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İ . ULUER 1 , H . R . YAZAR 2 , S . YAŞAR 1 , V . ÜNALOĞLU 1

Low spin states of odd-mass Xenon Isotopes. İ . ULUER 1 , H . R . YAZAR 2 , S . YAŞAR 1 , V . ÜNALOĞLU 1 1 Kırıkkale University , Faculty of Art & Science, Kırıkkale, Turkey 2 Nevşehir University , Faculty of Art & Science , Nev ş ehir, Turkey. Over View.

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İ . ULUER 1 , H . R . YAZAR 2 , S . YAŞAR 1 , V . ÜNALOĞLU 1

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  1. Low spin states of odd-mass XenonIsotopes İ.ULUER1, H.R. YAZAR2, S. YAŞAR1, V. ÜNALOĞLU1 1Kırıkkale University, Faculty of Art & Science, Kırıkkale, Turkey 2Nevşehir University, Faculty of Art & Science, Nevşehir, Turkey

  2. Over View Interacting Boson Model(IBM) Dynamic symmetries IBM Phase Triangle IBM-I Hamiltonian IBFM-I Hamiltonian

  3. In This Work . We analyse the positive parity of states of odd-mass nucleus within the framework of interacting boson fermion model. The result of an IBFM-1 multilevel calculation with the 2d5/2, 1g7/2, 3s1/2, 2d3/2, and 1h11/2, single particle orbits is reported for the positive parity states of the odd mass nucleus 125-129Xe. Also, an IBM-1 calculation is presented for the low-lying states in the even-even 124-128Xe core nucleus. The energy levels and B(E2) transition probabilities were calculated and compared with the experimental data. It was found that the calculated positive parity low spin state energy spectra of the odd-mass 125-129Xeisotopes agree quite well with the experimental data.

  4. Interacting Boson Model (IBM) A general theory which explains the structure and different properties of a nuclei has not been developed yet. In order to explain the experimental results, which are obtained by different methods, various nuclei models have been developed. First of these models was the Bohr's model in 1930. However this model has not last very long due to its inefficiency in explaining the high stability of magic nuclei compared to neighbouring nuclei.

  5. Shell Model is not Enough !! In order to explain this phenomena in 1934 Elsasse and Guggenheimer developed the Shell Model. When nucleons have magic numbers, it was observed that the proton and neutron shells in the nuclei are full and these nuclei have special stability compared to the other nuclei. In addition, nuclei with the number of proton and neutron is equal to the magic numbers, have near zero quadrupole moment which supports the existence of closed shells with near spherical symmetry. However the major drawback of this model is that it can not explain the large quadrupole moments in the deformed areas. In addition possibility of electromagnetic transitions and low energy excitation spectrums also can not be explained by this model.

  6. A suggested solution is IBM In 1970, Arima and Lachello, developed the interacting boson model for explanation of low energy states in even- even nuclei. In this model, the low energy colective states of the even-even nuclei is defined by N interacting angular momentum and parite LP = 0+,monopole and LP = 2+ quadrupole as well as boson systems. Therefore valance, monopole and quadrupole bosons are determined by nucleon couples so that the total N boson number is determined by the total number of active proton-neutron couples and according to the nearest closedshell..

  7. Dynamic symmetries Hamiltonian matrix is diagonalizable in different numerical method in order to obtain energy Eigenvalues. But limit case is present. It means that energy spectra can be calculated closed analytic form. This special case is related to the dynamic symmetries. Because of nuclear states have very good angular momentum, in three dimension SO(3) rotational group includes all the sub groups. Under this reduction we may have three caseHere therelated dynamic symmetries as shown as U(5) , SU(3) and SO(6).

  8. IBM Phase Triangle Many nuclei show similar properties between the dynamic symmetries. Most general Hamiltonian form must be used to describe the transition region which are between the three dynamic symmetries. Eigen values and eigenvectors can be calculated with the help of the dynamic symmetries. SO(6) SU(3) U(5)

  9. IBM Phase Triangle • U(5) vibrational limit • SU(3) rotational limit (prolate ve oblatte deformation) • O(6) -soft limit. ( Xe nuclei )

  10. (i)- Energy eigenvalues in U(5) Limit Where n, v and L Quantum numbers. n quadrupole boson numbers v boson seniority L angular momentum. Ground level n = v = L = 0 E0 A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253 D. Brink et al., Phys. Lett. 19 (1965) 413

  11. (ii)–Energy eigenvalues in SU(3) Limit Where ,  and L are the label of main energy band levels. Spectrum which is labeled with (, ) can be characterized as rigid rotor model. Here energy band spaces are directly proportianal to the L(L+1). Ground level band (, ) =(2N,0), for the prolate and oblate rotor (, )=(0,2N). A. Arima & F. Iachello, Ann. Phys. (NY) 111 (1978) 201 A. Bohr & B.R. Mottelson, Dan. Vid. Selsk. Mat.-Fys. Medd. 27 (1953) No 16

  12. (iii) Energy eigenvalues in O(6) Limit Where ,  and L are characterize the label of main levels and  boson seniority labels,  is the same as v in U(5) limit.  includes monopole and quadrupole bosons generalized seniority. Energy spectrum that is labeled with the  consists of many vibrational multiplet series. Ground level  = N ,  = L = 0 and Ground level energy E0. A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468 L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788

  13. IBM-I Hamiltonyen The interacting boson model of Arima and Iachello [1, 2] has become widely accepted as a tractable theoretical scheme of correlating, describing and predicting low-energy collective properties of complex nuclei. In this model it was assumed that low-lying collective states of even-even nuclei could be described as states of a given (fixed) number N of bosons. Each boson could occupy two levels one with angular momentum L=0 (s-boson) and another with L=2 (d-boson). In the original form of the model known as IBM-1, proton- and neutron-boson degrees of freedom are not distinguished. In terms of s- and d-boson operators the most general IBM Hamiltoniancan be expressed as [13]

  14. IBM-I Hamiltonian Parameters Values of the interaction parameters in the IBM-1 Hamiltonian (in terms of code PHINT notation EPS,ELL,QQ,OCT and HEX) which gave the best fit to the experimental data are in MeV are given in Table 1.

  15. ENERGY LEVELS OF EVEN EVEN CORE124Xe Comparison of the calculated and experimental energy spectra of 124Xe. In each band the experimental data are plotted on the left and calculated values on the right.

  16. ENERGY LEVELS OF EVEN EVEN CORE126Xe Comparison of the calculated and experimental energy spectra of 126Xe. In each band the experimental data are plotted on the left and calculated values on the right.

  17. ENERGY LEVELS OF EVEN EVEN CORE128Xe Comparison of the calculated and experimental energy spectra of 128Xe. In each band the experimental data are plotted on the left and calculated values on the right.

  18. ELECTROMAGNETIC TRANSITION PROBABILITIES OF EVEN EVEN CORE124-128XeIBM-1 The B(E2) values were calculated by using the E2 operator. The E2 transition operator must be a hermitian tensor of rank two and therefore the number of bosons must be conserved. Since, with these constraints there are two operators possible in the lowest order, the general E2 operator can be written as [13]

  19. ELECTROMAGNETIC TRANSITION PROBABILITIES OF EVEN - EVEN CORE124-128XeIBM-1 Taking into account of the dynamic symmetry location of the even-even Xeonon nuclei at the IBM phase triangle where their parameter sets are at the O(6)region, we used the multiple expansion form of the Hamiltonian for our approximation. The predicted B (E2) values agree very well with the theoretical ones, which suggest that the wave functions obtained in this work are reliable. Some calculated B (E2) values from the ground state band are given in Table 2.

  20. IBFM - I Hamiltonyen In the IBFM, odd-A nuclei are described by the coupling of the odd fermionic quasiparticle to a collective boson core. The total Hamiltonian can be written as the sum of three parts : Where HB is the usual IBM-1 Hamiltonian for the even-even core, HF is the fermion Hamiltonianterms and VBF is the boson-fermion interaction that describes the interaction between the odd quasi-nucleon and the even-even core nucleus. VBF is dominated by three terms: a monopole interaction characterised by the parameter A0 which plays a minor role in actual calculations; the most important arise from the quadrupole interaction [4] characterised by 0 and the exchange of the quasiparticle with one of the two fermions forming a boson [14] characterised by 0.

  21. IBFM - I

  22. IBFM - I The first term in VBF is a monopole interaction which plays a minor role in actual calculations and the dominant term are the second and third, which arise from the quadrupole interaction. The third term represents the exchange of the quasiparticle with one of the two fermions forming a boson;

  23. IBFM - I These two remaining parameters in equation can be related to the BCS occupation probabilities, of the single-particle orbits,

  24. BCS occupation probabilities

  25. BCS occupation probabilities

  26. IBFM - I The Hamiltonian was diagonalised by means of the computer program ODDA [19] in which the IBFM parameters are identified as: A0= BFM, 0 = BFQ and 0= BFE. The best agreement with experiment for the level calculations of Xenon isotopes is found by means of slightly varying the occupation probabilities to allows a better fit with the experiment. ( see Table 3). The boson-fermion parameters used in ODDA are A0 = -0.11, 0 = 0.41 and 0 = 0.23 MeV for 125Xenon, and A0 = -0.16 0 = 0.37 and 0 = 0.25 MeV for 129Xenon and gave a good agreement with the experimental data as shown in Figures below.

  27. ENERGY LEVELS OF125Xe IBFM - I Comparation of some calculated energy levels for pozitive parity with experimental data of 125Xe.

  28. ENERGY LEVELS OF129Xe IBFM - I Comparation of some calculated energy levels for pozitive parity with experimental data of 129Xe.

  29. ELECTROMAGNETIC TRANSITION PROBABILITIES OF EVEN ODD CORE 125-129Xe IBFM - I In general, the electromagnetic transition operators can be written as sums of two terms, the first of which acts only on the boson part of the wave function and second of which acts only on the fermion part. In the IBFM, the E2 operator is

  30. ELECTROMAGNETIC TRANSITION PROBABILITIES OF EVEN ODD CORE IBFM - I

  31. SUMMARY AND CONCLUSION • In this paper we have carried out an analysis for the odd mass Xenon isotope based on the IBFM-1. • The boson core parameters have been obtained from an IBM-1 analysis and the main results for energy levels and quadrupole transition probabilities agree very well with experiment. • The boson-boson interaction parameters were fixed by the calculations on the boson core nuclei and the boson-fermioon monopole interaction was omitted (A0 = 0.0), there are only two (0 and 0) free varying boson-fermion interaction parameters for the odd even Xenon isotopes.

  32. SUMMARY AND CONCLUSION • The IBFM was extended to include a multilevel calculation for 125-129Xe. The present study has shown that the IBFM provides a successful description for the energy level properties of the transitional Xenon nucleus, for which four single-particle levels plays a major role. • In general, the calculated values agree with the experimental data reasonably well. The B (E2)values depend quite sensitively on the wave functions, which suggest that the wave functions obtained in this work are reliable. The model may be applied to many other even-odd nuclei and its many other nuclear properties.

  33. List of references for further reading • [1] A. Arima, F. Iachello, Ann. Phys. (N.Y) 99, 253 (1976). • [2] A. Arima, F. Iachello, Ann. Phys. (N.Y) 123, 468 (1979). • [3] A. Arima, F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). • [4] F. Iachello, P. Van Isacker, The Interacting Boson-Fermion Model. Cambridge, Cambridge University Press 1991. • [5] R. L. Gill, R. F. Casten, W. R. Phillips, B. J. Varley, C. J. Lister, J. L. Durell, J.A. Shannon, and D. D. Warner, Phys. Rev. C. Vol. 54, 2276 (1996). • [6] I. Alfter, E. Bodenstedt, W. Knichel and J. Schüth, Nucl.Phys.A635, 273 (1998). • [7] N. Minkov, S. B. Drenska, P. P. Raychev, R. P. Roussev and D. Bonatsos, Phys. Rev. C, Vol. 60, 034305 (1999). • [8] B. R. Barrett, S. Kuyucak, P. Navrátil, and P. Van Isacker, Phys. Rev. C. Vol. 60, 037302 (1999). • [9] R. S. Guo and L. M. Chen, J. Phys. G, Nucl. Part. Phys. 26, 1775 (2000). • [10] H. R. Yazar, I. Uluer, Pramana J. Phys. Vol. 65, No.3, 393 (2005). • [11] O. Scholten, Computer code PHINT, KVT, Groningen Holland 1980. • [12] De Voight and M. J. A. Dudek, Rev. Mod. Phys., 55, 949 (1983). • [13] F. Iachello, O. Scholten, Phys. Rev. Lett. 43, 679 (1979). • [14] I. Talmi, Interacting Bose-Fermi System in Nuclei. Iachello, F.(ed), p 329. New York, Plenum Press 1981. • [15] O. Scholten, Progress in Partivle and Nuclear Physics ed A Faessler 14, 189 (1985). • [16] O. Scholten, PhD dissertation, University of Groningen 1980. • [17] J. Bardeen, L. N. Cooper, J. R. Schriefer, Phys. Rev. 108, 1175 (1957). • [18] B. S. Reehal, R. A. Sorensen, Phys. Rev. C2, 819 (1970). • [19] O. Scholten, Internal Report KVI 252 Computer Code ODDA, University of Groningen 1980. • [20] Nuclear Data Sheets, http://www.nndc.bnl.gov/ensdf/ • [21] R. G. Helmer, Nuclear Data Sheets 72, 83 (1994).

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