Empirical Scaling Rule Research Articles

Size-Dependent Phenomena in Angle-Resolved Measurements of Submicron Sn Particle Scattering from a Molybdenum Surface

Nanoparticle scattering dynamics play a critical role in a wide range of astrophysical, industrial, and ambient environments; however, experimental data to guide theoretical models that predict this behavior are lacking. The experiments reported here examine these phenomena using single mass-selected, charged, submicron solid tin particles covered with an oxide layer of ∼10 nm thickness that are accelerated with varying energies onto a highly polished molybdenum surface. The scattering angle and speed for each backscattering event were measured and analyzed, revealing notable size-dependent trends in the coefficient of restitution and onset of sticking and charge transfer over the range from 150 to 500 nm diameter. The experimental results are interpreted using a mechanical model of the measured impact behavior, extending particle scattering measurements into a new intermediate size range, important for understanding the transport of submicron tin particles. An empirical scaling rule is also presented that normalizes the size-dependent behavior in terms of the ratio of the incident kinetic energy and impact contact area.

Nonperturbative scaling behavior for net ionization of biologically relevant molecules by multiply charged heavy-ion impact

A recently developed model to describe proton collisions from molecules involving basic atoms such as hydrogen, carbon, nitrogen, oxygen and phosphorus (H, C, N, O, P) is extended to treat collisions with multiply charged ions. The ion-atom collisions are computed using the two-center basis generator method (TC-BGM), which has a proven track record of yielding accurate total cross sections for electron capture and ionization. The atomic net ionization cross sections are then used to assemble two models for ion-molecule collisions: an independent atom model (IAM) that follows the Bragg additivity rule (labeled IAM-AR), and also the so-called pixel-counting method (IAM-PCM). The latter yields reduced cross sections relative to IAM-AR near the maximum, since it takes into account the overlapping nature of effective cross sectional areas. The IAM-PCM for higher-charge projectiles leads to strong reductions of net ionization cross sections relative to the IAM-AR method, and is computed directly for projectile charges $Q=1, 2, 3$. The scaling behavior of the IAM-PCM is investigated over a wide range of energies $E$, and at high $E$ it converges towards the IAM-AR. An empirical scaling rule is established which allows to reproduce these results based on proton impact calculations. Detailed comparisons are provided for the uracil target ($\rm C_4 H_4 N_2 O_2$), for which other theoretical as well as experimental results are available. Based on the scaling model derived from the IAM-PCM data it is shown how the experimental data for uracil and water bombarded by multiply charged ions can be reduced to effective $Q=1$ cross sections respectively, and these are compared to proton impact data.

Pi ^{0} and eta meson production in proton-proton collisions at sqrt{s}=8 TeV

An invariant differential cross section measurement of inclusive pi ^{0} and eta meson production at mid-rapidity in pp collisions at sqrt{s}=8 TeV was carried out by the ALICE experiment at the LHC. The spectra of pi ^{0} and eta mesons were measured in transverse momentum ranges of 0.3<p_{ text{ T }} <35 text{ GeV/c } and 0.5<p_{ text{ T }} <35 text{ GeV/c }, respectively. Next-to-leading order perturbative QCD calculations using fragmentation functions DSS14 for the pi ^{0} and AESSS for the eta overestimate the cross sections of both neutral mesons, although such calculations agree with the measured eta /pi ^0 ratio within uncertainties. The results were also compared with PYTHIA 8.2 predictions for which the Monash 2013 tune yields the best agreement with the measured neutral meson spectra. The measurements confirm a universal behavior of the eta /pi ^0 ratio seen for NA27, PHENIX and ALICE data for pp collisions from sqrt{s}=27.5 GeV to sqrt{s}=8 TeV within experimental uncertainties. A relation between the pi ^{0} and eta production cross sections for pp collisions at sqrt{s}=8 TeV is given by m_{ text{ T }} scaling for p_{ text{ T }} >3.5 text{ GeV/c }. However, a deviation from this empirical scaling rule is observed for transverse momenta below p_{ text{ T }} <3.5 text{ GeV/c } in the eta /pi ^0 ratio with a significance of 6.2sigma .

Scaling laws for electron loss from ion beams

Empirical scaling rules for projectile stripping induced by collisions with argon were extracted by analyzing a large database of experimental cross-sections. By scaling both the cross-sections and the impact velocities, data for single and multiple electron loss from negative ions, neutral particles, and singly or multiply charged positive ions could all be fitted with a single universal curve. At high energies, the scaled data are consistent with a v −1 impact velocity dependence up to scaled velocities of 10. For higher scaled velocities, limited data imply that the velocity dependence becomes v −2. Using our fitted curve, cross-sections are predicted for total and single electron loss from low-charge-state heavy ions of interest for heavy ion fusion and high-energy density research projects. Using a v −2 or a v −1 dependence to extrapolate to energies of 10–20 MeV/ u does not strongly influence the predicted magnitude of the total loss cross-sections but does influence the multiple/single loss ratios.

Charge-changing cross sections for 8–40-MeV O^{7,8+}+(Ar, Ne) collisions

Absolute cross sections for single- and double-electron capture, and single-electron loss, have been determined for O{sup 7,8+} ions colliding with Ar and Ne in the energy range 8{endash}40 MeV. Double-electron-capture cross sections were found to be factors of 5{endash}10 smaller than the single-electron capture cross sections. The present results for electron capture from Ar are systematically larger by about 30{percent} than the corresponding cross sections determined previously by Macdonald and Martin [Phys. Rev. A {bold 4}, 1965 (1971)]. The present single-electron-capture cross sections agree well with the empirical scaling rule of Schlachter {ital et al.} [Phys. Rev. A {bold 27}, 3372 (1983)]. {copyright} {ital 1997} {ital The American Physical Society}

Electronic and optical properties of strained wurtzite GaN

Band structures of wurtzite GaN (α-GaN) under strains in the region -5%−5% are calculated in a tight-binding framework. The empirical scaling rule has been used for considering the effects of hydrostatic strains. The scaling indexes are determined by fitting the deformation-potential constants with other theoretical values. The band gap at Γ point increases with the absolute value of strains. GaN turns to be of indirect band gap when strains reach 5%. The density of states and the imaginary part of dielectric function (2 (ω)) are studied. Both the shape and energy position of the highest peak in the 2 (ω) spectrum successively change with the strains. The real part of dielectric function, refractive index and the effects of the strains on them are also shown.

Structural aspects of dense solid halogens under high pressure studied by x-ray diffraction—Molecular dissociation and metallization

We have performed structural studies of solid bromine under ultrahigh pressures by using advanced synchrotron radiation X-ray diffraction techniques incorporated with a diamond-anvil high pressure cell. Reliable intensity measurements in its molecular phase up to 80 GPa, where it undergoes the molecular-to-monatomic phase transition (molecular dissociation), have enabled us to determine pressure dependence of molecular positions. By combining the previous results of iodine and chlorine, we have found an empirical scaling rule with respect to the crystal structure scaled with its intramolecular bond length for these three isostructural systems. By applying the Herzfeld criterion, we have also found a scaling rule for the metallization process prior to molecular dissociation.

Double-atom excitation in heavy-particle collisions induced by the electron-electron interaction

The authors present an analytical method for rapid calculation of matrix elements for double-atom excitation induced by the electron-electron interaction. The method is applied to calculation of probabilities and cross sections for double-atom n=2 excitation in H(1s)-H(1s) and various A(Zp-1)+(1s)-H(1s) collisions at intermediate and high projectile energies. Based on first-order perturbation theory an empirical scaling rule for the double-atom excitation cross sections is derived. The calculated cross sections are shown to be in good agreement with this scaling relation.

Electronic band structure of coherently strained GexSi1-x alloys on Si(001) substrates.

Electronic band structure of coherently strained ${\mathrm{Ge}}_{\mathit{x}}$${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ alloys grown on Si(001) substrates are calculated in a tight-binding framework. The empirical scaling rule has been used for considering the effects of hydrostatic strains on the energy band structures. The scaling index is determined by fitting the deformation-potential constants with their experimental values. The energy gaps and the valence-band structures of ${\mathrm{Ge}}_{\mathit{x}}$${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$/Si(001) as a function of the alloy composition x are given and the effects of the strains on the constant-energy surfaces at the top of the valence band and the effective masses of electrons and holes are discussed. Finally, the imaginary part of dielectric constant ${\mathrm{\ensuremath{\epsilon}}}_{2}$, the density of states for ${\mathrm{Ge}}_{\mathit{x}}$${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$/Si(001), and the effects of the strains on them are shown.

OPTICAL PROPERTIES OF COHERENTLY STRAINED ALLOYS GexSi1-x ON Si (001) SUBSTRATES

The optical constants were calculated for coherently strained alloys GexSi1-x on Si (001) substrates with tight - binding method. The effects of strains on the electronic energy band structures are taken into account by modifying the tight-binding parameters according to the variations of the bond angle cosines and the bond lengths. For the changes of the bond lengths, an empirical scaling rule is used, in which the scaling indexes are determined through fitting the calculated values of deformation potential constants to their experimental values for Ge and Si. The momentum matrix elements in the formula for calculating the imaginary part of dielectric constants are determined by fitting the ε2 of Ge and Si. The calculated results for the optical constants , including the imaginary part of dielectric constant, the refractive index, the absorptance and the reflectance, are given both for the strained alloys and bulk alloys GexSi1-x as x=0.2 and 1. By comparison of them, the effects of strains on the optical constants are discussed.

Single and double electron capture in collisions of highly ionized, decelerated Ge ions with Ne

Experimental cross sections for non-radiative single and double electron capture from Ne target into decelerated H-like Ge ions at collision energies of (4-12) MeV u-1 are presented. The results are compared with theoretical calculations and an empirical scaling rule. Information concerning the impact parameter dependence of electron capture is extracted using classical trajectory Monte Carlo calculations.

Single electron capture by high velocity bare and one-electron projectiles in collisions with molecular hydrogen

Total cross-sections for single electron capture by 0.5–2.4 /amu bare and hydrogen-like F and O projectiles on molecular hydrogen have been measured. The measured cross sections are compared with empirical scaling rules and are found to scale as E −4.82 and q 3.75.

Single-electron capture and loss cross sections versus target Z for 1 MeV/u oxygen ions incident on gases.

In an ion-atom collision electron capture and loss depend, in general, on the beam energy, projectile charge state, and target Z. While most fundamental theories of capture and loss are formulated in terms of a single electron moving in the fields of the pertinent nuclei, various scaling laws have been developed to account for partially stripped ions and multielectron targets. The purpose of the present work is to systematically examine the dependence of capture and loss cross sections on target Z. Cross sections for single-electron capture and loss for 1 MeV/u oxygen ions passing through gas targets were measured as a function of target Z for incident projectile charge states of 5+, 6+, 7+, and 8+. The targets used were ${\mathrm{D}}_{2}$, He, Ne, Ar, and Kr. The electron-capture measurements are generally in reasonable agreement with existing theoretical and empirical scaling rules. The electron-loss cross sections differ appreciably from predictions of the plane-wave Born approximation, particularly for the heaviest targets studied.

A determination of the relative bulk moduli of GaInAsP and InP

Abstract Relative spectroscopic measurements of the bulk modulus of GaIn As P lattice-matched to InP yield a value within 1% of that of InP. This surprising result is not in agreement with accepted interpolated values but is predicted by Keyes' empirical scaling rule for elastic constants.

Electron capture and loss for 2.5-200-MeV 16S

Electron capture and loss cross sections have been measured for highly charged (q=13+) sulfur ions with energies 2.5--200 MeV colliding with helium. Electron capture varies by nearly six orders of magnitude over the energy range investigated, while electron loss varies by only about a factor of 2. The capture cross sections are in reasonable agreement with classical and empirical scaling rules, while the loss cross sections agree well with the plane-wave Born approximation.

Resonant transfer and excitation (rte) and nonresonant electron transfer and excitation (nte) in Si 11+ on He collisions

Recently, crossed electron-ion beam experiments, involving 2s-2p or 3s-3p excitations designed to measure the scale for dielectronic recombination (DR) cross sections have been reported. At TUNL, the scale for DR cross sections in 1s-2p excitations is obtained in a manner similar to that of Tanis et al. [1]. Si 11+ ion beams with energies varying from 15–100 MeV are incident on He and Si K X-rays in coincidence with Si 10+ ions are recorded so that cross sections can be extracted. At 85 MeV, RTE is the dominant process. We conclude that Hahn's [2] DR cross section calculations are somewhat too high. At 20 MeV NTE dominates. Impact parameter dependent models of capture and excitation describe the shape the the NTE feature quite well. The energy dependence of Si K X-ray excitation cross sections is reasonably well described by a semiclassical calculation with a straight line trajectory and screened hydrogenic wavefunctions. The capture cross sections are compared with an impact parameter dependent Bohr-Lindhard model, scaled OBK, and empirical scaling rules.

Electron capture for fast highly charged ions in gas targets: An empirical scaling rule

A universal empirical scaling rule for electron-capture cross sections is reported for fast, highly charged ions in atomic and molecular targets. Projectiles range in energy from 0.3 to 8.5 MeV/amu, with charge states as high as 59+. This rule permits prediction of electron-capture cross sections for a wide variety of projectile-target combinations.

On the scaling of crater dimensions: 2. Impact processes

Existing theoretical and empirical scaling rules for cratering phenomena are commonly based upon implicit assumptions regarding the roles of the impactor conditions and the target properties. It is common, for example, to assume that only the kinetic energy of the impactor and the work against gravity in the target govern large impact craters. A more general case is considered here. A spherical impactor is characterized by its kinetic energy, momentum, and various material properties of both the impactor and the target. Under these general assumptions, scaling among both similar and nonsimilar events is studied. Similar events always satisfy the cube‐root mass scaling rule. The dependence on energy varies according to how the similarity is achieved. For constant mass or constant velocity tests, variations in gravity are required, and cube‐root energy scaling is obtained. For a strengthless material only, quarter‐root energy scaling is possible, but only for restricted variations in both impactor mass and velocity, which imply that the mass is proportional to the three‐quarters power of the energy. Scaling rules for the more common and more useful application to nonsimilar events (e.g., variable energy and/or velocity at constant gravity) are shown to be bounded by certain well defined limits. Dimensional analysis is used with plausible assumptions on the sign of the partial derivative of the crater volume with regard to each of the independent variables. Distinctions are made among different forms of cube‐root scaling obtained from strength‐dominated phenomena. Quarter‐root scaling is shown to be a distinct limit to gravity domination, occurring only when there is no dependence upon impactor velocity and target strength. Special cases of the general scaling obtained correspond to degenerate forms common in the literature. Of these, energy scaling, momentum scaling, strength scaling, and gravity scaling are shown to be the bounds for all permissible scaling rules. This result is shown to be in agreement with a comprehensive collection of data obtained from a survey of the literature. Included are experimental results for both geological materials and metals, analytical models, and finite difference code calculations.

Empirical determination of thermal expansion in insulators with no experimental input

The localized-continuum model of thermal expansion which was applied to insulators in the previous paper is further refined by empirical means to allow the calculation of thermal expansion in insulating materials with no input other than the chemical formula. Estimation of the Madelung constants is made by using the empirical methods of Kapustinsky and Templeton while the atomic nearest-neighbour distances are estimated by using atomic radii tables. Approximate values of repulsive exponents are obtained by using an empirical scaling rule and a table of monovalent repulsive exponents. The Debye temperature is approximated by a simple formula involving the estimated parameters of the interatomic potential. Using this approach, thermal expansion is estimated in a group of binary and complex ternary materials and a group of complex salts. The agreement with experiment is generally good, although the elimination of experimental input appears to increase the probable error in the calculations by 15% to 20%. The results indicate that this approach is capable of predicting with reasonable accuracy the thermal expansion in a wide range of insulating materials with no experimental input and no adjustable parameters. The limitations of this method for certain cases is also discussed.

Comments on an empirical scaling rule related to x-ray emission from superheavy quasimolecules

A scaling rule related to maximum contributing impact parameters for inner shell Coulomb ionization probabilities is studied by an adiabatic perturbation approach. Qualitative arguments are given for an extended validity of the scaling law to very heavy collision systems.