1. A Wigner crystal is a triangular lattice of electrons in a two dimensional plane. The...

Transcribed Image Text:

1. A Wigner crystal is a triangular lattice of electrons in a two dimensional plane. The longitudinal vibration modes of this crystal are bosons with dispersion relation w = a√k. Show that, at low temperatures, these modes provide a contribution to the heat capacity that scales as CT¹. 2. Use the fact that the density of states is constant in d = 2 dimensions to show that Bose-Einstein condensation does not occur no matter how low the temperature. 3. Consider N non-interacting, non-relativistic bosons, each of mass m, in a cubic box of side L. Show that the transition temperature scales as TN2/3/mL² and the 1-particle energy levels scale as E₁ x 1/mL². Show that when T < Te, the mean occupancy of the first few excited 1-particle states is large, but not as large as O(N). 4. Consider an ideal gas of bosons whose density of states is given by g(E) = CE-1 for some constants C and a > 1. Derive an expression for the critical temperature Tes below which the gas experiences Bose-Einstein condensation. In BEC experiments, atoms are confined in magnetic traps which can be modelled by a quadratic potential of the type discussed in Question 9 of Example Sheet 2. Determine Te for bosons in a three dimensional trap. Show that bosons in a two dimensional trap will condense at suitably low temperatures. In each case, calculate the number of particles in the condensate as a function of T < Te 5. A system has two energy levels with energies 0 and e. These can be occupied by (spinless) fermions from a particle and heat bath with temperature T and chemical potential. The fermions are non-interacting. Show that there are four possible microstates, and show that the grand partition function is Z(μ, V.T) = 1+ 2+ ze¬Be +2²e-Be where z = e. Evaluate the average occupation number of the state of energy €, and show that this is compatible with the result of the calculation of the average energy of the system using the Fermi-Dirac distribution. How could you take account of fermion interactions? 6. In an ideal Fermi gas the average occupation numbers of the single particle state [r) is n.. Show that the entropy can be written as S = Ə ƏT S = -kB [(1 - n,) log(1 − n) + n, log n,] (KBT log Z)μ.V. Find the corresponding expression for an ideal Bose gas. = Show that (An,)² n, (1 n,) for the ideal Fermi gas. Comment on this result, especially for very low T. What is the corresponding result for an ideal Bose gas? - nc 7. As a simple model of a semiconductor, suppose that there are N bound electron states, each having energy -A <0, which are filled at zero temperature. At non-zero temperature some electrons are excited into the conduction band, which is a continuum of positive energy states. The density of these states is given by g(E)dE = AVEDE where A is a constant. Show that at temperature T the mean number ne of excited electrons is determined by the pair of equations N e(μ+A)/kBT + 1 = 5.² 0 Show also that, if ne << N and kBT <A and e/knT <1, then g(E) dE e(E-H)/kBT + 1 2μA+KBT log 2N A√(KBT) ³ 8. Consider an almost degenerate Fermi gas of electrons with spin degeneracy 9, = 2. At high temperatures, show that the equation of state is given by pV = NkBT (1 and the average energy is At low temperatures, show that the chemical potential is (1-1/2 (KBP) EF E + = Ep 1 X³N 4√2g,V 2 + + 3.NE (1+5 (²) * -..) 2 (kBT 5 12 EF 1. A Wigner crystal is a triangular lattice of electrons in a two dimensional plane. The longitudinal vibration modes of this crystal are bosons with dispersion relation w = a√k. Show that, at low temperatures, these modes provide a contribution to the heat capacity that scales as CT¹. 2. Use the fact that the density of states is constant in d = 2 dimensions to show that Bose-Einstein condensation does not occur no matter how low the temperature. 3. Consider N non-interacting, non-relativistic bosons, each of mass m, in a cubic box of side L. Show that the transition temperature scales as TN2/3/mL² and the 1-particle energy levels scale as E₁ x 1/mL². Show that when T < Te, the mean occupancy of the first few excited 1-particle states is large, but not as large as O(N). 4. Consider an ideal gas of bosons whose density of states is given by g(E) = CE-1 for some constants C and a > 1. Derive an expression for the critical temperature Tes below which the gas experiences Bose-Einstein condensation. In BEC experiments, atoms are confined in magnetic traps which can be modelled by a quadratic potential of the type discussed in Question 9 of Example Sheet 2. Determine Te for bosons in a three dimensional trap. Show that bosons in a two dimensional trap will condense at suitably low temperatures. In each case, calculate the number of particles in the condensate as a function of T < Te 5. A system has two energy levels with energies 0 and e. These can be occupied by (spinless) fermions from a particle and heat bath with temperature T and chemical potential. The fermions are non-interacting. Show that there are four possible microstates, and show that the grand partition function is Z(μ, V.T) = 1+ 2+ ze¬Be +2²e-Be where z = e. Evaluate the average occupation number of the state of energy €, and show that this is compatible with the result of the calculation of the average energy of the system using the Fermi-Dirac distribution. How could you take account of fermion interactions? 6. In an ideal Fermi gas the average occupation numbers of the single particle state [r) is n.. Show that the entropy can be written as S = Ə ƏT S = -kB [(1 - n,) log(1 − n) + n, log n,] (KBT log Z)μ.V. Find the corresponding expression for an ideal Bose gas. = Show that (An,)² n, (1 n,) for the ideal Fermi gas. Comment on this result, especially for very low T. What is the corresponding result for an ideal Bose gas? - nc 7. As a simple model of a semiconductor, suppose that there are N bound electron states, each having energy -A <0, which are filled at zero temperature. At non-zero temperature some electrons are excited into the conduction band, which is a continuum of positive energy states. The density of these states is given by g(E)dE = AVEDE where A is a constant. Show that at temperature T the mean number ne of excited electrons is determined by the pair of equations N e(μ+A)/kBT + 1 = 5.² 0 Show also that, if ne << N and kBT <A and e/knT <1, then g(E) dE e(E-H)/kBT + 1 2μA+KBT log 2N A√(KBT) ³ 8. Consider an almost degenerate Fermi gas of electrons with spin degeneracy 9, = 2. At high temperatures, show that the equation of state is given by pV = NkBT (1 and the average energy is At low temperatures, show that the chemical potential is (1-1/2 (KBP) EF E + = Ep 1 X³N 4√2g,V 2 + + 3.NE (1+5 (²) * -..) 2 (kBT 5 12 EF

Answer rating: 100% (QA)

Ans 1 The ... View the full answer