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kBTc≈1.13 ℏωD e−1/N(0)Vk_B T_c \approx 1.13\, \hbar\omega_D \, e^{-1/N(0)V}

Why this formula appears here

BCS theory gives condensed-matter physics one of its few genuinely load-bearing equations, and it is worth writing down because it exposes a real, falsifiable assumption rather than decorating the prose. In the weak-coupling limit, the transition temperature TcT_c is related to the strength of the electron-phonon coupling N(0)V (density of states at the Fermi level times the pairing interaction) and the characteristic phonon energy ℏ\hbarωD\omega_D (the Debye frequency) by: kBTc≈1.13 ℏωD e−1/N(0)Vk_B T_c \approx 1.13\, \hbar\omega_D \, e^{-1/N(0)V}. The exponential is the important feature, not decoration: it says TcT_c depends on the coupling strength in a way that is extraordinarily sensitive at weak coupling and saturates at strong coupling, and it…

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kBk_B

Symbol k_B

kBk_B is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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TcT_c

Symbol T_c

TcT_c is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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e−1/N(0)Ve^{-1/N(0)V}

Symbol e^-1/N(0)V

e−e^-1/N(0)V is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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Published contexts (1)

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kBTc≈1.13 ℏωD e−1/N(0)Vk_B T_c \approx 1.13\, \hbar\omega_D \, e^{-1/N(0)V}

Equation 4 · Physics

From Origins to Frontier: A History of Quantum and Condensed-Matter Systems

This equation gives an approximation: it relates the quantities while allowing an approximation.

BCS theory gives condensed-matter physics one of its few genuinely load-bearing equations, and it is worth writing down because it exposes a real, falsifiable assumption rather than decorating the prose. In the weak-coupling limit, the transition temperature TcT_c is related to the strength of the electron-phonon coupling N(0)V (density of states at the Fermi level times the pairing interaction) and the characteristic phonon energy ℏ\hbarωD\omega_D (the Debye frequency) by: kBTc≈1.13 ℏωD e−1/N(0)Vk_B T_c \approx 1.13\, \hbar\omega_D \, e^{-1/N(0)V}. The exponential is the important feature, not decoration: it says TcT_c depends on the coupling strength in a way that is extraordinarily sensitive at weak coupling and saturates at strong coupling, and it…

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