Momentum analysis
A uniform magnetic field is often used in making a “momentum analyzer,” or “momentum spectrometer,” for high-energy charged particles. Suppose that charged particles are shot into a uniform magnetic field at the point $A$ in Fig. 29–2(a), the magnetic field being perpendicular to the plane of the drawing. Each particle will go into an orbit which is a circle whose radius is proportional to its momentum. If all the particles enter perpendicular to the edge of the field, they will leave the field at a distance $x$ (from $A$) which is proportional to their momentum $p$. A counter placed at some point such as $C$ will detect only those particles whose momentum is in an interval $\Delta p$ near the momentum $p=qBx/2$. It is, of course, not necessary that the particles go through $180^\circ$ before they are counted, but the so-called “$180^\circ$ spectrometer” has a special property. It is not necessary that all the particles enter at right angles to the field edge. Figure 29–2(b) shows the trajectories of three particles, all with the same momentum but entering the field at different angles. You see that they take different trajectories, but all leave the field very close to the point $C$. We say that there is a “focus.” Such a focusing property has the advantage that larger angles can be accepted at $A$—although some limit is usually imposed, as shown in the figure. A larger angular acceptance usually means that more particles are counted in a given time, decreasing the time required for a given measurement. By varying the magnetic field, or moving the counter along in $x$, or by using many counters to cover a range of $x$, the “spectrum” of momenta in the incoming beam can be measured. [By the “momentum spectrum” $f(p)$, we mean that the number of particles with momenta between $p$ and $(p+dp)$ is $f(p)\,dp$.] Such measurements have been made, for example, to determine the distribution of energies in the $\beta$-decay of various nuclei.  There are many other forms of momentum spectrometers, but we will describe just one more, which has an especially large solid angle of acceptance. It is based on the helical orbits in a uniform field, like the one shown in Fig. 29–1. Let’s think of a cylindrical coordinate system—$\rho,\theta,z$—set up with the $z$-axis along the direction of the field. If a particle is emitted from the origin at some angle $\alpha$ with respect to the $z$-axis, it will move along a spiral whose equation is \begin{equation*} \rho=a\sin kz,\quad\theta=bz, \end{equation*}  where $a$, $b$, and $k$ are parameters you can easily work out in terms of $p$, $\alpha$, and the magnetic field $B$. If we plot the distance $\rho$ from the axis as a function of $z$ for a given momentum, but for several starting angles, we will get curves like the solid ones drawn in Fig. 29–3. (Remember that this is just a kind of projection of a helical trajectory.) When the angle between the axis and the starting direction is larger, the peak value of $\rho$ is large but the longitudinal velocity is less, so the trajectories for different angles tend to come to a kind of “focus” near the point $A$ in the figure. If we put a narrow aperture of $A$, particles with a range of initial angles can still get through and pass on to the axis, where they can be counted by the long detector $D$. Particles which leave the source at the origin with a higher momentum but at the same angles, follow the paths shown by the broken lines and do not get through the aperture at $A$. So the apparatus selects a small interval of momenta. The advantage over the first spectrometer described is that the aperture $A$—and the aperture $A'$—can be an annulus, so that particles which leave the source in a rather large solid angle are accepted. A large fraction of the particles from the source are used—an important advantage for weak sources or for very precise measurements. One pays a price for this advantage, however, because a large volume of uniform magnetic field is required, and this is usually only practical for low-energy particles. One way of making a uniform field, you remember, is to wind a coil on a sphere, with a surface current density proportional to the sine of the angle. You can also show that the same thing is true for an ellipsoid of rotation. So such spectrometers are often made by winding an elliptical coil on a wooden (or aluminum) frame. All that is required is that the current in each interval of axial distance $\Delta x$ be the same, as shown in Fig. 29–4.