j Á E
h
i ¼
1
ffiffi ffi
2
p π 3=2
1
n e λ
3
De
Z
2 n i m
X 1
n¼1
nω
ð Þ
2
Z 1
0
dk
k
3
1
ε RPA k, nω
ð
Þ
j
j
2
 exp À nω
ð Þ
2 = 2v
2
e k
2
À
Á
h
i
exp À ħk
ð Þ
2 = 8m
2 v
2
e
À
Á
h
i
Â
sinh nħω=2T
ð
Þ
nħω=2T
ð
Þ
Z 1
0
dzJ
2
n
eE 0 k
mω 2 z
ð3:4:5Þ
The collision frequency by the IB process ν
IB
ei is derived using the relation (2.6.13)
as:
ν
IB
ei ¼
ω
2
ω 2
pe
ν E ,
ν E ¼
j Á E
h
i
ε 0 E
2
ð3:4:6Þ
In the classical mechanics, ħ ! 0, this expression tends to the classical ones (3.4.2)
and (3.4.3) in the both limits derived by Silin [12]. Decker et al. [11] have obtained
such expression in the framework of the nonlinear Dawson-Oberman kinetic theory.
However, the classical formulation has the well-known problem of a divergence at
large k that is solved by Landau cut procedure. In contrast, the quantum formula of
(3.4.5) has no divergence in the integral to k.
Quantum effects, indicated by ħ, appear in the following three places in (3.4.5).
The first place is one of the exponential functions in (3.4.5) describing the quantum
diffraction effect at large momenta k. The exponential function ensures the convergence of the integral by k. It is noted that the first exponential vanishes at small k,
and the second one does at large k, and the divergence is avoided at both limits. This
is approximately expressed with Coulomb log in the classical calculation. The
second place is the term with the sinh function that is connected with the Bose
statistics of multiple-photon emission and absorption. Finally, the quantum effects
are also included in the dielectric function derived with random phase approximation
(RPA).
Equation (3.4.5) is solved numerically for hydrogen with the standard parameters
given above and compared with the classical ones. In Fig. 3.18, the solid line is the
result from (3.4.5) for T ¼ 30 eV [13]. It is clear that the nonlinear effects become
dominant for v os /v e > 1, and the collision frequency reduces roughly in proportion to
(v os /v e )
À3 as predicted by Silin [12]. It is constant in the linear collision phase for v os /
v e < 1 , and the difference of the value comes from how to evaluate the Coulomb log
in the classical formulations. It is seen that the quantum formulation provides 3–5
times larger values compared to the two classical results.
In order to see explicitly the breakdown of Coulomb log obtained by the cuts at
both k limits, the collision frequency dependence to the coupling parameter Γ
defined in (2.5.14) is studied for a fixed v os /v e ¼ 0.2 and 10 by varying the
temperature T. In Fig. 3.19, the results for the case of v os /v e ¼ 10 are plotted and
compared to two classical models. The result from (3.4.5) continuously increases
100
3 Ultra-Short Pulse and Collisionless Absorption
h
i ¼
1
ffiffi ffi
2
p π 3=2
1
n e λ
3
De
Z
2 n i m
X 1
n¼1
nω
ð Þ
2
Z 1
0
dk
k
3
1
ε RPA k, nω
ð
Þ
j
j
2
 exp À nω
ð Þ
2 = 2v
2
e k
2
À
Á
h
i
exp À ħk
ð Þ
2 = 8m
2 v
2
e
À
Á
h
i
Â
sinh nħω=2T
ð
Þ
nħω=2T
ð
Þ
Z 1
0
dzJ
2
n
eE 0 k
mω 2 z
ð3:4:5Þ
The collision frequency by the IB process ν
IB
ei is derived using the relation (2.6.13)
as:
ν
IB
ei ¼
ω
2
ω 2
pe
ν E ,
ν E ¼
j Á E
h
i
ε 0 E
2
ð3:4:6Þ
In the classical mechanics, ħ ! 0, this expression tends to the classical ones (3.4.2)
and (3.4.3) in the both limits derived by Silin [12]. Decker et al. [11] have obtained
such expression in the framework of the nonlinear Dawson-Oberman kinetic theory.
However, the classical formulation has the well-known problem of a divergence at
large k that is solved by Landau cut procedure. In contrast, the quantum formula of
(3.4.5) has no divergence in the integral to k.
Quantum effects, indicated by ħ, appear in the following three places in (3.4.5).
The first place is one of the exponential functions in (3.4.5) describing the quantum
diffraction effect at large momenta k. The exponential function ensures the convergence of the integral by k. It is noted that the first exponential vanishes at small k,
and the second one does at large k, and the divergence is avoided at both limits. This
is approximately expressed with Coulomb log in the classical calculation. The
second place is the term with the sinh function that is connected with the Bose
statistics of multiple-photon emission and absorption. Finally, the quantum effects
are also included in the dielectric function derived with random phase approximation
(RPA).
Equation (3.4.5) is solved numerically for hydrogen with the standard parameters
given above and compared with the classical ones. In Fig. 3.18, the solid line is the
result from (3.4.5) for T ¼ 30 eV [13]. It is clear that the nonlinear effects become
dominant for v os /v e > 1, and the collision frequency reduces roughly in proportion to
(v os /v e )
À3 as predicted by Silin [12]. It is constant in the linear collision phase for v os /
v e < 1 , and the difference of the value comes from how to evaluate the Coulomb log
in the classical formulations. It is seen that the quantum formulation provides 3–5
times larger values compared to the two classical results.
In order to see explicitly the breakdown of Coulomb log obtained by the cuts at
both k limits, the collision frequency dependence to the coupling parameter Γ
defined in (2.5.14) is studied for a fixed v os /v e ¼ 0.2 and 10 by varying the
temperature T. In Fig. 3.19, the results for the case of v os /v e ¼ 10 are plotted and
compared to two classical models. The result from (3.4.5) continuously increases
100
3 Ultra-Short Pulse and Collisionless Absorption