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Note on stopping power and statistics of particle penetration
.__ __ LB
cm
Nuclear Instruments and Methods in Physics Research B 95 (1995)
477-479
Beam Interactions with Materials&Atoms
ELSEVIER
Note on stopping power and statistics of particle penetration Peter Sigmund Received 23 August 1994
In a recent publication in this journal, Trickey et al. [ 1] express their views on fundamental aspects of particle penetration, in particular the central concept of stopping power. These views, they assert, are at variance with common definitions and descriptions. While possible direct implications of these assertions on the practice of measurements and calculations in the area were not really spelled out, the paper may well leave the reader with the impression of a lack of rigor in the fundamentals of particle penetration. Such a handicap, if correct, would be unfortunate in view of the large effort spent by numerous groups in measurements and calculations, some of which aim at high precision. A major development in the fundamental concepts may of course occur at any instant in any field of science. However, the most fundamental questions raised by Trickey and coworkers have either well-defined answers within established knowledge or have at least been asked many times before. The response given below to several central points does not invoke novel arguments and will therefore be kept brief and supported by pertinent references. The main problem concerns the conventional definition of stopping power, or energy loss per pathlength, -g
=lim E, X=0 x
(1)
It is said [ 1 ] that for this concept to be applicable, the statistical distribution of energy loss AE should need to be narrow and symmetric, the thickness x be a continuous parameter and the limiting process in Eq. (1) meaningful. It is then asserted that the length scale specifying x in Eq. ( 1) should be intrinsically quanta1 and it is proposed, therefore, to specify the thickness of a thin foil via the spatial extension of its electron cloud. In this manner, Trickey and coworkers arrive at the following alternative definition of stopping power, dE --_= dx
(A&r(v))) r(v)
’
(2)
where r(v) is an effective thickness, defined via the extent of the electron cloud, of a thin solid foil made up by v monolayers, and (AE( t(v))) the mean energy loss to the foil. Because of surface effects, r(y) will only become proportional to Y in the limit of large ZJ, i.e., for fairly thick foils. From their definition, the authors arrive, therefore, at 0168-583X/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSD10168-583X(94)00852-3
the conclusion that the limit x = 0 in Eq. ( 1) should be replaced by something close to the opposite. In most experimental techniques applied in particle stopping, target thickness enters in terms of atoms/area or weight/area. This is independent of the stopping mechanism, i.e. whether stopping is due to processes involving target or projectile electrons, target nuclei, or radiation. Thus, it is the number of nuclei per area, and not the electron cloud, that specifies the thickness. If target thickness were defined in accordance with Eq. (2)) its precise value would depend on the stopping mechanism under investigation, as well as on the type, energy, and direction of the incident beam, in addition to the problem of defining a monolayer. Consequently, target thickness would be both ambiguous and hard to measure. Conversely, weight per area is measured directly by weighing, the error being caused by and large by the measurement of the area [2]. Alternatively. the number of atoms per area can be determined by ion beam techniques such as Rutherford backscattering. Some techniques in particle stopping, such as the Doppler shift attenuation technique [ 31, record the rate of energy loss per unit time and thus circumvent target thickness altogether. The stopping power is related to the mean energy loss but otherwise unrelated to the detailed shape of the spectrum [ 41. The latter is important in attempts to relate the mean energy loss to the most probable energy loss, a point of frequent interest [ 51 but not addressedin the present context. In particular, the spectrum does not need to be narrow or symmetric for the stopping power to be a well-defined, meaningful, measurable, and calculable quantity. Indeed, for thin targets, under conditions where particles undergo at most one noticeable collision event, the energy loss spectrum is Rutherford-like, i.e., far from narrow or symmetric [ 61. Ignoring this feature when significant would be a more severe limitation than ignoring the spatial extension of an electron cloud beyond the target surface. It is convenient to divide up the following discussion into dilute and dense targets. For a gaseous target, the mean energy loss by a beam particle to a layer is given by [4] (AE) = NxS,
(3)
where Nx is the mean number of target atoms per area (N being the number density) and S, the stopping cross section,
an atomic parameter. The following essential assumptions underly this formula: (i) the state of the projectile does not change during passage, (ii) the loss of energy does not cause noticeable variations in the stopping cross section across the foil, and (iii) the increase in travelled pathlength through the layer due to angular deflection is insignificant. All three assumptions imply an upper limit on target thickness N.x. A theoretical scheme avoiding the first limitation in rigorous terms has been found recently [ 4,7 1. The second and third limitation are overcome within conventional transport theory [4]. For a multicomponent target, NxS would have to be replaced by xi N;xSi, with i denoting the species present. For an inhomogeneous target, this would amplify into c, so* Ni(x’) dx’ S,. It is noted in particular that Eq. (3) does not invoke a limiting step toward x = 0. At the same time, Eq. (3) remains valid even for very thin targets, far below the monolayer limit N lOI atoms/cm* discussed in Ref. [ I]. Thus, the limiting step x = 0 in Eq. ( 1) ensures small enough target thickness to make effects like (i) - (iii) mentioned above insignificant, in case this should be desirable. This is consistent with the notion in Ref. [ 11 of x = 0 being a limit on a macroscopic scale. For stopping due to electronic excitation of the target, the stopping cross section can be written in the form S=
Cqgj,
(4)
where j denotes the excitation levels of a target atom or molecule, Tj the excitation energy, and cj the cross section associated with a transition from the ground state. Notations expressing the right-hand side by an integral are equivalent. This description, valid for a gas but already from the early days of particle penetration designed for dense targets too, is in fact more general. Take first the case where the pertinent events can be split into interactions between the projectile and one target atom at a time. Then, very little change occurs when Eqs. (3) and (4) are applied to a solid film. Indeed, if the trajectory does not undergo systematic changes due to deflection from the atoms in a crystal lattice, i.e., if the motion is ungoverned [8], both relations remain valid as they stand [ 4,8,9]. Systematic changes in mean energy loss in case of governed motion (e.g. channeling), interesting as they may be, are not central to the discussion in Ref. [ I] or the present comment. The interaction between a swift projectile and the electrons of a solid target extends over several interatomic distances, dependent on projectile speed, and thus typically comprises a large number of electrons at any time. The cumulative effect is characterized by a superposition of momentum transfers due to interaction between the dynamically screened projectile charge and an individual target electron. In addition, collective degrees of freedom (plasma modes) may be
excited. Therefore, the excitation spectrum or the mean energy transfer for two monolayers does not necessarily resemble twice these quantities for one monolayer. The authors of Ref. [ I ] try to address this important feature, but mixing it up with a discussion of effective target thickness blurs the essential issue. Moreover, careful distinction would be desirable between the energy loss to an isolated thin layer and the differential change in energy loss when a similar layer is deposited on a substrate. The conventional description of stopping in a dense medium [ IO] concerns a stationary state characterized by a projectile moving uniformly through a target with fairly general properties but infinite spatial dimensions. Actual calculations, with few exceptions (e.g. Ref. [ 111)) usually refer to a free electron gas. It has long been known that this description needs to be expanded in the presence of a surface [ 121, and this holds true even more for penetration through films in the nanometer range [ 131. Target excitation while the projectile is still (or again) moving in vacuum needs to be considered, in particular for oblique incidence, as is evident from numerous current experimental and theoretical studies [ 141. It is not obvious whether a theoretical treatment of such effects is within the range of applicability of local-density approximations of the type advocated in Ref. [ I 1. And in any case, surface effects of this kind must be far more pronounced than the effective thickness entering Eq. (2). A final remark is triggered by the attention given in Ref. [ 1] and elsewhere [ 151 to the problem of energy loss to films in the monolayer range, Free, self-supporting films have been produced with thicknesses of few nanometers [ 161. Energy loss of ions penetrating such films may be interesting, but it appears both desirable and more convenient to concentrate on energy loss spectra as opposed to stopping power. The latter draws vita1 contributions from high excitations which have low intensity and are consequently hard to measure. From a computational point of view, a thin-layer spectrum is easier to evaluate than a mean energy loss since the bulk of the spectrum is dominated by frequent, low excitations while the tail. well approximated by Rutherford’s law, may be extremely hard to compute by the tools of quantum chemistry since numerous high excitations are involved [ 171. From an applications point of view, monolayer stopping is of interest for adsorbates, oxide coverages and alike. Stopping powers tell very little in this context while the detailed analysis of spectra provides interesting information
1181.
References
[I] S.B. Trickey, J.Z. Wu. and J.R. &bin, Nucl. Instr. and Meth B 93 (1994) 186. [ 7,) H.H. Andersen, Studies of Atomic Collisions in Solids by Means of Calorimetric Techniques, Thesis, University of Aarhus ( 1974). 131 W. Pietsch, U. Hauser, and W. Neuwirth. Nucl. Instr. and Meth. 132 f 1976) 19.
P. Sigmund/Nucl. [ 41 P. Sigmund,
in: Interaction
of Charged Particles
Instr. and Meth. in Phys. Res. B 95 (1995) 477-479
with Solids
and
Surfaces, eds. A. Gras-Marti et al. NATO AS1 Series B 271 (1991) 73. 1.51 P. Sigmund and K.B. Winterbon, Nucl. Ins@. and Meth. B 12 (1985) 161 L. Landau, J. Phys. USSR 8 ( 1944) 201. 171 P Sigmund, Nucl. Instr. and Meth. B 69 ( 1992) 113;Phys. Rev. A 50 ([994) 3197. [ 8 I J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (14) ( 1965)
[ 1 I] [ 121 [ 131 [ 141
[ 151 [ 161
[ 171 19 1 P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40 (5) ( 1978) I.
10 I J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28 (8) ( 1954)
[ 181
479
P. Sigmund and A. Belkacem, Nucl. Instr. and Meth. B 48 ( 1990) 29. R.H. Ritchie, Phys. Rev. 106 (1957) 874. CC. Sung and R.H. Ritchie, J. Phys. C 14 (1991)
2409.
See, e.g., several papers in Nucl. Instr. and Meth. B 90 ( 1994) Section III. J.Z. Wu, S.B. Trickey, and J.R. Sabin, Int. J. Quantum Chem. Symp. 27 (1993) 219. G. Both, E.P. Kanter, Z. Vager, B.J. Zabransky. and D. Zajfman. Rev. Sci. Instr. 58 (1987) 424. E.H. Mortensen, J. Oddershede, and J.R. Sabin, Nucl. Ins&. and Meth. B 69 ( 1992) 24. I.C. Vi&ridge and G. Amsel, Nucl. Ins% and Meth. B 85 ( 1994) 566.
cm
Nuclear Instruments and Methods in Physics Research B 95 (1995)
477-479
Beam Interactions with Materials&Atoms
ELSEVIER
Note on stopping power and statistics of particle penetration Peter Sigmund Received 23 August 1994
In a recent publication in this journal, Trickey et al. [ 1] express their views on fundamental aspects of particle penetration, in particular the central concept of stopping power. These views, they assert, are at variance with common definitions and descriptions. While possible direct implications of these assertions on the practice of measurements and calculations in the area were not really spelled out, the paper may well leave the reader with the impression of a lack of rigor in the fundamentals of particle penetration. Such a handicap, if correct, would be unfortunate in view of the large effort spent by numerous groups in measurements and calculations, some of which aim at high precision. A major development in the fundamental concepts may of course occur at any instant in any field of science. However, the most fundamental questions raised by Trickey and coworkers have either well-defined answers within established knowledge or have at least been asked many times before. The response given below to several central points does not invoke novel arguments and will therefore be kept brief and supported by pertinent references. The main problem concerns the conventional definition of stopping power, or energy loss per pathlength, -g
=lim E, X=0 x
(1)
It is said [ 1 ] that for this concept to be applicable, the statistical distribution of energy loss AE should need to be narrow and symmetric, the thickness x be a continuous parameter and the limiting process in Eq. (1) meaningful. It is then asserted that the length scale specifying x in Eq. ( 1) should be intrinsically quanta1 and it is proposed, therefore, to specify the thickness of a thin foil via the spatial extension of its electron cloud. In this manner, Trickey and coworkers arrive at the following alternative definition of stopping power, dE --_= dx
(A&r(v))) r(v)
’
(2)
where r(v) is an effective thickness, defined via the extent of the electron cloud, of a thin solid foil made up by v monolayers, and (AE( t(v))) the mean energy loss to the foil. Because of surface effects, r(y) will only become proportional to Y in the limit of large ZJ, i.e., for fairly thick foils. From their definition, the authors arrive, therefore, at 0168-583X/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSD10168-583X(94)00852-3
the conclusion that the limit x = 0 in Eq. ( 1) should be replaced by something close to the opposite. In most experimental techniques applied in particle stopping, target thickness enters in terms of atoms/area or weight/area. This is independent of the stopping mechanism, i.e. whether stopping is due to processes involving target or projectile electrons, target nuclei, or radiation. Thus, it is the number of nuclei per area, and not the electron cloud, that specifies the thickness. If target thickness were defined in accordance with Eq. (2)) its precise value would depend on the stopping mechanism under investigation, as well as on the type, energy, and direction of the incident beam, in addition to the problem of defining a monolayer. Consequently, target thickness would be both ambiguous and hard to measure. Conversely, weight per area is measured directly by weighing, the error being caused by and large by the measurement of the area [2]. Alternatively. the number of atoms per area can be determined by ion beam techniques such as Rutherford backscattering. Some techniques in particle stopping, such as the Doppler shift attenuation technique [ 31, record the rate of energy loss per unit time and thus circumvent target thickness altogether. The stopping power is related to the mean energy loss but otherwise unrelated to the detailed shape of the spectrum [ 41. The latter is important in attempts to relate the mean energy loss to the most probable energy loss, a point of frequent interest [ 51 but not addressedin the present context. In particular, the spectrum does not need to be narrow or symmetric for the stopping power to be a well-defined, meaningful, measurable, and calculable quantity. Indeed, for thin targets, under conditions where particles undergo at most one noticeable collision event, the energy loss spectrum is Rutherford-like, i.e., far from narrow or symmetric [ 61. Ignoring this feature when significant would be a more severe limitation than ignoring the spatial extension of an electron cloud beyond the target surface. It is convenient to divide up the following discussion into dilute and dense targets. For a gaseous target, the mean energy loss by a beam particle to a layer is given by [4] (AE) = NxS,
(3)
where Nx is the mean number of target atoms per area (N being the number density) and S, the stopping cross section,
an atomic parameter. The following essential assumptions underly this formula: (i) the state of the projectile does not change during passage, (ii) the loss of energy does not cause noticeable variations in the stopping cross section across the foil, and (iii) the increase in travelled pathlength through the layer due to angular deflection is insignificant. All three assumptions imply an upper limit on target thickness N.x. A theoretical scheme avoiding the first limitation in rigorous terms has been found recently [ 4,7 1. The second and third limitation are overcome within conventional transport theory [4]. For a multicomponent target, NxS would have to be replaced by xi N;xSi, with i denoting the species present. For an inhomogeneous target, this would amplify into c, so* Ni(x’) dx’ S,. It is noted in particular that Eq. (3) does not invoke a limiting step toward x = 0. At the same time, Eq. (3) remains valid even for very thin targets, far below the monolayer limit N lOI atoms/cm* discussed in Ref. [ I]. Thus, the limiting step x = 0 in Eq. ( 1) ensures small enough target thickness to make effects like (i) - (iii) mentioned above insignificant, in case this should be desirable. This is consistent with the notion in Ref. [ 11 of x = 0 being a limit on a macroscopic scale. For stopping due to electronic excitation of the target, the stopping cross section can be written in the form S=
Cqgj,
(4)
where j denotes the excitation levels of a target atom or molecule, Tj the excitation energy, and cj the cross section associated with a transition from the ground state. Notations expressing the right-hand side by an integral are equivalent. This description, valid for a gas but already from the early days of particle penetration designed for dense targets too, is in fact more general. Take first the case where the pertinent events can be split into interactions between the projectile and one target atom at a time. Then, very little change occurs when Eqs. (3) and (4) are applied to a solid film. Indeed, if the trajectory does not undergo systematic changes due to deflection from the atoms in a crystal lattice, i.e., if the motion is ungoverned [8], both relations remain valid as they stand [ 4,8,9]. Systematic changes in mean energy loss in case of governed motion (e.g. channeling), interesting as they may be, are not central to the discussion in Ref. [ I] or the present comment. The interaction between a swift projectile and the electrons of a solid target extends over several interatomic distances, dependent on projectile speed, and thus typically comprises a large number of electrons at any time. The cumulative effect is characterized by a superposition of momentum transfers due to interaction between the dynamically screened projectile charge and an individual target electron. In addition, collective degrees of freedom (plasma modes) may be
excited. Therefore, the excitation spectrum or the mean energy transfer for two monolayers does not necessarily resemble twice these quantities for one monolayer. The authors of Ref. [ I ] try to address this important feature, but mixing it up with a discussion of effective target thickness blurs the essential issue. Moreover, careful distinction would be desirable between the energy loss to an isolated thin layer and the differential change in energy loss when a similar layer is deposited on a substrate. The conventional description of stopping in a dense medium [ IO] concerns a stationary state characterized by a projectile moving uniformly through a target with fairly general properties but infinite spatial dimensions. Actual calculations, with few exceptions (e.g. Ref. [ 111)) usually refer to a free electron gas. It has long been known that this description needs to be expanded in the presence of a surface [ 121, and this holds true even more for penetration through films in the nanometer range [ 131. Target excitation while the projectile is still (or again) moving in vacuum needs to be considered, in particular for oblique incidence, as is evident from numerous current experimental and theoretical studies [ 141. It is not obvious whether a theoretical treatment of such effects is within the range of applicability of local-density approximations of the type advocated in Ref. [ I 1. And in any case, surface effects of this kind must be far more pronounced than the effective thickness entering Eq. (2). A final remark is triggered by the attention given in Ref. [ 1] and elsewhere [ 151 to the problem of energy loss to films in the monolayer range, Free, self-supporting films have been produced with thicknesses of few nanometers [ 161. Energy loss of ions penetrating such films may be interesting, but it appears both desirable and more convenient to concentrate on energy loss spectra as opposed to stopping power. The latter draws vita1 contributions from high excitations which have low intensity and are consequently hard to measure. From a computational point of view, a thin-layer spectrum is easier to evaluate than a mean energy loss since the bulk of the spectrum is dominated by frequent, low excitations while the tail. well approximated by Rutherford’s law, may be extremely hard to compute by the tools of quantum chemistry since numerous high excitations are involved [ 171. From an applications point of view, monolayer stopping is of interest for adsorbates, oxide coverages and alike. Stopping powers tell very little in this context while the detailed analysis of spectra provides interesting information
1181.
References
[I] S.B. Trickey, J.Z. Wu. and J.R. &bin, Nucl. Instr. and Meth B 93 (1994) 186. [ 7,) H.H. Andersen, Studies of Atomic Collisions in Solids by Means of Calorimetric Techniques, Thesis, University of Aarhus ( 1974). 131 W. Pietsch, U. Hauser, and W. Neuwirth. Nucl. Instr. and Meth. 132 f 1976) 19.
P. Sigmund/Nucl. [ 41 P. Sigmund,
in: Interaction
of Charged Particles
Instr. and Meth. in Phys. Res. B 95 (1995) 477-479
with Solids
and
Surfaces, eds. A. Gras-Marti et al. NATO AS1 Series B 271 (1991) 73. 1.51 P. Sigmund and K.B. Winterbon, Nucl. Ins@. and Meth. B 12 (1985) 161 L. Landau, J. Phys. USSR 8 ( 1944) 201. 171 P Sigmund, Nucl. Instr. and Meth. B 69 ( 1992) 113;Phys. Rev. A 50 ([994) 3197. [ 8 I J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (14) ( 1965)
[ 1 I] [ 121 [ 131 [ 141
[ 151 [ 161
[ 171 19 1 P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40 (5) ( 1978) I.
10 I J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28 (8) ( 1954)
[ 181
479
P. Sigmund and A. Belkacem, Nucl. Instr. and Meth. B 48 ( 1990) 29. R.H. Ritchie, Phys. Rev. 106 (1957) 874. CC. Sung and R.H. Ritchie, J. Phys. C 14 (1991)
2409.
See, e.g., several papers in Nucl. Instr. and Meth. B 90 ( 1994) Section III. J.Z. Wu, S.B. Trickey, and J.R. Sabin, Int. J. Quantum Chem. Symp. 27 (1993) 219. G. Both, E.P. Kanter, Z. Vager, B.J. Zabransky. and D. Zajfman. Rev. Sci. Instr. 58 (1987) 424. E.H. Mortensen, J. Oddershede, and J.R. Sabin, Nucl. Ins&. and Meth. B 69 ( 1992) 24. I.C. Vi&ridge and G. Amsel, Nucl. Ins% and Meth. B 85 ( 1994) 566.