tv2'(k, w) = k4(k, w) jP)( Q(l))

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When applied to an electron gas [5], with zero-point motion neglected, eq. (1) yields a Zf correction that is negative at large momentum transfers, i.e., near the.
Nuclear

Instruments and Methods in Physics Research

B48 (1990) 8-9 North-Holland

THE 2; CONTRIBUTION TO THE STOPPING POWER OF PROTONS AND ANTIPROTONS IN SILICON: TWO THEORETICAL PREDICTIONS H .H . MIKKELSEN

‘), H. ESBENSEN

2, and P. SIGMUND

1,2)

” Odense University, 5230 Odense, Denmark ‘I Argonne National Luborarory Argonne, IL 60439, USA

In a recent Letter, Andersen et al. [l] reported measured energy losses of 0.5-3.0 MeV protons and antiprotons in silicon and extracted a Z: (or Barkas) correction to the stopping power. They compared their results to calculations that are valid for distant collisions and are based on a perturbation expansion for the excitation of a classical harmonic oscillator. The authors mentioned that there has been a long-standing discussion about the contribution from close collisions to the Z: correction, but that no agreement had been reached in view of lacking quantitative calculations. In this brief communication we report on the results of recent theoretical evaluations of the Z: correction in two different models, a spherical harmonic oscillator [2] and a static electron gas [3], and apply them to the data in ref. [l]. The calculations are quanta1 and include the effect of both close and distant collisions. For the oscillator, the Born series has been evaluated up to second order, leading to the energy loss versus impact parameter and the stopping power up to third order in Z, [2]. This calculation is the first to preserve shell corrections in the Z: term. The Barkas correction to the energy loss was found nonzero at zero impact parameter. The Barkas correction to the stopping power was found positive for protons, except at velocities well below the stopping maximum. The calculation for the electron gas has been based on a general expression for the self-consistent polarization field in a dense medium, as derived in ref. [4]. In Fourier space, it reads

tv2’(k, w) =

4=

k4(k, w)

jP)( Q(l)),

(North-Holland)

by dE dx

47rZfe4 -----NZ,{L,+L,+...}, mu2

where .I+,, is independent of and L, proportional to Z,. Curve 1 represents L, for a single oscillator with a resonance frequency Z/ti and Z = 165 eV. Curve 2 represents the electron gas with a plasma frequency Z/A. The dashed curve is based on the local density

o 6.9pm A 2.9,um

(1)

where c(k, w) is the dielectric function, and @(‘)(k, o) is the self-consistent polarization field to first order [5]. The induced charge to second order, p(‘)(@), is determined by second-order quantal perturbation theory. the Z: The gradient of @(‘) in real space determines correction to the stopping power. Eq. (1) can be applied to a dilute, random distribution of one-electron atoms in the Hartree approximation. The resulting stopping power agrees completely with the one found directly by perturbation theory. 0168-583X/90/$03.50 0 Elsevier Science Publishers

When applied to an electron gas [5], with zero-point motion neglected, eq. (1) yields a Zf correction that is negative at large momentum transfers, i.e., near the maximum value 2mv. The same was found for the oscillator. However, the region of negative values is narrow at high velocities, where the contributions from close and distant collisions become similar in magnitude. It was demonstrated in ref. [2] that for the oscillator, the quanta1 results deviate drastically from the corresponding classical ones at all pertinent impact parameters. Fig. 1 shows the stopping power corrections found from the two approaches in terms of the quantity L, discussed in ref. [l]. It is related to the stopping power

B.V.

Fig. 1. See text.

Si Si

H. H. Mikkelsen

et al. / Stopping power

approximation for silicon using the Lenz-Jensen approximation to the electron density. Experimental points [l] actually show the sum of I,, + L, + . . . for two foil thicknesses. Provided that L, and higher-order terms are small, there is good agreement between the theoretical and experimental results. For the electron gas, this agreement is better than 20% except for the lowest point. While questions may be raised about the quantitative accuracy of either model to characterize the stopping cross section of a silicon atom, the two curves 1 and 2 represent extreme cases with regard to the treatment of electron binding and polarization for close collisions. The two underlying theoretical models predict a sizable contribution from close collisions to the Barkas correction. We take this, together with the close agreement with experiment, as clear evidence in favor of the importance of close collisions to the Barkas effect, as predicted in ref. [6].

of protons and antiprotons in Si

9

The work at Argonne was supported by the U.S. Department of Energy, Nuclear Physics Division and Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.

References

PI L.H. Andersen,

P. Hvelplund, H. Knudsen, S.P. Meller, J.O.P. Pedersen, E. Uggerhoj, K. Elsener, and E. Morenzoni, Phys. Rev. Lett. 62 (1989) 1731. PI H.H. Mikkelsen and P. Sigmund, Phys. Rev. A40 (1989) 101. 131H. Esbensen and P. Sigmund, submitted to Ann. Phys. [41 H. Esbensen, Ph.D. Thesis, University of Aarhus, Denmark (1977), unpublished. f51 J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28 (1954) no. 8. WI J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [71 R.H. Ritchie and W. Brand& Phys. Rev. Al7 (1978) 2102.

I. EXCITATION,

STOPPING