Numerical Analysis of the Ground-State Magnetic Moments of 143,145,147Sm Isotopes

The description of the ground-state magnetic properties of odd-mass nuclei is very informative in understanding of the complex structure of the deformed nuclei. The ground-state magnetic moments of most of the odd-A deformed nuclei have been measured by various experimental studies and there are numerous studies in the literature. However, many of the theoretical studies on magnetic moments and spin polarization effects affecting them are far from explaining these measured values. In this paper, the magnetic moments and effective spin g factors of 143,145,147Sm isotopes in the lanthanides region of the periodic table were investigated within the framework of the Quasiparticle-Phonon Nuclear Model (QPNM) based on the Lagrange Multiplier Method for the first time. Spin-spin interaction parameters (χ) were determined by comparing theoretical and experimental values of magnetic moments of the related isotopes and these interactions were found to have an isovector character q = −1. It has been observed that the ground-state structures of the studied isotopes are weakly affected by quasiparticle⊗phonon interactions and the contribution of these interactions (G K0v iµ values) to the ground-state wave functions is quite small (around 0.01%). Theoretical explanation of the renormalization of spin gyromagnetic factor is one of the most important problems of nuclear structure physics. The results obtained in this study for the effective spin gyromagnetic factor g ef f. s g τ s also agree with the phenological value g ef f. s = (0.5 − 0.7)g τ s

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[1] Yakut, H., Guliyev, E., Guner, M., Tabar, E., Zenginerler, Z.: QPNM calculation for the ground state magnetic moments of odd-mass deformed nuclei: 157−167Er. Nuclear Physics A. 888, 23-33 (2012).

[2] Yakut, H., Tabar, E., Kuliev, A.A., Zenginerler, Z., Kaplan, P.: Ground state magnetic properties of odd neutron Dy isotopes. International Journal of Modern Physics E. 22 (10), 1350076(13) (2013).

[3] Yakut, H., Tabar, E., Kuliev, A.A., Guliyev, E.: The ground-state magnetic moments of odd-mass Hf isotopes. Central European Journal of Physics. 12, 843-850 (2014).

[4] Yakut, H., Tabar, E., Ho¸sgör, G.: Effects of the Isoscalar and Isovector Interaction on the Ground-State Magnetic Moments of the Odd-Mass 137−145Ce Nuclei. Canadian Journal of Physics. 97 (11), 1187-1190 (2019).

[5] Tabar, E., Yakut, H., Kuliev, A.A., Quliyev, H., Ho¸sgör, G.: Magnetic Moments and g factors in odd-mass Ho isotopes. Chinese Physics C. 41 (7), 074101 (2017).

[6] Tabar, E.: Magnetic properties of K =1/2 states in deformed odd-mass nuclei. Nuclear Physics A. 986, 150-166 (2019).

[7] Tabar, E.: A theoretical study on the ground and low-energy magnetic dipole characteristics of 239Pu nucleus. Nuclear Physics A. 987, 202-221 (2019).

[8] Tabar, E.: A theoretical investigation of the magnetic dipole moment of the ground- and excited-states below 600keV in well-deformed 183,185W, 185,187Re and 187,189Os . Int. J. Mod. Phys. E 28(6), 1950040 (2019).

[9] Arima, A., Horie, H.: Configuration mixing and magnetic moments of odd nuclei. Prog. Theor. Phys. 12(5), 623–641 (1954).

[10] Enguang, Z.: Recent progress in theoretical studies of nuclear magnetic moments. Chin. Sci. Bull. 57(34), 4394–4399 (2012).

[11] Shimizu, K., Ichimura, M., Arima, A.: Magnetic moments and GT-type β-decay matrix elements in nuclei with a LS doubly closed shell plus or minus one nucleon . Nucl. Phys. A. 226(2), 282–318 (1974).

[12] Towner, I.S., Khanna, F.C.: Corrections to the single-particle M1 and Gamow-Teller matrix elements. Nucl. Phys. A. 339, 334–364 (1983).

[13] Arima, A.: A short history of nuclear magnetic moments and GT transitions. Sci. China Phys. Mech. Astron. 54, 188–193 (2011).

[14] Bochnacki, Z., Ogaza, S.: Spin polarization effect and the magnetic moments of odd-mass deformed nuclei. Nucl. Phys. 69 (2), 186–192 (1965).

[15] Kuliev, A.A., Pyatov, N.I.: Spin polarization effects in odd-mass deformed nuclei. Phys. Lett. B 28 (7), 443–445 (1969).

[16] De Boer, J., Rogers, J.D.: Concerning the magnetic properties of deformed nuclei in region 153 ≤ A ≤ 187. Phys. Lett. 3 (6), 304–306 (1963).

[17] Bochnacki, Z., Ogaza, S.: Spin polarization effect on the fast allowed beta transitions between deformed odd-mass nuclei. Nuclear Physics A 102 , 529-533(1967).

[18] Kuliev, A.A., Pyatov, N.I.: Magnetic dipole interactions in deformed nuclei. Sov. J. Nucl. Phys 9 (2), 313–323 (1969).

[19] Yakut, H., Kuliev, A., Guliyev, E.: Investigations of the gK-factors in the 175,177,179Hf isotopes. AIP Conf. Proc. 1072, 258–261 (2008).

[20] Yakut, H., Kuliev, A.A., Guliyev, E., Yıldırım, Z.: Intrinsic gK factors of odd-mass 167−179Lu isotopes. Pramana–J. Phys 73, 829-837 (2009).

[21] Yakut, H.: Nadir toprak deforme çekirdeklerinde kolektif dipol seviyelerinin elektrik ve manyetik dipol özelliklerinin incelenmesi. Ph.D. thesis. Sakarya University (2009).

[22] England, J., Grant, I., Griffith, J., Evans, D., Eastham, D., Newton, G., Walker, P.: Isotope shifts and hyperfine splitting in 144−154SmI . Journal of Physics G: Nuclear and Particle Physics, 16 (1), 105–123 (1990).

[23] Letokhov, V., Mishin, V., Sekatsky, S., Fedoseyev, V., Alkhazov, G., Barzakh, A., Denisov, V., Starodubsky, V.: Laser spectroscopic studies of nuclei with neutron number N < 82 (Eu, Sm and Nd isotopes). Journal of Physics G: Nuclear and Particle Physics, 18 (7), 1177–1173 (1992).

[24] Kaplan, M., Blok, J., Shirley, D.: Magnetic Moment of Sm145 and Attenuation Following the Decay of Oriented Sm145 . Phy.Rev. 184 (4), 1177–1180 (1969). [25] Woodgate, G.K.: Hyperfine structure and nuclear moments of samarium. The Royal Society, Proc. R. Soc. Lond. A, 293, 117-144 (1966).

[26] Childs, W.J., Goodman, L.S.: Reanalysis of the Hyperfine Structure of the 4f66s2 7 F Multiplet in 147,149Sm, Including Measurements for the 7F6 State. Physical Review A, 6 (4), 2011-2021 (1972).

[27] Soloviev, V.G.: Microscopic description of vibrational states in deformed nuclei. Prog. Part. Nucl. Phys. 28 (C), 49-74 (1992).

[28] Soloviev, V.G. Sushkov, A.V., Shirikova, N.Y.: Gamma-ray transitions between excited states in 168Er. J. Phys. G: Nucl. Part. Phys. 20, 113-134 (1994).

[29] Soloviev, V.G. Sushkov, A.V., Shirikova, N.Y.: Description of low-lying vibrational and two-quasiparticle states in 166Er. Phys. Rev. C. 51(2), 551-558 (1995).

[30] Malov, L.A., Nesterenko, V.O., Soloviev, V.G.: Low-energy octupole resonances in deformed nuclei. J. Phys. G: Nucl. Phys. 3(9), 219-222 (1977).

[31] Pyatov, N.I., Salamov, D.I.: Conservation laws and collective excitations in nuclei. Nukleonika. 22(1), 127-140 (1977).

[32] Bohr, A., Mottelson, B.: Nuclear structure. Vol. 1, Benjamin, New York and Amsterdam (1969).

[33] Soloviev, V.G., Sushkov, A.V., Shirikova, N.Y.: Low-lying magnetic dipole strength in 163Dy. Phys. Rev. C. 53, 1022-1024 (1995).

[34] Luenberger, D.G.: Optimization by Vector Space Methods. John Wiley Sons. New York (1969). [35] Bussotti, P.: On the Genesis of the Lagrange Multipliers. Journal of Optimization Theory and Applications 117, 453–459 (2003).

[36] Bauske, I., Arias, J.M., Brentano, P.Von., Frank, A., Friedrichs, H., Heil, R.D., Herzberg, R.D., Hoyler, F., Van Isacker, P., Kneissl, U.: First observation of scissors mode states in an odd-mass nucleus. Phy. Rev. Lett. 71, 975-978 (1993).

[37] Raman, S., Nestor, C.W., Tikkanen, P.: Transition probability from the ground to the first-excited 2 + state of even–even nuclides. Atomic Data and Nuclear Data Tables 90, 1-128 (2001).

[38] Soloviev, V.G.: Theory of complex nuclei. Pergamon Press New York (1976).

[39] Stone, N.J.: Table of nuclear magnetic dipole and electric quadrupole moments. Atomic Data and Nuclear Data Tables 78, 75-176 (2005).

[40] Bow, Y.F.: Magnetic dipole transition probabilities of deformed odd-mass nuclei. Phys. Rev. C 2(5), 1608–1611 (1970).

[41] Soloviev, V.G.: Theory of atomic nuclei: Quasiparticles and phonons. Institute of Physics Publishing Bristol and Philadelphia (1992).

[42] Balasubramanian, K.: Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. Model., 1(3), 158 - 180 (2018).

[43] Li, R.: Energy conditions for Hamiltonian and traceable graphs, Univers. J. Math. Appl. 2(1), 33-35 (2019).

[44] Thangamuthu, M., Thippan, j.: Numerical solution for Hybrid fuzzy differential equation by fifth order Runge-Kutta Nystrom method, J. Math. Sci. Model., 2(1), 39-50 (2019).

[45] Fujita, Jun-I., Ikeda, K.: Existence of isobaric states and beta decay of heavier nuclei. Nucl. Phys. 67(1), 145–1771 (1965).

[46] Gabrakov, S.I., Kuliev, A.A., Pyatov, N.I.: 0 + and 1 + unlike particle-hole states in deformed odd-odd nuclei and ß-strength functions. Phys. Lett. B. 36(4), 275-277 (1971).

[47] Kuliev, A.A., Akkaya, R., Ilhan, M., Guliyev, E., Salamov, C., Selvi, S.: Rotational-invariant model of the states with Kπ=1+ and their contribution to the scissors mode. Int. JMod. Phys. E 9(3), 249-261 (2000).

[48] Kuliev, A.A., Guliyev, E., Gerçeklioglu, M.: The dependence of the scissors mode on the deformation in the 140−150Ce isotopes. J. Phys. G: Nucl. Phys. 28(3), 407-414 (2002).

[49] Guliyev, E., Ertugral, F., Kuliev, A.A.: Low-lying magnetic dipole strength distribution in the γ-soft even-even 130,136Ba. Eur. Phys. J. A. 23(7), 313-320 (2006).

[50] Guliyev, E., Kuliev, A.A., Ertugral, F.: Low-lying dipole excitations in the deformed even-even isotopes 154,160Gd. Acta. Phys. Pol. B. 40(3), 653-656 (2009).

[51] Guliyev, E., Kuliev, A.A., Ertugral, F.: Systematic investigation of the low-energy dipole excitations in 176,178,180Hf within rotational, translational and Galilean invariant quasiparticle RPA. Nucl. Phys. A. 915, 78-89 (2013).

[52] Zenginerler, Z., Guliyev, E., Kuliev, A.A., Yakut, H., Soluk, G. : Systematic investigation of the low-lying dipole excitations in even-even 124,136Ba isotopes. Eur. Phys. J. A. 49(9), 1-7 (2013).

[53] Tabar, E., Yakut, H., Kuliev, A.A.: Magnetic dipole response of the 169Tm nucleus. Nuclear Physics A. 981, 130-146 (2019).

[54] Tabar, E., Yakut, H., Kuliev, A.A.: Low-energy dipole strength in even–even 152,164Dy isotopes within the quasiparticle random phase approximation including symmetry restoring interactions. Nuc.Phy. A 979, 143-164 (2018).

[55] Tabar, E., Yakut, H., Kuliev, A.A.: Microscopic description of low-lying M1 excitations in odd-mass actinide nuclei. Nuc.Phy. A 957, 33–50 (2017).

[56] Tabar, E., Yakut, H., Kuliev, A.A.: Microscopic description of ground state magnetic moment and low-lying magnetic dipole excitations in heavy odd-mass 181Ta nucleus. Inter. J. Modern Phy. E- Nuclear Physics 25(8), 1650053 (2016).