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09statistics2 - Boson and Fermion Gases If free/quasifree...

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P461 - Quan. Stats. II 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…. dE e e E D dE E D E n N kT E = = 0 0 / 1 ) ( ) ( ) ( α 1 ) 2 ( 4 / 2 / 1 2 / 1 3 0 3 = kT E e e dE E m h V N α π .... ) 2 1 1 ( ) 2 ( 2 / 3 3 2 / 3 α α π + = e e h V mkT N
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P461 - Quan. Stats. II 2 Boson Gas Solve for e −α by going to the classical region (very good approximation if 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 ftn). See liquid He discussion 2 / 3 3 ) 2 ( mkT h V N e π α = 5 . 0 ) 0001 . 4 2 ( ) 1240 ( / 10 2 / 3 3 3 22 eV GeV eVnm cm e π α ...... ) 1 ( ) ( ) ( / ) ( ) ( 2 / 3 3 2 / 5 ) 2 ( 2 1 2 3 0 0 mkT h V N kT E dE E D E n dE E D E En E π = =
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P461 - Quan. Stats. II 3 Fermi Gas Repeat for a Fermi gas. Add factor of 2 for S=1/2. Define Fermi Energy E F = - α 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 ftn). Pauli exclusion forces to higher energy and often much larger ...... ) 1 ( ) ( ) ( / ) ( ) ( 2 / 3 3 2 / 5 ) 2 ( 2 1 2 3 0 0 mkT h V N kT E dE E D E n dE E D E En E π + = = ) 1 ( ) 2 ( 8 ) ( ) ( / ) ( 3 2 / 1 2 / 1 3 + = kT E E F e h E m V E D E n π
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P461 - Quan. Stats. II 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 barb2right need tighter packing for degeneracy to occur degenerate 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 is independent of T) - white dwarves and Iron core of massive stars Neutron and proton examples - nuclei with Fermi momentum = 250 MeV/c - neutron stars 3 / 1 = n separation p h particle λ
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P461 - Quan. Stats. II 5 Degenerate vs non-degenerate
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P461 - Quan. Stats. II 6 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 =
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