International Conference “Nuclear Science and its Application”, Samarkand, Uzbekistan, September 25-28, 2012
Fig. 1. Dependence of specific empiric function Wigner's /)( AAa from the mass number A.
From the expression (4) it is evident that for nuclei with indefinitely high mass number ( A ~ )
aAAa 1./)( (6) The value /)( AAa is an effective mass of nucleon in nucleus [2]. Therefore coefficient a1 can be interpreted as effective mass of nucleon in indefinite nuclear matter. The parameter a1 is numerically very close to the universal atomic unit of mass u keV)7(009.931494 [4]. Difference could be even smaller (~1 MeV), if we take into account that for definition of u, used mass of neutral atom 12C.
The value of a1 can be used as an empiric unit of nuclear mass that has natural origin. The translation coefficient between universal mass unit u and empiric nuclear mass unit a1 is equal to:
au 1 ).9(004434.1/ (7)
1. A.M. Nurmukhamedov, Physics of Atomic Nuclei, 72 (3), 401 (2009). 2. A.M. Nurmukhamedov, Physics of Atomic Nuclei, 72, 1435 (2009). 3. Yu.V. Gaponov, N.B. Shulgina, and D.M. Vladimirov, Nucl. Phys. A 391, 93 (1982). 4. G. Audi, A.H. Wapstra and C. Thibault, Nucl.Phys A 729, 129 (2003).
PAIR ENERGY OF PROTON AND NEUTRON IN ATOMIC NUCLEI
Nurmukhamedov A.M. Institute of Nuclear Physics, Tashkent, Uzbekistan
The work [1] demonstrated that the structure of Wigner’s mass formula contains pairing of nucleons. Taking into account that the pair energy is playing significant role in nuclear events and in a view of new data we would like to review this issue again. The aim of the given paper is to describe the number of experimental facts justifying Wigner’s definition of odd-even parameter basing on 1) features of empiric universal Wigner’s
105 Section I. Physics of Particles and Nuclei (theory and experiment)
International Conference “Nuclear Science and its Application”, Samarkand, Uzbekistan, September 25-28, 2012
functions, 2) experimental data on the energy of spin-orbit interaction for nuclei with even mass number and 3) systematization of energy “gap” in even-even nuclei. Experimental values of the pair energy of protons p and neutrons n have to be defined through experimental values of the sequence of isotones and isotopes respectively. There are three kinds of formula applied for calculation of the pair energy of protons (neutrons) [2,3]. These formulas (presented below) are differing by the counted number of nuclear masses. All these formulas developed with assumption that the nuclear mass is changing smoothly like functions Z and N while pair effect does not taken into account. The biggest shortcoming of these formulas is that they are neglecting the shell structure of nucleus [1]. Thus we can come to the conclusion that there is no exact method to define the pair energy of protons and neutrons in the contemporary physics. It also questions validity of theoretical calculations in the framework of microscopic approach to nuclear theory.
The Wigner’s mass formula [1] contains traditionally defined pair energypair ,( NZE ) . Taking into account explicit view of the Casimir’s operator [1] can be presented as follows: 2 z z Coul sl ZANZEZAETTAbAaZAM ),(),(),()4)((5.0)(),( , (1) where AbZA )(5.0),( pair NZE ),( .
The Wigner mass formula (1) in relation to sl ,( NZE ) can be presented as following: 2 (2) sl NZE nucl z z z Coul ZAZAETTAbAuAaTA ),(),())((5.0])([),(),( where nucl z ),(),( AuZAMTA is a surplus of nuclear mass, and u – is the relative or universal isotope unit of mass. Figures 1 and 2 present dependencies sl ,( NZE ) from number of neutrons N for nuclei with mass number A 100 and 184 respectively calculated using formula (2). Calculation of energy of spin-orbit interaction was developed fore each mass number provided that pair NZE ),( 0 and
pair NZE ),( : pair NZE ),( for odd-odd nuclide and pair NZE ),( for even-even nuclide. Figures 1 and 2 show that regular relative shifts of sl ,( NZE ) for even-even and odd-odd nuclei are not presented only when pair NZE ),( 0. Absence of shifts when pair NZE ),( 0 justify assumption that the Wigner’s formula (1) does not need term pair ,( NZE ) . We should notice that the odd-even effect in the formula (1) was taken into account by the term AbZA .)(5.0),(
Fig. 1. Dependence sl NZE ),( from number of Fig. 2. Dependence sl NZE ),( from number of neutrons for mass number A .100 neutrons for mass number A .184
106 Section I. Physics of Particles and Nuclei (theory and experiment)
International Conference “Nuclear Science and its Application”, Samarkand, Uzbekistan, September 25-28, 2012
The Figure 3 demonstrates experimental values of energy gap 2exp [4] depending on A mass number in the range of 80 A 210 for all possible cases. The curve in Figure 3 presents dependence of the minimal distance between parabolas from the mass number A for even-even and odd-odd nuclei which is according to Wigner is equal to Ab )(5.1 . As it seen from the Figure the mutual agreement of 2exp and (5.1 Ab ) stay qualitatively on the general trend of change of calculated data. The bad quantitative agreement we explain by influence of shell effects on calculated values of 2exp and also by the uncertainty of structure of excited states. Nevertheless the obtained data confirm our finding that the free term of the explicit view of Casimir operator C2 determines existing odd-even effect.
Fig. 3. Dependence 2exp from A mass number.
Thus we can state that the odd-even effect is an immediate sequence of Wigner mass formula based on SU(4)-symmetry. It assumes that existence of pair interaction cannot be considered as an evidence of violation of Wigner’s SU(4)-symmetry. Based on the abovementioned, we propose to use the term Ab )(5.1 as odd-even parameter for different calculations.
1. A.M. Nurmukhamedov, Physics of Atomic Nuclei, 72 (3), 401 (2009). 2. A. Bohr and B. Mottelson, Nuclear Structure, vol. 1: Single-Particle Motion, (Benjamin, New York, 1969). 3. D.G. Madland and J.R. Nix, Nucl. Phys. A 476, 1 (1988). 4. L. Valentin, Physique Subatomique: Noyaux et Particules, vol. 1, 2 (Hermann Paris, 1983).
TO A PROBLEM OF |ΔN|=2-INTERACTION IN ODD DEFORMED NUCLEI
Sharonov J.A., Kolpakov P. A. Samarkand State University, Samarkand, Uzbekistan
Within the framework of "rotor+particle" model (RPM) states in odd-neutron deformed nuclei of the rare-earth elements are bound only with Coriolis interaction, which is weak for
107 Section I. Physics of Particles and Nuclei (theory and experiment)