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Microscopic-macroscopic approach to the nuclear fission process

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Microscopic-macroscopic approach to the nuclear fission process

Aleksandra Kelić and Karl-Heinz Schmidt

GSI-Darmstadt, Germany

http://www.gsi.de/charms/

- 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

- Fission reactors

- Accelerator-driven systems:

Fragmentation method, ISOL method

Data measured at GSI*

* Ricciardi et al, PRC 73 (2006) 014607; Bernas et al., NPA 765 (2006) 197; Armbruster et al., PRL 93 (2004) 212701; Taïeb et al., NPA 724 (2003) 413; Bernas et al., NPA 725 (2003) 213

www.gsi.de/charms/data.htm

TransU elements ? 1)

r-process endpoint ? 2)

Fission cycling ? 3, 4)

S. Wanajo et al., NPA 777 (2006) 676

1) Cowan et al, Phys. Rep. 208 (1991) 267

2) Panov et al., NPA 747 (2005) 633

3) Seeger et al, APJ 11 Suppl. (1965) S121

4) Rauscher et al, APJ 429 (1994) 49

Fission corresponds to a large-scale collective motion:

N. Carjan et al, NPA452

Both static (e.g. potential) and dynamic (e.g. viscosity) properties play important role

Excellent tool to study:

- Viscosity of nuclear matter
- Nuclear structure effects at large deformations
- Fluctuations in charge polarisation

Daughter

Fission competition in de-excitation of excited nuclei

- • Height of fission barriers
- Fragment distributions
- • Level densities
- • Nuclear viscosity

- • Height of fission barriers
- Fragment distributions
- • Level densities
- • Nuclear viscosity

- Available experimental information

- Modelisation

- GSI model

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 www.gsi.de/charms/)

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)

- 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 survey at GSI by use of secondary beams

K.-H. Schmidt et al., NPA 665 (2000) 221

Empirical systematicson A or Z distributions - Problem is often

too complex

Theoretical models - Way to go, but not always precise enough and

still very time consuming

Encouraging progress in a full microscopic description of fission:

H. Goutte et al., PRC 71 (2005) Time-dependent HF calculations with GCM:

Semi-empirical models - Theory-guided systematics

208Pb

238U

N~90

N=82

- Transition from single-humped to double-humped explained by

macroscopic and microscopic properties of the potential-energy landscape near outer saddle.

Macroscopic part: property of CN

Microscopic part: properties of fragments*

* Maruhn and Greiner, Z. Phys. 251 (1972) 431, PRL 32 (1974) 548; Pashkevich, NPA 477 (1988) 1;

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(ZCN2/ACN))

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(Nf,Zf))

Dynamics:

Approximations based on Langevin calculations (P. Nadtochy)

τ (mass asymmetry) >> τ (saddle scission): decision near saddleτ (N/Z) << τ (saddle scission) : decision near scission

Population of available states with statistical weight (near saddle or scission)

- 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.

- For each fission fragment we get:
- Mass
- Nuclear charge
- Kinetic energy
- Excitation energy
- Number of emitted particles

92U

91Pa

142

140

141

90Th

138

139

89Ac

132

131

133

134

137

135

136

Fission of secondary beams after the EM excitation:

black - experiment

red - calculations

How does the model work in more complex scenario?

238U+p at 1 A GeV

Experimental data:

Model calculations (model developed at GSI):

300U

260U

276Fm

Usually one assumes:

a) symmetric split: AF1 = AF2

b) 132Sn shell plays a role: AF1 = 132, AF2 = ACN - 132

But! Deformed shell around A=140 can play in some cases a dominant role!

Influence of nuclear structure (shell corrections, pairing, ...)

LDM

LDM+Shell

Relative uncertainty: >10-2

Available data on fission barriers, Z ≥ 80 (RIPL-2 library)

Fission barriers

Relative uncertainty: >10-2

GS masses

Relative uncertainty: 10-4 - 10-9

Courtesy of C. Scheidenberger (GSI)

- Experimental sources:
- Energy-dependent fission probabilities
- Extraction of barrier parameters:
- Requires assumptions on level densities

Gavron et al., PRC13 (1976) 2374

Extraction of barrier parameters:

Requires assumptions on level densities.

Gavron et al., PRC13 (1976) 2374

- 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

- Another approach microscopic-macroscopic models (e.g. Möller et al; Myers and Swiatecki; Mamdouh et al; ...)

Bjørnholm and Lynn, Rev. Mod. Phys. 52

Dimensionality (Möller et al, PRL 92) and symmetries (Bjørnholm and Lynn, Rev. Mod. Phys. 52) of the considered deformation space are very important!

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.

Neutron-induced fission rates for U isotopes

Panov et al., NPA 747 (2005)

Limited experimental information on the height of the fission barrier

Kelić and Schmidt, PLB 643 (2006)

Experimental saddle-point mass

Macroscopic saddle-point mass

Predictions of theoretical models are examined by means of a detailed

analysis of the isotopic trends of saddle-point masses.

Usad Empirical saddle-point shell-correction energy

Very schematic!

SCE

Neutron number

What do we know about saddle-point shell-correction energy?

1. Shell corrections have local character

2. Shell-correction energyat SP should be very small (e.g Myers and Swiatecki PRC 60 (1999); Siwek-Wilczynska and Skwira, PRC 72 (2005))

1-2 MeV

If an model is realistic Slope ofUsad as function of N should be ~ 0

Any general trend would indicate shortcomings of the model.

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) 2039.

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) 0 14606-1

A. Mamdouh et al, NPA 679 (2001) 337

Usadas a function of a neutron number

A realistic macroscopic model should give almost a zero slope!

Slopes of δUsadas a function of the neutron excess

Themost 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)

- 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

Experiment

Calculations

(experimental resolution not included)

K.-H. Schmidt et al., NPA 665 (2000) 221

Calculations done by Pashkevich

Mass distributionCharge distribution

Z

nth + 235U (Lang et al.)

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(ZCN2/ACN))

The figure shows the second derivative of

the mass-asymmetry dependent potential, deducedfrom the widths of the mass distributions withinthe statistical model compared to different LD

model predictions.

Figure from Rusanov et al. (1997)

Ternary fission less than 1% of a binary fission

Open symbols - experiment

Full symbols - theory

Rubchenya and Yavshits, Z. Phys. A 329 (1988) 217

Mass and charge distributions in neutrino-induced fission of

r-process progenitors

Phys. Lett. B616 (2005) 48