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Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei
Accepted Manuscript Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei
G.L. Zhang, Y.J. Yao, M.F. Guo, M. Pan, G.X. Zhang, X.X. Liu
PII: DOI: Reference:
S0375-9474(16)30025-2 http://dx.doi.org/10.1016/j.nuclphysa.2016.03.039 NUPHA 20664
To appear in:
Nuclear Physics A
Received date: Revised date: Accepted date:
8 February 2016 17 March 2016 21 March 2016
Please cite this article in press as: G.L. Zhang et al., Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei, Nucl. Phys. A (2016), http://dx.doi.org/10.1016/j.nuclphysa.2016.03.039
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Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei G. L. Zhang 1
1,2
, Y. J. Yao 2 , M. F. Guo 2 , M. Pan2 , G. X. Zhang2 , X. X. Liu2
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China 2
Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China
Abstract Half-lives of large cluster radioactivity of even-even nuclei calculated by using fourteen proximity potentials are compared to experimental data. The results show that the results of BASS77 and Denisov potentials are most agreeable with the experimental data. Christensen and Wither 1976 potential gives the smallest halflives. In comparison with the distributions of different proximity potentials and the distributions of total potentials when the values of total potentials are more than the released energy Qc , it is found that at the small distances the large differences of proximity potentials do not affect the calculation results. The different distributions of total potentials affect the penetration probability of large cluster radioactivity, and then affect the half-life of large cluster radioactivity. PACS: 23.60.+e, 24.10.-i Key words: large cluster radioactivity, proximity potential, half-life
1
Introduction
In 1896, the natural radioactivity was first observed by Becquerel, then in 1908 Rutherford found α radioactivity in the first experimental observation of α decay for nuclei [1,2]. Gamow [3] and Gurdney and Condon [4,5] used quantum-tunneling model to successfully explain α radioactivity. From that Email address: [email protected] (G. L. Zhang
1,2 ).
time people began to explore the nuclear decay. In 1980, one pointed to the possibility of large cluster radioactivity [6] which is heavier than α particle. As a result, in 1984 14 C radioactivity was experimentally found for 232 Ra nucleus [7,8]. At present the emitted 14 C, 24,26 Ne, 28,30 Mg and 32,34 Si clusters from heavy nuclei were observed and the half-lives were measured. After the decay of these large cluster radioactivities from the parent nuclei, the residual daughter nuclei are almost close to the closed-shell nuclei in trans-lead region. It shows that the shell effect and paring effect play an important role for large cluster radioactivities. Theoretically, the different models were used to describe the cluster radioactivity, for example shell model, cluster model [9], liquid drop model [10] and double folding model [11,12,13] etc. These models can calculate the nuclear potential. Generally cluster tunneling model was used in which the penetration probability was calculated by using Wentzel-Kramers-Brillouin (WKB) approximation assuming the cluster radioactivity tunneling through the potential barrier between cluster radioactivity and the daughter nucleus in the parent nucleus. The cluster is assumed to form before it penetrates the barrier. Besides, proximity potential model is also an available way and has been generally used [14,15,16]. Shi and Swiatecki [17] were the first to use the proximity potential in an empirical manner and later on, Gupta et al [18,19] have been using the same, quite extensively for over a decade in the preformed cluster model (PCM). Further, the proximity potential has been subjected to several modifications, and Dutt and Puri [20,21] have been using different versions of proximity potentials for studying the fusion cross sections of different target-projectile combinations. Santhosh et al., have been using the proximity potential as the nuclear potential in the Coulomb and proximity potential model (CPPM) [22], and Coulomb and proximity potential model for deformed nuclei (CPPMDN) [23] for various studies on the cluster decay [24,25,26], alpha decay [27,28,29]. Therefore, by using it, the nuclear potential can be easily determined, especially between cluster radioactivity and different target nuclei. So it can provide some valuable supports for studying nuclear decay. In proximity potential model, the expression of the nuclear potential is mainly composed of two parts. One depends on the shape and geometry of two nuclei, and the other is the universal function φ(s) only related to the short separation distance s between two nuclei. The universal function is independent of the shapes of two nuclei and geometry of nuclear system. The idea of the universal function is fundamental advantage of proximity potential model. At present, there are different versions of proximity potentials. The main difference is for the different function φ(s0 ) and the different parameter values among them. It is necessary to explore the systematic behaviors of proximity potentials for the cluster radioactivity. In this paper we will select fourteen proximity potentials to calculate the nuclear potentials, then give the 2
half-lives of cluster radioactivity of nuclei. In comparison with the experimental data, we can find which proximity potential is fit for the calculation of the decay of large cluster radioactivity of nuclei and can explore the reason. This paper is organized as follows. In Section 2 the calculation process is introduced including several proximity potential models. In Section 3 the calculation results and discussion are displayed. The conclusion is given in Section 4.
2
The calculation process
Theoretically, the total interaction potential V(R) between projectile and target nuclei is given by V (R) = VN (R) + VC (R) +
l(l + 1)2 , 2μR2
(1)
where the first term VN (R) is the nuclear potential and the last term is the md centrifugal potential. μ = mmcc+m represents the reduced mass of cluster rad dioactivity and the daughter nucleus, mc and md denote the masses of cluster radioactivity and the daughter nucleus in the unit of M eV /c2 , respectively. l represents the angular momentum carried away by the emitted cluster radioactivity. VC (R) is Coulomb potential which is calculated by 2
⎧ ⎪ ⎨
VC (R) = Z1 Z2 e ⎪ ⎩
1 R 1 2Rc
3 − ( RRc )2
1/3
(R > Rc )
.
(2)
(R < Rc ) −1/3
Here, Rc = R1 + R2 , Ri = 1.28Ai − 0.76 + 0.8Ai (i=1,2) corresponds to the radii of cluster radioactivity and the daughter nuclei. For the nuclear potential VN (R) calculation fourteen different versions of proximity potentials are selected. The function of VN (R) of the proximity potential Prox77 (p77) and Prox88 (p88) and Prox00 (p00) is given by ¯ VN (R) = 4πγbRφ(s),
(3)
¯ is the mean curvature radius. The surhere φ(s) is the universal function and R 2 face energy coefficient γ = γ0 [1−ks I ] with the coefficient γ0 =0.9517 MeV/fm2 and the surface asymmetry constant ks =1.7826 [30] for Prox77, γ0 =1.2496 MeV/fm2 and ks =2.3 [31] for Prox88, γ0 =1.65 MeV/fm2 and ks =2.3 [32] for the modified proximity potential 1988 (mp88). I = NA−Z with N=N1 +N2 , Z=Z1 +Z2 , and A=A1 +A2 . Ni , Zi and Ai (i = 1,2) refer to the neutron, proton, and mass numbers of projectile and target nuclei, respectively. Their de3
tailed formulas and processes are shown in Ref. [14]. In the proximity potential Prox00 [15], the matter radius Ci has the form C i = ci +
Ni ti (i = 1, 2). Ai
(4)
Here the half-density radius ci of the charge distribution has the form ci = R00i (1 −
7b2 49b4 − )(i = 1, 2), 2 4 R00i 8R00i
(5)
with the nuclear charge radius 1/3
Ai − 2Zi )), Ai
(6)
](i = 1, 2),
(7)
R00i = 1.256Ai (1 − 0.202( and neutron skin of nucleus −1/3
1 gZi Ai 3 JIi − 12 ti = r0 [ −1/3 2 Q + 9 Ai 4
where r0 is 1.14 fm, the nuclear symmetric energy coefficient J=32.65 MeV and g=0.757895 MeV, the neutron skin stiffness coefficient Q=35.4 MeV. The surface energy coefficient γ is given by γ=
1 (t2 + t2 ) [18.63 − Q 1 2 2 ]. 2 4πr0 2r0
(8)
The universal function φ(ξ) is written as ⎧ ⎪ ⎨ −0.1353 + 5
φ(ξ) = ⎪
n=0 [cn /(n
+ 1)](2.5 − ξ)n+1
⎩ −0.0955exp( 2.75−ξ ) 0.7176
(0 < ξ ≤ 2.5) (ξ ≥ 2.5)
.
(9)
Here ξ = (r − C1 − C2 )/b and the width parameter b is close to unity. The different values of constant cn are: c0 =-0.1886, c1 =0.2628, c2 =-0.15216, c3 =¯ is cal0.04562, c4 =0.069136, and c5 =-0.011454. The mean curvature radius R culated by ¯ = C 1 C2 . R (10) C 1 + C2 For the generalized proximity potential 1977 (gp77) [33], the universal function is given by φ(ξ) =
⎧ ⎪ ⎨ −1.7817 + 0.927ξ + 0.0169ξ 2 − 0.05148ξ 3 ⎪ ⎩ −4.41exp(−
ξ ) 0.7176
4
(0 ≤ ξ ≤ 1.9475)
. (ξ > 1.9475) (11)
The nuclear radius parameter R0i is given by R0i = Ri − 1/3
b2 . Ri
(12)
−1/3
(i=1,2) corresponds to the radii of large Here Ri = 1.28Ai − 0.76 + 0.8Ai cluster radioactivity and the daughter nuclei. The definitions of ξ, the width b and the other parameters are same as that of Prox77. The parameter ξ=(RC1 -C2 )/b with Ci = Ri [1 − ( Rbi )2 ], Ri corresponds to the radii of large cluster radioactivity and the daughter nuclei, and is same as the above definition. For the modified proximity potential 2000 (mp00) [34], the nuclear charge radius R00i is given by 1/3
R00i = 1.2332Ai [1 +
2.348443 Ai − 2Zi − 0.151541 ]. Ai Ai
(13)
Here Ai denotes the masses of large cluster radioactivity and the daughter nucleus. The other parameters are same as that of Prox00. For Denisov potential (DP) [35], the nuclear potential VN is calculated by R1 R2 φ(r − R1 − R2 − 2.65) R1 + R2 A1 A2 3/2 × [1 + 0.003525139( + ) − 0.4113263(I1 + I2 )], A2 A1
VN (r) = − 1.989843
where Ii =
Ni −Zi . Ai
(14)
Ri is given by
Ri = Rip (1 −
3.413817 0.4Ai ) + 1.284589(Ii − ), 2 Rip Ai + 200
(15)
with
1.646 Ai − 2Zi − 0.191( )]. Ai Ai The universal function φ(s = r − R1 − R2 − 2.65) is given by 1/3
Rip = 1.24Ai [1 +
(16)
⎧ ⎪ ⎪ 1 − s/0.7881663 + 1.229218s2 − 0.2234277s3 − 0.1038769s4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − R1 R2 (0.1844935s2 + 0.07570101s3 ) ⎪ ⎪ ⎨ R1 +R2
φ(s) =
2
3
+(I1 + I2 )(0.04470645s + 0.0334687s ) (−5.65 ≤ s ≤ 0) ⎪ ⎪ ⎪ ⎪ ⎪ R2 s ⎪ (1 − s2 (0.05410106 RR11+R exp(− 1.76058 ) − 0.539542(I1 + I2 ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ s ⎩ exp(− ))) × exp(− s ) (s > 0) 2.424408
.
0.7881663
(17) 5
For Christensen and Wither 1976 (CW76) [36], the nuclear potential VN is given by R1 R2 VN (r) = −50 φ(r − R1 − R2 ), (18) R 1 + R2 1/3
where Ri = 1.233Ai
−1/3
− 0.978Ai
1 −R2 , and φ(s) = exp(− r−R0.63 ).
For Broglia and Winther 1991 (BW91) [31], the nuclear potential VN is written by V0 VN (r) = − . (19) 0 1 + exp( r−R ) 0.63 1/3
R2 γa with a=0.63 fm. R0 = R1 +R2 +0.29. Ri = 1.233Ai − Here V0 = 16π RR11+R 2 −1/3
0.98Ai
. The surface energy constant γ has the form γ = γ0 [1 − ks (
N1 − Z1 N2 − Z2 )( )], A1 A2
(20)
where the coefficients γ0 =0.95MeV/fm2 and ks =1.8. For Aage Withner (AW95) potential [37], only a= 1/3
Ri = 1.2Ai BW91.
1 −1/3
1.17(1 + 0.53(A1
−1/3
+ A2
))
,
(21)
− 0.09, R0 = R1 + R2 . The other parameters are same as that of
For Ngˆo 1980 potential (Ng80) [38], the nuclear potential VN is calculated ¯ ¯ form is same as that of Prox77. However, Ri = Ni Rni +Zi Rpi by VN =Rφ(s). R Ai 1/3
1/3
with Rpi = r0pi Ai and Rni = r0ni Ai . r0pi = 1.128 f m and r0ni = 1.1375 + 1.875 × 10−4 Ai . φ(s) is given by
φ(s) =
⎧ ⎪ ⎨ −33 + 5.4(s − s0 )2
(s < s0 )
⎪ ⎩ −33exp(− 1 (s − s )2 ) 0 5
(s ≥ s0 )
.
(22)
Here s0 =-1.6. For Bass 1973 (B73) [39,40], the nuclear potential VN is calculated by 1/3
VN (r) = − 1/3
1/3
das A1 A2 r − R12 exp(− ), R12 d
1/3
where R12 =1.07(A1 + A2 ), d=1.35 fm and as = 17M eV . 6
(23)
For Bass 1977 (B77) [41], the nuclear potential VN is given by VN (s) = − 1/3
R1 R2 φ(r − R1 − R2 ), R1 + R2
(24)
−1/3
where Ri =1.16Ai -1.39Ai . The universal function φ(s = r − R1 − R2 ) has the form s s (25) φ(s) = [Aexp( ) + Bexp( )]−1 , d1 d2 with A=0.030 MeV−1 fm, B=0.0061 MeV−1 fm, d1 =3.30 fm and d2 =0.65fm. For Bass 1980 (B80) [31], the form of nuclear potential VN is same as that of the above Bass77, only φ(s = r − R1 − R2 ) is given by φ(s) = [0.033exp( with Ri =Rs (1 −
0.98 ). Rs2
s −1 s ) + 0.007exp( )] , 3.5 0.65
1/3
−1/3
Rs =1.28Ai -0.76+0.8Ai
(26)
.
The half-life T1/2 of large cluster radioactivity of the parent nucleus can be calculated by πln2 T1/2 = [1 + 1/P ], (27) P0 where P0 is the preformation factor and P0 = 10−(0.4A1 −2) [9] where A1 denotes the mass number of large cluster radioactivity. P is the penetration probability which is calculated in the WKB approximation by
2 Rb 2μ(V (R) − Qc )dR , P = exp − Ra
(28)
where μ is the reduced mass of large cluster radioactivity and the daughter nucleus. Qc is the released energy when the parent nucleus decays the large cluster radioactivity, here we select the experimental released energies for the emission of large cluster radioactivity. Ra and Rb are the two turning points of WKB approximation integral determined from the equation V (Ra ) = Qc = V (Rb ).
3
(29)
Results and discussion
Here we select the emissions of large cluster radioactivities for even-even nuclei. Since the selected large cluster radioactivities are the even-even nuclei, as a result, the residual daughter nuclei are also the even-even nuclei. Finally, during the emission process of the even-even large cluster radioactivity for the 7
Table 1. The values of the obtained half-lives of the large cluster radioactivities through Prox77 potential (p77, the fourth column), Prox88 potential (p88, the fifth column), Prox00 potential (p00, the sixth column), generalized Prox77 potential (gp77, the seventh column), modified Prox88 potential (mp88) and modified Prox00 potential (mp00) which correspond to the eighth and the ninth columns, respectively. The second and third columns are for released energies [42] for emission of large cluster radioactivities and the experimental half-lives [43], respectively. parent
Qc
exp LogT1/2
p77 LogT1/2
p88 LogT1/2
p00 LogT1/2
gp77 LogT1/2
mp88 LogT1/2
mp00 LogT1/2
nuclei
(MeV)
(s)
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
33.05
11.22
12.96
10.24
13.63
11.19
8.50
9.43
224 Ra
30.54
15.86
17.99
15.21
18.39
15.92
13.28
14.18
226 Ra
28.20
21.34
23.29
20.58
23.48
20.98
18.40
19.24
228 Th
44.72
20.72
23.37
20.24
22.56
20.76
17.58
19.0
230 Th
57.76
24.61
26.61
23.85
24.55
23.70
19.92
21.38
232 Th
54.67
>29.20
31.81
29.63
29.54
28.24
25.04
26.33
232 Th
55.91
>29.20
31.38
27.95
28.33
28.04
24.10
25.83
230 U
61.39
19.57
22.15
19.92
20.72
19.22
15.93
17.71
232 U
62.31
21.08
22.30
18.80
20.44
19.23
15.56
17.24
232 U
74.32
>22.26
26.53
24.35
24.20
23.57
19.56
21.39
234 U
58.83
25.92
27.66
24.99
24.79
24.27
20.74
22.31
234 U
59.42
25.92
28.30
24.43
25.41
24.71
20.92
22.62
234 U
74.11
27.54
26.76
24.59
23.60
23.81
19.72
21.56
236 U
55.95
>25.90
32.49
29.92
29.38
28.86
25.50
26.90
236 U
56.70
>25.90
32.61
29.67
29.82
29.10
25.44
27.02
236 U
70.73
27.58
31.63
29.68
28.22
27.95
24.57
25.52
236 U
72.28
27.58
30.83
28.12
27.38
27.38
23.19
24.71
236 Pu
79.67
21.67
22.30
19.11
19.37
19.13
15.08
16.69
238 Pu
75.91
25.70
27.26
24.88
24.04
23.79
19.94
21.33
238 Pu
76.80
25.70
27.42
24.04
24.12
23.68
19.63
21.43
238 Pu
91.19
25.27
26.76
24.41
23.39
23.47
19.11
21.06
240 Pu
91.03
>25.52
28.15
25.20
23.76
24.46
19.91
21.58
242 Cm
96.51
23.15
24.65
20.71
20.47
21.19
16.26
18.25
8
Table 2. Same as Table 1, but for AW95 (the second column), CW76 (the third column), Ng80 (the fourth column), BW91 (the fifth column), DP (the sixth column) potentials and three types of Bass potentials which correspond to Bass73 (B73, the seventh column), Bass77 (B77, the eighth column) and Bass80 (B80, the last column), respectively. AW 95 LogT1/2
CW 76 LogT1/2
N g80 LogT1/2
BW 91 LogT1/2
DP LogT1/2
B73 LogT1/2
B77 LogT1/2
B80 LogT1/2
nuclei
(s)
(s)
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
9.35
6.84
12.62
9.16
9.69
11.89
10.82
10.02
224 Ra
14.25
11.37
17.71
14.19
14.43
16.82
15.76
14.98
226 Ra
19.50
16.29
23.25
19.87
19.63
22.07
21.03
20.25
228 Th
19.43
15.0
24.11
19.28
20.42
23.09
20.69
19.97
230 Th
23.23
16.65
27.95
21.84
24.33
26.88
23.70
23.09
232 Th
28.50
21.35
33.29
27.05
30.15
32.16
29.06
28.49
232 Th
27.38
20.65
32.31
26.17
28.68
31.44
27.81
27.18
230 U
19.35
12.83
23.78
17.84
20.62
22.73
20.04
19.44
232 U
18.48
12.55
23.56
17.47
19.66
22.50
19.23
18.53
232 U
23.59
15.82
28.57
21.53
25.63
28.02
24.09
23.57
234 U
24.22
17.36
28.84
22.88
25.53
27.92
24.74
24.12
234 U
24.14
17.65
29.25
23.07
25.17
28.34
24.68
24.0
234 U
23.72
15.96
28.71
21.68
25.73
28.16
24.18
23.60
236 U
29.20
21.74
33.74
27.69
30.60
32.81
29.70
29.12
236 U
28.82
21.84
34.01
27.78
30.12
33.04
29.37
28.74
236 U
28.76
20.29
33.68
26.51
30.95
33.13
29.30
28.82
236 U
27.01
19.25
32.66
25.36
29.24
31.99
27.58
26.99
236 Pu
18.66
11.73
24.16
17.15
20.58
23.49
19.40
18.74
238 Pu
23.76
16.12
29.27
22.13
25.92
28.64
24.64
24.04
238 Pu
23.50
15.94
28.65
21.91
25.63
28.50
24.01
23.35
238 Pu
23.49
15.01
29.14
21.25
26.21
28.99
24.24
23.65
240 Pu
24.43
15.89
30.34
22.23
26.70
29.99
24.75
24.09
242 Cm
20.36
12.59
26.48
18.40
22.86
26.32
21.01
20.26
parent
9
Table 3. The standard deviations σ of logarithm values of the calculated half lives for different proximity potentials in comparison with the experimental data. σ p77
σ p88
σ p00
σ gp77
σ mp88
σ mp00
σ AW 95
2.129
1.558
1.889
1.610
5.079
3.581
2.112
σ CW 76
σ N g80
σ BW 91
σ DP
σ B73
σ B77
σ B80
7.951
3.348
3.221
1.373
2.753
1.374
1.955
Fig. 1. (Color online) The distributions of total potentials for 238 Pu - 32 Si+206 Hg when the values of total potentials are more than the released energy Qc .
10
even-even parent nuclei, the angular momentum l carried away by large cluster radioactivity is equal to 0, which corresponds to zero for the centrifugal potential. The half-lives provided by the above mentioned proximity potentials are presented in Table 1 which includes Prox77, Prox88, Prox00, gp77, mp88, mp00 potentials, in Table2 which includes AW95, CW76, Ng80, BW91, DP, BASS73, BASS77, and BASS80 potentials. The standard deviations σ of logarithm values of the calculated half lives for these proximity potentials in comparison with the experimental data are shown in Table 3. In comparison with Table 1 and Table2 as well as the σ values of Table 3, we can see that when comparing the experimental data, the values of DP and BASS77 are closest to the experimental data, moreover, their σ values are also smallest. the values of Prox88, gp77 and BASS80 are closer to the experimental data. The values of other proximity potentials have the difference in the whole range. The calculation results of mp00 and BW91 are nearly close, but less about four order magnitude than the experimental data. Those of Ng80 and BASS73 are nearly close, but more about one order magnitude than the experimental data. In all calculation, those of CW76 are smallest and those of mp88 are smaller. The calculation results of Prox77 are more almost two order magnitudes than the experimental data, and are close to those of Ng80 and BASS73 in the range of light large cluster radioactivities, however, in the range of heavy large cluster radioactivities there is the difference among them, those of Prox77 are closer to the experimental data. Except that the several values of AW95 for 14 C and 20 O cluster radioactivities are less than those of Prox00, the other ones are close between them. In Fig. 1 the distributions of total potentials V are shown when V is more than the released energy Qc . Here we only select 238 Pu - 32 Si+206 Hg as a example, the others give the same distributions of V. We can see that the distance is more than 8 fm when V is more than Qc . Therefore, in refer to the figure 1 of Ref. [46], the large difference at small distances for different proximity potentials does not affect the values of half-lives of large cluster radioactivities. Only at the large distances there is a little difference to affect the values of half-lives. From Fig. 1, at more than 14 fm each potential gives the almost same values. However, at less than 13 fm there is the large differences among the different proximity potentials. Even some also show the different distribution. CW76 gives the thinnest distribution and mp88 give the thinner distribution, so their integrals of Eq. (28) are smallest and smaller, respectively. Then the calculated half-lives are smallest and smaller, respectively. So we can know the reason why the results of CW76 are smallest and that of mp88 is smaller in comparison with the experimental data in Table 1 and 2. BW91 also shows a thin distribution, so the values of its half-lives are also small. B73 and Ng80 show the wider distribution, then their integrals of Eq. (28) are larger, which corresponds to the larger values of half-lives. The distributions of B77 and DP are similar, then their half-lives are close. So in Table 1 and 2 the values of B77 and DP show the similarity. The distributions 11
Table 4. The values of the obtained half-lives of the large cluster radioactivities through BASS77 potential (B77, the third column) are compared with experimental data (the second column), those of liquid drop model (LDM, the fourth column) [10], ref1 (the fifth column) [43], ref2 (the sixth column) [44] and ref3 (the seventh column) [45]. exp LogT1/2
B77 LogT1/2
LDM LogT1/2
ref 1 LogT1/2
ref 2 LogT1/2
ref 3 LogT1/2
nuclei
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
11.22
10.82
10.59
10.44
9.59
11.81
224 Ra
15.86
15.76
16.59
15.17
14.34
16.55
226 Ra
21.34
21.03
22.51
20.39
19.37
21.54
228 Th
20.72
20.69
21.61
21.86
19.89
21.90
230 Th
24.61
23.70
25.451
26.13
23.39
24.99
232 Th
>29.20
29.06
28.76
31.19
26.66
28.09
232 Th
>29.20
27.81
29.72
30.76
27.70
29.91
230 U
19.57
20.04
21.40
21.77
19.48
20.26
232 U
21.08
19.23
19.99
21.89
19.35
20.75
232 U
>22.26
24.09
25.74
25.42
234 U
25.92
24.74
26.54
27.05
24.59
25.76
234 U
25.92
24.68
25.91
27.75
24.91
26.83
234 U
27.54
24.18
25.90
25.53
24.03
25.48
236 U
>25.90
29.70
32.18
31.91
30.33
236 U
>25.90
29.37
31.48
32.40
31.20
236 U
27.58
29.30
29.34
30.35
27.24
28.51
236 U
27.58
27.58
29.28
29.48
27.21
29.43
236 Pu
21.67
19.40
20.0
21.07
19.93
21.05
238 Pu
25.70
24.64
26.34
25.95
24.72
25.69
238 Pu
25.70
24.01
24.83
26.05
24.13
25.99
238 Pu
25.27
24.24
25.73
26.66
24.52
24.91
240 Pu
>25.52
24.75
26.08
27.71
25.34
27.25
242 Cm
23.15
21.01
21.11
24.23
22.14
23.70
parent
12
25.43
of p88 and B80 are nearly close to those of B77 and DP, then if the values of B77 and DP are close to the experimental data, those of p88 and B80 are also close to the experimental data. The distribution of p77 is thinner than those of B73 and Ng80, but is higher than those of B77 and DP, as a result, the calculated half-lives of p77 are larger than the experimental data and close to the experimental data with respect to those of B73 and Ng80. The same conclusion is also obtained from Table 1 and 2. Therefore, depending on Fig. 1 we can know the reason for the conclusion of Table 1 and 2. From the above calculation for even-even nuclei, we know that B77 and DP are most suitable for the calculation of half-life of large cluster radioactivity. In order to further check B77 and DP, in Table 4 we list the results of B77 in comparison with the experimental data, those of liquid drop model [10], ref1 [43], ref2 [44] and ref3 [45]. we can see that the values of B77 are similar as those of those models, and closer to that of ref2. With respect to the experimental data, sometimes the values of B77 are closer in comparison with those of those models. It is indicated that B77 can calculate the half-life of large cluster radioactivity very well. As a result, it verifies that B77 and DP can also calculate the half-lives of large cluster radioactivity for even-even nuclei.
4
Conclusion
The fourteen different proximity potentials are used to calculate the halflives of large cluster radioactivity for even -even nuclei with from light large cluster radioactivity to heavy large cluster radioactivity in comparison with the experimental data. It is shown that BASS77 (B77) potential and Denisov potential (DP) are suitable for calculating the half-life of large cluster radioactivity. The calculation results of Prox88 and BASS80 are nearly close to the experimental data. CW76 and mp88 give the smallest and smaller half-lives, respectively, and the remains give values with a difference with respect to the experimental data. It is the reason that the distributions of different proximity potentials at large distances with more than 8 fm lead to the different half-lives, the difference is not from the different distributions of proximity potentials at small distances where they have a large difference. This comparison provides a reference how to select the proximity potentials to calculate the half-life of large cluster radioactivity. In comparison with the results of the other models for even-even nuclei, the results of BASS77 are closer to the experimental data. Therefore, it further verifies that the BASS77 potential and Denisov potential can calculate the half-lives of large cluster radioactivity for even-even nuclei very well. Acknowledgement 13
This work is supported by the National Nature Science Foundation of China under Grant Nos. 11475013, 11035007, 11175011, 11375266 and State Key Laboratory of Software Development Environment (SKLSDE-2014ZX08) as well as by the Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences.
References [1] E. Rutherford, H. Geiger, Proc. R. Soc. A 81, 141 (1908). [2] E. Rutherford, T. Royds, Philos. Mag. 1, 281 (1908). [3] G. Gamow, Z. Phys. 51, 204 (1928). [4] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). [5] R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929). [6] A. Sandulescu, D. N. Poenaru, W. Greiner, Sov. J. Part. Nucl. 11, 528 (1980). [7] H. J. Rose, G. A. Jones, Nature 307, 245 (1984). [8] D. V. Aleksandro et al, JETP Lett. 40, 909 (1984). [9] Z. Z. Ren, C. Xu, Z. J. Wang, Phys. Rev. C 70, 034304 (2004). [10] G. Royer, R. Moustabchir, Nucl. Phys. A 683, 182 (2001). [11] L. Zheng, G. L. Zhang, J. C. Yang, W. W. Qu, Nucl. Phys. A 915, 70 (2013). [12] G. L. Zhang, X. Y. Le, Nucl. Phys. A 848, 292 (2010). [13] G. L. Zhang, H. Liu and X. Y. Le, Acta Physica Sinica 58, 2300 (2009). ´ [14] J. Blocki, J. Randrup, W. J. Swiatecki, C. F. Tsang, Ann. Phys. (NY) 105, 427 (1977). ´ [15] W. D. Myers, W. J. Swiatecki, Phys. Rev. C 62, 044610 (2000). [16] P. M¨ oller, J. R. Nix, Nucl. Phys. A 361, 117 (1981). [17] Y. J. Shi, W. J. Swiatecki, Nucl. Phys. A 438, 450 (1985). [18] S. S. Malik, Raj K. Gupt, Phys. Rev. C 39, 1992 (1989). [19] S. S. Malik, S. Singh, R. K. Puri, S. Kumar, R. K. Gupta, Pramana-J. Phys. 32, 419 (1989). [20] I. Dutt, R. K. Puri, Phys. Rev. C 81, 064608 (2010). [21] I. Dutt, R. K. Puri, Phys. Rev. C 81, 064609 (2010). [22] K. P. Santhosh, Antony Joseph, Pramana-J. Phys. 58, 611 (2002).
14
[23] K. P. Santhosh, Sabina Sahadevan, Jayesh George Joseph, Nucl. Phys. A 850, 34 (2011). [24] K. P. Santhosh, Sabina Sahadevan, B. Priyanka, M. S. Unnikrishnan, Nucl. Phys. A 882, 49 (2012). [25] K. P. Santhosh, B. Priyanka, M. S. Unnikrishnan, Nucl. Phys. A 889, 29 (2012). [26] K. P. Santhosh, B. Priyanka, Nucl. Phys. A 929, 20 (2014). [27] K. P. Santhosh, B. Priyanka, Int. J. Mod. Phys. E 22, 1350081 (2013). [28] K. P. Santhosh, Indu Sukumaran, B. Priyanka, Nucl. Phys. A 935, 28 (2015). [29] K. P. Santhosh, B. Priyanka, Nucl. Phys. A 940, 21 (2015). ´ [30] W. D. Myers, W. J. Swiatecki, Nucl. Phys. 81, 1 (1966). [31] W. Reisdorf, J. Phys. G: Nucl. Part. Phys. 20, 1297 (1994). [32] R. Kumar, M. K. Sharma, Phys. Rev. C 85, 054612 (2012). [33] K. P. Santhosh, B. Priyanka, Phys. Rev. C 87, 064611 (2013). [34] G. Royer, R. Rousseau, Eur. Phys. J. A 42, 541 (2009). [35] V. Y. Denisov, Phys. Lett. B 526, 315 (2002). [36] P. R. Christensen, A. Winther, Phys. Lett. B 65, 19 (1976). [37] A. Winther, Nucl. Phys. A 594, 203 (1995). [38] H. Ngˆ o, C. Ngˆo, Nucl. Phys. A 348, 140 (1980). [39] R. Bass, Phys. Lett. B 47, 139 (1973). [40] R. Bass, Nucl. Phys. A 231, 45 (1974). [41] R. Bass, Phys. Rev. Lett. 39, 265 (1977). [42] G. Audi et al., Chin. Phys. C 36, 1287 (2012); M. Wang et al., ibid. 36, 1603 (2012). [43] A. Bhagwat and R. J. Liotta, Phys. Rev. C 92, 044312 (2015). [44] D. N. Basu, Phys. Lett. B 566, 90 (2003). [45] L. Zheng, G. L. Zhang, J. C. Yang, W. W. Qu, Nucl. Phys. A 915, 90 (2013). [46] Y. J. Yao, G. L. Zhang, W. W. Qu, J. Q. Qian, Eur. Phys. J. A 51, 122 (2015).
15
G.L. Zhang, Y.J. Yao, M.F. Guo, M. Pan, G.X. Zhang, X.X. Liu
PII: DOI: Reference:
S0375-9474(16)30025-2 http://dx.doi.org/10.1016/j.nuclphysa.2016.03.039 NUPHA 20664
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Nuclear Physics A
Received date: Revised date: Accepted date:
8 February 2016 17 March 2016 21 March 2016
Please cite this article in press as: G.L. Zhang et al., Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei, Nucl. Phys. A (2016), http://dx.doi.org/10.1016/j.nuclphysa.2016.03.039
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Comparative studies for different proximity potentials applied to large cluster radioactivity of nuclei G. L. Zhang 1
1,2
, Y. J. Yao 2 , M. F. Guo 2 , M. Pan2 , G. X. Zhang2 , X. X. Liu2
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China 2
Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China
Abstract Half-lives of large cluster radioactivity of even-even nuclei calculated by using fourteen proximity potentials are compared to experimental data. The results show that the results of BASS77 and Denisov potentials are most agreeable with the experimental data. Christensen and Wither 1976 potential gives the smallest halflives. In comparison with the distributions of different proximity potentials and the distributions of total potentials when the values of total potentials are more than the released energy Qc , it is found that at the small distances the large differences of proximity potentials do not affect the calculation results. The different distributions of total potentials affect the penetration probability of large cluster radioactivity, and then affect the half-life of large cluster radioactivity. PACS: 23.60.+e, 24.10.-i Key words: large cluster radioactivity, proximity potential, half-life
1
Introduction
In 1896, the natural radioactivity was first observed by Becquerel, then in 1908 Rutherford found α radioactivity in the first experimental observation of α decay for nuclei [1,2]. Gamow [3] and Gurdney and Condon [4,5] used quantum-tunneling model to successfully explain α radioactivity. From that Email address: [email protected] (G. L. Zhang
1,2 ).
time people began to explore the nuclear decay. In 1980, one pointed to the possibility of large cluster radioactivity [6] which is heavier than α particle. As a result, in 1984 14 C radioactivity was experimentally found for 232 Ra nucleus [7,8]. At present the emitted 14 C, 24,26 Ne, 28,30 Mg and 32,34 Si clusters from heavy nuclei were observed and the half-lives were measured. After the decay of these large cluster radioactivities from the parent nuclei, the residual daughter nuclei are almost close to the closed-shell nuclei in trans-lead region. It shows that the shell effect and paring effect play an important role for large cluster radioactivities. Theoretically, the different models were used to describe the cluster radioactivity, for example shell model, cluster model [9], liquid drop model [10] and double folding model [11,12,13] etc. These models can calculate the nuclear potential. Generally cluster tunneling model was used in which the penetration probability was calculated by using Wentzel-Kramers-Brillouin (WKB) approximation assuming the cluster radioactivity tunneling through the potential barrier between cluster radioactivity and the daughter nucleus in the parent nucleus. The cluster is assumed to form before it penetrates the barrier. Besides, proximity potential model is also an available way and has been generally used [14,15,16]. Shi and Swiatecki [17] were the first to use the proximity potential in an empirical manner and later on, Gupta et al [18,19] have been using the same, quite extensively for over a decade in the preformed cluster model (PCM). Further, the proximity potential has been subjected to several modifications, and Dutt and Puri [20,21] have been using different versions of proximity potentials for studying the fusion cross sections of different target-projectile combinations. Santhosh et al., have been using the proximity potential as the nuclear potential in the Coulomb and proximity potential model (CPPM) [22], and Coulomb and proximity potential model for deformed nuclei (CPPMDN) [23] for various studies on the cluster decay [24,25,26], alpha decay [27,28,29]. Therefore, by using it, the nuclear potential can be easily determined, especially between cluster radioactivity and different target nuclei. So it can provide some valuable supports for studying nuclear decay. In proximity potential model, the expression of the nuclear potential is mainly composed of two parts. One depends on the shape and geometry of two nuclei, and the other is the universal function φ(s) only related to the short separation distance s between two nuclei. The universal function is independent of the shapes of two nuclei and geometry of nuclear system. The idea of the universal function is fundamental advantage of proximity potential model. At present, there are different versions of proximity potentials. The main difference is for the different function φ(s0 ) and the different parameter values among them. It is necessary to explore the systematic behaviors of proximity potentials for the cluster radioactivity. In this paper we will select fourteen proximity potentials to calculate the nuclear potentials, then give the 2
half-lives of cluster radioactivity of nuclei. In comparison with the experimental data, we can find which proximity potential is fit for the calculation of the decay of large cluster radioactivity of nuclei and can explore the reason. This paper is organized as follows. In Section 2 the calculation process is introduced including several proximity potential models. In Section 3 the calculation results and discussion are displayed. The conclusion is given in Section 4.
2
The calculation process
Theoretically, the total interaction potential V(R) between projectile and target nuclei is given by V (R) = VN (R) + VC (R) +
l(l + 1)2 , 2μR2
(1)
where the first term VN (R) is the nuclear potential and the last term is the md centrifugal potential. μ = mmcc+m represents the reduced mass of cluster rad dioactivity and the daughter nucleus, mc and md denote the masses of cluster radioactivity and the daughter nucleus in the unit of M eV /c2 , respectively. l represents the angular momentum carried away by the emitted cluster radioactivity. VC (R) is Coulomb potential which is calculated by 2
⎧ ⎪ ⎨
VC (R) = Z1 Z2 e ⎪ ⎩
1 R 1 2Rc
3 − ( RRc )2
1/3
(R > Rc )
.
(2)
(R < Rc ) −1/3
Here, Rc = R1 + R2 , Ri = 1.28Ai − 0.76 + 0.8Ai (i=1,2) corresponds to the radii of cluster radioactivity and the daughter nuclei. For the nuclear potential VN (R) calculation fourteen different versions of proximity potentials are selected. The function of VN (R) of the proximity potential Prox77 (p77) and Prox88 (p88) and Prox00 (p00) is given by ¯ VN (R) = 4πγbRφ(s),
(3)
¯ is the mean curvature radius. The surhere φ(s) is the universal function and R 2 face energy coefficient γ = γ0 [1−ks I ] with the coefficient γ0 =0.9517 MeV/fm2 and the surface asymmetry constant ks =1.7826 [30] for Prox77, γ0 =1.2496 MeV/fm2 and ks =2.3 [31] for Prox88, γ0 =1.65 MeV/fm2 and ks =2.3 [32] for the modified proximity potential 1988 (mp88). I = NA−Z with N=N1 +N2 , Z=Z1 +Z2 , and A=A1 +A2 . Ni , Zi and Ai (i = 1,2) refer to the neutron, proton, and mass numbers of projectile and target nuclei, respectively. Their de3
tailed formulas and processes are shown in Ref. [14]. In the proximity potential Prox00 [15], the matter radius Ci has the form C i = ci +
Ni ti (i = 1, 2). Ai
(4)
Here the half-density radius ci of the charge distribution has the form ci = R00i (1 −
7b2 49b4 − )(i = 1, 2), 2 4 R00i 8R00i
(5)
with the nuclear charge radius 1/3
Ai − 2Zi )), Ai
(6)
](i = 1, 2),
(7)
R00i = 1.256Ai (1 − 0.202( and neutron skin of nucleus −1/3
1 gZi Ai 3 JIi − 12 ti = r0 [ −1/3 2 Q + 9 Ai 4
where r0 is 1.14 fm, the nuclear symmetric energy coefficient J=32.65 MeV and g=0.757895 MeV, the neutron skin stiffness coefficient Q=35.4 MeV. The surface energy coefficient γ is given by γ=
1 (t2 + t2 ) [18.63 − Q 1 2 2 ]. 2 4πr0 2r0
(8)
The universal function φ(ξ) is written as ⎧ ⎪ ⎨ −0.1353 + 5
φ(ξ) = ⎪
n=0 [cn /(n
+ 1)](2.5 − ξ)n+1
⎩ −0.0955exp( 2.75−ξ ) 0.7176
(0 < ξ ≤ 2.5) (ξ ≥ 2.5)
.
(9)
Here ξ = (r − C1 − C2 )/b and the width parameter b is close to unity. The different values of constant cn are: c0 =-0.1886, c1 =0.2628, c2 =-0.15216, c3 =¯ is cal0.04562, c4 =0.069136, and c5 =-0.011454. The mean curvature radius R culated by ¯ = C 1 C2 . R (10) C 1 + C2 For the generalized proximity potential 1977 (gp77) [33], the universal function is given by φ(ξ) =
⎧ ⎪ ⎨ −1.7817 + 0.927ξ + 0.0169ξ 2 − 0.05148ξ 3 ⎪ ⎩ −4.41exp(−
ξ ) 0.7176
4
(0 ≤ ξ ≤ 1.9475)
. (ξ > 1.9475) (11)
The nuclear radius parameter R0i is given by R0i = Ri − 1/3
b2 . Ri
(12)
−1/3
(i=1,2) corresponds to the radii of large Here Ri = 1.28Ai − 0.76 + 0.8Ai cluster radioactivity and the daughter nuclei. The definitions of ξ, the width b and the other parameters are same as that of Prox77. The parameter ξ=(RC1 -C2 )/b with Ci = Ri [1 − ( Rbi )2 ], Ri corresponds to the radii of large cluster radioactivity and the daughter nuclei, and is same as the above definition. For the modified proximity potential 2000 (mp00) [34], the nuclear charge radius R00i is given by 1/3
R00i = 1.2332Ai [1 +
2.348443 Ai − 2Zi − 0.151541 ]. Ai Ai
(13)
Here Ai denotes the masses of large cluster radioactivity and the daughter nucleus. The other parameters are same as that of Prox00. For Denisov potential (DP) [35], the nuclear potential VN is calculated by R1 R2 φ(r − R1 − R2 − 2.65) R1 + R2 A1 A2 3/2 × [1 + 0.003525139( + ) − 0.4113263(I1 + I2 )], A2 A1
VN (r) = − 1.989843
where Ii =
Ni −Zi . Ai
(14)
Ri is given by
Ri = Rip (1 −
3.413817 0.4Ai ) + 1.284589(Ii − ), 2 Rip Ai + 200
(15)
with
1.646 Ai − 2Zi − 0.191( )]. Ai Ai The universal function φ(s = r − R1 − R2 − 2.65) is given by 1/3
Rip = 1.24Ai [1 +
(16)
⎧ ⎪ ⎪ 1 − s/0.7881663 + 1.229218s2 − 0.2234277s3 − 0.1038769s4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − R1 R2 (0.1844935s2 + 0.07570101s3 ) ⎪ ⎪ ⎨ R1 +R2
φ(s) =
2
3
+(I1 + I2 )(0.04470645s + 0.0334687s ) (−5.65 ≤ s ≤ 0) ⎪ ⎪ ⎪ ⎪ ⎪ R2 s ⎪ (1 − s2 (0.05410106 RR11+R exp(− 1.76058 ) − 0.539542(I1 + I2 ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ s ⎩ exp(− ))) × exp(− s ) (s > 0) 2.424408
.
0.7881663
(17) 5
For Christensen and Wither 1976 (CW76) [36], the nuclear potential VN is given by R1 R2 VN (r) = −50 φ(r − R1 − R2 ), (18) R 1 + R2 1/3
where Ri = 1.233Ai
−1/3
− 0.978Ai
1 −R2 , and φ(s) = exp(− r−R0.63 ).
For Broglia and Winther 1991 (BW91) [31], the nuclear potential VN is written by V0 VN (r) = − . (19) 0 1 + exp( r−R ) 0.63 1/3
R2 γa with a=0.63 fm. R0 = R1 +R2 +0.29. Ri = 1.233Ai − Here V0 = 16π RR11+R 2 −1/3
0.98Ai
. The surface energy constant γ has the form γ = γ0 [1 − ks (
N1 − Z1 N2 − Z2 )( )], A1 A2
(20)
where the coefficients γ0 =0.95MeV/fm2 and ks =1.8. For Aage Withner (AW95) potential [37], only a= 1/3
Ri = 1.2Ai BW91.
1 −1/3
1.17(1 + 0.53(A1
−1/3
+ A2
))
,
(21)
− 0.09, R0 = R1 + R2 . The other parameters are same as that of
For Ngˆo 1980 potential (Ng80) [38], the nuclear potential VN is calculated ¯ ¯ form is same as that of Prox77. However, Ri = Ni Rni +Zi Rpi by VN =Rφ(s). R Ai 1/3
1/3
with Rpi = r0pi Ai and Rni = r0ni Ai . r0pi = 1.128 f m and r0ni = 1.1375 + 1.875 × 10−4 Ai . φ(s) is given by
φ(s) =
⎧ ⎪ ⎨ −33 + 5.4(s − s0 )2
(s < s0 )
⎪ ⎩ −33exp(− 1 (s − s )2 ) 0 5
(s ≥ s0 )
.
(22)
Here s0 =-1.6. For Bass 1973 (B73) [39,40], the nuclear potential VN is calculated by 1/3
VN (r) = − 1/3
1/3
das A1 A2 r − R12 exp(− ), R12 d
1/3
where R12 =1.07(A1 + A2 ), d=1.35 fm and as = 17M eV . 6
(23)
For Bass 1977 (B77) [41], the nuclear potential VN is given by VN (s) = − 1/3
R1 R2 φ(r − R1 − R2 ), R1 + R2
(24)
−1/3
where Ri =1.16Ai -1.39Ai . The universal function φ(s = r − R1 − R2 ) has the form s s (25) φ(s) = [Aexp( ) + Bexp( )]−1 , d1 d2 with A=0.030 MeV−1 fm, B=0.0061 MeV−1 fm, d1 =3.30 fm and d2 =0.65fm. For Bass 1980 (B80) [31], the form of nuclear potential VN is same as that of the above Bass77, only φ(s = r − R1 − R2 ) is given by φ(s) = [0.033exp( with Ri =Rs (1 −
0.98 ). Rs2
s −1 s ) + 0.007exp( )] , 3.5 0.65
1/3
−1/3
Rs =1.28Ai -0.76+0.8Ai
(26)
.
The half-life T1/2 of large cluster radioactivity of the parent nucleus can be calculated by πln2 T1/2 = [1 + 1/P ], (27) P0 where P0 is the preformation factor and P0 = 10−(0.4A1 −2) [9] where A1 denotes the mass number of large cluster radioactivity. P is the penetration probability which is calculated in the WKB approximation by
2 Rb 2μ(V (R) − Qc )dR , P = exp − Ra
(28)
where μ is the reduced mass of large cluster radioactivity and the daughter nucleus. Qc is the released energy when the parent nucleus decays the large cluster radioactivity, here we select the experimental released energies for the emission of large cluster radioactivity. Ra and Rb are the two turning points of WKB approximation integral determined from the equation V (Ra ) = Qc = V (Rb ).
3
(29)
Results and discussion
Here we select the emissions of large cluster radioactivities for even-even nuclei. Since the selected large cluster radioactivities are the even-even nuclei, as a result, the residual daughter nuclei are also the even-even nuclei. Finally, during the emission process of the even-even large cluster radioactivity for the 7
Table 1. The values of the obtained half-lives of the large cluster radioactivities through Prox77 potential (p77, the fourth column), Prox88 potential (p88, the fifth column), Prox00 potential (p00, the sixth column), generalized Prox77 potential (gp77, the seventh column), modified Prox88 potential (mp88) and modified Prox00 potential (mp00) which correspond to the eighth and the ninth columns, respectively. The second and third columns are for released energies [42] for emission of large cluster radioactivities and the experimental half-lives [43], respectively. parent
Qc
exp LogT1/2
p77 LogT1/2
p88 LogT1/2
p00 LogT1/2
gp77 LogT1/2
mp88 LogT1/2
mp00 LogT1/2
nuclei
(MeV)
(s)
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
33.05
11.22
12.96
10.24
13.63
11.19
8.50
9.43
224 Ra
30.54
15.86
17.99
15.21
18.39
15.92
13.28
14.18
226 Ra
28.20
21.34
23.29
20.58
23.48
20.98
18.40
19.24
228 Th
44.72
20.72
23.37
20.24
22.56
20.76
17.58
19.0
230 Th
57.76
24.61
26.61
23.85
24.55
23.70
19.92
21.38
232 Th
54.67
>29.20
31.81
29.63
29.54
28.24
25.04
26.33
232 Th
55.91
>29.20
31.38
27.95
28.33
28.04
24.10
25.83
230 U
61.39
19.57
22.15
19.92
20.72
19.22
15.93
17.71
232 U
62.31
21.08
22.30
18.80
20.44
19.23
15.56
17.24
232 U
74.32
>22.26
26.53
24.35
24.20
23.57
19.56
21.39
234 U
58.83
25.92
27.66
24.99
24.79
24.27
20.74
22.31
234 U
59.42
25.92
28.30
24.43
25.41
24.71
20.92
22.62
234 U
74.11
27.54
26.76
24.59
23.60
23.81
19.72
21.56
236 U
55.95
>25.90
32.49
29.92
29.38
28.86
25.50
26.90
236 U
56.70
>25.90
32.61
29.67
29.82
29.10
25.44
27.02
236 U
70.73
27.58
31.63
29.68
28.22
27.95
24.57
25.52
236 U
72.28
27.58
30.83
28.12
27.38
27.38
23.19
24.71
236 Pu
79.67
21.67
22.30
19.11
19.37
19.13
15.08
16.69
238 Pu
75.91
25.70
27.26
24.88
24.04
23.79
19.94
21.33
238 Pu
76.80
25.70
27.42
24.04
24.12
23.68
19.63
21.43
238 Pu
91.19
25.27
26.76
24.41
23.39
23.47
19.11
21.06
240 Pu
91.03
>25.52
28.15
25.20
23.76
24.46
19.91
21.58
242 Cm
96.51
23.15
24.65
20.71
20.47
21.19
16.26
18.25
8
Table 2. Same as Table 1, but for AW95 (the second column), CW76 (the third column), Ng80 (the fourth column), BW91 (the fifth column), DP (the sixth column) potentials and three types of Bass potentials which correspond to Bass73 (B73, the seventh column), Bass77 (B77, the eighth column) and Bass80 (B80, the last column), respectively. AW 95 LogT1/2
CW 76 LogT1/2
N g80 LogT1/2
BW 91 LogT1/2
DP LogT1/2
B73 LogT1/2
B77 LogT1/2
B80 LogT1/2
nuclei
(s)
(s)
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
9.35
6.84
12.62
9.16
9.69
11.89
10.82
10.02
224 Ra
14.25
11.37
17.71
14.19
14.43
16.82
15.76
14.98
226 Ra
19.50
16.29
23.25
19.87
19.63
22.07
21.03
20.25
228 Th
19.43
15.0
24.11
19.28
20.42
23.09
20.69
19.97
230 Th
23.23
16.65
27.95
21.84
24.33
26.88
23.70
23.09
232 Th
28.50
21.35
33.29
27.05
30.15
32.16
29.06
28.49
232 Th
27.38
20.65
32.31
26.17
28.68
31.44
27.81
27.18
230 U
19.35
12.83
23.78
17.84
20.62
22.73
20.04
19.44
232 U
18.48
12.55
23.56
17.47
19.66
22.50
19.23
18.53
232 U
23.59
15.82
28.57
21.53
25.63
28.02
24.09
23.57
234 U
24.22
17.36
28.84
22.88
25.53
27.92
24.74
24.12
234 U
24.14
17.65
29.25
23.07
25.17
28.34
24.68
24.0
234 U
23.72
15.96
28.71
21.68
25.73
28.16
24.18
23.60
236 U
29.20
21.74
33.74
27.69
30.60
32.81
29.70
29.12
236 U
28.82
21.84
34.01
27.78
30.12
33.04
29.37
28.74
236 U
28.76
20.29
33.68
26.51
30.95
33.13
29.30
28.82
236 U
27.01
19.25
32.66
25.36
29.24
31.99
27.58
26.99
236 Pu
18.66
11.73
24.16
17.15
20.58
23.49
19.40
18.74
238 Pu
23.76
16.12
29.27
22.13
25.92
28.64
24.64
24.04
238 Pu
23.50
15.94
28.65
21.91
25.63
28.50
24.01
23.35
238 Pu
23.49
15.01
29.14
21.25
26.21
28.99
24.24
23.65
240 Pu
24.43
15.89
30.34
22.23
26.70
29.99
24.75
24.09
242 Cm
20.36
12.59
26.48
18.40
22.86
26.32
21.01
20.26
parent
9
Table 3. The standard deviations σ of logarithm values of the calculated half lives for different proximity potentials in comparison with the experimental data. σ p77
σ p88
σ p00
σ gp77
σ mp88
σ mp00
σ AW 95
2.129
1.558
1.889
1.610
5.079
3.581
2.112
σ CW 76
σ N g80
σ BW 91
σ DP
σ B73
σ B77
σ B80
7.951
3.348
3.221
1.373
2.753
1.374
1.955
Fig. 1. (Color online) The distributions of total potentials for 238 Pu - 32 Si+206 Hg when the values of total potentials are more than the released energy Qc .
10
even-even parent nuclei, the angular momentum l carried away by large cluster radioactivity is equal to 0, which corresponds to zero for the centrifugal potential. The half-lives provided by the above mentioned proximity potentials are presented in Table 1 which includes Prox77, Prox88, Prox00, gp77, mp88, mp00 potentials, in Table2 which includes AW95, CW76, Ng80, BW91, DP, BASS73, BASS77, and BASS80 potentials. The standard deviations σ of logarithm values of the calculated half lives for these proximity potentials in comparison with the experimental data are shown in Table 3. In comparison with Table 1 and Table2 as well as the σ values of Table 3, we can see that when comparing the experimental data, the values of DP and BASS77 are closest to the experimental data, moreover, their σ values are also smallest. the values of Prox88, gp77 and BASS80 are closer to the experimental data. The values of other proximity potentials have the difference in the whole range. The calculation results of mp00 and BW91 are nearly close, but less about four order magnitude than the experimental data. Those of Ng80 and BASS73 are nearly close, but more about one order magnitude than the experimental data. In all calculation, those of CW76 are smallest and those of mp88 are smaller. The calculation results of Prox77 are more almost two order magnitudes than the experimental data, and are close to those of Ng80 and BASS73 in the range of light large cluster radioactivities, however, in the range of heavy large cluster radioactivities there is the difference among them, those of Prox77 are closer to the experimental data. Except that the several values of AW95 for 14 C and 20 O cluster radioactivities are less than those of Prox00, the other ones are close between them. In Fig. 1 the distributions of total potentials V are shown when V is more than the released energy Qc . Here we only select 238 Pu - 32 Si+206 Hg as a example, the others give the same distributions of V. We can see that the distance is more than 8 fm when V is more than Qc . Therefore, in refer to the figure 1 of Ref. [46], the large difference at small distances for different proximity potentials does not affect the values of half-lives of large cluster radioactivities. Only at the large distances there is a little difference to affect the values of half-lives. From Fig. 1, at more than 14 fm each potential gives the almost same values. However, at less than 13 fm there is the large differences among the different proximity potentials. Even some also show the different distribution. CW76 gives the thinnest distribution and mp88 give the thinner distribution, so their integrals of Eq. (28) are smallest and smaller, respectively. Then the calculated half-lives are smallest and smaller, respectively. So we can know the reason why the results of CW76 are smallest and that of mp88 is smaller in comparison with the experimental data in Table 1 and 2. BW91 also shows a thin distribution, so the values of its half-lives are also small. B73 and Ng80 show the wider distribution, then their integrals of Eq. (28) are larger, which corresponds to the larger values of half-lives. The distributions of B77 and DP are similar, then their half-lives are close. So in Table 1 and 2 the values of B77 and DP show the similarity. The distributions 11
Table 4. The values of the obtained half-lives of the large cluster radioactivities through BASS77 potential (B77, the third column) are compared with experimental data (the second column), those of liquid drop model (LDM, the fourth column) [10], ref1 (the fifth column) [43], ref2 (the sixth column) [44] and ref3 (the seventh column) [45]. exp LogT1/2
B77 LogT1/2
LDM LogT1/2
ref 1 LogT1/2
ref 2 LogT1/2
ref 3 LogT1/2
nuclei
(s)
(s)
(s)
(s)
(s)
(s)
222 Ra
11.22
10.82
10.59
10.44
9.59
11.81
224 Ra
15.86
15.76
16.59
15.17
14.34
16.55
226 Ra
21.34
21.03
22.51
20.39
19.37
21.54
228 Th
20.72
20.69
21.61
21.86
19.89
21.90
230 Th
24.61
23.70
25.451
26.13
23.39
24.99
232 Th
>29.20
29.06
28.76
31.19
26.66
28.09
232 Th
>29.20
27.81
29.72
30.76
27.70
29.91
230 U
19.57
20.04
21.40
21.77
19.48
20.26
232 U
21.08
19.23
19.99
21.89
19.35
20.75
232 U
>22.26
24.09
25.74
25.42
234 U
25.92
24.74
26.54
27.05
24.59
25.76
234 U
25.92
24.68
25.91
27.75
24.91
26.83
234 U
27.54
24.18
25.90
25.53
24.03
25.48
236 U
>25.90
29.70
32.18
31.91
30.33
236 U
>25.90
29.37
31.48
32.40
31.20
236 U
27.58
29.30
29.34
30.35
27.24
28.51
236 U
27.58
27.58
29.28
29.48
27.21
29.43
236 Pu
21.67
19.40
20.0
21.07
19.93
21.05
238 Pu
25.70
24.64
26.34
25.95
24.72
25.69
238 Pu
25.70
24.01
24.83
26.05
24.13
25.99
238 Pu
25.27
24.24
25.73
26.66
24.52
24.91
240 Pu
>25.52
24.75
26.08
27.71
25.34
27.25
242 Cm
23.15
21.01
21.11
24.23
22.14
23.70
parent
12
25.43
of p88 and B80 are nearly close to those of B77 and DP, then if the values of B77 and DP are close to the experimental data, those of p88 and B80 are also close to the experimental data. The distribution of p77 is thinner than those of B73 and Ng80, but is higher than those of B77 and DP, as a result, the calculated half-lives of p77 are larger than the experimental data and close to the experimental data with respect to those of B73 and Ng80. The same conclusion is also obtained from Table 1 and 2. Therefore, depending on Fig. 1 we can know the reason for the conclusion of Table 1 and 2. From the above calculation for even-even nuclei, we know that B77 and DP are most suitable for the calculation of half-life of large cluster radioactivity. In order to further check B77 and DP, in Table 4 we list the results of B77 in comparison with the experimental data, those of liquid drop model [10], ref1 [43], ref2 [44] and ref3 [45]. we can see that the values of B77 are similar as those of those models, and closer to that of ref2. With respect to the experimental data, sometimes the values of B77 are closer in comparison with those of those models. It is indicated that B77 can calculate the half-life of large cluster radioactivity very well. As a result, it verifies that B77 and DP can also calculate the half-lives of large cluster radioactivity for even-even nuclei.
4
Conclusion
The fourteen different proximity potentials are used to calculate the halflives of large cluster radioactivity for even -even nuclei with from light large cluster radioactivity to heavy large cluster radioactivity in comparison with the experimental data. It is shown that BASS77 (B77) potential and Denisov potential (DP) are suitable for calculating the half-life of large cluster radioactivity. The calculation results of Prox88 and BASS80 are nearly close to the experimental data. CW76 and mp88 give the smallest and smaller half-lives, respectively, and the remains give values with a difference with respect to the experimental data. It is the reason that the distributions of different proximity potentials at large distances with more than 8 fm lead to the different half-lives, the difference is not from the different distributions of proximity potentials at small distances where they have a large difference. This comparison provides a reference how to select the proximity potentials to calculate the half-life of large cluster radioactivity. In comparison with the results of the other models for even-even nuclei, the results of BASS77 are closer to the experimental data. Therefore, it further verifies that the BASS77 potential and Denisov potential can calculate the half-lives of large cluster radioactivity for even-even nuclei very well. Acknowledgement 13
This work is supported by the National Nature Science Foundation of China under Grant Nos. 11475013, 11035007, 11175011, 11375266 and State Key Laboratory of Software Development Environment (SKLSDE-2014ZX08) as well as by the Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences.
References [1] E. Rutherford, H. Geiger, Proc. R. Soc. A 81, 141 (1908). [2] E. Rutherford, T. Royds, Philos. Mag. 1, 281 (1908). [3] G. Gamow, Z. Phys. 51, 204 (1928). [4] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). [5] R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929). [6] A. Sandulescu, D. N. Poenaru, W. Greiner, Sov. J. Part. Nucl. 11, 528 (1980). [7] H. J. Rose, G. A. Jones, Nature 307, 245 (1984). [8] D. V. Aleksandro et al, JETP Lett. 40, 909 (1984). [9] Z. Z. Ren, C. Xu, Z. J. Wang, Phys. Rev. C 70, 034304 (2004). [10] G. Royer, R. Moustabchir, Nucl. Phys. A 683, 182 (2001). [11] L. Zheng, G. L. Zhang, J. C. Yang, W. W. Qu, Nucl. Phys. A 915, 70 (2013). [12] G. L. Zhang, X. Y. Le, Nucl. Phys. A 848, 292 (2010). [13] G. L. Zhang, H. Liu and X. Y. Le, Acta Physica Sinica 58, 2300 (2009). ´ [14] J. Blocki, J. Randrup, W. J. Swiatecki, C. F. Tsang, Ann. Phys. (NY) 105, 427 (1977). ´ [15] W. D. Myers, W. J. Swiatecki, Phys. Rev. C 62, 044610 (2000). [16] P. M¨ oller, J. R. Nix, Nucl. Phys. A 361, 117 (1981). [17] Y. J. Shi, W. J. Swiatecki, Nucl. Phys. A 438, 450 (1985). [18] S. S. Malik, Raj K. Gupt, Phys. Rev. C 39, 1992 (1989). [19] S. S. Malik, S. Singh, R. K. Puri, S. Kumar, R. K. Gupta, Pramana-J. Phys. 32, 419 (1989). [20] I. Dutt, R. K. Puri, Phys. Rev. C 81, 064608 (2010). [21] I. Dutt, R. K. Puri, Phys. Rev. C 81, 064609 (2010). [22] K. P. Santhosh, Antony Joseph, Pramana-J. Phys. 58, 611 (2002).
14
[23] K. P. Santhosh, Sabina Sahadevan, Jayesh George Joseph, Nucl. Phys. A 850, 34 (2011). [24] K. P. Santhosh, Sabina Sahadevan, B. Priyanka, M. S. Unnikrishnan, Nucl. Phys. A 882, 49 (2012). [25] K. P. Santhosh, B. Priyanka, M. S. Unnikrishnan, Nucl. Phys. A 889, 29 (2012). [26] K. P. Santhosh, B. Priyanka, Nucl. Phys. A 929, 20 (2014). [27] K. P. Santhosh, B. Priyanka, Int. J. Mod. Phys. E 22, 1350081 (2013). [28] K. P. Santhosh, Indu Sukumaran, B. Priyanka, Nucl. Phys. A 935, 28 (2015). [29] K. P. Santhosh, B. Priyanka, Nucl. Phys. A 940, 21 (2015). ´ [30] W. D. Myers, W. J. Swiatecki, Nucl. Phys. 81, 1 (1966). [31] W. Reisdorf, J. Phys. G: Nucl. Part. Phys. 20, 1297 (1994). [32] R. Kumar, M. K. Sharma, Phys. Rev. C 85, 054612 (2012). [33] K. P. Santhosh, B. Priyanka, Phys. Rev. C 87, 064611 (2013). [34] G. Royer, R. Rousseau, Eur. Phys. J. A 42, 541 (2009). [35] V. Y. Denisov, Phys. Lett. B 526, 315 (2002). [36] P. R. Christensen, A. Winther, Phys. Lett. B 65, 19 (1976). [37] A. Winther, Nucl. Phys. A 594, 203 (1995). [38] H. Ngˆ o, C. Ngˆo, Nucl. Phys. A 348, 140 (1980). [39] R. Bass, Phys. Lett. B 47, 139 (1973). [40] R. Bass, Nucl. Phys. A 231, 45 (1974). [41] R. Bass, Phys. Rev. Lett. 39, 265 (1977). [42] G. Audi et al., Chin. Phys. C 36, 1287 (2012); M. Wang et al., ibid. 36, 1603 (2012). [43] A. Bhagwat and R. J. Liotta, Phys. Rev. C 92, 044312 (2015). [44] D. N. Basu, Phys. Lett. B 566, 90 (2003). [45] L. Zheng, G. L. Zhang, J. C. Yang, W. W. Qu, Nucl. Phys. A 915, 90 (2013). [46] Y. J. Yao, G. L. Zhang, W. W. Qu, J. Q. Qian, Eur. Phys. J. A 51, 122 (2015).
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