David Mermin on his life in science: funny, insightful, and significant

 David Mermin has posted a preprint with the modest title, Autobiographical Notes of a Physicist

There are many things I enjoyed and found interesting about his memories. A few of the stories I knew, but most I did not. He reminisces about his interactions with Ken Wilson, John Wilkins, Michael Fisher, Walter Kohn, and of course, Neil Ashcroft.

Mermin is a gifted writer and can be amusing and mischievous. He is quite modest and self-deprecating about his own achievements.

He explains why we should refer to the Hohenberg-Mermin-Wagner theorem, not Mermin-Wagner.

One of his Reference Frame columns in Physics Today, stimulated Paul Ginsbarg to start the arXiv.

I was struck by how Mermin's career belongs to a different era. The community was smaller and more personal. Doing physics was fun. Time was spent savouring the pleasure of learning new things and explaining them to others. Colleagues were friends rather than competitors. His research was curiosity-driven. This led to Mermin making significant contributions to quantum foundations. And, he only published about two papers per year!

Teaching was valued, enjoyable, and stimulated research. It was also a way to learn a subject, regardless of the level at which it was taught. For eight years, Mermin and Ashcroft spent half their time writing their beautiful textbook!

I look forward to hearing others' reflections.

What materials have a transition from a metal to a band insulator?

 A metal and a band insulator are distinct states of matter. 

A suitable order parameter is the Drude weight, defined as the integral over frequency of the frequency-dependent conductivity at zero temperature.

How might a single material undergo a transition between a band insulator and a metal? 

The schematic below (from Kittel) illustrates three distinct possibilities for the band structure and band fillings.

(a) Band insulator. The bands do not overlap and the lower band is full. This occurs if there is an integer number of electrons per primitive cell in the crystal. 

(b) Metal I. The bands overlap and the system is a metal regardless of the number density of electrons.

(c) Metal II. The bands do not overlap. There is a non-integer number of electrons per primitive cell.

Suppose the number of electrons is fixed, (i.e. there is no chemical doping).

How can a metal-insulator transition occur when some physical parameter (such as pressure) changes?

Scenario A.

There is no structural phase transition associated with the metal-insulator transition.

A transition between (a) and (b) can occur. This means that at the transition the volume of the Fermi surface will be zero.

This may be an example of a Lifshitz transition (where there is a change in the topology of the Fermi surface), based on this 1959 paper by I.M. Lifshitz.

[Aside: I find this nomenclature confusing because there is also a Lifshitz point, where there is a phase transition between commensurate ordering (such as Neel antiferromagnet) and incommensurate ordering (such as a spiral antiferromagnet).

This is E.M. Lifshitz, co-author with Landau of A Course in Theoretical Physics, and I.M.'s brother.]

Scenario B.

The transition is accompanied by a structural phase transition, such as the doubling of the size of the primitive cell in the crystal.

One example of the latter is the Peierls transition in a one-dimensional metal with one electron per lattice site. Dimerising the lattice produces an energy gap at the Fermi wavevector leading to an insulator.

Another example is the transition from graphite to diamond that occurs at about 15 kbar of a pressure. The crystal structure changes and consequently there is a transition from a semi-metal to an insulator.

I have a few questions for readers. The discussion above involves basic solid state physics but I have not seen it clearly set out before. 

1. Do I have the physics correct?

2. Do you know somewhere this is discussed?

3. For scenario A, are the metallic and insulating states adiabatically connected?

4. Do you know of any specific materials where scenario A. actually occurs?

Key ideas in solid state physics

I have had some interesting discussions with an editor at Oxford University Press about the Very Short Introductions series. The upshot is that I have been asked to write a VSI Condensed Matter Physics. I find it amazing and concerning that after 500 titles there wasn't one about CMP. There are excellent ones on Magnetism, Superconductivity, Complexity, and Crystallography.
I am very happy about this and will post more about it later. At first, we discussed a VSI on Solid State Physics. Here is my outline for that.

1. Introduction
    Solid state physics
   - is central to technology (diodes, transistors, LEDs, photovoltaic cells, and computer memories)
   - provides important lessons in scientific model building
   - is one of the largest fields of physics
   - is a rich source of ideas and concepts that have cross-fertilised with other fields of science

2. Solids are quantum matter
Solids are made of atoms (nuclei and electrons).
Electrons are waves. Electrons are fermions. Quantum degeneracy
How is a metal like a white dwarf star?

3. Symmetry matters
Crystal structures. Think in reciprocal space, not in real space.
Why is it possible to determine a crystal structure from x-ray diffraction?
Internal symmetries of electrons: spin, gauge symmetries.

4. Electron waves in a crystal
Bragg scattering. Extended states.
Energy gaps: metals, semiconductors, and insulators
Why is copper a metal while diamond is an insulator?
Why can an electron go through a crystal and pass millions of atoms without being scattered?

5. Multitudes of solid phases
Phase diagrams. Allotropes.
When is graphite less stable than diamond?
Magnetic and superconducting phases
Classifications of phases through "broken symmetry"

6. Emergence
Quasi-particles: electrons and holes, phonons, magnons
How does structure (chemical and crystal) determines electronic and structural properties?
Why does magnesium seem to have positively charged electrical currents?

7. Beyond perfect infinite crystals
a. Impurities, disorder, localisation, glasses: the value of imperfection
b. Flatland. Surfaces and dimensionality

8. Topology matters
Quantum Hall effects, Topological insulators, Quantum magnetism

9. Solid state technology
 Diodes, transistors, LEDs, photovoltaic cells, and computer memories

10. Solid concepts
What have we learned about scientific model building?

This is too much. But what would you add or subtract?

The role of the Platonic ideal in solid state physics

In the book Who Got Einstein's Office?, about the Institute for Advanced Study at Princeton, the author Ed Regis, mocks it as the "One True Platonic Heaven" because he claims its members are Platonic idealists, who are interested in pure theory, and disdain such "impurities" as computers and applied mathematics.

Platonic solids

This stimulated me to think about the limited but useful role of pure mathematics, Platonic idealism, and aesthetics in solid state theory. People seem particularly excited when topology and/or geometry plays a role.

The first example I could think of is the notion of a perfect crystal.

Then comes Bloch's theorem, which surely is the central idea of introductory solid state physics.

The icosahedron is central to quasi-crystals.

Beautiful examples where advanced pure maths plays are role are
Chern-Simons theory of edge states in the Quantum Hall Effect
and topological terms in the action for quantum spin chains, as elucidated by Haldane.

As I have said before I think topological insulators is a beautiful, fascinating, and important topic. However, I am concerned by the disproportionately large number of people working on the topic and the associated hype. I wonder if some of the appeal and infatuation is driven by Platonic idealism.

For a classic example of how Platonism leads to imperfect theory is Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596).

Good theory finds a balance between beauty and the necessity of dirty details.

Can you think of other examples where Platonic idealism plays a positive role in condensed matter theory?

Is this a reasonable exam?

I struggle to set good exam questions. One wants to test knowledge and understanding in a way that is realistic within the constraints of students abilities and backgrounds.

I do not have a well-defined philosophy or approach, except for often recycling my old questions...
I think I do have a prejudice towards two goals.

A. Testing higher level skills [e.g. relating theory to experiment, putting things in context, ...] as much as specific technical knowledge [e.g. state Bloch's theorem or solve the Schrodinger equation for a charged particle in constant magnetic field].

B. Testing general and useful knowledge. For basic undergraduate courses [e.g. years 1 to 3] the question should be one that another faculty member could do, even if they have not taught the course. Sometimes, colleagues write questions that I cannot do. You have to have done the problem before, e.g. in a tutorial. We seeming to be testing whether someone has done this course, not "essential" knowledge.

However, I am not sure I really go anywhere near reaching these goals.
Here is a recent mid-semester exam I set for my solid state class of fourth-year undergraduates.
Is it reasonable?

How do you set exam questions?
Do you have a particular approach?

Science is told by the victors and Learning to build models

A common quote about history is that "History is written by the victors". The over-simplified point is that sometimes the losers of a war are obliterated (or at least lose power) and so don't have the opportunity to tell their side of the story. In contrast, the victors want to propagate a one-sided story about their heroic win over their immoral adversaries. The origin of this quote is debatable but there is certainly a nice article where George Orwell discusses the problem in the context of World War II.

What does this have to do with teaching science?
The problem is that textbooks present nice clean discussions of successful theories and models that rarely engage with the complex and tortuous path that was taken to get to the final version.
If the goal is "efficient" learning and minimisation of confusion this is appropriate.
However, we should ask whether this is the best way for students to actually learn how to DO and understand science.

I have been thinking about this because this week I am teaching the Drude model in my solid state physics course. Because of its simplicity and success, it is an amazing and beautiful theory. But, it is worth thinking about two key steps in constructing the model; steps that are common (and highly non-trivial) in constructing any theoretical model in science.

1. Deciding which experimental observables and results one wants to describe.

2. Deciding which parameters or properties will be ingredients of the model.

For 1. it is Ohm's law, Fourier's law, Hall effect, Drude peak, UV transparency of metals, Weidemann-Franz, magnetoresistance, thermoelectric effect, specific heat, ...

For 2. one starts with only conduction electrons (not valence electrons or ions), no crystal structure or chemical detail (except valence), and focuses on averages (velocity, scattering time, density) rather than standard deviations, ...

In hindsight, it is all "obvious" and "reasonable" but spare a thought for Drude in 1900. It was only 3 years after the discovery of the electron, before people were even certain that atoms existed, and certainly before the Bohr model...

This issue is worth thinking about as we struggle to describe and understand complex systems such as society, the economy, or biological networks. One can nicely see 1. and 2. above in a modest and helpful article by William Bialek, Perspectives on theory at the interface of physics and biology.

What are the biggest discoveries in solid state electronic technology?

Watching an excellent video about the invention of the transistor stimulated to me to think about other big discoveries and inventions in solid state technology.

Who would have thought that huge device would become the basis of an amazing revolution (both technological, economic, and even social...)?

In particular, which are the most ubiquitous ones?
For which devices did both theory and experiment play a role, as they did for the transistor?

I find it worthwhile to think about this for two reasons. First, this semester I am again teaching solid state physics and it is nice to motivate students with examples.
 Second, there is too much hype about basic research in materials and device physics, that glosses over the formidable technical and economic obstacles, to materials and devices becoming ubiquitous. Can history give us some insight as to what is realistic?

Here is a preliminary list of some solid state devices that are ubiquitous.

transistor

inorganic semiconductor photovoltaic cell

liquid crystal display

semiconductor laser

optical fiber

giant magnetoresistance used in hard disk drives

blue LED used in solid state lighting

lithium battery

Some of these feature in a nice brochure produced by the USA National Academy of Sciences.

Here are a few that might be on the list but I am not sure about as I think they are more niche applications with limited commercial success. Of course, that may change...

thermoelectric refrigerators

organic LEDs

superconductors (in MRI magnets and as passive filters in mobile phone relay towers )

Is graphene in any commercial device?

What would you add or subtract from the list?

A good video on the discovery of the transistor

I am on the lookout for good videos that meet roughly the following criteria:
-available free online
-on condensed matter or chemistry
-accessible and interesting to a popular audience
-represent the science in a reasonable and helpful way
-lack hype

This is motivated by the following experience. I knew that there were people (mostly young) who will spend endless hours watching trashy videos (whether B-grade movies or silly antics) on Youtube. However, I only recently learned that there are also people who will spend hours watching videos on serious subjects (science, politics, history, religion, ...)

I am keen to find such material so I can recommend it as alternatives to the kind of thing featuring Michio Kaku or string theory propaganda from Brian Greene.

This is why I recently watched Forces of Nature with Brian Cox. Unfortunately, I think only the first episode is available for free.

In my search, I came across this nice history of the discovery of the transistor narrated by Ira Flatow and produced by PBS.

It nicely brings out the healthy interaction between Bardeen's theoretical work and Brattain's experimental work, even when one or both did not work out. They were a great stimulus to one another.

It also discusses the mixed legacy of Shockley, the Broken Genius.




What videos can you recommend that meet the criteria above?

Why are quantum gases called degenerate?

In my recent tutorial on bad metals at IISER Pune a student asked me a basic question that I could not answer:
"Why is the degenerate Fermi gas called "degenerate"?
Is it anything to do with degenerate energy levels?"

So, I went in search for answers.

The Wikipedia entry on Degenerate matter is a bit rambling and I found it unhelpful.

I then went to the library and looked at a few textbooks and found a range of answers. Some books use the term "degenerate" without any elaboration.

In the discussion below it seems looking at the Oxford dictionary is helpful:

Having lost the physical, mental, or moral qualities considered normal and desirable; showing evidence of decline. 

technical: Lacking some usual or expected property or quality, in particular.

Here are a few entries

degenerate. (This use of the word is completely unrelated to its other use to describe a set of quantum states that have the same energy).

Daniel V. Schroeder, An Introduction to Thermal Physics, page 272.
[my favourite undergraduate text on statistical mechanics]

... a bit of terminology. At low temperature, quantum ideal gases behave very differently from the way a classical ideal gas behaves. ..... The quantum gases are said to be degenerate at low temperatures. This is not a moral judgement. Rather, the word "degenerate" is used in the sense of departing markedly from the properties of an "ordinary" classical gas.

Ralph Baierlein, Thermal Physics, page 192.

Gas degeneration proper. The quantitative study of the deviations from the classical gas laws when xi is not very small ...

Erwin Schrodinger, Statistical Thermodynamics (1936)

Here xi is the product of the particle density and the thermal deBroglie wave length cubed.

The definition of a quantum gas is one where xi becomes larger than one. (This occurs for "high" densities and "low" temperatures).

But then there is an interpretation in terms of degeneracy of energy levels because one can consider the case where each energy level has degeneracy g and condition for a non-degenerate gas (i.e. Maxwell-Boltzmann statistics to apply) is
g >> n_i ~ exp ((mu-Ei)/kB T) = number of particles in level i

I welcome comments.

A basic quantum concept: energy level repulsion (avoided crossings)

When I learnt and later taught basic quantum mechanics I don't think the notion of energy level repulsion (or equivalently avoided crossings) was emphasised (or even discussed?).

Much later I encountered the idea in advanced topics in theoretical physics such as random matrix theory and in theoretical chemistry  (non-adiabatic transitions and conical intersections).

Yet level repulsion is a very simple phenomena that can be illustrated with just a two by two matrix describing two coupled quantum states, as nicely discussed on the Wikipedia page.

Last semester when I was teaching Solid State Physics I realised just how central and basic the phenomena is and that the students did not appreciate this.

Level repulsion is the origin of several key phenomena in chemistry and physics.

In solid state physics, it is the origin of the appearance of band gaps at the zone boundary and thus the all important distinction between metals and insulators.

Previously, I posted how Chemistry is quantum science because chemical bonding (the lowering of energy due to interacting atoms) arises due to the superposition principle. This could also be viewed as level repulsion.

Another key idea in chemistry is that of transition states and activation energies for chemical reactions. When one uses a diabatic state picture, particularly as emphasised by Shaik and Warshel, the transition state emerges naturally in terms of level repulsion.

The figure is taken from here.

Can you think of any other nice examples?

Incorporating scientist biographies into lectures

A few years ago I decided I wanted to include brief biographies of relevant great scientists in my undergraduate lectures. I posted (5 years ago!) about how I started with Landau but I lost momentum. This year I have put more effort into it. I just taught my second year undergraduate thermo class about Gibbs free energy and so I profiled Gibbs.

In solid state physics I have profiled Drude, Sommerfeld, von Laue, and Bloch.
I have found this quite enjoyable for myself and hopefully for the students. I have learnt quite a bit, just by reading the relevant Wikipedia pages. It also introduces students to the human dimension of science. For example, Drude died by suicide and so it is a good opportunity to flag mental health issues. Sommerfeld was a mentor of many great scientists. von Laue actively opposed the Deutsche Physik of the Nazis. Bloch was the first Director General of CERN.

Has anyone else experience at doing similar things? Any suggestions?

How Ashcroft and Mermin quickly became irrelevant and then relevant

This week in my Solid State Physics class I taught covered weak periodic potentials (including higher Brilloiun zones and Fermi surface reconstruction) and the tight binding model. I closely follow chapters 9 and 10 in Aschroft and Mermin, which was published in 1975.
This topic is somewhat iconic in that it features on the front and back cover of the book.

I think for the first time I understood the higher Brilloiun zones (rather than being overwhelmed by the geometrical complexity) and how this leads to the complex hole Fermi surfaces for metals of valence 2, 3, and 4. The key to visualising this better is just to do the problem in two dimensions first.

This got me wondering: why do we teach this stuff to students?

First, there is the intellectual beauty of the subject: how simple analytical and geometrical models can capture the complex band structures and Fermi surfaces of elemental metals.
However, today almost no one cares about elemental metals, or at least does research on them.
Note that most physics undergraduates and graduates don't ever take a course on nuclear physics. My department does not even teach one! Yet, the subject is a beautiful one and of great historical importance (both intellectually and politically!). The reason for this is that there is now very little research in basic nuclear physics. (I think this is a bad thing, but that is another story..)

But, metal physics is different.
A compelling reason I teach it in detail is that it provides a foundation to understand so much condensed matter research today: particularly how strongly correlated electron materials do (and do not) deviate from the Fermi liquid paradigm. Otherwise, I think Ashcroft and Mermin type courses would have been eventually sent off to the electrical engineering and materials engineering departments.

One can argue that almost happened. For the decade (1975-1985) following publication of the book, the content (not just on metals but also superconductivity) must have been largely of historical interest or considered only of interest to those working in "applied physics". But, the discovery of superconducting cuprates, heavy fermions, organic charge transfer salts, and iron pnictides, changed all that....

A helpful question to include in any student homework assignment

It is good to encourage students to be reflective in a concrete way about how they are going in a course. It is even better if the teacher knows what they are thinking.

I recently stumbled across the following idea. Include the following questions on a homework problem set.

(a) What is the most interesting thing that you have learnt in the course so far? 

(b) What is the concept that you understand the least? 

(c) What specific action are you going to take to address (b)? 

(d) List some of the skills required in this course. Which one do you think that you most need to improve? How might you do that?

I recently did this for my undergraduate solid state physics course.

I found the answers to (a) interesting. Sometimes the things that we think are interesting or boring are not necessarily the same as students. Many of my students said they really liked learning about crystal structures.

(c) is good because it makes the student actually reflect on what they might do and putting it down in writing hopefully creates some pressure and accountability for them to do it.

I think (d) is particularly important for my course. One reason why students find it so difficult is that it not only draws together many areas of physics (quantum, stat. mech., electromagnetism), but also uses a range of mathematics skills (calculus, sketching graphs, series expansions,...), and general scientific skills (model building, comparing experiment and theory, approximations, critical thinking, estimating orders of magnitude,...., keeping track of physical units, ....

Do you have any experience with similar exercises to encourage students to be self-reflective and take responsibility for addressing weaknesses?

Simple analytical models for crystal structure energetics

I am currently teaching my solid state class the basics of crystal structures.
For any simple material a basic question is:
can you construct a simple analytical model that can quantitatively predict (actually postdict) the following?

• the most stable crystal structure (e.g. FCC vs. BCC)
• the lattice constant 
• the binding energy of the crystal 
• the bulk modulus (i.e. compressibility) ?

Sometimes people make a big deal about the fact that computations based on Density Functional Theory approximations (with the "right" functional!) do reasonably well at post-dicting the above. However, it is important to acknowledge that

* there are very simple analytical models that do well too
* the relative energy differences between different structures are very small and may be quite sensitive to the choice of approximation.

Previously, I have posted about the challenge of crystal structure prediction for organic molecules.

In past years I gave a lecture about the predictions of simple analytical models, but lately I struggle to fit it into the course (a mistake?).
Here are my old slides, which closely follow Ashcroft and Mermin and Marder.

I find it quite striking how well these simple theories work and that they show how the energy difference between different crystal structures is quite small. For example, for inert gases modelled by a Lenard Jones potential the relative energy difference between FCC and hexagonal close packed is 0.1 per cent.
This subtle competition between different phases shows that this is not a unique feature of strongly correlated electron systems.

Teaching students to think like a condensed matter physicist

Yesterday I heard Carl Wieman give a talk at UQ, Taking a scientific approach to science education. I hope I will say more about it later. Here I just want to highlight one helpful point he made concerning relating teaching to the psychology and practise of "becoming an expert".
We need to teach students to "think like a physicist". This is quite different to imparting (memorising) information in textbooks.

Later in the day I taught my class PHYS4030 Condensed Matter Physics which is really the basics of solid state physics, a la Ashcroft and Mermin. I led a discussion with the students about
"What is the conceptual strategy that we are following in this course?"
We came up with something like the following.

1. Define the simplest possible model.
2. Calculate some properties of materials that are predicted by the model.
3. Compare the predicted properties with experimental results. What are the successes and failures of the model?
4. Refine the model in the hope of better agreement with experiment.
5. Repeat the process.

This is what we are doing as we go from Drude to Sommerfeld to Bloch models, and then consider the role of electron-electron interactions.

I stressed that this is not just what we do in this course but this is the general research strategy in condensed matter physics.

Note this strategy is quite different to how one teaches most physics courses; e.g., quantum mechanics and electromagnetism.
The latter is largely an exercise in the applied mathematics of Maxwell's equations. One never really considers whether they are right or not, or need to be modified.

The Sommerfeld model is a Pauling point

A basic question that comes up in introductory solid state physics is:
Why does the Sommerfeld model for metals work so well?
It assumes that electrons are non-interacting fermions. Yet if you calculate the first order correction (in e^2 where e is the electronic charge) in the Coulomb energy you find it is comparable to the kinetic energy associated with the ground state.

Aside: the success of Sommerfeld is such a puzzle that Wigner mentioned it (for the wrong reasons in my view) at the end of his famous 1962 essay, The Unreasonable Effectiveness of Mathematics in the Physical Sciences.

The standard answer we give students is screening  plus Landau's Fermi liquid theory.
However, an interesting question is what happens if you try to actually do some sort of systematic many-body expansion with respect to the Coulomb interaction. Can you get the calculation to converge to experiment and see why Sommerfeld is good?

In Telluride last (northern) summer I heard a nice talk by Timothy Berkelbach that is relevant to this issue. The message I took away was that the Sommerfeld model is a Pauling point, i.e. by accident it gets the right answer for the wrong reasons.

The relevant paper has now appeared on the arXiv.

Spectral Functions of the Uniform Electron Gas via Coupled-Cluster Theory and Comparison to the GW and Related Approximations 
James McClain, Johannes Lischner, Thomas Watson, Devin A. Matthews, Enrico Ronca, Steven G. Louie, Timothy C. Berkelbach, Garnet Kin-Lic Chan

A key graph is below. It shows the quasi-particle dispersion relation for a uniform electron gas with r_s=4, the value relevant to sodium.

For comparison the "binding energy" [i.e. the k=0 energy] deduced from experiment is -2.6 eV, and the values for Sommerfeld and LDA are about -3.1 eV. Hartree-Fock gives -7.3 eV!

As you increase the "level of theory" [i.e. the sophistication of treatment] of electron correlations you go from Sommerfeld to HF to HF+GW to CCSD [Coupled Cluster Singles and Doubles].
Then one sees the answer at first gets worse and then improves and you almost get back to where you started!

Aside: This also illustrates how LDA is a Pauling point too!

Don't confuse necessary and sufficient conditions

In Carl Caves recent UQ Quantum Science Seminar "Quantum metrology meets Quantum Information Science" as an aside he also made an important side point.

People often erroneously assume that the converse of a statement is true (i.e.  A implies B means that B implies A). This came up because a few referees had said that some of the results he presented in the seminar were "obvious". Roughly speaking, this concerns the issue of trying to determine what input quantum state to an interferometer will produce a "physical" output state. He found that the input state had to be "physical" (by some well-defined technical criteria). Showing this is non-trivial. However, it is obvious a physical input state is sufficient to produce a physical output state. But, that does not mean it is necessary. Showing this turned out to be quite non-trivial.

I can immediately think of two other cases where scientists made similar errors of conflating necessary and sufficient conditions.

The first case is the existence of quasi-crystals. It was well known that a periodic array of atoms is sufficient to produce sharp diffraction peaks. However, many people erroneously assumed that this was also necessary. I emphasise this point when I teach undergraduates about quasi-crystals.

The second concerns Angle-dependent MagnetoResistance Oscillations in quasi-two-dimensional metals. Not long after their experimental discovery AMRO was explained in terms of a coherent interlayer transport and a three-dimensional Fermi surface. It was subsequently more or less assumed that observing AMRO was evidence for a three-dimensional Fermi surface.  However, in 1998, Perez Moses and I showed that a 3D Fermi surface was not necessary for the existence of AMRO.

Effective tutorials, II.

I think one of the weakest aspects of my teaching is running tutorials. In Australia, for most upper level undergraduate courses there is a weekly one hour tutorial [problem solving session] that is run by the lecturer.

Mostly I have run these tutorials according to a traditional format. There are a set of problems that the students are meant to attempt before the session. At the tutorial I then work through the solutions on the board. There are many problems with this approach. Students often don't attempt the problems beforehand because they are not assessed. It is just like a lecture. Students are hesitant to ask questions and just write down what you write on the board. It is somewhat boring. I am not sure the students get much out of it.

Previously, I posted about a different approach that my colleague Joel Corney introduced for a large second year class we were co-teaching. I thought this was quite effective. But, it also required TA's (grad. student tutors) to help.

For PHYS4030 [a solid state physics class with 15 fourth year undergrads] I finally did something I have wanted to do for a long time. Each week I have assigned two students in the class to run the tutorial. They can opt out if they want. They are meant to attempt them beforehand. They then stand at the board and do what they can. Other students offer suggestions and ask questions. I only speak up when essential.

I think it is going well. The students seem more engaged.  Furthermore, it is very helpful for me to see what they find difficult or are confused about; sometimes things that I think are basic and gloss over too quickly. On the other hand, I think you do need a critical mass of motivated and engaged students. Unfortunately, not every class has this.

I welcome other ideas.

Enhanced teaching of crystal structures

This past week I taught my condensed matter class about crystal structures and their determination by X-rays. This can be a little dry and old. Here, are few things I do to try and make things more interesting and relevant. I emphasise that many of these developments go beyond what was known or anticipated when Ashcroft and Mermin was written. Furthermore, significant challenges remain.

Discuss whether the first X-ray crystallography experiment the most important experiment in condensed matter, ever?

Take crystal structure "ball and stick" models to the lectures.

Give a whole lecture on quasi-crystals.

Use the bravais program in Solid State Simulations to illustrate basic ideas. For example, the equivalence of each reciprocal lattice vector to an X-ray diffraction peak, to a family of lattice planes in real space, and to a Miller indice.

Show a crystal structure for a high-Tc cuprate superconductor and an organic charge transfer salt. Emphasize the large number of atoms per unit cell and how small changes in distances can totally change the ground state (e.g. superconductor to Mott insulator). Furthermore, these small changes may be currently beyond experimental resolution. This is very relevant to my research and that of Ben Powell.

Very briefly mention the Protein Data Bank, and its exponential growth over the past few decades. It now contains more than 100,000 bio-molecular structures. Mention the key concept that Structure determines Property determines Function. Mention that although many structures resolve bond lengths to within 0.2-0.6 Angstroms, that this just isn't good enough to resolve some important questions about chemical mechanisms related to function. I am currently writing a paper on an alternative "ruler" using isotopic fractionation. 

Next year I may something about the importance of synchrotrons and neutron sources, and crystallographic databases such as the Cambridge Structural Database, which contains more than 700,000 structures for small organic molecules and organometallics.

Teaching students to be more critical

One on the many disturbing things I find about science today is people claiming that because a particular theory agrees with a particular experiment that the theory must be valid.
Little consideration is given to the possibility that the agreement may just be an accident. The "correct" theory may actually be quite different. They may be getting the "right" answer for the "wrong" reasons.
I am never sure if the people who make these kind claims are sincere, naive, and/or just engaging in marketing.
Students need to be taught to be more critical.

I am currently teaching an advanced undergraduate course on solid state physics, PHYS4030. It follows Ashcroft and Mermin closely.

I have just taught the Drude and Sommerfeld model. Drude provides a nice example of getting the "right" answer for the "wrong" reasons. In both models the thermal conductivity is given by the following expression from kinetic theory

where c_p is the specific heat capacity and u_f^2 denotes the average kinetic energy of the heat carriers.

In Drude, the first factor is independent of temperature and "large", being of order k_B.
The second factor is proportional to temperature, and "small".

However, in the Sommerfeld model, which gets the physics correct, the specific heat is proportional to temperature, "small", and of order k_B T/T_F, where T_F is the Fermi temperature.
The average kinetic energy is independent of temperature and "large", being proportional to the Fermi energy.

The different terms in the Drude model are off by a factor of order one thousand, but these errors cancel beautifully so it gives an answer that agrees with Sommerfeld and with experiment to within a factor of two!

I stressed to the students that this is a good example how sometimes you get the right answer for the wrong reason. The fact your theoretical model agrees with a particular experiment does not prove it is correct.

This underscores the need for the method of multiple working hypotheses.