Variation After Projection With Time-odd Hartree-fock Mean Field

The nucleus is a very complicated many-body quantum system.Solving Schr?dinger's equation with nuclear Hamiltonian is still a very difficult task.In this thesis,we developed a method called as Variation After Projection(VAP),which we expect it to be a good approximation to the full Shell Model(SM).We first introduced the theoretical background of Variation After Projection(VAP),in which the basic idea of the interacting shell model and the mean field approximated method are presented.The advantages of the two theoretical models as well as their difficulties encountered in their application are discussed.The interacting shell model can be used to describe various properties of the nuclei.However,it can not handle the heavy nuclear system due to the huge configuration space.On the other hand,the mean field approximation can be applied to the whole nuclear region.The widely used self-consistent mean field methods are the Hartree-Fock and Hartree-Fock-Bogoliubov methods.Unfortunately,the mean field methods ignore some important physical effects that can not be included in the mean field.So the mean field wave function can only describe the global properties of nuclei.However,beyond mean field methods may take the advantages of both Shell Model and the mean field methods,and may present better description of various nuclear properties even without much computational time.Among various beyond-mean field methods,there is a type call as Variation After Projection(VAP).The main idea of VAP is that,one can start from a HFB mean field state,and do the projection on the HFB vacuum state to get the projected states,which have the symmetries that the nuclear Hamiltonian obeys.Then one performs variation of the projected HFB vacuum state.A typical example of VAP is VAMPIR.Although VAMPIR is an very good approximation substituting the shell model,VAMPIR also needs a great deal of computational time.This makes this method little progress in recent years.Based on the newly developed theories,we started to develop our own VAP method.In this paper,we extend the VAP calculations to the case that the adopted mean field is time-odd.In the previous VAP,the symmetry of time reversal was imposed.This restricts the VAP calculation only-to the even-even nuclei.Now,the present VAP can be naturally extended to all the yrast states in even-even,odd-odd and odd-A nuclei.The results show that the approximate values given by our VAP method are very close to the exact values given by the shell model.This clearly shows that our VAP method is a very effective approximation method of shell model.Since only the spin projection was done in the present work,we believe that the spin projection is the key to obtaining a good shell model approximation.Additionally,in the present VAP calculation,we calculated the exact Hessian matrix of the projected energy for the first time.It can be used not only to speed up the convergence,but also to test whether the obtained energy value reaches the minimum.Comparing with the previous VAP method,the current VAP method looks more simple and more efficient.The present algorithm may be a good starting point for further development and improvement of the VAP.We expect that the VAP method can be extended to the heavy nuclei so that we can study the structure of that region more microscopically.If it is possible,we hope to extend our VAP method to the calculation of the non-yrast nuclear states.