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Focusing of an Ion Beam and Ion Energy in a Cyclotron
A. I. Murin, Leningrad
Since the first work of Cockcroft and Walton on the artificial disintegration of nuclei by a beam of fast protons, work on developing methods for producing fast ions has proceeded continuously in numerous physics laboratories. Of all the devices created for this purpose, the most remarkable is undoubtedly the cyclotron invented by Lawrence and Livingston.
The principle of operation of the cyclotron is simple: multiple acceleration of ions moving in a magnetic field in resonance with a variable high-frequency electric field. In a cyclotron the ions pass many times through the accelerating, relatively small, potential difference—of the order of 50 kV—and thus attain an enormous energy.
Fig. 1. A and B are the dees of a cyclotron.
a—lines of force of the electric field, b—lines of the magnetic field.
Between two hollow semicylindrical electrodes—the dees, which have the shape of a box for a cake cut across—a variable potential difference is applied, the source of which is a high-frequency generator. Perpendicular to the planes of the covers of the dees, and consequently perpendicular to the electric field, a strong uniform magnetic field is produced. Fig. 1 schematically shows the distribution of the electric and magnetic fields in a cyclotron.
The ion source is located near the center of the cyclotron. The ions are accelerated in the electric field between the dees. Inside the dees, however, the electric field is equal to zero, and the ions, if one assumes in a first approximation that their trajectories lie in a plane parallel to the covers of the dees, move under the influence of the magnetic field along circular orbits.
After traversing a semicircle, the ion enters an electric field whose frequency has been chosen so that during the time of motion of the ion inside the dee the field changes its direction; more precisely, the period of the electric field is equal to the period of circular motion
of an ion; therefore the ions are repeatedly accelerated by the field between the dees (Fig. 2).
The period of revolution of an ion along the circumference is
\[ T=\frac{2\pi m}{eH}, \tag{1} \]
where \(m\) is the mass of the ion, \(e\) is the charge in electromagnetic units, and \(H\) is the strength of the magnetic field. The radius of the circumference described by the ion will be:
\[ \rho=\frac{mv}{eH}, \tag{2} \]
where \(v\) is the velocity of motion of the ion. The energy of the ion \(W\) is then equal to:
\[ W=\frac{mv^2}{2}=\frac{m}{2}\cdot\frac{4\pi^2\rho^2}{T^2}=\frac{e^2}{2m}\rho^2H^2. \tag{3} \]
As the ion’s velocity increases, it will move along circles of ever larger radius and, finally, will reach the edge of the dees. Here the ion beam can be used either inside the dee chamber itself, or, with the aid of a transverse electric field, deflected and led out through a platinum window. The energy of the ions at the exit from the chamber can be calculated from formula (3), if, instead of \(\rho\)—the orbit radius—the chamber radius is substituted. For the large cyclotron in Berkeley (USA), with a chamber radius of 60 cm and a magnetic field of 18,000 oersteds, we obtain the following limiting values for the energy of protons, deuterons, and \(\alpha\)-particles:
\[ \begin{aligned} \text{protons and } \alpha\text{-particles} &\ —\ 58\ \text{MeV},\\ \text{deuterons} &\ —\ 29\ \text{MeV}. \end{aligned} \]
Fig. 2. Trajectory of an ion in a cyclotron; the electric field is directed parallel to the \(x\)-axis. \(A\) and \(B\)—dees.
The intensity of the ion beam at the exit from the cyclotron is comparatively large. The ion current at the exit of the new Berkeley cyclotron already reaches \(90\ \mu\text{A}\), which is far from being a possible limit.
If one also takes into account that the width of the ion beam at the exit can be made less than 1 mm, it becomes entirely clear that the ion beam emerging from the cyclotron is an ideal object for a number of experiments on the splitting of atomic nuclei.
Already in their first work Lawrence and Livingston drew attention to the fact that the intensity of the ion beam at the exit constitutes a very considerable fraction of the intensity of the ion source. Lawrence and Livingston worked with a cyclotron giving protons an energy of \(1\,220\,000\ \text{eV}\). The maximum potential difference on the dees was \(4\,000\ \text{V}\), so that, in order to attain the final energy, the ions had to undergo at least three hundred accelerations. With a magnetic-field strength of \(10\,000\) oersteds, the period of revolution of the ion along the circumference [see equation (1)] was equal to
$0.7\cdot 10^{-7}$ sec.; thus, the time of stay of an ion in the cyclotron was of the order of $0.7\cdot 10^{-7}\cdot \dfrac{300}{2}=10^{-5}$ sec. The ions were formed (by bombarding hydrogen molecules with electrons) throughout the entire space between the dees. The height of the dees (the distance between the covers) was $1$ cm.
In order that, during $10^{-5}$ sec., an ion be displaced in the direction of the covers of the dees by $1$ cm, it is sufficient that the component of its velocity perpendicular to the covers be $\dfrac{1}{10^{-5}}=10^{5}$ cm/sec. The thermal velocity of a proton (at room temperature) is of the order of $3\cdot 10^{5}$ cm/sec. If one also takes into account the scattering action of the space charge, contact emf’s, and the nonuniformity of the magnetic and accelerating electric fields, then at first it seems obvious that the “diffusion” of ions toward the covers of the dees will be so strong that the beam intensity at the exit will become entirely negligible and practically simply equal to zero (the ions, it would seem, should strike the covers of the dees before reaching the exit slit). But, as was immediately pointed out by Lawrence and Livingston, as a result of the nonuniformity of the electric and magnetic fields in the cyclotron, the Lorentz force $\mathbf{F}=e(\mathbf{E}+[\mathbf{vH}])$ has a component directed toward the central plane1 of the chamber. The electric field between the dees, like an electron lens, focuses the ion beam, gathering it toward the central plane. A detailed description of this effect is given below. The magnetic field acts analogously, but its focusing action is considerably stronger. The magnetic field of all cyclotrons now in operation weakens toward the edge of the chamber. The reason for this is the saturation of the iron of the pole pieces of the cyclotron magnet. Figure 3 shows the form of the magnetic lines of force; it is seen that they are curved toward the center of the cyclotron. As is easy to see, the Lorentz force $e[\mathbf{vH}]$ in this case has a component always directed toward the central plane. The existence of this component causes the energetic focusing of the ion beam, thanks to which the attainment of high ion energies and large intensities becomes possible.
Fig. 3. Magnetic field of the cyclotron; the arrows indicate the direction of the Lorentz force
Naturally, the question of the motion of ions in the cyclotron and, in particular, the question of focusing the ion beam soon attracted the attention of numerous investigators, both theoreticians and experimentalists. It is sufficient to point to the works of Bethe, Rose, Khurgin, Thomas, and other investigators. Moreover, in the very first works (Bethe, Rose, and Khurgin) attention was drawn to the following. As indicated above, the period of revolution of an ion in
circumference must coincide with the period of the electric field, or, in other words, the motion of the ion must be in resonance with the oscillations of the electric field. As the ion velocity increases, its mass increases according to the law
\[ m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\simeq m_0\left(1+\frac{1}{2}\frac{v^2}{c^2}\right), \tag{4} \]
where \(m_0\) is the rest mass. Consequently [see equation (1)], in order that the period of revolution of the ion \(T\) be a constant quantity—and only under this condition is exact maintenance of resonance possible—the magnetic field \(H\) must not be homogeneous, but must increase with radius toward the edge of the chamber, just as the ion mass does, i.e. as \(\left(1+\frac{1}{2}\frac{v^2}{c^2}\right)\). But for focusing an ion beam, without which operation of the cyclotron is altogether impossible, the magnetic field must not increase but, on the contrary, decrease toward the periphery of the chamber1.
Thus, ideal resonance between the variation of the electric field and the motion of the ion, i.e. complete coincidence of the field period and the ion revolution period, cannot exist—only approximate equality is possible. As was shown by Bethe, Rose, and, simultaneously, by Khurgin, this leads to the fact that in any magnetic field focusing the ion beam the number of possible accelerations of the ion is limited, and, consequently, there exists a limit to the attainable ion energy.
Further work (Rose, Thomas, Schiff, Wilson) was devoted to a detailed study of the influence of magnetic fields of various types on the focusing of an ion beam and on the limiting ion energy in a cyclotron. Very interesting results were obtained, especially by Thomas, who pointed out the possibility of using axially asymmetric magnetic fields in a cyclotron; these give a peculiar focusing effect and make it possible to satisfy the resonance condition almost ideally exactly.
Let us now pass to a more detailed exposition. Since the interaction of the electric and magnetic fields in a cyclotron may be neglected—in other words, since one may neglect the magnetic fields induced by displacement and conduction currents—we shall analyze separately the action of the electric and magnetic fields.
Electric focusing is significant only in the inner region of the cyclotron, i.e. for distances from the center considerably smaller than the radius of the dees. In this region one may take \(E_y=0\) and \(\frac{\partial E_x}{\partial y}=\frac{\partial E_z}{\partial y}=0\) (the coordinate system is the same as in Figs. 2 and 3). Thus, in this region the electric-field potential \(U\) does not depend on \(y\) and is equal to
\[ U_t=U(x,z)\cos(\omega t+\theta). \tag{5} \]
FOCUSING BY AN ELECTRIC FIELD
The potential difference at the dees \(V\) varies with time \(t\) according to the same law
\[ V = V_0 \cos(\omega t + \theta). \]
We shall reckon the time \(t\) from the moment when the ion passes through the center of the accelerating region between the dees.
The electric field and its phase, and consequently also the potential difference \(V\), do not have time to change appreciably during the time in which the ion passes through the gap between the dees, and in calculating the increment of the ion energy as a result of one acceleration we may regard this difference as constant. But then the increment of the ion energy \(\Delta W\) will simply be equal to:
\[ \Delta W = eV_0[\cos(\omega t+\theta)_{t=0}] = eV_0 \cos\theta. \tag{6} \]
Here, as everywhere below, \(\theta\) is the phase of the electric field at the moment of acceleration. \(\Delta W>0\) corresponds to acceleration of the ion, which will occur for
\[ -\frac{\pi}{2}<\theta<\frac{\pi}{2}; \]
for
\[ \frac{\pi}{2}<\theta<\frac{3}{2}\pi \]
\(\Delta W<0\) and the ion is decelerated.
\(\theta=0\) corresponds to the maximum possible acceleration.
In the region where electric focusing is appreciable, i.e., near the center of the chamber, the magnetic field is very homogeneous and the Lorentz force due to it has no component directed toward the central plane. Consequently, in this region the magnetic field does not focus the ion beam. Focusing here is effected exclusively by the electric field between the dees, whose lines of force are shown in Fig. 4. We see that the accelerating region acts as an electric lens. When the ions enter the lens, they are deflected toward the central plane—the ion beam is focused; when the ions leave the lens, they are deflected away from the central plane—the beam is defocused.
Fig. 4. Motion of an ion in the gap between the dees. The dashed lines show the lines of force of the electric field.
If we neglect the change of the electric field during the ion’s passage through the lens and the change in the particle’s velocity in the process, then the focusing and defocusing mutually compensate one another. Suppose now that the electric field weakens during the time of acceleration of the ion \(\left(0<\theta<\frac{\pi}{2}\right)\); it is clear that such a field must focus the beam, since the focusing when the ions enter the lens is then stronger than the defocusing when they leave. Conversely, an increasing electric field \(\left(-\frac{\pi}{2}<\theta<0\right)\) acts defocu-
FOCUSING OF AN ION BEAM IN A CYCLOTRON
in a defocusing manner. The deflection of ions associated with the change in their velocity when passing through a lens always produces focusing, since an ion entering the lens has a velocity smaller than when leaving it and, consequently, remains in the focusing field longer than in the defocusing one.
The focusing action of the electric field associated with its variation in time must play the most significant role when the field changes rapidly, i.e., when \(\dfrac{dV}{dt}\) reaches a maximum, which will occur at
\[ \theta=\frac{\pi}{2}. \]
Focusing arising as a result of the change in the velocity of the ions while passing through the lens is, on the contrary, substantial at the maximum value of the accelerating field, i.e., at values of \(\theta\) close to zero. For negative values of \(\theta\) close to zero, focusing by the “change in ion velocity” dominates over defocusing by the “change of the field in time”; therefore, a focused ion beam is obtained in an interval of \(\theta\) somewhat greater than half the cycle of the electric field, i.e., for
\[ -a<\theta<\frac{\pi}{2}\quad \left(0<a\ll\frac{\pi}{2}\right). \]
Fig. 5. Deviation of an ion from the central plane as its energy \(W\) increases
Rose, investigating the motion of ions under repeated acceleration in a homogeneous magnetic field, came to the conclusion that for positive \(\theta\) we are dealing with oscillations about the central plane with slowly increasing amplitude—the amplitude grows proportionally to the fourth root of the ion energy (Fig. 5)\(^{1}\). For negative \(\theta\) sufficiently far from zero, the ion motion loses its oscillatory character and \(z\) increases exponentially with the number of revolutions, i.e., we have energy defocusing of the ion beam.
FOCUSING BY A MAGNETIC FIELD
For ions that have undergone a considerable number of accelerations, focusing by the electric field should no longer play a significant role, since the relative change in velocity in a single acceleration becomes small, and during the time the ions pass through the lens (the region between the dees) the electric field does not have time to change substantially in time.
\(^{1}\) In this case
\[ z \simeq A\left(\frac{W}{\sin\theta}\right)^{\frac14} \cdot \sin\left[ \sqrt{\frac{\pi}{2}} \int_{0}^{n} dn \left(\frac{eV_{0}\sin\theta}{W}\right)^{\frac12} \right], \]
where \(A\) is a constant, and \(n\) is the number of accelerations.
A. N. MURIN
The situation is different with magnetic focusing. First of all, the ion is subjected to the action of the magnetic field throughout its entire path, and not only in the interval between the duants, as is the case for the electric field. The focusing action of the magnetic field is therefore not a differential effect, i.e., not the difference between focusing and defocusing. In addition, the Lorentz force increases with the velocity of the particle. If cylindrical coordinates are introduced with the axis \(OZ\) directed along the axis of the cyclotron, then an estimate of the order of magnitude of the ion deflection in the direction of the \(Z\) axis in one revolution gives:
\[ \Delta z \sim \frac{evH_r}{m}\cdot \frac{4\pi^2 r^2}{2v^2} \cong \frac{eH_r}{m}\frac{\pi r}{\nu}, \tag{7} \]
where \(H_r\) is the component of the magnetic field directed toward (or away from) the \(OZ\) axis, and \(\nu\) is the frequency of oscillation of the electric field. \(H_r\) usually increases toward the edge of the magnet; consequently, as the ions move away from the center of the chamber, i.e., as \(r\) increases, magnetic focusing should play an ever greater role.
In a cyclotron the pole pieces of the magnet are always arranged symmetrically with respect to the central plane and axially symmetrically with respect to the \(OZ\) axis. The magnetic-field intensity therefore satisfies the following relations:
\[ \begin{aligned} H_z(r,z)&=H_z(r,-z),\\ H_r(r,z)&=-H_r(r,-z),\\ H_\varphi&=0,\\ \operatorname{rot}\mathbf{H}&=0. \end{aligned} \tag{8} \]
For the condition of resonance to be fully satisfied, taking into account the relativistic change of the ion mass, the magnetic field must increase from the cyclotron axis \(OZ\) toward the edge of the chamber; but for a magnetic field increasing with increasing \(r\) and satisfying equalities (8), the field lines are deflected as shown in Fig. 6. Ions whose orbits do not lie in the central plane will be deflected away from it, and the ion beam will be defocused; this effect is proportional to the relative change of the magnetic field and, thus, proportional to
\[ \frac{v^2}{c^2}. \]
[See equations (1) and (4).]
Fig. 6. Defocusing magnetic field of an ion beam; the arrows indicate the direction of the Lorentz force
For focusing of the ion beam the magnetic field must decrease with radius. In this case the magnetic field lines are bent toward the center of the cyclotron and the Lorentz force has a component directed toward the central plane (Fig. 3). But in the case of such a field the time of revolution of the ion around the axis, as is seen from equation (1)
(H should be replaced by \(H_z\)), increases with increasing velocity (\(m\) increases, while \(H\) decreases toward the edge of the chamber). Thus, the resonance condition between the oscillations of the electric field and the rotation of the ion can be satisfied only approximately, and the phase \(\theta\) will change from acceleration to acceleration. After a certain number of accelerations the change in \(\theta\) may become so considerable that the ion will fall not into the accelerating field, but into the decelerating field.
Let us emphasize once more that, as both experiment and theory show, without magnetic focusing it is impossible to obtain a beam of ions at the exit from the cyclotron. The impossibility of simultaneously satisfying resonance and focusing compels us to make a certain compromise, choosing such a magnetic field as would focus the beam sufficiently well and at the same time would not change the phase of the electric field too strongly from one acceleration to the next.
Fig. 7
\(H\) — effective field; \(H_{\text{res}}\) — field giving ideal resonance
The maximum total change of phase \(\theta\) during all accelerations that we may allow ourselves, if we do not want the ion to fall into the decelerating field, is equal to \(2\pi\), and not \(\pi\), as might appear at first glance. The point is that one can make the magnetic field at the center of the chamber greater than the resonance field, and at the edge smaller (Fig. 7); then at the beginning of its motion the ion will outrun the change of the electric field—\(\theta\) will decrease from acceleration to acceleration, and after the ion’s trajectory crosses the resonance field, the ion will lag behind the field—\(\theta\) will increase. Under this condition the phase may change from \(\frac{\pi}{2}\) to \(-\frac{\pi}{2}\) in the first and from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in the second part of the path, without the ion falling into the decelerating field. This gives a total phase change of \(2\pi\). Fig. 8 schematically explains this reasoning: \(t_1, t_2, t_3\), etc., are the moments of successive accelerations of the ion. \(V = V_A - V_B\); at the times \(t_1, t_3, t_5, t_7, t_9\) the ion moves from deuteron \(A\) to deuteron \(B\), and at the times \(t_2, t_4, t_6, t_8, t_{10}\) from \(B\) to \(A\). At the moment \(t_6 = t_{\text{res}}\) the ion’s trajectory crosses the resonance field. It goes without saying that the true number of accelerations is many times larger, and \(\theta\) changes from acceleration to acceleration much more slowly.
The greater the focusing of the ion beam that we wish to achieve, the sharper must be the decrease of the magnetic field toward the edge of the cyclotron, the faster \(\theta\) will change, the smaller will be the number of revolutions at our disposal, and the smaller limiting ion energies we shall be able to obtain at the exit. If we want to obtain a high intensity of the emerging beam, then we cannot confine ourselves to ions
with an initial phase $\theta$ equal or very close to $\dfrac{\pi}{2}$, i.e., to use the ion source during an insignificant fraction of the full cycle of the electric field. If, however, we wish to use all ions with initial phases $\theta_0$ such that $\theta'_0 - \theta_0 < \dfrac{\pi}{2}$, then the total change of phase of the ion must be no greater than $\pi + 2\theta'_0$ (Fig. 8, b), i.e., the number of possible accelerations, and correspondingly the final energy, become smaller. In short, if we want to obtain better focusing and greater intensity, we have to sacrifice the energy of the ions and, conversely, in striving for the greatest possible energies, we lose in intensity.
Fig. 8. The figure shows how the phase $\theta$ changes from one acceleration to the next.
In Fig. 8, b—the initial phases of the ions (see the points on segment 1) lie between $\theta'_0$ and $\dfrac{\pi}{2}$. At the instants of time represented by the points on segment 5, the ion trajectories cross the resonance field. From a comparison of Fig. 8, a and b, it is clear that in case b the number of permissible accelerations has become smaller. The figure is entirely schematic.
Let us illustrate all these arguments on the simple example of a uniform magnetic field1; in doing so we shall follow Rose’s work.
Such a magnetic field does not focus the ion beam. Focusing and defocusing are effected exclusively by the electric field; consequently, we must avoid phases $\theta$ close to $-\dfrac{\pi}{2}$, since then the electric field defocuses the beam (see above). If we require that the phase of the electric field $\theta$ remain positive, i.e., that the electric field decrease while the ion is passing between the duants, then we shall have focusing, though admittedly rather weak, since the amplitude of the ions’ oscillations along the $Z$ axis will nevertheless increase slowly, as $W^{1/4}$.
The condition \(\theta>0\) allows us to dispose of the total change of phase \(\theta\) by a maximum of \(\pi\). With the aid of Fig. 8 this case can also be explained—we need only “discard” the part of the drawing lying between the straight lines \(MM\) and \(NN\). In the first part of the path, before the ion trajectory intersects the resonance field, \(\theta\) decreases from \(\frac{\pi}{2}\) to zero; in the second part it increases from zero to \(\frac{\pi}{2}\); \(t_4=t_8=t_{\mathrm{res}}\). If, while satisfying the condition \(\theta>0\), we wish to use all ions with an initial phase between \(\theta'_0\) and \(\frac{\pi}{2}\) \(\bigl(0<\theta'_0<\frac{\pi}{2}\bigr)\), then the total change of phase must be no greater than \(2\theta'_0\); but, of course, there is no point in making this change smaller, since the latter would mean that we are not using the full number of all possible accelerations and thereby are not attaining the maximum possible ion energy. As is easy to see, in this case the initial phase \(\theta_0\) and the final phase (at the exit) \(\theta_1\) are equal, i.e. \(\theta_0=\theta_1\) (for any of the ions used).
Indeed, in the first “pre-resonance” part of the path the phase decreases by \(\theta'_0\), and in the second it increases by the very same amount. We shall use the condition \(\theta_0=\theta_1\) in the further calculations.
Let us denote the number of accelerations which the ion has undergone up to some instant of time by \(n-1\). Then the change of the ion phase \(\Delta\theta\) during the \(n\)-th half-turn is equal to:
\[ \Delta \theta = \frac{d\theta}{dn} = \pi \left( \frac{H_{\mathrm{res}}}{H_z} - 1 \right) \tag{9} \]
(\(H_{\mathrm{res}}\) is the resonance field, different at different distances from the center because of the relativistic dependence of mass on velocity).
Let us represent \(H_z\) in the form:
\[ H_z = H_0(1+h), \tag{10} \]
where \(H_0\) is the resonance field at the center of the cyclotron, and \(h>0\), since it follows from our arguments that, in order to obtain ions of the greatest possible energy, the field at the center must be greater than the resonance field.
The change in the ion energy \(\Delta W\) in the \(n\)-th acceleration will be:
\[ \Delta W=\frac{dW}{dn}=eV_0^{*}\cos\theta . \tag{11} \]
\(V_0^{*}\) is the amplitude of the oscillations of the potential difference on the dees. From (9) and (11) we have:
\[ \frac{d\sin\theta}{dW} = \frac{\pi}{eV_0} \left( \frac{H_{\mathrm{res}}}{H_z}-1 \right), \tag{12} \]
but since \(H_{\mathrm{res}}\) must increase toward the edge of the cyclotron like the ion mass, i.e. as
\[ \left(1+\frac{1}{2}\frac{v^2}{c^2}\right) = \left(1+\frac{1}{2}\frac{m_0v^2}{m_0c^2}\right) = 1+\frac{W}{m_0c^2}, \]
then
\[ H_{\mathrm{res}}=H_0\left(1+\frac{W}{m_0c^2}\right). \tag{13} \]
Using equations (10) and (13), instead of (12) we obtain
\[ \frac{d\sin\theta}{dW}=\frac{\pi}{eV_0}\left(\frac{W}{m_0c^2}-h\right). \tag{14} \]
Integrating equation (14), we have:
\[ \sin\theta=\frac{\pi}{eV_0}\left(\frac{W^2}{2m_0c^2}-hW\right)+\sin\theta_0; \tag{15} \]
here \(\theta_0\) is the initial phase.
Equating \(\dfrac{d\sin\theta}{dW}\) to zero, we find that \(\sin\theta\) reaches a minimum at
\[ W_1=hm_0c^2 \tag{16} \]
at the same point where the magnetic field is equal to the resonance field. According to equation (15),
\[ \sin\theta_{\min}=\frac{\pi}{eV_0}\left(\frac{W_1^2}{2m_0c^2}-hW_1\right)+\sin\theta_0. \tag{17} \]
For given \(h\) and \(V_0\), \(\sin\theta_{\min}\) is equal to zero for some value \(\theta'_0\) of the initial phase \(\theta_0\), i.e.
\[ 0=\frac{\pi}{eV_0}\left(\frac{W_1^2}{2m_0c^2}-hW_1\right)+\sin\theta'_0 . \tag{18} \]
Wishing to obtain a focused beam of ions, we required that \(\theta\) and, consequently, \(\sin\theta\) be greater than zero; therefore, for us all ions with an initial phase between \(-\dfrac{\pi}{2}\) and \(\theta'_0\) are “suitable.” Not wishing to lose either in energy or in intensity, we, as stated above, must set the final phase \(\theta_1\) equal to the initial phase \(\theta_0\). Then from the conditions \(\theta_1=\theta_0\) and equation (15), written for the end of the path,
\[ \sin\theta_1=\frac{\pi}{eV_0}\left(\frac{W_{кр}^2}{2m_0c^2}-hW_{кр}\right)+\sin\theta'_0, \tag{19} \]
where \(W_{кр}\) is the final energy of the ions, we obtain:
\[ \frac{W_{кр}^2}{2m_0c^2}-hW_{кр}=0, \tag{20} \]
i.e.
\[ W_{кр}=2m_0c^2h. \tag{21} \]
Comparing (21) and (16), we see that
\[ W_{кр}=2W_1. \tag{22} \]
Thus (18) can be rewritten as follows:
\[ 0=\frac{\pi}{eV_0}\left(\frac{W_{кр}^2}{8m_0c^2}-\frac{hW_{кр}}{2}\right)+\sin\theta'_0. \tag{23} \]
From (21) and (23) one can determine both \(W_{кр}\) and \(h\):
\[ W_{кр}=4\cdot\sqrt{\frac{1}{2\pi}\,eV_0m_0c^2\sin\theta'_0}, \tag{24} \]
\[ h=\sqrt{\frac{2eV_0\sin\theta'_0}{\pi m_0c^2}}. \tag{25} \]
For \(V_0=50\ \mathrm{kV}\) and \(\theta'_0=-\frac{\pi}{2}\), i.e., for “zero” intensity, these equalities give \(10.9\ \mathrm{MeV}\) for protons, \(15.4\ \mathrm{MeV}\) for deuterons, and \(30.8\ \mathrm{MeV}\) for \(\alpha\)-particles. For \(\sin\theta'_0=0.9\) these energies must be reduced by \(5\%\). In this case \(\theta'_0=64^\circ\), and we use the ion source for \(30\%\) of the total time of its operation.
\(h\) turns out to be equal to \(0.55\cdot 10^{-2}\) for protons, and \(0.39\cdot 10^{-2}\) for deuterons and \(\alpha\)-particles. We see that the magnetic field must exceed the resonant value at the center by only a fraction of a percent. It should be noted that the resulting energy is extremely sensitive to the exact value of \(h\). Suffice it to say that for \(h=0\) the final energy is reduced by half compared with that indicated. Let us also note that if, in solving our problem, we had sacrificed focusing, i.e., had considered also values of \(\theta\) between zero and \(-\frac{\pi}{2}\) admissible, then for the limiting possible energies in a homogeneous magnetic field we would have obtained a value \(\sqrt{2}\) times larger.
This calculation would have to be carried out in exactly the same way as above; only the smallest value of \(\sin\theta\) should be set equal to \(-1\), and instead of equation (18) one should write:
\[ -1=\frac{\pi}{eV_0}\left(\frac{W_1^2}{2m_0c^2}-hW_1\right)+\sin\theta'_0 . \tag{26} \]
Then for the ion energy in a homogeneous magnetic field we would obtain:
\[ W_{\mathrm{cr}}=4\sqrt{\frac{1}{2\pi}\,eV_0m_0c^2\,(1+\sin\theta'_0)}, \tag{27} \]
\[ h=\sqrt{\frac{2eV_0(1+\sin\theta'_0)}{\pi m_0c^2}} . \tag{28} \]
These formulas, in a somewhat different form, were first obtained by Khurgin.
For a magnetic field that provides focusing and, consequently, permits a complete variation of \(\theta\) up to \(2\pi\) (for example, for a field decreasing quadratically with distance from the center so as to compensate the defocusing by the electric field near \(\theta=-\frac{\pi}{2}\)), we obtain values close to those calculated from formula (27), i.e., in the limit, about \(15.5\ \mathrm{MeV}\) for protons, \(22\ \mathrm{MeV}\) for deuterons, and \(43\ \mathrm{MeV}\) for \(\alpha\)-particles. We omit the detailed calculation and rigorous justification of this result.
It must be noted that the limiting ion energy increases with the voltage on the dees as \(\sqrt{V_0}\); therefore the development of the cyclotron is proceeding in the direction of increasing this voltage; in particular, the voltage on the dees of the Berkeley cyclotron reaches \(200\ \mathrm{kV}\). Let us note, in conclusion, that in the outer part of the ion orbit, where magnetic focusing acts, the amplitude of the ion oscillations along the \(Z\)-axis is proportional to \(-\frac{d\lg H}{d\lg r}\), i.e., the focusing effect depends on
of the relative change of the magnetic field with radius and, if the gradient of the magnetic field falls off sufficiently rapidly toward the edge of the dees, as is usually the case, the height of the ion beam will decrease in the peripheral region of the cyclotron chamber.
FOCUSING ACCORDING TO THOMAS
Until now we have assumed that the intensity of the magnetic field is a function only of \(r\) and \(z\)1 and does not depend on the azimuthal angle \(\varphi\). In this case magnetic focusing is possible only when the field weakens toward the edge of the cyclotron, and we have no possibility of satisfying the resonance condition over the entire length of the ion path. But, as Thomas first showed, by varying the fields with azimuth we can create a force that focuses the ion beam and, with an appropriately chosen angular and radial variation of the field, achieve both focusing and an almost ideally exact fulfillment of resonance, i.e. equality of the frequency of rotation of the ion about the cyclotron axis to the frequency of the alternating electric field.
Fig. 9. \(H_z\) is directed toward the reader
Let us first consider the path of a charged particle moving with constant velocity in a magnetic field that varies as a function of the azimuthal angle \(\varphi\) and does not depend on \(r\). In the case under consideration we no longer require axial symmetry of the field with respect to the axis \(OZ\), restricting ourselves to the conditions of symmetry of the magnetic lines with respect to the central plane. The intensity of this field satisfies the following equations:
\[ \left. \begin{aligned} H_z(\varphi,z) &= H_z(\varphi,-z),\\ H_\varphi(\varphi,z) &= -H_\varphi(\varphi,-z),\\ H_r &= 0,\\ \operatorname{rot}\mathbf H &= 0. \end{aligned} \right\} \tag{29} \]
Let us consider the motion of an ion in such a field. We shall measure the angle \(\varphi\) in the direction of rotation of the ion. In Fig. 9 a closed orbit of the ion lying in the central plane is shown. The trajectory of the particle must have greater curvature where the field is stronger, and if the ion path is a closed curve, then at this point the ion must move farther away from the center than at those points of the field where it is weaker. In Fig. 9 near point \(A\) the field weakens with increasing azimuthal angle \(\varphi\). The magnetic-field lines are turned by their convexity toward decreasing magnetic field; consequently, near \(A\), in the direction of increasing \(\varphi\), i.e. \(H_\varphi<0\) above the central plane and \(H_\varphi>0\) below it. Let us now suppose that
the path of the ion does not lie in the central plane, and in Fig. 9 the projection of the ion path onto the central plane is shown. For definiteness we shall assume that at point \(A\) the ion is moving above the central plane, i.e. at point \(A\), \(H_\varphi<0\). Then, referring to the figure, we see that the Lorentz force has a component directed toward the central plane,
\[ F_z=e[\mathbf{v}H_\varphi]^{1}), \]
and, consequently, the ion beam is focused at this point. As is easy to see, focusing also occurs where the field increases with the angle. This focusing effect is proportional to the product of the relative change in the distance from the center and the relative change in the magnetic field and, consequently, is ultimately proportional to the square of the relative change in the magnetic field with azimuthal angle.
Thus, without changing the magnetic field along the radius, we can, by means of Thomas focusing, obtain a focused ion beam. We have seen that the field which gives resonance increases along the radius and gives a defocusing effect proportional to \(\frac{v^2}{c^2}\). It is natural to try to compensate this defocusing by varying the magnetic field with azimuth so that its relative change (say, over an angle of one radian) is proportional to \(\frac{v}{c}\). Thomas, and after him Schiff, succeeded in indicating such fields, which give focusing and resonance in any case up to such ion velocities that, in the calculation, quantities of order \(\frac{v^3}{c^3}\) may be neglected in comparison with \(\frac{v}{c}\) and \(\frac{v^2}{c^2}\). We shall give here the formula for these fields, without presenting all the calculations by which they were obtained. These are fields of the form:
\[ H=-\frac{m_0\omega c}{e}\left[1+A\frac{\omega r}{c}\cos n\varphi+B\left(\frac{\omega r}{c}\right)^2\right] \tag{30} \]
\[ (n=3\ \text{or}\ 4), \]
where \(e\) is the ion charge in electrical units, and \(\omega\) is the cyclic frequency of rotation of the ion; \(A\) and \(B\) are constants which must satisfy the following resonance conditions:
for \(n=3\)
\[ B=\frac{1}{2}-\frac{A^2}{8}, \tag{31} \]
for \(n=4\)
\[ B=\frac{1}{2}-\frac{A^2}{15}. \tag{32} \]
To obtain fields of the form (30), the magnetic pole pieces must have a wavy surface\(^{2}\), like the pole pieces shown in Fig. 10.
\(^{1}\) In calculating the focusing force one may assume that the ion velocity is parallel to the central plane.
\(^{2}\) Of course, to obtain such a surface one must use “shims” (see below).
For \(A>A_0\), where
\[ A_0=\sqrt{\frac{4}{3}} \tag{33} \]
for \(n=3\), and
\[ A_0=\sqrt{\frac{30}{19}} \tag{34} \]
for \(n=4\), the motion of the ion along the \(z\)-axis is oscillatory in character. For \(A<A_0\), the distance of the ion from the central plane increases exponentially. The first case gives focusing, the second defocusing. The focusing will be the better the larger \(A\) is and, consequently, the smaller \(B\) is. As a measure of focusing we may take the number of revolutions with mean radius \(a\), described during one complete oscillation along the \(z\)-axis. This number will be the smaller the stronger the focusing. When the resonance condition is fulfilled, the number of revolutions is equal to
\[ \frac{c}{a\omega}\left[\left(\frac{A}{A_0}\right)^2-1\right]^{-1/2} \]
both for the case \(n=3\) and for the case \(n=4\).
Fig. 10. Magnet pole tips with a wavy surface
In formula (30), attention is drawn to the fact that the period of variation of the magnetic field in azimuth here is equal to \(\frac{2\pi}{n}\), where \(n=3\) or \(4\).
Fig. 11. Trajectory of an ion in a magnetic field of the form described by equation (35) \((\mathbf{v}\perp\mathbf{H})\)
This circumstance is connected with the question of the stability of ion orbits. For simplicity of reasoning let us for the time being forget the accelerating electric field and suppose that the magnetic field has the form
\[ H=\frac{m_0\omega c}{e}\left[1+A\frac{\omega r}{c}\cos\varphi\right]. \tag{35} \]
The term proportional to \(\cos\varphi\) increases the curvature of the trajectory where \(\cos\varphi>0\), and decreases it where \(\cos\varphi<0\). As a result, the orbit will have the form shown in Fig. 11, and we see that the “center” of the orbit shifts in the direction of the \(X\)-axis.
If one takes into account the term \(B\left(\frac{\omega r}{c}\right)^2\) in (30), which depends on the radius \(r\), and the accelerating electric field of the cyclotron, then the form of the orbit will, of course, change, but the main feature of the phenomenon under consideration—the continuous drift of the center, increasing with the ion velocity—will remain.
In a magnetic field of the form
\[ H=\frac{m_0\omega c}{e}\left(1+A\frac{\omega r}{c}\cos 2\varphi\right) \tag{36} \]
there exists a family of closed orbits whose center coincides with the center of the field, but these orbits are not stable, i.e., if some small perturbation displaces the orbit so that its center no longer coincides with the center of the field, then the center of the orbit will begin to shift in a manner analogous to what occurred in the case of a field containing a term proportional to \(\cos\varphi\). The introduction of an electric field and of a part of the magnetic field depending on the radius again does not change the essence of the phenomenon.
It is clear that the part of the magnetic field proportional to \(\cos\varphi\) and \(\cos 2\varphi\) disturbs the regularity of the ion’s motion through the cyclotron, causing particles to reach the outer part of the chamber along paths of different radius and with different energies. If sufficiently large, these terms so detune the resonance between the ion’s period of rotation and the variation of the electric field that the particles never attain large energies. Components of the magnetic field proportional to \(\cos n\varphi\), where \(n=3\) or \(4\), or, more generally, fields with period \(\frac{2\pi}{3}\), \(\frac{2\pi}{4}\), give stable orbits, i.e., a small accidental perturbation of the orbit does not cause slipping of the center of the orbit, but only makes it perform small oscillations about the center of the chamber.
Thus, the fields indicated by Thomas and Schiff give stable, focused orbits that are in resonance with the electric field.
We see that the form of the cyclotron’s magnetic field has a significant influence on the ion beam passing through it. Even before this question was investigated by the theorists Bethe, Rose, Thomas, Khurgin, and others, experimenters, in the person of Lawrence, discovered the extraordinary sensitivity of the ion beam to any, even insignificant, change in the magnetic field. Already in his first works Lawrence used “shims,” i.e., iron spacers between the pole pieces of the magnet and the covers of the dees. These “shims” change the form of the magnetic field in the chamber; by using them, it is possible to increase considerably the energy and intensity of the ion beam emerging from the cyclotron. The dimensions, shape, and arrangement of these shims are usually selected purely experimentally. In this area, much unquestionably remains to be done both for the theorist and for the experimenter. In particular, the experimental realization of the magnetic fields indicated by Schiff and Thomas is extremely difficult and even scarcely feasible, but, while retaining Thomas’s basic idea—the variation of the magnetic field with the azimuthal angle—it is possible to attempt to find magnetic fields that are experimentally easier to realize and that, as in the Thomas–Schiff case, provide focusing, resonance, and stability.
For Soviet physicists, the problems connected with the operation of the cyclotron become all the more urgent since at the present time
in addition to the operating cyclotron of the Radium Institute of the Academy of Sciences of the USSR, a powerful cyclotron of the Physico-Technical Institute of the Academy of Sciences of the USSR is being built, and cyclotrons are planned for construction by the Ukrainian Academy of Sciences and the Physics Institute of the Academy of Sciences of the USSR in Moscow.
References
- Ya. L. Khurgin, DAN SSSR, 19, 237, 1938.
- M. E. Rose, Phys. Rev., 53, 392, 1938.
- L. I. Schiff, Phys. Rev., 54, 1114, 1938.
- L. H. Thomas, Phys. Rev., 54, 580, 1938.
- R. R. Wilson, Phys. Rev., 53, 408, 1938.
- Bethe, Rose, Phys. Rev., 52, 1255, 1937.