Microscopic-macroscopic approach to the nuclear fission process Aleksandra Kelić and Karl-Heinz Schmidt GSI-Darmstadt, Germany

Overview Why studying fission? What is the needed input? Mass and charge distributions in fission  GSI semi-empirical model Saddle-point masses  Macroscopic-microscopic approaches Conclusions

Why is fission interesting?

Energy production - Fission reactors - Accelerator-driven systems:

RIB production Fragmentation method, ISOL method Data measured at GSI* * Ricciardi et al, PRC 73 (2006) ; Bernas et al., NPA 765 (2006) 197; Armbruster et al., PRL 93 (2004) ; Taïeb et al., NPA 724 (2003) 413; Bernas et al., NPA 725 (2003) 213

Astrophysics S. Wanajo et al., NPA 777 (2006) 676 2) Panov et al., NPA 747 (2005) 633 TransU elements ? 1) Fission cycling ? 3, 4) r-process endpoint ? 2) 3) Seeger et al, APJ 11 Suppl. (1965) S121 4) Rauscher et al, APJ 429 (1994) 49 1) Cowan et al, Phys. Rep. 208 (1991) 267

Basic science Fission corresponds to a large-scale collective motion: Excellent tool to study:   Viscosity of nuclear matter   Nuclear structure effects at large deformations   Fluctuations in charge polarisation N. Carjan et al, NPA452 Both static (e.g. potential) and dynamic (e.g. viscosity) properties play important role

What do we need? Height of fission barriers Fragment distributions Level densities Nuclear viscosity Fission competition in de-excitation of excited nuclei Daughter Height of fission barriers Fragment distributions Level densities Nuclear viscosity

Mass and charge division in fission - Available experimental information - Modelisation - GSI model

Experimental information - High energy In cases when shell effects can be disregarded, the fission-fragment mass distribution is Gaussian  Data measured at GSI: M. Bernas et al, J. Pereira et al, T. Enqvist et al (see Width of mass distribution is empirically well established (M. G. Itkis, A.Ya. Rusanov et al., Sov. J. Part. Nucl. 19 (1988) 301 and Phys. At. Nucl. 60 (1997) 773)

Experimental information - Low energy Particle-induced fission of long- lived targets and spontaneous fission Available information: - A(E*) in most cases - A and Z distributions of light fission group only in the thermal- neutron induced fission on the stable targets EM fission of secondary beams at GSI Available information: - Z distributions at "one" energy

Experimental information - Low energy K.-H. Schmidt et al., NPA 665 (2000) 221 Experimental survey at GSI by use of secondary beams

How can we describe experimental data? Encouraging progress in a full microscopic description of fission: H. Goutte et al., PRC 71 (2005)  Time-dependent HF calculations with GCM:  Empirical systematics on A or Z distributions - Problem is often too complex  Semi-empirical models - Theory-guided systematics  Theoretical models - Way to go, but not always precise enough and still very time consuming

Macroscopic-microscopic approach - Transition from single-humped to double-humped explained by macroscopic and microscopic properties of the potential-energy landscape near outer saddle. * Maruhn and Greiner, Z. Phys. 251 (1972) 431, PRL 32 (1974) 548; Pashkevich, NPA 477 (1988) 1; N~90 N= Pb 238 U Macroscopic part: property of CN Microscopic part: properties of fragments*

Basic assumptions Macroscopic: Potential near saddle from exp. mass distributions at high E* (Rusanov) Macroscopic potential is property of fissioning system ( ≈ f(Z CN 2 /A CN )) Microscopic: Assumptions based on shell-model calculations (Maruhn & Greiner, Pashkevich) Shells near outer saddle "resemble" shells of final fragments (but weaker) Properties of shells from exp. nuclide distributions at low E* Microscopic corrections are properties of fragments (= f(N f,Z f )) Dynamics: Approximations based on Langevin calculations (P. Nadtochy) τ (mass asymmetry) >> τ (saddle scission): decision near saddle τ (N/Z) << τ (saddle scission) : decision near scission Population of available states with statistical weight (near saddle or scission) Basic ideas of our macroscopic-microscopic fission approach (Inspired by Smirenkin, Maruhn, Pashkevich, Rusanov, Itkis,...)

Macroscopic-microscopic approach For each fission fragment we get: Mass Nuclear charge Kinetic energy Excitation energy Number of emitted particles Fit parameters: Curvatures, strengths and positions of two microscopic contributions as free parameters These 6 parameters are deduced from the experimental fragment distributions and kept fixed for all systems and energies.

Comparison with EM data 89 Ac 90 Th 91 Pa 92 U Fission of secondary beams after the EM excitation: black - experiment red - calculations

Comparison with high-energy data

Comparison with data How does the model work in more complex scenario? 238 U+p at 1 A GeV Model calculations (model developed at GSI) : Experimental data:

Applications in astrophysics 260 U 276 Fm 300 U Usually one assumes: a) symmetric split: A F1 = A F2 b) 132 Sn shell plays a role: A F1 = 132, A F2 = A CN But! Deformed shell around A=140 can play in some cases a dominant role!

Saddle-point masses

How well do we understand fission? Influence of nuclear structure (shell corrections, pairing,...) LDM LDM+Shell

Fission barriers - Experimental information Relative uncertainty: >10 -2 Available data on fission barriers, Z ≥ 80 (RIPL-2 library)

Fission barriers - Experimental information Fission barriers Relative uncertainty: >10 -2 GS masses Relative uncertainty: Courtesy of C. Scheidenberger (GSI)

Experiment - Difficulties Experimental sources: Energy-dependent fission probabilities Extraction of barrier parameters: Requires assumptions on level densities Gavron et al., PRC13 (1976) 2374

Experiment - Difficulties Extraction of barrier parameters: Requires assumptions on level densities. Gavron et al., PRC13 (1976) 2374

Theory Another approach  microscopic-macroscopic models (e.g. Möller et al; Myers and Swiatecki; Mamdouh et al;...) Recently, important progress on calculating the potential surface using microscopic approach (e.g. groups from Brussels, Goriely et al; Bruyères- le-Châtel, Goutte et al; Madrid, Pèrez and Robledo;...): - Way to go! - But, not always precise enough and still very time consuming

Theory - Difficulties Dimensionality (Möller et al, PRL 92) and symmetries (Bjørnholm and Lynn, Rev. Mod. Phys. 52) of the considered deformation space are very important! Bjørnholm and Lynn, Rev. Mod. Phys. 52 Reflection symmetric Reflection asymmetric Limited experimental information on the height of the fission barrier  in any theoretical model the constraint on the parameters defining the dependence of the fission barrier on neutron excess is rather weak.

Open problem Limited experimental information on the height of the fission barrier Neutron-induced fission rates for U isotopes Panov et al., NPA 747 (2005) Kelić and Schmidt, PLB 643 (2006)

Idea Predictions of theoretical models are examined by means of a detailed analysis of the isotopic trends of saddle-point masses.  U sad  Empirical saddle-point shell- correction energy Macroscopic saddle-point mass Experimental saddle-point mass

Idea If an model is realistic  Slope of  U sad as function of N should be ~ 0 Any general trend would indicate shortcomings of the model. SCE Neutron number Very schematic! What do we know about saddle-point shell-correction energy? 1. Shell corrections have local character 2. Shell-correction energy at SP should be very small (e.g Myers and Swiatecki PRC 60 (1999); Siwek-Wilczynska and Skwira, PRC 72 (2005)) 1-2 MeV

Studied models 1.) Droplet model (DM) [Myers 1977], which is a basis of often used results of the Howard-Möller fission-barrier calculations [Howard&Möller 1980] 2.) Finite-range liquid drop model (FRLDM) [Sierk 1986, Möller et al 1995] 3.) Thomas-Fermi model (TF) [Myers&Swiatecki 1996, 1999] 4.) Extended Thomas-Fermi model (ETF) [Mamdouh et al. 2001] W.D. Myers, „Droplet Model of Atomic Nuclei“, 1977 IFI/Plenum W.M. Howard and P. Möller, ADNDT 25 (1980) 219. A. Sierk, PRC33 (1986) P. Möller et al, ADNDT 59 (1995) 185. W.D. Myers and W.J. Swiatecki, NPA 601( 1996) 141 W.D. Myers and W.J. Swiatecki, PRC 60 (1999) A. Mamdouh et al, NPA 679 (2001) 337

Example for uranium  U sad as a function of a neutron number A realistic macroscopic model should give almost a zero slope!

Results Slopes of δU sad as a function of the neutron excess  The most realistic predictions are expected from the TF model and the FRLD model  Further efforts needed for the saddle-point mass predictions of the droplet model and the extended Thomas-Fermi model Kelić and Schmidt, PLB 643 (2006)

Conclusions - Good description of mass and charge division in fission based on a macroscopic-microscopic approach, which allows for robust extrapolations - According to a detailed analysis of the isotopic trends of saddle- point masses indications have been found that the Thomas-Fermi model and the FRLDM model give the most realistic predictions in regions where no experimental data are available - Need for more precise and new experimental data using new techniques and methods  basis for further developments in theory - Need for close collaboration between experiment and theory

Additional slides

Comparison with data - spontaneous fission Experiment Calculations (experimental resolution not included)

Macroscopic-microscopic approach Calculations done by Pashkevich K.-H. Schmidt et al., NPA 665 (2000) 221

Comparison with data n th U (Lang et al.) Z Mass distributionCharge distribution

Needed input Basic ideas of our macroscopic-microscopic fission approach (Inspired by Smirenkin, Maruhn, Pashkevich, Rusanov, Itkis,...) Macroscopic: Potential near saddle from exp. mass distributions at high E* (Rusanov) Macroscopic potential is property of fissioning system ( ≈ f(Z CN 2 /A CN )) The figure shows the second derivative of the mass-asymmetry dependent potential, deduced from the widths of the mass distributions within the statistical model compared to different LD model predictions. Figure from Rusanov et al. (1997)

Ternary fission Rubchenya and Yavshits, Z. Phys. A 329 (1988) 217 Open symbols - experiment Full symbols - theory Ternary fission  less than 1% of a binary fission

Applications in astrophysics - first step Mass and charge distributions in neutrino-induced fission of r-process progenitors  Phys. Lett. B616 (2005) 48