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Stimulated scattering and frequency diffusion of photons in plasmas
J. Quant. Spectrosc. Radiat. Transfer Vol. 55, No. 6, pp. 809-813, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
Pergamon
S0022-4073(%)00049-0
0022~4073/96
$15.00 + 0.00
STIMULATED SCATTERING AND FREQUENCY DIFFUSION OF PHOTONS IN PLASMAS V. N. TSYTOVICH,“t
R. BINGHAM,”
U. DE ANGELIS,’
and A. FORLANI*
“Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX1 1 OQX, U.K. and bDepartment of Physical Sciences, University of Naples, Italy (Received
7 September
1995)
Abstract-Stimulated scattering on electrons and frequency diffusion in the transport equation for photons in plasmas are considered including the collective plasma effects. The results are applied to the solar interior (where collective plasma effects are not negligible for a wide range of frequencies) to find the effect on the solar opacity. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION This is the third in a series of papers dealing with relativistic corrections to scattering of photons
in plasmas and their effects on the solar opacity. In the first two papers in the series’,* the corrections to spontaneous scattering were found. Here we consider the induced scattering of photons by the plasma electrons and the effect of frequency diffusion taking into account the collective plasma effects. The transport equation for the distribution function Nk of photons of wavenumber k_ and frequency wk = w(k) can be written, in the quasi-classical approximation, as’
+
s
Nk Nk' Wj$h(k
where
-‘-
af $1 d’p d’k ’
- &‘).--
8P
WI6
(1)
and $=
C&$,
2
E(O)=
( 1 l-2
is the plasma frequency, _P@) is the electrons’ (a = e) or is the group velocity, ape = J4ne*n,/m, ions’ (LX= i) distribution function in the plasma and Wpl, is the probability of scattering on electrons or ions, respectively. The first two terms on the rhs of (1) represent the spontaneous processes and the last term gives the contribution of stimulated scattering on electrons (we neglect stimulated scattering on ions which is smaller by the factor m,/mi). tpermanent
address: General Physics Institute, Russian Academy of Sciences, Moscow, Russia. 809
810
V. N. Tsytovich et al
In the present paper we evaluate the contributions of stimulated scattering on electrons [last term of Eq. (l)] and of the frequency diffusion terms to the transport equation (Sec. 2). The effects on the Rosseland opacity in the solar interior are found in Sec. 3. We shall consider the corrections up to second order in vTe/c, where vTe= (r, /m,)‘j2 is the electron thermal velocity. The presence of k - k’ in the stimulated term makes this term already a 1st order term in (vrC/c) (from the Doppler shift due to the conservation law, expressed through the a-function in the probability, see Sec. 2). As a consequence we only need the zero order scattering probability and we can neglect the relativistic correction toS@‘(p) (which are also of order v&/c’) and assume a Maxwellian distribution in the following. The relativistic corrections to the distribution function and to the scattering probabilities were considered in the first two papers.‘,’ 2. THE
TRANSPORT
The zero order scattering probability
EQUATION
on electrons can be written as’ l+C@2 , 5G
6(R-q.u) _ -
(2)
where q =k -k’, R=o --co’, O,,p is the angle of the k and k’ vectors, x = cos Okkk.,xl,, are the ion susceptibilities, 6ko the plasma-dielectric function. In Eq. (1) we perform the integration over the component of the electron velocity perpendicular to 4, neglecting the ions and use the properties of the Maxwellian distribution then Eq. (1) becomes
where
!I’!!
y-
$PTe is the dimensionless component cross-section gT given by
of the velocity parallel to 4. We have introduced the Thomson 8
e4
and changed the k ‘-integration using the dispersion relation for electromagnetic d3k ’ r -=2x JU
waves
I r dx J-I Jo
UJ
C-J-I
y
dw’w’m.
Let us consider, now, the usual expansion for N& namely [1,2] Nk = N”w + cos 8, SN, n
(5)
where N$)= {exp(h$)
- I}-’
is the thermal photon distribution, SN, the deviation responsible for the flux of radiation, n_ the unit vector in the direction of flux propagation and T the thermal energy.
811
Stimulated scattering and frequency diffusion of photons
After substitution of Eq. (5) in Eq. (3) we obtain the linearized transport equation (for steady-state) in the following form [the thermal distribution (6) gives exactly zero in the rhs of Eq.
(I)1 &$E$
- ne(c7”SN, - r_Pcm,)
(7)
where the first term was obtained in Paper I and is given by
and the second term (stimulated scattering) follows from Eq. (3) and is given by
It is important to stress that in the zero-order (non relativistic) solution to these equations CO’= w, since the Doppler shift (introduced by the d-function [1]) is of order VJC. If the Doppler corrections are included expanding NE) and 6N,. in powers of w’ - w, crtr and 0” become differential operators in frequency so that the transport equation contains diffusion terms in frequency. From the a-function in Eqs. (8,9) we have the Doppler shift sz=w -wg=q.y=Jiqv,,y
(10)
where
Solving this equation to second order in vTe/c we obtain W’ -= W
1-2%yJ1-x
+2
*y2(1 -X).
? (
(12)
)
Then in the integrand of Eqs. (8,9) we can use
+2-g.y)=
1
(13)
w~+2$?(1-x)w2~
(15)
1-3?~-+2$~~‘(3-2~)
S(w’(w)-w’),
and
where w’(w) is given by Eq. (12),
Nf+N~-Zf+cl
l-?.Vfi [
1
and SN,. = SN, +
-2~yfi+2($2(l
-x)]wT
+2$y’(l
-x)w2$.
(16)
Then, the substitution of (13-16) in (8,9) leads to the following expression for the cross-sections (performing the w’ integration using the delta-function) a’VN, = (0; + 6”)6N,
(17)
V. N. Tsytovich et al
812
1(18)
l-3~yJi7+2$y’(3-2x)
x(1+x2)(1-x) with the operator given by d”=f0~j_~~e-“j~,’ X
,F(xdi,r),2(l
+x2)x
T);JE iC
-
4$y2(1x) CO& -
$y’(l
(19)
- x)&$ 1
1
and &SN, = (a; + 8”‘)6N,
(20)
with
and we have introduced the notations hiw
(23)
z=-
T F(x, Y, z) =
$fl
Id=w’(o).
(24)
The explicit calculation of F(x, y, z) has been done in Ref. 1, neglecting the ion contribution and taking into account the corrections due to collisions, Doppler shift and relativistic effects. These results conclude the aim of the present paper: the zero order transport equation namely
(25) can now be written, taking into account stimulated scattering, as:
where I a;’
=
i*,
UT” C
s
_] ,F(xd;,z),2(l
+x2)x&= x
I
I-2TyJiF dx
(1 +x2)x(1 s -1 lF(x,y, z)12
( 2+ S)]
(27)
- x).
(28)
Stimulated scattering and frequency diffusion of photons 3. THE
OPACITY
IN THE
SOLAR
813
INTERIOR
In the solar interior the transport equation is usually written in terms of the Planck function B,, and the energy flux 9, defined as
and takes the form
where geK is the total cross-section including inverse bremsstrahlung Using perturbation theory we solved Eq. (30) assuming
and line absorption.
S,=F-o,+SFC, where p Co= -
4n
aBw dT
3n,~r,~ aT
dr
(31)
is the zero order flux and the correction is due to the presence of stimulated scattering and frequency diffusion. The solution for 69,, integrated over all frequencies, is used to calculate the correction to the opacity given by (32) where p is the mass density, (33)
f(z) =
&
and the zero order Rosseland mean opacity is given by 1 -=4
z4eZ (34) PKO 4l.c s wpe(e’ - l)‘~,a,,~~. We have calculated numerically the effect of the new terms using Eq. (32) neglecting line-absorption and using the zero order cross-sections for scattering and inverse bremsstrahlung.’ At the solar centre we found a reduction of the continuous opacity of 4.5%. The reduction of the total opacity of course is smaller, it is known that absorption from iron ions at the solar centre accounts for almost one-third of the total opacity.3 This result however is a part of the total correction to scattering of photons in the solar interior. Other corrections were found in the first two papers.‘.’ Taken all together these corrections amount to a decrease of opacity of order 5% at the solar centre, which could be significant for the neutrino flux prediction from Standard Solar Models. 15
a
REFERENCES I. V. N. Tsytovich, R. Bingham, U. de Angelis, and A. Forlani, JQSRT 55, 787 (1996). 2. V. N. Tsytovich, R. Bingham, U. de Angelis, and A. Forlani, JQSRT 55, 797 (1996). 3. W. F. Huebner, in Physics ofthe Sun, P. A. Sturrock et al, eds., Vol. 1, pp. 33-75, D. Reidel. Dordrecht (1986).
Pergamon
S0022-4073(%)00049-0
0022~4073/96
$15.00 + 0.00
STIMULATED SCATTERING AND FREQUENCY DIFFUSION OF PHOTONS IN PLASMAS V. N. TSYTOVICH,“t
R. BINGHAM,”
U. DE ANGELIS,’
and A. FORLANI*
“Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX1 1 OQX, U.K. and bDepartment of Physical Sciences, University of Naples, Italy (Received
7 September
1995)
Abstract-Stimulated scattering on electrons and frequency diffusion in the transport equation for photons in plasmas are considered including the collective plasma effects. The results are applied to the solar interior (where collective plasma effects are not negligible for a wide range of frequencies) to find the effect on the solar opacity. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION This is the third in a series of papers dealing with relativistic corrections to scattering of photons
in plasmas and their effects on the solar opacity. In the first two papers in the series’,* the corrections to spontaneous scattering were found. Here we consider the induced scattering of photons by the plasma electrons and the effect of frequency diffusion taking into account the collective plasma effects. The transport equation for the distribution function Nk of photons of wavenumber k_ and frequency wk = w(k) can be written, in the quasi-classical approximation, as’
+
s
Nk Nk' Wj$h(k
where
-‘-
af $1 d’p d’k ’
- &‘).--
8P
WI6
(1)
and $=
C&$,
2
E(O)=
( 1 l-2
is the plasma frequency, _P@) is the electrons’ (a = e) or is the group velocity, ape = J4ne*n,/m, ions’ (LX= i) distribution function in the plasma and Wpl, is the probability of scattering on electrons or ions, respectively. The first two terms on the rhs of (1) represent the spontaneous processes and the last term gives the contribution of stimulated scattering on electrons (we neglect stimulated scattering on ions which is smaller by the factor m,/mi). tpermanent
address: General Physics Institute, Russian Academy of Sciences, Moscow, Russia. 809
810
V. N. Tsytovich et al
In the present paper we evaluate the contributions of stimulated scattering on electrons [last term of Eq. (l)] and of the frequency diffusion terms to the transport equation (Sec. 2). The effects on the Rosseland opacity in the solar interior are found in Sec. 3. We shall consider the corrections up to second order in vTe/c, where vTe= (r, /m,)‘j2 is the electron thermal velocity. The presence of k - k’ in the stimulated term makes this term already a 1st order term in (vrC/c) (from the Doppler shift due to the conservation law, expressed through the a-function in the probability, see Sec. 2). As a consequence we only need the zero order scattering probability and we can neglect the relativistic correction toS@‘(p) (which are also of order v&/c’) and assume a Maxwellian distribution in the following. The relativistic corrections to the distribution function and to the scattering probabilities were considered in the first two papers.‘,’ 2. THE
TRANSPORT
The zero order scattering probability
EQUATION
on electrons can be written as’ l+C@2 , 5G
6(R-q.u) _ -
(2)
where q =k -k’, R=o --co’, O,,p is the angle of the k and k’ vectors, x = cos Okkk.,xl,, are the ion susceptibilities, 6ko the plasma-dielectric function. In Eq. (1) we perform the integration over the component of the electron velocity perpendicular to 4, neglecting the ions and use the properties of the Maxwellian distribution then Eq. (1) becomes
where
!I’!!
y-
$PTe is the dimensionless component cross-section gT given by
of the velocity parallel to 4. We have introduced the Thomson 8
e4
and changed the k ‘-integration using the dispersion relation for electromagnetic d3k ’ r -=2x JU
waves
I r dx J-I Jo
UJ
C-J-I
y
dw’w’m.
Let us consider, now, the usual expansion for N& namely [1,2] Nk = N”w + cos 8, SN, n
(5)
where N$)= {exp(h$)
- I}-’
is the thermal photon distribution, SN, the deviation responsible for the flux of radiation, n_ the unit vector in the direction of flux propagation and T the thermal energy.
811
Stimulated scattering and frequency diffusion of photons
After substitution of Eq. (5) in Eq. (3) we obtain the linearized transport equation (for steady-state) in the following form [the thermal distribution (6) gives exactly zero in the rhs of Eq.
(I)1 &$E$
- ne(c7”SN, - r_Pcm,)
(7)
where the first term was obtained in Paper I and is given by
and the second term (stimulated scattering) follows from Eq. (3) and is given by
It is important to stress that in the zero-order (non relativistic) solution to these equations CO’= w, since the Doppler shift (introduced by the d-function [1]) is of order VJC. If the Doppler corrections are included expanding NE) and 6N,. in powers of w’ - w, crtr and 0” become differential operators in frequency so that the transport equation contains diffusion terms in frequency. From the a-function in Eqs. (8,9) we have the Doppler shift sz=w -wg=q.y=Jiqv,,y
(10)
where
Solving this equation to second order in vTe/c we obtain W’ -= W
1-2%yJ1-x
+2
*y2(1 -X).
? (
(12)
)
Then in the integrand of Eqs. (8,9) we can use
+2-g.y)=
1
(13)
w~+2$?(1-x)w2~
(15)
1-3?~-+2$~~‘(3-2~)
S(w’(w)-w’),
and
where w’(w) is given by Eq. (12),
Nf+N~-Zf+cl
l-?.Vfi [
1
and SN,. = SN, +
-2~yfi+2($2(l
-x)]wT
+2$y’(l
-x)w2$.
(16)
Then, the substitution of (13-16) in (8,9) leads to the following expression for the cross-sections (performing the w’ integration using the delta-function) a’VN, = (0; + 6”)6N,
(17)
V. N. Tsytovich et al
812
1(18)
l-3~yJi7+2$y’(3-2x)
x(1+x2)(1-x) with the operator given by d”=f0~j_~~e-“j~,’ X
,F(xdi,r),2(l
+x2)x
T);JE iC
-
4$y2(1x) CO& -
$y’(l
(19)
- x)&$ 1
1
and &SN, = (a; + 8”‘)6N,
(20)
with
and we have introduced the notations hiw
(23)
z=-
T F(x, Y, z) =
$fl
Id=w’(o).
(24)
The explicit calculation of F(x, y, z) has been done in Ref. 1, neglecting the ion contribution and taking into account the corrections due to collisions, Doppler shift and relativistic effects. These results conclude the aim of the present paper: the zero order transport equation namely
(25) can now be written, taking into account stimulated scattering, as:
where I a;’
=
i*,
UT” C
s
_] ,F(xd;,z),2(l
+x2)x&= x
I
I-2TyJiF dx
(1 +x2)x(1 s -1 lF(x,y, z)12
( 2+ S)]
(27)
- x).
(28)
Stimulated scattering and frequency diffusion of photons 3. THE
OPACITY
IN THE
SOLAR
813
INTERIOR
In the solar interior the transport equation is usually written in terms of the Planck function B,, and the energy flux 9, defined as
and takes the form
where geK is the total cross-section including inverse bremsstrahlung Using perturbation theory we solved Eq. (30) assuming
and line absorption.
S,=F-o,+SFC, where p Co= -
4n
aBw dT
3n,~r,~ aT
dr
(31)
is the zero order flux and the correction is due to the presence of stimulated scattering and frequency diffusion. The solution for 69,, integrated over all frequencies, is used to calculate the correction to the opacity given by (32) where p is the mass density, (33)
f(z) =
&
and the zero order Rosseland mean opacity is given by 1 -=4
z4eZ (34) PKO 4l.c s wpe(e’ - l)‘~,a,,~~. We have calculated numerically the effect of the new terms using Eq. (32) neglecting line-absorption and using the zero order cross-sections for scattering and inverse bremsstrahlung.’ At the solar centre we found a reduction of the continuous opacity of 4.5%. The reduction of the total opacity of course is smaller, it is known that absorption from iron ions at the solar centre accounts for almost one-third of the total opacity.3 This result however is a part of the total correction to scattering of photons in the solar interior. Other corrections were found in the first two papers.‘.’ Taken all together these corrections amount to a decrease of opacity of order 5% at the solar centre, which could be significant for the neutrino flux prediction from Standard Solar Models. 15
a
REFERENCES I. V. N. Tsytovich, R. Bingham, U. de Angelis, and A. Forlani, JQSRT 55, 787 (1996). 2. V. N. Tsytovich, R. Bingham, U. de Angelis, and A. Forlani, JQSRT 55, 797 (1996). 3. W. F. Huebner, in Physics ofthe Sun, P. A. Sturrock et al, eds., Vol. 1, pp. 33-75, D. Reidel. Dordrecht (1986).