Energy Straggling Of Protons Research Articles

Investigation on energy straggling of protons via Fokker–Planck equation

The Fokker–Planck equation (FPE) governing the behavior of a monoenergetic proton beam passing through an absorbing element was solved by using the separation of variable technique in which the energy distribution of protons obtained from our analysis satisfies the Vavilov theory. Most noteworthy, the accuracy of the method was benchmarked by comparison our results with those of Reynolds experiments and Hines in which a satisfactory agreement was found. The variation of the distribution function in time, the interval of the primary energy of the protons traversing an element as well as the density of the medium for which the energy distribution of protons follows a Gaussian function were discussed. Moreover, on the one hand, it was shown that the broadening of the distributions increases by increasing the initial energy of protons traveling through a specified element. On the other hand, we made the conclusion that the average broadening is the same for monoenergetic protons passing through the elements having different densities. It is worth noting that solving the FPE for energy straggling of protons penetrating through an element has been carried out for the first time using the analytical approach based on the separation of variable technique.

Total variation superiorization schemes in proton computed tomography image reconstruction

Iterative projection reconstruction algorithms are currently the preferred reconstruction method in proton computed tomography (pCT). However, due to inconsistencies in the measured data arising from proton energy straggling and multiple Coulomb scattering, the noise in the reconstructed image increases with successive iterations. In the current work, the authors investigated the use of total variation superiorization (TVS) schemes that can be applied as an algorithmic add-on to perturbation-resilient iterative projection algorithms for pCT image reconstruction. The block-iterative diagonally relaxed orthogonal projections (DROP) algorithm was used for reconstructing GEANT4 Monte Carlo simulated pCT data sets. Two TVS schemes added on to DROP were investigated; the first carried out the superiorization steps once per cycle and the second once per block. Simplifications of these schemes, involving the elimination of the computationally expensive feasibility proximity checking step of the TVS framework, were also investigated. The modulation transfer function and contrast discrimination function were used to quantify spatial and density resolution, respectively. With both TVS schemes, superior spatial and density resolution was achieved compared to the standard DROP algorithm. Eliminating the feasibility proximity check improved the image quality, in particular image noise, in the once-per-block superiorization, while also halving image reconstruction time. Overall, the greatest image quality was observed when carrying out the superiorization once per block and eliminating the feasibility proximity check. The low-contrast imaging made possible with TVS holds a promise for its incorporation into future pCT studies.

Stopping cross-section and energy straggling of protons in SiC obtained by nuclear resonant reaction of protons with 12C and 28Si

Abstract In order to evaluate stopping cross-section and energy straggling of protons in compound material SiC and its constituents C and Si, resonant backscattering spectra have been measured using proton beams in an energy range 4.9–6.1 MeV per a 100 keV step. We have observed two sharp nuclear resonances at proton energies of 4.808 MeV by 12C and 4.879 MeV by 28Si. By systematic analyses of the resonance peak profiles, i.e., energy shift of the peak position and broadening of the peak width, the values of the stopping cross-section and the energy straggling have been deduced to be compared with SRIM-2006 and Bohr’s prediction.

Stopping power and energy straggling of protons in graphite and amorphous carbon obtained from a resonance in BS spectra

Backscattering (BS) spectra with a sharp 4.8-MeV resonance for carbon targets have been measured using proton beams in an energy range 4.85–6.1 MeV per 100-keV step. By systematic analyses of the resonance peak profiles, values of stopping power and energy straggling have been deduced for proton energies from 0.8 to 3.4 MeV which corresponds to a penetration depth of 88 μm. In particular, to investigate the difference in stopping power and straggling caused by target inhomogeneity, we used two target materials which were highly oriented pyrolytic graphite (HOPG, 2.26 g/cm 3) as a homogeneous material and amorphous carbon (1.73 g/cm 3) as an inhomogeneous material. We describe a method of measuring stopping power and straggling using a resonance in the BS spectra. The stopping powers obtained are compared with the values determined by SRIM-2006. Moreover, collision straggling and a density straggling due to the inhomogeneity of the target materials are evaluated from the width broadening of resonance peaks.

A study of energy loss and straggling of MeV protons in biological samples

In order to study the energy loss and its straggling of energetic ions in biological samples, onion membrane samples with different thicknesses were irradiated with 1.8 and 2.8 MeV protons, respectively. For comparison, Mylar membrane and filter membrane were also used in this experiment. The energy loss and straggling of these protons after penetrating through the biological and polymer samples were measured by transmission energy spectrometry. The experimental results showed that the average energy losses of MeV protons in the biological samples are consistent with SRIM simulation but the energy straggling of the protons is different from the simulation results. The possible mechanisms of these new phenomena are discussed in this paper.

Proton Energy Straggling Measurement in Metal Foils by Using a Si(Li) Detector

Measurements of energy loss and transmitted energy straggling were performed using a Si(Li) detector for 28.6 MeV protons incident on thin foil targets of Al, Cu and Au in air. The straggling widths were studied as functions of the target thickness and the target species. The initial proton beam energy from the MC-50 cyclotron was 45 MeV and it was decreased to 28.6 MeV by passing it through a 2-mm-thick Al window, 34.5 cm of air, a 2-mm-thick Al degrader and 144.5 cm of air. This 28.6 MeV proton beam was incident on Al, Cu and Au foils of various thicknesses installed in front of a Si(Li) detector. As a result, the energy loss strongly depended on the target thickness and species and the transmitted energy straggling was not related to the target species. Compared with the calculated results based on a simple Bohr's expression, the measured values agreed with the calculated ones in the low energy loss region less than 60 % of the incident proton beam energy. In the high-loss region, there was big mismatch of more than 25 %.

Theoretical aspects of energy–range relations, stopping power and energy straggling of protons

The Bragg–Kleeman rule R CSDA = AE 0 p provides a connection between the initial energy E 0 of a proton and the range R CSDA in a medium, if the continuous-slowing-down approximation (CSDA) is assumed. The rule results from a generalized (nonrelativistic) Langevin equation; its integration also yields information on the residual energy E ( z ) or d E ( z ) / d z of a proton at position z. A relativistic extension of the generalized Langevin equation leads to the formula R CSDA = A ( E 0 + E 0 2 / 2 Mc 2 ) p . Since the initial energy E 0 of therapeutic protons satisfies E 0 ⪡ 2 Mc 2 , relativistic contributions can be treated as correction terms. Besides this phenomenological aspect, a complete integration of Bethe–Bloch equation (BBE) is presented, which provides the determination of R CSDA , E ( z ) , d E ( z ) / d z and works without any empirical parameters. The results of these different methods are compared with Monte Carlo calculations (GEANT4). Since the energy transfer from proton to the environmental atomic electrons regarded in the CSDA-framework has to account for local fluctuations, an analysis of the Gaussian convolution and the Landau–Vavilov distribution function is performed on the basis of quantum-statistical mechanics. The Landau tail can be described as a Hermite polynomial correction of a Gaussian convolution.

Experimental energy straggling of protons in inhomogeneous materials

The energy straggling of proton beams in inhomogeneous materials has been experimentally investigated by measurements of the nuclear resonance for Rutherford backscattering spectroscopy. The dependence of the energy straggling on bulk densities of powdery carbon material is studied with a peak profile for the sharp 4.8-MeV resonance for incident 5.5-MeV proton. The targets used were three powdery samples made from carbon of coarse, medium and fine particles for inhomogeneous material. Two carbon crystals of a highly oriented pyrolytic graphite and a diamond were used for the homogeneous material. The powdery carbons were pressed into bulk densities of 0.63, 0.98 and 1.48 g/cm 3, according to the grain sizes. Although the positions of the peaks do not change at all, we found that the peak profiles strongly depend on bulk densities of the inhomogeneous targets. The peak broadening observed in the present work cannot be explained by the Bohr energy straggling process. It is necessary to take into account the other contribution, i.e. an additional straggling caused by the target inhomogeneities. The analysis of the peak profile was performed by using our simulation code recently developed. The present results strongly suggest new applications for the study of the interior structure of powdery materials.

Energy straggling of protons through thin solid foils

The energy straggling of protons penetrating thin foils is theoretically and experimentally investigated for intermediate and high impact velocities. We calculate separately the contributions due to the interaction of the projectile with valence and core electrons. The energy straggling originated in the valence band is evaluated within the dielectric formalism, using the Mermin-Lindhard dielectric response function. The contribution coming from the core ionization is calculated with the continuum-distorted-wave--eikonal-initial-state (CDW-EIS) approximation. The predictions of the model are compared with recent measurements for Al, Zn, and Au targets. In addition, experimental results of straggling for Si are presented. The transmission method is used to measure the straggling, and experimental data are corrected to consider roughness effects. Theoretical values are in good agreement with the experiments in the whole range of energies considered. At low energies the binary interaction with valence electrons is the dominant mechanism, while the inner-shell contribution becomes the most important one as the energy increases.

Electronic Energy Loss and Straggling of Protons in Metals**The project supported by China Postdoctoral Science Foundation

By making use of the linear-response dielectric theory and local electron density approximation, we calculate the electronic stopping cross section and energy straggling of protons in metals. The correlation and exchange interaction of electron gas in metals is taken into consideration by introducing a local-field-correction function in the dielectric function. The theoretical results are compared with experimental and empirical data, as well as with other previous theoretical results.

Influence of energy straggling on quantitative PIXE analysis

Examples of numerical calculations aimed to estimate the effect of proton energy straggling on quantitative PIXE are presented. Special attention has been taken to low energy PIXE and external beam PIXE. X-ray yield calculations including beam energy straggling for beam energies ranging from 0.3 to 3 MeV were carried out and the X-ray yield variations due to the energy straggling were determined. Proton energy straggling was calculated using the uncorrected Bohr's formula. K-and L-shell ionization cross sections variations were considered for the X-ray yields calculations. Results indicate that in some cases the error induced on the produced X-ray yield may be up to a few percent on low energy PIXE. For external beam PIXE and high energy PIXE the X-ray yield variations are very small (less than 0.5%).

Energy straggling of protons in solids

By use of the linear-response dielectric theory and local electron density approximation, we calculate the energy straggling of protons in solids. The correlation and exchange interaction of electron gas in solids is taken into consideration by introducing a static local-field-correction function in the dielectric function. The theoretical results are compared with experimental and empirical data, as well as with other previous theoretical results.

Proton energy straggling measurements in aluminum, titanium, silver and tungsten foils

Measurements of the transmitted energy straggling are made using a silicon surface barrier detector for 2 to 7 MeV protons incident on thin foil targets of Al, Ti, Ag and W. The straggling widths are studied as a function of target thickness, proton energy and target species. The combination of several incident energies with many target thicknesses provides data for the investigation of energy straggling in the region where Tschalär's theory is applicable. The theoretical estimates are found to agree with the measured values for higher energy beams, but are about 10% lower than the data for lower incident energies. A simple empirical expression for the straggling is given which agrees with the experimental data for energy losses up to 60%.

Energy straggling of energetic light ion beams

The energy straggling of protons and helium ions is investigated based on the dielectric function theory of Lindhard and Winther using two local electron density models (LEDM). Not only point charges (H +, He 2+), but also a partially stripped ion (He +), are treated. At low energies, the collisional straggling of point charges is proportional to the energy, and at high energies it approaches the value predicted by Bohr. The solid LEDM calculation yields larger straggling values than the atomic LEDM does. The straggling due to change state fluctuations of a 250 keV/amu helium beam results in an enhancement of the well-known Z 2 (target atomic number)-oscillation.

Energy straggling of light-ion beams.

The dielectric function method is applied to investigate the energy straggling of protons and helium ions using the atomic and the solid local-electron-density models. A partially stripped ion is treated as well as point charges. At low energies, the conventional stragglings of a proton (${\ensuremath{\Omega}}_{{\mathrm{H}{}^{+}}^{2})}$ and of helium ions (${\ensuremath{\Omega}}_{{\mathrm{He}{}^{+}}^{2}}$,${\ensuremath{\Omega}}_{\mathrm{He}}^{2+}$ $^{2}$) due to the fluctuation in electronic excitations are proportional to the kinetic energies of the ions even when the local-electron-density models are adopted, and at high energies they approach the values predicted by Bohr. Here we do not consider the bunching term since we assume the probability of exciting an electron is small. The straggling ratio ${\ensuremath{\Omega}}_{{\mathrm{He}{}^{+}}^{2}/{\ensuremath{\Omega}}_{{\mathrm{H}{}^{+}}^{2}}}$ shows a remarkable feature that it is nearly constant up to \ensuremath{\sim}50 keV/amu, and increases gradually beyond this energy for solid targets. The estimation of the collisional straggling of helium-ion beams is performed using the charge-state fractions, resulting in displaying ${Z}_{2}$ (target atomic number) oscillations similar to those of ${\ensuremath{\Omega}}_{{\mathrm{He}{}^{+}}^{2}}$ and ${\ensuremath{\Omega}}_{{\mathrm{He}{}^{2+}}^{2}}$. The straggling caused by charge-state fluctuations, which enhance the ${Z}_{2}$ oscillations of the total straggling of helium-ion beams more sharply at the energies considered, is also estimated.

The energy straggling of protons in thin carbon foils

This work describes the measurement of the energy straggling of protons in the energy range 50–150 keV, passing through a thin carbon foil of a surface density ranging from 3 to 6 μg/cm2. The results were found to be in agreement with the theoretical model of Lindhard and Scharff. The composition of the carbon foils was established by means of a Rutherford-backscatter technique, using 2 MeV He+-ions. This procedure yielded a carbon content of about 96%.

Energy loss and energy straggling of protons and pions in the momentum range 0.7 to 115 GeV/c

The energy-loss distribution for pions and protons, with momenta between 0.7 and 115 GeV/c, has been measured in an ion-implanted detector of 300 \ensuremath{\mu}m thickness. The data are well described by an empirical energy-loss distribution.

Stopping power and straggling of 0.2-2.0 MeV protons and 0.3-3.1 MeV 4He ions in erbium

2014 The stopping power cross section and energy straggling of protons and 4He ions in erbium films evaporated in vacuum, were determined using a backscattering technique in the energy range from 0.2 to 2.0 MeV for protons, and 0.3 to 3.1 MeV for 4He ions. The experimental stopping power data were fitted by means of Brice’s formula, and agree with Ziegler’s semiempirical calculations. The accuracy of the measurements was estimated to be ± 3 %. The energy straggling results for protons in erbium are in agreement with the predictions of Bethe and Livingston. Disagreement between the experimental and theoretical points for energy straggling of 4He ions is explained by the influence of the thickness non uniformity in the target. J. Physique 43 (1982) 485-491 1 MARS 1982, Classification Physics Abstracts 61.80M

Energy straggling of protons in nickel

Straggling of protons in nickel in the energy range between 0.4 MeV and 2.6 MeV was measured for different absorber thickness. No energy dependence of the straggling width could be detected for thin absorbers and the results are in good agreement with the Bohr theory. For a thick absorber the dependence on the energy loss was found to be appreciably stronger than expected. This latter effect might be due to nuclear collisions which are neglected in the theory.

Energy straggling of protons in Be, C, Al, and Si

Ion backscattering from solid targets of elements with prominent structure in their ($p, p$) scattering cross section has been used to measure the energy straggling of protons in Be, C, Al, and Si. Single-crystal targets were used in order to minimize porosity, channeling, and surface-roughness effects. In order to obtain straggling parameters from the experimental data, corrections for contributions, other than those mentioned above, to the broadening of the observed spectra were made. The energy dependence of the stopping power of the target for the protons makes a significant contribution to the broadening of the observed backscattering spectra, and must be properly accounted for if precise values of the straggling parameters are to be obtained. A procedure is described for accomplishing this. Using the experimental technique and analysis procedure described, proton straggling was evaluated in Be, C, Al, and Si in the energy range 100-3000 keV. The results are summarized by defining $f\ensuremath{\equiv}\frac{\ensuremath{\Omega}}{{\ensuremath{\Omega}}_{B}}$, where ${\ensuremath{\Omega}}_{B}$ is the straggling predicted by the Bohr theory. The results are: $\mathrm{Be}({E}_{\mathrm{res}}=2.530 \mathrm{MeV}), f={1.0}_{\ensuremath{-}1.0}^{+0.3}$ $\mathrm{C}({E}_{\mathrm{res}}=0.47 \mathrm{MeV}), f={0.0}_{\ensuremath{-}0.0}^{+1.5}$ $\mathrm{C}({E}_{\mathrm{res}}=1.73 \mathrm{MeV}), f=0.9\ifmmode\pm\else\textpm\fi{}0.1$ $\mathrm{Al}({E}_{\mathrm{res}}=2.480 \mathrm{MeV}), f=1.25\ifmmode\pm\else\textpm\fi{}0.2$ $\mathrm{Al}({E}_{\mathrm{res}}=2.555 \mathrm{MeV}), f=1.25\ifmmode\pm\else\textpm\fi{}0.2$ $\mathrm{Si}({E}_{\mathrm{res}}=1.682 \mathrm{MeV}), f=1.15\ifmmode\pm\else\textpm\fi{}0.2$ $\mathrm{Si}({E}_{\mathrm{res}}=2.099 \mathrm{MeV}), f=1.4\ifmmode\pm\else\textpm\fi{}0.3$ In addition to the straggling results, relative elastic scattering cross sections were determined for protons on Be (2.25-2.85 MeV, ${\ensuremath{\theta}}_{\mathrm{lab}}=170\ifmmode^\circ\else\textdegree\fi{}$) and on Al (2.4-2.6 MeV, ${\ensuremath{\theta}}_{\mathrm{lab}}=170\ifmmode^\circ\else\textdegree\fi{}$).