Stability of heavy elements against alpha and cluster radioactivity

Transcription

1 CHAPTER III Stability of heavy elements against alpha and cluster radioactivity The stability of heavy and super heavy elements via alpha and cluster decay for the isotopes in the heavy region is discussed in the present chapter. The cluster radioactivity calculations suggest an interesting possibility that it involves the interplay of close shell effects of parent and daughter nuclei. The study of alpha decay and cluster decay has been used for identifying the shell closure effects including even the very weak sub-shell closures by calculating the decay half lives and both the cases of large and small half lives are important. A small value of decay half life indicates the presence of shell closure in the daughter nucleus whereas high value on the other hand implies the shell closure of parent nucleus. The spherical/deformed shell closures observed this way may belong to the already known or may predict the existence of new ones. In the cluster decay of trans-lead nuclei observed [5] so far, the end product is doubly magic lead or its neighboring nuclei. This chapter presents the calculations, not only to search the spherical and/or deformed magicity, i.e. not to just study the stability/instability of the concerned nuclei, but also for looking the most probable cluster decay modes. We have studied the alpha and cluster radioactivity of Cf and even-even 44- Fm isotopes [115, 116] in the heavy region. The branching ratios of cluster decay have also been studied for various nuclei. In this context, we used the Coulomb and proximity potential model [11-15] for computing the half lives and other characteristics of the decay process. The details of the model are given in this chapter. 4

2 3.1 The Coulomb and proximity potential model In Coulomb and proximity potential model the potential energy barrier is taken as the sum of Coulomb potential, proximity potential and centrifugal potential for the touching configuration and for the separated fragments. For the pre-scission (overlap) region, simple power law interpolation as done by Shi and Swiatecki [35] is used. The schematic representation of pre-session and post session regions are shown in Figure 3.1.The inclusion of proximity potential reduces the height of the potential barrier, which closely agrees with the experimental result [13, 15]. The proximity potential was first used by Shi and Swiatecki [35] in an empirical manner and has been quiet extensively used over a decade by Gupta et al [3] in the preformed cluster model (PCM) which is based on pocket formula of Blocki et al [117] given as: 1 Φ( ε ) = ( ε.54) 0.085( ε. 54 ) 3, for ε (3.1) ε Φ( ε ) = exp, for ε > (3.) 0.75 where Φ is the universal proximity potential. In the present model, another formulation of proximity potential [118] is used as given by equations (3.6) and (3.7). In this model cluster formation probability is taken as unity for all clusters irrespective of their masses, so the present model differs from PCM by a factor P 0, the cluster formation probability. In the present model assault frequency, ν is calculated for each parent-cluster combination which is associated with zero point vibration energy. But Shi and Swiatecki [119] get ν empirically, unrealistic values for even A parent and 0 for odd A parent. 43

3 (A) Pre-scission (B) Post-scission configuration Figure 3.1 The pre-scission and post-scission configuration of a nucleus by The interacting potential barrier for a parent nucleus exhibiting exotic decay is given V = Z1Ze r h l( l + 1) + Vp( z) + μr, for z > 0 (3.3) Here Z 1 and Z are the atomic numbers of the daughter and emitted cluster, z is the distance between the near surfaces of the fragments, r is the distance between fragment centers. The term l represents the angular momentum, μ the reduced mass and V P is the proximity potential. The distances z and r are displayed in Figure Pb z r 14 C Figure 3. The pictorial representation of the distance between the near surfaces of the fragments z and the distance between fragment centers r. 44

4 The proximity potential V P is given by Blocki et al [117] as C1C Φ z V p ( z) = 4πγ b (3.4) ( C1 + C ) b With the nuclear surface tension coefficient, γ = [ ( N Z ) / A ] MeV/fm (3.5) where N, Z and A represent neutron, proton and mass number of parent, Φ represents the universal the proximity potential [118] given as ε / ( ε ) = 4.41 Φ e, for ε (3.6) 3 ( ε ) = ε ε ε Φ, for 0 ε (3.7) with ε = z/b, where the width (diffuseness) of the nuclear surface b 1 and Siissmann central radii C i of fragments related to sharp radii R i is C i = R i b Ri (3.8) For R i we use semi empirical formula in terms of mass number A i as [117] R (3.9) 1/ 3 1/ 3 i = 1.8Ai Ai The potential for the internal part (overlap region) of the barrier is given as V ) n = a0 ( L L0, for z < 0 (3.) here L = z + C1 + C and L0 = C, the diameter of the parent nuclei. The constants a0 and n are determined by the smooth matching of the two potentials at the touching point. Using one dimensional WKB approximation, the barrier penetrability P is given as 45

5 b Ρ = exp μ( V Q) dz h a (3.11) Here the mass parameter is replaced by μ = ma A / A, where m is the nucleon mass and A 1, A are the mass numbers of daughter and emitted cluster respectively. The turning points a and b are determined from the equationv ( a) = V ( b) = Q. The above integral can be evaluated numerically or analytically, and the half life time is given by 1 ln ln T1 / = = (3.1) λ υp ω E where, υ = = v represent the number of assaults on the barrier per second and π h λ the decay constant. E v, the empirical zero point vibration energy is given as [] ( 4 ) A E = exp v Q, for A 4 (3.13).5 For alpha decay, A = 4 and the empirical zero point vibration energy becomes, E v = Q (3.14) 3. Decay of californium isotopes The Californium nucleus offers interesting possibilities for heavy cluster decay studies, since the closed effects of the doubly magic 48 Ca nucleus are expected to come in to play. So far it was the closed shell effects of the daughter nucleus ( 08 Pb or 0,13 Sn) that has been observed [5, ] or predicted [11-14]. Another interesting possibility is that of 5 Cf since both spontaneous binary and ternary fission has already been observed by triple Gamma coincidence technique with Gamma sphere having 7 Gamma-ray detectors [15-19]. In the fission of 5 Cf, about 0 different final fragments are produced [15]. During the fission process two primary fragments along with several neutrons and/or light clusters are emitted. Another point is the theoretical confirmation of the 46

6 existence of two distant regions of 5 Cf cold fission. The first extending from the mass split 96/156 upto 114/138 and the second one comprising only a narrow mass range around the mass split /13. Here the shell closure of neutrons or protons does not seem to be involved. Although shell effects should play an important role in the odd-even differences by enhancing the odd-odd mass splits with respect to the even-even one, this result emphasizes that the fragments are emitted with deformations corresponding to that of the ground state. The spherical region gives only a hint of the importance of the magic nucleus 13 Sn which is susceptible to be produced in a heavy cluster process, similar to that of light clusters [1]. In the case of cold alpha-ternary fission [131] of 5 Cf, the correlation between two even-z fragments with sum of charges Z=96 and sum of masses A=48 (For eg. 146 Ba and Zr) is involved. In the present chapter the calculations are done by the potential barrier determined by two sphere approximation, as the sum of Coulomb and nuclear proximity potential [13, 133] for the touching and separated configurations (z > 0). The possibility to have a cluster decay process is, Q = M (A, Z) M (A 1, Z 1 ) M (A, Z ) > 0 (3.15) where M (A, Z), M (A 1, Z 1 ) and M (A, Z ) are the atomic masses of the parent, daughter and cluster respectively. The concept of cold valley was introduced in relation to the structure of minima in the so called driving potential, which is defined as the difference between the interaction potential and the decay energy (Q value) of the reaction. Q values are computed using experimental binding energies of Audi and Wapstra [1]. The Q value is very sensitive to the half life calculations. An increase in Q value by 1% decreases the barrier height (increases barrier penetrability by 85%) which results in the reduction of half life by 46%. The sensitivity of half life time with Q value and nuclear radius has already been pointed out by 47

7 Poenaru et al (Figure 5 of Ref. [135]) in the case of 1 C emission from 114 Ba. The interaction potential is a function of distance of separation of interacting fragments. The driving potential of the compound nucleus is calculated for all possible cluster-daughter combinations as a A1 A Z1 Z function of mass and charge asymmetries, η = and η Z = for the touching A + A Z + Z 1 1 configuration of the fragments. i.e the distance between the fragments is r = C 1 +C, where C 1 and C are the Siissmann central radii. The charges of the fragments are fixed by minimizing the driving potential for fixed η and r. In the charge asymmetric coordinate η z that is for every fixed mass pair (A 1, A ) a single pair of charges is determined among all possible combinations. The occurrence of cold valleys for mass asymmetric combinations is due to the shell effect of one or both fragments. Figures represent the plot of driving potential vs A, mass of one fragment for 48 Cf to 54 Cf, with and without including proximity potential. The minima in potential energy curve represent the most probable decay. From these potential energy curves, it is clear that inclusion of proximity potential does not change the position of minima, but become deeper. Figures represent the the plot of computed log (T 1/ ) vs A, mass of one fragment for Cf. The angular momentum l carried away in decay process, appearing in equation (3.3) is often high which is decided by the spin parity conservation. In the present work calculations are done assuming zero angular momentum transfers. Because the angular momentum l carried away in the cluster decay process, appearing in equation (3.3) is very small ( 5ћ) and its contribution to half-life time are shown to be small [13, ] which is decided by the spin parity conservation. Our study reveals that Cf nuclei are stable against light clusters but unstable against heavy clusters ( A ). The calculated alpha decay half life time values for Cf match the experimental values within two orders of magnitude (e.g. in 48 Cf experimental log (T 1/ ) = 7.4 and present value = 8.91; in 49 Cf 48

8 experimental log (T 1/ ) =.04 and present value = 9.4). The experimental alpha half lives are taken from [1]. S, 46 Ar from 48 Cf ; 4 S, 46 Ar, Ca from Cf ; 46,47,48 Ar, 48,49, Ca from 5 Cf ; 46,47,48 Ar,,51,5 Ca from 54 Cf are most probable for decay with T 1/ < s. Another point is that odd A clusters are also probable from odd A parents (e.g. 41 S, 45 Ar, 49,51 Ca from 49 Cf; 45,47 Ar, 49,51 Ca from 51 Cf and 47 Ar, 49 Ca, 51 Ca from 53 Cf). We would like to point out that in cluster radioactivity the only odd A cluster so far observed is 3 F from 31 Pa [137]. The computed half life time values for various clusters from 49,5 Cf isotopes are in agreement with the values reported by Balasubramaniam et al [138] based on PCM and with microscopic super asymmetric fission model [139]. The computed half life time and other characteristics for the most probable cluster emission (with T 1/ < s) and branching ratio with respect to alpha decay, from various californium nuclei are shown in Tables The branching ratio with respect to alpha decay for all clusters is given by λ B = λ cluster alpha T = T alpha 1/ cluster 1/ (3.16) We would like to point out that with presently available experimental techniques, branching ratio up to -19 are possible for measurement [1]. Table 3.3 gives the details of the most probable spontaneous symmetric fission. The first isotope of californium that we considered is 48 Cf. The deepest minimum corresponds to the splitting of S + 08 Pb. The cluster is predicted by the macroscopicmicroscopic model [141] to be prolate deformed (β = 0.54) which sensitively lowers the barrier. The lowest half life time (deepest minima in cold valley) for 41 S from 49 Cf and 4 S from Cf shows the role of doubly magic 08 Pb daughter and lowest half life time for 46 Ar from 51,5 Cf, 47 Ar from 53 Cf and 48 Ar from 54 Cf shows the role of near doubly magic 06 Hg daughter. From Tables , 48 Ca from all Californium isotopes are favorable for emission 49

9 with T 1/ between 7 sec to 3 sec, which are near and with in the present limits for experiments. This stresses the role of doubly magic cluster (N= 8, Z=0) in decay process. Minima for 46 Ar (near doubly magic cluster with N=8, Z=18) from 49 Cf (T 1/ ~ 9 s) and from Cf (T 1/ ~ 6 s) show that these clusters are also favorable for emission. Branching ratio calculation also reveals that 46 Ar, Ca are favorable for emission from these parents. Our decay calculation shows that symmetric fission is also probable in 51 Cf (e.g.,1 Cd + 131,19 Sn), 53 Cf (e.g. 119 Pd + 1 Te, 11,1 Cd + 13,131 Sn) and 54 Cf (e.g. 1,14 Cd + 13,1 Sn). This also stresses the role of doubly or near doubly magic 13 Sn nuclei and is shown in Table 3.3. Figure represents Geiger Nuttal plots of log (T 1/ ) Vs ln P for 4 He, -4 S, Ar and 48-5 Ca emitted from various californium isotopes. These plots are also found to be linear. We would like to point out that Geiger-Nuttal law is for pure Coulomb potential but the inclusion of proximity potential does not produce much deviation to its linear nature, which agrees with our earlier observations [1, 13]. Figure represents Geiger - Nuttal plots of log (T 1/ ) vs Q -1/ for 4 He, -4 S, Ar and 48-5 Ca emitted from Cf isotopes. Geiger - Nuttal plots for all clusters are found to be linear with different slopes and intercepts. From the observed linear nature of these plots, we arrived at an equation for logarithm of half life time as X log ( T 1/ ) = + Y (3.17) Q The values of slope X and intercept Y for different clusters are given in Table 3.4. Using the above equation we have calculated the half life time for all clusters from various californium isotopes are in good agreement with theoretical values.

10 Driving potential (Mev) Cf S+ 08 Pb 44 Ar+ 04 Hg 4 He+ 44 Cm Ca+ 198 Pt 48 Ca+ 00 Pt 46 Ar+ 0 Hg 8 Ge+ 166 Dy 1 Ru+ 138 Xe V = V c +V p V = V c A Figure 3.3 The plot of driving potential vs. A mass of one fragment for 48 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. Driving potential ( MeV ) Cf S+ 09 Pb 41 S+ 08 Pb 4 S+ 07 Pb 4 He+ 45 Cm Ca+ 199 Pt 48 8 Ge+ 167 Dy Ca+ 01 Pt 46 Ar+ 03 Hg 116 Pd+ 133 Te 44 Ar+ 05 Hg V = V c V = V c +V p A Figure 3.4 The plot of driving potential vs. A mass of one fragment for 49 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. 51

11 90 Driving potential ( MeV ) 70 0 Cf S+ Pb 4 S+ 08 Pb 44 Ar+ 06 Hg 4 He+ 46 Cm V= V c Ca+ 00 Pt 116 Pd+ 1 Te 48 Ca+ 0 Pt 8 Ge+ 168 Dy 46 Ar+ 04 Hg V = V c + V p A Figure 3.5 The plot of driving potential vs. A mass of one fragment for Cf isotope. The calculations are made for touching configuration, r = C 1 + C. Driving potential ( MeV ) Cf 4 S+ 09 Pb Ca+ 01 Pt 48 Ca+ 03 Pt 46 Ar+ 05 Hg 4 He+ 47 Cm V = V c Cd+ 131 Sn 1 Cd+ 19 Sn V = V c + V p A Figure 3.6 The plot of driving potential vs. A mass of one fragment for 51 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. 5

12 Driving potential (MeV ) Cf 4 S+ Pb Ca+ 0 Pt 46 Ar+ 06 Hg 4 8 Ar+ 04 Hg 4 He+ 48 Cm V = V c 118 Cd+ 1 Sn Cd+ 13 Sn V = V c + V p A Figure 3.7 The plot of driving potential vs. A mass of one fragment for 5 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. Driving potential ( MeV ) Cf V = V c 44 S+ 09 Pb 78 Zn+ 175 Er 119 Pd+ 1 Te 47 Ar+ 06 Hg 8 Ge+ 171 Dy 11 Cd+ 13 Sn 51 Ca+ 0 Pt V = V c + V p 4 He+ 49 Cm A Figure 3.8 The plot of driving potential vs. A mass of one fragment for 53 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. 53

13 Cf V = V c Driving potential ( MeV ) S+ Pb 46 5 Ar+ 08 Hg Ca+ 0 Pt 48 Ar+ 06 Hg 4 He+ Cm 78 Zn+ 176 Er 1 Cd+ 13 Sn 14 Cd+ 1 Sn V = V c + V p A Figure 3.9 The plot of driving potential vs. A mass of one fragment for 54 Cf isotope. The calculations are made for touching configuration, r = C 1 + C. log (T 1/ ) 0 48 Cf Present Expt A Figure 3. Plot of the computed log (T 1/ ) vs. mass A for 48 Cf isotopes. 54

14 log (T 1/ ) 49 Cf 0 Present Expt Cf 0 Present Expt A Figure 3.11 Plot of the computed log (T 1/ ) vs. mass A for 49, Cf isotopes. Figure taken from [14] log (T 1/ ) 51 Cf 0 Present Expt Cf 0 Present Expt A Figure 3.1 Plot of the computed log (T 1/ ) vs. mass A for 51,5 Cf isotopes. Figure taken from [14] 55

15 log (T 1/ ) 0 53 Cf Present Expt Cf 0 Present Expt A Figure 3.13 Plot of the computed log (T 1/ ) vs. mass A for 53,54 Cf isotopes. Figure taken from [14] He S 4 41 S 4 S 38 log (T 1/ ) ln P Figure 3.14 Geiger Nuttal plots of log (T 1/ ) vs. ln P for 4 He and -4 S emitting from various californium isotopes. 56

16 log (T 1/ ) Ar Ar Ar Ar 48 Ar ln P Figure 3.15 Geiger Nuttal plots of log (T 1/ ) vs. ln P for Ar emitting from various californium isotopes Ca 49 Ca 3 Ca Ca 5 Ca log (T 1/ ) ln P Figure 3.16 Geiger Nuttal plots of log (T 1/ ) vs. ln P for 48-5 Ca emitting from various californium isotopes. 57

17 4 He 4 S S 37 4 S log (T 1/ ) Q -1/ Figure 3.17 Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 4 He, -4 S emitting from various californium isotopes 44 Ar 45 Ar 46 Ar Ar Ar 38 log (T 1/ ) Q -1/ Figure 3.18 Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for Ar emitting from various californium isotopes. 58

18 log (T 1/ ) Ca Ca Ca Ca 4 5 Ca Q -1/ Figure 3.19 Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 48-5 Ca emitting from various californium isotopes. 59

19 Table 3.1 Logarithm of predicted half life time, Branching ratio and other characteristics of Cf isotopes decaying by the emission of most probable clusters. T 1/ is in seconds ===================================================================== Parent Nuclei Emitted Cluster Daughter Nuclei Q-value (MeV) Penetrability P Decay constant(s -1 ) log (T 1/ ) Branching Ratio ===================================================================== 48 Cf 4 He 44 Cm E E S 08 Pb E E E-0 49 Cf Cf 44 Ar 46 Ar 48 Ca Ca 4 He S 41 S 4 S 45 Ar 46 Ar 47 Ar 48 Ca 49 Ca Ca 51 Ca 4 He S 4 S 44 Ar 45 Ar 46 Ar 48 Ca Ca 04 Hg E E E-5 0 Hg E E E-4 00 Pt E E E Pt E E E-6 45 Cm E E Pb E E E-1 08 Pb E E E-0 07 Pb E E E-1 04 Hg E E E- 03 Hg E E E-0 0 Hg E-55.7 E E-3 01 Pt E E E- 00 Pt E E E- 199 Pt E E E Pt E E E- 46 Cm E E Pb E E E-4 08 Pb E E E-1 06 Hg E E E-3 05 Hg E E E-5 04 Hg E E E-18 0 Pt E E E-3 00 Pt E E E-3 51 Cf 4 He 4 S 45 Ar 46 Ar 47 Cm E E Pb E E E-1 06 Hg E E E Hg E E E Ar 04 Hg E E E-18 ======================================================================

20 Table 3. Logarithm of predicted half life time, Branching ratio and other characteristics of Cf isotopes decaying by the emission of most probable clusters. T 1/ is in seconds ===================================================================== Parent Nuclei Emitted Cluster Daughter Nuclei Q-value (MeV) Penetrability P Decay constant(s -1 ) log (T 1/ ) Branching Ratio ====================================================================== 51 Cf 48 Ar 03 Hg E E E- 5 Cf 48 Ca 49 Ca Ca 51 Ca 4 He 4 S 46 Ar 03 Pt E E E-17 0 Pt E E E-0 01 Pt E E E-0 00 Pt E-54.4 E E-1 48 Cm E E Pb E E E-5 06 Hg E E E-17 47Ar 05 Hg E E E- 48 Ar 48 Ca 49 Ca Ca 51 Ca 04 Hg E-5.4 E E- 04 Pt E E E-0 03 Pt E E E-1 0 Pt E E E-3 01 Pt E E E-5 53 Cf 4 He 46 Ar 47 Ar 48 Ar 48 Ca 49 Ca Ca 51 Ca 49 Cm E E Hg E E E Hg E E E-16 Hg E E E Pt E E E- 04 Pt E E E Pt E E E-17 0 Pt E E E-1 54 Cf 4 He 46 Ar 47 Ar 48 Ar 48 Ca 49 Ca Ca 51 Ca 5 Ca Cm E E Hg E E E-1 07 Hg E E E- 06 Hg E E E Pt E E E-3 05 Pt E E E- 04 Pt E E E Pt E E E-18 0 Pt E E E- ====================================================================== 61

21 Table 3.3 Logarithm of predicted half life time and other characteristics of most probable spontaneous symmetric fission from Cf isotopes. T 1/ is in seconds. =========================================================== Parent Nuclei Decay mode Q-value (MeV) Penetrability P Decay constant(s -1 ) log (T 1/ ) =========================================================== 51 Cf Cd Sn E E Cd + 19 Sn E E Cf 53 Cf 118 Pd + 1 Te E E Pd + 1 Te E E Cd + 13 Sn E E Cd Sn E E Cf 1 Cd+ 13 Sn E E Cd + 1 Sn E E-1 0. Pd + 1 Te.5.37 E- 1.4 E ============================================================ Table 3.4 Slopes and intercepts of Geiger - Nuttal plots for different clusters emitted from various Cf isotopes. ================================== Emitted Cluster Slope X Intercept Y ================================== 4 He S S S Ar Ar Ar Ar Ar Ca Ca Ca Ca Ca =================================== 6

22 3.3 Decay of fermium isotopes Fermium is one of the transuranium elements, which lies beyond uranium in the periodic table. All isotopes of fermium are radioactive. Fermium does not occur naturally in the Earth's crust. The production and other characteristics of fermium isotopes are discussed in section In this section we would like to explore the stability of 44- Fm isotopes against alpha and heavy cluster decays. Figures represent the plots of driving potential vs. A, mass of one fragment for 44- Fm isotopes with and without inclusion of proximity potential. From these plots it is clear that the inclusion of proximity potential does not change the position of minima but minima become deeper. The minima in driving potential represent the most probable decay which is due to the shell closure of one or both fragments. In these figures it is obvious that 4 He, 8, Be, 1,14,16 C, 18,0,0 O, 8, Mg, 4,44,46 Ar, 48, Ca etc. have minimum driving potential. The present work Q values are computed using experimental binding energies of Audi and Wapstra [143] and some masses are taken from the table of KTUY [144]. In Figure 3.0, as we move towards the heavy cluster region we can see a deep region consisting of several comparable minima, which is centered on 08 Pb (for e. g. 38 Ar + 06 Pb, Ar + 04 Pb) showing the role of double magicity of the daughter nuclei at N=16 and Z=8. As we move towards the symmetric fission region, we can see that the driving potential decreases with increase in mass number (A ). i. e. in this region there is a chance for symmetric fission to occur (for e. g. Sn + 14 Sn). This also stresses the role of double or near double magicity of the fragments 13 Sn. From figures , we can see a deep valley consisting of several comparable minima and is centered on 08 Pb same as that of Figure 3.0 (for e. g. 4 Ar + 08 Pb in 46 Fm, 4 Ar + 06 Pb in 48 Fm, 44 Ar + 06 Pb in Fm, 44 Ar + 08 Pb in 55 Fm, 46 Ar + 08 Pb in 54 Fm, 48 Ar + 08 Pb in 56 Fm, Ar + 08 Pb in 58 Fm, 48 Ar + 1 Pb in Fm etc.) which stresses the 63

23 role of double magicity of the daughter nuclei with N =16 and Z = 8. In the symmetric fission region, we can see the driving potential decreases with increase in mass number (A ). The minima in driving potential is obtained around 13 Sn (for e.g. Sn + 16 Sn in 46 Fm, 1 Sn + 16 Sn in 48 Fm, 1 Sn + 18 Sn in Fm, 1 Sn + 1 Sn in 5 Fm, 1 Sn + 13 Sn in 54 Fm, 14 Sn + 13 Sn in 56 Fm, 16 Sn + 13 Sn in 58 Fm, 18 Sn + 13 Sn in Fm) which again stresses the role of double magicity of the fragments 13 Sn with N=8 and Z =. Tables 3.5 and 3.6 give the computed half life time values (Present I), barrier penetrability and other characteristics of 44- Fm isotopes with half life time T. 1/ s We would like to point out that with the presently available experimental techniques, the half life time up to s are possible for measurement. The present calculations are done assuming zero angular momentum transfers. The computed alpha decay half lives are compared with experimental data taken from Royer [1] and the standard deviation is found to be 1.6. We have improved our alpha decay half life predictions by changing Q value as Q+E v in the expression for barrier penetrability (eqn. 3.11) and with new turning points determined by V(a) = V(b) = Q + E v. The predicted half life times are in good agreement with experimental data (e.g. in Fm experimental log (T 1/ ) = 3.5 and present value = 3.99; in 54 Fm experimental log (T 1/ ) = 4.19 and present value = 4.99) and the estimated standard deviation is found to be We have extended this calculation to all clusters up to 70 Ni and these are presented in Tables 3.5 and 3.6 as Present II. Our study reveals these parents are instable against alpha and heavy cluster (for 46 Ar, 48, Ca) emissions and stable against light cluster emission, excluding 8 Be from Fm isotopes. In alpha and cluster radioactivity it is found that the half life time has the minimum value for those decays which lead to doubly magic daughter []. We have identified that 44 Ar ( log T1/ = s ) from 5 Fm, 46 Ar ( log T1/ = 0. 6s ) from 54 Fm and 48 Ar ( log T1/ = 1. 74s ) from 56 Fm, 44 Ar ( log T1/ = s ) from 5 Fm have the smallest half life time values, which is due to the 64

24 presence of the near doubly magic clusters with N 8, Z 0 and it indicates the role of doubly magic 08 Pb daughter (N=16, Z =8). The lowest half life time for 48 Ca ( log T1/ = s ) from 54 Fm, Ca ( log T1/ =. 181s ) from 56 Fm and 5 Ca ( log T1/ = 4. 55s ) from 58 Fm are obtained, which is due to the presence of doubly or near doubly magic clusters 48 Ca (N=8, Z=0) and stress the role of near double magicity of the 06 Hg daughter nuclei (N=16, Z 8). Figures represent the Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 4 He, 8, Be, 14 C, 6 Ne, Mg, Si and 38 S, 4,46 Ar, 56 Cr and 6 Fe clusters emitted from various fermium isotopes. Geiger-Nuttal plots for all clusters are found to be linear with different slopes and intercepts. From the observed linear nature of these plots, we arrived at an equation for logarithm of half life time as given in eqn The values of slope X and intercept Y for different clusters are given in Table 3.7. Using the above equation we have calculated the half life time for all clusters from various fermium isotopes and they are in good agreement with predicted values. Nuclear shell effects (through Q value) are obvious from the variation in slopes and intercepts for the various clusters. The difference in slopes and intercepts for each cluster is associated with the fact that each cluster has different barrier penetrability. Figure 3.3 shows the variation of logarithm of barrier penetrability with mass number for various clusters emitted from 44- Fm isotopes. It is found that in the case of 46 Ar and 48, Ca barrier penetrability has a large value (compared to other clusters) and all these cases refer to doubly magic 08 Pb daughter or neighboring one. In the case of 46 Ar emission penetrability increases as it approaches the doubly closed shell 08 Pb daughter. Similar result is obtained in the case for 48, Ca emission which stresses the role of near doubly magic 06 Hg daughter with N=16 and Z 8. That is shell structure effects are 65

25 evident in these plots in terms of maxima (largest penetrability value) or coming up of all plots compared to other clusters. Figures represent Geiger Nuttal plots of log (T 1/ ) vs. ln P for 4 He, 8, Be, 14 C, 0 O, 6 Ne, Mg, Si, 38 S, 4 Ar, 48 Ca and 56 Cr emission from 44- Fm isotopes. These plots are found to be linear. We would like to point out that Geiger-Nuttal law is for pure Coulomb potential but the inclusion of proximity potential does not produce much deviation to its linear nature. The difference in slope and intercept of Geiger-Nuttal plots arises due to the presence of nuclear (proximity) potential in the expression for potential (eqn 3.3). i.e. structure effect of nuclear proximity potential is evident from the Geiger-Nuttal plots of each cluster. Figure 3. represents the computed half-life time versus neutron number of daughter nuclei for 44- Fm isotopes emitting clusters ranging from 4 He to 70 Ni. The computed alpha decay half life times are in good agreement with the experimental data. Experimental alpha half life time values are taken from Royer [1]. In this plot we have seen that a dip is obtained at neutron number N = 16, which indicates the neutron shell closure of daughter nuclei at N = 16. The figures are plotted using half life values (Present II) are taken from Tables We have computed the branching ratio with respect to alpha decay for all clusters using equation We would like to mention that using the presently available techniques in cluster radioactivity half life times up to sec and branching ratio as low as -19 are possible for measurement [1]. Branching ratio calculations also reveal that 44,46 Ar, 48 Ca from 5 Fm; 46 Ar, Ca from 54 Fm; 46,48 Ar from 56 Fm; 48 Ar,,5 Ca from 58 Fm and 48 Ar,,5 Ca from Fm are faourable for measurements in an online experiments. For e.g. 48 Ar (B = 8.57x -16 ) from 56 Fm and 5 Ca (B=3.89x -15 ) from 58 Fm. Radioactive beams could be used to produce these parent nuclei. 66

26 When the decay of 44 Ar from 5 Fm is compared with that from heavier isotopes up to Fm, it is evident that the logarithm of half life time values increases from 4.065s (for 5 Fm, Q = MeV) to s (for Fm, Q =.493MeV). All these cases refer to the doubly or near doubly magic 08 Pb daughter. Also 46 Ar emission from 54 Fm is compared with that from heavier isotopes up to Fm, it is evident that the logarithm of half life time values increases from 0.6s (for 54 Fm, Q = MeV) to.13s (for Fm, Q = MeV). A similar result is obtained for 48 Ar emission also. The logarithm of half lives increases from 1.74s (for 56 Fm, Q = 1.995MeV) to 9.614s (for Fm, Q = MeV). These cases also refer to the doubly or near doubly magic 08 Pb daughter. From these we would like to point out that neutron excess in the parent nuclei slows down the cluster decay process. 67

27 Driving Potential (MeV) Fm V = V C Ar+ 04 Pb 8 Se+ 16 Dy 4 Ne+ 0 Th 14 C+ Pu 48 Ca+ 196 Hg Sn+ 14 Sn 8 Be+ Cm V = V C +V P 4 He+ Cf A Figure 3.0 The plot of driving potential vs. A mass of one fragment for 44 Fm with and without including proximity potential Driving Potential (MeV) Fm S+ Po 14 C+ 3 Pu 8 Be+ 38 Cm 4 He+ 4 Cf 48 Ca+ 198 Hg 4 Ar+ 04 Pb 84 Se+ 16 Dy V = V C +V P V = V C Sn+ 16 Sn A Figure 3.1 The plot of driving potential vs. A mass of one fragment for 46 Fm with and without including proximity potential 68

28 48 Fm V = V C Driving Potential (MeV) 70 0 S+ 1 Po 84 Se Dy Ca+ 00 Hg 14 C+ Pu 4 Ar+ 06 Pb 8 1 Sn+ 16 Sn Be+ Cm V = V C +V P 4 He+ 44 Cf A Figure 3. The plot of driving potential vs. A mass of one fragment for 48 Fm with and without including proximity potential Fm V = V C Driving Potential (MeV) 0 48 Ca+ 0 Hg 14 C+ Pu 4 4 Ar+ 06 Pb 8 Be+ 4 Cm 4 He+ 46 Cf 84 Se+ 166 Dy V = V C + V P 1 Sn+ 18 Sn A. Figure 3.3 The plot of driving potential vs. A mass of one fragment for Fm with and without including proximity potential 69

29 Driving Potential (MeV) Fm S+ 1 Po 48 Ca+ 04 Hg 14 C+ Pu Be+ 4 Cm 44 Ar+ 08 Pb 4 He+ 48 Cf V = V C 84 Se+ 168 Dy 1 Sn+ 1 Sn V = V C + V P A Figure 3.4 The plot of driving potential vs. A mass of one fragment for 5 Fm with and without including proximity potential 54 Fm V = V C Driving Potential (MeV) 70 0 S+ 14 Po 14 C+ Pu Be+ 44 Cm 4 He+ Cf 48 Ca+ 06 Hg 46 Ar+ 08 Pb 8 Ge+ 17 Er V = V C + V P 1 Sn+ 13 Sn A Figure 3.5 The plot of driving potential vs. A mass of one fragment for 54 Fm with and without including proximity potential 70

30 Driving Potential (MeV) Fm V = V C S+ 16 Po 5 Ca+ 04 Hg 14 C+ 4 Pu 48 Ar+ 08 Pb 14 Sn Sn Be+ 46 Cm Ge+ 174 Er V = V C +V P 4 He+ 5 Cf A Figure 3.6 The plot of driving potential vs. A mass of one fragment for 56 Fm with and without including proximity potential Driving Potential (MeV) Fm 4 S+ 16 Po 14 C+ 44 Pu Be+ 48 Cm 4 He+ 54 Cf 5 Ca+ 06 Hg 48 Ar+ Pb 8 Ge+ 176 Er V = V C + V P V = V C 16 Sn+ 13 Sn A Figure 3.7 The plot of driving potential vs. A mass of one fragment for 58 Fm with and without including proximity potential 71

31 Fm V = V C Driving Potential (MeV) 70 0 S+ 0 Po 14 5 Ca+ 08 C+ 46 Pu Hg 48 Ar+ 1 Pb Be+ Cm 8 Ge+ 178 Er 18 Sn+ 13 Sn V = V C + V P 4 He+ 56 Cf A Figure 3.8 The plot of driving potential vs. A mass of one fragment for Fm with and without including proximity potential 4 He 4 8 Be 68 Be C log (T 1/ ) Q -1/ Figure 3.9 Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 4 He, 8, Be and 14 C emitting from various fermium isotopes. Half life time is in seconds. 7

32 log (T 1/ ) Ne Mg 48 Si Q -1/ S Figure 3. Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 6 Ne, Mg and,38 Si emitting from various fermium isotopes. Half life time is in seconds Ar 3 46 Ar 56 Cr Fe log (T 1/ ) Q -1/ Figure 3.31 Geiger-Nuttal plots of log (T 1/ ) vs. Q -1/ for 4,46 Ar, 56 Cr and 6 Fe emitting from various fermium isotopes. Half life time is in seconds. 73

33 -0 4 He 8 Be - Be log (P) C 6 Ne Mg 38 S 4 Ar 46 Ar 48 Ca Ca Mass Number Figure 3.3 Variation of logarithm of barrier penetrability with mass number of the parent for various clusters emitting from 44- Fm isotopes. 4 He 8 Be 68 Be 14 C log (T 1/ ) ln P Figure 3.33 Geiger Nuttal plots of log (T 1/ ) vs. ln P for 4 He, 8, Be and 14 C emitting from various fermium isotopes. Half life time is in seconds. 74

34 5 0 O 54 6 Ne 48 Mg 48 Si 44 log (T 1/ ) ln P Figure 3. Geiger Nuttal plots of log (T 1/ ) vs. ln P for 0 O, 6 Ne, Mg and Si emitting from various fermium isotopes. Half life time is in seconds S 55 4 Ar 48 Ca 56 Cr log (T 1/ ) ln P Figure 3.35 Geiger Nuttal plots of log (T 1/ ) vs. ln P for 38 S, 4 Ar, 48 Ca and 56 Cr emitting from various fermium isotopes. Half life time is in seconds. 75

35 log (T 1/ ) Expt. 4 He 8 Be Be 14 C 16 C 0 O 6 Ne Mg Si 38 S 4 Ar 48 Ca Ca 46 Ar 5 Ti 56 Cr 6 Fe Neutron Number of Daughter Nuclei 70 Ni Figure 3. Computed half-life time versus neutron number of daughter nuclei for 44- Fm isotopes emitting clusters ranging from 4 He to 70 Ni. The computed alpha decay half life time is compared with experimental data [1]. Horizontal dotted line represents the experimental limit. Half life time is in seconds. 76

36 Table 3.5 The computed half life time values, barrier penetrability and other characteristics of 44- Fm isotopes. T 1/ is in seconds ===================================================================== Parent Nuclei Emitted Cluster Daughter Nuclei Q-value (MeV) Penetrability P Decay constant(s -1 ) log (T 1/ ) Present I Present II ====================================================================== 44 Fm 4 He Cf E Be 1 C 3 Si Si S 38 S 4 Ar 48 Ca Cm E E Pu.64.7E-.3E Rn E E Rn E E Po E-47 1.E Po E E Pb E E Hg E-51 3.E Fm 4 He 8 Be 1 C 3 Si Si S 38 S 48 Ca 4 Cf E-3 3.E Cm E E Pu 9.79.E-5.00E Rn E E Rn E E Po E- 3.06E Po E- 1.5E Hg E- 3.11E Fm 4 He 8 Be 38 S S 46 Ar 48 Ca Ca 44 Cf E E Cm E- 1.E Po E-5.91E Po E-5 3.E Pb E E Hg E-49 3.E Hg E-53.0E Fm 4 He 8 Be S 4 Cl 44 Ar 46 Ar 48 Ca 46 Cf E-6 3.E Cm E E Po E E Bi.73 1.E E Pb E E Pb E- 9.57E Hg E E Ca 00 Hg E E ====================================================================== 77

37 Table 3.6 The computed half life time values, barrier penetrability and other characteristics of 5- Fm isotopes with zero angular momentum transfers T 1/ is in seconds =================================================================== Parent Emitted Daughter Q-value Penetrability Decay log (T 1/ ) Nuclei Cluster Nuclei (MeV) P constant(s -1 ) Present I Present II =================================================================== 5 Fm 4 He 48 Cf E E Ar 46 Ar 48 Ca Ca 08 Pb E E Pb 1.3.3E E Hg E E Hg E E Fm 4 He 44 Ar 46 Ar 48 Ca Ca 5 Ca Cf E E Pb E E Pb E E Hg E E Hg E-46 4.E Hg E E Fm 4 He 46 Ar 48 Ar Ca 5 Ca 54 Ti 5 Cf E E Pb E E Pb E E Hg E E Hg E E Pt E-5 5.E Fm 4 He 46 Ar 48 Ar Ca 5 Ca 54 Ti 54 Cf E-9 4.0E Pb E-51.E Pb E E Hg E- 6.08E Hg E E Pt E E Fm 4 He 8 Be 48 Ar Ca 5 Ca 54 Ca 56 Ti 56 Cf E E Cm E-5 1.7E Pb E E Hg E E Hg E E Hg E E Pt E E ==================================================================== 78

38 Table 3.7 Slopes and intercepts of Geiger - Nuttal plots for different clusters emitted from 44- Fm isotopes. ============================= Emitted Cluster Slope X Intercept Y ============================= 4 He Be Be C C O Ne Mg Si S Ar Ca Ca Ti Cr Fe ============================= 79

39 3.4 Conclusion Based on the concept of cold valley in cold fission and fusion, the alpha and cluster decay of Cf and even-even 44- Fm isotopes are investigated in this chapter. The details of the Coulomb and proximity potential model are also given in this chapter, which is based on the potential barrier consisting of the sum of Coulomb potential, proximity potential and the centrifugal potential. From the cold valley plots for driving potential versus mass number A of Cf and 44- Fm isotopes there arises a deep valley consisting of several comparable minima and is centered on the doubly magic 08 Pb, the lead radioactivity. In the symmetric fission region, we can see that the driving potential decreases with increase in mass number (A ).The minima in driving potential are obtained around 13 Sn, the tin radioactivity. It is evident from the studies that most of these parents are instable against alpha and heavy cluster ( 46 Ar, 48, Ca) emissions and stable against light cluster emission, excluding 8 Be from Fm isotopes. The most probable clusters from these parents are predicted to be 46 Ar, 48, Ca which indicate the role of doubly or near doubly magic clusters in cluster radioactivity. The branching ratios with respect to alpha decay for all the possible cluster emissions different californium isotopes are also computed. Geiger-Nuttal plots were studied for various clusters and are found to be linear with varying slopes and intercepts. It is found that inclusion of proximity potential will not produce any deviation to the linear nature of the plots. Nuclear structure effects and shell effects are evident from the observed variation in slope and intercept of Geiger-Nuttal plots.