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					State of the Art in Nuclear Cluster Physics : SOTANCP2 (Bruxelles, May 25-28, 2010)



   Analytical relations for alpha decay half-lives
               and barrier heights and positions
                             G. Royer, C. Schreiber, H. Saulnier
       Laboratoire Subatech (IN2P3/CNRS, Université et Ecole des Mines de Nantes, France)




- Alpha decay potential barriers from a Liquid Drop Model
- Alpha decay half-lives from a Liquid Drop Model
- Analytical formulae for total alpha decay half-lives
- Introduction of the angular momentum in the analytical
expressions to reproduce the partial alpha-decay half-
lives
        Nantes Liquid Drop Model energy
               ELDM=Evol+Esurface+ECoulomb +Eproximity

                                                                              Surf
E vol  a v (1  k v I 2 )A           E surf  a s (1  k s I 2 )A 2 / 3 
                                                                              4R 0
                                                                                  2


                                  9e 2 Z 2           dd'
                      E Coul                  
                                 16  2 R 60         r  r'


          I  ( N  Z) / A
          a v  15.494 MeV                 a s  17.9439 MeV
          k v  1.8            k s  2.6
          R 0  1.28A1/ 3  0.76  0.8A 1/ 3
                 Proximity energy

  Additional energy to the surface energy taking into
account the nuclear interaction between opposite
nucleons in a neck or a gap between fragments (~ - 27
MeV for 264Hs alpha decay at the contact point)
                         h max
     E proximity (r )  2   D(r, h ) / b2hdh
                         h min
       Analytical proximity energy for the alpha emission

EProximity from the contact point between an alpha particle and the daughter nucleus :

E pr                                          0.172                                                     
     e 1.38( r  R cont ) 0.6584 A 2 / 3   1 / 3  0.4692 A1 / 3 r  0.02548 A1 / 3r 2  0.01762 r 3 
4                                          A                                                          
One-body compact and creviced (or like-cluster) shapes

            2 2                                                 a
            a sin ( )  c12 cos2 ( )      0          s1 
                                                  2              c1
 R( ) 2  
           a 2 sin 2 ( )  c 2 cos2 ( )                s2 
                                                                    a
                                                
           
           
                              2
                                             2      
                                                                    c2




                                    Elliptic lemniscatoïds
            Barrier against Alpha emission




Alpha decay potential barrier for      108Te   and   264Hs   nuclei
----- no proximity energy
___ with proximity energy


The experimental Q value is taken into account.
 Analytic expressions for the barrier characteristics




Distance between the mass centres at the barrier top :


    R  2.536  1.1157 [41 / 3  ( A  4)1 / 3 ]    fm


Barrier height against alpha decay (fission barrier) :

                       e 2  2  ( Z  2)
  E  1.43                                   Q    (MeV)
              2.536  1.1157 [4  ( A  4) ]
                                  1/ 3    1/ 3
        Theoretical alpha-decay half-life


Decay constant of the parent nucleus :                 0P
Assault frequency :             0  1020 s1


                                  2 R out                       
Penetrability :           P  exp R in 2B(r )( E (r )  E )dr 
                                                               
R in  R daughter  R        R out  e2Zd Z / Q

Inertia : B(r)  
                                 ln 2
Partial half-life : T1/ 2 
                                  
Experimental and theoretical (LDM) alpha decay half-lives




                                                            


    decay half-lives around 208Pb and for the heaviest elements

rms deviation between the theoretical (LDM) and experimental values :
0.35 for 131 e-e nuclei, 0.71 for 106 e-o, 0.57 for 86 o-e, 0.99 for 50 o-o and
0.63 for the whole set of 373 alpha emitters.
         Formulas for the alpha decay half-lives
(adjusted on 373 total alpha decay half-lives, J. Phys. G 26 (2000) 1149 )

For 131 even (Z) – even (N) emitters, with                = 0.28
           
log10 T1 2 (s)  25.31  1.1629 A1 / 6 Z 
                                              1.5864 Z
                                                  Q
For 106 even (Z) – odd (N) emitters, with                 = 0.39
log10 T(s)  26.65  1.0859 A1 6 Z 
                                          1.5848 Z
                                              Q
For 86 odd (Z) – even (N) emitters : with                 = 0.36
log10 T(s)  25.68  1.1423 A1 6 Z 
                                              1.592 Z
                                                 Q
For 50 odd (Z) – odd (N) emitters, with                   = 0.35
log10 T(s)  29.48  1.113 A1 6 Z 
                                          1.6971 Z
                                             Q

 : rms deviation of log10 T(s) between the values derived from the formulas
and the experimental data.
          Recent alpha decay half-lives for Z < 111
                                      (PRC 77 (2008) 037602)

Nucleus    Tformula(s)   Texp(s)    Nucleus   Tformula(s)   Texp(s)    Nucleus   Tformula(s)   Texp(s)
 105Te      0.7×10-6     0.4×10-6    156Er     2.3×1010     1.6×1010    158Yb     4.3×106      2.7×106
 160Hf      1.9×103      1.9×103     174Hf     6.3×1022     4.5×1023    158W      1.5×10-3     1.2×10-3
 168W       1.6×106      3.1×106     162Os     1.9×10-3     2.3×10-3    164Os     4.2×10-2     2.2×10-2
 166Pt      3.0×10-4     2.9×10-4    168Pt     2.0×10-3     2.2×10-3    170Pt     1.4×10-2     2.0×10-2
 172Hg      4.2×10-4     2.7×10-4    174Hg     2.1×10-3     2.0×10-3    188Hg     5.3×108      2.0×108
 178Pb      2.3×10-4     2.1×10-4    180Pb     5.0×10-3     2.7×10-3    184Pb     6.1×10-1     3.6×10-1
 186Pb      1.2×101      4.7×100     188Pb     2.7×102      1.3×102     190Pb     1.8×104      8.7×103
 192Pb      3.6×106      2.1×106     194Pb     9.8×109      1.3×109     188Po     4.0×10-4     1.1×10-4
 189Po      5.0×10-3     3.0×10-3    190Po     2.5×10-3     1.5×10-3    192Po     2.9×10-2     2.2×10-2
 210Po      1.2×107      1.0×106     196Rn     4.4×10-3     1.4×10-2    198Rn     6.5×10-2     9.6×10-2
 202Ra      2.6×10-3     3.6×10-3    204Ra     5.9×10-2     5.5×10-2    210Th     1.7×10-2     1.3×10-2
 212Th      3.6×10-2     2.4×10-2     218U     5.1×10-4     4.0×10-4     220U     6.0×10-8     5.8×10-8
  224U      7.0×10-4     8.2×10-4     226U     5.0×10-1     5.7×10-1    228Pu     2.0×10-1     5.1×10-1
 230Pu      1.0×102      2.7×102     238Cm     2.3×105      3.3×105     258No     1.2×102      5.4×101
 258Rf      9.2×10-2     1.0×10-1    260Rf     1.0×100      1.0×100     271Sg     1.1×102      1.4×102
 272Bh      1.7×101      9.8×100     266Hs     2.1×10-3     2.3×10-3    270Hs     1.0×101      2.2×101
 275Hs      1.9×100      0.15×10     275Mt     3.2×10-3     9.7×10-3    276Mt     6.5×10-1     7.2×10-1
                            0
 270Ds      4.3×10-2                 279Ds     6.5×10-1     1.8×10-1
                         7.3×10-2

  The experimental Qalpha is taken into account in the formulas.
          Exp. and theor. alpha-decay half-lives
                       for Z > 110
     Z             A            Qexp           T1/2,exp        T1/2 GLDM      /     Formula
                               (MeV)
   111            279          10.52           170 ms             12.4 ms            10.9 ms
   111            280          9.87             3.6 s             0.69 s              3.1 s
   112            283          9.67             4.0 s             0.95 s              9.6 s
   112            285          9.29             34 s              13.22 s            127 s
   113            283          10.26           100 ms             222 ms             234 ms
   113            284          10.15           0.48 s             0.43 s              2.4 s
   114            286          10.35           0.16 s             0.05 s              0.11 s
   114            287          10.16           0.51 s             0.16 s              1.79 s
   114            288          10.09            0.8 s             0.22 s              0.52 s
   114            289          9.96             2.7 s             0.52 s              6.1 s
   115            287          10.74           32 ms              46 ms              53 ms
   115            288          10.61           87 ms              94.7 ms            582 ms
   116            290          11.00           15 ms              3.47 ms            8.9 ms
   116            291          10.89           6.3 ms             6.35 ms           89 ms
   116            292          10.80           18 ms              10.45 ms         27 ms
   116            293          10.67           53 ms              22.81 ms         308 ms
   118            294          11.81           1.8 ms             0.15 ms         0.39 ms


GLDM : H.F. Zhang, W. Zuo, J.Q. Li, G. Royer, Phys. Rev. C 74 (2006) 017304
          Predictions for other superheavy elements
Nucleus    Q(MeV)   Tformula    Nucleus   Q(MeV)   Tformula   Nucleus   Q(MeV)   Tformula

 293118     12.30    187 s      292117    11.60   6.47 ms     291117    11.90   0.32 ms
 291115     10.00      4.8 s     290115    10.30     4.2 s     289116    11.70   1.05 ms
 289115     10.60    113 ms      287113     9.34     99.4 s    286113     9.68    61.5 s
 285114     11.00     12 ms      285113    10.02     1.0 s     284112     9.30    25.1 s
 283111      8.96    5.5 min     282112     9.96     0.30 s    282111     9.38    99.8 s
 281112     10.28      0.2 s     281111     9.64     2.72 s    281110     8.96   4.6 min
 280112     10.62    25.4 ms     280111     9.98     1.43 s    279112    10.96   3.88 ms
 279109      8.70   7.72 min     278112    11.38   0.083 ms    278111    10.72   12.5 ms
 278110     10.00    51.8 ms     278109     9.10     143 s     277112    11.62   0.12 ms
 277111     11.18    0.28 ms     277110    10.30    39 ms      277109     9.50    1.48 s
 277108      8.40   65.25 min    276111    11.32   0.39 ms     276110    10.60   1.47 ms
 276108      8.80     40.6 s     275111    11.55    42.3 s    275110    11.10   0.43 ms
 274111     11.60    88.1 s     274110    11.40    19.5 s    274109    10.50   9.84 ms
 274108      9.50      0.3 s     274107     8.50   48.4 min    273111    11.20   0.29 ms
 273110     11.37    0.11 ms     273109    10.82    0.5 ms     273108     9.90   101 ms
 273107      8.90     21.1 s     272110    10.76    0.7 ms     272109    10.60   5.74 ms
 272108     10.10    6.9 ms      272106     8.30   6.38 min    271110    10.87   1.79 ms
 271109     10.14    29.9 ms     271108     9.90   109.7 ms    271107     9.50   0.338 s

Qalpha from 2003 A.M.E of Audi, Wapstra, Thibault (Nucl. Phys. A 729 (2003) 337).
                        New Data : Z = 117
        (Oganessian et al, PRL 104 (April 2010) 142502)



  A      293      289      285     294      290       286    282      278    274
  Z      117      115      113     117      115       113    111      109    107
          21 ms   0.32 s   7.9 s   0.11 s   0.023 s   28.3 s 0.74 s   11 s   1.3 min
 Texp
          15 ms   0.28 s   2.5 s   0.28 s   17.9 s    34.3 s 6.85 s 1.8 s    1.5 min
TFor


Comparison betweeen the experimental and the theoretical half-lives
     Formulas for the partial alpha decay half-lives
New data set of 344 partial alpha decay half-lives of
ground-state-to-ground-state transitions and angular momentum dependence
(V. Yu. Denisov and A. A. Khudenko, ADNDT 95 (2009) 815)

For 136 even (Z) – even (N) emitters, with                  = 0.328
           
log10 T1 2 (s)  25.752  1.15055 A1 / 6 Z 
                                                1.5913 Z
                                                   Q
For 84 even (Z) – odd (N) emitters, with                    = 0.956
log10 T(s)  34.156  0.87487 A
                                               1.6923 Z
                                    16
                                            Z
                                                  Q
For 76 odd (Z) – even (N) emitters : with                   = 0.889
log10 T (s)  32.623  1.0465 A1 6 Z 
                                               1.7495 Z
                                                  Q
For 48 odd (Z) – odd (N) emitters, with                     = 0.908
log10 T(s)  31.186  0.98047 A1 6 Z 
                                               1.6744 Z
                                                  Q
         Formulas for the even – even nuclei




For the 59 even – even heaviest emitters
with N > 126 and Z > 82 :                                  = 0.187

log10 T1 2 (s)  27.69  1.0441 A1 / 6
                                               1.5702 Z
                                            Z
                                                   Q


For the other 77 even – even lightest emitters :           = 0.266

log10 T(s)  28.786  1.0329 A
                                               1.6127 Z
                                      16
                                            Z
                                                   Q
L-dependent formulas for the alpha decay half-lives

 For 84 even (Z) – odd (N) emitters : with                    = 0.555

 log10 T(s)  27.750  1.1138 A1 / 6 Z 
                                                1.6378 Z
                                                   Q
   1.7383 10 6 A N Z [l(l  1)]1 / 4
                                       0.002457A[1  (1) l ]
                 Q
 For 76 odd (Z) – even (N) emitters, with                    = 0.666

 log10 T(s)  27.915  1.1292 A1 / 6 Z 
                                                1.6531Z
                                                   Q
   8.9785 10 7 A N Z [l(l  1)]1 / 4
                                       0.002513 [1  (1) l ]
                                                 A
                 Q

 For 48 odd (Z) – odd (N) emitters, with                     = 0.681

 log10 T(s)  26.448  1.1023 A1 / 6 Z 
                                               1.5967 Z
                                                  Q
   1.6961 10 6 A N Z [l(l  1)]1 / 4
                                       0.00101 [1  (1) l ]
                                                A
                 Q
  L-dependent formulas for the alpha decay half-lives
         proposed in PRC 79, 054614 (2009)
For 84 even (Z) – odd (N) emitters : with                    = 0.616

log10 T(s)  29.989  1.067 (A  4)1 / 6 Z 
                                                       1.675 Z
                                                          Q
  0.7182A1 / 6 [l(l  1)]1 / 2
                               0.6932[1  (1) l ]
           Q
For 76 odd (Z) – even (N) emitters, with                     = 0.675

log10 T(s)  29.769  1.0817 (A  4)1 / 6 Z 
                                                         1.6758Z
                                                            Q
  0.2322A1 / 6 [l(l  1)]1 / 2
                               0.6476[1  (1) l ]
           Q

For 48 odd (Z) – odd (N) emitters, with                     = 0.679

log10 T(s)  30.597  0.9839(A  4)1 / 6 Z 
                                                       1.6439 Z
                                                          Q
  0.5901A1 / 6 [l(l  1)]1 / 2
                               0.2889[1  (1) l ]
           Q
            Viola-Seaborg-Sobiczewski formula
                  (re-adjusted on 344 partial alpha decay half-lives
                    of ground-state-to-ground state transitions)
For even (Z) – even (N) emitters

log10        
        T1 2 (s)  31.392  0.22783Z 
                                        1.5872 Z  1.3456
                                            Q  (MeV)
                                                            0.349

For even (Z) – odd (N) emitters

log10        
        T1 2 (s)  48.445  0.04826 Z 
                                         1.4125 Z  20.649
                                                 Q
                                                             0.950

For odd (Z) – even (N) emitters

log10        
        T1 2 (s)  40.248  0.17826 Z 
                                         1.69185Z  3.3357
                                                Q
                                                             0.912

For odd (Z) – odd (N) emitters

             
log10 T1 2 (s)  40.410  0.13858Z 
                                      1.5344 Z  9.463
                                             Q
                                                         0.870
                 Alpha decay half-lives
          for subbarrier excitation energies E*

For even (Z) – even (N) emitters, with  = 0.28

            
log10 T1 2 (s)  (25.31  1.1629A1 / 6 Z 
                                                1.5864Z
                                                               )(1  4.5182  10 4 E*2 )
                                                 Q  E      *

For even (Z) – odd (N) emitters, with  = 0.41
                                           1.5848Z
log10 T(s)  (26.65  1.0859A1 6 Z                   )(1  1.117  10 2 E*  1.4903 103 E*2 )
                                            Q  E   *

For odd (Z) – even (N) emitters : with  = 0.27
                                        1.592Z
log10 T(s)  (25.68  1.1423A1 6 Z           )(1  8.9617  103 E*  1.3446  103 E*2 )
                                        Q  E *

For odd (Z) – odd (N) emitters, with  = 0.5
                                                1.6971Z
log10 T(s)  (29.48  1.113A1 6 Z                          )(1  8.8806  103 E*)
                                                 Q  E      *

 : deviation between the theoretical predictions and the formula values

The introduction of the excitation energy does not diminish the accuracy of
the formulae.
     Half-lives for the cluster emission from                 209Pb


Dependence of the partial half-lives on the emitted light nucleus and the
excitation energy for a 209Pb nucleus (formed and excited by the
absorption of a neutron). The Q value is positive only for the alpha
emission.
                                      Summary
   - A liquid drop model taking into account both the proximity
energy, the mass and charge asymmetry has been used to
describe the alpha decay. The potential barrier top
corresponds to two separated fragments : the alpha particle
and its daughter nucleus. The alpha decay half-lives can be
reproduced using the experimental Q value.
   - Previously proposed analytical formulas remain reliable to
reproduce recent new alpha decay half-lives and predictions
are given for other possible superheavy nuclei.
   - New expressions depending on the angular momentum of
the alpha particle are proposed to reproduce a new data set of
344 partial alpha decay half-lives and angular momentum of
ground state to ground state transitions .
                                 Thank you
J. Phys. G 26 (2000) 1149, Nucl. Phys. A 683 (2001) 266, Phys. Rev. C 77 (2008) 037602.

                       SOTANCP2 (Bruxelles, May 25-28, 2010)
                    Be, Li, He and H decay half-lives
                   for subbarrier excitation energies



Similar formulas than for the alpha decay but for :
9Be,    with     s = 0.48
7Li,   with s = 0.38
6Li,   with s = 0.36
6He,    with s = 0.35
3He,    with s = 0.24
3H,    with     s = 0.14
2H,    with s    = 0.16
1H,    with s = 0.08
s : deviation between the theoretical predictions and the formula values
                              Nuclear radius

                                  4                     
R is the equivalent sharp radius : R 3 f (bulk )  4 0 f (r )r 2 dr .
                                  3
(radius of a uniform sharp distribution having the same value in the bulk )
                                                    1/ 2
Q is the equivalent rms radius Q  5 / 3 r 2
C is the central radius : C  R 1 / 2 (half  value of f (r ))




                                                                                Z
                                                Q2  R 2 
                                                       Z
                                                                 4
                                                                 35   0.0156          R 2  5b 2
                                                                                        Z
                                                                               A1 / 3
                                             (W.D. Myers, K.H. Schmidt, Nucl. Phys. A 410(1983) 61)
                   Values of r0 in the models
*Liquid Drop Model, D.L. Hill, J.A. Wheeler, [Phys. Rev. [89 (1953) 1102]
r0 = 1.216 fm
*Liquid Drop Model, W.D. Myers, W.J. Swiatecki [Arkiv för Fysik, 36 (1966) 343]
r0 = 1.2249 fm
*Liquid Drop Model, J.R. Nix, W.J. Swiatecki, [Nucl.Phys 71 (1965) 1 ]
r0 = 1.216 fm
* Droplet Model, W.D Myers (1977)
r0 = 1.18 fm
* Finite Range Liquid Drop Model, A.J. Sierk [Phys. Rev. C 33 (1986) 2039]
 r0 = 1.2249 fm
 * Finite Range Liquid Drop Model, P. Möller, J.R. Nix, W.D. Myers, W.J.
    Swiatecki, [ADNDT 59 (1995) 185]
r0 = 1.16 fm
* Thomas-Fermi Model, W.D. Myers, W.J. Swiatecki, [LBL 36803 (1994), Nucl. Phys.
   A 601 (1996) 141]
r0 = 1.14 fm
* Liquid Drop Model, K. Pomorski, J. Dudek [Phys. Rev. C 67 (2003) 044316]
r0 = 1.2172 fm
                  Values of as in the models
*Liquid Drop Model, D.L. Hill, J.A. Wheeler, [Phys. Rev. [89 (1953) 1102]
as = 14. MeV
*Liquid Drop Model, W.D. Myers, W.J. Swiatecki [Arkiv för Fysik, 36 (1966) 343]
as = 17.9439 MeV
*Liquid Drop Model, J.R. Nix, W.J. Swiatecki, [Nucl.Phys 71 (1965) 1 ]
as = 17.8 MeV
* Droplet Model, W.D Myers (1977)
as = 20.69 MeV

* Finite Range Liquid Drop Model, A.J. Sierk [Phys. Rev. C 33 (1986) 2039]
as = 17.9439 MeV
* Finite Range Liquid Drop Model, P. Möller, J.R. Nix, W.D. Myers, W.J. Swiatecki,
    [ADNDT 59 (1995) 185]
as = 21.1847 MeV

* Thomas-Fermi Model, W.D. Myers, W.J. Swiatecki, [LBL 36803 (1994), Nucl. Phys. A
    601 (1996) 141]
as = 18.63 MeV

* Liquid Drop Model, K. Pomorski, J. Dudek [Phys. Rev. C 67 (2003) 044316]
as = 16.9707 MeV
                                       Nuclear radius
                                                                  2 1/ 2
Equivalent rms charge radius : R 0  5 / 3 r
Experimentally, for a given isotopic serie, a decrease of the charge (and mass)
reduced rms radius r0 is observed with increasing mass : (I. Angeli ADNDT 87, 185 (2004))
R 0 / A1 / 3  1.312 fm for   40
                                   Ca , R 0 / A1 / 3  1.234 fm for    48
                                                                            Ca
R 0 / A1 / 3  1.217 fm for   190
                                    Pb , R 0 / A1 / 3  1.201 fm for   214
                                                                             Pb


Coefficients of the different formulae given R0 from a fit on 782 ground state
rms charge radii (N,Z > 7)
R 0  1.22572 A1 / 3 fm        with          0.124 fm.
R 0  1.0996 A1 / 3  0.653 fm           with        0.066 fm.
R 0  1.1718 A1 / 3  1.4069 / A1 / 3 fm          with       0.064 fm.
R 0  1.1818 A1 / 3  0.089  1.5938 / A1 / 3 fm           with          0.064 fm.
R 0  1.1769 A1 / 3  1.2046 / A1 / 3 1.5908 / A fm           with            0.064 fm.
R 0  1.2332 A1 / 3  2.8961 / A 2 / 3  0.18688 A1 / 3 I fm           with         0.052 fm.
Emission of 3He from   108Te   and   14C   from   223Ra
Contributions to the total deformation energy

                           160
                                 Dy 80As  80As




       Without proximity
       energy term
            Exp. and theor. alpha-decay half-lives for Z > 110
       Z           A          Qexp         T1/2,exp     T1/2 DDM3Y           T1/2 GLDM        /     Formula    T1/2 VSS
                             (MeV)
      111         279         10.52        170 ms           9.6 ms          12.4 ms                10.9 ms     45.3 ms
      111         280          9.87          3.6 s           1.9 s          0.69 s                3.1 s          5.7 s
      112         283          9.67          4.0 s           5.9 s          0.95 s                9.6 s         41.3 s
      112         285          9.29          34 s            75 s           13.22 s               127 s         592 s
      113         282         10.63         73 ms                                                 43 ms
      113         283         10.26        100 ms          202 ms           222 ms                234 ms        937 ms
      113         284         10.15         0.48 s          1.55 s          0.43 s                2.4 s          4.13 s
      114         286         10.35         0.16 s          0.14 s          0.05 s                0.11 s         0.19 s
      114         287         10.16         0.51 s          1.13 s          0.16 s                1.79 s         7.24 s
      114         288         10.09          0.8 s          0.67 s          0.22 s                0.52 s         0.98 s
      114         289          9.96          2.7 s           3.8 s          0.52 s                6.1 s          26.7 s
      115         287         10.74         32 ms           49 ms           46 ms                  53 ms        207 ms
      115         288         10.61         87 ms          409 ms           94.7 ms                582 ms       997 ms
      116         290         11.00         15 ms          13.4 ms          3.47 ms                8.9 ms      15.2 ms
      116         291         10.89        6.3 ms          60.4 ms          6.35 ms               89 ms        336.4 ms
      116         292         10.80         18 ms           39 ms           10.45 ms              27 ms          49 ms
      116         293         10.67         53 ms          206 ms           22.81 ms              308 ms       1258 ms
      118         294         11.81        1.8 ms          0.66 ms          0.15 ms               0.39 ms      0.64 ms

Density Dependent M3Y effective interaction : P. Roy Chowdhury, C. Samanta, D.N. Basu, Phys. Rev. C 73 (2006) 014612
GLDM : H.F. Zhang, W. Zuo, J.Q. Li, G. Royer, Phys. Rev. C 74 (2006) 017304
Viola Seaborg Sobiczewski formulae : A. Sobiczewski, Z. Patyk, S. Cwiok, Phys. Lett. B 224 (1989) 1.
P. Roy Chowdhury, C. Samanta and D.N. Basu approach

 Decay constant :            P  E v P /( h / 2)


 Zero point vibration energy Ev :
 Ev = 0.1045 Q for even-even parent nucleus
 Ev = 0.0962 Q for odd Z-even N
 Ev = 0.0907 Q for even Z-odd N
 Ev = 0.0767Q for odd-odd

 Total interaction energy :

 E(R )  VN (R )  VC (R )   2 l(l  1) /(2R 2 )

 Nuclear interaction potential :
                                                
  VN ( R )   1 (r1 ) 2 (r2 ) v r2  r1  R d 3 r1d 3 r2

                 
  v r2  r1  R : DDM3Y effective interaction
         Viola-Seaborg-Sobiczewski formula

For even (Z) – even (N) emitters

           
log 10 T1 2 (s)  33.9069  0.20228 Z 
                                           1.66175 Z  8.5166
                                                Q  (MeV)
For even (Z) – odd (N) emitters

           
log 10 T1 2 (s)  33.9069  0.20228 Z 
                                           1.66175 Z  8.5166
                                                   Q
                                                               1.066

For odd (Z) – even (N) emitters

           
log 10 T1 2 (s)  33.9069  0.20228 Z 
                                           1.66175 Z  8.5166
                                                   Q
                                                               0.772

For odd (Z) – odd (N) emitters

           
log 10 T1 2 (s)  33.9069  0.20228 Z 
                                           1.66175 Z  8.5166
                                                   Q
                                                               1.114

				
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