عنوان البحث(Papers / Research Title)
Study of Phase Transitions and Critical Points in the Rare-Earth Region in the Framework of the Interacting Boson Model (IBM-1)
الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)
فؤاد عطية مجيد
Citation Information
فؤاد,عطية,مجيد ,Study of Phase Transitions and Critical Points in the Rare-Earth Region in the Framework of the Interacting Boson Model (IBM-1) , Time 5/11/2011 4:32:56 AM : كلية التربية للعلوم الصرفة
وصف الابستركت (Abstract)
Phase transition and critical points in the rare-earth region
الوصف الكامل (Full Abstract)
Study of Phase Transitions and Critical Points in the Rare-Earth Region in the Framework of the Interacting Boson Model (IBM-1)
Fouad A. MAJEED 1and Kalid S. JASSIM 2
1,2 Department of Physics, Collegeof Education/Ibn Hayyan, Babylon University,
P.O.Box 4, Hilla-Babel, Iraq.
1 E-mail: fouadalajeeli@yahoo.com, 2 E-mail: dr.kalid_phy@yahoo.com
Abstract
A systematic study of isotope chains in the rare–earth region is presented. The energy levels, E2 transition rates, and two–neutron separation energies are described by using the most general IBM-1 Hamiltonian for the chains. For each isotope chain a general fit is performed in such a way that all parameters but one are kept fixed to describe the whole chain.
In this region nuclei evolve from spherical to deformed shapes and a method based on catastrophe theory, in combination with a coherent state analysis to generate the IBM-1-1 energy surfaces, is used to identify critical phase transition points.
Keywords: Gamma transitions and level energies, Shell model
الخلاصة
أجريت دراسة منظّمة لسلسلة النظائر المشعة في منطقة العناصر الأرضية النادرة. تمّت عملية وصف لــــ مستويات الطاقة، نسب الأنتقال (E2) وطاقات الفصل بين نيوترونيين بأستخدام الشكل الأكثر عمومية لهاملتونيان نموذج البوزونات المتفاعلة (IBM-1) للسلاسل <!--[if !لكل سلسلة من سلاسل العناصر المشعة قيد الدراسة أجريت عملية ملائمه شمولية بطريقة، حيث أن كل المتغيرات (parameters) ما عدا واحد تَبْقى ثابتة لوَصْف السلسلةِ الكاملةِ.
في منطقة العناصر الأرضية النادرة، تَتطوّرُ النوى مِنْ الأشكال الكروية إلى الأشكالِ المُشَوَّهةِ وأستعملت الطريقة المعتمدة على (catastrophe theory) بالتمازج مع (coherent state analysis) لتوليد سطوح طاقة الـ (IBM-1) لتَمييز نقاطِ تحوّل المرحلةِ الحرجةِ.
Introduction
In the last few years, an interest for the study of phase transitions and phase coexistence in atomic nuclei has been revived [1, 2, 3, 4].
A new class of symmetries that applies to systems localized at the critical points has been proposed. In particular, the “critical symmetry” E(5) [5] has been suggested to describe critical points in the phase transition from spherical to ?-unstable shapes while X(5) [6] is designed to describe systems lying at the critical point in the transition from spherical to axially deformed systems. These are based originally on particular solutions of the Bohr-Mottelson differential equations, but are usually applied in the context of the Interacting Boson Model (IBM-1) [7], since the latter provides a simple but detailed framework in which first and second order phase transitions can be studied.
In the IBM-1 language, the symmetry E(5) corresponds to the critical point between the U(5) and O(6) symmetry limits, while the X(5) symmetry should describe the phase transition region between the U(5) and the SU(3) dynamical symmetries, although the connection is not a rigorous one. The O(6) limit itself has also been proposed to correspond to a critical point [8].
The IBM-1 analyses of phase transitions have been carried out using schematic Hamiltonians in which the transition from one phase to the other is governed by a single parameter. It is thus necessary to see how much these predictions vary when a more general Hamiltonian is used. The global approach was first used by Casta?os et al[9] for the study of series of isotopes [10, 11, 12]. An alternative procedure is provided by the use of the consistent Q formalism (CQF) [13]. In this case, although the Hamiltonian is simpler than the general one, the main ingredients are included.
Within this scheme a whole isotope chain is described in terms of few parameters that change smoothly from one isotope to the next. Because of the possible non-uniqueness of such nucleus by nucleus fits and the restricted parameter space, it is important to study under what circumstances the prediction of the location of critical points in a phase transition is robust. In this paper we follow Refs. [10, 11, 12, 14,15] and use a more general one– and two–body IBM-1 Hamiltonian to obtain the model parameters from a fit to energy levels of chains of isotopes.
In this way a set of fixed parameters, with the exception of one that varies from isotope to isotope, is obtained for each isotope chain and the transition phase can be studied in the general model space.
The fit to a large data set in many nuclei diminishes the uncertainties in the parameter determination. A possible problem arising from working with such a general Hamiltonian, however, is the difficulty in determining the position of the critical points. Fortunately, the methods of catastrophe theory [16] allow the definition of the essential parameters needed to classify the shape and stability of the energy surface [14, 15].
In this paper we analyze diverse spectroscopic properties of several isotope chains in the rare-earth region, in which shape transition from spherical to deformed shapes is observed. We combine this study with a coherent-state analysis and with catastrophe theory in order to localize the critical points and test the X(5) predictions. Since the introduction of the E(5) and X(5) symmetries, only a small number of candidates [17, 18, 19, 20, 21, 22, 23, 24] have been proposed as possible realizations of such critical point symmetries. In this paper we show that the critical points can be clearly identified by means of a general theoretical approach [14, 15].
The paper is structured as follows. In section 2 we present the IBM-1 Hamiltonian used. In section 3 the results of the fits made for the different isotope chains are presented. Comparisons of the theoretical results with the experimental data for excitation energies, E2 transition rates and two-neutron separation energies are shown. In section 4 the intrinsic state formalism is used to generate the energy surfaces produced by the parameters obtained in the preceding section. In addition, the location of the critical point in the shape transition for each isotope chain is identified by using catastrophe theory. Also in this section, the alternative description provided by the CQF for the rare-earth region is briefly discussed. Finally, section 5 is devoted to summarize and to present our conclusions.
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this work is published in special issue for the kufa’s for the first conference for physics, 6-7 October, pp. 138-154, ISSN 2077-5830, 2010.
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