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Sputtering of plasma facing material by simultaneous bombardment with carbon and deuterium ions
r "T'
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IL ill
ELSEVIER
Journal of Nuclear Materials 220-222 (1995) 993-996
Sputtering of plasma facing material by simultaneous bombardment with carbon and deuterium ions D. Naujoks, W. Eckstein Maxoplanck-Institut fiir Plasmaphysik, EURA TOM Association, D-85748 Garching, Germany
Abstract
The sputtering of material (C, Si, and W) exposed to a plasma with an electron temperature of 40 eV containing deuterium ions and carbon ions has been investigated using the computer program TRIDYN. The sputtering yields for the various elements were calculated dependent on the plasma ion fluence and the carbon concentration in the plasma. The question, whether erosion of the target material or deposition of the carbon atoms occurs, strongly depends on the carbon concentration in the plasma and the target atom species. Also a transition from a erosion phase to a deposition phase (and also reversed) during the exposure was observed under certain conditions. Steady state conditions were achieved only in a time scale which is of the same order as the discharge time in today's fusion experiments. An analytical model has been developed and the results agree well with those obtained by computer simulations.
1. Introduction
The main erosion process for a material exposed in fusion plasma experiments is physical sputtering (except for carbon under certain conditions). This process has been widely investigated under laboratory conditions, i.e. well defined conditions such as bombardment of monoatomic, clean materials with one type of monoenergetic ions at a fixed angle of incidence [1,2]. In fusion experiments the wall material is simultaneously bombarded with ions of various species and ionization stages which impinge on the surface with a certain energy and angular distribution. The plasma impurity ions such as carbon are implanted and alter the surface composition and therefore also the sputtering yields of the modified target material. It was shown by erosion experiments performed in JET, T E X T O R and ASDEX-Upgrade that a linear superposition of the sputtering yields (i.e. summarized sputtering of material, assumed as pure, by C and D ions) cannot explain the observed erosion [3-5]. In order to study the material erosion by physical sputtering under the simultaneous bombardment with various ion species the Monte Carlo program TRIDYN [6] was used. This enabled us to simulate the altering
of the surface composition and, consequently, the change of the erosion during the bombardment. Using simplified assumptions about of the sputtering yields for material with altered surface layers an analytical model is developed to describe and to understand the simulation results.
2. Simulation model and results
The erosion was calculated with the program TRIDYN (version 40.1). This program is described in Ref. [6]. The program assumes an amorphous target and is based on the binary collision approximation. The "krypton-carbon" potential [7] is applied as interaction potential. The electronic energy loss is chosen as an equipartition of the continuous Lindhard-Scharff [8] and the local P e n - R o b i n s o n [9] interactions. The binding of surface atoms is described by a planar potential with the heat of sublimation as the value of the surface binding energy. The surface binding energies are taken from Table 6.1 in Ref. [10]: C (7.42 eV), Si (4.70 eV), W (8.68 eV). Projectiles and target atoms are followed as long as their energy is above the surface binding energy. Each projectile corresponds to
0022-3115/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved $SDI 0022-3115(94)00459-5
994
D. Naujoks, 144 Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996
a fluence of about 4 x 1017 atoms/m e in most calculations. Version 40.1 allows the incidence of two ion species as version 40 [11] but allows a Maxwellian distribution and different charge states of the projectiles. The two projectiles are chosen randomly according to the composition of the incident flux. The bombardment is performed using a Maxwellian distribution of deuterium with a temperature of 40 eV; carbon concentrations, J'c, are varied from 0 to 10% applying the same Maxwellian distribution but assuming triply charged carbon ions. The energy of the projectiles is increased and the isotropic angular distribution of the projectiles is changed to a more normal incidence by the sheath potential. The sheath potential of 120 V has a stronger influence on the carbon ions than on the deuterium ions due to the higher charge of the carbon ions. The program determines the thickness removed or deposited as a function of fluence which is proportional to time. In addition sputtering yields, reflection coefficients, surface composition and the depth distributions of the different species are calculated versus the incident ion fluence. If the carbon concentration in the incident flux is zero only erosion occurs. With an increasing carbon flux carbon deposition is found and the erosion is reduced. For a carbon target deposition takes place for carbon concentrations in the incident flux above about 3%, see Fig. 1. All the erosion and deposition curves show a nearly linear relationship with fluence. For a silicon substrate these curves are not linear for a few % of carbon; initially erosion occurs but then at larger fluences the curves change their slopes and carbon is
TRIDYN simulation analytical theory
--
5%/"
./"
A
4%.
40
i , ~ , i. -
TRIDYN simulation analytical theory
°
20 J
c~
,y) o ~"
20
4
Q< -
0
~
4
-
(*/0
2e
8 m
10
e)
Fig. 2. A s in Fig. 1, for a s i l i c o n t a r g e t .
deposited as demonstrated in Fig. 2. The result is an interplay between sputtering, refection, and surface composition change. The most interesting system discussed in this paper is the bombardment of tungsten. Here carbon is deposited for low carbon concentrations at low fluences until finally the target is eroded. Above 3.8% carbon in the incident flux the deposition of carbon rises at first, then the thickness of the carbon layer stays constant for some fluence until it finally rises, see Fig. 3. As another example the sputtering yields of the different species due to the two projectile species are shown in Fig. 4. For a high carbon concentration (10%) in the incident flux the sputtering of carbon by carbon dominates whereas for low concentrations (3.5%) the deuterium sputtering of carbon dominates. The sputtering of tungsten by carbon is always smaller and by deuterium to small to be seen in the Fig. 4. A similar behaviour of the sputtering of tungsten by C was early observed in a monoenergetic bombardment [12]. The
_
'
'
'I , ' ' t
'i/'
....
,
/
"
/"
38% /
///~:<_
e~
t
"
6
fluence
/
0%
1%
0%
4O
10
~"~--
' '
"
4%
--
_
g co ..a
3
6
4 [luence
(*t022
8 rn
10
2)
Fig. 1. S u r f a c e r e c e s s i o n a n d c a r b o n d e p o s i t i o n vs f l u e n c e as for t h e e x p o s u r e o f c a r b o n to a p l a s m a c o n t a i n i n g d e u t e r i u m and triply c h a r g e d c a r b o n i o n s in d e p e n d e n c e o n t h e c a r b o n c o n c e n t r a t i o n in t h e p l a s m a , w h i c h is i n d i c a t e d in t h e f i g u r e . T h e p r o j e c t i l e s a r e a s s u m e d to be M a x w e l l i a n d i s t r i b u t e d a n d e x p e r i e n c e a s h e a t h p o t e n t i a l o f 120 V.
e~
TRIDYN simulation analytical theo~'y S
i
0
i
i
i
I~b
i
i
I
."
3% ~--
L b ,
i
I
10
20
30
f[uence
(~i0 2~ m
z)
,
i
i
Fig. 3. As in Fig. 1, for a tungsten target.
40
D. Naujoks, W. Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996 yields presented in Fig. 4 are used in simplified approximations in the analytical model. It should be kept in mind that the absolute values for the very sensitive behaviour for W are model dependent.
995
.025 .020 .015
3. Analytical model and discussion
.010
The change of the eroded amount of the target material, n l, during the bombardment by carbon and deuterium ions can be expressed as following: /)nI ~t
.005
"-'"Y~,c -~ w z
e~ ~.
"fDFeYD~ I -- f c Fel)c --.I'
electron density near the target plate; c~ being the sound speed of the plasma ions and a being the angle between the surface and the magnetic field line. The concentrations of species, i, in the plasma with respect to the electron density are denoted by f/ with the definition Y~iffqi = 1, where qi being the charge states. The sputtering yields of species, j, by projectiles, i, are given by ~ ~ j and depend on the amount of carbon deposited onto surface of the metal, I, and can be expressed in a first approximation by the formulae
Yc_~I=~'c_~I(t)=Yc~
, 1
n0CA'
nCoAI
5
10
15
20
25
30
35
40
.05
(1) .04
F~ = nec ~ sin(a) is the electron flux and n~ denotes the
]~D__,I=I')D~I(t)=YD~ 1 1
.0
35~ c
.03 .02 .01 ", C ~ W "4-, 2
10% C
4 6 FLUENCE (x 1022 m -2)
8
10
Fig. 4. Sputtering yields of the different target species due to the projectile species. Same conditions as in Fig. 1. ions and the carbon ions themselves is represented by the second and the third term. The sputtering yields 12//~f are assumed as
,and
,
^ nC(t) YD~C = l?O ~ c ( t ) = YD-~C nCAc , and
with the conditions ]~D ~ i = 0 ,
nC(t)
nC(/) by n-----~---___A i,
^
]~c --*c = ])c--, c ( / ) = Y c ~ c n0CAc , with the conditions
nC(t)
Yc_.i=0,
by n---~-->_A~.
The sputtering yields Y~~ j are defined as the average number of removed atoms, j, per incident ion, i, in the case of pure material, j (i.e. without implanted atoms). If the amount of deposited carbon is larger than the parameter A, which is expressed as a thickness A = n C ( t ) / n c, then the corresponding sputtering yields remain constant. The change of the deposited amount of carbon, n c, can be calculated as following i)nC ot
=/cro(1
-K) - fDF¢gD ~ C -- fcF~I~c. c •
(2)
The first term on the right hand side of Eq. ( 2 ) d e scribes the deposition of the incoming carbon ions which stick onto the surface with a probability unity minus the reflection coefficient /~. The sputtering of the carbon atoms from the surface by the deuterium
^ YDoc=YD~c
by
l)c+c=Vc~ c
by
nC(t) n°c > A c, nC(/) --_>A
c,
where n0c is the atomic density of carbon The reflection coefficient R depends also on the carbon concentration in the metal, I:
l~ = I~c -~ I.c( t ) = R c ~ , - ( R c -~ , - R c ~ c ) nnC ( tA) whereas /~ = R c _, c for nC(t)/nCo >_ A a. Eqs. (1,2) have the solutions
nC(t) for
n - ~ < A c : nC(t) R
=-ff(1-e-m)+nC(t=O)e
-m,
(3)
996
D. Naujoks, W. Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996
[ SR h i ( t ) = [ --fi- - Q ) t -
SR ~-y(1 - e - e ' )
+
SnC(t = O) P (1-e
P') + n l ( t = 0).
(4)
where p = fDFeYD ~ C + )~cCYc ~ c n~A C nCoAc
fcFe(Rc~l-
Rc-+c)
nCAR
Q = f D F ~ Y D ~ I + f c F ~ Y c ~ l ; g = fcF~(1 - g c ~ i ) ; S=
foCr~ n~A I
+
f~r.Y~i n c .a l
The initial conditions are nC(t = O) = nl(t = O) = O. In the case of n C ( t ) / n c > A c and nC(t)/nCo > A R we have the same form of the solution as in Eqs. (3,4) but with
fCre(Rc~l -Rc+c) nCa ~
p=
and R = f c F ~ ( 1 - R c ~ l ) - f D F e Y D ~ c - f c l ' e Y c + c
.
The initial conditions have to be changed consistently. If n C ( t ) / n c > A t then n I ( t ) = const., and at least, if nC(t)/nCo > A R then an c = (fcr~(1
- Rc ~ c )
-fcFeYc~c)At.
-fororo~c (5)
Using the described above analytical model calculations were performed and the results presented as the deposited/eroded thickness, d: HC
HI
d = n°c + n i ,
ions. At higher fluences more and more carbon atoms are implanted in the near-surface layers and prevent the target atoms from being sputtered. The plasma temperature, i.e. the D ion energies, are too low for this specific carbon concentration, fc, to sputter the deposited carbon atoms. In the tungsten target (Fig. 3) deposition of carbon atoms starts already even at low carbon concentration, fc due to the fact that tungsten is really not sputtered by the deuterium ions. Sputtering of deposited carbon atoms is initially low because of the low C concentration in the near-surface layer. With higher fluences carbon concentration accumulates at the surface, the sputtering of the C atoms increases, these atoms are removed and thus the tungsten atoms can be sputtered. The carbon concentration, fc, influences very sensitively (as shown in Fig. 3) the steady state behaviour at higher fluences. The analytical theory shows the correct tendency but the absolute values are not in full agreement with the values calculated with the TRIDYN program. Taking n e = 1 × 1019 m -3 and a = 2° the ion fiuence of 1 × 1023 m -2 corresponds to a exposure time of about 5 s which is of the order of the discharge times in today's fusion experiments. Therefore, the described effects have to be taken into account analysing experimental studies of material erosion. Further, strong decrease of material erosion can be expected if the carbon concentration in the divertor plasma is higher than approximately 3%.
(6)
and are compared with those obtained by computer simulation with TRIDYN; here n I denotes the atomic density of the target material. The sputtering yields Yi~i and the reflection coefficients R i ~ j (i.e. for the case of pure material) were taken from TRIDYN simulations. The results are shown in Figs. 1-3 for for C, Si, and W respectively by dashed lines. They are in good agreement with the TRIDYN simulation results for all three materials. For silicon the transition of a erosion phase at the beginning of the bombardment to a deposition phase at higher fluences for a carbon concentration of about 4% (Fig. 2) is also obtained in the analytical calculations and can be explained as following. At lower fluences we still have a surface containing only a small amount of carbon and the target material is sputtered by the impinging carbon and deuterium
References
[1] R. Behrisch, ed., Sputtering by Particle Bombardment, I, Topics in Applied Physics, (Springer, Heidelberg, 1981). [2] W. Eckstein, C. Garcia-Rosales, J. Roth and W. Ottenberger, Report IPP 9/82 (1993). [3] D. Naujoks, R. Behrisch, J.P. Coad and L. deKock, Nucl. Fusion 33 (1993) 581. [4] D. Naujoks, R. Behrisch, V. Philipps and B. Schweer, in Plasma Physics and Controlled Fusion (Proc. 20th Eur. Conf. Lisboa) vol. 17C, part II (1993) p. 651. [5] D. Nanjoks, J. Roth, K. Krieger et al., J. Nucl. Mater. (1994) in press. [6] W. Mfller, W. Eckstein and J. Biersack, Comp. Phys. Comm. 51 (1988) 355. [7] W. Mrller, D. Bouchier, O. Burat and V. Stambouli, Surf. Coat. Technol. 51 (1992) 190. [8] J. Lindhard and M. Scharff, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 27 (1953) 15. [9] O.S. Oen, M.T. Robinson, Nucl. Instr. and Meth. 132 (1976) 647. [10] W. Eckstein, Computer Simulation of Ion-Solid Interactions (Springer, Berlin, 1991). [11] W.D. Wilson, L.G. Haggmark and J.P. Biersack, Phys. Rev. B 15 (1977) 2458. [12] W. Eckstein, J. Roth, Nucl. Instr. and Meth. B 53 (1991) 279.
"
~I
"
IL ill
ELSEVIER
Journal of Nuclear Materials 220-222 (1995) 993-996
Sputtering of plasma facing material by simultaneous bombardment with carbon and deuterium ions D. Naujoks, W. Eckstein Maxoplanck-Institut fiir Plasmaphysik, EURA TOM Association, D-85748 Garching, Germany
Abstract
The sputtering of material (C, Si, and W) exposed to a plasma with an electron temperature of 40 eV containing deuterium ions and carbon ions has been investigated using the computer program TRIDYN. The sputtering yields for the various elements were calculated dependent on the plasma ion fluence and the carbon concentration in the plasma. The question, whether erosion of the target material or deposition of the carbon atoms occurs, strongly depends on the carbon concentration in the plasma and the target atom species. Also a transition from a erosion phase to a deposition phase (and also reversed) during the exposure was observed under certain conditions. Steady state conditions were achieved only in a time scale which is of the same order as the discharge time in today's fusion experiments. An analytical model has been developed and the results agree well with those obtained by computer simulations.
1. Introduction
The main erosion process for a material exposed in fusion plasma experiments is physical sputtering (except for carbon under certain conditions). This process has been widely investigated under laboratory conditions, i.e. well defined conditions such as bombardment of monoatomic, clean materials with one type of monoenergetic ions at a fixed angle of incidence [1,2]. In fusion experiments the wall material is simultaneously bombarded with ions of various species and ionization stages which impinge on the surface with a certain energy and angular distribution. The plasma impurity ions such as carbon are implanted and alter the surface composition and therefore also the sputtering yields of the modified target material. It was shown by erosion experiments performed in JET, T E X T O R and ASDEX-Upgrade that a linear superposition of the sputtering yields (i.e. summarized sputtering of material, assumed as pure, by C and D ions) cannot explain the observed erosion [3-5]. In order to study the material erosion by physical sputtering under the simultaneous bombardment with various ion species the Monte Carlo program TRIDYN [6] was used. This enabled us to simulate the altering
of the surface composition and, consequently, the change of the erosion during the bombardment. Using simplified assumptions about of the sputtering yields for material with altered surface layers an analytical model is developed to describe and to understand the simulation results.
2. Simulation model and results
The erosion was calculated with the program TRIDYN (version 40.1). This program is described in Ref. [6]. The program assumes an amorphous target and is based on the binary collision approximation. The "krypton-carbon" potential [7] is applied as interaction potential. The electronic energy loss is chosen as an equipartition of the continuous Lindhard-Scharff [8] and the local P e n - R o b i n s o n [9] interactions. The binding of surface atoms is described by a planar potential with the heat of sublimation as the value of the surface binding energy. The surface binding energies are taken from Table 6.1 in Ref. [10]: C (7.42 eV), Si (4.70 eV), W (8.68 eV). Projectiles and target atoms are followed as long as their energy is above the surface binding energy. Each projectile corresponds to
0022-3115/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved $SDI 0022-3115(94)00459-5
994
D. Naujoks, 144 Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996
a fluence of about 4 x 1017 atoms/m e in most calculations. Version 40.1 allows the incidence of two ion species as version 40 [11] but allows a Maxwellian distribution and different charge states of the projectiles. The two projectiles are chosen randomly according to the composition of the incident flux. The bombardment is performed using a Maxwellian distribution of deuterium with a temperature of 40 eV; carbon concentrations, J'c, are varied from 0 to 10% applying the same Maxwellian distribution but assuming triply charged carbon ions. The energy of the projectiles is increased and the isotropic angular distribution of the projectiles is changed to a more normal incidence by the sheath potential. The sheath potential of 120 V has a stronger influence on the carbon ions than on the deuterium ions due to the higher charge of the carbon ions. The program determines the thickness removed or deposited as a function of fluence which is proportional to time. In addition sputtering yields, reflection coefficients, surface composition and the depth distributions of the different species are calculated versus the incident ion fluence. If the carbon concentration in the incident flux is zero only erosion occurs. With an increasing carbon flux carbon deposition is found and the erosion is reduced. For a carbon target deposition takes place for carbon concentrations in the incident flux above about 3%, see Fig. 1. All the erosion and deposition curves show a nearly linear relationship with fluence. For a silicon substrate these curves are not linear for a few % of carbon; initially erosion occurs but then at larger fluences the curves change their slopes and carbon is
TRIDYN simulation analytical theory
--
5%/"
./"
A
4%.
40
i , ~ , i. -
TRIDYN simulation analytical theory
°
20 J
c~
,y) o ~"
20
4
Q< -
0
~
4
-
(*/0
2e
8 m
10
e)
Fig. 2. A s in Fig. 1, for a s i l i c o n t a r g e t .
deposited as demonstrated in Fig. 2. The result is an interplay between sputtering, refection, and surface composition change. The most interesting system discussed in this paper is the bombardment of tungsten. Here carbon is deposited for low carbon concentrations at low fluences until finally the target is eroded. Above 3.8% carbon in the incident flux the deposition of carbon rises at first, then the thickness of the carbon layer stays constant for some fluence until it finally rises, see Fig. 3. As another example the sputtering yields of the different species due to the two projectile species are shown in Fig. 4. For a high carbon concentration (10%) in the incident flux the sputtering of carbon by carbon dominates whereas for low concentrations (3.5%) the deuterium sputtering of carbon dominates. The sputtering of tungsten by carbon is always smaller and by deuterium to small to be seen in the Fig. 4. A similar behaviour of the sputtering of tungsten by C was early observed in a monoenergetic bombardment [12]. The
_
'
'
'I , ' ' t
'i/'
....
,
/
"
/"
38% /
///~:<_
e~
t
"
6
fluence
/
0%
1%
0%
4O
10
~"~--
' '
"
4%
--
_
g co ..a
3
6
4 [luence
(*t022
8 rn
10
2)
Fig. 1. S u r f a c e r e c e s s i o n a n d c a r b o n d e p o s i t i o n vs f l u e n c e as for t h e e x p o s u r e o f c a r b o n to a p l a s m a c o n t a i n i n g d e u t e r i u m and triply c h a r g e d c a r b o n i o n s in d e p e n d e n c e o n t h e c a r b o n c o n c e n t r a t i o n in t h e p l a s m a , w h i c h is i n d i c a t e d in t h e f i g u r e . T h e p r o j e c t i l e s a r e a s s u m e d to be M a x w e l l i a n d i s t r i b u t e d a n d e x p e r i e n c e a s h e a t h p o t e n t i a l o f 120 V.
e~
TRIDYN simulation analytical theo~'y S
i
0
i
i
i
I~b
i
i
I
."
3% ~--
L b ,
i
I
10
20
30
f[uence
(~i0 2~ m
z)
,
i
i
Fig. 3. As in Fig. 1, for a tungsten target.
40
D. Naujoks, W. Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996 yields presented in Fig. 4 are used in simplified approximations in the analytical model. It should be kept in mind that the absolute values for the very sensitive behaviour for W are model dependent.
995
.025 .020 .015
3. Analytical model and discussion
.010
The change of the eroded amount of the target material, n l, during the bombardment by carbon and deuterium ions can be expressed as following: /)nI ~t
.005
"-'"Y~,c -~ w z
e~ ~.
"fDFeYD~ I -- f c Fel)c --.I'
electron density near the target plate; c~ being the sound speed of the plasma ions and a being the angle between the surface and the magnetic field line. The concentrations of species, i, in the plasma with respect to the electron density are denoted by f/ with the definition Y~iffqi = 1, where qi being the charge states. The sputtering yields of species, j, by projectiles, i, are given by ~ ~ j and depend on the amount of carbon deposited onto surface of the metal, I, and can be expressed in a first approximation by the formulae
Yc_~I=~'c_~I(t)=Yc~
, 1
n0CA'
nCoAI
5
10
15
20
25
30
35
40
.05
(1) .04
F~ = nec ~ sin(a) is the electron flux and n~ denotes the
]~D__,I=I')D~I(t)=YD~ 1 1
.0
35~ c
.03 .02 .01 ", C ~ W "4-, 2
10% C
4 6 FLUENCE (x 1022 m -2)
8
10
Fig. 4. Sputtering yields of the different target species due to the projectile species. Same conditions as in Fig. 1. ions and the carbon ions themselves is represented by the second and the third term. The sputtering yields 12//~f are assumed as
,and
,
^ nC(t) YD~C = l?O ~ c ( t ) = YD-~C nCAc , and
with the conditions ]~D ~ i = 0 ,
nC(t)
nC(/) by n-----~---___A i,
^
]~c --*c = ])c--, c ( / ) = Y c ~ c n0CAc , with the conditions
nC(t)
Yc_.i=0,
by n---~-->_A~.
The sputtering yields Y~~ j are defined as the average number of removed atoms, j, per incident ion, i, in the case of pure material, j (i.e. without implanted atoms). If the amount of deposited carbon is larger than the parameter A, which is expressed as a thickness A = n C ( t ) / n c, then the corresponding sputtering yields remain constant. The change of the deposited amount of carbon, n c, can be calculated as following i)nC ot
=/cro(1
-K) - fDF¢gD ~ C -- fcF~I~c. c •
(2)
The first term on the right hand side of Eq. ( 2 ) d e scribes the deposition of the incoming carbon ions which stick onto the surface with a probability unity minus the reflection coefficient /~. The sputtering of the carbon atoms from the surface by the deuterium
^ YDoc=YD~c
by
l)c+c=Vc~ c
by
nC(t) n°c > A c, nC(/) --_>A
c,
where n0c is the atomic density of carbon The reflection coefficient R depends also on the carbon concentration in the metal, I:
l~ = I~c -~ I.c( t ) = R c ~ , - ( R c -~ , - R c ~ c ) nnC ( tA) whereas /~ = R c _, c for nC(t)/nCo >_ A a. Eqs. (1,2) have the solutions
nC(t) for
n - ~ < A c : nC(t) R
=-ff(1-e-m)+nC(t=O)e
-m,
(3)
996
D. Naujoks, W. Eckstein /Journal of Nuclear Materials 220-222 (1995) 993-996
[ SR h i ( t ) = [ --fi- - Q ) t -
SR ~-y(1 - e - e ' )
+
SnC(t = O) P (1-e
P') + n l ( t = 0).
(4)
where p = fDFeYD ~ C + )~cCYc ~ c n~A C nCoAc
fcFe(Rc~l-
Rc-+c)
nCAR
Q = f D F ~ Y D ~ I + f c F ~ Y c ~ l ; g = fcF~(1 - g c ~ i ) ; S=
foCr~ n~A I
+
f~r.Y~i n c .a l
The initial conditions are nC(t = O) = nl(t = O) = O. In the case of n C ( t ) / n c > A c and nC(t)/nCo > A R we have the same form of the solution as in Eqs. (3,4) but with
fCre(Rc~l -Rc+c) nCa ~
p=
and R = f c F ~ ( 1 - R c ~ l ) - f D F e Y D ~ c - f c l ' e Y c + c
.
The initial conditions have to be changed consistently. If n C ( t ) / n c > A t then n I ( t ) = const., and at least, if nC(t)/nCo > A R then an c = (fcr~(1
- Rc ~ c )
-fcFeYc~c)At.
-fororo~c (5)
Using the described above analytical model calculations were performed and the results presented as the deposited/eroded thickness, d: HC
HI
d = n°c + n i ,
ions. At higher fluences more and more carbon atoms are implanted in the near-surface layers and prevent the target atoms from being sputtered. The plasma temperature, i.e. the D ion energies, are too low for this specific carbon concentration, fc, to sputter the deposited carbon atoms. In the tungsten target (Fig. 3) deposition of carbon atoms starts already even at low carbon concentration, fc due to the fact that tungsten is really not sputtered by the deuterium ions. Sputtering of deposited carbon atoms is initially low because of the low C concentration in the near-surface layer. With higher fluences carbon concentration accumulates at the surface, the sputtering of the C atoms increases, these atoms are removed and thus the tungsten atoms can be sputtered. The carbon concentration, fc, influences very sensitively (as shown in Fig. 3) the steady state behaviour at higher fluences. The analytical theory shows the correct tendency but the absolute values are not in full agreement with the values calculated with the TRIDYN program. Taking n e = 1 × 1019 m -3 and a = 2° the ion fiuence of 1 × 1023 m -2 corresponds to a exposure time of about 5 s which is of the order of the discharge times in today's fusion experiments. Therefore, the described effects have to be taken into account analysing experimental studies of material erosion. Further, strong decrease of material erosion can be expected if the carbon concentration in the divertor plasma is higher than approximately 3%.
(6)
and are compared with those obtained by computer simulation with TRIDYN; here n I denotes the atomic density of the target material. The sputtering yields Yi~i and the reflection coefficients R i ~ j (i.e. for the case of pure material) were taken from TRIDYN simulations. The results are shown in Figs. 1-3 for for C, Si, and W respectively by dashed lines. They are in good agreement with the TRIDYN simulation results for all three materials. For silicon the transition of a erosion phase at the beginning of the bombardment to a deposition phase at higher fluences for a carbon concentration of about 4% (Fig. 2) is also obtained in the analytical calculations and can be explained as following. At lower fluences we still have a surface containing only a small amount of carbon and the target material is sputtered by the impinging carbon and deuterium
References
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