BINDING ENERGIES AND RADII OF THE NUCLEI WITH
BINDING ENERGIES AND RADII OF THE NUCLEI WITH N ≥ Z IN AN ALPHA_CLUSTER MODEL G. K. Nie 1 Institute of Nuclear Physics, Academy of Sciences, Tashkent, Uzbekistan
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 • Main steps in developing the model • Binding energies and radii of the nuclei with N ≥ Z • Conclusions
Main steps in developing the model
Formulas for the symmetrical nuclei with even Z=N ≤ 29 The binding energy Eb = N +3(N - 2) Eb 1 = Eb + pn 3 + pn Thus, the nuclear force binding energy E = N nuc + 3(N - 2) nuc Enuc 1 = Enuc + nucpn 3 + pn The empirical values of the Coulomb energy Ec = ∑ ∆Epn The empirical values of surface tension energy EST = Eexp - Enuc + Ec The radius of the last cluster position Rp=2. 168(N – 4)1/3.
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/Rc is used, where RC = 1. 869 N 1/3) The formula for the surface tension energy EST = (Nα + 1. 7)( Nα - 4)2/3 The binding energy of the excess neutron pairs E∆N = (21. 93 – 0. 762 Nnn 2/3) Nnn
Binding energies of beta-stable nuclei The values of Eb and EWz calculated for the A(Z)
Radii of beta-stable nuclei
Published in • • • 1. Uzbek Journal of Physics, V 6(1), 1(2004) 2. Bull. Rus. Acad. Sci. Phys. V 69, 100 (2005) 3. Uzbek Journal of Physics, V 7, N 3, 175 -183 (2005). 4. Изв. РАН, сер. физ. , Т 70 № 5 (2006) с. 718 - 722 5. G. K. Nie, Mod. Phys. Lett. A, (2006) V. 21. N 24. P. 1889 -1900. 6. G. K. Nie, Mod. Phys. Lett. A, V. 22, N 3 (2007) 227 -242.
Binding energies and radii of the nuclei with N ≥ Z Nucleus consists of a core made of and a nuclear molecule of alpha-clusters.
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(Z 1, 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.
Radii of the nuclei for 147 isotopes A(Z, N) A 1(Z 1, N+2) with Z≥ 6 for 116 isotopes A(Z, N) A 1(Z 1, N+2) with Z≥ 24 , , for 147 isotopes A(Z, N) A 1(Z 1, N+2) with Z≥ 6 for 116 isotopes A(Z, N) A 1(Z 1, 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 Me. V, 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, ar. Xiv: 0707. 4291 v 3 • G. K. Nie, ar. Xiv: 0806. 3552 v 2
- Slides: 17