A Circulating Charged Particle — Question 6

Question 6

Semicircular mass analysis converts a small mass difference into detector separation

A velocity selector sends singly charged ions of common speed v into an analyzing field B. Each ion follows a semicircle and hits a detector at distance s=2r from its entrance. Two masses differ by a small delta m. Detector spots are resolved when their centers differ by at least a.

1. Derive the mass resolution condition at fixed v and express it using s.

2. For s=.200 m and a=1.00 mm, find the smallest fractional mass difference. Compare the first-order result if the ions instead have equal accelerating voltage and charge.

Question 6 - Solution

Semicircular mass analysis converts a small mass difference into detector separation

(a) Method and calculation

$$s=\frac{2mv}{qB},\quad\Delta s=\frac{2v\Delta m}{qB}=s\frac{\Delta m}{m}.$$$$\Delta s\ge a\Rightarrow\frac{\Delta m}{m}\ge\frac as.$$

(b) Interpretation and further calculation

$$a/s=.001/.200=.00500\quad\text{at fixed }v.$$$$qU=\tfrac12mv^2\Rightarrow s\propto\sqrt m,\quad\Delta s/s\simeq\tfrac12\Delta m/m;\ \Delta m/m\gtrsim.0100.$$

Physical check

Equal speed and equal accelerating voltage are different beam preparations and give different mass-dispersion laws.