P460 - Quan. Stats. III1 Boson and Fermion “Gases” If free/quasifree gases mass > 0 non-relativistic P(E) = D(E) x n(E) --- do Bosons first let N(E) = total number of particles. A fixed number (E&R use script N for this) D(E)=density (~same as in Plank except no 2 for spin states) (E&R call N) If know density N/V can integrate to get normalization. Expand the denominator….

P460 - Quan. Stats. III2 Boson Gas Solve for e  by going to the classical region (very good approximation as m and T both large) this is “small”. For helium liquid (guess) T=1 K, kT=.0001 eV, N/V=.1 g/cm 3 work out average energy average energy of Boson gas at given T smaller than classical gas (from BE distribution fntn). See liquid He discussion

P460 - Quan. Stats. III3 Fermi Gas Repeat for a Fermi gas. Add factor of 2 for S=1/2. Define Fermi Energy E F = -  kT change “-” to “+” in distribution function again work out average energy average energy of Fermion gas at given T larger than classical gas (from FD distribution fntn). Pauli exclusion forces to higher energy and often much larger

P460 - Quan. Stats. III4 Fermi Gas Distinguishable Indistinguishable Classical degenerate depend on density. If the wavelength similar to the separation than degenerate Fermi gas larger temperatures have smaller wavelength --> need tighter packing for degeneracy to occur electron examples - conductors and semiconductors - pressure at Earth’s core (at least some of it) -aids in initiating transition from Main Sequence stars to Red Giants (allows T to increase as electron pressure independent of T) - white dwarves and Iron core of massive stars Neutron and proton examples - nuclei with Fermi momentum = 250 MeV/c - neutron stars

P460 - Quan. Stats. III5 Conduction electrons Most electrons in a metal are attached to individual atoms. But 1-2 are “free” to move through the lattice. Can treat them as a “gas” (in a 3D box) more like a finite well but energy levels (and density of states) similar (not bound states but “vibrational” states of electrons in box) depth of well V = W (energy needed for electron to be removed from metal’s surface - photoelectric effect) + Fermi Energy at T = 0 all states up to E F are filled W EFEF V Filled levels

P460 - Quan. Stats. III6 Conduction electrons Can then calculate the Fermi energy for T=0 (and it doesn’t usually change much for higher T) Ex. Silver 1 free electron/atom n E T=0

P460 - Quan. Stats. III7 Conduction electrons Can determine the average energy at T = 0 for silver ---> 3.3 eV can compare to classical statistics Pauli exclusion forces electrons to much higher energy levels at “low” temperatures. (why e’s not involved in specific heat which is a lattice vibration/phonons)

P460 - Quan. Stats. III8 Conduction electrons Simu\ilarly, from T-dependent the terms after the 1 are the degeneracy terms….large if degenerate. For silver atoms at T=300 K not until the degeneracy term is small will the electron act classically. Happens at high T The Fermi energy varies slowly with T and at T=300 K is almost the same as at T=0 You obtain the Fermi energy by normalization. Quark-gluon plasma (covered in 461) is an example of a high T Fermi gas