-
The PCM is based on quantum mechanical fragmentation theory (QMFT) [50−53], where the cluster is supposed to be in a preformed state inside the parent nucleus. The model mainly operates in terms of the mass asymmetry coordinate and relative separation coordinate R. The probability of this preformation state of a cluster with respect to other probable clusters is called the preformation probability, which can be calculated by solving Schrödinger's wave equation in the η (mass asymmetry) co-ordinate and written as
{−ℏ22√Bηη∂∂η1√Bηη∂∂η+VR(η,R)}ψν(η)=Eνψν(η),
(1) where
ν=0,1,2,3,... refer to ground state(ν=0) and excited state(ν≠0) solutions. For the case of the ground state, the preformation probabilityPo is given asPo=|ψ[η(Ai)]|2√Bηη2ACN.
(2) The mass parameters
Bηη (η) are the classical hydrodynamical masses of Kröger and Scheid [53] given for homogeneous radial mass flow. The potential in Eq. (1) is given asVR(η,R)=2∑i=1B(Ai,Zi)+VC(R,Zi,βλi,θi)+VP(R,Ai,βλi,θi).
(3) Here,
B(Ai,Zi ) are the binding energies of decaying fragments and are taken from Ref. [49], and when not available, are taken from [54]. The second and third terms inVR represent the Coulomb and nuclear proximity potentials, respectively. For the case of deformed and oriented interacting nuclei, the expression for the Coulomb potential isVC(Zi,βλi,θi,αi)=Z1Z2e2R+3Z1Z2e2∑λ,i=1,212λ+1Rλi(αi)Rλ+1×Y(0)λ[βλi+47β2λiY(0)λ(θi)].
(4) Moreover, the expression for two interacting spherical nuclei in Eq. (4) can be written as
VC(R)=Z1Z2e2R . The nuclear proximity potential is given asVPij(s)=4π¯RγbΦ(s).
(5) A detailed explanation of the proximity potential can be found in Ref. [55]. Furthermore, the barrier penetration probability probability P is calculated using the WKB integral and written as
P=PaWiPb.
(6) The penetrability is calculated in three steps:
(i) The penetrability
Pa fromRa toRi Pa=exp[−2ℏ∫RiRa{2μ[V(R)−V(Ri)]}1/2dR],
(7) (ii) the inner de-excitation probability
Wi atRi (taken to be unity in reference to [56]),Wi=exp(−bEi),
(8) (iii) the penetrability
Pb fromRi toRb Pb=exp[−2ℏ∫RbRi{2μ[V(R)−Q]}1/2dR].
(9) This three step penetration process is shown in Fig. 3. It may be noted that R=
Ra is the first (inner) turning point, and R=Rb is the second (outer) turning point calculated from the conditionV(Rb )=Q-Value. This means that tunneling begins at R=Ra and terminates at R=Rb .Ra (Q) and Ra (ΔR ) are the points obtained using this relation (discussed in Sec. III). Here, theRa point plays an important role in the decay analysis and is calculated asRa =R1 +R2 +ΔR (=Rt +ΔR ), whereRt is the relative separation at the touching configuration. The fragment radii can be calculated viaRi(αi)=R0i[1+∑λβλiY(0)λ(αi)].
(10) Figure 6. (color online) Scattering potential for the decay of
236 Pu for cluster radioactivity using Ra (Q).The deformation parameters are taken from Ref. [54], and
R0i can be written asR0i=[1.28A1/3i−0.76+0.8Ai] fm,
(11) where i = 1, 2 are the radii of the fragments, and
βλi is taken to be zero for spherical choice of nuclei. Furthermore,ΔR is the relative separation distance between two fragments or clustersAi and is supposed to assimilate the neck formation effects. Hence, it is referred to as the neck length parameter [26]. The decay half-lifeT1/2 and decay constant λ are calculated asT1/2=ln2λ,λ=νoP0P.
(12) Here,
νo is the barrier assault frequency, calculated asνo=velocityR0=(2E2/μ)1/2R0,
(13) where
R0 is the radius of the parent nucleus, μ is the reduced mass, andE2 , the kinetic energy related to the Q-value, is given asE2=(A1/A)Q . -
This section represents the theoretical investigation of different ground state decay mechanisms (α, CR, HPR, and SF) using the PCM for nuclei with Z = 89−102. First, in Sec. III.A, the Q-value dependent first turning point
Ra (Q) is obtained, and an alpha decay analysis of the chosen set of isotopes is conducted using this relation. The overall trends of the preformation probability (Po ) and penetrability (P) are analyzed as functions of the mass of the parent or daughter nucleus. In Secs. III.B and III.C, the decay half-lives are estimated in reference to CR and HPR usingRa (Q) of the respective mode, and a comparison is made with the available literature. Finally, an SF analysis is conducted for the chosen set of nuclei in Sec. III.D. -
In this section, an alpha decay analysis of the heavy mass region (Z = 89−102, A = 207−258) is carried out using the PCM. As discussed earlier, the nuclei in this mass range are highly radioactive, and alpha decay is the prominent decay mode in general [57, 58]. Therefore, it will be interesting to explore the alpha decay mechanism and study its behavior with respect to an increase in the mass or neutron number of the parent nucleus. α disintegration from a radioactive parent nucleus is the emission of a stable α nucleus along with a complementary daughter. The shell closure effects associated with the parent or daughter nucleus are the main reason for radioactive disintegration. The final product of radioactive disintegration is basically the decaying fragments associated with nearby proton or neutron shell closure. Hence, shell effects significantly affect alpha decay or other radioactive decay mechanisms (CR, HPR, and SF), and the associated decay constant or decay half-life is influenced accordingly.
As a first step, the α decay half-lives of actinium isotopes (
207Ac ,209Ac ,211Ac ,213Ac ,215Ac ,217Ac , and221Ac ) are calculated using the PCM. Calculations are performed using the optimum choice of the neck length parameter (ΔR ). In reference to available data, a comparison between the calculated and experimental half-lives [59] is conducted and shown in Fig. 1(a). The calculated half-lives are in good agreement with the experimental data. It may be noted from the figure that the decay half-life for N = 128 (with daughter nucleusN2= 126) is the shortest among the isotopes. Hence, nuclei withN2= 126 are more stable due to the neutron shell closure effect. After the calculation of the decay half-lives, the Q-value of the decay channel, i.e., Qα , is plotted with respect to the neutron numberN2 of the daughter nucleus, as shown in Fig. 1(b). Note thatQα has the highest magnitude for the nucleus with N = 128 (N2 = 126) compared with its neighboring nuclei. As a result,Qα also demonstrates the significance of shell closure effects. As mentioned earlier, the PCM operates in terms of the mass asymmetry coordinate (η) and relative separation coordinate (R). Similarly, in a prior investigation of the PCM [60], it was found that the relative separation R and neck length parameter (ΔR) play key roles in the determination of different decay properties associated with radioactive nuclei.Figure 1. (color online) Variation in the (a) PCM calculated logarithmic half-lives for α decay and experimental data, as a function of neutron number (N
2 ) of the daughter nucleus, (b) Q-value for alpha decay, (c) variation in the first turning pointRa (=R1 +R2 +ΔR ).The calculations of the preformation probability and barrier penetrability are carried out at the first turning point R
a , which is the sum ofRt and ΔR, i.e., Ra =Rt +ΔR . Hence, to observe the role of the first turning point, theRa point is plotted as a function of the neutron numberN2 of the daughter nucleus, as shown in Fig. 1(c). It may be noted thatRa has the highest magnitude atN2 =126, in accordance with Qα and the decay half-life effect. This motivates us to obtain a Q-value dependentRa relation, i.e.,Ra (Q). Therefore, to constructRa (Q), two ranges of nuclei, Z = 89−93 and Z = 94−102, are chosen and a graph is made between the ratio (Ra /Rt ) against Qα , as shown in Fig. 2(a) and (b), whereRt is the sum of the radiiR1 andR2 . TheRa points here are calculated by fitting the decay half-lives with respect to the available experimental data [59, 61]. Hence, using Fig. 2(a) and (b), we can obtain the polynomialsRa (Q) for the two ranges of nuclei.Figure 2. (color online) Variation in
Ra/Rt with respect to Qα for the chosen set of nuclei (a) Z = 89−93, (b) Z = 94−102, where the fitted polynomial is represented by the solid line.Ra(Q)=Rt(A1+A2Qα−A3Q2α)89≤Z≤93,
(14) Ra(Q)=Rt(B1+B2Qα−B3Q2α)94≤Z≤102.
(15) Here,
A1 =1.10413,A2 =0.00768, andA3 =4.65x10−4 , and similar constants for heavier nuclei areB1 =1.08702,B2 =6.657x10−4 , andB3 =7.184x10−5 . Hence, we may use Eqs. (14) and (15) to calculate theRa point for lighter and heavier nuclei, respectively. Validation of the aboveRa (Q) relations (Eqs. (14) and (15) is performed by calculating the alpha decay half-lives of the same set of lighter and heavier nuclei. A comparison of both types ofRa points, i.e., Ra(ΔR) andRa(Q) , is shown in Table 1. A comparison of the calculated decay half lives is shown with the available experimental data.Parent nucleus Decay channel Ra(ΔR)/fm Ra(Q)/fm logT1/2[Ra(fm)(ΔR)]/sec logT1/2[Ra(Q)]/sec logT1/2/s(Exp.) Z = 89−93 212 Ac4 He+208 Fr9.73 9.89 0.00 −0.03 −0.02 219 Ac4 He+215 Fr10.01 10.01 −3.93 −3.93 −4.92 222 Ac4 He+218 Fr10.05 10.02 2.28 2.23 0.73 213 Th4 He+209 Ra9.94 9.91 −0.80 −0.84 −0.85 220 Th4 He+216 Ra10.02 10.02 −4.91 −4.91 −5.01 222 Th4 He+218 Ra10.05 10.03 −2.11 −2.12 −2.55 221 Pa4 He+217 Ac10.04 10.05 −4.99 −5.00 −5.23 224 Pa4 He+220 Ac10.07 10.05 1.03 1.00 −0.10 225 Pa4 He+221 Ac10.08 10.06 1.07 1.03 0.38 223 U4 He+219 Th10.06 10.06 −2.78 −2.78 −4.74 224 U4 He+220 Th10.07 10.07 −3.44 −3.44 −3.05 225 U4 He+221 Th10.08 10.07 0.24 0.21 −1.00 225 Np4 He+221 Pa10.08 10.08 −3.20 −3.21 > -5.7227 Np4 He+223 Pa10.11 10.09 −0.15 −0.14 −0.29 230 Np4 He+226 Pa10.14 10.12 5.22 5.24 ≤ 3.96Z = 94−102 228 Pu4 He+224 U9.72 9.70 −0.01 0.16 −0.70 230 Pu4 He+226 U9.74 9.73 2.75 2.80 ≥ 2.30232 Pu4 He+228 U9.76 9.75 4.85 4.88 4.20 234 Pu4 He+230 U9.78 9.78 6.71 6.74 5.72 235 Pu4 He+231 U9.80 9.79 8.91 8.91 7.75 236 Pu4 He+232 U9.81 9.80 8.93 8.93 8.11 239 Pu4 He+235 U9.84 9.84 12.63 12.63 12.00 232 Am4 He+228 Np9.76 9.75 3.72 3.77 3.59 237 Am4 He+233 Np9.82 9.82 7.70 7.70 7.18 240 Cm4 He+236 Pu9.85 9.85 7.24 7.24 6.52 242 Cm4 He+238 Pu9.87 9.88 8.40 8.41 7.28 243 Cm4 He+239 Pu9.88 9.89 9.21 9.21 8.96 244 Cm4 He+240 Pu9.89 9.90 9.89 9.89 8.88 250 Cf4 He+246 Cm9.96 9.97 9.87 9.87 8.69 252 Cf4 He+248 Cm9.98 9.99 9.71 9.71 8.00 251 Es4 He+247 Bk10.02 9.98 9.33 9.32 7.48 253 Es4 He+249 Bk9.94 10.00 7.73 7.63 6.30 254 Es4 He+250 Bk10.05 10.01 9.12 9.13 7.38 250 Fm4 He+246 Cf9.96 9.97 3.96 3.90 < 3.30252 Fm4 He+248 Cf9.98 9.99 5.73 5.71 5.04 255 Md4 He+251 Es10.01 10.02 3.28 3.24 3.21 260 Md4 He+256 Es10.06 10.08 8.64 8.62 > 6.98254 No4 He+250 Fm10.00 10.01 2.28 2.28 1.86 To observe the relevance of the polynomials in Eqs. (14) and (15), in the context of alpha decay within the PCM, the standard rms deviation is calculated using the equation
σ=√∑[log10(Ti/Texpt.)]2/n−1,
(16) where n is the total number of parent nuclei under consideration. The standard deviation of the decay half-lives using Eqs. (14) and (15) is 1.14 and 1.05, respectively, whereas we obtain 1.15 and 1.05 for the optimized
ΔR values, respectively. Furthermore, the scattering potential of236 Pu is plotted with respect to the internuclear separation distance R by taking theRa point from the Q dependent relation (Eq. (15)) and that shown in Fig. 3. We also compare the Ra point by taking the optimum choice ofΔR and obtain a similar result. Hence, we may use these Q-dependent relations to calculate the decay half-lives of radioactive nuclei because they provide a good approximation to experimental data. After this, an investigation of several nuclei is carried out from Z = 82−93 and Z = 94−102, and a comparison is conducted between the half-lives calculated usingRa (Q) and the experimental values, as shown in Table 2 and Table 3. As shown in these tables, theRa (Q) calculated decay half-lives are in good agreement with the experimental data, except for a few cases. The difference between the experimental alpha decay half-lives and the calculated values may arise in the case of certain nuclei, particularly neutron-deficient and neutron-rich isotopes. These isotopes may exhibit deviations from the predicted decay half-lives owing to specific nuclear structure effects and underlying physics. Moreover, these results are compared with the observations of other theoretical models, i.e., the ASAFM [62] and effective liquid drop model (ELDM) [63], which further reveals a reasonable agreement. Hence, Eqs. (14) and (15) can be used to calculate the decay half-lives of radioactive nuclei. After calculating the decay half-lives, the overall trend of the Q-value, penetration probability, preformation probability, and decay half-lives is studied, where all of these values are calculated using theRa (Q) relation. In Fig. 4(a), the calculated Qα using theRa(Q) relation is plotted against the neutron number of the parent nuclei. At N = 128 (N2= 126), Qα is maximum owing to the shell closure effects of the daughter nucleus. Such nuclei are relatively unstable; therefore, they have higher Q-values and the decay rate is relatively fast. Furthermore, as one approaches from N = 126 to N = 142, i.e., away from the shell closure, Qα starts decreasing, and this trend is compared with Fig. 1(a) of Ref. [64], which is shown in the inset of Fig. 4 (a). Moreover, in Fig. 4(b), Qα decreases sharply around A = 217−235. This is due to the effect of the neutron shell closure of the daughter nuclei for A = 217−219 (N = 128,N2 = 126). It begins to decrease as we move away from the shell closure. Note that there are few stable nuclei (against alpha decay) at mass number A≥ 230 (here, the q-value is approximately 4 MeV, and the respective half-life is very large, that is, 1019 ). Beyond this, Qα starts increasing, which may be due to the presence of the next shell closure afterN2= 126.Parent nucleus Decay channel Qα /MeV Ra/fm (Poly.) logT1/2/s (Poly.) logT1/2/s ASAFM [62] logT1/2/s ELDM [63] logT1/2/s (Exp) [59] 207 Ac4 He+203 Fr7.84 9.82 −3.62 3.30 −1.60 −1.66 208 Ac4 He+204 Fr7.72 9.84 −1.18 4.50 −1.16 −1.02 209 Ac4 He+205 Fr7.72 9.85 −2.78 − −1.18 −1.00 210 Ac4 He+206 Fr7.60 9.86 −0.41 − −0.80 −0.44 211 Ac4 He+207 Fr7.62 9.88 −2.06 − −0.88 −0.60 213 Ac4 He+209 Fr7.49 9.90 −1.15 − −0.48 −0.10 215 Ac4 He+211 Fr7.74 9.93 −1.33 − −1.36 −0.77 217 Ac4 He+213 Fr9.83 10.02 −6.94 −6.90 −6.99 −7.16 218 Ac4 He+214 Fr9.37 10.02 −4.19 −6.50 −5.96 −7.16 210 Th4 He+206 Ra8.06 10.01 −3.91 − −1.85 −5.96 212 Th4 He+208 Ra7.95 9.89 −3.41 −1.20 −1.57 −1.52 214 Th4 He+210 Ra7.82 9.92 −2.10 −0.80 −1.19 −1.00 216 Th4 He+212 Ra8.07 9.95 −2.73 −1.50 −2.00 −1.55 218 Th4 He+214 Ra9.85 10.03 −7.37 − −6.71 −6.96 219 Th4 He+215 Ra9.51 10.03 −4.66 −5.96 − −5.96 224 Th4 He+220 Ra7.29 10.05 1.08 0.00 0.43 0.11 226 Th4 He+222 Ra6.45 10.07 4.85 3.30 3.79 3.38 228 Th4 He+224 Ra5.52 10.12 9.53 7.90 8.38 7.92 230 Th4 He+226 Np4.76 10.17 14.13 12.50 13.05 12.49 232 Th4 He+228 Ra4.08 10.22 19.41 − 18.45 17.76 213 Pa4 He+209 Ac8.39 9.92 −4.22 − −2.53 −2.28 214 Pa4 He+210 Ac8.27 9.93 −2.04 − −2.19 ≥ −1.77215 Pa4 He+211 Ac8.24 9.94 −3.48 −1.80 −2.12 −1.82 216 Pa4 He+212 Ac8.09 9.95 −1.36 1.30 −1.68 −0.54 217 Pa4 He+213 Ac8.48 9.97 −3.43 −2.30 −2.89 −2.47 218 Pa4 He+214 Ac9.80 10.03 −5.32 −3.70 −6.25 −3.74 219 Pa4 He+215 Ac10.09 10.06 −7.59 −6.90 −6.92 −7.28 220 Pa4 He+216 Ac9.65 10.05 −4.65 −5.50 −6.37 −6.11 227 Pa4 He+223 Ac6.58 10.09 4.67 3.70 3.68 3.73 218 U4 He+214 Th8.77 9.99 −4.54 − −3.35 −2.82 222 U4 He+218 Th9.43 10.07 −5.97 − −5.27 −6.00 226 U4 He+222 Th7.70 10.08 −0.02 −0.30 −0.21 −0.70 228 U4 He+224 Th6.80 10.10 3.73 2.90 3.17 < 2.76229 U4 He+225 Th6.47 10.11 6.34 4.30 4.57 4.43 230 U4 He+226 Th5.99 10.13 7.83 6.40 6.85 6.43 233 U4 He+229 Th4.90 10.20 14.63 12.70 13.18 12.77 226 Np4 He+222 Pa8.19 10.08 0.03 − −1.37 −1.15 228 Np4 He+224 Pa7.30 10.10 2.86 − 1.23 2.18 229 Np4 He+225 Pa7.01 10.11 3.38 2.30 2.78 < 2.66231 Np4 He+227 Pa6.36 10.14 6.45 5.40 5.52 5.18 Table 2. PCM-calculated alpha decay half-lives LogT
1/2 of 31 nuclei for Z = 89−93. Using Eq. (14), the results are compared with the available experimental data.Parent nucleus Decay channel Qα/MeV Ra/fm (Poly.) logT1/2/s (Poly.) logT1/2/s ASAFM [62] logT1/2/s ELDM [63] logT1/2/s (Exp.) [59] 233 Pu4 He+229 U6.41 9.77 7.00 5.80 5.72 6.00 238 Pu4 He+234 U5.59 9.83 10.73 9.50 9.88 9.59 240 Pu4 He+236 U5.25 9.85 12.42 11.40 11.88 11.45 242 Pu4 He+238 U4.98 9.88 14.08 − 13.64 13.18 238 Cm4 He+234 Pu6.67 9.83 5.63 4.90 5.64 ≥ 4.93246 Cm4 He+242 Pu5.47 9.92 12.34 − 11.50 11.26 248 Cm4 He+244 Pu5.16 9.95 14.48 − 13.46 13.15 250 Cm4 He+246 Pu5.16 9.97 14.68 − 13.38 12.45 240 Cf4 He+236 Cm7.71 9.85 2.30 − 2.00 1.81 242 Cf4 He+238 Cm7.51 9.87 3.24 2.40 2.70 2.41 244 Cf4 He+240 Cm7.32 9.90 3.96 3.10 3.43 3.18 246 Cf4 He+242 Cm6.86 9.92 5.97 − 5.34 5.20 248 Cf4 He+244 Cm6.36 9.95 8.52 − 7.63 7.54 251 Cf4 He+247 Cm6.17 9.98 10.68 10.90 8.52 10.45 254 Cf4 He+250 Cm5.92 10.02 11.27 9.30 9.79 9.30 256 Cf4 He+252 Cm5.56 10.04 13.48 − 11.90 10.87 252 Es4 He+248 Bk6.78 9.99 9.67 7.60 5.99 7.61 245 Fm4 He+241 Cf8.43 9.99 1.73 − 0.28 0.62 246 Fm4 He+242 Cf8.37 9.91 0.80 − 0.50 0.08 248 Fm4 He+244 Cf7.99 9.94 2.16 − 1.70 1.65 254 Fm4 He+250 Cf7.30 10.01 5.61 4.10 4.22 4.15 256 Fm4 He+252 Cf7.02 10.04 6.79 5.10 5.36 5.15 247 Md4 He+243 Es8.76 9.92 −0.10 − −0.86 0.46 256 Md4 He+252 Es7.73 10.03 5.28 3.90 2.36 3.66 257 Md4 He+253 Es7.55 10.05 5.02 3.70 3.60 4.30 258 Md4 He+254 Es7.27 10.06 7.96 5.80 4.73 6.65 259 Md4 He+255 Es7.10 10.07 7.10 − 5.39 > 5.28250 No4 He+246 Fm8.94 9.96 −0.32 − −0.65 −0.30 252 No4 He+248 Fm8.54 9.98 1.00 − 0.58 0.62 255 No4 He+251 Fm8.42 10.02 1.33 2.80 0.88 2.27 256 No4 He+252 Fm8.58 10.03 1.58 0.50 0.41 0.56 258 No4 He+254 Fm8.14 10.05 3.10 1.70 1.83 2.08 Table 3. PCM-calculated alpha decay half-lives LogT
1/2 of 32 nuclei for Z = 94−102. Using Eq. (15) the results are compared with the available experimental data.Figure 4. (color online) (a) Calculated Q
α as a function of the neutron number of the parent nucleus, the inset of the figure is taken from Ref. [64]. The scales of the x axis and y-axis represent the same quantities for both the graphs (i.e., Qα and neutron number (N)) of the parent nucleus. (b) Calculated Qα as a function of mass number, (c) penetration probability P as a function of A2 , and (d) decay decay half-lives (logT1/2) as a function of A. All the above values are calculated for α decay data using Ra (Q) relations (Eqs. (14) and (15)).The penetration probability P is calculated and shown in Fig. 4(c). The penetration probability (the three step process is explained in Eq. (6)) and role of Q
α in penetrability can be visualized from Fig. 3. The penetration probability is maximum forA2= 213−215 (N2= 126); therefore, it is easier for the alpha particle to escape across these daughter nuclei. This is due to the neutron shell closure at N = 126. After the analysis ofQα and the penetrability, the PCM calculated alpha decay half-lives are plotted with respect to mass number A in Fig. 4(d). The decay half-lives depend on the decay constant, which further depend on the assault frequency (nearly constant), penetration probability, and preformation probability. Here, the penetration probability starts decreasing until A = 217 and then sharply increases between A = 217−235 because of the influence of shell effects, as mentioned earlier. In the next step, the preformation probability of the nuclei is calculated using Eqs. (14) and (15) and shown in Fig. 5. From this figure, we find that the preformation factor decreases with an increase in the mass number of the parent nuclei. In this study, the calculations are performed using collective clusterization, which treats all decay processes (such as α, cluster, heavy mass fragment, and fission decay) on equal footing. Therefore, the preformation probabilities of all binary fragments are obtained by solving the Schrödinger equation in the mass asymmetry coordinate η. Because the preformation probability is calculated in preview of the collective clusterization process, P0 of one decay channel is affected by the relative contribution of the remaining fragments in the binary exit channel. Hence, alpha preformation is affected by other competing decay modes. A similar analysis of preformation probability using the GLDM for α decay is presented in Ref. [65]. -
In the previous section, alpha decay of different nuclei is investigated using the
Ra (Q) relation, and the calculated half-lives are found to be in good agreement with the available data. In the present section, the cluster emission is studied for the chosen set of isotopes using theRa (Q) relation. Previously, in Refs. [66, 67], cluster decay analysis was performed using older mass tables or proximity potentials within the framework of the PCM. It is important to mention that the current calculations are performed using the mass table of Audi et al. 2017 [49]. The decay half-lives are calculated for the experimentally observed clusters via the optimization of the neck length parameterΔR , and the appropriate value of the Ra point is fixed. The obtained Ra value is further used to get the Ra (Q) relation for the spherical and deformed choice of the decaying fragments using the same procedure as mentioned in the previous section for the α decay case. The Ra (Q) relations for the CR process read asRa(Q)=Rt(C1−C2Qcluster+C3Q2cluster),
(17) Ra(Q)=Rt(D1−D2Qcluster+D3Q2cluster),
(18) where Eq. (17) is for the spherical choice of the decaying fragments, and Eq. (18) is for the deformed choice of the decaying fragments. Q
cluster is the q-value of the decay channel. In Eq. (17),C1 =1.05502,C2 =1.17016×10−4 , andC3 =3.424×10−7 . Similarly, the constants for Eq. (18) areD1 =1.0573,D2 =2.07598×10−4 , andD3 =8.8151×10−7 . Eqs. (17) and (18) can be employed to give the decay half-lives in a cluster decay mechanism. The validity of these equations can be tested via the comparison of theRa (Q) calculated results with the corresponding experimental data. The comparison of theRa (Q) calculated decay half-lives and the experimental results for the spherical and deformed choice of the decaying fragments is shown in Table 4. In this table,Ra is obtained using theRa (Q) relation. It may be noted from the table that the calculatedRa points and corresponding decay half-lives are in good agreement with experimental data. We find that the deformed choice of the decaying fragment offers relatively better agreement with the experimental data compared to the spherical case. In addition, to check the validity of the polynomials used, the standard deviation (SD) is calculated with respect to the experimental results. For the spherical choice of the decaying fragments, the SDs are 4.20 and 4.21 forRa andRa (Q), respectively. An SD of more than 3 with data is considered not good. Hence, the spherical clusters do not impart good agreement. After analyzing the spherical choice, the SDs are calculated for the deformed choice of clusters, which are 2.75 and 2.77 forRa andRa (Q), respectively. Furthermore, the calculated half-lives are compared with the data of theoretical models, i.e., the ASAFM [62] and mean-field approximation in the Hartree-Fock-Bogoliubov (HFB) [40], which shows a reasonable agreement with the predicted data. Similar to the alpha decay analysis, a scattering plot for the CR of236 Pu is plotted with respect to internuclear separation distance (R) by taking the first turning point Ra from Eq.(18), as shown in Fig. 6. Then, the first turning point is compared by taking the optimum choice ofΔR .Parent nucleus Decay channel Q cluster /MeVPCM Spherical Deformed ASAFM log T1/2 /sHFB log T1/2 /s [62]Expt. log T1/2 /s [40]Ra /fm (Poly.)logT 1/2/s (Poly.)Ra /fm (Poly.)logT 1/2 /s (Poly.)223 Ac14 C +209 Bi33.06 10.12 17.06 10.16 14.77 12.7 − 12.60 223 Ac15 N +208 Pb39.47 10.17 18.85 10.17 16.91 14.1 − > 14.76225 Ac14 C +211 Bi30.47 10.15 21.54 10.19 20.77 17.8 − 17.16 226 Th14 C +212 Po30.54 10.16 22.29 10.18 21.26 17.7 − > 15.3226 Th18 O +208 Pb45.72 10.35 22.43 10.37 21.70 18.0 17.31 > 15.3228 Th20 O +208 Pb44.72 10.47 23.54 10.48 22.79 21.9 19.53 20.87 230 Th24 Ne +206 Hg57.76 10.64 26.90 10.70 25.96 25.2 25.08 24.61 231 Pa23 F +208 Pb51.88 10.62 25.25 10.86 24.29 25.9 − 26.02 231 Pa24 Ne +207 Tl60.40 10.65 23.37 10.71 22.65 23.3 − 23.23 230 U22 Ne +208 Pb61.38 10.56 23.90 11.22 20.70 20.4 20.49 19.57 230 U24 Ne +206 Pb61.81 10.64 23.09 10.69 22.28 22.4 − > 18.2232 U24 Ne +208 Pb62.30 10.66 21.09 10.72 20.28 20.8 23.35 21.05 232 U28 Mg +204 Hg74.31 10.79 26.01 11.52 21.91 − − > 22.26233 U24 Ne +209 Pb60.48 10.68 25.77 10.76 24.76 25.2 − 24.84 233 U25 Ne +208 Pb60.70 10.71 25.08 10.82 24.15 25.7 − 24.84 233 U28 Mg +205 Hg74.22 10.81 27.50 11.42 24.26 27.4 − > 27.59234 U24 Ne +210 Pb58.82 10.69 28.26 10.74 27.14 26.1 27.24 25.92 234 U25 Ne +209 Pb57.79 10.73 29.49 10.86 28.31 28.8 − 25.92 234 U26 Ne +208 Pb59.41 10.76 26.39 11.02 24.93 27.0 28.02 25.92 234 U28 Mg +206 Hg74.11 10.82 26.81 11.42 23.35 25.9 25.85 25.54 235 U24 Ne +211 Pb57.36 10.70 32.16 10.76 30.24 − − 27.62 235 U25 Mg +210 Pb57.68 10.74 45.97 10.84 29.20 − − 27.62 235 U28 Mg +207 Hg72.42 10.83 30.66 11.46 26.32 − − > 28.09235 U29 Mg +206 Hg72.46 10.86 30.38 11.37 26.61 − − > 28.09236 U28 Mg +208 Hg70.73 10.85 32.37 11.45 28.45 − 33.68 27.58 236 U30 Mg +206 Hg72.27 10.91 28.78 11.17 26.46 29.8 33.1 27.58 238 U34 Si +204 Pt85.18 11.04 30.63 11.02 29.85 − − 29.04 237 Np30 Mg +207 Tl74.79 10.92 26.41 11.18 23.88 28.3 − > 26.93236 Pu28 Mg +208 Pb79.66 10.84 20.36 11.45 16.19 21.1 20.13 21.67 238 Pu28 Mg +210 Pb75.91 10.87 26.94 11.48 22.65 26.2 29.42 25.70 238 Pu30 Mg +208 Pb76.79 10.93 24.48 11.19 22.02 26.2 29.52 25.70 238 Pu32 Si +206 Hg91.18 10.98 25.25 11.12 22.84 26.1 28.23 25.27 240 Pu34 Si +206 Hg91.02 11.06 24.57 11.05 22.73 27.4 26.96 > 25.52241 Am34 Si +207 Tl93.92 11.07 21.80 11.06 20.16 25.8 − > 22.71242 Cm34 Si +208 Pb96.50 11.08 19.44 11.07 17.98 23.5 24.9 23.24 252 Cf46 Ar +206 Hg126.75 11.48 16.41 11.66 14.98 26.5 − > 15.89252 Cf48 Ca +204 Pt138.17 11.52 18.61 11.52 18.27 − − > 15.89252 Cf50 Ca +202 Pt138.31 11.56 17.73 11.71 15.95 − − > 15.89Table 4. Comparison of decay half-lives of various experimentally detected clusters with those calculated using prox 1977 with in the PCM for spherical and deformed choices of nuclei. The corresponding Q
cluster are calculated using the binding energies of Audi et al. [49] given for each decay.This comparison give similar results. Therefore, it is clear from the above calculations that the SD is reduced for the deformed choice of fragments. Therefore, the polynomial in Eq. (18) gives a reasonable approximation and can be used to calculate the decay half-lives of other sets of nuclei belonging to the heavy/superheavy region.
-
Following the analysis of the α and cluster decays, HPR is studied in this section. In a previous study [4, 6], one of us and collaborators predicted the decay half-lives of the heavy clusters emitted by certain heavy nuclei. In this study, we extend the previous work to the full range of actinide series. The HPR results are shown in Table 5. Here, the most probable HPR fragments are identified using the minima in the fragmentation potential. The calculated Q-values of the decay channel, i.e.,
QHPR , and the decay half-lives are also shown in Table 5.Ra (Q) for the HPR case is given asParent nucleus Decay channel Q HPR /MeVR a (Poly.)log T1/2 /s Poly.232 U82 Ge+150 Nd173.70 10.92 30.32 234 U82 Ge+152 Nd173.71 10.95 30.54 236 U82 Ge+154 Nd173.68 10.98 29.40 238 U82 Ge+156 Nd173.19 11.01 30.07 240 Pu82 Ge+158 Sm180.79 11.04 28.07 242 Pu82 Ge+160 Sm180.36 11.07 28.19 244 Pu82 Ge+162 Sm179.75 11.10 28.32 246 Pu82 Ge+164 Sm178.90 11.12 28.93 238 Cm84 Se+154 Sm198.75 11.03 23.05 240 Cm84 Se+156 Sm197.03 11.05 24.00 242 Cm84 Se+158 Sm196.00 11.08 25.27 244 Cm82 Ge+162 Gd188.14 11.10 25.85 246 Cm82 Ge+164 Gd187.80 11.12 26.06 248 Cm82 Ge+166 Gd187.33 11.15 25.93 250 Cm82 Ge+168 Gd186.76 11.18 26.37 244 Cf84 Se+160 Gd205.36 11.11 21.37 246 Cf84 Se+162 Bk204.30 11.14 22.26 248 Cf84 Se+164 Gd202.95 11.17 23.55 250 Cf82 Ge+168 Dy195.14 11.18 24.07 252 Cf82 Ge+170 Dy195.10 11.20 23.68 254 Cf82 Ge+172 Dy194.76 11.23 22.71 Table 5. PCM calculated half-lives for the heavy particle radioactivity of Z=89−102 parent systems. The choice of the outgoing fragment is based on the most probable fragment with the highest preformation probability. Note that in each case, the spherical choice of fragments is considered, and calculations with
l=0ℏ andT=0 MeV with only the optimizedΔR as −0.25 is set of all isotopes.Ra=Rt(E1+E2QHPR−E3Q2HPR).
(19) Here,
E1 =0.9484,E2 =3.055×10−4 , andE3 =7.867×10−7 The calculations of the first turning point (Ra ) by taking the optimum value of the parameterΔR and using Eq. (19) are shown in Columns 4 and 5, respectively. The half lives are represented in the last two columns. The predicted fragments are isotopes of Ge because these nuclei exhibit shell effects around N = 50. These heavy clusters are not yet experimentally detected; therefore, it will be of future interest to verify the validity of such predictions. -
In the previous sections, an analysis of α emission, CR, and HPR is conducted using their respective
Ra (Q) relations. It is concluded that the first turning point plays a crucial role in exploring the dynamics of these decay modes. It is known that alpha decay and SF are the most observed decay modes. In this section, a comprehensive analysis is performed among the α decay and SF modes using the PCM. It is relevant to note that within the PCM approach, all the decay modes are explored on an equal footing. The most probable decaying fragments of a radioactive nucleus can be identified using the fragmentation structure and preformation probability. In Fig. 7, the fragmentation structure and preformation probabilityP0 are compared for the230 Th and256 No nuclei (extreme nuclei of the chosen mass region). The trend of the fragmentation structure for the230 Th and256 No nuclei (see Fig. 7(a) and (c)) remains similar for both modes (α and SF), but changes in the magnitude of the fragmentation potential are observed. The minima in the fragmentation potential suggest the most probable decay fragments. The identified fission fragments using the minima of the fragmentation potential for alpha decay and SF for the230 Th and256 No nuclei are96 Sr+134 Te and124 Sn+132 Te, respectively. After the fragmentation structure, the preformation probability is calculated and plotted with respect to fragment massAi and shown in Fig. 7(b) and (d). The trend of the preformation probability is also similar for the230 Th and256 No nuclei in the α and SF decay modes. The fission region is found to change from the asymmetric to symmetric mode when going from the230 Th to256 No nucleus. It will be interesting to explore the transition point where the mass distribution changes from the asymmetric to symmetric mode.Figure 7. (color online) Fragmentation potential and preformation probability of nuclei for (a), (b)
230 Th and (c), (d)256 No for alpha and spontaneous fission by taking the emitting fragments as spheres only.To gain a better insight into this transition, we take two more isotopes (
252 Cf and256 Fm) from the chosen range of nuclei, and the preformation probability is plotted as a function of fragment massAi (i = 1, 2), as shown in Fig. 8. For the SF analysis, the decay half-lives are calculated for the230 Th,252 Cf,256 Fm, and256 No nuclei first for the spherical case. Then, the deformation effects are included up to quadrupole (β2 ) deformations with hot optimum orientations of the decay fragments. A comparison between the calculated decay half-lives and the experimental data [68] is shown in Table 6. The calculated results are in reasonable agreement with the experimental data, except for the spherical choice fragments for the256 Fm nucleus because the calculated half-lives are found to underestimate the experimental SF data. It is relevant to note that the half-life increases with decreasingΔR value. However, in the case of256 Fm,ΔR cannot be reduced further as V(Ra ) becomes less than Qsf (Q-value of the decaying channel). On the other hand, we have a reasonable scope of theΔR extension for256 No because the barrier characteristics become modified as Coulomb repulsion increases with increasing atomic number. Interestingly, the barrier height can be altered for the256 Fm nucleus by adding deformations. The calculated half life of256 Fm becomes modified by taking a hot-compact choice of fragments, as shown in Table 6.Figure 8. (color online) Preformation yield
Po as a function of fission fragments (a)230 Th, (b)252 Cf, (c)256 Fm, and (d)256 No for spherical fragments as well asβ2 deformed hot compact configurations.Parent nucleus Decay channel Ra /fmlog10TSF1/2 /s (PCM)log10TSF1/2 /s (Exp.)Spherical 230 Th96 Sr+134 Te10.82 24.25 24.79 252 Cf122 Cd+130 Sn11.33 9.00 9.43 256 Fm128 Sn+128 Sn12.20 −7.36 4.01 256 No124 Sn+132 Te11.49 2.31 2.73 Deformed 230 Th96 Sr+134 Te11.34 24.16 24.79 252 Cf124 Cd+128 Sn12.12 9.40 9.43 256 Fm128 Sn+128 Sn12.24 4.25 4.01 256 No124 Sn+132 Te12.37 2.09 2.73 Table 6. PCM-calculated half life for spontaneous fission along with the fitted neck-length parameter
ΔR . The choice of the outgoing fragment is based on the most probable fragment with the highest preformation probability. Note that in each case, spherical as well asβ2 -deformed (hot) orientations are taken into consideration. The calculated half-lives are also compared with experimental data [68].The identified most probable fission fragments for spherical as well as a hot-compact choice of fragments are also shown in Table 6. For the
230 Th nucleus, the asymmetric fission fragments96 Sr+134 Te (N = 82) are identified, which lie near N = 82 and the deformed magic number Z = 38. For the252 Cf nucleus, asymmetric fragments (122 Cd+130 Sn) are identified, which lie near Z = 50. The symmetric fission fragments of256 Fm also belong to the spherical magic shell closure Sn(Z=50) and near the neutron shell closure N = 82. Similarly, symmetric fragments of256 No are identified near Z = 50. This indicates that proton and neutron shell closures play an important role in the SF region. These results are in fair agreement with Refs. [69, 70]. Furthermore, the SF half-life is calculated for a chosen set of nuclei Z∼ 89−102, whose experimental data are available. For these cases, the neck length parameter (ΔR ) ranges from −0.5 to 0.8 fm. A comparison between the calculated half-life and experimental data is shown in Fig. 9. The calculated results show good agreement with experimental data [71−73].Figure 9. Logarithmic half-lives for 34 nuclei undergoing spontaneous fission in the range Z=89−102 compared with experimental half-lives by optimizing
ΔR .In addition, the TKE is also studied for these nuclei because it plays an important role in SF dynamics. It alters its behavior with respect to the symmetry/asymmetry of the decaying fragments. The TKE can be computed using the relation taken from Herbach et al. [74] as follows:
TKE=0.2904(Z1+Z2)2A1/31+A1/32−(A1+A2)1/3A1A2(A1+A2)2,
(20) where
A1 andA2 are the mass numbers of the emitted fragments. First, the TKE is calculated for the nuclei whose experimental data are available, as shown in Table 7. Because the fragments appearing at the minima in Eq. (3) correspond to the most probable fragments, the calculated results are found to be in good agreement with experimental data. Second, experimental analyses of certain nuclei are not yet available for SF. Therefore, an attempt is made to predict the TKE for cases where experimental data are absent, as shown in Fig. 10. It is evident from this figure that the TKE of the fragments increases with an increase in the fissility parameter (Z2 /A1/3 ). Because of an increase in the fissile nature of the nuclei, the mass of the decaying fragments increases, leading to an increase in the TKE values. It will be of future interest to compare the calculated TKE values with experimental data.Parent nucleus Decay channel TKE/MeV Calc. Expt. 238 Pu104 Mo+134 Te174.26 177.00±0.5 240 Pu106 Mo+134 Te174.05 179.40±0.5 242 Pu110 Ru+132 Sn174.23 180.70±0.5 246 Cm116 Pd+130 Sn181.37 183.90±1.0 248 Cm116 Pd+132 Sn180.77 182.20±0.9, 179.00 ± 2.0248 Cf120 Cd+128 Sn188.79 188.70±1.3 250 Cf120 Cd+130 Sn188.21 187.00±1.0, 185.00 ± 3.0252 Cf122 Cd+130 Sn187.79 185.9±1.0, 185.70±0.1, 183.00±0.5 254 Cf122 Cd+132 Sn187.20 186.90±1.0, 185.00 ± 2.0253 Es125 In+128 Sn191.50 188.00±3.0 254 Fm126 Sn+128 Sn195.15 195.1±1.0, 189.00±2.0 256 Fm126 Sn+130 Sn194.65 197.9±1.0 252 No122 Sn+130 Sn203.43 202.4±0.0, 194.30±3.0
Study of various ground state decay mechanisms of Actinide nuclei
- Received Date: 2023-05-29
- Available Online: 2023-10-15
Abstract: The special property of the actinide mass region is that nuclei belonging to this group are radioactive and undergo different ground state processes, such as alpha decay, cluster radioactivity (CR), heavy particle radioactivity (HPR), and spontaneous fission (SF). In this study, the probable radioactive decay modes of the heavy mass region (Z = 89−102) are studied within the framework of the preformed cluster model (PCM). In the PCM, the radioactive decay modes are explored in terms of the preformation probability (