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the collective magnetic dipole mode ( m1 ) was one of the central subjects in the last two decades . the interest has grown with time due to the wealth of experimental data provided by several groups performing either @xmath2 experiments at backward angles with high resolution or @xmath3 measurements using brehmstralung radiation . the theoretical work in this field preceded the experiment . indeed lo iudice and palumbo @xcite predicted the existence of this state from an anti - phase angular oscillation of proton and neutron axially symmetric and rigid systems . the model is known under the name of two rotor model ( trm ) , and the predicted mode as scissors mode . later on iachello predicted the existence of such a mode at lower energy within the interacting boson approximation ( iba ) @xcite . the experimental discovery of this state in @xmath4gd , by achim richter and his collaborators @xcite , in an @xmath2 experiment , was a stimulus for both experimental @xcite and theoretical @xcite intensive activities . thus , many descriptions have been proposed and some authors reactivated some old own papers devoted in the past to closely related topics . both phenomenological@xcite and microscopic @xcite descriptions have been proposed . the phenomenological formalisms provide an intuitive basis for the description of the new mode , although many details such as the fragmentation and the magnitude of the total strength can not be accounted for . that is the reason why several improvements have been proposed along the time . the rigid motion of the proton - neutron system was replaced by a share motion@xcite or the motion of two liquid drops@xcite . the interacting boson approximation was extended by distinguishing between the proton and neutron bosons @xcite and pointed out that the f spin symmetry plays an important role for the structure of the magnetic state . also the coherent state model@xcite was generalized to gcsm@xcite , which provides a new boson description of m1 states . the microscopic formalisms use the qrpa approach within a deformed single particle basis and predict indeed that the m1 strength is distributed among many qrpa states . the nice feature brought by the microscopic theories is that the low lying states with relatively large m1 transition probability are excited mostly by the orbital part of the transition operator . such type of states are conventionally called scissors like states since their structure confirm the hypothesis of the trm model that the mode is due to the convection current created by the protons motion .    early microscopic description of the state as a doorway state orthogonal to the spurious rotational modes predicted fragmentation @xcite the fragmentation effect was confirmed experimentally and as a consequence a new question was opened , namely , whether there are orbital states in the region beyond 4 mev where the spin - flip excitations dominate . a long standing debate was about the mixture in the predicted mode , of spurious components of isoscalar nature@xcite . several projection procedures were proposed@xcite to purify the theoretical state from spurious components . many review articles discussing the results on this topic appeared in journals or proceedings of conferences and summer schools . since we focus our attention on a specific description , we might unavoidably omit some important contributions to the field from our list of references and therefore , we advise the reader to consult the most recent reviews from refs.@xcite . the present paper is devoted to the study of the orbital and spin excitations in the even - even isotopes of sm , using a projected single particle basis @xcite . some ingredients like the projected states and the model hamiltonian are similar to those used in ref.@xcite . the new point of the present investigation consists of the definition of the mean field . here the mean field of the single particle motion is defined by averaging a particle core hamiltonian on a coherent state describing a phenomenological deformed core . the coupling term comprises a monopole - monopole and a quadrupole - quadrupole interaction . the monopole term is calculated by exploiting the restriction for the volume conservation . in this way the single particle energies , approximated by the average of the mean field hamiltonian on the projected states , have a complex dependence on the deformation parameter . we recall that in ref . @xcite the single particle energies depend linearly on deformation . therefore our aim is to investigate whether by considering more realistic single particle energies one obtains an improvement of the quantitative description given in ref . @xcite . as we shall see at the end of this paper many details shown by the experimental data are nicely described by our approach which in fact confirm our expectation that in order to describe the non - global features of the system one should define properly the mean field of the single particle motion . the results are described according to the following plan . in section 2 , we present briefly the projected single particle basis . also , here we describe the mean field and derive analytical expressions for the single particle energies . the model hamiltonian and the qrpa states are introduced in section 3 . the reduced probability for exciting these states from the ground state is also given . applications to the even - even isotopes of sm are described in section 4 . the main results are summarized in section 5 technical details about the construction of the angular momentum projected single particle basis appropriate for the description of the single particle motion in a deformed mean field generated by the particle core interaction , are given in ref . aiming at a self - consistent presentation , we give here the main ingredients of the method used in the above quoted reference .    the single particle mean field is determined by a particle - core hamiltonian : @xmath5 here @xmath6 denotes the spherical shell model hamiltonian while @xmath7 is a harmonic quadrupole boson ( @xmath8 ) hamiltonian associated to a phenomenological core . the interaction of the two subsystems is accounted for by the third term of the above equation , written in terms of the shape coordinates @xmath9 . the quadrupole shape coordinates are related to the quadrupole boson operators by the canonical transformation : @xmath10 where @xmath11 is an arbitrary c number . the monopole shape coordinate is to be determined from the volume conservation condition . in the quantized form , the result is : @xmath12.\ ] ] averaging @xmath13 on the eigenstates of @xmath6 , hereafter denoted by @xmath14 , one obtains a deformed boson hamiltonian whose ground state is described by a coherent state @xmath15|0\rangle_b,\ ] ] with @xmath16 standing for the vacuum state of the boson operators and @xmath17 a real parameter which simulates the nuclear deformation . on the other hand , the average of @xmath13 on @xmath18 is similar to the nilsson hamiltonian @xcite . due to these properties , it is expected that the best trial functions to be used to generate , through projection , a spherical basis are :    @xmath19    the upper index appearing in the l.h . side of the above equation suggests that the product function is associated to the particle core system the projected states are obtained in the usual manner by acting on these deformed states with the projection operator @xmath20 a certain subset of projected states is orthogonal :    @xmath21.\ ] ]    the main properties of these projected spherical states are : a ) they are orthogonal with respect to i and m quantum numbers . b ) although the projected states are associated to the particle - core system , they can be used as a single particle basis . indeed when a matrix element of a particle like operator is calculated , the integration on the core collective coordinates is performed first , which results in obtaining a final factorized expression : one factor carries the dependence on deformation and one is a spherical shell model matrix element . c ) the connection between the nuclear deformation and the parameter @xmath17 entering the definition of the coherent state ( 2.4 ) is readily obtained by requiring that the strength of the particle - core quadrupole - quadrupole interaction be identical to the nilsson deformed term of the mean field : @xmath22 here @xmath23 and @xmath24 denote the frequencies of nilsson s mean field related to the deformation @xmath25 by : @xmath26 the average of the particle - core hamiltonian @xmath27 on the projected spherical states defined by eq.(2.7 ) has the expression @xmath28 \frac{(\omega^2_{\perp}-\omega^2_z)}{90}.\end{aligned}\ ] ] here we used the standard notation for the clebsch gordan coefficients @xmath29 . @xmath30 stands for the following integral @xmath31^kexp[d^2p_2(x)]dx , ~k=0,1\ ] ] it is worth mentioning that the norms for the core s projected states as well as the matrix elements of any boson operator on these projected states can be fully determined once the overlap integrals defined in ( 2.11 ) , are known @xcite . since the core contribution does not depend on the quantum numbers of the single particle energy level , it produces a shift for all energies and therefore is omitted in eq.(2.10 ) . however when the ground state energy variation against deformation is studied , this term should be considered as well . the first term from ( 2.10 ) is , of course , the single particle energy for the spherical shell model state @xmath14 . according to our remark concerning the use of the projected spherical states for describing the single particle motion , the average values @xmath32 may be viewed as approximate expressions for the single particle energies in deformed nilsson orbits . we may account for the deviations from the exact eigenvalues by considering , later on , the exact matrix elements of the two body interaction when a specific treatment of the many body system is applied . it is worth mentioning that the dependence of the new single particle energies on deformation is similar to that shown by the nilsson model . this is clearly seen in fig . 1 and fig . 2 described in section 4 . we remark that the energies shown in the above mentioned plot depend on deformation in a different manner than those obtained by one of the present authors in ref . @xcite . indeed therein , they depend linearly on deformation while here non - linear effects are present . the difference between the two sets of energies is caused by the fact that here the volume conservation condition was used for the monopole shape coordinate while in ref.@xcite this term was ignored and moreover there is no account of the volume conservation condition . in this respect , one expects that this difference in the single particle energies will induce some modifications of the results of ref.@xcite concerning the fragmentation process of the m1 strength . although the energy levels are similar to those of the nilsson model , the quantum numbers in the two schemes are different . indeed here we generate from each j a multiplet of @xmath33 states as i , which plays the role of the nilsson quantum number @xmath34 , runs from 1/2 to j. on the other hand , for a given i there are @xmath35 degenerate sub - states while the nilsson states are only double degenerate . as explained in ref.@xcite , the redundancy problem can be solved by changing the normalization of the model functions : @xmath36 due to this normalization , the states @xmath37 used to calculate the matrix elements of a given operator should be multiplied with the weighting factor @xmath38 . concluding , the projected single particle basis is defined by eq.(2.7 ) . the projected states are thought of as eigenstates of an effective hamiltonian , with the corresponding energies given by eq.(2.10 ) . in the states introduced in the previous section , we consider a system of z protons and n neutrons interacting among themselves in the following manner . the nucleons of similar isotopic charge interact through pairing interaction , dipole - dipole and quadrupole quadrupole interaction . protons and neutrons interact only through dipole and quadrupole two body interaction . the many body hamiltonian describing such a system is written in second quantization as :    @xmath39    note that although the model hamiltonian is formally identical to that used in ref.@xcite , the mean field parts of the two schemes are different from each other . therefore even if the procedures used are similar , the final results are different . before applying the qrpa treatment , it is convenient to rewrite the qq interaction by re - coupling the initial and final nucleons in a different way and keep in the final result only the dipole - dipole term . by doing this , the new expression is no longer separable . the many body hamiltonian is written first in quasiparticle representation , by using the bogoliubov - valatin transformation , defined by the coefficients @xmath40 and @xmath41 . at this stage one can define the qrpa dipole phonon operator which creates , by acting on the vacuum state @xmath42 , the qrpa states :    @xmath43|0\rangle\equiv\gamma_{1\mu}^+(\lambda)|0\rangle,\ ] ]    where @xmath44 and @xmath45 denote the dipole operator constructed out of two quasiparticle creation operators @xmath46 and @xmath47 and two annihilation operators @xmath48 and @xmath49 , respectively @xmath50 the phonon amplitudes are obtained by solving the qrpa equations with the normalization condition :    @xmath51    the solutions of the qrpa equations are labeled by the index @xmath52 defined by the ordering relation for the corresponding energies ( @xmath53 ) : @xmath54 the qrpa states can be excited from the ground state , the vacuum state of the phonon operator , by the transition operator : @xmath55.\ ] ] where the effective gyromagnetic factors are obtained by accounting for the renormalization induced by the angular momentum carried by the core system : @xmath56 the reduced transition probability is defined by : @xmath57 the involved matrix element has the expression : @xmath58,~~ { \hat i}=\sqrt{2i+1}.\end{aligned}\ ] ] with obvious notations for the reduced matrix elements for the orbital angular momentum ( l ) and spin operator ( @xmath59 ) . the qrpa equations were solved for @xmath60sm isotopes . the single particle energies are those given by eq . ( 2.10 ) , with the spherical shell model energies @xmath61 determined by the frequency @xmath62 and the strengths c and d for the spin orbit and @xmath63 terms , respectively @xmath64 where @xmath65 for protons and @xmath66 for neutrons @xcite . the single particle energies can be considered either as functions of the nuclear deformation @xmath67 by means of the frequencies @xmath23 , @xmath24 , or as functions of the deformation parameter @xmath17 due to eq . ( 2.8 ) . in the later situation , an additional parameter is introduced,@xmath11 , which remained undetermined when the shape coordinates were transformed in boson variables . in order to compare the present results with those reported in ref.@xcite it is useful to define the strength of the particle core qq interaction so that the corresponding matrix elements have similar expressions in the two descriptions . it results @xmath68 for illustration , proton and neutron single particle energies are presented as functions of the deformation parameter @xmath17 in the figures 1 and 2 , respectively , for @xmath69sm . the particle - core coupling constant @xmath70 was taken equal to that given in ref.@xcite . in principle the deformation parameter should be determined by requiring that the system of z protons and n neutrons filling the energy levels in figs . 1 and 2 respectively , in accordance to the pauli exclusion principle , has a minimum energy with respect to changing the deformation . by this procedure , one obtains the equilibrium value of @xmath17 specified by the dashed vertical line . however in the generalized coherent state model ( gcsm)@xcite , which is also based on coherent states description , the deformation parameter was fixed by a fit of the experimental e2 strength to the lowest @xmath71 states . it turns out that this procedure works well for deformed nuclei but not unambiguously for near spherical nuclei . the value obtained for the case of @xmath69sm , by using this method , is mentioned by vertical full lines in figs . 1 , 2 . the deviation of this value from the equilibrium deformation might be corrected in two ways ; either by modifying the strength @xmath70 ( or , equivalently , the parameter @xmath11 ) so that the two values are the same , or by considering the normalization of the single particle energies due to the motion of the core . indeed , we recall that the term of the model hamiltonian which corresponds to the core motion was set to zero when the average on the coherent state describing the phenomenological core was performed . the average of the core hamiltonian is a function of @xmath17 which influences , to a large extent , the equations for energy minimum . the fermi energies for protons and neutrons are pointed by arrows in the corresponding graphs . the strength parameters of the model hamiltonian which were used in our calculations are collected in table 1 . as we have already mentioned , we want to outline the effect on the m1 strength coming from the fact that we use a different set of single particle energies than in ref.@xcite . such an effect could be best determined if the strength parameters used in the two descriptions were close to each other . indeed the parameters from table 1 are , with few exceptions , practically the same as in ref . we modified the pairing strengths so that the gap parameters produced with the new set of single particle energies were the same as in the previous treatment . moreover these are the same as those used by hamamoto and magnusson in ref@xcite . only for @xmath72sm and @xmath69sm one notices a deviation of the proton gap of about 100kev . also the strength of the qq interaction for @xmath73sm and @xmath74sm is smaller in the present case by a factor of about 2 . the strengths of the pairing interaction are consistent with the results obtained by nilsson @xcite while the strengths of dipole and quadrupole interactions are close to the parameters used by de coster and heyde @xcite and by hamamoto and magnusson@xcite . the results of the bcs treatment are shown in terms of quasiparticle energies for @xmath69sm , in figs . 3 and 4 . from these figures , one sees that despite the fact that the considered nucleus is a deformed one , the quasiparticle energies exhibit a shell like structure manifested by some energy gaps . the quasiparticle energies were used to solve the qrpa equations . further , the transition operator given by eq . ( 3.5 ) is used to excite the resulting states . the reduced transition probability for such excitation to take place is calculated by means of relation ( 3.8 ) which contains in a factorized form the contribution coming from the bv and qrpa transformation amplitudes as well as from the matrix elements of the orbital and spin operators
a many body hamiltonian comprising pairing , quadrupole - quadrupole and spin - spin interaction is treated within a projected spherical basis with the aim of describing the detailed structure of the magnetic states of orbital and spin - flip nature in @xmath0sm . the mean field for the single particle motion takes care of the volume conservation of a phenomenological core which results in having a complex dependence on deformation for single particle energies . 
 several experimental features like the total orbital and total spin strengths , the shape of the orbital strength distribution , the hump structure of the spin strength distribution , the quadratic dependence of the orbital strength on nuclear deformation , the saturation effect of the orbital strength with respect to the casten s factor p are very well described within a quasiparticle random phase approximation formalism . 
 comparison of present results with those obtained in some previous treatments are also given . 
 # 1@xmath1#1