Lecture 5: Heat Capacity

We define the heat capacity, and calculate the phonon heat capacity in the high and low temperature limits. We also introduce the density of states.


Technical difficulties meant that this lecture did not get recorded this year.
In its place, I post last year's lecture 5:

Lecture Audio

Lecture 4: Diatomic Chain

We discuss generalities of phonon spectra. These include: frequency goes to zero at the reciprocal lattice vectors; group velocity goes to zero at the zone edge; frequency goes linear in k for small frequency; all physical modes are contained in the first Brillouin zone. We derive the dispersion relation for a linear chain of 2 distinct atom types. We discuss the quantization of phonons, and embark on a lightning fast review of the harmonic oscillator. A great java app showing how the harmonic oscillator eigenstates actually connect back to oscillatory motion is at:
http://www.fi.uib.no/AMOS/MOV/HO/

Audio

Lecture 3: Reciprocal Lattice

We review lattice planes, and talk about how to construct the corresponding Miller indices. We define the reciprocal lattice: Think of this as the Fourier wavevectors of the original lattice. It turns out that the reciprocal lattice of a Bravais lattice is itself a Bravais lattice. We define the first Brillouin zone. We calculate the dispersion (frequency vs. wavevector) for phonons in a 1D crystal in the harmonic approximation, and encounter our first instance of Goldstone's theorem: Since crystals break the continuous translational symmetry of free space, there is a corresponding Goldstone mode (the phonons, or vibrations of the crystal), which has the important property that the frequency goes continuously to zero.

Be sure and play with the Solid State Simulations package by Cornell, available at: http://www.ruph.cornell.edu/sss/sss.html

Demos: Today, we used the "Bravais" SSS program in class.
Visual Aid: The squishy crystal model -- actual balls and springs.

Audio

Lecture 2: Bravais Lattices

A lattice is a regular arrangement of an infinite set of points in space. A Bravais lattice is one where every point looks the same as every other point. You can build any lattice from a Bravais lattice by "decorating" it, in which case we call it a lattice with a basis. We show how to construct the Wigner-Seitz cell, a particular type of unit cell. Roger Penrose, mathematician, came up with a way to tile space that has (in a manner of speaking) five fold symmetry, and never repeats. The patterns are beautiful. Be sure and google Penrose tiles.

Audio

Lecture 1: The Failure of Reductionism

Reductionism is the idea that by breaking things into their smallest constituents, we will learn all about them. For example, we might want to learn about solids by breaking them into atoms, then learn about the atoms by breaking them into the constituent electrons and nuclei, and so on. But reductionism is merely a philosophy handed down to us by the Greeks -- is it really correct? New ideas in the field point toward the failure of reductionism, and lead to "emergence" as a better paradigm for gaining knowledge in condensed matter/solid state physics. Emergence is the idea that when many particles get together, new phenomena appear which are not encoded in the microscopic laws, and in fact are independent of the microscopic laws. For example, all solids are hard, regardless of what atoms are in them. Likewise, if we changed the microscopic laws, by changing, say, the shape of the Coulomb potential, we'd still get solids that are hard. That means hardness (and many properties like it) is not caused by the microscopic laws, but rather is caused by deeper physics. It turns out in fact that it's related to symmetry.

Audio