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Boris V. Vasiliev Supercondustivity Superfluidity

Superconductivity and Superfluidity This band of energy can be filled by N ∆ particles: ∫ EF N ∆ = 2 F (E)D(E)dE. (7.15) E F −∆ Where F (E) = 1 e E−µ τ +1 is the Fermi-Dirac function and D(E) is number of states per an unit energy interval, a deuce front of the integral arises from the fact that there are two electron at each energy level. To find the density of states D(E), one needs to find the difference in energy of the system at T = 0 and finite temperature: ∆E = ∫ ∞ 0 ∫ EF F (E)ED(E)dE − ED(E)dE. (7.16) 0 For the calculation of the density of states D(E), we must note that two electrons can be placed on each level. Thus, from the expression of the Fermi-energy Eq.(6.12) we obtain where is the Sommerfeld constant 1 . D(E F ) = 1 2 · dn e dE F = 3n e 4E F = 3γ 2k 2 π 2 , (7.17) γ = π2 k 2 n e = 1 ( π ) ( ) 3/2 2 k 4E F 2 · m e n 1/3 e (7.18) 3 Using similar arguments, we can calculate the number of electrons, which populate the levels in the range from E F − ∆ to E F . For an unit volume of material, Eq.(7.15) can be rewritten as: ∫ 0 n ∆ = 2kT · D(E F ) − ∆ 0 kTc dx (e x + 1) . (7.19) By supposing that for superconductors ∆0 kT c = 1.86, as a result of numerical integration we obtain ∫ 0 − ∆ 0 kTc dx (e x + 1) = [x − ln(ex + 1)] 0 −1.86 ≈ 1.22. (7.20) 1 It should be noted that because on each level two electrons can be placed, the expression for the Sommerfeld constant Eq.(7.18) contains the additional factor 1/2 in comparison with the usual formula in literature [44]. 84 Science Publishing Group

Chapter 7 The Condensate of Zero-Point Oscillations and Type-I Superconductors Thus, the density of electrons, which throw up above the Fermi level in a metal at temperature T = T c is ( ) 3γ n e (T c ) ≈ 2.44 k 2 π 2 kT c . (7.21) Where the Sommerfeld constant γ is related to the volume unit of the metal. From Eq.(6.6) it follows L 0 ≃ λ F πα and this forms the ratio of the condensate particle density to the Fermi gas density: (7.22) n 0 n e = λ3 F L 3 0 ≃ (πα) 3 ≃ 10 −5 . (7.23) When using these equations, we can find a linear dimension of localization for an electron pair: L 0 = Λ 0 2 ≃ 1 . (7.24) πα(n e ) 1/3 or, taking into account Eq.(6.16), we can obtain the relation between the density of particles in the condensate and the value of the energy gap: ∆ 0 ≃ 2π 2 α 2 m e n 2/3 0 (7.25) or n 0 = 1 L 3 0 = ( me ) 3/2 2π 2 α 2 ∆ 0 . (7.26) It should be noted that the obtained ratios for the zero-point oscillations condensate (of bose-particles) differ from the corresponding expressions for the bose-condensate of particles, which can be obtained in many courses (see eg [43]). The expressions for the ordered condensate of zero-point oscillations have an additional coefficient α on the right side of Eq.(7.25). The de Broglie wavelengths of Fermi electrons expressed through the Sommerfelds constant are shown in Table 7.4. λ F = 2π p F (γ) ≃ π 3 · k2 m e 2 γ (7.27) Science Publishing Group 85

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