Thin-quadrupole energy dependence ($f(E)=f_0[1+\delta /{\beta }_0^2]+O({\delta }^2)$)

At fixed gradient and magnet length, the signed thin-lens focal length is proportional to particle momentum. For total energy error $\delta =(E-E_0)/E_0$, differentiating $p^2{c}^2=E^2-m^2{c}^4$ gives $d\ln p/d\ln E=1/{\beta }^2$. The ultrarelativistic result is $f=f_0(1+\delta )$ to first order. If energy means kinetic energy instead, the coefficient is $(K+m{c}^2)/(K+2m{c}^2)$.