Cyclotrons and Synchrotrons — Question 9

Question 9

Equal magnetic rigidity does not imply equal revolution frequency for two ion species

Two nonrelativistic ion species have masses m and 4m and positive charges q and 2q. They circulate on the same radius in the same magnetic field, so they have equal p/q.

1. Find the ratios of their momenta, speeds, kinetic energies and revolution frequencies.

2. If both must use one RF frequency, find the ratio of the harmonic numbers h2/h1 required for synchronous timing.

Question 9 - Solution

Equal magnetic rigidity does not imply equal revolution frequency for two ion species

(a) Method and calculation

$$p_2/p_1=q_2/q_1=2,\quad v_2/v_1=(p_2/p_1)/(m_2/m_1)=1/2.$$$$K_2/K_1=\frac{p_2^2/m_2}{p_1^2/m_1}=1,\quad f_2/f_1=v_2/v_1=1/2.$$

(b) Interpretation and further calculation

$$f_{\mathrm{RF}}=h_1f_1=h_2f_2.$$$$h_2/h_1=f_1/f_2=2.$$

Physical check

A shared bending orbit constrains momentum per charge; timing also depends on mass and speed.