Open Journal of Microphysics, 2013, 3, 85105 http://dx.doi.org/10.4236/ojm.2013.33015 Published Online August 2013 (http://www.scirp.org/journal/ojm) The Coulomb Resonances, the QuasiReal Photons and ElectroDisintegration of Nuclei by HighEnergy Electrons A. A. Pasichnyi1, O. A. Prygodiuk2 1Institute for Nuclear Research, Kyiv, Ukraine 2Taras Shevchenko Kyiv National University, Kyiv, Ukraine Email: apasichny@kinr.kiev.ua Received October 3, 2012; revised November 5, 2012; accepted November 15, 2012 Copyright © 2013 A. A. Pasichnyi, O. A. Prygodiuk. This is an open access article distributed under the Creative Commons Attribu tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Various aspects of the influence of the quasireal photons and the Coulomb resonances on the formation of the cross section of inelastic scattering of high energy electrons on atomic nuclei are investigated. Emiss is the energy that disap pears in the processes of knockingon of protons in the reactions . A new hypothesis that interprets the origin of the energy losses is proposed. Specific experiments that can confirm or refute this hypothesis are proposed as well. The “regularized” crosssections of electrodisintegration of nuclei by highenergy electrons are calculated in the framework of the nuclear shell model. It is shown that for the experimental verification of the exis tence of Coulomb resonances, it is necessary to investigate the processes at relatively small angles of scattering. The peculiarities of numerical methods that are crucial in the investigation of inelastic scattering of highenergy elec trons on nuclei in the framework of the nuclear shell model are analyzed in this work as well. The crosssections of the scattering of highenergy electrons on the angle are calculated. It is shown that the orthogonality of the wave functions of a knockedon proton in the initial and final states plays an important role in the interpretation of this proc ess. ()( ,Aeep A ′− () ,ee ′ ) 1 () , reg σωθ ′ 0 θ ′= Keywords: Coulomb Resonance; QuasiReal Photon; Inelastic Scattering; HighEnergy Electron; ElectroDisintegration; Nuclear Shell Model; Numerical Method; Inelastic Scattering; QuasiElastic Peak 1. Introduction: The ElectroDisintegration of Atomic Nuclei by HighEnergy Electrons Due to the relative weakness of the electromagnetic and weak interactions of electrons, positrons, muons and neu trinos with nuclei, the structure investigation of atomic nuclei in the processes of elastic and inelastic scattering of these particles on nuclei [127] provides the most re liable information on various aspects of the structure of atomic nuclei. The above statement is based on the rela tively high reliability of the information about the prop erties of electromagnetic and weak interactions, as well as the relatively high accuracy of the perturbation theory, in the framework of which (impulse approximation) we interpret the inelastic scattering of highenergy leptons by nuclei. The technical perfection reached at the mo ment in forming the monoenergetic highenergy elec trons beams, as well as in registering these particles in nuclear experiment, played a decisive role in the choice of particlesprojectiles as means of external influence on the atomic nucleus for studying the structure and proper ties of nuclei: they were certainly the highenergy elec trons. It is important to note that the structure of nuclei is in vestigated in this paper in the framework of the nuclear shell model (LScoupling, independent particles). In the framework of this nuclear shell model, we will study and interpret the features of such unusual phenomena as the Coulomb resonances and the quasireal photons in the aspect of their influence on the dynamics of electrodis integration of nuclei. In other words, the aim of this pa per is the investigation of possibilities of visualization and identification of Coulomb resonances in experimen tal studies of nuclear electrodisintegration. C opyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 86 It must be emphasized that the Coulomb resonances are the something more than merely the Coulomb reso nances. First, the Coulomb resonances present a practi cally essential part of the quantummechanical theory of quasidiscrete spectrum in nonrelativistic quantum me chanics [12,25,27]. Second, the Coulomb resonances are, first and foremost, a natural extension of the nuclear shell model to the continuous spectrum region [12,25,27]. Third, it is the Coulomb resonances that will provide additional and very useful insight into our understanding of gigantic dipole resonance phenomenon in the framework of the shell model [12,25,27], etc. Finally, the reader may re member [12,25,27] that at this stage the Coulomb reso nances and the quasireal photons are investigated exclu sively poorly. However, in both theory of inelastic high energy electron scattering and theory of atomic nuclei, the Coulomb resonances and the quasireal photons are phenomena that can manifest itself in many phenomena and experiments of nuclear physics [12,25,27]. We suppose that the study of the reactions of proton knockout and neutron knockout from various atomic nuclei presents a particular interest just in the framework of the nuclear shell model. From this point on, we shall assume that the process of inelastic scattering of a highenergy electron at a nucleus is accompanied by transfer of energy (; and are the initial and final energies of the scattered electron), and momentum to the nucleus. and are the electron momenta before and after the act of inelastic collision of the electron and the nucleus. We also suppose that the process of inelastic collision of the electron with the nucleus in the investi gated region of the kinematic variables is caused mainly by a collision of the scattered electron with a single nucleon of the nucleus. ()( ,Aeep A ′− () 1A − ε ′ k′ k ) 1 () ,Aeen ′ >0 ωε ωεε ′ =− ′ =− kqk , ω q As a result of this collision, one of the nucleons of the atomic nucleus, having received the required energy in this act, overcomes the action of attractive nuclear forces and flies out from the atomic nucleus A with mo mentum ω ≡ K ( p= and n= in the cases of knocking out of a proton and a neutron, respectively) and energy EE≡: 2 E= x K. Note that according to the conservation laws of mo mentum and energy, the exact expression for the distri bution of the transferred electron energy between the nucleus ω 1 − and the knockedout nucleon has the fol lowing form (here and in the following [28]): 1=c= () 2 2 ,1 . 22 xlxl ApA p A ww AM M νν ω − − =+ +≡++ q K qTT (1) The new quantities appearing in Equation (1) are as follows: l w is the energy required for ejection of nu cleons from the xvlshell of an infinitely heavy nucleus (the separation energy of the nuclear xvlnucleon); A is the energy of motion of the center of mass of the target nucleus T after its collision with the scattering electron; is the energy of relative motion of the knocked ( ,1pA T− ) out proton and the residual nucleus 1 −; = p A1M − is the reduced mass of the proton. We point out that Equation (1) could be written in the following form as well: 2 , 2 eff l p w M ω =+ K (2) where the effective separation energy eff l w is defined by the following apparent formula: 22 2 22 p eff xl xlAM AM AM ww νν ⋅ =+ +− qq Kq (3) It is important to point out that, according to Equation (3), the effective separation energy eff l w of the xvl nucleon depends somewhat on the kinematics of experi ment. In the limit case of 1 we have: () 2 ,1 ; 2 eff xl xl pA TE ww ν −== = K. (4) The effective separation energy eff l w in the approxi mation ( 1, p ) MM= will be identified in the fu ture numerical calculations with the binding energy of the nucleon l ε in the nuclear shell. There is a good reason to believe that the calculations of the cross sec tions of nuclear electrodisintegration of heavy, medium and even light nuclei, which are performed in this ap proximation, will be quite acceptable for preliminary conclusions. A nucleon knocked out from a nucleus moves in the average field of this nucleus in both the bound state and the state of continuous spectrum. In the shell model the knockedout nucleon dynamics in the bound state is presented by the wave functions calculated in average field ; here () Ax Vr () ) ψϕ = 1, ( xlm ν rr 2,3, () Ax Vr = () Ax Vr () ) ψϕ = () is the radial quantum number; l = 0(s), 1(p), 2(d), 3(f), ··· is the orbital quantum number. It is reasonable to calcu late the continuous spectrum wave functions K in the same average field In this case the bound state wave functions and the continuous spectrum wave functions are orthogonal to each other. () = () = ( ψψ r ( ψψ r ) r ) r ± ± ( xlm ν rr <0E K Note that the wave functions of the discrete and continuous spectra are solutions of the SturmLiouville problem based on the nonrelativistic singleparticle Schrödinger equation: ( >0E ) ()()() 2 . 2Ax Vr E M −Δ+ Ψ=Ψ rr 5) ( Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 87 The bound state solutions of Equation (5) must satisfy the following integrability conditions: () 2 20,> 0; r r δ δ + →∞ Ψ→r (6) in the latter case, the condition (6) must be re following condition: placed by () () () () e e, iKr i K f r ψ ± ± Ψ=→ + Kr K rr n . K r →∞ =K nK (7) One can recall that the vector K appearing in Equation (7) is the wave vector of the knockedout nucleon: 2. ME==K 2. The Cross Sections of the Nucleus ElectroDisintegration om the In this paper we restrict ourselves to those processes in which the knockout of protons and neutrons fr nuclei is associated with relatively small transfers of energy ω and momentum q from the scattered elec trons to the atomic nucleus : 1 ω ; 1 q ; 1 x K ; In the present approximonteraf can be quite accurately described the quasirela ati the inction o a relativistic electron with a nonpoint nucleon of the nucleus tivistic Hamiltonian of McVoyVan Hove [1,2], which takes into account empirically the electromagnetic struc ture of the knockingout nonpoint nucleon in the form of relativistic corrections. After carrying out simple calcula tions (perturbation theory, the impulse approximation), the cross section () ,, xl ν σ ′ kk K of knocking out of an nucleon from the l shell of the nucleus by an inelastically scattered ultrarelativistic electron is pre sented in the follow expression [5,7,21,25]: () ing factorized () ()( )() 5 d ,, dd d xl xl ν ν σ σε ′′′ ΩΩ ≡kk K 42 2 2 4,, , ,, xl xx xl MK eN FqPSG νμ ν ′′ =×kk kk KqK k (8) where is the proton charge,e l N is the number of nucleons in the nuclear l shell, , , 0, '0, π, θθθθθ ′′ ′ ′≡ === kKk q (9) 1.791 , ≡≡ k 1.9 xp xn δδ =− (10) (11) () () () () 2 2 22 2 1 0.055; x Fq FqFmq μμ μ − ≡=+ () () () () ()() [] () 2 2 22 2 2 22 2 2 ,, 12 11 24 4 2 2 4 4 x x xp x S kkM M q MM q M μ μ γ ω δεε εε γ ′ − ′′ =++ + ′ ′ + ′′ −++ ′+ + kk K qkk KkKkK Kk k kk (12) is a dimensionless, positive definite and continuous function of the vector arguments is in this function that the structural features of the interaction of the scattered relativistic electron and ockedout nucle tonian o tially di () () ,, 1 x S′kkK ,, ′ kk K. It the knon in the quasirelativistic Hamil f McVoyVan Hove are reflected. Two essen fferent functions appearing in Equation (8), which depend on kinematic variables of the process of the elec trodisintegration of nuclei, ()() () 22 2 2 ,PP q μ θ ′ ′′ ≡= kk kk (13) and ()() () ()()() () 3 2 *3 1 (,) 212π expd , xvl ml xlm ml G l i ν ψϕ =− =− =× + K qK rqrrr (14) exert most comprehensive and determinative [25,2 fluence on the interpretation of various aspects processes electrodisintegration of nuclei. The function 7] in of the () ,qK xl G ν depending on the kinematic parameters and q defines the distorted momentum distribution of nucleons in the l shell of the atomic nucleus. It is () , xl G ν qK that contains the most complete information abouynamics of the nucleon in the nucleus force field () () : Axx l Vr ν ϕ r appearing in Equation (14) are the wave function of the nucleon in the bound state; t the d m () − () uous spectrum field on final state, the factorization of expression (8) for the cross x ψ Kr is the wave function of the nucleon in the state of contin. If we take into account the influence of the nuclear the motion of the knockedon nucleon in the section () ,, σ ′ kk K is approxim xl ν ate. The factorized expression (8) becomes exact only in the planewave approximation, which is valid when the energy of the knockingout protons is quite large. In this case () () ψ ±≈ KrKr , and () , xl G ν qK transforms into () 0 xl G ν −qK, which determines the momentum distribu ( ) exp i Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 88 tion of the nucleons in the xvlshell: () () () () ( ( ) ) 01 212π exp ml G l i ϕ = + ⋅ x (15) 3 2 3 d . xl ml xl m ν ν =− −= − qK qKxx Here and below the exclusive and inclusive cross tions of the nucleus electrodisintegration calculate the planewave approximation will be labeled, if neces sary, in the graphs, tables and formulas by the symbol 0, an of sec d in d the similar values calculated with taking into account the interaction of the knockedout nucleon in the final state will be labeled by the symbol d. Formula (8) speci fies the initial exclusive cross section of electrodisinte gration of the atomic nucleus. Experimental verification of (8) requires fairly laborious experiments in which both the inelastically scattered electron and the knockedout proton are simultaneously registered or identified. At the moment, a large number of less laborious experiments are realized. In these experiments one investigates the energy distributions of inelastically scattered electrons at certain scattering angles and initial energies of the scat tered electrons. There are carefully developed methodologies taking into account the inelastically scattered electrons that have lost their energy in a variety of quantumelectrody namical [28,30] processes such as bremsstrahlung, birth electronpositron pairs, etc. If we subtract these elec trons from other scattered ones, we obtain the inclusive cross section () , σωθ ′ of inelastic scattering of high energy electrons in the process of collisions of ultra relativistic electrons and nuclei: () ()() ,, ,, u xl xl xl N νν ν σωθσ ωθσ ωθ ′′ ′ == (16) where xl xl ν ν l N is the number of xnucleons in xvlnuclear shell, () , xl ν σωθ ′ and () () ,, u lxlxl N νν θ σωθ ′′ = are thlete and specific knockout cross se a xnthe σω e comp ucleon f ctions of rom l shell of the atomic nucle partially, filled shells of the investigated us, respectively. The summation in Equation (9) is applied to all, ful nucleus, and () , xl ν σωθ ′ is calculated in 0 and dap proximations by direct numerical integration of the differential cross section (8) over the total solid angle Ω≡Ω ly or of propagation of the knockedout nucleons: () () d ,,,d. dd xl xl xl ν νν σ σωθ σ ε Ω ′′ ≡= Ω ′′ ΩK kk K (17) ing the reaction ()() ,1Aeen A ′−, we can extract 3 Study realistic information about tic nuclei in the framework of an investigated model by co the calculated and measured exclusive (8) and inclusive (1 sections of t shell model then the calculations of these cross sections he structure of atom mparing 7), (17) differential crosshe electrodisin tegration of nuclei. If we restrict ourselves to the nuclear are associated with laborintensive computing of the over lap integrals: () () () ()()() *3 ,e expd . x i xlm xx Ixlm ixlm ν ν ψϕ ν − ≡ = −∞ qr K qK K rqr rr (18) These integrals determine the distorted momentum distributions (Equation (14)) of nucleons in the filled xvlshells of atomic nuclei. The deductions an sions of this paper depend strongly on the numerical values of the calculated electrodisintegration cross sec tio of nuclear forces is re st the knockedout nucleon m d conclu ns of various nuclei. Because of this, the reliability and correctness [33] of applied numerical methods of the electrodisintegration crosssection calculation will be dis cussed in Appendix A in detail. Note that the electrodisintegration processes of nuclei are relatively easy interpreted in the approximation pre sented below by the quantumelectrodynamical Feyn man diagram of second order. If we suppose that the range ricted then we can assert that the wave function of the knockedout () >0E nucleon [29] has the asymptotic form given by Equation (7). The condition (7) mentioned above once more affirms that oving in the region of the residual nucleus () 1A− scatter itself elastically on this one. These processes of the nucleus electrodisintegration are presented on the above Feynman diagram. On this diagram, we can distinguish the initial electron with 4 momentum kk ≡, which acquires the status of scat tered electron a () kk ′′ with the residua rese ≡ ts is p after emission of a virtual pho ton. The virtual photon q is absorbed by one of the nucleons of the target nucleus and provokes the ejection of this nucleon from the nucleus. The knockedout nu cleon interacl nucleus in the final state. This interactionnted on the diagram by a hypo thetical exchange meson with 4momentum π. In this paper we estimate the dependence of the so called χsections (see Equation (35)) of nuclear electro disintegration from the processes represented below by the sum of Feynman diagrams of higher order: Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 89 In these diagrams we see new additional participants of the more comprehensive theoretical interpretation of the inelastic scattering of highenergy electrons on nuclei. First and foremost, it is a quasireal () 00 qq ≡ oton of the virtual photon [25], which is absorb by a prnucleus ed and provokes the transition of this proton in the qua sidiscrete state Kr of the nucleus or 1 −. The knockedout protons and r interact in the final state with the residual nucleus. This in shown schematically in the diagrams by 4momenta of ππ teraction is = hypothetical mesons. The knockedout nucleons ex change by such a meson with the daughter nucleus A − 2. The Feynman diagram(diag.2diag.2b) describe the processes of the twoproton knockout from the atomic nucleus, which are predicted by the nuclear shell As we will see later, these processes are also capable to provide a nontrivial impact on the results of experimen tal studies of reactions ()() ,1AeepA ′−. It is well known that 4vertex quantumelectrody namical processes (diag.2a, diag.2b) in general case (if we substitute 000 ,qqqqq ′ →≠ ≠ in diag.2a, diag.2b) are weakly visible against the backgrounds of the 2 vertex processes (diag of s a, model. .1). It is use s (diag.2a and dia ful to remind once again that 4vertex quantumelectrodynamical processes (diag.2a, diag.2b) become apparent only in the case when in both diagramg.2b) and in, at least, one of two left vertex of each diagram the electron emits a quasireal photon 0 q (0 θ ′=, “00”scattering). It is the quasireal photon 0 q that, as we will see subse quently, is capable to excite with high probability the Coulomb resonance in the atomic nucleus. It is necessary to note that the nucleon knocked out from the nucleus an participate in the processes of inelastic scattering on th residual () 1A− nucleus. For example, this nucleon can spend a part of its energy for the excitation of the discrete state of the res c e idual nucleus. C c ng int is olliding with another nucleon of the residual nucleus, the knockedout nuleon is capable to increase the num ber of knockedout nucleons. Takio account the process of inelastic scattering of the knockedout nucleon on the residual nucleus can essentially complicate the interpretation of the electrodisintegration processes in the coincidence experiments. 3. QuasiDiscrete Spectra of Atomic Nuclei Let us calculate the cross sections (8) and (16) of the knockout of a nucleon (reactions ()() ,1AeepA ′−). In the nuclear shell model (LScoupling) we approximate the potential () Ax Vr by a sum of two terms: () ()() AxWSxpC VrVr Vr δ =+ . The first term () WS Vr the shortrange WoodsSaxon potential: () () () 0 0 00 , 1exp >0, Ax Ax WS Ax x Vbr V Vr rR rR VV Θ− =− ≈− −− ≡ 1e xpaa ++ (19) where: () () 00 1 122 x xx xx δ + Θ=− + δ is the Heaviside unit function; (20) 3 020br Aa≈⋅ +; 0Ax V, a and 3 o Rr A=× are parameters of the Saxon potential. The second term is the lon lomb potential: Woods () xp C Vr δ grange Cou () () () ()() 2. 22rrR Rr R 2 1 3 C Vr Ze 2 21 Ze rR − Θ− − =− +Θ− (21) The wave functions of continuous spectrum can be obtained in the form of the following series: n. (22) () () () ()( )() 111 1 11 11 1 111 * 0 4π lml l lmrlmK Kl lml iR rYY ψ =∞ =± ± ==− = Krn () () ()() 111 1 1 11 1 1 111 0 * 4π lml Kl l lmr lm Kl lml Zr iAY Y r =∞ =± ==− = nn (23) 1K The radial functions the halfopen space real and bounded soer wave equation with () ()( , Kl Kl ZrAZ rE λ =∀ 0<r≤∞ in our model are always tions of the radial Schröding e real potential () Ax Vr: ) >0. in lu th () () ) () ( () 2 0 . ll rZ r + − = pect (7) de termines the asymptotic behavior of the radial functions 22 d1 2 dAx Zr mE V rr λ λ +− (24) It is useful to remind [31] that the asymptotic behavior of the wave functions of continuous srum () r λ : () () () () () () () () () () () () () () ; 2 Kl Kl Kl llll AZrrb Zr Sg ifg if i λ ;rb Kr ρ − ⋅∀ ≥= −− ∀≤ =+−− (25) Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 90 () l g and () l f () r ρ = oulomb funct Here: in Equation (25) are then [3ions wellknow () 0≠: 4,35] C () C Vr ()() ()() ,, ,, ll ll gG fF η η == (26) where () 2 12 ZeE η =− (27) mb pais the Coulorameter. If , then () 0 C Vr= ()()( ) ()()( ) 0 0 ,; lim ,, lim ll l ll l gnG fj F η η η ρρρ ηρ → → =− = == where () l n and () l j are the N utions (25ion (24) at the point , we find the coefficients eumann and Bessel spherical functions. Tailoring [12,25,27] the sol) of Equat rb= () () () KL L AE: () − −− ≡ ()() () 1, LL AE wEiwE − =+ LG ZFZ where (28) ()) () (() () dd, dd Kl l L GZ Kl Zr G wE Zr rr ηρ ρ =− , l G rb η = ; (29) ()( )() ()() dd, , dd Kll . l Kl Zr F F Zr rr rb ηρ ηρ =− = (30 Simple mathematical analysis of the amplitude expres sion L FZ wE ) () () l E − ing conclusions. omb reson in Equation (28) [12] leads to the fol One can determine the energies of the Coulances low r NL EE= from the condition [12] () () () () dd, dd KL L Zr G rr Z ηρ , KL L rG rb rb ηρ = == (31) and the halfwidth r Lp from the equality [12] () () () 1 d. 2d L r GZ pNL L FZ r L wE wE E EE γ − = = Direct mathematical analysis of the expression (28) for (32) the amplitude () L () E reveals [1 magnitude of the prot − of the separate radial compo nent 2,25,27] the cause of the abrupt increase of the cross section in th on resonance energy () Kl Rr of the e vicinity () , σωθ ′ r NL e in the EE=. We lities (31) and (32) are tru case when the strong inequality recall that the equa () () () () 22 22 d, d, ,, dd LL GF GF ηρ ηρ ηρ ηρ ρρ +>>>+ is valid. When the energy r LL NL EE= of Coulo nance approaches the Coulomb barrier height () mb reso 2 3 C r pNL Vb E ≥ , the mentioned strong inequality re laxes to a more delicate inequality: () () (() ) 22 2 d, d, , L GF 2 , dd LL L GF ηρ ηρ ηρ he Coulhe status of incipient resonanc apparent when one calculates spectra of inelastically peaks of resonances have moderate heights and rather large halfwidths ηρ ρρ ++ . In that case tomb resonances acquire t es. Such Coulomb resonances become scattered electrons. In this case the r l . )(3As an example of application of Equations (24 we present the calculated main characteristics (the ener 2), gies Lx E, the halfwidths Lx and the wave function amplitudes Lx A) of quasidiscrete levels of some atomic nuclei in Table 1. It is important to keep in mind that these characteristics can depend essentially [12,27] on the parameters of the WoodsSaxon potential. Ne otic th the c mental aical inv ockout reac tio 9,25,27] that t functions of transmitted energy r shell width) changed due to the small at the parameters r0 and a of the WoodsSaxon poten tial in the calculations of this article are kept invariable: 01.24r=Fm and 0.55a= Fm. This simplification does not influenceonclusions of this paper. Note also that the analysis of the quasidiscrete spectra (jjcoupling) of the light, medium and heavy nuclei per mits to affirm that the nucleus quasidiscrete spectrum is the natural extension of the nuclear shell structure to the continuous spectrum region. For this reason, the experi nd theoretestigations of quasidiscrete spectra properties in ()() ,1AeepA ′−kn ns of protons are, at the same time, investigations of the nuclear shell model. 4. The CrossSections of Excitation of Coulomb Resonances It is well known [12,13,1heoretical inves tigations of the inelastic scattering of highenergy elec trons predict the resonance structure of inclusive (Equa tion (9)) crosssections as ω (50 ω ≤ MeV) in the framework of the nuclea model. The dynamic characteristics (height, half resonance peaks substantiallyof variations of the parameters of WoodsSaxon potential. For instance, the maximum values of () , lNL ν σω θ →′ and the halfwidths lNL ν → of resonance crosssections () , lNL ν σω θ →′ can undergo enormous quantitative ang huge variations of characteristics of reso nance peaks undoubtedly require additional investiga tions of this phenomenon. ches. Such Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK Copyright © 2013 SciRes. OJM 91 Table 1. The principal physical characteristics of the quasidiscrete spectra of some atomic nuclei. NL x ENLx NLx γNLx NLx ENLx NLx γNLx 56Fe; (0; 0)=(63.3 ; 50.55) pn VMeV 58Ni; (0; 0)= (62.0 ; 49.3415) pn VMeV 2 1.36 10− × 5 3.82 10× 12 3.05 10− × 13n 3 2.02 10− × 7 1.67 10× 14 N 16 6.06 10− × 14 P 2 1.3 − × 6 1044 6.85 10× 90 1.071− × 014p 1 5.9 − × 7 106 7.151× 014 6.33 10− × 22 P 15 p 1 81Y; MeV 119Sn; 2.78 15.43 2 1.48 10− × 3.350.76 0.47 () () 0;0 =54.13 ;41.73V4 pn () 0 () ;0 =63.3 ; 50.55 pn VMeV 14 P 3 8.83 10− × 78 3.27 10× 158 3.5110− × 23p 2.26 3 6.0932 10× 7 1.06 10− × 22 N 2 4.40 10− × 31.58 22 P 16p 3.99 10.31 27 MeV 40Ca; 4 3.15 10− × 31p 3.96 1 2.6415 10× 3 5.09 1× 0− 3.48 22.43 3 8.17 10− × 7.20 2 2.4490 10×4 1.82 10− × 30 P 4.12 1 1.60 10− × 17n 11.48 5.961 3.34 10− × 15 P 11.18 2 9.52 10− × 17p 17.64 9.98 1 1.60 10− × Al; () () 0;0 =63.5 ; 54. pn V0 () V () 0;0 =53.9 ; 42.352 pn MeV 13 P 0.455 3 09.17 1× 8 3.3172 10− × 13n 3− 5.27 10× 4 03.79 1× 10 1.76 10− × 13 N 0.748 31.20 3 3.2143 10− × 13p 3− 6.06 10× 51 3.93 10× 104 2.02 10− × 21 P 21 p 208Pb; MeV 198Au; MeV 0.802 22.633 3.0319 10− × 1.56 1 1.77 10× 3 7.54 10− × () () 0;0 =57.48 ;47.4125 pn V () () 0=57.48 ;47.4125 pn V;0 17 N 3− 8.17 10× 10 9.7110× 23 3.57 10− × 17    23 P 2 2.70 10− × 100 5.75 10× 202 1.31 10− × 23p 1 6.721 − × 015 3.09 10× 31 2.28 10− × 16 P 17 n 1.98 9 1.30 10× 18 3.22 10− × 1.39 3 1.88 10× 6 1.25 10− × 31 P 2.39 5 4.36 10× 11 1.66 10− × 16p 2.86 6 6.8210 ×13 1.41 0− × 1 24 P 7.74 77.53 3 1.16 10− × 31p 2.99 4 1.90 10× 9 9.65 10− × 18 N 8.30 17.372 3.38 10− × 24p 8.49 1 3.67 10× 3 5.35 10− × 32 P 9.83 5.001 2.12 10− × 18n 9.81 11.142 8.78 10− × 17 P 10.06 2 3.7110× 5 8.92 10− × 17p 11.14 2 1.73 10× 4 4.3110− × 18 P 18.53 22.39 2 3.23 10− × 18p 19.79 15.882 6.67 10− × te tht in this cas the location of peaks re s [25,27] practically unchanged thample, the microscopical mfica MeV) of depth lNL ν ωω → = gain tions (4 10− ≤ No ae main . Note once a at, for exodi 0 V f of WoodsSaxon poten to tions of the nutial para of Coulom almost insurmountable difficulties in the inter pr of thisonance does exist in the crtion given Eat(16) [27]. Indeed, the predicted values of the rnaross sectins (which are me s ured in experiment) prt depend on the a nucleus wh tial can lead reso of varia an increase of the Coulomb resonance peak () () , lNL ν σω θ →′ of order of 100200 10 10− times and a corresponding decrease of the halfwidth of this nance of order o100 200 10 10 −− − times. In this connec tion it is necessary to investigate the influence clear shell potenmeters on the theoretically predicted measurable values of crosssec tions of excitationsb and centrifugal reso nances. It is worth noting that the absence of intercompensa tive relation between the increase of height of any peak and the decrease of its halfwidth would mean the exis tence of resoss secby qu ion esonce co etation of processes of nuclei electrodisintegration in the framework of the nuclear shell model. However, we will see that a correlation of high order between the in crease (decrease) of the height of Coulomb resonance and the “adequate” decrease (increase) of the halfwidth height and halfwidth of Coulomb resonances, as we will see below. Let us consider an example of the influence of so called quasireal photons [25,27] on the scattering of () , lNL ν σω θ → actically do no ′a high energy electrons on nuclei. Let us recall that the quasireal photons are the result of such collision of a high energy electron anden the direction of movement of the electron is practically unchanged after it (0 θ ′=, “00”  scattering): ′ ≈ kk . The energy ω ′ kk transmitted at such a frontal collision and the value of the transmitted momentum ′ =−qkk (the energy and mo mentum of the quasireal photon 0 qq= on the Feyn man diagrams (the diag.2a and diag.2b)) are approxi mately equal. Let us recall also [25,27] that the cross section () , σωθ ′ as a function of the electron scattering angle sharp maximume point, θ ′ have a at th 0 θ ′=
A. A. PASICHNYI, O. A. PRYGODIUK 92 especially in the region where td energy ω is not l () ωε ′ . Moreover, ry confirms [25, ransmitte arge our theo retica e 27] that it is the quasireal photons that gives the main contribution to the knockout of the protons in the reac tion ()() ,1AeepA ′− in the investigated kinematic re gion. Scattering of electrons to large angles is barely no ticeable in the proton knockout mentioned above. The theol interpretation of the inclusive cross section () , σωθ ′ as a function of ω allows us to ′ confirm that at arbitrary electron scattering angl θ () 0 θπ ′ ≤≤ or some values of transmitted energy lNL ν ωω → = one can see sharp and high (10300 b/MeV/sr and more) peaks on the plots of () , σωθ ′. The half width and f lNL ν → of such peaks may be abnormally small (300 10− MeV and less). These peaks correspond to the excitation of socalled Coulomb () p= and centrifu gal () n= resonances in atomic nuclei. Note that the excitation ef the resonances is equal to the sum of the binding energy of the nucleon nergy o l ε in the nuclear l shel resona l an nce d the energy of the Coulomb (centrifugal) NL E: r lxNLxlxNL E νν ωε →=+. Taking into account properties of the funon () , xl ν σωθ ′ and properties of the inelastic electron scat tering with lNL ν ωω → = and 0 θ ′≈, one can conclude that the integral () 02π 00 ,dd, , E r rur plpNLpll NL E r r NL N θ ωδ ννν ωδ σσσωθω γ ′ + →→ − ′′ ≡× =Ω (33) determines le cti E δ the lower limit of the compte ( r ) lpNL σ → ν fic () ur and speci lpNL ν σ → crosssections of excitation of the pNLCoulomb resonance of the investigated atomic nucleus quite accurately. The limit of integration over the angle, upper , appearing in Eq is a small quwe restrict ou value of 0 θθ ′′ = antity. As a ru 1 le, uation (33) rselves to the 0 π 60 θ ≈ during the numerical integration. The total crosssection of excitation of the Coulomb NL resonance r L σ is equal to the sum r LplNL pl ν ν σ → = (34) In the case of low Coulomb NLresonances this cross section is defined the only term corresponding to the dipole transition: , σ mainly by l N = 1Ll=+. For instanceet us consider the process of inela scattering the energy 500 ε = MeV , lstic of electrons with on th tion of the Ca with the quantum num d in Table 2. The prot es of exciton of pNL C f the in on e nucleus 40Ca. The results for specific crosssections N L of excita of the Coulomb reso 40 ur l ν σ → nucleus are presente from di this nu nances bers 13,21NL = ons are knocked out fferent occupied shells () 10,11,12, 20l ν = of cleus. The dominant position in the excitation of Table 2. The theoretical predictions of specific crosssec tions → ν σ ur lNL and energi ati → ν ω r lNL oulomb resonances with quantum numbers NL = 13.21 in the nucleus 40Ca. The initial energy oelastically scattered electrons is ε = 500 MeV. The halfwidths of that resances are 13 r = 1.964 × 10−117 MeV and 21 r = 7.51 × 10−3 MeV. The depth parameter of the W potential is V0p = 53.9020 MeV. l oodsSaxon 13 ru l ν σ →, b 13 r l ν ω →, MeV 21l ν →,b 21l ν →, MeV ru σ r ω 10 2.288 10 × 33.59 8.414 10× 35.54 3−3− 11 1 10 − × 23.59 2 9.12 10− × 25.31 1215.19 12.20 2.19 13.96 20 5 7.55 10− × 9.57 17.98 11.29 1.265 Coulomb resones belongs to the dipole transitions: anc 1213 ,2021→→ , as follows from the results given in Table 2. In this case assumthe radiquantum bers are equal e w to each ot e that al num her: lL N = resonance . One can see thexcitation oous cusedthe qupole that by e f Clomb a uadr () 11 13→, octupole () 10 1→ tion again ike to ] for the 3, etc. trnfthe background on, for exam chosen that the abov ansitios is very diff of dpole o ple, that [][ ur e inequality rbitrary icult or obs We wo [][ ur σσ →→ 1 . ervast ines.uld lmenti ] θ ′ 12 1310 13 parameters of the model. At the same time, it is possible may be strongly weakened or even violated for a Let us assume that the radial quantum numbers are not equal now: lL N ≠ (usually,1 l L N =+). In this case the magnitude of ur lNL ν σ → is significantly smaller as compared wi lL N th the case of =. The effect considered above may be a starting point for the interpretation of the phenomenon of gigantic di pole resonance in the nuclear shell model. Thus, the investigation of the inelastic scattering of highenergy electrons in the region where the trans mitted energy is not large e he framework of t can bused for studying the ph that thelts have to comm nd on e choice of the nu cl enomenon of gigantic dipole resonance. It is interest ing to note above resu some extent general character andon quantum nature. These results do not depeth ear target and the energy of the scattered ultrarelativ istic electrons. For instance, let us consider the excitation of the Coulomb resonance with the quantum numbers [NL] = [12] in the nucleus 12C when a proton is knocked out from one of two filled shells () 10,11l ν =. The initial energy of the scattered electrons is 2020 ε = MeV. It follows from the data of Table 3 that the excita tion crosssection [][] 11 12 ru pp σ → substantially exceeds the crosssection [][] 10 12 ru pp σ →. Let us note that the cross  se ctions of exitation of Coulomb resonances as well as the role of quasireal photons increase significantly with an increase of the initial energee ε of the scattered Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK Copyright © 2013 SciRes. OJM 93 Table 3. Main physical characteristics (energies 12 r , maximum amplitudes 12palfwidths 12 r A, h r ) of the Coulomb resonance NL = 12, specific crosssec energies→ ν ω lNL of excitation of the Coulomb resonance NL = [12] in the nucleus 12C versus the depth V0p of the Woodsteal. The initial energy of the inelastically sattered electrons is ε = 2020 MeV. 12 r p E, MeV tions Saxon pontic → ν σ r lNL and 12 r p 12 r p , MeV 11 ru σ →1210 1211 12 , b , b , MeV ru σ → r ω →0MeV p V 0.0062 13 6.85 10× 29 6.36 10− × 540.4 2.705 16.71 61.412 0.182 750.8 2.81 539.98 2.654 16.48 60.5 2.090 5.23 0.169 570.2 2.52 15.24 56.0 6 8 10− × electrons [25,27]. We note lso that the enon of the aus increase of the scattering crosssection caused an increase of the initialy of the electroeam may have significan influe the results oinvestigation ofrious as pects of nuclear structure. For instance, the results of do not require additional comments. i re determin aphenom nomalo by quasireal photons with energ t n b nce onf va experimental measurements of different crosssections of inelastic electron scattering can essentially depend on the target thickness. As an additional illustration of the above statements, in the Table 4 we present the calculated values of specific crosssections of the Coulomb resonances excitation in the heavy nucleus 198Au for two initial energies of scat tered electrons: ε = 2020 MeV and ε = 3365 MeV. The results of Table 4 We would like to attract attention the fact that the inclusive crosssections () , σωθ ′ have the form of sharp resonance peaks at certain values of the transmitted energy r lNL ν ωω → =. Note that the theoretically predicted enormous heights and insignificantly small halfwidths of () , r lNL ν σω θ →′ give no possibility to determine d ctly the shape of the Coulomb resonances in experi ment. That is why we cane only integral char acteristics of the Coulomb resonances, such as the cross sections r lpNL ν σ → the physical experim of excitation of these resonances, in ent. Let unce again [25,27] that the physical cha racteristics (the halfwidths r s recall o Lp and the amplitudes ( Lp A) of the wave functions) of Coulomb resonances can undergo essential changes. For example, the sharp decrease of r Lp and the equally drastic increase of Lp A) is possible even at nee changes of parame te cr on of we pende gligibl rs of nuclear shell model potential. This uncertainty raises the question whether theosssections of excita ti Coulomb resonances undergo essential changes too. Table 5 gives an answer to this question. In Table 5 present the calculated functional de nce of the most important specific crosssections r lpNL ν σ → of excitation of resonances of nucleus 40Ca. These crosssections are interpreted as functions of the halfwidth of Coulomb resonances, which varies with the change of the depth 0 V of the WoodsSaxon potential. used by the dle transition[13], [20] →21] h, in tuused by ine astic scattef ultrarelac electronsnu leus 40Ca. The resulted in ble 5 do quire extensive comments as well. The specific crosssections ca ipos [12]→[ whic l rn, are ca ring o quasireal pho tivisti tons in on the c ts presenthe Ta not re Note that the specific crosssections mentioned above are of the Coulomb resonance excitation ru lpNL ν σ → and the n energies r lNL ν ω → are practically invariant when the halfwidths and the amplitudes of the wave function of Coulomb resonances change substantially. This result is very imrtant. It essentially increases the chances of success for the nuclear shell model in the excitatio po lusive CrossSections lfwidths of the b and cenifu ly less thahe hac peaks. At the scattered el recall that the crosssection at the arbitrary interpretation of inelastic electron scattering experiments aimed at studying the nuclear structure. 5. The Inc () ,′ σωθ and the Coulomb Resonances One of the toppriority tasks of this paper is to study the manifestation of Coulomb and centrifugal resonances in experiments on inelastic scattering of highenergy elec trons. In particular, it is of interest to investigate the pos sibilities to disclose and identify Coulomb and centrifu gal resonances in the spectra of highenergy electrons inelastically scattered on various atomic nuclei. Comparing the Coulomb resonances and the quasi elastic peaks, one can state that the ha theoretically calculated peaks of Coulomtr gal resonances are, as a rule, considerabn t lfwidths of the calculated quasielasti same time, the height of a Coulomb resonance peak is significantly larger than the height of a quasielastic peak. For convenience, such peaks of () , σωθ ′ should be “cut off” on the plots of () , σωθ ′ versus transmitted energy ω. For this reason, we lose clarity and important information about the observability of Coulomb reso nances in the measured spectra of inelastically ectrons. Taking the logarithm of that function adds very little information since the microscopical halfwidth of a Coulomb resonance can hardly be represented on the plot. In order to investigate the possibility of detection of Coulomb resonances in physical experiment, one should () , σωθ ′
A. A. PASICHNYI, O. A. PRYGODIUK 94 Table 4. Specific crosssections [][] → ν σ ru lNL of excitation of the pNLCoulomb resonances with the quantum numbers NL in the nucleus 198Au. The initial energiese inelastically scatteredectrons are ε = 2020 MeV and ε = 3595 MeV. The parameters of the WoodsSaxon potential are V0p = 51.13 MeV and V0n = 41.734 MeV. 202 ε = of th el 0 MeV NL → 23 16 31 24 17 18 l ν 23 ru l ν σ →, b 16 ru l ν σ →, b ru l ν σ 31→24 17 18 , b , b , b , b ru l ν σ → ru l ν σ → ru l ν σ → 10 2 2.42 10− × 6 1.28 10− × 2 1.97 10− × 3 1.89 10− × 6 3.73 10− × 4 1.45 10− × 11 1 2.57 10− × 5 6.04 10− × 1 3.67 10− × 2 2.4110− × 6 5.97 10− × 5 4.28 10− × 12 1.20 3 2.91 10− × 1 3.00 10− × 1 2.29 10− × 4 2.12 10− × 4 7.32 10− × 20 13 21 6. 5. 8. M 2 9.82 10− × 8 8.81 10− × 1.88 3 4.54 10− × 5 8.49 10− × 3 1.42 10− × 3 .821 1.31− 10×1 3.10 − 10×1 7.22 − 10×3 6.50 − 10×3 2.620− 1× 4.08 6 0− 1.03 1× 5.71 1 0− 1.16 1× 4− 1.20 10× 3 0− 3.59 1× 14 45.3 733 3.19 10− × 4.49 1 2.21 10− × 2 1.49 10− × 22 498.0 5 1.04 10− × 108.3 394 7.4110− × 3 7.67 10− × 30 3 1.81 10− × 12 3.60 10− × 596.3 3 1.23 10− × 3 2.10 10− × 3 9.15 10− × 15 1 2.02 10− × 665.5 7 5.22 10− × 43.53 891 3.57 10− × 3595 ε = eV 10 0.077 6 4.07 10− × 2 6.27 1× 0−3 6.02 10− × 5 1.19 10− × 4 4.66 10− × 11 0.817 4 1.92 1× 0−1.17 2 7.68 10− × 5 1.91 10− × 4 1.37 10− × 12 3.81 3 9.22 10− × 1− 9.52 10× 1 7.27 10− × 4 6.75 10− × 3 2.34 10− × 20 0.311 7 2.79 10− × 5.95 2 1.44 10− × 4 2.70 10− × 3 4.53 10− × 13 12.1 1− 4.15 10× 1 9.81 10− × 2.29 2 2.06 10− × 3 8.36 10− × 21 28 12.9 6 3.25 10− × 18.0 1 3.69 10− × 4 3.81 10− × 2 1.14 10− × 14 143.0 21.2 2 1.00 10− × 14.2 1 7.00 10− × 2 4.75 10− × 22 1566.0 5 3.28 10− × 341.1 17.03 2.34 10− × 2 2.43 10− × 30 3 5.67 1× 0−11 1.13 10− × 1876.0 3 3.87 10− × 3 6.63 10− × 2 2.90 10− × 15 0.636 2093.06 1.64 10− × 137.3.1 1.13 Table 5ecific crosssections and enegies ocitation of the Coulomb resonans with the quaum number3,21] in the nucleselascat electrons versus te resonance halfwth . Sp → ν σ ru lNL 40Ca at in r→ ν ω lNL tering of f excent s [1utic shid r NLp . The initial energy e electrons is ε = 500 MeV. The angle of scattering is θ , MeV of th' = =0. 12 σ 13 ru →, b12 13 r ω → 13 r p , MeV , Me 20 21 ru σ →, b 2021 r ω →V 21 r p , MeV 0MeV p V 1 12.20 17.98 11.30 53.908 5.19 294 1.116 10− × 3 7.41 1× 0−4 15.19 12.20 53.9050 15.19 12.20 117 1.964 10− × .98 11.29 3 7.51 10− × 53.9020 15.20 12.19 38 2.935 10− × 17.98 11.27 3 8.46 10− × 53.8400 15.22 12.18 25 6.10510− × 17.97 151 4.688 10− × 17.98 11.29 17 11.48 53.7500 15.31 12.10 18.00 11.19 53.2000 3 7.46 10− × 2 1.00 10− × 11 10− ×3.392 2 10− × 2.41 15.52 11.87 6 3.680 10− × 18.47 10.65 2 9.22 10− × 52.0000 15.60 11.87 5 2.917 10− × 18.90 10.64 1 1.40 10− × 51.5000 point ) is meaxperimentally by means of ”av” proced ( , ωθ ′sured e eraging'ure: () ()() ′ 0E ω +Δ 0E x ω −Δ 1,dd 2 x xx x E σςωθσωθω Ω ′′ =Ω Δ⋅Ω (35) Here ,, ; 2 ωθ ′ E is the spectral widtof gap  eter. Note that Δh of the scat tered electron spectrom () , ω fficiency of registration of sc we su assu θ ′ is the ctionat determines the eof d electrons at various anles ne . Hereinafter in this papwe fun the scattere th rgie ) 1 = g attering θ ′ that the and es ε ′=− εω her on ppose ( , ςωθ ′. Furter me that Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 95 strong inequality 2 NLx x E Δ vy stance is connected wi t ( 0.5 MeV γ h 2 is true. worth n nuclei in our model. This circumth the existence [27] of the socalled incipien ) Coulomb resonances. When tsible that the inequality It isoting that this inequality may be broken in the case of medium and hea 2 MeV NLx ≤≤ ey take place, it is quite pos Lx x E ≈Δ mentioned above ma brnd e y be oken aven inverted: 2 Lxx E ≥Δ . In the calcu lated crosssections the parameters of Wxon po tential were chosen so that the inequality 2 oodsSa Lx x E Δ was true for the most of investigated cases. Consequentlyrons losing their energy lNL ν ω → 00 < , the elect () < lNL x E ν ωω →+Δ during the excitation of a Coulomb resonagistered a E nce will be re ω −Δ the en s electrons with ergies in the interval 00 x EE εωεεω ′ −−Δ≤≤−+Δ . In this case the “regu larized” crosssection () , reg σωθ ′ can be theoretically determined as () () 1 ,,dd. 2 x E reg xx E ω ω σωθ σωθω +Δ ′′′ =Ω Δ⋅Ω (36) It is necessary to note that the proposed above regu larization practically does not change the fu regions of its smooth variation. In this co xx E−Δ Ω nctions in the nnection, for exam nces and may have values comparable with the height of th be done i solated Coulomb resonance in the specified energy interval ple, we have: () () () 00 ,, reg σωθσωθ ′′ ≈; >60MeV. This “regularized” crosssection has resonant form in the area of Coulomb resona e quasielastic peak. Our further conclusions willn the assumption that we have only one i () () , dregd σωθ ′ ≈ () ,for: σωθω ′ 2 EΔ .0 r these co . In the following suppose that MeV for all experiment various kinenditions we get pre di es of inelastic scattering of electrons. In this case photons emitted by rel pola one studie electrons is is the id ented plot that we s with 21 x EΔ= matics. Unde ctive plots of experimental crosssections () , reg σωθ ′ of inelastic scattering of highenergy electrons on nuclei. In the examples presented below we will investigate relatively large angl () 01 θ ′≈ the virtual ativistic electrons have relatively large () 22 2 m ω −q imaginary masses. This fact manifest itself, first of all, in reduction and disappearance of the dominant role of di pole transitions during the excitation of Coulomb reso nances. Dipole, quadrupole, octopole transitions and tran sitions of even higher multirity should be taken into account whens the spectra of inelastic large angle scattering of highenergy electrons. Very often the scattering of highenergy investigated with a carbon (12C) target. This light nucleus selected as standard for comparison of the efficiency of different kinematics inentification of Coulomb resonances in the spectra of inelastic scattered electrons (Figure 1). One can see from the pres, using the kinematics [15], one cannot observe the Cou Figure 1. The inclusive and “regularized” crosssections of inelastic scattering of electrons on nuclei 12C initial energy of scattered electrons ε and the scattering . The kinematics of scattering is in fulfilledriments [15]: ε = 500 MeV, θ' = MeV,' = 37˚ (b); ε = 730 MeV, θ' = 37˚ MeV, = 50.1˚ (d). The parameters of WoodsSaxon potential are: V0p/MeV = 61.412, V0n/MeV = 55.684. Here and hereinafter the solid and the dashed curves were calculated, respectively, with allowance for the finalstate interaction (regularized crosssectins, [(14),(36)]), and in the planewave approximation (15). lomb resonance with quantum numbers , which is theoretically predicted by the nul, in be empha corre sponds to the height of theperirmined () ′ reg σω,θ versus the angle of selected as 60˚ (a); ε = (c); ε = ′ θ expe θ θ' 537 779.5 12NL = clear shell mode the spectra of inelastically scattered electrons. It should sized that this conclusion is wrong in the case of other kinematics presented on Figure 1. It is also pertinent to note that for the disclosure and identification of the resonances of the nucleus 12C with the quantum numbers 12NLx p= and 12NLx n=, it is necessary to carry out more thorough experimental measurements in the spectral region of interest, having essentially reduced the step of the argument ω Δ. It is worth noting that the height of the regularized re sonance peak on the graphs of reg σ () , ωθ ′ mentally dete ex crosssection if the energy gap width of the recording device of scattering electrons is, as indicated above, 21 x EΔ= MeV. At the same time, the width of the re sonance peak is equal to the theoretically calculated width of the investigated resonance. It is necessary to empha Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 96 size that the height of the Coulomb resonance essentially depends on the width of the energy gap of the recording device of for instwidth of the energy gap of spectrometer is equal to 5 MeV, then the possibil ity of manifestation of Coulomb resonances in the spec tra of inelastically scatterlec electrons. If,ance, the ed etrons strongly decreases. tering heights o One can conclude from Figure 1 that decreasing the initial energy ε and the angle of scatof elec trons improves the conditions for observation of Cou lomb resonances in the spectra of inelastically scattered electrons. Let us remind that the theoretically predicted f Coulomb and centrifugal resonances are com parable with the height of the quasielastic peak. Note that the WoodsSaxon potential depth parameters (0 θ ′ V and 0n V) giving the highest accuracy were used, as a rule, in our calculations. The halfwidth γ12 of a Cou lomb resonance essentially increases (see Table 3) with a decrease of the depth of the WoodsSaxon potential. In this case the theoretically calculated Coulomb resonance 12NL = is widened so much that its halfwidth can considerably exceed the spectral energy gap of the spec trometer 2 EΔ. In this case the height of inciient Cou lomb resonance decreases and this resonance can be iden tified directly in spectra of inelastically scattered electrons. Figure 2 shows theoretically calculated regularized crosssections () , reg σωθ ′ of inelastical scattering of electrons and experimentally measured cross p sections ne c heavy nucle () , ωθ ′ of electrodisintegration on different nuclei () 27 58198 ,Al,i,Au. The kinematics of the experi ments is borrowed from [20]. Analysing the calculated () , reg σωθ ′ (solid line) and () 0, σωθ ′ (dashed line; plane wave approximation), we can observe some peculiarities of identification of Cou onances in the spectra of inelastically scattered electrons in the framework of nuclear shell model. For example, an conclude that the Coulomb resonances may be registered in all considered cases. But careful and painstaking measurements in the kinematics region in σ 12 C N lomb res o ter width ν esting for us were not presented in [20]. Note also that in the case of i we have a large number of Coulomb, centrifugal, and incipient (with large half 12 lNL → the theoreti ≈÷ MeV) [27] resonances. In this case of the full resonance picture of crosssection () , reg σωθ ′ beco cal analys Figure 2. Regularized crosssection σreg(ω, θ') of inelastic scattering of electrons on the nuclei 12C, 27Al, 56Fe, 198Au versus the transmitted energy ω. The initial energy of the scattered electrons is ε = 2020 MeV, the angle of scattering is θ' = 15˚. is complex. sections mes more presented Finally at the Figure 3 arecalculated cross () 0 , reg σωθ ′ for different nuclei (A1, Mg, Ca, Ni) and for the different kinematics: the initial energies of scattered electrons 779.5;500;545;500 ε = MeV and the angles of scattering θ ′ = 50.1˚; 60˚; 45˚; 60˚. The results of calculations presented on Figure 3, in general, do not contradict to the conclusions based on analysis of results presented at Figures 1 and 2. We can, however, state a fact that qualitative and quantitative comparisons of theoretical calculations and experimental data and Figure 3. The crosssections σreg(ω, θ') of inelastic scattering of highenergy electrons on nuclei 27Al, 24Mg, 40Ca, 58Ni as a functions of the initial energy of scattered electrons ε and angle of scattering θ'. Kinematics of scattering agrees with kinematics of experiments [9,15,22]: MeV, θ' = 50.1˚ (Al), ε = 500 MeV, θ' = 60˚ (MgMeV, ε = 545 (Ca). The parameters of WoodsSaxon potential are: Al: V0p,0n = 63.5; 54.0 (MeV); Mg: V0p,0n = 68.674; 60.1 (MeV); Ca: V0p,0n = 53.9; 42.352 (MeV); Ni: V0p,0n = 62.85; 49.342 (MeV). = 779.5 ε ,Ni); ε = 545 Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 97 moreover the conclusions from such comparison are im possible at the moment. The reason is that experimental data and theoretical investigations are practically absent for the declared above range of transmitted energies. 6. Quasielastic Peaks and Orthogonality of Wave Functions The interpretation of inelastic scattering of highenergy electrons and the reaction ) considered here in the frames of nul, despite of simplicity, remains selfconsistent quantummechanical problem in the theory of nuclei. The main reason and decisive argument for this statement is such wellknown property as the mutual orthogonality of the wave func tions [7] of discrete and continuous, including the quasi discrete, spectra. Notice that the term “orthogonal func tions” in aspect of interpretation of processes of elec trodisintegration of nuclei by the highenergy electrons was unambiguously used in [7]. In addition to stated in [7] one may note that it is very difficult to imagine quan tummechanical theory of quantum transitions as well as based on it theory of inelastic scattering withoum ()( ,1AeepA ′− clear shell mode t co plete orthogonal basis of wave functions of investigated quantummechanical system. We emphasize also that the conclusions below are based only on the postulates of the nuclear shell model and may be used for the investigation of different aspects of this model. In order to illustrate the role of orthogonality of wave functions in the theory of electrodisintegration of nuclei, let us consider the process of smallangle (θ' = 15˚) and very smallangle scattering (quasireal photons phenome non: 1 m ω θεε ′′ ). Let us now calculate the cross sections () ,1 σωθθ ′′ of inelastic scattering of high 12 al at the dominant position in the iables, energy electrons on nucleus C for two values of initi energy of electrons, ε = 500 MeV and ε = 2020 MeV. First, let us remind th expression for the exclusive cross section belongs to two functions of kinematical var ) (Equation (13))) and () , xl G ν qK (Equation (14))). It is more than appropriate mention here that the latter one is determined by overlap integral: () () ()() () ( ,P′ kk () *3 ,expd xlm xxlm x Ii νν ψφ − −∞ = qKrqrr r K. Taking into consideration the formula cited above, in an early stage of our analysis let us suppose that 0=q. Therefore, () exp 1i=qr . Then, due to the orthogonality condition of the wave functions of discrete and continu ous spectra, () 0 ,0 xlm x I ν == q qK . It should be noted that this case () 0=q is unacceptable. If 0=q then ine lastic (and elastic) scattering is impossible. Second, let us now suppose that 1 R− =q. In this case () exp 1iqR ≈ (R is the radius of the nucleus) and () 32 ,1 xlm x RI ν −qK . This latter case () 0≠q is possible. In this case the losses of momentum and energy of the scattered electron must be minimal. If the value of transmitted energy ω is fixed then the minimal value of transmitted momentum takes place at the minimum angle of scattering (0 θ ′=, quasireal photons). As this is the case, ω ≈q. And, as a rule, the inequality 22 1 ω −q takes place as well [25,27]. 2 The minime of the energy necessary for the ofr ω excitationesonances is practically defined al va Coul lu omb ω by the minimal excitation energy of the lowest Coulomb resonance, pNLx l E ε +, where l ε is the proton en ergy in the highest filled nuclear shell. Hence, the ine quality 1 R− q may be satisfied with reasonable ac curacy. The numerical estimates for ttigated nu clei lead to the fog result: he inves llowin 1 min 0.05 m− ≈q. So, it is natural that the inequality 1 R− q, in turn, leads to the inequality () 32 ,1RI −qK , which has already been ment xl ν g ou f in is ph m x ioned. Thus, restrictinrselves only with the analysis of ph ease o ction oe d small scattering a hat th because the p enomenon of orthogonality of wave functions of dis crete and continuous spectra, we may predict decrf the crossselastic scattering of highenergy electrons. This decrease takes place at small transmitted energies ω anngles () 1 θ ′ of elec trons. Note tenomenon is cauecreasing of distorted momentum distribution ) xl G ν . In the case of planewave approximation this state sed by d () ( , dqK ment is wrong lane wave () exp i r is essentially nonor function of bounded e proton in the considered kinematical region. Let us recall that the expression for the crosssection () , σωθ ′ has dimensionless factor () ,P′ kk that, as it well known [25,27], is large in the mentioned above region. Just the function () ,P′ kk due to nonorthogo nality of () exp i thogonal to the w of thave state r and () xlm ν ϕ r will lead to unac ceptabl values of crosssection in planey hugewave ap proximation. A to draw an unambiguous conclficance of ta of ulations rons ri (NL = 12 sually ob ctions calculated in fily exceed calculated with taking into account of inknocked nd just this factor allows us usion concerning the signi king into account of orthogonality wave functions of discrete and continuous spectra. The phenomenon of orthogonality of wave functions in our calcrepresented on Figure 4, where plotted the crosssections () , σωθ ′ of inelastic scatter ing of elect on nucleus 12C. The parameters of WoodsSaxon potential were chosen in such way that the Coulomb (NL = 20, 12) and centfugal ) reso nances have relatively large viserved halfwidth. The crosssethe planewave ap proximation signi ) () is cant ( , d σωθ ′ teraction of the Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK Copyright © 2013 SciRes. OJM 98 Figure 4. The crosssections σ(ω, θ') of inelastic scattering of electrons on the nucleus 12C versus the transmitted energy ω. The parameters of the WoodsSaxon potential: V0p = 53.7 MeV, V0n = 51.0 MeV; the initial energies of the electrons: ε = 500 MeV (a, b); ε = 2020 MeV (c); the angles of scattering are: θ' = 15˚ (a); [] − ′′ 9 310 mω εε θ =× (b, c).
A. A. PASICHNYI, O. A. PRYGODIUK 99 out nucleon in the final state, see Figure 4. Note that after some doubts and speculations the reader cannot disturb himself and fear of unusual units (Gb, Tb) that appear at the Figure 4. These units will be usual and natural units in the near future. The planewave crosssections have huge, nonphysical magnitudes in the region of the quasielastic peak. This fact is a convincing proof of the necessity of taking into account the interaction of the knockedout nucleon with the residual nucleus in the final state. 7. Coulomb Resonances, QuasiReal Photons and χExperiments In this section we will attempt to interpret some new aspects of experiments on investigation of exclusive cross sections [10,15,23] basing on the results above. These new crosssections () 6 d ddd d miss EE σ χε =′′ ΩΩ (37) depend on the energy lost in processes of scatter ing and rescatteringinelastically scattered elec tron and the knockedoughenergy proton in the initial and final states. In sueriments one fixes the cases of simultaneous regiof scattered electron with the energy and knoc proton with the energy MeV). As ts place, we can ea perimente lost energy : . Thes ofesented weaknce pl e nucle mum convenient interpretation of the minimal lost energy is is the binding energy at the upper s miss E of the t hi ch exp stration kedout his take ally th plot resona us 12 at E 40 M 0 miss rk of n ε ′ 0 ine ex ,23] the case of t har mum a () s in th E sily in e, in () miss ooth ost (10E≈ determ εε ′ =+ [10, 15 have a s max i  mis χ With miss E () pr For exam nction MeV and a sm same tim . ode miss EE + have h p maxi t E 0 <EE≈∀ e miss E χ nature. the fu 17 At the MeV clear shell m C miss eV. 17E≈ u E χ e, l the m ≈ miss ≈ miss framewo 11 h miss E ε =, filled shell If E 11 >0 h ε of nucleu h 12 C. > miss l ε , then additional losses of energy h lmiss miss EE ε Δ= or several p re: strong he − one he ena a in with tresidual nucleus h miss missl EE ε Δ= − knockedout proton in strong interaction [15,16] of the with the residual nucleus in the finite state. Note that in this case we suppose that the probabil ity of inelastic rescattering of the weakly interacting highenergy electron on nucleons of the residual nucleus or another nucleus of the target is negligible. Let us suppose that one of the protons of an upper shell of the nucleus receives the energy after an act of collision with a highenerg This proton with the energy ωεε ′ =− y electron. h l E ωε =− scattering on ords, the knoc t the part of acqu ttering s. The knockin ent, for instan out nucleons i cleons, etc. Let u ering of knocki portant role in at these processe retation of processe c nuclei in co propose anothe ditional losses can participate in residual A kingout stro ired energy on that 1A− out strong ce, the pa residual nucleus or inc the processes s suppo ngout proto our investig s can esse s of electro ncidence ex r alternative in of energy in the processes of inelastic nucleus too. In other w interacting proton spen processes of inelastic resca other A atomic nucleug teracting proton may sp energy for the excitation of the number of knockingn collisions with other nu the processes of rescatt the finite state play an im Then it is evident th complicate the interp disintegration of atomii ments. We intend to pretation of origin of ad () 1− ngly in the () or ly in rt of its rease of se that ns in ation. ntially peri ter  anoma used experiments [10,15,2proceed from lously large values of thoss by quasireal photons ( We noticed above that in the analyzed experiments the coincidence of two particles was recorded. These parti cles are: the inelastically scattered electron and knockingout (it is very desirable that just by t elec tron) proton Using simple empirical concept one can state thatockingout of proton from someeus unambusly testify that the scattere e insider in the immediate vicinity) of thnu s. It is nifficult estimate the length mean path of istic electron inside ofleus ing thosssection of excitation of b re ce in s by electrons with trgy As a sult of the estimate we can assert hierg in the prollision leus re or after of knockooton te tb resonance is (or another) nucleus with large probability. Previously developed theory [12,25,27] of Coulomb resonances and presented above estimates based on the calculated data of crosssections of excitation of Cou lomb resonances stimulate us to state such hypothesis in contrary to primordially widespread opinion: the lost energy in experiments [10,15,23] is the energy that is u for the excitation of discrete, quasidiscrete and continuous spectra of investigated nucleus but namely 3]. Let us e scattering cr 0 ′=, “00” the sections ca  scattering). θ k′ his ion nucl d electron is of nuc om e ene of pr th K. kn iguo (o ot d relativ e cr nucleu MeV. ghen 12C he C 12 C resid cleu free know sonan ε = that su with may λ this Coul h cess of co ut in 12C re y electron befo om 779 ch nuc exci oul 12NL = may be interpreted as the result of nomena listed below. These phenom teraction of the knockedout proton 1 − in the final state [15,16], which is accompanied by excitation of this nucleus; the knockingout of protons out of deeper filled nuclear shells; losses of energy of the scattered electron in initial state; accidental coincidences that are caused by the existence of intensive background of protons, the source of which is [25] the scattering of highenergy electrons caused by quasireal photons; the dispersion of the initial energies of scattered electrons in the incident beams; etc. At the initial stage of investigations it is reasonable to look for the cause of additional energy losses miss E sed Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 100 by the highenergy electrons. Note that in this case the main role belongs to quasireal photons (0 θ ′=, “00”  scattering). In connection with the hypothesis preseove we ncesquasireal ceirmation. ent [15] where the iusive cross section of electrodisintegratio 12C by hi (MeV) el tigated. Tmeasuremrely the ed n ditiona elimi nary estimate of tsed aux e e = 100 get contai 22 nucleon are protons. The Table 3 contains calculatsections of 1pshell of this nuc MeV) electronse the Cmb reso nted ab and ncl n of nu ectr is me cross that excit should say the following. Let us suppose for a while that the results presented in Tables 15 are unknown. In this case the hypothesis stated above is, at least, a striking and very glaring example of absurdity, irresponsibility, and nonsense. The origin of the hypothesis stated above is in results of works [12,25,27] and in abovepresented calculations of the crosssections of the excitation of Coulomb reso nances. In other words, the hypothesis mentioned above is undoubtedly based on the processes that are introduced to our theory by the Coulomb resona photons. It is worth noting that the experiment can verify or refute this hypothesis, but in both cases we will re ve equally important info In order to confirm or to refute the hypothesis pre sented above, we suggest the modification of experi ments [10,15,23] already mentioned above. Consider, for instance, the experim () miss E χ ghenergy he cleus ons is inves 779.5 ε = ent of tende () E χ miss first stage of slightly exd experiment. The final stage of suggested experiment is practically the same. The physicist must measure the crosssection () miss E χ in slightly changed configuration. The important but practically negligible and easily im plement modification of presented configuration is i adl auxiliary carbonic target installed at the very entry of spectrometer of scattered electrons. For pr hickness of suppoiliary carbonic target we may use the following speculations. Let we have 1 cm2 of target of thtalon density (D mg/cm2). This carbonic tarns 100 × 10−3/(1.66 × 10−24) ≈ 6 × 10s. A half of these nucleons ed excitation of Coulomb resonances of the nucleus 12C. According to this Table, each of four protons of the leus is an impermeable shield for the highenergy (700 ε ≈ oulonce na 21NL = of the investigate f this shield is 24 2 202010cmb− =× . Thus the total impermeable shield of 1 cm2 of our etalon target will be 242222 2010210cm0.4 cm − ×××≈ . The obtained result testifies tout 40% of electrons moving through the target and having the direction of movement unchanged will have the energy less by 17 MeV than in the incident beam. Thus, according to the calculations, the density D of our additional target must be 2 240260mg/cmD≈−. If the hypothesis about the origin of energy losses shifted to the right along the axis miss E by the value 17 miss EΔ≈ MeV. Note that the simplest version of d nu cleus. The are crosssection he only difference be the two cros is that the latter will be instance, by varying the th ion of ine la nces and ativistic me arized he di  i h resolutio al state f high energy a o hat ab miss E in our measurements is valid then in the modified configuration we must obtain practically the same picture of themiss E. T () χ ssectionstween modification of experiment [15], which was proposed above, may be diversified and complicated, for ickness of the additional target on the entry slit of the electron/proton spectrometer or by choos ing another sort of nuclei for the additional target. At the moment, a more detailed analysis of such variations is, at least, premature. Finally, we can conclude that the investigat stic scattering of highenergy electrons and, particularly, the experimental study of Coulomb resonaquasi real photons is of general theoretical importance, espe cially, for quantum nonrelchanics. 8. Conclusions The main results of present paper can be summand briefly stated in the following way: • Coulomb resonances are the direct theoretical pro longation of the nuclear shell structure to the conti nuous spectrum region. In the framework of the one particle(!) theory of Coulomb resonances one can readily explain many features of such wellknown phenomenon as dipole (quadrupol, octopol,...) giant resonance. It is a real possibility to interpret the phe nomenon of dipole giant resonance as real experi mental confirmation of existence of the Coulomb re sonances in the atomic nuclei. The brief theory of Coulomb resonances and calculated quasidiscrete spectra of some atomic nuclei are presented in Sec tions 2 and 3. • The regularized cross sections calculated in this paper convince of the theoretical possibility of direct mani festation of Coulomb resonances in the spectra of in elastically scattered high energy electrons. T pre sented calculations allow us to suggest the best kine matic contions for observation of such manifesta tion: nitial energies of high energy electrons, ε ≈ 300  500 MeV;  electron scattering angles, 1025 θ ′≈− degrees. The necessary condition for such manifestation is sub stantially painstaking measurement of cross sections of inelastic scattering ofigh energy electrons with high n in the region of transmitted energy, 5 MeV ≤ ω ≤ 60 MeV (Section 3). • To emphasize the significance of the wave functions orthogonality (both initial and fins of nuclei) one may investigate the scattering o Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 101 electrons at very small angles 1 θ ′ (Section 4). • The main kinematic peculiarity of virtual quasireal photons is the approximate equality of their quasi momentum q and energy ω : 22 q ω ≈. This ap proximate equality determines the effective capability of queal photons for knockingout of protons of high nuclear shells into quasidiscrete states of nuclei. In other words, quasireal photons excite Coulomb resonances of atomic nuclei with high efficiency. The cross sections of such exciting processe result in huge, alm asir an ost tremendous val be equal to dgements . s c hat mayues t hundreds and thousands of barns depending on the initial value of energy ε of scattering electrons. Such processes can be discovered in experiments proposed in this paper (Section 5). 9. Acknowle Authors wish to thank Dr. V. V. Lutsenko and Dr. Yu. V. Yakovenko for their sincere efforts in improvement of the style of the language of submitted text of this article. 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A. A. PASICHNYI, O. A. PRYGODIUK 103 Appendix A. Calculation of the ElectroDisintegration Cross Sections and Testing of Numerical Programs The accurate calculation of the cross sections of inelastic electron scattering is connected with the accurate calcu lation of the overlap integrals (18), which are the most important components of Equation (8). That problem will be investigated in detail below. Note that the inte grand in Equation (18) is the product of the exponent () exp iqr , the boundstate wave function () xlm ν ϕ r and ± the wave function of continuous spectrum () ψ Kr. For the WoodsSaxon potential, the wave function of the bound state () xlm ν ϕ r is well known [6]. This function an be written as c ()()( )()() () 22 0 ; d1. xl xlmxllm rxllmr xl xl Zr RrY AY r AZ rr ν νν ν νν ϕ ∞ =≡ = rn n (A1) The form and the content of the expressions presented bove for and together with the quiremene crosations are of igh precision predetermine the choice of the integration ethod in Equation (18). To calculate the overlap integral (18), we use the well nown planewave expansion [31,32]: a () xlm ν ϕ r t that th () () ψ − Kr ssection calculre h m k () () () 222 2 22222 222 * 0 e4π, r lml l llmlmqq lml iij qrYY =∞ = ==− == q qr nnn q (A2) After putting Equations (22), (A1), and (A2), into quation (18), we presentin the follow orm: E () ,. xlm x I ν qK ing f () () () ()() () () () ()() () 11122 21 12 111222 12 12 12 2 00 2 21 1 22 11 ,4π1 2121000 4π21 ,, lmllmll ll xlm xlmllml ll ll lmm Ii ll lll l lmlmlmAqKYY ν ν =∞ ==∞= + ==−==− =− ++ ×+ ×Kq qK nn (A3) here w ()()( )() 121 2 2 0 ,d lllKllxl; qKRrjqrRr rr νν ∞ = (A4) nd a () 22 11 lmlmlm are the ClebshGordon [31,32] coef cients. The overlap integral is presented in Equation (A3) in ector form, that is to say, Equation (A3) is valid in any rbitrarily chosen reference frame. We can take advan fi v a tage of this freedom of choice and simplify Equation (A3). With this purpose, we recall that () 0 21 4π zm l m Yl δ + == nne (A5) If we superpose vector e with vector q or then, according to Equation (A5), the summa tion in Equation (A3) is essentially simplified. We restrict ourselves to the case ofz eq and make use of wellknown [32] properties of the ClebshGordon coefficients. After that, the overlap integral takes the form () () () () () () () ()()() 1 100 1 1 ,4π21121 ljl j l xlm xlj Iil l ν =∞ = − == =+− + qK () 1 12 11 1 1 21 20 00 20 ,:2 and l ll l mz ll jllll jlmlm AqKY lllj ν ×− + ×−+ ×∀=−+ K ne q (A6) Notwithstanding the achieved facilitation, the overlap integral (A6) remains a very complicated expression in the form of a twofold series, the convergence of which depends essentially on the kinematic parameters. It is well known [33] that the computer programming of such problems is always associated with a risk to receive an erroneous result because of “natural” errors, which easily and freely “penetrate” into initial formula and programs. To avoid such errors, we propose a few tests [8], which substantially raise the level of trust to the obtained results. Note that it is the planewave approximation that pro vides useful and invaluable aid at this stage of investiga tions. Note also that the tests proposed below retain their validity in the case of relativistic models as well. Let us recall that the distorted momentum distributions turn into planewave ones if we substitute r . After that substitution the plane can be calculated with two differ rst method is to perform the limit (18) → Equation(A6), using the . According to Equation ()( ) exp i ψ → KrK wave overlap integral ent methods. The fi transition of Equation substitution () Kl Rr (A6), we obtain: ( ) 11 l jKr→ () () i lm r 003 ,e eed ii xlm xxx Ixlm νν νϕ − −∞ ≡= qrKr qr qK K ()() ()() () r 7) (A ()() ()() () 1112 221 12 1112 22 12 12 12 22 00 1 0 21 2211 2121 4π14π21 00 0,, lmll mll ll lmll ml ll ll lmm ll il llllmlmlmA qKYY ν =∞ ==∞=+ ==−==− ++ − + ⋅× Kq nn Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK 104 (A8) here . w () ()()() 121 2 02 0 ,d lllllxl qKjKrjqr Rr rr νν ∞ = (A9) d method is to calculate of the overlap ion (A7) in another way: The secon integral Equat () ( 00 ,e ee iii xlm xxxlm Ixlm νν νϕ − −∞ ≡= qrKr qr qK Kr ) 3 dr (A10) () () () () () () () 3 2 0 ed 4πd, i xlm ll mll qK iYsjsrRr rr ν ν ϕ − −∞ ∞ − = =× qKr rr (A11) where =−sqK, =ss . Note that the succession of operations Equations 11) can be continued. All one ha is to remember that the planewave momentum diution Equation (15) is proportional to the sum ( (A10), (As to do strib ) 2 00 ,, ml lxlm ml SI ν = =− =qK . In this case one can as well as x to recall a wellknown equality [32]: () 221 4π ml l m ml l Y = =− + = ner m . After that, the summation ov in can be performed analytically: 0 l S ()( ) () () () 2 0 0 4π21 d. ll SljsrRrrr ∞ →+ It is useful to note that for a large number 1 l () 120l≥ the strong inequality () xl ν ( ( ) ) 2 11 12 Ax llrMVr+ takes place in the region at for we have the approxality h is of high accreforthe conditions of convergence of the planewave approxima tion series and those of the series with distorted ov in stablishe o values of the planewave overlap integral calculated with the two different methods. This means that we have obtained a confirmation of reliable conver gence of both series Equation (A7) and Equation (18). Note that in this way we can, at the same time, verify the convergence of expansions of the plane wave in te spherical functions [Equation (25)] and the distorted wave function [Equation (A2)] in the investigated kin ematical region. calculating the overlap integrals (one after an other) by two different (Equations (A8) and (A11)) meth ods and comparing the former and the latter resu have established the following. r small energ ≤ E ≤ 50 MeV, medium nuclei) it is relatively easy to achieve agreement between the planewave cross sec tions calculated by the two mentioned above methods with an accuracy ofsignificant figures in man tissa. Note that withosed comparison we achieve also a reliableck of Equations (A3) and check alsouracy of such specialfunc tio ) the accuracy of calcu [the parameters of gration step, the n (A resul rounds for the following assertions: • The probability of errors and inaccuracies in our programmes and transformations of formulas in the investigated kinematic region is insignificant. • The convergence of series in Equations (A3) and (A6 with empirically chosen boundaries of summation over quantum numbers is quite sat isfactory for the investigate of ele of omic nucleus () rR≤. It is evident that when the mentioned inequality takes place then1>20l ( ) Kr, whic that imate equ uracy. The () 11 Kll Rr j≈ e, we can affirm erlap tegrals are identical. Let us assume that we have ed the agreement between tw rms of After lts, we Foies of the knockedon protons (1 MeV (A8); we 14 10÷ the prop accuracy che the acc lations he calcul ber of ite n subroutines as () () ,l lm jx Px, coefficients of Clebsh Gordon, as well as the accuracy of direct integration me thods in the overlap integrals, etc. When the transmit ted energy ω and, consequently, the energy E of the knockedout protons increase ( E ω ≈,300450 MeV÷, r invar proce in the sum in Equation ults coin unde tation um ms iable conditions ss such as inte 7), etc. do not change] decreases: the two res cide only with an accuracy of 65÷ significant figures in mantissa. The good agreement between two ts obtained with two different methods with high accuracy and in a wide interval of variation of the kinematic parameters gives g ) ctro 12 ,ll () 12 ,50ll≤ d kinematics disintegration of medium and heavy nuclei. Let us consider one more test, which also essentially raises the reliability of the calculated numerical values of the nucleus electrodisintegration cross sections. In this case we test the process of solution of the radial Schrödinger equation [Equation (24)] and the process of tailoring of the solutions () () Kl Rr − at the point rb≈. To do this test, we should implement the special case of ()() 0>0 Ax Vr E=∀ in the subroutine solving the ra dial Schrödinger Equation (24). One way to do this is to put 00 x V= and 10Z−= for the potentials () WS Vr and () C Vr) correspondingly. In this case, if the pro gram is correct, we must finally realize the conversions: ()( ) Kl l Rr jKr=; () () e 0 i rVr − Ψ= = Ax r K. Hence, the cide planewave momentum distribution calculated so must coin with two distributions [Equations (A3) and (A7)] calculated by the traditional methods mentioned above. In our tests we obtained agreement of all distributions with an accuracy of 126÷ significant figures in man tissa in diverse reof kiatic parameters. Therefore, we o suppose that the cross sections of nuclea gions nem have a right t r electrodisintegration in the approxi mation of distorted waves are calculated in our computer program with sufficient accuracy. Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK Copyright © 2013 SciRes. OJM 105 We have considered two tests that substantially raise the reliability of final results of our computer program. Let us suppose that we deal with large mass numbe r and large energy of the knockedon proton. It structive to mention about one more mathematical trick [8], which in this case improves the convergence of se ries appearing in calculations of the distorted momentum distributions. One can represent the overlap integral Eq uation (11) in the following form: d ν (A12) Finally, ν ϕ (A13) We calculate the first term of Equation (A13) with a e in Equation (A6) but with su ou ap method as rational as possible, for instance, like in Equa tions (A3)(A11). As to the second term, we calculate it with the help of series lik E is in bstantially improved convergence. For this purpose, one needs to perform the following substitution in Equa tion (A6): [] − () () ()()() () () () () *3 * ,exp exp exp xlm xxlm i ii ν ψϕ ψ − −∞ − = =−−+− K K qKrqrr r rKrKr () () ()()( )() 12 12 112 2 0 ,, d. ll lll l Klllx l AqK AqK RrjKrjqrRrrr νν ν ∞ → =− This method gives a possibility to extend essentially the possibilities of numerical simulation of electrodis integration processes of heavy nuclei for large trans ferred energies ω. Note that all the tests described above have been taken into accnt,proved and implemented practically as early as in paper [7]. It is a cause for regret that all these tests do not were published so far elsewhere. I () () 3 exp d . xl m i ν ϕ −∞ ⋅ qr r r () () () () () ()()()() 3 *3 ,expd exp expd xlm xxlm xlm Ii ii ν ν ϕ ψ −∞ − −∞ =− +−− K qKqKrrr rKrqrrr
