Cooperative Internal Conversion Process by Proton Exchange

Home , Internal conversion

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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. The rest masses of the two initial nuclei N = (2λ + 1) i . (4) 1λ 0 0 0 of mass numbers A1 and A2 are m1 = A1m0 (particle 2) and m = A m (particle 4) where m c2 = 931.494 2 2 0 0 The parenthesized expression is Wigner 3j symbol. (The MeV is the atomic energy unit. For the nuclear potential suffix sh denotes that the quantity is calculated in the a rectangular potential well is assumed, i.e. V = V st 0 one particle spherical shell model.) (x R ) and V = 0 (x > R ) where x is− the 2 ≤ A2 st 2 A2 2 magnitude of vector x2, which is the relative coordinate 1 of the neutron in the second nucleus and RA2 is its radius. Sλ = f (x) gλ (x) h1 (x) h2 (x) dx, (5) Direct proton capture may be assumed at the surface of Z0 the second nucleus (of A ). The effective volume in which 2 2 2 2 2λ+1 (k0b1) x 2 strong interaction induces proton capture can be consid- 1 x x e− J 1 (xk0RA2 ) − lf + 2 ered as a shell of a sphere of radius RA2 and of thickness f (x)= 2 2 , (6) L, where L is the mean free path of the ingoing proton 1+ ∆Bi (1 x2) A1a12 x2 +1+ ξ EB A1 1 − − in the nucleus [5]. h i h i Introducing the wave vectors ke and k1, k2 of the free x = k2/k0, ξ = (∆p ∆ + EBi) /∆Bi and ∆Bi = ∆ A1 1 A2+1 − − − E . J 1 is a Bessel-function of the first kind. In electron and particles Z1 −1 V (particle 2′) and Z2+1 W Bi lf + 2 − 3

Eq.(5) gλ (x) = 1 if λ = li + 1 and Isotope P roducts ∆−(MeV ) ∆+(MeV ) ∆(MeV ) 19 18 20 2 F O, Ne −0.705 5.555 4.850 g (x)=(2l + 1) 2 (2l +1)(k b x)2 + (k b x)4 (7) λ i − i 0 1 0 1 23Na 22Ne,24 Mg −1.505 4.404 2.899 27 26 28 − if λ = l 1. h (x) = d (x)/ [exp(d (x)) 1], j = 1, 2 Al Mg, Si 0.982 4.296 3.314 i j j j 31 30 32 with − − P Si, S −0.008 1.575 1.567 45Sc 44Ca,46 T i 0.400 3.056 3.456 2 55 54 56 − 1 (A1 + A2) m0c Mn Cr, Fe 0.778 2.895 2.117 d1(x)=2π(Z1 1)αf (8) 59Co 58F e,60 Ni −0.075 2.245 2.170 − x s2A1 (A2 +1)∆Bi 103Rh 102Ru,104 Pd 1.076 1.369 2.445 127 126 128 and I T e, Xe 1.083 0.873 1.956 133Cs 132Xe,134 Ba 1.204 0.879 2.083 2 1 (A1 + A2) m0c d2(x)=2πZ2αf . (9) x s2 (A1 1) (A2 +1)∆Bi TABLE I: Data for cooperative internal conversion process by − proton exchange. (Data to reaction (11).) In the first column αf denotes the fine structure constant. In the numerical the initial stable isotope (of unity relative natural abundance) calculation V0 = 50 MeV is used [4]. and in the second column the reaction products can be found. The differential cross section dσ0bf,sh/dE2 of the pro- For the definition of ∆−, ∆+ and ∆ see the text. cess can be determined with the aid of 2λ ~ ~ √ 2 N1λ (k0b1) f (x) gλ (x) h1 (x) h2 (x) k2 = k0x = x 2m0∆Bia12. It gives E3 = x ∆BiA1/2 P (x)= since a = A /2 if A = A and near below x = 1 the Γ λ + 3 x 12 1 1 2 λ=li 1 2 X± value of E3 is large enough to result moderately small (10) Coulomb factors. as dσ0bf,sh/dE2 = Kbf [P (x)]x=√z / (2E20) where z = For a gas of atomic Al and of number density E2/E20 with E20 = (A1 1) ∆Bi/ (A1 + A2), which is n the transition probability per unit time λ1 = the possible maximum of− the kinetic energy E of parti- 2 cn A2 rA2 σ0bf,sh = cnσ0bf,sh [1] since the relative natu- A2+1 27 cle Z2+1 W (particle 5) created in the process, Kbf stands ral abundance r of the initial Al isotope equals unity. P A2 13 for the whole factor which multiplies the sum in (3). λ1 is estimated as λ1 > λ1(K), which is the transition dσ0bf,sh/dE2 has accountable values near below z = 1, probability per unit time of the bound-free CICP-PE i.e. if E2 E20. from the K shell of Al (λ1(K)= cnσ0bf,sh(K)), resulting ∼ 16 1 3 3 1 The differential cross section dσ0bf,sh/dEe = λ1 > 1.92 10− s− and rtot > 5.09 10 cm− s− for × 19× 3 Kbf [P (x)]x=√1 z / (2∆Bi) can also be determined with a gas of normal state (n =2.652 10 cm− , T = 273.15 − × the aid of P (x) where z = Ee/∆Bi, Ee is the ki- K, p = 100 kPa and rtot = nλ1, which is the total rate netic energy of the electron and Kbf is defined above. per unit volume of the sample, in this case since rA1 =1 dσ0bf,sh/dEe has accountable values near above z = 0, [1]). i.e. if Ee 0. In Table I. the ∆ , ∆+ and ∆ data of some cooperative ∼ − It is a special case of (2) if the two initial nuclei are internal conversion processes by proton exchange (data to identical. In this case the CICP-PE reads as reaction (11) can be found. In the first column the initial stable isotope of relative natural abundance unity and in A1 A1 A1 1 A1+1 − e1 + Z1 X + Z1 X e1′ + Z1 1 V + Z1+1 W + ∆. (11) the second column the reaction products are listed. → − There are other possibilities to realize CICP, when a 27 27 For example of such a case the reaction e+ 13Al+ 13Al charged heavy particle (such as d, t, 3He and 4He) is 26 28 → 2 2 e′+ 12Mg+ 14Si + ∆ is considered when the reaction exchanged instead of proton exchange. The process is starts from the K shell. The initial and final nuclear called cooperative internal conversion process by heavy states are supposed to be 0d spherical shell model states charged particle exchange (CICP-HCPE) and it can be of li = lf =2,∆=3.31362 MeV . The electron binding visualized with the aid of Fig.1 too. Denoting the in- energy in the K shell is EBi = 1.5596 keV and ∆ = A3 − termediate particle (particle 3 in Fig. 1) by z3 w, which 0.98235 MeV . In this case 2E20 = 3.1894 MeV and is exchanged, the cooperative internal conversion process − 35 2 1 Kbf / (2E20)=2.41 10− cm MeV − ). σ0bf,sh(K) = by heavy charged particle exchange reads as 46 2 × 2.41 10− cm− is obtained in the case of bound-free × A1 A2 A1 A3 A2+A3 CICP from the K shell of Al. If one compares this result e + Z1 X + Z2 Y e′ + Z1 −z3 V + Z2+z3 W + ∆. (12) 45 2 → − with σ0bf,sh(K)=8.25 10− cm− obtained in case of CICP by neutron exchange× in Ne one can recognize Here e and e′ denote bound and free electron and ∆ that the ratio of the two cross sections is only 0.030. is the energy of the reaction, i.e. the difference be- A1 A2 At first sight it seems to be larger when expected since tween the rest energies of initial Z1 X +Z2 Y and fi- two Coulomb factors appeare in the cross section. But A1 A3 A2+A3 nal Z1 −z3 V + Z2+z3 W states. ∆ = ∆ + ∆+, with as it was said earlier the intermediate proton has wave − − k ~2k2 A1 A1 A3 A2 A2+A3 A1 vector 2 and thus its energy E3 = 2/ (2m0) with ∆ = ∆Z1 ∆Z1 −z3 and ∆+ = ∆Z2 ∆Z2+z3 . ∆Z1 , − − − − − 4

Isotope P roducts ∆−(MeV ) ∆+(MeV ) ∆(MeV ) Isotope τ(y) P roducts ∆−(MeV ) ∆+(MeV ) 19F 16O,22 Ne 3.250 6.537 9.787 99T c 2.11 × 105 98Mo,100 Ru 0.789 1.896 23Na 20Ne,26 Mg −2.488 6.685 4.197 129I 1.57 × 107 128T e,130 Xe 0.491 1.378 27Al 24Mg,30 Si −3.263 7.236 3.973 135Cs 2.3 × 106 134Xe,136 Ba 0.538 1.305 31P 28Si,34 S −2.948 5.491 2.543 137Cs 30.07 136Xe,138 Ba −0.126 1.716 45Sc 42Ca,48 T i −2.522 7.418 4.896 155Eu 4.7611 154Sm,156 Gd 0.637 0.717 55Mn 52Cr,58 Fe −2.294 4.443 2.149 59Co 56F e,62 Ni −1.623 4.519 2.896 103 100 106 TABLE III: Data for cooperative internal conversion process Rh Ru, Pd 1.197 1.882 3.079 by proton exchange of long lived nuclear fission products. 127 124 130 I T e, Xe 1.536 0.893 2.429 (Data to reaction (11).) Products are the two stable final 133Cs 130Xe,136 Ba 1.806 0.816 2.622 izotopes, τ is the half-life of the fission product in y units. For the definition of ∆− and ∆+ see the text. TABLE II: Data for cooperative internal conversion process by triton exchange. (Data to reaction (14).) In the first col- A. Appendix - Coulomb factors F2′3 and F34 umn the initial stable isotope (of unity relative natural abundance) and in the second column the reaction products can be found. For the definition of ∆−, ∆+ and ∆ see the text. Since particles 2′, 3 and 4 all have positive charge, furthermore they are all heavy, the two essential Coulomb factors, which appear in the cross section, are F2′3 and A1 A3 A2 A2+A3 F34. Since Coulomb factors F2′3 and F34 determine the ∆Z1 −z3 , ∆Z2 , ∆Z2+z3 are the energy excesses of neu- − tral atoms of mass number-charge number pairs A1, Z1; order of magnitude of the cross section of the process ′ A1 A3, Z1 z3; A2, Z2; A2 + A3, Z2 + z3, respectively (as it is proportional to F2 3F34) we treat them in more [3].− − detail in the case of CICP-PE in the following. In (12) the initial bound electron (particle 1) Coulomb We adopt the approach standard in nuclear physics A1 when describing the cross section of nuclear reactions of interacts with the nucleus Z1 X (particle 2). A free elec- A3 heavy, charged particles j and k of like positive charge of tron (particle 1′), the intermediate particle z3 w (particle A1 A3 charge numbers zj and zk and of relative kinetic energy 3) and the nucleus − V (particle 2 ) are created due Z1 z3 ′ E. The cross section of such a process can be derived − A3 to this interaction. The intermediate particle z3 w (par- applying the Coulomb solution ϕ(r), ticle 3) is captured due to the strong interaction by the ik r A2 A2+A3 ϕ(r)= e · f(k, r)/√V, (15) nucleus Z2 Y (particle 4) forming the nucleus Z2+z3 W A1 (particle 5) in this manner. So in (12) the nucleus Z1 X which is the wave function of a free particle of charge A3 (particle 2) looses a particle z3 w which is taken up by number zj in a repulsive Coulomb field of charge number A2 zk [7], in the description of relative motion of projectile the nucleus Z2 Y (particle 4). The process is energetically forbidden if ∆ < 0. and target. In (15) V denotes the volume of normaliza- r k As an example we deal with the tion, is the relative coordinate of the two particles, is the wave number vector in their relative motion and A1 A2 A1 3 A2+3 − e + Z1 X + Z2 Y e′ + Z1 1 V + Z2+1 W + ∆ (13) k r πηjk /2 k r → f( , )= e− Γ(1 + iηjk)1F1( iηjk, 1; i[kr ]), − − − · reaction in which a triton is exchanged. It is called CICP (16) by triton exchange (CICP-TE). Special type of reaction where 1F1 is the confluent hypergeometric function and Γ r πηjk/2 (13) is is the Gamma function. Since ϕ( ) e− Γ(1+iηjk), the cross section of the process is proportional∼ to A1 A1 A1 3 A2+3 e + X + X e′ + − V + W + ∆. (14) 2 Z1 Z1 Z1 1 Z2+1 πηjk /2 2πηjk (E) → − e− Γ(1 + iη ) = = F (E), jk exp[2πη (E)] 1 jk In Table II. the ∆ , ∆ and ∆ data of some cooperative jk − − − (17) internal conversion processes by triton exchange (data to which is the so-called Coulomb factor. Here reaction (14) can be found. In Table III. some long lived fission products are listed m c2 η (E)= z z α a 0 (18) which can take part in CICP-PE. The values of ∆ and jk j k f jk 2E − r ∆ indicate that each pair of the listed isotopes can pro- is the Sommerfeld parameter in the case of colliding parti- duce− CICP-PE since ∆ + ∆ = ∆ > 0 in every case. cles of mass numbers A , A and rest masses m = A m , Consequently similarly− to those− long lived fission prod- j k j j 0 m = A m . m c2 = 931.494 MeV is the atomic energy ucts which can decay through CICP by neutron exchange k k 0 0 unit, α is the fine structure constant and E is taken in [1], it seems to stand also a practical chance to accelerate f the center of mass (CM) coordinate system. the decay of the listed isotopes if they were collected in appropriately high concentration and density in atomic Aj Ak ajk = (19) state. Aj + Ak 5 is the reduced mass number of particles j and k of mass in case of proton exchange. Here the k2 = k0x substi- numbers Aj and Ak. tution is also used. Calculating the Coulomb factor F34, If initial particles have negligible initial momentum the energy E3 of particle 3 is now given in the laboratory then in the final state k = k (k ′ = frame of reference since particle 4 is at rest. In the CM 1 − 2 particle,2 kparticle,5) because of momentum conservation. (It was system of particles 3 and 4 the energy E3(CM) is −obtained [1] that the process has accountable cross section if the momentum of the final electron can be ne- Aparticle,4 glected.) In this case the momentum and energy of E3(CM)= E3. (21) (Aparticle,3 + Aparticle,4) the virtual particle 3 (e.g. proton) are kparticle,3 = k ′ = k k and E = ~2k2/ (2m ), − particle,2 particle,5 ≡ 2 3 2 3 where ~ is the reduced Planck-constant. Calculating the Substituting it into (18) Coulomb factor F2′3 [see (17)] between particles 2′ and 3 the energy E3 is given in their CM coordinate system 2 (since k = k ′ ) thus E can be substi- 1 (A1 + A2) m0c particle,3 − particle,2 3 η = Z α (22) tuted directly in (18) producing 34 2 f x 2 (A 1) (A +1)∆ s 1 − 2 Bi 2 1 A1 + A2 m0c η2′3 = (Z1 1) αf . (20) in case of proton exchange. − xsA1 (A2 + 1) 2∆Bi

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