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) • 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 π ....) 2 1 1 ( ) 2 ( 2 / 3 3 2 / 3 + = 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 ...... ) 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 = =
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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
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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 b 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
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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

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