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Boson and Fermion “Gases”

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)

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Boson and Fermion “Gases”

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  1. 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. III

  2. Boson Gas • Solve for e-a 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/cm3 • 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. III

  3. Fermi Gas • Repeat for a Fermi gas. Add factor of 2 for S=1/2. Define Fermi Energy EF = -akT 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. III

  4. 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. III

  5. 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 EF are filled W V EF Filled levels P460 - Quan. Stats. III

  6. Conduction electrons T=0 n • 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 E P460 - Quan. Stats. III

  7. 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. III

  8. 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 P460 - Quan. Stats. III

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