Lesson 4 Objectives

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Lesson 4 Objectives

• Neutron energy spectrum
• Fast energy region
• Slowing-down region
• Resonance concepts
• Doppler broadening
• Practical width
• Narrow resonance approximation
• Wide resonance approximation
• Bondarenko
• Thermal energy region

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Analytical approxns of energy spectrum

• Returning to the first section of Chap. 4, for
  reactor problems we need approximate analytical
  solution of
• to use in forming the multigroup cross
  sections.
• We solve it in three energy regimes
• Fast, slowing-down, thermal

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Analytical approxn Fast

• In the fast range, we assume that the most
  important source is fission neutrons
• Therefore, in fast region, flux spectrum looks
  like the fission neutron production spectrum

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Watt spectrum (from MCNP5 manual)
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Watt spectrum (from MCNP5 manual)
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Analytical approxn Slowing-down

• In the slowing-down range, we assume that the
  most important source is elastic scattering and
  absorption can be ignored
• Therefore, in slowing-down region, flux spectrum
  looks like 1/E (in the absence of absorption)

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Resonance concepts

• Now we want to fit resonance processing into the
  overall picture and get some concepts under our
  belt
• Remember
• j and c functions
• Doppler broadening
• Resonance Integral
• Resonance escape probability
• Multigroup cross section
• Practical width
• NR and WR approximations

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j and c functions

• Absorption and scattering

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Doppler Broadening

• Effective resonances get shorter and broader with
  temperature increases
• Total integral remains the same (approx.)
• Resulting reaction rates tend to rise
• See Fig. 4.5 in the text
• Very important to reactor safety because of
  negative temperature coefficients

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Practical width

• Energy range width over which the resonance cross
  section is larger than the non-resonance cross
  section
• Sometimes thousands of times the true width (but
  still not very far)

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NR approximation

• We will return to basic balance condition
• Assume flux on RHS is 1/E
• Resulting flux is

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WR approximation

• Again, from basic balance condition
• Assume flux on RHS is 1/E BUT that the energy
  loss from scattering collisions with RESONANCE
  material does not reduce energy (sometimes called
  NRIA)
• Resulting flux is

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IR approximation

• Iintermediate Some resonance scattering
• ffractional effectiveness of resonance
  scattering

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Bondarenko method

• With NR, WR, or IR you need MACROSCOPIC cross
  sections
• Bottom line Cross sections cannot be
  preprocessed (i.e., as isotopes) because you must
  know how much the medium will depress the flux
• So, resonance processing must wait until the
  problem materials are known
• Alternate approach to resonance processing is the
  Bondarenko method which greatly reduces (but does
  not eliminate) the resonance processing

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Bondarenko method (2)

• Break down the formation the total cross sections
  in NR method into two pieces
• Contribution of resonance nuclide
• Contribution from other nuclides (scattering)
• Multigroup isotopic total cross section becomes

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Bondarenko method (4)

• Procedure
• Pre-calculate the group cross sections (for
  resonance isotopes in resonance groups) for a
  variety of sigma-0 values (0, 1, 10, 100, 1000,
  infinity)
• When the problem materials are known, calculate
  the value of sigma-0
• Interpolate between the precalculated values
• Art of the deal Choice of sigma-0s
  interpolation used

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Analytical approxn Thermal

• In the thermal range, we have upscatter as well
  as down-scatter, so the neutron density (ignoring
  absorption) assumes a shape similar to a
  Maxwellian free-gas solution
• Perturbations 1/E scattering source from above,
  1/v absorption
• Handled with neutron temperature

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Mental experiment Non-absorbing, source-less
thermal problem

• A useful mental experiment is to assume only
  scattering and drop 100 neutrons into a medium
  at given T
• Result They will come into thermal equilibrium
  with the medium

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Maxwellian shapes

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Mental experiment (2)

• Note that we know the result and it does not
  depend on scattering cross sections!
• If we form the balance equation
• and substitute the answer

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Mental experiment (3)

• we get
• This is an equation for the cross sections
• Note that the equation is satisfied by
• which is the detailed balance
  relationship of down- and up-scatter between two
  energies

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Mathematical example

• Equation 12.1

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Mathematical example (2)

• For monatomic gas
• In cold target limit, back to elastic
  down-scatter equation
• Proton gas model given by Eqn. 12.10

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Mathematical example (3)

• Where
• Abramowitz and Stegun

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Homework 4

• For an assumed Fission-1/E-Maxwellian shape with
  energy breakpoints of Eth and Ef, find the
  normalization constants on the Watt spectrum and
  Maxwellian shapes to maintain continuity.
• Show that the proton gas model satisfies detailed
  balance.