Helmut Paul

Fitted empirical reference cross sections for K-shell ionization by protons

On the basis of our recent review of experimental cross sections for K-shell ionization by light ions, we present a table of best-fitted cross sections for protons on all elements from 4Be to 92U. Experimental cross sections are first normalized by the ECPSSR theory of Brandt and Lapicki; then the normalized values are averaged, interpolated, and used to produce reference cross sections (19 values per energy decade). Particular attention is given to assigning a reasonable uncertainty to the reference values: the error consists of a calculated random contribution and an estimated systematic contribution which describes both the influence of a small deficiency of the theory and the influence of possible bias in the data. A complete list of experimental data is given.

An empirical approach to the stopping power of solids and gases for ions from 3Li to 18Ar - Part II

A large collection of stopping power data for projectiles from 3Li to 18Ar is investigated as a possible basis for producing a table of stopping powers. We divide the experimental stopping powers for a particular projectile (nuclear charge Z1) by those for alpha particles in the same element, as given in ICRU Report 49. With proper normalization, we then obtain experimental stopping power ratios Srel that lie approximately on a single curve, provided we treat solid and gaseous targets separately, and provided we exclude H2 and He targets. For every projectile, this curve is then fitted by a 3-parameter sigmoid function Srel=Srel(a,b,c). We find that the three parameters a, b and c depend smoothly on Z1 and can themselves be fitted by suitable functions af,bf and cf of Z1, separately for solid and gaseous targets. The low energy limit (coefficient a) for solids agrees approximately with the prediction by Lindhard and Scharff. We find that agas<asol in almost all cases. Introducing the coefficients af , bf and cf in Srel, we can calculate the stopping power for any ion (3⩽Z1⩽18), and for any element (except H2 and He) and any mixture or compound contained in the ICRU table.

Review of experimental cross sections for K-shell ionization by light ions

We review experimental K-shell ionization cross sections using a data file containing about 7800 total X-ray or Auger production cross sections taken from the literature for which Z1/Z2<0.3, where Z1 and Z2 are the atomic numbers of projectile and target. We compare various recent collections of K-shell fluorescence yields ω, and we use Krauses tables to convert the data to ionization cross sections. For every projectile, we normalize these data using the theoretical cross section due to Brandt and Lapicki (ECPSSR). We show them plotted versus log ξ (where ξ is the scaled projectile velocity) in the appendix, and we average them in equal intervals ° log ξ. A statistical criterion is used to exclude references with discrepant data. We find that the average normalized cross section s is mostly close unity (i.e., ECPSSR describes the data well), but that there are also significant deviations at certain values of ξ. For almost all projectiles, s decreases below unity for log ξ<-0.6. For protons, we find in addition a small dependence of s upon Z2 for small and for large log ξ. We approximate s by analytical functions of log ξ and thus produce “reference” cross sections for selected proton energies and targets. For heavier projectiles (and also for protons on light targets), additional systematic deviations develop: a maximum of s around log ξ=−0.6 and a minimum around log ξ=−0.3. Above log ξ=−0.1, the influence of multiple ionization and of electron capture by the projectile becomes noticeable with increasing Z1. X-ray cross sections for light solid or gaseous targets (Z2 5).

An analytical cross-section formula for K X-ray production by protons

Abstract We present an analytical cross-section formula for the K X-ray production cross section: σax = scσf, where σf approximates σx = ωkσecpssr where ωK is Krauses fluorescence yield; ECPSSR refers to the theoretical ionization cross section by Brandt and Lapicki. sc is a correction factor close to unity, based upon a comparison of about 3200 experimental data points taken from the literature; it is found to depend essentially only on the scaled velocity ξ and can be approximated by an analytical function. We plot log (σxZ22.2) as a function of e = log( E 1 Z 2 2 ) , where Z2 is the atomic number of the target and E1 is the proton energy in MeV, and approximate this function by a sixth order polynomial in e; the expansion coefficients in turn are rational functions of Z2. We thus obtain an analytical cross section for 11 ⩽ Z2 ⩽ 90 and − 3.7 ⩽ e ⩽ − 1.4 which is accurate to 2.5% (standard deviation) for 21 ⩽ Z2 ⩽ 30 if − 3.5 ⩽ e ⩽ −1.8 and less accurate in other regions.

K-shell ionization by protons: A quantitative comparison between published cross sections and theories

Abstract The cross sections for K-shell ionization by protons available in the literature are compared to each other and to the ECPSSR theory by Brandt and Lapicki. For low scaled velocities ξ, we find that a “Coulomb correction factor” C K = exp(− λx ) can explain the data, with λ increasing with target atomic number Z 2 , where the variable x is proportional to ξ −3 . This factor C K probably contains a residual part of the relativistic correction. For medium to high scaled velocities ξ, we average normalized experimental ionization cross sections s = σ ex σ ECPSSR over all targets, within various small intervals Δξ. The average cross sections are found to agree well with σ ECPSSR . After rejecting the data sets containing the most discrepant data, using a statistical criterion, the remaining data are almost compatible with each other. No clear evidence of a dependence of s upon Z 2 found. Empirical reference cross sections for Cu, Ag and Au targets are calculated at a few selected energies. When comparing four different SCA calculations for the ionization probability vs. impact parameter, for 2 MeV protons on Cu, we find that the curves disagree by at most 30% and that they bracket the two sets of data available.

Uncertainties in thick-target PIXE analysis

Abstract Thick-target PIXE analysis involves uncertainties arising from the calculation of thick-target X-ray production in addition to the usual PIXE uncertainties. The calculation demands knowledge of ionization cross-sections, stopping powers and photon attenuation coefficients. Information on these is reviewed critically and a computational method is used to estimate the uncertainties transmitted from this data base into results of thick-target PIXE analyses with reference to particular specimen types using beams of 2–3 MeV protons. A detailed assessment of the accuracy of thick-target PIXE is presented.

Judging the reliability of stopping power tables and programs for heavy ions

Using our large collection of electronic stopping power data for ions from 3Li to 36Kr taken from the literature (0.001–1000 MeV/nucleon), we compare these data to stopping power tables and computer codes by Steward, Northcliffe and Schilling, Ziegler et al. (SRIM), Hubert et al., Konac et al., Grande and Schiwietz, Sigmund and Schinner, Hiraoka and Bichsel, Paul and Schinner (MSTAR) and the Geant Collaboration. Using either representative Z1–Z2-combinations, or all the data from our collection, we determine the average relative difference between experiment and table, and its standard deviation. On the basis of these numbers, we estimate the reliability of the various tables and codes. We find that SRIM and MSTAR are best in the entire energy region, but MSTAR is better for gaseous targets.

Reference cross sections for K-shell ionization by light ions

Abstract Reference cross sections, i.e., accurate cross sections obtained by averaging all the relevant published data points, are useful for practical applications and for comparison with theories. It is shown how such cross sections can be calculated for one type of projectile by using the data from a large range of target atomic numbers, after normalizing the data by the Brandt-Lapicki theory. For protons, an analytical curve is given that describes the empirical deviation from that theory. This deviation is found to be small except at low scaled velocities. As an example, σ = (95.2±1) b is suggested as reference ionization cross section for 2 MeV protons on Cu; this is compared to recent theoretical values. For heavier projectiles up to chlorine, the deviation between the Brandt-Lapicki theory and experiment is found to be slightly larger, but qualitatively similar.

K-Shell ionization by light ions: A graphical comparison of cross sections

Abstract Experimental K -shell ionization cross sections from the 1978 compilation by Gardner and Gray, and from some more recent publications, are presented graphically, for projectiles from 1 H up to 19 F. The cross sections were normalized by dividing the experimental results by values obtained with the PWBA theory, corrected for binding, polarization, Coulomb, and relativistic effects, according to Brandt and Lapicki. The normalized cross sections are plotted versus projectile energy or versus the scaled velocity variable ξ. In spite of some obvious discrepancies between different experiments, it is found that almost all the cross sections agree with the corrected PWBA theory within 60%. The majority of those measured with H, He, or Li projectiles are between 30% below and 10% above that theory. Many of the plots versus ξ show small systematic deviations from the theory, within those 60% limits: a minimum at ξ = 0.6, a maximum at ξ = 0.3, and a steep decrease below ξ = 0.2. The last feature might be due to a deficiency of the Coulomb correction.

Reference stopping cross sections for hydrogen and helium ions in selected elements

Abstract This article presents new fits to all the available published experimental data on electronic stopping for hydrogen projectiles (10 to 2500 keV/nucleon) on Au, Ag, Cu, Ni, and Al, and for helium projectiles (60 to 7500 keV) on Au and Ag. In all of these cases, we also have data measured in our own group. The data from 13 (Out of 123) publications are rejected a priori, mostly because the measurements are relative. We use a least-squares fit to a four-parameter function of the energy, and we reject discrepant data using two statistical criteria. Our fit results are shown graphically together with all the data points and with various fits by other authors. For hydrogen projectiles, we find that the stopping maximum has a lower value than given by Andersen and Ziegler, except in the case of Al, where almost all the fits agree well. For helium projectiles, the agreement of the various data sets and of the fits is generally better. For both types of projectiles, the fitted cross sections for Ag and Au do not cross in the energy region considered, which is not true of fits by other authors.