Cyclotrons and Synchrotrons — Question 1

Question 1

Cyclotron timing matches one voltage reversal to each semicircular transit

A nonrelativistic positive ion has mass $3.34\times10^{-27}$ kg and charge $1.60\times10^{-19}$ C in B=1.20 T. It crosses the accelerating gap twice per revolution, gaining qV0 at each crossing with V0=20.0 kV. Gap transit times are negligible.

1. Find the revolution frequency and the required fundamental alternating-voltage frequency.

2. Find the energy gained per revolution and the average energy-gain rate of one ion under these assumptions.

Question 1 - Solution

Cyclotron timing matches one voltage reversal to each semicircular transit

(a) Method and calculation

$$T=\frac{2\pi m}{qB},\quad f_{\mathrm{rev}}=\frac{qB}{2\pi m}=9.14903\,\mathrm{MHz}.$$$$t_{\mathrm{half}}=T/2\Rightarrow f_{\mathrm{RF}}=f_{\mathrm{rev}}\quad\text{for fundamental operation}.$$

(b) Interpretation and further calculation

$$\Delta K_{\mathrm{turn}}=2qV_0=6.40\times10^{-15}\,\mathrm J=40.0\,\mathrm{keV}.$$$$\left\langle\frac{dK}{dt}\right\rangle=2qV_0f_{\mathrm{rev}}=5.85538\times10^{-8}\,\mathrm W.$$

Physical check

The RF period is a full orbital period; the field polarity reverses after half of that period.