Trapped and Passing Particles
The motion of a charged particle in the presence of a uniform magnetic field consists of
gyration in the plane perpendicular to the local magnetic fieldline combined with a steady drift along the fieldline [18].
Suppose, however, that the magnetic field is nonuniform, meaning that there are spatial gradients in its direction and strength.
As is well known, if the characteristic gradient scalelength of the field is much larger than the particle gyroradius then the motion
still consists of gyration in the plane perpendicular to the local fieldline, combined with drift along the fieldline (which
is no longer straight). However, the magnetic moment,

(2.73) 
of the particle is a conserved quantity during this motion [18,40].
Here,
is the particle's perpendicular (to the magnetic field) velocity, and the local magnetic fieldstrength. Now, it is clear from Table 2.1
that the gyroradii of both electrons and ions in a tokamak fusion reactor are much smaller than the dimensions of the reactor. Assuming that the characteristic
gradient scalelength of the reactor's magnetic field is comparable to its dimensions, we conclude that the motions of both ions and electrons
in such a reactor consist of rapid (see Table 2.1) gyration perpendicular to magnetic fieldlines, combined with drift along
magnetic fieldlines at constant magnetic moment. However, these motions are interrupted by occasional collisions (i.e., scattering events).
Figure 2.1:
Charged particle motion around a magnetic fluxsurface of circular poloidal crosssection.

Consider the gyroaveraged motion of a charged particle around an idealized magnetic fluxsurface of circular poloidal crosssection.
Let us set up a righthanded cylindrical coordinate system, , , , whose axis corresponds to the symmetry axis
of the tokamak. Let and be the major and minor radii of the fluxsurface, respectively. As shown in Figure 2.1,
we can specify the location of the charged particle in the poloidal plane in terms of an angular coordinate, , which is
zero on the outboard midplane (i.e., , ). In fact, the particle's coordinates in the poloidal plane are
Now, if we neglect the influence of the comparatively small poloidal currents flowing in the plasma on the toroidal magnetic field then we
expect the overall magnetic fieldstrength (which is dominated by the toroidal component of the field) to vary as

(2.76) 
Here, is the toroidal magnetic fieldstrength on the magnetic axis (i.e., , ). Note that the overall magnetic fieldstrength
varies slightly around the fluxsurface, being larger at smaller major radii, and vice versa.
Let us suppose that the parallel
(to the magnetic field) electric fieldstrength,
, is comparatively weak, as is indeed the case in a hightemperature tokamak
plasma (otherwise the field would generate an absurdly large parallel current). (See Section 2.9.) If this is the case then (neglecting collisions, for the moment) our charged particle
drifts around the magnetic fluxsurface with a constant kinetic energy (recall that a magnetic field cannot do work on the particle). In other words,

(2.77) 
is a constant of the motion. Here,
is the particle's parallel velocity. However, the particle's magnetic moment, which is defined in Equation (2.73),
is also a constant of the motion. Equations (2.73) and (2.77) can be combined to give

(2.78) 
Hence, we conclude that the particle is excluded from regions of the fluxsurface in which
(because
clearly cannot be imaginary). In fact, if the
particle reaches a socalled bounce point, characterized by
, where
, then its parallel motion must reverse direction (i.e., the sign in the previous equation must flip).
Let
and
be the particle's parallel and perpendicular velocities at the outermost point of the fluxsurface (i.e., ), where the
magnetic field is weakest. It follows that and both take the constant values
respectively.
The previous three equations can be combined to give

(2.81) 
Finally, Equation (2.76) yields

(2.82) 
where
is the inverse aspectratio of the fluxsurface, and we have made use of the large aspectratio approximation
.
The previous equation, combined with the requirement that
not be imaginary, leads to the conclusion that (neglecting collisions) there are two populations of charged particle on a magnetic fluxsurface. The first population
satisfies

(2.83) 
where

(2.84) 
is known as the pitch angle (on the outboard midplane).
This first population of particles are called passing particles because they all have sufficiently large parallel
velocities that they can circulate freely around the fluxsurface. The second population satisfies

(2.85) 
These socalled trapped particles have small enough parallel velocities that they are trapped on the outer part of the fluxsurface, oscillating between
bounce points located at
, where

(2.86) 
(See Figure 2.2.)
Assuming that the charged particles at the outermost point on the fluxsurface have a Maxwellian velocity distribution, all possible values of
are equally likely. Thus, we conclude that the fraction of the particles on the fluxsurface that are trapped is

(2.87) 
[A more exact expression for is specified in Equation (2.202).]
Given that
, it is clear that the trapped particle fraction grows as we move from the innermost (i.e., ) to the outermost (i.e., ) magnetic fluxsurface in the plasma.
Consider a trapped particle oscillating between bounce points located at
. Because the particle is drifting parallel to the magnetic field, its motion
is characterized by
. [See Equation (1.76).] Let represent pathlength along a magnetic fieldline. It
follows that
. The particle's parallel equation of motion is

(2.88) 
where use has been made of Equations (2.82), (2.84), and (2.86). Hence, we obtain

(2.89) 
Taking the derivative of the previous equation with respect to time, and making use of Equations (2.84) and (2.86), we get

(2.90) 
where

(2.91) 
is known as the bounce frequency [52]. Here, we have
made the large aspectratio approximation that, on average,
for a trapped particle. Obviously, Equation (2.90)
has the same form as the equation of motion of a simple pendulum.
It follows that a deeply trapped particle, characterized by
, executes simple harmonic motion in ,
between bounce points located at
, at the angular frequency
.
In this case, the appropriate solution of Equation (2.90) is

(2.92) 
A more accurate expression for the bounce frequency, which does not assume that the particle is deeply trapped, is

(2.93) 
where
is a complete elliptic integral of the first kind [1].
Note that the bounce frequency decreases with increasing amplitude of the angular motion, eventually approaching zero
logarithmically as
.
Let us now take collisions into account. In the absence of collisions, the value of the pitch angle on the outboard midplane,
, is a constant of a given particle's
motion. Thus, we conclude that if the particle is initially trapped then it remains trapped at all subsequent times. However, collisions cause the
value of
to diffuse in velocity space. (Note that Coulomb collisions in a high temperature plasma are dominated by
smallangle scattering events [18], which means that each collision only causes a small change in
.)
It takes a time of order (where is the collision time) to change
by order unity.
However, it is only necessary to change
by order
in order to detrap a trapped particle. [See Equation (2.85).] The time
required for collisional detrapping is thus

(2.94) 
Now, if
then a trapped particle is collisionally detrapped long before it has time to complete an oscillation between its
bounce points. Obviously, in this case, the plasma is sufficiently collisional that it is meaningless to draw a distinction between trapped and passing particles.
On the other hand, if
then collisions are too infrequent to interfere with the oscillations of a trapped particle between
its bounce points.
Table: 2.4
Neoclassical parameters in a lowfield and a highfield tokamak reactor. Here, is the toroidal
magnetic fieldstrength, the fraction of trapped particles,
the electron collisionality parameter,
the ion collisionality parameter,
the electron banana orbit width,
the ion banana orbit width, the electron toroidal magnetization parameter, the
ion toroidal magnetization parameter,
the electron poloidal magnetization parameter,
the ion poloidal magnetization parameter,
the
electron collisionality parameter, and
the ion collisionality parameter. All quantities are calculated with and . (See Table 2.1.)

LowField 
HighField 

5.0 
12.0 

0.81 
0.81 































It is helpful to define the dimensionless collisionality parameter [64],

(2.95) 
Note that
.
It follows that the criterion for species trapped particles to exist is
.
Table 2.4 shows that
and
are both much less than unity in a tokamak fusion reactor, implying that populations of
trapped electrons as well as trapped ions exist in such reactors. Moreover, it is clear that the fraction of trapped particles, which is the same for both plasma species, is about on the
outermost magnetic fluxsurfaces.
Figure: 2.2
A gyroaveraged trapped particle orbit (solid line) on a magnetic fluxsurface (dashed line) of circular poloidal crosssection. The orbit parameters are
,
, and
.

Let us examine the motion of a trapped particle slightly more accurately. In an axisymmetric tokamak plasma, the gyroaveraged motion of the particle
takes place at constant toroidal canonical angular momentum:

(2.96) 
where denotes magnetic vector potential.
Note that

(2.97) 
Suppose that the particle is trapped on a magnetic fluxsurface of minor radius . We can write

(2.98) 
in the immediate vicinity of the fluxsurface, where use has been made of Equation (2.97). Now,

(2.99) 
where use has been made of Equations (2.84), (2.86), and (2.88). Thus, employing Equations (2.96) and (2.98),

(2.100) 
which implies that the gyroaveraged particle orbit satisfies
where

(2.102) 
Here, we have again used the large aspectratio approximation that, on average,
for a trapped particle. Note that the signs in Equation (2.101) correspond to motion in the
and
directions, respectively. As is illustrated in Figure 2.2, a trapped particle that oscillates between its
bounce points makes radial excursions from the guiding magnetic fluxsurface that are of amplitude
. In fact, the gyroaveraged orbit's poloidal crosssection looks a
little like a banana. Hence, such orbits are known as banana orbits, and
is known as the banana width [52]. [Note that
is the banana width of a barely trapped particle (i.e.,
). Deeply
trapped particles (i.e.,
) have much narrower banana widths.] Finally, as is clear from Tables 2.1 and 2.4, although the electron and
ion banana widths in a tokamak fusion reactor are much greater than the corresponding gyroradii, they are much smaller than the minor radius of the plasma.