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
«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. Accordingly, we shall use the strict relativistic form of the beta decay hadron current J which in Blin-Stoyl's notation /9/ looks as follows : if + Here fA » Xfy. As shown in /S/ the expressions for %he nonrela- tivis'tic dominant vector and iixial vector formfactor coefficients are J.I,-1 J£»OIf 0;I F"° 7 I l Ri°i Vf SV°KKO IkvcW^V (13J 1 1 J =0I ACFCN) _ C, .A-K~i/T) X " ţ r i !* Cr f f , D ,. i 3 ,0J , _nI.T |i JCAT " " t f * S^ Rfa£ ^fWp f°' f || ik^KN)SaiRi; [Vn)Ji;Ir where the summations are running over i , j , I , j , R-, Rr, a-, a,, ana J-. Using the angular momentum a-lgebra the expres sions of (13) reduce to single particle matrix elements «VPIIMKOIKV Csee ref. /4/) Similarly, the relativistic formfactor coefficients con necting small and large Dirac spinor components are J-I-=l J =OI V J K n rf U1f KA« Y - (-) ' l/£ Î c c I. R^ 0j m l 0 H) ' ****£'• lW £ °i iV * «iVW^V (14) Ap(N) c f f fKKO ~ I cR a K Ri*i f f N) Situations may be figured when thf formfactor coetficients F k mnp and F mnp are anofflalousi KK0 ( e ^ KAY ^e ^ y small and 8 then the complete relativistic expression of the beta decay hadron current (eq. 12)) should b included in the calculation. The formfactors fv and f. characterize the strengths of vec tor and axial vector coupling , f corresponds to the induced weak magnetisjn , fc coincide? Ln its form with the induced sea- lar interaction, -fp is proportional with induced pseudoscalar interaction and may have different origins. Assuming conservation of vector current we have in the limit q +0 (see ref. 19/). fvC0) - 1 ; fwC0) * - |^ ; fs - 0 (15) The situation is no longer easy in case of axial current terms because this current is only partially conserved. Weinberg /10/ studied the behaviour of weak current V -A under G-parity transformation G = C e iwt2 (16) v where C is the charge conjugation operator and e 2 iS the charge symmetry operator (a rotation about the 2-axis in the iso- <:pin space through an angle w ). Using (16) the beta current (12) can be separated in a part invariant under G - transforma tion, with the formfactors f„, f., f and fp (first class terms) and a part noninvariant containing the terms fg and £j (second class terms). As f<, = 0 (see (15)) the only axial vector current term noninvariant under transformation (16) remains f~. There are general arjuments /ll/ indicating that if the nucleons are considered free particles the only remaining terms are the first class ones (i.e. f~ = 0). However, Wilkinson /12/ has pointed out that presently available data of beta transi - tions in mirror riuclei are consistent with relationship (ft) • . f s a rjr/r - i - i -y- cw0 * wo) (i7) yielding a £j = 2 x 10 value. This mir.-or asymmetry in beta decay of ligh.t nuclei was discussed by Blin-Stoyle /13/, /9/ Wilkinson /12/ and Blomqvist /14/ in terms of differences in th wavefunctions of mirror nuclei and off-mass shell effects. More- 9 - over, nuclear structure effects as for instance those related to change in isospin multiplet configuration, higher order re- lativistic corrections, or possibly charge asymmetry of nucle ar forces which have not been investigated thoroughly so far, could also produce fT f 0 in the axial vector currents. In the present paper we considered the Wilkinson value f- - 2 x 10 as an empirical data and introduced it in calculation taking no care about its origin. The pseudoscalar term was exactly computed although as suggested in /9/ its contribution might be significant only in case of 1st order forbidden transition with 4J • 0 or in tran sitions of higher order. Now, from the complete relativistic expression C12), we derived the final expressions for the nonrelativistic formfac- tors coefficients as follows; MS x T + 2N * 101 (T/& [2NI(r)Tl»(r)]BT121 k mnp) + Ml! ( e * *\S(ke«np) *T in/îjcji) [(3*2N)I(r)Tl'(r)] 2N x »T101 - p-J(|> [ZNI(r) * rI'(r)1.8Tin • (f) I(r)[W0R*oZU(r)3gY5Tin U8) 10 1 V Vw(N) r. „ , 3.7 (7 r ZN* . FWckemnp). ff Cj) [ZNI(r)Tl-Cr)J x 6Y5Tnrf^'N -wCV* oZUCr^ &Tioi[ A A 2N : F^Jckemnp) -- X M^'{ Ckemnp)*fT J||/f (£) [2NI (r)*rl (r)] x 6T + R oZU 22l "J $ W K> * W]6r5T211 + f N p*/I jV !>2NICT)*rI'Cr)]BT 5T220 , Assuming an" uniform charge distribution in the nucleus 0€ took U(r « 0) - 1.5 and then we got : 1 f r 1+2N J J 4) Ukemnp;r) [W^ • .ZUCO] er5Tm„ . [ „^jSigJo^l + 0 1 JJCj) Hkemn^;r) [w^aZUCr)]^^- -[V^Mll^e* * The relativistic matrix elements/ ST101 and / T101 can be calculated either directly or invoking the CVC theory which yields straightforward relationships between relativisticJ T,Q, and nonrelativistic JT,0, matrix elements. Indeed the 4-diver- gence of the vector current should be zero except for effects from violation of exact conservation of isospin due to nuclear Coulomb interactions. Using Dirac equation one can show that in the nonrelativistic limit the relativistic matrix elements should have the same order of magnitude : Koi * />Tioi * Vjt 120) 11 - where vn denotes the average nuclear velocity inside nucleus. The d.rect calculation based on our model wavefunctions yielded small values : T 6 7 ioi MJVr10! \- io' - io" •\ dve to ver/ strong cancellation. Much larger values, TO- 3 - -104 were obtained by ir.cans of CVC-based approach. Those v . re thought as more reliable because the CVC relations a odel independent. A lower hound was estimated with the help of certainty rela - ticns. indeed, M and [: being the nuclear mas? and radius respec tively, we have in cnse of °Tm nucleus T = 2 i|»iU mi0^l jI L1 j! — Ii - 3.1733.17,4 x 10~ (21; Thus, to evaluate the most important induced terms in CIS) we can use the lower bounds : I^OT/1&" «loiii^Mira (22) 3 7 f r 2N r - 3*7I WoR * -~HcE> Here again U(r=0) « 1.5 and U17°Tm) - (—) - 14.8 2R Tn In /4/ we neglected the pseudoscalar interaction term and also the terms containing the expression t2NI(r) • rI'(*)J . In * the present paper we calculated them using the formulas (14) of paper /4/ with the following single particle matrix elements : r- K+2N , J«**«-°« n* p xW. n» n Jp' pţBp nn ^J L - 12 f r hK+2 *"N 2 x 8fCr,X£)%) •Cr)fiCr,xi)r dr+signCxjOG^^-C-xr.Xil "» f r K+2N 2 X| fj^Cr.AfK^) *Cr) gfCT,Xf)r drfV; Pt P u~a, 4 n £23) 3rf 3h «*; 'l a + 2N 2 x j gf Cr,xf) Cj) *(r)£i(r,xi)r dr+sign(xf)GKAyr-xf,xi) * f A+2N | fCr.xf) (f) • WgiCrvx^r dr f *i » u, » 'p p 'n n In the nbnrelativistic limit of the Dirac equation the snail radial wavefunction f(r,x) can be expressed in tenu of the large g(r,x) a^i fC'»x) -[fe ^>^] gcr.x ) (24) The gC r.x) function can be understood as solution of the Schrodinger equation. By GxAr^xi'xf^ we denoted tne spin-angu- 'af part of the saigle particle reduced nuclear matrix elenent: 13 i. +a_+A n ? 1 y 1 GKAYCxf,Xi) - CD V & VpVt ^C^)" jp r V*f> *pCxf) &n(Xi) A Jn I *nC*i) C25) 0 0 ) Here l(x) 3x if X>0 and *(x)a ixl"l if x < 0. The parity quantum number is defined as |x|* j+ j (if X > ° *nen x'* * and if x < ° then "X * l+i)• The other notations her» have the same meaning as in paper /4/. 3. RESULTS AND DISCUSSION By means of the formfactor coefficients we calculated the reduced JiaIf lives, the spectrum shape functions CCWg) and the beta-gamma angular correlation coefficients e2(We), accord ing to the relations In 2 5.982 x 10" ft - sec C26) s; ccwe) ccwe) g T (27) l ^ " t"i.. <*i M.M'. r'" * + b e2CWe, I"C«2*CY)0 ) - c(WJLn ^Tt LL' f^, (2; 1,2,0) ® o L, L (28) ,The beta decay transition amplitudes Tj^Mr were computed by the fcelp of the matrix elements of the beta decay current (12) while the C(We) functions were obtained by means of a qua dratic form in- Mk(ke,k) and mk(ke,k) which in turn are linear 14 combinations of the formfactois coefficients /8/. Finally, the coefficients appearing in the beta gamma angular corre lation e c C0S e) ut«re,e) * l * I n W n=even 21 were calculated as functions of the quantities T.., and b*(\ , as defined by Schopper /17/. In the calculation of the formfactor coefficients we = used the equilibrium deformation parameters B2 0.28, (J.=-0.02 and Y • 12.3°» which fit best the experimental data in the region A'* 170 /18/, /19/. The moments of inertia in the rota tional part of the Hamiltonian were obtained, using the VMI approach, from the rotational states of the even-even nucleus 16ft — Er. The nucleon configuration giving the 1. ground states of °Tm (.denoted by I) was build upon the orbitals : 02d5/2)Olg7/2)(Tr2d3/2) x Cv2f?y2) (v3p3/2) It yielded a ground state energy of 2.57641 MeV, only slightly differing from the value 2.4887 MeV given by the simpler con - figuration (denoted by II) : Uihn/2) x C'vii13/2). In* the table 1 there are given the values of the form- A(0),(i) . (0),(i) , nn A ro) m A A factor coefficients FX11 ' Fm . Umj, AFIOJ,UJ and AF2iiUlll) computed with the configuration I and with (a) without (b) induced interaction terms neglected in paper /4/ and also with no induced Co). Locking at rows (c) and (d) in this Table one sees first that when the entire induced interaction is switched on Cease (a) relative to case (o)) the axial vector formfactor coefficients generally decrease (more, steeply when K-l as compared to K«2) and the effect of the additional terns Cease (a) relative to case (b)) amounts vp to a few percents. Secondly, one can remark J markedly effect of the addi tional terms ior the l" •*• 2+ transition while when K • 2 the effect of thef.e terms is even opposite in sign'as compared with - 15 - the effect of all the other interactions taken together. Thirdly, from a general overview on this Table it cculd be suggested that the ground state and the first ex cited state wavefunctions given by the model used are un - equally appropriate in describing the beta decay matrix elements. The table 2 shows a comparison of calculated ft- values with the experimental data. At a first iook one sees that introducing the entire induced interaction entails slight departure from the 'agreement with experimjnr. and thj agreement is scil.\ better than in case when no induced !:•- ttraction is implied (case (c)j. In fact the agri.-:Mieiit wir:- experimental log ft - values should he considered In CASC (a] as excellent because the core polarization effects en reduced life times, here neglected, contribute positively to iog it - values by an amount < 0.6. The results for the spectrum shape coefficients CO" ) are given for the transition 1* -* 0* and i"" -<- 2 f in Figs. 2 and 3, respectively. Here- the full line represents tiie. results obtained in this paper, whereas the dashed line represents the previous results /4/, when pseudoscalar in duced interaction term and other integrals were neglected. The relative agreement with the experimental data is some - what worse in case of the l"-*- 0 transition against the case of the 1" •> 2 transition. Thus one can state that the ground state wavefunction describing the beta decay could be more inappropriate as compated to that describing the 2 excited state. Finally in Fig; 4 the beta-gamma angular correlation functions e2(We) are shown. The theoretical values appear to match better the experimental points when the pseudoscalar term and the additional integrals are introduced (full line) than when tr. *.y are neglected in the induced axial vector current (dotted line). As concerning the contribution of the pseiidoscalar ter« alone, it appears to be zero for the l~-»-0 transition While for the transition l" + 2 its order mag - nitud© is only about 10" . Thus at least for the case of the beta decay of Tm the contribution of this interaction term is negligible. Consequently our results give no indication on the existence of the seconjs) class currents. The value - 16 - f„ = 2 x 10~ used in our paper nay just indicate that the small terms occuring from the various nuclear effects men tioned above can simulate a fT nonzero. In conclusion one can st^te that taking Into account the weak magnetism and tensor-irtteraction terms in the ha - iron beta decay current contribute substantially in getting ;.greeraent' with experimental data. In view of rhis good agree- nent, it should be worth investigating further and to bring .;iouc i mere detailed fit of theoretical e-clir-ate with the ex- •;orime;:tai data concerning the dependence O-L t>w W^ energy, and also to improve tno wavefunction descrying the ground •tate of the even-even daughter nucleus. 17 - Table 1. Formfactor coefficients calculated with configuraţiei I with (a)• without (b) additional interaction terms and with no induced interaction terms (o). In rows (c) and Cd) the J GO-(o) |/Co) % and | (b) - Co)|/Co) % ratios are given with arrows 1 CI) indicating increase ( decre ase "L relative wO case Co). —• 1 ApCo) ApCl) A Trans; 'ill Flll ^J J (1111) F}}JCIIH) N r-o* a -l.o45.6xlo""2 -1.6827xlo"2 -1.155xlo'2 -1.164xlo"2 b -l.o43 xlo"2 -1.121 xlo"2 -l.o48xlo"2 -1.151xlo'2 o -o.l517xlo"2 -0.2369xlo"2 -O.1571xlo"2 -O.2697xlo"2 c |S89.2 (610.3 j 635.2 J331.7 d 1587,5 \ 373.2 J 567.0 v326.7 r-z* a -1.9404xlo*Z -1.4954xlo"Z -2.153xlo~Z -1.279xlo"Z b -1.446 xlo*2 -1.483xlo"2 -l.S45xio"2 -1.571xlo"2 o -0.5972xlo"2 -0.6384xlo"2 -0.7066x10~2 -0.7347xlo"Z c ?229.9 j134.2 {204.7 i 74.0 d Î 142.1. i132.2 ill8.6 T113.8 Table 1. Continuation \ fFC ApCO) ApCl) A ( A TransS^ r211 F211 P2 JJ(1111) 1» r-o* 0.0 0.0 0.0 0.0 • + r-2 a. 4.156xlo"2 3.153xlo"2 2.4195xlo"2 3.5695xlo"2 b 1.86Sxlo"2 2.631xlo"2 2.019 *lo"2 2.978 xlo"2 o 2.078xlo"2 2.931xlo"2 2.249 xlo"2 3.318xlo "2 c f 100 f 7.5 T 7.6 • 7.5 d 1100.5 .J 10.2 j 10.2 {10.2 - 18 - Table 2. Reduced life tines of Ta calculated in configuration I with Ca)» without, (b) additional interaction teras and with no induced interaction tera (0). lQ ft Transition < * >theor. (log ft)exp a b c 8.897 8.95 11.07 9.0 r + 2* 8.84 9.16 10.73 9.3 - 19 - REFERENCES 11/ Bogdan D. et al.: Nifcl.Phys. A119, 113 (u*68j. Behrens H. and Bogdan D. : Nuc 1.Phys. AI43, 463 (19,70). 11/ Faessler A. and Toki H.: Phys.Lett. 5;>B, 106 (I97S) . Toki H.et al.: Phys.Lett. 66B, 510 (li.-""";; Phys.Lett. 71B, 1 (1977); Faessler A., Poggel G. and Schmid K.W.: Z.Phys. A299, 55 (1931). 11/ Stephens F.S. et al.:Phys.Rev.Lett. 29, 348 (1972). Stephens F.S.: Revs. Mod. Phys. 47, 43 (1975). /4/ Bogdan D., Faessler A., Schmid K.V. and Holan S.: J.Phys. (G) (1982). /5/ Bogdan D., Faessler A. et al.: J.Phys.(G) 6, 993 (198o). /6/ Meyer-ter-Vehn J.: Nucl.Phys- A249, 111; 141 (1975). Toki H, and Faessler A.: Nucl.Phys. 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I\M Blomqvist J.: Phys.Lett. 35B, 375 (1971). /15/ Buhring W. and Schulner L.: Nucl.Phys. 65, 369 (1965). /16/ Stech B. and Schulke L.: Z.Phys. 179, 314 (1964). /17/ Schopper H.F.; Nuclear Beta Decay and Weak Interactions, North Holland» Amsterdam, 1966. /18/ Soloviev V.G.: Teoria Slojnikh Jader, Nauka, Moskva,1973. /19/ Gareev F.A., Ivanova S.P., Soloviev V.G. and"Fedotov S.J.: Physics of Elementary Particles and At.Nucl. 4; 357(1973). /20/ Van der Werf S.Y. et al.: Nucl.Phys. A134, 215 (1969). Van der Werf S.Y.: Ph.D.Thesis, Univ.of Groningen, 1971. FIGURE CAPTIONS Figure 1. The beta decay scheme of °Tm Figure 2. The spectrum shape coefficients for the transi tion 1~ •*• 0+. The experimental data are taken from ref. /20/. The full line represents the present results with total induced interaction taken into account while the dashed curve represents earlier re sults IM when additional induced interaction terms were neglected. Figure 3. The spectrum shape coefficients for the transi tion 1 •* 2 , See explanations to Fig. 2. Figure 4. The calculated and experimental values of the oeta-gamma angular correlation coefficients + + c2(We) of the 1~(B)2 (Y)0 cascade. The full line represents the present results while the dashed line represents calculations reported earlier J A/. - 21 Q«-=968keV E loqft (key) * 2* 883 9.3- Sk.25 0+ 965 8.9' 0 170. 70 Yb Fij. 1 - 22 - no Tm r-+o+ 11 I 8i 11 i ai ^ :/ * a * / ' * /.J*< ,' «* ^ 1 w 1.5 2.0 IS Jtf Fig. 2 - 23 *vTmM* fj- 1.5 2.0 2.5 3.0 2 We [me C ] rig. 3 - 24 - 170 . . Tm 1+2 + Ang corr an H. Dulaney J.C. Manthurufhii } A.Khoyyoom 008 QM- £ <0 -OM- 000 - - 012 - 2 Wjmec J Fig. 4