Feasibility Studies of Alpha-Channeling in Mirror Machines Page: 4 of 12
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II. ALPHA-CHANNELING IN MIRROR
The a-channeling effect occurs when a particles born
through fusion reactions diffuse along one-dimensional
paths in the phase space (F, p) due to resonant interac-
tion with electromagnetic waves. If the diffusion induced
along the path is suppressed at high energy, whereas at
low-energy there is an effective particle "sink", the inter-
action with the waves will result in the ejection of the cold
a particles from the system and the simultaneous transfer
of their initial energy to the waves, thereby accomplish-
ing the so-called "a-channeling" effect . The quick
a particle ejection leads to fusion ash removal, while by
coupling the amplified wave to the background plasma
species, it is possible to redirect extracted energy to the
plasma, thus suppressing the instabilities and increasing
the effective fusion reactivity compared to the typical sce-
nario, in which a particles heat plasma by slowing down
collisionally on electrons .
The a-channeling effect might be achieved in a mirror
machine by arranging several rf regions along the device
length. If the wave frequency w is approximately equal
to the nth harmonic of the a particle cyclotron frequency
Q, the wave will be resonantly coupled to the a particles
having a parallel velocity vll ' / B1B = v res given by:
vll res (1)
where v is an a particle velocity and B is the magnetic
If the wave amplitude is large enough, the resonant a
particle dynamics becomes stochastic. Assuming a slab
geometry approximation, one can show using the canon-
ical perturbation theory  that the corresponding par-
ticle random walk is constrained to a one-dimensional
curve in the unperturbed action variable space :
where it is assumed that the perpendicular (to the mag-
netic field) component k1 of the wave vector k is di-
rected along yo, X and Y are the particle gyrocenter
coordinates, B is directed along z, kl= k - B/ B and p
is the magnetic moment of the particle. Note that the
particle excursion given by Eq. (2) is directed along the
resonance surface -nQ - kllvl =0 only if kll = 0. In
0 1 2 3 4 WIIo, MeV
FIG. 1: (Color online) Arrangement of the diffusion paths
(bars intersecting at W1I ~ WI res) in the midplane energy
space. Born through the fusion reaction a particles dif-
fuse along the paths limited at high energies (limitation is
schematically shown with the short segments) and leave the
device through the loss cone at the intersection points with
the paths [with the energy W ~ Wrs - (m/2)(w-nQ)2/k ].
other words, if kl = 0, the resonance condition is satis-
fied even if the change of the particle energy is substan-
tial. For simplicity in choosing diffusion paths, namely
in order to be sure that the particles in resonance re-
main in resonance, we proceed in the following assuming
that k < kL. However, this condition is not strictly
necessary for a-channeling, particularly when there are
multiple rf regions in inhomogeneous fields.
Thus, the particle parallel velocity change will be small
compared to the characteristic perpendicular velocity
change if k l< kL. Since the wave-particle interaction
occurs in the region where the resonance condition is sat-
isfied, but not necessarily in the midplane, the diffusion
path equation in the midplane energy space turns out to
be given by :
where WI and W are the particle's parallel and perpen-
dicular energies at the midplane correspondingly, and Rrf
is the mirror ratio at the rf region location. If the a par-
ticle resonant velocity is nonzero, such a diffusion path
intersects the loss cone (see Fig. 1). Since the particle
heating along the path can be limited for kip 1 ,
where p is the a particle gyroradius, particle cooling ac-
(Rrf- - 1)-i
(R, f - 1) -1
Wl 2 k + WL (Rrf
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Zhmoginov, A. I. & Fisch, N. J. Feasibility Studies of Alpha-Channeling in Mirror Machines, report, March 19, 2010; Princeton, New Jersey. (digital.library.unt.edu/ark:/67531/metadc1013916/m1/4/: accessed February 15, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.