Formula For Cross Section Research Articles

New semi-empirical formulae for (n,d) cross sections at 14–15 MeV

The aim of this letter is to suggest new semi-empirical formulae for the prediction of (n,d) cross sections with a neutron energy of 14–15 MeV. In this context, the cross sections are calculated using systematic that is function only of the reaction Q-value and mass number A. The (n,d) reactions are analyzed by using the systematics based on the statistical theory. Thus, the present semi-empirical formulae compared with the cross section data in literature give a good fit for the prediction of the (n,d) cross sections at 14–15 MeV.

Radiative transfer in atmospheres with a large chaotic magnetic field

ABSTRACT We derive the radiative transfer equations for all Stokes parameters of continuum radiation in atmospheres with any value of homogeneous magnetic field $\boldsymbol{B}$. The explicit formulas for cross-sections and the phase shifts are given with allowance for absorption effects. We consider the scattering of non-polarized radiation in an optically thin envelope with a dipole magnetic field. The presented theory is valid for magnetic fields B ≤ 1010G. In general, a magnetic field consists of the mean value $\boldsymbol{B_0}$ and the chaotic part $\boldsymbol{ B^{\prime }}$. The latter is assumed to have an isotropic distribution over directions and a Gaussian-type distribution over the value B′. It is shown that for B0(G)λ(μm) ≪ 108, the fluctuations play a dominating role. This case is considered in detail. First of all, we derive the system of transfer equations for observed averaged Stokes parameters. The averaging procedure consists of two stages: the averaging of fluctuations $\boldsymbol{B }^{\prime }$ over values and the averaging of these over all directions. The averaging over Gaussian fluctuations B′ is carried out using the exponential Fourier transform of polarizability tensor components and the known formula for the averaged exponential. This technique is available for arbitrary values of a magnetic field, both large and small. The system of transfer equations for four averaged Stokes parameters, I, Q, U and V, splits up into two independent systems – for I, Q and V, U parameters. The form of equations for the case of large magnetic fluctuations differs strongly from the Thomson scattering. These equations describe the large decrease of linear and circular polarization of observed radiation.

Accurate Gaunt Factors for Nonrelativistic Quadrupole Bremsstrahlung

Abstract The exact result for nonrelativistic quadrupole bremsstrahlung in a Coulomb field was established only recently in Pradler & Semmelrock. It requires the evaluation and integration of hypergeometric functions across a wide range of parameters and arguments, which, in practice, is unfeasible. Here we provide a highly accurate tabulation of the Gaunt factor for quadrupole radiation, its thermal average in a Maxwellian plasma, and the associated cooling function over the entire kinematically relevant range. In addition, we provide a simple approximate formula for the emission cross section that works to within a few percent accuracy for all practical purposes. The results can be applied to the scattering of electrons with themselves, for which quadrupole radiation is the dominant process.

Factorization and transverse phase-space parton distributions

We first revisit impact-parameter dependent collisions of ultra-relativistic particles in quantum field theory. Two conditions sufficient for defining an impact-parameter dependent cross section are given, which could be violated in proton-proton collisions. By imposing these conditions, a general formula for the impact-parameter dependent cross section is derived. Then, using soft-collinear effective theory, we derive a factorization formula for the impact-parameter dependent cross section for inclusive hard processes with only colorless final-state products in hadron and nuclear collisions. It entails defining thickness beam functions, which are Fourier transforms of transverse phase-space parton distribution functions. By modelling non-perturbative modes in thickness beam functions of large nuclei in heavy-ion collisions, the factorization formula confirms the cross section in the Glauber model for hard processes. Besides, the factorization formula is verified up to one loop in perturbative QCD for the inclusive Drell-Yan process in quark-antiquark collisions at a finite impact parameter.

New empirical formula for nucleon-induced nonelastic cross sections based on two physical effects

A simple universal parametrization of nucleon-induced nonelastic cross sections is presented for a wide range of targets that is valid for the entire energy range from zero to a few gigaelectronvolts. We review several early studies by Letaw et al., Pearlstein, Shen, Niita et al., and Tripathi et al., and our proposed approach differs completely from the formulas therein. The present formula is constructed based on recently discovered physical effects involving Coulomb repulsion and the discrete-level constraint and is based on the assumption that cross sections are continuous in both incident energies and targets. Our formula is given by a set of smooth functions of the mass number, which differs from the best formula to date by Tripathi et al. To compare our formula precisely with that by Tripathi et al., we proposed the relative error index, which indicates the relative error between experimental data and predicted values. For the $^{12}\mathrm{C}, ^{27}\mathrm{Al}, ^{56}\mathrm{Fe}, ^{\mathrm{nat}}\mathrm{Ag}, ^{\mathrm{nat}}\mathrm{Cd}, ^{\mathrm{nat}}\mathrm{Sn}, ^{197}\mathrm{Au}$, and $^{208}\mathrm{Pb}$ targets used in this paper, the corresponding relative error indices show clearly that our formula is superior to that by Tripathi et al.

Empirical Neutron Cross Section Formulae for (n, p) and (n, d) Reactions of He-3 Target

In this work, new cross section formulae for (n, p) and (n, d) reactions of He-3 have been investigated as a function of incident neutron energy. The new Empirical formulae were produced by using the least squares method to the experimental cross sections data, which were taken from Experimental Nuclear Reaction Data EXFOR Database Version of 2021. Several functions were examined to choose the best one that fits the data. Statistical hypothesis testing was used to insure how well the suggested equations fit the set of data. Two statistical indicators were used for each case for goodness-of-fit. Very high compatibility was found between the empirical values and the experimental data for both reactions.

Semi-empirical formula for pair production cross-section of gamma in nuclear and electric field

We have proposed the semi-empirical formula for pair production cross-section in nuclear and electric field of atoms with atomic number [Formula: see text] in the energy range of [Formula: see text]. To validate the present formula, the values produced by the proposed formula are compared with the available experiments. The percentage of deviation of present formula pair production was found to be less than [Formula: see text]. The present formula is useful in the fields of radiation, particle and Nuclear Physics.

Double-Differential FRaGmentation (DDFRG) models for proton and light ion production in high energy nuclear collisions

A new set of Double-Differential FRaGmentation (DDFRG) models for proton and light ion production from high energy nucleus–nucleus collisions, relevant to space radiation, is introduced. The proton model employs thermal production from the projectile, central fireball and target sources, as well as quasi-elastic direct knockout production. The light ion model uses a hybrid coalescence model. The data show a prominent quasi-elastic peak at small angles which becomes highly suppressed at large angles. The models are able to describe this wide range of experimental data with only a limited set of model parameters. Closed form analytic formulas for double-differential energy and angle cross-sections as well as single-differential spectral cross-sections are developed. These analytic formulas enable highly efficient computation for radiation transport codes.

Total Cross Section Phenomenological Formulas for X-Ray and Gamma Radiation Interaction With Matter for Different Energies and Absorber Types

Abstract The in-function log-polynomial regression method for empirical approximations of physical relationships was presented and applied in practice. Thanks to that the single phenomenological equation for the total cross section for X-ray and gamma radiation interaction with matter was presented. The proposed equation depends on the photon energy (from 1 keV to 100 GeV) and absorber type (atomic number from Z = 1 to 100) and takes into considerations all important physical effects of photon interaction, namely, coherent (Rayleigh) and incoherent (Compton) scattering, photo-electric effect and pair/triplet production. Finally, the single formula for the total cross section of photon interaction with matter was presented in the simple form easy to apply in practice, e.g., in radiation protection or in radiation shields designing.

One-loop jet functions by geometric subtraction

In factorization formulae for cross sections of scattering processes, final-state jets are described by jet functions, which are a crucial ingredient in the resummation of large logarithms. We present an approach to calculate generic one-loop jet functions, by using the geometric subtraction scheme. This method leads to local counterterms generated from a slicing procedure; and whose analytic integration is particularly simple. The poles are obtained analytically, up to an integration over the azimuthal angle for the observable- dependent soft counterterm. The poles depend only on the soft limit of the observable, characterized by a power law, and the finite term is written as a numerical integral. We illustrate our method by reproducing the known expressions for the jet function for angularities, the jet shape, and jets defined through a cone or kT algorithm. As a new result, we obtain the one-loop jet function for an angularity measurement in e+e− collisions, that accounts for the formally power-suppressed but potentially large effect of recoil. An implementation of our approach is made available as the GOJet Mathematica package accompanying this paper.

Photoproduction of axionlike particles in the NA64 experiment

Axion-like particles $a$ (ALPs) that couple to the Standard Model (SM) gauge fields could be observed in the high-energy photon scattering $\gamma N\to N a$ off nuclei followed by the $a\to \gamma\gamma$ decay. In the present paper we describe the calculation of the ALP production cross-section and the properties of this production. The cross section formulas are implemented in the program for the simulation of events in the NA64 experiment, the active electron beam dump facility at the CERN SPS. We study the prospects of the NA64 experiment to search for ALP in the $10\, \mbox{MeV} \lesssim m_a\lesssim 100$ MeV mass range for the statistics corresponding to up to $5\times 10^{12}$ electrons on target (EOT).

Bound-free pair production in relativistic nuclear collisions from the NICA to the HE LHC colliders

We consider the electron-positron pair production in relativistic heavy ion collisions, in which the produced electron is captured by one of the nuclei resulting, thus, in the formation of a hydrogen--like ion. These ions emerge from the collision point and hit the vacuum chamber wall inside superconducting magnets. Therefore, this process may be important for the problems of beam life time and for the quenching the irradiated magnet. A theoretical investigation for such a bound-free pair production (BFPP) at the colliders from NICA to HE LHC is presented. We obtain an approximate universal formula for the total cross section of the process. We compare it with the results of available numerical calculations and estimate that an accuracy of our calculations is better than $30$ % at the energies of the NICA collider and becomes of the order of a few percent for the RHIC and HE LHC colliders. Based on the obtained results, the detailed calculations are performed for future experiments at the NICA collider. We find that the expected BFPP cross sections for the Au$^{79+}$-Au$^{79+}$ and Bi$^{83+}$-Bi$^{83+}$ collisions are in the range from $10$ to $70$ barn, while for the p-Au$^{79+}$ and p-Bi$^{83+}$ collisions they are in the range of a few mbarn.

A new empirical formula for cross-sections of (n,nα) reactions

Studies on the cross-sections of (n,n[Formula: see text]) reactions which are energetically possible, about 14 MeV neutrons are quite scarce. In this paper, the cross-sections of (n,n[Formula: see text] nuclear reactions at [Formula: see text]14–15 MeV are analyzed by using a new empirical formula based on the statistical theory. We show that neutron cross-sections are closely related to the [Formula: see text]-value of nuclear reaction, in particular for (n,n[Formula: see text]) channels. Results obtained with this empirical formula show good agreement with the available measured cross-section values. We hope that the estimations on the cross-sections using the present formalism may be helpful in future studies in this field.

The empirical cross section behavior of (d, 3n) reaction for 20±1.5 MeV energy

Many nuclear reaction cross-section formalism has been developed for particle induced reaction up to date. In this study, a new empirical (d, 3n) cross section formulae systematic have been developed at 45≤A≤112 mass range for 20 ± 1.5 MeV incident deuterium energy. Also, (d, 3n) systematic have been grouped according to Even-N and Odd-Z numbers around 45≤A≤209 mass range. The Levkovskii asymmetry (N-Z/A) parameter have been used to improve empirical (d, 3n) cross section formula systemic. By using obtained new (d, 3n) empirical formulae systematics, theoretical (d, 3n) cross section calculations are made and the results have been found good agreement with the EXFOR data.

The Effect of an Alternating Magnetic Field and Silver Nanoparticles on the Spectral Characteristics of an Aqueous Solution of Human Serum Albumin

In this paper, we discuss experimental data obtained during study of the effect of a low-frequency alternating magnetic field on the conformational transitions of human serum albumin in the presence of silver nanoparticles using fluorescence spectroscopy at different pH values. A sharp increase in the albumin fluorescence intensity was detected in the presence of silver nanoparticles; this effect changed when exposed to a low-frequency alternating magnetic field. The resulting calculation formula for cross section of light scattering by a spherical nanoparticle of a certain radius allows calculation of the sizes of nano-microstructures based on nanoparticles and natural biopolymers of the “core–shell” type, whose physicochemical properties can be controlled by a magnetic field. The proposed model for the formation of protein–nanoparticle associates in the solution by the dipole–dipole mechanism is in good agreement with the experimental data.

Peculiarities of a helium interaction with hydrogen and free electrons from the point of view of the exchange perturbation theory solutions

We consider an adiabatic potential of a Hydrogen-Helium interaction by using a new approach based on the exchange perturbation theory (EPT). This method allows to present results analytically. We find the solution as an expansion in an antisymmetric basis thanks to the proved completeness property for a nonorthogonal set of antisymmetric wave functions. We use the developed invariant form with respect to interatomic electron permutations for both the Hamiltonian describing an unperturbed system and the perturbation operator. The invariant form for both operators allows one to obtain properly antisymmetric corrections to the antisymmetric wave function of the zero approximation. The probability of electron scattering with electron capture by the helium atom is discussed. We use an EPT formalism developed for the time-dependent perturbations. A formalism of the Time-dependent EPT has a form of an invariant perturbation theory that accounts for intercentre electron permutations among the overlapping states. The expressions for the matrix elements of an S-matrix and a T-matrix account for interatomic electron permutations. We use the obtained differential cross section formulas for the description of the electron capture during the scattering processes. The differential cross section of the electron capture by Helium atom has a complex shape of peak for the scattering energy around 0.7 atomic units, associated with the adiabatic creation of the negative ion He(-1).

Approximate Formula for the Total Cross Section for a Moderately Small Eikonal Function

We study the eikonal approximation of the total cross section for the scattering of two unpolarized particles and obtain an approximate formula in the case where the eikonal function χ(b) is moderately small, |χ(b)| ≲ 0.1. We show that the total cross section is given by a series of improper integrals of the Born amplitude AB. The advantage of this representation compared with standard eikonal formulas is that these integrals contain no rapidly oscillating Bessel functions. We prove two theorems that allow relating the large-b asymptotic behavior of χ(b) to analytic properties of the Born amplitude and give several examples of applying these theorems. To check the effectiveness of the main formula, we use it to calculate the total cross section numerically for a selection of specific expressions for AB, choosing only Born amplitudes that result in moderately small eikonal functions and lead to the correct asymptotic behavior of χ(b). The numerical calculations show that if only the first three nonzero terms in it are taken into account, this formula approximates the total cross section with a relative error of O(10−5).

Material-dependent d + d cross-section in dense and metallic mesh wire environments at very low energies

New [Formula: see text] resonance parameters have been extracted and interpreted in terms of the experimental neutron–proton branching ratio of [Formula: see text] reactions in strontium and tantalum metallic lattices. The material-dependent cross-section formula, which includes the new [Formula: see text] resonance parameters, the screening effect, and the effective mass, have been applied to [Formula: see text] and [Formula: see text] reactions in metallic environments. The contributions of the [Formula: see text] resonance and effective electron mass are crucial to the understanding of the discrepancy between the theoretical and experimental screening values in metallic environments. As a different and connected part of this research, we have proposed a new target material: deuterated metal wire mesh structures. Experimentally confirmed numerical results for Pendry’s effective electron mass for aluminum metal wire have been used to calculate the cross-section and screening energy.

Resonance cross-section formula for low-energy elastic scattering

A simple analytical formula for partial wave cross section describing low-energy elastic scattering in the presence of a shape resonance located near the elastic threshold is proposed. The formula is based on the extension of the regularized method of analytical continuation in the coupling constant (RAC) recently proposed by Hor\'a\ifmmode \check{c}\else \v{c}\fi{}ek and co-workers. The RAC method provides the complex energies of resonances and virtual states requiring as input information a set of real bound state energies of a perturbed system for which standard highly sophisticated numerical codes may be employed. No scattering calculations are needed. These parameters define the low-energy partial wave resonance scattering cross section. Cross sections are calculated for the $2{s}^{2}\ensuremath{\varepsilon}p\phantom{\rule{0.16em}{0ex}}^{2}P^{0}$ state of beryllium and $3{s}^{2}\ensuremath{\varepsilon}p\phantom{\rule{0.16em}{0ex}}^{2}P^{0}$ of magnesium. The results compare well with recently published data.

Real scalar dark matter: relativistic treatment

A stable real scalar provides one of the simplest possibilities to account for dark matter. We consider the regime where its coupling to the Standard Model fields is negligibly small. Due to self-coupling, the scalar field can reach thermal or at least kinetic equilibrium, in which case the system is characterized by its temperature and effective chemical potential. We perform a fully relativistic analysis of dark matter evolution, thermalization conditions and different freeze-out regimes, including the chemical potential effects. To this end, we derive a relativistic Bose-Einstein analog of the Gelmini-Gondolo formula for a thermal averaged cross section. Finally, we perform a comprehensive parameter space analysis to determine regions consistent with observational constraints. Dark matter can be both warm and cold in this model.