New ground-state band energy formula for transitional nuclei

Transcription

1 New ground-state band energy formula for transitional nuclei Shuifa Shen,, *, Yupeng Yan,, Jiejie Shen, Jing Song, Mengyun Cheng, Lijuan Hao Key Laboratory of Neutronics and Radiation Safety, nstitute of Nuclear nergy Safety Technology, Chinese Academy of Sciences, Hefei, Anhui, People s Republic of China School of Physics, nstitute of Science, Suranaree University of Technology, Nakhon Ratchasima, Thailand Thailand Center of xcellence in Physics (ThP), Commission on Higher ducation, 8 Si Ayutthaya Road, Ratchathewi, Bangkok, Thailand nstitute of Pharmaceutical Biotechnology, College of Pharmaceutical Sciences, Zhejiang University, Hangzhou, Zhejiang 58, People s Republic of China Abstract: A new two parameter formula for ground-state bands in even-even nuclei, which is deduced for soft rotors or transitional nuclei, is proposed. This new formula blends those of very soft nuclei and large deformed nuclei. PACS number(s):..re,.6.v Over the last five decades or more, there has been continuing interest in the interpretation of energies of ground-state bands in even-even nuclei. A variety of theoretical and phenomenological approaches has been developed, ranging from the simplest ( ) dependence of the ideal rotor and the dependence of the ideal harmonic vibrator to numerous more sophisticated expressions designed to account for deviations from these paradigms. n the work done by Brentano et al. [], a two parameter formula for yrast energies in soft rotors or transitional nuclei was proposed, that is ( ), where the moment of inertia depends linearly on ) spin and excitation energy, but it was not pointed out by them how it was deduced, we tend to believe that their formula lacks a clear physical significance, so * shuifa.shen@fds.org.cn

2 in the present work a new formula for soft rotors or transitional nuclei is tried to derive. n the variable-moment-of-inertia (VM) model [], the level energy is given by and the equilibrium condition ( ) [ ) / ] C(, () ( ) () ) determines the moment of inertia (given in units of ) for each states with spin. is a parameter defined as the ground-state moment of inertia and C is the restoring force constant. From () and () we obtain which is equivalent to the cubic equation { [ ) C ]}, () [ ) C]. () This cubic equation can be solved analytically and it has one real root for any finite positive value of and C, the real root of this equation is [ ( ) 7 C ) 7 C / / / ) ) 7[ ] C C ) ) 7[ ] C C / ] /. q. () combined with q. () yields the following expression for the energy of the state with spin : [ ) ]{ [ ) C ]}. (5) f by defining r and by dividing q. () by, we obtain r r ), (6) where. For rigid-body nucleus (the adiabatic limit) C, so, C and hence r. On the other hand, in the limit of very soft nuclei,, the left

3 side of q. (6) will be r ( r ) r r r r r, so from q. (6) / r [ ( )] is obtained, i.e., / [ )]. (7) q. (5) then becomes ( ) [ ) ] / [ )] / ). On the other hand, when discussing the rotation-vibration coupling energy spectrum in the rotation-vibration model (RVM), an approximate expression for the effective moment of inertia has ever been obtained as follows: eff [ b )]. (8) For ideal rotor Bohr formula gives: ), (9) therefore. () eff C C xperimental data on many large deformed nuclei indicate that the linear relationship between the effective moment of inertia and the excitation energy is fairly well established. For transitional nuclei between spherical and deformed limits where neither the vibrator nor rotor limit is very apt. n the present work, a new formula is thus proposed which is tried to fit the level energies of ground-state bands in this kind of nuclei. The basis of this expression is utterly simple: it is the ideal rotor expression ) (, ), () but where the moment of inertia depends linearly on expression ( / ( )) of spin and excitation energy. That is / (, ) ( ( )) ), () where and are adjustable parameters and sets the overall scale. This moment of

4 inertia is the linear superposition of q. (7) and q. (). The idea of our present work is just like that of the work done by jiri et al. [], they have given the empirically based expression ( ) a b ) that blends vibrator and rotor concepts. Fits to yrast (ground-state band) data can be done either by using the + and + energies to fix the parameters, in which case q. () predicts all the higher spin levels, or by doing a least squares fit to the entire (pre-backbending or alignment portion of the) band or quasiband. n the work done by Brentano et al. [], a two parameter formula for yrast energies in soft rotors or transitional nuclei was also proposed, that is ( ). Although in our formula at high spin the term ) / / )), but it is still different from ( ; although may be not very far from / ( )). n addition, in the work done by Brentano et al. [], the energy-dependent term is dropped, since their test fits show that it is very small in the transitional region, but in our formula it may not be the case. n fact, the term / ( )) and term should be in the same status. Acknowledgments The project is supported by the National Natural Science Foundation of China under Grant No. 65, the Foundation of the ducation Department of Jiangxi Province under Grant No. GJJ7 and Suranaree University of Technology under contract Nos. 5/55 and 8/55. We thank the members of FDS team for their helpful advice and support. References []P. von Brentano, N. V. Zamfir, R. F. Casten et al., Phys. Rev. C69, () []M. A. J. Mariscotti, G. Scharff-Goldhaber, and B. Buck, Phys. Rev. 78, 86(969) []H. jiri, M. shihara, M. Sakai, K. Katori, and T. namura, J. Phys. Soc. Jpn.,

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