Detailed Modeling of Fission Ramona Vogt (LLNL) LLNL-PRES-661908"
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344"
Fission Nomenclature!
§  Compound nucleus: created from n + A, mass A0 = A + 1, charge Z0
§  Scission: when the neck connecting the pre-fragments snaps
§  Fragments: the two excited nuclei immediately after scission,
AH + AL = A0, ZH + ZL = Z0 assuming binary fission, in general Af and Zf
§  Products: two types
•  Fission products: after initial neutron emission
(detected in fission experiments)
•  Cumulative products: after both prompt and delayed
emission has occurred
§  Prompt emission: in the first 10-13 s, neutrons emitted
first, followed by photon emission during initial de-excitation,
only Af changes (is reduced), not Zf, as fragments become
products
§  Delayed emission: late in time, after prompt emission
products can still emit neutrons and photons (small
compared to prompt emission) and can beta decay, emitting
neutrinos and electrons (Af reduced by delayed neutron
emission, Zf can increase by beta decay, n àp + e- + νe)
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2
More Nomenclature!
§  Total Kinetic Energy, TKE: kinetic energies of the two fragments
§  Excitation energy, E*: excitation energy of fragments during prompt emission
§  Neutron separation energy, Sn: energy needed to remove one neutron,
if E* > Sn, a neutron can be emitted, if E* < Sn, it can’t and photons
emitted preferentially
§  Rotational energy, ER: energy in spinning fragments
§  Fission Fragment Yields, Y(Af,Zf,TKE): probabilities for fragment
production in fission events, normalized to 2 fragments
§  Fission Product Yields: probabilities after prompt emission
§  Neutron multiplicity: number of neutrons per fission event,
_
expressed as ν, average multiplicity as ν, can be presented as
average, as a function of A or TKE, or as probability P(ν)
§  Photon multiplicity: number of photons per fission event, Nγ
§  Total photon energy: energy of photons emitted per fission
event, Eγ
§  Statistical photons: continuous energy spectrum, no effect
on rotational energy
§  Yrast photons: quadrupole transitions, reduces nuclear spin
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3
And Yet More Nomenclature!
Binary fission: split into 2 fragments
•  Symmetric fission: both fragments are near equal mass
•  Asymmetric fission: one fragment is heavy and the other is light
§  Ternary fission: split into 3 fragments
§  Spontaneous fission: internal excitation energy high enough for
nucleus to fission without any external energy applied, A0 = A, E* = 0
§  Neutron-induced fission: fission caused by incident neutron of
sufficiently high energy to overcome fission barrier, A0 = A + 1,
E* = Bn + En
§  Photofission: fission induced by incident photon with high enough
energy to overcome barrier, A0 = A, E* = Bn + Eγ
§  Multichance fission: for sufficiently high incident energies (not
for spontaneous fission), one or more neutrons could be emitted
before fission – 0n, 1st chance; 1n, 2nd chance; 2n, 3rd chance; etc.
§  Pre-equilibrium emission: incident neutron does not equilibrate
and is emitted again, endpoint of neutron energy in this case is
Eend = En - Bn
§ 
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4
Fission Product Yields as a function of mass A
10
Y(A)
10
10
10
10
10
Y(A)
Yields are asymmetric but become more symmetric and
broader at higher energies, shapes similar for same Z (= 94)
-2
thermal
-3
Bottom: illustrates the peak shift for the light fragment with
initial A. Note that the heavy fragment mass yield peak does
not move while that of the light fragment does
-4
10
10
0
-1
10
10
Left: yields for Pu isotopes at different energies –
Thermal = 0.025 eV; Fast ~ 1 MeV; High energy ~ 14 MeV
1
0
-1
-2
1
10
fast
-3
241
Pu
Am
245
Cm
249
Cf
237
Np
241
0
Y(A)
10
10
10
10
10
-4
high energy
0
238
-1
Pu
239
Pu
Pu
240
Pu
242
Pu
-2
80
100
120
Th
U
235
10
-3
10
-4
10 60
227
-2
241
-3
-1
10
Y(A)
10
140
160
180
Fragment Mass A
Lawrence Livermore National Laboratory
-4
10
80
100
120
140
160
180
A
5
Fission Product Yields as a function of charge Z
Points are a calculation based on shape evolution of a potential
energy surface as it approaches fission
The model, by Randrup and Moller, has shown good agreement
with a wide range of actinide data
25
240
Calc. (6.84 MeV)
Exp. 239Pu(n,f)
Pu
236
Calc. (6.54 MeV)
Exp. 235U(n,f)
U
20
a
15
b
Yield Y(Zf) (%)
10
5
0
25
234
Calc. (6.54 MeV)
Exp. 233U(n,f)
U
234
Calc. (11.0 MeV)
Exp. 234U(γ,f)
U
20
c
d
15
10
5
0
30
40
50
60
30
Fragment Charge Number Zf
40
50
60
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6
Total Kinetic Energy of fragments
§ 
§ 
Most energy of fission goes into motion of the fragments away from each other
Average TKE tends to decrease slowly with incident neutron energy (left) but
energy dependence of shape with heavy fragment mass TKE(AH) is less well known
Right (top): TKE vs AH for
several isotopes, highest TKE
for most stable AH, ~132
Right (bottom): Kinetic energy
for single fragments from 239Pu
Both at thermal energies
Lawrence Livermore National Laboratory
190
180
170
239
Tsuchiya Pu(n,f)
239
Nishio Pu(n,f)
239
Wagemans Pu(n,f)
252
Hambsch Cf(sf)
235
Nishio U(n,f)
240
Schillebeeckx Pu(sf)
244
Schmitt-Henschel Cm(sf)
238
Ivanov U(sf)
160
150
140
120
130
140
150
160
Heavy fragment mass number AH
120
Fragment kinetic energy (MeV)
Left (top): fragment energy,
integrated over A;
Left (bottom): product energy,
integrated over A
Note the higher TKE for the
fragments than for products
and the steeper slope on the
products
Both for 235U(n,f)
Total kinetic energy TKE (MeV)
200
110
100
90
80
Tsuchiya (2000)
70
Nishio (1995)
60
50
80
90
100
110
120
130
140
Fragment mass number A
150
7160
Neutron multiplicity distributions
Data on neutron multiplicity distributions are compared to a Poisson distribution
The data differ from a Poisson because not only is the neutron kinetic energy
removed with each neutron but also its separation energy
0.4
P( )
0.3
0.2
0.1
239
Pu(n,f)
240
Pu(sf)
0
0.4
P( )
Cf(sf)
Poisson
Holden-Zucker
Boldeman
Huanqiao
Vorobiev
Franklyn
Diven
0.3
0.2
0.1
252
244
Cm(sf)
235
U(n,f)
238
U(sf)
0
0 12 3 4 5 6 7 8 0 12 3 4 5 6 7 8 0 12 3 4 5 6 7 8 9
Neutron multiplicity
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Energy dependence of neutron multiplicity!
Evaluations of average neutron multiplicity as a function of incident neutron energy"
"
‘Evaluations’ involve compiling data and deciding which are ‘better’; there is also"
Some tweaking going on because the evaluations have to match certain criteria in applications"
"
In some cases where there isn’t much data, the evaluations are mainly educated guesses"
9
241
Average neutron multiplicity
8
7
6
Pu
Am
245
Cm
249
Cf
237
Np
241
ENDF-B/VII"
227
Th
U
235
5
4
3
2
0
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5
10
15
Incident neutron energy En (MeV)
20
9
Neutron spectral shapes: comparison to a Maxwellian!
A Maxwell distribution (Maxwellian), dN/dE ≈ √E exp(−E/T), with constant temperature T,"
T = 1.42 MeV is often compared to the data, easier to see differences in a ratio on a linear"
Scale than individual distributions on a logarithmic scale"
"
Note that the ‘kink’ in the Boykov 14.7 MeV data could be attributed to pre-equilibrium emission "
nth + 239Pu"
Ratio to Maxwellian (T = 1.42 MeV)
1.5
1.25
1
0.75
0.5
Adams 0.5 MeV
Staples 0.5 MeV
Staples 1.5 MeV
Conde 1.5 MeV
Knitter 1.5 MeV
0.25
1.25
n + 235U"
1
0.75
0.5
0.25
0.1
Staples 2.5 MeV
Boykov 2.9 MeV
1
Boykov 14.7 MeV
10 0.1
1
Outgoing neutron energy (MeV)
10
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10
Trends in neutron and photon A dependence similar
Both ν(A) and Eγ(A) show a sawtooth shape but the slopes of the ‘teeth’ are not necessarily the same:"
Eγ for 252Cf(sf) seems to be flatter while Nγ seems to have a stronger A dependence than"
Eγ for 235U(n,f) while Eγ is more similar to ν(A), within large uncertainties"
4
Pu(n,f)
2
1
0
3
(a)
252
Cf(sf)
2
1
235
U(n,f)
0
80
(A)
2
Maslin
Nishio
Shengyao
Vorobiev
Zakharova
Nardi
4
239
(A), E /1.75 (MeV)
(A)
3
100
120
140
160
Mass number A
(A), E /2.5 (MeV), N /2.5
1
0
4
(A)
Sawtooth shape"
of ν(A) reflects"
shell structure:"
A = 132 is doubly-"
closed shell so"
hard to excite and"
thus few neutrons"
emitted in any"
case shown"
"
Tooth is ‘sharper’"
for larger A0"
"
Dependence of"
shape on neutron"
energy not well"
known"
Tsuchiya
Nishio
Apalin
3
Shangyao
Vorobiev
Zakharova
252
Cf(sf)
2
1
0
Lawrence Livermore National
80 Laboratory
100
120
140
Fragment Mass A
160
3
(b)
235
U(nth,f)
Maslin
Nishio
Pleasonton E
Pleasonton N
2
1
0
80
100
120
Mass number A
140
160
11
Photon multiplicity distributions!
§ 
Left: Photon multiplicity distribution measured in 252Cf(sf) at LANSCE in Los Alamos"
•  Photon distributions are more Poisson-like because there is no separation energy"
Right: Photon multiplicity distribution for 2 neutron emission (solid points) and 4 neutron
emission (open points) measured at LBNL in 252Cf(sf) "
•  If fewer photons are emitted with higher neutron multiplicity, then there should be
an anti-correlation in multiplicity and the ν = 4 distribution should shift to the left of ν
= 2, if there is a correlation of more photons emitted at higher neutron multiplicity,
shift should be to the right; not possible to distinguish"
0.14
0.12
106
144
106
142
2
4
Mo+
Mo+
Counts (Normalized)
§ 
Ba (ν=2)
Ba (ν=4)
0.10
0.08
0.06
0.04
0.02
0
0
6
8
10
12
14
Detected photon multiplicity
16
18
20
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Data are often insufficient for comprehensive
modeling of fission process!
§  Fission experiments have most often focused on measuring only a
single type of observable, e.g. the fragment mass and/or charge or the
number and/or energy of prompt neutrons or prompt photons"
§  Such inclusive data provides only limited guidance for fission modeling,
in contrast to more exclusive data, e.g. prompt neutrons and/or prompt
photons together with the fragment mass and/or charge"
§  Fission experiments largely focus on just a few cases, further limiting the
experimental basis for modeling"
§  Some (n,f) data, such as Y(Af), TKE(AH) and ν(Af), have been measured
only at low incident energies"
§  Codes like FREYA can be used to study interdependencies and
sensitivities, thus helping identify which further measurements might be
most informative"
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13
Perils of comparing models to data!
§ 
§ 
§ 
§ 
Single particle observables such as average neutron multiplicity are sort of easy –
they just require counting neutrons, sometimes within a given angle or energy bin
but are basically counting experiments (these are good for many applications which
require only average quantities"
Going beyond these measurements requires measuring e.g. neutrons and
fragments"
Experiments measure products (the fission fragments reach the detectors after
emitting neutrons and are thus ‘products’) but report measurement of fragments"
Thus any reported inclusive measurement such as ν(A), TKE(A) has already had
some reverse analysis done on it to go back to fragments from products"
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14
Fission models!
§ 
§ 
§ 
§ 
There is no real ‘theory’ of fission: it’s difficult to do a fully many-body calculation,
writing down the anti-symmetrized wavefunction of 230+ nucleons – some have tried
but even going from some trial wavefunction to two distinct fragments is extremely
difficult and more than marginal success with application to more than a single
isotope is still years away"
Three things have been very important for application codes: the average neutron
multiplicity, the prompt fission neutron spectrum (PFNS), and the fission cross
section. The rest of the event is ignored, energy and momentum are not conserved
and the same spectrum is sampled for all neutrons emitted in an event."
The average multiplicity as a function of incident neutron energy is evaluated and
tabulated in databases. For some isotopes it is very well known (claimed to be
known better than 0.1%) and regarded as sacrosanct."
The PFNS is also an evaluated quantity, based on the “Los Alamos” model. As we
already saw, uncertainties in certain energy regions can still be large so a great deal
of experimental effort has been aimed at reducing these uncertainties. (Hint:
measuring neutron energies accurately is REALLY hard) "
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15
Los Alamos model (Madland & Nix, 1982)!
§ 
§ 
§ 
§ 
§ 
§ 
This approach has been the ‘gold standard’ in PFNS evaluations since it was first
developed."
It assumes that the light and heavy fragments are the ‘average’ ones, those that are the
most probable. It also assumes an average value of the separation energy, Sn, based
on the identities of the most probable fragments."
They also make assumptions about the average fission Q value and the average total
kinetic energy, TKE, to obtain the average total excitation energy. The average neutron
multiplicity is then <ν> = (<E*> − <Eγ>)/(<Sn> + <E>) since the average energy emitted
by photons is subtracted."
The Weisskopf-Ewing spectral shape, dN/dE ~ E exp(−E/Tmax), is used with Tmax the
maximum temperature of the daughter nucleus, obtained for E = 0, giving an average
neutron kinetic energy of <E> = 2T"
The average neutron spectrum is obtained from this spectral shape folded with a
triangular temperature distribution, P(T) = 2T/(Tmax)2 for T ≤ Tmax; 0 for T > Tmax; the
average neutron spectrum is then"
with an average energy of"
<E> = (4/3)Tmax "
Many variants of this model exist but all provide smooth PFNS for all incident energies"
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16
FREYA (Fission Reaction Event Yield Algorithm) developed at LLNL !
§  Model of complete fission events, just one example, there are others, as we’ll see
§  FREYA developed in collaboration with J. Randrup (LBNL)
§  FREYA references: Phys. Rev. C 80 (2009) 024601, 044611; 84 (2011) 044621;
85 (2012) 024608; 87 (2013) 044602; 89 (2014) 044601; User Manual LLNLTM-654899.
§  General reference: R. Vogt and J. Randrup, “Nuclear Fission”, Chapter 5 of 100
Years of Subatomic Physics, World Scientific, 2013.
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How do complete event treatments differ from traditional
fission models?!
Fission model in frequently used"
simulation code MCNP:"
0
1
10
239
239
Pu(n,f)
Pu(n,f)
-1
10
-2
10
0.6
-3
10
0.4
-4
10
0.5 MeV
14 MeV
0.2
-5
10
fn (E) (1/MeV)
P( )
0.8
-6
10
0
P( )
•  In ‘average’ models, fission is a black box, "
neutron and gamma energies sampled from"
same average distribution, regardless of "
multiplicity and energy carried away by each"
emitted particle; fluctuations and correlations "
cannot be addressed"
"
•  Models like FREYA generate complete fission
events: energy & momentum of neutrons,
photons, and products in each individual fission
event; correlations are automatically included"
U(n,f)
0.6
-1
Blue: 0.5 MeV"
Red: 14 MeV"
235
U(n,f)
10
-2
10
-3
10
0.4
10
0.2
10
0
10
0
5
10
15
20
Outgoing neutron energy (MeV)
-4
-5
-6
1
2
3
4
5
Neutron multiplicity
6
•  Traditionally, neutron multiplicity"
sampled between nearest values"
to get correct average value"
•  All neutrons sampled from same"
spectral shape, independent of"
multiplicity"
fn (E) (1/MeV)
235
0.8
Event-by-event modeling is efficient framework for
incorporating fluctuations and correlations!
Goal(s): Fast generation of (large) samples of complete fission events
""
Complete fission event: Full kinematic information on all final particles
Two product nuclei: ZH , AH , PH and ZL , AL , PL
ν neutrons: { pn }, n = 1,…,ν
Nγ photons: { pm }, m = 1,…,Nγ
Advantage of having samples of complete events:
Straightforward to extract any observable,
including fluctuations and correlations,
and to take account of cuts & acceptances
Advantage of fast event generation:
Can be incorporated into transport codes
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19
Fragment mass and charge distribution!
No quantitative models for P(Af) exists yet, so …
P(Af) is sampled either from the measured mass distribution
or from five-gaussian fits to data:
[W. Younes et al: PRC 64 (2001) 054613]
Mass
number
P (Af ) =
m=+2
!
m=+2
!
N|m| Gm (Af )
m=−2
N|m| = 1
m=−2
2
1
2
)− 2 e−(Af −Āf −D|m| )/2σ|m|
Gm (Af ) = (2πσ|m|
Dependence on En:
N1,2 (En ) =
2
PAf (Zf ) ∼ e−(Zf −Z̄f )/2σZ
Charge
number
Z0
Af
A0
σZ = 0.38 − 0.50
252Cf
1.0
Fission probability
2nd: 35%
1st chance
2nd chance
3rd chance
4th chance
0.6
0.4
0.2
Lawrence
Livermore National Laboratory
rd
3 : 21%
Ẽ ≈ 1 MeV
e(En −Ê)/Ẽ + 1
[W. Reisdorf et al: NPA 177 (1971) 337]
0.0
0
5
10
240Pu
Fragment Yield (%)
Z̄f =
0.8
44%
Ê ≈ 10 MeV
10
En = 14 MeV:
1st:
0
N1,2
10
10
10
10
1
0
-1
-2
Thermal
14 MeV
-3
-4
10 60
15
Incident neutron energy (MeV)
20
80
100
120
140
Fragment Mass A
160
180
20
Pre-equilibrium neutron emission!
Based on the two-component exciton model by Koning & Duijvestin [NPA744]
Master equation
for P(pπ hπ pν hν):
Emission rate:
0.06
0.2
n+
239
0.1
Multiplicity
0.0
0.6
With pre-eq
0.4
Excitons:
(pπ hπ pν hν)
0.04
0.03
En=14 MeV
0.02
No pre-eq
0.01
0.2
0.0
0
0.05
Pu
dσ/dE (b/MeV)
Probability
0.3
5
10
15
Incident neutron energy En (MeV)
20
0
0
5
10
Sum
(1,1,1,0)
(2,2,1,0)
(3,3,1,0)
(4,4,1,0)
(5,5,1,0)
(0,0,2,1)
(1,1,2,1)
(2,2,2,1)
(3,3,2,1)
(4,4,2,1)
(0,0,3,1)
(1,1,3,1)
(2,2,3,1)
(3,3,3,1)
(0,0,4,1)
(1,1,4,1)
(2,2,4,1)
(0,0,5,1)
(1,1,5,1)
(0,0,6,5)
(0,0,1,0)
(0,0,2,1)
(0,0,3,2)
(1,1,1,0)
(1,1,2,1)
(2,2,1,0)
15
Outgoing neutron energy (MeV)
Calculations made by Erich Ormand
Lawrence Livermore National Laboratory
Vogt, Randrup, Brown, Descalle, Ormand:
Physical Review C 80 (2009) 024601
21
Multichance fission based on fission-evaporation
competition, transition state theory!
Swiatecki et al:
PR C 78 (2008) 054604
Xf = E - Vf
Xn = E - Vn
ν = 4; fifth chance fission"
"
ν = 3; fourth chance fission"
"
ν = 2; third chance fission"
"
ν = 1; second chance fission"
"
ν = 0; first chance fission"
1.0
P(εn)
Q *n
0.8
εf
S n A−1
εn
A
Lawrence Livermore National Laboratory
Fission probability
εi
1st chance
2nd chance
3rd chance
4th chance
0.6
0.4
0.2
0.0
0
5
10
15
Incident neutron energy (MeV)
20
22
H: heavy
L: light
No models for TKE(Af) exists yet, so …
Total Fragment Kinetic Energy (MeV)
Fission fragment kinetic energies!
we adjust TKE to exp data:
190
Tsuchiya
Nishio
Wagemans
180
170
239Pu(n,f)
160
150
Average TKE versus heavy
fragment mass number AH
140
130
120
130
140
150
160
170
Heavy fragment mass number AH
TKE = TKEdata - dTKE(En)
with an adjustable shift
to reproduce the mean
neutron multiplicity <ν>(En)
Fragment kinetic energy (MeV)
120
Average fragment kinetic energy
versus fragment mass number Af
110
100
239Pu(n,f)
90
80
Tsuchiya (2000)
70
Nishio (1995)
60
50
80
90
100
110
120
130
140
Fragment mass number Af
150
160
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23
Fragment excitation energies!
H: heavy
L: light
Q value:
QLH = M(240Pu*) – ML - MH
Mean thermal excitation: E* = QLH - TKE = EL* + EH*
Common temperature:
T = [E*/(aL+aH)]1/2
Excitation is shared:
EL* : EH* = aL : aH => Ef* = afT2
Thermal fluctuations:
σ 2(Ef*) = 2Ef*T
Fragment excitation:
Ef* = Ef* + δEf*
Small adjustment:
EL* -> x EL* (x>1) - dist?
Thermal equilibrium:
Fragment momenta then follow from
energy & momentum conservation:
*)
=> δEf*
TKE = TKE - δEL* - δEH*
pL + pH = 0
*) aA(E*) from Kawano et al, J. Nucl. Sci. Tech. 43 (2006) 1
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24
Temperature of daughter nuclei after neutron emission!
Distribution of maximum temperature of fragment daughter nucleus after emission of one or more"
neutrons – cases that emit more neutrons on average are visibly hotter"
Shape is not triangular (as assumed by Madland and Nix)"
0.07
0.06
239
Pu(n,f)
0.05
P(T)
0.03
all ν
ν=1
ν=2
ν=3
0.02
0.01
Cf(sf)
0.05
0.04
0.07
0.06
0.03
0.01
0.01
0.07
0.07
0.04
0.02
0.01
0.01
1
1.5
all ν
ν=1
ν=2
ν=3
0.03
0.02
0.5
238
2
T (MeV)
0
0
U(sf)
0.05
P(T)
0.03
all ν
ν=1
ν=2
ν=3
0.06
Cm(sf)
0.05
0.04
0.03
0.02
244
P(T)
P(T)
0.05
U(n,f)
0.04
0.02
0.06
Pu(sf)
235
0.05
0.04
240
0
0
0.06
252
P(T)
0.06
P(T)
0.07
0.07
0.04
0.03
0.02
0.01
0.5
1
T (MeV)
1.5
2
0
0
0.5
1
1.5
2
T (MeV)
Lawrence Livermore National Laboratory
25
Energy dependence of daughter temperature distribution!
Distribution of maximum temperature of fragment daughter nucleus after emission of one or more
neutrons – higher energy, more neutron emission, is visibly hotter, even for 3 emitted neutrons"
Shape is even less like a triangle"
0.07
0.07
n(0.5 MeV) +
0.06
239
Pu
0.05
0.05
0.04
0.04
0.03
P(T)
P(T)
0.06
all ν
ν=1
ν=2
ν=3
0.02
0.01
235
U
all ν
ν=1
ν=2
ν=3
0.01
0.07
0.06
n(14 MeV) +
0.06
239
Pu
0.05
0.05
0.04
0.04
P(T)
P(T)
0.03
0.02
0.07
0.03
0.02
0.01
0.01
0.5
1
T (MeV)
1.5
Lawrence Livermore National Laboratory
2
n(14 MeV) +
235
U
0.03
0.02
0
0
n(0.5 MeV) +
0
0
0.5
1
1.5
2
T (MeV)
26
Angular momentum at scission: Rigid rotation plus fluctuations
Rigid rotation:
Si = (Ii/I)S0 + δSi
Wriggling:
I+ = (IH+IL)I/IR
Bending:
I- = IHIL/(IH+IL)
I = IL + IH + IR; IR = µR2; R = RL – RH; µ = mNALAH/(AL + AH)
The dinuclear rotational modes (+ & -) have thermal fluctuations governed by an
adjustable “spin temperature” TS = cS Tsc, where Tsc is the scission temperature
Lawrence Livermore National Laboratory
27
Fluctuations Contribute to Fragment Rotational Energy!
H: heavy
Scission induces statistical agitation
of dinuclear rotation modes –
wriggling (s+) and bending (s-)
s± = (s±x,s±y,0):
L: light
SH
P(s±) ~ exp(-s±2/2I±TS)
SL
TS: related to scission temperature by TS = cSTsc
(used cS = 0,0.1,1)
Fluctuating angular momentum components of fragments,
δSLk = (IL/I+)s+k + s-k; δSHk = (IH/I+)s+k - s-k;
Total angular momenta of fragment i are then Si’ = Si + δSi
with orbital angular momentum L’ = L – δSL – δSH; contribution to dinuclear
rotation modes δErot = S+2/2I+ + S-2/2I-, as well as rigid rotation part Erot,
is not available for statistical excitation
Mean statistical excitation is reduced correspondingly and shared between fragments:
E* = QLH - TKE - Erot - δErot = EL* + EH*
Photon observables are very sensitive to fragment spin while neutrons are not
Lawrence Livermore National Laboratory
28
Neutron evaporation from fragments!
Mi∗ = Migs + εi
P(εn)
εi
Q *n εf
S n A−1
Mf∗ = Mfgs + εf
Mi∗ = Mf∗ + mn + ϵ
εn
Q∗n = εi + Qn = εi − Sn
Qn ≡ Q∗n (εi = 0) = Migs − Mfgs − mn = −Sn
A
ϵ + εf =
d3 p ∼
√
Tfmax
ϵ dϵ dΩ
Mi∗
−
Mfgs
∗∗
−
ϵ +mεnf ==QQ
nn
!
!
= εmax
/af = Q∗n /af
f
=
!
εmax
f
ϵmax
(non-relativistic)
√ −ϵ/T max √
d3 N 3
−ϵ/Tfmax
f
ϵ dϵ dΩ
d
p
∼
ϵ
e
ϵ
dϵ
dΩ
=
e
d3 p
Lorentz boost both ejectile and daughter motion from emitter frame to laboratory frame
Neutron energy spectrum:
Neutron (and photon) emission in FREYA based on Weisskopf-Ewing theory which conserves only"
energy, charge, and mass number"
Lawrence Livermore National Laboratory
29
Neutron evaporation from rotating fragments
ω = S/I
vn = v0 + ω x r
S’ = S – r x pn
ωxr
Usual thermal emission from the moving surface element, v0 ,
subsequently boosted with the local rotational velocity ω x r .
Conserves energy as well as linear & angular momentum.
Lawrence Livermore National Laboratory
30
Photon emission follows neutron emission!
After neutron evaporation has ceased, E* < Sn , the remaining
excitation energy is disposed of by sequential photon emission …
… first by statistical photon cascade
down to the yrast line …
ABCDE&
ABCDE&
ABCDE&
&
Ini5al&&
fragment&
E*&
Sn&
E*max&
Sta5s5cal&&
neutron&
Eyrast&
Sta5s5cal&γ&
Discrete&γ&
J&
d3 Nγ 3
d pγ ∼ c e−ϵ/Ti ϵ2 dϵ dΩ
3
d pγ
<=
d3 pγ ∼ ϵ2 dϵ dΩ
(ultra-relativistic)
Ef∗ = Ei∗ − ϵγ
… then by stretched E2 photons
along the yrast line …
Sf = Si − 2
ϵγ = Si2 /2IA − Sf2 /2IA
2
IA = 0.5 × 52 AmN RA
Each photon is Lorentz boosted from
the emitter to the laboratory frame
Lawrence Livermore National Laboratory
31
External parameters in FREYA which can be adjusted to data!
§ 
§ 
In addition to isotope-specific inputs such as Y(A) and TKE(AH), there are also intrinsic
parameters such as nuclear masses (Audi and Wapstra for experimentally-measured
masses, supplemented by masses calculated by Moller, Nix, Myers and Swiatecki),
barrier heights, pairing energies and shell corrections"
There are also external parameters that can be adjusted, either universally or per
isotope"
•  Shift in total kinetic energy, dTKE, adjusted to give the evaluated average neutron
multiplicity"
•  Asymptotic level density parameter, e0, ai ~ (A/e0)[1+ (δWi/Ui)(1 – exp(-γUi))] where
Ui = E*i – Δi, γ = 0.05, and the pairing energy, Δi, and shell correction, δWi, are
tabulated (if δWi ~ 0 or Ui is large so that 1 – exp(-γUi) ~ 0, ai ~ A/e0)"
•  Excitation energy balance between light and heavy fragment, x"
•  Width of thermal fluctuation, σ 2(Ef*) = 2cEf*T, c is adjustable (default = 1)
•  Multiplier of scission temperature, cS, that determines level of nuclear spin
•  Energy where neutron emission ceases and photon emission takes over, Sn + Qmin
•  Default values: e0 ~ 10/MeV, c = 1, cS = 1, Qmin = 0.01 MeV
•  Specific to 252Cf(sf): x = 1.3, dTKE = 0.5 MeV
Lawrence Livermore National Laboratory
32
Other Monte Carlo fission codes!
§ 
§ 
§ 
§ 
§ 
§ 
§ 
§ 
CGMF (Talou et al), LANL"
FIFRELIN (Serot et al), France"
GEF (Schmidt and Jurado), Germany"
FREYA is based on Weisskopf-Ewing theory and thus emission is based only on
charge, mass number and energy conservation, a fast procedure more useful for
neutron transport codes"
CGMF and FIFRELIN are based on Hauser-Feshbach theory which includes
angular momentum and parity as well so involves a sum over all outcomes that
result in the same total angular momentum and parity: this is a very slow procedure
and is so far limited to fewer nuclei than FREYA"
CGMF, FIFRELIN and FREYA use similar inputs of Y(A,Z) and TKE to ultimately
extract the excitation energy"
Since CGMF and FIFRELIN have concentrated on fewer isotopes, they have tried
to tune their results to some inclusive data such as ν(A); FREYA has focused more
on using single global parameters to be somewhat more predictive"
GEF is a somewhat different beast, it models the potential energy surface of the
fissioning (compound) nucleus to directly obtain the fragment yields and excitation
energies, does not really address the TKE "
Lawrence Livermore National Laboratory
33
ν(A) for FREYA spontaneous and thermal fission!
Mean neutron multiplicity as a function of fragment mass; agrees with sawtooth shape of data
Not all cases have data to compare to FREYA, smoothness of sawtooth dependent on
quality of yield and TKE data
4
4
ν(A)
3
2
1
Shangyao
Vorobiev
Zakharova
252
Cf(sf)
3
ν(A)
FREYA
Tsuchiya
Nishio
Apalin
3
2
n(0.5 MeV) +
n(0.5 MeV) +
Maslin
Nishio
235
U
2
1
1
239
Pu
0
0
240
Pu(sf)
2
ν(A)
ν(A)
3
FREYA
Schmitt-Henschel
uncorrected
0
4
238
U(sf)
244
Cm(sf)
3
ν(A)
ν(A)
4
2
FREYA
2
1
1
0
80
100
120
140
160
Fragment mass A
0
80
1
100
120
Fragment mass A
140
160
0
80
100
120
140
160
Fragment mass A
Lawrence Livermore National Laboratory
34
Model comparison: ν(A)!
4
FREYA
Tsuchiya
Nishio
Apalin
3
(A)
CGMF"
2
1
n(0.5 MeV) +
0
80
239
100
Pu
120
140
160
Fragment mass A
(A)
3
R4.5
=1.25 , E*lim=Sn+Erot(Jrig.)
T
RT=1.25 , E*lim=Sn+Erot(0.5 Jrig.)
Budtz-Jorgensen
(1988)
4.0
Hambsch (2009)
Bowman (1963)
252
Cf(sf)
252Cf(sf)"
2
1
3.0
2.5
2.0
1.5
1.0
FIFRELIN"
0.5
0
80
100
120
140
GEF"
3.5
Neutron multiplicity
4
Shangyao
Vorobiev
Zakharova
FREYA
160
252Cf(sf)"
0.0
Mass number A
80
90
100
110
120
130
140
150
160
170
Mass number A
Lawrence Livermore National Laboratory
35
Correlations between neutron number and energy!
Spectral shapes shown, all normalized to unity for better comparison"
Most cases show considerable softening of the spectrum for increased neutron multiplicity"
0
10
10
-1
-2
Cf(sf)
10
10
all ν
ν=1
ν=2
ν=3
ν=4
ν=5
ν=6
-2
10
0
10
10
2
4
10
10
12
U(sf)
-1
-2
ν
ν
8
238
fn (E) (1/MeV)
fn (E) (1/MeV)
ν
fn (E) (1/MeV)
10
10
6
-2
Cm(sf)
all ν
ν=1
ν=2
ν=3
ν=4
ν=5
ν=6
0
-1
244
-1
-3
10
-3
14
Outgoing neutron energy E (MeV)
10
10
10
0
2
4
U
-3
Pu(sf)
-2
10
235
10
0
10
240
10
n(0.5 MeV) +
-1
-3
-3
10
0
10
10
0
252
Pu
ν
10
10
239
fn (E) (1/MeV)
ν
fn (E) (1/MeV)
n(0.5 MeV) +
0
ν
10
and 238U(sf) more tightly correlated"
fn (E) (1/MeV)
252Cf(sf)
6
8
10
12
Outgoing neutron energy E (MeV)
14
-1
all ν
ν=1
ν=2
ν=3
ν=4
ν=5
ν=6
-2
-3
0
2
4
6
8
10
12
14
Outgoing neutron energy E (MeV)
Lawrence Livermore National Laboratory
36
Energy dependence of prompt fission neutron spectra !
‘Pile-up’ in 14 MeV spectra at En-Bf due to energy conservation: there can be no pre-fission neutrons
with energy greater than En-Bf, otherwise fission would not occur"
"
Bump shows single pre-fission neutron emission, strongest for ν=1, reduced when evaporated
neutrons are also present, high energy tail is from first chance fission"
10
0
10
0
10
239
Pu
-2
10
10
235
U
-1
-2
ν
10
n(0.5 MeV) +
-1
fn (E) (1/MeV)
ν
fn (E) (1/MeV)
n(0.5 MeV) +
10
-3
-3
10
0
10
10
0
-2
10
-3
Pu
all ν
ν=1
ν=2
ν=3
ν=4
ν=5
ν=6
Lawrence0 Livermore National
Laboratory
5
10
Outgoing neutron energy E (MeV)
10
10
10
235
U
-1
all ν
ν=1
ν=2
ν=3
ν=4
ν=5
ν=6
-2
ν
10
n(14 MeV) +
-1
239
fn (E) (1/MeV)
10
ν
fn (E) (1/MeV)
n(14 MeV) +
-3
0
5
10
Outgoing neutron energy E (MeV)
15
37
Probability for prompt neutron emission !
Since each neutron emitted reduces the excitation energy not only by its kinetic energy εn"
(<εn> = 2Tfmax) but also by the (larger) separation energy Sn, the neutron multiplicity "
distribution is narrower than a Poisson "
n(0.5 MeV) +
Pu
FREYA
Poisson
Holden-Zucker
0.4
252
Cf(sf)
0.2
Vorobiev
0.2
0.1
0.1
Pu(sf)
0.4
Cm(sf)
0.2
0
0
0.2
0.1
0.1
1
2
3
4
5
6
7
8
9
0
0
Neutron multiplicity ν
238
U(sf)
0.3
FREYA
Poisson
Huanqiao
P(ν)
P(ν)
P(ν)
Boldeman
Huanqiao
Franklyn
Boldeman
Diven
0.2
0.4
244
0.3
0.3
U
0
0.4
240
235
0.1
0
0
0.5
n(0.5 MeV) +
0.3
0.3
P(ν)
0.3
P(ν)
0.4
239
P(ν)
0.4
FREYA
Poisson
Holden-Zucker
0.2
0.1
1
2
3
4
5
6
Neutron multiplicity ν
7
8
9
0
0
1
2
3
4
5
6
7
8
9
Neutron multiplicity ν
Lawrence Livermore National Laboratory
38
Energy dependence of neutron multiplicity distribution !
As incident neutron energy increases, P(ν) is peaked at higher multiplicity and"
distribution broadens, reflecting higher probabilities of fluctuations with large multiplicities "
0.4
Pu
n(0.5 MeV) +
FREYA
Poisson
Holden-Zucker
0.2
0.1
0
0
0.4
n(14 MeV) +
Pu(sf)
n(14 MeV) +
235
U
0.3
P(ν)
P(ν)
FREYA
Poisson
240
0.3
0.2
0.2
0.1
0.1
0
0
U
Franklyn
Boldeman
Diven
0.2
0.1
0.4
235
0.3
P(ν)
0.3
P(ν)
0.4
239
n(0.5 MeV) +
1
2
3
4
5
6
Neutron multiplicity ν
7
Lawrence Livermore National Laboratory
8
9
0
0
1
2
3
4
5
6
7
8
9
Neutron multiplicity ν
39
Model comparison: neutron multiplicity distribution!
Neutron-induced fission of 239Pu is calculated in FREYA (left) and CGMF (right)"
"
FREYA result is similar to RT = 1.1 in CGMF, not surprising because RT(A) is tuned to"
ν(A) data "
"
Holden-Zucker (Holden) is compilation of data on a number of isotopes, e.g. for 239Pu"
it represents an average over several data sets"
0.4
n(0.5 MeV) +
Pu
FREYA
Poisson
Holden-Zucker
0.3
P( )
239
0.2
CGMF"
0.1
0
0
1
2
3
4
5
6
7
8
9
Neutron multiplicity
Lawrence Livermore National Laboratory
40
Model comparison: neutron kinetic energy vs mass!
2.5
2.50
2.25
Average Neutron Kinetic Energy <Ecm> (MeV)
Average neutron kinetic energy <ε> (MeV)
Neutron kinetic energy was measured by Tsuchiya et al as a function of fragment mass"
"
FREYA (left) and CGMF (right) compared to data, note different scales on the y-axes"
"
FREYA and RT = 1.1 in CGMF are very similar except near symmetry"
"
Note that tuning RT(A) to n(A) generally decreases agreement with data, especially at"
A < 90 and 120 < A < 135 – latter region is because fewer neutrons are emitted at closed"
Shell and, in CGMF, these neutrons are less energetic"
FREYA
Tsuchiya
2.00
1.75
1.50
1.25
1.00
n + 239Pu"
0.75
0.50
0.25
0.00
80
90
100
110
120
130
Mass number A
140
150
160
Tsuchiya, 2000
Monte Carlo, RT(A)
Monte Carlo, RT=1.1
nth+239Pu
2
1.5
1
0.5
80
90
100
110
120
130
140
150
160
Fission Fragment Mass (amu)
Lawrence Livermore National Laboratory
41
Other inclusive ‘observables’!
(Left) Residual excitation energy is how much is left after neutron emission stops, depends on"
Qmin, the more neutrons emitted, the more energy they take away; increasing Qmin would "
increase residual excitation energy – essentially what is left over for photons, equivalent to Eγ
(Right) Neutron multiplicity vs TKE: Budtz-Jorgensen data and model calculations are averaged"
Over Y(A,Z) while Bowman data are for yields of specific fragment pairs; models are consistent"
8
9
252Cf(sf)"
8
7
6
5
4
3
FREYA"
1
0
6
5
4
3
2
2
0
1
2
FREYA
FIFRELIN
Budtz-Jorgensen
Bowman
7
Neutron mutliplicity
Residual excitation energy (MeV)
10
252Cf(sf)"
1
3
4
5
6
Neutron multiplicity
7
8
9
0
140
150
160
170
180
190
200
210
220
Total Kinetic Energy (MeV)
Lawrence Livermore National Laboratory
42
Two-neutron angular correlations reflect emitter source!
n1
Correlations between neutrons when exactly"
2 neutrons with En > 1 MeV are emitted:"
"
One from each fragment (blue) back to back;"
both from single fragment emitted in same "
direction, tighter correlation when both from "
light fragment (green) than from heavy (red);"
open circles show sum of all possibilities"
φ12
Correlations of neutrons with energies above"
a specified threshold energy"
"
Yield forward and backward is more symmetric"
for higher-energy neutrons"
n2
6
4
239
Pu(nth,f)
En > 1.0 MeV
3
ν=2
(1,1)
5
Correlation yield
Correlation yield
En > 1.5 MeV
En > 0.5 MeV
ν=2 (1.0 MeV)
2
239
Pu(nth,f)
(2,0)
(0,2)
4
3
2
1
1
0
0
30
60
90
120
Relative angle φ12 (degrees)
150
180
0
0
30
60
90
120
Relative angle φ12 (degrees)
150
180
Lawrence Livermore National Laboratory
43
Examples of two-neutron correlations in FREYA!
Back-to-back correlation is stronger for 240Pu(sf) and 238U(sf) since both emit about 2 neutrons on average"
Cases with larger multiplicities, like 252Cf(sf), show weaker two-neutron correlations"
4
3
n(0.5 MeV) +
3
239
Pu
252
Cf(sf)
n(0.5 MeV) +
235
U
3
2
2
0
240
Pu(sf)
3
En > 0.5 MeV
En > 1.0 MeV
En > 1.5 MeV
2
0
3
244
Cm(sf)
En > 0.5 MeV
En > 1.0 MeV
En > 1.5 MeV
2
1
1
0
0
1
20
40
60
80 100 120 140 160 180
θ12 (degrees)
0
0
n-n Angular Correlation
n-n Angular Correlation
1
n-n Angular Correlation
2
1
0
238
U(sf)
3
En > 0.5 MeV
En > 1.0 MeV
En > 1.5 MeV
2
1
20
40
60
80 100 120 140 160 180
θ12 (degrees)
0
0
20
40
60
80 100 120 140 160 180
θ12 (degrees)
Lawrence Livermore National Laboratory
44
Energy dependence of two-neutron angular correlations!
Back-to-back direction of emission is reduced at higher energies since two energetic neutrons
can come from same fragment as often as one from each fragment"
Correlations get washed out because more neutrons are emitted"
4
3
n(0.5 MeV) +
239
n(0.5 MeV) +
Pu
235
U
3
2
1
0
3
n(14 MeV) +
239
Pu
E > 0.5 MeV
E > 1.0 MeV
E > 1.5 MeV
2
1
0
0
n-n Angular Correlation
n-n Angular Correlation
2
1
0
n(14 MeV) +
3
235
U
E > 0.5 MeV
E > 1.0 MeV
E > 1.5 MeV
2
1
20
40
60
80 100 120 140 160 180
Lawrence LivermoreθNational
Laboratory
(degrees)
12
0
0
20
40
60
80 100 120 140 160 180
θ12 (degrees)
45
Sensitivity of correlations to input parameters!
n-n correlation
n-n correlation
Changing Qmin, cS, e0 and c does not have a strong effect on the shape of the n-n correlations"
"
Only changing x strongly modifies the correlation shape: x < 1.3 default reduces the correlation"
at θnn = 0° while leaving that at 180° unchanged; x > 1.3 (giving more excitation to light fragment)"
produces a significantly stronger correlation at θnn = 0°"
"
Correlation shape is relatively robust with respect to model parameters"
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0
cS = 1, x = 1.3
x=1
x = 1.6
x = 0.75
cS = 1, Qmin = 0.01 MeV
cS = 0.1
Qmin = 1 MeV
(a)
(b)
cS = 1, e0 = 10/MeV
e0 = 8/MeV
e0 = 12/MeV
cS = 1, c = 1.0
c = 1.2
c = 0.8
(c)
30
60 90 120 150
θnn (degrees)
Lawrence Livermore National Laboratory
(d)
0
30
60 90 120 150 180
θnn (degrees)
46
Effect of changing input parameters on other observables!
(Left) changing x reduces agreement with ν(A) in the range of highest yield, 100 < A < 140;"
x = 1.3 gives best agreement in this range, x = 0.75 gives too much energy to the heavy "
fragment, x = 1 does somewhat better for A < 100 but is bad everywhere else, x = 1.6 is far off"
"
(Right) changing the width of the thermal distributions reduces the agreement of FREYA with "
the Vorobiev P(ν) data, increasing c makes P(ν) too broad, decreasing c makes it too narrow"
4
Shangyao
Vorobiev
Zakharova
x = 1.3
x=1
x = 1.6
x = 0.75
252
Cf(sf)
252
Cf(sf)
2
1
0
80
Vorobiev
FREYA
Poisson
c = 1.2
c = 0.8
0.3
P(ν)
ν(A)
3
0.4
0.2
0.1
100
120
Fragment mass A
140
160
0
0
1
2
3
4
5
6
Neutron multiplicity ν
7
8
9
Lawrence Livermore National Laboratory
47
Default version of FREYA gives rather good
agreement with angular correlation data!
En = 0.425 MeV
En = 0.550 MeV
En = 0.800 MeV
En = 1.200 MeV
En = 1.600 MeV
FREYA
4
3
2
1
Gagarski et al 252Cf(sf), 2008"
0
20
40
60
80
100
120
θnn (degrees)
En > 2.50 MeV
4
140
160
180
1.4
2
0
6
n-n correlation (arb. units)
n-n angular correlation (arb. units)
6
5
n-n correlation (arb. units)
•  All experiments took measurements at different"
angles, discriminating between photons and "
neutrons by timing, Gagarski et al used time of"
flight, others used pulse shape discrimination"
•  Newer data seems to show higher back-to-back"
correlation, more consistent with FREYA, than"
older data"
•  Higher Qmin might bring data and calculations"
closer together at lower energies and θnn > 120°"
where calculation and data are most discrepant"
En > 1.75 MeV
4
2
0
6
FREYA
235
U data
4
En > 1.00 MeV
2
0
0
1.2
1
0.8
0.6
252
FREYA Cf(sf)
Pringle and Brooks 1975"
Gagarski et al
0.4
0.2
20
40
60
80
100
120
140
160
180
Lawrence Livermore National
Laboratory
θnn (degrees)
Franklyn et al, 1978"
0
0
20
40
60
80
100 120
θnn (degrees)
140
160
180
48
Correlation between neutron and light fragment!
1
0.9
neutrons from light fragment
neutrons from heavy fragment
all neutrons
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100 120
θnL (degrees)
Lawrence Livermore National Laboratory
140
160
180
neutron - light fragment correlation (arb. units)
neutron - light fragment correlation (arb. units)
Neutron emission can also be correlated with individual fragments"
"
(Left) Angle of neutrons emitted by either the light or heavy fragment or both fragments"
with respect to the direction of the light fragment: neutrons from light fragment emitted "
preferentially toward θnL = 0°; neutrons from heavy fragment are typically moving opposite the "
light fragment in the lab frame, θnL = 180°; correlation becomes more tightly peaked for "
higher neutron kinetic energies, here En > 0.5 MeV"
"
(Right) FREYA result is compared to data, light fragment is determined and correlation is made"
with all measured neutrons, as in black curve at left; good agreement is seen "
1
0.9
Bowman
FREYA En > 0.5 MeV
Gagarski
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100 120
θnL (degrees)
140
160
180
49
Other possible neutron correlation observables!
Neutron-induced fission endows compound nucleus"
with small initial angular momentum S0, giving the"
fragments non-vanishing angular momentum along"
S0 in addition to that acquired from fluctuations;"
fragment angular momentum modified by each "
neutron emission"
angle between initial angular momentum of compound"
nucleus and fragment after evaporation is the"
dealignment angle ∆θ (Si’·S0 = Si’S0 cos Δθ)"
Angular distribution dν/dcos θn
1.10
1.08
1.06
252
Cf(sf)
1.04
1.02
1.00
0.98
cS = 0.0
0.96
cS = 0.1
0.94
cS = 1.0
0.92
0.90
-1.0
-0.5
0.0
cosLaboratory
θn
Lawrence Livermore National
0.5
1.0
Dealignment distribution Pdealign(cos ∆θ)
Angular distribution of neutrons evaporated from"
rotating nucleus acquires oblate shape –"
rotational boost enhances emission in plane"
perpendicular to angular momentum of emitter"
centrifugal effect quantified by 2nd Legendre "
moment"
〈P2(cos θ)〉 = 〈P2(p·S/|p||S|)〉"
0 for isotropic emission; + for prolate (polar);"
- for oblate (equatorial) – small effect overall"
3.0
2.5
239
cS = 0.0
Pu(nth,f)
cS = 0.1
cS = 1.0
2.0
1.5
1.0
0.5
-1.0
-0.5
0.0
0.5
1.0
cos(∆θ)
50
Summary!
§  Event-by-event treatment shows significant correlations between
neutrons that are dependent on the fissioning nucleus
§  FREYA agrees rather well with most neutron observables for
several spontaneously fissioning isotopes and for neutron-induced
fission
§  Comparison with n-n correlation data very promising
§  Photon data do not present a very clear picture – clearly more
experiments with modern detectors needed to verify older data
§  Refined modeling of photon emission in FREYA is planned
§  Incorporation of FREYA into MCNP6, FREYA1.0 with neutrons,
released as open source in July 2013, is in progress
§  FREYA1.0 is available from
http://nuclear.llnl.gov//simulation/main2.html
§  User manual also available online
Lawrence Livermore National Laboratory
51
Homework!
§  Prove that <E> = 2T for the Weisskopf-Ewing spectrum
and the average energy of the PFNS in the Los Alamos
model is <E> = (4/3)Tmax "
§  Download FREYA and run example problems."
Lawrence Livermore National Laboratory
52