DETERMINATION OF THE MOMENTUM OF A CHARGED PARTICLE IN A WILSON CHAMBER WITH A NONUNIFORM MAGNETIC FIELD*)

The momentum of a charged particle moving in the magnetic field of a Wilson chamber is determined from the curvature of its trajectory by the well-known formula \(pc = ZeHr\) only in the case where the magnetic field is uniform throughout the entire space of the Wilson chamber. This necessary condition greatly limits experimental possibilities in work with chambers, since it is rather difficult to obtain uniform fields of large dimensions.

The proposed method for determining the momentum of charged particles is free of this shortcoming, which makes it possible to work with chambers of, in principle, arbitrarily large dimensions. Only one condition is imposed on the nonuniform magnetic field: it must be symmetric with respect to the axis of the chamber.

The trajectory of a charged particle in a magnetic field is determined by the variational equation

\[ \delta \int \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) ds = 0, \tag{1} \]

where \(\mathbf{p}\) is the momentum of the particle, \(e\) its charge, \(c\) the speed of light, \(\mathbf{A}\) the vector potential, and \(ds\) an element of length along the particle trajectory. Let the magnetic field be directed parallel to the \(z\)-axis and be symmetric with respect to

*) A. M. Sagask, Rev. Scient. Instrum. 24, No. 3, 232 (1953).

this axis. Then the components of the vector potential \(\mathbf A\) in cylindrical coordinates are equal to:

\[ A_r=A_z=0;\qquad A_\theta=\frac{1}{r}\int_0^r \rho H(\rho)\,d\rho, \tag{2} \]

where \(H(\rho)\) is the magnetic-field intensity as a function of the radius \(\rho\). If the trajectory of the charged particle lies in a plane perpendicular to the \(z\) axis, then \(ds^2=dr^2+r^2d\theta^2\) and equation (1) takes the form:

\[ \delta\int\left[p(1+r^2\theta'^2)^{1/2}-\frac{e}{c}\,rA_\theta\theta'\right]dr=0, \tag{3} \]

where \(\theta'=\dfrac{d\theta}{dr}\). The corresponding Euler equation can be immediately integrated, and as a result we obtain:

\[ pr^2\theta'(1+r^2\theta'^2)^{-1/2} -\frac{e}{c}\int_0^r \rho H(\rho)\,d\rho=\mathrm{const}. \tag{4} \]

Equation (4) is the differential equation of the motion of a particle in the magnetic field \(H(\rho)\).

Let us denote by \(O\) the center of the chamber and the coincident center of the magnetic field (see figure). \(AB\) is the trajectory of the particle. If we denote by \(\psi_i\) the angle between the tangent to the particle trajectory and the radius vector at the point where the particle intersects the circle of radius \(R_i\), then

\[ \sin\psi_i=R_i\theta'(1+R_i^2\theta'^2)^{-1/2}. \tag{5} \]

If \(\psi_1\) and \(\psi_2\) are the values of \(\psi_i\) for the points of the path at the intersections of the circles with radii \(R_1\) and \(R_2\), respectively, then, combining (4) and (5), we obtain:

\[ R_2\sin\psi_2-R_1\sin\psi_1= \frac{e}{pc}\int_{R_1}^{R_2}\rho H(\rho)\,d\rho; \tag{6} \]

if \(\Phi(R_1,R_2)\) is the magnetic flux between the circles with radii \(R_1\) and \(R_2\), and \(p\) is the particle momentum in units of \(H\rho\), then equation (6) may be written in the form:

\[ 2\pi p\,(R_2\sin\psi_2-R_1\sin\psi_1)=\Phi(R_1,R_2). \tag{7} \]

On a photograph of particle tracks one may draw a large number of circles of arbitrary radii, which makes it possible to determine the values \(R_i\) sufficiently accurately. The magnetic-field intensity can likewise be measured with great accuracy, which accordingly ensures the accuracy of calculating \(\Phi(R_1,R_2)\) for any pair of values \(R_1\) and \(R_2\). The error in determining \(p\) from equation (7) therefore depends exclusively on the accuracy of measuring \(\psi_1\) and \(\psi_2\). From the symmetry of the track with respect to some radius it is possible

to find very accurately the value \(R_i\) for which \(\psi=\dfrac{\pi}{2}\). The corresponding point can then be used for pairing with other points in order to obtain several measurements of the momentum, which makes it possible to compensate errors in determining the individual values of \(\psi_i\).

R. B.