= Solution
{c}
Use the stipulated noninteracting <impurity band> approximation, with one donated <electron> per donor and two available <spin> states per donor orbital. The <energy band> is half filled. Approximating its spin-summed <density of states> per volume near the <Fermi energy> by $D_v(E_F)\simeq2n/W$ gives the zero-<temperature> <Pauli paramagnetism>
$$
\chi_{\rm imp}=\mu_0\left(\frac{|g|\mu_B}{2}\right)^2D_v(E_F)
\simeq\frac{\mu_0g^2\mu_B^2n}{2W}.
$$
With $\alpha=1$,
$$
\boxed{\chi_{\rm imp}(n)\sim\frac{\mu_0g^2\mu_B^2}{2E_b}\,n\exp\left(\frac1{a_bn^{1/3}}\right)>0.}
$$
A factor of order one depends on the <density of states> shape; using the <conduction band> $g$-factor rather than silently setting $g=2$ matters here.
To locate the comparison with the magnitude of the undoped <Larmor-Langevin susceptibility>, put $x=n^{-1/3}/a_b>1$. Numerically,
$$
\chi_{\rm imp}\simeq P\frac{e^x}{x^3},\qquad
P=\frac{\mu_0g^2\mu_B^2}{2E_ba_b^3}=1.179\times10^{-8}.
$$
For the stated atomic-scale estimate $R=a_0$ in part (b), equality with $|\chi_0|=2.599\times10^{-5}$ requires
$$
x-3\ln x=\ln\frac{|\chi_0|}{P},\qquad x\simeq16.02.
$$
Thus the dilute-branch crossover is
$$
\boxed{n_*\simeq2.3\times10^{20}\,\mathrm{m^{-3}}
=2.3\times10^{14}\,\mathrm{cm^{-3}},}
$$
well below $a_b^{-3}=9.45\times10^{23}\,\mathrm{m^{-3}}$. Within this model the positive <spin> response exceeds the negative background magnitude for \b[lower] <dopant densities> on this dilute branch: the exponential narrowing of $W$ wins over the loss of donors. Indeed $d\ln(e^x/x^3)/dx=1-3/x>0$ for $x>3$. This is not a claim of <ferromagnetism>.
The degenerate <Pauli paramagnetism> calculation requires $k_BT\ll W$, with an unimportant order-one filling factor. At the estimated crossing,
$$
W=E_be^{-16.02}\simeq6.04\times10^{-10}\,\mathrm{eV},\qquad
\boxed{T\ll W/k_B\simeq7.0\times10^{-6}\,\mathrm K.}
$$
The weak-field condition also requires $|g|\mu_BB\ll W$. For a different assumed atomic $R$, use $|\chi_0|=2.599\times10^{-5}(R/a_0)^2$ in the displayed equation for $x$. An order-of-magnitude background $|\chi_0|\sim10^{-5}$ gives the same density order and a crossover scale of order $10^{-5}\,\mathrm K$. At higher <temperature> than this extremely narrow <bandwidth> permits, the $n/W$ enhancement cannot be used; the independent-donor <Curie law> is the appropriate thermal scale if interactions remain negligible. At very small <dopant density>, disorder, localization and <Coulomb interactions> can also invalidate the assumed noninteracting <impurity band>. The formal divergence as $n\to0$ is therefore a limit of the stated zero-<temperature> band model, not a prediction for isolated donors at a fixed nonzero <temperature>.
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