urr energy ranges

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


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)


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


4

URR Energy Ranges


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.


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


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)


8

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.


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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.


10

σn

(barns)

E(eV)

σγ

(barns)

E(eV)

174Hf Capture Cross Section174Hf Scattering Cross Section


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


12

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


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


14

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.


15

Temperature Dependence of 174Hf Capture Cross Section

0 K273 K

2000 K


16

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.)


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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


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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)

urr energy ranges

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