Monday, March 10, 2014

Getting a feel for orders of magnitude

Each time I teach a course I realise there is some particular intellectual challenge for students that I take for granted because the issue has become so second nature to me.

As an undergraduate I don't think I really learnt, or was taught, to make orders of magnitude estimates and then consider their consequences. I never took a course in solid state physics. I learnt every subject in a precise manner, more like applied mathematics. Perhaps, physics was not taught that way. But that is certainly how I learnt it. It was only when I went to graduate school in the US, that I had to learn to deal with orders of magnitude estimates. Indeed in the General Exam [qualifying Ph.D exam after 2 years] at Princeton there was a whole section called General Physics that did this kind of stuff. You can see some of the questions in this book. I actually think that learning to solve these type of problems was one of the most useful things I learnt during my whole Ph.D. This is the first step in theoretical model building.

Dealing with orders of magnitude and the associated approximations is one of the reasons why condensed matter is so hard for undergraduates.

Even in the first few weeks of a solid state physics course students are confronted with a plethora of energy, time, and length scales:

Size of an atom, separation of atoms in a crystal, mean free path, Fermi wavelength, wavelengths of different types of electromagnetic radiation, ...

Thermal energy [k_B T], Fermi energy, Coulomb repulsion between electrons, uncertainty due to scattering, ...

Scattering time, Cyclotron period, ...

Students need to get a feel for all these scales and remember them.
But this is not just an exercise in mindless memorisation, such as a random historical dates or the Latin names of different flora and fauna. Rather, they need to learn and understand the significance and implications of the relative magnitudes of these numbers. Here are a few examples of increasing profundity.

1. For the electrons in an elemental metal the Fermi temperature is orders of magnitude larger than room temperature.

Consequently, the electrons can be treated as a degenerate gas of fermions and most of their thermodynamic and transport properties are determined by the properties of the Fermi surface.

2. At low temperatures the electronic mean free path can be orders of magnitude larger than the spacing of atoms in a crystal.

This is completely inconsistent with the Drude and Sommerfeld models where the electrons scatter off the ions in the crystal. How can they "miss" thousands of atoms? This problem is resolved by the Bloch model: Bloch states do not scatter off the periodic potential of the crystal. The Bloch wavevector is a "good quantum number."

3. In elemental metals the average electronic kinetic energy is comparable to the Coulomb repulsion energy between electrons.

Yet, the Drude, Sommerfeld, and Bloch models all ignore interactions between the electrons. So, why do they work so well? This turns out to because of Landau's Fermi liquid theory.

4. The thermal energy at the superconducting transition temperature is orders of magnitude smaller than other energy scales in the problem [phonon energies, Fermi energy, ...].

This turns out to be because superconductivity is an emergent phenomena that leads to a new emergent energy scale, also reflecting the non-pertubative nature of the problem.

Besides emphasising the above issues in lectures and assigning relevant homework problems are their particular ways to help students learn this important skill and concept?


  1. "For the electrons in an elemental metal the Fermi temperature is orders of magnitude larger than room temperature."

    Since I worked on 2DEGs, I have to mention that this is not quite true for 2DEGs such as doped Si at the 'usual' doping levels (1E12 cm-2) where the Fermi temperatures are around ~100 K.


    More generally, the scales for 2D are different to those for 3D. Since I work on 2D stuff, I don't really have a good feel for the numbers in 3D. I reckon that people who work on more 1D stuff may also have a less developed feel for 2D and 3D scales.

    1. Thanks for the comment.
      I agree the statement is not true for the two-dimensional electron gas associated with semiconductor heterostructures.
      This is a nice problem I give for students to work through the 2D case and use real numbers for AlGaAs heterostructures.

      I think if people had appreciated that the bare Fermi temperature can be of order 10-100 K in 2DEGs they would have got quicker to the heart of the issues involved in the metal-insulator transition for these systems. They would have been less pre-occupied with issues of localisation and disorder and focused more on correlation effects. See for example,

  2. I think the article by Weisskopf is a wonderful source of showing the significance of understanding order of magnitudes:

    1. Sounds interesting. Unfortunately, I cannot get the link to work. Please send the page reference or the DOI.

    2. Here it is. Weisskopf VF (1975) Of atoms, mountains, and stars: A study in qualitative physics. Science 187(4177):605–612

    3. Thanks for recommending it. It is a beautiful article.