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Use $N_0={\nu }_s$ for the normal-state single-spin density of states per unit energy and volume at the Fermi surface. Write the attraction as $-V/\mathcal{V}$, with $V>0$, and define $\lambda =N_{0}V$.^[ref]

Since $V/\mathcal{V}$ has units of energy, $V$ has units of energy times volume, making $\lambda$ dimensionless. Set ${\Omega }_D=\hslash {\omega }_D$, an energy cutoff using the Debye frequency ${\omega }_D$, rather than a frequency itself. These conventions also fix the meaning of the earlier constant-shell BCS gap equation.

Here is the reduction of the BCS gap equation in these conventions. Substituting the zero-temperature BCS anomalous average givesThe $k$ sum counts one spin species. For $\Delta >0$, cancelling $\Delta$ and replacing this sum by an integral with constant $N_0$ givesThe factor two from the symmetric integral cancels the $1/2$ in the BCS anomalous average; it is not an extra spin degeneracy. Substituting $\xi =\Delta \sinh t$ evaluates the integral as $\mathrm{arsinh}({\Omega }_D/\Delta )$. Solving givesThe first equality is exact within the constant-density of states, sharp-cutoff mean-field approximation; the last is the weak coupling asymptote. Extending the first equality to large $\lambda$ does not make this simplified model an accurate description of a strongly coupled material.

For an original numerical example, choose $\hslash {\omega }_D=20\mathrm{meV}$ and $\lambda =0.25$. The exact constant-shell BCS gap equation gives $\Delta (0)=0.732871\mathrm{meV}$; the weak coupling formula gives $0.732626\mathrm{meV}$. Its fractional underestimate, measured against the exact result, is exactly $e^{-2/\lambda }=e^{-8}\simeq 0.03355\%$. At $\lambda =1$ that error is $e^{-2}\simeq 13.53\%$, so the two formulas are no longer interchangeable. No material parameters have been fitted here.

At nonzero temperature, substitute the finite-temperature BCS anomalous average instead. The same cancellation yieldsAt the superconducting transition temperature take the continuous $\Delta \to 0$ limit. The integrand at $\xi =0$ then has the finite limiting value $1/(2k_B{T}_c)$, so no divergent endpoint needs to be inserted into a numerical quadrature. For $k_B{T}_c\ll {\Omega }_D$, the resulting integral gives $k_B{T}_c\simeq 1.134{\Omega }_D{e}^{-1/\lambda }$ and $2\Delta (0)/(k_B{T}_c)\simeq 3.528$. These ratios concern the isotropic weak coupling model.^[ref]

The BCS quasiparticle spectrum is the positive excitation energy obtained by diagonalizing the paired particle-hole block. Here the weak coupling shell model has $\Delta \ll {\Omega }_D$: the minimum lies inside the pairing shell, and states outside it cost more. To derive it for a uniform isotropic real $\Delta >0$, measure the normal-state energy from the chemical potential, ${\xi }_k={ϵ}_k-\mu$, and use the same sign convention as the BCS coherence factors:Its eigenvalues are ${\epsilon }_{\pm }=\pm E_k$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }^2}$. The positive $E_k$ is the cost of creating a quasiparticle above the BCS ground state. The lower branch $-E_k$ is its particle-hole partner in this doubled matrix description, not an excitation that lowers the ground state energy without bound.

The location and value of the minimum follow directly:The unique minimum as a function of $\xi$ is $E(0)=\Delta$, at the normal-state Fermi surface. On an isotropic three-dimensional Fermi surface it occurs for every direction of $k$, rather than at a unique point of momentum space. Near that minimum,Farther away, $E(\xi )\simeq |\xi |$ when $\Delta \ll |\xi |\ll {\Omega }_D$. For the illustrative $\lambda =0.25$ example, a state at $\xi =\Delta$ costs $\sqrt{2}\Delta \simeq 1.03644\mathrm{meV}$ rather than the minimum $0.732871\mathrm{meV}$.

An attractive interaction $V_{k,k^{\prime }}=-V/\mathcal{V}$ within $|{\xi }_k|,|{\xi }_{k^{\prime }}|<{\Omega }_D$ gives a constant gap inside that energy shell and zero outside. With approximately constant single-spin density of states ${\nu }_s$ per volume,The weak coupling limit is $\Delta \simeq 2{\Omega }_D{e}^{-1/({\nu }_{s}V)}$. A density counting both spin species is twice ${\nu }_s$ and must be divided by two. The shell cutoff is a relative-energy condition, even when every Cooper pair has zero total momentum.

David Gross and Frank Wilczek, and independently H. David Politzer, published their one-loop order Non-Abelian gauge theory results in adjacent 1973 Physical Review Letters papers. Both established a high-energy route to weak coupling; extrapolating into the low-energy region was a separate dynamical problem.^[ref][ref]

The three shared the 2004 Nobel Prize in Physics for discovering asymptotic freedom in the theory of the strong interaction. David Gross's Nobel lecture describes the unexpected sign and their comparison of independent calculations.^[ref] In his lecture The Dilemma of Attribution, H. David Politzer emphasizes that the prize marks a wider community's work and that a tidy prize-centered history hides how discovery actually proceeds.^[ref]

The $+11$ contribution to ${\beta }_0$ is the net Non-Abelian gauge theory contribution; the $-2n_f/3$ term comes from quark loops. Thus quarks screen color charge, opposing asymptotic freedom. It is the net gluon contribution that produces antiscreening: the inferred QCD coupling decreases toward shorter distances. Unlike the electrically neutral photon, a gluon carries color charge and has self-interactions. Describing this as vacuum polarization must not imply that all vacuum contributions have the screening sign of quantum electrodynamics.^[ref]

Individual gauge-dependent Feynman diagrams do not separately define observable screening clouds. A proof of the coefficient sums the gauge fixing, Faddeev-Popov ghost and matter contributions and extracts the charge renormalization; the final one-loop order coefficient is independent of the gauge choice. The beta function is negative near zero for integer $0\le n_f\le 16$. This is a statement about weak coupling, not a determination of the low-energy particle spectrum.^[ref]

In each factor of the BCS ground state, the BCS anomalous average is $⟨b_k⟩=u_k{v}_k={\Delta }_k/(2E_k)$. Self-consistency therefore gives the zero-temperature BCS gap equationPut ${\Omega }_D=\hslash {\omega }_D$. The specified constant attractive interaction makes ${\Delta }_k$ independent of $k$ inside the energy shell and zero outside it:The printed ${\delta }_{k,0}$ is incompatible with this interaction: $k$ labels relative pair momentum, not the total momentum of a Cooper pair. Every pair here has zero total momentum, while many relative momenta contribute.

For the nonzero solution, the constant-shell BCS gap equation becomes $1=(V/L^3){\sum }_{|{\xi }_k|<{\Omega }_D}(2\sqrt{{\xi }_k^2+{\Delta }^2}{)}^{-1}$. Let ${\nu }_s$ denote the approximately constant single-spin density of states per unit volume at the Fermi level. ThenIn weak coupling, $\Delta \simeq 2\hslash {\omega }_D{e}^{-1/({\nu }_{s}V)}$. The displayed answer in the question uses $\nu ={\nu }_s$. If “total electronic density of states” includes both spin species, ${\nu }_{\mathrm{tot}}=2{\nu }_s$, the argument instead reads $2/({\nu }_{\mathrm{tot}}V)$. If the density counts the whole box rather than unit volume, divide it by $L^3$ before using this formula.

The superconducting isotope effect is the dependence of the superconducting transition temperature on isotopic mass $M$ at fixed chemical composition. Its discovery in 1950 was a crucial clue that phonons participate in conventional superconductivity.^[ref]

The model prediction can be derived from two assumptions. For harmonic motion with a fixed restoring-force constant $K$, a phonon frequency scales as $\omega \propto \sqrt{K/M}$. If the BCS dimensionless coupling is independent of $M$, the weak coupling result $k_B{T}_c\simeq 1.134\hslash {\omega }_D{e}^{-1/\lambda }$ then gives $T_c\propto M^{-1/2}$ and $\alpha =1/2$. Changes in the effective interaction can change this exponent; it is not a universal law for every material exhibiting superconductivity. For a stipulated mass ratio $M_2/M_1=1.04$, the ideal model predicts $T_{c,2}/T_{c,1}=1/\sqrt{1.04}\simeq 0.98058$. This is an original scaling example, not a pair of measured transition temperatures.

For a normal transition dipole of magnitude $D$, the Fermi's golden rule, one-polarization mode density $\omega /(2\pi c^{2}d)$ and single-photon squared field $\hslash \omega /(2{ϵ}_{0}V)$ give $A_{\perp }=D^2{\omega }_0^2/(2{ϵ}_0\hslash c^{2}d)$. Dividing by the free-space electric-dipole spontaneous emission rate gives the title. The factor three relative to the scalar density-of-states ratio comes from the isotropic free-space polarization average. This ideal weak coupling formula assumes wide plates and unperturbed atomic structure, with atomic size much smaller than $d$.

Weak coupling means a suitably dimensionless interaction strength is small. In BCS theory, ${\nu }_{s}V\ll 1$ gives a gap exponentially small compared with the pairing-shell energy cutoff; the expansion parameter is not the dimensional quantity $V$ alone.