Lecture 3

Nuclear Data

22.106 Neutron Interactions and Applications Spring 2010

Common Misconceptions

• It’s just a “bunch” of numbers

• J ust give me the right value and stop changing it.

Traditional evaluation method

Exp. data

Analysis &

line fitting

Exp.

x- sections

Comparison &

mer ging

Evaluation

Model param.

Model calculations

Theo.

x- sections

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Theoretical model

• C omputer code SAMMY

– U s ed for analysis of neutron and charged particle cross-section data

– U s e s Bayes method to find parameter values

• G eneralized least squares

– U s es R-matrix theory to tie experimental data to theoretical models

• R eich-Moore approximation

• Breit-Wigner theory

– T reats most types of energy -differential cross sections

• T reats energy and angle differential distributions of scattering

– F its integral data

– Generates covariance and sensitivity parameters for resolved and unresolv ed resonance region

Three energy regions

• R esolved resonance range

– E xperimental resolution is smaller than the width of the resonances; resonances can be distinguished. Cross-section representation can be made by resonance parameters

– R -matrix theory provides for the general formalism that are used

• U nresolved energy range

– Cross-section fluctuations still exist but experimental resolution is not enough to distinguish multiplets.

Cross-section representation is made by average resonance parameters

– F ormalism

• Statistical models e.g. Hauser-Feshbac h model combined with optical model

• lev el dens ity models, ….

• P robability tables

- E ne rg y r a nge ov e r w hi c h re s ona nc e s a r e s o n a rro w a nd c l os e t oge t he r t ha t t h e y c a nnot be e xp e ri m e nt a l l y re s ol ve d .

- A c om bi na t i on o f e xpe r i m e nt a l m e a s ure m e nt s of t he a v e ra ge c ros s s e c t i on a nd t he ore t i c a l m ode l s y i e l ds di s t ri bu t i on fun c t i on s fo r t he s pa c i ngs a n d w i dt h s .

- T he d i s t ri but i ons m a y be us e d t o c om put e t h e ‘d i l ut e -a v e ra ge ’ c ro s s s e c t i ons :

 4  2  2  g  2  

  E 

  

 2 l  1  s i n 2  

  J

n, l , J

 2 

s i n 2   

l   k 2

l k 2

J  D l , J

 l , J

n, l , J l

  

 c  E 

 2  2 g

k D

 n, l , J   , l , J



l J

 2  2

l , J

g J

l , J

 n, l , J  f , l , J

f  E 

 k 2  D 

l J l , J

l , J

l = orbi t a l a ngul a r m om e nt um qua nt um no. , J = s pi n of t h e c om pound nuc l e us

k = w a ve num be r , g J = s pi n s t a t i s t i c a l fa c t or,  l

= ph a s e s hi ft

 n , l , J ,   , l , J ,  f , l , J ,  l , J

= n e ut ron , c a pt ure , f i s s i on, a nd t ot a l w i d t hs

D l , J

= r e s ona nc e s pa c i ng, ⋯ de not e s a v e ra gi ng ove r t he di s t ri but i on(s )

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The Pr obability T able Method

- Concept developed in the early 1970s by Levitt (USA) and Nikolae v , et al . (USSR).

- Uses the distributions of resonance widths and spacings to infer distributions of cross section values.

- Basic idea:

- Compute the probability p n that a cross section in the URR lies in band n

& defined as  ^ n -1 <   ^ 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 ener gy 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.

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- p t (  , E ) d   proba bi l i t y t h a t t he t ot a l c ros s s e c t i on l i e s i n d  a bout  a t e ne rg y E

- A ve ra ge t ot a l c r os s s e c t i on:

 t ( E )

  d   p t (  , E )

- Ba nd proba b i l i t y:

p n ( E ) 

 ˆ n

 ˆ n  1

d  p t

(  , E )

- Ba nd-a ve r a g e t o t a l c ros s s e c t i on:

 t , n

( E ) 

1

p n ( E )

 ˆ n

 ˆ n  1

d   p t

(  , E )

- q  (   , E

 ) d  

 c ondi t i ona l p roba bi l i t y t ha t t h e pa r t i a l c ros s s e c t i on of t y p e  l i e s i n

d   a bou t   gi v e n t h a t t he t ot a l c ros s s e c t i on ha s t he va l u e 

- Ba nd -a ve r a ge pa rt i a l c ros s s e c t i o n: 

( E )  1

 ˆ n

d  p ( 

   q

(   , E  )

 , n

p n ( E )

  ˆ

n  1

t , E )  0 d 

U nf or t unat e l y , c om put i ng

p t (  , E )

and

q  (   , E  )

di r e c t l y i s an i nt r ac t abl e pr obl e m .

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- Use ENDF/B parameters to create probability distribution functions (PDFs) for resonance widths (W igner distribution) and spacings (chi-squared distributions).

- Randomly sample widths and spacings from PDFs to generate 'fictitious' sequences (realizations)

of resonances about the ener gy E for which the table is being created.

- Use single-level Breit-W igner formulae to compute sampled cross section values at E:

  E   

 E   4    2 l  1  s i n 2 

s s, sm oot h

k 2 l

 4 

g  n r

   c os  2 

   1   n r

     , X

  s i n  2 

    , X  

k 2   J    

l     r r l r r  

l J r  R lJ

r   

 r    

   E     , sm oot h

 E   4  g

k 2 J

 n r   r   



r , X r

    c , f

l J r  R lJ r

  , sm oot h = tabulated background cross section,   s , c , f

 n , r ,   , r ,  f , r ,  r = neutron, capture, fission and total widths for resonance r

R l J = set of sampled resonances for quantum number pair ( l,J )

 ,  = Doppler functions,  r   r 4 k B T E A , X r  2  E  E r   r , A = atomic mass

- Use the sampled cross sections to compute band averages and probabilities.

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• H igh energy region

– N o cross-section fluctuations exist. Cross- sections are represented by smooth curves.

– F o r m a l i s m

• Statistical models e.g. Hauser-Feshbach

• I ntra-nuclear cascade model

• Pre-equilibirum model

• Evaporation model

Cross section measurements

Cross section evaluation

Cross section processing

Point data libraries

Multigr oup libraries

Data testing using transport m ethods and integral experimen ts

T ransport methods

Cro ss section proces sing methods

Sensitivity and uncertainty a nalyses

Cross sections for user applications

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ORELA

• H igh flux (10 14 n/s )

• E xcellent Resolution ( Δ t = 2-30 ns)

– f aciliates better evaluations

• “ White” neutron spectrum from 0.01eV to 80 MeV

– R educes systematic uncertainties

• M easurement systems and background well understood

– Very accurate data

• S imultaneous measurements in different beams lines

• M easurements performed on over 180 isotopes

Figures removed due to copyright restrictions.

ORELA Target

• H igh energy electrons hitting a tantalum target produce bremsstrahlung (photon) spectrum. Neutrons are generated by photonuclear reactions, Ta(gamma, n), Ta(gamma, 2n), …

Figure removed due to copyright restrictions.

Bayesian Inference

• B ayesian inference is statistical inference in which evidence or observations are used to update or to newly infer what is known about underlying parameters or hypotheses.

Cost of evaluations

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Covariance Matrix

• T he covariance matrix or dispersion matrix is a matrix of covariances between elements of a random vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.

E[( X 1 -  1 ) ( X 1 -  1 )]

E[( X 2 -  2 ) ( X 1 -  1 )]

E[( X 1 -  1 ) ( X 2 -  2 )]

E[( X 2 -  2 ) ( X 2 -  2 )]

...

...

E[( X 1 -  1 ) ( X n -  n )]

E[( X 2 -  2 ) ( X n -  n )]

E[( X n -  n ) ( X 1 -  1 )]

E[( X n -  n ) ( X 2 -  2 )]

...

E[( X n -  n ) ( X n -  n )]

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• E ach type of data comes from a separate measurement

– Cross-sections are measured independently

– H owever, the various types are highly interrelated

• D ata include measurement-related effects

– F inite temperature

– F inite size of samples

– F inite resolution

– …

• Measured data may look very different from the underlying true cross-section

– T hink Doppler broadening

Advantages of evaluated data

• Incorporate theoretical understanding

– Cross-section shapes

– R elationships between cross-sections for different reactions

• Incorporate all available experimental data and all available uncertainty

• A llow extrapolation

– Different temperatures

– Different energies

– Different reactions

• G enerate artificial “experimental” points from ENDF resonance parameters

– Include Doppler and resolution broadening

• M ake reasonable assumptions regarding experimental uncertainties

– S tatistical (diagonal terms)

– S ystematic (off-diagonal terms)

• N ormalization, background, broadening, …

• R un models with varying resonance parameters with an assumed distribution

• Include systematic uncertainties for measurement-related quantities

• P erform simultaneous fit to all data

– A ll experimental uncertainty is thus propagated

Computational cost

• Large cases require special care

– U -235 has ~3000 resonances

• 5 parameters for each resonance need to be varied

• Very time consuming

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22.106 Neutron Interactions and Applications

Spring 2010

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