## Tuesday, December 07, 2010

### The tyranny of reciprocal space

I was again thinking about why it can be difficult to explain some solid-state physics ideas to the lay public, and I think part of the problem is what I call the tyranny of reciprocal space.  Here's an attempt to explain the issue in accessible language.  If you want to describe where the atoms are in a crystalline solid and you're not a condensed matter physicist, you'd either draw a picture, or say in words that the atoms are, for example, arranged in a periodic way in space (e.g., "stacked like cannonballs", "arranged on a square grid", etc.).  Basically, you'd describe their layout in what a condensed matter physicist would call real space.  However, physicists look at this and realize that you could be much more compact in your description.  For example, for a 1d chain of atoms a distance a apart from each other, a condensed matter physicist might describe the chain by a "wavevector" k = 2 \pi/a instead.  This k describes a spatial frequency; a wave (quantum matter has wavelike properties) described by cos kr would go through a complete period (peak of wave to peak of wave, say) and start repeating itself over a distance a.   Because k has units of 1/length, this wavevector way of describing spatially periodic things is often called reciprocal space.  A given point in reciprocal space (kx, ky, kz) implies particular spatial periodicities in the x, y, and z directions.

Why would condensed matter physicists do this - purely to be cryptic?  No, not just that.  It turns out that a particle's momentum (classically, the product of mass and velocity) in quantum mechanics is proportional to k for the wavelike description of the particle.  Larger k (shorter spatial periodicity), higher momentum.  Moreover, trying to describe the interaction of, e.g., a wave-like electron with the atoms in a periodic lattice is done very neatly by worrying about the wavevector of the electron and the wavevectors describing the lattice's periodicity.  The math is very nice and elegant.  I'm always blown away when scattering experts (those who use x-rays or neutrons as probes of material structure) can glance at some insanely complex diffraction pattern, and immediately identify particular peaks with obscure (to me) points in reciprocal space, thus establishing the symmetry of some underlying lattice.

The problem is, from the point of view of the lay public (and even most other branches of physics), essentially no one thinks in reciprocal space.  One of the hardest things you (as a condensed matter physicist) can do to an audience in a general (public or colloquium) talk is to start throwing around reciprocal space without some preamble or roadmap.  It just shuts down many nonexperts' ability to follow the talk, no matter how pretty the viewgraphs are.  Extreme caution should be used in talking about reciprocal space to a general audience!  Far better to have some real-space description for people to hang onto.

Zed said...

Funny you should raise this today -- I was at a talk earlier today by someone from Guifre Vidal's group, who was trying to use this entirely real-space variational scheme (MERA) to talk about the structure of Fermi liquids... a lot of their results were reminiscent of the renormalization group for interacting fermions, but it was essentially impossible to see any of that structure intuitively in real space.

Peter Armitage said...

When I try to explain this to new grad students or undergrads, I usually use the example of frequency decomposition of a musical note. i.e. the fact that a middle C sounds different if it is played on a piano or a trumpet and that this arises in the relative strength of their overtones (higher harmonics). For a note or sound one COULD specify the trace of Pressure vs. time for all times (e.g. real time), but of course it is easier to describe the fundamental note and the relative strength of the higher harmonic frequencies (e.g. reciprocal time).

Hey maybe that is it?... Instead of Hz or GHz or whatever, we could just start referring to Hz as "reciprocal time units"? RTUs!

Douglas Natelson said...

Peter - Ahh, you're just giving me a hard time b/c you've rewired your brain to think in k-space :-) I like the overtone analogy, and of course the whole root of all this is the discrete Fourier transform - writing anything with the spatial periodicity of the lattice as a linear combination of the "fundamental" (2 \pi/a) and "overtones" (integer multiples of 2 \pi/a) of the lattice.

While this is all well-defined and good physics, there is nothing that can make a non-CM audience's eyes glaze over faster than throwing up a plot of the Brillouin zone in k-space and talking about \pi-\pi scattering, Fermi surface nesting, or the evolution of some cut through a particular diffraction peak. It's possible to talk about all of these things, but serious guideposts and handholding are needed if non-specialists are to get anything out of it.

Don Monroe said...

Peter makes an important point, which is that the conceptual barrier seems a lot lower for time/frequency than for space/wavevector.

I've encountered the spatial version many times in writing for Physical Review Focus, and not just for periodic systems. Mostly we work hard to avoid reciprocal-space language if at all possible, because readers' attention is limited. In a recent example, this story would have been much easier to write (and probably clearer) if we could have explained the dispersion relation in terms of wavevector. For such continuous systems, though, it's usually possible to convey the idea of faster or slower spatial variation. For some of the solid-state examples you mention, that casual description would not be enough.

rob said...

in reciprocal space no one can hear you scream.

Douglas Natelson said...

@anon: Tonight, on Star Trek: Voyager, Captain Janeway and Commander Chakotay are trapped in Reciprocal Space, a terrifying realm where moving at all results in enormous changes in your velocity when you re-enter normal space.

DanM said...

Peter makes another interesting point. Reciprocal time has a handy (and familiar to the layman) unit, called "Hertz". We need a similar unit for inverse length - saying "inverse meters" is just plain cumbersome. Any suggestions for a name for the unit equal to one inverse meter? How about the "Natelson"?

Anonymous said...

You lost me at wavevector...

Douglas Natelson said...

Anon3:16, I was afraid of that.

Dan, a much more appropriate name would be the Bragg or the Laue, since this is all tied up with diffraction...

Anonymous said...

Hey, nice blog entry! I have to admit I'm still struggling to grasp k-space. I just can't help but shrug off the question "...yes, but what is it?"

The best I can get at the moment is that the vector k describes the periodicity of your wave in a particular direction. K-space describes its periodicity in all directions.

Does that even make sense!? :/