urr energy ranges
DESCRIPTION
URR Energy Ranges. 174 Hf Capture Cross Section. 174 Hf Scattering Cross Section. σ n (barns). σ γ (barns). E (eV). E (eV). Probability Table for 235 U at 2250 eV band 1: ≤ 12 b band 100: > 110 b 98 bands geometrically-spaced between 12 and 100 b. - PowerPoint PPT PresentationTRANSCRIPT
Probability Table Method for the Unresolved Resonance Range
1
Implementing the Probability Table Method for Treating the Unresolved Resonance Range in a Continuous-
Energy Monte Carlo Code
Thomas M. Sutton Advisory Scientist – Nuclear Methods and Computations
Knolls Atomic Power Laboratory
Probability Table Method for the Unresolved Resonance Range
2
The Unresolved Resonance Range (URR) Energy range over which resonances are so narrow and close together that they cannot
be experimentally resolved. A combination of experimental measurements of the average cross section and
theoretical models yields distribution functions for the spacings and widths. The distributions may be used to compute the ‘dilute-average’ cross sections:
22n, ,2 2
s n, ,2 2,,
4 22 1 sin 2 sinl JJ
l l J ll J l Jl J
gE l
k k D
2
n, , , ,c 2
,,
2 l J l JJ
l J l Jl J
gE
k D
2
n, , f, ,f 2
,,
2 l J l JJ
l J l Jl J
gE
k D
l = orbital angular momentum quantum no., J = spin of the compound nucleus k = wave number, Jg = spin statistical factor, l = phase shift
n, ,l J , , ,l J , f , ,l J , ,l J = neutron, capture, fission, and total widths
,l JD = resonance spacing, denotes averaging over the distribution(s)
Probability Table Method for the Unresolved Resonance Range
3
238U Total Cross Section Computed by LANL code NJOY. Shows use of dilute-average cross section in the unresolved resonance range (URR).
Energy (eV)
To
talC
ross
Se
ctio
n(b
)
10-2 10-1 100 101 102 103 104 105 106 107100
101
102
103
104
URR
Probability Table Method for the Unresolved Resonance Range
5
The Probability Table Method Concept developed in the early 1970s by Levitt (USA) and Nikolaev, et al. (USSR). Uses the distributions of resonance widths and spacings to infer distributions of cross
section values. Basic idea: Compute the probability pn that a cross section in the URR lies in band n defined as
1ˆ ˆn n . Compute the average value of the cross sections ( n ) for each band n. Following every collision (or source event) in a Monte Carlo calculation for which
the final energy of the neutron is in the URR, sample a band-averaged cross section with the computed probabilities and use that value for that neutron until its next collision.
Probability Table Method for the Unresolved Resonance Range
6
A Model Problem Due to Dermot E. (Red) Cullen of LLNL. Mono-directional beam perpendicularly incident on a purely-absorbing slab. Beam spectrum is uniform over the URR. Cross section probability distribution function is given by
2
0 0 0
0
33( ) 6 ;
2 2 2p
.
Mean value is 0 . Slab thickness is 20 mfp ( 020 ( )N ).
0.6 0.8 1.2 1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
20 mfp
( )p
Probability Table Method for the Unresolved Resonance Range
7
Model Problem (cont.)
0 5 10 15 20
1. 107
0.00001
0.001
0.1
Analytic Solutions to the Model Problem
exact 4 equiprobable bands 2 equiprobable bands
1 band (dilute average)
Probability Table Method for the Unresolved Resonance Range
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Mathematical Theory of the Probability Table Method t ( , )p E d probability that the total cross section lies in d about at energy E
Average total cross section: t t( ) ( , )E d p E
Band probability: 1
ˆ
tˆ( ) ( , )
n
nnp E d p E
Band-average total cross section: 1
ˆ
t, tˆ
1( ) ( , )
( )
n
nn
n
E d p Ep E
( , )q E d conditional probability that the partial cross section of type lies in
d about given that the total cross section has the value
Band-average partial cross section: 1
ˆ
, tˆ 0
1( ) ( , ) ( , )
( )
n
nn
n
E d p E d q Ep E
Unfortunately, computing t ( , )p E and ( , )q E directly is an intractable problem.
Probability Table Method for the Unresolved Resonance Range
9
Monte Carlo Algorithm for Generating the Tables Use ENDF/B parameters to create probability distribution functions (PDFs) for
resonance widths (Wigner distribution) and spacings (chi-squared distributions). Randomly sample widths and spacings from PDFs to generate ‘fictitious’ sequences
(realizations) of resonances about the energy E for which the table is being created. Use single-level Breit-Wigner formulae to compute sampled cross section values at E:
2s s, smooth 2
n n2
42 1 sin
4cos 2 1 , sin 2 ,
lJ
ll
r rJ l r r l r r
l J r R r r
E E lk
g X Xk
n, smooth 2 2
4,
lJ
r rJ r r
l J r R r
E E g Xk
, c, f
,smooth = tabulated background cross section, s, c, f
n,r , ,r , f ,r , r = neutron, capture, fission and total widths for resonance r
lJR = set of sampled resonances for quantum number pair (l,J)
, = Doppler functions, B4r r k TE A , 2r r rX E E , A = atomic mass
Use the sampled cross sections to compute band averages and probabilities.
Probability Table Method for the Unresolved Resonance Range
10
σn
(barns)
E(eV)
σγ
(barns)
E(eV)
174Hf Capture Cross Section174Hf Scattering Cross Section
Probability Table Method for the Unresolved Resonance Range
11
Pseudocode for Monte Carlo Algorithm for Generating the Tables loop over M realizations
randomly generate partial cross section values s for energy E ts s
determine band number n such that 1 tˆ ˆn ns 1n nm m , ,n ns s s
t , t , tn ns s s
end loop over realizations n np m M , ,n n ns s m
t , t ,n n ns s m
Probability Table Method for the Unresolved Resonance Range
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band-averaged cross section (b)
ba
nd
pro
ba
bili
ty
0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
Probability Table for 235U at 2250 eVband 1: ≤ 12 bband 100: > 110 b98 bands geometrically-spaced between 12 and 100 b
Probability Table Method for the Unresolved Resonance Range
13
Visualization of the Energy-Dependence of a Probability Table
U-235 at 293.6 K
0.00E+00
5.00E-01
1.00E+00
1.50E+00
2.00E+00
2.50E+00
3.00E+00
3.50E+00
4.00E+00
0.00E+00 5.00E+03 1.00E+04 1.50E+04 2.00E+04 2.50E+04 3.00E+04
E (eV)
rati
o o
f to
tal
ban
d-t
o-a
vera
ge
cro
ss
sect
ion
Probability Table Method for the Unresolved Resonance Range
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Notes on Temperature Dependence Tables are generated at multiple temperatures. The same randomly-generated sequence of resonances is used for all temperatures. Band assignment for all cross section types and temperatures is based on the sampled
band for the total cross section at one ‘master’ temperature.
Probability Table Method for the Unresolved Resonance Range
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Temperature Dependence of 174Hf Capture Cross Section
0 K273 K
2000 K
Probability Table Method for the Unresolved Resonance Range
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Using the Tables in Monte Carlo When a neutron first enters the URR of a nuclide, the band probabilities (interpolated
on energy) are used to sample the band number. The cross sections values corresponding to that band and the temperature of the region
are obtained by interpolating the tables on temperature and energy. o The dilute-average cross section is used to interpolate on energy:
, , PT , PTn n n nn
E E E p E
EPT = energy of nearest probability table E = ENDF/B dilute-average cross section
These cross section values are then used for that neutron history until its next collision. If the neutron enters a region at a different temperature, the cross sections are re-
interpolated on temperature. If, after a collision, the exit energy is still within the URR then the cross section values
are resampled.)
Probability Table Method for the Unresolved Resonance Range
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Example: ZEBRA8D Critical Assembly
Monte Carlo keff Results
Average URR Cross Sections
0.9724(6)
URR Probability Tables
0.9862(6)
Difference 0.0138(8)
numbers in parentheses are the 95% confidence intervals on the last digit
Probability Table Method for the Unresolved Resonance Range
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References Sutton, T. M. and F. B. Brown, “Implementation of the Probability Table Method in a
Continuous-Energy Monte Carlo Code System”, Proc. Int. Conf. Physics Nuc. Sci. Tech., 891, Long Island, New York, October 5-8, (1998)
Weinman, J. P., “Monte Carlo Testing of Unresolved Resonance Treatment for Fast and Intermediate Critical Assemblies”, Proc. Int. Conf. Physics Nuc. Sci. Tech., 1029, Long Island, New York, October 5-8, (1998)