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30 CHAPTER 2. SOMMERFELD THEORY This means that a small shift in the Fermi energy, δE F gives rise to a change in the number density δn = g V (E F )δE F . The Fermi energy shift, in turn, is linked (via Eqn. 2.28) to V tot as δE F = e(δV + V ext ), from which we obtain δn = eg V (E F )(δV + V ext ) . (2.30) Linearised Thomas-Fermi. When the added potential V ext is small, the induced number density δn is small, and therefore the number density n cannot differ very much from the density n 0 of the system without the potential (n = n 0 + δn). We may then express Eqn.2.24 in a linearised form with respect to the perturbing potential: ∇ 2 δV (r) = e2 g V (E F ) ɛ 0 (δV (r) + V ext (r)) (2.31) Density response. This is solved by Fourier transforms, for instance by assuming an oscillatory perturbing potential V ext = V ext (q)e iqr and a resulting oscillatory induced potential δV = δV (q)e iqr : δV (q) = − e2 g V (E F )/ɛ 0 V q 2 + e 2 ext (q) = − V g V (E F )/ɛ 0 q 2 + qT 2 ext (q) , (2.32) F where we have collected e 2 g V (E F )/ɛ 0 into the Thomas Fermi wave vector q T F = (e 2 g V (E F )/ɛ 0 ) 1 2 , which for the free electron gas is calculated as q2 T F qT 2 F = 1 me 2 π 2 ɛ 0 h¯ k 2 F = 4 k F = ( 2.95 √ Å −1 ) 2 . (2.33) π a B rs Here, a B = 4πh¯2ɛ 0 ≃ 0.53 Å is the Bohr radius and r me 2 S is the Wigner-Seitz radius, defined by (4π/3)rS 3 = n−1 . For the induced number density we obtain: n ind (q) = ɛ 0q 2 e V ext (q) [1 + q 2 /q 2 T F ] , (2.34) Dielectric permittivity. In general, this phenomenon is incorporated into electromagnetic theory through the generalised wavevector dependent dielectric function ɛ(q). The dielectric function relates the electric displacement D to the electric field E, in the form ɛ 0 ɛ(q)E(q) = D(q). While the gradient of the total potential V tot = V 0 + δV + V ext = δV + V ext (V 0 = 0 for Jellium) gives the E− field, the gradient of the externally applied potential V ext gives the displacement field D. As E and D are related via the relative permittivity, ɛ, the potentials from which they derive are also connected by ɛ: V ext (q) = ɛ(q) (δV (q) + V ext (q)) (2.35) Using Eq. 2.32 we find q 2 V tot (q) = V ext (q) q 2 + qT 2 F , (2.36)
2.5. SCREENING AND THOMAS-FERMI THEORY 31 and for ɛ(q): ɛ T F (q) = 1 + q2 T F q 2 . (2.37) Screening. ɛ T F ∝ q −2 at small q (long distances), so the long range part of the Coulomb potential (also ∝ 1/q 2 ) is exactly cancelled. In real space, if v ext = Q/r is Coulombic (long range), V (r) = (Q/r)e −q T F r is a short-range Yukawa, or screened potential 3 . In a typical metal, r s is in the range 2 − 6, and so potentials are screened over a distance comparable to the interparticle spacing; the electron gas is highly effective in shielding external charges. 3 This form is originally due to P.Debye and E.Hückel, Zeitschrift für Physik 24, 185, (1923) and was derived for the theory of electrolytes; it appears also in particle theory under the name of the Yukawa potential; the physics in all cases is identical
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