Energy Loss Formula Research Articles

A study of nonlinear radiation damping by matching analytic and numerical solutions

A linear oscillator coupled to a nonlinear field was studied using a mixed analytic-numerical matching scheme. An approximate causal solution was constructed in the radiation zone and matched to a numerical solution obtained using a finite differencing scheme in the inner zone. The requirement that the two solutions agree in the overlap region determined the arbitrary function in the retarded solution and at the same time allowed one to extend the numerical scheme arbitrarily into the future. The late time behavior of the system was investigated using a variety of initial conditions. In all the cases studied this behavior was found to be independent of which initial conditions were used. Also, it was shown that the linearized “monopole energy loss formula” breaks down in cases involving either fast motions or strong nonlinearities.

The new formula for gravitational radiation energy loss Applications to the axisymmetric two-body problem and the binary pulsar

view Abstract Citations (3) References (24) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The New Formula for Gravitational Radiation Energy Loss: Applications to the Axisymmetric Two-Body Problem and the Binary Pulsar Cooperstock, F. I. ; Lim, P. H. Abstract By accounting for nonlinearities and applying the Gauss theorem for retarded functions, the formula for gravitational radiation energy loss in general relativity is developed. This replaces the quadrupole formula, while clarifying the evident minimal domain of validity of the quadrupole formula. The new formula is considered in detail for the axisymmetric two-body problem with the conventional equation of state, epsilon = epsilon (Rho), where the nonlinear contributions are shown to cancel and the time-retardation contributions vanish individually. An order-of-magnitude analysis is performed for the binary pulsar, whereby it is found that both types of new contributions have the potential to compete with the quadrupole term. Publication: The Astrophysical Journal Pub Date: May 1986 DOI: 10.1086/164205 Bibcode: 1986ApJ...304..671C Keywords: Axisymmetric Bodies; Binary Stars; Gravitational Waves; Pulsars; Relativity; Two Body Problem; Computational Astrophysics; Energy Dissipation; Equations Of State; Gauss Equation; Quadrupoles; Astrophysics; GRAVITATION; PULSARS; RADIATION MECHANISMS; RELATIVITY; STARS: BINARIES full text sources ADS | data products SIMBAD (1)

The new formula for gravitational-radiation energy loss and the axially symmetric two-body problem

We present the new formula for gravitational-radiation energy loss that replaces the familiar quadrupole formula. The new formula helps to clarify why the quadrupole formula works when it does. The origin of the correction tensor is discussed and then applied to the axially symmetric two-body problem. With the conventional equation of state, ε = ε(P), the correction tensor vanishes and the radiation is that of the bulk-motion quadrupole formula. This is to be compared with our earlier result with an unconventional equation of state giving more radiation and with that of critics claiming dominant radiation via internal motions with the quadrupole formula. An order-of-magnitude calculation is performed for a binary system, where it is found that there is the potential for contributions from nonlinear terms to be as significant as those from the linear terms after(Gm/α)−5/6 orbits. This occurs after ~102 years for the binary pulsar PSR1913 + 16.

New formula for gravitational radiation energy loss in general relativity.

Field contributions and the Gauss theorem for retarded functions lead to a new formula for gravitational radiation energy loss. It clarifies the applicability of the quadrupole formula for systems with ${\mathrm{T}}^{\mathrm{ik}}$-dominated motion. Applied to the axially symmetric two-body problem with equation of state \ensuremath{\epsilon}=\ensuremath{\epsilon}(P) for short free-fall times, the new formula yields the bulk-motion quadrupole-formula result. This contrasts our earlier larger flux with unconventional equation of state and that of critics claiming a quadrupole formula flux from internal motions. An order of magnitude estimate yields potential corrections of quadrupole formula size after \ensuremath{\sim}${10}^{2}$ yr for the binary pulsar.

Stopping power formula for intermediate energy electrons

An accurate electron energy loss formula, modified from that of F. Rohrlich and B.C. Carls (1954), is presented which is useful in the energy range from a few tens of electron volts to 10 keV. Higher order z3 corrections, which are similar to those for heavy charged particles, and z2 corrections, defined as deviations from the equipartition rule, are taken into account. For energies below 200 eV, the semiempirical effective charge of electrons, adjusted to fit the semiempirical formula of L.R. Peterson and A.E.S. Green (1968) is introduced. The calculated stopping powers are in good agreement with semiempirical formulae for intermediate energies.

The measurement of pre-equilibrium heavy ion energy loss

A method is described for the measurement of pre-charge-state-equilibrium energy losses of heavy ions in solid targets. Preliminary results for 130 MeV 35Cl incident on carbon targets between 3 and 130 μg/cm 2 thick are reported to demonstrate the techniques. The results are described in terms of the Bethe formula for energy loss.

Feynman graph derivation of the Einstein quadrupole formula

The one-graviton transition operator and, consequently, the classical energy-loss formula for gravitational radiation are derived from the Feynman graphs of helicity+or-2 theories of gravitation. The calculations are done both for the case of electromagnetic and gravitational scattering. The departure of the in and out states from plane waves owing to the long-range nature of gravitation is taken into account to improve the Born approximation calculations. This also includes a long-range modification of the graviton wavefunction which is shown to be equivalent to the classical problem of the true light cones deviating logarithmically at large distances from the flat space light cones. The transition from the S-matrix elements calculated graphically to the graviton transition operator is done by using complementarity of space-time and momentum descriptions. The energy-loss formula drove originally by Einstein is shown to be correct.

Modification of the theory of energy-angle distributions of low energy heavy particles after penetration through matter by Meyer, Klein, and Wedell

Abstract In their theory Meyer, Klein and Wedell consider the electronic stopping in an insufficient way, since both the nuclear and electronic energy loss in a single collision are assumed to be proportional to the square of the scattering angle. This assumption, however, is only fulfilled for the nuclear energy loss (in the small-angle approximation). In this paper, energy-angle distributions are calculated using more realistic distributions for the electronic energy loss-a Gaussian distribution and a distribution, calculated by means of the Landau-Vavilov equation and Firsov's formula for the electronic energy loss in a single collision.

Monte Carlo simulation of ion beam penetration in solids

Abstract The essential features of a Monte Carlo program which simulates the scattering and the retardation of energetic ions in amorphous targets are presented. The new experimentally derived universal scattering cross-section of Kalbitzer and Oetzmann is used to describe nuclear scattering events. For electronic energy loss, the Lindhard-Scharff and Bethe formulae are used. The program was primarily developed to study the spatial distribution of ion deposited energy for lithographic applications. However, for initial verification, the program has been used to study the fundamental characteristics of ion transport for various ion-target combinations. The theoretical predictions obtained using the above expressions for nuclear scattering and electronic energy loss are shown to agree very well with experimental results.

Gravitational Radiation Quadrupole Formula is Valid for Gravitationally Interacting Systems

An argument is presented for the validity of the quadrupole formula for gravitational radiation energy loss in the far field of nearly Newtonian (e.g., binary stellar) systems. This argument differs from earlier ones in that it determines beforehand the formal accuracy of approximation required to describe gravitationally self-interacting systems, uses the corresponding approximate equation of motion explicitly, and evaluates the appropriate asymptotic quantities by matching along the correct space-time light cones.

Fundamental Studies of the Interproximity Effect in Electron-Beam Lithography

We have developed a computer simulation program based on the Monte Carlo calculation for electron-beam lithography, which can take account of pattern geometry. The three-dimensional spatial distribution of the absorbed energy density in a resist was calculated for a 0.8-µm thick poly-methyl methacrylate film on a silicon substrate irradiated by 20-keV electrons. The pattern geometries investigated here are composed of the two adjacent elements of a 20-µm square and a 1.5×10 µm2 rectangle, where the proximity effect cannot be disregarded. Some exposure experiments under the same conditions were conducted using an electron beam exposure system in the dose range 0.72×10-4 to 5.43×10-4 C/cm2. The plane profiles of the developed resist-pattern agree well with those predicted by the simulation. Though the accuracy of the simulation decreases at low doses, there is a prospect of improvement by introducing the energy-loss formula proposed by Spencer and Fano instead of Bethe's formula used here.

Stopping power and straggling of 65–500 keV lithium ions in H 2, He, CO 2, N 2, O 2, Ne, Ar, Kr and Xe

Stopping power and straggling of lithium ions with velocities around the Bohr velocity (65–500 keV) have been measured for a number of simple gases (H 2, He, CO 2, N 2, O 2, Ne, Ar, Kr and Xe). In contrast to the results of recent extensive measurements in solids by the inverted Doppler-shift method, the measured stopping powers are not strictly proportional to velocity, and for all targets d E/d x increases significantly faster than E 0.5. At the Bohr velocity, the measured stopping powers do not agree much better with results of semiempirical calculations based on Hartree-Fock-Slater wavefunctions than with the simple Lindhard-Scharff formula. For lighter gases, the straggling data are in approximate quantitative agreement with Hvelplund's calculations based on Firsov's differential energy-loss formula.

Tachyons do not emit Čerenkov radiation in vacuum

1. Recently, many papers dealt with the so-called Cerenkov radiation in vacuum by tachyons. We arc referring in particular to the last one, appeared in Phys ica l Rev i ew D (1). We want to underline that tachyons are not expected to emit ~crenkov radiation in vacuum. Most o~ the previous pap~s (z'~) are characterized by ~n ~bitra~y, uncritical (*) extension of the usual energy-loss formula (4) for Superlumin~l velocities. Such an unjustified extension led of course to difficulties (~) or inconsistencies (3), one of them JONES was precisely engaged with (1). Even at first sight, that extension does not really appear to be sound, since---as is widely known (~)--~ercnkov radiation comes out not from the (~ radiating ~) particle itself, but from the (material) medium. So that, in the usual context, the very expression (~ Cerenkov radiation in vacuum ~ appears meaningless, unless one simultaneously provides a suitable theory about a possible (~ vacuum structure )). But most experimenters have been till now looking for tachyons through detection of Cerenkov radiation, assumed to exist in vacuum too, so that it is worth-while to criticize that assumption a bit more.

Corrections to the Bethe-Bloch formula for average ionization energy loss of relativistic heavy nuclei Close collisions

Corrections to the Bethe-Bloch formula for a average ionization energy loss of relativistic nuclei are calculated by numerically integrating the exact Mott cross section. Results are tabulated in the form of a correction term Φ( β, Z) for nuclei with Z = 6, 12, 26, 36, 52, 60, 75, 80, 92, 104 and 114 and velocities corresponding to β = 0.85, 0.90, 0.95, 0.97, 0.99 and 0.999. These results are compared with closed form corrections obtained from the second and third order Born approximations. Close agreement is found between the corrections from the Mott and third order Born approximations for Z ≤ 60 and β ≥ 0.85.

A fast analogue particle identification system

An analogue particle identifier is described which uses a four quadrant integrated circuit multiplier. The identification function generated in this identifier is based on the Bethe-Bloch energy loss formula. The output is produced with an overall delay in the region of 120 ns. The input pulses are standard RC shaped monopolar pulses and the output pulses are positive monopolar and positive leading bipolar. The time jitter introduced by the circuit as measured at the zero crossing point of the bipolar output pulse is less than 10 ns fwhm. The dynamic range of the circuit is from 2 MeV to 110 MeV and is covered by two forms of the identification function. Measurements with charged particle beams are presented to illustrate the operation of the system.

Z3Corrections to Energy Loss and Range

Higher-order corrections to the stopping power proportional to ${z}^{3}$ are evaluated. Both close and distant collisions are considered. The energy-loss formula can be written $\frac{\mathrm{dE}}{\mathrm{dx}}={z}^{2}I+{z}^{3}({J}_{c}+{J}_{d})$, where $I$ is the customary lowest-order energy loss and ${J}_{c}$ and ${J}_{d}$ are the close- and distant-collision parts of the ${z}^{3}$ term, respectively. The close-collision contribution ${J}_{c}$ is a relativistic effect, first estimated in unpublished work by Fermi. It has the simple form ${J}_{c}=\frac{\ensuremath{\pi}{\ensuremath{\alpha}}^{C}}{2\ensuremath{\beta}}$, where $C$ is the standard constant multiplying ${\ensuremath{\beta}}^{\ensuremath{-}2}$ times the Bethe-Bloch logarithm in $I$ and $\ensuremath{\alpha}$ is the fine-structure constant. At high energies ${J}_{c}$ gives a constant-${z}^{3}$ contribution to the energy loss and causes a range difference $\ensuremath{\Delta}R$ roughly proportional to the range $R$ for stopping particles of the same mass and energy, but opposite charge. For $2<\frac{P}{\mathrm{Mc}}<20$, $\frac{\ensuremath{\Delta}R}{R}$ changes by less than \ifmmode\pm\else\textpm\fi{}6% and depends only slightly on the stopping material, varying from 1.9 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}3}$ for carbon to 2.5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}3}$ for lead for $z=\ifmmode\pm\else\textpm\fi{}1$. The distant-collision effect is important only at low velocities. The calculation of this contribution is patterned after a recent work of Ashley, Ritchie, and Brandt, but differs from it in detail. Using a statistical model for the atom it is found that at low velocities the relative ${z}^{3}$ contribution can be written $\frac{{J}_{d}}{I}=\frac{F(V)}{{(Z)}^{\frac{1}{2}}}$, where $Z$ is the atomic number of the stopping medium and $F(V)$ is a universal function of the reduced velocity variable $V=\frac{137\ensuremath{\gamma}\ensuremath{\beta}}{{(Z)}^{\frac{1}{2}}}$ In the region where $\frac{{J}_{d}}{I}$ is appreciable $(1<V<10)$, $F(V)$ varies as ${V}^{\ensuremath{-}n}$ with $n\ensuremath{\simeq}2.0\ensuremath{-}2.5$. These results on the ${\mathcal{z}}^{3}$ effect at low velocities are in good agreement with available data on comparison of the energy loss of helium ions and protons of the same velocities. Range differences are calculated for carbon, copper, lead, and emulsion absorbers, including the effects of both close and distant collisions. The results are in rough agreement with data on slow-stopping pions and $\ensuremath{\Sigma}$ hyperons in emulsions and in good agreement with very recent measurements of fast positive and negative muons. The upper limit of the range of validity of the results is examined in some detail. It is found that the approximations begin to fail for dynamic reasons above $\ensuremath{\gamma}\ensuremath{\simeq}20$ for muons, and presumably also for other heavy particles.

Tachyov and Čerenkov Radiation

Lapedes and Jacobs1 give a formula for the gravitational Cerenkov energy loss per second of a tachyon and for the resulting “lifetime”. Their results are, however, invalid, as were earlier considerations of electromagnetic Cerenkov radiation by Alvager and Kreisler2. The reasons are given in ref. 3; I shall summarize them in this letter and describe the theory of Cerenkov radiation from tachyons.

The density effect and rate of energy loss in common plastic scintillators

The density effect term in the rate of ionization energy loss formula has been determined for common types of plastic scintillator following the well-known dispersion model of Sternheimer. The rate of energy loss and the most probable energy loss in thin detectors made of plastic scintillator have been calculated and tabulated systematically taking into consideration both density effect and shell correction terms. The values of fwhm as percentage of the most probable energy loss in detectors of three different thickness have been calculated and presented graphically covering the range of interest in both high energy physics and cosmic rays.

Energy Loss of High-Energy Cosmic Rays in Pair-Producing Collisions with Ambient Photons

General and essentially exact formulas for the energy-loss process $A+\ensuremath{\gamma}\ensuremath{\rightarrow}A+{e}^{+}+{e}^{\ensuremath{-}}$ are given for high-energy cosmic-ray nuclei traversing an isotropic radiation field of arbitrary energy spectrum. These formulas are obtained by integrating the appropriate differential cross section in the rest frame of the cosmic-ray particle. Results are then given for the case of a Planckian photon distribution at arbitrary temperature. The effect of this energy loss on a universal flux of cosmic rays is then investigated along with the possibility that the observed "knee" at high energies is due to pair production. Finally, the energy-loss formula is applied to quasistellar objects, and it is shown that if these objects are at cosmological distances, they are opaque to particles above a certain energy which depends upon the size of the object.

On the temperature rise in electron irradiated foils

Abstract The temperature rise in thin metallic foils under intense electron irradiation has been computed as a function of beam conditions, electron energy, and the physical properties of the foil. The model due to Gale and Hale was used in conjunction with the corrected Bethe electron energy loss formula for relativistic electrons. Although the nature of the assumptions made renders the results generally unapplicable to low thermal conductivity materials, a set of beam conditions etc. are specific for the particular case of a small temperature rise in a ceramic foil (UO2).