Mixed-symmetry states and the structure of 200 Hg

zyxwvuts zyxwv J. Phys. G: Nucl. Part. Phys. 15 (1989) 93-112. Printed in the UK Mixed-symmetry states and the structure of 200Hg zyxwvuts zyxw zyxwvu zyx S T Ahmadt, W D Hamiltont, P Van Isackert, S A Hamada? and S J Robinson$ t School of Mathematical and Physical Sciences, University of Sussex, Brighton BN19QH, UK $ Institute Laue-Langevin, BP 156, 38042 Grenoble, France Received 18 August 1988 Abstract. Mixed-symmetry assignments are made to the 1.570 (l;), 1.573 (2:) and 1.659 (3:) MeV levels in 2wHg.The characteristics of the 2: level are shared with nearby 2+ levels at 1.254 and 1.593 MeV. An interacting boson approximation (IBA-2) analysis examines the sensitivity of the mixed-symmetry states to the E, parameters of the Majorana term and finds that both the magnitude and sign of the d(E2iM1) mixing ratio of 2'-2: transitions are well reproduced at the values of the ttparameters used to obtain a level energy fit. The IBA-2 calculations also give a satisfactory fit to the energy levels and in general provide X(EOiE2) values close to the experimental data. These calculations agree with the known oblate deformation of the ground state which we find to be predicted by Kumar's dynamic deformation model. Experimental data from n, yy(0) experiments together with internal conversion coefficient (ICC) results allow 1+ levels to be unambiguously established at 1.570, 1.631, 1.718, 2.061, 2.370, 2.640 and 2.979 MeV. The measurements also provide precise 6 values of many transitions while event-by-event coincidence information permits the decay scheme to the developed with previous unlocated transitions now firmly assigned. NUCLEAR REACTIONS '99Hg(n; y)ZWHg;thermal neutrons, measured I,, y-ray coincidences, y y ( 8 ) ; deduced 6(E2/Ml), X(EOIE2), mixed-symmetry states, level scheme; enriched target, Ge(Li) detectors. 1. Introduction Mixed-symmetry states arise from the out-of-phase collective motion of protons and neutrons and are now a widely recognised feature of vibrational [l], rotational [2] and y-unstable [3] nuclei. However, their identification is not straightforward although their basic properties are well described by the interacting boson approximation (IBA) [4]. In vibrational nuclei the lowest lying mixed-symmetry state has spin-parity 2+ and the identified levels lie at 2 MeV and decay predominantly to the 2: level by an essentially pure M1 transition. It is now apparent that the mixed-symmetry character may be shared with other neighbouring 2+ levels [ 5 , 6 ] while the strength of the M1 component is sensitive to the difference of the effective g values of the neutron and proton. The lowest lying mixed-symmetry state in rotational nuclei is a 1' level lying typically at 3 MeV [7]. This level is the band head of the K"= 1' band but so far it has been difficult to identify higher lying band members as the strength of the next highest level, which is 2+, is spread over many 2' levels. - - 0954-38891891010093+ 20 $02.50 @ 1989 IOP Publishing Ltd 93 94 zyxwvutsr zyxwvuts zyxwvu zyx S T Ahmad et a1 Clearly it is of interest to identify mixed-symmetry levels other than the lowest lying one and to understand how these states may be accounted for. One of the noteable features of the IBA is that, in its IBA-2 version, it distinguishes between neutrons and protons and this additional degree of freedom compared with IBA-1 allows mixed-symmetry states to occur naturally. The IBA-:! is thus unlike other geometric models in which this distinction is not made, although it must be remarked that the possibility of such excitations has long been recognised [S, 91. However, these early calculations suggested that the mixed-symmetry states would occur at 10 MeV excitation and no experimental interest was shown at that time. The energy dependence of the mixed-symmetry states and the sharing of mixedsymmetry characteristics with regular states are governed by the parameters of the is poorly Majorana term [lo]. The role of the different Majorana parameters understood and usually the three parameters are taken to be equal and simultaneously adjusted to obtain a best fit to a set of levels which includes the lowest lying mixedsymmetry state. The effect of the Majorana term is to alter the energy of the mixedsymmetry levels; if they are raised sufficiently high in energy then the lowest levels obtained in IBA-2 correspond to those in IBA-1. The different coefficients present in the term influence the energy of mixed-symmetry levels having different spin [lo, 111 while the regular IBA-1 states are largely unaffected. This dependence of level energy on the Majorana term is a good indication that a level contains a mixed-symmetry contribution. Another indicator is provided by its F spin [12]; states which are symmetric have F,, = (N,+ N,)/2 and those with F< F,, are not symmetric in neutron and proton collective motion. It is important to investigate mixed-symmetry states to obtain a better understanding of the role played by the three parameters of the Majorana term. Already it has been shown [ 101that in rotational nuclei a O& level may be introduced at approximately half the energy of the l&level at 3 MeV by a particular choice of the parameters and a possible example has been identified in 17*Yb[ 111. A problem common to most investigations of mixed-symmetry states is that, since they often lie at relatively high energy where the level density is great, their characteristics may be shared with other nearby levels. It is thus important to choose an example where the mixed-symmetry characteristics are not spread over so many levels that they cannot be distinguished. In this study we have selected a nucleus that has predominantly vibrational-like properties and which has very different g, and g, values so that the B(M1) strength of the 2;,-2: transition is large and other mixed-symmetry features are enhanced. The nucleus ' W g has many low-spin states lying between 1.5 and 3 MeV and some of these may be expected to have mixed-symmetry character. Although they have been often studied in the past [13] multipole mixing-ratio data are sparse and in particular the spin and parity of possible of 1' states have not been unambiguously established nor have the decays of these states been thoroghly examined. The required data may be obtained by measuring yy directional correlations following neutron capture which is particularly suitable in this case since the target 19'Hg has spin-parity I"= and hence the levels populated following the decay of the capture states by primary transitions will predominantly have spins in the range 0 to 3 . These measurements will also provide y-ray multipole mixing ratios of AZ= 0, I f 0 transitions for which we have already measured internal conversion electron coefficients [14]. It should thus be possible to calculate X(EO/E2) values and, in suitable cases obtain the strength of EO contributions in transitions from a 2& state to regular 2+ levels. Internal conversion data is also available for most other transitions (15,161. - - +- zyx zyxw zy zyxwvut Mixed-symmetry states and the structure of 'O'Hg 95 A further interesting feature of this nucleus is that the quadrupole moment of the 2: level is positive [17] which indicates that * W g prefers a stable oblate deformation and this is a comparatively rare feature. Previous analyses suggest that the deformation is small and that the nucleus also has some characteristics of a y-unstable nature. Additional data may thus allow a more detailed analysis within the framework of the IBA models while the dynamic deformation model may be used to predict the potential energy surface and nuclear parameters. A preliminary account of some aspects of this work was presented at the Leuven Conference on neutron capture spectroscopy [181. 2. Experimental method The measurements were carried out at the ILL reactor in Grenoble. A target in the form of HgS containing 14 mg of mercury enriched to 67% of 199Hgwas placed in a thermal neutron beam of 5 X lo8n cm-'s-'. This beam was collimated to match the target size and LiF enriched in 6Li was used extensively to reduce scattered neutrons entering the detectors and surrounding material and consequently minimise y-ray background. Coincidences were recorded between two co-axial Ge(Li) detectors each of 18% efficiency and 2.5 keV FWHM resolution at 1.33 MeV which were positioned 9 cm from the target. One detector was fixed and the other moved through a range of five selected angles lying in the quadrant 90 to 180" with respect to the axis specified by the target and fixed detector. The event-by-event collection of data and their analyses were carried out as in previous measurements [ 191. - 3. Data analysis and evaluation Gamma-ray singles spectra were recorded with and without 56C0and 1 5 2 Ecalibration ~ sources in the vicinity of the target. Calibration data were taken from Lederer and Shirley [20] and Debertin [21] and a Gaussian fitting routine was used to obtain energies and intensities. These intensity measurements have a greater precision than the data used by Subber et aZ[14] in their evaluation of internal conversion coefficients from measurements of electron intensities, and in several cases a re-evaluation has been made. The correlation data were obtained using the 368 keV 2+-O+ ground-state transition as a gate. Most intense cascades contain this y ray as a component and from the point of view of subsequent analysis it is the most suitable one since the large A 2 and A4 coefficients of a 2-0 transition give great sensitivity to the correlation data. A further twelve gates were used to obtain coincidence data and a summary of established coincidence relations is listed in table 1. Figure 1 shows the decay scheme based on this table and on published data [13]. This scheme differs in some detail from those published previously since the extensive coincidence data allow many previously unplaced transitions to be uniquely located in the scheme. Four independent measurements were made of most correlations although for a few there were only two due to poor statistics. All were found to be self-consistent and the directional correlation coefficients given in table 2 are the weighted mean values. Level spins and transition multipole mixing ratios were made on the basis of tan-' 6 plots [22]. The error range on 6 values is generally at Ximit=X&,+ 1 [23], and an example is shown in figure 2. zyxw zyx 96 zyxwvutsrq zyxwvutsrq zyxwvut zyxwvuts S T Ahmad et a1 Table 1. The coincidence results of 'OHg. Gate (keV) 368 540 579 661 886 1202,1205 1225 1263 1273 1350 1363 1408 1570 Transitions (keV) 141, 148, 152, 165, 203, 215, 224, 226, 243, 289, 307, 309, 316, 320, 339, 341, 343, 368, 383, 387, 398, 408, 420, 430, 445, 449, 464, 477, 486, 491, 540, 557, 565, 579, 612, 628, 661, 676, 689, 701, 711, 757, 784, 787, 796, 807, 828, 851, 862, 873, 886, 898, 904, 935, 975, 993, 1003, 1009, 1027, 1034, 1042, 1055, 1121, 1147, 1158, 1164, 1202, 1205, 1225, 1237, 1247, 1263, 1267, 1273, 1284, 1291, 1323, 1342, 1350, 1363, 1367, 1385, 1408, 1432, 1467, 1489, 1504, 1515, 1525, 1557, 1589, 1604, 1624, 1693, 1706, 1711, 1722, 1734, 1759, 1783, 1811, 1821, 1860, 1879, 1906, 1921, 1963, 1976, 2002, 2020, 2094, 2123, 2161, 2180, 2252, 2259, 2721, 2346, 2568, 2648, 2727, 2818, 2901, 2921, 3032 148, 368, 491, 557, 661, 851, 904, 1408, 1504, 1963 152, 243, 307, 368, 464, 477, 784, 787, 828, 898, 1027, 1323, 1342, 1432 148, 152,224, 368, 491, 540,557, 565, 612, 689,701, 851, 904, 1323, 1408, 1504, 1557, 1624, 1734, 1963, 2259, 2347 316, 320, 368, 387, 449, 464, 477, 592, 628, 807, 873, 935, 975,993, 1034, 1042, 1158, 1237, 1557 148, 309, 368, 398, 445, 491, 557, 676, 851, 904, 1121, 1408 141, 289, 368, 467, 1860 226, 341, 368, 430, 486, 496, 1589, 1711 368, 1055, 1706, 1811, 2568 165, 343, 368, 408 152, 368, 383, 1323, 1557 368, 540, 661, 904, 1202, 1570 148, 491, 557, 851, 904, 1408, 1504, 1963, 2727 zyxwvut zyxwv zyxwvut Some comments on the results are necessary for the higher lying levels for which new data were obtained. We have used ICC data of Subber et a1 to assign level parities in several cases. The 1570 keV 1' level. The spin I = 1 was established from the 1202-368, 14081202, and 1408(1202)-368 keV correlations and a consistent and unique 6 value obtained for the 1202 keV transition. The 1631 keV 1 level. The 1263-368 level correlation coefficients confirm the 1' assignment and give 0.020<6<0.086 for the E2/M1 mixing ratio of the 1263 keV transition and the ICC data confirm the parity. + The 1642 keV2+ level. The established decay modes to 0' and 4+ levels allow only a 2' assignment with 0.017< 6 < 0.076 for the 1274 keV 2+-2+ transition. The 1718keV 1+ level. Spin assignments of 1 and 3 are permitted by the correlation data. The 1 = 3 value is ruled out by the observed ground-state decay which is shown by the ICC data to be predominantly M1. The 1' assignment gives - 0.070< 6 < - 0.022 for the 1350 keV 1+-2+ transition. The 1732 keV 2+ level. The result - 0.42 < 6 - 0.026 for the 1363 keV 2+-2+ transition is inconsistent with previous work [24]. Mixed-symmetry states and the structure of 'O0Hg zy zyxw 97 zyxwvutsr zyxwvutsrqpo m + +++<++++++++++++ -"NO""-"-"& + d & * x [OF')I O'OZP 101'91 E'OEP 101'21 9'5BP [02'2l 2'9b5 108'EI E'LOB [ O I ' I L L . 1 E'E68L zyxwvutsrqpo zyxwvut s rqponml k j i h gf e dcbaZYXWVU zyxwvutsrqponm zyxwvutsrqpo L.2'01 B'P9L 05'21 E'S92 OP'I 1 1 ' 8 2 8 02'61 L'PiPI [QP'OI 6'522 1 L'89Pi lL9.01 1'252 162'01 2 ' 2 L 2 106'21 P'869 OL'OI [06'21 6'991 [05'1I 9'9PEL h v zyxwvutsrqpon [OZ'OI L'PL.2 IEL'I 1 9'9lP [OB'S] P ' L O L [09'21 9'EgL [09'91I 6'29EI [08'01 0'9PL [09'01 L'EOZ [08'01 E'b9b i W [ O L ' B ] 6'988 [OZ'EI 1 1'09E1 I W [06'921 5 ' B I L L 109'0l 9'ILL 23 lOL'2l Z'I62I [ E Z ' I I E'LBE [Oi'EI 0'2l9 IOV'OI 2 ' P W [Ob'9El E'ELZI [ O P ' B S ] 9'2QZt I n [oe'i] s.0~91 [ O Q ' S E 1 E.5221. [ €0'01 9 ' 6 I E [OP'OV 1 9'9021. b w [MI'S1 S'OPP [OI'SEI I ' Z O Z L I W [Ob'6E] O'OLPL 23 [ I Z ' O l 8'90E [01'6P1 6 ' 9 8 8 23 [01.'15111 I'9PE zyxwvut m 98 zyxwvutsr zyxwvutsrq zyxwvu S T Ahmad et a1 zyxwvutsrqpon zyxwvut srqponmlkjihgfedcbaZYXWVUT 100'21 L ' L B L 1 E'OL91 IOL'L [06'LL 1 P'LL22 1 .-- , .. . . . . [05'01 L'EOZ [06'01 L ' E W h 9 w zyxwvutsrqpo zyxwvutsrqpon [OO'Zl E'EEL LOP'S1 L ' P E E l I 'E] 6'2561 5.21 1 ' 2 P O l 6'L 'OL 1 5'9521 I 0'9622 I P . 0 1 2'PSE I L ' O I E'PEOl I L ' L I 5'1bEl I P ' P I 1'0261 06'L I P'2V5 06'01 6'EOL W'Zl l'L5L OE'Ll 6'50SL Mixed-symmetry states and the structure of "'Hg zyx 99 zyxwvutsrqponm zyxwvutsrqponm zyxwvutsrqponmlkjih [OZ'61 5 LSQI [os'zl Q'SP8I I O Q ' Z l I'B99b IOE'LI s P I L I 100 zyxw zyxw zyxw zyxwvutsr zyxwvutsrq zyxwvu zyxwvuts S T Ahmad et a1 a8 M M s 8 M s zyxwvutsrqpon - : : w s zyxw h d N 2I h N I in r? 0 v 3 + +++ + 3 s MSE5 E V i w m 0 h m h IC. h ^ I I q 0 v I : 0 : CO- C- v ? \o F, v R 2 \c + + + ++++ E E EMSEES -E+ zyxwvutsrqpon k h m UVI 0 -5 5 v m m z 8 h 8 v ic 3 0 I no -5 = - 10 N- + N GG 21 ? 3 + 0 C A 0 0 + I +1 N N + I +1 O N + + + + 0 3 0 0 I l l 1 + i t - I I I I N"N + * + - S - N N 0 + 0 I - N N +i + t 3 + N t N + + - - + + - - + -I +I- t N 0 0 0 0 0 C N N I - 1 +I dN"" - 1 - 1 +I +I + ' +I r Q 0 N m N - I l l t + N + i I l l + + T i-- hhhh mr.r.r. d N 4 m vvvv N U N 9 mIr,m- NNNN 0 0 0 0 I l l 1 3 I 0 U .e .-.v1 Y z z % I z & i 0 N 2 U 5 ' i 3 h doc '3 0 G zyxw zyxwvut zyxwvut zyxw zyxw zyxwvutsrq zy Mixed-symmetry states and the structure of 2ooHg 101 The 1865 keV 0' level. The ICC data gave a 0' assignment and this is consistent with the present correlation measurements. The 2061 keV 1 level. The decay modes and the correlation data limit the spin to I = 1 and the ICC requires positive parity while the mixing ratio of the 1693 keV transition level to the 2' level is - 0.010 < 6 < 0.016. + The 2370 keV1' level. Again the only acceptable spin is I = 1 with a small 6 value, - 0.033 < 6 < 0.005, for the 2002 keV transition. The ICC gives positive parity. The 2640 keV I + level. The correlation data permit spins of 1 or 3 and the latter is excluded by the aKcoefficient of the transition to the ground state. The large 6 value of the 2272 keV transition, - 0.48< 6 < - 0.37, implies positive parity. The 2979 keV 1' level. The 1408 keV transition was previously considered to be from the 1776 keV 3' level while in the present work the coincidence conditions show that it is from the 2978 keV 1' level. A unique 6 value cannot be obtained for the 1+-1+transition from the correlation data, but we find that the ICC data are only consistent with the result d = 1.44 (21, - 10). 4. Discussion The relatively large number of low spin levels lying above 1.5 MeV is a particular feature of several nuclei in this transitional region where the nuclei have small oblate deformations, Subber et a1 [14] considered primarily the 0' levels and their E0 decays 1-2 3-2 2- 2 zyxw zyx 0.1Yo 1% 10 Yo -90 - 60 -30 0 30 60 1 90 tan-' 6 Figure 2. A xz against tan-' 6 plot for the 1225-368 keV cascade. The data give an unambiguous spin-2 assignment to the 1574 keV level. 102 zyxwvut zyxwvuts zyxwvu zyxwvuts S T Ahmad et a1 15.01 'b-0 0 02 04 06 B d 04 Oblote 02 , 0 02 Deformation P 04 Prolate zyxw 06 Figure 3. The potential energy surface obtained from the dynamic deformation model. The potential energy V ( p ,y ) (in MeV) shows a broad minimum at /3 = 0.1 and 'J = 60". ZPM indicates zero point motion. and concluded that, in the framework of the IBA model, a description of '""Hg is intermediate between the O(6) and U(5) limits. This y-unstable aspect is also apparent from the work of Koppel et a1 [25]who used the dynamic deformation model to obtain the potential energy function V ( p ,y ) . Their results suggest that the nucleus is y unstable at small deformations and in our plot of the potential energy surface, figure 3, we confirm this feature. 4.1. Interacting boson model calculations zyxwv Previous studies of the mercury isotopes within the framework of the IBA model have been reported by Morrison [26] and Barfield et a1 [ 2 7 ] . Morrison examined the occurrence of mixed-symmetry states in the mercury isotopes when using the O(6) limit but he excluded zooHgfrom his analysis. Barfield et a1 chose their model parameters with the requirement that they should be consistent with those used to describe the neighbouring isotones of W, Os and Pt without requiring a best fit to the zooHgexperimental data. They chose not to fit the 0: level at 1029 keV in "'Hg which they considered to lie outside the IBA model space and neither did they vary the Majorana parameters. Subber et al, as a starting point, used the parameters from Barfield et a1 and then adjusted these to give a best fit in which the 0; level occurred naturally. The new model parameters indicated that "'Hg was nearer to U(5) than to O(6). The fit of Subber et a1 failed however to give the correct sign for the quadrupole moment and we find also that the agreement between the predicted and measured E2 :M1 mixing ratio is poor. Since we now have many more data on "'Hg we propose that a better and more complete correspondence should be made between the experimental data and the IBA-:!. The dominant M1 decay modes of a number of the low-spin states and their zyxw zyxwv zy zyxwv zyxwv zyxwvut Mixed-symmetry states and the structure of 2ooHg Table 3. N” N, E K K, K,, XI, IBA 103 zyxwv zyxwv zyxw Hamiltonian parameters used for the *%g. 3 1 0.47MeV -0.15MeV -0.05 MeV -0.05 MeV 1.1 0.4 Xn CO, C2, C,, c, t2 63 -0.33MeV -0.13MeV -0.07MeV 0.07MeV 0.12 MeV 0.02 MeV e, e, g, g, p, pv yn 0.163eb 0.133eb 0.70~~ 0.05~~ 0.034fm2 - 0.020 fm2 -0.101 fm2 zyx possible interpretation as mixed-symmetry states requires that careful attention must be paid to the Majorana term. We use the proton-neutron IBA Hamiltonian [28] H = ~(ndz+ ndv)+ xQ,QV+ Vvv+ Mv, where the quadrupole operator is Q = [dts + s’d];) + x,[dtd]:) (1) (2) and the Majorana term may be written as [lo]: . . M,, = fE2(stdL- d:sL)(2) (svd, - d,,s,)(’) - &(dtdA)(k) (dVda)(k). (3) k=1,3 The choice of parameters (see table 3) takes account of the underlying shell structure which results in variations of K , X, and x, as the proton and neutron numbers change. The level fit to these parameters is shown in figure 4. We find that a best fit is obtained with xv= 1.1 and x,= 0.4. These positive values give a positive quadrupole moment Q(2:) = 0.80 eb which agrees well with the experimental value [17], Q(2:) = 0.96(11) eb. These xlvalues disagree with those obtained by Barfield et al, xv= 1.1and xn= - 0.4. Druce et a1 [29] comment that the sign of 2, is not well established for the mercury isotopes but it should be noted that, since e,=e,, the value of X, may be negative only if its magnitude is smaller than the magnitude of xv.This is because the main contribution to the quadrupole moment comes from the neutron term since Nv=3 while N,=l. 4.2. Mixed-symmetry states The energy fit to several levels is very sensitive to the parameters in the Majorana term which also strongly influence the magnitude and sign of the multipole mixing ratios of many transitions. In particular we find that the calculated energies of a number of states are affected in a very similar way and these might be considered to have a mixed-symmetry origin, or contain substantial mixed-symmetry components. Those with a mixed-symmetry origin have no counterpart in IBA-1. The energy dependence of the 2: and 24‘ levels is consistent with the mixed-symmetry character of the 2: level being shared with neighbouring states. The influence of the different parameters on these states is shown in figure 5 . The 5; term strongly affects the energies of all of the levels considered to have a mixed-symmetry character or to contain mixed-symmetry components (figure 5(b)). In obtaining this plot the El and t3terms were maintained at their best-fit values. 104 zyxwvutsrq zyxwvu zyxwvut zyxwvuts S T Ahmad et a1 1, -7 33 33 zyxwvutsrqpo o , ,0 / 2.5 6-2 -0 -3 -2 11- 0- 2.0 FE 41 100 -8 1- 1 33 57 100 o, 3-5 -2 38 48 93 34 35 -4 -0 3, -4 87 42 91 41 -2 100 -0 100 36 1 33 34 zyxwvuts zyxwvutsrq 2 36 & 1.5 i W C 1 .o 0.5 ------ 0 zyxwvutsrqp ---_-- Expt IBA-2 Figure 4. The IBM-2 ‘best-fit’ level spectrum compared with the experimental spectrum. The F spin, expressed as a percentage, is shown for the lower lying levels. The F-spin components in the 22, 23, 24 and O3 levels as a function of t2are shown in figure 6. Our & parameters, cf table 3, obtained from the level energy fit disagree with those obtained by Druce et al. They found t1and l3to be large and positive and t2=O.0-t. zyxwvut iIt should be noted that the definition of M,, used by Druce et a1 differs from that given in equation (3). Mixed-symmetry states and the structure of 2ooHg zyxw 105 I zyx zyx zyx zyxwvu -3 -7 0 m -0yn -2 -7 I zyxwvu I m 0 m 0 I I - a - m * _ m N N-N I I N I N 0 zyxwvutsrqpo zyxwvutsrqponm - - a -m c * N m N N N N 0 106 zyxwvuts zyxwvutsrq zyxwvu S T Ahmad et a1 zyxwvuts zyxwvutsrqp zyxwvutsrq -012 b -011 52 0'1 dz ' Figure 6 . The F-spin components in the 2*, 23, 24and O3 levels as a function of other parameters are as in table 3. t2when all The mixing ratio data, discussed in the following section have a strong dependence on t2and show that t2cannot be zero in our fit. The 1: level is strongly affected by changing El, figure 5 ( a ) , while the 3: level energy depends on the t3value as shown in figure 5(c). The 2: mixed-symmetry state and the predominantly symmetric 2: and 2: levels are largely unaffected by changing or t3in contrast to their dependence on t2. el, 4.3. E2lMl multipole mixing ratios The Majorana term parameters also influence strongly the magnitude and sign of the multipole mixing ratio of many transitions. In order to explore this dependence we must first define the appropriate operators. The E2 transition operator is given by T(E2) = e,Qx + e v e v . (4) The effective boson charges e, and e, may be derived from the experimental value of B(E2; 21+01) and we obtained the values e,=0.163 eb and e,=0.133 eb. The M1 transition operator may be written in the form [30] (z) 112 T(M1)= [gx(dtd)~)+g,,(dtd)~))] where g, and g, are the boson g factors. The reduced E2 and M1 matrix elements may now be evaluated and in figure 7 we show their dependence on the term for transitions from the &, 23 and 24 levels. The reduced matrix element ratio A(E2lM1) is directly related to the usual multipole amplitude mixing ratio 6 by the expression 6 = 0.835E,(MeV)A(eb/pN). (6) zyxwvu zyxw zyxw zyxwvuts Mixed-symmetry states and the structure of "OHg 107 Figure 8 shows the variation of 6 for the group of 2+-2: transitions and it is seen that both the magnitude and sign of 6 are correctly obtained for the three transitions in the vicinity of = 0.12 which is the energy fit value. Table 4 contains a summary of the results where the experimental data have sufficient precision for a useful comparison and also when there is no ambiguity in the nature of the levels. (At higher energies where the level density is great the order of the experimental levels may differ from the calculated order.) c2 4.4. EO transitions and X values A re-analysis of the EO data of Subber et a1 may now be undertaken and we write the monopole transition operator as + ynNn+ P,(d:d,)(') + Pn(d;dn)(') T(Eo) = y,N, (7) which is related to the p(E0) transition matrix by the expression zyxwvu zyxwvuts zyxwvu where R = l.2A1'3fm. The theoretical X(EO/E2) ratios are evaluated from the equation X(EO/E2, i 4 f)= e2R4p2(EO;i+ f)/B(E2; i + f ) . (9) The necessary parameters are derived from the values p(E0) = 0.021(3) obtained by Subber et a1 [14] for the 22-2, transition in 2ooHgand from the isomer shift results [31]. We obtain By= - 0.020 fm2and Pn= 0.034 fm2. The value y = 0.085 fm2is derived from the isotope shift data for 198-202H €5 Table 5 contains the experimental and calculated X values. In general there is good agreement except for the 02-01 and 0,-0, transitions but it is not possible to say if these disagreements may be attributed to the EO or E2 component in the ratio. The zyxwvutsrqpon 1 x10-l 6 40-' 1 8- 51 \ \ 52 zyxwvutsrqponmlkjih 52 Figure 7. The reduced M1 and E2 matrix elements for 2-2, transitions shown as a function of & when 6, and & have their 'best-fit' values. 108 zyxwvuts zyxwvutsrq zyxwvut S T Ahmad et a1 \ \ - 0.2 -01 zyxwvutsr zyxwvu zyxwvutsr zyxw 0 0'1 02 5, Figure 8. The multipole mixing ratio, 6, of 2-21 transition plotted against g 2 . The range of g2 which gives acceptable agreement with experiment is indicated. disagreement in the results for the 25-21 and 26-21 transitions could be removed by interchanging the ordering since for the higher lying states the correspondence between the experimental and theoretical levels is uncertain. It must also be remarked that the comparatively large Xvalues for transitions from the 2: mixed-symmetry state and from the 2; and 24' states indicate that substantial EO components occur in these decays from mixed-symmetry states. The EO matrix element describing such decay is proportional to pV-Px and, although the /3 values are small, their sign difference results in the EO matrix being greatest. In making this comparison we have assumed that all the identified 0' levels correspond to IBA states and that the experimental level ordering is the same as the calculated order. Some previous work [27], which attempted to fit levels in several mercury isotopes with a single set of regularly varying parameters, has not been too success€ul for 2oo€Igas this isotope is distinctly different from its neighbours; the 2: lies well above the 4: level at the two phonon energy. We also find good agreement between the calculated and experimental values for isotope shifts for all mercury isotopes (table 6) but the isomer shift result for 19%g is in poor agreement with the experimental value. Most experimentally observed low-spin levels, apart from 1' states below -2.5 MeV; have their counterpart in the IBA-2 level spectrum although the energy zyxw zyxw zy Mixed-symmetry states and the structure of ‘OOHg Table 4. Mixing ratios 6(E2/M1) of transitions in 2wHg. 109 d(E2/M1) Transition energy (keV) 885.9 1205.6 1225.3 1273.3 1362.9 1514.9 1604.0 1202.1 1262.8 1350.1 1693.3 1291.2 1366.8 711.7 787.5 828.3 I, I,, Expt IBA-2a - 2.20:; 0.31(3) -2.48’::; 0.047?:::: - 0.32:; 0.12t;g: 0.87:; i f 0.16(5) 0.053(33) - 0.036(24) 0.003(13) - 0.99 0.13 - 2.00 0.36 - 0.07’ - 0.23b 0.16’ -0.11 0.36 -0.47b 0.05’ -0.18 2.65‘ 0.27’ -0.16 1.89’ 0.15b zyxwvu zyxw zyxwvutsr - 0.043(52) * g V = 0.05yN,g,= 0.70pN,e”= 0.163eh, e,= 0.133eh. Transitions are labelled according to the experimental level sequence; these may differ from the sequence given by the IBA. zyxwvuts Table 5. X(EOIE2) ratios for A I = O transitions in ’OOHg. Initial level (keV) Transition energy (keV) 1029.4 1254.0 1515.1 1029.4 885.9 485.6 1515.1 319.5 1205.6 339.2 1225.3 387.3 1273.3 476.8 1362.9 341.8 827.0 1857.0 1515.0 1604.0 1573.7 1593.4 1641.4 1731.8 1857.3 1882.7 1972.1 a I, 1, Pray intensity 4.27(2) 3.52(2) 3.40(2) 3.16(3) 1.62(3) 0.80(5) 1.27(3) Electron intensity (x O.ll(1) 3.96(2) 0.18(2) 0.27(1) 0.42(4) 2.68(11) 0.36(3) 2.22(9) 3 .OO( 11) 2.11(8) 1.12(7) 0.85(4) 0.24(8) 0.11(4) 0.76(5) 0.44(2) 0.29(1) X(EOIE2) X 10’ aK(exp) ( x 100) 0.92(3) Expt 0.014(2) 0.586”;;;: 2.5(4) 1.33(4) 0.76( 11) 3.02(53) 0.68(9) 4.6’:; 0.67(9) 250 IBA-2’ 0.284 0.342 5.36 6.41 0.77 5.37 0.023 4.25 1.28 3.21 364 54.6 22.3 10.2 1.3 149 72.0 zy zyxwvu The electron intensities are taken from reference [14]. ’e, = 0.163eb, e, = 0.133eb, 8, = - 0.020 fm2 and Pn= 0.034 fm2. 0.52(5) 0.55(5) 0.23(4) 8.0.: 1 39.1(8) 9.3(6) 28.1(10) 32(13) 7.91:; 110 zyxwvutsr zyxwvutsrq S T Ahmad et a1 match is not good in every case. It also appears that we may identify the members of the family of mixed-symmetry states corresponding to the [ N - 1,1] representation [12]. The small E2/M1 mixing ratios are consistent with this interpretation but level lifetimes are required for a firmer identification. 4.5. Dynamic deformation model calculations The DDM model of Kumar [32] was used to calculate the ground-state potential energy surfaces of mercury isotopes. This work is similar in many respects to that reported by Koppel et a1 [25] and indeed the results which they obtained for lY8Hgare in close agreement with those obtained with the DDM. However, Koppel et a1 were chiefly concerned with 196Hgand potential energy surfaces were not published for other heavier mass isotopes. The potential energy surface for *OOHgis shown in a contour plot of V ( p ,y ) (figure 3). A minimum does occur for y = 60" indicating a stable oblate deformation with p =0.1. However, if we take account of the zero point motion we see from the shaded area that the calculated potential also indicates that we may expect some features of a y-unstable nucleus at small deformations where y is unrestricted. A t higher excitation the potential become more like that of a vibrational nucleus with energy contours symmetric about the origin. The DDM also gives the correct sign for the quadrupole moment of first excited 2' state, Q(2:) = 0.46 eb although its magnitude is about half that of the experimental value Q(2:) = 0.96(11) eb [17]. A further success of the DDM is shown in table 7 where the predictions of the B(E2) values and the experimental results are given. The agreement is particularly good in view of the parameter-free nature of the DDM. zy zyxwvut zy zy zyxwv 5. Conclusions The low-energy level structure of "'Hg offers a difficult challenge to several aspects of nuclear structure. The oblate deformation of the ground state is predicted by Kumar's DDM which also indicates that a vibrational-like structure occurs at higher excitation energy. The IBA-2 calculations provide a satisfactory framework for describing the nucleus with a structure lying between the U(5) and O(6) limits. A notable feature of the IBA-2 analysis is the occurrence of the mixed-symmetry states and their dependence on the tiparameters of the Majorana term. The sharing Table 6 . The observed and calculated isotope shifts A(?) in the Hg isotopes. A(r2)(fm') A A' Expt" IBA-Zb 194 196 198 198 198 198 200 202 0.170(22) 0.082(11) 0.096(10) 0.202(21) 0.190 0.096 0.086 0.178 a Reference [33]. Parameters used in the calculation are pz = 0.034 fm', pv= - 0.020 fm' and yv = 0.085 fm2. zyxw zy zyxw zyxwvutsr Mixed-symmetry states and the structure of "OHg 111 Table 7. Transition rates: B(E2) (in e%') and B(M1) (in p i ) . Cascade 1, 1, Expt DDM IBA-2a ~ ~ B(E2) B(E2) B(M1) 0.171 0.0019 0.0004 0.02 I(4) 0.0007(4) 0.017 1(22) 0.008 ~0.001 0.0026 0.010(3) 0.0007 0.0 128(4) 0.0044 0.015 0.015 0.0021 0.0149 0.0033(11) 0.074(17) 0.224 0.043 0.0313 0.002 0.0004 0.0018 0.0006 0.00 0.0016 0.0006 0.0038 0.190 0.0462 0.0193 0.0368 0.0826 0.0222 0.0004 0.0582 0.231 0.017 0.0162 0.022 0.00004 0.0055 0.0014 0.0028 0.079 0.297 0.0219 0.0231 0.258 0.00001 0.031 0.0211 0.0209 0.0007(4) 0.013(4) 0.016(7) 0.159 0.0015 0.0010 0.0077 G0.0135 22 22 B(M1) 0.0466 0.0003 0.055( 11) 0.001l(3) 23 31 B(E2) 0.03(1) 0.0016(7) 0.0013(6) 0.0015 0.0235 0.0236 0.052 0.0568 0.0507 0.0321 a e, = 0.133eb and e, = 0.163eb. bAbsolute B(E2). The others are normalised relative to the B(E2; 2,-01). zyx of the 2: mixed-symmetry characteristics with other neighbouring 2+ levels is clearly indicated by the energy dependence of these levels on the t2term. The model also accounts most successfully for the magnitudes and signs of 2+-2: multipole mixing ratios. References zyx zyxwv [1] Hamilton W D, Irback A and Elliott J P 1984 Phys. Rev. Lett. 53 2469 [2] Bohle D, Kuchler G , Richter A and Steffen W 1984 Phys. Lett. 148B 260 [3] Subber A R H, Park P, Hamilton W D , Kumar K, Schreckenbach K and Colvin G 1986 J . Phys. G: NUCLPhys. 12 881 [4] Scholten 0, Heyde K, Van Isacker P, Jolie J, Waroquier M and Sau J 1985 Nucl. Phys. A 438 41 [5] Eid S A, Hamilton W D and Elliott J P 1986 Phys. Lett. 166B 267 [6] Collins S P, Hamada S A, Hamilton W D , Holyer F and Robinson S J 1986 Symmetries and Nuclear Structure ed R A Meyer and V Paar (London: Harwood) [7] Bohle D, Richter A , Steffen W, Dieperink A E L, Lo Iudice N, Palumbo F and Scholten 0 1984 Phys. Lett. 137B 27 [U] Faessler A 1966 Nucl. Phys. 85 653 (91 Maruhn-Rezwani V, Greiner W and Maruhn J A 1975 Phys. Lett. 57B 109 [lo] Scholten 0, Heyde K, Van Isacker P and Otsuka T 1985 Phys. Rev. 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