Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S19 / 2

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 $\boxed{E_c\simeq 8.38\mathrm{keV}}$.

Doubling the number of identical-angle bends halves each angle. The minimum emittance scales as ${\theta }^3$, so it becomes $\boxed{0.240\mathrm{nm}\mathrm{rad}}$. At unchanged energy and unchanged individual magnet length, the bending radius doubles and field halves, giving $B=0.70$ T and $\boxed{E_c\simeq 4.19\mathrm{keV}}$.

At fixed stored current the loss per turn, proportional to $E^4\oint ds/{\rho }^2$, halves: twice as many equal-length bends each have one quarter of the original contribution. Total radiation power therefore halves. The spectrum softens by the factor-two critical-energy change. The characteristic high-energy vertical opening angle remains of order $1/\gamma$ at fixed energy, while each magnet's horizontal sweep halves. At a fixed absolute photon energy the normalized spectral position changes, so a claim that every angular distribution is identical would be too strong. Lower emittance also reduces the electron source size/divergence subject to the optical functions.

Equilibrium emittance obeys ${ϵ}_x=C_q{\gamma }^2{I}_5/(J_x{I}_2)$, with $I_5=\oint H(s)|\rho {|}^{-3}ds$, $H={\gamma }_x{\eta }^2+2{\alpha }_x\eta {\eta }^{\prime }+{\beta }_x{\eta }^{\prime 2}$. Thus reduce both bend angles and the dispersion invariant inside bends, rather than merely focusing to a small beam size elsewhere. The damping partition can also be adjusted within stability constraints.

A DBA uses two bends and quadrupoles to return dispersion and its slope to zero at the cell boundaries; this provides dispersion-free insertion straights, at some emittance cost. A multi-bend achromat distributes the total angle over more weak bends and carefully controls the interior dispersion. A non-achromatic theoretical-minimum-emittance cell permits nonzero boundary dispersion and chooses the beta function and dispersion near a bend's optimum; longitudinal-gradient or reverse bends offer further control of radiation integrals. Strong focusing and dispersion suppression can increase natural chromaticity and sextupole demands, compromising dynamic aperture and momentum acceptance.