IBA-1 calculation of E2M1 mixing ratios in 154Gd

IBA-1 calculation of E2M1 mixing ratios in 154Gd


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Volume 139B, number 1,2


3 May 1984

IBA-1 CALCULATION OF E2/M1 MIXING RATIOS IN 154Gd P.O. LIPAS 1, E. HAMMARI~N z and P. TOIVONEN Department of Physics, University of Jyviiskylii, SF-40100 Jyviiskylii, Finland Received 30 December 1983 Revised manuscript received 14 February 1984

The general theoretical scheme, with various special cases, is presented for calculating ~,-ray E2/M1 mixing ratios for even nuclei on the IBA-1 model. The M1 transition operator needed for unconstrained algebraic signs contains three parameters. Numerical results for lS4Gd are compared with Warner's simplified IBA treatment and with Kumar's model. Sign flexibility is demonstrated by 16SEr.

The 7-ray multipole mixing ratios 6(E2/M1), or the energy-independent quantities A(E2/M1), provide a sensitive test o f collective nuclear models. The reduced mixing ratio A(E2/M1) is the ratio o f the E2 and M1 reduced matrix elements, and it has both signs, reflecting the phase relations of wave-function components. In addition, M1 transitions are generally weak between collective states and they are described by the models as a second-order effect, since the leading term o f the M1 operator is proportional to the angular momentum. Arima and Iachello have given for their original interacting-boson approximation (IBA-1) the M 1 operator in the restricted case o f U(5) dynamic symmetry [1] and generally [2]. However, even when starting with the general operator, they derive the E2/M1 mixing ratio by neglecting the terms which break SU(3) symmetry [2]. It follows that the reduced mixing ratio is given by the same simple formula for both U(5) and SU(3) symmetries. The formula contains only one parameter and the initial and final spins. It yields the same sign and the same order o f magnitude for all transitions in a given nucleus. This is generally

1 Work done partly at Brookhaven National Laboratory, Upton, NY, USA, and Joint Institute for Nuclear Research, Dubna, USSR. 2 Present address: Institut for Theoretische Physik, Universit/it Tiibingen, D-7400 Ttibingen, Fed. Rep. Germany. 10

contrary to experimental data in all types o f nuclei; see e.g. the recent compilation by Lange et al. [3]. Also it is well to note in this context that the geometric model, as treated by Grechukhin [4], again gives the same simple mixing-ratio formula. Indeed, as will be discussed in some detail below, the universality o f the formula results from the tensorial properties of the operators. An SU(3)-violating term in the M1 operator is also neglected by Scholten et al. [5] in their study o f the transition from U(5) to SU(3) in terms of Sm nuclei. Then the M1 and E2 reduced matrix elements become proportional for J ~ J -+ 1 transitions, exactly as for all transitions in refs. [ 1,2,4]. The proportionality factor depends only on the spins o f the initial and final states and on an adjustable multiplicative constant. All A(J ~ J + 1) for a given nucleus thus come out with the same sign and of the same order o f magnitude, and their ratios are the same for all nuclei. This is contrary to experimental data where both signs do occur, though not very frequently, and the magnitudes vary widely [3]. The description o f ref. [5] contains two M 1 parameters, the one mentioned above and an additional one for J ~ J transitions, which require a complete IBA-1 numerical calculation in the scheme o f ref. [5]. Warner [6] has developed an IBA description o f the E2/M1 mixing ratio whose point o f departure is essentially the same as that of Scholten et al. [5]. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Again the M1 matrix element f o r J ~ J -+ 1 transitions is taken to be proportional to the E2 matrix element, although there is no restriction to SU(3). The same universal behaviour is immediately predicted for J J + 1 transitions as in ref. [5]. For J -+ J transitions in deformed nuclei Warner presents additional arguments which again result in the same J dependence but with changed normalization for t3 ~ g. Beyond the early tests in refs. [1,5],Warner [6] and Girit et al. [7] have compared the simple universal prediction with experimental data for 7 ~ g and 7 -+ 7 transitions in deformed nuclei. We here sketch the general IBA-1 theory of M1 matrix elements [2.5] and proceed to apply it in unrestricted form to the 5(E2/M1) ratios of 154Gd. This nucleus possesses the richest data known [3 $ - 9 ] with altogether 15~i values for transitions between levels of ground,/3 and 7 bands; we restrict our test to them. Also 168Er will be considered as a quantitative example. The basic rule of the IBA is to construct the electromagnetic operators as linear functions of the U(6) generators in the sd-boson realization. Thus the E2 tensor operator is written as

3 May 1984

transition operator [4] is written as [aL] 1 with c~the physical E2 operator, to within multiplicative constants. Such an assumption can be based on the IBA dynamic symmetries [1,2,5] or it can be made quite arbitrarily for simplicity [6]. As for dynamic symmetries, it appears, at least in the rotational region, that even when the Hamiltonian is close to a symmetry, the E2 operator is not [10]. Altogether there is little a priori justification for the proportionality assumption. When the proportionality assumption is made, quite apart from any dynamic symmetries, it follows for J-~ J' 4:J transitions that the theoretical reduced E2/M1 mixing ratio is of the form [6]

A (J-+ J') =_(J'IIT( E2 )IIJ )I(J' IIT(M1)IIJ) = - B -1 [~- (J+J'+ 3 ) ( J - J ' + 2 ) X (J'-J+2)(J+J'-

1)] -1/2 ,


where clm =-(-)md_m. Similarly the M1 operator would be just/31 [dt~] 1" However, such an M1 operator, as well as its geometric counterpart [4], in collective coordinates of form [a&] 1, is proportional to the angular-momentum operator L and thus produces no M1 transitions. To have M1 transitions, the IBA rule must be extended to second order in the U(6) generators [ 1,2,5]. Angular-momentum recoupling shows that the most general second-order M1 operator can then be written as

withB relating to eq. (2) asBa 2 =B 1. This result follows from the tensorial, i.e. geometric, properties of the operators and is completely independent of the detailed nature of the states. If the further assumption is made at C = 0, as is done for the U(5) and SU(3) dynamic symmetries [1,2], or if the contribution of the C term is considered negligible as argued by Warner [6] for J - ~ J, 3' -+ g transitions in deformed nuclei, then eq. (3) applies also to J ~ J. We note here in passing that the E2 and M1 operators for the remaining dynamic symmetry 0(6) [11] are the same as for U(5), i.e./32 -- 0,B2 = 0, C -- 0, so that eq. (3) holds for that case too. Eq. (3) is also Grechukhin's [4] geometric result for all J ' . The reduced ratio A(E2/M1) is related to the conventional experimental mixing ratio 8(E2/MI) according to

T(M1) = (gb + A N ) L +B 1 [Q1 L] 1 +B2 [Q2L] 1 + CndL,

8 = 0.835 (E.r/MeV) (A/eb/~N1).


Warner [6] argues further that eq. (3), with a different constant B which is proportional to C of eq.

T(E2) = a2(dts + std) +/32 [dt3] 2 ,




Arima and Iachello's general M1 operator [2,5] is the same as this, although somewhat differently expressed. For M1 transitions, only the B1, B 2 and Cterms contribute, and the C term only for J ~ J . The assumption made in refs. [1,2,5,6] is that/32/a2 = B2/ B1, which means that the physical E2 operator is used in the construction of the M1 operator. The same assumption is present when the geometric M1


(2), should hold approximately even for 13~ g transitions in deformed nuclei. His argument relies on an estimated dominance of the C term, on a strongcoupling (Bohr-Mottelson) rotational structure of the E2 matrix element and on the J independence of the IBA matrix element o f n d in the SU(3) limit when N ~ oo [2]. However, he does not present a test of eq. (3) for/3 -+ g transitions. 11

Volume 139B, number 1,2


We have already applied the complete M1 operator of eq. (2) to the nuclei 146,148Sm and 172yb, in a primarily experimental context [ 12]. Since both signs occur in the experimental values of the/5(J ~ J _+1) for 146,148Sm, the need in their description for two independently adjustable parameters B 1 and B 2 is obvious. However, they both have only six relevant/5 data, some o f even them nonunique. The fi data are rich for 172yb do not provide a clearcut test of the IBA description because of the uncertain identification o f the higher bands. The nucleus 154Gd serves as a good test object, not only because of its abundant/5 data but also because Warner [6] and Girit et al. [7] have applied the simple theory of eq. (3) to it. Moreover, Kumar et al. [ 13 ] have calculated its 6 values on the dynamic deformation theory. The one shortcoming of 154Gd for our purposes is that it does not display the sign variation discussed above. Indeed, as seen in table 1, all the experimental 6(3' +.g) are negative and all the/5(/3 + g) positive. This sign pattern, which occurs in many deformed nuclei [3], is consistent with Warner's arguments but impossible to reproduce with one and the same constant as indicated by Girit et al. Unfortunately there are not many unique 6(E2/M1) data for nuclei which do exhibit a greater sign variety. The best-known case with different signs for/5(J-~ J +-1) is probably 168Er [3,6] and we examine it briefly below. The standard IBA boson number for 154c,.~ 64,Ju90 is 11. However, as calculated by Scholten [ 14], the shell structure brings about a different effective boson number, in this case 7. This value is intermediate between 11 and the value 4 which would result from Z -- 64 as a closed shell. We also found a better fit with boson number 7, which is used here. Indeed we found an excellent energy fit to the ground,/3 and 7 bands with the following PHINT [15] parameters (in keV): EPS = 0, PAIR = - 2 . 8 8 , ELL = 8.98, QQ = - 3 8 . 6 , OCT = 7.03, HEX = - 2 . 2 5 . These paramteres are clearly rotational and remarkably different from those reported in ref. [7], which are vibrational-intermediate. The fit of Girit et al. [7] has an rms deviation of 80 keV while ours is 31 keV for the same levels; however, they put OCT = 0 = HEX and have only four free energy parameters. The E2 parameters o f e q . (1) were chosen as a 2 ~E2SD = 0.2391 eb and %/~/32 =--E2DD = - 0 . 1 8 2 2 e b from a fit to B(E2; 2g ~ 0g) and B(E2; 2.r ~ 0g). The 12

3 May 1984

three contributing M1 parameters of eq. (2) were then obtained by fitting the A(E2/M1) ratios for the 2~, 2g, 3 -+ 2g and 2¢ -+ 2g transitions. The resultant parameters are (in/.tN)B 1 = --0.0014, B 2 = 0.0148 and C = 0.0049. (One should note that the program PHINT/ FBEM [15] has a W/'J missing so that it calculates reduced matrix elements of [QL] 1Ix~3.) Our numerical results obtained with three M 1 parameters are given in the last column of table 1. For comparison we have also calculated the mixing ratios according to Warner's [6] simple formulation embodied in our eq. (3). These two sets of results are seen to be quite similar. In view of 10~ + 10g, the simple formulation is here actually somewhat better, particularly since it has altogether only two parameters. Because of the similarity of the results, one might think that the condition/32/a2 B2/B 1 is approximately fulfilled, but this is far from being true since our parameters yield (32/a2 = - 0 . 3 4 and B2/B 1 = - 1 0 . 6 . As for the mixing-ratio results of Kumar et al. [13], table 1 shows that they have the right signs but the magnitudes are for 3,-~g transitions typically an order of magnitude too large. Their 13-~ g results are good, while their value for 3.r ~ 2 v exceeds the experimental value by three orders of magnitude. Of course their theory is less phenomenological than IBA-1 and does not have freely adjustable parameters, but overall the IBA-1 results in both versions, Warner's simplified and the present complete, are superior. Finally we give a specific example of the J + J + 1 sign variation. In 168Er all the experimental A(E2/M1) values [3,9] with known sign are, in eb//aN, A(3 ~ 2 g ) = 25.5. ~-.~.+3"~'A(37 ~ 4g) = - 9 . 3 ( 6 ) and A(4.r ~ 4g) = - - 1 6 4 ~ . To fit the first two on IBA-1 ,one needs both B t and B 2. We have thus found, after a complete numerical calculation, the values B 1 = 0.1660~t N and B 2 = -0.1519/~ N. Warner's scheme fails here [6]. Fitting the remaining A value yields C = --0.4522~t N. More A data are needed for testing the fit. We make the following conclusions about the ability of the present phenomenological models to reproduce and predict E2/MI mixing ratios: ( I ) The simplest models with only one M1 parameter, i.e. the strict U(5), SU(3) and 0(6) limits of IBA-1 and Grechukhin's geometric formulation, fail by predicting the same sign for all 8(E2/M1) ratios in a given nucleus. They all give the same formula. (2) The models o f Scholten et al. and Warner, with =

Volume 139B, number 1,2

3 May 1984


Table 1 Experimental and various theoretical E2/M1 mixing ratios for lS4Gd Transition

E3" (keV)

23" -~ 2g 3~, -+ 2g 33" --*4g 43" --* 4g

873 1005 757 893

53" ~ 4g


-9.7 -7.5 -5.6 -4.1



(5) c) (4) c) (2) c) (3) c) + 1.2 d) -4.3 -2.6

-13.3 (7) -8.9(5) -8.9 (3) -5.5 (5) +1.4 -4.9 -2.9

53" ~ 6g


+4.2 d) -6.9 _~

+7.1 -11.6 - ~

63" ~ 6g


+1.3 d) -3.1 _ ~

+ 1.8 -4.2 - ~

73" ~ 6g


+0.46 d) -2.71-0.61

73. ~ 8g


2fl ~ 2g

2Xth(eb/uN) Kumar et al. a)

Warner b)

this work

-56 -153 -127 -16.6

-9.5 -8.9 f) -6.5 -5.0

-13.3 f) -8.9 f) - 11.1 -7. !









-2.97 +0.50 -0.67



~6; -3.17__ + ". d)

+1.2 -5.7 -1.8




+1.5 c) 8.3 -1.1

+2.6 14.4 -1.9


14.4 f)

14.5 f)

413~ 4g



5.3 (23)




6fl ~ 6g


1.33 -0.14

, , , + 0 . 1 8 c)

2.86 -0.26




8fl --* 8g


+0.4 c) 1.2 -0.3

+0.8 2.3 -0.6



1013~ 10g


+0.5 c) 1.i -0.3

+1.0 2.4 -0.7



33" ~ 23"


+0.4. e~ _+(1.7 _0.5 ) ~





+3 4~ _+(15.5-4j4 )


a) Ref. [13], version DPPQ. b) Obtained from our eq. (3) with B = 0.0725 ~N for 7 ~ g and 3, ~ 3' transitions and B = -0.0479# N for fl ~ g. c) Ref. [3]. d) Ref. [8]. e) Ref. [9]. f) Fitted. two M 1 parameters, predict only one sign for the J J -+ 1 transitions. The former can yield b o t h signs for J ~ J transitions, while the latter gives one sign for all 7 ~ g and3' -+ 7, and o n e , possibly different, sign for all fl -+ g mixing ratios. The former requires a numerical calculation f o r J - + J , while the complete result o f the latter is c o n t a i n e d in the universal formula o f the simplest models b u t with different constants for 7 ~ g, 7 and fl -+ g. For J ~ J transitions Warner's m o d e l relies o n the assumption o f rotational behaviour. For 154Gd it works remarkably well in view o f its restrictive assumptions. (3) The unrestricted numerical t r e a t m e n t o f the present work, with three M1 parameters, is necessary whenever b o t h signs are present in the J - + J + 1 mix-

ing ratios, as we demonstrated with 168Er. Other such cases are 146,148Sm [12] and 162,164Er [ 3 ] , b u t good relevant data are very few. F u r t h e r calculations, also b y IBA-2, will be reported elsewhere. This work has been supported b y the A c a d e m y o f F i n l a n d and b y the US D e p a r t m e n t o f Energy u n d e r contract DE-AC02-76CH00016.

References [ 1] A. Arima and F. Iachello, Ann. Phys. (NY) 99 (1976) 253. [2J A. Arima and F. Iachello, Ann. Phys. (NY) 111 (1978) 201. 13

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[3] J. Lange, K. Kumar and J.H. Hamilton, Rev. Mod. Phys. 54 (1982) 119. [4] D.P. Grechukhin, Nucl. Phys. 40 (1963) 422; Yad. Fiz. 4 (1966) 691 [Sov. J. Nucl. Phys. 4 (1967) 490]. [5 ] O. Scholten, F. Iachello and A. Arima, Arm. Phys. (NY) 115 (1978) 325. [6] D.D. Warner, Phys. Rev. Lett. 47 (1981) 1819. [7] C. Girit, W.D. Hamilton and C.A. Kalfas, J. Phys. G9 (1983) 797. [8] R.L. West, E.G. Funk and J.W. Mihelich, Phys. Rev. C18 (1978) 679. [9] K. Schreckenbach and W. GeUetly, Phys. Lett. 94B (1980) 298 ; W. Gelletly, private communication.


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[10] D.D. Warner and R.F. Casten, Phys. Rev. C25 (1982) 2019. [11] A. Arima and F. Iachello, Ann. Phys. (NY) 123 (1979) 468. [12] T.I. Krac~ov~ et al., J. Phys. G., to be published. [13] K. Kumar, J.B. Gupta and J.H. Hamilton, Austr. J. Phys. 32 (1979) 307. [14] O. Seholten, Phys. Lett. 127B (1983) 144. [ 15 ] O. Scholten, The program-package PHINT, KVI report no.63 (1979).