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The effective charge of He ions in metals
Nuclear Instruments & Methods in Physics Research
Nuclear Instruments and Methods in Physics Research B67 (1992) 138-141 North-Holland
Sect,on B
The effective charge of He ions in metals Ch. Eppacher and D. Semrad hzstitut fiir Experimentalphysik,Johannes-Kepler-Unicersitiit,A-4040 Lbaz, Austria
Measurements of the stopping cross sections of Ge, Sn. and Pb for hydrogen and helium projectiles are presented. The projectile velocities range from the Bohr velocity c o to 2.5co. Large deviations from available tables of up to 28% were found. The measured ratio of helium and hydrogen stopping powers are compared to available predictions. Even for isoelectronic elements of the same group of the periodic system, no satisfactory agreement can be found.
1. Introduction
If a simple relation between the stopping cross sections fl~r protons and He ions could be found, the data base to produce stopping power tabulations fi)r both, protons and He ions, would be considerably increased. This could be done by introducing an effective charge of the projectile, which takes into account the screening of the projectile's charge by bound electrons. Let: us assume that we have a reliable theory to calculate the stopping cross section e = Eth(Zj~fr, Z 2, t h ) for a point charge Zl~tt moving at speed c I through an elemental target of charge number Z 2, then the effective charge of the projectile with charge number Z I would be implicitly given by
eth( Z,ejf(v,), Z2,
cl) =e.,( Zl, Z~, cl).
(1)
Here, t' I is the projectile's velocity and Em is the measuredl stopping cross section for this projectiletarget combination. Most of the available theories treat the stopping process as a perturbation to first order, which results in a scaling with the square of Z~ff, Eth(Zlcff(/-'l), Z2, i'l) : Z ~ c f f ( / ' l ) ~th(l, Z 2, t'l). (2) If we define the ratio R of e~ for He ions and protons of equal velocity normalized to 1 for large velocities, then we find, using eqs. (1) and (2),
g(v~)
era(2, Z2, t'l)
Z2etf(He)
4era(l, Zz, t,i)
4ZZ fr(p ) .
(3)
In the literature one finds the following statements concerning R: according to Ziegler, Biersaek and Littmark [l] R does not depend on the target material and is given by a universal curve R(v,)=l-exp
[
-i
a,[In(mn:,'f/2
,1] '
.
(4)
Here, mac denotes the mass of He in units which give the kinetic energy in keV. The coefficients a i are given by a0=0.2865, a1=0.1266, a2---0.001429, a3 = 0.02402, a 4 = -0.01135 and a 5 = 0.001475. According to the theories of Kreussicr, Varclas and Brandt [2] and of Brandt and Kitagawa [3], which are primarily intended for heavier ions, R depends on target properties via the Fermi velocity c F of their plasma electrons. Striektly speaking, R depends on the mean relative velocity t,r between the projectile and the electrons, which, assuming a quasi-free plasma, give r~
3( 2c~ c r = ~-rl.- 1 + 3t,F
c4 ) 15t,aF ,
(5) t' I < t' F.
As can be seen from eq. (5) for velocities L,I larger than twice the Fermi velocity, R is again found to be almost independent of the target material. To test both statements, we have started a series of reliable measurements on the stopping cross section for hydrogen and helium projectiles. As a first step, we present measurements of R for three elements of the fourth group of the periodic table, i.e. Ge, Sn, and Pb. The aims of this investigation is to clarify, whether for these elements one plasma parameter is sufficient to describe R in the velocity range from co to 2.5t' 0, where c 0 is the Bohr velocity. The three elements have a very similar electronic structure: in all cases roughly four electrons per atom form the plasma (see table 1). As the atomic densities of Sn and Pb are very similar their plasmon energies are very close to each other (table 1). Ge has a higher atomic density which leads to a higher plasmon energy. Due to the different number of core electrons, the stopping cross section of these elements differ from each other, predominantly at higher energies.
0168-583X/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
Ch. Eppacher, D. Semrad / Effectice charge of He ions bz metals
139
2. Measurements
3. Discussion of the results
We have performed absolute measurements of Em for protons, deuterons, and He ions in the energy range from 80 to 700 keV. The stopping cross sections were obtained from backscattering spectra. A detailed description of our measuring procedure can be found in refs. [7-9]. The targets were evaporated in a HVchamber at a pressure between 3 x 10 -8 and 1 x 10 -7 mbar, with liquid nitrogen cooled metal sheets surrounding the backing. The mass evaporated onto a well defined area was determined by weighing. For all three elements the measurements were performed on four targets, each with different mass density. The targets were tilted by 8 ° with respect to the incoming beam to avoid any influence of preferential orientation of crystal channels perpendicular to the target surface due to texture. Measurements were performed on different parts of the target to detect possible inequality in thickness. The evaluation procedure was carried out as described in ref. [10]. The error of our results (to be understood as one standard deviation) is about 3% at higher energies and about 5% at lower energies. The main contributions to the error come from: the areal mass density of the target (1.8%), the energy calibration of the ion beam (0.3%), the conversion of MCA channels to energy (1.5% at lower energies and 0.5% at higher energies), the determination of the center of mass of the high and low energy edge of the backscattering spectrum (0.5%), and at very low energies the increased energy loss due to multiple scattering [11] (up to 1%). These errors are added algebraically. As some of these errors cancel when calculating the ratio, we consider the error of R to be also 5% at low velocities and 3% at high velocities.
In figs. l a - l c we show the Em of Ge, Sn, and Pb for hydrogen projectiles. Each measurement is represented by two sections of a curve corresponding to the two energy intervals which the projectiles backscattered at the rear surface cover within the target [10]. The solid line shows the fit to the data. The fit function [10] used is Em(l,i)- I = ao+alE-~L4+a.,Ea.25. + a3t~r0s..
(6)
Here, the proton energy E has to be inserted in keV to get the stopping cross section in units of 10-15 e V c m 2. The parameters a0 through a 3 are listed in table 1. The broken line gives the result of Andersen and Ziegler [12] (A&Z). Fig, 1 also shows the calculated contribution of the plasma electrons (dotted line). These calculations are based upon the linear response theory by Lindhard and Winther [13]. The plasma density and the number of free electrons per atom were deduced [14] from measured plasmon energies, assuming the plasma to be free, and from the atomic density of the bulk material (see also table 1). Though for Ge (fig. la) the calculated plasma contribution appears too peaked and at too small energies, it can be seen that below 100 keV ~ is almost entirely given by the plasma. The plasma contribution determines also the position and the height of the maximum. Only above 200 keV there is a significant contribution from the core electrons, mainly from the 3d electrons. This is due to the fact, that these electrons are part of the completed third shell and, therefore, are more strongly bound [15] than the corresponding 4d electrons of Sn. Compared to A & Z , our measurements are lower by about 7% except for the very highest energies shown.
Table I Relevant data for Ge, Sn and Pb: the plasmon energies are taken from ref. [4] (GeL ref. [5] (Sn), and ref. [6] (Pb), respectively. All plasma parameters are calculated assuming a free plasma
Parameters for e Protons a o at a2 a3 He ions a0 a1 a2 a3 Electron configuration Plasmon energy [eV] Plasma density [a.u.] One electron radius [a.u.] Number of plasma electrons/atom Fermi velocity (calculated) [a.u.]
Ge
Sn
Pb
-0.12575 0.3 ! 305 0.037441 - 0.00014623 0.068177 0.0010195 - 0.016632 0.00015782 Ar3dla4s24p2 15.5 0.0258 2.10 3.93 0.914
0.013226 0.064638 - 0.0017580 0.00019534 -0.0013501 0.11218 - 0.00095374 0.000033388 Kr4dt°5s25p 2 13.7 0.0202 2.28 3.69 0.842
0.041374 0.010333 - 0.0090079 0.00023294 0.019328 0.044094 - 0.0048105 0.000051335 Xe 4ft45dl°6s26p2 13.3 0.0190 2.32 3.90 0.826 il. CAPTURE AND LOSS
140
I(a) .
.
Ch. Eppadwr, D. Semrad / Effectice charge of He ions in metals .
.
.
.
.
.
.
.
.
.
.
.
0.85
.
.
....
...................
za[ / ' i ~ " , , .,'i
"V.:.- "-.-...
0.50 0.40
Sn,Pb
16
0.35
.k"",<, .'" "°. """ "'"...
7-... . . . .
(c)
./
35
30
"L
-.,,
.-
.": /l/ "............... . """. -."//
.":'/i
"-.. ..
.
.
.
.
•
l.O
1.2
1.4
1.6
l.O
z.o
2.2
2.4
2.6
Ion Velocity [a.u.] Fig. 2. Normalized ratio of He to proton stopping power for Ge, Sn, and Pb. The broken curve shows the universal function given by Ziegler, Biersack and Littmark [1] (Z,B&L). Also shown are the low-velocity limits according to Echenique, Nieminen and Ritchie [16].
:(b) . . . . . . . . . . . . . . .
-". ......
.
Ge
~
18
i
.
0.55
Ge
i/
.
~. 0.60
0.45 - -
"0
.
""" Z,B&L
o.7o
"%",
.
.
Sn,
o.v5
ue
,
~
0.80
.. "-..
E n e r g y [keV]
Fig. !. Measured proton stopping cross section of (a) Ge (b) Sn and (c) Pb in units of 10- ts eV cm2. Also shown is the fit function by ,Andersen and Ziegler [12] (broken line) and the contribution from the plasma as calculated by the theol~ of Lindhard and Winther [13] (dotted line).
F o r Sn and even more so for Pb, the maximum of e is determined by the contributions from the core electrons (4d electrons in the case of Sn and 4f electrons for Pb). The plasma contribution influences the shape of the stopping power curve only at low velocities leading to a broad ridge. As there were not enough low energy data available to A & Z to show this trend, we find rather large discrepancies with respect to A & Z : - 8 % at 30 keV and + 1 2 % at 100 keV in fig. lb and - 2 8 % at 30 keV and + 7 % at 200 keV in fig. lc. The normalized ratio R of the stopping cross sections is shown in fig. 2 together with the universal curve by Ziegler, Biersack and Littmark [1]. We find agreement between our results for Sn and Pb due to their almost identical plasma. The ratio R for Ge is smaller in the whole velocity region and merges into the curves for Sn and Pb above 2.3v o. For velocities of the order of the Bohr velocity it seems to level off. The curve by Ziegler, Biersack and Littmark [1] fits the data fairly well, the maximum deviation being 8%. However, the fact that we get different curves even for elements of the same group of the periodic system indicates, that one cannot expect a universal curve for all targets. According to Kreussler et al. [2] one should draw R as a function of t,r rather than of c t. However, this would increase even further the discrepancy seen in fig. 2: as Ge has the larger Fermi velocity (table 1), R for Ge will be shifted towards higher values of v r with respect to R for Sn and Pb. The different behavior of R at low velocities can be understood from the results of Echenique, Nieminen and Ritchie [6]. In fig. 3 we show R for velocities much smaller than the Fermi velocity as calculated by the
Ch. Eppacher, 1). Semrad / Effectice charge of He ions in metals 0.70 ~
141
targets. This project was supported by the Fonds zur F6rderung der wissenschaftlichen Forschung under contract no. P 8149-PHY.
.......................................
0;0o\
References
°o:i.......... • 1.0
1.5
2.0
2.5 3.0 3.5 4.0 4.5 One gleetrolt Radius
5.0
5.5
Fig. 3. Normalized ratio in the limit of low velocities based upon density-functional calculations by Echenique, Nieminen and Ritchie [16]. density-functional formalism taken from fig. 1 of ref. [16]. The abscissa is the one electron radius r s. One notes that R for Sn and Pb should approach the value of 0.39 for v I tending to 0, whereas R for Ge should converge towards 0.45. This behavior seems to be in agreement with our results in fig. 2. In conclusion one can say, that neither existing statement on the "effective charge" of He ions can explain R in a wide range of velocities• Furthermore, it seems very unlikely, that a single plasma parameter like the Fermi velocity determines the shape of R. However, density-functional calculations seem to be a good basis to explain the behavior of R at least in the limit of low velocities.
Acknowledgements
[1] J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Ranges of Ions in Solids (Pergamon, Oxford, 1985) vol 1. [2] S. Kreussler, C. Varelas and W. Brandt, Phys. Rev. B23 (1981) 82. [3] W. Brandt and M. Kitagawa, Phys. Rev. B25 (1982) 5631. [4] O. Sueoka, J. Phys. Soc. Jpn. 20 (1965) 2203. [5] E.A. Bakulin, L. Balahanova, M.M. Bredow and E.V. Stepin, Soy. Phys. Solid State 16 (1975) 1670. [6] T. Aiyama and K. Yada, J. Phys. Soc. Jpn. 36 (1974) 1554. [7] D. Semrad, M. Geretschl~igerand A. Rabler, Nucl. Instr. and Meth. 215 (1983) 348. [8] D. Semrad, P. Mertens and P. Bauer, Nucl. Instr. and Meth. B15 (1986) 86. [9] P. Bauer, Nucl. instr, and Meth. B27 (1987) 301. [10] Ch. Eppacher and D. Semrad, Nucl. Instr. and Meth. B35 (1988) 109. [11] H. Paul, D. Semrad and A. Seilinger, Nucl. Instr. ,~nd Meth. B61 (1991) 261. [12] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter (Pergamon, New York, 1977) vol. 3. [13] J. Lindhard and A. Winther, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (4) (1964). [14] D. Semrad and P. Bauer, Nucl. Instr. and Meth. BI2 (1985) 24. [15] J. Bearden and A. Burr, Rev. Mod. Phys. 39 (1967) 125. [16] P.M. Echenique, R.M. Nieminen and R.H. Ritchie, Solid State Commun. 37 (1981) 779.
We acknowledge the valuable help of Mr. J. Winkler in the preparation and in the calibration of the
II. CAPTURE AND LOSS
Nuclear Instruments and Methods in Physics Research B67 (1992) 138-141 North-Holland
Sect,on B
The effective charge of He ions in metals Ch. Eppacher and D. Semrad hzstitut fiir Experimentalphysik,Johannes-Kepler-Unicersitiit,A-4040 Lbaz, Austria
Measurements of the stopping cross sections of Ge, Sn. and Pb for hydrogen and helium projectiles are presented. The projectile velocities range from the Bohr velocity c o to 2.5co. Large deviations from available tables of up to 28% were found. The measured ratio of helium and hydrogen stopping powers are compared to available predictions. Even for isoelectronic elements of the same group of the periodic system, no satisfactory agreement can be found.
1. Introduction
If a simple relation between the stopping cross sections fl~r protons and He ions could be found, the data base to produce stopping power tabulations fi)r both, protons and He ions, would be considerably increased. This could be done by introducing an effective charge of the projectile, which takes into account the screening of the projectile's charge by bound electrons. Let: us assume that we have a reliable theory to calculate the stopping cross section e = Eth(Zj~fr, Z 2, t h ) for a point charge Zl~tt moving at speed c I through an elemental target of charge number Z 2, then the effective charge of the projectile with charge number Z I would be implicitly given by
eth( Z,ejf(v,), Z2,
cl) =e.,( Zl, Z~, cl).
(1)
Here, t' I is the projectile's velocity and Em is the measuredl stopping cross section for this projectiletarget combination. Most of the available theories treat the stopping process as a perturbation to first order, which results in a scaling with the square of Z~ff, Eth(Zlcff(/-'l), Z2, i'l) : Z ~ c f f ( / ' l ) ~th(l, Z 2, t'l). (2) If we define the ratio R of e~ for He ions and protons of equal velocity normalized to 1 for large velocities, then we find, using eqs. (1) and (2),
g(v~)
era(2, Z2, t'l)
Z2etf(He)
4era(l, Zz, t,i)
4ZZ fr(p ) .
(3)
In the literature one finds the following statements concerning R: according to Ziegler, Biersaek and Littmark [l] R does not depend on the target material and is given by a universal curve R(v,)=l-exp
[
-i
a,[In(mn:,'f/2
,1] '
.
(4)
Here, mac denotes the mass of He in units which give the kinetic energy in keV. The coefficients a i are given by a0=0.2865, a1=0.1266, a2---0.001429, a3 = 0.02402, a 4 = -0.01135 and a 5 = 0.001475. According to the theories of Kreussicr, Varclas and Brandt [2] and of Brandt and Kitagawa [3], which are primarily intended for heavier ions, R depends on target properties via the Fermi velocity c F of their plasma electrons. Striektly speaking, R depends on the mean relative velocity t,r between the projectile and the electrons, which, assuming a quasi-free plasma, give r~
3( 2c~ c r = ~-rl.- 1 + 3t,F
c4 ) 15t,aF ,
(5) t' I < t' F.
As can be seen from eq. (5) for velocities L,I larger than twice the Fermi velocity, R is again found to be almost independent of the target material. To test both statements, we have started a series of reliable measurements on the stopping cross section for hydrogen and helium projectiles. As a first step, we present measurements of R for three elements of the fourth group of the periodic table, i.e. Ge, Sn, and Pb. The aims of this investigation is to clarify, whether for these elements one plasma parameter is sufficient to describe R in the velocity range from co to 2.5t' 0, where c 0 is the Bohr velocity. The three elements have a very similar electronic structure: in all cases roughly four electrons per atom form the plasma (see table 1). As the atomic densities of Sn and Pb are very similar their plasmon energies are very close to each other (table 1). Ge has a higher atomic density which leads to a higher plasmon energy. Due to the different number of core electrons, the stopping cross section of these elements differ from each other, predominantly at higher energies.
0168-583X/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
Ch. Eppacher, D. Semrad / Effectice charge of He ions bz metals
139
2. Measurements
3. Discussion of the results
We have performed absolute measurements of Em for protons, deuterons, and He ions in the energy range from 80 to 700 keV. The stopping cross sections were obtained from backscattering spectra. A detailed description of our measuring procedure can be found in refs. [7-9]. The targets were evaporated in a HVchamber at a pressure between 3 x 10 -8 and 1 x 10 -7 mbar, with liquid nitrogen cooled metal sheets surrounding the backing. The mass evaporated onto a well defined area was determined by weighing. For all three elements the measurements were performed on four targets, each with different mass density. The targets were tilted by 8 ° with respect to the incoming beam to avoid any influence of preferential orientation of crystal channels perpendicular to the target surface due to texture. Measurements were performed on different parts of the target to detect possible inequality in thickness. The evaluation procedure was carried out as described in ref. [10]. The error of our results (to be understood as one standard deviation) is about 3% at higher energies and about 5% at lower energies. The main contributions to the error come from: the areal mass density of the target (1.8%), the energy calibration of the ion beam (0.3%), the conversion of MCA channels to energy (1.5% at lower energies and 0.5% at higher energies), the determination of the center of mass of the high and low energy edge of the backscattering spectrum (0.5%), and at very low energies the increased energy loss due to multiple scattering [11] (up to 1%). These errors are added algebraically. As some of these errors cancel when calculating the ratio, we consider the error of R to be also 5% at low velocities and 3% at high velocities.
In figs. l a - l c we show the Em of Ge, Sn, and Pb for hydrogen projectiles. Each measurement is represented by two sections of a curve corresponding to the two energy intervals which the projectiles backscattered at the rear surface cover within the target [10]. The solid line shows the fit to the data. The fit function [10] used is Em(l,i)- I = ao+alE-~L4+a.,Ea.25. + a3t~r0s..
(6)
Here, the proton energy E has to be inserted in keV to get the stopping cross section in units of 10-15 e V c m 2. The parameters a0 through a 3 are listed in table 1. The broken line gives the result of Andersen and Ziegler [12] (A&Z). Fig, 1 also shows the calculated contribution of the plasma electrons (dotted line). These calculations are based upon the linear response theory by Lindhard and Winther [13]. The plasma density and the number of free electrons per atom were deduced [14] from measured plasmon energies, assuming the plasma to be free, and from the atomic density of the bulk material (see also table 1). Though for Ge (fig. la) the calculated plasma contribution appears too peaked and at too small energies, it can be seen that below 100 keV ~ is almost entirely given by the plasma. The plasma contribution determines also the position and the height of the maximum. Only above 200 keV there is a significant contribution from the core electrons, mainly from the 3d electrons. This is due to the fact, that these electrons are part of the completed third shell and, therefore, are more strongly bound [15] than the corresponding 4d electrons of Sn. Compared to A & Z , our measurements are lower by about 7% except for the very highest energies shown.
Table I Relevant data for Ge, Sn and Pb: the plasmon energies are taken from ref. [4] (GeL ref. [5] (Sn), and ref. [6] (Pb), respectively. All plasma parameters are calculated assuming a free plasma
Parameters for e Protons a o at a2 a3 He ions a0 a1 a2 a3 Electron configuration Plasmon energy [eV] Plasma density [a.u.] One electron radius [a.u.] Number of plasma electrons/atom Fermi velocity (calculated) [a.u.]
Ge
Sn
Pb
-0.12575 0.3 ! 305 0.037441 - 0.00014623 0.068177 0.0010195 - 0.016632 0.00015782 Ar3dla4s24p2 15.5 0.0258 2.10 3.93 0.914
0.013226 0.064638 - 0.0017580 0.00019534 -0.0013501 0.11218 - 0.00095374 0.000033388 Kr4dt°5s25p 2 13.7 0.0202 2.28 3.69 0.842
0.041374 0.010333 - 0.0090079 0.00023294 0.019328 0.044094 - 0.0048105 0.000051335 Xe 4ft45dl°6s26p2 13.3 0.0190 2.32 3.90 0.826 il. CAPTURE AND LOSS
140
I(a) .
.
Ch. Eppadwr, D. Semrad / Effectice charge of He ions in metals .
.
.
.
.
.
.
.
.
.
.
.
0.85
.
.
....
...................
za[ / ' i ~ " , , .,'i
"V.:.- "-.-...
0.50 0.40
Sn,Pb
16
0.35
.k"",<, .'" "°. """ "'"...
7-... . . . .
(c)
./
35
30
"L
-.,,
.-
.": /l/ "............... . """. -."//
.":'/i
"-.. ..
.
.
.
.
•
l.O
1.2
1.4
1.6
l.O
z.o
2.2
2.4
2.6
Ion Velocity [a.u.] Fig. 2. Normalized ratio of He to proton stopping power for Ge, Sn, and Pb. The broken curve shows the universal function given by Ziegler, Biersack and Littmark [1] (Z,B&L). Also shown are the low-velocity limits according to Echenique, Nieminen and Ritchie [16].
:(b) . . . . . . . . . . . . . . .
-". ......
.
Ge
~
18
i
.
0.55
Ge
i/
.
~. 0.60
0.45 - -
"0
.
""" Z,B&L
o.7o
"%",
.
.
Sn,
o.v5
ue
,
~
0.80
.. "-..
E n e r g y [keV]
Fig. !. Measured proton stopping cross section of (a) Ge (b) Sn and (c) Pb in units of 10- ts eV cm2. Also shown is the fit function by ,Andersen and Ziegler [12] (broken line) and the contribution from the plasma as calculated by the theol~ of Lindhard and Winther [13] (dotted line).
F o r Sn and even more so for Pb, the maximum of e is determined by the contributions from the core electrons (4d electrons in the case of Sn and 4f electrons for Pb). The plasma contribution influences the shape of the stopping power curve only at low velocities leading to a broad ridge. As there were not enough low energy data available to A & Z to show this trend, we find rather large discrepancies with respect to A & Z : - 8 % at 30 keV and + 1 2 % at 100 keV in fig. lb and - 2 8 % at 30 keV and + 7 % at 200 keV in fig. lc. The normalized ratio R of the stopping cross sections is shown in fig. 2 together with the universal curve by Ziegler, Biersack and Littmark [1]. We find agreement between our results for Sn and Pb due to their almost identical plasma. The ratio R for Ge is smaller in the whole velocity region and merges into the curves for Sn and Pb above 2.3v o. For velocities of the order of the Bohr velocity it seems to level off. The curve by Ziegler, Biersack and Littmark [1] fits the data fairly well, the maximum deviation being 8%. However, the fact that we get different curves even for elements of the same group of the periodic system indicates, that one cannot expect a universal curve for all targets. According to Kreussler et al. [2] one should draw R as a function of t,r rather than of c t. However, this would increase even further the discrepancy seen in fig. 2: as Ge has the larger Fermi velocity (table 1), R for Ge will be shifted towards higher values of v r with respect to R for Sn and Pb. The different behavior of R at low velocities can be understood from the results of Echenique, Nieminen and Ritchie [6]. In fig. 3 we show R for velocities much smaller than the Fermi velocity as calculated by the
Ch. Eppacher, 1). Semrad / Effectice charge of He ions in metals 0.70 ~
141
targets. This project was supported by the Fonds zur F6rderung der wissenschaftlichen Forschung under contract no. P 8149-PHY.
.......................................
0;0o\
References
°o:i.......... • 1.0
1.5
2.0
2.5 3.0 3.5 4.0 4.5 One gleetrolt Radius
5.0
5.5
Fig. 3. Normalized ratio in the limit of low velocities based upon density-functional calculations by Echenique, Nieminen and Ritchie [16]. density-functional formalism taken from fig. 1 of ref. [16]. The abscissa is the one electron radius r s. One notes that R for Sn and Pb should approach the value of 0.39 for v I tending to 0, whereas R for Ge should converge towards 0.45. This behavior seems to be in agreement with our results in fig. 2. In conclusion one can say, that neither existing statement on the "effective charge" of He ions can explain R in a wide range of velocities• Furthermore, it seems very unlikely, that a single plasma parameter like the Fermi velocity determines the shape of R. However, density-functional calculations seem to be a good basis to explain the behavior of R at least in the limit of low velocities.
Acknowledgements
[1] J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Ranges of Ions in Solids (Pergamon, Oxford, 1985) vol 1. [2] S. Kreussler, C. Varelas and W. Brandt, Phys. Rev. B23 (1981) 82. [3] W. Brandt and M. Kitagawa, Phys. Rev. B25 (1982) 5631. [4] O. Sueoka, J. Phys. Soc. Jpn. 20 (1965) 2203. [5] E.A. Bakulin, L. Balahanova, M.M. Bredow and E.V. Stepin, Soy. Phys. Solid State 16 (1975) 1670. [6] T. Aiyama and K. Yada, J. Phys. Soc. Jpn. 36 (1974) 1554. [7] D. Semrad, M. Geretschl~igerand A. Rabler, Nucl. Instr. and Meth. 215 (1983) 348. [8] D. Semrad, P. Mertens and P. Bauer, Nucl. Instr. and Meth. B15 (1986) 86. [9] P. Bauer, Nucl. instr, and Meth. B27 (1987) 301. [10] Ch. Eppacher and D. Semrad, Nucl. Instr. and Meth. B35 (1988) 109. [11] H. Paul, D. Semrad and A. Seilinger, Nucl. Instr. ,~nd Meth. B61 (1991) 261. [12] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter (Pergamon, New York, 1977) vol. 3. [13] J. Lindhard and A. Winther, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (4) (1964). [14] D. Semrad and P. Bauer, Nucl. Instr. and Meth. BI2 (1985) 24. [15] J. Bearden and A. Burr, Rev. Mod. Phys. 39 (1967) 125. [16] P.M. Echenique, R.M. Nieminen and R.H. Ritchie, Solid State Commun. 37 (1981) 779.
We acknowledge the valuable help of Mr. J. Winkler in the preparation and in the calibration of the
II. CAPTURE AND LOSS