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Barkas effect at low velocities
10
Nuclear
Instruments
and Methods
in Physics Research
B48 (1990) B48 (1990) lo-13 North-Holland
BARKAS EFFECT AT LOW VELOCITIES Allan H. SORENSEN Instituie of Physics, Universiiy of Aarhus, DK-8ooO Aarhur C, Denmark
The energy loss rates of protons and antiprotons at energies below the stopping maximum are determined in an electron gas model. For silicon and germanium targets the stopping power for antiprotons is found to be less than half of the proton stopping power. An analytical second-order Born expression pertaining to electron scattering in a Yukawa potential is obtained and compared to results of exact calculations for a self-consistent potential.
An important quantity in the study of the penetration of charged particles through matter is the energy loss rate or, equivalently, the stopping power - d E/dx of the medium in question. If we consider heavy particles of low charge, as in the case below, the slowing down is due to inelastic collisions with target electrons. Usually, a perturbation treatment is applied to describe such collisions. In consequence, the stopping power turns out proportional to the square of the projectile charge and the stopping of particles with opposite sign of charge but otherwise identical is the same under equal conditions. An example of a relation describing such behaviour is the celebrated Bethe formula. However, deviations from exact invariance under change of sign of projectile charge do occur. Such deviations are known as Barkas corrections or simply as the Barkas effect [l]. At velocities high compared to those of the target electrons, the Barkas effect is relatively small and it decreases rapidly with increasing velocity. For reviews, see refs. [2] and [3]. A sizable effect shows up only at relatively low energies: A very recent experiment has demonstrated differences in stopping power for protons and antiprotons in silicon of up to as much as 20% at energies down to 0.5 MeV [4]. The lower limit on energy in this experiment is not far from the region where stopping is at its maximum, - 100 keV. Obviously, we shall therefore ask: What happens to the Barkas effect at still lower energies, i.e., near or below the stopping maximum? Does it keep growing? In the present paper some answers are provided by an evaluation of the stopping power for protons and antiprotons at velocities below the stopping maximum. We describe the target electrons as constituting a degenerate, homogeneous Fermi gas in which case the maximum appears near the Fermi velocity, vr, cf. ref. [5]. Due to the large difference in mass, the velocity changes of the electrons scattering on the penetrating projectile are much larger than the velocity changes of the pro0168-583X/90/$03.50 0 Blsevier Science Publishers B.V. (North-Holland)
jectile itself. Its rest frame may therefore be considered a system of inertia and summing up the projections of the momentum transfers along the projectile velocity u in all collisions, the stopping power is found to read
[6>71 -dE/dx
= nmvFq,(vF)v
(1)
in the limit v c vr. Here, n is the density of the gas of electrons and m the electron mass. The momentum transport cross section ctr( vr) is determined by the differential cross section ~(8, vF) for an electron of velocity vF to scatter through an angle 0 as (Jtr(%)
= @
- cosf?)o(B,
v,)217 sin0 de.
(2)
All we have to do is to determine utr. In a Thomas-Fermi description, the screened potential of a charge Z,e immersed in a dense Fermi gas of electrons is of Yukawa type [7,8],
Here X, = A/mu, and X = ( u,/~v,)‘/~ with va = e2/A being the Bohr velocity. The density parameter X is assumed less than 1, cf. ref. [7]. Since the effective relative velocity in collisions equals vr, the parameter distinguishing perturbative quanta1 from classical scattering reads for protons and antiprotons K = 2 1z, 1VO/VF= 2nxz.
(4) For X-values of interest, K is at most equal to about 2 and a Born expansion is expected to apply. In order to track down the Barkas effect, it is necessary to go at least to second order. To second order, the differential cross section for electrons of velocity vr scattering in the potential (3) reads [9]
u=~(~)2(l+4nZ,x~~ ,,-lip), (5)
A. H. Srensen
and t = sin28/2. Since the with s2 = t +4x2 +4x4 argument of the inverse tangent is less than both x (which is assumed small) and it’/’ (which is never larger than l/2), we expand this function only to lowest order. It is then straightforward to obtain the transport cross section through insertion into eq. (2) and, in turn, the stopping power through further insertion into eq. (1). If the latter is re-expressed as -- dE = $z:c(x)=$ dx
1’
c,=; ,,eg_L c
=
2
1+x*
X
271z1x3 4(1 3+4x2
1
+x2)
In
t +x*+x4 x2+x4
(6) _lnl
+x* x2
I.
The values for the first-order term C, in eq. (6) are slightly less than those obtained in a dietectric description f5]; at x = 0.1 the difference amounts to 0.58, at x=l,‘Jiij; to6% and at x = 0.6 to 15%. The correction due to C,, on the other hand, is larger. With C = C, + C,, our values for the stopping power (eq. (1’)) agree closely with those obtained numerically by Hu and Zaremba [lo] according to a different and considerably more involved scheme valid to second order; for proton impact the difference amounts to 2% at x = 0.4 rising to 8% at x = 0.7, our values being the highest. Note that C, and thereby the Barkas correction are independent of projectile speed. For small values of x. the Barkas correction reads asymptotically c2 c
6
=4nZ,x3=
Table 1 Average values of C, and C, in a local density approximation. The electron density is obtained in a self-consistent LMTO calculation [12]. Element
(C,)
(C,)/Z,
Si Ge Cr CU A8
0.297 0.284 0.387 0.397 0.381
0.274 0.274 0.264 0.260 0.263
0
where ~2~= ti2/me2 is the Bohr radius, the second-order Born expression for the stopping function C is C = C, + C2 with
[
11
/ Barkas ejlect at law velocities
fmu2,’
upon neglect of differences in logarithmic factors. The quantity Vi characterizes the constant deviation of the potential (3) from the unscreened Coulomb potential at small distances, k’, = Z,e2/a. Bearing in mind that the effective relative velocity in collisions is ur, the result (7) is seen to be identical (except for a factor of 3/2) to that obtained at high velocities by Lindhard [II]. However, in cases of practical interest, x is not exceedingly small. If we define an effective Fermi velocity from the position of the stopping m~mum, we get, for silicon, a value of approximately x = 0.42. For the valence electrons, x even assumes the value xv = 0.58. To estimate the C,-values for a real substance, we average C,(x) according to the actual electron densities encountered in the target. The results for a few elements are given in table 1. Within a few percent, the averaged values for
silicon and germanium are identical to those pertaining to the valence electron density, (C,) = C,(x,). From the table it is evident that a first Born estimate of dE/dx is very far from sufficient at low velocities. The large value of the second-order correction given in table 1 suggests that also higher-order terms of the Born series may be of significance for our determination of the stopping power. To investigate this point, let us perform an exact numerical computation for scattering in the potential (3). In terms of the phase shifts S,( or) of the indi~dual partial waves of angular momentum I, the C-function, which is proportional to the transport cross section, takes the form C=(Z,nx2)-2
E (l+l)
sin’(6,--a,,,)
I=0
The phase shifts are determined by numerical integration of the radial Schriidinger equation for each f-value and subsequent matching of the resulting wave function to its asymptotic form at large distances [13]. In the perturbation region corresponding to low values of x, the phase shifts are small and many angular momenta are of importance. At high x-values, the opposite is true. The effective number of I-values is estimated by (a numerical factor times) the ratio u/Xr, which equals 1/2x. The difference in phase shifts between protons and antiprotons increases with x, as expected. Fig. 1 shows the result for the C-function (normalized to the first-order expression C,) as computed from eq. (8) for protons and antiprotons. Also shown is the second-order Born prediction, eq. (6) which provides reasonable estimates of the stopping power out to x-values of, roughly, 0.4. Beyond this value the &-term leads to an overestimate of the Barkas correction for both the positively and negatively charged projectile. So far, the evaluation of the momentum transport cross section, and thereby the stopping function, has been based on electron scattering in the Yukawa potential (3) obtained in a semiclassical model. The final question we now want to ask is: How well does this potential approximate the actual static, screened Coulomb potential of an immersed proton or antiproton? One obvious failure of the Yukawa potential is I. EXCITATION, STOPPING
A.H. Smensen / Barkas effect at low velocities
12 2
I
I
I
I
I
\
01 0
I
I
I
I
0.4
0.2
I
\
\
z,=-1
\
‘1 0.6
X Fig. 1. Normalized C-function vs x for attractive (Z, = 1) and repulsive (Z, = - 1) Yukawa potential. The dashed curve is the second Born prediction 1+ Cz/C,, whereas the full-drawn curve is a result of the exact numerical calculation.
that it dictates an induced electron density which diverges at the position of the (anti)proton. Furthermore, we expect nonlinearities: Whereas there is no limit on the induced charge density in the attractive case (positive Z,), the magnitude of the induced density in the repulsive case (negative Z,) is limited by the unperturbed density corresponding to complete dilution. To determine the actual scattering potential, a self-consistent calculation is made. For a given static potential, the density of electrons is determined by computing numerically (by the partial wave method) the wave functions of all electrons in the gas, i.e., an average is made of norm squares, according to the Fermi distribution of electron momenta of the unperturbed gas. From the resulting density a new potential is generated, and the process is repeated. Starting from the Yukawa potential (3) and adding a local density estimate for exchange and correlation [14], a rapid convergence of the iteration procedure is assured by mixing in each step only a minor fraction of the new potential into the previous one. Fig. 2 shows the resulting self-consistent electron density and scattering potential obtained for protons and antiprotons for x = 0.6, which corresponds to valence electron densities in silicon and germanium. The nonlinearity of the density with projectile charge is striking. As regards the potential, it is significantly stronger than in the semiclassical model at those distances which are of most importance in the determination of C, r 5 2X,. The enhancement is partly due to inclusion of exchange and correlation: The contribution to the effective potential energy is negative with a magnitude which increases with electron density. in consequence, the strength of the effective potential is enhanced over that of the pure electrostatic potential both for positive Z, (with positive induced density and negative potential energy) and for negative Z, (with
r/X, Fig. 2. Self-consistent electron density (a) and potential (b) as functions of distance r (measured in units of X,) to a proton or an antiproton immersed in a Fermi gas with x = 0.6. The density is normalized to the value for the unperturbed gas, the potential to the Yukawa potential (3). Dashed curves in (b) display the result of a pure Hartree calculation.
negative induced density and positive potential energy). For comparison, the result of a self-consistent Hartree calculation is also shown in fig. 2b. The results for the C-function evaluated with the self-consistent potential are finally given in table 2. They are compared to the results of exact evaluations for the self-consistent Hartree potential and for the Yukawa potential (3) as well as to the first and second Born approximations for the latter, eq. (6). In the repulsive case (antiprotons), the computed C-value is close to C,. In the attractive case we find roughly c= c, + 2c,.
Table 2 Stopping functions for protons and antiprotons in a Fermi gas. Third column corresponds to eq. (6). Fourth to sixth columns give the exact results for, respectively, the Yukawa potential (3) and self-consistent potentials without and with exchange and correlation included. All C-values have been normalized to the first-order Born expression C,, eq. (6).
1 -1 1 -1
0.3 0.3 0.6 0.6
1.19 0.81 1.94 0.06
1.17 0.81 1.76 0.44
1.17 0.90 2.56 0.62
1.31 1.00 3.05 0.82
The self-consistent calculation is fairly safe for antiprotons. However, for protons some uncertainty exists which cannot be eliminated within the present scheme: The computed self-consistent potential corresponds to all electrons being unbound. On the other hand, in the case of x = 0.6, the Fridel-sum rule [S] yields for the potential of fig. 2 a value of 3 (within 0.2%) i.e. bound states do exist. To set a limit on the effect of binding an electron, we may as an extreme take the induced electron density equal to the density encountered in neutrat hydrogen. For x = 0.6, the corresponding C-value equals roughly C, + C,. Note that this value corresponds closely to the result obtained in the self-consistent (“ground state”) calculation by Echenique et al. [lS], a result which we may express as C, -+ 0.96C,. To which extent bound states actually are populated is difficult to judge, But it appears unlikely that an electron would survive for a long time in the ground state while drifting with the proton through the target, being perturbed, e.g., by the positive nuclei. For the attractive case we therefore expect C, + C, 5 C 5 C, + ZC,. To check with experiments, we have selected a few targets from ref. [16], where there is agreement between (nearly atl) experimental low-energy data and the fit presented to the stopping power. From the values of the fits we deduce for chromium, copper and silver targets experimental C-values of C = 0.76 k 0.01. From table 1 this is seen to be within the suggested range 0.65 5 0.01 c: C < 0.912 0.01 - and about twice as large as (C,). In conclusion, a significant velocity-independent Barkas effect at low velocities has been found. The simple analytical first-order Born expression C,, eq. (6), provides fairly reliable estimates for the stapping of antiprotons in most cases of practical interest. For the stopping of protons, the stopping function is estimated byC=C,+$C,.
The author is indebted to Jens Lindhard for enlightening discussions and to Axe1 Sane for qualified su-
pervision on the inclusion of exchange and correlation (LDA) and for supply of electron densities (LMTO). I also wish to acknowledge the interest in the work shown by J.U. Andersen and E. Bonderup.
References 111w. Barkas, W. Bimbaum and F.M. Smith, phys. Rev. 101 (1956) 778; W.H. Barkas, NJ. Dyer and H.H. He&man, Phys. Rev. Lett. I1 (I963) 26. [2] HI-I. Andersen, in: Semiclassical Descriptions of Atomic and Nuclear Co&ions, eds. J. Bang and J. De Boer (Elsevier, Amsterdam, 1985) p. 409. [3] G. Basbas. Nucl. Instr. and Meth. B4 (1984) 227. [4] L.H. Andersen, P. Hvelplund, H. Knudsen, S.P. Msller, J.O.P. Pedersen, E. Uggerhej, K. Elsener and E. Morenzoni, Phys. Rev. Lett. 62 (1989) 1731. [5] J. Lindhard and Aa. Winther, K. Dan. Vidensk. SeIsk. Mat. Fys. Medd. 34, no. 4 (1964). [6] E. Fermi and E. TeIIer. Phys. Rev. 72 (1947) 399. [i’] E. Bonderup, Penetration of Charged Particles through Matter, Lecture Notes (University of Aarhus, 1978) unpublished. IS] W.A. Harrison, Solid State Theory (McGraw-Hi& New York, 1970). [9] R.N. Da&z, Proc. R. Sot. (London) 200 (19.51) 509. [lo] CD. Hu and E. Zaremba, Phys. Rev. B37 (1988) 9248. [ll] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [12] A. Svane, J. Phys. C21 (1988) 5369; Phys. Rev. Lett, 60 (1988) 2693. [13] L.I. Schiff, Quantum Mechanics (McGraw-Hill Kogakusha, Tokyo, I%%?). [14] S.H. Vosko, L. WiIk and M. Nusair, Can. J. Phys. 58 (1980) 1200. [15] P.M. Echenique, R.M. Nieminen, J.C. Ashley and R.H. Ritchie, Phys. Rev. A33 (1986) 897. [16] H.H. Andersen and J.F. ZiegIer, Hydrogen Stopping Powers and Ranges (Pergamon, New York, 1977).
I. EXCITATION, STQPPING
Nuclear
Instruments
and Methods
in Physics Research
B48 (1990) B48 (1990) lo-13 North-Holland
BARKAS EFFECT AT LOW VELOCITIES Allan H. SORENSEN Instituie of Physics, Universiiy of Aarhus, DK-8ooO Aarhur C, Denmark
The energy loss rates of protons and antiprotons at energies below the stopping maximum are determined in an electron gas model. For silicon and germanium targets the stopping power for antiprotons is found to be less than half of the proton stopping power. An analytical second-order Born expression pertaining to electron scattering in a Yukawa potential is obtained and compared to results of exact calculations for a self-consistent potential.
An important quantity in the study of the penetration of charged particles through matter is the energy loss rate or, equivalently, the stopping power - d E/dx of the medium in question. If we consider heavy particles of low charge, as in the case below, the slowing down is due to inelastic collisions with target electrons. Usually, a perturbation treatment is applied to describe such collisions. In consequence, the stopping power turns out proportional to the square of the projectile charge and the stopping of particles with opposite sign of charge but otherwise identical is the same under equal conditions. An example of a relation describing such behaviour is the celebrated Bethe formula. However, deviations from exact invariance under change of sign of projectile charge do occur. Such deviations are known as Barkas corrections or simply as the Barkas effect [l]. At velocities high compared to those of the target electrons, the Barkas effect is relatively small and it decreases rapidly with increasing velocity. For reviews, see refs. [2] and [3]. A sizable effect shows up only at relatively low energies: A very recent experiment has demonstrated differences in stopping power for protons and antiprotons in silicon of up to as much as 20% at energies down to 0.5 MeV [4]. The lower limit on energy in this experiment is not far from the region where stopping is at its maximum, - 100 keV. Obviously, we shall therefore ask: What happens to the Barkas effect at still lower energies, i.e., near or below the stopping maximum? Does it keep growing? In the present paper some answers are provided by an evaluation of the stopping power for protons and antiprotons at velocities below the stopping maximum. We describe the target electrons as constituting a degenerate, homogeneous Fermi gas in which case the maximum appears near the Fermi velocity, vr, cf. ref. [5]. Due to the large difference in mass, the velocity changes of the electrons scattering on the penetrating projectile are much larger than the velocity changes of the pro0168-583X/90/$03.50 0 Blsevier Science Publishers B.V. (North-Holland)
jectile itself. Its rest frame may therefore be considered a system of inertia and summing up the projections of the momentum transfers along the projectile velocity u in all collisions, the stopping power is found to read
[6>71 -dE/dx
= nmvFq,(vF)v
(1)
in the limit v c vr. Here, n is the density of the gas of electrons and m the electron mass. The momentum transport cross section ctr( vr) is determined by the differential cross section ~(8, vF) for an electron of velocity vF to scatter through an angle 0 as (Jtr(%)
= @
- cosf?)o(B,
v,)217 sin0 de.
(2)
All we have to do is to determine utr. In a Thomas-Fermi description, the screened potential of a charge Z,e immersed in a dense Fermi gas of electrons is of Yukawa type [7,8],
Here X, = A/mu, and X = ( u,/~v,)‘/~ with va = e2/A being the Bohr velocity. The density parameter X is assumed less than 1, cf. ref. [7]. Since the effective relative velocity in collisions equals vr, the parameter distinguishing perturbative quanta1 from classical scattering reads for protons and antiprotons K = 2 1z, 1VO/VF= 2nxz.
(4) For X-values of interest, K is at most equal to about 2 and a Born expansion is expected to apply. In order to track down the Barkas effect, it is necessary to go at least to second order. To second order, the differential cross section for electrons of velocity vr scattering in the potential (3) reads [9]
u=~(~)2(l+4nZ,x~~ ,,-lip), (5)
A. H. Srensen
and t = sin28/2. Since the with s2 = t +4x2 +4x4 argument of the inverse tangent is less than both x (which is assumed small) and it’/’ (which is never larger than l/2), we expand this function only to lowest order. It is then straightforward to obtain the transport cross section through insertion into eq. (2) and, in turn, the stopping power through further insertion into eq. (1). If the latter is re-expressed as -- dE = $z:c(x)=$ dx
1’
c,=; ,,eg_L c
=
2
1+x*
X
271z1x3 4(1 3+4x2
1
+x2)
In
t +x*+x4 x2+x4
(6) _lnl
+x* x2
I.
The values for the first-order term C, in eq. (6) are slightly less than those obtained in a dietectric description f5]; at x = 0.1 the difference amounts to 0.58, at x=l,‘Jiij; to6% and at x = 0.6 to 15%. The correction due to C,, on the other hand, is larger. With C = C, + C,, our values for the stopping power (eq. (1’)) agree closely with those obtained numerically by Hu and Zaremba [lo] according to a different and considerably more involved scheme valid to second order; for proton impact the difference amounts to 2% at x = 0.4 rising to 8% at x = 0.7, our values being the highest. Note that C, and thereby the Barkas correction are independent of projectile speed. For small values of x. the Barkas correction reads asymptotically c2 c
6
=4nZ,x3=
Table 1 Average values of C, and C, in a local density approximation. The electron density is obtained in a self-consistent LMTO calculation [12]. Element
(C,)
(C,)/Z,
Si Ge Cr CU A8
0.297 0.284 0.387 0.397 0.381
0.274 0.274 0.264 0.260 0.263
0
where ~2~= ti2/me2 is the Bohr radius, the second-order Born expression for the stopping function C is C = C, + C2 with
[
11
/ Barkas ejlect at law velocities
fmu2,’
upon neglect of differences in logarithmic factors. The quantity Vi characterizes the constant deviation of the potential (3) from the unscreened Coulomb potential at small distances, k’, = Z,e2/a. Bearing in mind that the effective relative velocity in collisions is ur, the result (7) is seen to be identical (except for a factor of 3/2) to that obtained at high velocities by Lindhard [II]. However, in cases of practical interest, x is not exceedingly small. If we define an effective Fermi velocity from the position of the stopping m~mum, we get, for silicon, a value of approximately x = 0.42. For the valence electrons, x even assumes the value xv = 0.58. To estimate the C,-values for a real substance, we average C,(x) according to the actual electron densities encountered in the target. The results for a few elements are given in table 1. Within a few percent, the averaged values for
silicon and germanium are identical to those pertaining to the valence electron density, (C,) = C,(x,). From the table it is evident that a first Born estimate of dE/dx is very far from sufficient at low velocities. The large value of the second-order correction given in table 1 suggests that also higher-order terms of the Born series may be of significance for our determination of the stopping power. To investigate this point, let us perform an exact numerical computation for scattering in the potential (3). In terms of the phase shifts S,( or) of the indi~dual partial waves of angular momentum I, the C-function, which is proportional to the transport cross section, takes the form C=(Z,nx2)-2
E (l+l)
sin’(6,--a,,,)
I=0
The phase shifts are determined by numerical integration of the radial Schriidinger equation for each f-value and subsequent matching of the resulting wave function to its asymptotic form at large distances [13]. In the perturbation region corresponding to low values of x, the phase shifts are small and many angular momenta are of importance. At high x-values, the opposite is true. The effective number of I-values is estimated by (a numerical factor times) the ratio u/Xr, which equals 1/2x. The difference in phase shifts between protons and antiprotons increases with x, as expected. Fig. 1 shows the result for the C-function (normalized to the first-order expression C,) as computed from eq. (8) for protons and antiprotons. Also shown is the second-order Born prediction, eq. (6) which provides reasonable estimates of the stopping power out to x-values of, roughly, 0.4. Beyond this value the &-term leads to an overestimate of the Barkas correction for both the positively and negatively charged projectile. So far, the evaluation of the momentum transport cross section, and thereby the stopping function, has been based on electron scattering in the Yukawa potential (3) obtained in a semiclassical model. The final question we now want to ask is: How well does this potential approximate the actual static, screened Coulomb potential of an immersed proton or antiproton? One obvious failure of the Yukawa potential is I. EXCITATION, STOPPING
A.H. Smensen / Barkas effect at low velocities
12 2
I
I
I
I
I
\
01 0
I
I
I
I
0.4
0.2
I
\
\
z,=-1
\
‘1 0.6
X Fig. 1. Normalized C-function vs x for attractive (Z, = 1) and repulsive (Z, = - 1) Yukawa potential. The dashed curve is the second Born prediction 1+ Cz/C,, whereas the full-drawn curve is a result of the exact numerical calculation.
that it dictates an induced electron density which diverges at the position of the (anti)proton. Furthermore, we expect nonlinearities: Whereas there is no limit on the induced charge density in the attractive case (positive Z,), the magnitude of the induced density in the repulsive case (negative Z,) is limited by the unperturbed density corresponding to complete dilution. To determine the actual scattering potential, a self-consistent calculation is made. For a given static potential, the density of electrons is determined by computing numerically (by the partial wave method) the wave functions of all electrons in the gas, i.e., an average is made of norm squares, according to the Fermi distribution of electron momenta of the unperturbed gas. From the resulting density a new potential is generated, and the process is repeated. Starting from the Yukawa potential (3) and adding a local density estimate for exchange and correlation [14], a rapid convergence of the iteration procedure is assured by mixing in each step only a minor fraction of the new potential into the previous one. Fig. 2 shows the resulting self-consistent electron density and scattering potential obtained for protons and antiprotons for x = 0.6, which corresponds to valence electron densities in silicon and germanium. The nonlinearity of the density with projectile charge is striking. As regards the potential, it is significantly stronger than in the semiclassical model at those distances which are of most importance in the determination of C, r 5 2X,. The enhancement is partly due to inclusion of exchange and correlation: The contribution to the effective potential energy is negative with a magnitude which increases with electron density. in consequence, the strength of the effective potential is enhanced over that of the pure electrostatic potential both for positive Z, (with positive induced density and negative potential energy) and for negative Z, (with
r/X, Fig. 2. Self-consistent electron density (a) and potential (b) as functions of distance r (measured in units of X,) to a proton or an antiproton immersed in a Fermi gas with x = 0.6. The density is normalized to the value for the unperturbed gas, the potential to the Yukawa potential (3). Dashed curves in (b) display the result of a pure Hartree calculation.
negative induced density and positive potential energy). For comparison, the result of a self-consistent Hartree calculation is also shown in fig. 2b. The results for the C-function evaluated with the self-consistent potential are finally given in table 2. They are compared to the results of exact evaluations for the self-consistent Hartree potential and for the Yukawa potential (3) as well as to the first and second Born approximations for the latter, eq. (6). In the repulsive case (antiprotons), the computed C-value is close to C,. In the attractive case we find roughly c= c, + 2c,.
Table 2 Stopping functions for protons and antiprotons in a Fermi gas. Third column corresponds to eq. (6). Fourth to sixth columns give the exact results for, respectively, the Yukawa potential (3) and self-consistent potentials without and with exchange and correlation included. All C-values have been normalized to the first-order Born expression C,, eq. (6).
1 -1 1 -1
0.3 0.3 0.6 0.6
1.19 0.81 1.94 0.06
1.17 0.81 1.76 0.44
1.17 0.90 2.56 0.62
1.31 1.00 3.05 0.82
The self-consistent calculation is fairly safe for antiprotons. However, for protons some uncertainty exists which cannot be eliminated within the present scheme: The computed self-consistent potential corresponds to all electrons being unbound. On the other hand, in the case of x = 0.6, the Fridel-sum rule [S] yields for the potential of fig. 2 a value of 3 (within 0.2%) i.e. bound states do exist. To set a limit on the effect of binding an electron, we may as an extreme take the induced electron density equal to the density encountered in neutrat hydrogen. For x = 0.6, the corresponding C-value equals roughly C, + C,. Note that this value corresponds closely to the result obtained in the self-consistent (“ground state”) calculation by Echenique et al. [lS], a result which we may express as C, -+ 0.96C,. To which extent bound states actually are populated is difficult to judge, But it appears unlikely that an electron would survive for a long time in the ground state while drifting with the proton through the target, being perturbed, e.g., by the positive nuclei. For the attractive case we therefore expect C, + C, 5 C 5 C, + ZC,. To check with experiments, we have selected a few targets from ref. [16], where there is agreement between (nearly atl) experimental low-energy data and the fit presented to the stopping power. From the values of the fits we deduce for chromium, copper and silver targets experimental C-values of C = 0.76 k 0.01. From table 1 this is seen to be within the suggested range 0.65 5 0.01 c: C < 0.912 0.01 - and about twice as large as (C,). In conclusion, a significant velocity-independent Barkas effect at low velocities has been found. The simple analytical first-order Born expression C,, eq. (6), provides fairly reliable estimates for the stapping of antiprotons in most cases of practical interest. For the stopping of protons, the stopping function is estimated byC=C,+$C,.
The author is indebted to Jens Lindhard for enlightening discussions and to Axe1 Sane for qualified su-
pervision on the inclusion of exchange and correlation (LDA) and for supply of electron densities (LMTO). I also wish to acknowledge the interest in the work shown by J.U. Andersen and E. Bonderup.
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I. EXCITATION, STQPPING