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Condensed Matter Physics > Condensed Matter Physics > Flashcards

Condensed Matter Physics Flashcards

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1
Q

What is space group symmetry?

A

The constant combination of symmetry operations. All symmetry operations in the space group leave the structure unchanged.

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2
Q

What symmetry operations are there?

A
  • Point symmetry (rotations or reflections)
  • Translation symmetry (translation by a primitive lattice vector)
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3
Q

What is the Brillouin Zone?

A

The Wigner-Seitz cell in reciprocal space.

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4
Q

What is the Leonard-Jones potential?

A

A useful approximation of the potential for an interatomic interaction

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5
Q

Where is the classical equilibrium separation position in the Leonard Jones potential curve of the interaction between two atoms?

A

The minimum of the potential.

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6
Q

What is the harmonic approximation?

A

For small deviations of atoms from their equilibrium position, we can approximate the potential as a quadratic.

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7
Q

What is the adiabatic/Born-Oppenheimer approximation?

A

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.

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8
Q

For phonons in 3D space, how many modes of polarisation are there per q value?

A

3 (Two transverse and one longitudinal)

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9
Q

What do systems with more than one atom per unit cell have?

A

Optical modes at higher energies, where atoms vibrate out of phase and acoustic modes at lower energies, where atoms move in phase.

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10
Q

What probes can be used to measure the dynamics of solids?

A
  • Inelastic neutron scatterings
  • Inelastic X-ray scattering
  • Raman Scattering
  • Two-photon infrared absorption
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11
Q

What do scattering probes measure?

A

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.

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12
Q

What is the scattering cross section proportional to?

A

The number of particles.

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13
Q

How is inelastic neutron scattering used to measure the dynamics of solids?

A

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.

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14
Q

How is inelastic X-ray scattering used to measure the dynamics of solids?

A

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.

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15
Q

How is Raman scattering used to measure the dynamics of solids?

A

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.

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16
Q

what are Stokes and anti-Stokes scattering?

A

Phonon creation and phonon annihilation are called “Stokes” and “anti-Stokes” scattering respectively.

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17
Q

At high temperatures, what does every degree of freedom in a solid have?

A

An average energy of 1/2 k_B T

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18
Q

What did the Einstein model aim to explain?

A

Why the heat capacity fell below the Dulong Petit value at low temperatures.

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19
Q

How does the Einstein model each atom in a solid?

A

Each atom behaves as a quantum harmonic oscillator. This has two degrees of freedom (KE and PE).

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20
Q

What did the Debye model aim to explain?

A

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.

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21
Q

Why is the cut-off wavevector and frequency and Debye temperature introduced into Debye theory?

A

The Debye model assumes a linear phonon dispersion (sound). Since there are no Brillouin zones in the model, we introduce these terms.

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22
Q

What does the number of states (q-points) in the Brillouin zone equal?

A

The number of unit cells.

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23
Q

When does the Debye T cubed law apply?

A

A low temperature where only the small q phonons are excited.

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24
Q

What does the harmonic approximation predict?

A
  • No thermal expansions (atoms vibrate around fixed positions)
  • Phonons do not interact (scatter) with each other
  • Infinite thermal conductivity
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25
Q

What three processes give rise to the finite mean path in real crystals?

A
  • Phonon-phonon scattering (Umklapp processes)
  • Crystal imperfections (defects and impurities)
  • Crystal boundaries due to he finite size of the crystal
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26
Q

Explain the Umklapp process.

A

Because meaningful phonons q’s lie in the first Brillouin zone, any longer q produced in a collision must be brought back to the first zone by the addition of a G. An Umklapp process can involve a reversal of the sense of the total momentum. They allow equilibrium to be established

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27
Q

What is the temperature dependence of thermal conductivity?

A
  • As T tends to 0, average specific heat is proportional to T cubed. The mean free path depends on impurities, isotropic disorder, defects and boundaries as temperature as is temperature independent
  • At intermediate temperatures, Umklapp processes become allowed and proportional to exp(TD/2T)
  • At high T, the total number of excited phonons is proportional to T and mean free path is proportional to 1/T. The average heat capacity is constant (3R)
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28
Q

What are there characteristics of fermions?

A
  • Have half integer spin
  • Include electrons, protons, 3He atoms etc
  • Wavefunction is asymmetric wrt to particle exchange
  • Pauli exclusion principle applies
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29
Q

What are the characteristics of bosons?

A
  • Have integer spin
  • Include photons pi mesons etc
  • Wavefunction is symmetric wrt particle exchange
  • No Pauli exclusion principle
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30
Q

What distributions apply to fermions and bosons respectively?

A

Fermions: Fermi-Dirac distribution
Bosons: Bose-Einstein distribution

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31
Q

What is the key point of Bose-Einstein condensation?

A

Particles condense into the p = k = 0 ground state.

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32
Q

What is needed for the BE formula to make physical sense?

A

The chemical potential must be less than or equal to zero.

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33
Q

What determines the gas phase and the condensed phase in BE condensation?

A
  • For the gas phase: nλ3 (T) = g3/2(z)
  • For the condensed phase: z = 1
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34
Q

What does a Zeeman slower do?

A

A beam of atoms travels towards a laser in a spatially varying magnetic field B(z). The field shifts the frequency of the atomic transition used in the Doppler cooling through the Zeeman effect. These low and high velocity atoms become resonant with the laser and are slowed. A common approach is that the field decreases along z so that some atoms will undergo a constant deceleration.

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35
Q

What is evaporative cooling?

A

When the most energetic atoms are allowed to escape as the confining trap potential is changed.

36
Q

How does Laser/Doppler cooling of gases work?

A

Atoms are subjected to a beam of laser light with a frequency just below the atomic transition. Atoms moving towards the source of light will see photons with a frequency which is doppler shifted to a higher frequency, making it more likely to absorb a photon. When atoms absorb a photon, the component of momentum towards to laser is reduced slowing the atom. The atom decays back to the ground state and the photon is emitted in a random direction thus there is on average a net reduction of kinetic energy in the direction towards the laser. The frequency of the atomic absorption resonance and be shifted by a spatially-dependent magnetic field B(r). This is used to slow faster moving atoms where B(r) is large. The circular polarised laser light can also be used to select a particular atomic transition in the field. This allows is two lasers in opposing directions to be combined.

37
Q

How is a magnetic atom trap made?

A

If a local minimum in B is produced, this will act as a trap for atoms of sufficiently low energy. This trap may be combined with a non-uniform laser illumination to create a magneto-optical trap.

38
Q

How does a Magneto-optical trap work?

A
  • The spatially varying field and Zeeman effect shift the resonance so that as atoms move away from zero field region they become resonant and are laser cooled
  • Six apposing lasers in three spatial directions are used to slow atoms in all directions
  • Opposing lasers have opposite circular polarisation to select different resonances
39
Q

Why is He a quantum fluid while Ne is not?

A

He has a smaller mass and weaker interaction potential.

40
Q

What is a superfluid?

A

A fluid that will flow with a infinitesimal pressure difference implying that the viscosity is zero.

41
Q

What is superflow?

A

The movement of particles without dissipation. He-II will flow through narrow capillaries without any resistance due to viscosity.

42
Q

What is Film flow?

A

Capillary action caused surface adhesion combined with superflow causes containers of superfluid to empty.

43
Q

What is a fountain effect?

A

A small heated superleak

44
Q

How are vortices formed in superfluids?

A

When a rotating superfluid breaks up into a lattice.

45
Q

What does the Andronikashvilli experiment show?

A

Measurement of the moment of inertia of a stack of disks immersed in helium and suspended with a wire (Andronikashvilli) shows that a fraction of the helium contributes to the inertia and a fraction does not.

46
Q

What is the specific heat anomaly in BEC and liquid helium?

A

The BEC show a change in gradient at Tc, whereas the helium superfluid transition shows a λ-point anomaly with a divergent Cp ∝ |t|^−α at Tc.

47
Q

What is the momentum distribution of particles as T tends to zero in superfluids?

A

Only some of the superfluid particles are in the zero momentum state.

48
Q

What is a continuous or second order phase transition?

A

In the absence of an applied magnetic field, an Ising Ferromagnet will order spontaneously to a state with finite magnetic moment below a critical temperature Tc. When we lower the temperature through the critical point in this way, the system undergoes a continuous or second order phase transition.

49
Q

What is the order parameter of the Ising ferromagnet?

A

The magnetic moment

50
Q

What type of symmetry is broken in the uniaxial ferromagnet?

A

Rotational symmetry. This means that if we rotate the ordered system by π about an axis perpendicular to the quantisation axis of the moment then the total moment is inverted and the system is different.

51
Q

What thermodynamic properties diverge in a continuous phase transition?

A

Heat capacity and susceptibility.

52
Q

What is a first order phase transition?

A

A phase transition from a discontinuous change in the magnetisation at B = 0 from changing the applied magnetic field for T < Tc.

53
Q

In terms of the free energy derivatives, what are first order and second order phase transitions?

A

If there is a finite discontinuity in one or more of the first derivatives the transition is termed first-order. If the first derivatives are continuous but second derivatives are discontinuous or infinite, the transition is described as second order.

54
Q

What does a jump (discontinuity) in the entropy mean in a phase transition?

A

There is a latent heat:
∆S = ∆Q/T0.

55
Q

How do we classify the state of a phase transition?

A

By minimising the relevant free energy. If the system is at constant pressure this is the Gibbs free energy or at constant applied field B this the magnetic Gibbs energy. If there is no explicit p or B in the Hamiltonian, we use the Helmholtz free energy.

56
Q

What is mean field theory?

A

An approximation for the thermodynamic properties of a system treating the order parameter as spatially constant.

57
Q

What is Bragg-Williams theory?

A

A model with which we can investigate mean-field theory and phase transitions.

58
Q

What happens when the temperature of a substance is above the critical point?

A

There is no difference between liquid and gas.

59
Q

What order is the phase transition of solid-liquid?

A

First order

60
Q

What order is the phase transition of gas-liquid?

A

First order with a critical end point

61
Q

What will two coexisting phases have the same of?

A

p and T

62
Q

What is the Van der Waals equation?

A

The van der Waals equation of state provides a simple model that predicts a gas to liquid phase transition.

63
Q

What features are added to the ideal gas equation to create the vdW equation?

A

Intermolecular attraction and short range repulsion.

64
Q

What is the spinodal region in a graph of the vdW at the liquid-phase transition?

A

The van der Waals isotherms for T < Tc have an unphysical region bounded by the spinodal line. where dv/dp > 0

65
Q

What is the Maxwell construction?

A

An addition to the Van der Waals theory that is a physically consistent variation of the free energy with volume.

66
Q

What did Maxwell propose for the Maxwell construction?

A

In Maxwell’s description the system separates into two phases whose volume per particle are V1 and V2. At a point along the line such as X, the total free energy is a linear combination of F1 and F2.

67
Q

What is the law of corresponding states?

A

The scaled version of the van der Waals equation suggests that we can scale the behaviour of different gases on to universal curves near the critical point. The law of corresponding states says that the phase diagram of different substances should be the same when plotted in reduced coordinates.

68
Q

Why does the vdW theory not predict the correct critical exponent?

A

This difference is because vdW is a mean field theory which does not take account of critical fluctuations.

69
Q

When does an object or system have a certain symmetry?

A

When the object is unchanged under the
corresponding symmetry transformation.

70
Q

What phase transition has no change in symmetry?

A

liquid-gas

71
Q

How does the Euler Strut work?

A

With no load, the Euler strut has left right (mirror) symmetry. The mechanical equations describing the system also have left-right symmetry. When the vertical force exceeds a critical value the strut buckles, breaking the left right symmetry

72
Q

How is order parameter measured?

A

Order parameter is the “periodic density” which can be measured by the intensity of a new Bragg peak.

73
Q

Why do superfluids and superconductors break gauge symmetry?

A

Because there is coherence of the wavefunctions of different objects.

74
Q

What is Landau theory?

A

Theory based on the view that near a phase transition, the free energy can be expanded as a simple polynomial. The system is described by a continuous local order parameter.

75
Q

What is a condition for the free energy in the Landau expansion?

A

f must be invariant under symmetry operations that leave the disordered phase unchanged. e.g. If we have an Ising magnet, a π rotation would cause any odd terms to change sign so they are not allowed.

76
Q

In the presence of an external field, h, what happens to the free energy in Landau theory?

A

It is replaced with the Gibbs free energy

77
Q

What is the issue with mean field theories e.g. Landau theory?

A

They only describe phase transitions where the amplitude of fluctuations of the order parameter are sufficiently small. This depends on the dimensionality of the systems and range of the interactions.

78
Q

Why do many superconductors show a mean field transition?

A

The long range of the interactions responsible for the phase transition suppress spatial fluctuations in the order parameter making the phase transition appear mean field like.

79
Q

What is Ginzburg Landau theory?

A

An extension to Landau theory which incorporates variation
of the order parameter in space (but not time).

80
Q

What does Ginzburg Landau theory add to the order parameter?

A

A correlation length, ξ. In the disordered state, ξ the characteristic length of a correlated region diverges at Tc.

81
Q

How do fluctuations change around the critical point?

A

At temperatures far below or far above a critical point, the behaviour of the order parameter resembles the surface of a tranquil lake (height representing the order parameter). As we approach the critical temperature, fluctuations driven by thermal energy become important an the correlation length (wavelength) of the fluctuations diverges. At a second order phase transition, infinitely long-range critical fluctuations develop.

82
Q

What does universality mean in relation to critical behaviour?

A

Universality means that near a continuous phase transition, the physical behaviour becomes independent of microscopic detail provided physical quantities are scaled.

83
Q

What does critical behaviour depend on?

A

The dimensionality of the order parameter and the dimensionally of the system.

84
Q

What do systems with the same universality class have?

A

The same critical components

85
Q

What do universality classes depend on?

A

Space-dimensionality of system and dimensionality of order parameter.

86
Q

What is the nature of heat capacity anomaly determined by?

A

Universality class

Condensed Matter Physics (1 decks)

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