Lecture on Debye model for solids

 I just finished preparing my undergraduate lecture on the Debye model for solids. Here are my slides. The course is a third-year course on statistical mechanics. In the previous lecture, they learned about blackbody radiation and derived the Planck spectrum.

(Note. The derivations are not included, as I sketch them by hand on a document viewer. This slows down the pace of the lecture and engages the students as they detect mistakes I make. :) ) 

A few things I find interesting about the lecture content.

The Debye theory is very similar to Planck's theory of blackbody radiation. The main difference is that for Debye there is a high-frequency cutoff for the phonon spectrum. This defines the Debye temperature.

Given its simplicity, the success of Debye theory, qualitatively and quantitatively, is impressive. Arguably, this is because it is an effective theory that is constrained to be consistent with the long-wavelength physics (elasticity theory) and the classical limit (number of modes of lattice vibration).

The beautiful textbook by Schroeder is a bit confusing on page 312. He claims that the Debye temperature lead (88K) is much less than that for diamond (1860 K) because the former is "soft and dense" whereas the latter is "stiff and light".  The speed of sound in diamond is about ten times that in graphite. The mass density is not particularly relevant. However, the number density of atoms (which is what enters the Debye formula) in diamond is actually 5 times larger than lead because of the strong covalent bonding which makes the interatomic distance small.  

A few historical notes I found interesting.  

Einstein's model for solids (1907) was important because it provided significant evidence for quantum theory, i.e., that the energy of lattice vibrations was quantised.

Sommerfeld said that his biggest discovery was his student Peter Debye.

Around 2006, there were moves to "cancel" Debye because of alleged Nazi sympathies. However, because the evidence was ambiguous, these moves faltered.

Reflections of the 2026 Nobel Prizes

 Last week I enjoyed following the Nobel Prize announcements in Physiology or Medicine, Physics, and Chemistry. Reading the popular accounts behind the science is enjoyable, as they are well-written and accessible. Here are a few things I found interesting.

I found the Physics prize a bit surprising. On the one hand, the creativity and vision needed to construct the IceCube neutrino observatory is beautiful and impressive. They have clearly detected high-energy neutrinos, and there is potential for some important astrophysical discoveries. On the other hand, the key word is potential. This prize seems in contrast to earlier prizes for gravitational wave detectors (2017) and for observations of cosmic neutrinos (2002, 2015). For example, the latter solved fundamental issues in our understanding of neutrinos. But perhaps I am missing something.

A fun cultural side to the Physiology or Medicine prize to Karl Deisseroth is that there is a beautiful novel, Transcendent Kingdom, that centres on the life of a graduate student who works in a fictional lab at Stanford based on his. On the science side, a key aspect was the discovery of the protein channelrhodopsin. The essential photophysics is that the protein contains a chromophore, all-trans retinal, that upon photoexcitation undergoes a conformational change (isomerisation) to a 13-cis configuration. This is similar to the photophysics responsible for vision. In both cases, the conformational change of the chromophore triggers a conformational change of the whole protein and an electrical signal.

The figure is taken from here.

The chromophore conformational changes are associated with conical intersections between the potential energy surfaces of the ground and excited states. The essential physics can be understood in terms of a two-site Hubbard model.

The Chemistry Prize was for the development of new synthesis methods to selectively make molecules of definite chirality. The popular science background from the Nobel foundation mentions three important conditions for chiral synthesis that were stated in a short paper in 1953, by Charles Frank, a condensed matter theorist. He was a pretty amazing scientist. He showed in 1951 how topological defects determine the growth rates and strength of crystals and wrote an influential review on snow crystals. In 1958  Frank wrote down a free energy density for nematic liquid crystals, without reference to Landau, and long before Anderson emphasised the importance of generalised rigidity in condensed matter. (Albeit, and perhaps more remarkably, Frank's paper freely builds on a 1933 paper by Carl Wilhelm Oseen.

Aside. There is a fascinating and fundamental question about why chiral molecules do not spontaneously quantum tunnel backwards and forwards between the two possible chiralities. Briefly, this is because of quantum decoherence due to the environment. This question often comes up in discussions about emergence in chemistry.

Was there anything that you thought was particularly interesting about any of these prizes?

Nobel Prize Predictions for 2026

This week Nobel Prizes will be announced. Predicting the awardees is a fun annual exercise. It is also good to reflect on what has been achieved, including outside our own areas, and big advances from the past we may now take for granted.

Before writing this I looked at suggestions from readers of Doug Natelson's blog, Nanoscale views, an article in Physics World, predictions from Clarivate based on citations, and recent recipients of the Wolf Prize.

Not surprisingly, this post is not that different from last year's.

Although we know little about how the process actually works or the explicit criteria used, I have a few speculative suggestions and observations.

1. The Wolf Prize is often a precursor. According to Wikipedia, for physics, "from the 26 prizes awarded between 1978 and 2010, fourteen winners have gone on to win the Nobel Prize, five of those in the following year." For chemistry, 13 awardees have subsequently won a Nobel.

2. Every now and then, the Swedes surprise us. Sometimes this may be because we now take for granted something discovered thirty or forty years ago.

3. Every few years, the physics committee seems to go for something technological, sometimes arguably outside physics, perhaps to remind people how important physics is to modern technology and other areas of science.

4. They seem to spread the awards around between different areas of physics.

5. Theory only gets awards when it has led to well-established experimental observations and new experimental fields. Brilliant theoretical discoveries that motivate large research enterprises (more theory and unsuccessful or ambiguous experimental searches) are not good enough. This is why predictions based on citation numbers may be misleading. High citations (particularly for a specific paper) are usually a necessary, but not sufficient, condition for a prize.

6. Once an award has been made on one topic, it is unlikely that there will be another award for a long time, if ever, on that same topic. In other words, there is a high bar for a second award.

7. I don't think the logic is to pick an important topic and then choose who should get the prize for the topic. This approach works against topics where many researchers independently made contributions that were all important. The awardee needs to be a standout who won't be a debatable choice.

8. The prize can go to at most 3 individuals and does not go to groups. This works against the large collaborations in particle physics and cosmology that produce significant work.

What do you think of these principles?

For some of the above reasons, I discuss below why I am sceptical about some specific predictions.

My top prediction for physics is Metamaterials with negative refractive index, going to John Pendry (theory) and David Smith (experiment). I know little about this topic.

Is it just a matter of time before twisted bilayer graphene wins a prize? This might go to Allan MacDonald (theory) and Pablo Jarillo-Herrero (experiment). They recently received a Wolf Prize. One thing that convinced me of the importance of this discovery was a preprint on moiré WSe2 with beautiful phase diagrams such as this one.

The level of control is truly amazing. Helpful background is the recent Physics Today article by Bernevig and Efetov.

This is big enough to overcome 6. and the earlier prize for graphene.

The prediction of Berry and Aharonov for topological phases in quantum mechanics is reasonable, except for questions about historical precursors in optics and quantum chemistry.

The prediction of topological insulators goes against 6. and the award to Haldane in 2016.

Predictions of a prize for quantum algorithms (Shor, Deutsch, Brassard, Bennett), conformal field theories, and braiding statistics go against 5. 

Chemistry 

I don't know enough chemistry to make meaningful predictions. On the other hand, in 2019 I correctly predicted John Goodenough for lithium batteries.  I do like the prediction (last year) from Clarivate for Biomolecular condensates (Brangwynne, Hyman, and Rosen). I discussed them briefly in my review article on emergence.

What do you think about my "principles"?

What are your predictions?