- Email: [email protected]
Beam-plasma instabilities in a strongly coupled plasma
PHYSICS LETTERS
Volume 80A, number 2,3
BEAM-PLASMA
24 November 1980
INSTABILITIES IN A STRONGLY COUPLED PLASMA
K.I. GOLDEN ’ Department
of Electrical Engineering,
Northeastern
University, Boston, MA 02115,
USA
G. KALMAN ’ Department
of Physics and Center for Energy Research, Boston College, Chestnut Hill, MA 02167,
USA
and P. HAMMERLING La Jolla Institute, La Jolla, CA 92038,
USA
Received 29 September 1980
We adopt a model hydrodynamic-like dispersion relation to describe the interaction of a low density electron beam with a high density strongly coupled electron plasma. Growth rate expressions are derived and stability-instability boundaries are established.
Plasmon behavior is substantially affected by correlational effects in the strongly coupled (-r = Ke2/T> 1, K is the inverse Debye length) one-component plasma (ocp). A great deal of effort has already been directed at calculating the coefficients A(y) and B(y) in the long wave length plasmon dispersion relation e,(kw) = 0, where ep(k~) = 1 - wi/[(w c.+ = up{ 1 + A(y)k2}
+ iQ2 ,
,
+ (WE - oi)]
vk = o$?(r)P
,
(1) A(0) = 3/2 ,
up is the plasma frequency and k = k/K is a dimensionless wave number. The molecular dynamics (MD) simulations of Hansen, Pollock and McDonald [l-3] reveal that somewhere in the domain, probably in the upper range, 5 < yc < 52, the dispersion becomes negative, i.e., A(y) < 0 for y > yc. Comparison of the data from these MD experiments with recent numerical results [4] obtained from the Golden-Kalman (GK) approximation scheme [S] indicates that the GK theory reproduces the qualitative features of the data very well throughout the entire range of y values up to crystallization. These theoretical developments pave the way for studying instabilities arising from interactions of charged beams with strongly coupled plasmas. Such studies could be relevant to inertial confinement fusion experiments. The interest in this topic is by no means new. Some time ago [6] it was found that if, in a two component plasma, one models electron-ion collisions by incorporating a simple relaxation form of the collision frequency, vei, into the plasma dielectric function ep+), the resulting collisionless cold beam-collisional plasma dispersion relation indicates that, collisional dominance notwithstanding [i.e., Vei > (w~w~)~‘~, o.+, is the beam plasma frequency], the plasmons grow via the resistive instability mechanism at a rate (Im w),,, = ob(wp/2vei)1’2. Such instabilities will be seen to occur as well from strong coupling effects (B > 0) in the ocp under consideration in the present work. ’ Work partially supported by AFOSR Grant No. 76-2960.
149
Volume
80A, number
PHYSICS
2,3
24 November
LETTERS
1980
We consider the simplest possible magnetic field free configuration; an ocp consisting of a low density (tit < wt) beam of cold electrons drifting with velocity U through a strongly coupled electron plasma: the inert positive neutralizing background has no dynamical role whatsoever. We adopt the simple hydrodynamical structure eq. (1) for the plasma dielectric function [4] with the stipulation that the plasma electron-plasma electron (p-p) correlation coefficients A(y) and B(y) are to be provided by the calculations in ref. [4]. Beam electron-beam electron (b-b) coupling is assumed to be weak, while beam electron--plasma electron (b-p) correlations can be strong. We suppose that the latter can be modelled by incorporating a simple relaxation form of the collision frequency, ubp I into the cold beam polarizability. ob, i.e., + ivbp - kU)2
olb(kW) =-C&W
(2)
From simple physical considerations, it is reasonable to take vbp = w,B(y)/M3, Mach number. Eqs. (1) and (2) then combine to yield the dispersion relation E(kO) = E&O)
where M =
KU/W~
is the drift
+ olb(kU) = 0 ,
(3)
with k + 0 solution, w(k + 0) = (a;
+ c,J;)~/~ t 77 1+7)
(4)
To obtain solutions at k > 0, let w = kU ~ ivbp t 6. where, in the weak beam approximation ]6/ < kU, up, eq. (3) then becomes S2/o;
= l+ w;/[k2U2
(J/V = +,/c+,
- wz + 2kU(iC + 15)+ (iv + S)2] ,
< 1).
(5)
where i; = vx- - vbP. The task of solving eq. (5) will be facilitated by assuming kU % Vk, “bp + /i <(M/B),
k % (B/M4) ,
(6a.b)
w,~vv,,v,,-k2~(1/B),(B/M3)Ql.
(64
Resonant behavior. The resonant
k = kREs(r, M) = (1/2A)[M
beam-plasmon
interactions
correspond
to the matching condition
U = wk/k or
- (M2 - 4A)‘/‘]
(7)
Note that M2 Z 4A(y) for 0 < y yc. We consider resonant interactions in the two extreme limits (i) where collisional effects are non-dominant (I VI < ISI) and (ii) where they are dominant (IV1 9 IS]). Th e analysis over the entire range of permissable 1VI values is deferred to a more complete paper on this topic [7]. having the well known unstable root [6] (i) 151< 161: In this limit, eq. (5) simplifies to 6 3 z a2btip/2tik 2 6 Z [(ifi-
,
l)/2](c$w$2tik)1’3
whence Im ti Z Im 6 x (fl2) Evidently, ww3
s
(w~c$2wk)1’3
(8)
I VI < 16I amounts
to the requirement
IFI /wp =B l&S
- (1/M3)l .
that
(ii) Iti] %- 161: In this limit, eq. (5) reduces to the quadratic equation, 6 = [i-(v/lvl)]ObWp/[2(IV(wk)“2] indicates the possibility 150
15~= -iwtwi/(2iiwk),
,
of a resistive instability.
The resistive instability
can occur only if
whose root,
Volume 80A, number 2,3
Imwzwbwp/[2()FIwk)
PHYSICS LETTERS
24 November 1980
112]-vbp>o,
(9)
or (,7/2)113 > (n,/2)“3
~~/‘d,)“3(‘d&d,)1’3
= (~&,+,)2i3(2)
Since (9) and (10) are derived under the assumption bounded from above, viz., (71/2)‘/~ < (lVl/w,).
.
(10)
1fil S 16I = wbwP/(2 I PI wk)l/2, the beam strength is
Non-resonant behavior. When k f kREs, 0, one finds from eq. (5) that Im 6(x > 0) = c+, [(wi - x)/x] 1’2 ,
O
h6(X
k>kRES
< o)e tC_dbCd;k~/~X13'2(Cd; -X)1'2 ,
(11) (12)
’
where x E 6~: - k2 U2 f 0. The condition Im o = Im 6 - VbP > 0 dictates the ranges of it values for the occurrence of instability; these ranges are found from (11) and (12) to be -_ k, k,,s
k 0
k, =
2A2
(M2
M(M2 - 4A)‘/2 -
1+
2A2
4A2(q/G)
x=1-Mp3(M2-2A),
l/2
M2(M2 -4A) 1)
_12A)lj2 (’ +Mf2Ai(rlM2>1/6(x/2>1’3
G=q+(B2/M6),
l/2
a=lt-
(15)
’
M2y2A ,
[(a+ b1113 t (a - b>“3]t k(rlM2)1/2 2$!E2)
(M2r:A3
,
b = (2a - 1)1/2 .
Resonant and non-resonant behavior in the 1VI S (8 1 limit. When collisional effects are dominant, shown from (5) that 2 4 l/2 l/2 2xw; wP -------i-l Im&(x,y)z$ xsl+ (I x2 +y2 [ x2 ty2 I) I ’
(16)
it can be
(17)
where y = 2FkU. Note that eqs. (9) (11) and (12) can be readily recovered from (17). The fact that it is no longer necessary to make a distinction between the x # 0 and x = 0 cases in the I II S I6 I limit makes it possible to rigorously prove that both the resonant and non-resonant instabilities are quenched when n is decreased below nc [cf. eq. (1 O)].The proof is straightforward: since, by assumption, y 4 lx I # 0, it follows that for Q < rle, 0 > Im 6(x = O,y) > c’+,wplxl- 1/2>Im6(x>0,y=0). Conclusions. When a weak particle beam interacts with a strongly coupled plasma (with, necessarily, a fairly strong coupling between the beam and the plasma), and generates a beam-plasma instability, the effect of the coupling manifests itself through (i) the enhanced collisional damping of the resonant instability, (ii) a broad new spectrum of resistive instabilities, (iii) a relaxed resonance condition (due to negative dispersion) for the excitation of the resonant instability, and (iv) the enhanced scattering of the beam, which reduces the spectral domain in which the instability is excited. In this paper, we have studied all the above effects, with the exception of (iv), based on a newly developed formalism and calculation for the plasmon dispersion in strongly coupled ocp’s. Our results are qualitatively described by the sketches in fig. 1. When TV> ne, Im w(k) > 0 for ko < k
Volume
80A, number
PHYSICS
2,3
LETTERS
24 November
1980
Imw A
*k Pip. 1. Sketches of Im w(k) for I) > lie and Q = qc; M and y are fixed. Instability corresponds to Im w(k) > 0.
fastest growth modes occur near k = kREs where the growth rate is proportional to n1/3 or ~11~ depending on whether coupling effects are non-dominant or dominant. The bandwidth Ak = k, k, + 0 as 77+ q7, and for n < ne, the instabilities are entirely quenched. As we mentioned before, for A < 0, the drift Mach number M no longer has a lower bound. This is true at shorter wavelengths as well, where the general features of the dispersion curves at strong coupling have been reported by Baus and Hansen [3]. Therefore, even though we have adopted a long wavelength formulation of the plasma dielectric response function (which, in effect, would seem to impose a lower bound on M), our theory should, nevertheless, provide a reasonable qualitative description of bean-plasma instabilities in the strongly coupled ocp at shorter wavelengths and, consequently, at lower Mach numbers. A more detailed quantitative analysis with the inclusion of (iv) will be published later. Re.ferences [1] [2] [3] [4]
J.P. Hansen, E.L. Pollock and 1.R. McDonald, Phys. Rev. Lett. 32 (1974) 277. J.P. Hansen, I.R. McDonald and E.L. Pollock, Phys. Rev. Al 1 (1975) 1025. M. Baus and J.P. Hansen, Phys. Rep. 59 (1980) 1. K.I. Golden, G. Kalman and P. Carini, Proc. Intern. Conf. on Plasma physics (Nagoya, P. Carini, G. Kalman and K.I. Golden, Phys. Lett. 78A (1980) 450. [5] K.I. Golden and G. Kalman, Phys. Rev. Al9 (1979) 2112. [6] R.J. Briggs, in: Advances in plasma physics, Vol. 4, eds. A. Simon and W.B. Thompson [7] K.I. Golden, G. Kalman and P. Hammerling, to be published.
152
Japan,
April 1980);
(Interscience,
Vol. 1, p. 61:
New York,
1971) p. 43.
Volume 80A, number 2,3
BEAM-PLASMA
24 November 1980
INSTABILITIES IN A STRONGLY COUPLED PLASMA
K.I. GOLDEN ’ Department
of Electrical Engineering,
Northeastern
University, Boston, MA 02115,
USA
G. KALMAN ’ Department
of Physics and Center for Energy Research, Boston College, Chestnut Hill, MA 02167,
USA
and P. HAMMERLING La Jolla Institute, La Jolla, CA 92038,
USA
Received 29 September 1980
We adopt a model hydrodynamic-like dispersion relation to describe the interaction of a low density electron beam with a high density strongly coupled electron plasma. Growth rate expressions are derived and stability-instability boundaries are established.
Plasmon behavior is substantially affected by correlational effects in the strongly coupled (-r = Ke2/T> 1, K is the inverse Debye length) one-component plasma (ocp). A great deal of effort has already been directed at calculating the coefficients A(y) and B(y) in the long wave length plasmon dispersion relation e,(kw) = 0, where ep(k~) = 1 - wi/[(w c.+ = up{ 1 + A(y)k2}
+ iQ2 ,
,
+ (WE - oi)]
vk = o$?(r)P
,
(1) A(0) = 3/2 ,
up is the plasma frequency and k = k/K is a dimensionless wave number. The molecular dynamics (MD) simulations of Hansen, Pollock and McDonald [l-3] reveal that somewhere in the domain, probably in the upper range, 5 < yc < 52, the dispersion becomes negative, i.e., A(y) < 0 for y > yc. Comparison of the data from these MD experiments with recent numerical results [4] obtained from the Golden-Kalman (GK) approximation scheme [S] indicates that the GK theory reproduces the qualitative features of the data very well throughout the entire range of y values up to crystallization. These theoretical developments pave the way for studying instabilities arising from interactions of charged beams with strongly coupled plasmas. Such studies could be relevant to inertial confinement fusion experiments. The interest in this topic is by no means new. Some time ago [6] it was found that if, in a two component plasma, one models electron-ion collisions by incorporating a simple relaxation form of the collision frequency, vei, into the plasma dielectric function ep+), the resulting collisionless cold beam-collisional plasma dispersion relation indicates that, collisional dominance notwithstanding [i.e., Vei > (w~w~)~‘~, o.+, is the beam plasma frequency], the plasmons grow via the resistive instability mechanism at a rate (Im w),,, = ob(wp/2vei)1’2. Such instabilities will be seen to occur as well from strong coupling effects (B > 0) in the ocp under consideration in the present work. ’ Work partially supported by AFOSR Grant No. 76-2960.
149
Volume
80A, number
PHYSICS
2,3
24 November
LETTERS
1980
We consider the simplest possible magnetic field free configuration; an ocp consisting of a low density (tit < wt) beam of cold electrons drifting with velocity U through a strongly coupled electron plasma: the inert positive neutralizing background has no dynamical role whatsoever. We adopt the simple hydrodynamical structure eq. (1) for the plasma dielectric function [4] with the stipulation that the plasma electron-plasma electron (p-p) correlation coefficients A(y) and B(y) are to be provided by the calculations in ref. [4]. Beam electron-beam electron (b-b) coupling is assumed to be weak, while beam electron--plasma electron (b-p) correlations can be strong. We suppose that the latter can be modelled by incorporating a simple relaxation form of the collision frequency, ubp I into the cold beam polarizability. ob, i.e., + ivbp - kU)2
olb(kW) =-C&W
(2)
From simple physical considerations, it is reasonable to take vbp = w,B(y)/M3, Mach number. Eqs. (1) and (2) then combine to yield the dispersion relation E(kO) = E&O)
where M =
KU/W~
is the drift
+ olb(kU) = 0 ,
(3)
with k + 0 solution, w(k + 0) = (a;
+ c,J;)~/~ t 77 1+7)
(4)
To obtain solutions at k > 0, let w = kU ~ ivbp t 6. where, in the weak beam approximation ]6/ < kU, up, eq. (3) then becomes S2/o;
= l+ w;/[k2U2
(J/V = +,/c+,
- wz + 2kU(iC + 15)+ (iv + S)2] ,
< 1).
(5)
where i; = vx- - vbP. The task of solving eq. (5) will be facilitated by assuming kU % Vk, “bp + /i <(M/B),
k % (B/M4) ,
(6a.b)
w,~vv,,v,,-k2~(1/B),(B/M3)Ql.
(64
Resonant behavior. The resonant
k = kREs(r, M) = (1/2A)[M
beam-plasmon
interactions
correspond
to the matching condition
U = wk/k or
- (M2 - 4A)‘/‘]
(7)
Note that M2 Z 4A(y) for 0 < y
,
l)/2](c$w$2tik)1’3
whence Im ti Z Im 6 x (fl2) Evidently, ww3
s
(w~c$2wk)1’3
(8)
I VI < 16I amounts
to the requirement
IFI /wp =B l&S
- (1/M3)l .
that
(ii) Iti] %- 161: In this limit, eq. (5) reduces to the quadratic equation, 6 = [i-(v/lvl)]ObWp/[2(IV(wk)“2] indicates the possibility 150
15~= -iwtwi/(2iiwk),
,
of a resistive instability.
The resistive instability
can occur only if
whose root,
Volume 80A, number 2,3
Imwzwbwp/[2()FIwk)
PHYSICS LETTERS
24 November 1980
112]-vbp>o,
(9)
or (,7/2)113 > (n,/2)“3
~~/‘d,)“3(‘d&d,)1’3
= (~&,+,)2i3(2)
Since (9) and (10) are derived under the assumption bounded from above, viz., (71/2)‘/~ < (lVl/w,).
.
(10)
1fil S 16I = wbwP/(2 I PI wk)l/2, the beam strength is
Non-resonant behavior. When k f kREs, 0, one finds from eq. (5) that Im 6(x > 0) = c+, [(wi - x)/x] 1’2 ,
O
h6(X
k>kRES
< o)e tC_dbCd;k~/~X13'2(Cd; -X)1'2 ,
(11) (12)
’
where x E 6~: - k2 U2 f 0. The condition Im o = Im 6 - VbP > 0 dictates the ranges of it values for the occurrence of instability; these ranges are found from (11) and (12) to be -_ k,
k 0
k, =
2A2
(M2
M(M2 - 4A)‘/2 -
1+
2A2
4A2(q/G)
x=1-Mp3(M2-2A),
l/2
M2(M2 -4A) 1)
_12A)lj2 (’ +Mf2Ai(rlM2>1/6(x/2>1’3
G=q+(B2/M6),
l/2
a=lt-
(15)
’
M2y2A ,
[(a+ b1113 t (a - b>“3]t k(rlM2)1/2 2$!E2)
(M2r:A3
,
b = (2a - 1)1/2 .
Resonant and non-resonant behavior in the 1VI S (8 1 limit. When collisional effects are dominant, shown from (5) that 2 4 l/2 l/2 2xw; wP -------i-l Im&(x,y)z$ xsl+ (I x2 +y2 [ x2 ty2 I) I ’
(16)
it can be
(17)
where y = 2FkU. Note that eqs. (9) (11) and (12) can be readily recovered from (17). The fact that it is no longer necessary to make a distinction between the x # 0 and x = 0 cases in the I II S I6 I limit makes it possible to rigorously prove that both the resonant and non-resonant instabilities are quenched when n is decreased below nc [cf. eq. (1 O)].The proof is straightforward: since, by assumption, y 4 lx I # 0, it follows that for Q < rle, 0 > Im 6(x = O,y) > c’+,wplxl- 1/2>Im6(x>0,y=0). Conclusions. When a weak particle beam interacts with a strongly coupled plasma (with, necessarily, a fairly strong coupling between the beam and the plasma), and generates a beam-plasma instability, the effect of the coupling manifests itself through (i) the enhanced collisional damping of the resonant instability, (ii) a broad new spectrum of resistive instabilities, (iii) a relaxed resonance condition (due to negative dispersion) for the excitation of the resonant instability, and (iv) the enhanced scattering of the beam, which reduces the spectral domain in which the instability is excited. In this paper, we have studied all the above effects, with the exception of (iv), based on a newly developed formalism and calculation for the plasmon dispersion in strongly coupled ocp’s. Our results are qualitatively described by the sketches in fig. 1. When TV> ne, Im w(k) > 0 for ko < k
Volume
80A, number
PHYSICS
2,3
LETTERS
24 November
1980
Imw A
*k Pip. 1. Sketches of Im w(k) for I) > lie and Q = qc; M and y are fixed. Instability corresponds to Im w(k) > 0.
fastest growth modes occur near k = kREs where the growth rate is proportional to n1/3 or ~11~ depending on whether coupling effects are non-dominant or dominant. The bandwidth Ak = k, k, + 0 as 77+ q7, and for n < ne, the instabilities are entirely quenched. As we mentioned before, for A < 0, the drift Mach number M no longer has a lower bound. This is true at shorter wavelengths as well, where the general features of the dispersion curves at strong coupling have been reported by Baus and Hansen [3]. Therefore, even though we have adopted a long wavelength formulation of the plasma dielectric response function (which, in effect, would seem to impose a lower bound on M), our theory should, nevertheless, provide a reasonable qualitative description of bean-plasma instabilities in the strongly coupled ocp at shorter wavelengths and, consequently, at lower Mach numbers. A more detailed quantitative analysis with the inclusion of (iv) will be published later. Re.ferences [1] [2] [3] [4]
J.P. Hansen, E.L. Pollock and 1.R. McDonald, Phys. Rev. Lett. 32 (1974) 277. J.P. Hansen, I.R. McDonald and E.L. Pollock, Phys. Rev. Al 1 (1975) 1025. M. Baus and J.P. Hansen, Phys. Rep. 59 (1980) 1. K.I. Golden, G. Kalman and P. Carini, Proc. Intern. Conf. on Plasma physics (Nagoya, P. Carini, G. Kalman and K.I. Golden, Phys. Lett. 78A (1980) 450. [5] K.I. Golden and G. Kalman, Phys. Rev. Al9 (1979) 2112. [6] R.J. Briggs, in: Advances in plasma physics, Vol. 4, eds. A. Simon and W.B. Thompson [7] K.I. Golden, G. Kalman and P. Hammerling, to be published.
152
Japan,
April 1980);
(Interscience,
Vol. 1, p. 61:
New York,
1971) p. 43.