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Ft-216-1.982 CENTRAL INSTITUTE OF PHYSICS INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING Department of Fundamental Physics \tiu- FT-216-1.982 . September THE RELATIVE IMPORTANCE OF RELATIVIST1C INDUCED INTERACTIONS IN THE BETA DECAY OF 17oTm D.Bogdan, Amand Faessler , M.I. Cristu, Suzana HoIan ABSTRACT : The log ft-values, the spectrum shape functions, and the beta-gamma angular correlation coefficients of the 17o Tm beta decay are computed in the framework of relativistic formfactor formalism using asymmetric rotor model wavefunctions, Main vector and axi*l vector hadron currents being strongly bindered, the relative importance of induced interaction matrix elements is accurately estimated» Good agreement with experi - ment is obtained for the beta decay observables when the main induced interaction terms were taken into account. The contri­ bution of the pseudos'îalar term was found insignificant. *) Permanent address : Institut fur Theoretische PhysiJc, Universităt Tubingen, D-7400 Tubingen, West - Germany 1 - 1. INTRODUCTION The theoretical evaluation of observables in the first forbidden beta decay provides an excellent mean to assess the relative importance of the so-called induced interaction cor - rection terms contributing to the beta decay probability when vector and axial vector nuclear beta matrix elements are strongly hindered by either mutual cancellation or selection rules. In this paper the log ft-values, the shape of the beta spectrum and the beta-gamma angular correlation were computed for the l" -* 2 and l" •• 0* beta transitions of the Tm ground 170 state towards the first excited and the ground state of Yb nucleus, respectively. For the Tm nucleus which is strongly deformed ( 02 = = 0.28)*the dominant coupling scheme should be the strong coupling. The failure of some earlier calculations /1/ of the beta decay observables /20/ is thought to be due to the unsuit­ able choice of the odd-odd ground state wavefunction of J'°Tm. In the present calculations, the residual interaction between the odd particles, the gainma-asymmetry, and t£e competition /2/ between the strong coupling and decoupling scheme / 3 / which mix through Coriolis force the projection quantum number of the odd particles, were taken into accost in calculating the im - plied wavefunctions. Thus the odd-orîd ground state is no longer a sum oyer the single-particle angular momentum projection fl to the nucleus z-axis but also over the orbital angular momentum i and the total single*particle angular momentum j. In a recent paper /4/ dealing also with the beta decay of * uTm the odd - deformed ground state wavefunction was taken as a combination of spherical states |zjn> with only one pair of angular momenta t and j corresponding to the proton and the neutron respectively •'(*1лц/2) х (v 1*13/2) • Tn*s wave " function led to large formfactor coefficients, resulting in small log ft- values, small and negative beta-gamma angular cor­ relation, and low beta spectrum shapes with respect to the expe­ rimental data. What became even more important was that these results could not be improved through addition to the beta decay -currents of mere and more small Induced interaction terms. These - 2 - terms are ail too small to amount to a significant contribu­ tion. In the sanie paper better resultr were reported by adopt­ ing both competition between strong coupling and decoupling schemes and a more realistic structure of the I'^Tm ground state:- [w 2d5y2) t* lg?/2) (it2d3/2) x O 2f 1j2) Cv 3p3y2) In this case due to the strong cancellation among the matrix elements the formfactor coefficients were strongly re­ duced and although the disagreement with experiment persisted, the way to improvement was open now because the main (Vu-Av) matrix elements got reduced down to the order of magnitude of the induced weak current terms, namely fw and f-r terms cor­ responding to the weak magnetism component of the vectorial current and to the induced interaction term in the axial current respectively. The present paper is intended to investigate further within the same^ nuclear model, the behaviour of beta decay ob- servables when all induced interaction terms are accurately computed with the pseudoscalar interaction terms also added to the calculation of the formfactor coefficients. Like in the previous work /4/, /5/ the main ingredients adopted are the following : • A variant of the asymmetric rotor model adapted to include the strong coupling scheme /6/ was used. • The CVC thjeory was assiimed to hold true for the beta decay vector current. • In the nonrelativistic limit the CVC estimate of the relativistic matrix element jTiol w*s shown,using the Dirac equation, to have about the seme order of magnitude as the nonrelativistic estimate of the \0Tlol matrix element. • No off-shell meson exchange nuclear effects were taken into account so that the fj"2xlO"3 value- was used in calculation as an empirical value. An accurate calculation of fj was thus postponed for a future search of still more detailed agreement between theory and experiment. 3 - • Exact expressions were taken for both the Nilsson wavefunctions and the integral factor [2NI(r)+rl*(r)j entering the induced interaction components of the formfactor expressions. • The S~ - transition matrix elements were obtained in the quasi-weak representation in agreement with charge conservation and charge conjugation laws (the quasi neutrons get transformed in quasi pro - tons]. 2. THE 6£TA HAMILTONIAN The competition between the strong coupling 11/ and decoupling scheme /3/ as worked out by Toki et al. HI in their asymmetric rptor model was taken into account in calculating the terms of model Hamiltonian. H = tL„^ + H , ,., + H, c * H . + H (1) rot she11. def pair res By diagonalizing the total matrix H one gets the wave- functions c". , . _ and the energies En (jp3nJJ;Ra * I |n;IM> - I CnI |Ra ;(j j )J;I> C2) UoJn)J;Ra,nl «p3n)J-.R« Here n stands for the ordering number of the states of same total angular momentum I. The basis wavefunctions are given in the quasi-weak coupling representation by iRa;UDJn)J;I>- I CCRJI^MjM) C(j JnJ;m mnMj) x MJtMR,M (3) oR ^ R X Bi i m Kim lBCS> I BN SM N Vp"p Vnmn all N NR MRNR (even) The core wavefunctions used in (3) are expressed by 4 - jRaMn> %.^VR*™\-»j-aJNOSRNR C4, Cevenj (even) where RaR R .oR v, ~7 AN„>0 H lC5T"Cl+fc S J V ' R'° J.OTT R aR c-r R = 2R+1 NR>0 'Ihe creation operators 8„. generating the sp\erical r Ijm quasi-particle states, defined in the laboratory frame are given as functions of quasi-particle operators o- defined in the nucleus associated frame which in turn are given by the Bogolinbov-Valatin transformation as follows 3 a+ (5) ijm I X "\-o L mp. ijfl l «fi + u a v a l i i i i In our model for odd-odd nuclei extended to include the competition between strong and decoupling schemes, the odd deformed orbital is no more a combination of spherical states |AjO> with only one orbital angular momentul l and one to - tal single particle angular momentum j. Instead, now the par­ ticle creation operator is expressed by I W1 r * (6) so that the summation extends not only over the single partdcle angular p1 Section u to the nucleus z-axis but also 6ver % and j.'N- the Ha»i"*>nian (1) reads : 1 - 5 - 3 R2 r2Y. z n-1 *nR ija t^?0. •^^„•Y^li-J'tt^c^c,^. (7) G C C e C 4w <a|Y,.. !Y> I ljQ lj5 *«j»o' l'j'5' " * Ml * '""XV *jO>0 Ap aST« «MY^NC^cJc^)) where R_ are the core_ angular momentum components relative to the nucleus frame and the momenti s of inertia 9 R for the asymmetric rotor are given by enR" T9oRsin2^-^n) '.5Î with 9 „ denoting the VMI parameter. The parameter k iii che deformation term of the Ha- nrltonian has a well known expression dependent on A and 0 -deformation : k . y-f. 206 A ji-far C9) r For the calculation of beta decay observables the form- factor formalism /8/ was adopted/where the following nuclear beta decay operators are used : % r A+2N * ' M Ck mnp) °XA>M e "& Kkemp;r)U-*Y5) Tj^t, DO) TKAy A K Ti C-D " * YKM» , T- 1 where e = V. or A, X = 1.2, k is the quantum number related to the angular momentum of the electron and m, n and p are the powers occuring in the radial electron wavefunct'ions m n p ( meR ) (We R ) ( az) The expressions ot radial integrals I(keranp;r) and the set of or­ thogonal tensors TÎÎ. are given in /8/. The tensors (10) give the reduced matrix elements of the beta transition via the formfactors Fj^. which aTe functions of the square momentum transfer q and the formfactor coefficients v FC« rKAr RAY v N (2N)11(2 A+2N-1)11 v RAY which carry the information about the nuclear structure. The ma­ trices occuring in eqs. (10) y =-iBa , y.~ -B , Tr = TTy^, a » 0Yr , o = oYc, and a = - •*! (Y Y - Y Y ) are the matrices 5 ' '5' jiv 2 ^v v v ii . associated to the Dirac equation (ap • em + W)<> s 0. In calculating the formfactor coefficients the assumption was been made that the beta decaying nucleons inside the nucleus interact with the leptons in the same way as the free nucleons. This is a drastic approximation which was not thoroughly tested so far. Thus meson exchange effects and other many-body processes are neglected.
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