What can we know about hypernuclei via analysis of bremsstrahlung photons?
aa r X i v : . [ nu c l - t h ] O c t What can we know about hypernuclei via analysis of bremsstrahlung photons?
Xin Liu (1 , , ∗ Sergei P. Maydanyuk (2 , , † Peng-Ming Zhang , ‡ and Ling Liu (1) § (1) College of Physical Science and Technology, Shenyang Normal University, Shenyang, 110034, China (2)
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China and (3)
Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kiev, 03680, Ukraine (Dated: October 30, 2018)We investigate possibility of emission of the bremsstrahlung photons in nuclear reactions withhypernuclei for the first time. A new model of the bremsstrahlung emission which accompaniesinteractions between α particles and hypernuclei is constructed, where a new formalism for themagnetic momenta of nucleons and hyperon inside hypernucleus is added. For first calculations, wechoose α decay of the normal nucleus Po and the hypernucleus
Po. We find that (1) emission forthe hypernucleus
Po is larger than for normal nucleus
Po, (2) difference between these spectrais small. We propose a way how to find hypernuclei, where role of hyperon is the most essential inemission of bremsstrahlung photons during α decay. As demonstration of such a property, we showthat the spectra for the hypernuclei Te and
Te are essentially larger than the spectra for thenormal nuclei
Te and
Te. Such a difference is explained by additional contribution of emissionto the full bremsstrahlung, which is formed by magnetic moment of hyperon inside hypernucleus.The bremsstrahlung emission formed by such a mechanism, is of the magnetic type. A new formulafor fast estimations of bremsstrahlung spectra for even-even hypernuclei is proposed, where role ofmagnetic moment of hyperon of hypernucleus in formation of the bremsstrahlung emission is shownexplicitly. Such an analysis opens possibility of new experimental study of properties of hypernucleivia bremsstrahlung study.
PACS numbers: 21.80.+a, 23.60.+e, 41.60.-m, 03.65.Xp, 24.10.Ht, 25.80.Pw, 23.20.JsKeywords: bremsstrahlung, alpha decay, Λ hypernuclei, magnetic emission, hyperon, photon, magneticmoment, microscopic model, Pauli equation, tunneling
I. INTRODUCTION
Physics of hypernuclei is an important branch of nuclear physics at low as well as at intermediate energies [1–7].Hypernucleus is a kind of nucleus with atomic weight A and atomic number Z , contained at least one hyperon (Λ,Σ, Θ, and perhaps Ω) except protons and neutrons. A hypernucleus is characterized by its spin, isospin, in thecase of Λ hypernuclei, a strangeness of − − α particles. α -Nucleus interactions are understood deeply and determined well (see investigations [32–36] providingthe accurate potential of interactions between the α particles and nuclei basing on existed experimental informationof α decay and α capture, reviews and databases [37–45], other approaches [46–55], approaches of sharp angularmomentum cut-off in α -capture [56, 57], quantum mechanical calculations of fusion [61, 62], experimental data for α -capture [58–60], evaluations of the α -particle capture rates in stars [63–65] ). By such a reason, α decay can beconsidered for proper test of new calculations with hypernuclei. Note that emission of α -particles from Be and
Bwas studied experimentally in Nuclotron accelerator at JINR, Dubna [66–68].Unfortunately, opportunities to study hypernuclei experimentally are restricted. In such a situation, we put atten-tion to bremsstrahlung emission of photons which accompany nuclear reactions. Such a topic is traditional in nuclearphysics which has been causing much interest for a long time (see reviews [69, 70]). This is because of bremsstrahlungphotons provide rich independent information about the studied nuclear process. Dynamics of the nuclear process,interactions between nucleons, types of nuclear forces, structure of nuclei, quantum effects, anisotropy (deformations) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] can be included in the model describing the bremsstrahlung emission. At the same time, measurements of suchphotons and their analysis provide information about all these aspects, and verify suitability of the models. So,bremsstrahlung photons is the independent tool to obtain experimental information for all above questions. So, inthis paper we investigate idea if is it possible to use bremsstrahlung photons, which can be emitted in reactionswith participation of hypernuclei, in order to obtain a new information about properties of these hypernuclei. Fromanalysis of literature we conclude, that this question has not been studied yet. On the other side, we can base ournew investigations on our current formalism of emission of bremsstrahlung photons in nuclear reactions [71–84].At present, we do not know anything about emission of bremsstrahlung photons during reactions with hypernu-clei. By such a reason, in order to perform the first estimation of bremsstrahlung spectra, we choose reaction withnuclei, where our bremsstrahlung formalism was the most successful (in description of existed experimental data ofbremsstrahlung). This is bremsstrahlung in α decay of nuclei (the experimental bremsstrahlung data obtained withthe highest accuracy are data [85, 86] for α decay of nucleus Po, the most accurate description of these data wereobtained in Refs. [78, 79]; also see calculations [87–89] and reference therein). So, in current research we will studybremsstrahlung in α decay of hypernuclei. Note that some investigations of properties of hypernuclei in α decay havebeen already performed [90]. This reinforces our motivation for current research (one can use parameters of potentialsbetween α -particles and hypernuclei obtained in Ref. [90] for new calculations of the bremsstrahlung spectra).Hyperon (with zero electric charge) has anomalous magnetic moment, which is essentially different from anomalousmagnetic moments of neutrons and protons. So, one can suppose that hyperon inside nucleus should form emissionof photons with another intensity (and another type of emission as possible) of the bremsstrahlung photons, incomparison with neutrons inside the same nucleus (protons also has non-zero electric charge, so by such a reasonthey are principally different from hyperons in formation of bremsstrahlung). In order to clarify this question, weneed in bremsstrahlung model which takes into account magnetic momenta of nucleons and hyperon. Note that thisformalism have not been constructed yet. Importance of investigations of magnetic emission in nuclear reactions instars (see Ref. [83]) reinforces motivation to create this bremsstrahlung formalism. Another point of application iscorrections of incoherent emission (after inclusion of anomalous magnetic moments of nucleons to model) which canbe not small in nuclear reactions [82].In this paper we focus on realization of ideas above. The paper is organized in the following way. In Sec. II anew bremsstrahlung model of bremsstrahlung emission during α -decay of normal nuclei and hypernuclei is presented.Here, we include a new formalism of emission of photons due to magnetic momenta of nucleons and hyperons insidenucleus. In Sec. III we study bremsstrahlung emitted during α decay of normal nuclei Po,
Te,
Te andhypernuclei
Po,
Te,
Te. We summarize conclusions in Sect. IV. Details of calculations of relative coordinatesand corresponding momenta, operator of emission in relative coordinates, electric and magnetic form-factors are givenin Appendixes A–C.
II. MODELA. Generalized Pauli equation for nucleons in the α –nucleus system and operator of emission of photons Let us consider α -particle interacting with nucleus (which can be hypernucleus). In order to describe evolutionof nucleons of such a complicated system in the laboratory frame (we have A + 4 nucleons of the system of nucleusand α -particle), we shall use many-nucleon generalization of Pauli equation (obtained starting from Eq. (1.3.6) inRef. [91], p. 33; this formalism if along Refs. [81–83], and reference therein) i ¯ h ∂ Ψ ∂t = ˆ H Ψ , (1)where hamiltonian isˆ H = X i =1 (cid:26) m i (cid:16) p i − z i ec A i (cid:17) + z i e A i, − z i e ¯ h m i c σ · rot A i (cid:27) ++ A X j =1 (cid:26) m j (cid:16) p j − z j ec A j (cid:17) + z j e A j, − z j e ¯ h m j c σ · rot A j (cid:27) + V ( r . . . r A +4 ) . (2)Here, m i and z i are mass and electromagnetic charge of nucleon with number i , p i = − i ¯ h d / dr i is momentum operatorfor nucleon with number i , V ( r . . . r A +1 ) is general form of the potential of interactions between nucleons, σ arePauli matrixes, A i = ( A i , A i, ) is potential of electromagnetic field formed by moving nucleon with number i , A insummation is mass number of a daughter nucleus .We rewrite hamiltonian (2) as ˆ H = ˆ H + ˆ H γ , (3)where ˆ H = X i =1 m i p i + A X j =1 m j p j + V ( r . . . r A +4 ) , ˆ H γ = X i =1 (cid:26) − z i em i c p i · A i + z i e m i c A i − z i e ¯ h m i c σ · rotA i + z i e A i, (cid:27) ++ A X j =1 (cid:26) − z j em j c p j · A j + z j e m j c A j − z j e ¯ h m j c σ · rotA j + z j e A j, (cid:27) . (4)Here, ˆ H is hamiltonian describing evolution of nucleons of the α -particle and nucleus in the studied reaction (withoutphotons), ˆ H γ is operator describing emission of bremsstrahlung photons in the α -nucleus reaction.We introduce magnetic momentum of particle with number i (which is given by Dirac’s theory for proton, seeEq. (1.3.8) in page 33 in Ref. [91]), defining it as µ (Dir) i = z i · e ¯ h / m i c . In order to go to anomalous magneticmomenta of particle µ (an) i , we use the following change: µ (Dir) i → µ (an) i . (5)We have the following anomalous magnetic momenta for proton, neutron and Λ-hyperon [92]: µ (an)p = 2 . µ N , µ (an)n = − . µ N , µ (an)Λ = − . µ N , (6)where µ N = e ¯ h/ (2 m p c ) = 3 . − MeV T − is nuclear magneton. Using change (5), neglecting terms at A j and A j, , using Coulomb gauge, operator of emission (4) is transformed toˆ H γ = X i =1 (cid:26) − z i em i c A i p i − µ (an) i σ · H i (cid:27) + A X j =1 (cid:26) − z j em j c A j p j − µ (an) j σ · H j (cid:27) , (7)where H = rot A = (cid:2) ∇ × A (cid:3) . (8)This expression is many-nucleon generalization of operator of emission ˆ W in Eq. (4) in Ref. [81] with includedanomalous magnetic momenta of nucleons. B. Formalism in space representation
Principle of uncertainty forms grounds of quantum mechanics. This gives us relations between space coordinatesand corresponding momenta. By such a reason, we need to obtain full formalism in space variables or momenta. Note that Eqs. (1)–(2) is modification of Pauli equation, which is obtained as the first approximation of Dirac equation. Wave functionfor Pauli equation is spinor Ψ (it has two components), while wave function of Dirac equation is bi-spinor Ψ (Dir) = ( χ, Ψ) (it has fourcomponents). In case of one-nucleon problem, another spinor component χ of bi-spinor wave function of Dirac equation has form [seeEq. (2) in Ref. [81]; also Eq. (1.3.4) in Ref. [91]]: χ = 12 mc σ (cid:16) p − zec A (cid:17) Ψ. This magnetic moment represents a potential energy of magnetic dipole inside external magnetic field H . But in this definition weinclude also the electric charge z i , and do not use Pauli matrixes, in contrast to Ref. [91]. For further convenience, we will rewrite the operator of emission (and perform all further calculations) in the spacerepresentation.Substituting the following definition for the potential of electromagnetic field: A = X α =1 , s π ¯ hc w ph e ( α ) , ∗ e − i k ph r , (9)we obtain: H = rot A = (cid:2) ∇ × A (cid:3) = s π ¯ hc w ph X α =1 , n − i e − i k ph r (cid:2) k ph × e ( α ) , ∗ (cid:3) + e − i k ph r (cid:2) ∇ × e ( α ) , ∗ (cid:3)o . (10)Here, e ( α ) are unit vectors of polarization of the photon emitted [ e ( α ) , ∗ = e ( α ) ], k ph is wave vector of the photonand w ph = k ph c = (cid:12)(cid:12) k ph (cid:12)(cid:12) c . Vectors e ( α ) are perpendicular to k ph in Coulomb calibration. We have two independentpolarizations e (1) and e (2) for the photon with impulse k ph ( α = 1 , h = 1 and c = 1, but we shall write constants ¯ h and c explicitly. Also we have properties: h k ph × e (1) i = k ph e (2) , h k ph × e (2) i = − k ph e (1) , h k ph × e (3) i = 0 , X α =1 , , h k ph × e ( α ) i = k ph ( e (2) − e (1) ) . (11)We substitute formulas (9) and (10) to formula (7) for operator of emission and obtain:ˆ H γ = s π ¯ hc w ph 4 X i =1 X α =1 , e − i k ph r i (cid:26) i µ N z i m p m αi e ( α ) · ∇ i + µ (an) i σ · (cid:16) i (cid:2) k ph × e ( α ) (cid:3) − (cid:2) ∇ i × e ( α ) (cid:3)(cid:17)(cid:27) ++ s π ¯ hc w ph A X j =1 X α =1 , e − i k ph r j (cid:26) i µ N z j m p m Aj e ( α ) · ∇ j + µ (an) j σ · (cid:16) i (cid:2) k ph × e ( α ) (cid:3) − (cid:2) ∇ j × e ( α ) (cid:3)(cid:17)(cid:27) , (12)where µ N is nuclear magneton defined after Eqs. (6). This expression coincides with operator of emission ˆ W in form(6) in Ref. [81] in limit case of problem of one nucleon with charge Z eff in the external field [taking Eqs. ( ?? ), (5), e ( α ) , ∗ = e ( α ) and ¯ h = 1 into account]. C. Transition to coordinates of relative distances
Let us rewrite formalism above via coordinates of relative distances. We define coordinate of centers of masses forthe α particle as r α , for the daughter nucleus as R A , and for the complete system as R : r α = 1 m α X i =1 m i r αi , R A = 1 m A A X j =1 m j r Aj , R = m A R A + m α r α m A + m α = c A R A + c α R α , (13)where m α and m A are masses of the α particle and daughter nucleus, and we introduced new coefficients c A = m A m A + m α and c α = m α m A + m α . Introducing new relative coordinate r , new relative coordinates ρ αi for nucleons of the α -particle,new relative coordinates ρ Aj for nucleons (with possible hyperon) for the daughter nucleus as r = r α − R A , ρ αi = r αi − r α , ρ Aj = r j − R A , (14)we obtain new independent variables R , r , ρ αj ( i = 1 , ,
3) and ρ Aj ( j = 1 . . . A − r αi , r Aj of nucleons via new coordinates ρ αi (see calculations in Appendix A): r αi = ρ αi + R + c A r , r Aj = ρ Aj + R − c α r . (15)For numbers i = n and j = A it is more convenient to use r αn = R + c A r − m n n − X k =1 m k ρ αk , r AA = R − c α r − m AA A − X k =1 m k ρ Ak . (16)We calculate momenta connected with new independent variables R , r , ρ αi , ρ Aj (at j = 1 . . . A − i = 1 . . . n − n = 4 for the α particle). We obtain (see Appendix A for details): p αi = m αi m A + m α P + m αi m α p + m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk at i = 1 . . . n − , p αn = m αn m A + m α P + m αn m α p − m αn m α n − X k =1 ˜p αk , p Aj = m Aj m A + m α P − m Aj m A p + m A − m Aj m A ˜p Aj − m Aj m A A − X k =1 ,k = j ˜p Ak at j = 1 . . . A − , p AA = m AA m A + m α P − m AA m A p − m AA m A A − X k =1 ˜p Ak . (17) D. Operator of emission with relative coordinates
Now we will find operator of emission in new relative coordinates. For this, we start from (12), rewriting thisexpression via relative momenta:ˆ H γ = − s πc ¯ hw ph 4 X i =1 X α =1 , e − i k ph r i (cid:26) µ N z i m p m αi e ( α ) · p αi + i µ (an) i σ · (cid:16) − ¯ h (cid:2) k ph × e ( α ) (cid:3) + (cid:2) p αi × e ( α ) (cid:3)(cid:17)(cid:27) −− s πc ¯ hw ph A X j =1 X α =1 , e − i k ph r j (cid:26) µ N z j m p m Aj e ( α ) · p Aj + i µ (an) j σ · (cid:16) − ¯ h (cid:2) k ph × e ( α ) (cid:3) + (cid:2) p Aj × e ( α ) (cid:3)(cid:17)(cid:27) . (18)Substituting formulas (17) to these expressions, we find (see calculations in Appendix B):ˆ H γ = ˆ H P + ˆ H p + ∆ ˆ H γ + ˆ H k , (19)whereˆ H P = − s πc ¯ hw ph µ N m p m A + m α e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 z i e − i k ph ρ αi + e i c α k ph r A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · P + − s πc ¯ hw ph im A + m α e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 µ (an) i m αi e − i k ph ρ αi σ + e i c α k ph r A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) ×× · (cid:2) P × e ( α ) (cid:3) , (20)ˆ H p = − s πc ¯ hw ph µ N m p e − i k ph R X α =1 , (cid:26) e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · p −− i s πc ¯ hw ph e − i k ph R X α =1 , (cid:26) e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ −− e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) · (cid:2) p × e ( α ) (cid:3) , (21)ˆ H k = i ¯ h s πc ¯ hw ph e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 µ (an) i e − i k ph ρ αi σ + e i c α k ph r A X j =1 µ (an) j e − i k ph ρ Aj σ (cid:27) · (cid:2) k ph × e ( α ) (cid:3) , (22)and ∆ ˆ H γ is calculated in Appendix B [see Eqs. (B11) and (B12) in this Section]. A summation of expression (21)and ˆ H k is many-nucleon generalization of operator of emission ˆ W in Eq. (6) in Ref. [81] with included anomalousmagnetic momenta for nucleons. E. Wave function of the α -nucleus system Emission of the bremsstrahlung photons is caused by the relative motion of nucleons of the full nuclear system.However, as the most intensive emission of photons is formed by relative motion of the α particle related to the nucleus,it is sensible to represent the total wave function via coordinates of relative motion of these complicated objects. Wefollow the formalism given in [82] for the proton-nucleus scattering, and we add description of many-nucleon structureof the α -particle as in Ref. [83]. Such a presentation of the wave function allows us to take into account the mostaccurately the leading contribution of the wave function of relative motion into the bremsstrahlung spectrum, whilethe many nucleon structure of the α particle and nucleus should provide only minor corrections . Before developinga detailed many-nucleon formalism for such a problem, we shall clarify first if the many-nucleon structure of the α nucleus system is visible in the experimental bremsstrahlung spectra. In this regard, estimation of many-nucleoncontribution in the full bremsstrahlung spectrum is well described task. Thus, we define the wave function of the fullnuclear system as Ψ = Φ( R ) · Φ α − nucl ( r ) · ψ nucl ( β A ) · ψ α ( β α ) + ∆Ψ , (23)where ψ nucl ( β A ) = ψ nucl (1 · · · A ) = 1 √ A ! X p A ( − ε pA ψ λ (1) ψ λ (2) . . . ψ λ A ( A ) ,ψ α ( β α ) = ψ α (1 · · ·
4) = 1 √ X p α ( − ε pα ψ λ (1) ψ λ (2) ψ λ (3) ψ λ (4) . (24)Here, β α is the set of numbers 1 · · · α particle, β A is the set of numbers 1 · · · A of nucleons ofthe nucleus, Φ( R ) is the function describing motion of center-of-mass of the full nuclear system, Φ α − nucl ( r ) is thefunction describing relative motion of the α particle concerning to nucleus (without description of internal relativemotions of nucleons in the α particle and nucleus), ψ α ( β α ) is the many-nucleon function dependent on nucleons of the α particle (it determines space state on the basis of relative distances ρ · · · ρ of nucleons of the α particle concerningto its center-of-mass), ψ nucl ( β A ) is the many-nucleon function dependent on nucleons of the nucleus. Summation inEqs. (11) is performed over all A ! permutations of coordinates or states of nucleons. One-nucleon functions ψ λ s ( s )represent the multiplication of space and spin-isospin functions as ψ λ s ( s ) = ϕ n s ( r s ) (cid:12)(cid:12) σ ( s ) τ ( s ) (cid:11) , where ϕ n s is the spacefunction of the nucleon with number s , n s is the number of state of the space function of the nucleon with number s , (cid:12)(cid:12) σ ( s ) τ ( s ) (cid:11) is the spin-isospin function of the nucleon with number s .In definition (23) of the wave function we have also included the new term ∆Ψ. It is correction, which shouldtake into account fully anty-symmetric formulation of wave function for all nucleons. However, in this work we shallneglect by this correction, supposing that it has minor influence on studied physical effects here. F. Matrix element of emission
We define matrix element of emission of the bremsstrahlung photons, using the wave functions Ψ i and Ψ f of thefull nuclear system in states before emission of photons ( i -state) and after such emission ( f -state), as h Ψ f | ˆ H γ | Ψ i i .In this matrix element we should integrate over all independent variables. These variables are space variables R , r , ρ αn , ρ Am . Here, we should take into account space representation of all used momenta P , p , ˜p αn , ˜p Am . Substitutingformulas (18) for operator of emission to matrix element, we obtain: h Ψ f | ˆ H γ | Ψ i i = s π c ¯ hw ph n M + M + M o , (25)where M = r ¯ hw ph πc (cid:28) Ψ f (cid:12)(cid:12)(cid:12)(cid:12) ˆ H P (cid:12)(cid:12)(cid:12)(cid:12) Ψ i (cid:29) == − m A + m α X α =1 , (cid:28) Ψ f (cid:12)(cid:12)(cid:12)(cid:12) µ N m p e − i k ph R (cid:26) e − i c A k ph r X i =1 z i e − i k ph ρ αi + e i c α k ph r A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · P ++ i e − i k ph R (cid:26) e − i c A k ph r X i =1 µ (an) i m αi e − i k ph ρ αi σ + e i c α k ph r A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) · (cid:2) P × e ( α ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) Ψ i (cid:29) , (26) M = r ¯ hw ph πc (cid:28) Ψ f (cid:12)(cid:12)(cid:12)(cid:12) ˆ H p (cid:12)(cid:12)(cid:12)(cid:12) Ψ i (cid:29) == − X α =1 , (cid:28) Ψ f (cid:12)(cid:12)(cid:12)(cid:12) µ N m p e − i k ph R (cid:26) e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · p ++ i e − i k ph R (cid:26) e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) · (cid:2) p × e ( α ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) Ψ i (cid:29) , (27) M = (cid:28) Ψ f (cid:12)(cid:12)(cid:12)(cid:12) ∆ ˆ H γ + ˆ H k (cid:12)(cid:12)(cid:12)(cid:12) Ψ i (cid:29) . (28) G. Integration over space variable R
In Eq. (23) we defined the wave function of the full nuclear system. Here, Φ( R ) is wave function describing evolutionof center of mass of the full nuclear system. We rewrite this wave function asΨ = Φ( R ) · F ( r , β A , β α ) , F ( r , β A , β α ) = Φ α − nucl ( r ) · ψ nucl ( β A ) · ψ α ( β α ) , (29)Now we assume approximated form for this wave function before and after emission of photons asΦ ¯ s ( R ) = e − i K ¯ s · R , (30)where ¯ s = i or f (indexes i and f denote the initial state, i.e. the state before emission of photon, and the final state,i.e. the state after emission of photon), K s is momentum of the total system [93]. Also we assume K i = 0 as the α -decaying nuclear system before emission of photons is not moving in the laboratory frame.Let us consider just contribution M to the full matrix element. We calculate M = − Z + ∞−∞ e i ( K f − k ph ) · R dR ×× X α =1 , (cid:28) F f (cid:12)(cid:12)(cid:12)(cid:12) µ N m p n e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj o e ( α ) · p ++ i n e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ o · (cid:2) p × e ( α ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) F i (cid:29) . (31)From definition of δ -function we have: Z + ∞−∞ e i ( K f − k ) · R dR = (2 π ) δ ( K f − k ) . (32)Then, from (31) we obtain: M = − (2 π ) δ ( K f − k ph ) ×× X α =1 , (cid:28) F f (cid:12)(cid:12)(cid:12)(cid:12) µ N m p n e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj o e ( α ) · p ++ i n e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ o · (cid:2) p × e ( α ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) F i (cid:29) . (33)In this formula, we have integration over space variables r , ρ αn , ρ Am . H. Electric and magnetic form-factors
We substitute explicit formulation (29) for wave function F ( r , β A , β α ) to the obtained matrix element (33). Wecalculate this matrix element and obtain (see Appendix C for details): M = i ¯ h (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) e i k ph r ×× (cid:26) µ N m p h e − i c A k ph r m α F α, el − e i c α k ph r m A F A, el i e − i k ph r · e ( α ) ddr ++ i h e − i c A k ph r m α F α, mag − e i c α k ph r m A F A, mag i e − i k ph r · h ddr × e ( α ) i (cid:27) · Φ α − nucl , i ( r ) dr . (34)Here, we introduce new definitions of electric and magnetic form-factors of the α -particle and nucleus as F α, el = X n =1 (cid:28) ψ α,f ( β α ) (cid:12)(cid:12)(cid:12)(cid:12) z n e − i k ph ρ αn (cid:12)(cid:12)(cid:12)(cid:12) ψ α,i ( β α ) (cid:29) ,F A, el = A X m =1 (cid:28) ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12)(cid:12) z m e − i k ph ρ Am (cid:12)(cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) (cid:29) , F α, mag = X i =1 D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) µ (an) i m αi e − i k ph ρ αi σ (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E , F A, mag = A X j =1 D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) µ (an) j m Aj e − i k ph ρ Aj σ (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E . (35) I. Introduction of effective electric charge and magnetic moment of the full system
Let us consider the first two terms inside the first brackets in Eq. (34). In the first approximation, electricalform-factors tend to electric charges of α -particle and the daughter nucleus. So, we write: e − i c A k ph r m α F α, el − e i c α k ph r m A F A, el = 1 m · h e − i c A k ph r m A m α + m A F α, el − e i c α k ph r m α m α + m A F A, el i , (36)where m = m α m A m α + m A . (37)Here, m is reduced mass of system of α -particle and the daughter nucleus.We introduce a new definitions of effective electric charge and effective magnetic moment of the full system as Z eff ( k ph , r ) = e i k ph r h e − i c A k ph r m A m α + m A F α, el − e i c α k ph r m α m α + m A F A, el i , (38) M eff ( k ph , r ) = e i k ph r h e − i c A k ph r m A m α + m A F α, mag − e i c α k ph r m α m α + m A F A, mag i . (39)So, from Eq. (36) we obtain e i k ph r h e − i c A k ph r m α F α, el − e i c α k ph r m A F A, el i = 1 m · Z eff ( k ph , r ) , (40) e i k ph r h e − i c A k ph r m α F α, mag − e i c α k ph r m A F A, mag = i = 1 m · M eff ( k ph , r ) . (41)Now we can rewrite expression (34) for M via effective electric charge and magnetic moment in a compact form as M = i ¯ h (2 π ) m δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) e − i k ph r ×× (cid:26) µ N m p · Z eff ( k ph , r ) · e ( α ) ddr + i M eff ( k ph , r ) · h ddr × e ( α ) i (cid:27) Φ α − nucl , i ( r ) dr . (42)So, we have obtained the final formula for the matrix element, where we have our new introduced effective electriccharge and magnetic momentum of the full nuclear system (of the α particle and nucleus).In order to connect the found matrix element (42) with our previous formalism, we use Eq. (20) in Ref. [83] andobtain (we use only term of M , ¯ h = 1 and c = 1): M = − em p · p full δ ( K f − k ) , (43)where p full = − m p e i ¯ h (2 π ) m · X α =1 , Z Φ ∗ α − nucl , f ( r ) e − i k ph r ×× (cid:26) µ N m p · Z eff ( k ph , r ) · e ( α ) ddr + i M eff ( k ph , r ) · h ddr × e ( α ) i (cid:27) Φ α − nucl , i ( r ) dr . (44) J. Dipole approximation of the effective charge
For first estimations of the bremsstrahlung emission for hypernuclei, one can consider the dipole approximationof effective charge (i.e. at k ph r → Z eff ( k ph , r ) → Z (dip)eff = m A m α + m A Z α − m α m α + m A Z A = m A Z α − m α Z A m α + m A . (45)For the effective magnetic moment, we obtain correspondingly M eff ( k ph , r ) → M (dip)eff ( k ph , r ) = m A m α + m A F α, mag − m α m α + m A F A, mag . (46)In particular, for even-even nuclei with additional hyperon (for example, for Po), we obtain: F (dip) α, mag = 0 , F (dip) A, mag = µ (an)Λ m Λ σ , (47)and from Eq. (46) we obtain: M (dip)eff ( k ph , r ) = m A m α + m A F (dip) α, mag − m α m α + m A F (dip) A, mag = − m α m α + m A µ (an)Λ m Λ σ = − µ (an)Λ m Λ µm A σ . (48)The matrix element in Eq. (42) is transformed to the following: p (dip)full = − Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m · X α =1 , Z Φ ∗ α − nucl , f ( r ) e − i k ph r ×× (cid:26) e ( α ) ddr + i M (dip)eff µ N m p Z (dip)eff · h ddr × e ( α ) i (cid:27) Φ α − nucl , i ( r ) dr . (49)In particular, for even-even nuclei with additional hyperon (for example, for Po), we obtain: p (dip)full = − Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m ×× X α =1 , Z Φ ∗ α − nucl , f ( r ) e − i k ph r (cid:26) e ( α ) ddr − i c σ · h ddr × e ( α ) i (cid:27) Φ α − nucl , i ( r ) dr , (50)0where we introduce a new factor: c = µ (an)Λ µ N m Λ m p Z (dip)eff mm A . (51)Substituting values for masses and magnetic moments for proton and hyperon [92], we calculate: c = − . · m α m A Z α − m α Z A . (52) K. Calculations of matrix elements of emission in multipolar expansion
We have to calculate the following matrix elements: (cid:28) k f (cid:12)(cid:12)(cid:12)(cid:12) e − i kr ∂∂ r (cid:12)(cid:12)(cid:12)(cid:12) k i (cid:29) r = Z ϕ ∗ f ( r ) e − i kr ∂∂ r ϕ i ( r ) dr . (53)Such matrix elements were calculated in Ref. [81] in the spherically symmetric approximation of nucleus. Accordingto Eqs. (24), (29) in Ref. [81], we have: (cid:28) k f (cid:12)(cid:12)(cid:12)(cid:12) e − i kr ∂∂ r (cid:12)(cid:12)(cid:12)(cid:12) k i (cid:29) r = r π X l ph =1 ( − i ) l ph p l ph + 1 X µ = ± ξ m µ × h p Ml ph µ − iµ p El ph µ i , (54)where [see Eqs. (38), (39) in Ref. [81]] p Ml ph ,µ = r l i l i + 1 I M ( l i , l f , l ph , l i − , µ ) · n J ( l i , l f , l ph ) + ( l i + 1) · J ( l i , l f , l ph ) o −− r l i + 12 l i + 1 I M ( l i , l f , l ph , l i + 1 , µ ) · n J ( l i , l f , l ph ) − l i · J ( l i , l f , l ph ) o ,p El ph ,µ = s l i ( l ph + 1)(2 l i + 1)(2 l ph + 1) · I E ( l i , l f , l ph , l i − , l ph − , µ ) · n J ( l i , l f , l ph −
1) + ( l i + 1) · J ( l i , l f , l ph − o −− s l i l ph (2 l i + 1)(2 l ph + 1) · I E ( l i , l f , l ph , l i − , l ph + 1 , µ ) · n J ( l i , l f , l ph + 1) + ( l i + 1) · J ( l i , l f , l ph + 1) o ++ s ( l i + 1)( l ph + 1)(2 l i + 1)(2 l ph + 1) · I E ( l i , l f , l ph , l i + 1 , l ph − , µ ) · n J ( l i , l f , l ph − − l i · J ( l i , l f , l ph − o −− s ( l i + 1) l ph (2 l i + 1)(2 l ph + 1) · I E ( l i , l f , l ph , l i + 1 , l ph + 1 , µ ) · n J ( l i , l f , l ph + 1) − l i · J ( l i , l f , l ph + 1) o , (55)and J ( l i , l f , n ) = + ∞ Z dR i ( r, l i ) dr R ∗ f ( l f , r ) j n ( k ph r ) r dr,J ( l i , l f , n ) = + ∞ Z R i ( r, l i ) R ∗ f ( l f , r ) j n ( k ph r ) r dr,I M ( l i , l f , l ph , l , µ ) = Z Y ∗ l f m f ( n r ) T l i l , m i ( n r ) T ∗ l ph l ph , µ ( n r ) d Ω ,I E ( l i , l f , l ph , l , l , µ ) = Z Y ∗ l f m f ( n r ) T l i l , m i ( n r ) T ∗ l ph l , µ ( n r ) d Ω . (56)Here, j n ( k ph r ) is spherical Bessel function of order n , T l ph l ′ ph ,µ ( n r ) are vector spherical harmonics . Vectors ξ − and ξ +1 are (complex) vectors of circular polarization of photon emitted with opposite directions of rotation which arerelated with vectors e α of polarization as (see Ref. [94], p. 42): ξ − = 1 √ e (1) − i e (2) ) , ξ +1 = − √ e (1) + i e (2) ) . (57)1Using representation (54), the matrix element (50) is simplified as p (dip)full = Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m · r π X l ph =1 ( − i ) l ph p l ph + 1 ×× (cid:26) X µ = ± h m (cid:16) µ − i c σ · (cid:2) ξ − × ξ ∗− (cid:3)(cid:17) (cid:2) p Ml ph µ − iµ p El ph µ (cid:3)(cid:27) , (58)where h − = 1 √ − i ) , h = − √ i ) , h − + h = − i √ . (59)Now we take into account that two vectors e (1) and e (2) are vectors of polarization of photon emitted, which areperpendicular to direction of emission of this photon defined by vector k . Modulus of vectorial multiplication (cid:2) e (1) × e (2) (cid:3) equals to unity. So, we have: n ph ≡ k ph (cid:12)(cid:12) k ph (cid:12)(cid:12) = (cid:2) e (1) × e (2) (cid:3) (60)and h ξ − × ξ ∗− i = − h ξ +1 × ξ ∗ +1 i = − (cid:2) ξ − × ξ +1 (cid:3) = i (cid:2) e × e (cid:3) = i n ph . (61)From here, we find property in Eq. (58): (cid:2) ξ m × ξ ∗ m (cid:3) = − µ · (cid:2) ξ − × ξ ∗− (cid:3) = − i µ · n ph , (62)and the matrix element (58) is simplified as p (dip)full = Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m · r π X l ph =1 ( − i ) l ph p l ph + 1 n X µ = ± h m (cid:0) µ + c σ · n ph (cid:1) (cid:2) p Ml ph µ − iµ p El ph µ (cid:3)o . (63)Analyzing such a form for the matrix element, now we introduce a new formula for fast approximated estimationsof the spectra (here, we include the largest contributions in summation): p (dip)full = Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m · (cid:0) (cid:12)(cid:12) c σ · n ph (cid:12)(cid:12) (cid:1) r π X l ph =1 ( − i ) l ph p l ph + 1 n X µ = ± h m (cid:2) p Ml ph µ − iµ p El ph µ (cid:3)o . (64)We see appearance of a new factor (cid:0) (cid:12)(cid:12) c σ · n ph (cid:12)(cid:12) (cid:1) in this formula. This factor characterizes explicitly influence ofthe magnetic moment of hyperon inside hypernucleus on matrix element of emission (and on the final bremsstrahlungspectrum). Here, parameter c is dependent on choice of hypernucleus (see Eq. (51)). L. Probability of emission of the bremsstrahlung photon
We define the probability of the emitted bremsstrahlung photons on the basis of the full matrix element p full inframeworks of our previous formalism (see Refs. [81–83], also Refs. and reference therein). Here, we choose our lastinvestigation [83] developed for bremsstrahlung in α -decay (see Eq. (22) in that paper). But, in current research,we are interesting in the bremsstrahlung probability which is not dependent on angle θ f . So, we have to integrateEq. (22) in Ref. [83] over this angle and we obtain: d Pdw ph = N · e π c w ph E i m k i (cid:12)(cid:12) p full (cid:12)(cid:12) , (65)where k i = 1¯ h p µ E i is wave number of the full nuclear system (i.e. the α -particle and nucleus) in the initial state(i.e. in state before emission of the bremsstrahlung photon), E i is energy of the full nuclear system in the initial state(it corresponds to kinetic energy of the α -particle in the asymptotic distance from nucleus), µ = m α m A / ( m α + m A ) is2reduced mass of system of the α -particle and nucleus. Here, we add an additional new factor N in order to normalizecalculations on experimental data. In this research, we find the factor N from the best agreement between theory andexperiment (this is a case of α decay of the normal nucleus Po). Then, we use the same found normalized factor N for all other calculated spectra. By such a way, we obtain a possibility to compare the calculated bremsstrahlungspectra for different nuclei and hypernuclei.Basing on Eq. (60), for fast estimations in computer Eq. (65) for even-even nuclei with possible inclusion of Λ-hyperon can be simplified as d P (dip) dw ph = n (cid:12)(cid:12) c σ · n ph (cid:12)(cid:12) o · d P (dip , no − Λ) dw ph , d P (dip , no − Λ) dw ph = N · e π c w ph E i m k i (cid:12)(cid:12) p (dip , no − Λ)full (cid:12)(cid:12) , (66) p (dip , no − Λ)full = Z (dip)eff · i ¯ h (2 π ) m p e µ N m p m · r π X l ph =1 ( − i ) l ph p l ph + 1 · X µ = ± h m µ (cid:2) p Ml ph µ − iµ p El ph µ (cid:3) . (67)Here, we express explicitly the bremsstrahlung probability P (dip , no − Λ) and the matrix element p (dip , no − Λ)full whichbelong to standard bremsstrahlung formalism without hyperons in nuclei.
III. DISCUSSIONS
For the first estimations of the bremsstrahlung spectra we have chosen two nuclei:
Po and
Po. We explainsuch a choice by the following. At present, there are experimental data for the bremsstrahlung in α decay for fournuclei: Po [85, 86, 95–97],
Po [79],
Ra [80] and
Cm. The experimental data obtained with the highestaccuracy are experimental data [85, 86] for nucleus
Po. In our approach [78, 79], we achieved the most accurateagreement with these data (see also Refs. [87–89]). So, we put main focus to this normal nucleus in our research.According to Ref. [90], the hypernucleus, which has the most close α –nucleus interaction with the normal nucleus Po, is
Po.We calculated bremsstrahlung of photons emitted in α decay of Po and
Po. Results of such calculations incomparison with analysis of magnetic momenta of hyperon and nucleons with experimental data [85, 86] are presentedin Fig. 1 (a). From such calculations we see that the bremsstrahlung spectrum in α decay of the hypernucleus Po
100 200 300 400 500 600 70010 -13 -12 -11 -10 -9 -8 d P / d E ( / ( ke V d ecay )) Photon energy, E (keV)
Boie 2007 alpha-decay of standard nucleus
Po alpha-decay of hyper nucleus Po (a)
100 200 300 400 50010 -12 -11 -10 -9 -8 d P / d E ( / ( ke V d ecay )) Photon energy, E (keV)
Boie 2007 alpha-decay of hyper-nucleus alpha-decay of standard nucleus (b)
FIG. 1: (Color online) The bremsstrahlung probabilities of photons emitted during α decay of the normal Po nucleus and the
Po hypernucleus in comparison with experimental data (Boie 2007: [85, 86]) [parameters of calculations: the bremsstrahlungprobability is defined in Eq. (65), parameters of potentials are taken in Ref. [90]]. Here, open circles are experimental data [85, 86]for α decay of Po, blue solid line is the calculated spectrum for α decay of normal nucleus Po [(figures (a), (b)], red dashedline is the calculated spectrum for α decay of hyper nucleus Po with magnetic momentums of nucleons and hypeon [figure(a)], brown dash-dotted line is the calculated spectrum for α decay of hyper nucleus Po without magnetic momentums ofnucleons and hyperon [figure (b)]. Panel (a): One can see small difference between the spectra for the normal and hyper nuclei.Such a difference is explained by additional contribution of emission of photons formed by the anomalous magnetic momentof hyperon inside the hyper nucleus
Po (which is absent in the normal nucleus
Po). Panel (b): Without inclusion of themagnetic moments to calculations, difference between the spectra for
Po and
Po is small also. (see red dashed line in figure) is above than the bremsstrahlung spectrum in α decay of normal nucleus Po (see blue3solid line in figure). Such a difference between the spectra is explained mainly by additional contribution to the fullbremsstrahlung emission, which is caused by magnetic moment of hyperon inside hypernucleus. The bremsstrahlungemission formed by such a mechanism, is of the magnetic type. However, as we estimate this contribution is reallysmall for all isotopes of Polonium. Before such calculations, we estimated emission of bremsstrahlung photons withoutinclusion of the magnetic moments of hyperon and nucleons. In any case, potentials of interactions are different, butsuch difference in small also [see Fig. 1 (b)].Analyzing formulas (52) and (66), one can find hypernuclei, for which role of hyperon is more essential in emission ofbremsstrahlung photons during α decay. The simplest idea is to look for nuclei at condition of Z (dip)eff → Z (dip)eff = 0). As demonstration of this property, we estimate the bremsstrahlungspectra for the normal nuclei Te and
Te in comparison with the hypernuclei
Te and
Te. Results of suchcalculations are presented in Fig. 2, they confirm property described above. More high tendency of spectra for Te
100 200 300 400 500 600 700 80010 -16 -15 -14 -13 -12 -11 -10 d P / d E ( / ( ke V d ecay )) Photon energy, E (keV) alpha-decay of standard nucleus
Te alpha-decay of hypernucleus Te alpha-decay of standard nucleus Te alpha-decay of hypernucleus Te FIG. 2: (Color online) The bremsstrahlung probabilities of photons emitted during α decay of the normal nuclei Te and
Te in comparison with the hypernuclei
Te and
Te [parameters of calculations: the bremsstrahlung probability is definedin Eqs. (66)–(67) we use Q = 4 .
290 MeV for
Te and
Te, Q = 3 .
450 MeV for
Te and
Te, Q -values for normal nucleiare taken from Table I in Ref. [39] ]. Here, blue solid line is the calculated spectrum for α decay of normal nucleus Te, reddashed line is the calculated spectrum for α decay of hypernucleus Te with magnetic momentums of nucleons and hyperon,green dash-dotted line is the calculated spectrum for α decay of normal nucleus Te, purple dash-double dotted line is thecalculated spectrum for α decay of hypernucleus Te with magnetic momentums of nucleons and hyperon. One can see thathere role of magnetic moment of hyperon inside hypernucleus in emission of bremsstrahlung photons is more essential, than inprevious calculations in Fig. 1 for
Po and
Po. and
Te in comparison with spectra for
Te and
Te is explained by higher Q -values for these nuclei. One cansee that emission in the α decay of hypernucleus has similar tendency as studied before emission in the α decay ofnormal nuclei, without existence any resonant peak in the spectrum. IV. CONCLUSIONS AND PERSPECTIVES
At first time, we investigate possibility of emission of the bremsstrahlung photons in nuclear reactions with hyper-nuclei. In such an analysis we focus on interactions between α -particles and nuclei. In order to perform this research,we construct a new model of the bremsstrahlung emission which accompanies interactions between α particles andhypernuclei. As example for calculations and analysis, we choose α decay of isotopes of Polonium. For the firstestimations of the bremsstrahlung spectra we have chosen two nuclei: the normal nucleus Po and the hypernucleus
Po. Motivation of such a choice of reaction is the following:1. For the first estimations of bremsstrahlung in reactions with hypernuclei, we need in the most testedbremsstrahlung model and calculations. The experimental bremsstrahlung data obtained with the highest accu-racy are data [85, 86] for α decay of nucleus Po. For this nucleus the best confirmation of the bremsstrahlungmodel and calculations were obtained (see Refs. [78, 79], also calculations [87–89])).2. α Decay of hypernuclei has been already studied theoretically (see Ref. [90] and reference therein). From sucha research, interacting potentials are known, needed for our calculations.3. Today, heavy hypernucleus Λ Pb has already been known experimentally (see Ref. [92]).44. According to Ref. [90], the hypernucleus, which has the most close α –nucleus interaction with the normal nucleus Po, is
Po.Our new contribution to the existed theory is the following:1. We generalize our previous many-nucleon bremsstrahlung model [83], including new formalism for the magneticmomenta of nucleons and hyperon.2. We introduce the new magnetic form-factors for nucleus and α particle. This characteristic can be related withanomalous magnetic momentum of hypernucleus. It can be useful for study properties of hypernuclei.3. Our approach allows to study and estimate role of anomalous magnetic momenta of nucleons and hyperon inemission of bremsstrahlung photons.4. Our approach allows can be used for next step for study of incoherent bremsstrahlung emission in reactions withhypernuclei.5. We propose a new formula for fast estimations of bremsstrahlung spectra for even-even hypernuclei in com-puter [see Eqs. (66)–(67)], where role of magnetic moment of hyperon in hypernucleus in formation of thebremsstrahlung emission is shown explicitly.Our new results in study of bremsstrahlung in α decay of hypernuclei are the following:1. We have not found any information in literature about emission of bremsstrahlung photons during reactionswith participation of hypernuclei. We performed the first estimations of emission of the bremsstrahlung photonsduring α -decay of hypernuclei, estimating the bremsstrahlung spectra for isotopes of Polonium.2. Hyperon has own magnetic moment (which is different from magnetic momenta for nucleons). By such areason, hyperon inside hypernucleus (which is under α -decay) forms additional bremsstrahlung emission. Thiscontribution of emission to the full bremsstrahlung spectrum is small, but it reinforces the full emission (seeFig. 1, where we calculated the spectra during α decay of the normal nucleus Po and the hypernucleus
Po).The bremsstrahlung emission formed by such a mechanism, is of the magnetic type.3. The bremsstrahlung emission in the α decay of hypernucleus has similar tendency as studied beforebremsstrahlung emission in the α decay of normal nuclei, without existence any resonant peak in the spec-trum (see Fig. 1).4. Before such calculations, we estimated emission of bremsstrahlung photons in α decay of Po and
Po withoutinclusion of the magnetic moments of hyperon and nucleons to calculations. α -Nucleus potentials of interactionsfor Po and
Po are different, but this difference in small [90]. As a result, we find that difference betweenthe bremsstrahlung spectra is small, if to neglect magnetic momenta of hyperon and nucleons [see Fig. 1 (b)].5. We propose a way how to find hypernuclei, where role of hyperon is the most essential in emission ofbremsstrahlung photons during α decay. The simplest idea is to look for nuclei at condition of Z (dip)eff → Z (dip)eff = 0). As example, we estimate the bremsstrahlung spectra for thenormal nuclei Te and
Te in comparison with the hypernuclei
Te and
Te (see Fig. 2).This opens perspective to study properties of strange nuclear matter via analysis of the bremsstrahlung emission,which accompanies reactions with hypernuclei.
Acknowledgements
S. P. Maydanyuk and Liu Xin thank the Institute of Modern Physics of Chinese Academy of Sciences for warmhospitality and support. This work was supported by the Major State Basic Research Development Program inChina (No. 2015CB856903), the National Natural Science Foundation of China (Grant Nos. 11575254, 11447105,11175215, 11575060, 11375062, 11505057 and 11647306), the Chinese Academy of Sciences fellowships for researchersfrom developing countries (No. 2014FFJA0003).5
Appendix A: Transition to coordinates of relative distances
In this Section we rewrite formalism via coordinates of relative distances. We start from definitions (13) forcoordinate of centers of masses for the α particle as r α , for the daughter nucleus as R A , and for the complete systemas R : r α = 1 m α X i =1 m i r αi , R A = 1 m A A X j =1 m j r Aj , R = m A R A + m α r α m A + m α = c A R A + c α R α , (A1)where m α and m A are masses of the α particle and daughter nucleus, and c A = m A m A + m α , c α = m α m A + m α . Also we usedefinitions (14) for relative coordinate r , relative coordinates ρ αi for nucleons of the α -particle, relative coordinates ρ Aj for nucleons (with possible hyperon) for the daughter nucleus as r = r α − R A , ρ αi = r αi − r α , ρ Aj = r j − R A . (A2)From here we find ( n = 4): R = m A R A + m α r α m A + m α = 1 m A + m α n A X j =1 m j r Aj + n X i =1 m i r αi o , r = r α − R A = 1 m α n X i =1 m i r αi − m A A X j =1 m j r Aj , ρ αi = r αi − r α = r αi − m α n =4 X k =1 m k r αk ( i = 1 . . . n − , ρ αn = − m n n − X k =1 m k ρ αk , ρ Aj = r Aj − r A = r Aj − m A A X k =1 m k r Ak ( j = 1 . . . A − , ρ AA = − m AA A − X k =1 m k ρ Ak . (A3)Vectors ρ αn and ρ AA are dependent on other ρ α . . . ρ αn − and ρ A . . . ρ AA − (as we define them concerning to centerof mass of studied fragment). So, one can rewrite them as ρ αn = r αn − r α = r αn − m α n X k =1 m k r αk , ρ AA = r AA − r A = r AA − M A X k =1 m k r Ak . (A4)We express old coordinates r α , R A via new coordinates R , r : (cid:26) R = c A R A + c α r α , r = r α − R A → (cid:26) R + c A r = c α r α + c A r α = ( c α + c A ) r α = r α , R − c α r = c α R A + c A R A = ( c α + c A ) R A = R A or r α = R + c A r , R A = R − c α r . (A5)Now, using (A2) and (A3), we rewrite old coordinates r αi , r Aj of nucleons via new coordinates ρ αi , : r αi = ρ αi + r α = ρ αi + R + c A r , r Aj = ρ Aj + R A = ρ Aj + R − c α r or r αi = ρ αi + R + c A r , r Aj = ρ Aj + R − c α r . (A6)For numbers i = n and j = A it is more convenient to use [from (A3) and (A6)] r αn = R + c A r − m n n − X k =1 m k ρ αk , r AA = R − c α r − m AA A − X k =1 m k ρ Ak . (A7)6From (A3) we shall calculate derivatives: dRdr αi = m i m A + m α , dRdr Aj = m j m A + m α , drdr αi = m i m α , drdr Aj = − m j m A . (A8)From (A3) and (A4) for ρ αi and ρ Aj we have (at i = 1 . . . n − j = 1 . . . A − d ρ αi dr αi = m α − m i m α , d ρ αi dr αk ( k = i, k = n =4) = − m k m α , d ρ αi dr αn = − m n m α , d ρ αi dr Aj = 0 , d ρ Aj dr Aj = m A − m j m A , d ρ Aj dr Ak ( k = j, k = A ) = − m k m A , d ρ Aj dr AA = − m AA m A , d ρ Aj dr αi = 0 . (A9)From (A3) and (A4) for ρ αn and ρ AA we have (at i = 1 . . . n − j = 1 . . . A − d ρ αn dr αi,i = n = − m i m α , d ρ αn dr αn = 1 − m n m α , d ρ AA dr αj,j = A = − m j m A , d ρ AA dr AA = 1 − m AA m A . (A10)Now we shall calculate momenta connected with new independent variables R , r , ρ αi , ρ Aj (at j = 1 . . . A − i = 1 . . . n − n = 4 for the α particle). From (A9) at i = 1 . . . n − p αi = − i ¯ h ddr αi = − i ¯ h dRdr αi ddR − i ¯ h drdr αi ddr − i ¯ h n − X k =1 d ρ αk dr αi dd ρ αk − i ¯ h A − X k =1 d ρ Ak dr αi dd ρ Ak == − i ¯ h m i m A + m α ddR − i ¯ h m i m α ddr − i ¯ h n − X k =1 , k = i d ρ αk dr αi dd ρ αk − i ¯ h n − X k =1 , k = i d ρ αk dr αi dd ρ αk == − i ¯ h m i m A + m α ddR − i ¯ h m i m α ddr − i ¯ h m α − m i m α dd ρ αi + i ¯ h n − X k =1 ,k = i m i m α dd ρ αk == m i m A + m α P + m i m α p + m α − m i m α ˜p αi − m i m α n − X k =1 ,k = i ˜p αk , (A11)and at i = n = 4 we have: p αn = − i ¯ h ddr αn = − i ¯ h dRdr αn ddR − i ¯ h drdr αn ddr − i ¯ h n − X k =1 d ρ αk dr αn dd ρ αk − i ¯ h A − X k =1 d ρ Ak dr αn dd ρ Ak == − i ¯ h m n m A + m α ddR − i ¯ h m n m α ddr + i ¯ h n − X k =1 m n m α dd ρ αk = m n m A + m α P + m n m α p − m n m α n − X k =1 ˜p αk . (A12)For nucleons of the nucleus we have the similar expressions, but need only change sign before momentum p . Summarizeall final formulas: p αi = m αi m A + m α P + m αi m α p + m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk at i = 1 . . . n − , p αn = m αn m A + m α P + m αn m α p − m αn m α n − X k =1 ˜p αk , p Aj = m Aj m A + m α P − m Aj m A p + m A − m Aj m A ˜p Aj − m Aj m A A − X k =1 ,k = j ˜p Ak at j = 1 . . . A − , p AA = m AA m A + m α P − m AA m A p − m AA m A A − X k =1 ˜p Ak . (A13)7 Appendix B: Operator of emission in relative coordinates
Now we will find operator of emission in new relative coordinates. For this, we start from (12), rewriting thisexpression via relative momenta:ˆ H γ = − s πc ¯ hw ph 4 X i =1 X α =1 , e − i k ph r i (cid:26) µ N z i m p m αi e ( α ) · p αi + i µ (an) i σ · (cid:16) − ¯ h (cid:2) k ph × e ( α ) (cid:3) + (cid:2) p αi × e ( α ) (cid:3)(cid:17)(cid:27) −− s πc ¯ hw ph A X j =1 X α =1 , e − i k ph r j (cid:26) µ N z j m p m Aj e ( α ) · p Aj + i µ (an) j σ · (cid:16) − ¯ h (cid:2) k ph × e ( α ) (cid:3) + (cid:2) p Aj × e ( α ) (cid:3)(cid:17)(cid:27) . (B1)Substituting here formulas (A13), we find:ˆ H γ = − s πc ¯ hw ph 4 X i =1 X α =1 , e − i k ph r i (cid:26) µ N z i m p m αi e ( α ) · h m αi m A + m α P + m αi m α p + m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk i i = n ++ µ N z i m p m αi e ( α ) · h m αn m A + m α P + m αn m α p − m αn m α n − X k =1 ˜p αk i i = n − i ¯ h µ (an) i σ · (cid:2) k ph × e ( α ) (cid:3) ++ i µ (an) i σ · h(cid:16) m αi m A + m α P + m αi m α p + m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk (cid:17) i = n × e ( α ) i ++ i µ (an) i σ · h(cid:16) m αn m A + m α P + m αn m α p − m αn m α n − X k =1 ˜p αk (cid:17) i = n × e ( α ) i(cid:27) −− s πc ¯ hw ph A X j =1 X α =1 , e − i k ph r i (cid:26) µ N z j m p m Aj e ( α ) · h m Aj m A + m A P − m Aj m A p + m A − m Aj m A ˜p Aj − m Aj m A A − X k =1 ,k = j ˜p Ak i j = A ++ µ N z j m p m Aj e ( α ) · h m AA m A + m α P − m AA m A p − m AA m A A − X k =1 ˜p Ak i j = A − i ¯ h µ (an) j σ · (cid:2) k ph × e ( α ) (cid:3) ++ i µ (an) j σ · h(cid:16) m Aj m A + m α P − m Aj m A p + m A − m Aj m A ˜p Aj − m Aj m A n − X k =1 ,k = i ˜p αk (cid:17) j = A × e ( α ) i ++ i µ (an) j σ · h(cid:16) m AA m A + m α P − m AA m A p − m AA m A n − X k =1 ˜p Ak (cid:17) j = A × e ( α ) i(cid:27) . (B2)In this expression, we combine terms with the similar momenta asˆ H γ = ˆ H γ + ˆ H γ + ˆ H γ + ˆ H γ , (B3)whereˆ H γ = − s πc ¯ hw ph 4 X i =1 X α =1 , e − i k ph r i (cid:26)h(cid:16) µ N z i m p m αi e ( α ) · m αi m A + m α (cid:17) i = n + (cid:16) µ N z i m p m αi e ( α ) · m αn m A + m α (cid:17) i = n i P ++ h(cid:16) i µ (an) i σ · m αi m A + m α (cid:17) i = n + (cid:16) i µ (an) i σ · m αn m A + m α (cid:17) i = n i (cid:2) P × e ( α ) (cid:3) ++ h(cid:16) µ N z i m p m αi e ( α ) · m αi m α (cid:17) i = n + (cid:16) µ N z i m p m αi e ( α ) · m αn m α (cid:17) i = n i p ++ h(cid:16) i µ (an) i σ · m αi m α (cid:17) i = n + (cid:16) i µ (an) i σ · m αn m α (cid:17) i = n i (cid:2) p × e ( α ) (cid:3)(cid:27) , (B4)8ˆ H γ = − s πc ¯ hw ph 4 X i =1 X α =1 , e − i k ph r i (cid:26)(cid:16) µ N z i m p m i e ( α ) · m α − m αi m α (cid:17) i = n ˜p αi ++ (cid:16) − µ N z i m p m i e ( α ) · m αi m α (cid:17) i = n n − X k =1 ,k = i ˜p αk − µ N z i m p m i e ( α ) · h m αn m α n − X k =1 ˜p αk i i = n + − i ¯ h µ (an) i σ · (cid:2) k ph × e ( α ) (cid:3) ++ i µ (an) i σ · h(cid:16) m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk (cid:17) i = n × e ( α ) i + i µ (an) i σ · h(cid:16) − m αn m α n − X k =1 ˜p αk (cid:17) i = n × e ( α ) i(cid:27) , (B5)ˆ H γ = − s πc ¯ hw ph A X j =1 X α =1 , e − i k ph r j (cid:26)h(cid:16) µ N z j m p m Aj e ( α ) · m Aj m A + m α (cid:17) j = A + (cid:16) µ N z j m p m Aj e ( α ) · m AA m A + m α (cid:17) j = A i P ++ h(cid:16) i µ (an) j σ · m Aj m A + m α (cid:17) j = A + (cid:16) i µ (an) j σ · m AA m A + m α (cid:17) j = A i (cid:2) P × e ( α ) (cid:3) −− h(cid:16) µ N z j m p m Aj e ( α ) · m Aj m A (cid:17) j = A + (cid:16) µ N z j m p m Aj e ( α ) · m AA m A (cid:17) j = A i p −− h(cid:16) i µ (an) j σ · m Aj m A (cid:17) j = A + (cid:16) i µ (an) j σ · m AA m A (cid:17) j = A i (cid:2) p × e ( α ) (cid:3)(cid:27) , (B6)ˆ H γ = − s πc ¯ hw ph A X j =1 X α =1 , e − i k ph r j (cid:26)(cid:16) µ N z j m p m Aj e ( α ) · m A − m Aj m A (cid:17) j = A ˜p Aj ++ (cid:16) − µ N z j m p m Aj e ( α ) · m Aj m A (cid:17) j = A A − X k =1 ,k = j ˜p Ak − µ N z j m p m Aj e ( α ) · h m AA m A A − X k =1 ˜p Ak i j = A −− i ¯ h µ (an) j σ · (cid:2) k ph × e ( α ) (cid:3) ++ i µ (an) j σ · h(cid:16) m A − m Aj m A ˜p Aj − m Aj m A A − X k =1 ,k = j ˜p Ak (cid:17) j = A × e ( α ) i + i µ (an) j σ · h(cid:16) − m AA m A A − X k =1 ˜p Ak (cid:17) j = A × e ( α ) i(cid:27) . (B7)We simplify this expression, analyzing summations over different indexes. After calculations, we obtain:ˆ H γ = ˆ H P + ˆ H p + ∆ ˆ H γ + ˆ H k , (B8)where ˆ H P = − s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i (cid:20) µ N z i m p m A + m α e ( α ) · P + i µ (an) i m αi m A + m α σ · (cid:2) P × e ( α ) (cid:3)(cid:21) ++ A X j =1 e − i k ph r j (cid:20) µ N z j m p m A + m α e ( α ) · P + i µ (an) j m Aj m A + m α σ · (cid:2) P × e ( α ) (cid:3)(cid:21)(cid:27) , (B9)ˆ H p = − s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i (cid:20) µ N z i m p m α e ( α ) · p + i µ (an) i m αi m α σ · (cid:2) p × e ( α ) (cid:3)(cid:21) −− A X j =1 e − i k ph r j (cid:20) µ N z j m p m A e ( α ) · p + i µ (an) j m Aj m A σ · (cid:2) p × e ( α ) (cid:3)(cid:27) , (B10)9∆ ˆ H γ = − s πc ¯ hw ph X α =1 , X i =1 e − i k ph r i (cid:26)(cid:16) µ N z i m p m i e ( α ) · m α − m αi m α (cid:17) i = n ˜p αi −− (cid:16) µ N z i m p m i e ( α ) · m αi m α (cid:17) i = n n − X k =1 ,k = i ˜p αk − µ N z i m p m i e ( α ) · h m αn m α n − X k =1 ˜p αk i i = n ++ i µ (an) i σ · h(cid:16) m α − m αi m α ˜p αi − m αi m α n − X k =1 ,k = i ˜p αk (cid:17) i = n × e ( α ) i − i µ (an) i m αn m α σ · h(cid:16) n − X k =1 ˜p αk (cid:17) i = n × e ( α ) i(cid:27) −− s πc ¯ hw ph X α =1 , A X j =1 e − i k ph r j (cid:26)(cid:16) µ N z j m p m Aj e ( α ) · m A − m Aj m A (cid:17) j = A ˜p Aj ++ (cid:16) − µ N z j m p m Aj e ( α ) · m Aj m A (cid:17) j = A A − X k =1 ,k = j ˜p Ak − µ N z j m p m Aj e ( α ) · h m AA m A A − X k =1 ˜p Ak i j = A ++ i µ (an) j σ · h(cid:16) m A − m Aj m A ˜p Aj − m Aj m A A − X k =1 ,k = j ˜p Ak (cid:17) j = A × e ( α ) i − i µ (an) j m AA m A σ · h(cid:16) A − X k =1 ˜p Ak (cid:17) j = A × e ( α ) i(cid:27) , (B11)and ˆ H k = s πc ¯ hw ph X α =1 , X i =1 e − i k ph r i (cid:26) i ¯ h µ (an) i σ · (cid:2) k ph × e ( α ) (cid:3)(cid:27) ++ s πc ¯ hw ph X α =1 , A X j =1 e − i k ph r j (cid:26) i ¯ h µ (an) j σ · (cid:2) k ph × e ( α ) (cid:3)(cid:27) . (B12)Let us simplify these expressions. For ˆ H P we obtain:ˆ H P = − s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i µ N z i m p m A + m α + A X j =1 e − i k ph r j µ N z j m p m A + m α (cid:27) e ( α ) · P + − s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i i µ (an) i m αi m A + m α + A X j =1 e − i k ph r j i µ (an) j m Aj m A + m α (cid:27) σ · (cid:2) P × e ( α ) (cid:3) == − s πc ¯ hw ph µ N m p m A + m α X α =1 , (cid:26) X i =1 e − i k ph r i z i + A X j =1 e − i k ph r j z j (cid:27) e ( α ) · P + − s πc ¯ hw ph im A + m α X α =1 , (cid:26) X i =1 e − i k ph r i µ (an) i m αi σ + A X j =1 e − i k ph r j µ (an) j m Aj σ (cid:27) · (cid:2) P × e ( α ) (cid:3) . (B13)For ˆ H p we obtain:ˆ H p = − s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i µ N z i m p m α − A X j =1 e − i k ph r j µ N z j m p m A (cid:27) e ( α ) · p −− s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i i µ (an) i m αi m α − A X j =1 e − i k ph r j i µ (an) j m Aj m A (cid:27) σ · (cid:2) p × e ( α ) (cid:3) == − s πc ¯ hw ph µ N m p X α =1 , (cid:26) m α X i =1 z i e − i k ph r i − m A A X j =1 z j e − i k ph r j (cid:27) e ( α ) · p −− i s πc ¯ hw ph X α =1 , (cid:26) m α X i =1 µ (an) i m αi e − i k ph r i σ − m A A X j =1 m Aj µ (an) j e − i k ph r j σ (cid:27) · (cid:2) p × e ( α ) (cid:3) . (B14)For ˆ H k we obtain:ˆ H k = i ¯ h s πc ¯ hw ph X α =1 , (cid:26) X i =1 e − i k ph r i µ (an) i σ + A X j =1 e − i k ph r j µ (an) j σ (cid:27) · (cid:2) k ph × e ( α ) (cid:3) . (B15)0Now we rewrite the found solutions in relative coordinates. Using Eqs. (A6) and (A7), from Eqs. (B13) and (B14) weobtain:ˆ H P = − s πc ¯ hw ph µ N m p m A + m α e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 z i e − i k ph ρ αi + e i c α k ph r A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · P + − s πc ¯ hw ph im A + m α e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 µ (an) i m αi e − i k ph ρ αi σ + e i c α k ph r A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) ×× (cid:2) P × e ( α ) (cid:3) , (B16)ˆ H p = − s πc ¯ hw ph µ N m p e − i k ph R X α =1 , (cid:26) e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj (cid:27) e ( α ) · p −− i s πc ¯ hw ph e − i k ph R X α =1 , (cid:26) e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:27) ×× (cid:2) p × e ( α ) (cid:3) . (B17)ˆ H k = i ¯ h s πc ¯ hw ph e − i k ph R X α =1 , (cid:26) e − i c A k ph r X i =1 µ (an) i e − i k ph ρ αi σ + e i c α k ph r A X j =1 µ (an) j e − i k ph ρ Aj σ (cid:27) · (cid:2) k ph × e ( α ) (cid:3) . (B18) Appendix C: Electric and magnetic form-factors
We substitute explicit formulation (29) for wave function F ( r , β A , β α ) to the obtained matrix element (33): M = − (2 π ) δ ( K f − k ph ) · X α =1 , (cid:28) Φ α − nucl , f ( r ) · ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12)(cid:12) ×× µ N m p n e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj o e ( α ) · p ++ i n e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ o · (cid:2) p × e ( α ) (cid:3) ×× (cid:12)(cid:12)(cid:12)(cid:12) Φ α − nucl , i ( r ) · ψ nucl , i ( β A ) · ψ α,i ( β α ) (cid:29) . (C1)We rewrite integration over variable r explicitly: M = − (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) · D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) ×× µ N m p n e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj o e ( α ) · p ++ i n e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ o · (cid:2) p × e ( α ) (cid:3) ×× (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) E · Φ α − nucl , i ( r ) dr . (C2)1We calculate this equation further as M = − (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) · (cid:26) × µ N m p D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) n e − i c A k ph r m α X i =1 z i e − i k ph ρ αi − e i c α k ph r m A A X j =1 z j e − i k ph ρ Aj o ×× (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) E · e ( α ) p ++ i D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) n e − i c A k ph r m α X i =1 µ (an) i m αi e − i k ph ρ αi σ − e i c α k ph r m A A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ o ×× (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) E(cid:2) p × e ( α ) (cid:3) (cid:27) · Φ α − nucl , i ( r ) dr (C3)or M = − (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) ×× (cid:26) µ N m p n e − i c A k ph r m α D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) X i =1 z i e − i k ph ρ αi (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) E −− e i c α k ph r m A D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) A X j =1 z j e − i k ph ρ Aj (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) Eo · e ( α ) p ++ i n e − i c A k ph r m α D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) X i =1 µ (an) i m αi e − i k ph ρ αi σ (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) E −− e i c α k ph r m A D ψ nucl , f ( β A ) · ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) · ψ α,i ( β α ) Eo (cid:2) p × e ( α ) (cid:3) (cid:27) ×× Φ α − nucl , i ( r ) dr . (C4)Here, we take into account that function ψ α,s ( β α ) is dependent of variables ρ αn (i.e. it is not dependent on variables ρ Am ), as the function ψ nucl , s ( β A ) is dependent on variables ρ Am (i.e. it is not dependent on variables ρ αn ). On sucha basis, we rewrite Eq. (C4) as M = − (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) ×× (cid:26) µ N m p (cid:20) e − i c A k ph r m α (cid:28) ψ α,f ( β α ) (cid:12)(cid:12)(cid:12)(cid:12) X i =1 z i e − i k ph ρ αi (cid:12)(cid:12)(cid:12)(cid:12) ψ α,i ( β α ) (cid:29) · D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E −− e i c α k ph r m A (cid:28) ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12)(cid:12) A X j =1 z j e − i k ph ρ Aj (cid:12)(cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) (cid:29) · D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E (cid:21) · e ( α ) p ++ i (cid:20) e − i c A k ph r m α (cid:28) ψ α,f ( β α ) (cid:12)(cid:12)(cid:12)(cid:12) X i =1 µ (an) i m αi e − i k ph ρ αi σ (cid:12)(cid:12)(cid:12)(cid:12) ψ α,i ( β α ) (cid:29) · D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E −− e i c α k ph r m A (cid:28) ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12)(cid:12) A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:12)(cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) (cid:29) · D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E (cid:21) (cid:2) p × e ( α ) (cid:3) (cid:27) ×× Φ α − nucl , i ( r ) dr . (C5)We take into account normalization condition for wave functions as D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E = 1 , D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E = 1 , (C6)2and Eq. (C5) is transformed to M = − (2 π ) δ ( K f − k ph ) · X α =1 , Z Φ ∗ α − nucl , f ( r ) ×× (cid:26) µ N m p (cid:20) e − i c A k ph r m α D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) X i =1 z i e − i k ph ρ αi (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E −− e i c α k ph r m A D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) A X j =1 z j e − i k ph ρ Aj (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E(cid:21) e ( α ) p ++ i (cid:20) e − i c A k ph r m α D ψ α,f ( β α ) (cid:12)(cid:12)(cid:12) X i =1 µ (an) i m αi e − i k ph ρ αi σ (cid:12)(cid:12)(cid:12) ψ α,i ( β α ) E −− e i c α k ph r m A D ψ nucl , f ( β A ) (cid:12)(cid:12)(cid:12) A X j =1 µ (an) j m Aj e − i k ph ρ Aj σ (cid:12)(cid:12)(cid:12) ψ nucl , i ( β A ) E (cid:21) (cid:2) p × e ( α ) (cid:3) (cid:27) ×× Φ α − nucl , i ( r ) dr . 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