Theory of Superconductivity

• p. 22 Eq. (5.54)

  λ_L(T)=(4π/Λ_Tc²)^½

• p. 23 Fig. 7 caption

  λ_L(T)=(Λ_Tc²/4π)^½

• p. 3 Eq. (1.2)

  λ_L²=Λc²/4π

• p. 22 Eq. (5.54)

  λL(T) = (4π/ΛTc²)½

Judge's note: I read all four places on the page images. Page 3 Eq. (1.2) gives λ_L² = Λc²/4π. On page 22 the sentence before (5.54) says K(q) is the constant 4π/Λ_Tc², and (5.54) then prints (4π/Λ_Tc²) raised to +½, with no minus sign visible in the image or in the text layer. The Fig. 7 caption on page 23 and (C37) on page 28 give the form consistent with (1.2); the minus in λ_L⁻² of (C37) is clearly visible at the same resolution. As printed, (5.54) is 1/λ_L; the argument should be inverted or the exponent should be −½. A minus sign too faint to survive the scan cannot be fully excluded.

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• p. 22 Sec. V, Penetration Depths, sentence after Eq. (5.54)

  value (mc²/4πne²) at T=0°K

• p. 22 Sec. V, sentence after Eq. (5.54)

  this reduces to the London value (mc²/4πne²) at T=0°K

• p. 3 Introduction, Eq. (1.2) and following

  this approach gives Λ=m/ne²

Judge's note: On the page 22 image the parenthesis closes with no exponent before 'at T=0°K'. Page 3 gives λ_L² = Λc²/4π and Λ = m/ne², so λ_L² = mc²/(4πne²) and the depth itself is (mc²/4πne²)^½. The printed expression has dimensions of length squared, while the sentence is about the penetration depth λ_L(T).

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• p. 8 Sec. II, paragraph after Eq. (2.42)

  −N(0)(ħω)²V, for the average interaction

• p. 8 Eq. (2.42)

  (2.42)

Only the authors can say what is right here.

Judge's note: Read (2.42) and the following text on the PDF page 8 image and crop; N(0) carries no superscript 2 in the quoted limit. My reworking: (2.42) is W0 = −2N(0)(ħω)²/(e^{2/N(0)V} − 1). For N(0)V≫1 the denominator tends to 2/(N(0)V), so W0 tends to −[N(0)]²(ħω)²V. The left-hand form of (2.42) gives the same with ε0 ≈ ħω·N(0)V from (2.40). N(0)V is dimensionless throughout (e.g. 2.39), so the printed expression is also dimensionally an energy squared. The printed limit is missing one factor of N(0).

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• p. 28 Appendix C, Eq. (C40)

  Λ/Λ_T=0.23β_c²ε0²(T)=2(1−t)

• p. 12 Eq. (3.31)

  ε0=3.2kT_c[1−(T/T_c)]^½

• p. 13 Eq. (3.49)

  10.2

• p. 28 Appendix C, Eq. (C40)

  Λ/ΛT=0.23βc²ε0²(T)=2(1−t)

Only the authors can say what is right here.

Judge's note: I read (C22) on page 27 (Λ_T/Λ = (1/ε0)d(βε0)/dβ), (3.49) on the page 13 crop (10.2/β_c³), (3.31) on page 12 (coefficient 3.2) and (C40) on page 28 (0.23, clear in both image and text layer). My reworking: near Tc, ε0² ≈ 10.2(β−β_c)/β_c³ ≈ 10.2(1−t)/β_c². From (C22), Λ_T/Λ = 1 + (β/2ε0²)dε0²/dβ ≈ 10.2/(2β_c²ε0²). So Λ/Λ_T = (2/10.2)β_c²ε0² = 0.196β_c²ε0² = 2(1−t). The equality with 2(1−t) holds for any gap coefficient, but the printed 0.23 does not match the paper's own 10.2: 0.23 × 10.2 = 2.35, not 2. For 0.23 to hold, the near-Tc gap coefficient squared would have to be about 8.7, not 10.2. The 0.23 is carried forward: 3/(2 × 0.23 × 1.75) = 3.73, matching the 3.75π in (C43). Only the authors can say whether 0.23 or the 10.2/3.2 of Sec. III is the intended value.

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• p. 13 Eq. (3.50)

  =1.52.

• p. 13 Eq. (3.49)

  10.2

• p. 13 Eq. (3.52)

  =19.4,

Only the authors can say what is right here.

Judge's note: Read (3.42), (3.47), (3.49), (3.50), (3.51) and (3.52) on the PDF page 13 image. The jump is kN(0)βc²·10.2/βc³ = 10.2N(0)k²Tc; dividing by γTc = (2/3)π²N(0)k²Tc gives 10.2/6.580 = 1.55. From (3.51), (1/γ)(dHc/dT)² = 4π × jump/(γTc): 1.52 would give 19.1, while the printed 19.4 corresponds to about 1.54. Rounding of 10.2 (10.15 to 10.25) gives 1.54 to 1.56, never 1.52. The Fig. 3 note added in proof (intercept 2.52, PDF page 14) follows the 1.52 value. The paper offers no explanation; which of 10.2, 1.52 and 19.4 is off is for the authors to say.

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• p. 13 Eq. (3.38), second line

  1−β²

Only the authors can say what is right here.

Judge's note: Read (3.33) and (3.37) on PDF page 12 and (3.38) on PDF page 13; no 6/π² factor appears in the brace. My reworking: Fn − Fs = −(π²/3)N(0)(kT)² + 2N(0)∫dε[(2ε²+ε0²)/E]f(βE) + N(0)(ħω)²{[1+(ε0/ħω)²]^½ − 1}. Factoring −(π²/3)N(0)(kT)² out of the first two terms leaves {1 − (6/π²)β²∫…}. Check at ε0 = 0 (T = Tc): β²∫0^∞ 2ε f(βε)dε = π²/6, so the correct bracket vanishes (Hc = 0), while the printed bracket gives 1 − π²/6 ≈ −0.645. Eq. (3.43) is consistent with the corrected form, so the slip is confined to (3.38) as printed. Read from the full page image only; no crop was available.

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• p. 14 Text after Eq. (3.55)

  where a=9.10 and

• p. 14 Fig. 4, curve label

  9.17 exp

• p. 14 text after Eq. (3.55)

  where a=9.10 and b=1.50

• p. 14 Fig. 4 curve label

  9.17 exp[-1.50 Tc/T]

Only the authors can say what is right here.

Judge's note: Crop C013-a shows both on PDF page 14: the text reads "a=9.10 and b=1.50" and the Fig. 4 curve label reads "9.17 exp[−1.50 Tc/T]". The caption says the simple exponential fits tin and vanadium for Tc/T>1.4, the same fit as in the text, so the prefactor is one quantity printed two ways. Which value is right cannot be settled from the paper. The separate point that b=1.50 differs from the 1.44 in (3.54) is not a defect: the text only claims "good agreement" between the empirical and theoretical values.

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• p. 28 Appendix C, Eq. (C36)

  3π²ε0

• p. 28 Appendix C, Eq. (C39)

  4qλ_L²(0)ξ0

Only the authors can say what is right here.

Judge's note: All equations read on the page 28 image. My reworking: (C26) has prefactor 6πε0²/(qc²Λħv0). Its first term, with ∫g(b)db/b = ½π² from (C29), gives 6πε0² × (1−2f(ε0))/ε0 × π²/2 = 3π³ε0(1−2f(ε0)). Its second term, using (C34) with ε1 = ½qħv0, gives 6πε0² × (8/qħv0) ln(πqξ0) = 48πε0² ln(πqξ0)/(qħv0). Dividing by 3π³ε0 gives 16ε0/(π²qħv0), exactly the term printed in the brace, so the brace presupposes π³ outside. Cross-check with (C39): 3π/(4qλ_L²(0)ξ0), with 1/λ_L² = 4π/(Λc²) and 1/ξ0 = πε0/ħv0, equals 3π³ε0/(qc²Λħv0), and the brace term becomes 16/(π³qξ0) as printed there. The printed prefactor is therefore short by a factor of π. I read the exponent on π in (C36) as 2 at page-image resolution, while the 3 in π³ of (C39) is distinguishable; the text layer is garbled there, so a higher-resolution scan would make the reading certain. The later results (C39) and (C43) are consistent with π³, so this looks like a misprint confined to (C36).

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• p. 12 Sec. III, sentence before Eq. (3.30)

  is given by combining (2.36) and (3.28):

• p. 12 Sec. III, sentence before Eq. (3.30)

  is given by combining (2.36) and (3.28)

• p. 8 Eqs. (2.36) and (2.40)

  (2.36)

• p. 12 text before Eq. (3.30)

  to kTc is given by combining (2.36) and (3.28):

Only the authors can say what is right here.

Judge's note: On the PDF page 8 image, (2.36) is the expression for [h_k(1−h_k)]^½; the T=0 gap is fixed only by (2.39)/(2.40), ε0 = ħω/sinh[1/N(0)V]. On PDF page 12, (3.28)/(3.29) give kTc = 1.14ħω exp[−1/N(0)V]. In weak coupling 2ε0 ≈ 4ħω exp[−1/N(0)V], so the ratio is 4/1.14 ≈ 3.5, which matches (3.30). The (3.28) half of the pointer is acceptable because (3.29) is its closed form; (2.36) looks like a slip for (2.40) or (2.39), but only the authors can say which was meant.

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• p. 12 Sec. III, paragraph after Eq. (3.27)

  Thus (3.26) can be used to determine the critical temperature and we find

• p. 11 Eq. (3.26)

  which is singular at the edges of the gap

• p. 12 paragraph before Eq. (3.28)

  Thus (3.26) can be used to determine the critical temperature

Judge's note: Crop C007-b of PDF page 11 shows (3.26) is dN(E)/dE = N(0)E/(E²−ε0²)^½, the modified density of states. On PDF page 12, (3.27) is the gap equation 1/N(0)V = ∫dε tanh[½β(ε²+ε0²)^½]/(ε²+ε0²)^½, and the preceding sentences all cite (3.27). Eq. (3.28) is (3.27) with ε0 = 0. The pointer should be (3.27).

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• p. 13 Sec. III, sentence introducing Eq. (3.38)

  The critical field is given by combining (3.31), (3.32), and (3.37):

• p. 12 Eqs. (3.31) and (3.33)

  (3.31)

Judge's note: On PDF page 12, (3.31) is ε0 = 3.2kTc[1−(T/Tc)]^½, (3.32) is Hc²/8π = Fn − Fs, (3.33) is Fn = −(π²/3)N(0)(kT)², and (3.37) is Fs. On PDF page 13, (3.38) explicitly contains the term (π²/3)N(0)(kT)² from (3.33) together with the terms of (3.37), and nothing from (3.31). The first equation number should be (3.33).

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• p. 23 Table IV, footnote a

  Ratios in columns 5 and 7 are taken from Fig. 6.

• p. 23 Sec. V text

  Fig. 7, have determined λ(0)/λ_L(0) for these two metals

• p. 23 Sec. V text below Table IV

  using Fig. 7, have determined

• p. 22 Fig. 6 caption

  The temperature variation of the penetration depth

Judge's note: I read the footnote on the crop. Columns 5 and 7 are λ/λ_L for specular reflection and random scattering. Fig. 6 on page 22 plots [λ∞(0)/λ∞(T)]² against t = T/Tc and contains no λ/λ_L versus ξ0/λ_L information. Fig. 7 on page 23 plots λ(T)/λ_L(T) against ξ0/λ_L(T) for both boundary conditions, and the body text names Fig. 7 as the source for Table IV. The footnote's figure number is wrong.

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