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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
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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
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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
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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
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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- . >
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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
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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
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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
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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
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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