Condensed Matter Physics Flashcards
What is space group symmetry?
The constant combination of symmetry operations. All symmetry operations in the space group leave the structure unchanged.
What symmetry operations are there?
- Point symmetry (rotations or reflections)
- Translation symmetry (translation by a primitive lattice vector)
What is the Brillouin Zone?
The Wigner-Seitz cell in reciprocal space.
What is the Leonard-Jones potential?
A useful approximation of the potential for an interatomic interaction
Where is the classical equilibrium separation position in the Leonard Jones potential curve of the interaction between two atoms?
The minimum of the potential.
What is the harmonic approximation?
For small deviations of atoms from their equilibrium position, we can approximate the potential as a quadratic.
What is the adiabatic/Born-Oppenheimer approximation?
Divide the system into 2 subsystems that can exchange energy (nuclei and electrons). If the electronic system remains in its ground state, we can assume energy is adiabatically interchanged between the KE of the nuclei and the electronic system. This assumes the velocities of the nuclei are much smaller than electrons due to the mass differences.
For phonons in 3D space, how many modes of polarisation are there per q value?
3 (Two transverse and one longitudinal)
What do systems with more than one atom per unit cell have?
Optical modes at higher energies, where atoms vibrate out of phase and acoustic modes at lower energies, where atoms move in phase.
What probes can be used to measure the dynamics of solids?
- Inelastic neutron scatterings
- Inelastic X-ray scattering
- Raman Scattering
- Two-photon infrared absorption
What do scattering probes measure?
The power spectrum of the density fluctuations corresponding to excitations of the system. These fluctuations are due to thermal agitation and zero point motion of the system.
What is the scattering cross section proportional to?
The number of particles.
How is inelastic neutron scattering used to measure the dynamics of solids?
Neutrons are produced by a reactor. Phonons are measured via the strong force i.e. interaction between the neutron and the nucleus. Neutrons have a relatively weak interaction with matter meaning that it is a bulk probe. Phonons are measured in many Brillouin zones. Thermal neutrons are usually used.
How is inelastic X-ray scattering used to measure the dynamics of solids?
Synchrotron X-rays are produced. The x-rays scatter from the electron cloud around the atom/ion. More sensitive to heavy atoms and their motion. Moderately surface sensitive. Penetration of 20-30 keV x-rays is microns into sample. Small sample can be used. Higher energy photons (x-rays) mean that many Brillouin zones can be investigated.
How is Raman scattering used to measure the dynamics of solids?
It is done with light from lasers. Interaction of visible light with solid occurs via the polarizability (χ = P/E) of valence electron. A prerequisite for the observation of a Raman excitation is that χ changes with the displacement of the phonon. Only certain (“Raman active”, even parity) phonons are observed. A symmetry analysis reveals which these are. Usually a surface probe. Can be used in a microscope.
what are Stokes and anti-Stokes scattering?
Phonon creation and phonon annihilation are called “Stokes” and “anti-Stokes” scattering respectively.
At high temperatures, what does every degree of freedom in a solid have?
An average energy of 1/2 k_B T
What did the Einstein model aim to explain?
Why the heat capacity fell below the Dulong Petit value at low temperatures.
How does the Einstein model each atom in a solid?
Each atom behaves as a quantum harmonic oscillator. This has two degrees of freedom (KE and PE).
What did the Debye model aim to explain?
The T cubed variation at low temperatures. He realised that oscillation of atoms is the same thing as sound, and sound is a wave (phonons), so it should be quantised the same way as Planck quantised light waves.
Why is the cut-off wavevector and frequency and Debye temperature introduced into Debye theory?
The Debye model assumes a linear phonon dispersion (sound). Since there are no Brillouin zones in the model, we introduce these terms.
What does the number of states (q-points) in the Brillouin zone equal?
The number of unit cells.
When does the Debye T cubed law apply?
A low temperature where only the small q phonons are excited.
What does the harmonic approximation predict?
- No thermal expansions (atoms vibrate around fixed positions)
- Phonons do not interact (scatter) with each other
- Infinite thermal conductivity