- Email: [email protected]
Model of beam formation in a glow discharge electron gun with a cold cathode
surface science ELSEVIER
Applied Surface Science 111 (1997) 288-294
Model of beam formation in a glow discharge electron gun with a cold cathode S.V. Denbnovetsky a, j. Felba b,* , V.I. Melnik
a,
I.V. Melnik
a
a National Technical Universi~, Electron Devices Department, korpus 12, Prospekt Peremogy 37, 252056 Kiev, Ukraine h Institute of Electronic Technology, Technical UniversiO' of Wroctaw, ul Grabiszyhska 97, 53-439 Wroctaw, Poland Received 30 June 1996; revised 2 September 1996; accepted 13 October 1996
Abstract The paper describes a method of computer simulation of electron beam formation in a high voltage glow discharge with a special shape of the electrode surfaces. Our objective was to use the modified current tube method for numerical calculation of the electric field between electrodes including the space charge of moving particles. We have taken into consideration the influence of accelerated ion interaction with atoms of the operating gas in a discharge cathode-fall region. All simulations have been performed for a diode electron gun with an axial symmetry forming a point-focus electron beam. The validity of the model has been confirmed by an experimental result, which corresponds closely to the calculation.
1. Introduction In an electron source, where the conventional thermionic cathode is replaced by a cold, secondary-electronemitting electrode, electron emission is stimulated by bombarding the cathode with high-energy ions. When a voltage Ua of several kV is applied between the anode and cathode of the gun (the gun being immersed in a gas atmosphere with a pressure of 0.5 Pa to 10 Pa), gas ionization takes place, which creates a plasma of high ion and electron concentration. The positive ions supplied by the plasma are accelerated in an electric field and bombard the cathode. The generated secondary electrons are also accelerated in the field between cathode and plasma to form an electron beam, the shape of which is related to the specific electrode geometry. The diversity of electron beam shapes makes them useful in various industrial applications such as the welding of thin-wall pieces [1,2], annealing and surface modification [3,4], producing high-quality hard ceramic coatings [5] etc. In technical publications, such electron beam sources are called glow discharge electron guns (GDEG), plasma-anode electron guns, glow discharge electron guns with an anode plasma or glow discharge electron guns with a cold cathode. In this paper we use the acronym GDEG. The physical processes of a glow discharge and of secondary emission of electrons are well-known. Moreover, numerical methods exist for the calculation of ion and electron trajectories [6-10]. For non-laminar flows, beam characteristics are usually obtained by considering the self-consistency of the particle trajectories
* Corresponding author. Tel.: + 48-71-611813; fax: + 48-71-213504; e-mail: [email protected]. 0169-4332/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. P11S0169-4332(96)00761-1
S. V. Denbnovets~ et al. / Applied Surface Science 111 (1997) 288-294
289
r [mm]~ =oo
cathode'~J~
|l
I /
[t
I I
I ~: :
,/
i
Ll_ J
o
10
.
t I ~:. : .
:::~/"-
'l.
5 .
.
[ [ 20
:,.:~ 30
40
,:_~ 50
...... 60
z [mm]
Fig. 1. The geometry of an axially-symmetrical GDEG forming a point-focus electron beam, with a plasma boundary and the distribution of the electric Field (calculated for an acceleration voltage of 12 kV and a discharge current of 350 mA).
and the fields. Unfortunately, such an approach often results in mathematical complexities even if simplifications are made in the model, what leads to discrepancies between calculated and experimental results. So, presently the GDEGs are improved mainly experimentally. Our objective was to propose a model for beam formation based on standard trajectory analysis.
2. Object of study We have considered a diode electron gun with axial symmetry forming a point-focus electron beam. The principle scheme of such a gun, which consists of a spherical cathode and a conical anode is given in Fig. 1. The plasma boundary is also shown in Fig. 1. The plasma, with the anode potential and a shape similar to that of the cathode, may be considered as the third electrode. The region between the cathode and the plasma boundary (cathode dark space), in which most of the potential drop occurs (known as the cathode fall), is very important for electron beam formation. The position of the plasma boundary depends on the value of the acceleration voltage U~. This dependence is shown in Fig. 2.
3. Basic model of high-voltage glow discharge A basic physical model of the flow of charged particles with interaction between them is based on the system of Poisson-Vlasov differential equations, which define a particle density distribution in phase space f,( p; 2', t) [6,7]: --+g~gradf,+ Ot
'
L = q~L(~)
--
|grad, f,--
m~
]
~
"
at lint
d~
P~_=Pi--Pe
E = - grad(q~) div grad(~p) = - - go
( 1)
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
290
a~[mmll
~ , U~ : 2 0 k V
~°°I
i15kV; lOkV'
75 50 25 i i
0
50
100
I 150
200
I,[mA]
Fig. 2. Dependence of the cathode-plasma distance dco on the discharge current 1d for different acceleration voltages Ua; the cathode radius
was 35 mm. with f, the particle density distribution, s the kind of particles, q~ the charge of one particle, m~, v~ its mass and speed, P.v, Pi, Pe the full charge density distribution and distribution of ions and electrons space charge, respectively, (Ofs/Ot)in t - a factor, which defines the interactions between the particles flows. As Liouville's theorem is defined in a stationary hydrodynamic regime, the trajectory analysis gives the complete flow characteristics, even with interaction between them. In this case solution of the system of Eq. (1) is not required for complex simulation of the GDEG characteristics and well-known methods of trajectory analysis, such as the particle in cell method and the current tube method, [7,9,10] can be used. The boundary conditions necessary to solve the system of Eq. (1) are defined by the geometry of the GDEG electrodes and by the physical conditions of the glow discharge which are determined experimentally. Such a simplified model allows to study the main properties of high-voltage discharge and to define the GDEG characteristics for the external current regime. The complete simulation of the GDEG beam requires definition of the plasma boundary geometry and its position (which depends on discharge regime and cathode and anode geometry), as well as the calculation of electron and ion trajectories in the electric field of the cathode-fall region, and taking into account the space charge as well as interaction between the flows and with the neutral gas atoms. This can be done by a standard calculation technique for trajectory analysis.
4. Analysis of cathode-fall region processes In axially-symmetric electrode systems the Poisson equation for defining the potential (last equation in the system of Eq. (1)) can be solved numerically by using a finite-difference method. A standard five-point iteration equation for finding the potential in this case can be written as [7]
U " ( i , k ) = t o [ C a U" l ( i + l , k ) + C b U
~ l(i,k+l)+CcU"(i-l,k)
+CdU"( i, k - 1) + p " - ' ( i, k ) / e 0 ] + ( 1 - t o ) U " - l ( i ,
k)
(2)
with n the current iteration, n - 1 the previous iteration, i and k the number of the current considered item at the longitudinal and radial axis, respectively, to the relaxation parameter which influences the rate of iteration process convergence. For a different geometry of the simulation region, to can be varied between 1 and 2. For the considered geometry, given in Fig. l, the relaxation parameter selected by calculation equals 1.115. Coefficients Ca, C b, Cc and Cd for the discretization step h r and h= at the radial and longitudinal axis are respectively defined by 0.25 Ca = Cc =
h2
Cb --
1 + 1/2k 2 hr
Ca
1 - l /2k * h7
(3)
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
291
for non-axial points, and 4 Ca = 6h2r
1 Cb = Cd = 6h~
Cc = 0
(4)
for axial points. The particle trajectories in the electric field can be defined by solving the Newton differential equation with standard Runge-Kutta methods. In some cases an additional finite-difference mesh with a fine pitch is required in near-electrode regions for increasing the accuracy of calculations [6,7]. In case of a GDEG the tiny finite-difference mesh is used near the cathode and near the plasma where the particle emission occurs. In our numerical experiments the pitch of this additional mesh on the longitudinal axis was half of that in the main one. In this way the distribution of space charge produced by the moving particles can be established including the influence of interaction of accelerated ions with atoms of the operation gas. First the plasma boundary position must be defined taking into account the experimental data (see Fig. 2). Next, the ion current density from the plasma boundary can be obtained by solving Eq. (1) which may be correctly applied to a high-voltage glow discharge using Ji = 0"4eNi
~ 2kT~ mi
(5)
with N i the ion concentration in the discharge plasma, Te the temperature of plasma electron gas, k Boltzmann's constant, e the electron charge, m i the ion mass. The secondary electron emission current due to ion bombardment of the cathode surface can be defined by the approximate equation Je = CeUa~Ji
(6)
with U~ the accelerating voltage, and C e and a empirical constants which depend on the cathode material. To calculate the space charge, we have modified the current tube method. In this approach space charge accumulated in the finite-difference cell is calculated with respect to the volume of this cell and input and output current including the ion interaction with gas atoms. This process can for most gases be described by a power function, which depends on the voltage in the considered point: f ( U ~ ) = ag Uag
(7)
where A g and g are empirical coefficients, depending on gas composition. Thus the numerical equation for defining space charge can be written in the form Pls
27rrtA r
~
+ Ur 2
Ar = r2 - r I
A rlh r = 2
Pi2 Ji2
'Ua(1
1 -- (Ni/27rNa(1
~z -
Uc=
E Pls n=l
hr
U/,k --~-U/_ i,k -4- U/.k_ 1 -1- Ui_ l,k_ 1 4
~)r2(Arlhr/Az)f(Uc))
Jil 1 - (N//27rNa(1 - s¢ ) r t 2 ( A r l h r / A z ) f ( U c ) ) ul
P2s
-
Url = Ui, k +
292
S. V. Denbnovetsk), et al. / Applied Surface Science l 11 (1997) 288-294 P~ = Pi
-
Pe
pz(i,k,i+ l,k+ l)+p,-(i,k,iPi,k =
I,k- l)+p,,(i,k,i-
1,k+ l)+pz(i,k,i+
1, k - l )
4
(8) with I~ the current of the tube, r t the diametral radius, r I and r e the input and output radius respectively, Urj and Ur2 the approximate potential values in the considered input and output points, ~ = N~/Na the relative level of gas ionization, Nt the number of particle trajectories, that pass through the cell volume, Ni the ion concentration, and N~ the atom concentration.
5. Focal beam parameters The main focal beam parameters are the focal distance, the focal beam radius and focal beam current density, from which the focal beam power density can be calculated. Knowing the approximate electron trajectories to the plasma region and taking into account the dispersion of electrons on plasma ions the above mentioned parameters can be easily calculated. The corresponding system of equations can be written in the form 10-424/3
tan(0.50mi . )
2"//32
e2Z, . tan(0"50m"x)- m~,~rb(t)
V[~eU~ ~'=
m
d L = h2~/1 + tan2(~) dO=
(y 2
_
)2 1
n0 dLln
dz=Rc-dcp+{(Rc-dcp)2-ro(k)
2
q)(i, k) = ~p(i, k - 1) + d O j;nax(k) = max [jb(i, k)] i~ "Qi
rb(k) = arg(0.7R "x) d b = 2 rain [ r b(k)] k~ -(2k
F b = dcp + argmin [rb(k)]
(9)
k~ 12k
with i, k the discretization parameters at the r and z axis, respectively, r b the beam radius at the considered
point, Jb the beam current density, U~ the acceleration voltage, Rc the cathode sphere radius, Z the nuclear number of the gas atoms, r e the electron radius with respect to the Bohr model, q~ the input angle of electron trajectories, and y, /3 the relativistic factors. The main difficulty in numerical analysis of Eq. (8) is a high convergence angle of the GDEG beams (maybe
293
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
0,' 0,'
0
5
10
15
20
rc[mm]
Fig. 3. Numerical (1) and experimental (2) results of the beam current density distribution on the cathode surface; the acceleration voltage was 12 kV and the discharge current 350 mA.
greater then 10°). Furthermore, electron trajectories in the anode plasma are non-paraxial and in this way the beam focus is not sharply defined.
6. Numerical results and experimental verification Numerical Eqs. (2)-(9) have been used to define the focal parameters of GDEG with an electrode shape as shown in Fig. 1. All results, described further, have been obtained for r and z discretization steps for field calculation of 0.05 mm, accuracy of field calculation of 0.1 V, number of ion trajectories 500 (each), and number of iterations for space charge 30. The distribution of the electric field, obtained as result of computer simulation on the last iteration is shown in Fig. 1 (by dotted lines). In most cases these lines are similar to the shape of the electrode surfaces. Some deviations from this shape are mainly caused by the space charge of positive ions, especially after its interaction with neutral gas atoms. The dependence of the electric field and the GDEG parameters on electron beam space charge in the analyzed regions is not very strong, which suggests a full beam compensation [1]. Furthermore, electrons are strongly accelerated in the near-cathode region and their mean free path becomes much longer than the gun length. The results of measurement and calculated data for the beam current density from a cold cathode surface, caused by ion and neutral atom bombardment, are shown in Fig. 3. The maximum difference between theoretical and experimental results is observed at the edge of the cathode. This can be explained by the influence of other particle flows, such as accelerated electrons reflected from the anode. Some results obtained for geometrical beam focal parameters are presented in Table 1. As has been pointed out, the electron trajectories in a high-voltage discharge are non-paraxial, and in this case a beam focus is not sharply defined. All numerical results were obtained for a discharge regime with a practically stable plasma position (see Fig. 2). In this case a change of calculated values of focal distance and diameter can be caused only by alteration of the
Table 1 Dependence of the beam focal distance, the beam diameter in the focal plane and the beam power on the cathode sphere radius Beam parameters Discharge current (mm) Cathode sphere radius (mm) Focal distance (mm) Beam diameter (ram) Beam power (kW) (measured)
300 400 500
70
75
77
80
80 4 ----
85 4 1.9 2.7 3,0
--2.25 2.9 3.25
103 6 2.1 2.75 3.1
294
s. ~ Denbnotetsl~v et a L / Applied Surface Science 111 (1997) 288-294
electric field distribution, which is mainly defined by the ion space charge. Experimental data for gun efficiency are also given in Table 1.
7. Conclusion Numerical results, obtained by using Eqs. (2)-(9), are in agreement with experiments for the high-current regime, when the plasma boundary position is stable under current changing. For example, the accuracy of defining the output focal beam parameters was 30% and better. Such a high range of accuracy was achieved as a result of improvement of the modified current tube numerical technique including analysis of the contribution of accelerated ions and gas atom collisions to the space charge. This gives a possibility of using calculation results tbr design of GDEG with a wide range of operating pressures and accelerating voltages without complex experimental research.
References [1] A.A. Novikov, Istochniki elektronov vysokovoltnogotleuschego razryada s anodnoy plasmoy (Energoatomizdat, Moscow, 1983). [2] R.A. Dugdale, Glow discharge material processing (London, Mills, Boon, 1971). [3] M. Balaceanu, L. Dinu and C. Popovichi, J. Phys. D 18 (1985) 835. [4] J. Felba, K. Friedel and K. Przybecki, Vacuum 41(7-9) (1990) 2177. [5] J. Vanflettern and A.A. van Calster, Thin Solid Films 139 (1986) 84. [6] R.W. Hockney and J.W. Eastwood, Computer simulation using particles (McCraw-Hill, 1981). [7] E. Kasper, Optic 68(4) (1984) 341. [8] J.H. Whelton, Nucl. Instrum. Methods 189 (1981) 55. [9] P. Spadtke and D. Ivens, Vacuum 28(10-1 l) (1989) 453. [10] S.V. Denbnovetsky, V.I. Melnik and I.V. Melnik, Avtomatizaciyaproektirovaniya v elektronike, Respublicanskiy megvedomstvenniy nauchno-tehnicheskiy sbornik, No. 43 (Tehnika, Kiev, 1991).
Applied Surface Science 111 (1997) 288-294
Model of beam formation in a glow discharge electron gun with a cold cathode S.V. Denbnovetsky a, j. Felba b,* , V.I. Melnik
a,
I.V. Melnik
a
a National Technical Universi~, Electron Devices Department, korpus 12, Prospekt Peremogy 37, 252056 Kiev, Ukraine h Institute of Electronic Technology, Technical UniversiO' of Wroctaw, ul Grabiszyhska 97, 53-439 Wroctaw, Poland Received 30 June 1996; revised 2 September 1996; accepted 13 October 1996
Abstract The paper describes a method of computer simulation of electron beam formation in a high voltage glow discharge with a special shape of the electrode surfaces. Our objective was to use the modified current tube method for numerical calculation of the electric field between electrodes including the space charge of moving particles. We have taken into consideration the influence of accelerated ion interaction with atoms of the operating gas in a discharge cathode-fall region. All simulations have been performed for a diode electron gun with an axial symmetry forming a point-focus electron beam. The validity of the model has been confirmed by an experimental result, which corresponds closely to the calculation.
1. Introduction In an electron source, where the conventional thermionic cathode is replaced by a cold, secondary-electronemitting electrode, electron emission is stimulated by bombarding the cathode with high-energy ions. When a voltage Ua of several kV is applied between the anode and cathode of the gun (the gun being immersed in a gas atmosphere with a pressure of 0.5 Pa to 10 Pa), gas ionization takes place, which creates a plasma of high ion and electron concentration. The positive ions supplied by the plasma are accelerated in an electric field and bombard the cathode. The generated secondary electrons are also accelerated in the field between cathode and plasma to form an electron beam, the shape of which is related to the specific electrode geometry. The diversity of electron beam shapes makes them useful in various industrial applications such as the welding of thin-wall pieces [1,2], annealing and surface modification [3,4], producing high-quality hard ceramic coatings [5] etc. In technical publications, such electron beam sources are called glow discharge electron guns (GDEG), plasma-anode electron guns, glow discharge electron guns with an anode plasma or glow discharge electron guns with a cold cathode. In this paper we use the acronym GDEG. The physical processes of a glow discharge and of secondary emission of electrons are well-known. Moreover, numerical methods exist for the calculation of ion and electron trajectories [6-10]. For non-laminar flows, beam characteristics are usually obtained by considering the self-consistency of the particle trajectories
* Corresponding author. Tel.: + 48-71-611813; fax: + 48-71-213504; e-mail: [email protected]. 0169-4332/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. P11S0169-4332(96)00761-1
S. V. Denbnovets~ et al. / Applied Surface Science 111 (1997) 288-294
289
r [mm]~ =oo
cathode'~J~
|l
I /
[t
I I
I ~: :
,/
i
Ll_ J
o
10
.
t I ~:. : .
:::~/"-
'l.
5 .
.
[ [ 20
:,.:~ 30
40
,:_~ 50
...... 60
z [mm]
Fig. 1. The geometry of an axially-symmetrical GDEG forming a point-focus electron beam, with a plasma boundary and the distribution of the electric Field (calculated for an acceleration voltage of 12 kV and a discharge current of 350 mA).
and the fields. Unfortunately, such an approach often results in mathematical complexities even if simplifications are made in the model, what leads to discrepancies between calculated and experimental results. So, presently the GDEGs are improved mainly experimentally. Our objective was to propose a model for beam formation based on standard trajectory analysis.
2. Object of study We have considered a diode electron gun with axial symmetry forming a point-focus electron beam. The principle scheme of such a gun, which consists of a spherical cathode and a conical anode is given in Fig. 1. The plasma boundary is also shown in Fig. 1. The plasma, with the anode potential and a shape similar to that of the cathode, may be considered as the third electrode. The region between the cathode and the plasma boundary (cathode dark space), in which most of the potential drop occurs (known as the cathode fall), is very important for electron beam formation. The position of the plasma boundary depends on the value of the acceleration voltage U~. This dependence is shown in Fig. 2.
3. Basic model of high-voltage glow discharge A basic physical model of the flow of charged particles with interaction between them is based on the system of Poisson-Vlasov differential equations, which define a particle density distribution in phase space f,( p; 2', t) [6,7]: --+g~gradf,+ Ot
'
L = q~L(~)
--
|grad, f,--
m~
]
~
"
at lint
d~
P~_=Pi--Pe
E = - grad(q~) div grad(~p) = - - go
( 1)
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
290
a~[mmll
~ , U~ : 2 0 k V
~°°I
i15kV; lOkV'
75 50 25 i i
0
50
100
I 150
200
I,[mA]
Fig. 2. Dependence of the cathode-plasma distance dco on the discharge current 1d for different acceleration voltages Ua; the cathode radius
was 35 mm. with f, the particle density distribution, s the kind of particles, q~ the charge of one particle, m~, v~ its mass and speed, P.v, Pi, Pe the full charge density distribution and distribution of ions and electrons space charge, respectively, (Ofs/Ot)in t - a factor, which defines the interactions between the particles flows. As Liouville's theorem is defined in a stationary hydrodynamic regime, the trajectory analysis gives the complete flow characteristics, even with interaction between them. In this case solution of the system of Eq. (1) is not required for complex simulation of the GDEG characteristics and well-known methods of trajectory analysis, such as the particle in cell method and the current tube method, [7,9,10] can be used. The boundary conditions necessary to solve the system of Eq. (1) are defined by the geometry of the GDEG electrodes and by the physical conditions of the glow discharge which are determined experimentally. Such a simplified model allows to study the main properties of high-voltage discharge and to define the GDEG characteristics for the external current regime. The complete simulation of the GDEG beam requires definition of the plasma boundary geometry and its position (which depends on discharge regime and cathode and anode geometry), as well as the calculation of electron and ion trajectories in the electric field of the cathode-fall region, and taking into account the space charge as well as interaction between the flows and with the neutral gas atoms. This can be done by a standard calculation technique for trajectory analysis.
4. Analysis of cathode-fall region processes In axially-symmetric electrode systems the Poisson equation for defining the potential (last equation in the system of Eq. (1)) can be solved numerically by using a finite-difference method. A standard five-point iteration equation for finding the potential in this case can be written as [7]
U " ( i , k ) = t o [ C a U" l ( i + l , k ) + C b U
~ l(i,k+l)+CcU"(i-l,k)
+CdU"( i, k - 1) + p " - ' ( i, k ) / e 0 ] + ( 1 - t o ) U " - l ( i ,
k)
(2)
with n the current iteration, n - 1 the previous iteration, i and k the number of the current considered item at the longitudinal and radial axis, respectively, to the relaxation parameter which influences the rate of iteration process convergence. For a different geometry of the simulation region, to can be varied between 1 and 2. For the considered geometry, given in Fig. l, the relaxation parameter selected by calculation equals 1.115. Coefficients Ca, C b, Cc and Cd for the discretization step h r and h= at the radial and longitudinal axis are respectively defined by 0.25 Ca = Cc =
h2
Cb --
1 + 1/2k 2 hr
Ca
1 - l /2k * h7
(3)
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
291
for non-axial points, and 4 Ca = 6h2r
1 Cb = Cd = 6h~
Cc = 0
(4)
for axial points. The particle trajectories in the electric field can be defined by solving the Newton differential equation with standard Runge-Kutta methods. In some cases an additional finite-difference mesh with a fine pitch is required in near-electrode regions for increasing the accuracy of calculations [6,7]. In case of a GDEG the tiny finite-difference mesh is used near the cathode and near the plasma where the particle emission occurs. In our numerical experiments the pitch of this additional mesh on the longitudinal axis was half of that in the main one. In this way the distribution of space charge produced by the moving particles can be established including the influence of interaction of accelerated ions with atoms of the operation gas. First the plasma boundary position must be defined taking into account the experimental data (see Fig. 2). Next, the ion current density from the plasma boundary can be obtained by solving Eq. (1) which may be correctly applied to a high-voltage glow discharge using Ji = 0"4eNi
~ 2kT~ mi
(5)
with N i the ion concentration in the discharge plasma, Te the temperature of plasma electron gas, k Boltzmann's constant, e the electron charge, m i the ion mass. The secondary electron emission current due to ion bombardment of the cathode surface can be defined by the approximate equation Je = CeUa~Ji
(6)
with U~ the accelerating voltage, and C e and a empirical constants which depend on the cathode material. To calculate the space charge, we have modified the current tube method. In this approach space charge accumulated in the finite-difference cell is calculated with respect to the volume of this cell and input and output current including the ion interaction with gas atoms. This process can for most gases be described by a power function, which depends on the voltage in the considered point: f ( U ~ ) = ag Uag
(7)
where A g and g are empirical coefficients, depending on gas composition. Thus the numerical equation for defining space charge can be written in the form Pls
27rrtA r
~
+ Ur 2
Ar = r2 - r I
A rlh r = 2
Pi2 Ji2
'Ua(1
1 -- (Ni/27rNa(1
~z -
Uc=
E Pls n=l
hr
U/,k --~-U/_ i,k -4- U/.k_ 1 -1- Ui_ l,k_ 1 4
~)r2(Arlhr/Az)f(Uc))
Jil 1 - (N//27rNa(1 - s¢ ) r t 2 ( A r l h r / A z ) f ( U c ) ) ul
P2s
-
Url = Ui, k +
292
S. V. Denbnovetsk), et al. / Applied Surface Science l 11 (1997) 288-294 P~ = Pi
-
Pe
pz(i,k,i+ l,k+ l)+p,-(i,k,iPi,k =
I,k- l)+p,,(i,k,i-
1,k+ l)+pz(i,k,i+
1, k - l )
4
(8) with I~ the current of the tube, r t the diametral radius, r I and r e the input and output radius respectively, Urj and Ur2 the approximate potential values in the considered input and output points, ~ = N~/Na the relative level of gas ionization, Nt the number of particle trajectories, that pass through the cell volume, Ni the ion concentration, and N~ the atom concentration.
5. Focal beam parameters The main focal beam parameters are the focal distance, the focal beam radius and focal beam current density, from which the focal beam power density can be calculated. Knowing the approximate electron trajectories to the plasma region and taking into account the dispersion of electrons on plasma ions the above mentioned parameters can be easily calculated. The corresponding system of equations can be written in the form 10-424/3
tan(0.50mi . )
2"//32
e2Z, . tan(0"50m"x)- m~,~rb(t)
V[~eU~ ~'=
m
d L = h2~/1 + tan2(~) dO=
(y 2
_
)2 1
n0 dLln
dz=Rc-dcp+{(Rc-dcp)2-ro(k)
2
q)(i, k) = ~p(i, k - 1) + d O j;nax(k) = max [jb(i, k)] i~ "Qi
rb(k) = arg(0.7R "x) d b = 2 rain [ r b(k)] k~ -(2k
F b = dcp + argmin [rb(k)]
(9)
k~ 12k
with i, k the discretization parameters at the r and z axis, respectively, r b the beam radius at the considered
point, Jb the beam current density, U~ the acceleration voltage, Rc the cathode sphere radius, Z the nuclear number of the gas atoms, r e the electron radius with respect to the Bohr model, q~ the input angle of electron trajectories, and y, /3 the relativistic factors. The main difficulty in numerical analysis of Eq. (8) is a high convergence angle of the GDEG beams (maybe
293
S. V. Denbnovetsky et al. / Applied Surface Science 111 (1997) 288-294
0,' 0,'
0
5
10
15
20
rc[mm]
Fig. 3. Numerical (1) and experimental (2) results of the beam current density distribution on the cathode surface; the acceleration voltage was 12 kV and the discharge current 350 mA.
greater then 10°). Furthermore, electron trajectories in the anode plasma are non-paraxial and in this way the beam focus is not sharply defined.
6. Numerical results and experimental verification Numerical Eqs. (2)-(9) have been used to define the focal parameters of GDEG with an electrode shape as shown in Fig. 1. All results, described further, have been obtained for r and z discretization steps for field calculation of 0.05 mm, accuracy of field calculation of 0.1 V, number of ion trajectories 500 (each), and number of iterations for space charge 30. The distribution of the electric field, obtained as result of computer simulation on the last iteration is shown in Fig. 1 (by dotted lines). In most cases these lines are similar to the shape of the electrode surfaces. Some deviations from this shape are mainly caused by the space charge of positive ions, especially after its interaction with neutral gas atoms. The dependence of the electric field and the GDEG parameters on electron beam space charge in the analyzed regions is not very strong, which suggests a full beam compensation [1]. Furthermore, electrons are strongly accelerated in the near-cathode region and their mean free path becomes much longer than the gun length. The results of measurement and calculated data for the beam current density from a cold cathode surface, caused by ion and neutral atom bombardment, are shown in Fig. 3. The maximum difference between theoretical and experimental results is observed at the edge of the cathode. This can be explained by the influence of other particle flows, such as accelerated electrons reflected from the anode. Some results obtained for geometrical beam focal parameters are presented in Table 1. As has been pointed out, the electron trajectories in a high-voltage discharge are non-paraxial, and in this case a beam focus is not sharply defined. All numerical results were obtained for a discharge regime with a practically stable plasma position (see Fig. 2). In this case a change of calculated values of focal distance and diameter can be caused only by alteration of the
Table 1 Dependence of the beam focal distance, the beam diameter in the focal plane and the beam power on the cathode sphere radius Beam parameters Discharge current (mm) Cathode sphere radius (mm) Focal distance (mm) Beam diameter (ram) Beam power (kW) (measured)
300 400 500
70
75
77
80
80 4 ----
85 4 1.9 2.7 3,0
--2.25 2.9 3.25
103 6 2.1 2.75 3.1
294
s. ~ Denbnotetsl~v et a L / Applied Surface Science 111 (1997) 288-294
electric field distribution, which is mainly defined by the ion space charge. Experimental data for gun efficiency are also given in Table 1.
7. Conclusion Numerical results, obtained by using Eqs. (2)-(9), are in agreement with experiments for the high-current regime, when the plasma boundary position is stable under current changing. For example, the accuracy of defining the output focal beam parameters was 30% and better. Such a high range of accuracy was achieved as a result of improvement of the modified current tube numerical technique including analysis of the contribution of accelerated ions and gas atom collisions to the space charge. This gives a possibility of using calculation results tbr design of GDEG with a wide range of operating pressures and accelerating voltages without complex experimental research.
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