Brownian shape mo.on: Fission fragment mass distribu.ons

Transcription

1 CNR*11: Interna/onal Workshop on Compound Nuclear Reac/ons and Related Topics Prague, Czech Republic, September 11 Brownian shape mo.on: Fission fragment mass distribu.ons Jørgen Randrup, LBNL Berkeley, California

2 Basic concepts in fission Nature 137 (1936) 351 quo.ng Niels Bohr: ``.. the energy of the incident neutron will be rapidly divided among all the nuclear par/cless.. N. Bohr & J.A. Wheeler, Phys Rev 56 (1939) 426: Niels Bohr, Nature 143 (1939) 33: ``.. any nuclear reac/on ini/ated by collisions or radia/on involves as an intermediate stage the forma/on of a compound nucleus in which the excita/on energy is distributed among the various degrees of freedom in a way resembling thermal agita/on..

3 5D deforma.on- energy surfaces for 5254 nuclei P. Möller et al, Phys Rev C79, 6434 (9) 8 Pb 24 Pu 264 Fm Five Essential Fission Shape Coordinates d " f1 " f2 M1 M2 Q 2 45 Q 2 ~ Elongation (fission direction) # 35! g ~ (M1-M2)/(M1+M2) Mass asymmetry # 15 " f1 ~ Left fragment deformation # 15 " f2 ~ Right fragment deformation 15 d ~ Neck $ grid points % 36 3 unphysical points $ physical grid points Macroscopic- microscopic method: E(Z, N, shape) = E macro (Z, N, shape) + E micro (Z, N, shape)

4 Example: deforma.on- energy surface for 258 Fm Fragment Elongation! (Units of R ) Fragment Elongation! (Units of R ) Potential energy for 258 Fm Family of shapes considered E (MeV) " 4 " Distance between Mass Centers r (Units of R Distance between Mass Centers r (Units ) of R ) Minima can be iden/fied by requiring U(this site) < U(any neighbor site): Saddles can be iden/fied by using the ``water- filling method :

5 How to use the poten.al- energy surfaces to elucidate the fission shape evolu.on? Random walk (Metropolis): JR: CNR*11 P(ΔU<) = 1 P(ΔU>) = exp(- ΔU/T)

6 P(A f ) from 24 Pu* and 236,234 U* 25 Calc. (6.84 MeV) Exp. 239 Pu(n,f) 24 Pu Calc. (6.54 MeV) Exp. 235 U(n,f) 236 U 15 a b 5 Yield Y(Z f ) (%) Calc. (6.54 MeV) Exp. 233 U(n,f) 234 U c Calc. (11. MeV) Exp. 234 U(!,f) 234 U d Fragment Charge Number Z f J. Randrup & P. Möller, PRL 6 (11) 13253

7 P(A f ) for 222,224,226,228 (γ,f) 25 Calc.(11 MeV) Exp. 222 (!,f) 222 Calc.(11 MeV) Exp. 224 (!,f) a b Yield Y(Z f )(%) Calc.(11 MeV) Exp. 226 (!,f) 226 c Calc.(11 MeV) Exp. 228 (!,f) 228 d Fragment Charge Number Z f J. Randrup & P. Möller, PRL 6 (11) 13253

8 JR, P Möller, AJ Sierk [nucl- th/ ] Restricted shape family? Five Essential Fission Shape Coordinates d " f1 " f2 M1 M2 5D - > 3D: V(i,j,m) = min kl [V(i,j,k,l,m)] Q Pu(n,f) 3D 5D 24 Pu (!,f) 3D 5D 45 Q 2 ~ Elongation (fission direction) # 35! g ~ (M1-M2)/(M1+M2) Mass asymmetry # 15 " f1 ~ Left fragment deformation # 15 " f2 ~ Right fragment deformation 15 d ~ Neck $ grid points % 36 3 unphysical points $ physical grid points U(n,f) 236 U 224 Yield Y(Z f ) (%) 233 U(n,f) 234 U Yield Y(Z f ) (%) U(!,f) 234 U Fragment charge number Z f Fragment charge number Z f => Must use sufficiently flexible shapes!

9 Nuclear shape dynamics: general Lagrange func/on: Rayleigh func/on: L( χ, χ) = 1 2 χ i M ij (χ) χ j U(χ) ij F( χ, χ) = 1 2 χ i γ ij (χ) χ j = 1 Q 2 ij π i = M ij χ j j d dt π i = L F χ i χ i? U: Poten/al energy of deforma/on M: Iner/al mass tensor for shape mo/on γ: Fric/on tensor Macroscopic- microscopic method Werner- Wheeler approxima.on to incompressible irrota.onal flow? v v V v =2V v One- body wall dissipa.on: Q = mρ v d 2 σ ṅ 2.. is strong => M is unimportant! J. Blocki, Y. Boneh, J. R. Nix, J. Randrup, M. Robel, A. J. Sierk, and W. J. Swiatecki, Ann Phys 113 (1978) 33: One- Body Dissipa.on and the Super- Viscidity of Nuclei

10 Strongly damped nuclear shape dynamics: Brownian mo.on Strong dissipa/on => creeping evolu/on => mass unimportant Brownian mo.on F pot = U/ χ Strongly damped EoM: F pot + F fric + F ran. F fric = F/ χ = γ χ = F ran (t) = Smoluchowski Equa.on Fi ran (t)fj ran (t ) = 2T γ ij δ(t t ) Solu/on: χ = µ (F pot + F ran ) => Average shape evolu/on: Direct numerical simula/on: χ i δχ i. = j = n Mobility tensor: µ ij F pot j χ (n) i µ γ 1 µ ij = n χ (n) i χ (n) j [ t χ (n) F pot + 2T t ξ n ] Isotropic mobility => simple random walk on the lance => Metropolis

11 Mobility tensor? v v V Dissipa/on rate: 1b wall formula Q = mρ v d 2 σ ṅ 2 v =2V v Q = ij χ i γ ij (χ) χ i = µν q µ g µν (q) q ν Fric/on tensor in the Nix variables: ρ g µν = π 2 2 mρ (z) v q µ ρ 2 (z) q ν [ ρ 2 (z)+ 1 4 ( ρ 2 z ) 2 ] 1 2 dz Five Essential Fission Shape Coordinates d " f1 " f2 Fric/on tensor γ in the lakce variables: At each lakce site IJKLM: γ ij (χ) = µν q µ χ i g µν (q) q ν χ j Renormalize γ(i,j,k,l,m) to ensure <γ n > = 1 γ(f) : γ n (f) γ n + f Isotropize: 1+f Mobility tensor: µ(f) = γ(f) 1 M1 Q 2 M2 45 Q 2 ~ Elongation (fission direction) # 35! g ~ (M1-M2)/(M1+M2) Mass asymmetry # 15 " f1 ~ Left fragment deformation # 15 " f2 ~ Right fragment deformation 15 d ~ Neck $ grid points % 36 3 unphysical points $ physical grid points f

12 Dependence of P(A f ) on the mobility tensor γ(f) : γ n (f) γ n + f 1+f µ(f) = γ(f) 1 => f =.2, 1., Pu(n,f) µ(f=.2) µ(f=1.) f >> 1 24 Pu 3 µ(f=.2) µ(f=1.) µ(f>>1) U(n,f) U 224 Yield Y(Z f ) (%) 233 U(n,f) 234 U Yield Y(Z f ) (%) U(!,f) 234 U Fragment charge number Z f Fragment charge number Z f

13 Summary so far: Shape mo.on is strongly damped => ignore iner.al masses => Smoluchowski equa.on (Brownian mo.on) χ = µ (F pot + F ran ) P(Af) depends only weakly on the dissipa.on tensor => Metropolis walk is a reasonable star.ng point e dependence on the dissipa.on anisotropy provides an es.mate of the uncertainty on the calculated P(Af) JR: CNR*11

14 Metropolis walk: e shape evolves un.l the neck has shrunk to a specified value c Shape diffusion: robust wrt c Sta.s.cal scission: Each scission configura.on is populated in propor.on to its sta.s.cal weight* Scission model: sensi.ve to c sc 24 Pu 239 Pu(n,f) 24 Pu 239 Pu(n,f) (!,f) c = 2.5 c = 2. c = (!,f) c sc = 2.5 c sc = 2. c sc = 1.5 Yield Y(Z f ) (%) (!,f) Yield Y(Z f ) (%) (!,f) (!,f) (!,f) Fragment charge number Z f Shape dynamics on the pre- scission poten.al landscape is important! Fragment charge number Z f * Fong, Phys Rev 2 (1956) 434; Wilkins et al, PRC 14 (1976) 1832

15 222 : Mass asymmetry is NOT determined at the saddle! Q 2 : I=7 I= Second minimum Outer saddle A f : M= M=8 Yield Y(Z f ) (%) Calc. (11 MeV) Exp. 222 (!,f) 222 Most probable fragment mass division 111 Re 111 Re M= Fragment Charge Number Z f Distribu/on of mass asymmetries at a specified fixed elonga/on Q 2 : P (A f ; Q 2 ) n δ(i n I) δ(m n M) Relative population P(Z f ;I) (E*=11 MeV) Prefragment charge number Z f

16 226 : Shape dependence of the Wigner term? E(Z, N, shape) = E macro (Z, N, shape) + E micro (Z, N, shape) E macro (Z, N, shape) = + W (shape) N Z A 25 One nucleus (E*= 11 MeV) µ(f =.2) Wigner (shape) Wigner ~ const e shape dependence of the Wigner term influences P(A f )! Yield Y(Z f ) (%) µ(f = 1.) 15 µ(f >> 1) Two nuclei Fragment charge number Z f

17 Dependence on level- density parameter? E = a A T 2 : ã A = A/e [ a A (E ; χ) =ã A 1+(1 e E /E damp ) δu ] sh(χ) E [Ignatyuk et al.] 234 U(!,f) E* = 11 MeV a(micro) Yield Y(Z f ) (%) 15 N=, M=1 a A = A/e e = 8 e = Fragment charge number Z f.. more refined studies needed, incl. pairing Δ(shape, excita.on)

18 24 Pu (E*= 6.84 MeV) Summary and perspec.ves Yield Y(Zf) (%) Novel model for calcula/ng fission fragment mass distribu/ons Required: Fragment charge number Zf 6 Augments the standard compound nucleus picture with explicit simula/on of the equilibra/on dynamics Conceptually very simple e deforma/on- energy landscape for a sufficiently rich family of fission shapes Very important e fric/on tensor for the shape mo/on Less crucial Unprecedented predic.ve power Useful tool Preliminary applica/on possible to >5, nuclei for which the 5D surfaces are already available Fission recycling in the r- process Requires only rela/vely modest compu/ng power 5, CPU hours vs ~s on a laptop A number of model aspects need further study Fric/on tensor, level density,.. Extensions an/cipated Spontaneous fission? fragment TKE? JR: CNR*11

19 ank you!