Saturday, May 18, 2019

Phonons in organic molecular crystals.

In any crystal the elementary excitations of the lattice are phonons. The dispersion relation for these quasi-particles relates their energy and momentum. This dispersion relation determines thermodynamic properties such as the temperature dependence of the specific heat and plays a significant role in electron-phonon scattering and superconductivity in elemental superconductors. A nice introduction is in chapter 13 of Marder's excellent text. [The first two figures below are taken from there].

The dispersion relation is usually determined in at least one of three different ways.

1. The classical mechanics of balls and harmonic springs, representing atoms and chemical bonds, respectively. One introduces empirical parameters for the strengths of the bonds (spring constants).

2. First-principles electronic structure calculations, often based on density functional theory (DFT). This actually just determines the spring constants in the classical model.

3. Inelastic neutron scattering.

The figure below shows the dispersion relations for a diamond lattice using parameters relevant to silicon, using method 1. I find it impressive that this complexity is produced with only two parameters.

Furthermore, it produces most of the details seen in the dispersion determined by method 3. (Squares in the figure below.) which compare nicely with method 2. (solid lines below).

What about organic molecular crystals?
The following paper may be a benchmark.

Phonon dispersion in d8-naphthalene crystal at 6K 
I Natkaniec, E L Bokhenkov, B Dorner, J Kalus, G A Mackenzie, G S Pawley, U Schmelzer and E F Sheka

The authors note that method 3. is particulary challenging for three reasons.
  • The difficulties in growing suitable single-crystal samples. 
  • The high energy resolution necessary to observe the large number of dispersion curves (in principle there are 3NM modes, where N is the number of atoms per molecule and M is the number of molecules per unit cell). 
  • The high momentum resolution necessary to investigate the small Brillouin zone (due to the large dimensions of the unit cell).
The figure below shows their experimental data for the dispersions. The solid lines are just guides to the eye.

The authors also compare their results to method 1. However, the results are not that impressive, partly because it is much harder to parameterise the intermolecular forces, which are a mixture of van der Waals and pi-pi stacking interactions. Hence, crystal structure prediction is a major challenge.

A recent paper uses method 2. and compares the results of three different DFT exchange-correlation functionals to the neutron scattering data above.
Ab initio phonon dispersion in crystalline naphthalene using van der Waals density functionals
Florian Brown-Altvater, Tonatiuh Rangel, and Jeffrey B. Neaton

What I would really like to see is calculations and data for spin-crossover compounds.

Thursday, May 16, 2019

Introducing phase transitions to a layperson

I have written a first draft of a chapter introducing phase diagrams and phase transitions to a layperson. I welcome any comments and suggestions. Feel free to try it out on your aunt or uncle!

Tuesday, May 7, 2019

Fun facts about phonons

Today we just take it for granted that crystals are composed of periodic arrays of interacting atoms. However, that was only established definitively one hundred years ago.
I have been brushing up on phonons with Marder's nice textbook, Condensed Matter Physics.
There are two historical perspectives that I found particularly fascinating. Both involve Max Born.

In a solid the elastic constants completely define the speeds of sound (and the associated linear dispersion relationship). In a solid of cubic symmetry, there are only three independent elastic constants, C_11, C_44, and C_12.
Cauchy and Saint Venant showed that if all the atoms in a crystal interact through pair-wise central forces then C_44=C_12. However, in a wide range of elemental crystals, one finds that C_12 is 1-3 times larger than C_44. This discrepancy caused significant debate in the 19th century but was resolved in 1914 by Born who showed that angular forces between atoms could explain the violation of this identity. From a quantum chemical perspective, these angular forces arise because it costs energy to bend chemical bonds.

The first paper on the dynamics of a crystal lattice was by Born and von Karman in 1912. This preceded the famous x-ray diffraction experiment of von Laue that established the underlying crystal lattice. In 1965, Born reflected
The first paper by Karman and myself was published before Laue's discovery. We regarded the existence of lattices as evident not only because we knew the group theory of lattices as given by Schoenflies and Fedorov which explained the geometrical features of crystals, but also because a short time before Erwin Madelung in Göttingen had derived the first dynamical inference from lattice theory, a relation between the infra-red vibration frequency of a crystal and its elastic properties.... 
Von Laue's paper on X-ray diffraction which gave direct evidence of the lattice structure appeared between our first and second paper. Now it is remarkable that in our second paper there is also no reference to von Laue. I can explain this only by assuming that the concept of the lattice seemed to us so well established that we regarded von Laue's work as a welcome confirmation but not as a new and exciting discovery which it really was.
This raises interesting questions in the philosophy of science. How much direct evidence do you need before you believe something? I can think of two similar examples from more recent history: the observation of the Higgs boson and gravitational waves. Both were exciting, and rightly earned Nobel Prizes.
However, many of us were not particularly surprised.
The existence of the Higgs boson made sense because it was a necessary feature of the standard model, which can explain so much.
Gravitational waves were a logical consequence of Einstein's theory of general relativity, which had been confirmed in many different ways. Furthermore, gravitational waves were observed indirectly through the decay of the orbital period of binary pulsars.

Wednesday, May 1, 2019

Emergence: from physics to international relations

Today I am giving a seminar for the School of Political Science and International Studies at UQ.
Here are the slides.

Thursday, April 25, 2019

Modelling the emergence of political revolutions

When do revolutions happen? What are the necessary conditions?
Here are the claims of two influential political theorists.

``a single spark can cause a prairie fire’’
Mao Tse Tung

 “it is not always when things are going from bad to worse that revolutions break out,... On the contrary, it often happens that when a people that have put up with an oppressive rule over a long period without protest suddenly finds the government relaxing its pressure, it takes up arms against it. … liberalization is the most difficult of political arts”
Alexis de Tocqueville (1856)

Is it possible to test such claims? What is the relative importance of levels of perceived hardship and government illegitimacy, oppression, penalties for rebellion, police surveillance, ...?

An important paper in 2002 addressed these issues.
Modeling civil violence: An agent-based computational approach 
Joshua M. Epstein

The associated simulation is available in NetLogo.
It exhibits a number of phenomena that can be argued to be emergent: they are a collective and are not necessarily unanticipated from the model.

Tipping points
There are parameter regimes at which there are no outbursts of rebellion.

Free assembly catalyzes rebellious outbursts
Epstein argues that this is only understood ex post facto.

Punctuated equilibrium
Periods of civil peace interspersed with outbursts of rebellion.

Probability distribution of waiting times between outbursts.
This distribution is not build explicitly into the model which involves only uniform probability distributions.
[Terminology here is analogous to biological evolution].

Salami corruption
Legitimacy can fall much further incrementally than it can in one jump, without stimulating large-scale rebellion.
[I presume the origin of Epstein's terminology is that salami is sliced something thinly... Maybe a clearer analogy would be the proverbial frog in a pot of slowly heated water].

de Tocqueville effect
Incremental reductions in repression can lead to large-scale rebellion. This is in contrast to incremental decreases in legitimacy.

Monday, April 22, 2019

Ten years of blogging!

I just realised that last month I had been blogging for ten years.
On the five year anniversary, I reflected on the influence that the blog has had on me.
I don't have much to add to those reflections. The second five years has not been as prolific but has been just as enriching and I am grateful for all the positive feedback and encouragement I have received from readers.

Wednesday, April 17, 2019

The emergence of social segregation

Individuals have many preferences. One is that we tend to like to associate with people who have some commonality with us. The commonality could involve hobbies, political views, language, age, wealth, ethnicity, religion, values, ... But some of us also enjoy a certain amount of diversity, at least in certain areas of life. We also have varying amounts of tolerance for difference.
A common social phenomenon is segregation: groups of people clump together in spatial regions (or internet connectivity) with those similar to them. Examples range from ethnic ghettos and teenage cliques to "echo chambers" on the internet.

The figure below shows ethnic/racist segregation in New York City. It is taken from here.

In 1971 Thomas Schelling published a landmark paper in the social sciences. It surprised many because it showed how small individual preferences for similarity can lead to large scale segregation. The context of his work was how in cities in the USA racially segregated neighbourhoods emerge.

One version of Schelling's model is the following. Take a square lattice and each lattice point can be black, white or vacant. Fix the relative densities of the three quantities and begin with a random initial distribution. A person is "unhappy" if only 2 or less of their 8 neighbours (nearest and next-nearest neighbours) on the lattice are like them. [They have a 25% threshold for moving]. They then move to a nearby vacancy. After many iterations/moves to an equilibrium is reached where everyone is "happy" but there is significant segregation.

The figure is taken from here.

There are several variants of the model that Schellman presented in later papers and an influential book Micromotives and Macrobehavior, published in 1978. He received the Nobel Prize in Economics in 2005 for work in game theory.

There is a nice simulation of the model in NetLogo. For example, you can see how if you set the individual preference for similarity at 30% one ends up with a local similarity of 70%.
In the Coursera, Model Thinking, Scott Page has a helpful lecture about the model.

This can be considered to be the first agent-based model. It is fascinating that Schellman did not use a computer but rather did his ``simulation'' manually on a checkerboard!

Physicists have considered variants of Schelling's model that can be connected to more familiar lattice models from statistical mechanics, particularly the Ising model. Examples include

Ising, Schelling and self-organising segregation 
D. Stauffer and S. Solomon

Phase diagram of a Schelling segregation model
L. Gauvin, J. Vannimenus, J.-P. Nadal
This connects to classical spin-1 models such as the Blume-Capel model.

A unified framework for Schelling's model of segregation 
Tim Rogers and Alan J McKane

Competition between collective and individual dynamics 
Sébastian Grauwin, Eric Bertin, Rémi Lemoy, and Pablo Jensen

Shelling's model is a nice example of emergence in a social system. A new entity [highly segregated neighbourhoods] emerges in the whole system thatwase not anticipated based on a knowledge of the properties of the components of the system.