Proceedings of ITC/ISHW2007

Thermal Barrier Formation for Plasma Electrons and Ions inKind of Connected Dip and Hump of Electrical Potential near
ECR Points in Cylindrical Trap
Vasyl .I. Maslov1, Ming Y. Yu2
1

NSC Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
2

Theoretical Physics I, Ruhr University, D-44780 Bochum, Germany(Received day month year / Accepted day month year should be centered 10-point type)The experimentally observed formation of localized solitary barriers for plasma electrons and ions near theelectron cyclotron resonance points near the ends of the cylindrical trap is investigated analytically. A new methodof confinement of plasma particles in cylindrical traps is proposed.
Keywords: barrier, ECR point, cylindrical trap, solitary perturbation, evolution equation.

1. Inroduction
In [1] the formation of thermal barrier of plasma
particles was observed near the location of electron
cyclotron resonance (ECR). In this paper, such thermal
barriers and their behavior in plasma are investigatedtheoretically. Mechanisms of thermal barrier formation in
the plasma near the two butt-ends of cylindrical magnetic
traps are considered. These barriers can enhance
confinement of the plasma electrons and ions incylindrical magnetic traps. We consider general case theplasma with negative ions.
21H043Fig. 1. Scheme of the thermal electrical barriers
formation for plasma electrons and ions in ECR points on
the edges of the magnetized cylindrical trap. Dotted line
and symbol H(z) show the nonhomogeneous magnetic
field. 1 is the qualitative scheme of the antenna forelectromagnetic wave injection into the plasma trap; 2 isthe cylindrical wall of the trap; 3 is the injected
electromagnetic wave; 4 is the thermal barrier in kind of
the dip and hump of the electric potential.
author’s e-mail: vmaslov@kipt.kharkov.ua

2. Thermal barrier for plasma electrons
Consider two electromagnetic waves injected into aplasma trap through its butt-ends (see Fig. 1). Near theelectron cyclotron resonance (ECR) points of theconfining magnetic well, the energy of the waves areconverted into that of the transverse motion of theelectrons. We shall show that this energy conversion canbe responsible for the observed [1] barriers of the plasma
particles. A thermal barrier is a self-consistent structure
consisting of a paired dip and hump of the electric
potential (Fig. 1). The electron and ion phase spaces atthe left barrier are shown in Figs. 2 and 3. The magnetic
well H(z) has a minimum at the center of the trap, and it
increases toward the edges of the trap. Near the ECR
point the transverse electron velocity V⊥ is increased upto V⊥o because of the resonant wave-particle interaction.
The electrons are reflected by the magnetic mirror.
Because of the inhomogeneous magnetic field H(z), theelectron transverse velocity V⊥(z)=V⊥o[H(z)/Ho]1/2, whereHo is the magnetic field at the ECR point, decreases infavorofits
longitudinalvelocity
V||(z)
1/2
V||(z)=V⊥o(1-H(z)/Ho) , which increases towards thecenter of the trap. This leads to an average electron flow
towards the center, and thus an uncompensated increaseof the electron density at the center of the trap.
The increase of the electron longitudinal velocity V||

Proceedings of ITC/ISHW2007
with respect to that of the ions near the ECR point leads
to a nonequilibrium state. The reflection of the electrons
with nonequilibrium velocity distribution from thepotential dip leads to a growth of the dip’s amplitude. So
these current-carrying electrons can excite the electric
potential dip with amplitude ϕo on an ion mode withvelocity Vc, close to zero, and are reflected from it.

Far from the dip the plasma is quasineutralne(z)⏐z→∞ =ne(z)⏐z→-∞=1-N-. Hence we derive, using (2),the expression for potential jump ∆φ near the dip(3)∆φ≈V||(2/π)1/2(1-exp(-φo))/β3,√φo2
β3≡[1-(2/√π)∫o dx exp(-x )]From hydrodynamic equations the expressions fordensities of positive and negative ions can be obtainedV

ni±=n±NL+n±τ , n±NL=no±/[1-(±q±)2ϕ/M±Vc2]1/2, (4)∂n±τ/∂z=±2(∂ϕ/∂t)(no±q±/M±Vc3)β4,β4≡[1-(±q±)ϕ/M±Vc2]/[1-(±q±)2ϕ/M±Vc2]3/2
Substituting (2), (4) in Poisson’s equation we canderive the nonlinear evolution equation∂3zφ+{β5+β6}2∂tφ/Vc3+(∂zφ/Vc2){β7+β8}-{exp(φ)-sign(z)V||(2/π)1/2{β9(5)-β10+(1-exp(-φo))β11β12/√π}}∂zφ=0,2 222 -3/222β5≡ Q+ V s+(1-2φQ+V s+/Vc ) (1-φQ+V s+/Vc ),β6≡Q-2N-V2s-(1+2φQ-V2s-/Vc2)-3/2(1+φQ-V2s-/Vc2),
β7≡Q+2V2s+(1-2φQ+V2s+/Vc2)-3/2,
β8≡Q-2N-V2s-(1+2φQ-V2s-/Vc2)-3/2,β9≡(φo/(φo+φ))1/2exp(-φo),β10≡∫√-φ√φo dy(1-2y2)exp(-y2)/(y2+φ)1/2,
β11≡[1-(2/√π)∫o√φo dxexp(-x2)]-1,β12≡[exp(-φo)/(φo+φ)1/2+2(φo+φ)exp(-φo)++4∫√-φ√φo dy y(y2+φ)1/2exp(-y2)].
Here ∂z≡∂/∂z, ∂t≡∂/∂t. (5) has been derived inapproximation of slow dip evolution, i.e. inapproximation of the small growth rate of the dip
amplitude γnl .
From the nonlinear equation (5) the dip is shown topropagate with a slow velocity Vc≈0. From (5) the growthrate of the dip’s small amplitude can also be obtainedγnl≈ωp+(V||/Vthe)3/2(q+/e)(n+/ne)1/2β13,(6)1/2β13≡{1+[1/3-(ne/n+)(e/q+)](eϕo/Te)(π/2) (Vthe/2V||)}.
One can see that the dip is formed at ratio of electron
current-carrying to thermal velocity V||/Vthe larger thanthe threshold. The threshold decreases at decreasing ofratio of electron and positive ion densities ne/n+ and isequal to zero at ne/n+<q+/3e. The threshold is maximum at
ne/n+=1.

zFig. 2. Electron phase space. Dotted line is the separatrix,
separating reflected and penetrated electrons. Arrows
specify a direction of the electron movement.
From the electron Vlasov equation and ionhydrodynamic equations one can derive the evolution
equation for the potential dip. In fact, we shall consider
the slow evolution of the dip for its description. Taking
into account that the resonant electrons are reflected fromthe dip, one can obtain from the Vlasov equation theexpression for the steady state electron distributionfunction fe
fe=foe(-[V2-2e(ϕ±∆ϕ)/me]1/2±V||),(1)
where foe is Maxwellian, ∆ϕ is the electric potential jumpnear the potential dip and has to be determined
self-consistently. The plus and minus signs are for V
greater and less than A(ϕ)sign(z), respectively. HereA(ϕ)≡[2e(ϕo+ϕ)/me]1/2.
In the following we shall use the normalizedquantities φ≡eϕ/Te, N-≡no-/no+, Ne≡noe/no+, Q±=q±/e, andVs±=(Te/M±)1/2. Furthermore, z is normalized by theelectron Debye radius rde, V|| by the electron thermalvelocity Vthe, t by the inverse ion plasma frequency ωp+-1,
and the speed Vc of the localized perturbation by theion-acoustic velocity (Te/M+)1/2. Here, Te, is thetemperature of electrons, no- and no+ are the unperturbeddensities of negative and positive ions, q± is the charge ofpositive and negative ions.
Integrating (1) on velocity, one can derive theelectron density ne in the first approximation on small V||ne≈noeexp(φ)[1-(2∆φ/√π)β1-2V||(2/π)1/2β2 , (2)β1≡∫oβdx exp(-x2) , β2≡∫oβdx (x2-φ)1/2exp(-x2)],β≡(φo+φ)1/2.

3. Barrier formation for plasma ions in kind ofelectric potential hump near the ECR point
As the electrons are reflected from the dip and theion flow passes with velocity Vo+ freely, theuncompensated volume charge of ions is formed after thedip, in the field of which the ions slow down and arereflected. This volume charge forms perturbation of theelectric potential hump. The ion flow enhances this humpof the electric potential. At first we describe theProceedings of ITC/ISHW2007
quasi-stationary properties of the hump, neglecting thenonequilibrium condition. Then taking into account thenonequilibrium condition we can obtain the hump’s
excitation, in other words the growth of the hump’s
amplitude. Further we will show that the electric potential
hump is almost fixed in space.
We consider general case, when flow of the positive
ions passes with velocity Vo+ in the plasma with electrons
and also with negative and motionless positive ions ofsmall densities.
VizFig. 3. Ion phase space. Dotted line is the separatrix,
separating reflected and penetrated ions. Arrows specify adirection of the ion movement.
In linear approximation the perturbation excitationby the positive ion flow, propagating relative to negativeand motionless positive ions of small densities, isdescribed by the following dispersion ratio:
1+1/(krde)2-ω2p+/(ω-Kvo+)2-ω2p-/ω2-ω2pq/ω2=0. (7)Here ω, k are the frequency and wave-vector of theperturbation; ωp± are the plasma frequencies of thenegative and the flow’s positive ions; ωpq is the plasma
frequency of the positive motionless ions; rde is the Debye
radius of electrons; Vo+ is the velocity of the positive ion
flow.
From (7) we show that one can select the flow
velocity such that following inequalities are correct
Vph=ω/k≈(Vo+/24/3)[(n-m+q-2/ n+m-q+2)+(n+qq+q2/n+q+2)]1/3<<Vs+,
λ=2π/k=2πrde/(Vs+2n+q+2/Vo+2nee2-1)1/2>>rde . (8)Hence the perturbation is almost motionless, that isVph<<Vs+. Vs+=(T/m+)1/2 is the ion-acoustic velocity of
the flow positive ions. Here n-, m-, q- (n+, m+, q+ ) are thedensity, mass and charge of the negative (positive) ions.
From (7) we derive the growth rate of theperturbation excitation:γ=(1.5)1/2(Vo+/rde)[(n-m+q-2/n+m-q+2)++(n+qq+q2/n+q+2)]1/3(Vs+2q+/Vo+2e-1)1/2.
(9)
On the non-linear stage of the instability
development the electric potential perturbation ϕrepresents the solitary hump of the finite amplitude ϕo.
The distribution function fe(v) of untrappedelectrons, that are arranged outside the separatrix, looks
like:fe(v)=[noe/Vte(2π)1/2]exp(eϕ/Te-mev2/2Te). (10)For trapped electrons, i.e. for electrons located inside
the separatrix, the distribution function does not dependon energy because of an adiabaticity of the evolution.
Integrating the electron distribution function onvelocity, we get following expression for electrondensity:ne=(no/(2π)1/2)(2/T)3/2∫∞odε(ε+eϕ)1/2exp(-ε/T). (11)From the hydrodynamic equations for positive ions
it is possible to get the following expression for theirdensity:(12)
n+=no+/[1-2q+ϕ/m+(Vo+-Vh)2]1/2.
Here Vh is the velocity of the solitary perturbation.
As a result from (11), (12) and the Poisson’sequation we have an equation for the spatial distributionof the electric potential φ of the perturbation of any
amplitude:φ’’=(2/√π)∫∞odae-a(a+φ)1/2-1/(1-2Qφ/voh2)1/2. (13)Q=q+/e, φ=eϕ/T, «’»=∂/∂x, x=z/rde, voh=(Vo+-Vh)/Vs+.
From the apparent condition that the electric field of
the electric potential hump is equal to zero for maximum
potential φ ’ | φ= φo=0 and from (13) we obtain the hump
velocity, voh:voh2/Q=(A-2)2/2(A-2-φo),A≡(8/3√π)∫∞odae-a(a+φo)3/2.

(14)

In the approximation of small amplitude from (13),(14) we obtain:(15)
voh2≈ Q, L≈[15√π/4(1-1/√2)]1/2φo-1/4.
If Vo+ is close to (q+/e)1/2Vs+, the perturbation isapproximately motionless.
Taking into account the small densities of negativeand motionless positive ions, we derive from the Poisson
equation the evolution equation:2ω2p+∂3ϕ/∂t3/(Vo+-Vh)3=-(ω2p-+ω2pq)∂3ϕ/∂z3. (16)
From (16) the growth rate of the non-linear
perturbation amplitude is derived:
γNL≈

(17)

≈ωp+(eϕo/T)1/2[(no-m+q2-/no+m+q2+)+(noqq2+q/no+q2+)]1/3.

Proceedings of ITC/ISHW2007

4. The electron mechanism of the barrier
formation for plasma ions near the ECR point
Let us consider the mechanism of the electric
potential hump formation by current-carrying (i.e. withV||≠0) plasma electrons near the dip of the electric
potential.
The potential jump ∆φ, formed near the dip,
accelerates electrons to the first front of the dip. Further
we assume that the current-carrying velocity V|| of theelectrons on the back front of the dip is close but smaller
than the electron thermal velocity Vth. Hence on the firstfront of the electric potential dip the electron
current-carrying velocity becomes more than electron
thermal velocity due to the flow continuity law. Hence on
the first front of the dip the Bunemann instability isdeveloped [2]. Due to Bunemann mechanism interactionof electron flow with this region an electric potential
hump is excited.
Let us describe the solitary perturbation in kind ofelectric potential hump. We will show that it represents a
nonlinear perturbation on a slow electron-sound mode.
As it is slow, resonant electrons can be trapped by suchperturbation.
From Vlasov’s equation the expression for theperturbation of the electron distribution function isderived. Integrating this expression on velocity in case ofa small amplitude of the solitary perturbation ϕo we getthe expression for the perturbation of electron densityδn'=∂tϕβ14+ϕ'R(y)+ϕϕ'β15,(18)222β14≡[y+(1-2y )(1-R(y))/y], β15≡[1-y +(3/2-y )(R(y)-1)],
R(y)≡1+(y/√π)∫-∞∞dtexp(-t2)/(t-y), y≡(V||-Vh)/Vth√2.
Here a point means derivative in time, and a prime is aspatial derivative. Vh, ϕ are the velocity and the potentialof soliton. φ≡eϕ/Te. Te is the electron temperature.
Substituting (18) in Poisson’s equation, we derive instationary approximation an equation, describing spatial
distribution of potential:(19)(φ')2 =φ2R(y)-[1+(2y2-3)R(y)]φ3/6.
The soliton width is followed from (19) to beapproximately equal ∆z=(48Te/ϕo)1/2. The soliton width
decreases with amplitude growth.
One can show that Vh≈0 if V||≈1.32Vth .
In case of large amplitudes, eϕo/Te>1, from Vlasovequation we have the expression for electron distributionfunction
f=fo[(u2-2eϕ/m)1/2+Vhsign(u)]for1/2⏐u⏐=⏐V-Vh⏐>(2eϕ/m) . Here fo is Maxwell distributionfunction. Thus we obtain the equation for the soliton
shape(φ')2=(20)=-φ+(2/√π)1/2∫-∞∞dt(t-y)2exp(-t2){[1+φ/(y-t)2]1/2-1}
From (20) we derive the soliton width∆z=[2eϕo/Te(√2-1)]1/2(21)From (21) we conclude that the soliton width grows with
ϕo. Therefore, it is necessary to take into account theelectrons, trapped by the soliton field. Assuming thedistribution of their density as ntr(z)=n2exp[eϕ(z)/Ttr], wederive similarly to (21), that width and velocity of thesoliton grow with amplitude growth (in difference fromthe case of small amplitudes of the solitary perturbation)
Here Ttr , n2 are the trapped electron temperature anddensity.
Thus, neglecting the ion mobility, this solitary
perturbation is stationary and an electron one. However attaking into account of the ion mobility it is necessary toexpect occurrence of slow growth of the perturbation’s
amplitude, as a result of Bunemann instabilitydevelopment. In the following order of the theory ofdisturbances from (18) we derive the correction of thenext order to a spatial derivative of electron density(22)
ne1'=∂tϕ[y+(1-2y2)(1-R(y))/y]This expression as follows from a spatial derivative fromPoisson’s equation must be equal to a spatial derivativefrom the ion density perturbation ni'. It is possible to findni' in a linear approximation from ion hydrodynamicequations(23)∂2tni=(me/mi)φ''Equating the second time derivative from (22) andthe first spatial derivative from (23), we obtain(24)∂3tφ=(6me/mi)φ'''The solution of (24) we search as
φ(z,t)=φo(t)η[z-∫-∞t dt1δvo(φo(t1))] .
(25)
η(z) is quasistationary shape of the perturbation. We
assume that ∂tφo(t)=γφo(t). In (25) the change of soliton
velocity with change of its amplitude is taken intoaccount.
Substituting ∂tφ for γφ-δvhφ’, we obtain from (24)(26)γ≈(me/mi)1/3φo1/2
In the case of the electron mechanism of the barrier
formation for plasma ions the inverse spatial sequence ofthe hump and dip, comparing with that one shown in Fig.
1, is realized.

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