Cooperative Internal Conversion Process by Proton Exchange

Cooperative internal conversion process by proton exchange Péter Kálmán∗ and Tamás Keszthelyi† Budapest University of Technology and Economics, Institute of Physics, Budafoki út 8. F., H-1521 Budapest, Hungary A generalization of the recently discovered cooperative internal conversion process is investigated theoretically. In the cooperative internal conversion process by proton exchange investigated the coupling of bound-free electron and proton transitions due to the dipole term of their Coulomb interaction permits cooperation of two nuclei leading to proton exchange and an electron emission. General expression of the cross section of the process obtained in the one particle spherical nuclear shell model is presented. As a numerical example the cooperative internal conversion process by proton exchange in Al is dealt with. As a further generalization, cooperative internal conversion process by heavy charged particle exchange and as an example of it the cooperative internal conversion process by triton exchange is discussed. The process is also connected to the field of nuclear waste disposal. PACS numbers: 23.20.Nx, 25.90.+k, 28.41.Kw, Keywords: internal conversion and extranuclear effects, other topics of nuclear reactions: specific reactions, radioactive wastes, waste disposal In a recent paper [1] a new phenomenon, the cooperative internal conversion process (CICP) is discussed which is a special type of the well known internal conversion process [2]. In CICP two nuclei cooperate by neutron exchange creating final nuclei of energy lower than the energy of the initial nuclei. The process is initiated by the coupling of bound-free electron and neutron transitions due to the dipole term of their Coulomb interaction in the initial atom leading to the creation of a virtual free neutron which is captured through strong interaction by an other nucleus. The process can be written as A1 A2 A1 1 A2+1 e + X + Y e′ + − X + Y + ∆, (1) FIG. 1: The graph of cooperative internal conversion process Z1 Z2 → Z1 Z2 by heavy charged particle (e.g. proton) exchange. Particle 1 A1 A2 A1 1 A2+1 (bound) and 1’(free) are electrons, particle 2 is the nucleus where X, Y and − X, Y are the initial and Z1 Z2 Z1 Z2 which looses the heavy charged particle (e.g. proton) and be- final nuclei, respectively, and e is an initial, bound elec- comes particle 2’. Particle 3 is the intermediate heavy charged A1 tron of atom containing Z1 X nucleus, and e′ is a free, final particle (e.g. proton). Particle 4 is the nucleus which absorbs electron. ∆ is the energy of the reaction. ∆ = ∆ + ∆+, the heavy charged particle (e.g. proton) and becomes particle − with ∆ = ∆A1 ∆A1 1 and ∆+ = ∆A2 ∆A2+1. ∆A1 , 5. The filled dot denotes (in case of proton the dipole term of) − − − − ∆A1 1, ∆A2 , ∆A2+1 are the energy excesses of neutral the Coulomb-interaction and the open circle denotes nuclear − atoms of mass numbers A1, A1 1, A2, A2 + 1, respec- (strong) interaction. tively [3]. The process is mainly− related to atomic state, i.e. it has accountable cross section if the initial nuclei arXiv:1602.05774v1 [nucl-th] 18 Feb 2016 are located in free atoms therefore the cross section was A1 determined in the case of free atoms (e.g. noble gases) atomic nucleus Z1 X, particle 2 is virtually excited into in one particle nuclear and spherical shell models. dip a free state (particle 3) due to the dipole term VCb of its In this work the process Coulomb-interaction (in electric dipole coupling the proton has effective charge q = (1 Z /A ) e [4]) with one p − 1 1 A1 A2 A1 1 A2+1 of the bound atomic electrons (e1) of the atom contain- 1 e1 + Z1 X + Z2 Y e1′ + Z1 −1 V + Z2+1 W + ∆, (2) A → − ing the Z1 X nucleus while the electron becomes free (e1′ ). The free, virtual proton is captured by an other nucleus called cooperative internal conversion process by proton A2 Y (particle 4) due to its nuclear potential V (created exchange (CICP-PE) is discussed theoretically in more Z2 st A2+1 detail. In this process (see Fig. 1) a bound proton of an by strong interaction) forming the nucleus Z2+1 W (particle 5) in this way. The sum of the rest energies of the initial nuclei is E0i and the sum of the the rest energies of the final nuclei is E . If E E = ∆ > 0, i.e. if 0f 0i − 0f ∗retired, e-mail: [email protected] E0i > E0f , then the process is energetically allowed and †retired, e-mail: [email protected] proton exchange is possible. The nuclear energy differ- 2 ence ∆, which is the reaction energy, is shared between (particle 5), respectively, the analysis of σbf shows that, the kinetic energies of the final, free electron and the two similarly to the CICP by neutron exchange [1], those pro- A1 1 final nuclei [Z1 −1 V (particle 2′) is the nucleus which has cesses give essential contribution to the cross section in lost the proton].− which k k and k k where k , k and k are e ≪ 1 e ≪ 2 e 1 2 The transition probability per unit time and the cross the magnitudes of the wave vectors of ke, k1 and k2. section σbf (A1, A2) of CICP-PE with bound-free (bf) (In this case as a consequence of momentum conserva- electron transitions can be determined with the aid tion k1 = k2, furthermore the intermediate proton has − of standard second order perturbation calculation of wave vector k2.) − quantum mechanics. The cross section has the form The initial and final nuclear states have the c form: φ (x ) = ϕ (x ) Y (Ω )/x and φ (x ) = σbf (A1, A2) = v σ0bf (A1, A2), where v is the relative i 1 i 1 limi 1 1 f 2 velocity of the two atoms, c is the velocity of light ϕf (x2) Ylf mf (Ω2)/x2 with ϕi (x1) /x1 and ϕf (x2) /x2 (in vacuum). (The cross section of CICP-PE with denoting the radial parts of the one particle shell- bound-bound electron transition is neglected since it model solutions of quantum numbers li, mi and lf , mf . has proved to be much smaller than σbf .) The cal- For ϕi (x1) and ϕf (x2) the corresponding part R0Λ = culation of σ is similar to the calculation of CICP 1/2 1/2 1/2 Λ+1 2 0bf bk− Γ(Λ+ 3/2)− 2 ρk exp( ρk/2) of the 0Λ one by neutron exchange [1]. The differences are the ap- particle spherical shell model states− [6] is applied. Here ′ 1/2 pearance of two Coulomb factors F2 3 and F34 which ~ 1/3 Z1 ~ − ρk = xk/bk, bk = m0ω and ωsh,k = 41Ak (in multiply the cross section and ∆n, A1 are changed sh,k Z1 MeV units, [4]) with k = 1, 2 corresponding to A1 and to ∆p, 1 A1 , respectively, in Eqs.(9)-(12) of [1]. − A2, and Γ(x) is the gamma function. The case of spher- ∆ = 7.288969 MeV is the energy excess of the pro- p ical shell model states of 0li initial nuclear state and of ton. (F ′ and F are determined in the Appendix.) 2 3 34 0lf final nuclear state is investigated. Here we repeat some essential features of the calcula- The initial electronic state is a 1s state of the form tion. It is carried out in one particle nuclear model. 3/2 Ri (xe)=2a− exp( xe/a) with a = a0/Zeff , where The motions of the centers of mass of the two atoms − a0 is the Bohr-radius, Zeff = EB /Ry and Ry is the are taken into account. Hydrogen like state of bind- Rydberg energy. In the Coulomb-corrected plane wave ing energy E and Coulomb-factor corrected plane wave Bi applied for the final free electronp the F (k )=2π/ (k a) are used as initial, bound and final, free electron states. Cb e e approximation is used, where FCb (ke) is the Coulomb The dipole term of the Coulomb interaction reads as 1 factor of the electron. Keeping the leading term of Je (ke) dip Z1 2 4π 2 m=1 + VCb = 1 A1 e 3 x1xe− m= 1 Y1∗m(Ωe)Y1m(Ω1), in [1] and in the case of li = even [li = 2; Al(5/2 , 0d)] − − where Z and A are charge and mass numbers of the first to be investigated one obtains 1 1 P nucleus, e is the elementary charge, x1, xe and Ω1,Ωe are 210π3 Z 2 V 2 b5L2 m magnitudes and solid angles of vectors x1, xe which are 1 0 1 0 σ0bf,sh = 1 2 2 a12 (3) the relative coordinates of the proton and the electron 3 − A1 ~ λea me ( c) 0 2 in the first atom, respectively and Y1m denotes spherical 2lf +3 ρ e ρf 2λ harmonics. (The order of magnitude of the cross section f − N1λ (k0b1) (2lf + 1) 1 3 Sλ. 2L 2 × Γ lf + Γ λ + produced by the L-th pole coupling is (R/r) − times 2 λ=li 1 2 smaller than the cross section produced by the dipole X± ~ coupling where R and r are the nuclear and atomic radii. Here λe = /(mec), me is the rest mass of the electron, Therefore the leading term to the cross section is pro- a12 = (A1 1) (A2 + 1) / (A1 + A2), ρf = RA2 /b2, k0 = − ~ duced by the dipole coupling.) The motion of the inter- √2m0∆Bia12/ , and mediate proton and the two final nuclei are also described 2 l 1 λ by plane waves.

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