Boson and Fermion “Gases”

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…. P461 - Quan. Stats. II

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 ftn). See liquid He discussion P461 - Quan. Stats. II

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 ftn). Pauli exclusion forces to higher energy and often much larger 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 (“proved” in 460) • 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 P461 - Quan. Stats. II

5. Degenerate vs non-degenerate 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 = 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 P461 - Quan. Stats. II

7. 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 P461 - Quan. Stats. II

8. 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) P461 - Quan. Stats. II

9. Conduction electrons P461 - Quan. Stats. II

10. Conduction electrons • Similarly, 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 later) is an example of a high T Fermi gas P461 - Quan. Stats. II

11. Fermi Gases in Stars • Equilibrium: balance between gravitational pressure and “gas” (either normal or degenerate) pressure • total gravitational Energy: • density varies in normal stars (in Sun: average is 1 g/cm3 but at r=0 is 100 g/cm3). More of a constant in white dwarves or neutron stars • will have either “normal” gas pressure of P=nkT (P=n<E>) or pressure due to degenerate particles. Normal depends on T, degenerate (mostly) doesn’t • n = particle density in this case P461 - Quan. Stats. II

12. Degenerate Fermi Gas Pressure • Start with p = n<E> • non-relativistic relativistic • P depends ONLY on density P461 - Quan. Stats. II

13. Degenerate Fermi Gas Pressure non-relativistic relativistic • P depends ONLY on density • Pressure decreases if, for a given density, particles become relativistic • if shrink star’s radius by 2  density increases by 8  gravitational E increases by 2 • if non-relativistic. <E> increases by (N/V)2/3 = 4 • if relativistic <E> increases by (N/V)1/3 = 2 •  non-relativistic stable but relativistic is not. can collapse P461 - Quan. Stats. II

14. Older Sun-like Stars • Density of core increases as H-->He. He inert (no fusion yet). Core contracts • electrons become degenerate. 4 e per He nuclei. Electrons have longer wavelength than He • electrons move to higher energy due to Pauli exclusion/degeneracy. No longer in thermal equilibrium with p, He nuclei • pressure becomes dominated by electrons. No longer depends on T • allows T of p,He to increase rapidly without “normal” increase in pressure and change in star’s equilibrium. • Onset of 3He->C fusion and Red Giant phase (helium flash when T = 100,000,000 K) P461 - Quan. Stats. II

15. White Dwarves • Leftover cores of Red Giants made (usually) from C + O nuclei and degenerate electrons • cores of very massive stars are Fe nuclei plus degenerate electrons and have similar properties • gravitational pressure balanced by electrons’ pressure which increases as radius decreases  radius depends on Mass of star • Determine approximate Fermi Energy. Assume electron density = 0.5(p+n) density • electrons are in this range and often not completely relativistic or non-relativistic  need to use the correct E2 = p2 + m2 relationship P461 - Quan. Stats. II

16. White Dwarves + Collapse • If the electron energy is > about 1.4 MeV can have: • any electrons > ET “disappear”. The electron energy distribution depends on T (average E) • the “lost” electrons cause the pressure from the degenerate electrons to decrease • the energy of the neutrinos is also lost as they escape  “cools” the star • as the mass increases, radius decreases, and number of electrons above threshold increases #e’s EF ET P461 - Quan. Stats. II

17. White Dwarves+Supernovas • another process - photodisentegration - also absorbs energy “cooling” star. Similar energy loss as e+p combination • At some point the not very stable equilibrium between gravity and (mostly) electron pressure doesn’t hold • White Dwarf collapses and some fraction (20-50% ??) of the protons convert to neutrons during the collapse • gives Supernovas P461 - Quan. Stats. II

18. White Dwarves+Supernovas P461 - Quan. Stats. II

19. Neutron Stars-approx. numbers • Supernovas can produce neutron stars - radius ~ 10 km - mass about that of Sun. always < 3 mass Sun - relative n:p:e ~ 99:1:1 • gravity supported by degenerate neutrons • plug into non-relativistic formula for Fermi Energy  140 MeV (as mass =940 MeV, non-rel OK) • look at wavelength • can determine radius vs mass (like WD) • can collapse into black hole P461 - Quan. Stats. II

20. Neutron Stars • 3 separate Fermi gases: n:p:e p+n are in the same potential well due to strong nuclear force • assume independent and that p/n = 0.01 (depends on star’s mass) • so need to use relativistic for electrons • but not independent as p <---> n • plus reactions with virtual particles • free neutrons decay. But in a neutron star they can only do so if there is an available unfilled electron state. So suppresses decay P461 - Quan. Stats. II

21. Neutron Stars • Will end up with an equilibrium between n-p-e which can best be seen by matching up the Fermi energy of the neutrons with the e-p system • neutrons with E > EF can then decay to p-e-nu (which raises electron density and its Fermi energy thus the balance) • need to include rest mass energies. Also density of electrons is equal to that of protons • can then solve for p/n ratio (we’ll skip algebra) • gives for typical neutron star: P461 - Quan. Stats. II