Cyclotrons and Synchrotrons — Question 7

Question 7

A fixed-radius synchrotron must ramp its bending field with momentum rather than kinetic energy alone

A synchrotron maintains bending radius rho for particles of rest mass m and charge magnitude q. Use the supplied relations $p=qB\rho$ and $p^2c^2=K(K+2mc^2)$.

1. Derive the field required at kinetic energy K.

2. Find the leading field scaling for K much smaller and much larger than mc squared, and compare the field changes when K doubles in each limit.

Question 7 - Solution

A fixed-radius synchrotron must ramp its bending field with momentum rather than kinetic energy alone

(a) Method and calculation

$$p=\frac1c\sqrt{K(K+2mc^2)}.$$$$B(K)=\frac{\sqrt{K(K+2mc^2)}}{qc\rho}.$$

(b) Interpretation and further calculation

$$K\ll mc^2:\quad B\simeq\frac{\sqrt{2mK}}{q\rho},\quad B(2K)/B(K)\simeq\sqrt2.$$$$K\gg mc^2:\quad B\simeq\frac K{qc\rho},\quad B(2K)/B(K)\simeq2.$$

Physical check

Magnetic rigidity is momentum per charge; equal kinetic-energy increments do not generally require equal field increments.