Relative kinetic energy correction to fission barriers 1. Motivation 2. Results for A= systems 3. A cluster model perspective 4. Prescription based on w.f. localization & some results 5. Conclusions

Nucleus A=100, E=T intr+V, T intr = T- T cm 100 nucleons, T intr = 0 A A relative kinetic energy problem 2 nuclei A=50, T intr(50+50)= T intr (100) – T rel T rel = T cm1 + T cm2 – T cm(1+2)

Within HF: Asymptotically: The last term is T rel and should be subtracted to have a correct barrier for fusion, J.F. Berger and D. Gogny, N.P. A 333 (1980) 302 Fission barriers: In macro-micro mathods, a deformation dependent Wigner term was introduced to improve agreement with the data for lighter systems, W.D. Myers and W.J. Swiatecki, N.P. A 612, 249 (1997) Problems with a transition from one- to two-piece systems are spelled out in: P. Moller, A.J. Sierk and A. Iwamoto, P.R.L. 92, (2004)

Our code assumes two symmetry planes, so we can study tip- and side collisions. The HF (BCS) problem is solved on a spatial mesh. Static fusion barrier (l=0): V(R) = E[1+2](R)+B1+B2, (1) Static fission barrier: V(R) = E[1+2](R)+B(1+2), (2) where B1&B2&B(1+2) positive binding energies and E[1+2] - negative HF(BCS) energy of the combined system. We use Skyrme forces SkM* and SLy6, include no pairing, or delta-pairing with cut-off, a la M. Bender et al., Eur. Phys. J. A 8, 59 (2000). The latter ensures smooth transition of pairing from a 2- to 1-piece shape.

To have V=0 at infinity, one has to replace: V by V-T rel, where the relative kinetic energy at infinity is: (A2*t1+A1*t2)/(A1+A2), with t1 & t2 c.m. kinetic energies of the fragments. Example: for the merging fragments 48Ca and 208Pb one has with SLy6: Ecm(48Ca)=8.2 MeV, Ecm(208Pb)=5.9 MeV, Ecm(256No)=5.7 MeV, So, asymptotically, one has to subtract 8.4 MeV from (1) or (2). Various forces have specific prescriptions to correct for the c.m. motion of the system: -one-body part of P^2/2Am – amounts to MeV (like SkM*) -total P^2/2Am (with two body part) – amounts to 5-8 MeV (like SLy6)

This subtraction is clear for light fusing systems, with barriers at nearly separated shapes. For heavier systems the barriers correspond to some density overlap, so the subtraction should be partial. When two fragments merge into one, their relative kinetic energy becomes a part of the potential energy. No exact treatment of this problem is known. A simple choice for relatively light systems: - apply subtraction of the symptotic term to obtain the fusion and fission barrier

P.R. C 74, (R) (2006)

[16]: T.S.Fan et al.., N.P.A 679, 121 (2000) [8]: K.X. Jing et al., N.P. A 645, 203 (2000)

48 Ca + 48 Ca 48 Ca Pb

Dispersion of particle number in the k-th orbital:

Gradual subtraction of E kin rel – no exact prescription available

48 Ca Pu : Z=114, A=292 Q =225 b, For SkM*: E constr – E neck = 25.7 MeV vs 15 MeV

Can one learn anything from the theory of light nuclei? R.G. Lovas et al., Phys. Rep. 294, 265 (1998) Overcomplete basis:

No prescription for T rel.

B corr = 4.2 MeV

Q=300 b

At Q=200 b, T rel = 1.9 MeV

Q=125 b Q=200 b