REGULARITIES OF THE MAGNETIC FIELD OF A BETATRON

B. N. RODIMOV

(Presented by the scientific seminar of the Physico-Technical Faculty)

Equations of motion of an electron in the magnetic field of a betatron

The equations of motion of an electron in the magnetic field of a betatron in cylindrical coordinates will have the form:

\[ F_r=\frac{e}{c}\,\dot{\Theta}\,\frac{\partial(rA)}{\partial r}+mr\dot{\Theta}^{2}, \]

\[ F_z=-\frac{e}{c}\,r\dot{\Theta}\,\frac{\partial A}{\partial z}, \]

\[ F_{\Theta}=\frac{e}{cr}\cdot\frac{d}{dt}(rA)=\frac{m}{r}\frac{d}{dt}(r^{2}\dot{\Theta}). \]

Here \(r, z, \Theta\) are the coordinates of the electron, \(A=A_{\Theta}(r,z)\) is the vector potential of the magnetic field, and \(m\) is the mass of the electron (for the time being we restrict ourselves to the nonrelativistic case).

The last of equations (1) is integrated directly:

\[ \frac{cm}{e}r^{2}\dot{\Theta}-rA=\frac{cm}{e}r_{0}^{2}\dot{\Theta}_{0}-r_{0}A_{0}=\mathrm{const}=C. \tag{2} \]

The subscripts “0” denote the initial values of the corresponding quantities.

Introduce the quantity

\[ V_{M}=\frac{e}{2mc^{2}}\left(A+\frac{C}{r}\right)^{2} \quad \text{or} \quad V_{M}=\frac{e}{2mc^{2}}\left(\frac{\dfrac{r^{2}H_{z}}{2}+C}{r}\right)^{2}, \]

where \(H_z=\dfrac{2A}{r}\) is the average magnetic-field intensity in a circle of radius \(r\).

In these notations:

\[ m\ddot r=-e\,\frac{\partial V_M}{\partial r}, \]

\[ m\ddot z=-e\,\frac{\partial V_M}{\partial z}, \tag{3} \]

\[ mr\dot\theta=mv_t=\frac{e}{c}\left(\frac{\dfrac{r^2H_z}{2}+C}{r}\right). \]

Over a sufficiently small interval of time the function \(V_M\) may be considered constant. Then equations (3) determine the motion of the electron in the \(r,z\) plane in the field of potential forces defined by the function \(V_M\).

As the condition for the existence of focusing forces for an electron, directed toward a certain circle of radius \(R\), the usual requirements for the existence of a minimum of a function of two variables are obtained for the function \(V_M\):

\[ \frac{\partial V_M}{\partial r}=0,\qquad \frac{\partial^2 V_M}{\partial r^2}>0,\qquad \frac{\partial V_M}{\partial z}=0,\qquad \frac{\partial^2 V_M}{\partial z^2}>0, \]

\[ \frac{\partial^2 V_M}{\partial r^2}\cdot \frac{\partial^2 V_M}{\partial z^2} -\left(\frac{\partial^2 V_M}{\partial r\partial z}\right)^2>0. \tag{4} \]

These requirements lead to the well-known condition for the radial decrease of the field in the vicinity of a circle of given radius:

\[ 0<-\frac{r}{H_z}\frac{\partial H_z}{\partial r}<1. \]

The quantity \(-\dfrac{r}{H_z}\dfrac{\partial H_z}{\partial r}\) is usually denoted by the letter \(n\).

In estimating the focusing properties of the field one is usually limited to studying the value of \(n\) at different points of the field. However, the distribution of \(n\) in the field does not make it possible to represent clearly the distribution of the focusing forces in the field and, from this point of view, to evaluate the quality of the field. For this purpose the more suitable quantity is \(V_M\)—the potential function of the focusing forces of the magnetic field of the betatron.

Relation of the quantity \(n\) to the potential function \(V_M\)

Equations (3) show that the tangential component of the velocity of an electron moving in the magnetic field of a betatron can be expressed in terms of the potential function \(V_M\) of this field:

\[ v_t=\sqrt{\frac{2e}{m}\,V_M}. \tag{5} \]

The focusing force in the direction of the \(Z\) axis may be written in the form

\[ m\ddot z=-e\,\frac{\partial V_M}{\partial z} =\frac{e}{c}\,v_tH_r. \]

Hence

\[ H_z=-c\sqrt{\frac{m}{e}}\, \frac{\dfrac{\partial V_M}{\partial z}}{\sqrt{2V_M}}. \]

The focusing force in the direction of the \(r\) axis is written as

\[ m\ddot r=-e\frac{\partial V_m}{\partial r}=-\frac{e}{c}v_t\Delta H_z . \]

Hence

\[ \Delta H_z=c\sqrt{\frac{m}{e}}\cdot \frac{\dfrac{\partial V_m}{\partial r}}{\sqrt{2V_m}} . \]

Here \(\Delta H_z=H_z-H_{zv}\), i.e., the difference between the magnetic field existing at the given point and the field required for circular motion of an electron at the given radius \(r\) with velocity \(v_t\). This difference also creates the focusing force in the direction \(r\). \(H_{zv}\) is found from the relation

\[ \frac{mv_t^2}{r}=\frac{e}{c}v_t H_{zv}. \]

Thus one obtains

\[ H_z=\frac{c}{r}\sqrt{\frac{2m}{e}V_m} +c\sqrt{\frac{m}{e}}\cdot \frac{\dfrac{\partial V_m}{\partial r}}{\sqrt{2V_m}} . \tag{7} \]

Since, by definition,

\[ n=-\frac{r}{H_z}\cdot\frac{\partial H_z}{\partial r}, \]

we obtain

\[ n=1- \frac{ 2\dfrac{\partial V_m}{\partial r} +r\left[ \dfrac{\partial^2 V_m}{\partial r^2} -\dfrac{1}{2V_m}\left(\dfrac{\partial V_m}{\partial r}\right)^2 \right] }{ \dfrac{2V_m}{r}+\dfrac{\partial V_m}{\partial r} }. \tag{8} \]

This is the required equation for \(n\).

At a point on the equilibrium orbit \(r=R,\ z=0,\ \dfrac{\partial V_m}{\partial r}=0\), i.e.,

\[ n_R=1-R^2 \left( \frac{\dfrac{\partial^2 V_m}{\partial r^2}}{2V_m} \right)_{r=R,\ z=0}. \]

From the condition

\[ \frac{\partial H_z}{\partial r}=\frac{\partial H_r}{\partial z} \]

one can obtain the relation

\[ n_R=R^2 \left( \frac{\dfrac{\partial^2 V_m}{\partial z^2}}{2V_m} \right)_{r=R,\ z=0}. \]

Derivation of the equation of the magnetic field of a betatron

Equation (8) can be rewritten as

\[ \frac{\partial^2 V_m}{\partial r^2} -\frac{1}{2V_m}\left(\frac{\partial V_m}{\partial r}\right)^2 +\frac{1+n}{r}\cdot\frac{\partial V_m}{\partial r} -(1-n)\frac{2V_m}{r^2}=0 . \tag{9} \]

If the distribution of \(n\) is known for a given value of \(z\), then this equation can serve to find the form of the dependence of \(V_m\) on \(r\) for the given \(z\).

One can obtain a similar equation for the dependence of \(V_m\) on \(z\). Taking the already known expressions for \(H_r\) and \(H_z\) and using Maxwell’s equation
\[ \frac{\partial H_z}{\partial r}=\frac{\partial H_r}{\partial z}, \]
we obtain
\[ \frac{\partial^2 V_m}{\partial r^2} -\frac{1}{2V_m}\left(\frac{\partial V_m}{\partial r}\right)^2 \cdot n\left[\frac{2V_m}{r^2}-\frac{1}{r}\frac{\partial V_m}{\partial r}\right]=0. \tag{10} \]

Adding the left-hand sides of equations (9) and (10), we shall have
\[ \frac{\partial^2 V_m}{\partial r^2} +\frac{1}{r}\cdot\frac{\partial V_m}{\partial r} +\frac{\partial^2 V_m}{\partial z^2} -\frac{1}{2V_m}\left[\left(\frac{\partial V_m}{\partial r}\right)^2 +\left(\frac{\partial V_m}{\partial z}\right)^2\right] -\frac{2V_m}{r^2}=0. \tag{11} \]

We shall call this equation the equation of the magnetic field of a betatron, since it encompasses all the principal properties of this field that are of interest from the standpoint of betatron theory.

As shown in [2], for \(V_m\) the relation
\[ 4\pi e\rho_{\mathrm{eq}} = \frac{\partial^2 V_m}{\partial r^2} +\frac{1}{r}\cdot\frac{\partial V_m}{\partial r} +\frac{\partial^2 V_m}{\partial z^2}, \tag{12} \]
holds, where \(\rho_{\mathrm{eq}}\) is the so-called equilibrium density of the electron beam. The field equation makes it possible to obtain a second expression for the equilibrium density,
\[ 4\pi e\rho_{\mathrm{eq}} = \frac{1}{2V_m} \left[ \left(\frac{\partial V_m}{\partial r}\right)^2 + \left(\frac{\partial V_m}{\partial z}\right)^2 \right] + \frac{2V_m}{r^2}. \tag{13} \]

The function \(V_m\) of the optimal field

The optimal field of a betatron must provide the greatest equilibrium charge in the space specified for the motion of electrons. Mathematically the problem reduces to finding the extremum of the following integral, representing the total equilibrium charge in the region of the focusing forces of the magnetic field (for electrons with that value of the constant \(C\) for which the function \(V_m\) is taken):
\[ J=\iiint 4\pi e\rho_{\mathrm{eq}}\, r\,dr\,d\theta\,dz = 2\pi e\iint \left[ \frac{1}{2V_m}\left(\frac{\partial V_m}{\partial r}\right)^2 + \frac{1}{2V_m}\left(\frac{\partial V_m}{\partial z}\right)^2 + \frac{2V_m}{r^2} \right]r\,dr\,dz . \]

The extremum condition for \(J\) will be the Euler equation for the function \(F\)
\[ F= \frac{r}{2V_m}\left(\frac{\partial V_m}{\partial r}\right)^2 + \frac{r}{2V_m}\left(\frac{\partial V_m}{\partial z}\right)^2 + \frac{2V_m}{r}, \]
i.e.,
\[ \frac{\partial F}{\partial V_m} - \frac{\partial}{\partial r} \left[ \frac{\partial F}{\partial\left(\dfrac{\partial V_m}{\partial r}\right)} \right] - \frac{\partial}{\partial z} \left[ \frac{\partial F}{\partial\left(\dfrac{\partial V_m}{\partial z}\right)} \right] =0. \]

It is easy to verify that Euler’s equation reduces again to the field equation (11) obtained by us. Thus, the specified region of focusing forces can be realized by us by only one single field. The choice of the optimal field reduces to the choice of the best contour for the region of focusing forces. This latter choice is made depending on the required intensity and on engineering and economic considerations.

Solution of the field equation and its application to field calculation

The expression for the potential function for electrons with an arbitrary value of the constant \(C\) is written in the form

\[ V_{\mathrm{mc}}=-\frac{e}{2mc^2}\left[\frac{\dfrac{r^2\overline{H}_z}{2}+C}{r}\right]^2 . \tag{14} \]

It can easily be expressed through the potential function of “zero” electrons, for which \(C=0\), i.e.

\[ V_{\mathrm{m0}}=-\frac{e}{2mc^2}\left[\frac{r\overline{H}_z}{2}\right]^2 , \tag{15} \]

in the form

\[ V_{\mathrm{mc}}=\left[\sqrt{V_{\mathrm{m0}}}+\frac{C}{r}\sqrt{\frac{e}{2mc^2}}\right]^2 . \tag{16} \]

Therefore it will be sufficient to find the solution of the field equation for zero electrons.

For the solution, let us return to the expression of the function \(V_{\mathrm{m0}}\) through the vector potential \(A\)

\[ V_{\mathrm{m0}}=\frac{eA^2}{2mc^2}. \tag{17} \]

Substituting the quantity \(A\) into the field equation in place of \(V_{\mathrm{m}}\), we obtain for \(A\) the equation

\[ \frac{\partial^2 A}{\partial r^2}+\frac{1}{r}\frac{\partial A}{\partial r} +\frac{\partial^2 A}{\partial z^2}-\frac{A}{r^2}=0. \tag{18} \]

This equation is solved by separation of variables. We set

\[ A=U(r)\cdot W(z). \]

We obtain two equations:

\[ \frac{d^2 W(z)}{dz^2}=k^2 W(z), \tag{19} \]

and

\[ \frac{d^2 U(r)}{dr^2}+\frac{1}{r}\frac{dU(r)}{dr} +\left(k^2-\frac{1}{r^2}\right)U(r)=0. \tag{20} \]

The condition \(\dfrac{dW}{dz}=0\) at \(z=0\) gives

\[ W(z)=ch\,(kz). \tag{21} \]

The solution of the second equation will be

\[ U(r)=A_0\left[aJ_1(kr)+bN_1(kr)\right], \tag{22} \]

where \(J_1(kr)\) and \(N_1(kr)\) are, respectively, Bessel and Neumann functions of the first order.

This solution will be the simplest and will require the smallest number of initial conditions. The fields realized in practice are more complicated in their analytic expression. It makes no sense to calculate precisely such fields from the very beginning, because of the necessity of specifying a large number of initial conditions, the choice of which is very difficult.

Let us consider an example of calculating the field.

Let us take as boundary conditions \(n=n_0\) at the radius of the equilibrium circle \(R_0\), and the value of the function \(A=A_0\) at this radius. In addition, let us require that \(A_0\) be minimal. We obtain:

\[ aJ_1(kR_0)+bN_1(kR_0)=1, \]

\[ aJ'_1(kR_0)+bN'_1(kR_0)=0, \tag{23} \]

\[ k^2aJ_1(kR_0)+k^2bN'_1(kR_0)=-\frac{n_0}{R_0^2}. \]

(The last expression is obtained from the relation \(\dfrac{\partial^2 V_{mo}}{\partial z^2}=\dfrac{2V_0}{R_0^2}n_0\).)

The first and third equations give

\[ k=\frac{\sqrt{n_0}}{R_0}. \]

The amplitudes are determined from the equations

\[ aJ_1(\sqrt{n_0})+bN_1(\sqrt{n_0})=1, \]

\[ aJ'_1(\sqrt{n_0})+bN'_1(\sqrt{n_0})=0. \tag{24} \]

For the case \(R_0=15\ \mathrm{cm}\) and \(n_0=0.5\) and \(0.75\), the following values of the amplitudes and of the coefficient \(k\) are obtained:

In Figs. (1, 2, 3, 4) the function \(V_{mo}=\dfrac{eA^2}{2mc^2}\) is presented for various values of \(r\) and \(z\) in the form of horizontal lines, and the profile of the potential well is given for \(z=0\). The curve \(n\) gives the values \(n(r)\), calculated from the formula

\[ n= \frac{ \dfrac{A}{r}-\dfrac{\partial A}{\partial r} +r\dfrac{\partial^2 A}{\partial r^2} }{ \dfrac{A}{r}+\dfrac{\partial A}{\partial r} }, \tag{25} \]

which is a modification of formula (8).

Fig. 1

Fig. 2

Fig. 3

Using these same data, the profile of the poles producing a non-skew field is calculated. As is known, \(H_r\) and \(H_z\) can be expressed in terms of the scalar magnetic potential \(V\):

\[ H_r=-\frac{\partial V}{\partial r}, \]

\[ H_z=-\frac{\partial V}{\partial z}. \tag{26} \]

Fig. 4

For an equipotential surface, which, in particular, may also be considered the pole surface, we have

\[ dV=\frac{\partial V}{\partial r}\,dr+\frac{\partial V}{\partial z}\,dz=0. \tag{27} \]

Hence, if we use expressions (6) and (7) for \(H_r\) and \(H_z\),

\[ \frac{dz}{dr} = \frac{\dfrac{\partial V_m}{\partial z}} {-\dfrac{2V_m}{r}+\dfrac{\partial V_m}{\partial r}}. \tag{28} \]

According to this equation, pole profiles have been constructed for the case under consideration (curves \(a\)—\(a\) in Figs. 2 and 4).

Such profiles would occur for poles infinite in radius. The finite dimensions of the poles require projections to be made along the edge of the poles, restoring, with one accuracy or another, the field calculated for poles infinite in radius. The form and dimensions of these projections can readily be obtained on a special model.

LITERATURE

1. J. A. Rajchman and W. H. Cherry, Jour. Frankl. Inst., 243, 261–285 (1947).
2. Rodimov B. N. On the mechanism of electron capture into acceleration in a betatron, I and II. Izvestiya TPI, vol. 87, 1957.