= Solution
The <electron-phonon interaction> lets one <electron> polarize the ionic lattice and another interact with that delayed deformation. <Phonon-mediated electron attraction> acts over <energy> transfers below a characteristic phonon scale. Pairing additionally requires a net attractive interaction in some allowed pairing channel after screened Coulomb repulsion is included. In the conventional weak-coupling situation, retardation separates the low phonon scale from the much larger electronic scale, reducing the effective repulsion. The <Cooper instability> then makes a <Fermi surface> unstable to an arbitrarily weak net attraction in the idealized channel; finite <temperature> cuts off its logarithmic enhancement.
In <BCS theory>, a momentum-independent pairing amplitude within the cutoff obeys a self-consistency equation containing $\tanh(E/2k_BT)/(2E)$, with <quasiparticle> <energy> $E=\sqrt{\xi^2+|\Delta|^2}$. At a continuous transition, take $\Delta\to0$. A constant effective <density of states> and coupling combine into $\lambda$, while the band bottom truncates the negative-energy side at $-\sigma$, with $\sigma=\min(\mu,\theta)$. This gives the supplied linearized gap equation. Its integrand is even, so with
$$
I(a,T)=\int_0^a\frac{d\xi}{\xi}\tanh\frac{\xi}{2k_BT},
$$
it becomes $1/\lambda=[I(\theta,T_c)+I(\sigma,T_c)]/2$. If both cutoffs are much larger than $k_BT_c$, the given large-cutoff integral gives $I(a,T_c)\simeq\ln(1.13a/(k_BT_c))$. Consequently the <asymmetric-cutoff BCS transition temperature> is
$$
\frac1\lambda\simeq\ln\frac{1.13\sqrt{\theta\sigma}}{k_BT_c},\qquad
\boxed{k_BT_c\simeq1.13\sqrt{\theta\sigma}\,e^{-1/\lambda}.}
$$
For $\mu>\theta$, this reduces to $1.13\theta e^{-1/\lambda}$; for $0<\mu<\theta$, it is $1.13\sqrt{\theta\mu}\,e^{-1/\lambda}$. The second limit requires $k_BT_c\ll\mu$ as well as $k_BT_c\ll\theta$. Weak $\lambda$ at fixed nonzero $\mu/\theta$ supplies that limit, but taking $\mu\to0$ afterward need not preserve it.
For a three-dimensional spin-degenerate parabolic <electron> band in the degenerate regime,
$$
\mu\simeq E_F=\frac{\hbar^2}{2m}(3\pi^2n)^{2/3}.
$$
Define
$$
n_\theta=\frac1{3\pi^2}\left(\frac{2m\theta}{\hbar^2}\right)^{3/2},\qquad
T_0=\frac{1.13\theta}{k_B}e^{-1/\lambda},\qquad u=\frac n{n_\theta}.
$$
Holding the other model parameters fixed gives
$$
\boxed{T_c(n)\simeq
\begin{cases}
T_0u^{1/3},&0<u<1,\\
T_0,&u\ge1.
\end{cases}}
$$
Thus the log-log slope is $1/3$ below the cutoff-density scale and zero above it. For the specified parameters,
$$
\boxed{n_\theta\simeq1.15\times10^{26}\,\mathrm{m^{-3}},\qquad T_0\simeq40.3\,\mathrm K.}
$$
This <density crossover of a fixed-coupling BCS cutoff model> is the requested prescribed model. A real three-dimensional parabolic band also has a changing <density of states>, so keeping $\lambda$ constant as density varies is an additional imposed assumption, not a consequence of that dispersion.
For noninteracting spinless bosonic pairs, <mass> $m_B=2m$ and density $n_B=n/2$, <homogeneous ideal Bose gas condensation> gives
$$
T_B=\frac{2\pi\hbar^2}{2mk_B}
\left[\frac{n/2}{\zeta(3/2)}\right]^{2/3}
=C_BT_F,\qquad T_F=E_F/k_B.
$$
Using the supplied integral $I_B=\int_0^\infty x^{1/2}/(e^x-1)\,dx\simeq2.3$ gives $C_B=\frac12[2/(3I_B)]^{2/3}\simeq0.219$. Its slight difference from the more precise $0.218$ is consistent with the rounded integral. Write $a=T_0/(\theta/k_B)=1.13e^{-1/\lambda}\simeq0.0403$. On the low-density branch,
$$
\frac{T_c}{T_B}=\frac{a}{C_B}u^{-1/3}.
$$
Since $a/C_B<1$, no part of the high-density plateau exceeds $T_B$. The formal crossing condition is
$$
\boxed{u<\left(\frac{a}{C_B}\right)^3\simeq0.00624,
\qquad n\lesssim7.2\times10^{23}\,\mathrm{m^{-3}}.}
$$
The crossing occurs near $T_c\simeq7.4\,\mathrm K$ and $T_F\simeq34\,\mathrm K$.
\Image[/past-exam-of-the-superconductivity-and-quantum-coherence-course-of-the-university-of-cambridge/2014/paper-1-transition-density.png]
{title=Fixed-coupling BCS cutoff prediction compared with ideal condensation of bosonic pairs}
The apparent low-density excess should not be read as a reliable transition above the ideal condensation scale of preformed pairs. On this extrapolation $k_BT_c/\mu=a u^{-1/3}$ grows without bound, violating the degenerate-gas and large-lower-cutoff approximations. It reaches one already near $7.5\times10^{21}\,\mathrm{m^{-3}}$; well before using an arbitrarily small density, the <chemical potential>, pairing, <density of states> and pair fluctuations need a self-consistent treatment. The asymptotic formula and ideal-boson curve compare two stipulated models, rather than solving the full <BCS-BEC crossover>.
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