To Study Triaxiality in Transitional Nuclei 122 134 Xenon with the help of Cubic Terms from Casimir Invariant Operators

Abstract

The Interacting Boson model (IBM) establishes the most appropriate Hamiltonian for newlinethe present 122-134Xe isotopes calculation. The boson numbers are found 9, 8,7, 6, 5, 4, and 3 newlinefor 122-134Xe isotopes, respectively. The nucleus is represented in the IBM in terms of newlineinteracting s and d bosons. The vibrational U(5) limit, the rotational SU(3) limit, and the and#947;- newlineunstable SO(6) limit are three separate types of analytical solutions or limits for such a newlinedevice. newlineGiven the microscopic understanding of bosons as correlated pairs of nucleons, a newlinenatural extension of the interacting Boson model-1(IBM-1) (the most basic version of the newlineIBM) is to assume two separate forms of bosons, neutron, and proton, resulting in the newlineneutron proton interacting boson model-2 (IBM-2). newlineWe conclude that, in the nucleus-by-nucleus fits that will be performed in this work, newlinethere is a single one-boson energy of relevance for the s-state, as well as five two-body and newlineten three-body interactions in the d-boson state. This number of independent terms is too high newlinefor practical applications and simplifications must be sought based on physical, empirical, or newlineformal arguments. newlineThe cubic terms are also formed in the Hamiltonian by the Casimir invariant newlineoperators. The ground state and the excited states of the nucleus all have triaxiality as a main newlineproperty. When we see some transitions from excited states to the ground V state, nuclei are newlinespherical or deformed with the help of operators in which differential operators are commutes newlinewith all the generators in the Lie algebra. newlineFor all isotopes, the agreement between estimated energy levels and experimental newlinemeasurements is slightly higher, especially for the higher excited states. This, we believe, is newlinedue to a variation in the angular momentum projection, which is mostly caused by band newlinecrossing. The boson numbers represent the number of collective pairs of valence nucleons. newlineThe nuclei valence proton number and neutron number have a fundamental correlation, newlineresulting in similar spectra. The number of valence proton Np and neutron Nn has a total N = newline(Np + Nn)/2 = nand#960; + nv bosons. newlineThe IBM-1 Hamiltonian parameters were calculated by fitting to experimental energy newlinelevels and permitting one parameter to change while the others remained constant. This newlinemethod was repeated until a satisfactory overall fit was attained. newlineThe symmetry shape of the nucleus can be predicted from the energy ratio R = E (41 newline+) newline/E(21 newline+), Where, E4 newline+ is the energy level at 41 newline+ and E2 newline+ is the energy level at 21 newline+ are good newlinev newlinecriteria for the shape transition. According to the Casten triangle, the energy ratio for newlinevibration nuclei U(5) = 2, and#947;- unstable nuclei O(6) = 2.5, and rotational nuclei SU(3) = 3.33. newlineWe studied at U(5) symmetry as a type of symmetry, R4/2 = 2.48, 2.39, 2.35, 2.45, 2.22, 2.17 newlineand 2.18 of 54Xe122, 54Xe124, 54Xe126, 54Xe128, 54Xe130, newline54Xe132 and 54Xe134 respectively. newlineVarious E2-reduced transition probabilities are examined experimentally. Theoretical newlineand experimental data for proton charge are compared. The theoretical B(E2) values agree newlinewith the experimental results within the specified errors. We have compared the ratio R = B newline(E2; 41 newline+ and#8594; 21 newline+) / B (E2; 21 newline+ and#8594; 01 newline+) of IBM-1 and the experimental values in the ground state newlinebands as a function of angular momentum L. As a measure to quantify the evolution, it is newlineshown that the results of R values increase with increasing the high spin states. newlineThe branching ratios B (E2; 41 newline+ and#8594; 21 newline+) / B (E2; 21 newline+and#8594; 01 newline+) is less than 2 representing newlineU(5) symmetry, less than 1.42 for O(6) symmetry, and zero for SU(3) symmetry. We newlineinvestigate U(5) symmetry as B (E2; 41 newline+ and#8594; 21 newline+) / B (E2; 21 newline+ and#8594; 01 newline+) = 1.78 (e2 b2 ), 1.74(e2 newlineb2), 1.71(e2 b2), 1.68(e2 b2), 1.60(e2 b2), 1.52(e2 b2), and 1.33(e2 b2) of 54Xe122, 54Xe124, newline54Xe126, 54Xe128, 54Xe130, newline54Xe132 and 54Xe134 respectively. newlineThe values of the boson effective charges for all isotopes are determined by newlineexperimental B (E2; 21 newline+ and#8594; 01 newline+). The B (E2; 21 newline+ and#8594; 01 newline+) and B (E2; 41 newline+ and#8594; 21 newline+) values newlinedecrease as the neutron number increases toward the middle of the shell. The value of B (E2; newline21 newline+ and#8594; 01 newline+) is small because this transition is from quasi beta to ground state band. The energy newlinelevels and B(E2) values in 122-134Xe isotopes, theoretical and experimental results are good in newlineagreement. In IBM-1, there is no signature of triaxiality in the 122-134Xe isotopes. newline