UDC 538.691

PHYSICS

A. I. ERSHKOVICH, V. D. PLETNEV, G. A. SKURIDIN

ON THE QUESTION OF THE MOTION OF CHARGED PARTICLES IN AN ACUTE-ANGLE TRAP

(Presented by Academician G. I. Petrov, January 24, 1966)

In works (¹, ²), regions of possible motion of particles in magnetic traps with opposing fields were studied. The magnetic field in such a trap, in the simplest case, has the form

\[ H_\rho=-A\rho;\qquad H_\varphi=0;\qquad H_z=2Az. \tag{1} \]

The equations of motion of a particle with mass \(m\) and charge \(e\) in the field (1) have the form

\[ \ddot{\rho}-\rho\dot{\varphi}^{2}=2A_0\rho z\dot{\varphi}, \tag{2a} \]

\[ 2\dot{\rho}\dot{\varphi}+\rho\ddot{\varphi}=-A_0(\rho\dot{z}+2z\dot{\rho}), \tag{2b} \]

\[ \ddot{z}=A_0\rho^{2}\dot{\varphi}, \tag{2c} \]

where \(A_0=Ae/mc\), \(m=m_0(1-v^2/c^2)^{-1/2}\) (in a stationary magnetic field \(v^2=\text{const}\)). System (2) is nonlinear, and it is not possible to obtain its general solution in analytic form. In (²) a number of trajectories obtained by numerical integration are presented. We shall show that, in addition to the trivial solutions \(\rho=\dot{\rho}=0\) and \(z=\dot{z}=0\), corresponding to uniform motion along the \(z\)-axis and in the radial direction in the plane of symmetry of the trap \(z=0\) (when the Lorentz force is equal to zero), system (2) admits one more exact particular solution, which corresponds to motion along the surface of the cone \(\rho^2=z^2\)*.

Integrating equation (2b), we have

\[ \rho^{2}\dot{\varphi}+A_0z\rho^{2}=\text{const}. \tag{3} \]

For trajectories passing through the origin, equation (3) gives

\[ \dot{\varphi}=-A_0z, \tag{4} \]

if \(\rho\ne0\). Obviously, (4) is also valid at the origin, since as \(z\to0\), \(\dot{\varphi}\to0\). Substituting (4) into (2a) and (2c), we see that the equations of motion (2) are compatible with the condition \(\rho^2=z^2\). Consequently, in the magnetic field (1) there is a family of trajectories lying on the surface of the cone \(\rho^2=z^2\) (it is not difficult to prove that trajectories not passing through the origin cannot lie on this cone). For \(\rho^2=z^2\), equations (2a) and (2c) reduce to the equation

\[ \ddot{z}+A_0^2z^3=0. \tag{5} \]

Let at \(t=0\), \(z=0\), \(\dot{z}=\pm v_1\) (\(v_1>0\)). Integrating (5), we obtain

\[ \dot{z}^{2}=v_1^{2}-A_0^{2}z^{4}/2. \tag{6} \]

Putting \(\dot{z}=0\), we determine from (6) the position of the reflection point

\[ z_{\max}^{2}=\rho_{\max}^{2}=\sqrt{2}\,v_1/A_0. \tag{7} \]

* In the two-dimensional problem \((H_x=Ay;\ H_y=Ax;\ H_z=0)\), exact particular solutions of the equations of motion in the planes \(x=0\) and \(y=0\) were obtained in work (³).

According to (6),

\[ t=\pm \frac{1}{v_1}\int_0^z \left(1-\frac{A_0^2z^4}{2v_1^2}\right)^{-1/2} dz . \]

Setting \(\sin x=(A_0/\sqrt{2}v_1)^{1/2}z=z/z_{\max}\) and integrating, we find (see (4))

\[ (\sqrt{2}A_0v_1)^{1/2}t=\pm F(\varphi,k), \tag{8} \]

where \(F(\varphi,k)\) is an elliptic integral of the first kind; the modulus is \(k=1/\sqrt{2}\); \(\sin\varphi=\sqrt{2}\sin x(1+\sin^2x)^{-1/2}=\sqrt{2}z(z_{\max}^2+z^2)^{-1/2}\). It is convenient to pass to Jacobi functions. Then (8) can be rewritten in the form

\[ z=\pm z_{\max} \frac{\operatorname{sn}\left(\sqrt{\sqrt{2}A_0v_1}\,t\right)} {\left[2-\operatorname{sn}^2\left(\sqrt{\sqrt{2}A_0v_1}\,t\right)\right]^{1/2}}, \tag{9} \]

where \(z_{\max}\) is determined by (7), \(\operatorname{sn}\) is the elliptic sine, and the plus and minus signs in (8) and (9) correspond to the different initial conditions \(\dot z(t=0)=\pm v_1\). From equation (4) we have

\[ d\varphi=-A_0 z\, dz/\dot z, \tag{10} \]

where \(\dot z\) is determined by expression (6). Put \(\varphi(t=0)=\varphi_0\).* Integrating (10), we obtain (see (4))

\[ \varphi=\varphi_0\mp \sqrt{2}\,\frac{\pi}{4} \pm \sqrt{2}\arcsin \frac{\operatorname{cn}\left(\sqrt{\sqrt{2}A_0v_1}\,t\right)} {\left[2-\operatorname{sn}^2\left(\sqrt{\sqrt{2}A_0v_1}\,t\right)\right]^{1/2}}, \tag{11} \]

where \(\operatorname{cn}\) is the elliptic cosine, and the upper sign corresponds to the initial condition \(\dot z(t=0)=+v_1\).

Let us determine the time interval \(T\) during which the particle travels from the vertex of the cone to \(z=z_{\max}\). Since \(\operatorname{sn}K=1\), it follows from (9) that

\[ T=K(1/\sqrt{2})/(\sqrt{2}A_0v_1)^{1/2}, \tag{12} \]

where

\[ K\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{4\sqrt{\pi}}\left[\Gamma\left(\frac14\right)\right]^2=1.8541 \]

is the complete elliptic integral of the first kind with modulus \(k=1/\sqrt{2}\).

Expressions (9), (11), and (12) make it possible to trace the trajectory of the particle over the course of the full period of its motion. Let, for definiteness, \(\dot z(t=0)=-v_1\). Using the periodicity properties of the Jacobi functions, we obtain: at \(t=T\), \(\varphi=\varphi_0+\sqrt{2}\pi/4\), \(z=-z_{\max}\); at \(t=2T\), \(\varphi=\varphi_{\max}=\varphi_0+\sqrt{2}\pi/2\), \(z=0\); at \(t=3T\), \(\varphi=\varphi_0+\sqrt{2}\pi/4\), \(z=z_{\max}\); finally, at \(t=4T\), \(\varphi=\varphi_0\), \(z=0\). Thus, the trajectory of a particle moving on the surface of the cone \(\rho^2=z^2\) resembles a figure eight, and the complete period of the motion is equal to \(4T\).

Received
18 January 1966

REFERENCES

1. G. Schmidt, Phys. Fluids, 5, No. 8, 994 (1962).
2. K. D. Sinelnikov, N. A. Khizhnyak et al., Plasma Physics and Problems of Controlled Thermonuclear Fusion, vol. 4, Kiev, 1965, p. 388.
3. E. Aström, Tellus, 8, No. 2, 260 (1956).
4. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Moscow, 1963, p. 187.

* Obviously, \(\varphi_0\) is the angle between the projection of the particle velocity vector onto the plane \(z=0\) and the \(x\)-axis at \(t=0\). Indeed, differentiating the relations \(x=\rho\cos\varphi\), \(y=\rho\sin\varphi\) with respect to time. Since, according to (4), \(\dot\varphi(t=0)=-A_0z(t=0)=0\), we have \(\tan\varphi_0=\dot y(t=0)/\dot x(t=0)\).