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1 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN LIBRARIES, GENEVA llllllllllllllllllllllllllllllllllllllllllillli CERN-PS CERN/ 94*3 H, / ji GY ELECTRON AND ION CONFINEMENT CONDITIONS IN THE OPEN MAGNETIC TRAP OF ECR ION SOURCE G. Shirl<ov* Abstract The improvement of "The Classical Model of Ion Confinement and Losses in ECR Ion Sources" is presented. It is shown that electron density in the source can be described with the balance equation for the electron component. Two balance equations are used here for the electron distribution with two different energy components. The mirror ratio is necessary for the determination of electron confinement time in the longitudinal magnetic field with plug configuration. The negative electrostatic potential regulates the rate of ion losses. The ion confinement time depends also on the ion energy. It is proposed to use a balance equation for the ion energy in this paper. This equation takes into account three main processes which can change the ion energy: the heating due to the elastic Coulomb collisions with high energy electrons; the heating resulting from the ionic charge state increasing in the potential well due to electron impact ionization; the cooling due to the loss of ions with an energy higher than the value of potential barrier in the trap. The results of this new approach to the problem of electron and ion confinement will be used for the understanding of the physical processes in the ECR ion sources and numerical simulations. ]oint Institute for Nuclear Research Dubna Geneva, Switzerland 11/7/94 OCR Output

2 The working region for ion production in the ECR ion source (ECRIS) is the plasma confined in the open magnetic trap. A system of permanent magnets and coils creates the compound magnetic field distribution in the source. The general view of one of the most successful ECRIS called MHQIMAFIOS [1] and the axial field distributions in it are shown in Fig. l. The permanent magnet hexapole is used for production the longitudinal magnetic field with azimuthal variations. Special coils (Sl S5 in this Fig. 1) make regions with increased magnetic field or magnetic mirrors on the ends of the source. The ions and electrons of plasma are generated from the neutral gas in the source chamber by electron impact ionization. A microwave field heats the electrons in the trap. The frequency of microwave field corresponds to the frequency of the electron Larmor rotation at surfaces in the magnetic field in the source. Two main physical processes detemiine the rate of electron and ion losses. The first is the classical processes connected with the motion of the charged particles in the electromagnetic fields and elastic scattering. The second is turbulence and different types of plasma instabilities. These later processes are stronger and may be a cause of a high rate of ion and electron losses. Following our previous papers [2,3], let us suppose that a thorough optimization of the magnetic field spatial distribution and a choice of all other source parameters in the real experimental conditions stabilize the plasma instabilities and turbulence. That is why we shall take into account here only the classical processes to study the electron and ion confinement. The magnetic mirrors reflect the charged particles and confine the plasma. Only the particles with velocity vectors in a small solid angle along the trap axis can be lost from the plasma. In the static case the elastic Coulomb collisions between the charged particles change the direction of particle movement and therefore there is a continuous loss of electrons and ions in the source. The heated electrons have much more energy and less probability of scattering than the ions. The relatively high rate of ion losses creates the negative potential of the plasma that regulates the ion losses and keeps the general neutrality of plasma. Thus, the mirror configuration of magnetic field confines the electrons and the negative potential of plasma, or potential well, confines the positive charged ions in the ECR ion source. This conclusion and the consideration of elastic and inelastic collisions in the plasma were assumed as a basis of "The Classical Model of Ion Confinement and Losses in ECR Ion Sources" [2,3]. The ionization process of neutral atoms and ions is described by a set of non-linear differential balance equations for all ionic charge states in the plasma [2,3]. These equations take into account all charge changing transitions in the source. In the static case, when all the processes in the source are stationary, the left sides of the equations can be chosen as equal to zero (dn,/dt = 0). Then this set of balance equations transforms into a set of algebraic equations. OCR Output

3 The main parameters of plasma, such as the electron and ion temperatures, the potential of plasma, the electron and ion confinement times, are included in the balance equations as parameters. Some special additional relationships connect the parameters of plasma to each other and to the ionic and electronic densities. The complete sets of balance equations with these additional equations are used for the numerical simulation of ionization processes in the ECR plasma. The parameters of plasma greatly influence the ionization process, particularly the charge distribution and the production of multiply charged ions. Therefore these additional equations and conditions for the plasma parameters are very important for the ECR source study and are the subject of this paper. Electron Confinement The electrons appear in the process of electron impact ionization of the atoms and ions in the plasma. The energy of newcomer electrons corresponds to the energy of ionization and has a range of some tens or hundreds electron volts. The charged particles rotate in the magnetic field B with cyclotron frequency. It can be defined for the electrons with: cu, = Q-. (1) Here me is the electron mass and e is the electrical charge of the electron. lf the electromagnetic wave has a frequency equal to the cyclotron frequency wc, then it can transfer the energy to the particle and heat it. The energy of heated electrons reaches the range of tens kev or sometimes hundreds kev according to some experimental data [4,5]. Only a small portion of the electrons has the resonance condition if the microwave source has a fixed frequency. These electrons move on a surface that has the resonance value of magnetic field B, All other electrons are not heated and remain at low energy. Therefore, the energy distribution function of electrons must be rather complicated. The value of electron energy is a very significant parameter for an ECR ion source. The probability of elastic scattering and loss is dependent on the energy of the charged particle. The cross section of electron impact ionization depends strongly on the electron energy. What is more, the electrons can ionize only those ions with an ionization potential lower than the electron kinetic energy. Hence, the maximum ionization state is limited by the electron energy. The assumption of two electron components with different energies is quite realistic from the physical point of view [6]. One component has a low energy that corresponds to the ionization energy of ions and atoms, the second has high energy electrons in kev region. It is possible to use a balance equation to determine the electron density in the numerical simulation of the ion charge state distribution in the ECR ion source [3]. Two balance equations are necessary in the case of two electron components: OCR Output

4 c C W 2 h 1 h 2 h nmj = Zi C[f(c)(ll2,[()()]tT these equations. /tlq lq+2 i 'léfnai + % lé' 2lQ +2 i 'VE ni l %- L <2> function of electron energy. confinement conditions. lt was shown [7] that with the so called "mirror ratio" ] dn;ng ng = dt Th Te movement of electron in the magnetic field, the value Th has a microsecond time scale. If one considers the supplementary injection of electrons into the ECR plasma then it will be possible to use an additional term related to the electron source in the one of Eqs. (2) or (3). The cross section 03 depends on the electron velocity Ve and it is necessary to use the average value <0 i Ve>. One can obtain it as the result of integration with the distribution It seems to be obvious that the value of magnetic mirror has an influence on the electron Te ~ lnr e h Here: ne are the electron densities; re are the electron confinement times; Ve are the velocities of electrons; o;(ve) are the cross sections of electron impact ionization as a function of electron velocity; Th is a characteristic time of electron heating with microwave power. o" and U2! correspond to the cross sections for single and double ionization, accordingly. All the upper indexes c correspond to the cold electron component and the indexes h to the hot electrons in The very important and, usually, unknown parameter is the time of cold electron heating 17,. The theory of electron heating with microwave power is out of range of our consideration for the moment. It is possible to assume that the electrons are only heated during not more than a few passes through the resonance surface in the source. Thus, taking into account the spiral R : ity; Bmin where Bmm and Bmax are the minimum and maximum values of magnetic field in the source. In the ECR ion sources R is usually in the range from 2 to 4. According to Pastukhov [8] the similar formula to evaluate the time of ion confinement is: 1 R + R/1 R T. = A (4) where ve is the frequency of electron collisions with all kinds of particles (electrons, ions and neutrals) in the plasma: OCR Output

5 Ve = VEB + véi + v60 The rates of electron scattering can be determined as usual [9]: Vee : trmcn L e e ee. %x/e me 4m*2m2c4. _ if E me _ 4 2 *7 no v0-. ><l0 Z 6 (7) where re is the classical radius 0f electron; c is the velocity of light; Te is the electron temperature; no is the density of neutrals in the plasma; Z is the atomic number; L are the so called Coulomb logarithms. The quantum formula for Coulomb logarithms is used for electron scattering: Lee = Lei = ln (8) Z &. EZ Two different electron energies (or temperatures) and densities must be taken into account in the expressions (4) to (8) if two electron components are to be used. Equations (2) to (8) describe the processes of the electron generation and confinement in the ECR plasma taking account of the mirror configuration of the magnetic field and the double component distribution for the electron energy. The electron heating rate with microwave power th can be chosen as an input or a fitting parameter. Ion Confinement The possible charge states of ions in the source are determined by the ion lifetime and electron density neq. This value is called "ionization factor". Therefore, the ion confinement time TQ is one of the most fundamental values for describing the plasma parameters in the ECR ion sources. The ion confinement conditions strongly depend on the ion energy in the plasma. The total ion energy can be described by the balance equation for the ion energy in the source [10]. Without plasma instabilities and turbulence we can obtain the balance equation for ion energy taking into account three main processes that change the ion energy in the ECR source: a) the electron heating due to the elastic Coulomb collisions; b) the heating due to the charge state changes in the electrostatic potential well; c) the cooling due to the losses of ions with the energy higher than the value of potential barrier ieu. OCR Output

6 It is also possible to suppose that the initial thermal energy of neutrals is negligible and does not influence on the total ion energy. Thus, the balance equation is: [Z-(" E )] de AE 4 i E" ; 4 JJ; hi : + - dz Eini dr Zi Ai dr Zi ri 9 ( ) where dei/dt is the rate of ion heating due to the collisions with electrons; AEi / Ai and dn; / dt are the corresponding energy increasing with the change of ionic charge state and the rate of ionization; is the average energy of the ions that leave the potential well. a) The rate of ion heating due to the elastic Coulomb collisions with electrons is [11]: dei 4s/27tneZ2r;32m3c41/me Lei dt Aivrr/if (10) where the ion energy Ei = 1.5T,; Z is the atomic number; A and M are the atom mass number and mass of a nucleon. b) For determination of the ion heating rate as the result of ionization in the negative electrostatic potential well, let us suppose that in the centre of the ECRIS, where the most parts of the ions appear and move that the density of electrons is close to a constant value. Then it is possible to use previous results [12-14] for ion energy in a negative potential well of an electron beam with constant density. The average energy of the ions Ei in a potential well U has a square root dependence on the charge state i: Ei:./{ eu, (11) and that it increases for every ion with the increased charge state due to the electron impact ionization. Then Mil - Ei Az 21 <12> The other possible way to obtain this result is by consideration of the average kinetic and potential energy of an ion in the potential well. If we suppose a square dependence of potential energy on the distance from the centre of well (it is so for our assumption of plasma constant density) then, according to the virial theorem [13], the average kinetic energy of ion oscillation in the potential well UQ is equal to the average potential energy U,. OCR Output

7 Hi : Us Z If the charge state of an ion increases by one unit due to ionization, then the potential energy increases by U,/i and the total ion energy increases by E,/2i correspondingly. This satisfies the relation (12) and its value is twice as large for double ionization. The rate of ionization is determined by the cross section of electron impact ionization Gi. 21 d;. =~ lévfn ni (13) and the corresponding energy increase is: AE; neu l Z`1 n T dn, Oi i i Z; KT? = el@ TGl+EEiGi+l U4) c) The ions with energy higher than the value of potential barrier ieu leave the well and remove energy from the plasma. The average energy of these ions can be defined as EF = ieu f (Ei)EdE f(ei)de <15> where f(e,) is a distribution function of the ion energy. If the rate of elastic ion ion collisions is high, then all ion components have the Boltzman energy distribution with equal, (or approximate equal) temperatures 7} [2,3]. The integrals in the equation (15) are equal to the incomplete gamma functions and the average energy of these high energy ions can be evaluated as with x = ieu/ti. * 3 E=7Q +1 i 2 x (16) r 1+x+\/x +(4x/zz:) Finally, the complete balance equation is: (17) [ ] Qt Z2Z gi de U 3 `n - n _ n Ei! 1` 2il +2)+ {i `? i% Zim Ee*"( i ")'ida ** iiz? OCR Output Here we take into account single and double ionization processes.

8 In the static case the left side of Eq. (17) is equal to zero and we can couple it with the condition of equal flows of electrons and ions from the source: c h 2-%-%-% I Ti TT e e <18> and G = The confinement times for ions Ti can be defined according to the Pastukhov theory for confinement of charged particles in the open magnetic trap [8]: with x = ieu/ti l is the effective length of the source working region in (19). The value vik is the ion collision frequency [1 1]: In Ref. [9] an approximate formula for the collision frequencies between ions and neutral atoms VK) 1S glvcn: It is now possible to define the potential of the plasma U and the temperature of ions Ti by solving Eqs. (17) and (18) in the static case. Naturally, the complete set of balance equations for all ionic charge states [3] is necessary to determine the total charge state distribution. If we study dynamic regime of an ECR ion source where the left sides of the balance equations are not equal to zero, then condition (18) may be disturbed. According to Ref. [3], for a cylindrical source working volume with length l and radius r the potential U can be evaluated as: x/e(r+1)1n(2r+ 2) 2R [l Ti = Rl + -Q + X)(Vik + Vio) Vk _ i _ 47rre2m3c4i2 E x/azajj2 jly 3 (20) /2 ' io 1.5>flino _;k4c 2f( ) 2 r U=ener1+ln exp(x), (19) (22) OCR Output where _ ini )" He

9 Equation (17) with (19) (22) in the dynamic regimes of the ECR ion source, or Eqs. (17) and (18) with (19), (20) in the static case, relate the plasma potential, the ionic average energy (temperature) and the continement times of ions in the source. Conclusions Consideration of the conditions of electron and ion confinement in the open magnetic trap of an ECR ion source is a further development of "The classical model of ion confinement and losses in ECR ion source" [3]. The balance equations for both low and high energy electron components are able to take into account the real energy distribution of electrons in the plasma. The proposed formulae for electron confinement time include the mirror ratio of magnetic field in the source. The balance equation for total ion energy in the trap describes three main processes that change the ion energy in the ECR source: the electron heating due to the elastic Coulomb collisions, the heating due to the charge state changes in the electrostatic potential well and the cooling due to the losses of ions with the energy higher then the value of potential barrier. These results will be used in the study and numerical simulations of the ionization processes in the ECR ion sources. References [1] R. Geller and B. Jacquot, MINIMAFIOS, Source d'ions Pulsés Fournissant des Faisceaux d'ions Complétement Epluchés (in French), Nucl. Instrum. Methods 202, 1982, p [2] G. Shirkov, Fundamental Processes Determining the Highly Charged Ion Production in ECR Ion Sources, Pre print JINR E , Dubna, 199, Nucl. Instrum. Methods A322, 1992, p [3] G. Shirkov, A Classical Model of lon Conjinement and Losses in ECR Ion Sources, Plasma Sources Sci. Technol. 2 (1993), p [4] C. Barrue, P. Briand, A. Girard, G. Melin, and G. Briffod, Hot Electron Studies in the MINIMAFIOS ECR Ion Source., Proc. of 4th Int. Conf. on lon Sources, Bensheim, Germany, 1991, edit. by B.l-l. Wolf, Rev. Sci. Instrum. (1992), vol. 63, No. 4, p [5] A. Girard, New Data and Comments on the ECR Source Behavior, Proc. of llth Int. Workshop on ECR Ion Sources, edit. by A.G. Drentje, 1993, p. 86. [6] H.I. West, Jr., Calculation of Ion Charge-State Distribution in ECR Ion Sources, UCRL 53391, Lawrence Livermore National Laboratory, California, OCR Output

10 [7] D.V. Sivuhin, Coulomb Collisions in the Fully Ionized Plasma (in Russian); in: Voprosu Tcorii Plasmu, v.4, Moscow, 1964, p. 81. [8] V.P. Pastukhov, The Classical Longitudinal Plasma Losses in the Open Aaliabatic Traps (in Russian); in: Voprosu Tcorii Plasmu, H13, Moscow, 1984, p [9] G.D. Shirkov, Computation of the Ion Charge-State Distribution in ECR Ion Sources, pro-print JINR P , Dubna, 1990 (in Russian); Sov. Phys. Tcch. Phys. 37(6), 1992, p. 610 (in English). [IQ] L. Spitzer, Physics of Fully Ionized Gases, J. Wilcy and Sons, Ncw York-London, [[1] K. Wicscmann, The Ion Temperature in a Plasma with Hot Electrons, [EEE Trans. Nucl. Scicncc, (1972), Vol. NS-19, 2, p [12] L.S. Laslctt, Prc-print ERAN -218, LBL Bcrkclcy (1972). [13] E.A. Pcrclstcin and G.D. Shirkov, On the Distribution Function of Ions; Prc-print JINR R , Dubna, 1982 (in Russian); Sov. Phys. Tcch. Phys. 29, 1984, p. 158 (in English). [[4] E.A. Pcrclstcin and G.D. Shirkov, Dynamics of Ion Storage Processes in Electron Beams and Rings, Sov. Joum. Part. Nucl. (1987), 18(1), p. 64 (in English). OCR Output

11 ?`;/4, \ L" ; / ````""` `` ` `" ` -, \ Ez E; T T EH CTW $ $S ;;am2<mvm { NM 6 v Fig.l. ECR ion source MINHVIAFIOS and distribution of the magnetic field B, along the axis and iield B, along the radius. The inked line corresponds to the operation with 10 GHz and the dotted line to the 16.6 GHz. S], S2, S7 - coils for the axial magnetic field; 1 - first stage of the source, 2 second stage, 3 - microwave field injection, 4 - gas injection, 5 - permanent magnet hexapole for the radial field variation, 6 - electrodes for the ion extraction, 7 - radiator.