 Solution

For identical uniform bends in an ideal double-bend achromat, the standard minimum is ${ϵ}_{x,\min }=C_q{\gamma }^2{\theta }^3/(4\sqrt{15}{J}_x)$, with bend angle $\theta =2\pi /48$ and $\gamma \simeq 3000/0.511$. Hence $\boxed{{ϵ}_{x,\min }\simeq 1.92\mathrm{nm}\mathrm{rad}}$. This uses the achromat constraint; the unconstrained theoretical-minimum-emittance coefficient is three times smaller.

The bending-magnet critical photon energy is $E_c=3\hslash c{\gamma }^3/(2\rho )\simeq 0.665E_{\mathrm{GeV}}^2{B}_{\mathrm{T}}$ keV, giving $\boxed{E_c\simeq 8.38\mathrm{keV}}$.