Mass Spectrometry of Potassium Isotopes · CAP 2012 #B1
Physics · modern-physics
Problem
Mass spectrometry is a way of measuring the mass-to-charge ratio of charged particles as well as distinguishing different components of a beam of particles. As an example of such an experiment, consider ions of two isotopes of Potassium, K1939 and K1941, with positive charge +e entering as a beam of particles into a region with magnetic field B in the z direction, as shown in the picture. The beam is injected at point O. Due to the magnetic field, all the ions travel in a circle with radius given by r=qBmv, in which v is the velocity of ions, m is the mass of ions and q is the charge. After these two types of ions travel through the magnetic field, they reach a sensitive film and record bright spots on it as show in the picture.
Consider the distance from point O to the bright spot produced by K1939 ion to be L1, and the distance to the spot recorded by K1941 ion to be L2. We use L for the average of L1 and L2.
(a) Assuming that the energy of both isotopes is the same, compute the ratio LL2−L1
(b) Since these ions have different masses, one can distinguish them by observing the two bright spots on the film. However, some experimental uncertainties can complicate the situation. For example, tuning the energy of the beam to a specific value is impossible. If the energy of particles in the beam varies from 7.9×10−17 J to 8.1×10−17 J, is it possible to distinguish these two types of ions?
(c) Furthermore, adjusting the ions to enter exactly in the y direction, as shown in the picture, is also impossible. Assuming that the beam enters at a small angle α from the y direction, compute L as a function of α.
(d) Now assume that the particles in the incident beam can have any angle α between 0° and 3° from the y direction, due to the uncertainty in the angle. Is it possible to distinguish the two types of ions in this case?