Magnetic Traps

configurations of a magnetic field that are capable of confining charged particles within a specific space over a prolonged period. The earth’s magnetic field is a natural magnetic trap; the extremely high number of high-energy cosmic charged particles (electrons and protons) that it captures and confines form the Van Allen radiation belt outside the atmosphere. Under laboratory conditions magnetic traps of various types are studied mainly as they apply to the problem of confining a mixture of a large number of positively and negatively charged particles (plasma). The improvement of magnetic traps for plasma is directed toward the accomplishment—using such traps—of a controlled thermonuclear reaction, in which the nuclear energy of light elements is liberated comparatively slowly, in the course of a process controlled and regulated by humans, rather than in the form of a powerful explosion.

To be a magnetic trap, a magnetic field must satisfy certain conditions. It is known that a magnetic field affects only moving charged particles. The velocity v of a particle at any point may always be represented in the form of the vector sum of two components—v₁, which is perpendicular to the field H at that point, and v₁, whose direction coincides with H. The force F of the field’s action on the particle, called the Lorentz force, is determined only by v₁ and is independent of V₁. In the cgs system of units, F is equal in absolute value to (e/c) v₁H, where c is the speed of light and e is the particle’s charge. The Lorentz force is always directed at right angles to both v₁ and V_ǀǀ and does not change the absolute value of the particle’s velocity, but it does change the direction of the velocity by curving the particle’s trajectory. The motion of a particle in a uniform magnetic field is the simplest case (H is everywhere identical in magnitude and direction). If the velocity of the particle is directed across such a field (v = v₁), then its trajectory will be a circle of radius R (Figure 1, a). In this case the Lorentz force plays the role of a centripetal force (equal to /wvi²/R, where m is the mass of the particle). This makes it possible to express R in terms of v₁ and H: R = v₁/ωH, where ωH = eH/mc. The circle about which a charged particle in a uniform magnetic field moves is called a Larmor orbit, its radius is the Larmor radius (/?/), and o># is the Larmor frequency. If the velocity of the particle is directed to the field at an angle that is not a right angle, then the particle has not only v₁ but also v_╟. Here the Larmor rotation is retained, but uniform motion along the magnetic field is added to it, so that the net trajectory will be a spiral line (Figure 1, b).

Figure 1. In a uniform magnetic field (H = const.), a charged particle moves in a circle if its velocity is directed across the field (a) and in a helical line if its velocity has a longitudinal component v₁ in addition to a transverse component V₁ (b); R is the radius of the circle (the Larmor radius)

Examination of even this simplest case of a uniform field makes possible formulation of one requirement for a magnetic trap: its dimensions must be great in comparison with R₁, or the particle will escape from the trap. Since R₁ decreases with an increase in H, this condition can be satisfied by an increase not only in the dimensions of the magnetic trap but also in the magnetic field intensity. Under laboratory conditions the second path is followed, whereas under natural conditions magnetic traps with extensive but comparatively weak fields (such as the Van Allen radiation belt) occur more often.

In addition, the smallness of R₁ ensures that the particle’s motion will be restricted across the field, but it also must be restricted in the direction of the field’s lines of force. A distintion is made between the toroidal and mirror (adiabatic) types of magnetic traps, depending on the method of restriction.

Toroidal traps. A method of preventing particles from leaving a magnetic trap in the direction of the field is to impart to the trap a configuration such that there are no “ends” in the space that it occupies; a torus is such a configuration. The first magnetic trap, which was proposed by I. E. Tamm and A. D. Sakharov in 1950 in connection with the problem of controlled thermonuclear reaction, was a trap of this type. A toroidal solenoid (Figure 2, a) is the simplest example of a magnetic trap of this type. However, particles are not confined very long in a trap with such a simple field geometry: during each revolution around the torus a particle is deflected a short distance across the field (toroidal drift). These displacements accumulate, and the particles ultimately strike the walls of the trap. To compensate for toroidal drift the field can be made nonuniform along the trap by “corrugating” it (Figure 2, b); however, it is more convenient to create a configuration in which the lines of force of the magnetic field are wound spirally onto closed surfaces that envelop one another. For example, if a conductor carrying a current passing through its center line is placed in a toroidal solenoid (Figure 2, c), the field’s lines of force will be wound onto the toroidal surfaces. Particles having a small R₁ will not be deflected very strongly from these surfaces. Similar configurations can be created by using external coils—for example, as suggested by the American scientist L. Spitzer (1951), by adding to the torus winding a spiral winding carrying alternatively directed current (Figure 2, a). Another method consists in twisting the torus into the shape of a figure eight (Figure 2, d). More complex configurations may also be used, combining different elements of “corrugated” and helical fields.

Figure 2. Configurations of toroidal magnetic traps, (a) Toroidal solenoid, in which the helical trajectory of a charged particle passes around the circular lines of force of the magnetic field. The trajectory is not closed; on each revolution around the torus the particle is displaced across it by a distance δ from its initial position (toroidal drift), (b) “Corrugated” torus, (c) Toroidal solenoid with a central conductor. The magnetic fields of the solenoid winding and the central conductor are added to form a field whose lines of force are wound spirally onto the toroidal surfaces, (d) “Twisted” torus.

Mirror traps. Another method of confining particles in a magnetic trap longitudinally (along the field) was proposed in 1952 by the Soviet physicist G. I. Budker and independently by the American scientists R. Post and H. York. It consists in the use of magnetic plugs, or magnetic mirrors—regions in which the magnetic field intensity increases rapidly but smoothly. Such regions can reflect charged particles that are “incident” on them along the field’s lines of force. Figure 3 shows the trajectory of a particle in a nonuniform magnetic field whose intensity changes along its lines of force. Reflection takes place because under certain conditions the particle’s transverse velocity v₁ and the related “transverse” energy, 1/2/m v₁², increase as it moves into the region where the field is stronger. However, the total energy of a charged particle E = 1/2mv₁² + 1/2mv₁² does not change during motion in the magnetic field, since the Lorentz force, which is perpendicular to the velocity, does not accomplish work. Therefore v₁ decreases with the increase in v₁. At some point V_ǀ may become equal to zero. At this point reflection of the particle from the “magnetic mirror” takes place. This mechanism of “pumping” the energy associated with v into the energy associated with v₁ (and vice versa) operates only if the magnetic field changes relatively little in one period of the particle’s helical motion.

Figure 3. Motion of a charged particle in a “mirror” magnetic trap: as the particle moves into the region where the field is strong, the radius of its trajectory decreases. The “magnetic mirror” from which the particle is reflected is located in the “throat” portion of the configuration.

Processes that take place during a comparatively slow change in external conditions are said to be adiabatic, as are magnetic traps that use “magnetic mirrors.” The simplest mirror (adiabatic) magnetic trap is created by two identical coaxial coils in which the current is flowing in the same direction (Figure 4). In this trap the regions of the strongest magnetic field within the coils are the magnetic mirrors.

Figure 4. A very simple adiabatic magnetic trap. The arrows indicate the directions of the current in the coaxial coils.

Adiabatic magnetic traps do not confine all particles: if v_∥ is sufficiently great in comparison with v₁, then the particles pass out of the magnetic mirrors. The higher the “mirror ratio”—the ratio of the maximum intensity of the magnetic field in “mirrors” to the field in the central part of the magnetic trap (between the mirrors)—the greater the maximum ratio V_ǀ/v₁ at which reflection still takes place. For example, the earth’s magnetic field attenuates as the cube of the distance from the center of the earth. Correspondingly, as a charged particle approaches the earth along a line of force that runs sufficiently far from the earth in the equatorial plane, the magnetic field increases very sharply. In this case the mirror ratio is great; the maximum ratio V₁/V₁. is also great (the fraction of particles that escape from the magnetic trap is low).

Magnetic plasma traps. If a magnetic trap is filled with particles of the same type (such as electrons), then the electric field generated by the particles increases as the particles accumulate. The force of electrostatic repulsion of like charges increases, and the effectiveness of the trap decreases. Therefore, a magnetic trap can be filled to a sufficiently high density only with a mixture of particles of unlike charges (such as electrons and protons) in a ratio such that their total charge is close to zero. Such a mixture of charged particles is called a plasma.

When the electric field in a plasma is so small that its effect on the motion of particles can be disregarded, the mechanisms of particle confinement in the trap do not differ from those examined above with respect to individual particles. Therefore, all the conditions stated above must be satisfied in a magnetic plasma trap. However, additional requirements associated with the necessity of stabilizing plasma instabilities—spontaneously occurring and abruptly increasing deviations of the electric field and particle density in the plasma from their average values— are imposed on such magnetic traps. The simplest instability, called chute instability, results from the diamagnetism of plasma, which causes plasma to be driven from the regions where the magnetic field is stronger. The process takes place as follows: first, the plasma surface becomes wavy—long chutes form that are directed along the field’s lines offeree (hence the name of the instability); the chutes then expand and the plasma breaks down into individual tubes that move toward the lateral boundaries of the volume occupied by the magnetic trap. For example, in a simple mirror magnetic trap (Figure 4), in which the field attenuates perpendicular to the common axis of the coils, plasma may be ejected in this direction. As first demonstrated in 1961 by Soviet physicists (M. S. loffe and others), chute instability can be stabilized by using additional current-carrying conductors that are mounted along the periphery of the magnetic trap. Here the magnetic field intensity reaches a minimum at some distance from the axis of the magnetic trap, but H increases once again at greater distances from the axis. Chute instability may also appear in toroidal magnetic traps; it can be stabilized by creating a configuration that has a magnetic field with a central minimum (along the line of force). Tokamak-type units, which are being studied by a team of Soviet physicists (directed until 1973 by L. A. Artsimovich), and also in many foreign laboratories, are an example of such magnetic traps. The name “Tokamak” is an acronym for the full name of the units#x2014;“toroidal chamber with an axial magnetic field.” In Tokamaks the toroidal magnetic field is generated by a solenoid such as that shown in Figure 2, a; a strong longitudinal current whose magnetic field, when added to the toroidal field, forms magnetic surfaces close to those described for Figure 2, c is passed through the plasma confined in the torus. Not only chute but also many other types of instability have been stabilized in these units, and comparatively long, stable confinement of high-temperature plasma has been achieved (hundredths of a second at a temperature of tens of millions of degrees). In magnetic traps called stellarators (in contrast to Tokamaks), configurations of the magnetic field in which the lines of force are wound onto toroidal surfaces (such as those turned into a figure eight, Figure 2, d) are generated only by external windings. Various modifications of stellarators are also under intensive study, with a view to using them to confine hot plasma.

Other mechanisms also exist for stabilizing chute instability. For example, in the Van Allen radiation belt it is stabilized by electrical contact between the plasma and the ionosphere: the charged particles of the ionosphere can compensate for the electric fields that arise in the radiation belts. Controlling chute and other types of plasma instability is one of the main tasks of laboratory research on magnetic traps.