diff --git a/docs/source/methods/physics.rst b/docs/source/methods/physics.rst index 67e9759eb..a03a7fffe 100644 --- a/docs/source/methods/physics.rst +++ b/docs/source/methods/physics.rst @@ -31,6 +31,9 @@ distribution is represented as either - An equiprobable distribution with 32 bins, or - A tabular distribution. +Isotropic Angular Distribution +++++++++++++++++++++++++++++++ + In the first case, no data needs to be stored on the ACE table, and the cosine of the scattering angle is simply calculated as @@ -41,6 +44,9 @@ of the scattering angle is simply calculated as where :math:`\xi` is a random number sampled uniformly on :math:`[0,1)`. +Equiprobable Bin Distribution ++++++++++++++++++++++++++++++ + For a 32 equiprobable bin distribution, the procedure to determine the scattering cosine is as follows. First, we select a random number :math:`\xi` to sample a cosine bin :math:`i` such that @@ -58,6 +64,9 @@ The same random number can then also be used to interpolate between neighboring \mu = \mu_i + (32\xi - i) (\mu_{i+1} - \mu_i) +Tabular Distribution +++++++++++++++++++++ + As the MCNP Manual points out, using an equiprobable bin distribution works well for high-probability regions of the scattering cosine probability, but for low-probability regions it is not very accurate. Thus, a more typical treatment @@ -131,7 +140,7 @@ values on the probability distribution function, we know that (\mu' - \mu_{l,k}) Solving for :math:`p(\mu')` in equation :eq:`pdf-interpolation` and inserting it -into :eq:`cdf-2`, we obtain +into equation :eq:`cdf-2`, we obtain .. math:: :label: cdf-linlin @@ -185,6 +194,132 @@ linear-linear interpolation: Sampling Secondary Energy and Correlated Angle/Energy Distributions ------------------------------------------------------------------- +For a reaction with secondary neutrons, it is necessary to determine the +outgoing energy of the neutrons. For anything other than elastic scattering, the +outgoing energy must be determined based on tabulated or parameterized data. The +`ENDF-6 Format`_ specifies a variety of ways that the secondary energy +distribution can be represented. ENDF File 5 contains uncorrelated energy +distribution where ENDF File 6 contains correlated energy-angle +distributions. The ACE format specifies its own representations based loosely on +the formats given in ENDF-6. In this section, we will describe how the outgoing +energy of secondary particles is determined based on each ACE law. + +One of the subtleties in the ACE format is the fact that a single reaction can +have multiple secondary energy distributions. This is mainly useful for +reactions with multiple neutrons in the exit channel such as (n,2n) or +(n,3n). In these types of reactions, each neutron is emitted corresponding to a +different excitation level of the compound nucleus, and thus in general the +neutrons will originate from different energy distributions. The first step in +sampling a secondary energy is to sample between multiple energy distributions +if more than one is present. + +Once a secondary energy distribution has been sampled, the procedure for +determining the outgoing energy will depend on which ACE law has been specified +for the data. + +ACE Law 1 - Tabular Equiprobable Energy Bins +++++++++++++++++++++++++++++++++++++++++++++ + + + +ACE Law 3 - Inelastic Level Scattering +++++++++++++++++++++++++++++++++++++++ + +It can be shown [Foderaro]_ that in inelastic level scattering, the outgoing +energy of the neutron :math:`E'` can be related to the Q-value of the reaction +and the incoming energy: + +.. math:: + :label: level-scattering + + E' = \left ( \frac{A}{A+1} \right )^2 \left ( E - \frac{A + 1}{A} Q \right ) + +where :math:`A` is the mass of the target nucleus measured in neutron masses. + +ACE Law 4 - Continuous Tabular Distribution ++++++++++++++++++++++++++++++++++++++++++++ + +ACE Law 7 - Maxwell Fission Spectrum +++++++++++++++++++++++++++++++++++++ + +One representation of the secondary energies for neutrons from fission is the +so-called Maxwell spectrum. A probability distribution for the Maxwell spectrum +can be written in the form + +.. math:: + :label: maxwell-spectrum + + p(E') dE' = c E'^{1/2} e^{-E'/T(E)} dE' + +where :math:`E` is the incoming energy of the neutron and :math:`T` is the +so-called nuclear temperature, which is a function of the incoming energy of the +neutron. The ACE format contains a list of nuclear temperatures versus incoming +energies. The nuclear temperature is interpolated between neighboring incoming +energies using a specified interpolation law. Once the temperature :math:`T` is +determined, we then calculate a candidate outgoing energy based on rule C64 in +the `Monte Carlo Sampler`_: + +.. math:: + :label: maxwell-E-candidate + + E' = -T \left [ \log (\xi_1) + \log (\xi_2) \cos^2 \left ( \frac{\pi + \xi_3}{2} \right ) \right ] + +where :math:`\xi_1, \xi_2, \xi_3` are random numbers sampled on the unit +interval. The outgoing energy is only accepted if + +.. math:: + :label: maxwell-restriction + + 0 \le E' \le E - U + +where :math:`U` is called the restriction energy and is specified on the ACE +table. If the outgoing energy is rejected, it is resampled using equation +:eq:`maxwell-E-candidate`. + +ACE Law 9 - Evaporation Spectrum +++++++++++++++++++++++++++++++++ + +Evaporation spectra are primarily used in compound nucleus processes where a +secondary particle can "evaporate" from the compound nucleus if it has +sufficient energy. The probability distribution for an evaporation spectrum can +be written in the form + +.. math:: + :label: evaporation-spectrum + + p(E') dE' = c E' e^{-E'/T(E)} dE' + +where :math:`E` is the incoming energy of the neutron and :math:`T` is the +nuclear temperature, which is a function of the incoming energy of the +neutron. The ACE format contains a list of nuclear temperatures versus incoming +energies. The nuclear temperature is interpolated between neighboring incoming +energies using a specified interpolation law. Once the temperature :math:`T` is +determined, we then calculate a candidate outgoing energy based on rule C45 in +the `Monte Carlo Sampler`_: + +.. math:: + :label: evaporation-E + + E' = -T \log (\xi_1 \xi_2) + +where :math:`\xi_1, \xi_2` are random numbers sampled on the unit +interval. The outgoing energy is only accepted according to a specified +restriction energy as in equation :eq:`maxwell-restriction`. + + +ACE Law 11 - Energy-Dependent Watt Spectrum ++++++++++++++++++++++++++++++++++++++++++++ + +ACE Law 44 - Kalbach-Mann Correlated Scattering ++++++++++++++++++++++++++++++++++++++++++++++++ + +ACE Law 61 - Correlated Energy and Angle Distribution ++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +ACE Law 66 - N-Body Phase Space Distribution +++++++++++++++++++++++++++++++++++++++++++++ + .. _rotate-angle: Transforming a Particle's Coordinates @@ -607,9 +742,9 @@ incorrect results. Probability tables in the ACE format are generated from the UNRESR module in NJOY following the method of Levitt. A similar method employed for the RACER and -MC21_ Monte Carlo codes is described in [Sutton]_. For the discussion here, we -will focus only on use of the probability table table as it appears in the ACE -format. +MC21_ Monte Carlo codes is described in a paper by `Sutton and Brown`_. For the +discussion here, we will focus only on use of the probability table table as it +appears in the ACE format. Each probability table for a nuclide contains the following information at a number of incoming energies within the unresolved resonance range: @@ -666,11 +801,8 @@ 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. -.. _NJOY: http://t2.lanl.gov/codes.shtml - -.. _PREPRO: http://www-nds.iaea.org/ndspub/endf/prepro/ - -.. _MC21: http://www.osti.gov/bridge/servlets/purl/903083-HT5p1o/903083.pdf +.. [Foderaro] Anthony Foderaro, *The Elements of Neutron Interaction Theory*, + MIT Press, Cambridge, Massachusetts (1971). .. [SIGMA1] Dermett E. Cullen and Charles R. Weisbin, "Exact Doppler Broadening of Tabulated Cross Sections," *Nucl. Sci. Eng.*, **60**, pp. 199-229 (1976). @@ -679,7 +811,8 @@ cross sections. National Laboratory (1979). .. [Williams] M. M. R. Williams, *The Slowing Down and Thermalization of - Neutrons*, North-Holland Publishing Co., Amsterdam (1966). + Neutrons*, North-Holland Publishing Co., Amsterdam (1966). This book can be + obtained for free from the OECD_. .. [Squires] G. L. Squires, *Introduction to the Theory of Thermal Neutron Scattering*, Cambridge University Press (1978). @@ -688,9 +821,18 @@ cross sections. Neutron Resonances in Monte Carlo Calculations," *Nucl. Sci. Eng.*, **49**, pp. 450-457 (1972). -.. [Sutton] Thomas M. Sutton and Forrest B. Brown, "Implementation of - the Probability Table Method in a Continuous-Energy Monte Carlo Code System," - *Proc. International Conf. on the Physics of Nucl. Sci. and Technology*, - October 5-8, Long Island, New York (1998). - .. |sab| replace:: S(:math:`\alpha,\beta`) + +.. _OECD: http://www.oecd-nea.org/dbprog/MMRW-BOOKS.html + +.. _NJOY: http://t2.lanl.gov/codes.shtml + +.. _PREPRO: http://www-nds.iaea.org/ndspub/endf/prepro/ + +.. _ENDF-6 Format: http://www-nds.iaea.org/ndspub/documents/endf/endf102/endf102.pdf + +.. _Monte Carlo Sampler: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-9721_3rdmcsampler.pdf + +.. _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