- Email: [email protected]
The influence of target composition on the specific energy loss measured in transmission geometry
Nuclear Instruments and Methods in Physics Research
NlllMlB
B 90 (1994) 45-48
North-Holland
Beam Interactions with Materials&Atoms
The influence of target composition measured in transmission geometry Robin Golser *, Andreas Schiefermiiller Johannes-Kepler-Universitiit,
Institut fiir fiperimentalphysik,
on the specific energy loss
and Dieter Semrad A-4040 Linz, Austria
At low projectile velocities, stopping cross section measurements by transmission methods with high directional correlation between incoming beam and detected projectiles and with a detector acceptance appreciably smaller than 4a, may suffer from experimental artifacts. Thickness inhomogeneities make the measured specific energy loss appear angle dependent. The influence of the target area1 density may be traced to the impact parameter dependence of both, the elastic and the inelastic stopping cross section. In this contribution we deal with a mixture of gases. A “true” deviation from Bragg’s rule may be due to charge changing processes. Here, we discuss an additional but only apparent violation of Bragg’s rule in gas mixtures: due to impact parameter selection by the experimental geometry, target atoms with different atomic numbers may contribute differently to the measured specific energy loss. Our arguments are based on experimental results and on Monte Carlo simulations.
1. Introduction
The stopping E =
cross section
E is defined
by
CTiUi, i
where ui is the cross section for a process leading to an energy transfer T, between projectile and target. Unfortunately, one can determine E according to Eq. (1) only in a very limited number of cases. If this is not possible, one has to look on stopping [l] as a penetration phenomenon rather than a collision phenomenon (Eq. (1)). It is well known, that E can be made accessible to experiment by introducing the probability Api for the ith process to happen. For NAxui +z 1 we get Api = NAxui; this allows one to define a mean energy (AT) transferred from the projectile to the target within a thin layer of area1 density NAx: (AT)
= NiXm cNA~~~~~=e,$rn
aNAx.
(2)
i
The stopping cross section can now be obtained [2] from (AT), or, as all the energy deposited in the target comes from the projectile kinetic energy, E can be obtained from the mean energy (AE) lost by the projectiles.
* Corresponding author, tel. +43 732 2468 510, fax +43 732 2468 9677, e-mail k342070@edvz,uni-linz.ac.at. 0168-583X/94/$07.00 0 1994 - Elsevier SSDI 0168-583X(93)E0661-Y
Science
However, when one measures (A E) problems might arise. To use Eq. (l), experimental conditions suitable for measuring the (total) cross sections oi are needed, e.g. the outgoing projectiles should not be selected according to their scattering angle (Y. But for measuring (AE) one necessarily does select projectiles, i.e. those which enter the detector. So in most experiments one determines a specific energy loss lo rather than the “true” stopping cross section E, where ln is given by
Eo = -
lim
(AE)o
NAx+O
NAx
(3)
’
The average ( ) is taken over the restricted sample of projectiles selected by the acceptance 0 of the detector; obviously, ln+, = E and - (AE)oc4= = (AT). en corresponds to angle-restricted stopping cross section defined in ICRU49 [3]. Due to the impact parameter dependence of electronic stopping, a difference between E and lo arises since the detected projectiles have probed different domains of the target atoms with different probabilities. For instance, in a transmission experiment with small detector solid angle the projectiles have not seen the inner region of target atoms, as close encounters would have scattered them out of the detector acceptance with high probability. By the kinematics of the scattering processes the limited detector acceptance is projected onto a limited range of impact parameters. It should be noted that backscattering experiments are less sensitive to impact parameter
B.V. All rights reserved
I. ENERGY
LOSS
R. Golser et al. /Nacl. Imtr. and Meth. in Phys. Res. 3 90 (19943 45-48
46
selection, and that the calorimetric method by Andersen [2] is quite immune to this problem. It is important to recognize that all transmission experiments measure en rather than E; statements of principle on this topic may be found in refs. [3,4]. In the case of thickness inhomogeneities of solid targets, projectiles that have passed the thicker parts of the foil have a higher probabilty to miss the detector due to effects of multiple scattering. Therefore, the contribution from projectiles which have suffered a comparatively large energy loss is reduced and the measured mean energy loss is too small [6,7]. The same error arises also for solid targets with constant thickness: projectiles which have probed the inner regions of the atom and hence have lost a comparatively large amount of energy, both to the nucleus and to the electrons, are scattered out of the detector acceptance f8,9]; thus, the measured mean energy loss is too small. Increasing the target thickness reduces this effect as projectiles will now be re-scattered into the detector with higher probability. This leads to an apparent thickness dependence of E [IO]. In the case of gaseous targets, we have measured the corresponding pressure dependence of elt in Ne 1111 and have explained it by assuming a simple model for the impact parameter dependence of the electronic processes. In this contribution we focus on a different type of experimental artifact that arises in mixtures of gases; the principle applies equally well to solid targets and should also be considered for compounds.
2. Bragg’s rule According to Eq. (11, for any combination materials A and B, Bragg’s rule [12] reads
of two
eBA,Bagg = cA c T,AuiA + ca z TiBqB = cAcA+ cseB, i
i
(4) where cA and cB are the atomic con~ntrations of the ~mponents A and B, respectively (c, + cB = 1). t$& as defined by Eq. (4) gives the stopping cross section per target atom. Evidently, Bragg’s rule is valid as long as there is no interaction between A and B. Therefore, we can distinguish between two basic cases where linear additivity (Bragg’s rule) has to fail: i> A direct interaction between A and B takes place, i.e. a compound is formed. This may lead to the well known deviations from Bragg’s rule [13]. As the compound has lower internal energy than the constituents, some of the values q will be increased, and some of the values ci will be affected, too. ii) An indirect interaction via the projectile takes place, predominantly via its charge state. This effect has been described recently [14]: the mean charge state
and the frequency of charge changing processes in a mixture might be changed in a nonlinear manner. Consider low velocity hydrogen projectiles in a mixture of He and H, as an example [14]: protons in He have a very small cross section for electron capture, wheras in H, they have a very large capture cross section; the opposite applies to the stripping cross sections. In the mixture, charge changing cycles are accelerated and the stopping cross section is increased considerably above the Bragg value. Both il and ii) can be considered “true” deviations from Bragg’s rule. In the following we discuss an apparent deviation from Bragg’s rule that is to be expected for en, since the projection of fi onto impact parameters depends on the composition of the gas in a nonlinear way. Our findings are based on Monte Carlo (MC) simulations of the transport of keV projectiles through pure gases and through a mixture of gases.
3. Monte Carlo calculation A detailed description of our MC program can be found in ref. [ll]. It allows to simulate both elastic and inelastic processes of hydrogen projectiles in dilute gases H,, D,, He, and Ne. We consider only binary collisions; the mean free path depends on gas density, and the distribution of free path lengths is assumed exponential. The next collision partner is determined by a random number according to the composition of the target. Two further random numbers give the impact parameter b and the azimuth of impact. The scattering angle cufb) and the corresponding elastic energy loss follow from a numerical solution of the scattering integral for a screened Coulomb potential. The electronic losses may either be derived from simple atomic models of our target atoms [ll] or from a continuous slowing down approximation. The geometric constraints follow the actual layout of our apparatus 1151:a well collimated beam enters the target along the chamber axis at x = 0; projectiles that leave the target at x = L = 210 cm through an exit aperture of radius r = 1 cm are detected; 50 anti-scattering baffles absorb projectiles scattered more than 3 cm off the axis. We therefore stop calculating the transport of a projectile when its radial distance p from the axis becomes larger than 3 cm. Projectiles with p s r at x = L are considered for further evaluation: we store the number of collisions separately for every target constituent, and also the co~esponding impact parameters.
4. Results Fig. 1 demonstrates for pure He and for pure Ne that at higher target pressures smaller impact parame-
47
R. Golser et al. / Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 45-48 ters are probed by the detected projectiles; their total number is denoted n. The results of the MC calculation are given as histograms: the number of interactions with impact parameters between b and b + Ab has been normalized by 2~ b Ab n, the number of interactions one would expect without impact parameter selection due to scattering (we call the normalized quantity “relative probability”). We first consider a projectile that is scattered exactly once. If it is detected, all impact parameters b it could have seen at position 0 IX IL are given implicitly by tan(a(b)) I r/(L -x). Assuming exactly one interaction with a neon atom (the scattering probability is taken constant along x) we get the rightmost curve in Fig. 1. The sharp edge of the curve for exactly one collision with He is found at tan(cu(b)) = r/L; from our geometry follows cr(b) = 0.273”. The corresponding impact parameter for scattering a 10 keV deuteron on He is b = OSa,, where a, = 1 atomic unit. At higher pressures (0.0075 and 0.03 mbar are shown in Fig. l), smaller impact parameters are possible, as there is a finite probability that a projectile already scattered out of the direction to the detector is re-scattered into the detector rather than hitting an anti-scattering baffle. The pressure dependence becomes more noticeable with increasing target atomic number Z,. It can be seen, too, that at higher pressures we get roughly the same distribution in probability for both target species: when the scattering atoms and the re-scattering atoms are of the same kind, the region of impact parameters probed by the projectiles is fairly independent of Z, and determined only by the experimental geometry. Now let us consider gas mixtures. An apparent deviation from Bragg’s rule may now arise, since here
1.0 0.8
-
Ne “lcoll.”
0.6
_._
0.02 0.03
0.05
0.1
0.2
0.3
0.5
1.0
2.0
Impact parameter (a.“.) Fig. 1. The relative probability for a 10 keV deuteron that enters our detector to have “seen” a certain impact parameter as function of gas pressure. The heavy solid line applies to He at 0.030 mbar, the heavy dotted line to He at 0.0075 mbar; the light solid line is for Ne at 0.030 mbar, and the light dashed line is for Ne at 0.0075 mbar. For the curves labelled “lcoll.” exactly one collision is assumed.
2
1.0 -
-
d 2
0.8 _
----.-.HeO.O15 pHeO.030
x 2 B d 2 5
He,5Neo5 0.030
0.6 0.4 -
.>
-1 I
0.3
Impact
parameter
0.5
1.0
2.0
(a.“.)
Fig. 2. The relative probability for a 10 keV deuteron to have seen a certain impact parameter in a collision with He. The curves correspond to a 1: 1 mixture of He and Ne at a total pressure of 0.030 mbar, and to pure He at 0.015 mbar and at 0.030 mbar, respectively.
we face different conditions for scattering: scattering is dominated by the heavier component, i.e. that with larger Z,. It is obvious that adding a heavier component significantly increases the probability for rescattering a projectile into the detector, that would have missed it if it had encountered one of the light atoms. Hence the inner region of the light atoms is probed more effectively. As mentioned in the introduction, such an indirect interaction between the target constituents leads to a failure of Bragg’s rule. But now, if we assume no further interaction between A and B, the failure is only due to the limited detector acceptance L?, i.e. it holds not for E but only for lR. The effect can be seen in Fig. 2: we show the relative probability that a 10 keV deuteron entering our detector has seen a certain impact parameter in a collision with He. The curves correspond to a gas mixture of 50% He and 50% Ne at total pressure of 0.03 mbar, and to pure helium at 0.015 and at 0.03 mbar, respectively. For the light component we find a clear difference between the pure gas and the mixture, that depends on target composition and on pressure in a nonlinear way. For the heavy component no significant effect is to be expected; this has been confirmed by calculation (the result is not shown). The explanation may be also put differently: at higher pressures, the scattering angles that lead to the detection of a projectile are the same for both components, but the corresponding impact parameters are different (contrary to the result for pure gases at higher pressures, cf. Fig. 1). How may one estimate the difference between E and ln? The impact parameter dependence of elastic losses may be calculated fairly easily, but angle restricted nuclear stopping can be neglected here [ll]. Electronic stopping dominates, but unfortunately, there are not enough data in the literature on the impact I. ENERGY
LOSS
R. Golser et al. /Nucl. Znstr. and Meth. in Fhys. Res. B 90 (1994) 45-48
48
meet the conditions defined by Eq. (1). In transmission experiments and at low projectile velocities the measured specific energy loss en might differ significantly from E, the difference being dependent on gas pressure and on target composition. A correct evaluation of experiments is only possible on the basis of a Monte Carlo simulation. The effects considered here for a pure gas and for a mixture apply, in principle, also to solids and to compounds.
Acknowledgement Heo,soNeO,sopressure (mbar) Fig. 3. The relative number of collisions of a 10 keV deuteron with He in a 1: 1 mixture of He and Ne as a function of the total gas pressure.
This work has been supported by the Austrian Fonds zur Fgrderung der wissenschaftlichen Forschung under Contract no. P8699-PHY.
References of electronic processes. In our measurement of the dependence of E~ on gas pressure for pure Ne 1111,we have found a 10% increase in the specific energy loss when changing the Ne pressure from 0.01 to 0.03 mbar; the effect is well reproduced by our MC simulation, assuming reasonable cross sections. From our measurements with molecular hydrogen targets [l&16] we can safely exclude a density effect on stopping beyond the 1%-level due to excited projectile states. Continuing the discussion of Fig. 2, we may expect a second effect closely related to impact parameter selection: Ne has the larger scattering cross section, so Ne is more effective than He to scatter a projectile in such a way that it finally misses the detector. Though we assume the same concentration for Ne and He, it is more likely that a projectile entering the detector has interacted with a helium atom than with a neon atom. In Fig. 3 we show the relative number of collisions with He in a 1: 1 mixture of He and Ne as a function of the total gas pressure (we arbitrarily assume that an interaction takes place if the impact parameter is smaller than 2a,). It can be seen that at low pressures the apparent ratio of the constituents is noticeably different from the true one, i.e. 53:47. parameter
dependence
5. Conclusion Determining the stopping cross section E by looking at the mean energy loss of projectiles usually wilI not
[l] P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40 (5) (1978). [2] H.H. Andersen and B.R. Nielsen, Nucl. Instr. and Meth. 191 (1981) 475, and references therein. 131 B. Fastrup, P. Hvelp~und and C.A. Sautter, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 3.5 (10) (1966). [4] P. Hvelplund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 38 (4) (1971). [5] Stopping Powers and Ranges for Protons and Alpha Particles, ICRU Report 49 (International Commission on Radiation Units and Measurement, 1993). [6] P. Mertens, Nucl. Instr. and Meth. B 27 (1987) 315 and 326. [7] R. Golser, Ch. Eppacher and D. Semrad, Nuel. Instr. and Meth. B 67 (1992) 69. [8] R. Ishiwari, N. Shiomi, T. Katayama-Kinoshita and F. Sawada-Yasue, J. Phys. Sot. Jpn. 39 (1975) 557; see also R. Ishiwari, N. Sbiomi and N. Sakamoto, Phys. Lett. A 75 (1979) 112. [9] N.M. Kabachnik, V.N. Kondratev and O.V. Chumanova, Phys. Status Solidi B 145 (1988) 103. [lo] L. Meyer, M. Klein and R. Wedell, Phys. Status Solidi B 83 (1977) 451. 1111A. Schieferm~Iler, R. Golser, R. Stohl and D. Semrad, Phys. Rev. A, in press. [12] W.H. Bragg and R. KIeeman, Philos. Mag. 10 (190.5) 318. [13] D.I. Thwaites, Nucl. Instr. and Meth. B 69 (1992) 53, and references therein. [14] R. Golser and D. Semrad, Phys. Rev. A 45 (1992) R4222. [15] R. Golser and D. Semrad, Phys. Rev. Lett. 66 (1991) 1831. [16] R. Golser and D. Semrad, Nucl. Instr. and Meth. B 69 (1992) 18.
NlllMlB
B 90 (1994) 45-48
North-Holland
Beam Interactions with Materials&Atoms
The influence of target composition measured in transmission geometry Robin Golser *, Andreas Schiefermiiller Johannes-Kepler-Universitiit,
Institut fiir fiperimentalphysik,
on the specific energy loss
and Dieter Semrad A-4040 Linz, Austria
At low projectile velocities, stopping cross section measurements by transmission methods with high directional correlation between incoming beam and detected projectiles and with a detector acceptance appreciably smaller than 4a, may suffer from experimental artifacts. Thickness inhomogeneities make the measured specific energy loss appear angle dependent. The influence of the target area1 density may be traced to the impact parameter dependence of both, the elastic and the inelastic stopping cross section. In this contribution we deal with a mixture of gases. A “true” deviation from Bragg’s rule may be due to charge changing processes. Here, we discuss an additional but only apparent violation of Bragg’s rule in gas mixtures: due to impact parameter selection by the experimental geometry, target atoms with different atomic numbers may contribute differently to the measured specific energy loss. Our arguments are based on experimental results and on Monte Carlo simulations.
1. Introduction
The stopping E =
cross section
E is defined
by
CTiUi, i
where ui is the cross section for a process leading to an energy transfer T, between projectile and target. Unfortunately, one can determine E according to Eq. (1) only in a very limited number of cases. If this is not possible, one has to look on stopping [l] as a penetration phenomenon rather than a collision phenomenon (Eq. (1)). It is well known, that E can be made accessible to experiment by introducing the probability Api for the ith process to happen. For NAxui +z 1 we get Api = NAxui; this allows one to define a mean energy (AT) transferred from the projectile to the target within a thin layer of area1 density NAx: (AT)
= NiXm cNA~~~~~=e,$rn
aNAx.
(2)
i
The stopping cross section can now be obtained [2] from (AT), or, as all the energy deposited in the target comes from the projectile kinetic energy, E can be obtained from the mean energy (AE) lost by the projectiles.
* Corresponding author, tel. +43 732 2468 510, fax +43 732 2468 9677, e-mail k342070@edvz,uni-linz.ac.at. 0168-583X/94/$07.00 0 1994 - Elsevier SSDI 0168-583X(93)E0661-Y
Science
However, when one measures (A E) problems might arise. To use Eq. (l), experimental conditions suitable for measuring the (total) cross sections oi are needed, e.g. the outgoing projectiles should not be selected according to their scattering angle (Y. But for measuring (AE) one necessarily does select projectiles, i.e. those which enter the detector. So in most experiments one determines a specific energy loss lo rather than the “true” stopping cross section E, where ln is given by
Eo = -
lim
(AE)o
NAx+O
NAx
(3)
’
The average ( ) is taken over the restricted sample of projectiles selected by the acceptance 0 of the detector; obviously, ln+, = E and - (AE)oc4= = (AT). en corresponds to angle-restricted stopping cross section defined in ICRU49 [3]. Due to the impact parameter dependence of electronic stopping, a difference between E and lo arises since the detected projectiles have probed different domains of the target atoms with different probabilities. For instance, in a transmission experiment with small detector solid angle the projectiles have not seen the inner region of target atoms, as close encounters would have scattered them out of the detector acceptance with high probability. By the kinematics of the scattering processes the limited detector acceptance is projected onto a limited range of impact parameters. It should be noted that backscattering experiments are less sensitive to impact parameter
B.V. All rights reserved
I. ENERGY
LOSS
R. Golser et al. /Nacl. Imtr. and Meth. in Phys. Res. 3 90 (19943 45-48
46
selection, and that the calorimetric method by Andersen [2] is quite immune to this problem. It is important to recognize that all transmission experiments measure en rather than E; statements of principle on this topic may be found in refs. [3,4]. In the case of thickness inhomogeneities of solid targets, projectiles that have passed the thicker parts of the foil have a higher probabilty to miss the detector due to effects of multiple scattering. Therefore, the contribution from projectiles which have suffered a comparatively large energy loss is reduced and the measured mean energy loss is too small [6,7]. The same error arises also for solid targets with constant thickness: projectiles which have probed the inner regions of the atom and hence have lost a comparatively large amount of energy, both to the nucleus and to the electrons, are scattered out of the detector acceptance f8,9]; thus, the measured mean energy loss is too small. Increasing the target thickness reduces this effect as projectiles will now be re-scattered into the detector with higher probability. This leads to an apparent thickness dependence of E [IO]. In the case of gaseous targets, we have measured the corresponding pressure dependence of elt in Ne 1111 and have explained it by assuming a simple model for the impact parameter dependence of the electronic processes. In this contribution we focus on a different type of experimental artifact that arises in mixtures of gases; the principle applies equally well to solid targets and should also be considered for compounds.
2. Bragg’s rule According to Eq. (11, for any combination materials A and B, Bragg’s rule [12] reads
of two
eBA,Bagg = cA c T,AuiA + ca z TiBqB = cAcA+ cseB, i
i
(4) where cA and cB are the atomic con~ntrations of the ~mponents A and B, respectively (c, + cB = 1). t$& as defined by Eq. (4) gives the stopping cross section per target atom. Evidently, Bragg’s rule is valid as long as there is no interaction between A and B. Therefore, we can distinguish between two basic cases where linear additivity (Bragg’s rule) has to fail: i> A direct interaction between A and B takes place, i.e. a compound is formed. This may lead to the well known deviations from Bragg’s rule [13]. As the compound has lower internal energy than the constituents, some of the values q will be increased, and some of the values ci will be affected, too. ii) An indirect interaction via the projectile takes place, predominantly via its charge state. This effect has been described recently [14]: the mean charge state
and the frequency of charge changing processes in a mixture might be changed in a nonlinear manner. Consider low velocity hydrogen projectiles in a mixture of He and H, as an example [14]: protons in He have a very small cross section for electron capture, wheras in H, they have a very large capture cross section; the opposite applies to the stripping cross sections. In the mixture, charge changing cycles are accelerated and the stopping cross section is increased considerably above the Bragg value. Both il and ii) can be considered “true” deviations from Bragg’s rule. In the following we discuss an apparent deviation from Bragg’s rule that is to be expected for en, since the projection of fi onto impact parameters depends on the composition of the gas in a nonlinear way. Our findings are based on Monte Carlo (MC) simulations of the transport of keV projectiles through pure gases and through a mixture of gases.
3. Monte Carlo calculation A detailed description of our MC program can be found in ref. [ll]. It allows to simulate both elastic and inelastic processes of hydrogen projectiles in dilute gases H,, D,, He, and Ne. We consider only binary collisions; the mean free path depends on gas density, and the distribution of free path lengths is assumed exponential. The next collision partner is determined by a random number according to the composition of the target. Two further random numbers give the impact parameter b and the azimuth of impact. The scattering angle cufb) and the corresponding elastic energy loss follow from a numerical solution of the scattering integral for a screened Coulomb potential. The electronic losses may either be derived from simple atomic models of our target atoms [ll] or from a continuous slowing down approximation. The geometric constraints follow the actual layout of our apparatus 1151:a well collimated beam enters the target along the chamber axis at x = 0; projectiles that leave the target at x = L = 210 cm through an exit aperture of radius r = 1 cm are detected; 50 anti-scattering baffles absorb projectiles scattered more than 3 cm off the axis. We therefore stop calculating the transport of a projectile when its radial distance p from the axis becomes larger than 3 cm. Projectiles with p s r at x = L are considered for further evaluation: we store the number of collisions separately for every target constituent, and also the co~esponding impact parameters.
4. Results Fig. 1 demonstrates for pure He and for pure Ne that at higher target pressures smaller impact parame-
47
R. Golser et al. / Nucl. Instr. and Meth. in Phys. Res. B 90 (1994) 45-48 ters are probed by the detected projectiles; their total number is denoted n. The results of the MC calculation are given as histograms: the number of interactions with impact parameters between b and b + Ab has been normalized by 2~ b Ab n, the number of interactions one would expect without impact parameter selection due to scattering (we call the normalized quantity “relative probability”). We first consider a projectile that is scattered exactly once. If it is detected, all impact parameters b it could have seen at position 0 IX IL are given implicitly by tan(a(b)) I r/(L -x). Assuming exactly one interaction with a neon atom (the scattering probability is taken constant along x) we get the rightmost curve in Fig. 1. The sharp edge of the curve for exactly one collision with He is found at tan(cu(b)) = r/L; from our geometry follows cr(b) = 0.273”. The corresponding impact parameter for scattering a 10 keV deuteron on He is b = OSa,, where a, = 1 atomic unit. At higher pressures (0.0075 and 0.03 mbar are shown in Fig. l), smaller impact parameters are possible, as there is a finite probability that a projectile already scattered out of the direction to the detector is re-scattered into the detector rather than hitting an anti-scattering baffle. The pressure dependence becomes more noticeable with increasing target atomic number Z,. It can be seen, too, that at higher pressures we get roughly the same distribution in probability for both target species: when the scattering atoms and the re-scattering atoms are of the same kind, the region of impact parameters probed by the projectiles is fairly independent of Z, and determined only by the experimental geometry. Now let us consider gas mixtures. An apparent deviation from Bragg’s rule may now arise, since here
1.0 0.8
-
Ne “lcoll.”
0.6
_._
0.02 0.03
0.05
0.1
0.2
0.3
0.5
1.0
2.0
Impact parameter (a.“.) Fig. 1. The relative probability for a 10 keV deuteron that enters our detector to have “seen” a certain impact parameter as function of gas pressure. The heavy solid line applies to He at 0.030 mbar, the heavy dotted line to He at 0.0075 mbar; the light solid line is for Ne at 0.030 mbar, and the light dashed line is for Ne at 0.0075 mbar. For the curves labelled “lcoll.” exactly one collision is assumed.
2
1.0 -
-
d 2
0.8 _
----.-.HeO.O15 pHeO.030
x 2 B d 2 5
He,5Neo5 0.030
0.6 0.4 -
.>
-1 I
0.3
Impact
parameter
0.5
1.0
2.0
(a.“.)
Fig. 2. The relative probability for a 10 keV deuteron to have seen a certain impact parameter in a collision with He. The curves correspond to a 1: 1 mixture of He and Ne at a total pressure of 0.030 mbar, and to pure He at 0.015 mbar and at 0.030 mbar, respectively.
we face different conditions for scattering: scattering is dominated by the heavier component, i.e. that with larger Z,. It is obvious that adding a heavier component significantly increases the probability for rescattering a projectile into the detector, that would have missed it if it had encountered one of the light atoms. Hence the inner region of the light atoms is probed more effectively. As mentioned in the introduction, such an indirect interaction between the target constituents leads to a failure of Bragg’s rule. But now, if we assume no further interaction between A and B, the failure is only due to the limited detector acceptance L?, i.e. it holds not for E but only for lR. The effect can be seen in Fig. 2: we show the relative probability that a 10 keV deuteron entering our detector has seen a certain impact parameter in a collision with He. The curves correspond to a gas mixture of 50% He and 50% Ne at total pressure of 0.03 mbar, and to pure helium at 0.015 and at 0.03 mbar, respectively. For the light component we find a clear difference between the pure gas and the mixture, that depends on target composition and on pressure in a nonlinear way. For the heavy component no significant effect is to be expected; this has been confirmed by calculation (the result is not shown). The explanation may be also put differently: at higher pressures, the scattering angles that lead to the detection of a projectile are the same for both components, but the corresponding impact parameters are different (contrary to the result for pure gases at higher pressures, cf. Fig. 1). How may one estimate the difference between E and ln? The impact parameter dependence of elastic losses may be calculated fairly easily, but angle restricted nuclear stopping can be neglected here [ll]. Electronic stopping dominates, but unfortunately, there are not enough data in the literature on the impact I. ENERGY
LOSS
R. Golser et al. /Nucl. Znstr. and Meth. in Fhys. Res. B 90 (1994) 45-48
48
meet the conditions defined by Eq. (1). In transmission experiments and at low projectile velocities the measured specific energy loss en might differ significantly from E, the difference being dependent on gas pressure and on target composition. A correct evaluation of experiments is only possible on the basis of a Monte Carlo simulation. The effects considered here for a pure gas and for a mixture apply, in principle, also to solids and to compounds.
Acknowledgement Heo,soNeO,sopressure (mbar) Fig. 3. The relative number of collisions of a 10 keV deuteron with He in a 1: 1 mixture of He and Ne as a function of the total gas pressure.
This work has been supported by the Austrian Fonds zur Fgrderung der wissenschaftlichen Forschung under Contract no. P8699-PHY.
References of electronic processes. In our measurement of the dependence of E~ on gas pressure for pure Ne 1111,we have found a 10% increase in the specific energy loss when changing the Ne pressure from 0.01 to 0.03 mbar; the effect is well reproduced by our MC simulation, assuming reasonable cross sections. From our measurements with molecular hydrogen targets [l&16] we can safely exclude a density effect on stopping beyond the 1%-level due to excited projectile states. Continuing the discussion of Fig. 2, we may expect a second effect closely related to impact parameter selection: Ne has the larger scattering cross section, so Ne is more effective than He to scatter a projectile in such a way that it finally misses the detector. Though we assume the same concentration for Ne and He, it is more likely that a projectile entering the detector has interacted with a helium atom than with a neon atom. In Fig. 3 we show the relative number of collisions with He in a 1: 1 mixture of He and Ne as a function of the total gas pressure (we arbitrarily assume that an interaction takes place if the impact parameter is smaller than 2a,). It can be seen that at low pressures the apparent ratio of the constituents is noticeably different from the true one, i.e. 53:47. parameter
dependence
5. Conclusion Determining the stopping cross section E by looking at the mean energy loss of projectiles usually wilI not
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