The danger of complaining

As in any institution, there a many things that could be done better in universities. Just beware of letting middle and upper management aware of your complaints. They will likely put you on a committee.

Provocative quote

Biology is the search for the chemistry that works.

RJP Williams, Lecture at Oxford (1996)

Why can undergraduates be taught nuclear physics?

Nuclear physics represents a very sophisticated quantum many-body problem. A nuclei is a dense droplet of very strongly interacting fermions. Furthermore, the interaction between a pair of nucleons is nothing like the simple Coulomb interaction between a pair of electrons in a solid or a molecule. Yet nuclear physics can be taught to undergraduates!

Why? Basically, because (as in most many-body systems) one can describe the low lying quantum states in terms of an effective Hamiltonian which describes weakly interacting quasi-particles (which have the same quantum numbers as protons and neutrons). The dynamics of an individual quasi-particle is determined by the mean field (average potential) of all the other nucleons. This is known as the nuclear shell model, first proposed in 1949 and recognised by the Nobel Prize in 1963.

A nice Physics article by John Schiffer is worth reading (it is just a page) as it puts in context a recent PRL which shows how the tensor component of the average potential can explain experimental data on a wide range of nuclei.

Mixed valence: physicists vs. chemists I

The chemical concept of valence is of great utility. The valence of an atom, M, determines the number of neighbouring atoms with which M can form chemical bonds. For most metal ions, the valence, V, is equal to the oxidation number, O. Deviations from this equality occur when delocalization of electrons occurs. The simplest case is where V and O differ by one because one electron from each of the M atoms is completely delocalized in the conduction band. Mixed valency of transition metal and rare-earth ions in solids and compounds is a question of fundamental interest in materials physics, chemistry, and molecular biophysics.

In 1967, Robin and Day (chemists) published a classification scheme for mixed-valence that is still widely used today. Class 1 describes systems with two crystallographic sites that are clearly distinct and and the two sites have integral but unequal valence. There is a large energy associated with transfer of electrons between sites. At the other extreme is Class 3 for which there are two sites which are not distinguishable, and one assigns a non-integral valence to both sites. The classic case of this is the Creutz-Taube ion. The valence electrons are delocalised between the two sites. Class 2 is the intermediate case where the environments of the two sites are distinguishable but not very different. The energy associated with electron transfer is sufficiently small that it can be thermally activated and be associated with significant optical absorption in the visible range. On the time scale of the vibrations of the atoms the electrons may appear to be delocalised.

Classes 1 and 2 correspond to what Varma (a physicist) termed inhomogeneous mixed valence, although, perhaps, inhomogeneous integral valence may be more appropriate. Class 3 corresponds to homogeneous mixed valence.

more to come.....

A promising career cut short?

He excelled academically at a young age. After graduating from Harvard, he completed a Ph.D in Mathematics at the University of Michigan. At aged 25, he became an Assistant Professor at University of California, Berkeley. He published a number of papers from his Ph.D, but according to Web of Science, they have rarely been cited. After two years, he resigned from Berkeley to pursue issues he was more passionate about. He wrote a pamphlet that some considered so important it was published in the New York Times and the Washington Post.

Where is he now?

In a US Federal Prison, serving a life sentence without the possibility of parole.

Who is he?

Theodore Kaczynksi, otherwise known as the Unabomber.

What went wrong?

A court-appointed psychiatrist diagnosed Kaczynski as suffering from paranoid schizophrenia.

What is my point?

Guard your mental health. The more brilliant you are the more vunerable you may be.

Desperately seeking spin liquids II

Kumar Raman has written a helpful comment on an earlier post where I made a conjecture which rules out spin liquid ground states for wide classes of Heisenberg antiferromagnetic spin models.

He points out that one can construct SU(2) invariant models with ground states similar to the spin liquid quantum dimer models, but concedes these models are not very physical.

Section 1.10 of Roderich Moessner and Kumar's review is a helpful description

of the procedure to construct a model.

So I have some questions?

Is the claim that the Hamiltonian (1.36) may have a spin liquid ground state?

In light of these points, how does my conjecture need to be sharpened to rule out such models?

How about no terms in Hamiltonian beyond first- and second-neigbour interaction and single plaquette, and first-neighbour must have the largest interaction?

Conjecture: Consider a spin-1/2 Heisenberg model on a two-dimensional lattice with short range antiferromagnetic exchange (both pairwise and ring exchange are allowed) interactions where the nearest neigbour interaction is the strongest. The Hamiltonian is invariant under SU(2) x L, where L is a space group and there is a total of spin-1/2 in the unit cell. Then the ground state spontaneously breaks at least one of the two symmetries SU(2) and L.

A signature of resonating valence bonds

In most molecules when you different vibrational modes have the highest frequency in the electronic ground state. Another way of looking at this is that it is chemical bonding which stabilises the ground state. These bonds will be strongest and stiffest in the ground state. Excited states are associated with less bonding and so most are associated with a reduction in vibrational frequencies.

An important except is benzene. In the lowest excited state the b2u vibrational mode increases in frequency by about 20 per cent. This has a natural explanation in terms of valence bond theory (see figure below) and is discussed in a Accounts of Chemical Research paper by Shaik, Zilberg, and Haas.

The ground state (which has A1g symmetry) can be described as a linear superposition of two valence bond structures, Kr + Kl.

The lowest excited state (which has B2u symmetry) can be described as the antisymmetric superposition, Kr - Kl.

The b2u distortion couples linearly to the energy of Kr and Kl near the degeneracy point. This leads to the curves shown above.

It is clear that the b2u vibrational mode will have a higher frequency in the excited state than the ground state.

High-level quantum chemistry calculations support this simple picture, which underscores the importance of using the appropriate Hilbert space to describe strongly correlated systems.