Propagation of Charged Particles through Helical Magnetic Fields

1
Propagation of Charged Particles through Helical
Magnetic Fields
C. Muscatello, T. Vachaspati, F. Ferrer Dept. of
Physics CWRU 10900 Euclid Ave., Cleveland, OH
44106
METHOD contd
In order to easily generate three components of
the magnetic field, we should want each of the
components to be independent of each other. If we
choose two mutually orthogonal vectors to k (say
l and m) then the off-diagonal components of the
first term in the brackets on the RHS of the
correlation function (eq.1) are zero. It then
turns out that bl and bm are simply generated
directly according to S(k) while bk is zero
(eq.3). In order to introduce helicity, two more
mutually orthogonal vector components of the
magnetic field should be generated bu and bv.
These two components are independent of k and can
be written in terms of bl and bm. bu and bv are
generated as Gaussian deviates with variances
S(k)A(k) and S(k)-A(k), respectively. Once bu
and bv are calculated, it is a trivial matter to
solve for bl and bm and equally trivial to
transform the b vector components to the home
(k1,k2,k3) Fourier basis. Using the
aforementioned method, we can generate magnetic
field vectors for a chosen range of k values
given some numerical spacing. To convert the
magnetic field into Cartesian space, the method
of Fourier transform is numerically implemented.
According to Fourier analysis, the magnetic
fields components are transformed independently
according to (where the index indicates the field
component), (4) The preceding
procedure is repeated for all three spatial
components using a 3-dimensional Fast Fourier
Transform (FFT) algorithm, and the values are
masked onto the numerical grid. ii.
Particle Propagation Particles propagate the
magnetic field, and their trajectories are
determined by the usual Lorentz force equation
where the electric field is assumed to be zero. A
slight manipulation is made to more easily
calculate the incremental change in momentum of
particles at each step through the mesh. The
following is calculated for every particle at
every step, (5) Because the
magnitude of the magnetic field is very small in
systems where this method is applicable, a
perturbative method is used to calculate the
total momentum of each particle on its exit from
the mesh. For each particle, the following is
calculated at each grid space i, (
6)
FFT
Figure 1. Above Left Magnetic field in Fourier
space with power spectra S(k)k2 and A(k) k2 /
3. Above Right Magnetic field in
Cartesian space. In both representations, similar
magnitude vectors share similar colors.
Figure 1. Left Depiction of helical field line
twisting around a toroidal axis1.
RESULTS AND CONCLUSIONS Figure 1 shows one
possible configuration of the magnetic field in
both Fourier and Cartesian spaces. Figure 2 left
shows how eq.6 is graphically represented for
each particle, and Figure 2 right shows the
momentum vector field for a flux of particles
traveling in the x2 direction. The code written
for this project allows a flux of particles to
travel in any one of the three principal
directions. By studying the change in momenta for
a collection of particles, we may be able to
determine if there is a general trend for
particle propagation direction through helical
fields. If this work were to be continued, the
next step would be to investigate the
relationship between the configuration of the
momentum vectors and the helicity of the field.
Overall, we have taken an active approach to
create a methodology to model particle
propagation through a magnetic field with given
analytic power spectra.
Figure 2. Above Left Diagram depicting how delta
momentum vectors for each particle trajectory
are determined. po is the unperturbed momentum
of the particle across the mesh, p is the
particles average momentum under the influence
of the magnetic field, and dp represents
the shift in the particles momentum across the
field (the difference between po and p) .
Above Right Delta momentum (dp) vectors of flux
of particles traveling parallel to the x2
direction (out of the page).
REFERENCES 1 Berger, Mitchell A 1999
Plasma Phys. 41 B167.