Open Journal of Microphysics, 2013, 3, 85-105 http://dx.doi.org/10.4236/ojm.2013.33015 Published Online August 2013 (http://www.scirp.org/journal/ojm) The Coulomb Resonances, the Quasi-Real Photons and Electro-Disintegration of Nuclei by High-Energy 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 quasi-real 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 knocking-on 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” cross-sections of electro-disintegration of nuclei by high-energy 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 high-energy elec- trons on nuclei in the framework of the nuclear shell model are analyzed in this work as well. The cross-sections of the scattering of high-energy electrons on the angle are calculated. It is shown that the orthogonality of the wave functions of a knocked-on 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; Quasi-Real Photon; Inelastic Scattering; High-Energy Electron; Electro-Disintegration; Nuclear Shell Model; Numerical Method; Inelastic Scattering; Quasi-Elastic Peak 1. Introduction: The Electro-Disintegration of Atomic Nuclei by High-Energy 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 [1-27] 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 high-energy leptons by nuclei. The technical perfection reached at the mo- ment in forming the monoenergetic high-energy elec- trons beams, as well as in registering these particles in nuclear experiment, played a decisive role in the choice of particles-projectiles as means of external influence on the atomic nucleus for studying the structure and proper- ties of nuclei: they were certainly the high-energy 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 (LS-coupling, 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 quasi-real photons in the aspect of their influence on the dynamics of electro-dis- 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 electro-disintegration. 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 quantum-mechanical theory of quasi-discrete spectrum in non-relativistic 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 quasi-real 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 quasi-real 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 high-energy 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 knocked-out 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 xvl-shell of an infinitely heavy nucleus (the separation energy of the nuclear xvl-nucleon); 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 electro-disintegration 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 knocked-out 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 Sturm-Liouville problem based on the non-relativistic single-particle 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 knocked-out nucleon: 2. ME==K 2. The Cross Sections of the Nucleus Electro-Disintegration 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 quasi-rela- ati the inction o a relativistic electron with a non-point nucleon of the nucleus tivistic Hamiltonian of McVoy-Van Hove [1,2], which takes into account empirically the electromagnetic struc- ture of the knocking-out non-point 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 ultra-relativistic 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 ocked-out nucle tonian o tially di () () ,, 1|| x S′kkK ,, ′ kk K. It the knon in the quasi-relativistic Hamil- f McVoy-Van Hove are reflected. Two essen- fferent functions appearing in Equation (8), which depend on kinematic variables of the process of the elec- tro-disintegration 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 electro-disintegration 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 knocked-on nucleon in the section () ,, σ ′ kk K is approxim xl ν ate. The factorized expression (8) becomes exact only in the plane-wave approximation, which is valid when the energy of the knocking-out 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 xvl-shell: () () () () ( ( ) ) 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 electro-disintegration calculate the plane-wave 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 knocked-out nucleon in the final state will be labeled by the symbol d. Formula (8) speci- fies the initial exclusive cross section of electro-disinte- gration of the atomic nucleus. Experimental verification of (8) requires fairly laborious experiments in which both the inelastically scattered electron and the knocked-out 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 quantum-electro-dy- namical [28,30] processes such as bremsstrahlung, birth electron-positron 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 x-nucleons in xvl-nuclear shell, () , xl ν σωθ ′ and () () ,, u lxlxl N νν θ σωθ ′′ = are thlete and specific knock-out cross se a x-nthe 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 d-ap- proximations by direct numerical integration of the differential cross section (8) over the total solid angle Ω≡Ω ly or of propagation of the knocked-out 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 electro-disin- tegration of nuclei. If we restrict ourselves to the nuclear are associated with labor-intensive 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 xvl-shells of atomic nuclei. The deductions an sions of this paper depend strongly on the numerical values of the calculated electro-disintegration cross sec- tio of nuclear forces is re- st the knocked-out nucleon m d conclu- ns of various nuclei. Because of this, the reliability and correctness [33] of applied numerical methods of the electro-disintegration cross-section calculation will be dis- cussed in Appendix A in detail. Note that the electro-disintegration processes of nuclei are relatively easy interpreted in the approximation pre- sented below by the quantum-electro-dynamical Feyn- man diagram of second order. If we suppose that the range ricted then we can assert that the wave function of the knocked-out () >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 electro-disintegration 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 knocked-out nu- cleon interacl nucleus in the final state. This interactionnted on the diagram by a hypo- thetical exchange meson with 4-momentum π. In this paper we estimate the dependence of the so- called χ-sections (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 high-energy electrons on nuclei. First and foremost, it is a quasi-real () 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- si-discrete state Kr of the nucleus or 1 −. The knocked-out protons and r interact in the final state with the residual nucleus. This in shown schematically in the diagrams by 4-momenta of ππ teraction is = hypothetical mesons. The knocked-out nucleons ex- change by such a meson with the daughter nucleus A − 2. The Feynman diagram(diag.2diag.2b) describe the processes of the two-proton 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 non-trivial impact on the results of experimen- tal studies of reactions ()() ,1AeepA ′−. It is well known that 4-vertex quantum-electro-dy- 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 4-vertex quantum-electro-dynamical 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 quasi-real photon 0 q (0 θ ′=, “0-0”-scattering). It is the quasi-real 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 knocked-out nuleon is capable to increase the num- ber of knocked-out nucleons. Takio account the process of inelastic scattering of the knocked-out nucleon on the residual nucleus can essentially complicate the interpretation of the electro-disintegration processes in the coincidence experiments. 3. Quasi-Discrete 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 (LS-coupling) we approximate the potential () Ax Vr by a sum of two terms: () ()() AxWSxpC VrVr Vr δ =+ . The first term () WS Vr the short-range Woods-Saxon 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 g-range 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 half-open 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 well-know () 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 half-width 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 half-widths 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 half-widths Lx and the wave function amplitudes Lx A) of quasi-discrete 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 Woods-Saxon 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 Woods-Saxon 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 quasi-discrete spectra (jj-coupling) of the light, medium and heavy nuclei per- mits to affirm that the nucleus quasi-discrete spectrum is the natural extension of the nuclear shell structure to the continuous spectrum region. For this reason, the experi- nd theoretestigations of quasi-discrete spectra properties in ()() ,1AeepA ′−-kn ns of protons are, at the same time, investigations of the nuclear shell model. 4. The Cross-Sections of Excitation of Coulomb Resonances It is well known [12,13,1heoretical inves- tigations of the inelastic scattering of high-energy elec- trons predict the resonance structure of inclusive (Equa- tion (9)) cross-sections 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 Woods-Saxon potential. For instance, the maximum values of () , lNL ν σω θ →′ and the half-widths lNL ν → of resonance cross-sections () , 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 quasi-discrete 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 Woods-Saxon 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 half-width 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 cross-sec- tions of excitationsb and centrifugal reso- nances. It is worth noting that the absence of inter-compensa- tive relation between the increase of height of any peak and the decrease of its half-width would mean the exis- tence of resoss secby qu ion esonce co etation of processes of nuclei electro-disintegration 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 half-width height and half-width of Coulomb resonances, as we will see below. Let us consider an example of the influence of so- called quasi-real 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 quasi-real 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 ′=, “0-0” - 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 quasi-real 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 quasi-real 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 knock-out 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 so-called 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 → cross-sections of excitation of the pNL-Coulomb 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 cross-section 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 NL-resonances 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 cross-sections 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 cross-sec- 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 half-widths 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 oods-Saxon 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 high-energy 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 ultra-relativ- 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 cross-section [][] 11 12 ru pp σ → substantially exceeds the cross-section [][] 10 12 ru pp σ →. Let us note that the cross - se ctions of exitation of Coulomb resonances as well as the role of quasi-real 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 12palf-widths 12 r A, h r ) of the Coulomb resonance NL = 12, specific cross-sec energies→ ν ω lNL of excitation of the Coulomb resonance NL = [12] in the nucleus 12C versus the depth V0p of the Woods-teal. 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 cross-section 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 quasi-real photons with energ t n b nce onf va experimental measurements of different cross-sections 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 cross-sections 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 cross-sections () , 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 half-widths 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 half-widths 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 theoss-sections 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 cross-sections r lpNL ν σ → of excitation of resonances of nucleus 40Ca. These cross-sections are interpreted as functions of the half-width of Coulomb resonances, which varies with the change of the depth 0 V of the Woods-Saxon potential. used by the dle transition[13], [20] →21] h, in tuused by ine- astic scattef ultra-relac electronsnu- leus 40Ca. The resulted in ble 5 do quire extensive comments as well. The specific cross-sections ca ipos [12]→[ whic l rn, are ca ring o quasi-real pho tivisti tons in on the c ts presenthe Ta not re Note that the specific cross-sections 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 half-widths 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 Cross-Sections lf-widths of the b and cenifu- ly less thahe hac peaks. At the scattered el recall that the cross-section 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 top-priority tasks of this paper is to study the manifestation of Coulomb and centrifugal resonances in experiments on inelastic scattering of high-energy 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 high-energy 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 lf-widths of the calculated quasi-elasti same time, the height of a Coulomb resonance peak is significantly larger than the height of a quasi-elastic 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 half-width 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 cross-sections [][] → ν σ ru lNL of excitation of the pNL-Coulomb 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 Woods-Saxon 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 cross-sections and enegies ocitation of the Coulomb resonans with the quaum number3,21] in the nucleselascat electrons versus te resonance half-wth . 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 so-called 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 cross-sections the parameters of Wxon po- tential were chosen so that the inequality 2 oods-Sa 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” cross-section () , 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” cross-section has resonant form in the area of Coulomb resona e quasi-elastic 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 cross-sections () , reg σωθ ′ of inelastic scattering of high-energy 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 multi-rity should be taken into account whens the spectra of inelastic large- angle scattering of high-energy electrons. Very often the scattering of high-energy 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” cross-sections 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 Woods-Saxon 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 final-state interaction (regularized cross-sectins, [(14),(36)]), and in the plane-wave 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 cross-section 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 quasi-elastic peak. Note that the Woods-Saxon potential depth parameters (0 θ ′ V and 0n V) giving the highest accuracy were used, as a rule, in our calculations. The half-width γ12 of a Cou- lomb resonance essentially increases (see Table 3) with a decrease of the depth of the Woods-Saxon potential. In this case the theoretically calculated Coulomb resonance 12NL = is widened so much that its half-width 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 cross-sections () , reg σωθ ′ of inelastical scattering of electrons and experimentally measured cross- p sections ne c heavy nucle () , ωθ ′ of electro-disintegration 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 cross-section () , reg σωθ ′ beco cal analys Figure 2. Regularized cross-section 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 cross-sections σreg(ω, θ') of inelastic scattering of high-energy 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 Woods-Saxon 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 high-energy electrons and the reaction ) considered here in the frames of nul, despite of simplicity, remains self-consistent quantum-mechanical problem in the theory of nuclei. The main reason and decisive argument for this statement is such well-known 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- tro-disintegration of nuclei by the high-energy electrons was unambiguously used in [7]. In addition to stated in [7] one may note that it is very difficult to imagine quan- tum-mechanical 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 quantum-mechanical 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 electro-disintegration of nuclei, let us consider the process of small-angle (θ' = 15˚) and very small-angle scattering (quasi-real 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 ′=, quasi-real 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 cross-selastic scattering of high-energy 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 plane-wave 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 cross-section () , 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 cross-section in plane-y 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 cross-sections () , σωθ ′ of inelastic scatter- ing of elect on nucleus 12C. The parameters of Woods-Saxon potential were chosen in such way that the Coulomb (NL = 20, 12) and centfugal ) reso- nances have relatively large viserved half-width. The cross-sethe plane-wave 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 cross-sections σ(ω, θ') of inelastic scattering of electrons on the nucleus 12C versus the transmitted energy ω. The parameters of the Woods-Saxon 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 plane-wave cross-sections have huge, nonphysical magnitudes in the region of the quasi-elastic peak. This fact is a convincing proof of the necessity of taking into account the interaction of the knocked-out nucleon with the residual nucleus in the final state. 7. Coulomb Resonances, Quasi-Real Photons and χ-Experiments 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 cross-sections () 6 d ddd d miss EE σ χε =′′ ΩΩ (37) depend on the energy lost in processes of scatter- ing and re-scatteringinelastically scattered elec- tron and the knocked-ough-energy 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 ked-out 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= − knocked-out 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 re-scattering of the weakly interacting high-energy 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 high-energ 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 king-out stro ired energy on that 1A− out strong ce, the pa residual nucleus or inc the processes s suppo ng-out 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 re-sca other A atomic nucleug- teracting proton may sp energy for the excitation of the number of knockingn collisions with other nu the processes of re-scatt 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 quasi-real 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 knocking-out (it is very desirable that just by t elec- tron) proton Using simple empirical concept one can state thatocking-out 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 thoss-section 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 cross-sections 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, quasi-discrete and continuous spectra of investigated nucleus but namely 3]. Let us e scattering cr 0 ′=, “0-0” 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. gh-en 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 knocked-out proton 1 − in the final state [15,16], which is accompanied by excitation of this nucleus; the knocking-out 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 high-energy electrons caused by quasi-real 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 high-energy electrons. Note that in this case the main role belongs to quasi-real photons (0 θ ′=, “0-0” - scattering). In connection with the hypothesis preseove we ncesquasi-real ceirmation. ent [15] where the iusive cross- section of electro-disintegratio 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 1p-shell 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 1-5 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 above-presented calculations of the cross-sections 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 gh-energy 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 cross-section () 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 high-energy (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 cross-section 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 s-sectionstween 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 high-energy 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 well-known 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 quasi-discrete 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 quasi-real 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 knocking-out of protons of high nuclear shells into quasi-discrete states of nuclei. In other words, quasi-real photons excite Coulomb resonances of atomic nuclei with high efficiency. The cross sections of such exciting processe result in huge, alm asi-r 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. REFERENCES [1] A. G. Sitenko and V. N. Gur’ev, “Inelastic Scattering of High Energy Electrons on Nuclei,” Soviet Physics—JETP, Vol. 12, No. 8, 1960, pp. 1228-1234 [2] K. W. McVoy and L. Van Hove, “Inelastic Electron- Nucleus Scattering and Nucleon-Nucleon Correlations,” Physical Review, Vol. 125, No. 3, 1962, pp. 1034-1043. doi:10.1103/PhysRev.125.1034 [3] W. Czylastic Electron Scattering from Nuclei and Single-Particle Ex Physical Review, Vol. 131, No. 5, 1963, pp. 2141-2148. doi:10.1103/PhysRev.131.2141 z, “Ine citations,” [4] V. Devanathan, “Quasi-Free Scattering of Electrons and Nuclear Shell Structure,” Annals of Physics, No. 1, 1967, pp.74-90. doi:10.1016/0003-4916(67)90292-8 Vol. 43, [5] tney et a , Vol. 9, No V. K. Tartakovskii and A. A. Pasichnyi, “Inelastic Scat- tering of High-Speed Electrons by Nuclei with Nucleon Knockout,” Ukrainian Journal of Physics, Vol. 13, No. 8, 1968, pp. 1361-1368. [6] A. A. Pasichnyi and V. K. Tartakovskii, “Coupled One-Particle States of Nucleons in a Spherical Potential,” Ukrainian Journal of Physics, Vol. 13, No. 12, 1968, pp. 2013-2019. [7] A. G. Sitenko, A. A. Pasichnyi and V. K. Tartakovskii, “On Electrodisintegration of Light Nuclei,” Soviet Jour- nal of Nuclear Physics, Vol. 12, 1970, pp. 663-672. [8] A. A. Pasichnyi, “Study of Nuclear Shell Structure in Inelastic High Energy Electron Scattering,” Ukrainian Journal of Physics, Vol. 17, No. 7, 1972, pp. 1130-1138. [9] R. R. Whil., “Quasielastic Electron Scattering,” Physical Review C. 6, 1974, pp. 2230-2235. doi:10.1103/PhysRevC.9.2230 [10] J. Moygey, et al., “Quasi-Free (e, e’p) Scattering on 12C, 28Si, 40Ca and No. 3, 1976, pp. 461-492. doi:10.1016/0375-9474(76)90510-8 58Ni,” Nuclear Physics A, Vol. 262, [11] J. S. O’Connell, et al., “Electromagnetic Excitation of the Delta Resonance in Nuclei,” Physical Review C, Vol. , No. 3, 1987, pp. 1063-1071. doi:10.1103/PhysRevC.35.1063 35 [12] ynamic s of Electrodisintegration A. A. Pasichnyi, “On D Phy Vol. 45 of Nuclei at Low Energies,” Soviet Journal of Nuclear Physics, Vol. 54, No. 12, 1991, pp. 944-955. [13] A. A. Pasichnyi, “Coulomb Resonances and Nuclear Disintegration,” Ukrainian Journal of Physics, Vol. 37, No. 4, 1992, pp. 487-497. [14] M. Zhongyu and F. Dachun, “Quasielastic Electron Scat- tering in Quasiparticle Approach,”sical Review C, , No. 2, 1992, pp. 811-818. doi:10.1103/PhysRevC.45.811 [15] G. Garino, et al., “Proton Propagation in Nuclei Studied in the (e,e’p) Reaction, ” Physical Review C, Vol. 45, No. 2, 1992, pp. 780-790. doi:10.1103/PhysRevC.45.780 [16] V. R. Pandharipande and S. C. Pieper, “Nuclear Trans- parency to Intermediate-Energy Nucleons from (e,e’p) Reactions,” Physical Review C, Vol. 45, No. 2, 1992, pp. 791-798. doi:10.1103/PhysRevC.45.791 [17] Y. H. Jin, D. S. Onley and L. E. Wright, “Electron Cou- lomb Effects in Quasielastic (e,e’p) Reactions,” Physical Review C, Vol. 45, No. 3, 1992, pp. 1311-1320. doi:10.1103/PhysRevC.45.1311 [18] A. A. Pasichnyi, “Dynamics of Electrodisintegration of nergies,” Proceedings of the 8th Seminar etic Nuclear Interactions at Low and In- pectra of Atomic Nu- Nuclei at Low E on Electromagn termediate Energies, Nauka, January 1992, pp. 296- 299. [19] A. A. Pasichnyi, “Quasi-Discrete S clei,” Ukrainian Journal of Physics, Vol. 38, No. 11, 1993, pp. 1619-1630. [20] D. B. Day, et al., “Inclusive Electron-Nucleus Scattering at High Momentum Transfer,” Physical Review C, Vol. 48, No. 4, 1993, 1849-1863. doi:10.1103/PhysRevC.48.1849 [21] A. I. Akhiezer, A. G. Sitenko and V. K. Tartakovskii, “Nuclear Electrodynamics,” Springer, Berlin, 1994. [22] C. F. Williamson et al., “Quasielastic Electron Scattering from 40Ca,” Physical Review C, Vol. 56, No. 6, 1997, pp. 3152-3172. doi:10.1103/PhysRevC.56.3152 [23] D. Dutta, et al., “Separated Spectral Functions for the Quasifree 12C(e,e’p) Reaction,” Physical Review C, Vol. 61, No. 6, 2000, Article ID: 061602. [24] A. Meucci, C. Giusti and F. D. Pacati, “Relativistic Cor- rections in (e,e’p) Knockout Reactions,” Physical Review C, Vol. 64, No. 1, 2001, Article ID: 014604. doi:10.1103/PhysRevC.64.014604 [25] A. A. Pasichnyi and O. A. Prigodyuk, “On the Electro- disintegration of Nuclei,” Physics of Atomic Nuclei, Vol. 68, No. 12, 2005, pp. 1988-2008. on-Nucleus Scattering,” Reviews of Modern Physics, 9- [2 Vol. 80, No. 1, 2008, pp. 18224. 6] O. Benhar, D. Day and I. Sick, “Inclusive Quasielastic Electr Copyright © 2013 SciRes. OJM
A. A. PASICHNYI, O. A. PRYGODIUK Copyright © 2013 SciRes. OJM 102 doi:10.1103/RevModPhys.80.189 [27] A. A. Pasichnyi, “The Coulomb Resonances and Dynam- ics of Electrodisintegration of Nuclei by High-Energy Electrons,” Physics of Particles and Nuclei, Vol. 41, No. 1, 2010, pp. 108-148. doi:10.1134/S1063779610010065 [28] J. J. Sakurai, “Advanced Quantum Mechanics,” Addison Wesley Publishing Company, Reading, 1967. [29] V. V. Balashov, “Quantum Theory of Collisions,” Mos- kow University, Moskow, 1985, in Russian. [30] V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, “Course of Theoretical Physics, Quantum Electro-dyna- mics,” Vol. 4, Pergamon, Oxford, 1982. [31] A. S. Davydov, “Theory of the Atomic Nucleus,” Nauka, Moscow, 1958 (in Russian). [32] D. A. Varshalovich, A. N. Moskalev and V. K. Herson- skiy, “Quantum Theory of Angular Momentum,” World , 1963, pp. 31-44. Scientific, Singapore, 1988. [33] S. E. Koonin, “Computational Physics,” Addison Wesley Publishing Company, London, 1986. [34] H. F. Lutz and M. D. Karvelis, “Numerical Calculation of Coulomb Wave Functions for Repulsive Coulomb Fields,” Nuclear Physics, Vol. 43, No. 1 doi:10.1016/0029-5582(63)90325-0 [35] M. Abramowitz and I. A. Stegun, “Handbook of Mathe- matical Functions,” National Bureau of Standarts: Appli- ed Mathematics Series 55, 1964.
A. A. PASICHNYI, O. A. PRYGODIUK 103 Appendix A. Calculation of the Electro-Disintegration 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 bound-state wave function () xlm ν ϕ r and ± the wave function of continuous spectrum () ψ Kr. For the Woods-Saxon 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 plane-wave expansion [31,32]: a () xlm ν ϕ r t that th () () ψ − Kr s-section 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 Clebsh-Gordon [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 well-known [32] properties of the Clebsh-Gordon 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 plane-wave 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 plane-wave 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 plane-wave 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 well-known 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 plane-wave approxima- tion series and those of the series with distorted ov in stablishe o values of the plane-wave 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 plane-wave 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 special-func- 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 knocked-on 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 knocked-out 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 electro-disintegration 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 plane-wave 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 electro-disintegration 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 knocked-on 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 electro-dis- 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
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