11/7/94 OCR Output

Geneva, Switzerland

]oint Institute for Nuclear Research Dubna

ECR ion sources and numerical simulations.confinement will be used for the understanding of the physical processes in the

The results of this new approach to the problem of electron and ionvalue of potential barrier in the trap.ionization; the cooling due to the loss of ions with an energy higher than theionic charge state increasing in the potential well due to electron impactCoulomb collisions with high energy electrons; the heating resulting from themain processes which can change the ion energy: the heating due to the elasticequation for the ion energy in this paper. This equation takes into account threeconfinement time depends also on the ion energy. It is proposed to use a balance

The negative electrostatic potential regulates the rate of ion losses. The ionconfiguration.electron confinement time in the longitudinal magnetic field with plugenergy components. The mirror ratio is necessary for the determination ofbalance equations are used here for the electron distribution with two differentcan be described with the balance equation for the electron component. Twoin ECR Ion Sources" is presented. It is shown that electron density in the source

The improvement of "The Classical Model of Ion Confinement and Losses

Abstract

G. Shirl

equations. OCR Output

zero (dn,/dt = 0). Then this set of balance equations transforms into a set of algebraic

processes in the source are stationary, the left sides of the equations can be chosen as equal to

take into account all charge changing transitions in the source. In the static case, when all the

differential balance equations for all ionic charge states in the plasma [2,3]. These equations

The ionization process of neutral atoms and ions is described by a set of non-linear

"The Classical Model of Ion Confinement and Losses in ECR Ion Sources" [2,3].

the consideration of elastic and inelastic collisions in the plasma were assumed as a basis of

potential well, confines the positive charged ions in the ECR ion source. This conclusion and

configuration of magnetic field confines the electrons and the negative potential of plasma, or

that regulates the ion losses and keeps the general neutrality of plasma. Thus, the mirror

than the ions. The relatively high rate of ion losses creates the negative potential of the plasma

in the source. The heated electrons have much more energy and less probability of scattering

the direction of particle movement and therefore there is a continuous loss of electrons and ions

plasma. In the static case the elastic Coulomb collisions between the charged particles change

particles with velocity vectors in a small solid angle along the trap axis can be lost from the

The magnetic mirrors reflect the charged particles and confine the plasma. Only the

take into account here only the classical processes to study the electron and ion confinement.

experimental conditions stabilize the plasma instabilities and turbulence. That is why we shall

magnetic field spatial distribution and a choice of all other source parameters in the real

Following our previous papers [2,3], let us suppose that a thorough optimization of the

These later processes are stronger and may be a cause of a high rate of ion and electron losses.

fields and elastic scattering. The second is turbulence and different types of plasma instabilities.

classical processes connected with the motion of the charged particles in the electromagnetic

Two main physical processes detemiine the rate of electron and ion losses. The first is the

the magnetic field in the source.

of microwave field corresponds to the frequency of the electron Larmor rotation at surfaces in

by electron impact ionization. A microwave field heats the electrons in the trap. The frequency

The ions and electrons of plasma are generated from the neutral gas in the source chamber

magnetic field or magnetic mirrors on the ends of the source.

with azimuthal variations. Special coils (Sl — S5 in this Fig. 1) make regions with increased

Fig. l. The permanent magnet hexapole is used for production the longitudinal magnetic field

successful ECRIS called MHQIMAFIOS [1] and the axial field distributions in it are shown in

compound magnetic field distribution in the source. The general view of one of the most

confined in the open magnetic trap. A system of permanent magnets and coils creates the

The working region for ion production in the ECR ion source (ECRIS) is the plasma

necessary in the case of two electron components: OCR Output

of the ion charge state distribution in the ECR ion source [3]. Two balance equations are

possible to use a balance equation to determine the electron density in the numerical simulation

ionization energy of ions and atoms, the second has high energy electrons in keV region. It is

the physical point of view [6]. One component has a low energy that corresponds to the

The assumption of two electron components with different energies is quite realistic from

electron kinetic energy. Hence, the maximum ionization state is limited by the electron energy.

more, the electrons can ionize only those ions with an ionization potential lower than the

cross section of electron impact ionization depends strongly on the electron energy. What is

probability of elastic scattering and loss is dependent on the energy of the charged particle. The

The value of electron energy is a very significant parameter for an ECR ion source. The

be rather complicated.

heated and remain at low energy. Therefore, the energy distribution function of electrons must

move on a surface that has the resonance value of magnetic field B, All other electrons are not

has the resonance condition if the microwave source has a fixed frequency. These electrons

hundreds keV according to some experimental data [4,5]. Only a small portion of the electrons

particle and heat it. The energy of heated electrons reaches the range of tens keV or sometimes

wave has a frequency equal to the cyclotron frequency wc, then it can transfer the energy to the

Here me is the electron mass and e is the electrical charge of the electron. lf the electromagnetic

(1)cu, = Q-.

field B with cyclotron frequency. It can be defined for the electrons with:

a range of some tens or hundreds electron volts. The charged particles rotate in the magnetic

the plasma. The energy of newcomer electrons corresponds to the energy of ionization and has

The electrons appear in the process of electron impact ionization of the atoms and ions in

Electron Confinement

study and are the subject of this paper.

equations and conditions for the plasma parameters are very important for the ECR source

charge distribution and the production of multiply charged ions. Therefore these additional

plasma. The parameters of plasma greatly influence the ionization process, particularly the

additional equations are used for the numerical simulation of ionization processes in the ECR

other and to the ionic and electronic densities. The complete sets of balance equations with these

parameters. Some special additional relationships connect the parameters of plasma to each

plasma, the electron and ion confinement times, are included in the balance equations as

The main parameters of plasma, such as the electron and ion temperatures, the potential of

neutrals) in the plasma: OCR Output

where ve is the frequency of electron collisions with all kinds of particles (electrons, ions and

(4)1 R + R/1 R T. = 1.48 3-A

similar formula to evaluate the time of ion confinement is:

the ECR ion sources R is usually in the range from 2 to 4. According to Pastukhov [8] the

where Bmm and Bmax are the minimum and maximum values of magnetic field in the source. In

BminR : ity;

with the so—called "mirror ratio"

Te ~ lnR

confinement conditions. lt was shown [7] that

It seems to be obvious that the value of magnetic mirror has an influence on the electron

function of electron energy.

average value . One can obtain it as the result of integration with the distribution

The cross section 03 depends on the electron velocity Ve and it is necessary to use the

be possible to use an additional term related to the electron source in the one of Eqs. (2) or (3).

If one considers the supplementary injection of electrons into the ECR plasma then it will

movement of electron in the magnetic field, the value Th has a microsecond time scale.

a few passes through the resonance surface in the source. Thus, taking into account the spiral

for the moment. It is possible to assume that the electrons are only heated during not more than

17,. The theory of electron heating with microwave power is out of range of our consideration

The very important and, usually, unknown parameter is the time of cold electron heating

these equations.

indexes c correspond to the cold electron component and the indexes h — to the hot electrons in

U2! correspond to the cross sections for single and double ionization, accordingly. All the upper

electron velocity; Th is a characteristic time of electron heating with microwave power. o" and

velocities of electrons; o;(Ve) are the cross sections of electron impact ionization as a function of

Here: ne are the electron densities; re are the electron confinement times; Ve are the

dt Th Te=—····

]dn;ng ng

e h

/tlQ¤lQ+2¤i 'léfnai + % lé'¤2lQ +2¤i 'VE ni—l ·%- L W · 2· h 1 · h 2· h nMj = Zi C[f(c)(ll2,[()()]tTc C

barrier ieU. OCR Output

c) the cooling due to the losses of ions with the energy higher than the value of potential

b) the heating due to the charge state changes in the electrostatic potential well;

a) the electron heating due to the elastic Coulomb collisions;

energy taking into account three main processes that change the ion energy in the ECR source:

Without plasma instabilities and turbulence we can obtain the balance equation for ion

total ion energy can be described by the balance equation for the ion energy in the source [10].

The ion confinement conditions strongly depend on the ion energy in the plasma. The

ion sources.

time TQ is one of the most fundamental values for describing the plasma parameters in the ECR

electron density neq. This value is called "ionization factor". Therefore, the ion confinement

The possible charge states of ions in the source are determined by the ion lifetime and

Ion Confinement

th can be chosen as an input or a fitting parameter.

component distribution for the electron energy. The electron heating rate with microwave power

the ECR plasma taking account of the mirror configuration of the magnetic field and the double

Equations (2) to (8) describe the processes of the electron generation and confinement in

in the expressions (4) to (8) if two electron components are to be used.

Two different electron energies (or temperatures) and densities must be taken into account

Lee = Lei = 23.64 — ln (8)Z —&—. EZscattering:

called Coulomb logarithms. The quantum formula for Coulomb logarithms is used for electron

temperature; no is the density of neutrals in the plasma; Z is the atomic number; L are the so

where re is the classical radius 0f electron; c is the velocity of light; Te is the electron

6 (7)_ 4 2 *7 no v0- . >

energy of ion oscillation in the potential well UQ is equal to the average potential energy U,. OCR Output

plasma constant density) then, according to the virial theorem [13], the average kinetic

potential energy on the distance from the centre of well (it is so for our assumption of

potential energy of an ion in the potential well. If we suppose a square dependence of

The other possible way to obtain this result is by consideration of the average kinetic and

Az 21Mil - Ei

ionization. Then

and that it increases for every ion with the increased charge state due to the electron impact

(11)Ei:./{ eU,

well U has a square root dependence on the charge state i:

of an electron beam with constant density. The average energy of the ions Ei in a potential

Then it is possible to use previous results [12-14] for ion energy in a negative potential well

parts of the ions appear and move that the density of electrons is close to a constant value.

electrostatic potential well, let us suppose that in the centre of the ECRIS, where the most

b) For determination of the ion heating rate as the result of ionization in the negative

number and mass of a nucleon.

where the ion energy Ei = 1.5T,; Z is the atomic number; A and M are the atom mass

dt Aivrr/if(10)

dEi __ 4s/27tneZ2r;32m3c41/me Lei

a) The rate of ion heating due to the elastic Coulomb collisions with electrons is [11]:

is the average energy of the ions that leave the potential well.

state and the rate of ionization;

AEi / Ai and dn; / dt are the corresponding energy increasing with the change of ionic charge

is the rate of ion heating due to the collisions with electrons;dEi/dt

where

9 ( ): + - dz Eini dr Zi Ai dr Zi ri[Z-("·E·)] dE AE 4 i E" ; 4 JJ; hi

Thus, the balance equation is:

does not influence on the total ion energy.

It is also possible to suppose that the initial thermal energy of neutrals is negligible and

Here we take into account single and double ionization processes.

Qt iiZ? OCR OutputZim Ee*"(°i “")'ida **dE· U ·· 3 `n· - · n _ n·Ei ! 1` 2il“+2)+ {i `Z2Z” gi? ”“i%[ ]

Finally, the complete balance equation is: (17)

with x = ieU/Ti.

r 1+x+\/x’+(4x/zz:)i 2xE=7Q—+1— (16)

* 3

can be evaluated as

equal to the incomplete gamma functions and the average energy of these high energy ions

equal, (or approximate equal) temperatures 7} [2,3]. The integrals in the equation (15) are

collisions is high, then all ion components have the Boltzman energy distribution with

where f(E,) is a distribution function of the ion energy. If the rate of elastic ion—ion

ieU

f(Ei)dE

EF =f (Ei)EdE

energy from the plasma. The average energy of these ions can be defined as

c) The ions with energy higher than the value of potential barrier ieU leave the well and remove

U4), Oi i i Z; KT? =”el@ TGl+EEiGi+ln§T·dnAE; neU l Z`1

and the corresponding energy increase is:

(13)d; =~€ lévfn ni.

The rate of ionization is determined by the cross section of electron impact ionization Gi

the relation (12) and its value is twice as large for double ionization.

increases by U,/i and the total ion energy increases by E,/2i correspondingly. This satisfies

If the charge state of an ion increases by one unit due to ionization, then the potential energy

. 21Hi : Us Z

ini " He_ )

where

U=ener1+ln (22) OCR Output£2f(£) 2 r

evaluated as:

for a cylindrical source working volume with length l and radius r the potential U can be

equations are not equal to zero, then condition (18) may be disturbed. According to Ref. [3],

If we study dynamic regime of an ECR ion source where the left sides of the balance

for all ionic charge states [3] is necessary to determine the total charge state distribution.

by solving Eqs. (17) and (18) in the static case. Naturally, the complete set of balance equations

It is now possible to define the potential of the plasma U and the temperature of ions Ti

io _;k4C1.5>flinoatoms VK) 1S glvcn:

In Ref. [9] an approximate formula for the collision frequencies between ions and neutral

/2 '3Vk _ i _ (20)47rre2m3c4i2 E x/AzAjj2”jLy

The value vik is the ion collision frequency [1 1]:

l is the effective length of the source working region in (19).

2Rand G =

x/E(R+1)1n(2R+ 2)

with x = ieU/Ti

+ X)(Vik + Vio)exp(x), (19)Ti = Rl — +———-Q[l

confinement of charged particles in the open magnetic trap [8]:

The confinement times for ions Ti can be defined according to the Pastukhov theory for

e e

2-%-%-% I Ti TTc h

condition of equal flows of electrons and ions from the source:

In the static case the left side of Eq. (17) is equal to zero and we can couple it with the

53391, Lawrence Livermore National Laboratory, California, 1982. OCR Output

[6] H.I. West, Jr., Calculation of Ion Charge-State Distribution in ECR Ion Sources, UCRL

Workshop on ECR Ion Sources, edit. by A.G. Drentje, 1993, p. 86.

[5] A. Girard, New Data and Comments on the ECR Source Behavior, Proc. of llth Int.

p. 2844.

Germany, 1991, edit. by B.l-l. Wolf, Rev. Sci. Instrum. (1992), vol. 63, No. 4,

MINIMAFIOS ECR Ion Source., Proc. of 4th Int. Conf. on lon Sources, Bensheim,

[4] C. Barrue, P. Briand, A. Girard, G. Melin, and G. Briffod, Hot Electron Studies in the

Plasma Sources Sci. Technol. 2 (1993), p. 250.

[3] G. Shirkov, A Classical Model of lon Conjinement and Losses in ECR Ion Sources,

A322, 1992, p. 161.

ECR Ion Sources, Pre—print JINR E9-92-33, Dubna, 199, Nucl. Instrum. Methods

[2] G. Shirkov, Fundamental Processes Determining the Highly Charged Ion Production in

1982, p. 399.

Faisceaux d'Ions Complétement Epluchés (in French), Nucl. Instrum. Methods 202,

[1] R. Geller and B. Jacquot, MINIMAFIOS, Source d'Ions Pulsés Fournissant des

References

processes in the ECR ion sources.

These results will be used in the study and numerical simulations of the ionization

barrier.

and the cooling due to the losses of ions with the energy higher then the value of potential

Coulomb collisions, the heating due to the charge state changes in the electrostatic potential well

processes that change the ion energy in the ECR source: the electron heating due to the elastic

in the source. The balance equation for total ion energy in the trap describes three main

The proposed formulae for electron confinement time include the mirror ratio of magnetic field

components are able to take into account the real energy distribution of electrons in the plasma.

losses in ECR ion source" [3]. The balance equations for both low and high energy electron

of an ECR ion source is a further development of "The classical model of ion confinement and

Consideration of the conditions of electron and ion confinement in the open magnetic trap

Conclusions

energy (temperature) and the continement times of ions in the source.

(17) and (18) with (19), (20) in the static case, relate the plasma potential, the ionic average

Equation (17) with (19) — (22) in the dynamic regimes of the ECR ion source, or Eqs.

and Rings, Sov. Joum. Part. Nucl. (1987), 18(1), p. 64 (in English). OCR Output

[[4] E.A. Pcrclstcin and G.D. Shirkov, Dynamics of Ion Storage Processes in Electron Beams

English).

R9-82-526, Dubna, 1982 (in Russian); Sov. Phys. Tcch. Phys. 29, 1984, p. 158 (in

[13] E.A. Pcrclstcin and G.D. Shirkov, On the Distribution Function of Ions; Prc-print JINR

[12] L.S. Laslctt, Prc-print ERAN -218, LBL Bcrkclcy (1972).

Scicncc, (1972), Vol. NS-19, 2, p. 156.

[[1] K. Wicscmann, The Ion Temperature in a Plasma with Hot Electrons, [EEE Trans. Nucl.

1962.

[IQ] L. Spitzer, Physics of Fully Ionized Gases, J. Wilcy and Sons, Ncw York-London,

1992, p. 610 (in English).

pro-print JINR P9—90—581, Dubna, 1990 (in Russian); Sov. Phys. Tcch. Phys. 37(6),

[9] G.D. Shirkov, Computation of the Ion Charge-State Distribution in ECR Ion Sources,

(in Russian); in: Voprosu Tcorii Plasmu, H13, Moscow, 1984, p. 160.

[8] V.P. Pastukhov, The Classical Longitudinal Plasma Losses in the Open Aaliabatic Traps

Tcorii Plasmu, v.4, Moscow, 1964, p. 81.

[7] D.V. Sivuhin, Coulomb Collisions in the Fully Ionized Plasma (in Russian); in: Voprosu

6 - electrodes for the ion extraction, 7 - radiator.

injection, 4 - gas injection, 5 - permanent magnet hexapole for the radial field variation,magnetic field; 1 - first stage of the source, 2 · second stage, 3 - microwave field

10 GHz and the dotted line to the 16.6 GHz. S] , S2 , S7 - coils for the axialthe axis and iield B, along the radius. The inked line corresponds to the operation withFig.l. ECR ion source MINHVIAFIOS and distribution of the magnetic field B, along

$S$;;am2