From 90d3ad89cf621585b3a03b7e54e1ed7071e16102 Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Wed, 25 Jul 2012 15:53:23 -0400 Subject: [PATCH] Added description of ACE Law 1 in documentation. --- docs/source/methods/physics.rst | 80 ++++++++++++++++++++++++++++----- 1 file changed, 70 insertions(+), 10 deletions(-) diff --git a/docs/source/methods/physics.rst b/docs/source/methods/physics.rst index a03a7fffea..dd1813480b 100644 --- a/docs/source/methods/physics.rst +++ b/docs/source/methods/physics.rst @@ -220,7 +220,55 @@ for the data. ACE Law 1 - Tabular Equiprobable Energy Bins ++++++++++++++++++++++++++++++++++++++++++++ +In the tabular equiprobable bin representation, an array of equiprobably +outgoing energy bins is given for a number of incident energies. While the +representation itself is simple, the complexity lies in how one interpolates +between incident as well as outgoing energies on such a table. If one does +simple interpolation between tables for neighboring incident energies, it is +possible for the resulting energies to violate laws governing the kinematics, +i.e. the outgoing energy may be outside the range of available energy in the +reaction. +To avoid this situation, the accepted practice is to use a process known as +scaled interpolation [Doyas]_. First, we find the tabulated incident energies +which bound the actual incoming energy of the particle, i.e. find :math:`i` such +that :math:`E_i < E < E_{i+1}` and calculate the interpolation factor :math:`f` +via :eq:`interpolation-factor`. Then, we intepolate between the minimum and +maximum energies of the outgoing energy distributions corresponding to +:math:`E_i` and :math:`E_{i+1}`: + +.. math:: + :label: ace-law-1-minmax + + E_{min} = E_{i,1} + f ( E_{i+1,1} - E_i ) \\ + E_{max} = E_{i,M} + f ( E_{i+1,M} - E_M ) + +where :math:`E_{min}` and :math:`E_{max}` are the minimum and maximum outgoing +energies of a scaled distribution, :math:`E_{i,j}` is the j-th outgoing energy +corresponding to the incoming energy :math:`E_i`, and :math:`M` is the number of +outgoing energy bins. Next, statistical interpolation is performed to choose +between using the outgoing energy distributions corresponding to energy +:math:`E_i` or :math:`E_{i+1}`. Let :math:`\ell` be the chosen table where +:math:`\ell = i` if :math:`\xi_1 > f` and :math:`\ell = i + 1` otherwise where +:math:`\xi_1` is a random number. Now, we randomly sample an equiprobable +outgoing energy bin :math:`j` and interpolate between successive values on the +outgoing energy distribution: + +.. math:: + :label: ace-law-1-intermediate + + \hat{E} = E_{\ell,j} + \xi_2 (E_{\ell,j+1} - E_{\ell,j}) + +where :math:`\xi_2` is a random number sampled uniformly on :math:`[0,1)`. Since +this outgoing energy may violate reaction kinematics, we then scale it to the +minimum and maximum energies we calculated earlier to get the final outgoing +energy: + +.. math:: + :label: ace-law-1-energy + + E' = E_{min} + \frac{\hat{E} - E_{\ell,1}}{E_{\ell,M} - E_{\ell,1}} + (E_{max} - E_{min}) ACE Law 3 - Inelastic Level Scattering ++++++++++++++++++++++++++++++++++++++ @@ -801,25 +849,35 @@ the unresolved range to get the actual cross sections. Lastly, the total cross section is calculated as the sum of the elastic, fission, capture, and inelastic cross sections. -.. [Foderaro] Anthony Foderaro, *The Elements of Neutron Interaction Theory*, - MIT Press, Cambridge, Massachusetts (1971). +---------- +References +---------- -.. [SIGMA1] Dermett E. Cullen and Charles R. Weisbin, "Exact Doppler Broadening - of Tabulated Cross Sections," *Nucl. Sci. Eng.*, **60**, pp. 199-229 (1976). +.. [Doyas] Richard J. Doyas and Sterrett T. Perkins, "Interpolation of Tabular + Secondary Neutron and Photon Energy Distributions," *Nucl. Sci. Eng.*, + **50**, 390-392 (1972). + +.. [Foderaro] Anthony Foderaro, *The Elements of Neutron Interaction Theory*, + MIT Press, Cambridge, Massachusetts (1971). **Note:** Students, faculty, and + staff at MIT can obtian a PDF copy of this book for free from the `MIT + Press`_. .. [Gelbard] Ely M. Gelbard, "Epithermal Scattering in VIM," FRA-TM-123, Argonne National Laboratory (1979). -.. [Williams] M. M. R. Williams, *The Slowing Down and Thermalization of - Neutrons*, North-Holland Publishing Co., Amsterdam (1966). This book can be - obtained for free from the OECD_. +.. [Levitt] Leo B. Levitt, "The Probability Table Method for Treating Unresolved + Neutron Resonances in Monte Carlo Calculations," *Nucl. Sci. Eng.*, **49**, + pp. 450-457 (1972). + +.. [SIGMA1] Dermett E. Cullen and Charles R. Weisbin, "Exact Doppler Broadening + of Tabulated Cross Sections," *Nucl. Sci. Eng.*, **60**, pp. 199-229 (1976). .. [Squires] G. L. Squires, *Introduction to the Theory of Thermal Neutron Scattering*, Cambridge University Press (1978). -.. [Levitt] Leo B. Levitt, "The Probability Table Method for Treating Unresolved - Neutron Resonances in Monte Carlo Calculations," *Nucl. Sci. Eng.*, **49**, - pp. 450-457 (1972). +.. [Williams] M. M. R. Williams, *The Slowing Down and Thermalization of + Neutrons*, North-Holland Publishing Co., Amsterdam (1966). **Note:** This + book can be obtained for free from the OECD_. .. |sab| replace:: S(:math:`\alpha,\beta`) @@ -836,3 +894,5 @@ cross sections. .. _MC21: http://www.osti.gov/bridge/servlets/purl/903083-HT5p1o/903083.pdf .. _Sutton and Brown: http://www.osti.gov/bridge/product.biblio.jsp?osti_id=307911 + +.. _MIT Press: http://hdl.handle.net/1721.1/1716