A defining characteristic of crystalline solids is that their constituent atoms are arranged in a spatially periodic way. In fancy lingo, the atomic configuration breaks continuous translational and rotational invariance (that is, it picks out certain positions and orientations in space from an infinite variety of possible choices), but preserves discrete translational invariance (and other possible symmetries).

The introduction of a characteristic spatial length scale, or equivalently a spatial frequency, is a big deal, because when other spatial length scales in the physical system coincide with that one, there can be big consequences. For example, when the wavelength of x-rays or electrons or neutrons is some integer harmonic of the (projected) lattice spacing, then waves scattered from subsequent (or every second or every third, etc.) plane of atoms will interfere constructively - this is called the Bragg condition, is what gives diffraction patterns that have proven so useful in characterizing material structures. Another way to think about this: The spatial periodicity of the lattice is what forces the momentum of scattered x-rays (or electrons or neutrons) to change only by specified amounts.

It gets better. When the wavelength of electrons bound in a crystalline solid corresponds to some integer multiple of the lattice spacing, this implies that the electrons strongly "feel" any interaction with the lattice atoms - in the nearly-free-electron picture, this matching of spatial frequencies is what opens up band gaps at particular wavevectors (and hence energies). Similar physics happens with lattice vibrations. Similar physics happens when we consider electromagnetic waves in spatially periodic dielectrics. Similar physics happens when looking at electrons in a "superlattice" made by layering different semiconductors or a periodic modulation of surface relief.

One other important point. The idea of a true spatial periodicity really only applies to infinitely large periodic systems. If discrete translational invariance is broken (by a defect, or an interface), then some of the rules "enforced" by the periodicity can be evaded. For example, momentum changes forbidden for elastic scattering in a perfect infinite crystal can take place at some rate at interfaces or in defective crystals. Similarly, the optical selection rules that must be rigidly applied in perfect crystals can be bent a bit in nanocrystals, where lattice periodicity is not infinite.

Commensurate spatial periodicities between wave-like entities and lattices are responsible for electronic and optical bandgaps, phonon dispersion relations, x-ray/electron/neutron crystallography, (crystal) momentum conservation and its violation in defective and nanoscale structures, and optical selection rules and their violations in crystalline solids. Rather far reaching consequences!

# nanoscale views

A blog about condensed matter and nanoscale physics. Why should high energy and astro folks have all the fun?

## Tuesday, April 15, 2014

## Sunday, April 13, 2014

### End of an era.

As long as we're talking about the (alleged) end of science, look at this picture (courtesy of Don Monroe). This is demolition work being done in Murray Hill, NJ, as Alcatel-Lucent takes down a big hunk of Building 1 of Bell Labs.

This building and others at the site were the setting for some of the most important industrial research of the 20th century. (Before people ask, the particular lab where the transistor was first made is not being torn down here.) I've written before about the near-demise of long-term basic research in the industrial setting in the US. While Bell Labs still exists, this, like the demise of the Holmdel site, are painful marks of the end of an era.

This building and others at the site were the setting for some of the most important industrial research of the 20th century. (Before people ask, the particular lab where the transistor was first made is not being torn down here.) I've written before about the near-demise of long-term basic research in the industrial setting in the US. While Bell Labs still exists, this, like the demise of the Holmdel site, are painful marks of the end of an era.

## Thursday, April 10, 2014

### John Horgan: Same old, same old.

John Horgan writes about science for National Geographic. You may remember him from his book, The End of Science. His thesis, 17 years ago, was that science is basically done - there just aren't going to be too many more profound discoveries, particularly in physics, because we've figured it all out and the rest is just details. Well, I'll give him this for consistency: He's still flogging this dead horse 17 years later, as seen in his recent column. I disagree with his point of view. Even if you limit yourself to physics, there are plenty of discoveries left to be made for a long time to come - things only look bleak if (a) you're only a reductionist; and (b) you limit your interest in physics to a narrow range of topics. In other words, possibly looking for supersymmetric partners at the LHC might not be a great bet, but that doesn't mean that all of science is over.

## Friday, April 04, 2014

### A video interview for an online nano course

Two of my colleagues (Dan Mittleman and Vicki Colvin) put together a Coursera class this past year, "Nanotechnology: The Basics", and as part of that they interviewed several Rice faculty about different bits and pieces. They spoke to me about nanoelectronics, but the conversation ended up ranging into a discussion of hype in science and the importance of communicating to a general audience. The video is now up online here, and the hype/science presentation discussion starts at around 18:38.

## Wednesday, April 02, 2014

### Recurring themes in (condensed matter/nano) physics: hybridization

Suppose I have two identical systems, such as two copies of a mass attached to a spring (anchored to an immovable wall). Each system by itself has some characteristic response, like a frequency of motion, and those responses are identical because the independent systems are identical. Now consider coupling the two systems together, such as linking the two masses by another (weak) spring, and ask what the total coupled system response looks like. With classical oscillators like our example, we would say that we find the "new normal modes" of the coupled system - instead of writing separate equations to describe Newton's laws for each mass separately, we can do some kind of change of variables and consider new coordinates that combine the motions of the two masses. When we do this, we end up again with two characteristic frequencies (basically two effectively independent oscillators), but now the frequencies differ a bit, one being higher and one being lower than the original independent oscillator frequency. You can generalize this to \(N\) oscillators and find \(N\) new normal modes with a

This coupling+splitting the modes crops up again and again. You can use this to make an electronic band pass filter by capacitively coupling together a bunch of \(LC\) circuits. You can make a mechanical band pass filter by elastically coupling together a bunch of mechanical oscillators. You can make a miniband of electronic states by coupling together a bunch of quantum wells that each individually would have identical bound states. You can make a band of electronic states by coupling together the single-particle atomic states of a periodic array of atoms. The stronger the coupling, the wider the resulting bandwidth. This basic idea (taking noninteracting systems, coupling them, and re-solving for new effectively noninteracting modes) is so powerful that when it doesn't work sometimes, it can be jarring.

*band*of frequencies, with the bandwidth determined by the strength of the couplings.This coupling+splitting the modes crops up again and again. You can use this to make an electronic band pass filter by capacitively coupling together a bunch of \(LC\) circuits. You can make a mechanical band pass filter by elastically coupling together a bunch of mechanical oscillators. You can make a miniband of electronic states by coupling together a bunch of quantum wells that each individually would have identical bound states. You can make a band of electronic states by coupling together the single-particle atomic states of a periodic array of atoms. The stronger the coupling, the wider the resulting bandwidth. This basic idea (taking noninteracting systems, coupling them, and re-solving for new effectively noninteracting modes) is so powerful that when it doesn't work sometimes, it can be jarring.

## Sunday, March 30, 2014

### Any advice: LaTeX and makeindex

Readers - For a long time now I have been working on a very large LaTeX document (actually built out of a number of sub-documents) that I will discuss further in later posts. I greatly desire to create an index for this document, and I know about the LaTeX package \makeindex. The question is, does anyone know of a good frontend application that can make the creation of the index less tedious? The brute force approach would require me to go through the document(s) by hand and insert a \makeindex tag every time a term that I wish to index appears. For an index containing a couple of hundred entries, this looks excruciating. What I would love is an application where I identify the terms for which I want index entries, and it then automatically inserts the appropriate tags (in a smart way, not putting tags inside LaTeX equations, for example). While this would be imperfect, it would be easier to start from an over-complete index and pare down or modify than to start from scratch. Yes, I am sure I could use perl or another scripting language to make something, but I'd rather not reinvent the wheel if someone has already solved this problem. Thanks for any suggestions.

## Friday, March 28, 2014

### Recurring themes in (condensed matter/nano) physics: boundary conditions

This is the first in a series of posts about tropes that recur in (condensed matter/nano) physics. I put that qualifier in parentheses because these topics obviously come up in many other places as well, but I run across them from my perspective.

Very often in physics we are effectively solving boundary value problems. That is, we have some physical system that obeys some kind of differential equations describing the spatial dependence of some variable of interest. This could be the electron wavefunction \( \psi(\mathbf{r})\), which has to obey the Schroedinger equation in some region of space with a potential energy \( V(\mathbf{r})\). This could be the electric field \( \mathbf{E}(\mathbf{r})\), which has to satisfy Maxwell's equations in some region of space that has a dielectric function \( \epsilon(\mathbf{r})\). This could be the deflection of a drumhead \( u(x,y) \), where the drumhead itself must follow the rules of continuum elasticity. This could be the pressure field \( p(z) \) of the air in a pipe that's part of a pipe organ.

The thread that unites these diverse systems is that, in the absence of boundaries, these problems allow a

Very often in physics we are effectively solving boundary value problems. That is, we have some physical system that obeys some kind of differential equations describing the spatial dependence of some variable of interest. This could be the electron wavefunction \( \psi(\mathbf{r})\), which has to obey the Schroedinger equation in some region of space with a potential energy \( V(\mathbf{r})\). This could be the electric field \( \mathbf{E}(\mathbf{r})\), which has to satisfy Maxwell's equations in some region of space that has a dielectric function \( \epsilon(\mathbf{r})\). This could be the deflection of a drumhead \( u(x,y) \), where the drumhead itself must follow the rules of continuum elasticity. This could be the pressure field \( p(z) \) of the air in a pipe that's part of a pipe organ.

The thread that unites these diverse systems is that, in the absence of boundaries, these problems allow a

*continuum*of solutions, but the imposition of boundaries drastically limits the solutions to a*discrete*set. For example, the pressure in that pipe could (within reasonable limits set by the description of the air as a nice gas) have any spatial periodicity, described by some wavenumber \(k\), and along with that it would have some periodic time dependence with a frequency \(\omega\), so that \( \omega/k = c_{\mathrm{s}}\), where \(c_{\mathrm{s}}\) is the sound speed. However, once we specify boundary conditions - say one end of the pipe closed, one end open - the rules that have to be satisfied at the boundary force there to be a discrete spectrum of allowed wavelengths, and hence frequencies. Even trying to have no boundary, by installing periodic boundary conditions, does this. This general property, the emergence of discrete modes from the continuum, is what gives us the spectra of atoms and the sounds of guitars.
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