Barkas Correction Research Articles

Stopping powers of Ta and Mo for MeV protons

The stopping powers of Ta and Mo were measured respectively for protons of 4.5–20.0 MeV and 4.0–12.5 MeV, with an accuracy of ±0.3%. The results for Mo agree very well with the calculated values in ICRU Report 49. By analyzing the experimental results using the modified Bethe-Bloch formula including the Barkas correction, we deduced the mean excitation energy, the I value. In this analysis, we used the Barkas correction of Ashley, Ritchie and Brandt and Bonderup's and Bichsel's shell corrections for the shell correction. The I values, respectively obtained with the Bonderup and Bichsel shell corrections, were 685.1 ± 30.2 eV and 743.1 ± 24.8 eV for Ta and 420.3 ± 14.8 eV and 426.9 ± 13.4 eV for Mo. These two I values for Ta differ and the Bonderup shell correction does not fit the experimental data well. On the other hand, the I values for Mo are very similar and the Bonderup and Bichsel shell corrections both reproduce the experimental situation nearly equally well.

Stopping powers of carbon for protons from 4 to 13 MeV

Stopping powers of carbon (graphite) have been measured for 4–13 MeV photons. The accuracy of the results is estimated to be ±0.35%. Analyzing the present stopping powers with the usual Bethe-Bloch formula, we have extracted the mean excitation energy, I, of carbon atoms. To estimate the shell correction we used both Bichsel's method and Bonderup's theory. The Barkas correction was evaluated by the theory of Ashley, Ritchie and Brandt. The I-values obtained are 79.6 eV for Bichsel's shell correction and 80.1 eV for Bonderup's shell correction. Experimental stopping powers and I-values obtained are compared with those of other authors.

Electron emission yield for impact of swift bare light ions on metals

The electron emission yield induced by 0.5–8 MeV H +, He 2+, Li 3+, B 5+ and C 6+ impact on aluminum and gold targets was measured. The stopping power of bare ions is investigated. Stopping power values calculated according to the Bethe term with and without Barkas and Bloch corrections and to the Z 2 scaling of hydrogen values are compared. The relation of electron emission yield and the stopping power is explored. A significant deviation of a proportionality of the yield to the stopping power is found. Using a — slightly modified — model by Borovsky and Suszcynsky [Phys. Rev. A 43 (1991) 1433] the experiments could be well represented. In this model the collective electric field generated along the projectile's path was taken into account. Consequently, the positive ion channel appears to be the dominant mechanism that leads beyond a projectile independent yield/stopping power ratio.

Impact-parameter dependence of Barkas correction in energy loss of charged particles

A method of calculation of Barkas correction to the charged particle energy loss for a given impact parameter, Δ E( b), is developed on the basis of local plasma approximation. It is shown that the anisotropy of wake potential should be accounted for adequate description of the difference between the proton and antiproton stopping cross sections. The results of calculation of Δ E( b) for Si and Au atoms are presented.

Stopping powers of Mylar for protons from 4 to 11.5 MeV

The stopping powers of Mylar for 4–11.5 MeV protons have been measured with an uncertainty of ±0.35%. The present results have been compared with experimental values obtained by Rauhala and Räisänen [Nucl. Instr. and Meth. B 35 (1998) 130] and with calculated values of the ICRU Report 49 and Janni [At. Data Nucl. Data Tables 27 (1982) 147]. These values agree well with each other within stated uncertainties. Requiring that experimental stopping powers are best reproduced by the Bethe-Bloch formula, we have extracted the mean excitation energy, I-value, for Mylar from the present results. In this analysis, we used the theory of Ashley et al. [Phys. Rev. B 5 (1972) 2393; Phys. Rev. A 8 (1973) 2402] for the estimation of the Barkas correction and regarded the I-value as an adjustable parameter. The I-values obtained are 80.2 eV if we adopted Bichsel's shell correction [UCRL Report No. 17538 (1967)] and 80.8 eV when we used Bonderup's shell correction [K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 (1967) no. 17]. Considering the accuracy of theories this discrepancy is not physically significant. The present I-values agree well with the recommended value of 78.7 eV in the ICRU Reports 37 and 49 but disagree with the value of 75.3±1.1 eV which Hiraoka et al. [Med. Phys. 20 (1993) 135] have recently obtained.

Nonlinear effects on charged particle interactions in matter

A survey is presented of the theoretical status in understanding some nonlinear aspects of charged particle interactions in matter. In particular, a diagrammatic analysis of the many-body interactions between a moving particle and the electron gas is made to investigate, within the random-phase-approximation, the Barkas correction to the electronic stopping power, the nonlinear wake potential generated by swift ions in matter, and processes involved in multiple excitations of plasmons.

Energy loss of heavy ions at high velocity

The slowing down of heavy ions by electronic stopping at high velocity is discussed. The ions are nearly fully stripped and have a well defined charge with relatively small fluctuations. Owing to the large charge of the ions, the classical Bohr formula applies instead of the Bethe formula, which is based on a quantum perturbation calculation. It is essential to include the Barkas effect in the description since it becomes quite large for heavy ions, especially in high-Z materials. In Lindhard's treatment [Nucl. Instr. and Meth. 132 (1976) l], the Barkas correction is viewed as an effect of dynamic screening of the ion potential in the initial phase of a collision with an electron, which reduces the relative velocity and therefore enhances the cross section. With inclusion of this enhancement factor for all impact parameters, as evaluated by Jackson and McCarthy for distant collisions [Phys. Rev. B 6 (1972) 4131], the description reproduces within a few percent measurements for 15 MeV/u Br on Si, Ni, and Au and for 10 MeV/u Kr on Al, Ni, and Au. The procedure is shown also to apply at lower velocities near the stopping maximum, albeit with less accuracy. The straggling in energy loss has been analyzed for a measurement on Si and it is well described by a combination of about equal contributions from fluctuations in the number of violent collisions with single electrons (Bohr straggling) and from fluctuations in ion charge state.

Cu, I and Au stopping powers in solid targets

We have measured the energy loss of Cu, I and Au ions in six elemental solids targets (C, Al, Cu, Ag, Ta and Au) at energies near and below the maximum of the stopping power curve. The low-velocity Barkas and Bloch correction terms are calculated and compared with values from four different formalisms applied to several projectiles (p, C, Al, Cu, I and Au) in various targets. The effective charge parameters deduced from stopping powers by using high-order approximations are found to be very sensitive to the model serving to determine the Barkas and Bloch effects. Finally, the formalism giving the best agreement is used to deduce from our measurements the effective charge parameter and to study its dependence on the stopping medium. Our results are compared with previous calculations that include z 3 and z 4 effects.

Axial channelling of low-energy antiprotons

Monte Carlo computer simulation of the passage of antiprotons and protons through a silicon crystal under axial channelling conditions is performed. The energy spectra and angular distribution of channelled protons and antiprotons moving along the (110) direction are calculated and compared. The projectile energy range is from 300 keV to several megaelectronvolts. It is shown that the energy loss corresponding to the maximum of the antiproton spectrum is close to the energy loss of particles moving in a random direction. Moreover, the most probable energy loss of the channelled part of the antiproton beam is slightly less than the random energy loss. The Barkas correction to the energy loss has little effect on the energy spectra of channelled particles and can hardly be studied with this type of experiment. The angular distributions of protons and antiprotons differ considerably. The characteristic features of the angular distribution may serve as evidence of channelling of low-energy antiprotons.

Stopping powers of metallic elements for high energy ions

Abstract Stopping powers of metallic elements have been measured for 55, 65 and 73 MeV protons and for 13 MeV/u 4He and 12C ions using a high resolution magnetic spectrograph. Analyzing experimental data for protons with the Bethe-Bloch formula, we deduced mean excitation energies for 10 metallic elements. The magnitude of Barkas correction was extracted from stopping power difference for 4He and 12C ions at the same velocity, which was found to be consistent with those measured in previous experiments.

Stopping power for alpha particles in organic liquids and vapours

The stopping power for 5.5 MeV alpha particles has been studied in liquid and vapour phases of water, methanol, ethanol, propanol, pentane, hexane, cyclohexane, heptane and octane. Experimental values for stopping powers in the range 5.5 MeV to 0.5 MeV have been extracted by fitting the range-energy data with polynomial and inverse stopping power functions, and compared with existing results. The effect of phase of the absorber on the stopping power has been clearly demonstrated, the stopping power being higher in the vapour phase. Mean values of the excitation potential have been deduced from the experimental stopping powers after allowing for shell, Barkas and Bloch corrections, which are then compared with the recommended values of Berger and Seltzer.

Barkas effect in a central collision: Exact numerical results and the tenth-order Born series.

The time-dependent Schr\"odinger equation for a central collision between a heavy point charge and a harmonically bound electron is solved numerically. This is done in a basis of energy eigenfunctions of the undisturbed electron. Our numerical method also provides the coefficients of the Born series for the collision. We compare the Born expansion with the exact results. In agreement with other calculations we find a significant Barkas effect for some collision parameter values. At low velocities the Barkas correction is shown to be very large, but at higher velocities it becomes very small even for ions with high charges.

Coupled-channel calculation of stopping powers for intermediate-energy light ions penetrating atomic H and He targets.

A single-center coupled-channel code based on an expansion in terms of atomic wave functions that includes dynamic curved projectile trajectories is applied to the calculation of stopping powers. Stopping powers and differential ionization cross sections are evaluated for p\ifmmode\bar\else\textasciimacron\fi{}, ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{2+}$, and ${\mathrm{Li}}^{3+}$ projectiles penetrating atomic H and He targets at energies of 10--500 keV/u. The results are compared to experimental data, to predictions of the first-order plane-wave Born approximation, and to results of calculation for excitation of a harmonic oscillator including Barkas corrections. The improvement of the present model to first-order or second-order perturbative treatments as well as the effect of polarization on the projectile trajectories is discussed.

Barkas effect in a dense medium: Stopping power and wake field

A general expression has been derived for the polarization field set up by a heavy charged particle penetrating through a random medium. This expression includes contributions of first and second order in the projectile charge, and it determines the stopping power up to the third order. For a dilute gas, the stopping-power formula reduces to the one that can be derived directly from perturbation theory, applied to an isolated atom or molecule. The interaction between a charged particle and the medium is shown to be characterized by two functions, the well-known dielectric function ε( k, ω) and a function Y( k, ω; k′, ω′) depending on two sets of wave number and frequency variables. Explicit expressions for Y have been derived, (i) for a random medium of harmonic oscillators from the classical equation of motion, (ii) for a classical electron gas from Boltzmann's transport equation, (iii) for a system of Hartree atoms from quantal perturbation theory, and (iv) for the Fermi gas. Considerable care has been taken to ensure that different approaches yield identical results in those limits where such is to be expected. The Barkas (or Z 1 3) effect on the stopping power has been evaluated for a medium of classical oscillators with a resonance frequency ω 0 at an arbitrary density, characterized by a plasma frequency ω P . The isolated classical oscillator and the classical electron gas emerge from this as simple limiting cases. The results can be described well by one effective resonance frequency α 0 = (ω 0 2 + ω P 2) 1 2 , as is suggested by classical dispersion theory. The classical description applies to distant collisions only. Quantitative results have been derived for a quantal electron gas with the zero-point motion disregarded (static electron gas). The results compare well with recent experimental data on silicon. The Barkas correction to the spectrum of momentum transfer is positive for positively charged projectiles up to fairly large momenta, but it becomes negative near the edge. The Barkas correction to the stopping power is also mainly positive for positively charged projectiles, but it turns negative at low speed, in agreement with existing results for the harmonic osicllator. The second-order contribution to the induced field (or wake field) contains a higher harmonic, i.e., a field oscillating at twice the standard wave number. This alters the wake potential dramatically behind the penetrating particle at low velocities. This effect is possibly an explanation for observed anomalies in molecular-ion energy-loss spectra. The relative significance of the Barkas term in the stopping power and the wake field increases with increasing electron density in the medium.

Stopping powers of A-150 tissue equivalent plastic for protons from 2.5 TO 7.5 MeV

The stopping powers of A-150 tissue equivalent plastic for protons having energies from 2.5 to 7.5 MeV have been measured with an uncertainty of ± 0.35%. The experimental results have been compared with the calculated stopping powers obtained by Bragg's additivity rule using the stopping power values for constituent elements calculated by Andersen and Ziegler's formula. The calculated values agree with the experimental data to within 0.5%. The Ivalue has been extracted by comparing the calculated values with the experimental data taking the Ivalue as an adjustable parameter. Using Andersen and Ziegler's shell correction, the I value has been determined as 62.5 eV. Using the Bonderup shell correction, the Barkas correction and the Bloch correction, the I value has been determined as 64.7 eV. It is considered that these two I values agree with each other and with the value 65.1 eV recommended in ICRU Report 37 within the uncertainties of the theories.

Stopping powers of Ag for 3–20 MeV protons

The stopping power of Ag for 3–20 MeV protons has been measured with an accuracy of ± 0.3% by using a surface-barrier silicon detector. The Ag sample used in this experiment was 33.081 mg/cm2. Since the energy loss, ΔE, was larger than 20% of the incident energy below 5.5 MeV, a correction was carried out using Andersen-Ziegler's empirical stopping powers. The results have been compared with the Risø and the Aarhus data of Andersen et al. The present results were in good agreement with the Aarhus data within the experimental uncertainties. The mean excitation energy has been extracted as 473 ±14 eV from the present stopping power data using the Bonderup shell correction, the Barkas correction and the Bloch correction. A set of parameters χ = 1.340 and b = 1.26 was used in the determinations of the Bonderup shell correction and the Barkas correction. It has been found that in the energy range from 3 to 19 MeV the Bonderup shell correction with an I-value of 473 eV reproduces the experimental situation very well.

Stopping power for low-energy protons

The stopping power and straggling of energy loss for protons of initial energies 418, 340 and 303 keV have been studied in ethylene, propylene, styrene and their polymers. Employing the inverse stopping power functions, cross sections have been obtained in the energy range 20-400 keV. The existence of a phase effect has been clearly demonstrated. The theoretical stopping cross section has been considered by using the Bethe-Bloch expression together with Shell, Bloch and Barkas corrections. I values thus obtained have been compared with the present experimental values and with those of others. Finally, the experimental straggling of energy loss has been compared with the theories of Tschalar (1968) and Bohr (1948).

Barkas effect in electronic stopping power: Rigorous evaluation for the harmonic oscillator

The Coulomb interaction between a heavy fast charged particle and an electron bound by a harmonic force has been evaluated accurately up to the second-order term in the Born series. The resulting mean energy transfer is given as a function of impact parameter, projectile velocity, and the resonance frequency of the oscillator. Integration over the impact parameter yields the stopping cross section up to third order in the projectile charge ${e}_{1}$ without introduction of an adjustable cutoff radius. Unlike previous treatments, the validity of the evaluation of the stopping cross section within the given physical model is not restricted to the high-speed limit. Numerical results cover the velocity range 0.1\ensuremath{\le}2${\mathrm{mv}}^{2}$/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\le}20. We find a significant ${e}_{1}^{3}$ (or Barkas) correction in the mean energy transfer at all impact parameters. The classical Barkas correction, as given for the harmonic oscillator in the high-speed limit by Ashley, Ritchie, and Brandt [Phys. Rev. B 5, 2393 (1972)], approaches the quantal result for large impact parameters, but the approach is slow in the velocity range covered, and large discrepancies are observed in the range of impact parameters that determine the stopping cross section. Stopping cross sections for protons on atomic hydrogen and helium have been estimated roughly by assigning a single resonance frequency to the target atom. The results cover a velocity range down to somewhat below the stopping maximum where the Barkas correction turns out to be large. This indicates that a properly executed Barkas correction plays an important role in the prediction of the height and the position of the stopping maximum.

Stopping powers of Al and Cu for protons from 3 TO 9 MeV

The stopping powers of Al and Cu for protons having energies from 3 to 9 MeV have been measured with an accuracy of ± 0.4%. The results have been compared with the Risø and the Aarhus data of Andersen et al. and with Tschalär's data for Al. The mean excitation energies have been extracted from the present results using the Bonderup shell correction, the Barkas correction and the Bloch correction as 167.4 ± 0.5 eV for Al and 326.3 ± 0.6 eV for Cu. A set of parameters χ = 1.340 and b = 1.26 was used in these determinations. By using the mean excitation energies thus determined, the Bonderup shell corrections, the Barkas corrections and the Bloch corrections, stopping powers have been calculated up to 30 MeV for Al and up to 20 MeV for Cu. The calculated stopping powers agree very well with the Aarhus data. For A1, the agreement between the calculated stopping powers and Tschalär's data is surprisingly good up to 30 MeV.

Stopping powers of Be, Al, Ti, V, Fe, Co, Ni, Cu, Zn, Mo, Rh, Ag, Sn, Ta, Pt and Au For 6.5 MeV protons

The stopping powers of Be, Al, Ti, V, Fe, Co, Ni, Cu, Zn, Mo, Rh, Ag, Sn, Ta, Pt and Au for 6.5 MeV have been measured using a surface barrier silicon detector. The accuracy of the results is ± 0.3%. The detail of the experimental procedures is described. The results have been compared with the Risø data and the Aarhus data of Andersen et al. The present results for Al, Cu and Ag agree very well with the Aarhus results at 6.5 MeV. The mean excitation energy has been extracted from the present results using the Bonderup shell correction, the Barkas correction and the Bloch correction. The parameters x and b which appear in the Bonderup shell correction and the Barkas correction have been newly determined as x = 1.358 and b = 1.3. The present mean excitation energy agrees very well with the I values obtained by other authors. It has been confirmed that the Bonderup shell correction is quantitatively correct for 6.5 MeV protons.