Degenerate Fermi gas ( 0< T ≪ T F )

1. Degenerate Fermi gas (0<T≪TF) T>0 When T≪TF, the # of “excited” fermions is: The extra thermal energy acquired by each fermion: characteristic behavior of electrons in metals.

2. The Sommerfeld expansion (pages 283-4) The chemical potential can be obtained from the condition:

3. Two types of bosons: • Composite particles which contain an even number of fermions. The number of these particles is conservedif the energy does not exceed the dissociation/separation energy. • (b) Particles associated with a field, of which the most important example is the photon. The number of these particles is not conserved: if the total energy of the field changes, particles would appear and disappear. The chemical potential of such particles is zeroin equilibrium, regardless of density. Bosons are special: any # of bosons can occupy one quantum state (energy level) A layman’s definition: Bose-Einstein Condensation (BEC) is a special macrostate with macroscopic # of bosons occupying one quantum state (often, the ground state) of a bosonic system. As the system temperature is cooled below certain temperature TC, BEC spontaneously forms. It is a phase transition purely driven by quantum (exchange) effect. Consider a special case of T = 0 K. What is the macrostate of a bosonic system? All the bosons (a macroscopic #) occupy the lowest energy level, i.e. the ground state, so that the system has lowest energy. 4 3 2 1

4.  (n,T) T TC T*: the temperature (energy) scale where quantum (exchange) effect becomes pronounced. On the other hand, 0 around T* according to Maxwell-Boltzmann distribution. Indeed, =0 right at T=TC and stay at 0 as T further decreases.

5. Let us perform the integration at TC, i.e., at =0. Critical temperature of BEC T>TC, the system can adjust (<0) to satisfy the constraint: What happens at T<TC?  is already 0 at TC. The right hand side decreases as T3/2… Resolving the paradox: The problem is caused by the behavior of the 3D density of states and our use of the continuum approximation. Because g()=0 at =0, our calculations of n ignored all the particles in the ground (=0) state. At low energies, we have to take into account the discrete nature of quantum states. Excited states only! n() g()  

6. T < TC g() n()  The expression for n(T) with  = 0 still works at T<TC for calculating the number of particles that are not in the ground state: huge number of particles in the lowest energy state Density of particles in the ground state: n>0 n n0 TC T Summary. We can discuss the ideal Bose gas in the terms of a phase transition. That is, at a critical value of temperature, TC,  (n,T) reaches the limit of  = 0 and stops increasing. Below TC, bosons begin to condense into the ground state. The abrupt accumulation of bosons in the ground state is called Bose-Einstein condensation.