A Fokker-Planck model for the interaction of fast ions with the thermal electrons in a quasineutral plasma is developed. When the fast ion population has a net flux (i.e., the distribution of fast ions is anisotropic in velocity space), the electron distribution function is perturbed from Maxwellian by collisions with the fast ions, even if the fast ion density is orders of magnitude smaller than the electron density. The Fokker-Planck model is used to derive classical electron transport equations (a generalized Ohm's law and a heat flow equation) that include the effects of the electron-fast ion collisions. It is found that these collisions result in a collisionally induced current term in the transport equations which can be significant. The new transport equations are analyzed in the context of a number of scenarios including α particle heating in inertial confinement fusion and magnetoinertial fusion plasmas as well as ion beam heating of dense plasmas.

1.
J.
Lindl
, “
Development of the indirect–drive approach to inertial confinement fusion and the target physics basis for ignition and gain
,”
Phys. Plasmas
2
,
3933
4024
(
1995
).
2.
I.
Lindemuth
and
R.
Kirkpatrick
, “
Parameter space for magnetized fuel targets in inertial confinement fusion
,”
Nucl. Fusion
23
,
263
(
1983
).
3.
G. A.
Wurden
,
S. C.
Hsu
,
T. P.
Intrator
,
T. C.
Grabowski
,
J. H.
Degnan
,
M.
Domonkos
,
P. J.
Turchi
,
E. M.
Campbell
,
D. B.
Sinars
,
M. C.
Herrmann
,
R.
Betti
,
B. S.
Bauer
,
I. R.
Lindemuth
,
R. E.
Siemon
,
R. L.
Miller
,
M.
Laberge
, and
M.
Delage
, “
Magneto-inertial fusion
,”
J. Fusion Energy
35
,
69
77
(
2016
).
4.
J. J.
Honrubia
and
M.
Murakami
, “
Ion beam requirements for fast ignition of inertial fusion targets
,”
Phys. Plasmas
22
,
012703
(
2015
).
5.
J.
Fernández
,
B.
Albright
,
F.
Beg
,
M.
Foord
,
B.
Hegelich
,
J.
Honrubia
,
M.
Roth
,
R.
Stephens
, and
L.
Yin
, “
Fast ignition with laser-driven proton and ion beams
,”
Nucl. Fusion
54
,
054006
(
2014
).
6.
A. L.
Velikovich
,
R. W.
Clark
,
J.
Davis
,
Y. K.
Chong
,
C.
Deeney
,
C. A.
Coverdale
,
C. L.
Ruiz
,
G. W.
Cooper
,
A. J.
Nelson
,
J.
Franklin
, and
L. I.
Rudakov
, “
Z-pinch plasma neutron sources
,”
Phys. Plasmas
14
,
022701
(
2007
).
7.
M.
Krishnan
, “
The dense plasma focus: A versatile dense pinch for diverse applications
,”
IEEE Trans. Plasma Sci.
40
,
3189
3221
(
2012
).
8.
W.
Cayzac
,
A.
Frank
,
A.
Ortner
,
V.
Bagnoud
,
M.
Basko
,
S.
Bedacht
,
C.
Bläser
,
A.
Blažević
,
S.
Busold
,
O.
Deppert
 et al, “
Experimental discrimination of ion stopping models near the Bragg peak in highly ionized matter
,”
Nat. Commun.
8
,
15693
(
2017
).
9.
J. A.
Frenje
,
P. E.
Grabowski
,
C. K.
Li
,
F. H.
Séguin
,
A. B.
Zylstra
,
M.
Gatu Johnson
,
R. D.
Petrasso
,
V. Y.
Glebov
, and
T. C.
Sangster
, “
Measurements of ion stopping around the Bragg peak in high-energy-density plasmas
,”
Phys. Rev. Lett.
115
,
205001
(
2015
).
10.
L. S.
Brown
,
D. L.
Preston
, and
R. L.
Singleton
 Jr
, “
Charged particle motion in a highly ionized plasma
,”
Phys. Rep.
410
,
237
333
(
2005
).
11.
D.
Michta
,
F.
Graziani
,
T.
Luu
, and
J.
Pruet
, “
Effects of nonequilibrium particle distributions in deuterium-tritium burning
,”
Phys. Plasmas
17
,
012707
(
2010
).
12.
B. E.
Peigney
,
O.
Larroche
, and
V.
Tikhonchuk
, “
Ion kinetic effects on the ignition and burn of inertial confinement fusion targets: A multi-scale approach
,”
Phys. Plasmas
21
,
122709
(
2014
).
13.
M.
Sherlock
and
S.
Rose
, “
The persistence of Maxwellian D and T distributions during burn in inertial confinement fusion
,”
High Energy Density Phys.
5
,
27
30
(
2009
).
14.

We use an average ion approximation for a thermal ion population containing both D and T ions.

15.

We use the subscript α to denote the fast ion population but remind the reader that our model can be generalized to any species of fast ions, not just α particles.

16.
E. M.
Epperlein
and
M. G.
Haines
, “
Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker–Planck equation
,”
Phys. Fluids
29
,
1029
1041
(
1986
).
17.
T. W.
Johnston
, “
Cartesian tensor scalar product and spherical harmonic expansions in Boltzmann's equation
,”
Phys. Rev.
120
,
1103
1111
(
1960
).
18.
T. W.
Johnston
, “
Cartesian tensor scalar product and spherical harmonic expansions in Boltzmann's equation
,”
Phys. Rev.
120
,
2277
2277
(
1960
).
19.
T. W.
Johnston
, “
Cartesian tensor scalar product and spherical harmonic expansions in Boltzmann's equation
,”
Phys. Rev.
122
,
1962
1962
(
1961
).
20.
I. P.
Shkarofsky
, “
Cartesian tensor expansion of the Fokker–Planck equation
,”
Can. J. Phys.
41
,
1753
1775
(
1963
).
21.
I. P.
Shkarofsky
,
T. W.
Johnston
, and
M. P.
Bachynski
,
The Particle Kinetics of Plasmas
(
Addison-Wesley Publishing
,
1966
).
22.
S. I.
Braginskii
, “
Transport processes in a plasma
,”
Rev. Plasma Phys.
1
,
205
(
1965
).
23.
B.
Appelbe
and
J.
Chittenden
, “
The production spectrum in fusion plasmas
,”
Plasma Phys. Controlled Fusion
53
,
045002
(
2011
).
24.
H.-S.
Bosch
and
G.
Hale
, “
Improved formulas for fusion cross-sections and thermal reactivities
,”
Nucl. Fusion
32
,
611
(
1992
).
25.
J.
Cordey
,
E.
Jones
,
D.
Start
,
A.
Curtis
, and
I.
Jones
, “
A kinetic theory of beam-induced plasma currents
,”
Nucl. Fusion
19
,
249
(
1979
).
26.
N. J.
Fisch
, “
Transport in driven plasmas
,”
Phys. Fluids
29
,
172
179
(
1986
).
27.
I. P.
Shkarofsky
,
I. B.
Bernstein
, and
B. B.
Robinson
, “
Condensed presentation of transport coefficients in a fully ionized plasma
,”
Phys. Fluids
6
,
40
47
(
1963
).
28.
A. R.
Hochstim
,
Kinetic Processes in Gases and Plasmas
(
Academic
,
New York; London
,
1969
).
29.
L.
Spitzer
and
R.
Härm
, “
Transport phenomena in a completely ionized gas
,”
Phys. Rev.
89
,
977
981
(
1953
).
30.

Numerical calculations show that similar values (discrepancies ∼1%) of ξeα and ζeα are obtained for fast ion distributions with a broad range of energies as for monoenergetic distributions with an energy equal to the mean of the energy range. Therefore, we assume that the monoenergetic distributions (29) and (30) are good approximations for a wide range of fast ion distributions if we use the mean energy of the distribution to calculate vα and ensure the same value of drift velocity.

31.
T.
Ohkawa
, “
New methods of driving plasma current in fusion devices
,”
Nucl. Fusion
10
,
185
(
1970
).
32.
S. L.
Pape
,
L. F.
Berzak Hopkins
,
L.
Divol
,
A.
Pak
,
E. L.
Dewald
,
S.
Bhandarkar
,
L. R.
Bennedetti
,
T.
Bunn
,
J.
Biener
,
J.
Crippen
,
D.
Casey
,
D.
Edgell
,
D. N.
Fittinghoff
,
M.
Gatu-Johnson
,
C.
Goyon
,
S.
Haan
,
R.
Hatarik
,
M.
Havre
,
D. D.-M.
Ho
,
N.
Izumi
,
J.
Jaquez
,
S. F.
Khan
,
G. A.
Kyrala
,
T.
Ma
,
A. J.
Mackinnon
,
A. G.
MacPhee
,
B. J.
MacGowan
,
N. B.
Meezan
,
J.
Milovich
,
M.
Millot
,
P.
Michel
,
S. R.
Nagel
,
A.
Nikroo
,
P.
Patel
,
J.
Ralph
,
J. S.
Ross
,
N. G.
Rice
,
D.
Strozzi
,
M.
Stadermann
,
P.
Volegov
,
C.
Yeamans
,
C.
Weber
,
C.
Wild
,
D.
Callahan
, and
O. A.
Hurricane
, “
Fusion energy output greater than the kinetic energy of an imploding shell at the national ignition facility
,”
Phys. Rev. Lett.
120
,
245003
(
2018
).
33.
J. K.
Tong
,
K.
McGlinchey
,
B.
Appelbe
,
C. A.
Walsh
,
A. J.
Crilly
, and
J. P.
Chittenden
, “
Burn regimes in the hydrodynamic scaling of perturbed inertial confinement fusion hotspots
,”
Nucl. Fusion
59
,
086015
(
2019
).
34.
S. Y.
Gus'kov
,
O. N.
Krokhin
, and
V. B.
Rozanov
, “
Transport of energy by charged particles in a laser plasma
,”
Sov. J. Quantum Electron.
4
,
895
898
(
1975
).
35.
C. A.
Walsh
,
J. P.
Chittenden
,
K.
McGlinchey
,
N. P. L.
Niasse
, and
B. D.
Appelbe
, “
Self-generated magnetic fields in the stagnation phase of indirect-drive implosions on the national ignition facility
,”
Phys. Rev. Lett.
118
,
155001
(
2017
).
36.
L. J.
Perkins
,
D. D.-M.
Ho
,
B. G.
Logan
,
G. B.
Zimmerman
,
M. A.
Rhodes
,
D. J.
Strozzi
,
D. T.
Blackfield
, and
S. A.
Hawkins
, “
The potential of imposed magnetic fields for enhancing ignition probability and fusion energy yield in indirect-drive inertial confinement fusion
,”
Phys. Plasmas
24
,
062708
(
2017
).
37.
L. J.
Perkins
,
B. G.
Logan
,
G. B.
Zimmerman
, and
C. J.
Werner
, “
Two-dimensional simulations of thermonuclear burn in ignition-scale inertial confinement fusion targets under compressed axial magnetic fields
,”
Phys. Plasmas
20
,
072708
(
2013
).
38.
S. A.
Slutz
and
R. A.
Vesey
, “
High-gain magnetized inertial fusion
,”
Phys. Rev. Lett.
108
,
025003
(
2012
).
39.
S. A.
Slutz
,
M. C.
Herrmann
,
R. A.
Vesey
,
A. B.
Sefkow
,
D. B.
Sinars
,
D. C.
Rovang
,
K. J.
Peterson
, and
M. E.
Cuneo
, “
Pulsed-power-driven cylindrical liner implosions of laser preheated fuel magnetized with an axial field
,”
Phys. Plasmas
17
,
056303
(
2010
).
40.
L.
Biermann
, “
Über den Ursprung der Magnetfelder auf Sternen und im interstellaren Raum (miteinem Anhang von A. Schlüter)
,”
Z. Naturforsch.
5
,
65
(
1950
).
41.
J. D.
Huba
,
Plasma Physics
(
Naval Research Laboratory
,
Washington, DC
,
2013
), pp.
1
71
.
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