σ = neμ = v D = E H, the Hall Field B Z E Y = ee y Determining n and μ: The Hall Effect V x, E x I, J x E y B z F = qe + qv B F y

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1 Detemining n and μ: The Hall Effect V x, E x E y I, J x F = qe + qv B F y = ev D B z F y = ee y B z In steady state, E Y = v D B Z = E H, the Hall Field Since v D =-J x /en, R H = ne E H = J x B Z = R H J X B Z ne σ = neμ

2 Expeimental Hall Results on Metals Valence= metals look like fee-electon Dude metals Valence= and 3, magnitude and sign suggest poblems Metal Valence -/R H nec Li Na K Rb Cs Cu Ag Au Be Mg In Al Hall coefficients of selected elements in modeate to high fields* * These ae oughly the limiting values assumed by R n as the field becomes vey lage (of ode 0 4 G), and the tempeatue vey low, in caefully pepaed specimens. The data ae quoted in the fom n 0 :n. Whee n 0 is the density fo which the Dude fom (.) agees with the measued R u :n 0 = -/R H ec. Evidently the alkali metals obey the Dude esult easonably well, the noble metals (Cu. Ag, Au) less well, and the emaining enties, not at all. Table by MIT OpenCouseWae.

3 Response of fee e- to AC Electic Fields Micoscopic pictue ty e - E = E e iωt Z O B=0 in conducto, dp(t) p(t) iωt = ee 0 e and F(E) >> F(B) dt τ p(t) = p 0 e iωt iωp 0 = p 0 ee 0 τ ee 0 p = ee 0 ω>>/τ, p out of phase with E 0 iω p 0 = iω ω, p 0 τ ω<</τ, p in phase with E p 0 = ee 0 τ

4 What if ωτ>>? When will J = σe beak down? It depends on electons undegoing many collisions, on the aveage a collision time τ apat. As long as thee ae many collisions pe cycle of the AC field (ωτ<<), the AC σ will be the DC σ. x x x xx x x x x x x x x x x x x x x x x But conside the othe limit: ωτ>>. Now thee will be many cycles of the field between collisions. In this limit, the electons will behave moe like electons in vacuum, and the elation between J and E will be diffeent x x

5 Complex Repesentation of Waves sin(kx-ωt), cos(kx-ωt), and e -i(kx-ωt) ae all waves e -i(kx-ωt) is the complex one and is the most geneal imaginay A θ Acosθ iasinθ eal e iθ =cosθ+isinθ

6 Response of e- to AC Electic Fields Momentum epesented in the complex plane imaginay p (ω >>/τ ) p p (ω <</τ ) eal Instead of a complex momentum, we can go back to macoscopic and ceate a complex J and σ iωτ nep 0 ne J (t) = J 0 e J 0 = nev = = m m( iω ) τ σ 0 ne τ σ =,σ 0 = iωτ m E 0

7 Response of e- to AC Electic Fields Low fequency (ω<</τ) electon has many collisions befoe diection change Ohm s Law: J follows E, σ eal High fequency (ω>>/τ) electon has nealy collision o less when diection is changed J imaginay and 90 degees out of phase with E, σ is imaginay Qualitatively: ωτ<<, electons in phase, e-iadiate, E i =E +E t, eflection ωτ>>, electons out of phase, electons too slow, less inteaction,tansmission ε=ε ε 0 ε = τ 0 4 sec,νλ = c,ν = 3x00 cm / sec 0 4 Hz 5000x0 8 cm E-fields with fequencies geate than visible light fequency expected to be beyond influence of fee electons

8 Response of light to inteaction with mateial Need Maxwell s equations fom expeiments: Gauss, Faaday, Ampee s laws second tem in Ampee s is fom the unification electomagnetic waves! SI Units (MKS) Gaussian Units (CGS) D = ρ D = 4πρ B = 0 B = 0 B B xe = xe = c t t 4π D D xh = J + xh = J + c c t t D = E + 4πP D = ε 0 E + P = εe B = H + 4πM B = μ 0 H + μ 0 M = μh μ = μ μ 0 ;ε = ε ε 0

9 Waves in Mateials Non-magnetic mateial, μ =μ 0 Polaization non-existent o swamped by fee electons, P=0 B xe = t E xb = μ 0 J + μ 0 ε 0 t xb x( xe) = t Fo a typical wave, E = E e i(k ϖt ) = E e ik e iϖt = E()e iϖt 0 0 E() = iϖμ 0 σe() μ 0 ε 0 ω E() E() = ω ε (ω )E() c iσ ε (ω ) = + ε 0 ω E = E [μ 0 J + μ 0 ε 0 ] t t E() = E 0 e ik E = μ ω 0 σ E E + μ 0 ε 0 t t k = ε (ω ) c ω c v = = k ε (ω ) Wave Equation

10 Waves in Mateials Waves slow down in mateials (depends on ε(ω)) Wavelength deceases (depends on ε(ω)) Fequency dependence in ε(ω) ε (ω) = + iσ iσ = + 0 ε 0 ω ε 0 ω( iωτ ) ε (ω) = + ω p = ne ε 0 m iω τ p ω iω τ Plasma Fequency Fo ωτ>>>, ε(ω) goes to Fo an excellent conducto (σ 0 lage), ignoe, look at case fo ωτ<< ε (ω) iω τ p ω iω τ iω τ p ω

11 Waves in Mateials ω ω σ 0 k = ε (ω ) = i c c ωε 0 ω + i σ σ 0 ω σ 0 ω k = 0 = + i c ωε 0 ε 0 c ε 0 c Fo a wave E = E 0 e i(kz ωt ) Let k=k eal +k imaginay =k +ik i The skin depth can be defined by E = E 0 e i[k z ωt ] e k i z ε c o δ = = = σω σ μ ω k i o o o δ

12 Waves in Mateials Fo a mateial with any σ 0, look at case fo ωτ>> ω p ε ( ω )= ω ω<ω p, ε is negative, k=k i, wave eflected ω>ω p, ε is positive, k=k, wave popagates R ω p ω

13 Success and Failue of Fee e- Pictue Success K/σ=themal conduct./electical conduct.~ct Metal conductivity Κ = c Hall effect valence= v v them τ 3 Skin Depth c v = = nk b ;v them = Wiedmann-Fanz law T v m E 3 3k b T Examples of Failue Κ = 3 3k nkb b T 3 nk τ = b Tτ Insulatos, Semiconductos 3 m m Hall effect valence> ne τ σ = Themoelectic effect m Colos of metals Κ 3 k b Theefoe : σ = e T Luck: c veal =c vclass /00; v eal =v class *00 ~C!

14 Wiedmann-Fanz Success 73K 373K Element Li Na K Rb Cu Ag Au Be Mg Nb Fe Zn Cd Al In Tl Sn Pb Bi Sb k (watt cm-k) k σ T (watt-ohm K ). x k (watt cm-k) k σ T (watt-ohm K ).43 x Expeimental themal conductivities and Loenz numbes of selected metals Themoelectic Effect Exposed Failue when cv and v ae not both in popety E = Q T c v nk b nk b Themopowe Q is Q = = = 3ne 3ne e 3 Table by MIT OpenCouseWae. Themopowe is about 00 times too lage!