Resonance Theory in Reactor Applications

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Resonance Theory in Reactor Applications
R. N. HwangNuclear Engineering Division Argonne
National Laboratory
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Outline

• Introduction
• Fundamentals
• Representations of Microscopic Cross Sections
• Doppler-Broadening of Cross Sections
• Applications in conjunction to multi-group
  approach
• Homogeneous media
• Heterogeneous media
• Unresolved resonance treatment
• Future challenges

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Historical Notes

• Resonance Structures in U238 Cross Sections
• Observations of Resonance Absorption in U238
  Attributable to Self shielding
• Reduction of absorption rate per atom of neutrons
  slowed-down in water/uranium mixture as
  concentration increased.
• A substantial reduction of resonance absorption
  was possible if natural uranium was made into a
  lump surrounded by moderator, a discovery
  signaling the beginning of nuclear era.

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Background

• Basic Issue
• Application of microscopic cross sections in
  macroscopic reactor systems.
• Two Quantities of Interest in Reactor Physics
• Self-Shielding Factor
• Can be construed as a measure of correlation
  between cross section and flux in energy and
  space at a a given temperature,

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Background (contd.)

• General Requirements
• Explicit description of microscopic cross section
  as a function of energy
• Concise representation of R-Matrix
  theory and approximations.
• High quality resonance parameters.
• Doppler-broadening of microscopic cross sections
  at 0K to form
• Determination of neutron flux and reaction rates
  in reactor lattices
• Multi group(/subgroup) methods via pre-computed
  multi group set
• Continuous energy Monte Carlo approach via
  point-wise cross sections.
• For high energy region where resonances are
  unresolved, the averages must be treated on the
  basis of statistical theory.

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Evolution of Resonance Theory

• Period of Earlier Thermal Reactor Development
• Focused on a handful of low-lying resonances of
  few actinides.
• Relied exclusively on the single level
  Breit-Wigner approximation.
• Utilizing the resonance integral concept
  exclusively.
• Assumption of two-region cell along with rational
  approximation.
• Emergence of Fast Reactor Program
• General shift of interest to higher energy
  region.
• Resonances of intermediate weight nuclides began
  to play an important role.
• Focused on reactivity coefficients (Doppler,
  sodium void, etc.)
• More important role for statistical treatment of
  unresolved resonances.
• Emphasis on issues pertinent to analysis of
  experiments of critical assemblies.

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Evolution of Resonance Theory(Contd.)

• Formation of CSWEG in 1966
• Provided the reference nuclear data base (ENDF/B
  Files).
• Motivated continuous improvement of evaluated
  resonance parameters.
• Dramatic Improvement of Computational Tools
• Made possible computation of many extremely
  complex problems.
• More frequent use of the Monte Carlo Approach.

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R-Matrix Representation

• Channel matrix and Level matrix formulations
• where

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R-Matrix Representation (contd.)

• Mathematical Properties
• Collision matrix is symmetric and unitary.
• Statistical Properties normally
  distributed ,
  Wigner/Dyson dist.
• Except for the explicit energy dependence that
  appears in and/or only three quantities
  are (weakly) energy-dependent, namely,
• - - a low order rational function of
  .
• - - Hard sphere phase shift is also a
  simple function of .
• Collision matrix must be single-valued and
  meromorphic in k-plane.
• Meromorphic in E-plane if and only if
  E-dependence of 3 factors is ignored.

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R-Matrix Representation (contd.)

• Four Practical Formalisms Currently in Use
• ENDF/B specified SLBW, MLBW, Adler-Adler, and
  Reich-Moore.
• Alternative Generalized Pole Representation.
• Meromorphic nature leads to the following
  expression
• where parameters are numerically deducible from
  any other representation.
• Consequences
• Exact Doppler-Broadening
• Keep Resonance integral concept intact.

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Doppler Broadening Based on Ideal Gas Model

• Broadening Via Maxwell-Boltzmann Distribution
• Solbrigs Kernel
• Let and
• Approximate Gauss Kernel in Energy Domain (For
  relatively high energy)
• Broadening Via Solution of Heat Equation
  (Semi-infinite cyl.)

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Earlier Approach Single-Level Breit-Wigner
Approx.

• Microscopic Cross Section at 0 K (Lorentzian
  Shape)
• Doppler-Broadened Line Shape (Approx. Gauss
  Kernel)
• where (complex
  probability integral) and .
  

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Computation of w-FunctionTwo-Distinct Examples

• Direct numerical approach (OShea and Thacher)
• Region of small Taylors Expansion.
• Intermediate region of Continued
  Fraction.
• Large Gauss-Hermite Quadrature.
• Cauchy integral approach Turing Method (as
  proposed by Bhat Lee-Whiting also used by
  Froehner).
• Consider an integral involving a meromorphic
  function .
• Contour integral via Cauchy integral theorem Let

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Computation of w-FunctionTwo-Distinct Examples
(contd.)

• Where
  
  
• Connection to w(z)
• Note that

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Numerical Approaches to Doppler-Broadening

• Kernel-Broadening Methods
• Piece-wise integration over the kernel.
• Where
  .
• Linearity approximation is
  linear between two mesh points.
• Generalized Gauss-Hermite Quadrature Analytical
  weights and abscissae. 2-point quadrature can
  produce benchmark quality results.
• Finite Difference Methods for Solving the Heat
  Equation
• Explicit Method
• Implicit Method

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Neutron Slowing-Down in Homogeneous Media

• Slowing-Down Equation
• Slowing-Down Density
• Neutrons slowed down past a given energy E.
• Differentiation with respect to and assuming
  yield
• Resonance Integral Concept

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Traditional Resonance Integral Approximations

• Narrow Resonance Approximation (NR)
• Extent of the res. .
• Infinite Mass Approximation (NRIM or WR)
• In the limit of ,
• where is the moderator cross section
  only.
• Intermediate Resonance Approximation (IR)
• The conjecture is
• where accounts for higher orders effects
  (NR) (WR).

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Traditional J-Integral Approach for SLBW
Resonances

• Earlier Version
• No Asymmetric Component
• Later Version
• With Asymmetric Component
• Overlap Correction for Neighboring Resonances

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Computation of Resonance Integral

• Rational Transformation
• Noting that . Let
• Gauss-Jacobi (Chebyshev) Quadrature
• Efficiency has been demonstrated in computations
  of resonance integral, cross section moments and
  analysis of self-indication measurements.

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Example Integrand of First Order Moment

• Set

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Neutron Slowing-Down in Heterogeneous Media

• Starting Point
• Coupled Integral Equations (2-Regions Cell)
• Reduction to Single Integral Equation
• Reciprocity Relation
• Reduced Integral Equation (assume
  )
• Further Simplifications
• NR-, WR-, and IR- Approximations.

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Brief Outline of Traditional Collision
Probability Approach

• Integral Transport Theory Origin
• Illustrative Example for Various Isolated Lumps

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Evolution of Collision Probability Methods

• Isolated Fuel Lumps
• Analytical Approaches for Simple Geometries
  Slabs, cylinders, spheres, ect.
• Mean-Chord Approximation
• Wigners Rational Approximation
• Closely-Packed/Repeated Cells
• Rational approximation with Dancoff
  Correction/Bells factor.
• Analytical approach for infinite slabs.
• Cylindricised two region cells.
• Interface current method
• Most effective for cell with multiple annuli.

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Evolution of Collision Probability Methods
(contd.)

• Two-dimensional collision probability approach
• Carlviks method not yet widely used in
  conjunction with resonance treatment.

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Two Approximations of Historical Importance

• Volume and Surface integral approach
• Pioneered by Wigner et al
• Equivalence relations resulting from rational
  approximation
• NR-example
• where equivalent cross section,
  .

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Philosophies of Resonance Cross Section Processing

• General
• Types of cross section libraries
• Pre-computed point-wise cross section libraries
  for various nuclides at a given T.
• Composition-dependent multi-group constant
  libraries.
• Nature of processing
• Deterministic Multi-group constants preparation
  in the resolved energy region.
• Probabilistic Unresolved resonance treatment
  sub-group constants.
• Rigor of resonance treatment
• How Doppler-broadening is treated
• How slowing-down equation is treated
• How integral transport equation is treated at
  unit cell level.

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Philosophies of Resonance Cross Section
Processing (contd.)

• Multi-group constants
• Representation of composition-dependent group
  constants
• Bondarenko Scheme
• Pre-compute self-shielding factors in terms of
  at T.
• Explicit composition-dependent processing.
• Group structure/Spatial dependence
  considerations
• Energy structure
• Ultra-fine, fine, and coarse groups.
• Spatial dependence
• Resonance related calculations at unit cell level
• Approximate methods based on rational
  approximation
• Integral transport theory approach.
• Region-dependent multi-group constants
• Computation of global spectrum of the system
  using average cell cross sections
• Collapsing to form desired multi-group constants.

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Unresolved Resonance Treatment

• Statistical theory fundamentals
• Basic Rule
• Statistically independent variables
• Addition Theorem
• Multiplication theorem (conditional
  probabilities)

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Unresolved Resonance Treatment (contd.)

• Known averages and statistical information
• ---- normal distribution with zero
  means and variance of unity
• Or partial width
  distribution.
• ---- Wigner
  distribution and long range correlation of
  Dyson.
• Expectation values of interest
• and
  .

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Unresolved Resonance Treatment (contd.)

• Two widely used approaches
• Direct Integration (Riemann integral)
• NR-approx. implicitly assumed.
• Probability Table Method (Lebesgue integral)
• Requires
  .
• These quantities are obtainable via (1) ladder
  method (2) Moment method (3) integral transform
  method.

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Illustration of Potential Issues in Ladder Method
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Future Challenges

• Nuclear Data
• Continuous efforts to improve resonance
  parameters and their representation
• More utilization of self-indication ratio
  measurements as means of data verification.
• Point-Wise Doppler-Broadening of Cross Sections
• More studies on potential effects of crystalline
  binding on broadening
• Optimization of point-wise cross section
  libraries at a given T.
• Multi-Group Cross Section Processing
• Improvement of current collision probability
  treatment at resonance level to accommodate more
  complex cells.
• Improvement of means of computing global
  weighting spectrum for cross section collapsing
  in multi-group constant codes.

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Future Challenges (contd.)

• Improvement of Unresolved Resonance Treatment
• Several new approaches under consideration are
• Analytical approach for computing probability
  tables via integral transform techniques
• Characteristic function approach for treating
  averages involving S-matrix (pioneered by
  Lukyanov et al)
• Maximization of information entropy (pioneered by
  Froehner).