SEPARATION OF ISOTOPES IN LARGE QUANTITIES BY ELECTROMAGNETIC METHODS*

L. Smith, W. Parkins, and A. Forrester

INTRODUCTION

Among the various methods used for separating isotopes in small quantities, particular attention has always been attracted by methods that make use of the action exerted on ions by suitable combinations of static or time-dependent electric and magnetic fields. This is due chiefly to the fact that, since only very small quantities are involved, such methods are suitable for separating the isotopes of almost all elements.

The application of this method to obtaining large quantities of the components being separated, while at the same time retaining the advantage just mentioned, encounters two principal difficulties. The first consists in the necessity of producing large ion currents carrying large quantities of substance. However, such great successes have been achieved in the construction of ion sources that the limitations connected with the impossibility of producing a large number of ions have at present completely disappeared. We shall therefore not consider the general principles of the design of ion sources, but shall confine ourselves only to those sources that are used in the separators described below. Of course, the possibility is not excluded that in the future sources will again impose a limitation on the quantity of substance separated, owing to the velocities and spatial distribution of the ions; however, this seems unlikely.

The second and main problem is the preservation of sufficient resolving power in the presence of the enormous space charge associated with large ion currents. It may be said that the success of applying electromagnetic methods to isotope separation on a large scale depends on finding ways to eliminate the action of the forces caused by the large space charge, or to prevent these large fo—

* Lloyd P. Smith, W. E. Parkins and A. T. Forrester, Phys. Rev. 72, 989 (1947). Translated by P. E. Kunin and I. M. Taksar.

distort the action of the electromagnetic forces used for separation.

Below are presented the analysis and experimental tests of the corresponding methods, carried out by the authors over the past several years and leading to the creation of an effective radial magnetic separator.

THE ORDINARY MAGNETIC MASS SPECTROGRAPH AND THE PROBLEM OF SPACE CHARGE

When an ordinary mass spectrograph, for example of the Dempster type, is used for separating large quantities of isotopes, a serious limitation arises with respect to the rate of their separation. It is caused by the presence of large forces produced by the space charge, if this space charge is not neutralized in some way. An estimate of this limitation may be obtained as follows. Consider, for example, a beam of ions in a semicircular Dempster mass spectrograph. The envelopes of the trajectories of the ions of both isotopes are shown in Fig. 1. The length of the beam in the direction perpendicular to the plane of the drawing is assumed to be very large. Let the change in curvature of the outermost ion trajectory be small in comparison with the change in position relative to the center of the beam. Then, owing to the presence of space charge, on ions moving along the outer trajectories from the inside there will act a force whose magnitude is found by a simple application of Gauss’s theorem. It is equal to

Fig. 1. Ion beams in a Dempster-type mass spectrograph.

\[ F = 2\pi \frac{Ie}{v}, \tag{1} \]

where \(I\) is the total current in the beam per unit length perpendicular to the plane of the drawing, and \(v\) is the mean velocity of the ions. It is obvious that the force acting on the outer ions does not depend on the width of the beam. Consequently, this outward-directed force will be constant along the entire length of the beam. The trajectories of the particles bounding the beam will therefore be circles of radii \(r_1\) and \(r_2\), which are determined from the relation

\[ \frac{Mv^2}{r_{1,2}} = Hev \pm \frac{2\pi Ie}{v}. \tag{2} \]

From this we obtain the width of the beam at an angle of \(180^\circ\)

\[ \Delta S = 2(r_2-r_1)=\frac{8\pi e}{M}\cdot \frac{I a^2}{v^3}, \tag{3} \]

where \(a\) is the unperturbed radius, i.e., the radius in the absence of space charges.

Even in the absence of a space charge, beams consisting of two different isotopes are focused in \(B\) at points very close to one another. Therefore one may assume that both beams nearly coincide along the entire path and that the outer ions in both beams are acted upon by one and the same force, due to the presence of space charges. Thus, both beams have one and the same width \(\Delta S\) near the point \(B\).

Neglecting the space charge, the distance between the points at which the beams of both isotopes are focused in \(B\) is equal to

\[ \Delta x=\frac{|M_2-M_1|}{M_1}\,a. \tag{4} \]

The maximum rate of deposition of completely separated isotopes is attained at such an ion current for which the width of each beam \(\Delta S\) is equal to the displacement \(\Delta x\). If the broadening of the beam is greater than the displacement, i.e. \(\Delta S>\Delta x\), then in \(B\) the area covered by each pure isotope per unit depth of the beam will be \(\Delta x\), and the rate of deposition of pure isotope per unit depth of the beam is proportional to \(\dfrac{\Delta x}{\Delta S} I\). But according to equation (3), \(\Delta S\) itself is proportional to \(I\). Thus, at currents greater than that corresponding to \(\Delta S=\Delta x\), nothing is gained. The magnitude of this limiting current \(I\) is obtained from expressions (3) and (4); it is equal to

\[ I=\frac{M_1}{8\pi e}\cdot \frac{v^3}{a}\cdot \frac{|M_2-M_1|}{M_1}. \]

Using the relation

\[ \frac{M_1 v^2}{a}=\frac{HeV}{c}, \]

which determines the radius of the mean ion trajectory, and the relation

\[ v^2=\frac{2eV}{M_1}, \]

we obtain the final expression for the limiting current:

\[ I=\frac{1}{4\pi}\cdot \frac{|M_2-M_1|}{M_1}\cdot \frac{e}{M_1 c}\,VH. \]

The mass \(\mathfrak{M}\) deposited per unit time is equal to

\[ \mathfrak{M}=\frac{\eta}{4\pi}\cdot \frac{|M_2-M_1|}{M_1}\cdot \frac{VH}{c}\ \text{g/sec}, \]

where \(\eta\) is the relative amount of the isotope under consideration. If \(V\) is expressed in volts and \(H\) in gauss, the number of milligrams of isotope that can be separated per hour is equal to

\[ \mathfrak{M}\ \text{mg/hour}=3.19\cdot 10^{-8}\eta\,\frac{|A_2-A_1|}{A_1}\,V\ (\text{volts})\cdot H\ (\text{gauss}), \tag{5} \]

where \(A_1\) and \(A_2\) are the mass numbers of the isotopes. In practical units, the limiting current expressed in terms of the mass numbers will be

\[ I(ma/cm)=0.855\cdot 10^{-6}\eta\,\frac{|A_2-A_1|}{A_1^2}\,VH. \tag{6} \]

From this relation it is immediately clear that the space charge imposes a severe limitation on the rate of separation even at high voltages and strong fields. For example, the rate of separation of Ca-44 from Ca-40 (a very favorable case, where \(\eta=1/20\)) is only \(12\ m^2\) per hour per centimeter even in the case when \(V=50\,000\ v\), and \(H=15\,000\) gauss. The total ion current in such a beam is \(1.6\ ma\) per centimeter. Even without touching upon the substantial problems associated with the use of such high voltages, one must conclude that this method is not efficient unless at least part of the space charge is neutralized.

When such high voltages are used, the accelerating electrode system must be such as to prevent the possibility of defocusing or overfocusing of the beam when it enters the region where separation takes place. It is also unclear whether ions of such high energy can be successfully collected on catching electrodes, owing to heating and sputtering of the latter.

NEUTRALIZATION OF THE SPACE CHARGE OF THE BEAM BY ELECTRONS

Because of the great mobility of electrons compared with ions, at first sight it might seem that neutralization of the large space charge of an intense ion beam is difficult to achieve. However, an ion beam in which the ions are not accelerated by an external electric field is capable of capturing low-energy electrons, since the latter begin to move along the axis of the beam. This is due to the fact that in an unneutralized ion beam the electric field is directed away from the beam axis and, consequently, the force acting on an electron is always directed toward the center of the beam. It can be said that the ion beam forms a “trough” of space charge, and slow electrons, having entered it, can no longer move far away from the axis. They can, however, move along the beam axis, but by means of an appropriate electrode arrangement one can obtain such a distribution of potentials at the ends of the ray that it will prevent slow electrons from leaving the beam. The distribution of potentials in an ion beam is shown in Fig. 2. If electrons can be captured efficiently, then ultimately such a number of them is collected that the space charge almost disappears.

In the presence of a magnetic field, for example in a Dempster-type mass spectrograph, electrons may be even more effective. They

migrate within the beam even in the presence of a strong magnetic field. Owing to the force acting on the electron in the direction toward the center of the beam, an electron initially at rest in the beam will move along small cycloid-like trajectories, as shown in Fig. 3. As a result of collisions accompanied by losses of kinetic energy, it will approach the center of the beam.

Fig. 2. Distribution of the potential over the entire cross section, the plane of which passes through the axis of the beam.

If a small retarding force is applied at the end of the beam, the electron will pass to the other side of the beam and begin to move in the opposite direction, as shown in the figure. Owing to this, the electrons may ultimately reach any part of the beam.

Fig. 3. Possible trajectories of electrons moving in an ion beam in an ordinary mass spectrograph, illustrating the capture of slow electrons.

The mean velocity of motion of an electron along the beam will be of the order of \(v_e = \dfrac{E}{H}c\), where \(E\) is the electric field directed perpendicular to the beam. In an unneutralized beam \(E\) may be of the order of a thousand volts per centimeter, and the migration of electrons takes place very rapidly. But even if it is assumed that the ion density is small, or that the beam has already been partially neutralized, and if \(E\) is taken to be only \(10\ \mathrm{V/cm}\), then in a field of 2000 gauss

\[ v_e \simeq 5 \cdot 10^5\ \mathrm{cm/sec}. \]

An ion with mass number 200 will have a velocity of the order of

\[ v_i \simeq 10^5 \sqrt{V}\ \text{(volts)}, \]

where \(V\) is the ion energy in electron-volts. It may be, for example, of the order of \(3600\) V. Then

\[ v_i \approx 6 \cdot 10^6\ \text{cm/sec}, \]

which is 10 times greater than the electron migration velocity. Consequently, such electrons can effectively neutralize the space charge. They can be introduced into the beam by ionizing gas atoms or by means of appropriately placed electron sources.

MODULATION METHODS

AND THE PROBLEM OF SPACE CHARGE

The influence of space charge is just as strong in methods of effective isotope separation that use electric fields varying in time. In these methods one usually seeks, either by modulating the velocity of a homogeneous ion beam or by periodically interrupting the beam, to use differences in velocities in order to achieve separation. In these cases the presence of space charge creates two difficulties. The first is connected with the rapid spreading of the beam’s space charge; the second, with the effect of shielding of the modulating electrodes by the surrounding ions.

The spreading of a beam having the form of a sheet of infinite width is essentially described by the expression already used above. If \(d\) is the distance along the beam, measured from the point at which the ions were moving along parallel trajectories, then for the spreading we have

\[ \frac{z}{z_0} = 1 + 2.04 \cdot 10^6 \frac{d^2 I}{z_0}\frac{A}{V^3}, \tag{7} \]

where \(z_0\) is the initial thickness of the beam, \(I\) is the current in amperes per unit width, \(V\) is the voltage, and \(A\) is the atomic weight of the ion.

If the beam has a cylindrical cross section, then the spreading will be given by a more complicated function of the distance \(d\) along the beam axis. This expression was obtained by various authors\(^1\) and may be written in the following form:

\[ d = 1.75 \cdot 10^{-3} r_0 \left(\frac{V^3}{A I^2}\right)^{1/4} \int_0^{\ln r/r_0} e^{y^2}\,dy, \tag{8} \]

where \(I\) is the current in the beam, \(V\) is the energy in electron-volts, \(r_0\) is the initial radius, \(r\) is the radius at distance \(d\), and \(A\) is the atomic weight of the ion. For convenience, a graph of this function is given in Fig. 4. With its aid one can estimate the order of magnitude of the spreading that may be expected in ordinary beams. For example, at \(10\) mA and \(10\,000\) V, a beam of ions with mass \(40\), initially having a radius of \(1\) cm, will have a radius of \(3\) cm at a distance of \(5.6\) cm.

L. SMITH, W. PARKINS AND A. FORRESTER

If the parallelism of the ion trajectories is substantial, then it is obvious that, at small initial voltages, the currents necessary for separation cannot be attained. The use of large voltages is associated with the difficulties discussed above. In addition, it is very difficult to modulate high-voltage beams, since in this case both the frequency and the amplitude of the modulating field must be large. In practical realization this is associated with special difficulties when a modulating voltage of nonsinusoidal form is required. Another difficulty connected with modulation schemes is the necessity of attaining, in some regions, sharp maxima of the potential gradients. Even when grids are used, this may prove difficult to carry out in regions where the space charge is large.

Fig. 4. Natural spreading of a circular cross-section beam, in which only radial forces are assumed to be present.

\(r_0\) — initial radius; \(r\) — final radius;
\(d\) — distance traversed; \(i\) — total current;
\(V\) — initial voltage; \(A\) — atomic weight of the particle. All quantities are in practical units.

The application of a magnetic field directed along the beam, as shown below, is not sufficiently effective for preventing spreading. One can calculate the maximum radius of a particle trajectory if the beam cross-section is a circle. As such a particle moves in the direction of the beam axis, it performs periodic oscillations between maximum and minimum radii. Apart from the motion along the axis, the conditions here are the same as in a magnetron. Hull\(^2\) showed that, for a charged particle moving in a uniform magnetic field \(H\) (directed along the \(z\)-axis) and in a radially symmetric electric field \(F\) (which is a function only of \(r\)), if the initial conditions at \(r=r_0\) are specified as follows:

\[ \mu_0=\left(\frac{dr}{dt}\right)_0,\qquad v_0=r_0\left(\frac{d\theta}{dt}\right)_0,\qquad w_0=\left(\frac{dz}{dt}\right)_0, \tag{9} \]

then the square of the radial velocity is equal to

\[ \left(\frac{dr}{dt}\right)^2 = \frac{2eV_r}{m} - H^2\left(\frac{e}{2m}\right)^2 r^2 \left(1-\frac{r_0^2}{r^2}\right)^2 - \]

\[ {}-H\frac{e}{m}r_0v_0 \left(1-\frac{r_0^2}{r^2}\right) + v_0^2\left(1-\frac{r_0^2}{r^2}\right) + \mu_0^2, \tag{10} \]

where

\[ V_r=\int_{r_0}^{r} F\,dr . \tag{11} \]

Setting \(\dfrac{dr}{dt}=0\), Hull obtained an implicit expression for the maximum radius \(r_m\) that the particle can reach:

\[ V_{r_m}=H^2\frac{e}{8m}r_m^2\left(1-\frac{r_0^2}{r_m^2}\right)^2+ \left(\frac{Hr_0v_0}{2}-\frac{v_0^2}{2e/m}\right) \left(1-\frac{r_0^2}{r_m^2}\right)-\frac{u_0^2}{2e/m}. \tag{12} \]

In order to obtain the maximum radius reached by a beam bounded by the presence of an axial magnetic field, we assume that the ion begins to move while situated on the surface of a beam of radius \(r_0\), and, as the beam expands, continues to remain at its boundary. By Gauss’s theorem, the effective radial electric field, expressed in terms of the beam current \(I\), is equal to

\[ F=\frac{2Ic^2}{rv_0}. \]

Using expression (11), we find

\[ V_{r_m}=\frac{2Ic^2}{v_0}\ln\frac{r_m}{r_0}. \tag{13} \]

Substituting (13) into equation (12) and setting \(v_0=0\), we obtain an equation from which the maximum radius can be determined:

\[ \frac{2Ic^2}{v_0}\ln\frac{r_m}{r_0} = \frac{H^2er_0^2}{8m} \left(\frac{r_m^2}{r_0^2}-2+\frac{r_0^2}{r_m^2}\right) -\frac{m}{2e}u_0^2 . \tag{14} \]

Putting \(\rho=\ln\dfrac{r_m}{r_0}\), we rewrite this equation in the form

\[ \frac{2Ic^2}{v_0}\rho= \frac{H^2er_0^2}{2m}\operatorname{sh}^2\rho -\frac{m}{2e}u_0^2 . \tag{15} \]

Passing in (15) from electromagnetic units to practical units, we obtain

\[ 2.72\cdot10^{10}\, \frac{A^{3/2}I}{(V_z)^{1/2}H^2r_0^2}\rho + 2.09\cdot10^{4}\, \frac{AV_R}{H^2r_0^2} = \operatorname{sh}^2\rho, \tag{16} \]

where \(A\) is the atomic weight of the singly ionized ion,

\[ V_z=\left|\frac{mv_0^2}{2e}\right| \quad\text{and}\quad V_R=\left|-\frac{mu_0^2}{2e}\right|. \]

The quantity \(r_m\) is most simply found by first solving graphically the equation for \(\rho\). As an illustration, let us give the value of the maximum radius for a definite concrete case. Let the velocity of the outer particle of a beam of diameter \(1\ \mathrm{cm}\) be directed along the \(z\)-axis. Let

then the current will be 10 ma, the voltage—10,000 V, and the ion mass—40. In a magnetic field of 5000 gauss the maximum radius is then equal to 18.6 cm. It is also important to know how rapidly the beam spreads out. For this it would be necessary to determine the time in which a particle reaches the maximum radius. However, for our purposes this can be avoided in the following way. It may be assumed that until the ion has reached approximately one half of its maximum radius, the magnetic force in the direction \(r\) is small in comparison with the electric force. It is therefore natural to suppose that until the ion has reached a radius of approximately \(\frac{1}{2} r_m\), the spreading will be almost the same as if it were caused only by the presence of space charge (see Fig. 4). Consequently, the magnetic field cannot effectively impede the spreading of the space charge of the ion beam.

The use of an axial magnetic field may prove useful only in the case when slow electrons are present in the beam. These electrons can be collected into a beam of any cross section. In this way the necessary negative charge can be created, preventing the spreading of the ions. This principle was used by Finkelstein\(^8\) in intense ion sources.

POSSIBLE APPLICATIONS OF SPREADING

In considering the difficulties caused by the presence of space charge in modulation devices, an interesting method suggests itself: the use of space-charge spreading for the separation itself. In the natural spreading of a group of charged particles, the lighter particles will be displaced over a greater radial distance than the heavier ones in the same interval of time. Therefore it is possible to arrange collectors in such a way that they collect particles of only one mass. Another method of using the same principle may also be applied, connected with applying certain potentials to a series of collectors and taking into account the fact that the light particles of successive beams also acquire greater energies during the spreading of the beams. As far as is known, separators based on these principles have not yet been used, but such a possibility is of interest. The principal difficulty associated with such a method lies in creating periodic space-charge beams before their spreading begins.

RADIAL MAGNETIC SEPARATOR

AND THE SPACE-CHARGE PROBLEM

Attempts to construct an instrument whose resolving power would, at least in a first approximation, not depend on space charge led to the creation of a radial magnetic separator. As shown in Fig. 5, ions are produced at the center

SEPARATION OF ISOTOPES IN LARGE QUANTITIES

of the apparatus and are accelerated radially by means of one or several cylindrical grids or slits. Under the action of an axial magnetic field, the particles, depending on their masses, move along different trajectories. The form of these trajectories is such that it proves possible to collect the heaviest ions on a cylindrical electrode whose diameter is large compared with the dimensions of the source. The remaining light fraction returns back toward the center and can be collected with electrodes of suitable geometry.

Fig. 5. Diagram of a radial magnetic separator.

The advantage of this method consists in the fact that the potential required for the return of a particle after it has moved away to some radial distance depends only on the radius. As can be seen from equation (12), it is entirely independent of the change of potential in the region where the particle is moving. Therefore, if the field produced by the space charge in the axial direction is not taken into account, it may be said that the presence of space charge plays no role until the magnitude of the field it produces becomes insufficient to stop some of the heavy ions before they reach the outer cylinder.

It should be emphasized that although this, as shown below, imposes a limitation on the magnitude of the current, the fact remains that only the very heaviest ions can reach the outer cylinder, while the remaining fraction, enriched in light ions, can be collected as it moves back toward the center.

If the length of the apparatus is small compared with its diameter, then the space charge will create a field in the axial direction. Since this affects the resolving power only insofar as radial velocities change, this effect is secondary.

It should be noted that only one ion source can be used to feed a powerful apparatus of this type.

An estimate of the limitation imposed on the current by the presence of space charge may be made as follows. Under specified conditions (the positions of the electrodes, their potentials relative to the source, and the magnetic-field strength), the magnitude of the current to the outer cylinder is limited by values satisfying the condition that the magnitude of the potential at all points between the collecting

cylinder and the source must be greater than the potential determined by equation (12):

\[ V_0=-\frac{1}{8c^2}\frac{e}{M}H^2r^2\left(1-\frac{a^2}{r^2}\right)^2 . \]

Here \(H\) is expressed in gauss, and the remaining quantities in CGSE units; \(a\) is the distance from the center to the place from which the particles begin to move. It is assumed that all particles begin their motion at one and the same distance from the center and that their initial velocity is zero. \(V_0\) is the potential relative to the point from which the motion of the particles begins. Since we are interested only in distances for which

\[ \frac{a}{r}\ll 1, \]

we may use the relation

\[ V_0=-\frac{1}{8c^2}\frac{e}{M}H^2r^2, \tag{17} \]

The essence of the problem can be clarified by considering Fig. 6. A source of small dimensions is surrounded by grids that create a potential sufficient to accelerate the ions and send them to the outer cylinder. When an arc source is used—which is practically necessary for obtaining large ion currents—the problem of drawing off a sufficient number of ions from the source to the first grid is solved automatically. The space charge in the arc must increase so rapidly that the majority of the ions formed diverge radially. We shall confine ourselves to consideration of the region between the most distant grid and the collecting cylinder, and shall seek the distribution of space charge corresponding to the critical conditions. From this one can find the minimum value of the potential that must be applied to a grid located at a fixed distance from the center in order to obtain a specified current to the outer cylinder.

Fig. 6. Distribution of potential illustrating the critical conditions on the collecting cylinder.

If the ions reach the cylinder with zero radial velocity, i.e., if

\[ V(R)=V_0(R), \tag{18} \]

then the condition determining the critical current will be

\[ V'(R)=V'_0(R). \tag{19} \]

In the case of symmetry of the problem, the coordinates \(z\) and \(\Phi\) drop out of Poisson’s equation, and we have

\[ \frac{1}{r}\frac{d}{dr}\left(r\frac{dV}{dr}\right)=-4\pi\rho. \tag{20} \]

Since

\[ I=2\pi r\rho v_r \]

represents the radial current per unit length, we obtain

\[ \frac{d}{dr}\left(r\frac{dV}{dr}\right)=-2\frac{I}{v_r}. \]

The equations for radial motion are obtained from Brillouin’s potential\(^4\) \(P\), i.e.

\[ M\frac{d^2 r}{dt^2}=-e\frac{\partial P}{\partial r}, \]

where

\[ P=V-V_0. \]

In these variables our differential equation has the form

\[ \frac{d}{dr}\left[r\frac{d}{dr}(P+V_0)\right] = \frac{-2I}{(-2eP/M)^{1/2}}, \]

whence it follows that

\[ \frac{d}{dr}\left(r\frac{dP}{dr}\right) -\frac{1}{2c^2}\frac{e}{M}H^2r = -I\left(\frac{2M}{eP}\right)^{1/2}. \tag{21} \]

From (18) and (19) we obtain the boundary conditions

\[ P(R)=0;\qquad P'(R)=0. \tag{22} \]

The differential equation can be simplified by using dimensionless variables

\[ \rho=\frac{r}{R},\qquad \Phi=\alpha P,\qquad \alpha=\left(\frac{e/M}{2I^2R^2}\right)^{1/2}. \]

Equation (21) then takes the form

\[ (-\Phi)^{1/2} \left\{ \frac{d}{d\rho}\left(\rho\frac{d\Phi}{d\rho}\right)-K\rho \right\} =-1, \tag{23} \]

where

\[ K= \left( \frac{(e/M)^{1/2}R^2}{4I} \right)^{2/3} \frac{H^2}{c^2}. \]

The boundary conditions for \(\Phi\) will be

\[ \Phi(\rho)=0,\qquad \Phi'(\rho)=0\quad \text{for } \rho=1. \tag{24} \]

If (23) is written in the form

\[ (-\Phi)^{1/2}\{\rho\Phi''+\Phi'-K\rho\}=-1, \]

then it becomes evident that \(\Phi''\to\infty\) as \(\rho\to1\). Therefore, as \(\rho\to1\), the solution of this equation tends to the solution of the equation

\[ \Phi_0''(-\Phi_0)^{1/2}=-1;\qquad \Phi_0(1)=0;\qquad \Phi_0'(1)=0. \]

The latter, as is easy to see, is satisfied if

\[ \Phi_0=-\left(\frac{9}{4}\right)^{2/3}(1-\rho)^{4/3}. \tag{25} \]

\(\Phi_0\) differs little from the exact solution (23) when

\[ \rho \simeq 1 \quad \text{and} \quad |\Phi'_0-K| \ll \Phi''. \]

For a given value of \(K\), the approximate solution may be used as the initial one in the numerical integration of equation (23). Knowing \(\Phi\), the potential can easily be found:

\[ V=P+V_0=\frac{\Phi}{a}+V_0 . \]

Such a solution is a lengthy procedure, and the numerical integration must be carried out for each value of \(K\). In order to find the solution in the form of a series, let us put

\[ u^2=-2^{2/3}\Phi=-2^{1/3}aP. \]

Then we obtain the differential equation

\[ u\frac{d}{d\rho}\left(\rho u\frac{du}{d\rho}\right)=1-K'\rho u, \tag{26} \]

where

\[ K'=\frac{K}{2^{1/3}}. \]

From \(\Phi_0\) we obtain a solution of this differential equation, valid for \(\rho\) close to unity:

\[ u_0=\left(\frac{9}{2}\right)^{1/3}(1-\rho)^{2/3}. \tag{27} \]

Hence it is seen that the solution should be sought in the form of the following series:

\[ u=\sum_{k=2}^{\infty} c_k(1-\rho)^{k/3}. \tag{28} \]

In equation (26) we put

\[ x=(1-\rho)^{1/3} \tag{29} \]

and shall seek the solution in the form

\[ u=c_2x^2+c_3x^3+c_4x^4+\cdots . \tag{30} \]

Let us give the values of the first few coefficients:

\[ \left. \begin{aligned} c_2&=\left(\frac{9}{2}\right)^{1/3},\\ c_3&=0,\\ c_4&=-\frac{K}{20}\,9^{2/3},\\ c_5&=\frac{4}{15}\left(\frac{9}{2}\right)^{1/3},\\ c_6&=-\frac{27}{14}\frac{K^2}{100}\cdot\frac{1}{2^{2/3}},\\ c_7&=-\frac{28}{405}\,9^{2/3}K,\\ c_8&=\left(\frac{9}{2}\right)^{1/3}\left\{\frac{67}{405}-\frac{827}{1008}\frac{K^3}{1000}\right\}. \end{aligned} \right\} \tag{31} \]

These series converge poorly, and therefore such a solution is not entirely satisfactory. In Fig. 7 a solution is shown, computed for a special case chosen so as to make it convenient to compare it with the available experimental data. Curve \(A_1\), representing the potential obtained from \(u_0\), and curve \(A_2\), representing the potential obtained from the series with all the terms for which the coefficients were calculated taken into account, are compared with the exact solution obtained numerically (curve \(A\)).

The requirement that the particles reach the outer cylinder with radial velocities equal to zero is too stringent, since in practice the particles are caught with some finite radial velocities. Suppose that for the potential of the outer cylinder precisely such a value has been chosen that ions of the lighter isotope are stopped. The computations were carried out for the case in which this isotope is \(\mathrm{Li}^6\). Then

\[ V(R)=-\frac{1}{8c^2}\frac{e}{M_6}H^2r^2, \tag{32} \]

where \(M_6\) is the mass of the \(\mathrm{Li}^6\) ions. As the second boundary condition, as before, we choose

\[ V'(R)=V'_0(R), \]

although in this case this condition is too stringent.

Fig. 7. Potential distribution for critical conditions as a function of the relative distance from the center to the collecting cylinder, for \(\mathrm{Li}^7\) ions at \(J=3.63\ \text{mA/cm}\), \(H=1350\ \text{gauss}\), and radius \(14\ \text{cm}\). \(A\) shows the potential distribution when the ions reach the outer cylinder with zero radial velocity; \(B\), when the potential is chosen so that \(\mathrm{Li}^6\) ions are stopped. Both curves were obtained by numerical integration; \(A_1\) is an approximation to \(A\), obtained using one term of the series; \(A_2\) is an approximation obtained with all computed terms of the series taken into account. The effect of \(\mathrm{Li}^6\) ions on the potential is neglected.

It would seem that one might expect ordinary power series to be suitable here as a solution; however, the latter converge so slowly that they are practically inapplicable. The results of the numerical integration are shown in Fig. 7 (curve \(B\)).

For the values of \(\rho\) shown in Fig. 7, \(A_1\) is a sufficiently good approximation to the exact solution. Therefore one can use the analytical expression for \(A_1\) and find how the potential depends on \(I\), \(e/M\), \(H\), and \(R\). It is reasonable to assume that the character of the variation of curve \(B\) is approximately the same as that of curve \(A\).

The expression for \(A\) is obtained from equation (25):

\[ V=V_0+\frac{\Phi_0}{\alpha} =-\left\{\frac{1}{8c^2}\frac{e}{M}H^2r^2+\frac{81}{8}\frac{I^2R^{1/3}}{e/M}(1-\rho)^{4/3}\right\}. \]

With the exception of the region where \(\rho\) is close to unity, the quantity \(V_0\) may be neglected in comparison with \(\Phi_0/\alpha\). Then, for a fixed value of \(\rho\), we obtain the following proportional relation:

\[ V\sim I^{2/3}R^{1/3}M^{1/3}. \tag{33} \]

It should be noted that, in this approximation, \(V\) does not depend on the magnetic field.

EXPERIMENTS WITH A MODULATION SEPARATOR

A study of the advantages and shortcomings of various methods of separation shows that the two schemes mentioned in the preceding discussion are of greatest interest for experimental investigation. The first of these is the modulation of velocities using electrons to neutralize the space charge.

Fig. 8. Schematic of an isotope separator using velocity modulation. The applied potentials can be obtained approximately from the energy diagram.

This apparatus is shown schematically in Fig. 8. The whole of it, with the exception of the mass-spectrographic analyzer, is placed inside a series of coaxial coils, which create a magnetic field of several hundred gauss. The ion source is copied from that described by Finkelstein \(^{3}\). The system is arranged so that the electrons which create ions are also used to neutralize the space charge. Leaving the cathode, they are accelerated, pass through the atomic beam, and reach the system of ion collectors, from which, owing to repulsive forces, they begin to move in the opposite direction. Subsequently, if their loss of energy is not too great, they are capable of forming ions again. If these electrons, on their way to the ion collector and back, lose a certain amount—

SEPARATION OF ISOTOPES IN LARGE QUANTITIES

…amount of energy, they will not be able to return to the cathode, but will be forced to move back and forth throughout the entire system, creating a negative space charge and ionizing atoms until they leave the beam. The energy of the electrons as a function of distance is shown approximately in Fig. 8. Owing to the small mass of the electron, each passage through the modulation region takes place so rapidly that only in rare cases will the energy of the electrons change as they pass through this region. The magnetic field makes it possible to hold the electrons in a strictly limited beam. The potential “trough” formed by the electrons, in turn, prevents the ions from spreading out.

Ions are formed in the beam of atoms by impacts of electrons. They enter the modulation region, from which they emerge with an energy depending on their mass. The frequency and amplitude of the modulating voltage, as well as the initial energy of the ion, are chosen so that the energy is increased only for ions of one mass. The ions then pass through a series of electrodes whose potentials successively increase. Because of this the ions are slowed down. In the end all ions are stopped except those whose energy has increased in the modulation region. The latter are the ions of the desired isotope. Most of the ions stopped before the last grid are scattered and are caught by the nearest electrodes. Some of them may return back, but their number will be small. In order to create better conditions for separation, the ions are passed through a small aperture in the collector, after which they are analyzed by a mass spectrograph.

The potentials must satisfy the condition that the voltage drop in the arc (between the cathode and the source of the atomic beam) must be greater than the initial voltage acting on the ions when they enter the modulator. This requires that the voltages be small until the ion beam spreads out. At low ion energies, modulation by a sinusoidal voltage cannot be sufficiently effective; however, at small amplitudes it is not difficult to obtain other forms of oscillation. A square-wave form is the most suitable here; it makes possible the separation of almost half of the ions of the desired type issuing from the source. It is precisely such a square-wave form of oscillation, produced by the generator described by Parkinson and Smith\(^5\), that is used. Since the ions issuing from the source have low energy, complete separation of the desired isotope is achieved with the aid of no fewer than three modulations. For this purpose three pairs of grids connected to the load resistance of the square-wave generator were used (see Fig. 9). This drawing also schematically shows the increase in the energy of ions of only one isotope.

The cathode, belonging to the dispenser type, is a modification of the cathode described by Hull\(^6\). A loop-shaped bag of molybdenum wire, containing the eutectic alloy BaO and Al\(_2\)O\(_3\), serves as the heater and dispenser. It is placed in front of a flat molybdenum plate that emits electrons. Ot-

hole in the screen in front of the cathode, which determines the beam diameter, is made \(1/4\) inch smaller than the molybdenum loop, so that bombardment of the bag cannot occur.

The furnace is a double-walled cylinder made of molybdenum sheets. A tungsten winding with refractory insulation is placed between the walls and serves to evaporate the substance whose isotopes are to be separated. In the experiments calcium was used, which in the metallic state was placed on the bottom of the furnace.

Labels in the figure: “Zero current to the plates”; “Constant current to the plates”; “Last cascade of the square-wave generator”; “Dependence of ion current on time for two isotopes. The pulse width determines the energy spread”; “Singly accelerated ions”; “Doubly accelerated ions.”

Fig. 9. Modulator and its principle of operation.

The grids were made of tantalum strips \(1/16\) inch wide, turned with their narrow edges toward the beam. The region of free ion motion is shielded by stainless-steel tubes \(1/2\) inch in diameter and of the corresponding length. The flat electrodes were made of 20-millimeter nonmagnetic stainless steel.

The entire unit, only 18 inches long, is mounted on three glass rods fastened to a flat brass base. Tungsten leads sealed into glass pass through the base, rubber gaskets being used, as shown in Fig. 10. To save space, eight low-current wires pass through each glass tube. All electrodes are mounted on a plate serving as the base, which is then placed inside the system. A seal is made on the rim of the base by means of a rubber gasket. The mass-spectrographic analyzer is an integral part of the system.

In the operation of this modulation device there were no difficulties connected with producing the arc and obtaining large ion beams.

SEPARATION OF ISOTOPES IN LARGE QUANTITIES

currents at the entrance to the modulator. However, only very small currents were measured at the collector. The difficulties in this scheme are connected with achieving effective neutralization of the ion beam by electrons. But it is precisely by this that the possibility of large transfer of matter by low-energy ion beams is at once determined. The causes of the ineffectiveness of electron neutralization are:

Fig. 10. Diagram of the instrument used in the real magnetic separator, showing the typical arrangement of the electrodes.

Labels visible in the diagram:

• Vacuum seals, insulating the wire into the upper part of the discharge.
• Air cooling.
• Region in which collisions and acceleration of ions occur.
• Slope.
• Resonance chamber.
• Five such plates in a glass tube serve for connection to each of the electrodes.
• Contours, free parts of the tube.
• The lower table consists of:
• a sloping screen of the metastable [[unclear: particles]];
• parallel wires;
• plates conducting away charge from the beam;
• plane drift space;
• knife edge;
• screen that divides the collectors;
• plates for [[unclear: emission]];
• metastability.
• Scale in inches: 0 1 2 3 4 5 6.
• Thermocouples.
• To the diffusion pump.
• Glass tube.
• Entry into the third stage.
• Tube of the first stage.
• To the diffusion pump.
• Electrode of equipotential space.
• Feathered [[unclear: plate]].
• Graphite emitter, fixed on two vertical supports of nickel, fastened to one of the glass tubes.
• Ion source from the second stage.
• Vacuum gauge.
• Rectangular region.
• Circular region.
• Region in which collisions and acceleration of the electrodes occur.
• Region of isotropic collisions and acceleration of the electrodes.
• Negative voltage is supplied by the high-frequency oscillator in the form of a V-wave.
• Lower plate.
• Voltmeter.
• Inner cylinder of the 3rd stage.
• Tungsten wire.
• Quartz rod.
• Electrons from heated thoria deposited on a platinum strip pass into the region.
• Beam neutralizer.
• From the diffusion pump.
• Oven of the negative ion source.
• Insulator of the carrier rod on the side where there is no platinum strip.
• Oven of the carrier rod.
• Manual holder.
• To the Faraday cup.
• Slit of the Faraday cup.

1. The presence of excessively large fluctuations in the energy of the electrons.

2. The voltage drop across the arc is so difficult to regulate that the production of electrons whose initial energy would be independently determined becomes impossible.

3. Too many electrons are lost at those points where they begin to move in the reverse direction. This occurs near the collector, since the increased negative space charge causes larger radii to be reached than those determined by equation (15). Near the cathode, losses of electrons occur because of scattering, since the effective collision cross section is very large for electrons of small energies.

EXPERIMENTS WITH A RADIAL MAGNETIC SEPARATOR

From equation (12) it can be seen that complete separation of two isotopes in a radial magnetic separator can be achieved only in the case when the ions are formed in a limited region where the radius and the potential vary little. If it is assumed that ions having zero energy begin their motion from points having one and the same radius, then, in order for the separation to be complete, it is necessary that the fluctuations of the potential in the source be no greater than the difference of the critical potentials for the ions of both isotopes. The energies of the ions at their formation can always be neglected. If the ions are formed in a region with constant potential, the permissible spread of initial radii can be found from

\[ \frac{r_m^2 + 2r_{01}^2 + r_{01}^2/r_m^2}{r_m^2 + 2r_{02}^2 + r_{02}^2/r_m^2} = \frac{M_1}{M_2}, \]

where \(M_1\) and \(M_2\) are the masses of the two types of ions, \(r_{01}\) and \(r_{02}\) are their initial radii, and \(r_m\) is the radius of the outer cylinder. This equation is obtained by equating the critical voltages. If \(r_0\) is fixed for ions of one mass, for example \(r_{01}\), then the two solutions for \(r_{02}\) determine the range of permissible initial radii for ions of the other mass under complete separation. When using a source in which both the initial radius and the potential vary, both effects must be taken into account in deriving the conditions for complete separation.

There are also other causes that can lead to mixing of isotopes. Among them the following may be mentioned:

1. A strong deviation of the applied electric field from a purely radial one.

2. Inhomogeneity of the magnetic field.

3. Asymmetry of the space-charge field.

4. Variation of the space-charge potential in the \(z\) direction in the separation region.

5. Oscillations of the space charge, which entail a change in the energy of the particles.

SEPARATION OF ISOTOPES IN LARGE QUANTITIES

The first cause can always be eliminated by improving the geometry of the electrodes and by an appropriate choice of potentials, using grids when necessary. The second cause gives rise to still fewer difficulties: a sufficiently homogeneous field is always easy to obtain. The last three circumstances always make themselves felt, since they are connected with the presence of space charge. It is difficult to estimate to what extent these effects influence the resolving power.

In experimental investigations of a radial magnetic separator, lithium was used. By means of a vertical magnetic field, a confined electron beam was produced. This made it possible to create an ion source of a definite radius. The cathode was placed on the axis of the system opposite the furnace from which the lithium evaporated. If the furnace is used as the anode, a low-voltage arc can be ignited, owing to which a considerable ion current in the radial direction is obtained. Dispenser and oxide cathodes were tested, but no advantages of such cathodes over ordinary tungsten ones were found. Because of the chemical activity of molten lithium and the necessity of using nonmagnetic materials throughout, the furnace was made of stainless steel. Attempts to heat the furnace in air proved not very successful. It was therefore necessary to place the furnace in an insulated evacuated chamber and heat it by means of a tungsten winding. In order to conserve lithium and prevent strong scattering, a collimated atomic beam was produced, the material that did not pass out returning back into the chamber. This was achieved by means of a long vertical furnace heated from below. The temperature at the top of the furnace was only slightly above the melting point of lithium, so that the greater part of the vapors striking the walls condensed and flowed downward. The furnace must have no constrictions anywhere, since in such places the lithium will adhere to the walls and close the opening. In order for the flow to occur sufficiently well, the inner walls of the furnace must be smooth. For this purpose it is desirable to coat them with platinum.

For collecting lithium isotopes, the catching surfaces in the operating chamber must be sufficiently cold so that the substance condenses well. In the initial experiments the first cylinder (numbering from the inside) was cooled with water in order to prevent its heating by the arc and by the collected current. However, this caused the loss of the possibility of regulating the potentials in the required way. Subsequently all the electrodes (three cylinders and two plates) were insulated from one another and from the metallic jacket in such a way that the potentials could be controlled independently and the currents measured. Various types of the first two cylinders were tested: experiments were carried out with different sizes of grids, slits, and radii. The entire system is shown schematically in Fig. 10.

To find the best operating conditions of the apparatus, graphs were plotted of the current to the second or third cylinder, as a function of

stresses on the third cylinder, with the arc conditions constant and the magnetic field unchanged. If this voltage is positive and too large, the ions will not reach the third cylinder, and most of them will be deposited on the second. If the third cylinder has a sufficiently negative potential, ions of the heavier isotope will be able to fall on it, which increases the current in the third cylinder and decreases it in the second. With an even greater negative voltage, such a change in the currents will also occur at the expense of ions of the lighter isotope. As a result, the graph of the current to each cylinder as a function of the voltage applied to the third cylinder will have horizontal sections corresponding to the presence in this current of ions of different isotopes. Confirmation of this is the fact that the difference of the currents corresponding to these horizontal sections is proportional to the relative content of the isotopes under consideration. In Fig. 11 the volt-ampere characteristic of the third cylinder is given.

Fig. 11. Volt-ampere characteristic of the outer cylinder of the radial separator, obtained with the arrangement of electrodes shown in Fig. 10.

These characteristics make it possible not only to determine the quality of the separation, but also provide an opportunity to clarify better the operation of the separator. In this connection the characteristics of other electrodes are also useful. In order to avoid the tedious construction of these curves point by point, an oscillograph may be used. In this case an alternating voltage with a frequency of 60 cycles is connected in series with the direct voltage applied to the third cylinder, and also to the horizontal plates of the oscillograph. The resistance through which the current to the electrode under consideration flows is connected to the vertical plates. Such an oscillograph is indispensable for checking the operating conditions over a long period of time while the separation is taking place. The alternating current interrupts the separation process for only a few seconds.

One of the greatest difficulties is the suppression of oscillations that arise spontaneously on all electrodes. The intensity of these oscillations could not be reduced by means of any combination of capacitive and inductive resistances. At the end of con-

SEPARATION OF ISOTOPES IN LARGE QUANTITIES

it was found that they arise in the region of the arc and can be suppressed only by creating the appropriate conditions in the space charge. The elimination of all oscillations is associated with removing the specific distribution of the space-charge potential in the region of the arc, the existence of which was detected by measurements of currents to various electrodes. This was achieved by maintaining a sufficiently low arc current (below \(0.5\) a) and by applying a negative potential (usually \(500—600\) v) to the first cylinder. The presence of the potential caused a considerable current to be drawn off from the arc, the magnitude of which is the greater, the larger the arc current. A very large negative potential of the first cylinder, naturally, ultimately extinguishes the arc. At the same time, however, a negative voltage on the first cylinder is necessary in order to increase the current entering the separation region. The ion currents that can practically be drawn off from the arc to the first cylinder in the separation region often reach one third of the magnitude of the arc current. That the currents measured in this way are ion currents is confirmed by the effect that the potential of the first cylinder has on the saturation current of the third cylinder. As a second confirmation of this circumstance it should be noted that the ratio of the current to the first cylinder to the current through the first slit is always approximately equal to the ratio of the effective length of the first cylinder to the length of the slit.

With a slit width in the first cylinder of \(3/8\) inch, a saturation current to the third cylinder of \(8\) ma corresponds to complete separation, as is seen from a characteristic similar to that shown in Fig. 11. Larger currents can also be obtained in the separation region. In this case, however, the space-charge distribution in the region of the arc changes greatly, and as a result the isotope separation ceases to be complete. With a slit width of \(3 \tfrac{1}{4}\) inches (only grids were used in the first and second cylinders), it is seen from the characteristic of the third cylinder that good separation is attained at a current of \(30\) ma. In this latter case the currents to the end electrodes are large. Both of the measurements mentioned were made at one and the same potential on the end electrodes and on the second and third cylinders.

Experiments were also carried out with a source in the form of a nickel cylinder coated with spodumene and placed near the slit concentrically with the axis of the system. When the cylinder is heated, good separation is obtained when the current through the \(3/8\)-inch slit reaches \(1\) ma. Owing to the presence of spodumene, a certain amount of sodium and potassium ions was observed. Because of the presence of the arc, other ions were also observed, in particular \(\mathrm{Li}_2^+\).

Figure 12 shows the apparatus proposed for separating \(\mathrm{Li}^6\). Measures were taken to use current from an expanded region of the arc. Bombardment of the first grid proved sufficient to prevent the lithium from condensing and clogging

holes, in the second cylinder, which was cooled, the use of grids was avoided. The ceiling and the bottom of the system served as end electrodes, which had the same potential as the second cylinder. It was assumed that the entire assembly would operate at the potential

Rings rigidly connected with a pipe of a small-diameter aqueous cooling unit (not shown). Lightweight isotope is collected on removable cylindrical sheets of stainless steel, protected from the substance scattered on the grid.

Furnace insulated from the lower plate.

Fig. 12. Modified apparatus proposed for separating Li\(^6\) in large quantities.

of the third cylinder. The third cylinder is insulated in order to make it possible to take volt–ampere characteristics.

COMPARISON OF THE CALCULATED AND TRUE LIMITATIONS IMPOSED BY SPACE CHARGE

The results of the calculations for space charge, given in the preceding paragraphs and presented in Fig. 7, correspond to the true lithium ion currents at which complete separation of Li\(^6\) from Li\(^7\) is achieved. According to these calculations, the potential at a point located midway between the source and the outer collecting cylinder should be approximately 2000 V lower than the po-

source potential, if a current of \(3.63\ \mathrm{ma/cm}\) is delivered to the outer cylinder. In reality, however, such currents are observed when the potential at this point is less than 600 V below the source potential. Since the length of the apparatus is practically limited, the influence of the end electrodes on the potential at the center cannot be estimated, because the distance between the end electrodes is comparable with the distance between the source and the outer cylinder. Moreover, since the ion current to the end electrodes is very small in comparison with the current to the outer cylinder, it may be concluded that the positive space charge in the separation region is not so large in comparison with the value it would have if there were positive ions here producing a current to the outer walls. Hence it may be concluded that the space charge is in some way neutralized. This may occur, for example, as a result of electron capture, discussed above.

Thus, neutralization of the space charge takes place even when no attempt is made to produce it artificially. This indicates that, when an electron is introduced into the beam, neutralization may be made still more effective by one of the many available methods.

It may also be concluded that at present there are no indications of limitations on the current used in this method that would be caused by the presence of a space charge.

SIZE OF THE APPARATUS

It is easy to see what changes must be made in order to make it possible to collect a large quantity of material and thus to separate large masses. Very large ion sources can be constructed, since the apparatus can be increased without limit in the axial direction, according to the scheme shown in Fig. 12. For large masses, the diameter of the apparatus should be increased several times and higher voltages and fields should be used. In accordance with the foregoing, the radial separator is a very promising means for separating large masses on a large scale.

REFERENCES

1. See, for example, L. P. Smith and P. Z. Hartman, J. Appl. Phys. 11, 220 (1940).
2. A. W. Hull, Phys. Rev. 18, 31 (1921).
3. A. T. Pinkelstein, Rev. Sci. Instr. 11, 94 (1940).
4. L. Brillouin, Phys. Rev. 60, 385 (1941).
5. W. E. Parkins and L. P. Smith, Phys. Rev. 57, 108 (1940).
6. A. W. Hull, Phys. Rev. 56, 86 (1939).