Relativistic Electron Beam Acceleration by Compton Scattering
of Lower-Hybrid Waves
R. Sugaya, T. Maehara and M. Sugawa
Department of Physics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan
Abstract. It has been proved theoretically and numerically that the highly relativistic
electron beam can be accelerated efficiently via the Compton scattering induced by nonlinear
Landau and cyclotron damping of the lower-hybrid waves.
1. INTRODUCTION
Acceleration and heating of a relativistic electron beam by the Compton scattering of
electrostatic waves propagating almost perpendicularly in a magnetized plasma are
investigated theoretically and numerically on the basis of the relativistic kinetic wave and
transport equations1-5. Two electrostatic waves interact nonlinearly with the relativistic
electron beam, satisfying the resonance condition for the Compton scattering (nonlinear
Landau and cyclotron damping)4,5,
k  k   (k  k )v  mce
,
(1)
where v
vb , vb is the parallel velocity of the relativistic electron beam, ce  eB0 /  b me c
(  b  (1  p 2 / me2c 2 )1/ 2 , p   b me v ) is the relativistic electron cyclotron frequency, and m is
an integer. The relativistic transport equations using the relativistic drifted Maxwellian
momentum distribution function of the relativistic electron beam were derived and analyzed.
2. BASIC EQUATIONS
The relativistic kinetic wave equations of the two electrostatic waves and the relativistic
transport equations for the relativistic electron beam are expressed as5
U k
  A0 Uk Uk 
,
t
U b k 

A U U ,
t
k 0 k k
U k 
 AU
1 kU k 
t
Pb k 

A0 Uk Uk 
t k
,
(2)
,
(3)
1  ( kk ) 
2

 , U b   d pnb b me c gb , Pb   d pnb pgb ,
8  k 
k

Ak ,k ,k   Im  Ck(b,k) ,k   Dk( b,k) ,k   .
A0  
Ak ,k ,k  , A1  k  A0 ,
Here, U k is the
4 k  k 
k
Ek  (k / k ) Ek is the wave electric field, the background stationary
wave energy density,
where
U k   k | Ek |2 ,
k 
-1-
uniform magnetic field B0  (0, 0, B0 ) is in the z-direction, k   k  k  ,
k  k  k =
( k , 0, k ) , k  (k , 0, k ) and k   (k , 0, k ) are in the x-z plane, the linear damping rate of
the electrostatic waves is assumed to be zero (  k   k   0 ),  k is the dielectric constant, U b
and Pb are the energy and momentum densities of the relativistic electron beam, and gb is
the momentum distribution function of the relativistic electron beam.
The matrix elements
(b )
(b )
Ck ,k ,k  and Dk ,k ,k  giving the nonlinear wave-particle coupling coefficients A0 and A1 are
described in Ref. 5 in detail, where it is needed to set vd  E0  0 in the matrix elements,
because the cross-field drift velocity and the cross-field electric field are absent. Equations
(2) and (3) yield the conservation laws for the total energy and momentum densities of the
electrostatic waves and the relativistic electron beam.
Next we assume the momentum distribution function of the relativistic electron beam
given by a relativistic drifted Maxwellian momentum distribution function,
gb 
where   1  vb2 / c 2 
1/ 2


 v v 
e x p   b  1 b z2  
3
4 m c  K 2 ( ) 
c 

3
e
,
(4)
,   me c 2 / k BTb with the beam temperature Tb , K r is a
modified Bessel function of the rth order.
the nonrelativistic limit ( 
1,

1)

In particular, it is easily found that g b in
becomes a usual drifted Maxwellian distribution

2
exp  me vx2  v y2   vz  vb  / 2k BTb  . Thus the energy and


momentum densities of the relativistic electron beam become as follows:
function2,
gb   2 me k BTb 
3 / 2
 K ( )
1 
U b  nb me c 2   3

2 
 K 2 (  )  
Here, it is found that
,
Pbz  nb me vb 
K3 ( )
,
K2 ( )
Pbx  Pby  0 .
(5)
U b and Pbz in the nonrelativistic limit are reduced to the usual forms
expressed by U b  nb  me c 2  me vb2 / 2  3k BTb / 2  and Pbz  nb mevb .
We investigate the acceleration and heating of the relativistic electron beam with the
nonrelativistic beam temperature of  1 .
By means of the asymptotic expansion of
K r (  )   / 2 
1/ 2
e  1   4r 2  1 / 8  ,
U b and Pb can be approximated as
Ub 


5
1 
5 
nb me c 2  1 

and Pbz  nb me vb  1 
Immediately we can get the
.
2 
 2  
 2 
simple relativistic transport equations with  ,  1 from Eqs. (3), and they are expressed in
the followings:


nb me c 2    k  A0U kU k 

t
k
,
 k  c2  k  bv k  

 2   0A kU k U
 nb kB Tb  
t
 k 
 k vb
-2-
(6)
.
(7)
Equation (6) shows the acceleration of the relativistic electron beam, and Eq. (7) shows its
heating and cooling. It can be proved from Ref. 5 that the following relation holds:
k (v 0  vb )  mce
,
(8)
A0 , A1  m 
1  v 0k  / k c 2 ce
where v 0  k  mce  / k  .
k (v 0  vb ) p2 / ce
In the case of m  0 ,
(  p  1  v 20 / c 2 
1/ 2
),
Eq. (8) becomes
A0 , A1  m 
and hence it can be stated that when
v 0  vb
( v 0  vb ) and k   0 , the relativistic electron beam can be accelerated (decelerated) via
m  0 scattering4,5. For m  0 , we find that the relativistic electron beam can be
accelerated (decelerated) always when m  0 ( m  0 ), because of m m | k c / m |
(v 0
vb
c )4,5.
3. NUMERICAL ANALYSIS FOR LOWER-HYBRID WAVES
In order to investigate the detailed behavior of the Compton scattering of the two
lower-hybrid waves, we performed the numerical analysis of the dimensionless nonlinear
wave-particle coupling coefficients  0  ( b / ci ) A0 and 1  ( b / ci ) A1 for m  0, 1 ,
where  b  nb me c 2 , ci  Zi eB0 / mi c , 0  k , 1  k  ,    k  , k0  k , k1  k  .
The numerical calculation was carried out under the plasma parameters of  pi2 / ci2  500 ,
2
 pb
/ ce2 0  0.01 , vte / c  0.05 , mi / me  1840 , Zi  1 , Te / Ti  1 , k 0 vti / ci  0.03 , and
Here, vts   2k BTs / ms  (s=e,i), and ce 0  eB0 / mec is the
nonrelativistic electron cyclotron frequency.
Furthermore, it was confirmed that
(b )
(b )
Im Dk ,k ,k  / Ck ,k ,k  1 , that is, the plasma shielding effect is negligibly small compared with
k 1vti / ci  0.028 .
the Compton scattering.
1/ 2
Figure 1(a) exhibits  0 and 1 versus  for  p /   1.02 ,
  1000 , m  0 , 0 / ci  26.46 , k0vti / ci  1.6 , 1 / ci  23.75 and k1vti / ci  1.83 .
The solid and dotted curves correspond to  0 and 1 , respectively. Figure 1(b) exhibits
 0 and 1 versus  for  p /   1 ,   1000 , m  1 , 0 / ci  24.39 , k0vti / ci  1.8 ,
1 / ci  20.83 ~ 22.67 and k1vti / ci  2.17 ~ 3.76 . The resonance condition Eq. (1) can
be satisfied when ce  ce0 /     2ci with   1000 . In Fig. 2(a), the absolute value
of  0 is shown versus  p /  for   2000 ,   100 , m  0 , 0 / ci  25.46 ,
k0vti / ci  1.6 , 1 / ci  23.75 and k1vti / ci  1.83 . The solid and dotted curves
correspond to  0  0 and  0  0 , respectively. In Fig. 2(b),  0 is shown versus  p / 
for   20000 ,   100 , m  1 , 0 / ci  25.46 , k0vti / ci  1.6 , 1 / ci  23.65 ~ 23.72
and k1vti / ci  1.83 ~ 1.86 .
4. CONCLUSION
It can be verified that the Compton scattering of the lower-hybrid waves can accelerate
efficiently the highly relativistic electron beam6. It can be available usefully to the
-3-
acceleration of the highly relativistic electron beam. This research was performed partially
under the collaborating Research Programs at the Institute of Laser Engineering, Osaka
University, and National Institute for Fusion Science.
Fig.1.
Here,  0 and 1 versus  are shown.
Fig. 2.
Here,  0 and  0 versus  p /  are shown.
References
[1] R. Sugaya, J. Phys. Soc. Jpn., 59, 3227 (1990); 60, 518 (1991).
[2] R. Sugaya, Phys. Plasmas, 1, 2768 (1994); 3, 3485 (1996).
[3] R. Sugaya, J. Plasma Phys., 56, 193 (1996); 64, 109 (2000); 66, 143 (2001).
[4] R. Sugaya, J. Plasma Phys., 6, 4333 (1999); 70, 331 (2004).
[5] R. Sugaya, Phys. Plasmas, 10, 3939 (2003).
[6] K. Kitagawa, Y. Sentoku, S. Akamatsu et al., Phys. Rev. Lett. 92, 205002 (2004).
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