Krane, K.S.

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Lawrence Berkeley National LaboratoryRecent WorkTitleE2/M1 Y RAY MULTIPOLE MIXING RATIOS IN EVEN-EVEN DEFORMED NUCLEIPermalinkhttps://escholarship.org/uc/item/9dn6r60cAuthorKrane, K.S.Publication Date1973eScholarship.org Powered by the California Digital LibraryUniversity of California.. Submitted to Physical LBL-1625 Review Letters Preprint<:' / E2/M1 y RAY MULTIPOLE MIXING RATIOS IN EVEN -EVEN DEFORMED NUCLEI K. S. Krane January 19 73 Prepared for the U. S. Atomic Energy Commission under Contract W -7405-ENG -48 For Reference Not to be taken from this room DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. i.) .. -1- LBL-1625 * E2/Ml y RAY MULTIPOLE MIXING RATIOS IN EVEN-EVEN DEFORMED NUCLEI Abstract: K. S. Krane Lawrence Berkeley Laboratory University of California Berkeley, California 94720 January 1973 A summary is presented of the magnitudes and phases of previously measured E2/Ml multipole mixing ratios of y-transitions de-exciting levels of the S-and y-vibrational bands to the ground-state band in even-even deformed nuclei. It is shown that none of the previously proposed theoretical interpretations is sufficient to explain both the magnitudes and relative phases of these mixing ratios. In the model of adiabatic vibrations of an ellipsoidally-deformed nucleus, magnetic dipole (Ml) transitions are forbidden to exist in y-transitions connecting rotational levels built on the vibrational excitations with those of the ground-state band; such transitions are expected to be pure electric quadrupole radiation (E2). However, non-vanishing Ml-admixtures are found in such transitions in even-even nuclei throughout the mass region 150 < A < 190; the Ml-intensity generally comprises 0.5% - 2% of the transition intensity. The measurement of y-r~ angular distributions or correlations is sensitive to interference effects between the Ml and E2 amplitudes, and thus -2- LBL-1625 depends on the relative phase of the Ml and E2 matrix elements. Some confusion has existed in the literature regarding the relationship of this phase to the observed angular distribution. This situation results from the various formalisms which have been proposed for interpreting angular correlation data. In the present work, the phase convention of Krane and Steffen1 is used; this choice permits a direct comparison of the deduced mixing ratio with theoretical predictions, and can be written in terms of the Bohr-Mottelson2 multipole operators as ( Ifll 112 (E2 )II I. ) = 0 · 835 ( IfJim (Ml )II I~ > (1) with the E2 matrix element in units of electron-barns (eb) and the Ml matrix element in units of nuclear magnetons (run); Ey is the energy of the transition in MeV. A comprehensive discussion of the electromagnetic transition operators is given in the work of Alder and Steffen. 3 Table I presents a summary of .the· results obtained from an analysis of the angular correlation literature in terms of the present phase convention. The tabulated value is the "reduced" mixing ratio o/E given by Eq. (l). The y quoted uncertainties are those arising from one standard deviation of the measured angular distribution coefficients. Transitions depopulating states of the 8 and y bands of I ~ 4 have been analyzed; the identification of the y-band is usually obvious, and the 8-band has generally been assigned as a K = 0+ excitation showing, for example, a large E2 excitation probability in a Coulomb excitation measurement. -) , -3- LBL-1625 The systematic behavior of the phase of the mixing ratio is apparent from an inspection of the table. With minor exceptions, transitions from the y-band have negative phase, while transitions from the 8-band seem to show the opposite phase. The magnitudes and phases of the mixing ratios m~ be predicted from a variety of different models. 1. ~K = 2 Band Mixing. This type of analysis takes into account mutual mixing of the ground-state, B, and y bands, and has been widely used with reasonable success to interpret deviations of the relative reduced transition probabilities of transitions from the y-band from the predictions of the adiabatic rotational model. The interpretation of transitions from the 8-band has met with considerably less success. The present notation for the band mixing parameters is that of Marshalek5 and of the Oak Ridge-Vanderbilt group. 6 A similar analysis has been done by Rud and Bonde Nielsen. 7 The Ml matrix elements are now given in terms of the static magnetic moments of the admixed intrinsic states, and the mixing ratios are given by ( .2.) -AQ I I 0 (2) = zy(gK- gR) + B y g E gR ZB Z8y ( .2.) -AQ Is - I 0 ( 3) = g E gH zy zl3y where A and B have the following values: 2 - 2 3 - 2 3y - 4 4 - 4 213- 2g 413 - 4g y g y g g y g A = 0.176 0.330 o.o48 0.092 0.029 0.001 B = 18 0 0 200 -4- LBL-1625 This calculation assumes that the intrinsic quadrupole moment Q and rotational 0 g-factor gR are constant for the three bands; gK is the intrinsic g-factor evaluated for the y-band. The ·band mixing parameters are in the notation of Ref. 6. With zy ~ 5 X 10-2 , this model gives 6-values for y-band transitions too large by at least an order of magnitude. Independent of ZY' the relative magnitudes of 0 for the four y-band de-excitations are not in agreement with experiment. The phases of they-band transitions. are not easily calculable, depending on the values of (gK - gR) and z6 , which numbers are not widely . y available for all the nuclei considered. 2. ~ = 1 Band Mixing. The first-order Coriolis interaction can mix K = 1 states into the K = 0 and K = 2 bands. The Ml matrix element resulting from such mixing is given by8 ( 11 K = Oli7Jz (Ml)li 12 K = 2 ) = where = - 1 <K=2lh IK=l) E :lE (OOII7Jt(Ml)llll) K = 1 K = 2 ( 4) (5) where ~· refers to the intrinsic system and h+l is the operator associated with 6K = l Coriolis mixing. . ~·.1 -~~ -., -5- LBL-1625 At present there exists insufficient knowledge of K = 1 excitations to predict either the coupling or Ml matrix elements of Eq. ( 5). · However, con-elusions are possible regarding the relative phases and magnitudes of the mixing ·{' ratios. The relative magnitudes are as follows: 1:0.94:0.68:0.44 These relationships are in better agreement with the observed values than are the relationships deduced above for ~K = 2 mixing. The relative phases of the mixing ratios are predicted to be the same, which is likewise in agreement with experiment. An estimate of the magnitude of the. required coupling strength indicates that the observed magnitudes of the mixing ratios require, for ( OOII71t (Ml )1111 ) 'V one single particle unit, a coupling matrix element ( K + llh+11K ) ~ 10 keV, which is not an unreasonably large value. 3. Microscopic Theory of the y-Band. Bes et al. 9 have considered the --microscopic structure of the x-vibrational state, in which the intrinsic state is treated as a superposition of quasiparticle pairs. The Ml amplitudes are obtained through Coriolis band-mixing of the y-band and ground-state band. The predictions of Bes et al. for the magnitudes of the E2/Ml mixing ratios are given in Table I. The phase of the mixing ratio is not uniquely determined in this model, but rather depends on the competition between the rotational motion and the orbital motion of the protons. If, as concl~ded by Beset a1.,9 the contribution from the rotational motion dominates, this model predicts 6 > 0, in disagreement with experiment, although the predicted magnitudes seem to be in good agreement with experimental values. -6- LBL-1625 A similar microscopic calculation was done by Tamura and Yoshida,10 who considered the magnitude and phase of the Ml matrix element in terms of the lowest-lying K = 2 two-quasiparticle states which can mix with both the y and ground-state bands. They estimated lol ~ 10, in reasonable agreement with observed values, and also o > 0. However, their 6 was defined in terms of absorption matrix elements, and the transformation to the presently-employed emission matrix elements requires a knowledge of the spatial and temporal symmetr,Y properties of the nuclear wave-functions and multipole operators used. 3 (A complete discussion of this problem is given by Alder and Steffen. ) If w.e assume the convention of Biedenharn-Rose1 was used, then in terms of the present convention, 6 < 0, in agreement with experiment. 4. g-Factor Variation. In the ~K = 2 band-mixing analysis given above, it was assumed that the gR-factors were identical for the 8-, y-, and ground-state bands. Relaxing this requirement gives rise to Ml transitions which depend on the variation of gR; however, this additional contribution to the Ml matrix element occurs only for ~I = 0 transitions. The 2 - 2 and 20 - 2 mixing y g ~ g ratios both require that the gR-factor difference between the ground and vibrational bands be which implies an increase in the ~-factor by 2-1/2 times in the excited bands. Such an increase seems highly unlikely. Greiner11 has discussed the lowering of gR-factors from the value Z/A in terms of a model in which the proton distribution is characterized by a somewhat smaller deformation than the neutrons. The Ml transition operator then obtains a tensor character dependent on the collective variables, and thus has non-vanishing __ , -7- LBL-1625 matrix elements between the coll~ctive bands. The predictions of this model for the magnitudes of the E2/Ml mixing ratios are given in Table I. This model is charact~rized by a smooth variation of o from nucleus to nucleus, and thus is unable to account for the sudden changes in o in the Er and Yb nuclei. The phase of the mixing ratio appears in this model to be positive for transitions from both the 8 andy bands; however, as discussed above, the absorption matrix elements experience a charge of phase when converted to emission matrix elements. Consequently, although the predicted phase of the y-band mixing ratio agrees with experiment, the identical phase predicted for the 8-band does not agree. 5. Pairing-Plus-Quadrupole Model. The apparent increase in o/E for the osmium nuclei comes about through a decrease in the energy of the K = 2+ level associated with the y-vibration, rather than through an increase in o. For these nuclei, which are in a region of transition from deformed to spherical equilibrium shapes, Kumar and Baranger12 have employed the pairing-plus-quadrupole model to predict energy levels and electromagnetic multipole moments. .. . . 13 The E2 and Ml moments were calculated by Kumar; and were found to be in good agreement with experimental E2/Ml mixing ratios (magnitude as well as phase) for Os nuclei, although the agreement is somewhat poorer for the (more deformed) W nuclei (see Refs. m-p of Table I) .• It can be concluded from this investigation that at present there is no satisfactory interpretation of both the magnitudes and phases of Ml admixtures in collective transitions in even-even deformed nuclei, although the ~K = 1 '-·· coupling through K = 1 + excitations seems to hold the most promise for a successful theory. Further insight into this problem must await studies of K = 1 + excitations, in order that the matrix elements entering into Eq. ( 5) m8iY be evaluated. Additionally, the agreement between the various theories and -8- LBL-1625 experiment seems to be poorest for the Er, Yb, and Hf nuclei, and it would thus be interesting to reduce the experimental uncertainty for the Er results and to obtain additional results for Yb and Hf nuclei. Finally, we note that, while most reasonable theories predict a unique phase for all mixing ratios of transitions depopulating the y-vibrational band, o(3y - 2g) in 168Er and o(2y ~ 2g) in 182w are at variance with the remainder of the cases studied. 182.._ While no explanation for the former case is apparent, -w also shows an anomalous phase and magnitude of o (28 - 2g). (While Refs. m and n of Table I chose the larger root for o, the directional correlation data of 14 Herzog et al. are more consistent with the smaller root.) Although the K = 0 ·t t· f 182w · t do 'b t' ·t · 1 d th t 1 exc1 a 1on o 1s no a goo ~-v1 ra 10n, 1 1s coup e ra er s rong y to they-vibration, owing primarily to the small energy spacing. 15 In the tuc= 2 formulism, the anomalous 2y - 2g value could arise from a contribution nd from the 2 term of the denominator of Eq. (2), and the 28 - 2g phase (compared with 184w) is consistent with the sign change of the z8y matrix element15 b.etween 182w and 184w. A measurement of o(4y - 4g) in 182w would shed considerable light on this problem. FOOTNOTES AND REFERENCES * Work performed under the auspices of the U. S. Atomic Energy Commission. 1. K. S. Krane and R. M. Steffen, Phys. Rev. C ~' 724 (1970). In this convention, the mixing ratio is defined in terms of emission matrix elements, and the phase of the mixing ratio is thus independent of the position of the y-ray in the cascade. The relationship to the Biedenharn-Rose convention [Rev. Mod. Phys. ~' 729 (1953)] for the cascade y1 - y2 is o(yl)KS = -o(yl)BR and o(y2)KS = o(y2)BR. 'I -9- LBL-1625 2. A. Bohr and B. R. Mottelson, Nuclear Structure (W. A. Benjamin, New York, 1969), Vol. I, p. 381 3. K. Alder and R. M. Steffen, in The Electromagnetic Interaction in Nuclear Physics, ed. by W. D. Hamilton (North-Holland, Amsterdam, 1972). 4. J. H. Hamilton, in Angular Correlation in Nuclear Disintegrations, ed. by H. van Krugten and B. van Nooijen (Rotterdam University Press, Wolters-Nordhoff Publishing, Groningen, The Netherlands, 1971), p. 181. 5. E. R. Marshalek, Phys. Rev. 158, 993 (1967). 6. L. L. Riedlinger, N. R. Johnson, and J. H. Hamilton, Phys. Rev. 179, 1214 (1969); J. H. Hamil ton et al. , Phys. Rev. C .2_, 899 ( 1972) ; P. Little et al. , Phys. Rev. C .2_, 252 (1972). 7. N. Rud and K. Bonde Nielsen, Nucl. Phys. Al58, 546 (1970). 8. A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II (to be published). 9. D. R. Bes, P. Federman, E. Maqueda, and A. Zuker, Nucl. Phys. 65, l (1965). 10. T. Tamura and H. Yoshida, Nucl. Phys. 30, 579 (1962). 11. W. Greiner, Nucl. Phys. 80, 417 (1966). 12. K. Kumar and M. Baranger, Nucl. Phys. Al22, 273 (1968). 13. K. Kumar, Phys. Letters 29B, 25 (1969). 14. P. Herzog, M. J. Canty, and K. D. Killig, Nucl. Phys. Al87, 49 (1972). 15. C. Glinther, P. Kleinheinz, R. F. Casten, and B. Elbek, Nucl. Phys. Al72 273 (1971). Table I. Reduced E2/Ml mixing ratios 6/Ey(MeV) of transitions from levels in 6- and Y-bands to levels in ground-state bandsa Nucleus 2 - 2 3 ~ 2 3 - 4 4 - 4 26 - 2g 48 - 4g y g y g y . g y g 152sm -9.5(2) b -8.0(9)c -7.0(3) b -2.8(3) b . 7 b .+(25: 4) +4.7(2l)b ( 8. 5; 12.5) (7.4; 11.7) (6.5; 8.5) (5.1; 6.5) (6.6) (J. 4) 154Gd -ll.6(ll)c -6.6(7)c -7.5(2) d -4.9(6)d +16(4)d +9(3)d (7. 3; 12.4) ( 6. 4; 11.6) (5.6; 8.4) (4.4; 6.4) (6.4) (3.4) 156Gd -17(3)e -(5.7 + 2.r)e - 1. (9.5; 14.1) (7.3) 160Dy -l2.5(19)f -9.4(25)g -(6 + 6)g - 2 (8.5; 13.6) (7.4; 12.7) (6.5; 9.3) 162Dy -(9 + oo)h - 7 -(3 + 6)i - 1 ( 10. 0; 14. 9) (6.0; 7.8) 164 . -(12 + oo)h Dy - 9 (12.0; 15.6) 166Er -(27 + 54)i - 13 -(5 + 4)i - 2 (9.0; 15.7) (5.4; 8.1) (continued) r . . \. ·--- --··· -- -------· - ... --·· ------· ···---· -------------- -----···- -------------------------- ---- -·-- ---· ··-·---------- ---. ------ --------- ·------- . --------·· -------···- -· -- -----~---- -------------- -------I f-' 0 I t"' tp 1 f-' 0\ 1\) V1 ,~-Nucleus 2 - 2 3 - 2 y g y g 168Er -(39 + 30)j - 12 +20(3)j (10.8; 15.9) (9.4; 14.8) 170Er + 00 i -(67 - 48) (11.2; 16.1) 172Yb -(7 ~ ~)g -(4 + 2)g - 1 (28; 16.5) (24; 15.4) 174Hf 178Hf -(30 + oo)R, -19 (3.0; 14.3) 182w +(19 : ~7)m ,...( 49 + Bl)m - 16 ( 4. 2; 13.4) ( 3.6; 12.5) 184 w -20(1) 0 -14.7(10) 0 (5.2; 13.0) (4.5; 12.1) Table I. (continued) 3 - 4 4 - 4 y g y g -7.7(5)j -(8 + 8)i . - 5 (8.3; 10~8) (6.5; 8.3) -(45 + 00 )i - 26 (6.7; 8.5) -9(2)m (3.2; 9.1) -13.2(12) 0 -( 8 + 4)o - 3 (4.0; 8.8) (3.1; 6.8) 28 - 2g < -4 k (8.2) -0.51 (5 )n (6.7) +2.3(6) 0 (6.5) #" ·-4(3 - 4 g -3(l)k (4.2) (continued) I I-' I-' I &; 1 I-' 0\ r\) Vl Nucleus 186w 1860s 1880s 1900s 2 - 2 y g -(18 + 6)n - 5 (5.2; 13.2) -(16 + 24)p - 6 (5.2; 14.0) -26(6)p ; 13.5) -23(3)p ; 13.7) 3 - 2 y g -(17 ~ ~2 )p (4.5; 13.1) -11(5)P ; 12.6) -16(3)P ; 12.8) Table I. (continued) 3 - 4 y g 4 - 4 y g ? - 2 -;s g +(15 + 8o)n 7 (,6.6) 4 - 4 B g aExplanation of Table: The subscripts y, B, and g refer to states of the y-, B-, and ground-state bands, respectively. The experimental uncertainty of the last place is given in parentheses follo~ing each entry. The two numbers under each entry give respectively the prediction for the magnitude of o as calculated according to the methods of Beset al. (Ref. 9) and Greiner (Ref. 11), except for -- . transitions from the S-band, where only the predictions of Greiner are given. bA. V. Ramayya et ~.,Angular Correlations in Nuclear Disintegration, ed. by H. van Krugten and B. van Nooijen (Groningen, The Netherlands, Rotterdam University Press, 1971), P• 247. cJ. Lange et al., Nucl. Phys. Al71, 92 (1971). -- -~.c. Whitlock et al., Phys. Rev. C }, 313 (1971). ( --·~ " c ce-nt i nued} I I-' 1\) I &; 1 I-' 0\ 1\) \.11 .. (' Table I. (continued) eJ. H. Hamilton et &·, Phys. Rev. C .2_, 899 (1972). f . . K. S. Krane and R. M. Steffen, Nucl. Phys. Al64, 439 (1971). gJ. Lange, Angular Correlations in Nuclear Disintegration, p" 242. h G. Engler and S. Lane, Bull. Am. Phys" Soc. 15, 100 (1970). i J. M. Domingos et al. , Nucl. Phys. Al80, 600 (1972). '• jL" M. Quinones et al., Bull. Am. Phys. Soc. 18, 37 (1973) and private communication from t-1. Behar (January 1973). ~. Ejiri and G. B. Hagemann, Nucl. Phys. Al6l, 449 (1970). ~ ---L. Varnell ~ ~· , Phys. Rev. C ,l, 1265 ( 1971). ~" S. Krane et ai:, Phys. Rev. C .2,, 1104 (1972). ~. T. Milner et al., Nucl. Phys. Al77, l (1971). --. -°K. S. Krane ~ &·, Phys. Rev. C 1 (to be published). PK" S. Krane and R. M. Steffen, Phys. Rev. C ,l, 240 (1971). I 1-' w I &; 1 1-' 0\ 1\) \.n i r-----------------LEGALNOTICE------------------~ This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. TECHNICAL INFORMATION DIVISION LAWRENCE BERKELEY LABORATORY UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720 ..., 

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E2/M1 Y RAY MULTIPOLE MIXING RATIOS IN EVEN-EVEN DEFORMED NUCLEI

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E2/M1 Y RAY MULTIPOLE MIXING RATIOS IN EVEN-EVEN DEFORMED NUCLEI

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