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Nature of trapped particle instability
Volume 53A. number 5
PHYSICS LETTERS
14 July 1975
NATURE OF TRAPPED PARTICLE INSTABILITY S. KRISHAN and V. KRISHAN Divisionof Physics and Mathematical Sciences, Indian Institute of Science, Bangalore 560012, India Received 9 April 1975 The amplification mechanism for the side bands which accompany a large amplitude electron wave on a plasma column are shown to arise due to’ two mode interaction between negative and positive energy waves.
In its wave frame, a finite amplitude monochromatic one-dimensional electron wave (frequency w, and wave number k,) traps electrons whose total energy is less than e9,, where (p. is the absolute maximum of potential 9 of the monochromatic wave (carrier wave). For k,x 4 1 the electrons then undergo oscillations of a harmonic oscillator about the minimum trough of each, where x is the displacement of the particle from the centre of the trough; otherwise they execute anharmonic oscillations. Now if a test wave of frequency o and wave vector 4 interacts with periodic density distribution of electrons, it causes field fluctuations of frequency + (w -n wo) and wave vector + (4 - nk,). This is shown in fig. la. Assuming that the untrapped particles are acting as a background medium, the test wave field is screened by the untrapped particles. Let us call VT as the bare interaction between two trapped particles. Then the effective interaction VTUT as screened by untrapped particles can be written as VTUT=
~~(EL[f(W--nGlo,f(q-nko)l}-l,
(1)
where eL is the linear dielectric function. for an electron gas. The potential vTuT is then further screened by trapped particles and is given by
‘TTT
= vTJ,T [ET{‘(W - nwo),
+(q - nko))]
-‘,
(2)
where eT is the dielectric function for the trapped particles given by
(3)
= 1 - ,2T [e L {[*(o-U
o)
qq-nko)u12-~~l-‘,
where u is the phase velocity of the carrier wave, G+ is the plasma frequency for the trapped particles, and we = (e90k~/m)1/2 is the electron bounce frequency. The modes of the oscillations will be given by eT =O. Ifw=ooandqako wesee(o-nw,)forn=O and -(o -nw,) for n = 2 are nearly equal, therefore one is tempted to add the two field fluctuations, since they also will have nearly the same momentum. Or alternatively, eq. (1) and eq. (2) are modified as written below ‘TUT+
v’TUT = VT [Ie,&
411-l
(4) + {~~(20~ -w, 2k, - q)}-1l, VTTT -+ vT”T = VT&.;.(a,
d
(9
where e;=1-
WI (w - 4v)2 - w;
(a)
[@,(U’ 4)P (6)
W
Fig. 1. a) Coulomb scattering caused by the test wave. b) Twowave interaction giving rise to instability.
+ (eL(2wo - w, 2ko - #I. The right hand side of (6) set equal to zero is the same dispersion relation as obtained before [ 11. Many 425
Volume 53A, number 5
PHYSICS LETTERS
of the later investigations are directly or indirectly are based on this model [2], though in some cases the trapped electrons have been considered as executing anharmonic oscillations [3,4]. Actually each term in (4) stands for coulomb scattering for a pair of particles. Strictly speaking, since energy is to be conserved at each end of the coulomb vertex, the final states of the particles corresponding to the first term is not identical with the final state of the particle corresponding to second term in (4), and therefore the two terms should not be added. However, in an approximate sense it seems alright. Eq. (6) has only a numerical solution. We shall take a different viewpoint which will bring out the physical mechanism involved in the side band instability and also it will lead to an analytical solution. We shall solve the dispersion relation ET = 0,
(7)
for n = 0 and 2 and then show that these two branches can interact via two-mode coupling. As 0~
+
(8)
w-nwo,q-nko)11'2, [w; +W$kL(
W2-nwo
=(q-nko)tJ
_ [w; t C+eL(w
--w,,
4 - nk,)]
1’2*
14 July 1975
side bands can now be calculated by the usual manybody techniques [6] by calculating the matrix element for two-wave interaction (fig. lb). Let or and ys be the temporal and spatial growth rates. Our calculations then yield, for each side band,
Yt,s=[(;w*)t,(p$
(1 Q))J
(IO)
where for ys the upper and lower signs refer to lower and upper side bands, respectively. For the side bands corresponding to ys the upper side band appears little suppressed than the lower one which agrees with the experiments [5]. For [7], wg/wo = 0.07, w+/w& = 2.5 X 1O-4 one finds rt/c+ = 8 X low3 which compares well with the experimental value of 5 X 10p3. For [5] ep,/T = 0.05, w,/2~ x 55.2 MHz, wpe = 2.6 X 108/sec and T = fs eV, ys * 6 X 10V2/cm which also compares well with the experimental value of 7 X 1OW2. The growth rates given by (10) increase with WB which agrees with the experiments, but our actual wB-dependence does not agree with it [5]. Perhaps, other class of theories [8] should be incorporated into the present theory to explain this behaviour. In conclusion, we have shown that the side band growth is due to two-mode coupling and that the growth rate can be calculated analytically, the growth rates are in fairly good agreement with the experiments.
(9) References
The two values of w given by eqs. (8) and (9) correspond to upper and lower side bands, respectively. We may remind again that even though the frequency of the test wave is w, the oscillations in the medium are taking place at f (o - nw,), + (4 - nk,). Therefore the sign of the mode - whether it is positive or negative energy mode - is given by a [* (w - nwo)eTeL]/a [“-(a - nw,)]. For ~7> 0, n = 0, and 2 eq. (8) gives positive and negative modes, respectively. These modes interact and give rise to upper side band instability. For n = 0,2, ~7> 0 eq. (9) gives negative and positive modes, respectively. The interaction of these two modes also leads to amplification. The growth rate for the lower and upper
426
[l] W.L. Kruer, J.M. Dawson and R.N. Sudan et al., Phys. Rev. Lett. 23 (1969) 838. [2] W.L. Kruer, Plasma Phys. Report, Princeton University, Matt. 887 (1972), and many references given therein. [3] R.N. Franklin et al., Phys. Rev. Lett. 28 (1972) 1114. [4] L.M. Al’tshul and V.I. Karponan, Soviet Phys. JETP 22 (1966) 361. [5] G. Jahns and G. Van Hoven, Phys. Rev. Lett. 31 (1973) 436. [6] E.G. Harris, A pedestrian approach to quantum field theory (John Wiley and Sons, Inc.). [7] C.B. Wharton et al., Phys. Fluids 11 (1968) 1761. [8] N.I. Bud’Ko, V.I. Karpman and D.R. Shklyar, Soviet Phys. JETP 34 (1972) 778; A.L. Brinca, J. Plasma Phys. 7 (1972) 385; T.M. O’Neil, Phys. Fluids 8 (1965) 2255.
PHYSICS LETTERS
14 July 1975
NATURE OF TRAPPED PARTICLE INSTABILITY S. KRISHAN and V. KRISHAN Divisionof Physics and Mathematical Sciences, Indian Institute of Science, Bangalore 560012, India Received 9 April 1975 The amplification mechanism for the side bands which accompany a large amplitude electron wave on a plasma column are shown to arise due to’ two mode interaction between negative and positive energy waves.
In its wave frame, a finite amplitude monochromatic one-dimensional electron wave (frequency w, and wave number k,) traps electrons whose total energy is less than e9,, where (p. is the absolute maximum of potential 9 of the monochromatic wave (carrier wave). For k,x 4 1 the electrons then undergo oscillations of a harmonic oscillator about the minimum trough of each, where x is the displacement of the particle from the centre of the trough; otherwise they execute anharmonic oscillations. Now if a test wave of frequency o and wave vector 4 interacts with periodic density distribution of electrons, it causes field fluctuations of frequency + (w -n wo) and wave vector + (4 - nk,). This is shown in fig. la. Assuming that the untrapped particles are acting as a background medium, the test wave field is screened by the untrapped particles. Let us call VT as the bare interaction between two trapped particles. Then the effective interaction VTUT as screened by untrapped particles can be written as VTUT=
~~(EL[f(W--nGlo,f(q-nko)l}-l,
(1)
where eL is the linear dielectric function. for an electron gas. The potential vTuT is then further screened by trapped particles and is given by
‘TTT
= vTJ,T [ET{‘(W - nwo),
+(q - nko))]
-‘,
(2)
where eT is the dielectric function for the trapped particles given by
(3)
= 1 - ,2T [e L {[*(o-U
o)
qq-nko)u12-~~l-‘,
where u is the phase velocity of the carrier wave, G+ is the plasma frequency for the trapped particles, and we = (e90k~/m)1/2 is the electron bounce frequency. The modes of the oscillations will be given by eT =O. Ifw=ooandqako wesee(o-nw,)forn=O and -(o -nw,) for n = 2 are nearly equal, therefore one is tempted to add the two field fluctuations, since they also will have nearly the same momentum. Or alternatively, eq. (1) and eq. (2) are modified as written below ‘TUT+
v’TUT = VT [Ie,&
411-l
(4) + {~~(20~ -w, 2k, - q)}-1l, VTTT -+ vT”T = VT&.;.(a,
d
(9
where e;=1-
WI (w - 4v)2 - w;
(a)
[@,(U’ 4)P (6)
W
Fig. 1. a) Coulomb scattering caused by the test wave. b) Twowave interaction giving rise to instability.
+ (eL(2wo - w, 2ko - #I. The right hand side of (6) set equal to zero is the same dispersion relation as obtained before [ 11. Many 425
Volume 53A, number 5
PHYSICS LETTERS
of the later investigations are directly or indirectly are based on this model [2], though in some cases the trapped electrons have been considered as executing anharmonic oscillations [3,4]. Actually each term in (4) stands for coulomb scattering for a pair of particles. Strictly speaking, since energy is to be conserved at each end of the coulomb vertex, the final states of the particles corresponding to the first term is not identical with the final state of the particle corresponding to second term in (4), and therefore the two terms should not be added. However, in an approximate sense it seems alright. Eq. (6) has only a numerical solution. We shall take a different viewpoint which will bring out the physical mechanism involved in the side band instability and also it will lead to an analytical solution. We shall solve the dispersion relation ET = 0,
(7)
for n = 0 and 2 and then show that these two branches can interact via two-mode coupling. As 0~
+
(8)
w-nwo,q-nko)11'2, [w; +W$kL(
W2-nwo
=(q-nko)tJ
_ [w; t C+eL(w
--w,,
4 - nk,)]
1’2*
14 July 1975
side bands can now be calculated by the usual manybody techniques [6] by calculating the matrix element for two-wave interaction (fig. lb). Let or and ys be the temporal and spatial growth rates. Our calculations then yield, for each side band,
Yt,s=[(;w*)t,(p$
(1 Q))J
(IO)
where for ys the upper and lower signs refer to lower and upper side bands, respectively. For the side bands corresponding to ys the upper side band appears little suppressed than the lower one which agrees with the experiments [5]. For [7], wg/wo = 0.07, w+/w& = 2.5 X 1O-4 one finds rt/c+ = 8 X low3 which compares well with the experimental value of 5 X 10p3. For [5] ep,/T = 0.05, w,/2~ x 55.2 MHz, wpe = 2.6 X 108/sec and T = fs eV, ys * 6 X 10V2/cm which also compares well with the experimental value of 7 X 1OW2. The growth rates given by (10) increase with WB which agrees with the experiments, but our actual wB-dependence does not agree with it [5]. Perhaps, other class of theories [8] should be incorporated into the present theory to explain this behaviour. In conclusion, we have shown that the side band growth is due to two-mode coupling and that the growth rate can be calculated analytically, the growth rates are in fairly good agreement with the experiments.
(9) References
The two values of w given by eqs. (8) and (9) correspond to upper and lower side bands, respectively. We may remind again that even though the frequency of the test wave is w, the oscillations in the medium are taking place at f (o - nw,), + (4 - nk,). Therefore the sign of the mode - whether it is positive or negative energy mode - is given by a [* (w - nwo)eTeL]/a [“-(a - nw,)]. For ~7> 0, n = 0, and 2 eq. (8) gives positive and negative modes, respectively. These modes interact and give rise to upper side band instability. For n = 0,2, ~7> 0 eq. (9) gives negative and positive modes, respectively. The interaction of these two modes also leads to amplification. The growth rate for the lower and upper
426
[l] W.L. Kruer, J.M. Dawson and R.N. Sudan et al., Phys. Rev. Lett. 23 (1969) 838. [2] W.L. Kruer, Plasma Phys. Report, Princeton University, Matt. 887 (1972), and many references given therein. [3] R.N. Franklin et al., Phys. Rev. Lett. 28 (1972) 1114. [4] L.M. Al’tshul and V.I. Karponan, Soviet Phys. JETP 22 (1966) 361. [5] G. Jahns and G. Van Hoven, Phys. Rev. Lett. 31 (1973) 436. [6] E.G. Harris, A pedestrian approach to quantum field theory (John Wiley and Sons, Inc.). [7] C.B. Wharton et al., Phys. Fluids 11 (1968) 1761. [8] N.I. Bud’Ko, V.I. Karpman and D.R. Shklyar, Soviet Phys. JETP 34 (1972) 778; A.L. Brinca, J. Plasma Phys. 7 (1972) 385; T.M. O’Neil, Phys. Fluids 8 (1965) 2255.