A current-carrying solenoid provides the simplest way to establish a cylindrically homogeneous magnetic field to contain a plasma. While such a device is relatively easy to construct and to operate, it suffers from a serious deficiency: leakage of plasma particles through the open ends constitutes a significant loss of fuel ions and thermal energy. A reduction of this end-leakage can be accomplished by establishing an increasing magnetic field at the two ends so that many of the charged particles are trapped because of the imposed constraints on particle motion with regards to conservation of energy and the magnetic moment as discussed in Secs. 5.7 and 5.8. This concept resulted in the name "magnetic mirror" and is illustrated in Fig. 9.3.

Fig. 9.3: Magnetic field lines in a magnetic mirror with an ion trajectory influenced by the mirror and grad-B effects also suggested.

Some important features of mirror fields can be described and analyzed on the basis of our preceding discussion of ion motion in homogeneous and inhomogeneous magnetic fields.

As discussed in Ch. 5, a positively charged ion spirals in a counterclockwise pattern about the direction of magnetic field lines; this is suggested in Fig. 9.3 for the case of a non-zero speed component of the ion parallel to the direction of the В-field. As the ion approaches the ends, it is increasingly subjected to drifts due to the inhomogeneity of the mirror field. However, note that these drifts occur in azimuthal directions and the particles are still bound to their magnetic surfaces discussed above. Thus they would be well confined in relation to the radial direction (_LB), if not for an occurrent perpendicular diffusion due to collisional and/or turbulent transport, Eq.(6.19), resulting in particle migration out of the contained plasma region.

The more important particle losses, however, will appear in the direction parallel to B, that is through the ends of the solenoid. How effectively these losses can be reduced by squeezing the В-lines at the ends with a magnetic field strength increase, is discussed next.

The concept of trapping the ion in the magnetic "bottle" of a mirror field configuration can be formulated by drawing upon particle kinetic energy conservation and the constancy of the magnetic moment. As in Sec. 5.8, we write for an individual ion the energy in terms of the parallel and perpendicular velocity components, as established by the В-field lines, by

(9.27a)

With E0 a constant of motion and in the absence of other force effects, we can therefore depict the kinetic state of the ion anywhere in a collisionless plasma on

a)

Ion exists on the E0 = constant line

vj* axis

Trapping range (v2)max forV2

Introducing the constant magnetic moment p, Eq.(5.76), the energy conservation reads as = + (9.27b) Depending upon where in the mirror an ion exists, it will sense a different magnetic field-from a Bmm in the mid-plane to a in the mirror plane, Fig. 9.3. This range of magnetic field variation can be introduced into Eq.(9.27b) by recognizing that, with E0 a constant, vy will be a maximum when В is a minimum, Ео = 1тЫ L + fc (9-28) and therefore (уц)^ occurs at the central plane, Fig. 9.3. Conversely, vj| will be a minimum when В is a maximum. The condition for trapping of particles-that is stopping their parallel motion before or at the B^-plane and thus not allowing them to escape through the ends is, in the absence of collisional effects, therefore : 0. (9.29)

(9.34)

VI sin в or, after some trigonometric arranging

sin 2e = — l—. (9.35)

vl

With the help of this expression, it becomes evident that Eq.(9.33) translates into the following statement: particles traveling in the weak magnetic field region, that is about the mid-plane, in directions such that

will be trapped, whereas the others moving within the so-called ‘loss cone’ with a polar angle

will escape from the plasma region. In Eq.(9.36) only the positive sign of the square root is retained because 0 can vary only between 0 and K. Note that Eq.(9.37) actually defines two values of 0O which apparently relates to the two opposite loss cones thus including both the motions parallel and antiparallel to B. Denoting the one solution which is less than 71/2 with 0„, then the second solution is К — 0O.

t

vz

The fraction of an initially isotropic distribution of particles at the mid-plane, f(v)= f(v)/(47tv2), which gets lost through both mirror ends, i. e. particles in both velocity cones, is then

jf(y)d

r _ double cone________________

J loss oo

jf(y)d

0

= 7-cos0o.

Evidently, then cos 0O represents the fraction of the particle distribution which is trapped, that is

f, raP = c™e’ = (9.39)

V В max

where the ratio Bmax/Bmm is called the mirror ratio determining the effectiveness of confinement. As a consequence of the occurring end losses, a mirror plasma is never isotropic. Taking collisions into consideration, it is possible for particles which are in the confined region before collision, to be scattered into the loss cone.

So far we have discussed the confinement of charged particles by a magnetic mirror field which, however, could not explicitly explain why particles are reflected in the increased field of the mirrors. For that, we consider a particle moving outside the loss cone in velocity space. Its parallel motion can be stopped by the field strength of the mirror coils, that is the conserved total kinetic energy

E0 = E± + Et

= {m(vi) +0 (9.40)

z V fmax

= constant

is made up only by the perpendicular motion. If now, by some virtual means, the particle were displaced by an infinitesimal distance into the region of higher B, then it would gyrate faster and thus increase its perpendicular velocity to v_L>(v_L)max. This, however, would violate the energy conservation, Eq.(9.39), if no additional energy was transferred; it follows therefore, that the only directions the particle is allowed to move into-if it is displaced from the plane where it possesses (vx)max-are those associated with vx < (vx)max and thus making room again for E||. Since vx < (vx)max requires a reduced В-field, the particle can regain

E|| only by moving opposite to VB and is thus reflected and returned to the weaker field region. Obviously, the force FM derived in Sec. 5.7 as FM = (1/2mvx2/B)V||B, Eq.(5.72), causes this mirror reflection.