We investigate exchange current contributions to elastic electron-deuteron scattering using exactly Poincaré invariant quantum mechanics with a null-plane kinematic symmetry. Our model exchange current is motivated by one-pion-exchange physics. Exact current conservation and current covariance are satisfied by using this current to compute a linearly independent set of current matrix elements. The remaining current matrix elements are generated from the independent matrix elements using the covariance and current conservation constraints. The presence of the exchange current increases the sensitivity to the choice of independent current matrix elements. Two choices of independent matrix elements that have distinct motivations lead to a good description of all of the elastic scattering observables for momentum transfers below 8 (GeV)$^2$.
The complex Kohn variational principle and the (correlated) hyperspherical harmonics method are applied to study the proton-deuteron elastic scattering at energies above the deuteron breakup threshold. Results for the elastic cross section and various elastic polarization observables have been obtained by fully taking into account the long-range effect of the Coulomb interaction and using a realistic nucleon-nucleon interaction model. Detailed comparison between the theoretical predictions and the accurate and abundant proton-deuteron experimental data can now be performed.
The Kohn variational principle is formulated for calculating elastic proton-deuteron scattering amplitudes at energies above the deuteron breakup threshold. The use of such a principle with an expansion of the wave function on the (correlated) hyperspherical harmonic basis is discussed.
Elastic lepton-deuteron scattering was considered within the limit of zero lepton mass. The deuteron wave function in the coordinate representation for the Reid93 potential was applied to numerical calculations. The angular-momentum dependence of values spin correlation coefficients $C_{xz}^{(0)}$, $C_{zz}^{(0)}$ and tensor asymmetries $A_{xx}^{(0)}$, $A_{xz}^{(0)}$, $A_{zz}^{(0)}$ have been evaluated in 3D format for Reid93 potential. These polarization observables are analyzed for different energies and scattering angles. Application of the obtained values allows better explanation and illus…
The general form of the cross section as well as the polarizations of electron-deuteron elastic scattering are given. Two-photon exchange effects are analyzed. Possible signatures of the two-photon exchange effects in the electron-deuteron elastic scattering are discussed.
We consider T-violating P-conserving proton-deuteron forward elastic scattering amplitude in a frame of the Glauber multiscattering theory, proceeding from T-odd, P-even N-N interaction.
A study of the deuteron structure in the framework of relativistic quantum mechanics is presented. Point-form (PF) relativistic quantum mechanics (RQM) is applied to the elastic eD scattering. The deuteron wave function and neutron form factors are fitted to the electromagnetic deuteron form factors. We also compare results obtained by different realistic deuteron wave functions stemming from Nijmegen-I, Nijmegen-II, JISP16, CD-Bonn, Paris, Argonne, Idaho and Moscow (with forbidden states) potentials. It is shown that the electromagnetic deuteron form factors may be described without exchange …
We present calculations of deuteron elastic and inelastic breakup cross sections and angular distributions at deuteron energies below 100 MeV obtained using the post-form DWBA approximation. The elastic breakup cross section was extensively studied in the past. Very few calculations of inelastic breakup have been performed, however. We also analyze the angular momentum - energy distributions of the cross section for formation of the compound nucleus after inelastic breakup.
Deuteron-proton elastic scattering is studied in the multiple scattering expansion formalism. The four contributions are taken into account: one-nucleon-exchange, single- and double scattering, and $Δ$-isobar excitation. The presented approach was applied to describe the differential cross sections at deuteron energies between 500 and 1300 MeV in a whole angular range. The obtained results are compared with the experimental data.
The deuteron-proton elastic scattering has been studied in the multiple scattering expansion formalism. The essential attention has been given to such relativistic problem as a deuteron wave function in a moving frame and transformation of spin states due to Wigner rotation. Parameterization of the nucleon-nucleon $t$-matrix has been used to take the off-energy shell effects into account. The vector, $A_y,$ and tensor, $A_{yy}$, analyzing powers of the deuteron have been calculated at two deuteron kinetic energies: 395 MeV and 1200 MeV. The obtained results are compared with the experimental d…
The reaction of the elastic electron-deuteron scattering at high momentum transfer is investigated within the Bethe-Salpeter approach. The relativistic covariant Graz II separable kernel of nucleon-nucleon interactions is used to analyze the deuteron structure functions, form factors and tensor of polarization components. The modern data for the electromagmetic nucleons structure from the double polarization experiments as well as some other models of the nucleon form factors are considered.
What range of momentum components in the deuteron wave function are available e d elastic scattering data sensitive to ? This question is addressed within the context of a model calculation of the deuteron form factors, based on realistic interactions and currents. It is shown that the data on the $A(q)$, $B(q)$, and $T_{20}(q)$ observables at $q \leq 6$ fm$^{-1}$ essentially probe momentum components up to $\simeq 4 m_π$.
A calculation of the deuteron polarization observables $A^d_y$, $A_{yy}$, $A_{xx}$, $A_{xz}$ and the differential cross-section for elastic nucleon-deuteron scattering at incident deuteron energies 270 and 880 MeV in lab is presented. A comparison of the calculations with two different deuteron wave-functions derived from the Bonn-CD $NN$-potential model and the dressed bag quark model is carried out. A model-independent approach, based on an optical potential framework, is used in which a nucleon-nucleon $T$-matrix is assumed to be local and taken on the energy shell, but still depends on the…
The general expressions for parity violation observables in elastic scattering of polarized and/or unpolarized electrons from unpolarized deuterons are given and are numerically evaluated for the kinematics of SAMPLE, PVA4 and G0 experiments. The dominant contribution from the interference of $γ$ and Z exchange as well as the smaller contributions from strangeness ($s{\bar{s}}$) components of the nucleon, parity odd admixtures in the deuteron wave function, anapole moments and radiative corrections are included and discussed in the context of parity violating electron scattering experiments of…
The elastic proton-deuteron backward reaction is analyzed within a covariant approach based on the Bethe-Salpeter equation with realistic meson-exchange interaction. Lorentz boost and other relativistic effects in the cross section and spin correlation observables, like tensor analyzing power and polarization transfer etc., are investigated in explicit form. Results of numerical calculations for a complete set of polarization observables are presented.
The general formalism for polarization observables in elastic electron deuteron scattering is extended to incorporate parity and time reversal violating contributions. Parity violating effects arise from the interference of $γ$ and $Z$ exchange as well as from the hadronic sector via a small parity violating component in the deuteron. In addition we have allowed for time reversal invariance violating contributions in the hadronic sector. Formal expressions for the additional structure functions are derived, and their decomposition into the various multipole contributions are given explicitly.
The eigenphase shifts and mixing parameters for elastic p-d scattering below the deuteron-breakup threshold are calculated for the AV14 two-body potential using two different numerical methods. The excellent agreement confirms that it is possible to perform accurate numerical studies of p-d scattering as well as n-d scattering for low energies. The numerical results can be considered as benchmarks for future calculations.
A calculation of vector and tensor polarization observables of a deuteron in an optical potential framework for elastic proton-deuteron scattering at the incident deuteron energy $E_d=880$ MeV is presented. The main aim is to point out the role of various effects in explaining the wide-range angle distribution of these observables. Under the investigation are a double-scattering mechanism in the optical potential, the influence of various $NN$-partial waves and as well a comparison of calculations with two different deuteron wave-functions derived from the Bonn-CD $NN$-potential model and dres…
The elastic backward proton-deuteron scattering is analyzed within a fully covariant approach based on the invariant expansion of the reaction amplitude. The method of calculation of any observables for any reaction mechanism is developed. The cross section and spin observables, like tensor analyzing power and polarization transfer coefficient etc., are investigated in explicit form within framework of the impulse approximation using different kinds of the relativistic deuteron wave function (DWF) with P-wave components. Our results of numerical calculation for complete set of polarization obs…
We calculate elastic electron-deuteron scattering in a chiral effective field theory approach for few-nucleon systems based on a modified Weinberg power counting. We construct the current operators and the deuteron wave function at next-to-leading (NLO) and next-to-next-to-leading (NNLO) order simultaneously within a projection formalism. The leading order comprises the impulse approximation of photons coupling to point-like nucleons with an anomalous magnetic moment. At NLO, we include renormalizations of the single nucleon operators. To this order, no unknown parameters enter. At NNLO, one f…
We investigate the effect of the deuteron breakup on the optical potential for the elastic scattering of 56 MeV deuterons from $^{51}$V. The breakup probabilities calculated within the post-form distorted wave Born-approximation theory of breakup reactions are fitted to derive the contribution to the optical potential from the breakup channels. The breakup potentials are found to have large real as well as imaginary parts. Thus the dynamical polarization potential due to the breakup process is expected to modify strongly the real part of the optical potential calculated by the double folding m…
The elastic backward proton-deuteron scattering is analyzed within a covariant approach based on the invariant expansion of the reaction amplitude. The relativistic invariant equations for all the polarization observables are presented. Within the impulse approximation the relation of the tensor analyzing power $T_{20}$ and the polarization transfer $κ_0$ to $P$-wave components of the deuteron wave function is found. The comparison of the theoretical calculations with experimental data is presented. An experimental verification of the reaction mechanism is suggested by constructing some combin…
Point-form relativistic quantum mechanics is applied to elastic electron-deuteron scattering. The deuteron is modeled using relativistic interactions that are scattering-equivalent to the nonrelativistic Argonne $v_{18}$ and Reid '93 interactions. A point-form spectator approximation (PFSA) is introduced to define a conserved covariant current in terms of single-nucleon form factors. The PFSA is shown to provide an accurate description of data up to momentum transfers of 0.5 ${\rm GeV}^2$, but falls below the data at higher momentum transfers. Results are sensitive to the nucleon form factor p…
A new computational method for solving the nucleon-deuteron breakup scattering problem has been applied to study the elastic neutron- and proton-deuteron scattering on the basis of the configuration-space Faddeev-Noyes-Noble-Merkuriev equations. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The Merkuriev-Gignoux-Laverne approach has been generalized for arbitrary nucleon-nucleon potentials and with an arbitrary number of partial waves. The nucleon-deuteron observables at the incident nucleon energy 3 …
The elastic backward proton-deuteron scattering is analyzed within a covariant approach based on the Bethe-Salpeter equation with realistic meson-exchange interaction. Contributions of the one-nucleon and one-pion exchange mechanisms to the cross section and polarization observables are investigated in explicit form. Results of numerical calculations for the cross section, tensor analyzing power and spin transfers are presented. The one-pion exchange contribution is essential for describing the spin averaged cross section, while in polarization observables it is found to be less important.
The role of the short-range part (repulsive core) of the proton-neutron ($pn$) potential in deuteron elastic breakup processes is investigated. A simplified one-range Gaussian potential and the Argonne V4' (AV4') central potential are adopted in the continuum-discretized coupled-channels (CDCC) method. The deuteron breakup cross sections calculated with these two potentials are compared. The repulsive core is found not to affect the deuteron breakup cross sections at energies from 40 MeV to 1 GeV. To understand this result, an analysis of the peripherality of the elastic breakup processes conc…
The elastic backward proton-deuteron scattering is analyzed including relativistic effects in the deuteron and the mechanism of this reaction which includes the graphs corresponding to the emission, rescattering and absorption of the virtual pion by a deuteron nucleon in addition to the one-nucleon exchange graph. It allows one to obtain a rather satisfactory description of all the experimental data on the differential cross section, tensor analyzing power of the deuteron and transfer polarization in this reaction.
We propose a novel approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically and provide analytical insight that the elastic proton-deuteron (pd) observables can be determined directly from the resulting on shell 3N amplitude increasing the screening radius. The screening limit exists without the need of renormalisation not only for observables but for the e…
The contributions to the reaction cross section from the elastic and inelastic breakup processes, calculated within the post-form distorted wave Born-approximation theory, are used as constraints to determine the contributions to the imaginary part of the deuteron optical potentials (IPDOP) due to the breakup channels. The Coulomb part of this potential due to the elastic breakup process is seen to account for the long range absorption in the optical potential. The nuclear parts of the IPDOP due to the elastic and inelastic breakup modes peak in different regions of the nuclear surface, with t…
We present the results of a calculation of differential cross sections and polarization observables for proton-deuteron elastic scattering, for proton laboratory energies from 2.5 to 22.5 MeV. The Paris potential parametrisation of the nuclear force is used. As solution method for the charged-composite particle equations the 'screening and renormalisation approach' is adopted which allows to correctly take into account the Coulomb repulsion between the two protons. Comparison is made with the precise experimental data of Sagara et al. [Phys. Rev. C 50, 576 (1994)] and of Sperison et al. [Nucl.…
Fully converged calculations of deuteron-deuteron elastic scattering observables are performed at energies above three- and four-body breakup threshold. Differential cross sections and analyzing powers are obtained using realistic nucleon-nucleon force models together with the Coulomb repulsion between protons. For all observables we find a very reasonable agreement with the available experimental data limited to deuteron beam energies up to 25.3 MeV.
We calculated response functions of the deuteron for charge exchange processes, including the final state interaction between two protons. Using them we evaluated the double differential cross section and polarization observables of $d(\vec{p},\vec{n})2p$ by means of plane wave impulse approximation with an optimal factorization. Calculation well reproduced the shape of the energy spectra of the cross section, though somewhat overestimated the magnitude. It also reproduced the spin observables well.
We investigate the importance of the three-nucleon (3N) force contact terms in elastic nucleon-deuteron (Nd) scattering by applying the N$^4$LO$^+$ chiral semi-local momentum space (SMS) regularized nucleon-nucleon (NN) chiral potential supplemented by N$^2$LO and all subleading N$^4$LO three-nucleon force (3NF) contact terms. Strength parameters of the contact terms were obtained by least squares fitting of theoretical predictions to cross section and analyzing powers data at three energies of the impinging nucleon. Although the N$^3$LO contributions to the 3N force were completely neglected,…
Spin-dependent total pbar-d cross sections are considered using the optical theorem. For this aim the full spin dependence of the forward pbar-d elastic scattering amplitude is considered in a model independent way. The single-scattering approximation is used to relate this amplitude to the elementary amplitudes of pbar-p and pbar-n scattering and the deuteron form factor. A formalism allowing to take into account Coulomb-nuclear interference effects in polarized pbar-d cross sections is developed. Numerical calculations for the polarized total pbar-d cross sections are performed at beam energ…
Perturbative calculations of nucleon-deuteron elastic scattering in chiral effective field theory
College of Physics, Sichuan University（四川大学物理学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Physics, Beihang University（北京航空航天大学物理学院）
We discuss the elastic ed scattering beyond Born approximation. It is shown that the reaction amplitude contains six generalized form factors, but only three linearly independent combinations of them (we call them generalized charge, quadrupole and magnetic form factors) contribute to the reaction cross section in the second order perturbation theory. We examine two-photon exchange and find that it includes two types of diagrams, when two virtual photons interact with the same nucleon and when the photons interact with different nucleons. Estimations based on nonrelativistic calculations with …
The electromagnetic properties of the deuteron are investigated within a Light-Front Hamiltonian Dynamics framework, with a current operator that contains both one-body and two-body contributions. In this work, we are considering new two-body contributions, with a dynamical nature generated within a Yukawa model and a structure inspired by a recent analysis of the current operator, that acts on the three-dimensional valence component and fulfills the Ward-Takahashi identity. Preliminary results for the magnetic moment are shown.
We solved the three-nucleon Faddeev equation including relativistic features such as relativistic kinematics, boost effects and Wigner spin rotations. As dynamical input a relativistic nucleon-nucleon interaction exactly on-shell equivalent to the AV18 potential has been used. The effects of Wigner rotations for elastic scattering observables were found to be small. The boost effects are significant at higher energies.They diminish the transition matrix elements at higher energies and lead in spite of the increased relativistic phase-space factor as compared to the nonrelativistic one to rathe…
The elastic backward proton-deuteron scattering is analyzed including both relativistic effects in the deuteron and the reaction mechanism. It is shown that inclusion of the graphs corresponding to the emission, rescattering and absorption of the virtual pion by a deuteron nucleon in addition to the one-nucleon exchange graph allows a rather satisfactory description of all the experimental data on the differential cross section, tensor analyzing power of the deuteron and transfer polarization in this reaction.
Goal of this work is the consistent description of elastic Compton scattering from the single nucleon and the deuteron. The theoretical framework chosen is Chiral Perturbation Theory, which is the low-energy formulation of Quantum Chromodynamics, where we extend the spectrum of active degrees of freedom from only pions and nucleons to also the Delta(1232) resonance. In the deuteron sector, we treat the nucleon-nucleon interaction non-perturbatively. Besides the Compton cross sections, our main concern is with the nucleon polarizabilities, which are a useful tool to describe the reaction of the…
Theoretical uncertainties of various types are discussed for the nucleon-deuteron elastic scattering observables at the incoming nucleon laboratory energies up to 200 MeV. We are especially interested in the statistical errors arising from uncertainties of parameters of a nucleon-nucleon interaction. The obtained uncertainties of the differential cross section and numerous scattering observables are in general small, grow with the reaction energy and amount up to a few percent at 200 MeV. We compare these uncertainties with the other types of theoretical errors like truncation errors, numerica…
The fixed-mass sum rules for the deuteron target have been derived by using the connected matrix element of the current anti-commutation relation on the null-plane. From these sum rules we obtain the relation between the pseudo-scalar meson deuteron total cross sections and the structure functions of the deuteron. We show that the nuclear effect on the mean hypercharge of the sea quark of the proton can be studied by this relation. Further we obtain the relation among the Born term, the resonances and the non-resonant background in the small $Q^2$ region, and as a new aspect of the spin 1 targ…
The deuteron break-up exclusive reaction $p+d \to N(180^{\circ}) + (NN)(0^{\circ})$ is considered in the framework of the one-nucleon-exchange mechanism at the incident proton kinetic energy 0.2--2 GeV and the relative energy of nucleons in the forward going final NN-pair $E_{NN}=0$--20 MeV, that is close to the kinematical conditions of the planned experiment at COSY. The off-energy-shell effects resulting, in particular, in the node of the half-off-shell $^1S_0$-state partial NN-scattering amplitude and the role of higher partial waves in the subprocess $pN\to pN$ are analysed and found to b…
The differential cross-sections for elastic and inelastic neutrino-deuteron scattering are calculated analytically using nucleon-nucleon effective field theory. For elastic scattering, the deuteron axial form factor and the deuteron strange magnetic moment form factor are computed to next-to-leading order, including two-body currents. For inelastic scattering, two neutral current processes nu+d->nu+n+p, nubar+d->nubar+n+p and one charged current process nubar+d->(e^+)+n+n are computed to next-to-leading order. These depend on an isovector axial two-body matrix element whose value is yet to be …
The longitudinal asymmetry due to $Z^0$ exchange is calculated in quasi-elastic electron-deuteron scattering at momentum transfers $|Q^2| \simeq 0.1$ GeV$^2$ relevant for the SAMPLE experiment. The deuteron and $pn$ scattering-state wave functions are obtained from solutions of a Schrödinger equation with the Argonne $v_{18}$ potential. Electromagnetic and weak neutral one- and two-nucleon currents are included in the calculation. The two-nucleon currents of pion range are shown to be identical to those derived in Chiral Perturbation Theory. The results indicate that two-body contributions to …
Three-nucleon forces, which compose an up-to-date subject in few-nucleon systems, provide a good account of the triton binding energy and the cross section minimum in proton-deuteron elastic scattering, while do not succeed in explaining spin observables such as the nucleon and deuteron analyzing powers, suggesting serious defects in their spin dependence. We study the spin structure of nucleon-deuteron elastic amplitudes by decomposing them into spin-space tensors and examine effects of three-nucleon forces to each component of the amplitudes obtained by solving the Faddeev equation. Assuming…
Pion elastic scattering on deuterium is studied in the KMT multiple scattering approach developed in momentum space. Using a Paris wave function and the same methods and approximations as commonly used in pion scattering on heavier nuclei excellent agreement with differential cross section data is obtained for a wide range of pion energies. Only for $T_π>250$ MeV and very backward angles, discrepancies appear that are reminiscent of disagreements in pion scattering on $^3$He, $^3$H, and $^4$He. At low energies the second order corrections have been included. Polarization observables are studie…
Expressions for the unpolarized differential cross section and for various polarization observables in the lepton-deuteron elastic scattering, $\ell+D\to \ell+D$, $\ell=e$, $μ$, $τ$, have been obtained in one-photon-exchange approximation, taking into account the lepton mass. Polarization effects have been investigated for the case of a polarized lepton beam and polarized deuteron target which can have vector or tensor polarization. Numerical estimations of the lepton mass effects have been done for the unpolarized differential cross section and for some polarization observables and applied to…
Two independent calculations of proton-deuteron elastic scattering observables including Coulomb repulsion between the two protons are compared in the proton lab energy region between 3 MeV and 65 MeV. The hadron dynamics is based on the purely nucleonic charge-dependent AV18 potential. Calculations are done both in coordinate space and momentum space. The coordinate-space calculations are based on a variational solution of the three-body Schrödinger equation using a correlated hyperspherical expansion for the wave function. The momentum-space calculations proceed via the solution of the Alt-G…
It is shown that the ratio of the deuteron and proton analysing powers in proton-deuteron elastic scattering at small angles is sensitive to subtle effects in a theoretical description. These include the transverse spin-spin term in the elementary nucleon-nucleon amplitudes and double-scattering corrections. On the other hand there is far less sensitivity to the spin-orbit amplitude and to binding or other kinematic effects associated with the use of the deuteron, as either target or projectile. The available data are in agreement with the results of a refined Glauber theory model.
The vector and tensor analysing powers in deuteron-proton elastic scattering have been measured in the forward hemisphere at deuteron kinetic energies of 1.2 GeV and 2.27 GeV using the ANKE spectrometer at the COSY storage ring. The results are compared with other experimental data and with predictions made within the framework of Glauber multiple scattering theory.
The neutron-deuteron (nd) scattering is solved in the Faddeev formalism, employing the energy-independent version of the quark-model baryon-baryon interaction fss2. The differential cross sections and the spin polarization of the elastic scattering up to the neutron incident energy $E_n=65$ MeV are well reproduced without reinforcing fss2 with the three-body force. The vector analyzing-power of the neutron, $A_y(θ)$, in the energy region $E_n$ < 25 MeV is largely improved in comparison with the predictions by the meson-exchange potentials, thus yielding a partial solution of the long-standing …
In this paper, we study the low-energy $d-α$ elastic scattering within the two-body cluster effective field theory (EFT) framework. The importance of the $d(α,α) d$ scattering in the $^6 \textrm{Li} $ production reaction leads us to study this system in an effective way. In the beginning, the scattering amplitudes of each channel are written in a cluster EFT with two-body formalism. Using the effective range expansion analysis for the elastic scattering phase shift of $S$, $P$ and $D$ partial waves, the unknown EFT low-energy coupling constants are determined and the leading and next-to-leadin…
Three-nucleon force effects in polarization transfers from the doubly spin-polarized initial neutron-deuteron state to the outgoing neutron in neutron-deuteron scattering
We extend our approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations from elastic proton-deuteron (pd) scattering to the breakup process. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger equation. We demonstrate numerically that the proton-deuteron breakup observables can be determined from the resulting on-shell 3N amplitudes increasing the screening radius. However, contrary to the pd elastic scattering, the screening limit exists only after r…
Faddeev equations for elastic Nd scattering have been solved using modern NN forces combined with the Tucson-Melbourne two-pion exchange three-nucleon force, with a modification thereof closer to chiral symmetry and the Urbana IX three-nucleon force. Theoretical predictions for the differential cross section and several spin observables using NN forces only and NN forces combined with three-nucleon force models are compared to each other and to the existing data. A wide range of energies from 3 to 200 MeV is covered. Especially at the higher energies striking three-nucleon force effects are fo…
The integrated cross section $\widetildeσ$ for a special type of double polarized proton-deuteron scattering constitutes a null test forvtime-invariance violating but P-parity conserving effects. Using the Glauber theory for the $pd$ elastic scattering and different types of phenomenological T-odd P-even NN-interactions we show that the contribution of the lowest mass meson exchange, i.e. the $ρ$-meson, to the null-test signal $\widetildeσ$ vanishes. Variation of the cross section $\widetildeσ$ due to strong hadronic and Coulomb interaction is studied and its energy dependence is calculated in…
