8.3. The Electron Gas in Metals History : Drude (1900) - Lorentz (1904-5) : Free e, MB statistics. Successes : Qualitative features of transport. Wiedemann-Franz law explained : ( K / = const ) Failures : Equipartition C = Cvib + Cel . Exp.: C Cvib. Paramagntism ( 1/T ). Exp.: indep of T & too small. Sommerfeld : Free e, FD statistics.
Sommerfeld Theory g = 2 : m* = effective mass of e. nA = # of atoms per unit cell ne = # of electrons per atom a = lattice constant ( length of unit cell edge ) Cubic lattice : Sodium (Na) : Mathematica Highly degenerate Fermi system ~ few percents of Cclassical
Low T : For Cu, c.f. 345K from elastic constants. Intercept of CV / T vs T 2 with the vertical axis gives c.f. 0.69 from DOS calculation. Paramagnetism, Lorenz number are likewise improved .
Thermionic & Photoelectric Electron Emission No spontaneous emission e in metal in potential well. Simplest model : square well Classically, only e’s with such that can escape. Low emission current Remaining e’s in quasistatic equilibrium. Treatment analogous to gas effusion .
8.3.A. Thermionic Emission ( Richardson Effect ) From § 6.4 : # of e emitted ( in the z > 0 direction ) per unit area per unit time is
= Work function ~ eV ~ 104 K Boltzmannian Thermionic current density :
J Classical statistics ( z << 1 ) : work function FD statistics ( z >> 1 ) :
Plot is straight line with slope W / k Plot is straight line with slope (W F ) / k e beam impinging on metal : refractive index of metal for e’s is W can be estimated from e diffraction experiments. W 13.5 eV for tungsten (W )
Effects of reflection at surface : J ( 1 r ) J Tungsten : = 4.5 eV Dotted Line : r = 0 Solid line : r = ½ For tungsten : W 13.5 eV, 4.5 eV, F 9 eV FD statistics For nickel : W 17 eV, 5.0 eV, F 11.8 eV FD statistics Effects of reflection at surface : J ( 1 r ) J Intercepts for most metal with clean surfaces range from 60 to 120 A cm2 K2 . c.f.
Schottky Effect Electric field surface : ( x = 0 at surface, x < 0 inside metal ) For an outside electron : Force from its image is Setting U(0) =0 , the corresponding potential energy is Potential energy due to E : By definition, potential energy of an inside e is . Potential energy of an outside e w.r.t. an inside one is
(xm) is a maximum E lowers the barrier by e3/2 F1/2 . Cold emission :
8.3.B. Photoelectric Emission ( Hallwachs Effect ) Condition for emission : c.f. § 8.3A can be ~ k T for ( Non-Boltzmannian ) where
Threshold = 0 :
Thermionic Pd ( = 4.97 eV )
8.4. Ultracold Atomic Fermi Gas MOT : ( same as Bose gas ) For 106 atoms in 100Hz trap : Ground state energy :
40K See §11.9 for BCS condensation
8.5. Statistical Equilibrium of White Dwarf Stars White dwarf stars ~ abnormally faint white stars. Reason: emitted light due NOT to fusion, but to gravitational contraction. Model : Star = Ball of He of Mathematica Ionization energy of He : 24.6 eV, 54.4 eV Ball = N e’s + ½ N (He nuclei) : dynamics of He ions negligible
Model e dynamics relativistic e gas completely degenerate Simplistic Model : Star = uniform, relativistic e gas Caution : n must varies for structural stability
Ground State Properties ( g = 2 ) Relativistic particle : with rest mass m = hamiltonian From § 6.4 : ( gas inside V )
where
P0 where Mathematica
Equilibrium Configuration Adiabatic change of spherical volume : Accompanying gravitational energy change : ~ 100 depends on the spatial variation of n. Equilibrium :
Equilibrium condition : Equilibrium condition : ( Mass-radius relationship ) QM+SR+Gr = Compton wavelength of electron
1. R >> 108 cm ( x << 1 ) : for M = 1033 g Mathematica 1. R >> 108 cm ( x << 1 ) : 2. R << 108 cm ( x >>1 ) : Chandrasekhar limit : as observed
Star collapse into neutron star or black hole Influx from companion binary star Type Ia supernovae (see Chap.9)
8.6. Statistical Model of the Atom Thomas-Fermi model : For a completely degenerate e gas Let the gas be under a Coulomb potential (r). For slowly varying n : Energy of e at top of Fermi sea : At boundary of system, pF = 0 & we can set = 0 so that = 0.
Poisson eq. : Thomas-Fermi eq. Spherical symmetry :
Let Set Dimensionless T-F eq. Bohr radius
Complete solution tabulated by Bush & Caldwell, PR 38,1898 (31) B.C. : For x , r 0 : For x , r r0 (boundary) : ( neutral atom ) For r0 : Sommerfeld, Z.Physik 78, 283 (32) Complete solution tabulated by Bush & Caldwell, PR 38,1898 (31)
Radial Distribution Function Hg atom Actual T-F
Binding Energy Mean ground state K.E.: Total ground state K.E.: Total ground state P.E.: nuclei other e’s cancels double counting Total ground state E:
E.Milne, Proc.Camb.Phil.Soc. 23,794 (1927) : Binding energy : EB of H atom Linear size of e cloud l 1 Z1/3 Classical regime :