Oscillations in mass asymmetry in second and third minima in actinides 1. Second & third minima in actinides 2. Barrier calculations: Micro-macro vs. selfconsistent calculations 3. Oscillations in mass asymmetry in the first & second minima 4. Vibrations in the III minima; different meaning of the K=0 mode 5. Conclusions M. Kowal & J. Skalski

Polikanov et al. (1962) Discovery of isomeric fission Strutinsky (1967) Calculated second minima Specht et al. (1972) Identification of the rotational band with large moment of inertia in 240Pu

At present: fission isomers in 34 nuclei, from U to Bk, in that 11 in Pu, 10 in Am half-lives: 6 ps to 14 ms (in 242Am) In some nuclei, there are double isomers (supposed spin isomeric states in II min.) Measured quadrupole moments: 236U 32(5) b 238U 29(3) b 236Pu 37(11) b 239Pu 36(4) b from optical isotope shift & hyperfine structure: 240Am 33.9 b 242Am 33.5(1) b 244Am 34.4 b

Third minima: Th,U First predicted: P. Moller, S.G. Nilsson and R.K. Sheline (1972) then Howard & Moller (1980) – rather shallow III-rd minima S.Cwiok et al. – rather deep III-rd minima some, not all, HF calculations give III-rd minima, BUT they often differ from macro-micro results Experiments: 1)Studies of microstructure in the resonances of fission probability found using (n,f), (t,pf) and (d,pf) reactions B.B. Back et al. (1972) J. Blons et al. (1975) recent claims of III-rd minima in 232,234,236U 2) Also observations of asymmetric angular distribution of light fission fragments around 232Th

Results (rather preliminary)

Difficulties with energy minimization with YPE

III min : 120 or 170 b

III min 140 or 195 b

ZNE ( I ) MINB ( I )E (II) MINB (II) E(III) a b3=0,6 E(III) b B3=0,3B(III) ,36, ,56,92.06, , ,74,53.05, , ,95,12.56, , ,05,92.06, , ,96,92.46, ,8

Vibrations in mass asymmetry: -Axially deformed, mass-symmetric I-st and II-nd minima Well defined K, but the phonon angular momentum is not well defined in the intrinsic frame. -We mix vibrations related to spherical harmonics 3,5 & 7 and fixed K=0,1,2,3. - Method: small vibrations det (C – omega^2 B) = 0, where C&B are 3 by 3 stiffness&mass matrices -Quadratic form C well approximates energy around the minima - For B we take cranking masses at the minimum (not very bright).

Parabolic fit to E(beta30) in the II-nd well

III-rd minima show large mass-asymmetry. One expects nearly degenerate, alternating parity gs band. The K=0 vibration means a different thing than in the I and II well: it is no a piori reason that it be small. Since the III-rd minima are axially symmetric, the scheme of calculating vibration energies is kept.

III MIN (b3=0,6) ZNAALL SQRT (C3i/B3i) OSCILLATION ,12154, ,12337, ,83055, ,63181, ,73297, ,83350,0 OSCILLATION ,02980, ,72988, ,62538, ,32751, ,72851, ,23044,8 OSCILLATION ,52145, ,82415, ,42736, ,22578, ,52886, ,32285,4 III MIN (b3=0,3) ZNAALL SQRT (C3i/B3i) OSCILLLATION ,32060, ,32161, ,32065, ,42239, ,32252, ,62122,8 OSCILLATION ,92271, ,92391, ,62208, ,62270, ,22287, ,52133,5 OSCILLATION ,02080, ,72128, ,92019, ,32023, ,52018, ,32173,8

Conclusions: 1.Within WS+YpE macro-micro approach one obtains rather deep & double mass-asymmetric III-rd minima. 2. Within WS+LSD they become single, but remain deep. 3. They often do not show in the HF calc., and are shallower in P. Moller’s results. ! So what is the truth? Is experiment clear enough? 4. Mass-asymmetry („octupole”) vibrations: - not a bad agreement in the I-st min (even the order of K is reproduced) - too high energies in the second min. in 240Pu – especially if K=0 is as low as suggested by expt. - no very low lying vibrational states in the III-rd well.