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.

Lecture on degenerate Fermi gases at low temperatures

Here are the slides for an undergraduate lecture I gave today. The slides also include the derivations I gave by hand on the document viewer.

For historical background, I found a book chapter from Out of the Crystal Maze, interesting and helpful.

Some fascinating context concerning Arnold Sommerfeld. Over a period of 32 years, he built up an amazing school of theoretical physics in Munich. He mentored seven future Nobel laureates. However, sadly, when he retired in 1934 from a Chair in Theoretical Physics, it took four years to replace him because of interference from the Nazi Party. Even worse, in the end, Wilhelm Muller, a Nazi loyalist, was appointed. According to Wikipedia

"Müller, an aerodynamicist, had not been thought of as a theoretical physicist before this time, and opposed the "new" theoretical physics promoted by scientists such as Albert Einstein. His appointment is seen by historians as political, and during his tenure he would teach only classical physics"

What is so amazing about the degeneracy pressure for Fermi liquids?

 I just finished preparing my undergraduate lecture on ideal Fermi gases at zero temperature. Here are my slides. 

(Note. The derivations are not included, as I sketch them by hand on a document viewer.) (Aside. Personally, I would prefer to do derivations on the whiteboard, but this is viewed unfavourably as the video recording of the lecture will not capture that.)

A few things I find amazing about the lecture content.

Degeneracy pressure is a macroscopic quantum effect, since it vanishes in the limit Planck's constant goes to zero.

Degeneracy pressure is relevant to astrophysics. I enjoyed learning more about how it prevents the collapse of white dwarfs and neutron stars. For the first time (to my memory) I derived the Chandrasekhar limit for white dwarfs. I also enjoyed seeing how this limit is essentially the same physics as the Tolman-Oppenheimer-Volkoff limit for neutron stars.

An earlier post discussed some things I find amazing about Chandrasekhar's expression for the upper.

A few historical notes I found interesting.

It is the 100th anniversary of Fermi-Dirac statistics and the distribution function. It is impressive how Ralph Fowler immediately saw how the degeneracy pressure could explain the stability of white dwarfs. He was the PhD supervisor of Dirac, Chandrasekhar, and Mott. He was also the first to define the zeroth law of thermodynamics.

Chandrasekhar's limit was not accepted for a long time because of the opposition of Arthur Eddington. In hindsight, some consider this was a setback for astrophysics, particularly interest in the possibility of black holes, for decades. This opposition to the profound new ideas of a young PhD student by a legendary senior figure reminded me of Bardeen's opposition to Josephson. However, in that case, within a few years, experiments showed that the establishment figure was wrong.

What is so amazing about the Fermi energy?

In two weeks, I am giving two lectures about degenerate Fermi gases in a third-year undergraduate course on statistical mechanics. The textbook is the beautiful book by Schroeder.

I want to highlight a few things that are amazing about the Fermi energy.

1. It is given by an incredibly simple expression.

E_F =

ℏ²

2m

(3π²n)^2/3

2. Besides fundamental constants, it is only determined by the number density of fermions n.

3. It is relevant to diverse systems: electrons in a metal, liquid 3He, electrons in a white dwarf star, neutrons in a neutron star, ultracold fermionic atoms such as 6Li. The table below shows that for all of these systems many of them are in the regime where the temperature is much less than the Fermi temperature. In other words, the Fermi energy (temperature) is so "large" that "low" temperatures can be very "high". 

4. Quantum effects can occur at surprisingly high temperatures. Most experiments that probe quantum effects are performed at low temperatures, often on the scale of a few Kelvin, or even less. Yet, inside a star, the quantisation of energy levels and quantum statistics can dominate the physics of the system, even at temperatures of millions or billions of Kelvin.

Note, how the temperatures of these systems span 18 orders of magnitude!

System	Typical Number Density (n/V)	Typical Temperature (T)	Fermi Temperature (TF)
Electrons in a metal	~1022 – 1023 cm-3	~300 K (Room temp)	~5 × 104 K
3He atoms in liquid 3He	1.6 ×1022 cm-3	< 1 K (Cryogenic)	~2 – 5 K
Nucleons in atomic nuclei	1.6 ×1038 cm-3	~0 K (Ground state)	4.4 × 1011 K
Electrons in a white dwarf star	~1029 – 1030 cm-3	~107 K (Interior)	~108 – 109 K
Neutrons in a neutron star	~1038 – 1039 cm-3	~108 – 109 K	~1011 – 1012 K
Fermionic atoms in an ultracold gas	~1012 – 1014 cm-3	~10 – 100 nK	~100 – 1000 nK

The table was made with Google Gemini.

Two asides.

A. An old blog post considered the interesting question of why people call these systems "degenerate."

B. A more profound question is why neglecting the interactions between the fermions (which involves energies comparable to the Fermi energy in most of these systems) can give a theory that is so successful, both qualitatively and reasonably quantitatively. Landau's Fermi liquid theory provides the answer.

Statistical mechanics in just one equation

 This semester, I am giving four lectures in a third-year undergraduate course on statistical mechanics. Last year, I gave a guest lecture on the Ising model.

The first thing I want to emphasise is that the whole course is built around just one equation.

exp(−F(T, V)/kT) = ∑_s exp(−E_s/kT) ≡ Z(T, V)

This connects macroscopic thermodynamic properties (contained in the Helmholtz free energy F(T,V)) to microscopic properties (the energies E_s of all possible states s of the system).

[Note that from equilibrium thermodynamics, partial derivatives of F(T,V) give the entropy (and specific heat capacity) and the pressure (equation of state)].

I think that we are so used to this equation that we may miss just how amazing and profound it is.

First, the equation is incredibly simple.

Second, it is universal. It applies to any system in thermodynamic equilibrium regardless of its chemical or physical composition.

Third, in the context of the theory of emergent phenomena, it is exceptional because it provides a robust, tested way to connect the microscopic to the macroscopic. Biology, neuroscience, economics, sociology, and computer science have nothing like it.

Fourth, although the above three points are impressive, the basis of its validity remains an outstanding problem (a mystery?). One can "derive" it and "justify" it by drawing on assumptions such as the fundamental postulate ("in an isolated system all accessible microstates are equally probable"), the principle of maximum entropy, and the validity of equilibrium thermodynamics. But why are they true?

Although the equation is simple, implementing it, particularly for systems of interacting particles, is challenging. This challenge can be broken into five steps. One can get stuck on any one of the steps. People build whole careers on them.

1. For a system of interest, propose microscopic states of each particle and a Hamiltonian for the whole system.

2. Enumerate all possible microscopic states of the system.

3. Evaluate the energy of each of these microstates.

4. Perform the sum over all states in the partition function Z.

5. Take the thermodynamic limit where the system size becomes infinite.

Undergraduates need to learn about the Ising model

A typical undergraduate course on statistical mechanics is arguably misleading because (unintentionally) it does not tell students several important things (related to one another).

Statistical mechanics is not just about how to calculate thermodynamic properties of a collection of non-interacting particles.

A hundred years ago, many physicists did not believe that statistical mechanics could describe phase transitions. Arguably, this lingering doubt only ended fifty years ago with Wilson's development of renormalisation group theory.

It is about emergence: how microscopic properties are related to macroscopic properties.

Leo Kadanoff commented, "Starting around 1925, a change occurred: With the work of Ising, statistical mechanics began to be used to describe the behaviour of many particles at once."

When I came to UQ 25 years ago, I taught PHYS3020 Statistical Mechanics a couple of times. To my shame, I never discussed the Ising model. There is a nice section on it in the course textbook, Thermal Physics: An Introduction, by Daniel Schroeder. I guess I did not think there was time to "fit it in" and back then, I did not appreciate how important the Ising model is. This was a mistake.

Things have changed for the better due to my colleagues Peter Jacobson and Karen Kheruntsyan. They now include one lecture on the model, and students complete a computational assignment in which they write a Monte Carlo code to simulate the model.

This year, I am giving the lecture on the model. Here are my slides  and what I will write on the whiteboard or document viewer in the lecture.

One hundred years of Ising

In 1925, Ising published his paper on the solution of the model in one dimension. An English translation is here.https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html

Coincidentally, next week I am giving a lecture on the Ising model to an undergraduate class in statistical mechanics. To flesh out the significance and relevance of the model, here are some of the interesting articles I have been looking at:

The Ising model celebrates a century of interdisciplinary contributions, Michael W. Macy, Boleslaw K. Szymanski and Janusz A. Hołyst

This mostly discusses the relevance of the model to understanding basic problems in sociology, including its relation to the classic Schelling model for social segregation.

The Ising model: highlights and perspectives, Christof Külske

This mostly discusses how the model is central to some work in mathematical physics and probability theory.

The Fate of Ernst Ising and the Fate of his Model, Thomas Ising, Reinhard Folk, Ralph Kennac, Bertrand Berche, Yurij Holovatche.

This includes some nice memories of Ising from his son, Thomas.

Aside: I wanted a plot of the specific heat for the one-dimensional model. According to Google AI "In a 1D Ising model with no external magnetic field, the specific heat is zero at all temperatures."

Amazing things about Chandrasekhar's white dwarf mass limit

This is ultra-condensed matter physics!

In 1931, Subrahmanyan Chandrasekhar published a seminal paper, for which he was awarded the Nobel Prize in 1983. He showed that a white dwarf star must have a mass less than 1.4 solar masses, otherwise it will collapse under gravity. White dwarfs are compact stars for which the nuclear fuel is spent and electron degeneracy pressure prevents gravitational collapse. 

The blog Galileo unbound has a nice post about the history and the essential physics behind the paper.

There are a several things I find quite amazing about Chandrasekhar's derivation  and the expression for the maximum possible mass. 

m_H is the mass of a proton. M_P is the Planck mass. The value of the mass limit is about 1.4 solar masses.

Relativity matters

If the electrons are treated non-relativistically then there is no mass limit. However, when the star becomes dense enough the Fermi velocity of the electrons approaches the speed of light. Then relativistic effects must be included.

A macroscopic quantum effect

Degeneracy pressure is a macroscopic quantum effect. The expression above involves Planck's constant.

Quantum gravity

The mass formula involves the Planck mass, M_P.  On the one hand, this phenomena does not involve quantum gravity because there is no quantisation of the gravitational field. On the other hand, the effect does involve the interplay of gravity and quantum physics.

A "natural" scale

Formula that involve Planck scales usually represent scales of length, energy, mass, time, and temperature that are "unreal", i.e., they are vastly different from terrestial and astrophysical phenomena. For example, the temperature of the Hawking radiation from a black hole of one solar mass is 60 nanoKelvin. 

In contrast, the limiting mass is on the same scale as that of our own sun!

It agrees with astronomical observations

Determinations of the masses of hundreds of white dwarfs show most have a mass of about 0.5 solar masses and the highest observed value is 1.3 solar masses.

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.

What is the chemical potential?

I used to find the concept of the chemical potential rather confusing.
Hence, it is not surprising that students struggle too.
I could say the mantra that "the chemical potential is the energy required to add an extra particle to the system" but how it then appeared in different thermodynamic identities and the Fermi-Dirac distribution always seemed a bit mysterious.

However, when I first taught statistical mechanics 15 years ago I used the great text by Daniel Schroeder. He has a very nice discussion that introduces the chemical potential. He considers the composite system shown below, where a moveable membrane connects two systems A and B. Energy and particles can be exchanged between A and B. The whole system is isolated by the environment and so the equilibrium state is the one which maximises the total entropy of whole system.

Mechanical equilibrium (i.e. the membrane does not move) occurs if the pressure of A equals the pressure of B.

Thermal equilibrium (i.e. there is no net exchange of energy between A and B) occurs if the temperature of A equals that of B. Thus, temperature is the thermodynamic state variable that tells us where two systems are in thermal equilibrium.

Diffusive equilibrium (i.e. there no net exchange of particles between A and B) occurs if the chemical potential of particles in A equals that in B, where the chemical potential is defined as

Starting with this one can then derive various useful relations such as those between the Gibbs free energy and the chemical potential (dG= mu dN and G=mu N).
Thus, the chemical potential is the thermodynamic state variable/function that tells us whether or not two systems are in diffusive equilibrium.

Doug Natelson also has a post about this topic. He mentions the American Journal of Physics article on the subject by Ralph Baierlein, drawing heavily from his textbook. However, I did not find that article very helpful, particularly as he mostly uses a microscopic approach, i.e. statistical mechanics. (Aside: the article does have some interesting history in it though).
I prefer to first  use a macroscopic thermodynamic approach before a microscopic one as, I discussed in my post, What is temperature?

Comparing statistical mechanics to real data

I have posted before that I think it is very important in teaching to present students with comparisons of theory with actual experimental data. It is disturbing that many teachers and textbook writers make little effort to do this. On a positive note, here is a particularly nice comparison.

In PHYS4030 Condensed Matter Physics this week I am teaching Paramagnetism and diamagnetism, closely following chapter 31 of Ashcroft and Mermin.
Consider non-interacting paramagnetic ions with total angular momentum J in a magnetic field B at thermal equilibrium at temperature T. Basic statistical mechanics can be used to derive an expression for the magnetisation, which is a universal function of B/T, known as the Brilloiun function. A&M do not compare this to experiment. However, I recalled that when I was an undergraduate we used a very nice book, Heat and Thermodynamics, [5th edition] by Mark Zemansky. It contains the comparison below, taken from a 1952 paper by Henry.

Measuring Planck's constant by thermodynamics

An incredibly important and useful equation is the Sackur-Tetrode equation which gives the absolute entropy of an ideal gas and shows how

-Planck's constant enters statistical mechanics in a way that is measurable
-to resolve the Gibbs paradox

I did not appreciate just how significant the equation was until I taught a statistical mechanics course.

The latest APS News has a beautiful article by Richard Williams on the equation in the This Month in Physics History column.

Did you know Tetrode as only 17 when he derived the equation from Boltzmann's equation (see below)!