NEW FISSION-BARRIER CALCULATION. T. Ichikawa (RIKEN) A. Iwamoto (JAEA) R. Bengtsson, Henrik Uhrenholt, and Sven Åberg (LUND)

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1 NEW FISSION-BARRIER CALCULATION P. Möller and A. J. Sierk (LANL) T. Ichikawa (RIKEN) A. Iwamoto (JAEA) R. Bengtsson, Henrik Uhrenholt, and Sven Åberg (LUND) 1

2 In fission, what are the shapes and related energies involved in the transition from a single ground-state shape to two separated fission fragments? 234 U E(def?)? E(def?)? 234 U : Asymmetric valley at Q 2 = 76 ε f1 =.1 ε f2 =.25 M H /M L = 135.7/ U : Symmetric valley at Q 2 = 76 ε f1 =.15 ε f2 =.1 M H /M L = 119.3/ I 98 Y 117 Pd 117 Pd

3 Fragment Elongation σ (Units of R ) Family of shapes considered Distance between Mass Centers r (Units of R )

4 Fragment Elongation σ (Units of R ) Potential energy for 258 Fm E (MeV) Distance between Mass Centers r (Units of R )

5 Five Essential Fission Shape Coordinates d ε f1 ε f2 M1 M2 Q 2 45 Q 2 ~ Elongation (fission direction) 35 α g ~ (M1-M2)/(M1+M2) Mass asymmetry 15 ε f1 ~ Left fragment deformation 15 ε f2 ~ Right fragment deformation 15 d ~ Neck grid points 36 3 unphysical points physical grid points

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9 1 A Saddle Search Strategies Illustrated B Y E (MeV) α 5 5 E D C X F G Θ

10 5 E One-Dimensional Paths Saddle point C Function Value 5 1 D Y X 15 A B F G Θ

11 1 Saddle Search Strategies Illustrated 5 α Θ Graphics by Peter Möller

12 Potential Energy of Deformation We use the macroscopic-microscopic method introduced by Swiatecki and Strutinsy: (1) The macroscopic term is calculated in a liquid-drop type model (for a specific deformed shape). The microscopic correction is determined in the following steps 1. A shape is prescribed 2. A single-particle potential with this shape is generated. A spin-orbit term is included. 3. The Schrödinger equation is solved for this deformed potential and single-particle levels and wave-functions are obtained 4. The shell correction is calculated by use of Strutinsky s method. 5. The pairing correction is calculated in the BCS or Lipkin-Nogami method.

13 Hexadecapole Deformation ε λ p = 34.8, a p =.8 fm Single-Proton Energy (MeV) Spheroidal Deformation ε 2

14 Fission Calculation Details 1. Fission Barriers of 5254 nuclei calculated for Several different parameterizations are used 2. 5D parameterization, energy for 5 different shapes are calculated for each nucleus 3. For small deformations a 3D parameterization is used Elongation, neck and axial-asymmetry shape degrees of freedom. 4. An improved determination of the ground-state energy and shape is done in a 4D space 5. When multiple minima are present a special strategy is used to establish which minimum and saddle define the barrier. (In practice this technique cannot be implemented in HFB)

15 6. Just the potential energies correspond to more than 25 Gb of information. 7. Saddles, minima, valleys are determined in a completely automated fashion. Compact result files are generated for each nucleus 8. Data sets, such as tables of barrier heights ( ) are generated, also by automated scripts. Results will be made available at URL (Capital M is essential) Results more complete than can be published will be available here in due course.

16 Energy Release Q α (MeV) Neutron Number N α-decay of RIKEN exp. (24) OTHER exp. FRDM (1992) FRLDM (1992) HFB-8 (Goriely) HFB-2 (Goriely) Proton Number Z

17 Wed Aug 3 1:43:15 MDT 26 Graphics Art by Peter Moller 6 118X 19 Scale.5 (MeV) γ Axial Asymmetry Spheroidal Deformation ε 2

18 1 Fission-Barrier and Associated Shapes for 242 Am (5) (4) (3) Potential Energy (MeV) 5 5 5D Asym. path β-par. (r constr) Separating ridge 5D Sym. path (1) (2) β-par. (β 2 constr) 5D Saddles 5D Minima (6) Nuclear Deformation (Q 2 /b) (1/2)

19 Barrier Energy E A (MeV) 1 5 Barrier Heights for U E II (MeV) Inner Isomer Barrier Energy E B (MeV) 5 Evaluated heights (RIPL, IAEA) Evaluated heights (Madland) Calculated heights Nucleon Number A Outer

20 Barrier Energy E A (MeV) 1 5 Barrier Heights for Pu E II (MeV) Inner Isomer Barrier Energy E B (MeV) 5 Evaluated heights (RIPL, IAEA) Evaluated heights (Madland) Calculated heights Nucleon Number A Outer

21 Asymmetric mode Symmetric mode Separating ridge 5 5 Graphics by Peter Möller Fission-Barrier Height (MeV) 1 Fission Barrier and Associated Shapes for 232Th Nuclear Deformation (Q2 / b)(1/2) 1

22 Level Density ρ (MeV -1 ) Outer Saddle Level Densities for 232 Th Relative to saddles gs. Asym. Saddle Sym. Saddle Excitation Energy (MeV)

23 !#"$!%&('*) +&,- ' "$. '/%$!" 132 "547687'9-!%:' 2 +&,,-'3!3 2 '9;$ $<#=-2>"$!?'2@/A 32A 2+&/ABA/A68/C'92%&D2E/%:'3/%> /A (FG"8!%8('9!!H ;$ %I68KJLM% +N,7F%O'9;$F3!MC3 P7QSR T2U%V7)>- '/AW"8"&(X$%8" PYR T2, -7)- '3/A "8"&(HU %8" Z 31!['2\';8K3!A 2%8"]%&"M'9;8"V"&"8,X 3!3+L!A '(!()X&T2,^_'+&.#;I68-+`!"aJNC3!XN3bc8Hedd&H f!%8('*)gc(' h2@c(' i 68A!6& jlk7m9n n o p qsrtvuxw o puqsrtvuxw P J&,% k7mn R Psy z R { k P}y z R P}y z R { k P}y z R H H Y Q H H Ŷ H Y Y Q H H ƒ Q Q H H H ƒ Q Q H Q H ƒ Ŷ H Q H H R H Ŷ Q ƒ Q H Q H Z R H H H H ƒ H ƒ Q ƒ H!%>!%]7)7'3F n7~n ; P7QSR, QS 5 O H ƒ Q Hvƒ O QS, n7~n ; PYR,, H ƒo QŠ H 2>"8">!%M7)>7'! n7~œ = P7QSR >H Y, QŠ H O C : nž k = P7QSR >H O YC Y >H Q, Hx ˆ nž 9~ = P7QSR Y, Q QS Y, H ƒ S 2>"8">2>"8"V7)>7'! n7~ = P7Q >H >Q! >H Q Y 5, n7~ = P7Q, QS Y YC Y QS Q H ƒ Y5 nž n = P7QKR >H >QŠ Y>Q H Q Hx ˆ

24 Calculated Energy Window for EC-Delayed Fission Proton Number Z Q EC (N,Z) B f (N + 1,Z 1) (MeV) ECdf Observed β-stable Neutron Number N

25 "!$# %'&( )*+(+&-,./(1, 2 345)&-,$687(+(+5)(:95);-9</ >=@?A%B&( (+C/5)&,EDF9(+G HI68KJL 3M5)&, 9N7O, &7 +(1- CPQ.R(5ST,UVJW X C/5)&, "!$# =@? "!Y#[Z =@? \^] *_a` \b] *_a` \^] *_a` cedf d*c jlk gih ww1d xyz. jlk gih w1yw x ES jlk gih w1y+ x ES jlk gih w1yd x Hˆ jlk gih wš w xx jlk gih wš xx jlk gih wš / xx jlk gih wš /d xx jlk gih cedf dfm ; nowqpp rywtsqn oqwtuv ww1d xw*{ pwt u }$WnKv ZFo~s< w1yw x+ Lƒ pwss vyw v n2wtu-} w1y+ x+ ƒ pw)n vywtsv oqw r w1yd x S pw pwtr- ZFo~)n } wš w xd Œ% }$Wt uywnku ZFo~rp wš xd Œ% pwqp } uywtur Z $~t Žp wš / xd Œ% vywur WnKu Z vy~qp$ wš /d xd Œ% $Wrs W Žp Z p~t u

26 ! "$# %&')('*+,(-./+') & "') & "' 9;:< =>?A@ B 9DCFE$G #IH 9;:$ =>I?A@KJ 9DCE<G L M N O 'P5 RQTS 5 O ')U5 VQTS 5 WX Y"ZUZ XU[U\ Y"Z&]^[TY Y$\T]^[W _4]Y$_ à]b[c` WdZ Y$[U_ XU[X efyp]bxuw eg[h]ixjy ekz]y$w WdZ Y"Zc\ XdZk Wj]bXX YY]lkam Z]imjY [4]bW[ Y+kUk Y<mUX XmUX \j]lkc\ X4]lkcW enyo]y"z e Xh]imdZ Y+kUk Y<mP_ XmP_ e à]^[oz efy+k]^wty WU\ Y<XP\ XXUX Wj]pZY ekzt]y<x WU_ Y<XP\ XXqZ YU]^\m ekzt]r`dk Y+kUk Y<XP\ XXP\ es[4]lkc[ e ma]ixuw WU\ Y<XP_ XXqZ egxh])y$\ eg_h]b[m WU\ Y$[dZ XU[ok efy+kt]i`p\ enyo]lza_ WU_ Y$[X XU[ok efy<mh]^w\ ekzt]imx YY$X Y$\m XÙ` egmh]^[c` enyyo]bwty enyyp]yy #H

27 Proton Number Z B f (MeV) Calculated Fission-Barrier Heights Neutron Number N