ISSN 10637842, Technical Physics, 2015, Vol. 60, No. 2, pp. 217–221. © Pleiades Publishing, Ltd., 2015. Original Russian Text © N.M. Gorshunov, E.P. Potanin, 2015, published in Zhurnal Tekhnicheskoi Fiziki, 2015, Vol. 85, No. 2, pp. 59–63.

PLASMA

Electron Temperature of a Plasma in a Calcium Plasma Source Based on an Electron Cyclotron Resonance Discharge in Vapor N. M. Gorshunova,b and E. P. Potanin*a,b a b

National Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia *email: [email protected] Received December 19, 2013; in final form, June 26, 2014

Abstract—We propose a method for estimating the plasma temperature in a source based on the electron cyclotron resonance (ECR) discharge. Calculations are made for the calcium plasma for the process of iso tope separation by the ion cyclotron resonance (ICR) method. Linearly polarized radiation from a gyrotron is fed from the side of the strong field. We assume that the resonance condition is observed for the extraordi nary wave for a small fraction of resonant electrons moving towards the wave. The longitudinal electron tem perature of the plasma is calculated and the concentration of resonant and nonresonant electrons is estimated on the basis of the assumption of smallness of the degree of ionization of the medium. DOI: 10.1134/S1063784215020085

INTRODUCTION The most important block of any ion cyclotron res onance (ICR) separation unit is the source of a low temperature plasma. One of the methods of producing intense metal plasma flows of a large cross section involves the ionization of the working substance in the electron cyclotron resonance (ECR) discharge [1–4]. Metal vapor is ionized by electrons heated by micro wave radiation. The transverse energy and electron density in an ECR source of a lowtemperature cal cium plasma was estimated in [5] in the case when the resonance zone is located in the region of a decrease of the main magnetic field in the ICR unit. A model for designing the source based on the assumption about the presence of two groups of “hot” and “cold” elec trons with strongly different concentrations was pro posed in [6]. Assuming that the longitudinal electron temperature is known from experimental data for ECR sources with close parameters, the authors of [6] estimated fraction β of the power lost for radiation in the discharge. In the present study, we calculate the characteristics of the plasma in the ECR source on the basis of the solution of the continuity equations for electrons as well as the energy conservation laws for the hot electron component, and the characteristics of the plasma in the ECR source are calculated for the entire discharge in the whole. A distinguishing feature of our analysis is that computations are performed without assumptions concerning longitudinal electron temperature Te||. The value of this quantity is calcu res

lated simultaneously with the number densities n e and ne of resonant and nonresonant electrons.

The scheme of the ICR unit was described in detail in [7, 8]. The plasma source is located in the end zone of the suppressed magnetic field in the separation unit. Thermally evaporated Ca is introduced into the zone of increasing magnetic field into which microwave radiation is directed via a special waveguide transmis sion line. Then the plasma enters the region of a uni form magnetic field (ICR zone). If the breakdown conditions for the vapor medium are observed, a discharge sustained by microwave radi ation is initiated in the vicinity of ECR. Electrons with a high “transverse” velocity relative to the magnetic field ionize the vapor of the substance being separated. Radial diffusion of electrons is hampered by the main magnetic field B0 of the setup. It is possible to create conditions impeding the departure of electrons towards weaker magnetic field by combining calcium evaporator with a disk electrode repelling electrons, to which a small negative potential (relative to the plasma) is applied. The electric field repels electrons towards the strong magnetic field, but unimpeded departure of all electrons to the ICR separation zone is hampered by the magnetic mirror. Hot electrons cap tured in such a combined trap practically do not leave the discharge zone and serve as a source of ionization of vapor by electron impacts. At the same time, the cold electron component can leave the trap in the direction of the ICR zone together with the neutral gas. Thus, the conditions for sustaining the discharge exist on both sides of the ECR zone.

217

218

GORSHUNOV, POTANIN Disc electrode

Discharge zone

B SHF

Ca R Ldis

Evaporator

ECRzone

Fig. 1. ECR zone and the discharge zone.

1. MICROWAVE HEATING OF ELECTRONS IN A NONUNIFORM MAGNETIC FIELD The behavior of the electron component in the dis charge type under investigation is extremely compli cated. A noncontradictory theory of such a discharge has not been constructed as of yet. We will try to clarify the main features of the process using a simplified model that makes it possible to determine the electron temperature as well as the plasma density. Since the extraordinary electromagnetic wave can propagate in the plasma only from the side of the strong magnetic field [9], we assume that radiation is introduced into the discharge zone from the side of the ICR zone in the direction quasiparallel to the magnetic field lines. Thermal (“chaotic”) velocities of electrons in the space of longitudinal velocities are usually much higher than hydrodynamic velocities. Therefore, we can assume that two oppositely directed and mutually compensating electron flows (in the direction of microwave radiation and in the opposite direction) exist in each cross section of the discharge column. In the steadystate regime, the electrons falling in reso nance with microwave radiation with allowance for the Doppler frequency shift are mainly heated. If the microwave power is limited, regimes can exist in which only a small part of electrons moving against micro wave radiation is heated (these electrons are the first to receive energy from the field). In the model proposed here, there are two components of the electron gas, which differ only in the transverse “temperature,” viz., resonant hot and nonresonant cold components. The longitudinal temperatures of the two components will be treated as identical. The entire space of the dis charge can be divided into a narrow ECR zone (bold arc in Fig. 1), in which the transverse energy of elec trons increases, and the discharge zone proper, which is conditionally shown by the dashed rectangle and in which the main ionization processes take place. Let us obtain estimates for the setup described in [7, 8] with a uniform magnetic field B = 1.5 T. The electron time of flight through the ECR zone can be estimated from the relation [10]

2πL* , τ ≈  v *z ω c

(1)

where v *z is the characteristic longitudinal velocity of electrons, ωc is the cyclotron frequency, and L* = B res   is the quantity characterizing the rate of ( gradB ) res spatial variation of the magnetic field in the resonance region. For the parameters of the calcium plasma of interest for the setup described in [7], the width of the resonance zone is defined by the relation 2πL*v * Δz ≈ v *z τ ≈ z , ωc

(2)

and amounts to about 2 mm for Te|| = 1 eV, L* = 0.65 m, and ωc = 2.3 × 1011 s–1. Time of flight τ of electrons through the ECR zone, which can be estimated using relation (1), is on the order of 5 × 10–9 s. Time τcol of collision of electrons with heavy particles for an elec tron energy of about 20 eV [6] turns out to be much larger than the value of τ for neutral particle concen tration nn < 1019 m–3 and degree of ionization α < 0.1. The characteristic value of the transverse energy acquired by a resonant electron flying through the ECR zone can be defined as 2˜2 2 eE τ (3) ε *⊥ = . 2m e Let us find the relation between electron tempera ture Te|| and the main parameters of the source and the properties of the plasma. This can be done considering the balance between the numbers of hot and cold elec trons in the discharge and the total energy conserva tion law. We assume that the density of excited parti cles is much lower than the density of atomic particles in the ground state. The difficulty in the application of energy conservation law is associated with the fact that the channels of energy supply to the discharge and for energy removal as a result of the flow of the plasma out of the source, as well as radiation loss channels, are separated by the intermediate process of energy trans fer from the group of resonant electrons with a high transverse energy to cold particles. In analysis of the balance of hot particles, the ionization of neutrals and elastic scattering of electrons from neutrals are the main processes. It should be noted that although elas tic scattering is not accompanied with energy loss, it leads to the conversion of the transverse energy of electrons into the longitudinal energy as a result of which electrons are not confined in the trap any longer and leave the discharge zone. Assuming that the entire power absorbed by resonant electrons is equal to power N of microwave radiation incident on the plasma, we obtain the following relation for the longitudinal flux density of hot resonant electrons passing from the ECR zone to the discharge zone: TECHNICAL PHYSICS

Vol. 60

No. 2

2015

ELECTRON TEMPERATURE OF A PLASMA IN A CALCIUM PLASMA SOURCE

N j e|| ≈  , * ε ⊥ S res

(4)

Using relations (4)–(7), we obtain t

where 2˜2

πe E L* ε *⊥ =  2 2m e kT e|| ω c is the characteristic value of the transverse energy acquired by a resonant electron during its flight through the ECR zone, which was obtained taking into account the results from [6], and Sres is the reso nance cross section of the plasma column. The esti mates obtained in [6] show that for powers N ~ 1 kW, the characteristic transverse energy of electrons in the setup under investigation is approximately 20 eV, which is substantially higher than their initial energy, and we can disregard the spread in the initial phases of electrons. 2. CALCULATION OF PLASMA CHARACTERISTICS IN THE ECR DISCHARGE ZONE Considering that the number of hot electrons entering the discharge zone should be equal to the number of hot electrons leaving the category of hot electrons as a result of elastic and ionization collisions with neutral particles, we obtain res j e|| S res = [ n n n e ( σ e + σ ion ) ]v *⊥ L dis S res , (5) where σe is the cross section of elastic collisions of hot resonant electrons with neutrals, σion is the cross section res

of ionization from the ground state, n e is the number density of resonant electrons in the discharge zone, Ldis 2ε *⊥  . me Equation (5) should be supplemented with the conti nuity condition for cold electrons in the form is the length of the discharge zone, and v *⊥ =

kT res n e e|| = n n n e σ ion v *⊥ L dis , (6) mi which expresses the equality of the number of cold electrons generated in the discharge to the number of electrons leaving the discharge zone as a result of lon gitudinal ambipolar diffusion. We assume that direct ionization of Ca atoms from the ground state is caused only by hot resonant electrons. The entire power received by the discharge from microwave radiation is spent on the acceleration of the ambipolar flow of electrons and ions, which, in addi tion, carries away ionization energy εi = eIi (Ii is the ionization potential), and to radiation βN, where β is the fraction of energy lost for radiation. We can write the equation for energy in the form kT N – βN = n e e|| ( 2kT e|| + ε i )S res . mi TECHNICAL PHYSICS

Vol. 60

No. 2

(7) 2015

219

3/2

1/2 ˜ 2 1 1/2 π ( 1 – β ) ( σ e + σ ion )e E L* +  t –   = 0, 3/2 2 4 2m e σ ion ω e I

(8)

kT where t = e|| . Let us estimate the value of β, assum εi ing that collisions of the “quenching” type can be dis regarded. Then, the radiation flux in spectral lines is determined by the number of exciting collisions. We take into account only radiation power Nc associ ated with the excitation of neutral atoms during their collisions with the main part of nonresonant electrons: N c = n n n e ΔEL dis SK ex ,

(9)

where ΔE is the excitation energy, Kex is the excitation constant defined as ∞

K ex =

∫σ

ex ( ε )f ( ε )

ΔE

2ε dε,  me

(10)

σex(ε) is the excitation cross section, and f(ε) is the Maxwellian function of electron distribution over lon gitudinal energies ε, which is defined as 2 ε  exp ⎛ –  ε ⎞ . f ( ε ) =  1/2 3/2 ⎝ kT e⎠ π ( kT e )

(11)

The energy dependence of the excitation cross sec tion can be written in the form [11] 4

2πe N a f 01 ln [ 1 + 0.5 ( x – 1 ) 1/2 ] σ ex   2 , 2 x+3 ( 4πε 0 ) ΔE

(12)

ε where x =  , Na is the number of valence electrons ΔE of the Ca atoms, and f01 is the oscillator strength for transition 41S0–41P1. Using relations (8)–(11), we obtain 3/2 3/2 ⎛ ΔE⎞ 2 2 β =   n n A ΔE   L dis S res n e 1 3/2 ⎝ εi ⎠ t N πm e ∞

(13)

1/2

x ln [ 1 + 0.5 ( x – 1 ) ] ΔE ×  exp ⎛ –  x⎞ dx, ⎝ x+3 εi t ⎠

∫ 1

where 4

2πe N a f 01 A 1 =  . 2 2 ( 4πε 0 ) ΔE Substituting Eq. (13) into (8) and using Eqs. (5) and (6), we obtain the following equation for deter mining the electron temperature

220

GORSHUNOV, POTANIN

The above assumption concerning weak ionization of the medium, which holds for

Te||, eV

m i m e σ ion cε 0 ω c    nn , 2 πe L* ( σ e + σ ion )

1.0

is satisfied for the above parameters with a margin. It should also be noted that the assumption

0.8

res

ne  ne

0.6

used in calculations is valid if 0.4

kT e||    1. m i σ ion n n v *⊥ L dis

0.2

0

0.2

0.4

0.6

0.8

1.0

1.2 N, kW

Fig. 2. Dependence of longitudinal electron temperature Te|| on power N. ∞

1/2 ⎛ 3/2 2 ln [ 1 + 0.5 ( x – 1 ) ] t + 1 t – M ⎜ t – R x 2 x+3 ⎝

∫

3

1

∫

(14)

ΔE ⎞ ⎞ × exp ⎛ –   x dx = 0, ⎝ ε i t ⎠ ⎟⎠

(17)

The value of Te|| depends on the power absorbed by electrons relatively weakly. If we assume that Te||  1 eV, relation (17) also holds for the above parameters of the problem. Using relations (15) and (16), we obtain N = 1 kW, res n e  2 × 1016 m–3, ne  1.3 × 1018 m–3, β  0.9, and Te||  1.1 eV. It should be noted that the value of longi tudinal temperature Te|| obtained in [6] for β = 0.9 is 0.8 eV. Figure 2 shows the value of Te|| calculated in the range of powers from 0.25 to 1.0 kW, indicating a rela tively weak increase in temperature with power N. This result is in qualitative agreement with the experimental data from [16], in which the electron temperature at the outlet of the ECR plasma source for various condi tions and gases varied in the interval 0.5–2 eV, increas ing slightly upon an increase in the power.

where 1/2 ˜ 2 π σ e + σ ion⎞ e E L* M =  ⎛    , 3/2 ⎝ ⎠ σ ion 4 2 me ωc I

CONCLUSIONS

m i ω c n n N a f 01 L dis e  16   R =  2 1/2 ˜ 2 ( 4πε 0 ) πΔE E L* 2

σ ion ⎞ ⎛ ΔE ⎞ 3/2 × ⎛  .   ⎝ σ e + σ ion⎠ ⎝ ε i ⎠ ˜ = 3.36 × 103 V/m), n = 1019 m–3, For N = 1 kW ( E n Ldis = 0.1 m, L* = 0.65 m, f01 = 1.5 [12], σe = 3 × 10–19 m2 [13, 14] (for an energy of 20 eV), σion = 5 × 10–20 m2 [15], ωc = 2.3 × 1011 s–1, Ii = 6.1 V [15], and ΔE = 2.9 eV, the solution of Eq. (14) gives Te||  1.1 eV for the temperature of the main electron component. Using expressions (4) and (5), we can estimate the res values of n e and ne from the equations res ne

= N  , ε *⊥ ( σ e + σ ion )S res n n v *⊥ L dis

N m i σ ion n e =  . ε *⊥ kT e|| ( σ e + σ ion )S res

(15) (16)

We have estimated the temperature and number density of electrons in the source of lowtemperature calcium plasma based on the ECR discharge in a non uniform magnetic field. We assumed that two groups of electrons with strongly differing transverse energies are formed in the discharge. As a result of analysis based on the solution of the continuity equations for electrons and the energy conservation laws for the hot electron component and for the entire discharge in the whole, the longitudinal electron temperature and the plasma concentration were estimated. The results have been compared with the results of earlier analysis and with available experimental data. It is shown that a more correct calculation performed here leads to bet ter agreement with experiment. ACKNOWLEDGMENTS The authors are grateful to A.V. Eletskii and A.V. Timo feev for fruitful discussions. This work was supported by the Russian Founda tion for Basic Research (project no. 120212020 ofi_m). TECHNICAL PHYSICS

Vol. 60

No. 2

2015

ELECTRON TEMPERATURE OF A PLASMA IN A CALCIUM PLASMA SOURCE

REFERENCES 1. P. Louvet, in Proceedings of the 2nd Workshop on Sepa ration Phenomena in Liquids and Gases, Versailes, France, 1989, pp. 5–104. 2. Yu. A. Muromkin, Itogi Nauki Tekh., Ser.: Fiz. Plazmy 12, 83 (1991). 3. A. I. Karchevskii and Yu. A. Muromkin, Isotope, Ed. by V. Yu. Baranov (IzdAT, Moscow, 2005), Vol. 1, p. 307. 4. M. Mussetto, T. E. Romesser, D. Dixon, et al., in Pro ceedings of the IEEE International Conference on Plasma Science, San Diego, CA, USA, 1983, p. 70. 5. V. S. Laz’ko, E. P. Potanin, and A. L. Ustinov, Tech. Phys. 57, 950 (2012). 6. E. P. Potanin and A. L. Ustinov, Plasma Phys. Rep. 39, 510 (2013). 7. N. M. Gorshunov, V. S. Laz’ko, Yu. A. Muromkin, E. P. Potanin, and A. L. Ustinov, Perspekt. Mater., No. 10, 72 (2011). 8. N. M. Gorshunov, D. A. Dolgolenko, Yu. A. Murom kin, E. P. Potanin, and A. L. Ustinov, Prib. Tekh. Eksp., No. 1, 105 (2011).

TECHNICAL PHYSICS

Vol. 60

No. 2

2015

221

9. A. V. Timofeev, Resonance Phenomena in Plasma Oscil lation (Plasma Physics) (Taylor, 2010). 10. E. V. Suvorov and M. D. Tokman, Sov. J. Plasma Phys. 15, 934 (1989). 11. A. V. Eletskii and B. M. Smirnov, in Encyclopedia of LowTemperature Plasma, Ed. by V. E. Fortov (Nauka, Moscow, 2000), p. 211. 12. A. A. Radtsig and B. M. Smirnov, A Handbook on Atomic and Molecular Physics (Atomizdat, Moscow, 1980). 13. V. A. Kelman, E. A. Svetlichnyi, and E. Yu. Remeta, Tech. Phys. 54, 1168 (2009). 14. V. I. Kelemen, E. Yu. Remeta, and E. P. Sabad, J. Phys. B 28, 1527 (1995). 15. Tables of Physical Quantities, Ed. by I. K. Kikoin (Atomizdat, Moscow, 1976), p. 201. 16. A. Compant La Fontaine, P. Louvet, P. Le Gourries, and A. Pailloux, J. Phys. D 31, 846 (1998).

Translated by N. Wadhwa