BINDING ENERGIES AND RADII OF THE NUCLEI WITH N ≥ Z IN

BINDING ENERGIES AND RADII OF THE NUCLEI WITH N ≥ Z IN

G.K.Nie

1 Institute of Nuclear Physics, Academy of Sciences, Tashkent, Uzbekistan

AN ALPHA_CLUSTER MODEL

The basic ideas of the model The basic ideas of the model

• The alpha-cluster model is based on pn-pair

interactions with using isospin invariance of nuclear force;

• Nucleus consists of a core with some nuclear

molecule on its surface. The core is a liquid drop;

• The number of excess neutron pairs fill out the free space in the core, which is determined by the

difference in the charge radius and the matter

radius of an alpha-cluster and the number of the

core clusters.

Contents of the report Contents of the report

• Main steps in developing the model

• Binding energies and radii of the nuclei with N ≥ Z

• Conclusions

Main steps in developing the model

28.296MeV

 

2.425MeV

 

1.925

c MeV

 

4.350

nuc MeV

 

The binding energy

E

= N



+3(N

- 2) 

E

= E

+ 

+ 

Thus, the nuclear force binding energy E = N 

+ 3(N

- 2) 

E

= E

+ 

+ 

The empirical values of the Coulomb energy E

= ∑ ∆E

The empirical values of surface tension energy E

= E

- E

+ E

The radius of the last cluster position

Rp=2.168(N – 4)^1/3.

Formulas for the symmetrical nuclei with even Z=N ≤ 29

Formulas for the nuclei with Z > 29

The formula for the Coulomb energy Ec = 1.848 Nα^5/3.

(The formula Ec = 3/5 (Ze)²/R^c is used, where R^C = 1.869N^1/3) The formula for the surface tension energy

E^ST = (Nα + 1.7)( Nα - 4)^2/3

The binding energy of the excess neutron pairs E^∆N = (21.93 – 0.762 N^{nn 2/3}) Nⁿⁿ

nuc ST C

E E E E E_N

The values of E_b and E_Wz calculated for the A(Z)

Binding energies of beta-stable nuclei

N N

E E _  E_

nuc ST c

EN_  E  E E

Radii of beta-stable nuclei

Published in

• 1. Uzbek Journal of Physics, V6(1), 1(2004)

• 2. Bull. Rus. Acad. Sci. Phys. V69, 100 (2005)

• 3. Uzbek Journal of Physics, V 7, N3, 175-183 (2005).

• 4. Изв. РАН, сер. физ., Т70 №5 (2006) с. 718 - 722

• 5. G.K. Nie, Mod. Phys. Lett. A, (2006) V. 21. N24. P. 1889-1900.

• 6. G.K. Nie, Mod. Phys. Lett. A, V. 22, N3 (2007) 227-242.

Binding energies and radii of the nuclei with N ≥ Z

( )

core core core N

E  E N_  E_

( ) 6

core core N core

E N

 E

 

3( N

 2) 3(  N

  2) 3( N

  2) 6 

ml core

N

 N

 N

_ c

N ml core core N ml

E E _  E  E _

N

N

Nucleus consists of a core made of

and a nuclear molecule of alpha-clusters^. alpha-clusters

2

_

2 2( ) /

ml

c

core N ml ml

E N N N e R







( 1.7)( )

E  N

 N

 N

2 12_C

( / 2 3) 6( 6) / E E   Z  v

  Z  e R

The binding energy of beta-stable isotopes. Dots indicate the experimental energies of the lightest and the heaviest even beta-stable isotopes. Two lines indicate the function (5) with Nαml = 3 and 5.

The number of excess neutrons calculated from two equations with taken into account that the number has

to be integer (solid line) in comparison with the actual number of excess neutrons in beta-stable nuclei (dots indicate the chain of the isotopes with A(Z,N) and A(Z1,N+2), N is even number, and the lightest and the heaviest isotopes). The calculation were made with Naml = 3 for Z < 78 , Naml = 4 for Z ≥ 78 , Naml =5 for Z ≥ 90 and Naml =6 for Z ≥ 96.

( )

N core core core

E

 N

v

  E N

(21.93 0.762

)

N nn nn

E

  N N

Radii of the nuclei R r N _{ }1/ 3 r_ 1.595fm

for 147 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 6

0.064 fm

 

0.030 fm

 

3 3 3

4He ml core

R  r N_ r N_{ } r_ 1.574 fm 0.040 fm

 

0.033 fm

 

for 116 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 24

for 147 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 6 for 116 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 24 2 _core( ( _core) )1/ 3 2 _ml 2

N R_  N_ r N_{ }  N_ R_ ^r ^1.595fm

,

0.038 fm

 

0.040 fm

 

2 3 3 3

4 (4 / ) (2 )

ml He core p n nn n

N R_  N r_ N_ r N r

/ 0.954

rp n  fm r_n 0.796fm

,

0.028 fm

 

0.037 fm

  for 147 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 6 for 147 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 6 for 116 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 24

for 116 isotopes A(Z,N) A1(Z1,N+2) with Z≥ 24

G. Fricke et al, Atomic Data and Nuclear Data Tables, 60, 207 (1995)

Fig. a) The values Eexp - Enuc - EC + EST for all beta-stable even-even nuclei with Z = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (41 square points), where Enuc - EC + EST is calculated on the alpha-cluster model. The solid line indicates the values

E∆N calculated on the Eq. The dashed line denotes the values E∆N (12) of this paper.

Fig b). The separation energy of excess nn-pairs Enn for the isotopes with A ≥ Ast in dependence on the number of nn-pairs Nnn for the nuclei with Z = 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 (spots and rectangles in a one). The solid line is the function E∆N.

conclusions

• The model implies that the shell effects come from the single not paired nucleon on the periphery of the nucleus interacting with the reduced amount of alpha-clusters and the single pn-pair (if there is one) of its close vicinity due to the short range nuclear force. In the case of

single proton there is also the long range Coulomb interaction with the other protons of the nucleus. This

consequences a conclusion that shell effects take only a few MeV, whereas the most part of the nuclear binding energy comes from the boson interactions in the core.

• In the framework of the alpha-cluster model the radii of the nuclei with N ≥ Z are calculated and for some

isotopes the binding energies are predicted, see

examples in [5].

Published in

• G.K. Nie, arXiv:0707.4291v3

• G.K. Nie, arXiv:0806.3552v2