- Email: [email protected]
The negative ionization of sputtered carbon atoms
Applied Surface Science 144–145 Ž1999. 208–211
The negative ionization of sputtered carbon atoms J.A.M.C. Silva ) , C.M.R. Henriques, O.M.N.D. Teodoro, A.M.C. Moutinho Departamento de Fısica, Faculdade de Ciencias e Tecnologia, UniÕersidade NoÕa de Lisboa, P-2825 Monte de Caparica, Portugal ´ ˆ
Abstract The formation and survival probability of negative carbon ions sputtered from a graphite sample was determined. The energy distribution of Cy ions was measured experimentally and the energy distribution of all sputtered carbon atoms was obtained from a computer simulation with the program TRIM.SP. From the two distributions, we obtained the probability that a carbon atom would capture an electron from the graphite and subsequently survive as it moves away from the surface. The results show two different situations. At low energy Žbelow 25 eV., the sputtered particles are all negatively ionized when the affinity level crosses the graphite Fermi level and afterwards, the population decay follows a semi-classical rate equation. At higher energy Žabove 25 eV., a velocity-dependent fraction reaches the crossing distance as neutral atoms. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Ionization; Carbon; Energy
1. Introduction The bombardment of a solid by energetic atoms or ions results in the emission of a variety of secondary particles from the surface. When a neutral atom is sputtered, its electron affinity increases due to the interaction of the affinity level with the surface and the capture of an electron by resonant charge transfer Ža one-electron process. from the surface to the atom to form a negative ion becomes possible. In order to calculate the formation and survival probabilities of the negative carbon ions sputtered from a graphite sample, we performed an experimental determination of their energy distribution and a computer simulation to obtain the theoretical energy distribution of all sputtered carbon atoms. )
Corresponding author. Tel.: q351-1-295-4464 ext. 0520; Fax: q351-1-294-8549; E-mail: [email protected]
2. Experimental The experiments were performed in a home-built multi-technique UHV system, which has been described in detail w1x. In this work, a 4.0 keV Arq beam was used to sputter a rigid graphite sample ŽGoodfellow C000430. at an angle of incidence of 418. The ion current density was 20 nA mmy2 . The secondary ions were mass selected by a triple filter quadruple mass spectrometer. Energy analysis was accomplished by a Bessel Box type filter w2x with 0.7 eV FWHM. A careful calibration of the energy filter was performed with Csq ions which were injected at constant current and variable energy in the filtering system. During the energy analysis of the sputtered Cy ions, the region in front of the sample was kept field-free. The presence of an extraction field in front of the sample would cause an energy-dependent
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 7 9 8 - 3
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
209
collection efficiency and a severe distortion of the energy spectrum. In this way, only the ions which are emitted along the surface normal will reach the first grid and be accepted by the filtering system.
3. Computer simulation The energy distribution of all carbon atoms sputtered from the graphite sample, along with its dependence on polar and azimuthal angle of emission, was obtained from a computer simulation. For that purpose, we used the Monte Carlo program TRIM.SP w3x. This program is based on the binary collision approximation, uses a randomized target structure and assumes that atoms leaving the sample have to overcome a planar surface barrier. For the elastic collisions between projectile and target atoms, the Krypton–Carbon potential was used. For the inelastic energy losses, an equipartition between the continuous Lindhard–Scharff w4x and the local Oen– Robinson w5x models were used. The primary beam energy Ž4.0 keV., incidence angle Ž418. and sample density Ž1.84 g cmy3 . were chosen to exactly match those of the experiment. As for the surface binding energy, the sublimation heat of graphite Ž7.41 eV. was used. A total of 10 7 projectiles have been simulated.
4. Charge transfer model The charge transfer process, which occurs when an atom or molecule is scattered by a metal surface, has been the subject of many theoretical and experimental studies w6,7x. Fig. 1 schematically shows the variation of the binding energy of a negative ion placed in front of a metal surface as a function of ion–surface distance. At large distances, the interaction of the negative ion with the surface is well represented by the interaction of the active electron with its image charge. The affinity level energy, Ea , varies as EaŽ Z . s EaŽ`. y Ž4Z .y1 , where Z is the distance between the atom and the image reference plane, causing an increase of the electron binding energy. As the distance decreases, the affinity level continues to shift downwards until it crosses the Fermi level at a distance
Fig. 1. Schematic representation of the variation of the affinity level, Ea as a function of the atom–surface distance, Z. As Z decreases, Ea shifts down in energy and acquires a width G . EF is the Fermi level and f the surface work function. For small Z, an electron may tunnel through the potential barrier separating the conduction band and the unoccupied affinity level to form a negative ion. For Z) Zc , the negative ion may decay into a neutral state by losing its electron to the conduction band.
Zc . For Z lower than Zc , the affinity level is degenerate with occupied metal states Žat T s 0 K. and thus, an electron can tunnel from the metal into the atom through the potential barrier separating the conduction band and the atomic level to form a negative ion. For Z greater than Zc , the affinity level is in front of unoccupied states Žat T s 0 K. of the metal valence band. The negative ion can decay into a neutral atom by electron transfer to the metal’s valence band. Graphite has a semi-metallic nature w8x. The density of states of single crystal graphite is very low near the Fermi level and rises sharply both above and below E F . In this work, the sample is a polycrystalline rigid graphite and, apart from the point defects which frequently occur in the production of graphite, sputtering creates vacancies and interstitial atoms. A study of the electronic structure of carbon atoms intercalated between graphite layers showed that, for a small concentration, the interstitials involve a large increase of the density of states near the graphite Fermi level w9x. It is, therefore, reasonable to consider that the electronic properties of our sample are closer to those of a metal than those of a semi-metal. The formation and destruction probability of a negative ion is determined via a semi-classical rate
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
210
equation describing the time evolution of the negative ion population Py w6x: d Py
s Ž 1 y Py . Gc y Py G l , Ž 1. dt where Gc and G l , the electron capture and loss rates, respectively, are related to the affinity level width, G.
5. Results and discussion It has been shown both experimentally and theoretically that the energy distribution of sputtered carbon atoms depends on both the polar and the azimuthal angles of emission w10x. For this reason, we extracted from the simulation results, the energy distribution corresponding to atoms ejected along the surface normal inside a cone of aperture 38, which corresponds approximately to the effective acceptance angle of the filtering system. This distribution was then convoluted with the energy window. The result is very similar to the original distribution due to the small width of the energy window. In linear cascade theory w11,12x, the energy distribution of sputtered atoms is given by: f Ž E. s
E
Ž E q Uo .
3q 2 m
.
Ž 2.
Applying Eq. Ž2. to the simulated distribution yields a correct fit with parameters Uo s 7.74 eV and 3 q 2 m s 2.93. The computed distribution, the fit performed with the linear cascade theory equation and the experimental negative ion energy distribution are plotted in Fig. 2. The main difference between the two distributions is the location of the maximum yield, at 4.0 eV for the simulation and around 15 eV for the experiment. The simulation results represent the energy distribution of all carbon atoms leaving the surface, independently of their charge state. As was discussed above, when a neutral atom is ejected from the surface, it can capture an electron by resonant charge transfer to form a negative ion, as long as its binding energy is greater than the surface work function. At T s 0 K and for Z - Zc , Eq. Ž1. reduces to d Pys Ž1 y Py. Gc d t. The metal states, being all occupied,
Fig. 2. Normalized energy distributions for carbon atoms Žboxes., obtained from a computer simulation, fitted with Eq. Ž2., and for negative carbon ions Žcircles. measured experimentally.
cannot accommodate an extra electron and a negative ion will not decay in the region Z - Zc . For Z ) Zc , the affinity level is degenerate with the unoccupied valence band states of the metal and there is a finite probability for the electron to transfer irreversibly from the ion to the metal. The rate equation is now d Pys yPyG l d t. Substituting d t s d ZrÕ H Ž Õ H is the velocity along the perpendicular to the surface, which, in the present case, coincides with the total velocity. and integrating from Zc to ` yields for the negative ion population: `
y yŽ1r Õ H .H G l d Z Pys PZs . Zc Z ce
Ž 3.
This equation predicts an exponential dependence of the negative ion population on Õ H Žwhich is assumed to be constant over the integration path.. Fig. 3Ža. shows the overall probability of formation and survival of a negative ion, obtained as the ratio between the measured negative ion energy distribution and the convolution of the fit to the computer simulation with the energy window. This probability shows an interesting feature: an exponential dependence on the inverse perpendicular velocity above 5 = 10y5 my1 s Žthis corresponds to energies below 25 eV.. This means that, below 25 eV, the negative ion population at Z s Zc is independent of the energy of the ions. The straight line in Fig. 3Ža. is a fit to the low energy region and represents the survival probability of the negative ions formed in the region Z - Zc as they go from Zc to `. This is the exponential term in Eq. Ž3..
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
211
close to the surface, the potential barrier separating the negative ion level and the valence band of graphite is small and the negative ion state has a short lifetime. Low energy ions spend a comparatively long time in the region where the electron transfer rate is high and thus, have a low survival probability. As the sputtered atom energy increases, so does its negative ion survival probability. 6. Conclusions
Fig. 3. Ža. Probability of formation and survival of a negative ion Žcircles. with fit Žsolid line. to the low energy region as a function of the inverse perpendicular velocity. Žb. Negative ion population Žsquares. at Zc .
The ratio between Py and the survival probability gives the negative ion population at Zc , PZyc , depicted in Fig. 3Žb.. This is also the formation probability of a negative ion since a sputtered atom can only become a negative ion by capturing an electron from the metal after being ejected from the surface. This is an interesting result, showing that, for E - 25 eV, all sputtered carbon atoms were negatively ionized during their path between the surface and Zc . For short atom–surface distances, the potential barrier separating atom and metal states is very small. The width of the affinity level can be of the order of 1 eV w7x; this corresponds to a very short lifetime and the charge transfer process is very fast. This way, negative ionization of sputtered carbon atoms moving slowly is highly efficient. As the energy of sputtered atoms increases, the time spent close to the surface decreases and the ionization efficiency drops below 1 for E ) 25 eV. After Zc , the negative ion population decays according to Eq. Ž3.. Since the affinity level crosses the Fermi level
The negative ionization of carbon atoms sputtered from a graphite sample was studied. The energy distribution of sputtered negative ions was determined experimentally and the energy distribution of all carbon atoms leaving the graphite surface was obtained theoretically, with the use of a computer simulation. From the two distributions, we obtained the probability that a sputtered neutral carbon atom would capture an electron from the graphite conduction band by resonant charge transfer and subsequently, survive neutralization. The calculated negative ion population at the distance where the affinity level crosses the Fermi level shows that, at Zc , all atoms with energy below 25 eV are negative ions. Above 25 eV, a velocity-dependent fraction remains neutral. The negative ion energy distribution and hence, its sputtering yield, was seen to be a function of resonant charge transfer between the surface and the sputtered atom. References w1x O.M.N.D. Teodoro, J.A.M.C. Silva, A.M.C. Moutinho, Vacuum 46 Ž1995. 1205. w2x J.H. Craig, J.L. Hock, J. Vac. Sci. Tech. 17 Ž1980. 1360. w3x J.P. Biersack, W. Eckstein, Appl. Phys. 34 Ž1984. 73. w4x J. Lindhard, M. Scharff, Phys. Rev. 124 Ž1961. 128. w5x O.S. Oen, M.T. Robinson, Nucl. Instr. Meth. 132 Ž1976. 647. w6x J. Los, J.J.C. Gerlings, Phys. Rep. 190 Ž1990. 133. w7x J.P. Gauyacq, A.G. Borisov, D. Teillet-Billy, in: V. Esaulov ŽEd.., Negative Ions, Cambridge Univ. Press, Cambridge, in press. w8x R.C. Tatar, S. Rabii, Phys. Rev. B 25 Ž1982. 4126. w9x C. Priester, G. Allan, J. Conrad, Phys. Rev. B 26 Ž1982. 4680. w10x W. Eckstein, E. Mashkova, Nucl. Instr. Meth. Phys. Res. B 62 Ž1992. 438. w11x M.W. Thompson, Philos. Mag. 18 Ž1968. 377. w12x P. Sigmund, Phys. Rev. 184 Ž1969. 383.
The negative ionization of sputtered carbon atoms J.A.M.C. Silva ) , C.M.R. Henriques, O.M.N.D. Teodoro, A.M.C. Moutinho Departamento de Fısica, Faculdade de Ciencias e Tecnologia, UniÕersidade NoÕa de Lisboa, P-2825 Monte de Caparica, Portugal ´ ˆ
Abstract The formation and survival probability of negative carbon ions sputtered from a graphite sample was determined. The energy distribution of Cy ions was measured experimentally and the energy distribution of all sputtered carbon atoms was obtained from a computer simulation with the program TRIM.SP. From the two distributions, we obtained the probability that a carbon atom would capture an electron from the graphite and subsequently survive as it moves away from the surface. The results show two different situations. At low energy Žbelow 25 eV., the sputtered particles are all negatively ionized when the affinity level crosses the graphite Fermi level and afterwards, the population decay follows a semi-classical rate equation. At higher energy Žabove 25 eV., a velocity-dependent fraction reaches the crossing distance as neutral atoms. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Ionization; Carbon; Energy
1. Introduction The bombardment of a solid by energetic atoms or ions results in the emission of a variety of secondary particles from the surface. When a neutral atom is sputtered, its electron affinity increases due to the interaction of the affinity level with the surface and the capture of an electron by resonant charge transfer Ža one-electron process. from the surface to the atom to form a negative ion becomes possible. In order to calculate the formation and survival probabilities of the negative carbon ions sputtered from a graphite sample, we performed an experimental determination of their energy distribution and a computer simulation to obtain the theoretical energy distribution of all sputtered carbon atoms. )
Corresponding author. Tel.: q351-1-295-4464 ext. 0520; Fax: q351-1-294-8549; E-mail: [email protected]
2. Experimental The experiments were performed in a home-built multi-technique UHV system, which has been described in detail w1x. In this work, a 4.0 keV Arq beam was used to sputter a rigid graphite sample ŽGoodfellow C000430. at an angle of incidence of 418. The ion current density was 20 nA mmy2 . The secondary ions were mass selected by a triple filter quadruple mass spectrometer. Energy analysis was accomplished by a Bessel Box type filter w2x with 0.7 eV FWHM. A careful calibration of the energy filter was performed with Csq ions which were injected at constant current and variable energy in the filtering system. During the energy analysis of the sputtered Cy ions, the region in front of the sample was kept field-free. The presence of an extraction field in front of the sample would cause an energy-dependent
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 7 9 8 - 3
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
209
collection efficiency and a severe distortion of the energy spectrum. In this way, only the ions which are emitted along the surface normal will reach the first grid and be accepted by the filtering system.
3. Computer simulation The energy distribution of all carbon atoms sputtered from the graphite sample, along with its dependence on polar and azimuthal angle of emission, was obtained from a computer simulation. For that purpose, we used the Monte Carlo program TRIM.SP w3x. This program is based on the binary collision approximation, uses a randomized target structure and assumes that atoms leaving the sample have to overcome a planar surface barrier. For the elastic collisions between projectile and target atoms, the Krypton–Carbon potential was used. For the inelastic energy losses, an equipartition between the continuous Lindhard–Scharff w4x and the local Oen– Robinson w5x models were used. The primary beam energy Ž4.0 keV., incidence angle Ž418. and sample density Ž1.84 g cmy3 . were chosen to exactly match those of the experiment. As for the surface binding energy, the sublimation heat of graphite Ž7.41 eV. was used. A total of 10 7 projectiles have been simulated.
4. Charge transfer model The charge transfer process, which occurs when an atom or molecule is scattered by a metal surface, has been the subject of many theoretical and experimental studies w6,7x. Fig. 1 schematically shows the variation of the binding energy of a negative ion placed in front of a metal surface as a function of ion–surface distance. At large distances, the interaction of the negative ion with the surface is well represented by the interaction of the active electron with its image charge. The affinity level energy, Ea , varies as EaŽ Z . s EaŽ`. y Ž4Z .y1 , where Z is the distance between the atom and the image reference plane, causing an increase of the electron binding energy. As the distance decreases, the affinity level continues to shift downwards until it crosses the Fermi level at a distance
Fig. 1. Schematic representation of the variation of the affinity level, Ea as a function of the atom–surface distance, Z. As Z decreases, Ea shifts down in energy and acquires a width G . EF is the Fermi level and f the surface work function. For small Z, an electron may tunnel through the potential barrier separating the conduction band and the unoccupied affinity level to form a negative ion. For Z) Zc , the negative ion may decay into a neutral state by losing its electron to the conduction band.
Zc . For Z lower than Zc , the affinity level is degenerate with occupied metal states Žat T s 0 K. and thus, an electron can tunnel from the metal into the atom through the potential barrier separating the conduction band and the atomic level to form a negative ion. For Z greater than Zc , the affinity level is in front of unoccupied states Žat T s 0 K. of the metal valence band. The negative ion can decay into a neutral atom by electron transfer to the metal’s valence band. Graphite has a semi-metallic nature w8x. The density of states of single crystal graphite is very low near the Fermi level and rises sharply both above and below E F . In this work, the sample is a polycrystalline rigid graphite and, apart from the point defects which frequently occur in the production of graphite, sputtering creates vacancies and interstitial atoms. A study of the electronic structure of carbon atoms intercalated between graphite layers showed that, for a small concentration, the interstitials involve a large increase of the density of states near the graphite Fermi level w9x. It is, therefore, reasonable to consider that the electronic properties of our sample are closer to those of a metal than those of a semi-metal. The formation and destruction probability of a negative ion is determined via a semi-classical rate
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
210
equation describing the time evolution of the negative ion population Py w6x: d Py
s Ž 1 y Py . Gc y Py G l , Ž 1. dt where Gc and G l , the electron capture and loss rates, respectively, are related to the affinity level width, G.
5. Results and discussion It has been shown both experimentally and theoretically that the energy distribution of sputtered carbon atoms depends on both the polar and the azimuthal angles of emission w10x. For this reason, we extracted from the simulation results, the energy distribution corresponding to atoms ejected along the surface normal inside a cone of aperture 38, which corresponds approximately to the effective acceptance angle of the filtering system. This distribution was then convoluted with the energy window. The result is very similar to the original distribution due to the small width of the energy window. In linear cascade theory w11,12x, the energy distribution of sputtered atoms is given by: f Ž E. s
E
Ž E q Uo .
3q 2 m
.
Ž 2.
Applying Eq. Ž2. to the simulated distribution yields a correct fit with parameters Uo s 7.74 eV and 3 q 2 m s 2.93. The computed distribution, the fit performed with the linear cascade theory equation and the experimental negative ion energy distribution are plotted in Fig. 2. The main difference between the two distributions is the location of the maximum yield, at 4.0 eV for the simulation and around 15 eV for the experiment. The simulation results represent the energy distribution of all carbon atoms leaving the surface, independently of their charge state. As was discussed above, when a neutral atom is ejected from the surface, it can capture an electron by resonant charge transfer to form a negative ion, as long as its binding energy is greater than the surface work function. At T s 0 K and for Z - Zc , Eq. Ž1. reduces to d Pys Ž1 y Py. Gc d t. The metal states, being all occupied,
Fig. 2. Normalized energy distributions for carbon atoms Žboxes., obtained from a computer simulation, fitted with Eq. Ž2., and for negative carbon ions Žcircles. measured experimentally.
cannot accommodate an extra electron and a negative ion will not decay in the region Z - Zc . For Z ) Zc , the affinity level is degenerate with the unoccupied valence band states of the metal and there is a finite probability for the electron to transfer irreversibly from the ion to the metal. The rate equation is now d Pys yPyG l d t. Substituting d t s d ZrÕ H Ž Õ H is the velocity along the perpendicular to the surface, which, in the present case, coincides with the total velocity. and integrating from Zc to ` yields for the negative ion population: `
y yŽ1r Õ H .H G l d Z Pys PZs . Zc Z ce
Ž 3.
This equation predicts an exponential dependence of the negative ion population on Õ H Žwhich is assumed to be constant over the integration path.. Fig. 3Ža. shows the overall probability of formation and survival of a negative ion, obtained as the ratio between the measured negative ion energy distribution and the convolution of the fit to the computer simulation with the energy window. This probability shows an interesting feature: an exponential dependence on the inverse perpendicular velocity above 5 = 10y5 my1 s Žthis corresponds to energies below 25 eV.. This means that, below 25 eV, the negative ion population at Z s Zc is independent of the energy of the ions. The straight line in Fig. 3Ža. is a fit to the low energy region and represents the survival probability of the negative ions formed in the region Z - Zc as they go from Zc to `. This is the exponential term in Eq. Ž3..
J.A.M.C. SilÕa et al.r Applied Surface Science 144–145 (1999) 208–211
211
close to the surface, the potential barrier separating the negative ion level and the valence band of graphite is small and the negative ion state has a short lifetime. Low energy ions spend a comparatively long time in the region where the electron transfer rate is high and thus, have a low survival probability. As the sputtered atom energy increases, so does its negative ion survival probability. 6. Conclusions
Fig. 3. Ža. Probability of formation and survival of a negative ion Žcircles. with fit Žsolid line. to the low energy region as a function of the inverse perpendicular velocity. Žb. Negative ion population Žsquares. at Zc .
The ratio between Py and the survival probability gives the negative ion population at Zc , PZyc , depicted in Fig. 3Žb.. This is also the formation probability of a negative ion since a sputtered atom can only become a negative ion by capturing an electron from the metal after being ejected from the surface. This is an interesting result, showing that, for E - 25 eV, all sputtered carbon atoms were negatively ionized during their path between the surface and Zc . For short atom–surface distances, the potential barrier separating atom and metal states is very small. The width of the affinity level can be of the order of 1 eV w7x; this corresponds to a very short lifetime and the charge transfer process is very fast. This way, negative ionization of sputtered carbon atoms moving slowly is highly efficient. As the energy of sputtered atoms increases, the time spent close to the surface decreases and the ionization efficiency drops below 1 for E ) 25 eV. After Zc , the negative ion population decays according to Eq. Ž3.. Since the affinity level crosses the Fermi level
The negative ionization of carbon atoms sputtered from a graphite sample was studied. The energy distribution of sputtered negative ions was determined experimentally and the energy distribution of all carbon atoms leaving the graphite surface was obtained theoretically, with the use of a computer simulation. From the two distributions, we obtained the probability that a sputtered neutral carbon atom would capture an electron from the graphite conduction band by resonant charge transfer and subsequently, survive neutralization. The calculated negative ion population at the distance where the affinity level crosses the Fermi level shows that, at Zc , all atoms with energy below 25 eV are negative ions. Above 25 eV, a velocity-dependent fraction remains neutral. The negative ion energy distribution and hence, its sputtering yield, was seen to be a function of resonant charge transfer between the surface and the sputtered atom. References w1x O.M.N.D. Teodoro, J.A.M.C. Silva, A.M.C. Moutinho, Vacuum 46 Ž1995. 1205. w2x J.H. Craig, J.L. Hock, J. Vac. Sci. Tech. 17 Ž1980. 1360. w3x J.P. Biersack, W. Eckstein, Appl. Phys. 34 Ž1984. 73. w4x J. Lindhard, M. Scharff, Phys. Rev. 124 Ž1961. 128. w5x O.S. Oen, M.T. Robinson, Nucl. Instr. Meth. 132 Ž1976. 647. w6x J. Los, J.J.C. Gerlings, Phys. Rep. 190 Ž1990. 133. w7x J.P. Gauyacq, A.G. Borisov, D. Teillet-Billy, in: V. Esaulov ŽEd.., Negative Ions, Cambridge Univ. Press, Cambridge, in press. w8x R.C. Tatar, S. Rabii, Phys. Rev. B 25 Ž1982. 4126. w9x C. Priester, G. Allan, J. Conrad, Phys. Rev. B 26 Ž1982. 4680. w10x W. Eckstein, E. Mashkova, Nucl. Instr. Meth. Phys. Res. B 62 Ž1992. 438. w11x M.W. Thompson, Philos. Mag. 18 Ž1968. 377. w12x P. Sigmund, Phys. Rev. 184 Ž1969. 383.