- Email: [email protected]
Measurement of the barkas effect using MeV antiprotons and protons and an active silicon target
Nuclear Instruments and Methods in Physics Research B48 (1990) l-7 North-Holland
Section I. Excitation
and stopping
MEASUREMENT OF THE BARKAS EFFECT USING MeV ANTIPROTONS AND AN ACTIVE SILICON TARGET
AND PROTONS
S.P. M0LLER Institute
of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark
The recently reported measurements of the Barkas effect are described in detail. The method, projectiles from the energy deposited in thin protons and alpha particles. The antiproton the energy range 3.0-0.538 MeV. The “2: measured antiproton stopping power to that
transmission silicon detectors, is tested stopping power of silicon is found to be contribution” to the stopping power, of protons. Finally, future experiments
1. Introduction
The theory of energy loss of fast charged particles in matter is based on the calculations by Bethe [l], the result being proportional to the projectile charge squared. It was thus a surprise when Barkas et al. [2] found that the range of negative pions was longer than that of positive pions of equal momentum. Later, Barkas et al. [3] suggested that the effect was due to a difference in the stopping power stemming from the opposite charge of the particles. The reduction in the stopping power responsible for the longer range of negative particles as compared to their positively charged antiparticles was later investigated with both sigma-hyperons [3], pions [4] and muons [5], but these measurements all suffer from the poor quality of the low-velocity particle/antiparticle beams used. This so-called Barkas effect has been interpreted as a polarization effect in the stopping material depending on the charge of the projectile Z,. It appears as the second term, proportional to Z:, and in the higher-order odd terms in the implied Born expansion of the energy loss. Deviations from a pure Zf dependence of the stopping power will also show up when stopping powers of light, bare nuclei are compared. This deviation was first observed in ref. [6] by comparing stopping powers for protons and alpha particles. Later, Li nuclei were also included as projectiles [7], whereby it became possible to extract the Zf, the Z:, and the Z,” term (the Bloch term) in the stopping power (neglecting higherorder terms). With the advent of LEAR at CERN, high-quality beams of antiprotons at low energies became available, permitting accurate measurements of the stopping power for antiprotons [8]. The present paper will describe these results in more detail. Also, the experimental 0168-583X/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
which infers the energy loss of the by reproducing well-known stopping powers for 3-1992 lower than for equivelocity protons over the Barkas term, is deduced by comparing the are briefly discussed.
method used will be presented in some detail, since the method is different from the conventional methods used in stopping-power measurements.
2. Theory The Bethe equation
for the stopping
power is given
by
_-=dE dx
4ne4NZ, z:L,
7
d
where v is the projectile velocity, N the target density and Z, the target atomic number. The Bethe stopping function, which is independent of the projectile charge Z,, may be written as
L, = In
2mv2
-P2-c/z,,
I(1 -P’> where fi = v/c, I is the mean ionization potential and C/Z, the so-called shell corrections. The Bethe equation, eq. (1) is derived in the first Born approximation and is hence proportional to Zf. Formally, one may generalize the Bethe equation to include higher-order Z, terms in the stopping function,
L=L,+z,L,+zfL,+...,
(3)
where L, and L, are the Z,-independent coefficients of the Zf and Zf terms in the stopping power and where higher-order terms are usually omitted. The oddZ, terms in eq. (3) give rise to the Barkas effect. Ashley, Ritchie and Brandt [9] first calculated a Zf contribution to the stopping power using a classical perturbation calculation for a harmonic oscillator. The effect originates in the non-negligible displacement of the atomic electron during the collision, which was included to first order in the calculations. Hence their I. EXCITATION,
STOPPING
2
S. P. Mder / Measurement of the Barkas effect
calculation only applies for distant collisions, but the authors assert that the close collisions are essentially those of free particles, giving an exact Zf dependence. The minimum impact parameter for distant collisions, of major importance for the result, was used as an adjustable parameter to fit expe~mental data and was modified in a later treatment [lo]. At the same time Jackson and McCarthy [ll] performed a similar, but relativistic, calculation arriving basically at the same conclusions, but they chose the minimum impact parameter as the radius of the harmonic oscillator in question. Subsequently Lindhard [12] argued that there is an equally important contribution to the Z: term from the close collisions, which are not exactly Coulomb-like due to dynamical screening of the projectile charge by the atomic electrons. This close-collision contribution was estimated to be comparable to that of distant collisions as calculated by Jackson and McCarthy. Lindhard 1121 also realized the significance of the 2: correction to the stopping power. This term includes, and is usually assumed to be equal to, the Bloch correction, which for - 2 ] Z, ] Z,(u,/o) < 1 is well approximated by Inclusion of the Bloch term ?,,, = -1.20(~,/2)*. and twice the Jackson and McCarthy Z: term improved the agreement with experimental data [12]. Later treatments also came from Arista [13] and Hill and Merzbacher [14], using a harmonic-oscillator model and from Sung and Ritchie [15], using an electron-gas model, all obtaining negligible Z: contributions from close collisions. This is also the view point in the most recent theoretical review of the field [16]. Nevertheless, the controversy about the significance of the close-collision contribution to the Barkas effect has remained. Very recently Mikkelsen and Sigmund (171 performed a calculation of the stopping power for a harmonic oscillator accurate up to the second term in the Born series. They found a significant 2: correction for all impact parameters, including the close collisions. Even more recent is the calculation by Sorensen 1181 who calculated the stopping power at energies below the stopping maximum in an electron-gas model. In this velocity-linear stopping region, the stopping power for antiprotons is found to be about half the proton stopping power, independent of velocity.
3. Experimental method The basic method to determine stopping powers is to measure the energy of the projectiles before and after the passage of the target. This technique was used in, e.g., ref. [19] to determine the stopping power of several gases for protons and alpha particles. An accuracy of + 2.5% was obtained. Such transmission measurements are implements by Rutherford backscattering (RBS)
measurements. For a recent stopping-power determination of silicon using both transmission and RBS techniques, see ref. 1201, where an accuracy of 12% was obtained. The most accurate stopping powers (&0.3%) have probably been obtained by the so-called calorimetric technique developed by Andersen 17,211. A similar precision has been obtained using thick silicon detectors to measure the proton-beam energies with and without absorbers [22]. These methods were, however, not found well-suited for an antiproton beam. Hence it was decided to use a silicon detector as an active target to determine the energy loss directly. This decision was also based on our experience with active silicon and germanium targets used in accurate stopping power measurements at very high energies [23]. 3. I. The energy-loss detectors When a charged particle traverses a semiconductor detector, the deposited energy results in the formation of electron-hole pairs. When the detector is biased, the charge formed can be extracted and measured by an amplifier system. The energy spent to produce an electron-hole pair, w, is to a large extent projectile-independent [24]. For highly charged ions, however, w is projectile-dependent (the pulse-height defect), and w increases with the projectile charge. On the contrary, for impact of low-energy alpha particles, w is found to be slightly lower than for protons [25]. Using the model of ref. ]25] a difference in w between MeV protons and antiprotons is predicted to be much less than 1%. There is a difference between the energy lost by the projectiles and the energy deposited in the silicon detector. This difference is extensively discussed in ref. [21] and is, for silicon in the present energy range, mainly caused by escape of g-rays. For proton energies less than 5 MeV, it is at most a correction of a few times 10T3 and is thus negligible. There is another effect which may strongly change the energy loss of charged particles traversing single crystals, namely channeling [26]. By tilting the silicon crystal detector slightly around perpendicular beam incidence, it was assured that the energy-loss spectra obtained were without significant influence of channeling. A ch~neling effect was observed for protons, but nor for antiprotons. Clearly, this does not prove that there is no channeling effect for antiprotons. In fact, there is a programme going on at LEAR to investigate the channeling effect for antiprotons [27]. The detectors used in the present work were commercial transmission silicon detectors, 6.9 and 2.9 urn thick with an active area of 10 mm*. The electronical resolution of the detector-amplifier system was 14.6 and 18.4 keV (FWHM) for the antiproton run. There is, in general, an additional contribution to the resolution due to the finite number of electron-hole pairs created.
S. P. M&er / Measurement of the Barkas effect Table 1 Energies of degraded beams Degrader thickness [pm AI1
P energy WV1
P energy [MeVl
0 25 50 59.3 68.6 73.6 77.9
3.073 2.446 1.721 1.397 1.045 0.785 0.607
3.400 2.865 2.178 1.549 _
This contribution could, in the present measurements, be neglected. The linearity of the amplifier system was checked with an electronic pulser, and the calibrations in the different runs were determined with a 2osPo alpha source. The stopping power is determined as the auerage energy loss divided by the target thickness, AE/Ax, at the mean energy i?= E, - AE/2, where E. is the incident particle energy. Multiple scattering can be neglected, as the average path length of the particles in the target is only about 0.2% larger than the actual target thickness in the worst case. 3.2. The antiproton, proton and alpha-particle beams The present experiment was performed with proton and antiproton beams from LEAR at CERN and proton and alpha-particle beams from the Aarhus tandem accelerator. The LEAR beam had a kinetic energy of 5.91 MeV (105.5 MeV/c), which is presently the minimum energy from LEAR. The total beam intensity was about lo4 s-‘, and the beam had a momentum spread of 10e3. Due to the strict vacuum procedures at LEAR, the beam exits the LEAR ultrahigh-vacuum system through a - 100 pm Be foil. The beam then traverses a few cm of air before it enters the experimental vacuum chamber through a 22 urn mylar foil. Next, the beam traverses a 100 pm scintillator (START) before it passes the silicon detector and finally stops in a thick scintillator (STOP) approximately 1 m downstream of the silicon detector. The two scintillators were used to measure the time of flight (TOF) and hence the energy of the beam particles, since the beam energy was unknown after the various foils. The energy of the antiproton and proton beams after the START scintillator was 3.07 and 3.40 MeV, respectively. To obtain lower energies, various aluminum degrader foils were inserted in the air between the two vacuum systems. For each degrader, TOF spectra were recorded with the STOP scintillator at two positions, exactly 0.500 m apart. The time resolution of the sys-
3
tem, determined as the FWHM of the TOF distribution of the 3.07 MeV beam, was 2.4 ns, corresponding to an energy resolution (FWHM) of 24% at 3 MeV and 9% at 0.6 MeV. The peak position of the TOF distribution, which was used to calculate the beam energy, could be deduced much better, leading to a determination of the beam energy to better than 1%. The aluminum degraders used and the obtained antiproton and proton energies are given in table 1. The measured proton energies are in agreement with calculations using the recommended stopping powers of ref. [28] taking into account the uncertainty of the degrader thickness. The energy of the proton and alpha-particle beam from the Aarhus tandem accelerator was inferred from the magnetic field in the analyzing magnet.
4. Experimental results Previous knowledge of the 2: contribution to the stopping power stems from experiments comparing the ranges and stopping power for particles and antiparticles [2-51 and for protons, alpha particles [19], and lithium ions [7]. Only the stopping-power measurements directly provide values for the Z: term, L,. Both the results with pions [4] and muons [5] are subject to large uncertainties, caused by the low quality of the particle beams used. Extraction of L, from the proton, alphaparticle and lithium-ion measurements is not straightforward, either, and assumptions about the Zp term, L,, were finally made to extract L, [7,19]. The estimated uncertainty in the extracted value of L, is + 25% in ref. [7]. See also the discussion in ref. [29]. 4. I. Proton and alpha-particle measurements Since the experimental method used in the present work has not previously been used to accurately mea-
E. KINETIC
ENERGY
(Me’/1
Fig. 1. Measured proton stopping powers of Si. The solid curve is the recommended stopping power for protons from ref. [28]. I. EXCITATION, STOPPING
S.P. Mder
4 06
I
II
4
g
p-69
pm 51, Tandem
.
u-69pm
St. Tandem
.
p-6
SI, LEAR
I
pa69pm
51. LEAR
A
p-29pm
SI, LEAR
9 pm
by the “trivial”
factors,
-p*_
X=ln
= In
P
-p2-
L.
02 P
_I
2
We note that this definition is different from the reduced stopping power introduced by Bichsel [30]. We have used the value I = 165 eV [28]. The theoretical reduced stopping power is now given as
_ q %--e P
& ti
o
;;ki:
: vI 9
is the stopping power reduced which we define by
I”“”
6 z fJ. OL
/ Measurement of the Barkas efject
I
pi
-P
P
_
oI 0
I 2
I
II L
11 6
1 6
” 10
X theor
E IMeVlomul
Fig. 2. Measured reduced stopping powers for antiprotons, protons and alpha particles. The solid curves show the reduced stopping power from ref. [28], the upper curve including the Barkas effect corresponding to twice the Jackson-McCarthy result [ll].
sure the stopping power of particles with velocities of a few atomic units, it was decided to try to reproduce known proton and alpha-particle stopping powers. In fig. 1 are shown the measured proton stopping powers with the 6.9 pm detector. The solid line is the recommended stopping power from Andersen and Ziegler [28]. Since no absolute calibration of the silicon detector can be performed, the measured stopping powers are scaled to fit the curve. The measurements reproduce the velocity dependence of the stopping power to high accuracy. To elucidate deviations of the measured stopping powers from the Bethe formula, we plot in fig. 2 the so-called reduced stopping power X, which
I
I
I
1
= c/z2 - z, L, - z:L2. (5) The lower full-drawn line in fig. 3 is the reduced stopping power from ref. (281. It is seen that the measurements reproduce the recommended stopping power within the experimental uncertainty, which is estimated to be about + 1 W. In fig. 2 are also plotted the measured alpha-particle stopping powers. The measured alpha-particle stopping powers are about 1% larger than the proton stopping power, in agreement with earlier findings [6], especially when a slightly smaller value of w for alpha particles [25] is taken into account. We note that a larger stopping power corresponds to a smaller reduced stopping power. One of the problems in stopping-power measurements is the fabrication of homogeneous targets of well-known thickness. Epitaxial silicon, used in the production of thin silicon detectors, can be well controlled in these respects. To investigate the thickness inhomogeneity of the silicon targets used, the energy straggling was also studied. The parameter characterizing the energy-loss distribution is K = t/E,,,,,. Here
1
I
I
1
b)
500
-I
I
OO
1
700
I
a)
50
rm
150
/
I
200
250
ENERGY CkeV)
Fig. 3. Energy-loss
spectrum
for 1.72 MeV antiprotons
300
600
lYoilo 50
100
150
200
ENERGY (keV)
traversing the 2.9 pm detector 6.9 pm Si detector (b).
(a) and for 3.01 MeV antiprotons
traversing
the
S. P. Mdler / h4easuremenrof the Barkas effect is a measure of the energy loss, energy transfer. and E,, = 2mv2 is the maximum When K z+ 1, there is a sufficiently large number of all energy transfers that the distribution is Gaussian [31]. At the highest energies in the present work, K is of the order of unity, and the distribution function is of the Vavilov type [31]. Examples of the slightly asymmetric energy-loss distributions are given in fig. 3 for the two detectors. In fig. 4 are shown the measured FWHM of the distributions from the proton and alpha-particle measurements. The FWHM is plotted relative to the width of the Gaussian as given by Bohr, Qi = 4=Z:e4Z2N Ax. The solid lines in fig. 4 are to guide the eye. The increase in the observed straggling for decreasing energy can be attributed to thickness inhomogeneities [32]. Assuming a Gaussian distribution of the foil thickness with standard deviation 6x, the additional contribution to the energy-loss straggling can be estimated as 52:, = (dE/dx)2Gx2. Since this term is proportional to the square of the stopping power, its magnitude can be estimated by comparing the straggling for the proton and alpha-particle beams. Strictly, the measured distributions are only approximately Gaussian at the very highest energies. Doing so, we estimate a thickness variation ax/Ax - 3%. This contribution has clearly no effect on the stopping-power measurements, and future straggling measurements could also easily be corrected for this contribution. In conclusion, the measurements of proton and alpha-particle stopping powers with an active silicon target have reproduced earlier data, giving confidence in the method and supporting the estimated accuracy.
5 = 2Te4AxNZ,/mv2
2-
5-
l-
Fig. 4. Measured
FWHM of the energy-loss
proton
and alpha-particle
distributions impact.
for
5
Table 2 Measured antiproton stopping powers and Barkas term (L,) E
(-dE/dx),
WV1
W’h
3.009 2.370 1.681 1.625 1.352 1.286 0.991 0.912 0.723 0.538
19.04 21.94 27.61 27.74 30.30 32.04 37.18 38.50 42.66 47.45
(-dE/dx);’
ml
L,
[keV/p ml 19.70 23.30 29.45 30.12 33.98 35.09 41.32 43.47 49.85 58.74
0.059 0.093 0.092 0.116 1.108 0.117 0.123 0.136 0.156 0.182
‘) Ref. [28].
4.2.
The antiproton measurements
The stopping power of silicon for antiprotons was measured between 3.01 and 0.538 MeV as given in table 2. To calibrate the silicon AE detector, proton stopping powers were also measured at a few energies with exactly the same experimental setup. The proton reference beams of 1.55-3.40 MeV were also obtained by degradation of a 105.5 MeV/c beam from LEAR. The measured reduced stopping powers for antiprotons and protons at LEAR are also shown in fig. 2. The measured antiproton stopping powers are lower than those of protons, as expected, and the difference in the stopping power corresponds to 3% at 3.01 MeV and 19% at 0.538 MeV. Note also the consistency between the measurements with the two detectors. The two curves in fig. 2 represent the reduced stopping power from ref. [28] (protons) and the same stopping power corrected for the Lindhard Zf term corresponding to twice the Jackson-McCarthy value. The proton measurements agree with the recommended curve [28] within the uncertainty (+l% in the stopping power), giving confidence in measurements with a degraded beam. The measured antiproton stopping powers are in reasonable agreement with the Lindhard result. Extracting the stopping power as described in section 3.1 requires a monoenergetic beam and AE K E,. The error caused by the finite energy loss is discussed by Andersen [21] and is less than 0.3% when AE < 0.2E,, which is fulfilled in the present experiment. The beam is, however, not monoenergetic due to the degradation process. At the lowest antiproton energy, 0.607 MeV, the energy straggling of the degraded beam is FWHM - 400 keV. It has been estimated that this leads to an increase in the extracted antiproton stopping power of about 4%, resulting in a value of L, about 10% too high. The data have, however, not been corrected for this effect, since the introduced error is only slightly larger than the experimental uncertainty. I. EXCITATION, STOPPING
S. P. Mder
6
2x1o-*4
I 5
I 6
1 7
1 6
I 9
I 10
/ Measurement
I 11
12
v/v, Fig. 5. The Z: contribution, tracted from the measurements.
L,, to the stopping
power
ex-
The full-drawn and the dashed curve correspond to the Jackson-McCarthy result and twice this value, respectively.
In fig. 5 we have extracted the Barkas term L, from the data, as one-half of the difference between the proton stopping power from ref. [28] and the measured antiproton stopping power. The results are here plotted as a function of the velocity in units of the Bohr velocity. Again, error bars stems from the + 1% uncertainty on the stopping-power measurement. The measured Barkas term is seen to be about a factor of 2 larger than that calculated by Jackson and McCarthy for the distant collisions only (full-drawn curve), and in close agreement with the estimate of the Barkas term by Lindhard [12] with roughly equal contributions from close and distant collisions (dashed line). We note, however, that by changing the minimum impact-parameter cutoff in the Ritchie-Brandt theory [9], equally good agreement with the data can be obtained. Earlier stopping-power experiments [4,5,7,19], on targets other than silicon, have inferred Z; contributions to the stopping power of equal or slightly larger magnitude than reported here, but with large uncertainties.
5. Future measurements In the light of the success of the initial measurements and motivated by the calculations of refs. [17,18], a continuation of the antiproton stopping-power measurements to energies lower than 0.5 MeV is presently being pursued. This requires at least two developments. First of all, a silicon detector of thickness about 1 Pm is required. We are currently trying to make and/or purchase such a detector. Secondly, it is necessary to restrict the antiproton energies to a narrow energy bin by TOF due to the large straggling in the degradation
of the Barkas effect
process. Such a system has already been used in our experiments at LEAR [33]. The data taking will then consist of recording for each antiproton the time of flight between the two scintillators and the energy loss in the silicon detector. In this way it seems feasible to measure antiproton stopping powers down to the proton stopping-power maximum, i.e. 100 keV. Furthermore, energy-straggling measurements can be performed. The accuracy of the present method to measure antiproton stopping powers, and in particular to infer L,, requires knowledge about the stopping power of silicon for protons obtained by other methods. Indeed, such measurements have recently been performed below 1 MeV [20]. We note that for the high-energy measurements in ref. [20], there is good agreement with the old Andersen-Ziegler curve [28], used in the present work to extract L,. For energies below - 500 keV there are, however, significant deviations from the same curve, compiled when no good silicon data existed. Finally, the uncertainty in the absolute proton stopping-power measurements only enters into the accuracy of present results with reduced weight owing to the relative measurements. The technique developed in the present work can, in principle, also be applied for other active semiconductor targets, e.g. Ge, but the fabrication of such micron-thin detectors presents a nontrivial problem. Let us finally mention that there is a development programme at LEAR aiming at antiproton beams of energy lower than 5 MeV.
6. Conclusions In the present work it is shown that the use of an active silicon absorber in the determination of stopping powers possesses a high intrinsic precision. This was done by performing measurements with proton and alpha-particle beams. The method is also well tailored to an antiprotonic beam. Antiproton stopping powers have been determined with an accuracy of about 1%. In this way, the Barkas correction to the stopping power has been measured unambiguously with high precision between 0.5 and 3 MeV. These measurements alone do not provide evidence for a significant close-collision contribution to the Barkas effect, but the data allow detailed comparisons with theory.
References [l] H.A. Bethe, Ann. Phys. (Leipzig) 5 (1930) 325; U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [2] W.H. Barkas. W. Bimbaum and F.M. Smith. Phys. Rev. 101 (1956) 778.
S. P. Mder
/ Measurement
[3] W.H. Barkas, N.J. Dyer and H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [4] H.H. Heckman and P.J. Lindstrom, Phys. Rev. Lett. 22 (1969) 871. [5] W. Wilhelm, H. Daniel and F.J. Hartmann, Phys. Lett. B98 (1981) 33. [6] H.H. Andersen, H. Simonsen and H. Sorensen, Nucl. Phys. Al25 (1969) 171. [7] H.H. Andersen, J.F. Bak, H. Knudsen and B.R. Nielsen, Phys. Rev. Al6 (1977) 1929. [S] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, J.O.P. Pedersen, E. Uggerhoj, K. Elsener and E. Morenzoni, Phys. Rev. Lett. 62 (1989) 1731. [9] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B5 (1972) 2393; ibid. A8 (1973) 2402. [lo] R.H. Ritchie and W. Brandt, Phys. Rev. Al7 (1978) 2102. [ll] J.D. Jackson and R.L. McCarthy, Phys. Rev. B6 (1972) 4131. [12] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [13] N.R. Arista, Phys. Rev. A26 (1982) 209. [14] K.W. Hill and E. Merzbacher, Phys. Rev. A9 (1974) 156. [15] C.C. Sung and R.H. Ritchie, Phys. Rev. A28 (1983) 674. [16] G. Basbas, Nucl. Instr. and Meth. B4 (1984) 227. [17] H.H. Mikkelsen and P. Sigmund, Phys. Rev. A40 (1989) 101. [18] A.H. Sorensen, these Proceedings (12th Int. Conf. on Atomic Collisions in Solids, Aarhus, Denmark, 1989) Nucl. Instr. and Meth. B48 (1990) 10. [19] F. Besenbacher, H.H. Andersen, P. Hvelplund and H. Knudsen, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40, no. 3 (1979). [20] P. Mertens and P. Bauer, Nucl. Instr. and Meth. B33 (1988) 133.
of the Barkas effect
7
[21] H.H. Andersen, Risa Report no. 93 (1965). [22] R. Ishiwari, N. Shiomi and N. Sakamoto, Nucl. Instr. and Meth. 194 (1982) 61. [23] J.F. Bak, A. Burenkov, J.B.B. Petersen, E. Uggerhoj, S.P. Moller and P. Siffert, Nucl. Phys. B288 (1987) 681; S.P. Moller, Ph.D. Thesis, Aarhus University (1986) unpublished. [24] Average energy required to produce an ion pair ICRU Report 31, Washington (1979). [25] W.N. Lennard, H. Geissel, K.B. Winterbon, D. Phillips, T.K. Alexander and J.S. Forster, Nucl. Instr. and Meth. A248 (1986) 454. [26] H. Esbensen et al., Phys. Rev. B18 (1978) 1039. [27] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, J.O.P. Pedersen, A.H. Sorensen, E. Uggerhoj, K. Elsener and E. Morenzoni, CERN/PSCC/89-5, PSCC/P64 Add. 4. [28] H.H. Andersen and J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York, 1977). [29] H.H. Andersen, in: Semiclassical Descriptions of Atomic and Nuclear Collisions, eds. J. Bang and J. de Boer (Elsevier Science, 1985). [30] H. Bichsel, in: Studies in penetration of charged particles in solids, ed. U. Fano, Nat. Acad. Sci. Publ. no. 1133, Washington, DC (1964) p. 17. [31] H. Bichsel, Rev. Mod. Phys. 60 (1988) 663. [32] F. Besenbacher, J.U. Andersen and E. Bonderup, Nucl. Instr. and Meth. 168 (1980) 1 and ref. [40] therein. [33] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Msller, J.O.P. Pedersen, S. Tang-Petersen, E. Uggerhoj, K. Elsener and E. Morenzoni, Phys. Rev. A40 (1989) in press.
I. EXCITATION,
STOPPING
Section I. Excitation
and stopping
MEASUREMENT OF THE BARKAS EFFECT USING MeV ANTIPROTONS AND AN ACTIVE SILICON TARGET
AND PROTONS
S.P. M0LLER Institute
of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark
The recently reported measurements of the Barkas effect are described in detail. The method, projectiles from the energy deposited in thin protons and alpha particles. The antiproton the energy range 3.0-0.538 MeV. The “2: measured antiproton stopping power to that
transmission silicon detectors, is tested stopping power of silicon is found to be contribution” to the stopping power, of protons. Finally, future experiments
1. Introduction
The theory of energy loss of fast charged particles in matter is based on the calculations by Bethe [l], the result being proportional to the projectile charge squared. It was thus a surprise when Barkas et al. [2] found that the range of negative pions was longer than that of positive pions of equal momentum. Later, Barkas et al. [3] suggested that the effect was due to a difference in the stopping power stemming from the opposite charge of the particles. The reduction in the stopping power responsible for the longer range of negative particles as compared to their positively charged antiparticles was later investigated with both sigma-hyperons [3], pions [4] and muons [5], but these measurements all suffer from the poor quality of the low-velocity particle/antiparticle beams used. This so-called Barkas effect has been interpreted as a polarization effect in the stopping material depending on the charge of the projectile Z,. It appears as the second term, proportional to Z:, and in the higher-order odd terms in the implied Born expansion of the energy loss. Deviations from a pure Zf dependence of the stopping power will also show up when stopping powers of light, bare nuclei are compared. This deviation was first observed in ref. [6] by comparing stopping powers for protons and alpha particles. Later, Li nuclei were also included as projectiles [7], whereby it became possible to extract the Zf, the Z:, and the Z,” term (the Bloch term) in the stopping power (neglecting higherorder terms). With the advent of LEAR at CERN, high-quality beams of antiprotons at low energies became available, permitting accurate measurements of the stopping power for antiprotons [8]. The present paper will describe these results in more detail. Also, the experimental 0168-583X/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
which infers the energy loss of the by reproducing well-known stopping powers for 3-1992 lower than for equivelocity protons over the Barkas term, is deduced by comparing the are briefly discussed.
method used will be presented in some detail, since the method is different from the conventional methods used in stopping-power measurements.
2. Theory The Bethe equation
for the stopping
power is given
by
_-=dE dx
4ne4NZ, z:L,
7
d
where v is the projectile velocity, N the target density and Z, the target atomic number. The Bethe stopping function, which is independent of the projectile charge Z,, may be written as
L, = In
2mv2
-P2-c/z,,
I(1 -P’> where fi = v/c, I is the mean ionization potential and C/Z, the so-called shell corrections. The Bethe equation, eq. (1) is derived in the first Born approximation and is hence proportional to Zf. Formally, one may generalize the Bethe equation to include higher-order Z, terms in the stopping function,
L=L,+z,L,+zfL,+...,
(3)
where L, and L, are the Z,-independent coefficients of the Zf and Zf terms in the stopping power and where higher-order terms are usually omitted. The oddZ, terms in eq. (3) give rise to the Barkas effect. Ashley, Ritchie and Brandt [9] first calculated a Zf contribution to the stopping power using a classical perturbation calculation for a harmonic oscillator. The effect originates in the non-negligible displacement of the atomic electron during the collision, which was included to first order in the calculations. Hence their I. EXCITATION,
STOPPING
2
S. P. Mder / Measurement of the Barkas effect
calculation only applies for distant collisions, but the authors assert that the close collisions are essentially those of free particles, giving an exact Zf dependence. The minimum impact parameter for distant collisions, of major importance for the result, was used as an adjustable parameter to fit expe~mental data and was modified in a later treatment [lo]. At the same time Jackson and McCarthy [ll] performed a similar, but relativistic, calculation arriving basically at the same conclusions, but they chose the minimum impact parameter as the radius of the harmonic oscillator in question. Subsequently Lindhard [12] argued that there is an equally important contribution to the Z: term from the close collisions, which are not exactly Coulomb-like due to dynamical screening of the projectile charge by the atomic electrons. This close-collision contribution was estimated to be comparable to that of distant collisions as calculated by Jackson and McCarthy. Lindhard 1121 also realized the significance of the 2: correction to the stopping power. This term includes, and is usually assumed to be equal to, the Bloch correction, which for - 2 ] Z, ] Z,(u,/o) < 1 is well approximated by Inclusion of the Bloch term ?,,, = -1.20(~,/2)*. and twice the Jackson and McCarthy Z: term improved the agreement with experimental data [12]. Later treatments also came from Arista [13] and Hill and Merzbacher [14], using a harmonic-oscillator model and from Sung and Ritchie [15], using an electron-gas model, all obtaining negligible Z: contributions from close collisions. This is also the view point in the most recent theoretical review of the field [16]. Nevertheless, the controversy about the significance of the close-collision contribution to the Barkas effect has remained. Very recently Mikkelsen and Sigmund (171 performed a calculation of the stopping power for a harmonic oscillator accurate up to the second term in the Born series. They found a significant 2: correction for all impact parameters, including the close collisions. Even more recent is the calculation by Sorensen 1181 who calculated the stopping power at energies below the stopping maximum in an electron-gas model. In this velocity-linear stopping region, the stopping power for antiprotons is found to be about half the proton stopping power, independent of velocity.
3. Experimental method The basic method to determine stopping powers is to measure the energy of the projectiles before and after the passage of the target. This technique was used in, e.g., ref. [19] to determine the stopping power of several gases for protons and alpha particles. An accuracy of + 2.5% was obtained. Such transmission measurements are implements by Rutherford backscattering (RBS)
measurements. For a recent stopping-power determination of silicon using both transmission and RBS techniques, see ref. 1201, where an accuracy of 12% was obtained. The most accurate stopping powers (&0.3%) have probably been obtained by the so-called calorimetric technique developed by Andersen 17,211. A similar precision has been obtained using thick silicon detectors to measure the proton-beam energies with and without absorbers [22]. These methods were, however, not found well-suited for an antiproton beam. Hence it was decided to use a silicon detector as an active target to determine the energy loss directly. This decision was also based on our experience with active silicon and germanium targets used in accurate stopping power measurements at very high energies [23]. 3. I. The energy-loss detectors When a charged particle traverses a semiconductor detector, the deposited energy results in the formation of electron-hole pairs. When the detector is biased, the charge formed can be extracted and measured by an amplifier system. The energy spent to produce an electron-hole pair, w, is to a large extent projectile-independent [24]. For highly charged ions, however, w is projectile-dependent (the pulse-height defect), and w increases with the projectile charge. On the contrary, for impact of low-energy alpha particles, w is found to be slightly lower than for protons [25]. Using the model of ref. ]25] a difference in w between MeV protons and antiprotons is predicted to be much less than 1%. There is a difference between the energy lost by the projectiles and the energy deposited in the silicon detector. This difference is extensively discussed in ref. [21] and is, for silicon in the present energy range, mainly caused by escape of g-rays. For proton energies less than 5 MeV, it is at most a correction of a few times 10T3 and is thus negligible. There is another effect which may strongly change the energy loss of charged particles traversing single crystals, namely channeling [26]. By tilting the silicon crystal detector slightly around perpendicular beam incidence, it was assured that the energy-loss spectra obtained were without significant influence of channeling. A ch~neling effect was observed for protons, but nor for antiprotons. Clearly, this does not prove that there is no channeling effect for antiprotons. In fact, there is a programme going on at LEAR to investigate the channeling effect for antiprotons [27]. The detectors used in the present work were commercial transmission silicon detectors, 6.9 and 2.9 urn thick with an active area of 10 mm*. The electronical resolution of the detector-amplifier system was 14.6 and 18.4 keV (FWHM) for the antiproton run. There is, in general, an additional contribution to the resolution due to the finite number of electron-hole pairs created.
S. P. M&er / Measurement of the Barkas effect Table 1 Energies of degraded beams Degrader thickness [pm AI1
P energy WV1
P energy [MeVl
0 25 50 59.3 68.6 73.6 77.9
3.073 2.446 1.721 1.397 1.045 0.785 0.607
3.400 2.865 2.178 1.549 _
This contribution could, in the present measurements, be neglected. The linearity of the amplifier system was checked with an electronic pulser, and the calibrations in the different runs were determined with a 2osPo alpha source. The stopping power is determined as the auerage energy loss divided by the target thickness, AE/Ax, at the mean energy i?= E, - AE/2, where E. is the incident particle energy. Multiple scattering can be neglected, as the average path length of the particles in the target is only about 0.2% larger than the actual target thickness in the worst case. 3.2. The antiproton, proton and alpha-particle beams The present experiment was performed with proton and antiproton beams from LEAR at CERN and proton and alpha-particle beams from the Aarhus tandem accelerator. The LEAR beam had a kinetic energy of 5.91 MeV (105.5 MeV/c), which is presently the minimum energy from LEAR. The total beam intensity was about lo4 s-‘, and the beam had a momentum spread of 10e3. Due to the strict vacuum procedures at LEAR, the beam exits the LEAR ultrahigh-vacuum system through a - 100 pm Be foil. The beam then traverses a few cm of air before it enters the experimental vacuum chamber through a 22 urn mylar foil. Next, the beam traverses a 100 pm scintillator (START) before it passes the silicon detector and finally stops in a thick scintillator (STOP) approximately 1 m downstream of the silicon detector. The two scintillators were used to measure the time of flight (TOF) and hence the energy of the beam particles, since the beam energy was unknown after the various foils. The energy of the antiproton and proton beams after the START scintillator was 3.07 and 3.40 MeV, respectively. To obtain lower energies, various aluminum degrader foils were inserted in the air between the two vacuum systems. For each degrader, TOF spectra were recorded with the STOP scintillator at two positions, exactly 0.500 m apart. The time resolution of the sys-
3
tem, determined as the FWHM of the TOF distribution of the 3.07 MeV beam, was 2.4 ns, corresponding to an energy resolution (FWHM) of 24% at 3 MeV and 9% at 0.6 MeV. The peak position of the TOF distribution, which was used to calculate the beam energy, could be deduced much better, leading to a determination of the beam energy to better than 1%. The aluminum degraders used and the obtained antiproton and proton energies are given in table 1. The measured proton energies are in agreement with calculations using the recommended stopping powers of ref. [28] taking into account the uncertainty of the degrader thickness. The energy of the proton and alpha-particle beam from the Aarhus tandem accelerator was inferred from the magnetic field in the analyzing magnet.
4. Experimental results Previous knowledge of the 2: contribution to the stopping power stems from experiments comparing the ranges and stopping power for particles and antiparticles [2-51 and for protons, alpha particles [19], and lithium ions [7]. Only the stopping-power measurements directly provide values for the Z: term, L,. Both the results with pions [4] and muons [5] are subject to large uncertainties, caused by the low quality of the particle beams used. Extraction of L, from the proton, alphaparticle and lithium-ion measurements is not straightforward, either, and assumptions about the Zp term, L,, were finally made to extract L, [7,19]. The estimated uncertainty in the extracted value of L, is + 25% in ref. [7]. See also the discussion in ref. [29]. 4. I. Proton and alpha-particle measurements Since the experimental method used in the present work has not previously been used to accurately mea-
E. KINETIC
ENERGY
(Me’/1
Fig. 1. Measured proton stopping powers of Si. The solid curve is the recommended stopping power for protons from ref. [28]. I. EXCITATION, STOPPING
S.P. Mder
4 06
I
II
4
g
p-69
pm 51, Tandem
.
u-69pm
St. Tandem
.
p-6
SI, LEAR
I
pa69pm
51. LEAR
A
p-29pm
SI, LEAR
9 pm
by the “trivial”
factors,
-p*_
X=ln
= In
P
-p2-
L.
02 P
_I
2
We note that this definition is different from the reduced stopping power introduced by Bichsel [30]. We have used the value I = 165 eV [28]. The theoretical reduced stopping power is now given as
_ q %--e P
& ti
o
;;ki:
: vI 9
is the stopping power reduced which we define by
I”“”
6 z fJ. OL
/ Measurement of the Barkas efject
I
pi
-P
P
_
oI 0
I 2
I
II L
11 6
1 6
” 10
X theor
E IMeVlomul
Fig. 2. Measured reduced stopping powers for antiprotons, protons and alpha particles. The solid curves show the reduced stopping power from ref. [28], the upper curve including the Barkas effect corresponding to twice the Jackson-McCarthy result [ll].
sure the stopping power of particles with velocities of a few atomic units, it was decided to try to reproduce known proton and alpha-particle stopping powers. In fig. 1 are shown the measured proton stopping powers with the 6.9 pm detector. The solid line is the recommended stopping power from Andersen and Ziegler [28]. Since no absolute calibration of the silicon detector can be performed, the measured stopping powers are scaled to fit the curve. The measurements reproduce the velocity dependence of the stopping power to high accuracy. To elucidate deviations of the measured stopping powers from the Bethe formula, we plot in fig. 2 the so-called reduced stopping power X, which
I
I
I
1
= c/z2 - z, L, - z:L2. (5) The lower full-drawn line in fig. 3 is the reduced stopping power from ref. (281. It is seen that the measurements reproduce the recommended stopping power within the experimental uncertainty, which is estimated to be about + 1 W. In fig. 2 are also plotted the measured alpha-particle stopping powers. The measured alpha-particle stopping powers are about 1% larger than the proton stopping power, in agreement with earlier findings [6], especially when a slightly smaller value of w for alpha particles [25] is taken into account. We note that a larger stopping power corresponds to a smaller reduced stopping power. One of the problems in stopping-power measurements is the fabrication of homogeneous targets of well-known thickness. Epitaxial silicon, used in the production of thin silicon detectors, can be well controlled in these respects. To investigate the thickness inhomogeneity of the silicon targets used, the energy straggling was also studied. The parameter characterizing the energy-loss distribution is K = t/E,,,,,. Here
1
I
I
1
b)
500
-I
I
OO
1
700
I
a)
50
rm
150
/
I
200
250
ENERGY CkeV)
Fig. 3. Energy-loss
spectrum
for 1.72 MeV antiprotons
300
600
lYoilo 50
100
150
200
ENERGY (keV)
traversing the 2.9 pm detector 6.9 pm Si detector (b).
(a) and for 3.01 MeV antiprotons
traversing
the
S. P. Mdler / h4easuremenrof the Barkas effect is a measure of the energy loss, energy transfer. and E,, = 2mv2 is the maximum When K z+ 1, there is a sufficiently large number of all energy transfers that the distribution is Gaussian [31]. At the highest energies in the present work, K is of the order of unity, and the distribution function is of the Vavilov type [31]. Examples of the slightly asymmetric energy-loss distributions are given in fig. 3 for the two detectors. In fig. 4 are shown the measured FWHM of the distributions from the proton and alpha-particle measurements. The FWHM is plotted relative to the width of the Gaussian as given by Bohr, Qi = 4=Z:e4Z2N Ax. The solid lines in fig. 4 are to guide the eye. The increase in the observed straggling for decreasing energy can be attributed to thickness inhomogeneities [32]. Assuming a Gaussian distribution of the foil thickness with standard deviation 6x, the additional contribution to the energy-loss straggling can be estimated as 52:, = (dE/dx)2Gx2. Since this term is proportional to the square of the stopping power, its magnitude can be estimated by comparing the straggling for the proton and alpha-particle beams. Strictly, the measured distributions are only approximately Gaussian at the very highest energies. Doing so, we estimate a thickness variation ax/Ax - 3%. This contribution has clearly no effect on the stopping-power measurements, and future straggling measurements could also easily be corrected for this contribution. In conclusion, the measurements of proton and alpha-particle stopping powers with an active silicon target have reproduced earlier data, giving confidence in the method and supporting the estimated accuracy.
5 = 2Te4AxNZ,/mv2
2-
5-
l-
Fig. 4. Measured
FWHM of the energy-loss
proton
and alpha-particle
distributions impact.
for
5
Table 2 Measured antiproton stopping powers and Barkas term (L,) E
(-dE/dx),
WV1
W’h
3.009 2.370 1.681 1.625 1.352 1.286 0.991 0.912 0.723 0.538
19.04 21.94 27.61 27.74 30.30 32.04 37.18 38.50 42.66 47.45
(-dE/dx);’
ml
L,
[keV/p ml 19.70 23.30 29.45 30.12 33.98 35.09 41.32 43.47 49.85 58.74
0.059 0.093 0.092 0.116 1.108 0.117 0.123 0.136 0.156 0.182
‘) Ref. [28].
4.2.
The antiproton measurements
The stopping power of silicon for antiprotons was measured between 3.01 and 0.538 MeV as given in table 2. To calibrate the silicon AE detector, proton stopping powers were also measured at a few energies with exactly the same experimental setup. The proton reference beams of 1.55-3.40 MeV were also obtained by degradation of a 105.5 MeV/c beam from LEAR. The measured reduced stopping powers for antiprotons and protons at LEAR are also shown in fig. 2. The measured antiproton stopping powers are lower than those of protons, as expected, and the difference in the stopping power corresponds to 3% at 3.01 MeV and 19% at 0.538 MeV. Note also the consistency between the measurements with the two detectors. The two curves in fig. 2 represent the reduced stopping power from ref. [28] (protons) and the same stopping power corrected for the Lindhard Zf term corresponding to twice the Jackson-McCarthy value. The proton measurements agree with the recommended curve [28] within the uncertainty (+l% in the stopping power), giving confidence in measurements with a degraded beam. The measured antiproton stopping powers are in reasonable agreement with the Lindhard result. Extracting the stopping power as described in section 3.1 requires a monoenergetic beam and AE K E,. The error caused by the finite energy loss is discussed by Andersen [21] and is less than 0.3% when AE < 0.2E,, which is fulfilled in the present experiment. The beam is, however, not monoenergetic due to the degradation process. At the lowest antiproton energy, 0.607 MeV, the energy straggling of the degraded beam is FWHM - 400 keV. It has been estimated that this leads to an increase in the extracted antiproton stopping power of about 4%, resulting in a value of L, about 10% too high. The data have, however, not been corrected for this effect, since the introduced error is only slightly larger than the experimental uncertainty. I. EXCITATION, STOPPING
S. P. Mder
6
2x1o-*4
I 5
I 6
1 7
1 6
I 9
I 10
/ Measurement
I 11
12
v/v, Fig. 5. The Z: contribution, tracted from the measurements.
L,, to the stopping
power
ex-
The full-drawn and the dashed curve correspond to the Jackson-McCarthy result and twice this value, respectively.
In fig. 5 we have extracted the Barkas term L, from the data, as one-half of the difference between the proton stopping power from ref. [28] and the measured antiproton stopping power. The results are here plotted as a function of the velocity in units of the Bohr velocity. Again, error bars stems from the + 1% uncertainty on the stopping-power measurement. The measured Barkas term is seen to be about a factor of 2 larger than that calculated by Jackson and McCarthy for the distant collisions only (full-drawn curve), and in close agreement with the estimate of the Barkas term by Lindhard [12] with roughly equal contributions from close and distant collisions (dashed line). We note, however, that by changing the minimum impact-parameter cutoff in the Ritchie-Brandt theory [9], equally good agreement with the data can be obtained. Earlier stopping-power experiments [4,5,7,19], on targets other than silicon, have inferred Z; contributions to the stopping power of equal or slightly larger magnitude than reported here, but with large uncertainties.
5. Future measurements In the light of the success of the initial measurements and motivated by the calculations of refs. [17,18], a continuation of the antiproton stopping-power measurements to energies lower than 0.5 MeV is presently being pursued. This requires at least two developments. First of all, a silicon detector of thickness about 1 Pm is required. We are currently trying to make and/or purchase such a detector. Secondly, it is necessary to restrict the antiproton energies to a narrow energy bin by TOF due to the large straggling in the degradation
of the Barkas effect
process. Such a system has already been used in our experiments at LEAR [33]. The data taking will then consist of recording for each antiproton the time of flight between the two scintillators and the energy loss in the silicon detector. In this way it seems feasible to measure antiproton stopping powers down to the proton stopping-power maximum, i.e. 100 keV. Furthermore, energy-straggling measurements can be performed. The accuracy of the present method to measure antiproton stopping powers, and in particular to infer L,, requires knowledge about the stopping power of silicon for protons obtained by other methods. Indeed, such measurements have recently been performed below 1 MeV [20]. We note that for the high-energy measurements in ref. [20], there is good agreement with the old Andersen-Ziegler curve [28], used in the present work to extract L,. For energies below - 500 keV there are, however, significant deviations from the same curve, compiled when no good silicon data existed. Finally, the uncertainty in the absolute proton stopping-power measurements only enters into the accuracy of present results with reduced weight owing to the relative measurements. The technique developed in the present work can, in principle, also be applied for other active semiconductor targets, e.g. Ge, but the fabrication of such micron-thin detectors presents a nontrivial problem. Let us finally mention that there is a development programme at LEAR aiming at antiproton beams of energy lower than 5 MeV.
6. Conclusions In the present work it is shown that the use of an active silicon absorber in the determination of stopping powers possesses a high intrinsic precision. This was done by performing measurements with proton and alpha-particle beams. The method is also well tailored to an antiprotonic beam. Antiproton stopping powers have been determined with an accuracy of about 1%. In this way, the Barkas correction to the stopping power has been measured unambiguously with high precision between 0.5 and 3 MeV. These measurements alone do not provide evidence for a significant close-collision contribution to the Barkas effect, but the data allow detailed comparisons with theory.
References [l] H.A. Bethe, Ann. Phys. (Leipzig) 5 (1930) 325; U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [2] W.H. Barkas. W. Bimbaum and F.M. Smith. Phys. Rev. 101 (1956) 778.
S. P. Mder
/ Measurement
[3] W.H. Barkas, N.J. Dyer and H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [4] H.H. Heckman and P.J. Lindstrom, Phys. Rev. Lett. 22 (1969) 871. [5] W. Wilhelm, H. Daniel and F.J. Hartmann, Phys. Lett. B98 (1981) 33. [6] H.H. Andersen, H. Simonsen and H. Sorensen, Nucl. Phys. Al25 (1969) 171. [7] H.H. Andersen, J.F. Bak, H. Knudsen and B.R. Nielsen, Phys. Rev. Al6 (1977) 1929. [S] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, J.O.P. Pedersen, E. Uggerhoj, K. Elsener and E. Morenzoni, Phys. Rev. Lett. 62 (1989) 1731. [9] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B5 (1972) 2393; ibid. A8 (1973) 2402. [lo] R.H. Ritchie and W. Brandt, Phys. Rev. Al7 (1978) 2102. [ll] J.D. Jackson and R.L. McCarthy, Phys. Rev. B6 (1972) 4131. [12] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [13] N.R. Arista, Phys. Rev. A26 (1982) 209. [14] K.W. Hill and E. Merzbacher, Phys. Rev. A9 (1974) 156. [15] C.C. Sung and R.H. Ritchie, Phys. Rev. A28 (1983) 674. [16] G. Basbas, Nucl. Instr. and Meth. B4 (1984) 227. [17] H.H. Mikkelsen and P. Sigmund, Phys. Rev. A40 (1989) 101. [18] A.H. Sorensen, these Proceedings (12th Int. Conf. on Atomic Collisions in Solids, Aarhus, Denmark, 1989) Nucl. Instr. and Meth. B48 (1990) 10. [19] F. Besenbacher, H.H. Andersen, P. Hvelplund and H. Knudsen, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40, no. 3 (1979). [20] P. Mertens and P. Bauer, Nucl. Instr. and Meth. B33 (1988) 133.
of the Barkas effect
7
[21] H.H. Andersen, Risa Report no. 93 (1965). [22] R. Ishiwari, N. Shiomi and N. Sakamoto, Nucl. Instr. and Meth. 194 (1982) 61. [23] J.F. Bak, A. Burenkov, J.B.B. Petersen, E. Uggerhoj, S.P. Moller and P. Siffert, Nucl. Phys. B288 (1987) 681; S.P. Moller, Ph.D. Thesis, Aarhus University (1986) unpublished. [24] Average energy required to produce an ion pair ICRU Report 31, Washington (1979). [25] W.N. Lennard, H. Geissel, K.B. Winterbon, D. Phillips, T.K. Alexander and J.S. Forster, Nucl. Instr. and Meth. A248 (1986) 454. [26] H. Esbensen et al., Phys. Rev. B18 (1978) 1039. [27] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Moller, J.O.P. Pedersen, A.H. Sorensen, E. Uggerhoj, K. Elsener and E. Morenzoni, CERN/PSCC/89-5, PSCC/P64 Add. 4. [28] H.H. Andersen and J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York, 1977). [29] H.H. Andersen, in: Semiclassical Descriptions of Atomic and Nuclear Collisions, eds. J. Bang and J. de Boer (Elsevier Science, 1985). [30] H. Bichsel, in: Studies in penetration of charged particles in solids, ed. U. Fano, Nat. Acad. Sci. Publ. no. 1133, Washington, DC (1964) p. 17. [31] H. Bichsel, Rev. Mod. Phys. 60 (1988) 663. [32] F. Besenbacher, J.U. Andersen and E. Bonderup, Nucl. Instr. and Meth. 168 (1980) 1 and ref. [40] therein. [33] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Msller, J.O.P. Pedersen, S. Tang-Petersen, E. Uggerhoj, K. Elsener and E. Morenzoni, Phys. Rev. A40 (1989) in press.
I. EXCITATION,
STOPPING