CHAPTER-2 ONE-PARTICLE PLUS ROTOR MODEL FORMULATION

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1 CHAPTE- ONE-PATCLE PLUS OTO MODEL FOMULATON. NTODUCTON The extension of collective models to odd-a nuclear systems assumes that an odd number of pons (and/or neutrons) is coupled to an even-even core. The most usual assumption is to take the core to consist of all the nucleons; however we could assume the core as nearly doubly magic nucleus to which odd particle is coupled. To describe the interplay between the motion of particles and the collective ation, Bohr and Mottelson proposed the theory of the ation of axially symmetric nuclei, to take into account only a few so called valence particles, which move more or less, independently in deformed well of the core.[bohr,95; Bohr and Mottelson, 95]. The one Particle plus axially symmetric otor Model (PM) consists of a odd particle/nucleon coupled to a or/core and is widely used to discuss the low lying spectra of odd-a nuclei and at the same time, illustrate various general features of ating systems [ Horth et al., 97]. t has therefore played an important role in the development of theory of nuclear ational spectra. The PM makes use of an approximation that one-particle motion can be separated from the collective vibrations and ations of the nucleus. This in turn implies that energies of intrinsic states of vibrations, and of ations are different enough so that intrinsic motion can follow adiabatically the vibration which in turn follows the ations. However, this approximation often becomes poor and there is mixing of these basic modes. n PM approach, the motion of odd nucleon in a potential formed by the core, is seen from the reference frame fixed in the nucleus. This situation is identical with that of nucleon moving in a similar but spatially fixed potential. The consideration of motion in the body fixed frame simplifies the problem, otherwise motion of particle in ating potential leads to complexities in calculations. The shape of an orbit of single particle depends on the extent to which its motion is coupled to that of the core i.e. it depends on the separation of single particle and ational degrees of freedom. A rigorous separation is not possible but an approximate separation is made, known as adiabatic approximation ( ω > ω ). nucleon nucleus 8

2 Mathematically the separation of the single particle and ational motion greatly simplifies the calculations and is the principal reason for evaluation of single nucleon motion first and later superimposes the ational motion in body fixed frame of reference.. THE MODEL AND METHODOLOGY.. Total Hamiltonian Formulation Let be a stationary coordinate frame (lab frame), in whichh coordinates of any point are denoted by r ( X, Y, Z ) and Κ be a coordinate frame moving or ating with deformed nucleus, in which coordinates of any point are denoted by r ( X, Y, Z ). The orientation of ating coordinate frame with respect to stationary frame (the origins of two frames coincides) is denoted by three Euler angles ΦΘand, Ψ shown in Fig... These Euler angles can be regarded as dynamical variables describing the ation of the nucleus as whole. The coordinates characterizing internal degrees of freedom of the nucleus are denoted by r ( X, Y, Z ). Fig..: otation of co-ordinate system ( X, Y, Z ) w.r.t coordinate frame ( X, Y, Z ) by Euler angles ( Φ, Θ, Ψ ). Thus, in PM formulation we consider an odd-a nucleus with last odd nucleon coupled to even-even core/orr [erman, 956 ]. The Collective Hamiltonian for or is given by Η=Η v T r (.) i i Η= Β ( β β γ ) Cβ k = ω k (.) 9

3 i i where Η v = Β ( β β γ ) Cβ represents vibrational energy consisting of β vibration and γ vibration. Η v remains unaffected by ation due to length invariance in ation and Tr = = k= ω k= is ational kinetic energy. The, andω respectively, are the ational angular momentum of core, moment of inertia w.r.t space fixed axis and angular velocity of principal axis w.r.t space fixed axis. For simplicity, we omit β - and γ - vibrational part of Hamiltonian, keeping only ational part for even-even core then adding it to the Hamiltonian of last odd nucleon ( Ηsp ) and its coupling with core ( Η PC ). Thus, the total Hamiltonian for one Particle plus axially symmetric otor Modal can be written as [Horth et al., 97; aczarowski, 977]: Η=Η sp Η Η res (.) where nucleus and Η sp describes the internal degrees of freedom, Η characterizes the ation of the Η res denotes the contribution arising from the residual interaction such as pairing. This may also comprise of the interaction of the odd nucleon with the quadrupole and octupole vibrations of the core, also a contribution from the coupling of an odd nucleon with or/core. The interaction of the odd nucleon with the quadrupole and octupole vibrations of the core has been neglected here. Let be angular momentum of valence particle and be angular momentum of ation of the whole nucleus. Thus total angular momentum of the nucleus is equal to the sum of and. = (.4) For axially symmetric nucleus, here we consider rd axis of the body fixed frame as symmetry axis (Fig..), the proection of internal angular momentum along the symmetry axis of nucleus is a constant of motion. Let Μ be the proection of total angular momentum of an odd-a nucleus on laboratory Z-axis, and and Ω respectively, are the proections of total angular momentum of an odd-a nucleus and odd nucleon angular momentum on body fixed -axis. 0

4 f system possesses an axis of symmetry then Κ and Ω will also be good quantum number. n such a case axis of symmetryy will be taken as -axis so that Κ and Ω will be appropriate proections on the symmetry axis. Since for axially symmetric nucleus, the proection of the internal angular momentum along the symmetry axis of the nucleus is a constant of motion. Fig..: Schematic Diagram for angular Momentum in deformed nuclei Thus, the ational part of total Hamiltoniann can be written as: Η = For spheroidal deformation of the core = = ; Thus Η k = ω = [ = k = = ] = Η = = ( ) ( foraxially symmetric nucleus = 0 ) ( ) = and = = (.4) (.5) (.6) Using raising and lowering operators, = ι, = ι and = ι = ι

5 We get,, = = and = = ι ι Hence = ( )( ) = ( ) and 4 = ( )( ) = ( ) ( ) = ι ι 4ι 4 This gives: = = and ( ) = ( ) = Therefore, the ational part of the total Hamiltonian becomes (.7) (.8) Η = ( ) Thus, total Hamiltonian can be written as Η=Η Η Η sp res ( ) (.9) = Hsp Hres (.0) = Η sp Η res ( ) ( ) ( ) o Η Η Η where o PC Η PC represents or particle coupling and also known as Coriolis term ( H Corriolis )... Calculation of Total Wave Function The nuclear wave functions are the Wigner D-functions which are the transformation functions for the spherical harmonics under finite ations. The simultaneous Eigen functions of,, and are denoted by Ω [oy, 967] and eigen values of these operators are given as z

6 ( ) Ω = Ω z Ω =Μ Ω (.) Ω = Κ Ω Ω = Ω Ω The wave functions corresponding to Hamiltonian given by eqn (.0) when ( Κ 0 ) can be ( r) and the Wigner function written as the product of intrinsic wave function ( ) ( M (,, )) D θ φϕ which is basis state of a ating system. Ψ Ω = Ω = χω( r) D M,, 8π ( θφϕ) The above factorization of ator wave function (,, ) ( ( r) ) χ Ω χ Ω ( M ) (.) D θ φϕ and particle wave function is possible in adiabatic or no coupling approximation and total angular momentum and ator angular momentum are both labeled by. Because of axial symmetry, the nuclear wave function should be invariant with respect to ation ( θ θφ φ α),, of body fixed frame about symmetry axis by an arbitrary amountα. Thus M Κ ι α ( θ, φϕ, ) = ( θφϕ,, ) D e D χ Ω = e Ω ι α χ Ω M (.) The exponents have opposite signs because the wave function D (of the body fixed frame) with respect to laboratory system, where χ Ω is the wave function with respect to body fixed frame. Ω Ω Ψ = Ψ is satisfied provided Κ =Ω. The additional invariance which must be considered is ( θ π θ φ π φ ).Under ιπ ( ) ( Κ,, ) D θ φϕ = e D ( θφϕ,, ) M M, M,, To investigate the transformation of χ under Ω, we expand χ Ω in terms of wave functions χ Ω which are eigenstates of the angular momentum, χ = a χ (.4) Ω Ω Ω

7 The wave function χ Ω (in body system) is related to the wave function Ω D Ω ΩΩ Ω χ Ω (in laboratory system) by the relation χ = χ (.5) Under (.6) ιπ ( Ω) ιπ ( Ω) χ Ω = χ = χ = χ Ω ΩΩ Ω Ω, Ω, Ω Ω Ω D e D e Thus normalized nuclear wave function which is invariant under the operations and, is given for following two cases: Case. For Κ=Ω 0 Ψ = = χ ( ) Κ rdm ( θφϕ,, ) ( ) χκ ( rd ) M, ( θφϕ,, ) 6π Case. For Κ=Ω= 0 Ψ Μ0 = Μ = χ 0( ry ) Μ( θφ, ) π (.7) (.8).. Calculation of Energy Eigen Values The Eigen values of the total Hamiltonian given by equation (.0) can be calculated by using total wave function following two cases: Ψ given by equations (.7) & (.8). We consider the Case : For otor Particle Coupling Term ( Η ) =0 PC f the particle is tightly bound to the or so that energy-level separations of Η 0 are large compared to the ational energies. The Eigen values Ε of ational spectrum) Η Η 0 0 are given by (the Ε =Ε 0 ( ) Κ with =Κ, Κ,... (.9) Case : Diagonal and Non-Diagonal Contributions of otor Particle Coupling Term: Calculations of various matrix elements The matrix elements corresponding to various terms given in equation (.0) within basis function M Ω ) are given below: 4

8 Matrix elements of rational term = CΚ ( ) = ( ) Matrix elements of otational term ( ( ) ) 0 Η = ( z ) = ( ) δ Matrix elements of the PC term δ (.0) (.) Calculationof H PC = = ( r) D ( ) ( r) D Κ Κ Μ, Κ χ χ 6π D χ ( r) ( ) D Κ Μ, Κ χκ ( r) = 6π (.) = = D ( r) ( ) ( r) D Κ Κ Μ, Κ χ χ 6π χ χ Κ Μ, Κ Κ = D ( r) ( ) D ( r) 6π (.) HCorriolis = HPC = [ ] (.4) H PC ( D χ ( r) ( ) D χ ( r) Κ Μ, Κ Κ ) = 6π ( ) 6 ( D χ ( r) ( ) D, χ ( r) π Κ Μ Κ Κ ) (.5) H PC ( D χ ( r) ( ) D χ ( r) Κ Μ, Κ Κ ) = 6 π ( ) ( D χκ( r) ( ) DΜ, Κ χκ( r) ) (.6) 5

9 H PC ( D χ ( r) ( ) D χ ( r) Κ Μ, Κ Κ ) = ( ) 6 ( D χ ( r) ( ) D, χ ( r) π Κ Μ Κ Κ ) ( D χ ( r) ( ) D χ ( r) Κ Μ, Κ Κ ) ( ) ( D χκ( r) ( ) DΜ, Κ χκ( r) 6π ) (.7) Calculations of various parts of first term of equation (.7) D D χ ( r) χ ( r) Κ Κ = ( Κ)( Κ) D D χ ( r) χ ( r), Κ Κ 8π = ( Κ)( Κ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) D D χ ( r) _ χ ( r) Κ Κ = ( ) ( Κ)( Κ) D D χ ( r) χ ( r), Κ Κ 8π = ( ) ( Κ)( Κ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) D D χ ( r) χ ( r), Κ _ Κ = ( ) ( Κ)( Κ) D D χ ( r) χ ( r ),, Κ Κ 8π = ( ) ( Κ)( Κ) δ χ ( r ) χ ( r ) Κ, Κ Κ Κ ( ) D D χ ( r) χ ( r),, Μ Κ Κ _ Κ = ( ) ( Κ)( Κ) D D χ ( r) χ ( r),, Κ Κ 8π = Κ Κ ( ) ( )( ) δ χ ( r) Κ, Κ Κ χκ( r) 6

10 Calculations of various parts of second term of equation (.7) D D χ ( r) χ ( r) Κ Κ 8π = ( Κ)( Κ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) D D χ ( r) χ ( r) Μ, Κ Κ Κ 8π = ( ) ( Κ)( Κ ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) D D χ ( r) χ ( r), Κ Κ 8π = ( ) ( Κ)( Κ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) D D χ ( r) χ ( r),, Μ Κ Κ Κ 8π = Κ Κ ( ) ( )( ) δ χ ( r) Κ, Κ Κ χκ( r) Using these contributions in equation (.7) we get, HPC = ( Κ)( Κ ) δ χ ( r) ( r), χκ Κ Κ Κ = ( Κ)( Κ ) δ χ ( r) χ ( r) Κ, Κ Κ Κ ( ) ( Κ)( Κ ) δ χ ( r) χ ( r) Κ, Κ Κ Κ The last term in above equation contributes only for 7 (.8) Κ = bands and has dual contribution diagonal as well as non-diagonal. t will have diagonal contribution, when given by: Κ= and this is ( ) ( ) ( ) ( )( ) χ( r) χ ( r) a Κ Κ =, where a is called the decoupling parameter since it corresponds to partial decoupling of the particle motion from the ator.

11 . CONCLUSON n the present thesis, our focus is the theoretical understanding of high spin features, especially signature effects of one quasiparticle (qp) bands in A=80 mass region using a Particle plus otor Model(PM) approach. Although there are many other approaches such as cranking model (CM) [Bengtsson et al.,984], Tilted axis cranking (TAC) model [Pearson et al., 997] Hartee-Fock-Bogoliubov (HFBV) [Szymanski,98; Voigt et al., 98 ] model etc. existing in literature for understanding of high spin features of qp bands. But the (PM) approach is in terms of an angular momentum, a physical observable in experiments, so that direct comparison of PM results with experimental data can be made. The role of angular momentum dependence of inertia parameter and ational correction term, respectively, in explaining the zero and pronounced staggering observed in qp bands has been discussed. 8