Relativistic Potential Function of the Magnetic Field and Its Application to Calculating the Density of the Equilibrium Charge in a Betatron

P. A. CHERDANTSEV

(Presented by the scientific seminar of the Faculty of Physics and Technology)

We shall regard the magnetic field of the betatron as cylindrically symmetric. Then it can be described by a vector potential with one component \(A_\theta = A(r,z)\). To formulate the equations of motion, let us take the relativistic Lagrange function in the form

\[ L=-m_0 c^2 \sqrt{1-\frac{v^2}{c^2}}-\frac{e}{c}\,\vec A \vec v . \]

The equations of motion take the form:

\[ \frac{d}{dt}(m\dot r)=m r \dot\theta^{\,2}-\frac{e}{c}\dot\theta\,\frac{\partial rA}{\partial r}. \tag{1} \]

\[ \frac{d}{dt}(m\dot z)=-\frac{e}{c}\dot\theta\,\frac{\partial rA}{\partial z}, \tag{2} \]

\[ \frac{d}{dt}\left(m r^2\dot\theta-\frac{e}{c}Ar\right)=0. \tag{3} \]

Here

\[ m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}},\qquad v^2=r^2\dot\theta^{\,2}+\dot r^{\,2}+\dot z^{\,2}. \]

From equation (3), after integration, we obtain

\[ m r^2\dot\theta=\frac{e}{c}(Ar+C), \tag{4} \]

where \(C\) is a constant of integration. The right-hand sides of equations (1) and (2) can be expressed through \((Ar+C)\), using relation (4). Then the system of equations is written as follows:

\[ \frac{d}{dt}(m\dot r) = e\sqrt{1-\frac{v^2}{c^2}}\, \frac{\partial}{\partial r} \frac{e}{2m_0c^2} \left(\frac{Ar+C}{r}\right)^2 = e\sqrt{1-\frac{v^2}{c^2}}\, \frac{\partial V_{\mathrm{mc}}}{\partial r}, \tag{5} \]

\[ \frac{d}{dt}(m\dot z)=-e\sqrt{1-\frac{v^2}{c^2}}\, \frac{\partial}{\partial z}\frac{e}{2m_0c^2}\left(A+\frac{C}{r}\right)^2 = \]

\[ =-e\sqrt{1-\frac{v^2}{c^2}}\frac{\partial V_{mc}}{\partial z}. \tag{6} \]

Here \(V_{mc}=\dfrac{e}{2m_0c^2}\left(A+\dfrac{C}{r}\right)^2\) is the nonrelativistic potential function [1].

From equations (5) and (6) it is seen that now the force is determined not by the potential \(V_{mc}\), but by some other relativistic potential \(V_{\text{rel}}\), satisfying the equations

\[ \frac{\partial V_{\text{rel}}}{\partial r} = \sqrt{1-\frac{v^2}{c^2}}\, \frac{\partial V_{mc}}{\partial r}, \tag{7} \]

\[ \frac{\partial V_{\text{rel}}}{\partial z} = \sqrt{1-\frac{v^2}{c^2}}\, \frac{\partial V_{mc}}{\partial z}. \tag{8} \]

Let us now relate the velocity \(v\) to the function \(V_{mc}\). We note that at relativistic velocities \(v\) is almost equal to the component \(v_\theta=r\dot\theta\). From equation (4) we have

\[ v_\theta=\frac{e}{c}\left(A+\frac{C}{r}\right) \frac{\sqrt{1-\dfrac{v_\theta^2}{c^2}}}{m_0} \tag{9} \]

or, expressing \(v_\theta/c\) through \(\left(A+\dfrac{C}{r}\right)\) or through \(V_{mc}\), we obtain for

\[ \sqrt{1-\frac{v_\theta^2}{c^2}} \]

the following expression:

\[ \sqrt{1-\frac{v_\theta^2}{c^2}} = \frac{1}{\left(1+\dfrac{2eV_{mc}}{m_0c^2}\right)^{1/2}}. \tag{10} \]

Substituting, instead of \(\sqrt{1-\dfrac{v^2}{c^2}}\), the root almost equal to it, \(\sqrt{1-\dfrac{v_\theta^2}{c^2}}\), we obtain

\[ \frac{\partial V_{\text{rel}}}{\partial r} = \left(1+\frac{2eV_{mc}}{m_0c^2}\right)^{-1/2} \frac{\partial V_{mc}}{\partial r}, \]

\[ \frac{\partial V_{\text{rel}}}{\partial z} = \left(1+\frac{2eV_{mc}}{m_0c^2}\right)^{-1/2} \frac{\partial V_{mc}}{\partial z}. \]

Carrying out the integration of the equations, we arrive at the following form of the relativistic magnetic potential:

\[ V_{\text{rel}}=\frac{m_0c^2}{e} \left(1+\frac{2eV_{mc}}{m_0c^2}\right)^{1/2} \tag{11} \]

or

\[ V_{\text{rel}}=\frac{m_0c^2}{e} \left[ 1+\frac{e^2}{m_0^2c^4}\left(A+\frac{C}{r}\right)^2 \right]^{1/2}. \tag{12} \]

Using the potential \(V_{\mathrm{rel}}\), one can calculate the density of the equilibrium charge in the region of the focusing forces of a betatron, if the beam consists of electrons with a single \(C\). From Poisson’s equation we obtain

\[ \rho_{\mathrm{rel}}=-\frac{1}{4\pi}\nabla^2 V_{\mathrm{rel}} . \tag{13} \]

After carrying out the operation \(\nabla^2\), we have

\[ \rho_{\mathrm{rel}}= \frac{\rho_{\mathrm{nr}}}{\left(1+\dfrac{2eV_{\mathrm{mc}}}{m_0c^2}\right)^{1/2}} + \]

\[ +\frac{e}{4\pi m_0c^2} \left(1+\frac{2eV_{\mathrm{mc}}}{m_0c^2}\right)^{-3/2} \left\{ \left(\frac{\partial V_{\mathrm{mc}}}{\partial r}\right)^2+ \left(\frac{\partial V_{\mathrm{mc}}}{\partial z}\right)^2 \right\}, \tag{14} \]

where \(\rho_{\mathrm{nr}}\) is the density calculated from \(V_{\mathrm{mc}}\).

If the squares of the derivatives are replaced by the quantity equal to them, determined from the equation for \(V_{\mathrm{mc}}\) [2], then we obtain another formula:

\[ \rho_{\mathrm{rel}}= \left\{ \rho_{\mathrm{nr}} - \frac{eV_{\mathrm{mc}}^2}{\pi m_0c^2r^2} \right\} \left( 1+\frac{2eV_{\mathrm{mc}}}{m_0c^2} \right)^{-3/2}. \tag{15} \]

In the particular case \(v_\omega \approx c\) (the ultrarelativistic case), \(\dfrac{eV_{\mathrm{mc}}}{m_0c^2}\gg 1\), \(\rho_{\mathrm{nr}}\), in order of magnitude, coincides with \(\dfrac{V_{\mathrm{mc}}}{r^2}\) and, consequently, is small in comparison with \(\dfrac{eV_{\mathrm{mc}}^2}{\pi m_0c^2r^2}\); therefore

\[ \rho_{\mathrm{rel}}\simeq -\left(\frac{m_0c^2}{2e}\right)^{1/2} \frac{V_{\mathrm{mc}}^{1/2}}{2\pi r^2}. \tag{16} \]

If the equilibrium beam consists of electrons with different \(C\), then in this case, in order to calculate \(\rho_{\mathrm{rel}}\), it is necessary to find the relativistic effective potential \(V_{\mathrm{eff}}^{(p)}\). The method of finding \(V_{\mathrm{eff}}^{(p)}\) is the same as in paper [1].

Let us set up the equations for \(V_{\mathrm{eff}}^{(p)}\):

\[ \frac{\partial V_{\mathrm{eff}}^{(p)}}{\partial r} = \left[ \frac{\partial V_{\mathrm{rel}}}{\partial r} \right]_{x_c=f(x_0,z)}, \tag{17} \]

\[ \frac{\partial V_{\mathrm{eff}}^{(p)}}{\partial z} = \left[ \frac{\partial V_{\mathrm{rel}}}{\partial z} \right]_{x_c=f(x_0,z)} . \tag{18} \]

Dividing one equation by the other and canceling identical factors leads to equation (7) of paper [1], i.e., to the equation

\[ \frac{\partial V_{\mathrm{eff}}^{(p)}}{\partial r}\, \frac{\partial A}{\partial z} - \frac{\partial V_{\mathrm{eff}}^{(p)}}{\partial z} \left( \frac{\partial A}{\partial r} - \frac{C_{x_c=f(x_0,z)}}{r^2} \right) =0. \tag{19} \]

The solution of the equation is obtained in the form

\[ V_{\mathrm{eff}}^{(p)}=\Phi_p\bigl(u(x_0,z)\bigr), \]

where \(u(x_0,z)\) has the same form as function (13) of paper [1].

Let us represent \(V_{\text{eff}}^{(p)}\) in the form of a series in powers of \(u(x_0,z)\)

\[ V_{\text{eff}}^{(p)}=\Phi_{p0}+\Phi'_{p0}u \]

and determine the quantity \(\Phi_{p0}\) by substituting \(V_{\text{eff}}^{(p)}\) into (18). As a result we obtain

\[ \Phi'_{p0}= \frac{V_0}{\left(1+\dfrac{2eV_0}{m_0c^2}\right)^{\frac12}}. \]

All the remaining coefficients have their former form. \(\Phi_0\) may be taken equal to \(\Phi'_{p0}\). Thus, finally, we have:

\[ V_{\text{eff}}^{(p)}= \frac{V_0}{\left(1+\dfrac{2eV_0}{m_0c^2}\right)^{\frac12}} \left\{ 1+\frac{n_0z^2}{R_0^2}+ \right. \]

\[ \left. +(1-n_0)\varphi\delta_0^2 +a\frac{\delta_0^3}{3} +\frac{(1-n_0)}{R_0^3}\eta z^2\delta_0 \right\}. \tag{20} \]

\(V_{\text{eff}}^{(p)}\) can be expressed in terms of the nonrelativistic expression \(V_{\text{eff}}\).

\[ V_{\text{eff}}^{(p)}= \frac{V_{\text{eff}}}{\left(1+\dfrac{2eV_0}{m_0c^2}\right)^{\frac12}}. \tag{21} \]

Now the charge density of the equilibrium beam can be found from the formula

\[ \rho_{\text{rel}}= -\frac{1}{4\pi}\nabla^2 V_{\text{eff}}^{(p)} = \frac{\rho_{nr}}{\left(1+\dfrac{2eV_0}{m_0c^2}\right)^{\frac12}}. \]

Thus, the relativistic density \(\rho_{\text{rel}}\) is \(\left(1+\dfrac{2eV_0}{m_0c^2}\right)^{\frac12}\) times smaller than the density \(\rho_{nr}\), calculated by the nonrelativistic formula (starting from \(V_{\text{eff}}\)).

The question of the stability of the relativistic beam is solved in the same way as in paper [1]. The equilibrium condition has the same form as the stability condition (19) of paper [1].

The calculations were carried out using the expansion of the vector potential up to quantities of the third order of smallness, which is quite sufficient. If necessary, the expansion can be continued, and this will lead to more cumbersome calculations, almost without changing the principal results.

LITERATURE

1. Cherdantsev P. A. On the stability of an equilibrium electron beam in a betatron. Izvestiya TPI, vol. 87, 1957.

2. Rodimov B. N. Regularities of the magnetic field of a betatron. Izvestiya TPI, vol. 87, 1957.