physics.cpp

#include "openmc/physics.h"

#include "openmc/bank.h"
#include "openmc/bremsstrahlung.h"
#include "openmc/constants.h"
#include "openmc/distribution_multi.h"
#include "openmc/eigenvalue.h"
#include "openmc/endf.h"
#include "openmc/error.h"
#include "openmc/material.h"
#include "openmc/math_functions.h"
#include "openmc/message_passing.h"
#include "openmc/nuclide.h"
#include "openmc/photon.h"
#include "openmc/physics_common.h"
#include "openmc/random_dist.h"
#include "openmc/random_lcg.h"
#include "openmc/reaction.h"
#include "openmc/secondary_uncorrelated.h"
#include "openmc/search.h"
#include "openmc/settings.h"
#include "openmc/simulation.h"
#include "openmc/string_utils.h"
#include "openmc/thermal.h"
#include "openmc/tallies/tally.h"

#include <fmt/core.h>

#include <algorithm> // for max, min, max_element
#include <cmath> // for sqrt, exp, log, abs, copysign

namespace openmc {

//==============================================================================
// Non-member functions
//==============================================================================

void collision(Particle& p)
{
  // Add to collision counter for particle
  ++(p.n_collision());

  // Sample reaction for the material the particle is in
  switch (p.type()) {
  case ParticleType::neutron:
    sample_neutron_reaction(p);
    break;
  case ParticleType::photon:
    sample_photon_reaction(p);
    break;
  case ParticleType::electron:
    sample_electron_reaction(p);
    break;
  case ParticleType::positron:
    sample_positron_reaction(p);
    break;
  }

  // Kill particle if energy falls below cutoff
  int type = static_cast<int>(p.type());
  if (p.E() < settings::energy_cutoff[type]) {
    p.alive() = false;
    p.wgt() = 0.0;
  }

  // Display information about collision
  if (settings::verbosity >= 10 || p.trace()) {
    std::string msg;
    if (p.event() == TallyEvent::KILL) {
      msg = fmt::format("    Killed. Energy = {} eV.", p.E());
    } else if (p.type() == ParticleType::neutron) {
      msg = fmt::format("    {} with {}. Energy = {} eV.",
        reaction_name(p.event_mt()), data::nuclides[p.event_nuclide()]->name_,
        p.E());
    } else if (p.type() == ParticleType::photon) {
      msg = fmt::format("    {} with {}. Energy = {} eV.",
        reaction_name(p.event_mt()),
        to_element(data::nuclides[p.event_nuclide()]->name_), p.E());
    } else {
      msg = fmt::format("    Disappeared. Energy = {} eV.", p.E());
    }
    write_message(msg, 1);
  }
}

void sample_neutron_reaction(Particle& p)
{
  // Sample a nuclide within the material
  int i_nuclide = sample_nuclide(p);

  // Save which nuclide particle had collision with
  p.event_nuclide() = i_nuclide;

  // Create fission bank sites. Note that while a fission reaction is sampled,
  // it never actually "happens", i.e. the weight of the particle does not
  // change when sampling fission sites. The following block handles all
  // absorption (including fission)

  const auto& nuc {data::nuclides[i_nuclide]};

  if (nuc->fissionable_) {
    auto& rx = sample_fission(i_nuclide, p);
    if (settings::run_mode == RunMode::EIGENVALUE) {
      create_fission_sites(p, i_nuclide, rx);
    } else if (settings::run_mode == RunMode::FIXED_SOURCE &&
      settings::create_fission_neutrons) {
      create_fission_sites(p, i_nuclide, rx);

      // Make sure particle population doesn't grow out of control for
      // subcritical multiplication problems.
      if (p.secondary_bank().size() >= 10000) {
        fatal_error("The secondary particle bank appears to be growing without "
        "bound. You are likely running a subcritical multiplication problem "
        "with k-effective close to or greater than one.");
      }
    }
  }

  // Create secondary photons
  if (settings::photon_transport) {
    sample_secondary_photons(p, i_nuclide);
  }

  // If survival biasing is being used, the following subroutine adjusts the
  // weight of the particle. Otherwise, it checks to see if absorption occurs

  if (p.neutron_xs(i_nuclide).absorption > 0.0) {
    absorption(p, i_nuclide);
  } else {
    p.wgt_absorb() = 0.0;
  }
  if (!p.alive())
    return;

  // Sample a scattering reaction and determine the secondary energy of the
  // exiting neutron
  scatter(p, i_nuclide);

  // Advance URR seed stream 'N' times after energy changes
  if (p.E() != p.E_last()) {
    p.stream() = STREAM_URR_PTABLE;
    advance_prn_seed(data::nuclides.size(), p.current_seed());
    p.stream() = STREAM_TRACKING;
  }

  // Play russian roulette if survival biasing is turned on
  if (settings::survival_biasing) {
    russian_roulette(p);
    if (!p.alive())
      return;
  }
}

void
create_fission_sites(Particle& p, int i_nuclide, const Reaction& rx)
{
  // If uniform fission source weighting is turned on, we increase or decrease
  // the expected number of fission sites produced
  double weight = settings::ufs_on ? ufs_get_weight(p) : 1.0;

  // Determine the expected number of neutrons produced
  double nu_t = p.wgt() / simulation::keff * weight *
                p.neutron_xs(i_nuclide).nu_fission /
                p.neutron_xs(i_nuclide).total;

  // Sample the number of neutrons produced
  int nu = static_cast<int>(nu_t);
  if (prn(p.current_seed()) <= (nu_t - nu)) ++nu;

  // Begin banking the source neutrons
  // First, if our bank is full then don't continue
  if (nu == 0) return;

  // Initialize the counter of delayed neutrons encountered for each delayed
  // group.
  double nu_d[MAX_DELAYED_GROUPS] = {0.};

  // Clear out particle's nu fission bank
  p.nu_bank().clear();

  p.fission() = true;
  int skipped = 0;

  // Determine whether to place fission sites into the shared fission bank
  // or the secondary particle bank.
  bool use_fission_bank = (settings::run_mode == RunMode::EIGENVALUE);

  for (int i = 0; i < nu; ++i) {
    // Initialize fission site object with particle data
    SourceSite site;
    site.r = p.r();
    site.particle = ParticleType::neutron;
    site.wgt = 1. / weight;
    site.parent_id = p.id();
    site.progeny_id = p.n_progeny()++;
    site.surf_id = 0;

    // Sample delayed group and angle/energy for fission reaction
    sample_fission_neutron(i_nuclide, rx, p.E(), &site, p.current_seed());

    // Store fission site in bank
    if (use_fission_bank) {
      int64_t idx = simulation::fission_bank.thread_safe_append(site);
      if (idx == -1) {
        warning("The shared fission bank is full. Additional fission sites created "
            "in this generation will not be banked.");
        skipped++;
        break;
      }
    } else {
      p.secondary_bank().push_back(site);
    }

    // Set the delayed group on the particle as well
    p.delayed_group() = site.delayed_group;

    // Increment the number of neutrons born delayed
    if (p.delayed_group() > 0) {
      nu_d[p.delayed_group() - 1]++;
    }

    // Write fission particles to nuBank
    p.nu_bank().emplace_back();
    NuBank* nu_bank_entry = &p.nu_bank().back();
    nu_bank_entry->wgt              = site.wgt;
    nu_bank_entry->E                = site.E;
    nu_bank_entry->delayed_group    = site.delayed_group;
  }

  // If shared fission bank was full, and no fissions could be added,
  // set the particle fission flag to false.
  if (nu == skipped) {
    p.fission() = false;
    return;
  }

  // If shared fission bank was full, but some fissions could be added,
  // reduce nu accordingly
  nu -= skipped;

  // Store the total weight banked for analog fission tallies
  p.n_bank() = nu;
  p.wgt_bank() = nu / weight;
  for (size_t d = 0; d < MAX_DELAYED_GROUPS; d++) {
    p.n_delayed_bank(d) = nu_d[d];
  }
}

void sample_photon_reaction(Particle& p)
{
  // Kill photon if below energy cutoff -- an extra check is made here because
  // photons with energy below the cutoff may have been produced by neutrons
  // reactions or atomic relaxation
  int photon = static_cast<int>(ParticleType::photon);
  if (p.E() < settings::energy_cutoff[photon]) {
    p.E() = 0.0;
    p.alive() = false;
    return;
  }

  // Sample element within material
  int i_element = sample_element(p);
  const auto& micro {p.photon_xs(i_element)};
  const auto& element {*data::elements[i_element]};

  // Calculate photon energy over electron rest mass equivalent
  double alpha = p.E() / MASS_ELECTRON_EV;

  // For tallying purposes, this routine might be called directly. In that
  // case, we need to sample a reaction via the cutoff variable
  double prob = 0.0;
  double cutoff = prn(p.current_seed()) * micro.total;

  // Coherent (Rayleigh) scattering
  prob += micro.coherent;
  if (prob > cutoff) {
    double mu = element.rayleigh_scatter(alpha, p.current_seed());
    p.u() = rotate_angle(p.u(), mu, nullptr, p.current_seed());
    p.event() = TallyEvent::SCATTER;
    p.event_mt() = COHERENT;
    return;
  }

  // Incoherent (Compton) scattering
  prob += micro.incoherent;
  if (prob > cutoff) {
    double alpha_out, mu;
    int i_shell;
    element.compton_scatter(alpha, true, &alpha_out, &mu, &i_shell, p.current_seed());

    // Determine binding energy of shell. The binding energy is 0.0 if
    // doppler broadening is not used.
    double e_b;
    if (i_shell == -1) {
      e_b = 0.0;
    } else {
      e_b = element.binding_energy_[i_shell];
    }

    // Create Compton electron
    double phi = uniform_distribution(0., 2.0*PI, p.current_seed());
    double E_electron = (alpha - alpha_out)*MASS_ELECTRON_EV - e_b;
    int electron = static_cast<int>(ParticleType::electron);
    if (E_electron >= settings::energy_cutoff[electron]) {
      double mu_electron = (alpha - alpha_out*mu)
        / std::sqrt(alpha*alpha + alpha_out*alpha_out - 2.0*alpha*alpha_out*mu);
      Direction u = rotate_angle(p.u(), mu_electron, &phi, p.current_seed());
      p.create_secondary(p.wgt(), u, E_electron, ParticleType::electron);
    }

    // TODO: Compton subshell data does not match atomic relaxation data
    // Allow electrons to fill orbital and produce auger electrons
    // and fluorescent photons
    if (i_shell >= 0) {
      const auto& shell = element.shells_[i_shell];
      element.atomic_relaxation(shell, p);
    }

    phi += PI;
    p.E() = alpha_out * MASS_ELECTRON_EV;
    p.u() = rotate_angle(p.u(), mu, &phi, p.current_seed());
    p.event() = TallyEvent::SCATTER;
    p.event_mt() = INCOHERENT;
    return;
  }

  // Photoelectric effect
  double prob_after = prob + micro.photoelectric;
  if (prob_after > cutoff) {
    for (const auto& shell : element.shells_) {
      // Get grid index and interpolation factor
      int i_grid = micro.index_grid;
      double f = micro.interp_factor;

      // Check threshold of reaction
      int i_start = shell.threshold;
      if (i_grid < i_start) continue;

      // Evaluation subshell photoionization cross section
      double xs = std::exp(shell.cross_section(i_grid - i_start) +
        f*(shell.cross_section(i_grid + 1 - i_start) -
        shell.cross_section(i_grid - i_start)));

      prob += xs;
      if (prob > cutoff) {
        double E_electron = p.E() - shell.binding_energy;

        // Sample mu using non-relativistic Sauter distribution.
        // See Eqns 3.19 and 3.20 in "Implementing a photon physics
        // model in Serpent 2" by Toni Kaltiaisenaho
        double mu;
        while (true) {
          double r = prn(p.current_seed());
          if (4.0*(1.0 - r)*r >= prn(p.current_seed())) {
            double rel_vel = std::sqrt(E_electron * (E_electron +
              2.0*MASS_ELECTRON_EV)) / (E_electron + MASS_ELECTRON_EV);
            mu = (2.0*r + rel_vel - 1.0) / (2.0*rel_vel*r - rel_vel + 1.0);
            break;
          }
        }

        double phi = uniform_distribution(0., 2.0*PI, p.current_seed());
        Direction u;
        u.x = mu;
        u.y = std::sqrt(1.0 - mu*mu)*std::cos(phi);
        u.z = std::sqrt(1.0 - mu*mu)*std::sin(phi);

        // Create secondary electron
        p.create_secondary(p.wgt(), u, E_electron, ParticleType::electron);

        // Allow electrons to fill orbital and produce auger electrons
        // and fluorescent photons
        element.atomic_relaxation(shell, p);
        p.event() = TallyEvent::ABSORB;
        p.event_mt() = 533 + shell.index_subshell;
        p.alive() = false;
        p.E() = 0.0;
        return;
      }
    }
  }
  prob = prob_after;

  // Pair production
  prob += micro.pair_production;
  if (prob > cutoff) {
    double E_electron, E_positron;
    double mu_electron, mu_positron;
    element.pair_production(alpha, &E_electron, &E_positron,
      &mu_electron, &mu_positron, p.current_seed());

    // Create secondary electron
    Direction u = rotate_angle(p.u(), mu_electron, nullptr, p.current_seed());
    p.create_secondary(p.wgt(), u, E_electron, ParticleType::electron);

    // Create secondary positron
    u = rotate_angle(p.u(), mu_positron, nullptr, p.current_seed());
    p.create_secondary(p.wgt(), u, E_positron, ParticleType::positron);

    p.event() = TallyEvent::ABSORB;
    p.event_mt() = PAIR_PROD;
    p.alive() = false;
    p.E() = 0.0;
  }
}

void sample_electron_reaction(Particle& p)
{
  // TODO: create reaction types

  if (settings::electron_treatment == ElectronTreatment::TTB) {
    double E_lost;
    thick_target_bremsstrahlung(p, &E_lost);
  }

  p.E() = 0.0;
  p.alive() = false;
  p.event() = TallyEvent::ABSORB;
}

void sample_positron_reaction(Particle& p)
{
  // TODO: create reaction types

  if (settings::electron_treatment == ElectronTreatment::TTB) {
    double E_lost;
    thick_target_bremsstrahlung(p, &E_lost);
  }

  // Sample angle isotropically
  Direction u = isotropic_direction(p.current_seed());

  // Create annihilation photon pair traveling in opposite directions
  p.create_secondary(p.wgt(), u, MASS_ELECTRON_EV, ParticleType::photon);
  p.create_secondary(p.wgt(), -u, MASS_ELECTRON_EV, ParticleType::photon);

  p.E() = 0.0;
  p.alive() = false;
  p.event() = TallyEvent::ABSORB;
}

int sample_nuclide(Particle& p)
{
  // Sample cumulative distribution function
  double cutoff = prn(p.current_seed()) * p.macro_xs().total;

  // Get pointers to nuclide/density arrays
  const auto& mat {model::materials[p.material()]};
  int n = mat->nuclide_.size();

  double prob = 0.0;
  for (int i = 0; i < n; ++i) {
    // Get atom density
    int i_nuclide = mat->nuclide_[i];
    double atom_density = mat->atom_density_[i];

    // Increment probability to compare to cutoff
    prob += atom_density * p.neutron_xs(i_nuclide).total;
    if (prob >= cutoff) return i_nuclide;
  }

  // If we reach here, no nuclide was sampled
  p.write_restart();
  throw std::runtime_error{"Did not sample any nuclide during collision."};
}

int sample_element(Particle& p)
{
  // Sample cumulative distribution function
  double cutoff = prn(p.current_seed()) * p.macro_xs().total;

  // Get pointers to elements, densities
  const auto& mat {model::materials[p.material()]};

  double prob = 0.0;
  for (int i = 0; i < mat->element_.size(); ++i) {
    // Find atom density
    int i_element = mat->element_[i];
    double atom_density = mat->atom_density_[i];

    // Determine microscopic cross section
    double sigma = atom_density * p.photon_xs(i_element).total;

    // Increment probability to compare to cutoff
    prob += sigma;
    if (prob > cutoff) {
      // Save which nuclide particle had collision with for tally purpose
      p.event_nuclide() = mat->nuclide_[i];

      return i_element;
    }
  }

  // If we made it here, no element was sampled
  p.write_restart();
  fatal_error("Did not sample any element during collision.");
}

Reaction& sample_fission(int i_nuclide, Particle& p)
{
  // Get pointer to nuclide
  const auto& nuc {data::nuclides[i_nuclide]};

  // If we're in the URR, by default use the first fission reaction. We also
  // default to the first reaction if we know that there are no partial fission
  // reactions
  if (p.neutron_xs(i_nuclide).use_ptable || !nuc->has_partial_fission_) {
    return *nuc->fission_rx_[0];
  }

  // Check to see if we are in a windowed multipole range.  WMP only supports
  // the first fission reaction.
  if (nuc->multipole_) {
    if (p.E() >= nuc->multipole_->E_min_ && p.E() <= nuc->multipole_->E_max_) {
      return *nuc->fission_rx_[0];
    }
  }

  // Get grid index and interpolatoin factor and sample fission cdf
  int i_temp = p.neutron_xs(i_nuclide).index_temp;
  int i_grid = p.neutron_xs(i_nuclide).index_grid;
  double f = p.neutron_xs(i_nuclide).interp_factor;
  double cutoff = prn(p.current_seed()) * p.neutron_xs(i_nuclide).fission;
  double prob = 0.0;

  // Loop through each partial fission reaction type
  for (auto& rx : nuc->fission_rx_) {
    // if energy is below threshold for this reaction, skip it
    int threshold = rx->xs_[i_temp].threshold;
    if (i_grid < threshold) continue;

    // add to cumulative probability
    prob += (1.0 - f) * rx->xs_[i_temp].value[i_grid - threshold]
            + f*rx->xs_[i_temp].value[i_grid - threshold + 1];

    // Create fission bank sites if fission occurs
    if (prob > cutoff) return *rx;
  }

  // If we reached here, no reaction was sampled
  throw std::runtime_error{"No fission reaction was sampled for " + nuc->name_};
}

void sample_photon_product(int i_nuclide, Particle& p, int* i_rx, int* i_product)
{
  // Get grid index and interpolation factor and sample photon production cdf
  int i_temp = p.neutron_xs(i_nuclide).index_temp;
  int i_grid = p.neutron_xs(i_nuclide).index_grid;
  double f = p.neutron_xs(i_nuclide).interp_factor;
  double cutoff = prn(p.current_seed()) * p.neutron_xs(i_nuclide).photon_prod;
  double prob = 0.0;

  // Loop through each reaction type
  const auto& nuc {data::nuclides[i_nuclide]};
  for (int i = 0; i < nuc->reactions_.size(); ++i) {
    const auto& rx = nuc->reactions_[i];
    int threshold = rx->xs_[i_temp].threshold;

    // if energy is below threshold for this reaction, skip it
    if (i_grid < threshold) continue;

    // Evaluate neutron cross section
    double xs = ((1.0 - f) * rx->xs_[i_temp].value[i_grid - threshold]
      + f*(rx->xs_[i_temp].value[i_grid - threshold + 1]));

    for (int j = 0; j < rx->products_.size(); ++j) {
      if (rx->products_[j].particle_ == ParticleType::photon) {
        // For fission, artificially increase the photon yield to account
        // for delayed photons
        double f = 1.0;
        if (settings::delayed_photon_scaling) {
          if (is_fission(rx->mt_)) {
            if (nuc->prompt_photons_ && nuc->delayed_photons_) {
              double energy_prompt = (*nuc->prompt_photons_)(p.E());
              double energy_delayed = (*nuc->delayed_photons_)(p.E());
              f = (energy_prompt + energy_delayed)/(energy_prompt);
            }
          }
        }

        // add to cumulative probability
        prob += f * (*rx->products_[j].yield_)(p.E()) * xs;

        *i_rx = i;
        *i_product = j;
        if (prob > cutoff) return;
      }
    }
  }
}

void absorption(Particle& p, int i_nuclide)
{
  if (settings::survival_biasing) {
    // Determine weight absorbed in survival biasing
    p.wgt_absorb() = p.wgt() * p.neutron_xs(i_nuclide).absorption /
                     p.neutron_xs(i_nuclide).total;

    // Adjust weight of particle by probability of absorption
    p.wgt() -= p.wgt_absorb();
    p.wgt_last() = p.wgt();

    // Score implicit absorption estimate of keff
    if (settings::run_mode == RunMode::EIGENVALUE) {
      p.keff_tally_absorption() += p.wgt_absorb() *
                                   p.neutron_xs(i_nuclide).nu_fission /
                                   p.neutron_xs(i_nuclide).absorption;
    }
  } else {
    // See if disappearance reaction happens
    if (p.neutron_xs(i_nuclide).absorption >
        prn(p.current_seed()) * p.neutron_xs(i_nuclide).total) {
      // Score absorption estimate of keff
      if (settings::run_mode == RunMode::EIGENVALUE) {
        p.keff_tally_absorption() += p.wgt() *
                                     p.neutron_xs(i_nuclide).nu_fission /
                                     p.neutron_xs(i_nuclide).absorption;
      }

      p.alive() = false;
      p.event() = TallyEvent::ABSORB;
      p.event_mt() = N_DISAPPEAR;
    }
  }
}

void scatter(Particle& p, int i_nuclide)
{
  // copy incoming direction
  Direction u_old {p.u()};

  // Get pointer to nuclide and grid index/interpolation factor
  const auto& nuc {data::nuclides[i_nuclide]};
  const auto& micro {p.neutron_xs(i_nuclide)};
  int i_temp =  micro.index_temp;
  int i_grid =  micro.index_grid;
  double f = micro.interp_factor;

  // For tallying purposes, this routine might be called directly. In that
  // case, we need to sample a reaction via the cutoff variable
  double cutoff = prn(p.current_seed()) * (micro.total - micro.absorption);
  bool sampled = false;

  // Calculate elastic cross section if it wasn't precalculated
  if (micro.elastic == CACHE_INVALID) {
    nuc->calculate_elastic_xs(p);
  }

  double prob = micro.elastic - micro.thermal;
  if (prob > cutoff) {
    // =======================================================================
    // NON-S(A,B) ELASTIC SCATTERING

    // Determine temperature
    double kT = nuc->multipole_ ? p.sqrtkT() * p.sqrtkT() : nuc->kTs_[i_temp];

    // Perform collision physics for elastic scattering
    elastic_scatter(i_nuclide, *nuc->reactions_[0], kT, p);

    p.event_mt() = ELASTIC;
    sampled = true;
  }

  prob = micro.elastic;
  if (prob > cutoff && !sampled) {
    // =======================================================================
    // S(A,B) SCATTERING

    sab_scatter(i_nuclide, micro.index_sab, p);

    p.event_mt() = ELASTIC;
    sampled = true;
  }

  if (!sampled) {
    // =======================================================================
    // INELASTIC SCATTERING

    int j = 0;
    int i = 0;
    while (prob < cutoff) {
      i = nuc->index_inelastic_scatter_[j];
      ++j;

      // Check to make sure inelastic scattering reaction sampled
      if (i >= nuc->reactions_.size()) {
        p.write_restart();
        fatal_error("Did not sample any reaction for nuclide " + nuc->name_);
      }

      // if energy is below threshold for this reaction, skip it
      const auto& xs {nuc->reactions_[i]->xs_[i_temp]};
      if (i_grid < xs.threshold) continue;

      // add to cumulative probability
      prob += (1.0 - f)*xs.value[i_grid - xs.threshold] +
        f*xs.value[i_grid - xs.threshold + 1];
    }

    // Perform collision physics for inelastic scattering
    const auto& rx {nuc->reactions_[i]};
    inelastic_scatter(*nuc, *rx, p);
    p.event_mt() = rx->mt_;
  }

  // Set event component
  p.event() = TallyEvent::SCATTER;

  // Sample new outgoing angle for isotropic-in-lab scattering
  const auto& mat {model::materials[p.material()]};
  if (!mat->p0_.empty()) {
    int i_nuc_mat = mat->mat_nuclide_index_[i_nuclide];
    if (mat->p0_[i_nuc_mat]) {
      // Sample isotropic-in-lab outgoing direction
      p.u() = isotropic_direction(p.current_seed());
      p.mu() = u_old.dot(p.u());
    }
  }
}

void elastic_scatter(int i_nuclide, const Reaction& rx, double kT,
  Particle& p)
{
  // get pointer to nuclide
  const auto& nuc {data::nuclides[i_nuclide]};

  double vel = std::sqrt(p.E());
  double awr = nuc->awr_;

  // Neutron velocity in LAB
  Direction v_n = vel*p.u();

  // Sample velocity of target nucleus
  Direction v_t {};
  if (!p.neutron_xs(i_nuclide).use_ptable) {
    v_t = sample_target_velocity(*nuc, p.E(), p.u(), v_n,
      p.neutron_xs(i_nuclide).elastic, kT, p.current_seed());
  }

  // Velocity of center-of-mass
  Direction v_cm = (v_n + awr*v_t)/(awr + 1.0);

  // Transform to CM frame
  v_n -= v_cm;

  // Find speed of neutron in CM
  vel = v_n.norm();

  // Sample scattering angle, checking if angle distribution is present (assume
  // isotropic otherwise)
  double mu_cm;
  auto& d = rx.products_[0].distribution_[0];
  auto d_ = dynamic_cast<UncorrelatedAngleEnergy*>(d.get());
  if (!d_->angle().empty()) {
    mu_cm = d_->angle().sample(p.E(), p.current_seed());
  } else {
    mu_cm = uniform_distribution(-1., 1., p.current_seed());
  }

  // Determine direction cosines in CM
  Direction u_cm = v_n/vel;

  // Rotate neutron velocity vector to new angle -- note that the speed of the
  // neutron in CM does not change in elastic scattering. However, the speed
  // will change when we convert back to LAB
  v_n = vel * rotate_angle(u_cm, mu_cm, nullptr, p.current_seed());

  // Transform back to LAB frame
  v_n += v_cm;

  p.E() = v_n.dot(v_n);
  vel = std::sqrt(p.E());

  // compute cosine of scattering angle in LAB frame by taking dot product of
  // neutron's pre- and post-collision angle
  p.mu() = p.u().dot(v_n) / vel;

  // Set energy and direction of particle in LAB frame
  p.u() = v_n / vel;

  // Because of floating-point roundoff, it may be possible for mu_lab to be
  // outside of the range [-1,1). In these cases, we just set mu_lab to exactly
  // -1 or 1
  if (std::abs(p.mu()) > 1.0)
    p.mu() = std::copysign(1.0, p.mu());
}

void sab_scatter(int i_nuclide, int i_sab, Particle& p)
{
  // Determine temperature index
  const auto& micro {p.neutron_xs(i_nuclide)};
  int i_temp = micro.index_temp_sab;

  // Sample energy and angle
  double E_out;
  data::thermal_scatt[i_sab]->data_[i_temp].sample(
    micro, p.E(), &E_out, &p.mu(), p.current_seed());

  // Set energy to outgoing, change direction of particle
  p.E() = E_out;
  p.u() = rotate_angle(p.u(), p.mu(), nullptr, p.current_seed());
}

Direction sample_target_velocity(const Nuclide& nuc, double E, Direction u,
  Direction v_neut, double xs_eff, double kT, uint64_t* seed)
{
  // check if nuclide is a resonant scatterer
  ResScatMethod sampling_method;
  if (nuc.resonant_) {

    // sampling method to use
    sampling_method = settings::res_scat_method;

    // upper resonance scattering energy bound (target is at rest above this E)
    if (E > settings::res_scat_energy_max) {
      return {};

    // lower resonance scattering energy bound (should be no resonances below)
    } else if (E < settings::res_scat_energy_min) {
      sampling_method = ResScatMethod::cxs;
    }

  // otherwise, use free gas model
  } else {
    if (E >= FREE_GAS_THRESHOLD * kT && nuc.awr_ > 1.0) {
      return {};
    } else {
      sampling_method = ResScatMethod::cxs;
    }
  }

  // use appropriate target velocity sampling method
  switch (sampling_method) {
  case ResScatMethod::cxs:

    // sample target velocity with the constant cross section (cxs) approx.
    return sample_cxs_target_velocity(nuc.awr_, E, u, kT, seed);

  case ResScatMethod::dbrc:
  case ResScatMethod::rvs: {
    double E_red = std::sqrt(nuc.awr_ * E / kT);
    double E_low = std::pow(std::max(0.0, E_red - 4.0), 2) * kT / nuc.awr_;
    double E_up = (E_red + 4.0)*(E_red + 4.0) * kT / nuc.awr_;

    // find lower and upper energy bound indices
    // lower index
    int i_E_low;
    if (E_low < nuc.energy_0K_.front()) {
      i_E_low = 0;
    } else if (E_low > nuc.energy_0K_.back()) {
      i_E_low = nuc.energy_0K_.size() - 2;
    } else {
      i_E_low = lower_bound_index(nuc.energy_0K_.begin(),
        nuc.energy_0K_.end(), E_low);
    }

    // upper index
    int i_E_up;
    if (E_up < nuc.energy_0K_.front()) {
      i_E_up = 0;
    } else if (E_up > nuc.energy_0K_.back()) {
      i_E_up = nuc.energy_0K_.size() - 2;
    } else {
      i_E_up = lower_bound_index(nuc.energy_0K_.begin(),
        nuc.energy_0K_.end(), E_up);
    }

    if (i_E_up == i_E_low) {
      // Handle degenerate case -- if the upper/lower bounds occur for the same
      // index, then using cxs is probably a good approximation
      return sample_cxs_target_velocity(nuc.awr_, E, u, kT, seed);
    }

    if (sampling_method == ResScatMethod::dbrc) {
      // interpolate xs since we're not exactly at the energy indices
      double xs_low = nuc.elastic_0K_[i_E_low];
      double m = (nuc.elastic_0K_[i_E_low + 1] - xs_low)
        / (nuc.energy_0K_[i_E_low + 1] - nuc.energy_0K_[i_E_low]);
      xs_low += m * (E_low - nuc.energy_0K_[i_E_low]);
      double xs_up = nuc.elastic_0K_[i_E_up];
      m = (nuc.elastic_0K_[i_E_up + 1] - xs_up)
        / (nuc.energy_0K_[i_E_up + 1] - nuc.energy_0K_[i_E_up]);
      xs_up += m * (E_up - nuc.energy_0K_[i_E_up]);

      // get max 0K xs value over range of practical relative energies
      double xs_max = *std::max_element(&nuc.elastic_0K_[i_E_low + 1],
        &nuc.elastic_0K_[i_E_up + 1]);
      xs_max = std::max({xs_low, xs_max, xs_up});

      while (true) {
        double E_rel;
        Direction v_target;
        while (true) {
          // sample target velocity with the constant cross section (cxs) approx.
          v_target = sample_cxs_target_velocity(nuc.awr_, E, u, kT, seed);
          Direction v_rel = v_neut - v_target;
          E_rel = v_rel.dot(v_rel);
          if (E_rel < E_up) break;
        }

        // perform Doppler broadening rejection correction (dbrc)
        double xs_0K = nuc.elastic_xs_0K(E_rel);
        double R = xs_0K / xs_max;
        if (prn(seed) < R) return v_target;
      }

    } else if (sampling_method == ResScatMethod::rvs) {
      // interpolate xs CDF since we're not exactly at the energy indices
      // cdf value at lower bound attainable energy
      double m = (nuc.xs_cdf_[i_E_low] - nuc.xs_cdf_[i_E_low - 1])
        / (nuc.energy_0K_[i_E_low + 1] - nuc.energy_0K_[i_E_low]);
      double cdf_low = nuc.xs_cdf_[i_E_low - 1]
            + m * (E_low - nuc.energy_0K_[i_E_low]);
      if (E_low <= nuc.energy_0K_.front()) cdf_low = 0.0;

      // cdf value at upper bound attainable energy
      m = (nuc.xs_cdf_[i_E_up] - nuc.xs_cdf_[i_E_up - 1])
        / (nuc.energy_0K_[i_E_up + 1] - nuc.energy_0K_[i_E_up]);
      double cdf_up = nuc.xs_cdf_[i_E_up - 1]
        + m*(E_up - nuc.energy_0K_[i_E_up]);

      while (true) {
        // directly sample Maxwellian
        double E_t = -kT * std::log(prn(seed));

        // sample a relative energy using the xs cdf
        double cdf_rel = cdf_low + prn(seed)*(cdf_up - cdf_low);
        int i_E_rel = lower_bound_index(&nuc.xs_cdf_[i_E_low-1],
          &nuc.xs_cdf_[i_E_up+1], cdf_rel);
        double E_rel = nuc.energy_0K_[i_E_low + i_E_rel];
        double m = (nuc.xs_cdf_[i_E_low + i_E_rel]
              - nuc.xs_cdf_[i_E_low + i_E_rel - 1])
              / (nuc.energy_0K_[i_E_low + i_E_rel + 1]
              -  nuc.energy_0K_[i_E_low + i_E_rel]);
        E_rel += (cdf_rel - nuc.xs_cdf_[i_E_low + i_E_rel - 1]) / m;

        // perform rejection sampling on cosine between
        // neutron and target velocities
        double mu = (E_t + nuc.awr_ * (E - E_rel)) /
          (2.0 * std::sqrt(nuc.awr_ * E * E_t));

        if (std::abs(mu) < 1.0) {
          // set and accept target velocity
          E_t /= nuc.awr_;
          return std::sqrt(E_t) * rotate_angle(u, mu, nullptr, seed);
        }
      }
    }
    } // case RVS, DBRC
  } // switch (sampling_method)

  UNREACHABLE();
}

Direction
sample_cxs_target_velocity(double awr, double E, Direction u, double kT, uint64_t* seed)
{
  double beta_vn = std::sqrt(awr * E / kT);
  double alpha = 1.0/(1.0 + std::sqrt(PI)*beta_vn/2.0);

  double beta_vt_sq;
  double mu;
  while (true) {
    // Sample two random numbers
    double r1 = prn(seed);
    double r2 = prn(seed);

    if (prn(seed) < alpha) {
      // With probability alpha, we sample the distribution p(y) =
      // y*e^(-y). This can be done with sampling scheme C45 from the Monte
      // Carlo sampler

      beta_vt_sq = -std::log(r1*r2);

    } else {
      // With probability 1-alpha, we sample the distribution p(y) = y^2 *
      // e^(-y^2). This can be done with sampling scheme C61 from the Monte
      // Carlo sampler

      double c = std::cos(PI/2.0 * prn(seed));
      beta_vt_sq = -std::log(r1) - std::log(r2)*c*c;
    }

    // Determine beta * vt
    double beta_vt = std::sqrt(beta_vt_sq);

    // Sample cosine of angle between neutron and target velocity
    mu = uniform_distribution(-1., 1., seed);

    // Determine rejection probability
    double accept_prob = std::sqrt(beta_vn*beta_vn + beta_vt_sq -
      2*beta_vn*beta_vt*mu) / (beta_vn + beta_vt);

    // Perform rejection sampling on vt and mu
    if (prn(seed) < accept_prob) break;
  }

  // Determine speed of target nucleus
  double vt = std::sqrt(beta_vt_sq*kT/awr);

  // Determine velocity vector of target nucleus based on neutron's velocity
  // and the sampled angle between them
  return vt * rotate_angle(u, mu, nullptr, seed);
}

void sample_fission_neutron(int i_nuclide, const Reaction& rx, double E_in,
  SourceSite* site, uint64_t* seed)
{
  // Determine total nu, delayed nu, and delayed neutron fraction
  const auto& nuc {data::nuclides[i_nuclide]};
  double nu_t = nuc->nu(E_in, Nuclide::EmissionMode::total);
  double nu_d = nuc->nu(E_in, Nuclide::EmissionMode::delayed);
  double beta = nu_d / nu_t;

  if (prn(seed) < beta) {
    // ====================================================================
    // DELAYED NEUTRON SAMPLED

    // sampled delayed precursor group
    double xi = prn(seed)*nu_d;
    double prob = 0.0;
    int group;
    for (group = 1; group < nuc->n_precursor_; ++group) {
      // determine delayed neutron precursor yield for group j
      double yield = (*rx.products_[group].yield_)(E_in);

      // Check if this group is sampled
      prob += yield;
      if (xi < prob) break;
    }

    // if the sum of the probabilities is slightly less than one and the
    // random number is greater, j will be greater than nuc %
    // n_precursor -- check for this condition
    group = std::min(group, nuc->n_precursor_);

    // set the delayed group for the particle born from fission
    site->delayed_group = group;

  } else {
    // ====================================================================
    // PROMPT NEUTRON SAMPLED

    // set the delayed group for the particle born from fission to 0
    site->delayed_group = 0;
  }

  // sample from prompt neutron energy distribution
  int n_sample = 0;
  double mu;
  while (true) {
    rx.products_[site->delayed_group].sample(E_in, site->E, mu, seed);

    // resample if energy is greater than maximum neutron energy
    constexpr int neutron = static_cast<int>(ParticleType::neutron);
    if (site->E < data::energy_max[neutron]) break;

    // check for large number of resamples
    ++n_sample;
    if (n_sample == MAX_SAMPLE) {
      // particle_write_restart(p)
      fatal_error("Resampled energy distribution maximum number of times "
        "for nuclide " + nuc->name_);
    }
  }

  // Sample azimuthal angle uniformly in [0, 2*pi) and assign angle
  // TODO: account for dependence on incident neutron?
  Direction ref(1., 0., 0.);
  site->u = rotate_angle(ref, mu, nullptr, seed);
}

void inelastic_scatter(const Nuclide& nuc, const Reaction& rx, Particle& p)
{
  // copy energy of neutron
  double E_in = p.E();

  // sample outgoing energy and scattering cosine
  double E;
  double mu;
  rx.products_[0].sample(E_in, E, mu, p.current_seed());

  // if scattering system is in center-of-mass, transfer cosine of scattering
  // angle and outgoing energy from CM to LAB
  if (rx.scatter_in_cm_) {
    double E_cm = E;

    // determine outgoing energy in lab
    double A = nuc.awr_;
    E = E_cm + (E_in + 2.0*mu*(A + 1.0) * std::sqrt(E_in*E_cm))
          / ((A + 1.0)*(A + 1.0));

    // determine outgoing angle in lab
    mu = mu*std::sqrt(E_cm/E) + 1.0/(A+1.0) * std::sqrt(E_in/E);
  }

  // Because of floating-point roundoff, it may be possible for mu to be
  // outside of the range [-1,1). In these cases, we just set mu to exactly -1
  // or 1
  if (std::abs(mu) > 1.0) mu = std::copysign(1.0, mu);

  // Set outgoing energy and scattering angle
  p.E() = E;
  p.mu() = mu;

  // change direction of particle
  p.u() = rotate_angle(p.u(), mu, nullptr, p.current_seed());

  // evaluate yield
  double yield = (*rx.products_[0].yield_)(E_in);
  if (std::floor(yield) == yield) {
    // If yield is integral, create exactly that many secondary particles
    for (int i = 0; i < static_cast<int>(std::round(yield)) - 1; ++i) {
      p.create_secondary(p.wgt(), p.u(), p.E(), ParticleType::neutron);
    }
  } else {
    // Otherwise, change weight of particle based on yield
    p.wgt() *= yield;
  }
}

void sample_secondary_photons(Particle& p, int i_nuclide)
{
  // Sample the number of photons produced
  double y_t =
    p.neutron_xs(i_nuclide).photon_prod / p.neutron_xs(i_nuclide).total;
  int y = static_cast<int>(y_t);
  if (prn(p.current_seed()) <= y_t - y) ++y;

  // Sample each secondary photon
  for (int i = 0; i < y; ++i) {
    // Sample the reaction and product
    int i_rx;
    int i_product;
    sample_photon_product(i_nuclide, p, &i_rx, &i_product);

    // Sample the outgoing energy and angle
    auto& rx = data::nuclides[i_nuclide]->reactions_[i_rx];
    double E;
    double mu;
    rx->products_[i_product].sample(p.E(), E, mu, p.current_seed());

    // Sample the new direction
    Direction u = rotate_angle(p.u(), mu, nullptr, p.current_seed());

    // In a k-eigenvalue simulation, it's necessary to provide higher weight to
    // secondary photons from non-fission reactions to properly balance energy
    // release and deposition. See D. P. Griesheimer, S. J. Douglass, and M. H.
    // Stedry, "Self-consistent energy normalization for quasistatic reactor
    // calculations", Proc. PHYSOR, Cambridge, UK, Mar 29-Apr 2, 2020.
    double wgt;
    if (settings::run_mode == RunMode::EIGENVALUE && !is_fission(rx->mt_)) {
      wgt = simulation::keff * p.wgt();
    } else {
      wgt = p.wgt();
    }

    // Create the secondary photon
    p.create_secondary(wgt, u, E, ParticleType::photon);
  }
}

} // namespace openmc