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Time behaviour of electron cyclotron harmonic waves excited by helical electron beams in a plasma
Volume 60A, number 2
PHYSICS LETTERS
7 February 1977
TIME BEHAVIOUR OF ELECTRON CYCLOTRON HARMONIC WAVES EXCITED BY HELICAL ELECTRON BEAMS IN A PLASMA R. SUGAYA, M. SUGAWA and H. NOMOTO Department ofPhysics, Faculty ofScience, Ehime Univeristy, Matsuyama, Japan Received 11 January 1977 Time behaviour of instabilities of electron cyclotron harmonic waves due to the anisotropy of the velocity distribution of electrons has been investigated. Measured growth rate was compared with the calculated ones.
It has been predicted theoretically that anisotropies in velocity space lead to high frequency electrostatic instabilities in a plasma [1]. Particularly, it has been predicted theoretically that the electrostatic electron cyclotron harmonic waves can become absolutely unstable in a magnetoplasma with a ring electron distribution or a mixture of ring and Maxwellian electron distributions, where the ring distribution describes monoenergetic electrons, moving only in the plane perpendicular to the magnetic fields, and uniformly distributed in velocity space on a circle with a certain radius [2,3]. We have reported the experimental observation of instabilities of electron cyclotron harmonic waves in the magnetoplasma with a mixture of ring and Maxwellian electron distribution, which were produced by the injection of the electron beam perpendicular to the magnetic field [4]. In this letter, we report an experimental result, where the time behaviour of the excited strong spontaneous emission was investigated and compared with the calculated dispersion relation and growth rate. The experimental apparatus used in this experiment is the same as that described in the ref. [4]. A monoenergetic electron beam with energy Ub = 490 eV, current = 10.8 mA, density ~b ~ X iO~cm~3,and width along the homogeneous magnetic field D 30mm is continuously injected perpendicularly to the magnetic field (10 50 gauss) into a beam-generated plasma of n~, 2 X 108 cm3, Te 7eV and momentum transfer collision frequency between electrons and neutral atoms 6 MHz at the argon gas pressure of 0.7 mtorr. The electrons of the_injected beam are moving with the speed u~(—sJ2Ub/m) in the plane perpendicular to the magnetic field. Two coaxial antenna probes, radially movable, are placed in the plasma in order to -~
122
100 )J. s.c
i—i
1 Fig. 1. Output signal of the spontaneous emission at the second 27r = 121 MHz, Wc/21r = cyclotron 56.5 MHz,harmonic. ~p/21r 130 uJ.o/utp MHz and 12,wb/21r 4 60MHz.
detect the wave. The beam-plasma system is linearly unstable, and a strong spontaneous emission growing up to 60 dB above background noise appears at the cyclotron harmonics. This instability which occurs spontaneously cannot be launched or synchronized by external modulation. It does not have an exponential growth in space; it has an almost constant amplitude in the region of the beamgenerated plasma. An output signal of the spontaneous emission from the field-intensity meter is shown in fig. 1. It is found that the amplitude is not steady in time, but switches off and on in time intervals of 10 to 100 psec. It grows exponentially in time initially, and decays after saturation [5]. We analyzed the present instabilities, using the dispersion equation of electrostatic waves pro-
Volume 60A, number 2
PHYSICS LETTERS
7 February 1977
sion relation is approximately given by
2.~ (a)
~j~i__L~IIIIII l—E~ ~1
0tb 2
2.C
~ d3J~(~))] 1dJ~(p) 2n2w~ dp dp3
1.5
where
0.05-
3
4
5
6
4wc of the spontaneous emission versus Fig. 2. (a) c~/w~. The Frequency vertical bars indicate a half-width ~w/wc of the emission. Experimental parameters are the same as those of fig. 1. The solid curve is the calculated dispersion relation with a 0.76, ujO/utp 12, ~z= 3.8, andy = 18. (b) Measured growth rate ~~/WC of the emissionof fig. 2(a). The solid curves show the calculated growth rate, where the values of a, vj.o/vtp and ~j are the same as those of the calculated dispersion curve of fig. 1(a), and the values of y, 30, 18 and 14 correspond to Tb 1, 3, and 5eV, respectively,
pagating perpendicularly to the magnetic field. It is assumed that ions cold and infinitely massive and electron distribution function is given by the following equation, 1 (\ 2~u~ /
3/2
fe(t)i,t~i)”flp +
nbA exp
[
ex
+
~
E 2V~p
2 + v2]
1
The calculated dispersion curve obtained from eq. (l)with p = 3.8 which corresponds to the maximum growth rate of the instabilities is plotted in fig. 2(a), where heavy and thin solid curves indicate unstable and stable waves, respectively. We selected the value of a (~ w~/w~) for the calculated curve so that it most fits the experimental dispersion, and the value of a agrees with that estimated from ‘b within the several percent. In fig. 2(b), the corresponding growth rate calculated from eq. (1) is also shown by the solid curves for the several values of y(=u 10/Vtb). The calculated values are larger by 2 to 5 times than the experimental ones, even though the energy spread of the beam is taken into account. We consider that this is ascribed to the experimental errors, the oblique propagation effects and the fact that the practical velocity distribution function of the beam electrons is not completely isotropic in the plane perpendicular to the magnetic field. In conclusion, we have presented experimental electron results oncyclotron the time harmonic behaviourwaves of thedue instability to the anisoof the
(u1 2V~b vjo) —
where the energy spread of the ring distribution (kBTb = mv~b)is taken into account, and A is the normalization constant *‘. When V1o/Vtb ~ 1, the disperA is determined by 27rAf~ +v~[2v~i,l= 1.
and ~‘~bare the plasma frequencies of the
plasma and beam electrons, respectively, Wc the electron cyclotron frequency, X = k~v?p/w~, p= k 1u1o/w~, k1 the component of wave number perpendicular to the magnetic field, and J,~and ‘n are the Bessel and modified Bessel functions of the nth order, respectively. In fig. 2(a) and (b), the frequency and meas~zredgrowth rate are plotted versus electron density w~/w~ (w~= + w~)for the fixed value ViO/vtp (=V2UbITe). The growth rate was obtained from the initial time evolution of the emission, and the values and arespontaneous kept constant on the dispersion curveof(varying
0.1
0’
=0, (~+iw~)2—n2w~
I
(b)
a
(1)
2
00dv11f~’ duj ojexp [—(vj— vJO)
tropy of the electron velocity distribution. The growth rate obtained experimentally can be explained qualitatively by the calculated dispersion relation. The detailed analysis including the numerical calculations of the growth rate taking into account the oblique propaga. tion effects is under study. 123
Volume 60A, number 2
PHYSICS LETTERS
Computer calculations are made using Facom 23075 of the Data Processing Center, Kyushu University, and Facom 230-28 of the Data Processing Center, Ehime University. This work was partially supported by a Grant-in-Aid from the Ministry of Education.
124
7 February 1977
References [1] E.G. Harris, Phys. Rev. Lett. 2 (1959) 34. 121 R.A. Dory, G.E. Guest and E.G. Harris, Phys. Rev. Lett. 14 (1965) 131. [31 J.A. Tataronis and F.W. Crawford, J. Plasma Phys. 4 (1970) 231,249. [41 R. Sugaya, M. Sugawa, and H. Nomoto, Phys. Lett. 56A (1976) 458. [51D. Boyd et al., Phys. Rev. Lett. 30 (1973) 1296.
PHYSICS LETTERS
7 February 1977
TIME BEHAVIOUR OF ELECTRON CYCLOTRON HARMONIC WAVES EXCITED BY HELICAL ELECTRON BEAMS IN A PLASMA R. SUGAYA, M. SUGAWA and H. NOMOTO Department ofPhysics, Faculty ofScience, Ehime Univeristy, Matsuyama, Japan Received 11 January 1977 Time behaviour of instabilities of electron cyclotron harmonic waves due to the anisotropy of the velocity distribution of electrons has been investigated. Measured growth rate was compared with the calculated ones.
It has been predicted theoretically that anisotropies in velocity space lead to high frequency electrostatic instabilities in a plasma [1]. Particularly, it has been predicted theoretically that the electrostatic electron cyclotron harmonic waves can become absolutely unstable in a magnetoplasma with a ring electron distribution or a mixture of ring and Maxwellian electron distributions, where the ring distribution describes monoenergetic electrons, moving only in the plane perpendicular to the magnetic fields, and uniformly distributed in velocity space on a circle with a certain radius [2,3]. We have reported the experimental observation of instabilities of electron cyclotron harmonic waves in the magnetoplasma with a mixture of ring and Maxwellian electron distribution, which were produced by the injection of the electron beam perpendicular to the magnetic field [4]. In this letter, we report an experimental result, where the time behaviour of the excited strong spontaneous emission was investigated and compared with the calculated dispersion relation and growth rate. The experimental apparatus used in this experiment is the same as that described in the ref. [4]. A monoenergetic electron beam with energy Ub = 490 eV, current = 10.8 mA, density ~b ~ X iO~cm~3,and width along the homogeneous magnetic field D 30mm is continuously injected perpendicularly to the magnetic field (10 50 gauss) into a beam-generated plasma of n~, 2 X 108 cm3, Te 7eV and momentum transfer collision frequency between electrons and neutral atoms 6 MHz at the argon gas pressure of 0.7 mtorr. The electrons of the_injected beam are moving with the speed u~(—sJ2Ub/m) in the plane perpendicular to the magnetic field. Two coaxial antenna probes, radially movable, are placed in the plasma in order to -~
122
100 )J. s.c
i—i
1 Fig. 1. Output signal of the spontaneous emission at the second 27r = 121 MHz, Wc/21r = cyclotron 56.5 MHz,harmonic. ~p/21r 130 uJ.o/utp MHz and 12,wb/21r 4 60MHz.
detect the wave. The beam-plasma system is linearly unstable, and a strong spontaneous emission growing up to 60 dB above background noise appears at the cyclotron harmonics. This instability which occurs spontaneously cannot be launched or synchronized by external modulation. It does not have an exponential growth in space; it has an almost constant amplitude in the region of the beamgenerated plasma. An output signal of the spontaneous emission from the field-intensity meter is shown in fig. 1. It is found that the amplitude is not steady in time, but switches off and on in time intervals of 10 to 100 psec. It grows exponentially in time initially, and decays after saturation [5]. We analyzed the present instabilities, using the dispersion equation of electrostatic waves pro-
Volume 60A, number 2
PHYSICS LETTERS
7 February 1977
sion relation is approximately given by
2.~ (a)
~j~i__L~IIIIII l—E~ ~1
0tb 2
2.C
~ d3J~(~))] 1dJ~(p) 2n2w~ dp dp3
1.5
where
0.05-
3
4
5
6
4wc of the spontaneous emission versus Fig. 2. (a) c~/w~. The Frequency vertical bars indicate a half-width ~w/wc of the emission. Experimental parameters are the same as those of fig. 1. The solid curve is the calculated dispersion relation with a 0.76, ujO/utp 12, ~z= 3.8, andy = 18. (b) Measured growth rate ~~/WC of the emissionof fig. 2(a). The solid curves show the calculated growth rate, where the values of a, vj.o/vtp and ~j are the same as those of the calculated dispersion curve of fig. 1(a), and the values of y, 30, 18 and 14 correspond to Tb 1, 3, and 5eV, respectively,
pagating perpendicularly to the magnetic field. It is assumed that ions cold and infinitely massive and electron distribution function is given by the following equation, 1 (\ 2~u~ /
3/2
fe(t)i,t~i)”flp +
nbA exp
[
ex
+
~
E 2V~p
2 + v2]
1
The calculated dispersion curve obtained from eq. (l)with p = 3.8 which corresponds to the maximum growth rate of the instabilities is plotted in fig. 2(a), where heavy and thin solid curves indicate unstable and stable waves, respectively. We selected the value of a (~ w~/w~) for the calculated curve so that it most fits the experimental dispersion, and the value of a agrees with that estimated from ‘b within the several percent. In fig. 2(b), the corresponding growth rate calculated from eq. (1) is also shown by the solid curves for the several values of y(=u 10/Vtb). The calculated values are larger by 2 to 5 times than the experimental ones, even though the energy spread of the beam is taken into account. We consider that this is ascribed to the experimental errors, the oblique propagation effects and the fact that the practical velocity distribution function of the beam electrons is not completely isotropic in the plane perpendicular to the magnetic field. In conclusion, we have presented experimental electron results oncyclotron the time harmonic behaviourwaves of thedue instability to the anisoof the
(u1 2V~b vjo) —
where the energy spread of the ring distribution (kBTb = mv~b)is taken into account, and A is the normalization constant *‘. When V1o/Vtb ~ 1, the disperA is determined by 27rAf~ +v~[2v~i,l= 1.
and ~‘~bare the plasma frequencies of the
plasma and beam electrons, respectively, Wc the electron cyclotron frequency, X = k~v?p/w~, p= k 1u1o/w~, k1 the component of wave number perpendicular to the magnetic field, and J,~and ‘n are the Bessel and modified Bessel functions of the nth order, respectively. In fig. 2(a) and (b), the frequency and meas~zredgrowth rate are plotted versus electron density w~/w~ (w~= + w~)for the fixed value ViO/vtp (=V2UbITe). The growth rate was obtained from the initial time evolution of the emission, and the values and arespontaneous kept constant on the dispersion curveof(varying
0.1
0’
=0, (~+iw~)2—n2w~
I
(b)
a
(1)
2
00dv11f~’ duj ojexp [—(vj— vJO)
tropy of the electron velocity distribution. The growth rate obtained experimentally can be explained qualitatively by the calculated dispersion relation. The detailed analysis including the numerical calculations of the growth rate taking into account the oblique propaga. tion effects is under study. 123
Volume 60A, number 2
PHYSICS LETTERS
Computer calculations are made using Facom 23075 of the Data Processing Center, Kyushu University, and Facom 230-28 of the Data Processing Center, Ehime University. This work was partially supported by a Grant-in-Aid from the Ministry of Education.
124
7 February 1977
References [1] E.G. Harris, Phys. Rev. Lett. 2 (1959) 34. 121 R.A. Dory, G.E. Guest and E.G. Harris, Phys. Rev. Lett. 14 (1965) 131. [31 J.A. Tataronis and F.W. Crawford, J. Plasma Phys. 4 (1970) 231,249. [41 R. Sugaya, M. Sugawa, and H. Nomoto, Phys. Lett. 56A (1976) 458. [51D. Boyd et al., Phys. Rev. Lett. 30 (1973) 1296.