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Branching ratios of α-decay to ground and excited states of Fm, Cf, Cm and Pu
Accepted Manuscript Branching ratios of α-decay to ground and excited states of Fm, Cf, Cm and Pu
H. Hassanabadi, S.S. Hosseini
PII: DOI: Reference:
S0375-9474(18)30058-7 https://doi.org/10.1016/j.nuclphysa.2018.03.006 NUPHA 21182
To appear in:
Nuclear Physics A
Received date: Revised date: Accepted date:
8 January 2018 20 March 2018 21 March 2018
Please cite this article in press as: H. Hassanabadi, S.S. Hosseini, Branching ratios of α-decay to ground and excited states of Fm, Cf, Cm and Pu, Nucl. Phys. A (2018), https://doi.org/10.1016/j.nuclphysa.2018.03.006
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Branching ratios of Į-decay to ground and excited states of Fm, Cf, Cm and Pu H. Hassanabadi1 and S. S. Hosseini*1 1
Faculty of Physics, Shahrood University of Technology, Shahrood, Iran *
Corresponding author, Tel.:+98 232 4222522; fax: +98 273 3335270 Email: [email protected]
Abstract We use the well-known Wentzel–Kramers–Brillouin (WKB) barrier penetration probability to calculate α-decay branching ratios for ground and excited states of heavy even-even nuclei of Fermium (248-254Fm), Californium (244-252Cf), Curium (238-248Cm) and Plutonium (234-244Pu) with 94Zp100. We obtained the branching ratios for the excited states of daughter nucleus by the Į-decay energy (Qα), the angular momentum of Į-particle (Ɛα), and the excitation probability of the daughter nucleus with the excitation energy of state Ɛ in the daughter nucleus (i.e. E*Ɛ). α-Decay half-lives have been evaluated by using the proximity potential model for the heavy eveneven nuclei. We have reported the half-lives and compared the results with the experimental data. The theoretical branching ratios of α-transitions in our calculation are found to agree with the available experimental data well for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+, 0+ĺ 6+ and 0+ĺ 8+ Į-transitions. Keywords. Alpha Decay Modes, Branching ratios, even-even Nuclei, WKB technique, half-life. PACS: No 23.60. +e, 21.10.Re, 24.10.Ѹi, 27.90. +b
1 Introduction Alpha-decay is one of the most important decay channels of unstable nuclei in nuclear physics, therefore, There are many theoretical and experimental approaches which investigate the Į cluster radioactivity and alpha-like models [1-19]. Many attempts have been done on both experimental and theoretical fronts for understanding the physics of cluster radioactivity. Recently the concept of heavy-particle radioactivity is further explored by Poenaru et al. [20]. Also the half-life for the emission of various clusters from even–even isotopes of barium in the ground state and as an excited states were studied using the CPPM by Santhosh et al [21]. Hans Geiger and John Mitchell Nuttall formulated in 1911 a relation between the alpha-particle energy and the halflife (Geiger–Nuttall (GN)) [22], which in its modern form is written ln λ = −a1 Z / Q + a 2 , Where Ȝ is the decay constant, Z the atomic number, Q the total kinetic energy (of the alpha particle and the daughter nucleus), and a1 and a2 are constants. The law is best for nuclei with even atomic number and even atomic mass. The trend is still there for even-odd, odd-even, and odd-odd nuclei but not as pronounced. The GN law has even been extended to describe cluster decays, decays where atomic nuclei are larger than Helium are released. Denisov and Khudenko [23] studied the Į-decay half-lives for transitions between ground states in 344 nuclei and the Įcapture cross sections of 40Ca, 44Ca, 59Co, 208Pb, and 209Bi have recently been well described in the framework of the unified model for Į decay and Į capture. Į-Decay half-lives and the branching ratios to ground (gs) and excited states (es) of deformed nuclei with 222 A 252 and 88 Z 102 for even-even Į-transitions were analyzed in the framework of the unified model for Į decay and Į capture was analyzed by Denisov and Khudenko [24]. Xu and Ren [25, 26] evaluated the α-decay branching ratios for members of ground-state rotational band of the α-decay chain (Fm → Cf → Cm → Pu → U →Th) by a barrier penetration approach. Now we apply this model to the description of the Į-decay branching ratios in various states of daughter nuclei for heavy even-even nuclei of Fm, Cf, Cm and Pu and we calculate systematically the alpha decay half-lives with the proximity potential model. Wang et.al [27] considered branching ratios of Į decay to members of the ground state rotational band and excited 0+ states of even-even nuclei are calculated in the framework of the generalized liquid drop model (GLDM). In this paper, we consider the coulomb and proximity potential model and calculate the half-lives of some nuclei the ground state to exited state α-transitions. In the next section, we will review on our calculation of the important quantity as well as the branching ratio for the even-even nuclei in section 3. We report the half-lives calculated theoretically for this decay processes. We discuss on our numerical results in section 4. Finally, section 5 includes conclusions. 2 Alpha decay and Penetrability We have considered the modified potential for even-even nuclei, [28]
°a0 + a1r + a2 r 2 + Kr 3 , V (r ) = ® °¯V C (r ) +V Pr ox (z ) +V " (r ),
(1a ) (1b )
R p ≤ r ≤ Ct r ≥ Ct
The potential V (r) is parameterized simply as a polynomial for r Ct, and for Rp r Ct, the interacting potential barrier is defined as the sum of the nuclear, the coulomb and the centrifugal potentials for z>0 where z is the distance between the near surfaces of the fragments. Also, the constants a0, a1 and a2 have been determined considering R=Ra =RP, V (R) =V (Ct) andV ′( R ) = V ′ (C t ) in which V (R) =Q and R= Ct [29]. Ct is the touching configuration of two nuclei defined via [28], C t = C d + C α . The Sussmann central radii Ci of the daughter nuclei and the emitted alpha particle are referred to sharp radii (Ri) via, C i = R i − b 2 / R i where i = 1, 2 represent the daughter nucleus and the alpha particle, respectively. R i = 1 .2 8 A i1/ 3 − 0 .7 6 + 0 .8 A i− 1 / 3 ( f m ) . The coefficient K has been obtained by fitting of theoretical predictions to experimental data of Į decay. In the fitting process, we have used the standard deviation as our estimator. The fitting parameter has been found such that the standard deviation obtained the least possible value. In the following more details have been discussed. VC(r) is the Coulomb potential, in the form V C ( r ) = Z d Z α e 2 / r (MeV) With Zd and Zߙ being the atomic numbers of daughter nuclei and alpha particle, respectively. The centrifugal potential energy VƐ(r) is V l ( r ) = ! 2 l ( l + 1) / 2 μ r 2 (MeV), where Ɛ is the angular momentum (orbital quantum number) carried by the
emitted Į cluster and ȝ is the reduced mass of the disintegrated system, μ = mA d A α / ( A d + A α ) and m, Ad, and Aߙ denote the nucleon mass, mass number of daughter nuclei and alpha particle, respectively. Vp (r) is the nuclear proximity potential taking the form (see Blocki et al [28]) V P r o x ( z ) = 4 π γ b (C d C α / C d + C α ) Φ ( z / b ) (MeV), with the nuclear surface tension coefficient for this potential is defined by (the Lysekil mass formula of Ref. [30]) γ = γ 0 [1 − k s I 2 ] (MeVfm-2), in the present version, the coefficients Ȗ0 and ks were taken to be Ȗ0=0.9517 MeV/fm2 and ks=1.7826, respectively, Where I= (N-Z)/A, and N, Z and A are the neutron number, proton number and mass number of the parent for the combined system of the two interacting nuclei respectively. ĭ is the universal function proximity potential which is given as [31] Φ 1 ( ε ) = − 1.7817 + 0.9270 ε + 0.0169 ε 2 − 0.05148ε 3 , f or 0 ≤ ε ≤ 1.9475, ( 2a ) Φ 2 ( ε ) = − 4.41 exp( − ε / 0.7176),
Where
ε
ε ≥ 1.9475, (2b ) = z / b is the overlap distance in units of b between the colliding surfaces. According to the WKB f or
approximation, the penetration probability is obtained via [33], P = exp( − 2 !
Rb
³
2 μ (V ( r ) − Q α )dr ) , Where
Ra
the turning points (Ra, Rb) corresponding to the potential energies are defined as V (Ra) = V (Rb) = Qα. The partial half-life is related to the decay constant λ by ln 2 ln 2 = T 1/ 2 ( s ) = . (3 ) λ νP Where Ȟ is the frequency of assaults on the barrier per second (the zero-point vibration energy) [28, 29], ν = 2 E ν ib / ! = ω / 2 π . For Aα4, the empirical vibration energy Evib is given as [34], E v ib = Q [0 .0 5 6 + 0 .0 3 9 ex p ( 4 − A α / 2 .5)] that Shi and Swiatecki got empirically unrealistic values of Ȟ as
1022 for even-A parents and 1020 for odd-A parents. These calculations and the comparisons have also been comprehended in Table1. 3. The α-decay Branching Ratios Now we consider the branching ratios for both the ground and excited states. Xu and Ren [25] calculated the alpha decay branching ratios to members of ground-state rotational band of heavy even-even nuclei of Fm, Cf, Cm, Pu, U and Th [25]. The penetration probability to excited states depend on the alpha decay energy of the ground-state transition (Qα), the excitation energy ( E "* ) of state Ɛ and the angular momentum (Ɛ) of the Į-
particle. The penetration probability of the α-particle using the WKB method is given as [35]
Pα (Q α , E "* , " ) ∝
ψ III ψI
2
R out
= exp[ − 2
³
(4)
k ( r )dr ],
R0
Where R0 is the radius of the daughter nucleus ( R
0
= 1 / 2 A d1 / 3 ), the outgoing (ࣜI) and incoming (ࣜIII) are wave
functions and Rout is the second turning point [35]. The wave number K(r) is given by
k (r ) =
2μ (V ( r ) − Q α ) , !2
2 μ Z α Z d e 2 ! 2 " ( " + 1) [ + − (Q α − E "* )]1/ 2 , (5 ) 2μ r 2 !2 r Where ȝ is the reduced mass. Zα and Zd are the charge numbers of the α-particle and the daughter nucleus, respectively. The Alpha emission from a nucleus obeys the spin-parity selection rule [36] " I i − I f ≤ " ≤ I i + I f , π i / π f = ( − 1) Where Ii, ʌi, and If, ʌf are the spin, and parity of the parent and
daughter nuclei, respectively. If ʌi= ʌf, Ɛ must be even and ʌi് ʌf Ɛ must be odd. The angular momentum Ɛ carried by the alpha particle in a ground state to ground state transition of an even-even nucleus is zero, i.e. the Į-particle spin and parity are ʌĮ = +1 and ƐĮ = 0 . In odd-even or odd-odd nuclei, Ɛ could be nonzero. Denisov and Khudenko [29] purposed the effect of atomic electrons on the energy of the Į-particle to calculate the reaction energy values. The energy released in Į-transition between gs of the parent and the daughter nuclei is calculated using a recent evaluation of atomic mass data [38]. We take into account the penetrating the electron shell (electron screening effect) [21] on the reaction energy values, [13] Q = Δ M p − Δ M d − Δ M α + k ( Z pε − Z d ε ), ( M eV ) (6 ) Where ¨Mp and ¨Md, are, correspondingly, the mass excess of parent and daughter nuclei and ¨MĮ= 2.42498783 MeV for Į particle [37]. The quantity kZε represents the total binding energy of the Z electrons in the atom, where the values of k=8.7 eV and ε= 2.517 for nuclei with Z60 and k=13.6 eV, ε= 2.408 for nuclei with Z < 60 have been derived from data reported by Huang et al. [39]. We used the energy released at α-decay (Qn) from the ground-state (0+) of the parent nucleus to the excited states (n) of the daughter nucleus, therefore the excitation energy (E*) should be also taken into account [23] Q n ≡ Q 0 + → n + = Q 0 + → 0 + − E nd , ( M eV ) (7 ) Therefore, the tunneling probability of the Į particle, is given by [25, 40] Pα (Q α , E "* , " ) = exp[ − χ ] × exp[ − ρ ],
(8 )
With
χ=
2 μ Z α Z d e 2π , ! 2 (Q α − E "* )]1/ 2
( 9a )
ρ=
!2 2 " ( " + 1) , 2 μ ( Z α Z d e 2 R 0 )1/ 2
( 9b )
The term χ have to include the effects of the excitation energy E "* on the penetration factor in the daughter
nucleus and ρ represents the influence of the non-zero angular momentum Ɛ in the atom. The Boltzmann distribution hypothesis is proposed for daughter states in its excited states (2+, 4+, 6+ …) to simulate the internal effect of nuclear states on alpha formation [25]. The excitation energy of state Ɛ ( E "* ) effects are included
through a Boltzmann distribution function,
(10 )
ω " ( E "* ) = exp[ −cE "* ],
Where E "* , is the excitation energy of state Ɛ and c is a free parameter (The value of parameter c is 1.5) [25]. The
total probability of α-transition (IƐ+) can be obtained as the product of the penetration factor and the excitation probability [26]
(11)
I " + = ω " ( E "* ) Pα (Q α , E "* , " ), +
The branching (B) ratio of Į-decay to the excited 0 state of the daughter nucleus is given by [26] I B gn. +s . % = 2 n n + × 100%, n = 0, 2, 4,... I ¦ n+
(12 )
n =0
Where the sum n is going over all excited states for each isotopes. The results are reported in Table 1. 4 Results and discussions In the present work the mode of decay of 4, 5, 6 and 6 isotopes of Zp = 100, 98, 96 and 94 of Fm, Cf, Cm and Pu within the range 234 Ap 254 for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+, 0+ĺ 6+ and 0+ĺ 8+ Į-transitions have been studied by evaluating the alpha decay half-lives, respectively. we calculated the branching ratios of Į decays by taking
into account the influence of α-decay energy, the angular momentum of α-particle, and the excitation probability of the daughter nucleus and compared with the values based on the unified model for Į decay and Į capture (UMADAC) model by Denisov and Khudenko [23] and the barrier penetration approach by Xu and Ren [25]. Also, we observed good description of available data in our approaches with the experimental data. We presented the α-decay half-lives and the branching ratio of even-even nuclei 248-254Fm, 244-252Cf, 238-248Cm and 234-244 Pu, In Table 1. The T1/2 values are given in s. The first column of Table 1 denotes the α-decay mode for each isotope. The second, third and fourth columns denote the atomic number of the parent nucleus, the ground to ground and excited-states α-decay energy (Qα) in MeV and the minimal possible values of the orbital momentum of the Į particle (Ɛmin), respectively. The experimental, theoretical, Xu and Ren (XR) [25] and Denisov and Khudenko (DK) [23] α-decay branching ratios are listed in the fifth, sixth, seventh and eighth columns, respectively. The ninth column of Table 1 shows the excitation energy of state Ɛ in the daughter nucleus (E*Ɛ). In the last three columns are, correspondingly, the experimental Texp, theoretical Ttheo and Denisov and Khudenko TDK values of the Į-decay half-lives. The standard deviation (σ) is computed via 2 1 N theor exp (13 ) ¦ ª log10 T(1/2 α )i -log 10 T(1/2 α )i º ¼ . N i =1 ¬ The standard deviation of our calculation and DK approach are presented in Table 2, which indicate the acceptability of the results. In Figs. 1-4, we plotted the experimental (Diamond-solid) [41], theoretical (Boxsolid) and Ref. [23] (circles-solid) values of log10T1/2 (s) of Fm, Cf, Cm and Pu of 0+, 2+, 4+, and 6+ states for αdecays in even–even parent nuclei for Ɛ=0, 2, 4 and 6 transitions, respectively. Fig. 5 depicts the difference between the experimental, theoretical and Ref [23] values of log10T1/2 (s) of 236Pu, 238Pu, 240Pu, 242Cm, 244Cm, 252 Cf for α-decays for transitions with Ɛ = 8, which indicate the accuracy is better for the CPPM method than for the other formulae for alpha-decay transitions with Ɛ = 8. We compared the calculation branching ratios and the HLs with the two models, the results indicate that, generally, modified CPPM (MCPPM) is the best for eveneven nuclei of Fm, Cf, Cm and Pu. The alpha decay energy (Qα) and the logarithm of the half-lives (log10T1/2(s)) as functions of Ap are plotted in Fig. 6. Alpha decay half-lives for Zp = 94 to Zp = 100 isotopes for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+ and 0+ĺ 6+ Į-transitions are shown in Fig. 6. (a), (b), (c) and (d), respectively, the red, blue, black and cyan boxes (solid) indicate the results of Fm, Cf, Cm and Pu isotopes. The peaks seen for fermium (252Fm) and californium (250Cm) isotopes are due to the deformed neutron shell at Np = 152. The plot shows linear relationships for the neutron numbers of the parent nuclei above the magic number (Np=156). More agreement is achieved with experimental half-lives, in general agreement with experimental data is better than in ref. [23], specifically for the 0+ → 4+, 0+ → 6+ and 0+ → 8+ transitions.
σ =
5 Conclusion In the present paper, we calculated the Į-decay half-lives of even-even nuclei of Fm, Cf, Cm and Pu with atomic numbers 94Zp100 within the range 234 A254 and in the framework of the CPPM and determined the Įdecay branching ratios by taking into account the angular momentum of the Į particle, Į-decay energy and the excitation probability of the daughter nucleus. We compared values of the branching ratio in the framework of MCPPM and other models such as Xu and Ren (XR) and Denisove and Khodenko (DK). The calculated branching ratios to all excited 0+, 2+, 4+, 6+ and 8+ states of the daughter nucleus are also in agreement with the accessible experimental data.
Acknowledgment The authors take great pleasure in thanking the referee for his/her several suggestions and comments. References [1] G. Gamow, Z. Phys. 51, 204 (1928). [2] R.W. Gurney, E.U. Condon, Nature 122 (1928) 439. [3] B. Buck, A.C. Merchant, S.M. Perez, J. Phys. G 17 (1991) 1223; [4] A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 2, Benjamin, New York, (1975). [5] M. A. Preston, Phys. Rev. 71, 865 (1947). [6] P. Moller, J. R. Nix, and K.-L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997). [7] D.S. Delion, A. Insolia, R.J. Liotta, Phys. Rev. C 46 (1992) 1346; [8] G. Royer, J. Phys. G 26 (2000) 1149; [9] R. Moustabchir and G. Royer, Nucl. Phys. A, 683, 266 (2001). [10] M. Fujiwara, T. Kawabata, and P. Mohr, J. Phys. G 28, 643 (2002). [11] M. Gupta and T. W. Burrows, Nucl. Data Sheets 106, 251 (2005). [12] S. Peltonen, D. S. Delion, and J. Suhonen, Phys. Rev. C 78, 034608 (2008). [13] V. Yu. Denisov, H. Ikezoe, Phys. Rev. C 72 (2005) 064613.
[14] C. Xu, Z. Ren, Phys. Rev. C 73 (2006) 041301. [15] A. Sobiczewski, A. Parkhomenko, Phys. Atom. Nucl. 69 (2006) 1155. [16] G. Royer, R. K. Gupta, V. Y. Denisov, Nucl. Phys. A 632, 275 (1998). [17] J. Dong, H. Zhang, Y. Wang, W. Zuo and J. Li, Nucl. Phys. A, 832, 198 (2010). [18] V. Yu. Denisov, A. A. Khudenko, Phys. Rev. C, 80, 034603 (2009). [19] K. P. Santhosh, J. G. Joseph, S. Sahadevan, Phys. Rev. C 82, 064605 (2010). [20] D. N. Poenaru, R. A. Gherghescu, W. Greiner, Phys. Rev. Lett. 107, 062503 (2011); Phys. Rev. C 85, 034615 (2012). [21] K.P. Santhosh, P. V. Subha, B. Priyanka, Pramana J. Phys,86 4, 819-836 (2016). [22] H. Geiger, J.M. Nuttall, Philosophical Magazine, Series 6, vol. 22, no. 130, pages 613-621, (1911). [23] V. Yu. Denisov and A. A. Khudenko, Phys Rev. C 80, 034603 (2009). [24] V. Yu. Denisov and A.A. Khudenko, Atomic Data and Nuclear Data Tables, 95 (2009) 815–835. [25] C. Xu and Z. Ren, Int. J Mod. Phys. E, 17 29 (2008) 36. [26] C. Xu and Z. Ren, Nuclear Physics A, 778 (2006) 1 9. [27] Y. Z. Wang, H. F. Zhang, J. M. Dong, and G. Royer, Phys Rev. C, 79, 014316 (2009). [28] J. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. Phys. (N.Y.) 105 (1977) 427. [29] S. S. Malik and R. K. Gupta, Phys. Rev. C, 39 1992 (1989). [30] I. Dutt, et al. Phys.Rev. C, 81, 064608 (2010). [31] J. Block and W. J. Swiatecki. Ann. Phys. (N. Y.), 132 53, (1981). [32] G. Sussmann, Nuclear Physics, Lawrence Berkeley Laboratory University of California Berkeley, California 94720, May (1973). [33] P. Marmier, E. Sheldon. Physics of Nuclei and Particle. Vol. New York: Academic Press, 1969. 303. [34] D. N. Poenaru, M. Ivascu, A. Sandulescu, and W. Greiner, Phys. Rev. C 32, 572 (1985). [35] C. Xu and Z. Ren, Nucl. Phys. A, 753 (2005) 174; A760 (2005) 303. [36] I. Hamamoto, S. V. Lukavanov and X. Z. Zhang, Nucl. Phys. A 683 (2001) 255. [37] V. Yu. Denisov and A. A. Khudenko, Phys Rev. C, 82, 059902(E) (2010). [38] G. Audi, O. Bersillon, J. Blachot, and A. H.Wapstra, Nucl. Phys. A, 729, 3 (2003). [39] Huang K-N, Aoyagi M, Chen M H, Crasemann B and Mark H (1976), At. Data Nucl. Data Tables 18 243. [40] G. Gamov and C. L. Critchfueld, Theory of atomic nucleus and nuclear energy-sources, Vol. III, The Clarendon Press, Oxford, 1949. [41] NuDat2.4, http://www.nndc.bnl.gov (last update July 15, 2008).
Table 1. Experimental and calculated branching ratios of alpha-decay and half-life values. α-Decay
Fmĺ244Cf (0+)→(0+) (0+)→(2+) 250 Fmĺ246Cf (0+)→(0+) (0+)→(2+) 252 Fmĺ248Cf (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 254 Fmĺ250Cf (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 244 Cfĺ240Cm (0+)→(0+) (0+)→(2+) 246 Cfĺ242Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 248 Cfĺ244Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) 250 Cfĺ246Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 252 Cfĺ 248Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 238 Cmĺ234Pu (0+)→(0+) (0+)→(2+) 240 Cmĺ236Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 242 Cmĺ238Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 244 Cmĺ 240Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 246 Cmĺ242Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) 248 Cmĺ244Pu (0+)→(0+) (0+)→(2+)
Zp
Qα(MeV)
Ɛmin
B exp[41] (%)
B theo(cal.) (%)
B XR[25] (%)
BDK[23] (%)
EƐ*(MeV) [41]
Texp(s)
TCPPM(s)(cal.)
TDK(s)[23]
100
8.002 7.9610
0 2
79.60 20.00
79.63 20.36
-
60.20 39.80
0.0 0.041
48.6 1.94×102
147 1.90×102
35.2 53.2
100
7.556 7.512
0 2
83.30 16.70
81.55 18.44
61.90 38.10
0.0 0.044
2.64×103 1.32×104
6.45×103 8.73×103
1.83×103 2.97×103
0 2 4 6
84.00 15.00 0.970 0.023
81.41 18.03 0.550 0.002
73.3 24.7 1.98 0.0375
9.14×104 6.09×105 9.42×106 3.97×108
2.72×105 3.62×105 8.18×106 1.73×108
9.31×104 1.55×105 5.11×105 3.36×106
7.3070 7.2644 7.1652 7.0142
0 2 4 6
84.9 14.2 0.82 0.0066
81.35 18.12 0.523 0.0018
73.2 24.8 1.98 0.0363
1.16×104
5.61×104 -
-
98
7.3289 7.2909
0 2
53 18
80.09 19.90
-
2.20×103 6.47×103
8.97×103 1.14×104
1.25103 1.92103
98
6.8616 6.8196 6.7236 6.5766
0 2 4 6
79.30 40.60 0.15 1.6×10-2
82.23 17.31 0.455 0.0013
-
1.62×105 6.24×105 8.57×107 8.03×108
8.01×105 1.08×106 2.66×107 5.86×108
1.18105 1.99105 6.58105 4.64106
6.361 6.318 6.2186
0 2 4
81.5 18.1 0.40
84.56 15.17 0.2608
75.4 23.2 1.47
59.10 32.70 8.15
0.0 0.043 0.142
2.88×107 3.60×107 1.47×108
1.74×108 2.47×108 7.13×109
2.34107 4.24107 1.70108
6.128 6.08554 6.0281 5.83764
0 2 4 6
84.5 15.1 0.30 0.010
83.76 14.71 1.51 0.0003
75.9 22.7 1.35 0.0156
60.10 32.00 7.25 0.68
0.0 0.043 0.142 0.295
4.12×108 2.75 ×109 1.38×1011 4.13×1012
2.58×109 3.72×109 1.14×1011 3.46×1012
4.85×108 9.12×108 4.02×109 4.29×1010
98
6.21687 6.17347 6.1150 5.91817 5.71077
0 2 4 6 8
84.20 15.70 0.24 2.00×10-3 5.99×10-5
84.76 14.99 0.247 0.00032 2.39×10-8
-
1.02×108 5.49×108 3.63×1010 4.31×1012 1.44×1014
8.18×108 1.17×109 3.55×1010 1.12×1012 5.24×1012
1.99108 3.74108 1.64109 1.741010 4.771011
96
6.620 6.574
0 2
69.53 30.47
84.42 15.57
3.24×105 7.38×105
1.59×106 2.32×106
1.83105 3.29105
0 2 4 6
71.10 28.90 0.05 1.40×10-2
84.52 15.32 0.15 0.00003
-
3.29×106 8.10×106 4.50×109 1.67×1010
1.69×107 2.45×107 9.06×109 2.19×1011
1.95106 3.54106 1.44107 1.34108
0 2 4 6 8
74.08 25.92 3.50×10-2 4.60×10-3 2.00×10-5
83.90 15.95 0.14 0.00002 6.07×10-12
-
1.90×107 5.43×107 4.02×1010 3.06×1011 7.04×1013
1.26×108 1.76×108 1.14×1011 1.9×1012 8.31×1015
7.32106 1.34107 5.50107 5.27108 1.251010
76.4 23.6 0.022 0.00035 0.00004
85.53 14.26 0.204 0.0002 1.12.10-8
5.70×108[52] 2.47×109 2.80×1012 1.62×1013 1.43×1015
5.40×109 7.88×109 2.51×1011 8.7×1012 4.46×1014
3.70108 6.99108 3.12109 3.411010 9.661011
1.50×1011[52] 8.44×1011 -
1.56×1012 2.44×1012 -
1.46×1013 6.65×1013 1.57×1016
1.54×1014 2.51×1014 2.63×1016
248
100
100
98
98
96
96
96
96
96
7.1530 7.1115 7.0202 6.8702
6.397 6.3524 6.2049 5.8991
6.21556 6.17456 6.02856 5.72516 5.21156
5.90174 5.85894 5.76004 5.60744 5.40424
0 2 4 6 8
5.4748 5.4303 -
0 2 4
82.2 17.8 0.040
5.16173 5.11753 4.96253 4.64463
0 2 4 6
81.90 18.03 7.60×10-2 5.46×10-3
-
-
55.20 33.20 10.10 1.530
-
66.60 39.40
55.60 33.00 9.96 1.41
60.00 32.00 7.27 0.69 2.50×10-2
64.20 35.80
58.80 32.40 7.96 0.86
59.00 32.30 7.85 0.82 3.45×10-2
0.0 0.042 0.138 0.283
0.0 0.043 0.142 0.296
0.0 0.038
0.0 0.0421 0.137 0.288
0.0 0.0434 0.1436 0.2981 0.5050
0.0 0.046
0.0 0.0446 0.1475 0.3058
0.0 0.041 0.146 0.3034 0.5136
76.5 22.2 1.24 0.0130 -
60.30 31.90 7.14 0.65 0.023
0.0 0.043 0.142 0.294 0.4975
87.87 12.13 -
78.4 20.6 0.934
67.60 32.40 -
0.0 0.045 0.147
88.88 11.09 0.026 2.59×10-7
-
66.20 29.70 3.91
0.0 0.0442 0.155
-
1.261011 2.641011 1.371013 3.061013 2.331014
Puĺ230U (0+)→(0+) (0+)→(2+) (0+)→(4+) 236 Puĺ232U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 238 Puĺ234U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 240 Puĺ236U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 242 Puĺ238U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 244 Puĺ 240U (0+)→(0+) (0+)→(2+)
0.3179
2.20×1017
2.81×1019
5.321015
0.0 0.0517 0.1695
7.73×105 1.67×106 1.32×108
------9.70×106 6.01×108
3.73105 7.28105 3.45106
62.10 31.20 6.20 0.48 1.37×10-2
0.0 0.0476 0.156 0.3227 0.5407
1.31×108 2.93×108 3.92×1010 4.88×1012 6.94×1014
1.09×109 1.69×109 6.02×1010 2.45×1012 1.51×1014
5.71107 1.14108 5.72108 7.41109 2.581011
0.17
(0+)→(4+) (0+)→(6+) 234
94
6.310 6.2583 6.0888
0 2 4
68.30 31.70 0.40
86.81 13.01 0.08
-
94
5.86707 5.81947 5.71047 5.54437 5.32637
0 2 4 6 8
69.10 30.80 0.23 1.85×10-3 1.30×10-5
87.04 12.81 0.14 0.00009 2.63×10-9
-
5.5932 5.5497 5.4498 5.2971 5.0961
0 2 4 6 8
70.90 28.98 0.11 3.00×10-3 6.80×10-6
86.69 13.15 0.51 0.0001 3.77×10-9
-
62.00 31.20 6.28 0.492 1.44×10-2
0.0 0.0435 0.1434 0.2961 0.4970
3.90×109 9.55×109 2.64×1012 9.23×1013 4.07×1016
3.55×1010 5.37×1010 1.87×1012 7.34×1013 4.36×1015
1.83109 3.63109 1.811010 2.311011 7.881012
5.25575 5.2123 5.1081 4.9478 4.7355
0 2 4 6 8
72.8 27.1 0.0852 0.00108 4.7×10-5
88.28 11.63 0.087 0.00003 2.55×10-10
79.1 20.1 0.807 0.00487 4.30×10-6
65.00 30.00 4.83 0.26 4.32×10-3
0.0 0.045 0.149 0.310 0.522
2.07×1011 7.64×1011 2.46×1014 1.95×1016 4.50×1017
3.89×1012 6.10×1012 2.57×1014 1.44×1016 1.47×1018
1.931011 4.181011 2.601012 4.811013 2.901015
4.9847 4.9398 4.8363 4.6773
0 2 4 6
77.5 22.4 0.098 0.0013
89.61 10.32 0.060 0.00001
80.1 19.2 0.688 0.00346
67.20 28.70 3.88 0.16
0.0 0.045 0.148 0.307
1.18×1013 5.02×1013 3.84×1016 1.37×1018
2.39×1014 3.98×1014 1.91×1016 1.28×1018
2.871013 6.721013 4.981014 1.221016
4.6655 4.6215
0 2
80.60 19.40
90.70 9.29
-
71.50 28.50
0.0 0.044
3.18×1015 1.32×1016
2.39×1016 8.51×1016
6.371015 1.601016
94
94
94
94
61.70 31.60 6.68
Table 2. The standard deviations (σ) for formulas DK No. of Nuclei Transition MCPPM 0.237 20 0.852 0+→0+ 0.432 20 0.573 0+→2+ 1.977 14 0.606 0+→4+ 2.354 12 0.763 0+→6+ 3.178 6 1.173 0+→8+ 0.759 1.62 72 All
Fig 1. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=0 transitions.
Fig 2. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=2 transitions.
Fig 3. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=4 transitions.
Fig 4. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=6 transitions.
Fig 5. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=8 transition.
(a)
(b)
(c)
(d)
Fig. 6. The alpha decay energy (Q-1/2) and the half-lives for Zp = 94–100 of even-even isotopes as functions of the mass number of the parent nuclei (Ap).
H. Hassanabadi, S.S. Hosseini
PII: DOI: Reference:
S0375-9474(18)30058-7 https://doi.org/10.1016/j.nuclphysa.2018.03.006 NUPHA 21182
To appear in:
Nuclear Physics A
Received date: Revised date: Accepted date:
8 January 2018 20 March 2018 21 March 2018
Please cite this article in press as: H. Hassanabadi, S.S. Hosseini, Branching ratios of α-decay to ground and excited states of Fm, Cf, Cm and Pu, Nucl. Phys. A (2018), https://doi.org/10.1016/j.nuclphysa.2018.03.006
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Branching ratios of Į-decay to ground and excited states of Fm, Cf, Cm and Pu H. Hassanabadi1 and S. S. Hosseini*1 1
Faculty of Physics, Shahrood University of Technology, Shahrood, Iran *
Corresponding author, Tel.:+98 232 4222522; fax: +98 273 3335270 Email: [email protected]
Abstract We use the well-known Wentzel–Kramers–Brillouin (WKB) barrier penetration probability to calculate α-decay branching ratios for ground and excited states of heavy even-even nuclei of Fermium (248-254Fm), Californium (244-252Cf), Curium (238-248Cm) and Plutonium (234-244Pu) with 94Zp100. We obtained the branching ratios for the excited states of daughter nucleus by the Į-decay energy (Qα), the angular momentum of Į-particle (Ɛα), and the excitation probability of the daughter nucleus with the excitation energy of state Ɛ in the daughter nucleus (i.e. E*Ɛ). α-Decay half-lives have been evaluated by using the proximity potential model for the heavy eveneven nuclei. We have reported the half-lives and compared the results with the experimental data. The theoretical branching ratios of α-transitions in our calculation are found to agree with the available experimental data well for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+, 0+ĺ 6+ and 0+ĺ 8+ Į-transitions. Keywords. Alpha Decay Modes, Branching ratios, even-even Nuclei, WKB technique, half-life. PACS: No 23.60. +e, 21.10.Re, 24.10.Ѹi, 27.90. +b
1 Introduction Alpha-decay is one of the most important decay channels of unstable nuclei in nuclear physics, therefore, There are many theoretical and experimental approaches which investigate the Į cluster radioactivity and alpha-like models [1-19]. Many attempts have been done on both experimental and theoretical fronts for understanding the physics of cluster radioactivity. Recently the concept of heavy-particle radioactivity is further explored by Poenaru et al. [20]. Also the half-life for the emission of various clusters from even–even isotopes of barium in the ground state and as an excited states were studied using the CPPM by Santhosh et al [21]. Hans Geiger and John Mitchell Nuttall formulated in 1911 a relation between the alpha-particle energy and the halflife (Geiger–Nuttall (GN)) [22], which in its modern form is written ln λ = −a1 Z / Q + a 2 , Where Ȝ is the decay constant, Z the atomic number, Q the total kinetic energy (of the alpha particle and the daughter nucleus), and a1 and a2 are constants. The law is best for nuclei with even atomic number and even atomic mass. The trend is still there for even-odd, odd-even, and odd-odd nuclei but not as pronounced. The GN law has even been extended to describe cluster decays, decays where atomic nuclei are larger than Helium are released. Denisov and Khudenko [23] studied the Į-decay half-lives for transitions between ground states in 344 nuclei and the Įcapture cross sections of 40Ca, 44Ca, 59Co, 208Pb, and 209Bi have recently been well described in the framework of the unified model for Į decay and Į capture. Į-Decay half-lives and the branching ratios to ground (gs) and excited states (es) of deformed nuclei with 222 A 252 and 88 Z 102 for even-even Į-transitions were analyzed in the framework of the unified model for Į decay and Į capture was analyzed by Denisov and Khudenko [24]. Xu and Ren [25, 26] evaluated the α-decay branching ratios for members of ground-state rotational band of the α-decay chain (Fm → Cf → Cm → Pu → U →Th) by a barrier penetration approach. Now we apply this model to the description of the Į-decay branching ratios in various states of daughter nuclei for heavy even-even nuclei of Fm, Cf, Cm and Pu and we calculate systematically the alpha decay half-lives with the proximity potential model. Wang et.al [27] considered branching ratios of Į decay to members of the ground state rotational band and excited 0+ states of even-even nuclei are calculated in the framework of the generalized liquid drop model (GLDM). In this paper, we consider the coulomb and proximity potential model and calculate the half-lives of some nuclei the ground state to exited state α-transitions. In the next section, we will review on our calculation of the important quantity as well as the branching ratio for the even-even nuclei in section 3. We report the half-lives calculated theoretically for this decay processes. We discuss on our numerical results in section 4. Finally, section 5 includes conclusions. 2 Alpha decay and Penetrability We have considered the modified potential for even-even nuclei, [28]
°a0 + a1r + a2 r 2 + Kr 3 , V (r ) = ® °¯V C (r ) +V Pr ox (z ) +V " (r ),
(1a ) (1b )
R p ≤ r ≤ Ct r ≥ Ct
The potential V (r) is parameterized simply as a polynomial for r Ct, and for Rp r Ct, the interacting potential barrier is defined as the sum of the nuclear, the coulomb and the centrifugal potentials for z>0 where z is the distance between the near surfaces of the fragments. Also, the constants a0, a1 and a2 have been determined considering R=Ra =RP, V (R) =V (Ct) andV ′( R ) = V ′ (C t ) in which V (R) =Q and R= Ct [29]. Ct is the touching configuration of two nuclei defined via [28], C t = C d + C α . The Sussmann central radii Ci of the daughter nuclei and the emitted alpha particle are referred to sharp radii (Ri) via, C i = R i − b 2 / R i where i = 1, 2 represent the daughter nucleus and the alpha particle, respectively. R i = 1 .2 8 A i1/ 3 − 0 .7 6 + 0 .8 A i− 1 / 3 ( f m ) . The coefficient K has been obtained by fitting of theoretical predictions to experimental data of Į decay. In the fitting process, we have used the standard deviation as our estimator. The fitting parameter has been found such that the standard deviation obtained the least possible value. In the following more details have been discussed. VC(r) is the Coulomb potential, in the form V C ( r ) = Z d Z α e 2 / r (MeV) With Zd and Zߙ being the atomic numbers of daughter nuclei and alpha particle, respectively. The centrifugal potential energy VƐ(r) is V l ( r ) = ! 2 l ( l + 1) / 2 μ r 2 (MeV), where Ɛ is the angular momentum (orbital quantum number) carried by the
emitted Į cluster and ȝ is the reduced mass of the disintegrated system, μ = mA d A α / ( A d + A α ) and m, Ad, and Aߙ denote the nucleon mass, mass number of daughter nuclei and alpha particle, respectively. Vp (r) is the nuclear proximity potential taking the form (see Blocki et al [28]) V P r o x ( z ) = 4 π γ b (C d C α / C d + C α ) Φ ( z / b ) (MeV), with the nuclear surface tension coefficient for this potential is defined by (the Lysekil mass formula of Ref. [30]) γ = γ 0 [1 − k s I 2 ] (MeVfm-2), in the present version, the coefficients Ȗ0 and ks were taken to be Ȗ0=0.9517 MeV/fm2 and ks=1.7826, respectively, Where I= (N-Z)/A, and N, Z and A are the neutron number, proton number and mass number of the parent for the combined system of the two interacting nuclei respectively. ĭ is the universal function proximity potential which is given as [31] Φ 1 ( ε ) = − 1.7817 + 0.9270 ε + 0.0169 ε 2 − 0.05148ε 3 , f or 0 ≤ ε ≤ 1.9475, ( 2a ) Φ 2 ( ε ) = − 4.41 exp( − ε / 0.7176),
Where
ε
ε ≥ 1.9475, (2b ) = z / b is the overlap distance in units of b between the colliding surfaces. According to the WKB f or
approximation, the penetration probability is obtained via [33], P = exp( − 2 !
Rb
³
2 μ (V ( r ) − Q α )dr ) , Where
Ra
the turning points (Ra, Rb) corresponding to the potential energies are defined as V (Ra) = V (Rb) = Qα. The partial half-life is related to the decay constant λ by ln 2 ln 2 = T 1/ 2 ( s ) = . (3 ) λ νP Where Ȟ is the frequency of assaults on the barrier per second (the zero-point vibration energy) [28, 29], ν = 2 E ν ib / ! = ω / 2 π . For Aα4, the empirical vibration energy Evib is given as [34], E v ib = Q [0 .0 5 6 + 0 .0 3 9 ex p ( 4 − A α / 2 .5)] that Shi and Swiatecki got empirically unrealistic values of Ȟ as
1022 for even-A parents and 1020 for odd-A parents. These calculations and the comparisons have also been comprehended in Table1. 3. The α-decay Branching Ratios Now we consider the branching ratios for both the ground and excited states. Xu and Ren [25] calculated the alpha decay branching ratios to members of ground-state rotational band of heavy even-even nuclei of Fm, Cf, Cm, Pu, U and Th [25]. The penetration probability to excited states depend on the alpha decay energy of the ground-state transition (Qα), the excitation energy ( E "* ) of state Ɛ and the angular momentum (Ɛ) of the Į-
particle. The penetration probability of the α-particle using the WKB method is given as [35]
Pα (Q α , E "* , " ) ∝
ψ III ψI
2
R out
= exp[ − 2
³
(4)
k ( r )dr ],
R0
Where R0 is the radius of the daughter nucleus ( R
0
= 1 / 2 A d1 / 3 ), the outgoing (ࣜI) and incoming (ࣜIII) are wave
functions and Rout is the second turning point [35]. The wave number K(r) is given by
k (r ) =
2μ (V ( r ) − Q α ) , !2
2 μ Z α Z d e 2 ! 2 " ( " + 1) [ + − (Q α − E "* )]1/ 2 , (5 ) 2μ r 2 !2 r Where ȝ is the reduced mass. Zα and Zd are the charge numbers of the α-particle and the daughter nucleus, respectively. The Alpha emission from a nucleus obeys the spin-parity selection rule [36] " I i − I f ≤ " ≤ I i + I f , π i / π f = ( − 1) Where Ii, ʌi, and If, ʌf are the spin, and parity of the parent and
daughter nuclei, respectively. If ʌi= ʌf, Ɛ must be even and ʌi് ʌf Ɛ must be odd. The angular momentum Ɛ carried by the alpha particle in a ground state to ground state transition of an even-even nucleus is zero, i.e. the Į-particle spin and parity are ʌĮ = +1 and ƐĮ = 0 . In odd-even or odd-odd nuclei, Ɛ could be nonzero. Denisov and Khudenko [29] purposed the effect of atomic electrons on the energy of the Į-particle to calculate the reaction energy values. The energy released in Į-transition between gs of the parent and the daughter nuclei is calculated using a recent evaluation of atomic mass data [38]. We take into account the penetrating the electron shell (electron screening effect) [21] on the reaction energy values, [13] Q = Δ M p − Δ M d − Δ M α + k ( Z pε − Z d ε ), ( M eV ) (6 ) Where ¨Mp and ¨Md, are, correspondingly, the mass excess of parent and daughter nuclei and ¨MĮ= 2.42498783 MeV for Į particle [37]. The quantity kZε represents the total binding energy of the Z electrons in the atom, where the values of k=8.7 eV and ε= 2.517 for nuclei with Z60 and k=13.6 eV, ε= 2.408 for nuclei with Z < 60 have been derived from data reported by Huang et al. [39]. We used the energy released at α-decay (Qn) from the ground-state (0+) of the parent nucleus to the excited states (n) of the daughter nucleus, therefore the excitation energy (E*) should be also taken into account [23] Q n ≡ Q 0 + → n + = Q 0 + → 0 + − E nd , ( M eV ) (7 ) Therefore, the tunneling probability of the Į particle, is given by [25, 40] Pα (Q α , E "* , " ) = exp[ − χ ] × exp[ − ρ ],
(8 )
With
χ=
2 μ Z α Z d e 2π , ! 2 (Q α − E "* )]1/ 2
( 9a )
ρ=
!2 2 " ( " + 1) , 2 μ ( Z α Z d e 2 R 0 )1/ 2
( 9b )
The term χ have to include the effects of the excitation energy E "* on the penetration factor in the daughter
nucleus and ρ represents the influence of the non-zero angular momentum Ɛ in the atom. The Boltzmann distribution hypothesis is proposed for daughter states in its excited states (2+, 4+, 6+ …) to simulate the internal effect of nuclear states on alpha formation [25]. The excitation energy of state Ɛ ( E "* ) effects are included
through a Boltzmann distribution function,
(10 )
ω " ( E "* ) = exp[ −cE "* ],
Where E "* , is the excitation energy of state Ɛ and c is a free parameter (The value of parameter c is 1.5) [25]. The
total probability of α-transition (IƐ+) can be obtained as the product of the penetration factor and the excitation probability [26]
(11)
I " + = ω " ( E "* ) Pα (Q α , E "* , " ), +
The branching (B) ratio of Į-decay to the excited 0 state of the daughter nucleus is given by [26] I B gn. +s . % = 2 n n + × 100%, n = 0, 2, 4,... I ¦ n+
(12 )
n =0
Where the sum n is going over all excited states for each isotopes. The results are reported in Table 1. 4 Results and discussions In the present work the mode of decay of 4, 5, 6 and 6 isotopes of Zp = 100, 98, 96 and 94 of Fm, Cf, Cm and Pu within the range 234 Ap 254 for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+, 0+ĺ 6+ and 0+ĺ 8+ Į-transitions have been studied by evaluating the alpha decay half-lives, respectively. we calculated the branching ratios of Į decays by taking
into account the influence of α-decay energy, the angular momentum of α-particle, and the excitation probability of the daughter nucleus and compared with the values based on the unified model for Į decay and Į capture (UMADAC) model by Denisov and Khudenko [23] and the barrier penetration approach by Xu and Ren [25]. Also, we observed good description of available data in our approaches with the experimental data. We presented the α-decay half-lives and the branching ratio of even-even nuclei 248-254Fm, 244-252Cf, 238-248Cm and 234-244 Pu, In Table 1. The T1/2 values are given in s. The first column of Table 1 denotes the α-decay mode for each isotope. The second, third and fourth columns denote the atomic number of the parent nucleus, the ground to ground and excited-states α-decay energy (Qα) in MeV and the minimal possible values of the orbital momentum of the Į particle (Ɛmin), respectively. The experimental, theoretical, Xu and Ren (XR) [25] and Denisov and Khudenko (DK) [23] α-decay branching ratios are listed in the fifth, sixth, seventh and eighth columns, respectively. The ninth column of Table 1 shows the excitation energy of state Ɛ in the daughter nucleus (E*Ɛ). In the last three columns are, correspondingly, the experimental Texp, theoretical Ttheo and Denisov and Khudenko TDK values of the Į-decay half-lives. The standard deviation (σ) is computed via 2 1 N theor exp (13 ) ¦ ª log10 T(1/2 α )i -log 10 T(1/2 α )i º ¼ . N i =1 ¬ The standard deviation of our calculation and DK approach are presented in Table 2, which indicate the acceptability of the results. In Figs. 1-4, we plotted the experimental (Diamond-solid) [41], theoretical (Boxsolid) and Ref. [23] (circles-solid) values of log10T1/2 (s) of Fm, Cf, Cm and Pu of 0+, 2+, 4+, and 6+ states for αdecays in even–even parent nuclei for Ɛ=0, 2, 4 and 6 transitions, respectively. Fig. 5 depicts the difference between the experimental, theoretical and Ref [23] values of log10T1/2 (s) of 236Pu, 238Pu, 240Pu, 242Cm, 244Cm, 252 Cf for α-decays for transitions with Ɛ = 8, which indicate the accuracy is better for the CPPM method than for the other formulae for alpha-decay transitions with Ɛ = 8. We compared the calculation branching ratios and the HLs with the two models, the results indicate that, generally, modified CPPM (MCPPM) is the best for eveneven nuclei of Fm, Cf, Cm and Pu. The alpha decay energy (Qα) and the logarithm of the half-lives (log10T1/2(s)) as functions of Ap are plotted in Fig. 6. Alpha decay half-lives for Zp = 94 to Zp = 100 isotopes for 0+ĺ 0+, 0+ĺ 2+, 0+ĺ 4+ and 0+ĺ 6+ Į-transitions are shown in Fig. 6. (a), (b), (c) and (d), respectively, the red, blue, black and cyan boxes (solid) indicate the results of Fm, Cf, Cm and Pu isotopes. The peaks seen for fermium (252Fm) and californium (250Cm) isotopes are due to the deformed neutron shell at Np = 152. The plot shows linear relationships for the neutron numbers of the parent nuclei above the magic number (Np=156). More agreement is achieved with experimental half-lives, in general agreement with experimental data is better than in ref. [23], specifically for the 0+ → 4+, 0+ → 6+ and 0+ → 8+ transitions.
σ =
5 Conclusion In the present paper, we calculated the Į-decay half-lives of even-even nuclei of Fm, Cf, Cm and Pu with atomic numbers 94Zp100 within the range 234 A254 and in the framework of the CPPM and determined the Įdecay branching ratios by taking into account the angular momentum of the Į particle, Į-decay energy and the excitation probability of the daughter nucleus. We compared values of the branching ratio in the framework of MCPPM and other models such as Xu and Ren (XR) and Denisove and Khodenko (DK). The calculated branching ratios to all excited 0+, 2+, 4+, 6+ and 8+ states of the daughter nucleus are also in agreement with the accessible experimental data.
Acknowledgment The authors take great pleasure in thanking the referee for his/her several suggestions and comments. References [1] G. Gamow, Z. Phys. 51, 204 (1928). [2] R.W. Gurney, E.U. Condon, Nature 122 (1928) 439. [3] B. Buck, A.C. Merchant, S.M. Perez, J. Phys. G 17 (1991) 1223; [4] A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 2, Benjamin, New York, (1975). [5] M. A. Preston, Phys. Rev. 71, 865 (1947). [6] P. Moller, J. R. Nix, and K.-L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997). [7] D.S. Delion, A. Insolia, R.J. Liotta, Phys. Rev. C 46 (1992) 1346; [8] G. Royer, J. Phys. G 26 (2000) 1149; [9] R. Moustabchir and G. Royer, Nucl. Phys. A, 683, 266 (2001). [10] M. Fujiwara, T. Kawabata, and P. Mohr, J. Phys. G 28, 643 (2002). [11] M. Gupta and T. W. Burrows, Nucl. Data Sheets 106, 251 (2005). [12] S. Peltonen, D. S. Delion, and J. Suhonen, Phys. Rev. C 78, 034608 (2008). [13] V. Yu. Denisov, H. Ikezoe, Phys. Rev. C 72 (2005) 064613.
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Table 1. Experimental and calculated branching ratios of alpha-decay and half-life values. α-Decay
Fmĺ244Cf (0+)→(0+) (0+)→(2+) 250 Fmĺ246Cf (0+)→(0+) (0+)→(2+) 252 Fmĺ248Cf (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 254 Fmĺ250Cf (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 244 Cfĺ240Cm (0+)→(0+) (0+)→(2+) 246 Cfĺ242Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 248 Cfĺ244Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) 250 Cfĺ246Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 252 Cfĺ 248Cm (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 238 Cmĺ234Pu (0+)→(0+) (0+)→(2+) 240 Cmĺ236Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 242 Cmĺ238Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 244 Cmĺ 240Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 246 Cmĺ242Pu (0+)→(0+) (0+)→(2+) (0+)→(4+) 248 Cmĺ244Pu (0+)→(0+) (0+)→(2+)
Zp
Qα(MeV)
Ɛmin
B exp[41] (%)
B theo(cal.) (%)
B XR[25] (%)
BDK[23] (%)
EƐ*(MeV) [41]
Texp(s)
TCPPM(s)(cal.)
TDK(s)[23]
100
8.002 7.9610
0 2
79.60 20.00
79.63 20.36
-
60.20 39.80
0.0 0.041
48.6 1.94×102
147 1.90×102
35.2 53.2
100
7.556 7.512
0 2
83.30 16.70
81.55 18.44
61.90 38.10
0.0 0.044
2.64×103 1.32×104
6.45×103 8.73×103
1.83×103 2.97×103
0 2 4 6
84.00 15.00 0.970 0.023
81.41 18.03 0.550 0.002
73.3 24.7 1.98 0.0375
9.14×104 6.09×105 9.42×106 3.97×108
2.72×105 3.62×105 8.18×106 1.73×108
9.31×104 1.55×105 5.11×105 3.36×106
7.3070 7.2644 7.1652 7.0142
0 2 4 6
84.9 14.2 0.82 0.0066
81.35 18.12 0.523 0.0018
73.2 24.8 1.98 0.0363
1.16×104
5.61×104 -
-
98
7.3289 7.2909
0 2
53 18
80.09 19.90
-
2.20×103 6.47×103
8.97×103 1.14×104
1.25103 1.92103
98
6.8616 6.8196 6.7236 6.5766
0 2 4 6
79.30 40.60 0.15 1.6×10-2
82.23 17.31 0.455 0.0013
-
1.62×105 6.24×105 8.57×107 8.03×108
8.01×105 1.08×106 2.66×107 5.86×108
1.18105 1.99105 6.58105 4.64106
6.361 6.318 6.2186
0 2 4
81.5 18.1 0.40
84.56 15.17 0.2608
75.4 23.2 1.47
59.10 32.70 8.15
0.0 0.043 0.142
2.88×107 3.60×107 1.47×108
1.74×108 2.47×108 7.13×109
2.34107 4.24107 1.70108
6.128 6.08554 6.0281 5.83764
0 2 4 6
84.5 15.1 0.30 0.010
83.76 14.71 1.51 0.0003
75.9 22.7 1.35 0.0156
60.10 32.00 7.25 0.68
0.0 0.043 0.142 0.295
4.12×108 2.75 ×109 1.38×1011 4.13×1012
2.58×109 3.72×109 1.14×1011 3.46×1012
4.85×108 9.12×108 4.02×109 4.29×1010
98
6.21687 6.17347 6.1150 5.91817 5.71077
0 2 4 6 8
84.20 15.70 0.24 2.00×10-3 5.99×10-5
84.76 14.99 0.247 0.00032 2.39×10-8
-
1.02×108 5.49×108 3.63×1010 4.31×1012 1.44×1014
8.18×108 1.17×109 3.55×1010 1.12×1012 5.24×1012
1.99108 3.74108 1.64109 1.741010 4.771011
96
6.620 6.574
0 2
69.53 30.47
84.42 15.57
3.24×105 7.38×105
1.59×106 2.32×106
1.83105 3.29105
0 2 4 6
71.10 28.90 0.05 1.40×10-2
84.52 15.32 0.15 0.00003
-
3.29×106 8.10×106 4.50×109 1.67×1010
1.69×107 2.45×107 9.06×109 2.19×1011
1.95106 3.54106 1.44107 1.34108
0 2 4 6 8
74.08 25.92 3.50×10-2 4.60×10-3 2.00×10-5
83.90 15.95 0.14 0.00002 6.07×10-12
-
1.90×107 5.43×107 4.02×1010 3.06×1011 7.04×1013
1.26×108 1.76×108 1.14×1011 1.9×1012 8.31×1015
7.32106 1.34107 5.50107 5.27108 1.251010
76.4 23.6 0.022 0.00035 0.00004
85.53 14.26 0.204 0.0002 1.12.10-8
5.70×108[52] 2.47×109 2.80×1012 1.62×1013 1.43×1015
5.40×109 7.88×109 2.51×1011 8.7×1012 4.46×1014
3.70108 6.99108 3.12109 3.411010 9.661011
1.50×1011[52] 8.44×1011 -
1.56×1012 2.44×1012 -
1.46×1013 6.65×1013 1.57×1016
1.54×1014 2.51×1014 2.63×1016
248
100
100
98
98
96
96
96
96
96
7.1530 7.1115 7.0202 6.8702
6.397 6.3524 6.2049 5.8991
6.21556 6.17456 6.02856 5.72516 5.21156
5.90174 5.85894 5.76004 5.60744 5.40424
0 2 4 6 8
5.4748 5.4303 -
0 2 4
82.2 17.8 0.040
5.16173 5.11753 4.96253 4.64463
0 2 4 6
81.90 18.03 7.60×10-2 5.46×10-3
-
-
55.20 33.20 10.10 1.530
-
66.60 39.40
55.60 33.00 9.96 1.41
60.00 32.00 7.27 0.69 2.50×10-2
64.20 35.80
58.80 32.40 7.96 0.86
59.00 32.30 7.85 0.82 3.45×10-2
0.0 0.042 0.138 0.283
0.0 0.043 0.142 0.296
0.0 0.038
0.0 0.0421 0.137 0.288
0.0 0.0434 0.1436 0.2981 0.5050
0.0 0.046
0.0 0.0446 0.1475 0.3058
0.0 0.041 0.146 0.3034 0.5136
76.5 22.2 1.24 0.0130 -
60.30 31.90 7.14 0.65 0.023
0.0 0.043 0.142 0.294 0.4975
87.87 12.13 -
78.4 20.6 0.934
67.60 32.40 -
0.0 0.045 0.147
88.88 11.09 0.026 2.59×10-7
-
66.20 29.70 3.91
0.0 0.0442 0.155
-
1.261011 2.641011 1.371013 3.061013 2.331014
Puĺ230U (0+)→(0+) (0+)→(2+) (0+)→(4+) 236 Puĺ232U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 238 Puĺ234U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 240 Puĺ236U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) (0+)→(8+) 242 Puĺ238U (0+)→(0+) (0+)→(2+) (0+)→(4+) (0+)→(6+) 244 Puĺ 240U (0+)→(0+) (0+)→(2+)
0.3179
2.20×1017
2.81×1019
5.321015
0.0 0.0517 0.1695
7.73×105 1.67×106 1.32×108
------9.70×106 6.01×108
3.73105 7.28105 3.45106
62.10 31.20 6.20 0.48 1.37×10-2
0.0 0.0476 0.156 0.3227 0.5407
1.31×108 2.93×108 3.92×1010 4.88×1012 6.94×1014
1.09×109 1.69×109 6.02×1010 2.45×1012 1.51×1014
5.71107 1.14108 5.72108 7.41109 2.581011
0.17
(0+)→(4+) (0+)→(6+) 234
94
6.310 6.2583 6.0888
0 2 4
68.30 31.70 0.40
86.81 13.01 0.08
-
94
5.86707 5.81947 5.71047 5.54437 5.32637
0 2 4 6 8
69.10 30.80 0.23 1.85×10-3 1.30×10-5
87.04 12.81 0.14 0.00009 2.63×10-9
-
5.5932 5.5497 5.4498 5.2971 5.0961
0 2 4 6 8
70.90 28.98 0.11 3.00×10-3 6.80×10-6
86.69 13.15 0.51 0.0001 3.77×10-9
-
62.00 31.20 6.28 0.492 1.44×10-2
0.0 0.0435 0.1434 0.2961 0.4970
3.90×109 9.55×109 2.64×1012 9.23×1013 4.07×1016
3.55×1010 5.37×1010 1.87×1012 7.34×1013 4.36×1015
1.83109 3.63109 1.811010 2.311011 7.881012
5.25575 5.2123 5.1081 4.9478 4.7355
0 2 4 6 8
72.8 27.1 0.0852 0.00108 4.7×10-5
88.28 11.63 0.087 0.00003 2.55×10-10
79.1 20.1 0.807 0.00487 4.30×10-6
65.00 30.00 4.83 0.26 4.32×10-3
0.0 0.045 0.149 0.310 0.522
2.07×1011 7.64×1011 2.46×1014 1.95×1016 4.50×1017
3.89×1012 6.10×1012 2.57×1014 1.44×1016 1.47×1018
1.931011 4.181011 2.601012 4.811013 2.901015
4.9847 4.9398 4.8363 4.6773
0 2 4 6
77.5 22.4 0.098 0.0013
89.61 10.32 0.060 0.00001
80.1 19.2 0.688 0.00346
67.20 28.70 3.88 0.16
0.0 0.045 0.148 0.307
1.18×1013 5.02×1013 3.84×1016 1.37×1018
2.39×1014 3.98×1014 1.91×1016 1.28×1018
2.871013 6.721013 4.981014 1.221016
4.6655 4.6215
0 2
80.60 19.40
90.70 9.29
-
71.50 28.50
0.0 0.044
3.18×1015 1.32×1016
2.39×1016 8.51×1016
6.371015 1.601016
94
94
94
94
61.70 31.60 6.68
Table 2. The standard deviations (σ) for formulas DK No. of Nuclei Transition MCPPM 0.237 20 0.852 0+→0+ 0.432 20 0.573 0+→2+ 1.977 14 0.606 0+→4+ 2.354 12 0.763 0+→6+ 3.178 6 1.173 0+→8+ 0.759 1.62 72 All
Fig 1. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=0 transitions.
Fig 2. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=2 transitions.
Fig 3. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=4 transitions.
Fig 4. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=6 transitions.
Fig 5. Comparison between the experimental, theoretical and Ref [23] values of log10T1/2 for αdecays for Ɛ=8 transition.
(a)
(b)
(c)
(d)
Fig. 6. The alpha decay energy (Q-1/2) and the half-lives for Zp = 94–100 of even-even isotopes as functions of the mass number of the parent nuclei (Ap).