On the Question of a Magnetic Field with the Best Focusing Properties

P. A. Cherdantsev

In the process of injection and acceleration of electrons in a betatron chamber, there exists a considerable space charge moving with velocities close to the velocity of light. This charge creates an additional magnetic field acting on the beam. If an equilibrium beam is considered, that is, a beam in which the focusing electrostatic forces are compensated by magnetic focusing forces, then the state of equilibrium is determined, in particular, by the beam’s own magnetic field.

In paper [1] an equation for the field was obtained, taking the space charge into account, in the form

\[ \frac{\nabla^{2}U}{U} -\frac{1}{U^{2}-1} \left[ \left(\frac{\partial U}{\partial r}\right)^{2} + \left(\frac{\partial U}{\partial z}\right)^{2} \right] -\frac{U^{2}-1}{r^{2}} =0, \tag{1} \]

where

\[ U=\frac{eV_{p}}{m_{0}c^{2}}, \]

and \(V_{p}\) is the relativistic potential function of the magnetic field.

To simplify this equation, we introduce, instead of \(U\), a new function \(w\) by the relation

\[ U=\operatorname{ch}\frac{w}{2}. \]

Then we obtain the following equation

\[ \nabla^{2}w-\frac{\operatorname{sh}w}{r^{2}}=0. \tag{2} \]

This equation is nonlinear, and it cannot be solved in this form.

Let us linearize it by expanding \(\operatorname{sh}w\) in a series in powers of \(w\) about the value \(w_{0}\) on the equilibrium orbit. Restricting ourselves to the linear term, we obtain

\[ \operatorname{sh}w=\operatorname{sh}w_{0}+\operatorname{ch}w_{0}\,(w-w_{0}), \]

and

\[ \nabla^{2}(w-w_{0}) -\frac{\operatorname{sh}w_{0}}{r^{2}} -\operatorname{ch}w_{0}\frac{(w-w_{0})}{r^{2}} =0. \]

Let us denote \(\operatorname{ch} w_0\) by \(m^*\) and choose the solution in the form of the sum

\[ w-w_0=\xi+A, \]

where \(A=\text{const}\). Then, if we choose \(A=-\operatorname{th} w_0\), for \(\xi\) we obtain the equation

\[ \nabla^2 \xi - m^{*2}\frac{\xi}{r^2}=0. \tag{3} \]

The original equation for \(U\) must be solved under the boundary conditions specified on the contour \(\Gamma\), which bounds the region of the focusing forces,

\[ U\big|_{\Gamma}=U_0=\text{const}. \]

The same condition is imposed on \(\xi\),

\[ \xi\big|_{\Gamma}=\xi_0=\text{const}. \]

We shall choose the form of the boundary equipotential line in accordance with real conditions. Since the magnetic field of a betatron is produced by poles having a wedge-shaped profile, for this reason the accelerating chamber must have an oval cross-section with smaller curvature on the outer side, in order to use the region of focusing forces as completely as possible. It is necessary to find a field whose equipotential lines would be ovals inscribed in the chamber. In this case the chamber will be used with the greatest efficiency.

Separation of variables in Laplace’s equation

\[ \nabla^2 U=0 \]

can be carried out if the system of orthogonal curvilinear coordinates \(\alpha,\beta,\theta\) is determined by the equality

\[ r+iz=f(\alpha+i\beta) \]

and \(f(\zeta)\) satisfies the equation

\[ f''(\zeta)=\sum_{k=0}^{4}\lambda_k f^k(\zeta), \]

where \(\lambda_k\) are constants [2].

If one chooses \(\lambda_0=k'^2,\lambda_2=1+k'^2,\lambda_4=1,\lambda_1=\lambda_3=0\), then the coordinates \(\alpha\) and \(\beta\) are introduced by the transformation [3]

\[ r+iz=a\,dn\,(K'i-\alpha-i\beta), \tag{4} \]

where \(-K<\alpha<K,\ 0<\beta<K'\), \(K(k)\) is the elliptic integral of the first kind, \(K'=K(k')\), \(k^2+k'^2=1\), and \(dn\,u\) is the Jacobi elliptic function.

The curves \(\alpha=\text{const}\) and \(\beta=\text{const}\) form, in the \((r,z)\) plane, two orthogonal families. The coordinate line \(\beta=\beta_0\) represents the section of the ring. Moreover, \(0<\beta<\beta_0\) corresponds to the region outside the ring, and \(\beta_0<\beta<K'\) to the interior region.

The modulus \(k\) of the elliptic integral characterizes the shape of the lines \(\beta=\text{const}\). In particular, for \(k=0\) we obtain a ring of circular cross-section; \(k\ne0\) gives rings of oval cross-section, passing into plane rings

for \(\zeta_0 > K'\). The modulus \(k\) is related to the characteristics of the flat ring by the relation

\[ k=\sqrt{1-\left(\frac{b}{a}\right)^2}, \]

where \(a\) is the outer radius and \(b\) its inner radius.

Starting from the transformation (4), we obtain:

\[ r=a\,\frac{dn\,z\,sn'\,\zeta}{1-cn^2 z\,cn'^2\zeta}, \qquad z=a\,\frac{sn\,z\,cn\,z\,cn'\zeta\,dn'\zeta}{1-cn^2 z\,cn'^2\zeta}, \]

where \(sn z\), \(cn z\), \(dn z\), and the other functions are Jacobi elliptic functions, \(sn'\zeta = sn(\zeta,K')\), and similarly for all the other primed functions.

Let us pass in equation (3) from the coordinates \((r,z)\) to the coordinates \((\alpha,\zeta)\). Introduce a new function \(F\)

\[ \varphi=\frac{F}{\sqrt{r}} . \]

For \(F\) we obtain the equation in the coordinates \(\alpha,\zeta\):

\[ \frac{\partial^2 F}{\partial \alpha^2} + \frac{\partial^2 F}{\partial \zeta^2} + \frac{1}{4}(1-4m^2) \left[ \left(\frac{k^2 sn\,z\,cn\,z}{dn\,z}\right)^2 - \left(\frac{k^2 sn(K'i-i\zeta)\,cn(K'i-i\zeta)}{dn(K'i-i\zeta)}\right)^2 \right]F=0 . \tag{5} \]

We shall seek the solution of the last equation in the form

\[ F=A(\alpha)B(\zeta). \]

After separation of variables we obtain

\[ A''-\left\{\lambda+\frac{1}{4}(1-4m^2) \left(\frac{k^2 sn\,z\,cn\,z}{dn\,z}\right)^2\right\}A=0; \tag{6} \]

\[ B''-\left\{\lambda+\frac{1}{4}(1-4m^2) \left[ \frac{k^2 sn(K'i-i\zeta)\,cn(K'i-i\zeta)}{dn(K'i-i\zeta)} \right]^2\right\}B=0, \tag{7} \]

where \(\lambda\) is the separation constant.

For the inner region of the ring, the solutions of equation (6) have the form

\[ A_0=1-\frac{1}{2}q^2(1-4m^2)\cos 2x-\cdots, \]

\[ A_1=\cos x-\frac{1}{8}q^2(1-4m^2)\cos 3x-\cdots, \tag{8} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ A_n=\cos nx-(1-4m^2)q^2 \left[ \frac{\cos(n+2)x}{4(n+1)} - \frac{\cos(n-2)x}{4(n-1)} \right]+\cdots \]

\[ n=2,3,\ldots,\infty,\qquad x=\frac{\pi \alpha}{K},\qquad q=e^{-\pi K'/K}. \]

The solutions of equation (7) have the form

\[ Q_{mn}(\xi,K)= \int_{-K}^{K} \left| \frac{dn\,\xi\, dn\,(K'i-i\zeta)} {1-k^{2}sn^{2}\xi\, sn^{2}(K'i-i\zeta)} \right|^{m+\frac12} A_n(\zeta)\,d\zeta . \]

The general solution of equation (3) is

\[ \varphi=\sqrt{\frac{a}{r}}\,\varphi_0 \sum_{n=0}^{\infty} M_n Q_{mn}(\xi) A_n(\alpha), \]

where \(\varphi_0=\varphi(a,\zeta_0)\). From the boundary condition for \(\varphi\) we obtain

\[ M_n= -\frac{Q_{0n}(\zeta_0)} {Q_{mn}(\zeta_0)\displaystyle\int_{-K}^{K} A_n^2\,d\alpha}. \]

Thus,

\[ \begin{aligned} \varphi=\varphi_0 \left( \frac{dn\,\alpha\, dn\,(K'i-i\zeta)} {1-k^2 sn^2\alpha\, sn^2(k'i-i\zeta)} \right)^{-1/2} \cdot \sum_{n=0}^{\infty} \frac{Q_{0n}(\zeta_0)Q_{mn}(\zeta)} {\displaystyle\int_{-K}^{K} A_n^2\,d\alpha\, Q_{mn}(\zeta_0)} A_n(\alpha). \end{aligned} \tag{9} \]

The value \(\zeta_0\), through which the coefficients \(M_n\) are determined, depends on the dimensions of the region of the focusing forces. Let us specify on the contour \(\Gamma\) the coordinates \(R_{\min}\) and \(R_{\max}\) at \(z=0\):

\[ R_{\min} =a-\left. \frac{dn\,\alpha\, sn'\zeta_0} {1-cn^2\alpha\, cn'^2\zeta_0} \right|_{\alpha=K} =ak'\,sn'\zeta_0, \]

\[ R_{\max} =a-\left. \frac{dn\,\alpha\, sn'\zeta_0} {1-cn^2\alpha\, cn'^2\zeta_0} \right|_{\alpha=0} =\frac{a}{sn'\zeta_0}. \]

Then

\[ \frac{R_{\min}}{R_{\max}}=\eta^2=k'\,sn'^2\zeta_0, \]

or

\[ sn'\zeta_0=\frac{\eta}{k'^{1/2}}. \]

For small \(k\) we obtain

\[ \zeta_0=\operatorname{Arth}\eta\left(1+\frac{k^2}{4}\right). \tag{10} \]

The field found makes it possible to calculate the equilibrium density of the charge and the total maximum charge retained by this field. The total charge \(Q\) is an important characteristic of the betatron. It shows the maximum capabilities of the betatron field. Investigations have shown that the radiation intensity is proportional to the maximum charge. Therefore it is important to know the dependence of the total charge on the shape of the field. The optimum field will be that for which the charge is maximal. To compute the equilibrium density we shall find \(V_\rho\) as a function of \(\varphi\). We had

\[ U'=\frac{eV_p}{m_0c^2}=\operatorname{ch}\frac{\bar w}{2}. \]

Therefore

\[ \begin{aligned} V_p&=\frac{m_0c^2}{e}\operatorname{ch}\frac{1}{2}(-\operatorname{th} w_0+\bar w_0+\zeta)\\ &=V_0\left(1+\operatorname{th}\frac{1}{2}\left(\bar w_0+\operatorname{th}w_0\right)\frac{\zeta}{2}\right). \end{aligned} \]

Hence the charge density is

\[ \rho=-\frac{1}{4\pi}\nabla^2 V_p-\frac{N}{2\pi}\nabla^2\zeta, \]

where

\[ N=\frac{m_0c^2}{4e}\operatorname{sh}\frac{1}{2}(\bar w_0-\operatorname{th}w_0). \]

We take into account the equation for \(\zeta\). Then

\[ \rho=-\frac{N}{2\pi}m^2\frac{\zeta}{r^2}-\frac{\mathfrak A}{2\pi}\frac{\zeta}{r^2}. \]

The total charge in the chamber is equal to

\[ Q=\mathfrak A\int_{r_1}^{r_2}\int_{z_1}^{z_2}\frac{\zeta}{r}\,dr\,dz. \tag{11} \]

\(Q\) is a function of the field shape. In our case the field shape is characterized by the parameter \(k\), therefore

\[ Q=Q(k). \]

In formula (11) it is necessary to pass from integration with respect to \(r,z\) to integration with respect to \(z,\zeta\). The Jacobian of the transition from \(r,z\) to \(z,\zeta\) is equal to

\[ I^2=a^2\left|\frac{d\,\operatorname{dn}(k'z-\zeta)}{dz}\right|^2, \]

or, after calculating the derivatives,

\[ I^2=r^2\left\{\left(\frac{k^2\operatorname{sn} z\,\operatorname{cn} z}{\operatorname{dn} z}\right)^2+ \left(\frac{\operatorname{ch}'\zeta\,\operatorname{dn}'\zeta}{\operatorname{sn}'\zeta}\right)^2\right\}. \]

Thus, we obtain

\[ Q=\mathfrak A\int_{-K}^{K}\int_{K'}^{z_0}\frac{\zeta}{r}\left|\frac{df}{dz}\right|^2\,dz\,d\zeta =\mathfrak A\int_{-K}^{K}\int_{K'}^{z_0}\zeta r\,\{\ \}\,dz\,d\zeta. \tag{12} \]

Represent \(Q(k)\) in the form of a series in \(k\). In formula (12) one may pass from \(z\) to the new variable \(y=\dfrac{\pi a}{2K}\); then the integration with respect to \(z\) over the limits from \(-K\) to \(K\) will pass to integration with respect to \(y\) within limits

from \(-\dfrac{\pi}{2}\) to \(\dfrac{\pi}{2}\).

We expand the functions of \(x\) and \(\zeta\) in series in \(k\). The expression in braces under the integral sign takes the form

\[ \{\ \}=\frac{4}{\operatorname{sh}^{2}2x}\left(1-\frac{k^{2}}{2}\right), \]

where \(x=\dfrac{\pi \zeta}{2K}\); terms with \(k^{4}\) have been neglected.

Expanding \(r\) in \(k\) leads to the formula

\[ r=r_{0}\left(1-f_{0}\frac{k^{2}}{2}\right), \]

where \(r_{0}\) denotes \(r\) at \(k=0\), i.e.

\[ r_{0}=a\,\frac{\operatorname{sh}2x}{\operatorname{ch}2x-\cos 2y}, \]

\[ f_{0}=\frac{\cos 2y}{2}+ \frac{\cos^{2}y\left(\dfrac{1}{\operatorname{ch}^{2}x}-1-\sin 2y\right)} {1-\left(\dfrac{\cos y}{\operatorname{ch}x}\right)^{2}}. \]

The potential \(\zeta\) can be represented in the form

\[ \zeta=\zeta_{1}(x,y,0)+\zeta_{2}(x,y,0)\frac{k^{2}}{2}, \]

where \(\zeta_{1}(x,y,0)\) and \(\zeta_{2}(x,y,0)\) contain elliptic functions at \(k=0\).

As a result of substituting the expansions of the functions into (12), we obtain

\[ Q=4\Omega\left\{\iint \frac{r}{\operatorname{sh}^{2}2x} \left[\zeta_{1}+\left(2\zeta_{1}+f_{0}\zeta_{2}\right)\frac{k^{2}}{2}\right]\,dx\,dy\right\} \tag{13} \]

or

\[ Q=Q(0)+\left(\frac{\partial^{2}Q}{\partial k^{2}}\right)_{0}\frac{k^{2}}{2}. \]

The coefficients are functions of \(R_{\min}\) and \(R_{\max}\).

Thus, the total charge is an even function of \(k\), which leads to the fact that the maximum charge held by the field is obtained at \(k=0\). It follows from this that, for given \(R_{\min}\) and \(R_{\max}\), the best field will be a magnetic field with a falloff index \(n\) close to \(0.5\). In the absence of oscillatory motions the best field is the field with \(n=0.5\). But in the real case, electron oscillations may lead to resonance when the chamber is filled with charge. For this reason \(n\) may be taken to differ somewhat from \(0.5\). In ordinary betatrons, in order to avoid large stray fields, \(n\) is chosen on engineering grounds. To obtain high radiation intensities [4], the question of holding a large charge is brought to the fore; therefore fields with a large gap and with \(n\simeq 0.5\) should be created, although this is disadvantageous from the economic point of view.

References

1. N. A. Chernavskii. Izv. TPI, 87, 52, 1957.
2. R. Lagrange. Acta Math., 71, 3, 1939.
3. N. N. Lebedev. Techn. Phys. USSR, 4, 3, 1937.
4. B. N. Rodimov, N. A. Chernavskii, T. A. Medvedeva. Izv. vuzov, MVO, Fizika, 5, 6, 1959.