, in XRD (X-ray diffraction) it reveals itself as layer thickness/peak broadening, but it actually holds for any pair of primal/dual quantities.

  1. Introduction to Solid State Physics, Either Edition, Charles Kittel ↩ ↩2

  2. Solid State Physics, Neil W. Ashchroft and N. David Mermin ↩