Energy Loss Moments Research Articles

Contrast modes in a 3D ion transmission approach at keV energies

We present options for visualizing contrast maps in 3D ion transmission experiments. Simultaneous measurement of angular distributions and flight time of ions transmitted through self-supporting, single-crystalline silicon foils allows for mapping of intensity and different energy loss moments. The transmitted projectiles were detected mainly for random beam-sample orientation using pulsed beams of He ions and protons with incident energies 50 and 200 keV. Differences in contrast, observed when varying the projectile type and energy, can be attributed to sample nuclear and electronic structure and bear witness to impact parameter dependent energy loss processes. Our results provide a base for interpretation of data obtained in prospective transmission studies when for example using a helium ion microscope.

Extended wave-packet model to calculate energy-loss moments of protons in matter

Fil: Archubi, Claudio Dario. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Astronomia y Fisica del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomia y Fisica del Espacio; Argentina

A comparative study of threshold effects in the energy loss moments of protons, electrons and positrons using dielectric models for band-gap materials

A comparative study of the energy loss, mean free path and straggling of protons, positrons and electrons in an electron gas is performed using three dielectric models which represent the case of metals (Lindhard model for a free electron gas) and the cases of semiconductors and insulators (Levine and Louie model and Brandt and Reinheimer model for systems with a band gap). The properties of individual and collective contributions according to each model and for each of the particles are analyzed. In particular, the effects produced by the band gap of the material and by the properties of the incident particle are analyzed in detail. Significant differences related to the mass and to the indistinguishability (in the case of electrons) are described. Analytical expressions for the high-energy limit are derived in a simple way using the plasmon-pole approximation.

Calculation of average projected range and range straggling of charged particles in solids

Based on the KRAL equations (Ashworth et al., 1991. J. Phys. D 24, 1376–1380), which take into account second-order energy loss moments, projected ranges and range moments of ions in mono-atomic targets were calculated. For this purpose, the set of three coupled differential equations of second order was first reduced to a set of differential equations of first order, and solved afterwards numerically, using initial values with respect to the electronic stopping power according to Montenegro et al. (1981) which are valid at non-relativistic velocities in all solid targets and are consistent with experimental data for a wide range of energies. The numerical solution of the KRAL equations was performed by applying a continuous Runge–Kutta method of seventh–eighth order and a Fehlberg Runge–Kutta method of fourth–fifth order. The results with respect to the projected ranges of Xe ions in amorphous Si are in a good agreement with measured data, and those with respect to the projected ranges and range straggling of H or B ions in amorphous Si with data calculated by the standard theories TRIM, PRAL 97 and WS (Wang and Shi, 1990. J. Phys. D 23, 1282–1289).

Calculation of the first four moments of electronic energy loss of protons in aluminium

We present a novel scheme for calculating electronic energy losses and their higher moments which allows to fully account for the electronic structure of solids. Typical structures in solid materials are found in metals with their half filled conduction bands, and in insulators which exhibit a completely filled valence band, and above this a forbidden band gap. The conduction electrons in a metal resemble most closely a free electron gas which would fill a Fermi sphere in velocity space (or in momentum space) homogeneously from ν = 0 to ν = νF. The core electrons are separated from the freely available phase space by distinct band gaps. For protons slowing-down in metals, this means that conduction electrons can accept small energy transfers which are inacceptable for core electrons or valence electrons in insulators. This leads to a completely different stopping behavior of metals, particularly at low velocities. In metals we expect that all energy loss moments increase strictly proportional to νn starting at zero velocity, i.e., the energy loss proportional to ν1, the energy loss straggling proportional to ν 2 etc. We perform the stopping calculations first for the metal Al, where the theoretical input is well known, and where most measurements have been performed. The comparison of theory and experiments shows agreement within the experimental straggle.

Calculation of the first four moments of electronic energy loss of protons in insulators

A novel scheme is presented for obtaining energy loss distributions for protons slowing-down in metals and bandgap materials. This scheme is here applied mainly to compare electronic energy loss distributions at low velocities, where the bandgap is most effective (projectile velocity less than Fermi velocity, v<vF). In this velocity regime the electronic energy loss is governed by the Pauli principle which leads immediately to the results that the energy loss moments Mn,n=1,…,4, change proportionally to the nth power of projectile velocity, Wn=dMn/dx=Cn∗vn, in metals (no band gap), and they are due to the energy transfers to conduction electrons only. In bandgap materials, the smallest energy transfers cannot occur, and all energy loss moments Wn are thus reduced. Only at velocities near the Fermi velocity, the valence electrons reach nearly the same level as the conduction electrons in a metal, and at this velocity also the inner shell electrons begin to contribute, so that the stopping power (the first moment of energy loss) reaches its maximum above vF. At higher velocities, we find that the moments Mn increase proportionally to the (n−2)th power of projectile energy, Wn=dMn/dx=Cn∗En−2,n>1. The stopping power (n=1, first moment of energy loss) comes very close to the predictions of Bethe or Bloch. At low velocities, all energy transfers are reduced considerably by the presence of a bandgap, and differ greatly from metals. Despite of using some simplifications, the results obtained for lithium metal and for the bandgap material diamond agree quite well with recent experimental findings.

Impact parameter dependence of electronic energy loss

Abstract Energy-loss moments of the electronic energy loss are calculated as functions of impact parameter for protons of 1 to 7 MeV incident upon Be, Al, Cu, Ag, and Ta targets. The Lindhard dielectric-function formalism is used, in the local-density approximation. The target-atom electron densities are Lenz-Jensen, and the projectile is a point charge. The moments are fitted by a simple algebraic function, and the energy and Z2 dependence of the parameters of this function are given.

New projected range algorithm as derived from transport equations

A new PRojected RAnge ALgorithm — further referred to as PRAL — was presented at two earlier conferences: PRAL as derived from an angular diffusion model (Albany 1980), and a more plausible direct derivation (Sidney 1981). This algorithm is distinguished from other projected range calculations by (i) being simpler and faster on the computer, (ii) readily allowing approximate analytic solutions, (iii) taking into account higher energy loss moments, e.g. nuclear and electronic straggling, and (iv) dealing with all mass ratiosM 2/M 1 in the same manner, and hence allowing to treat any chemically composed target materials, e.g. alloys, metal oxides, -hydrides, new semiconductor materials, etc. — Since results of PRAL show good agreement with presently available results from transport equations (backward Boltzmann equations), it was suspected that both approaches are equivalent. This indeed is proved in the present paper by deriving PRAL from the Boltzmann equation.

The energy-loss spectra of fast charged particles passing through a thin metallic film

The energy-loss spectra of heavy, fast charged particles passing through a thin film are studied. The authors generalised the Landau transport theory to include the quantum mechanical transient effect caused by the finite dimension of a thin film. The energy-loss spectrum of a cluster of correlated charged particles is then considered. Since the effect becomes important when the thickness a <or approximately=V/ Omega p, where omega p is the plasma frequency and V is the velocity of the incident particle, the coupling constant between the charged particle and bulk plasmons also must be modified to account for the size dependence. They have included both effects in a derivation of the probability distribution function and the energy-loss moments. The correlation effect on the energy loss moments is emphasised.

Asymptotic formulae for slow neutron scattering by bound atoms

Using the Placzek-Wick method, asymptotic formulae have been obtained at high neutron energies for the mean cosine of the scattering angle, energy loss moments and conjugate energy loss moments. The latter differ from the ordinary moments in that integration is conducted not over the final neutron energy but the initial neutron energy.