At fixed gradient and magnet length, f∝p. If E is total energy,Here β0=v0/c, not one of the Courant-Snyder parameters. For ultrarelativistic electrons this becomes f≃f0(1+δ), or 1/f≃(1−δ)/f0: higher-energy particles are bent and focused less strongly. If the symbol E instead denotes kinetic energy K, the coefficient is (K+mc2)/(K+2mc2), approaching 1/2 in the nonrelativistic limit. Stating the energy convention avoids applying the ultrarelativistic result to a slow beam.
dlnEdlnp=E2−me2c4E2=β021,f(E)=f0(1+β02δ)+O(δ2).
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