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Address @paulromano comments on #1085
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@ -85,26 +85,26 @@ momentum transfer is traditionally expressed as
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.. math::
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:label: momentum-transfer
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x = a \kappa \sqrt{1 - \mu}
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x = a k \sqrt{1 - \mu}
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where :math:`\kappa` is the ratio of the photon energy to the electron rest
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where :math:`k` is the ratio of the photon energy to the electron rest
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mass, and the coefficient :math:`a` can be shown to be
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.. math::
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:label: omega
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a = \frac{m_e c^2}{\sqrt{2}hc} \approx 29.14329,
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a = \frac{m_e c^2}{\sqrt{2}hc} \approx 29.14329~\unicode{x212B},
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where :math:`m_e` is the mass of the electron, :math:`c` is the speed of light
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in a vacuum, and :math:`h` is Planck's constant. Using :eq:`momentum-transfer`,
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we have :math:`\mu = 1 - [x/(a\kappa)]^2` and :math:`d\mu/dx^2 =
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-1/(a\kappa)^2`. The probability density in :math:`x^2` is
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we have :math:`\mu = 1 - [x/(ak)]^2` and :math:`d\mu/dx^2 =
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-1/(ak)^2`. The probability density in :math:`x^2` is
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.. math::
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:label: coherent-pdf-x2
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p(x^2) dx^2 = p(\mu) \left | \frac{d\mu}{dx^2} \right | dx^2 = \frac{2\pi
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r_e^2 A(\bar{x}^2,Z)}{(a\kappa)^2 \sigma(E)} \left (
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r_e^2 A(\bar{x}^2,Z)}{(ak)^2 \sigma(E)} \left (
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\frac{1 + \mu^2}{2} \right ) \left ( \frac{F(x, Z)^2}{A(\bar{x}^2, Z)} \right ) dx^2
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where :math:`\bar{x}` is the maximum value of :math:`x` that occurs for
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@ -113,7 +113,7 @@ where :math:`\bar{x}` is the maximum value of :math:`x` that occurs for
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.. math::
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:label: xmax
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\bar{x} = a \kappa \sqrt{2} = \frac{m_e c^2}{hc} \kappa,
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\bar{x} = a k \sqrt{2} = \frac{m_e c^2}{hc} k,
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and :math:`A(x^2, Z)` is the integral of the square of the form factor:
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@ -168,12 +168,11 @@ the two authors who discovered it:
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.. math::
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:label: klein-nishina
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\frac{d\sigma_{KN}}{d\mu} = \pi r_e^2 \left ( \frac{\kappa'}{\kappa} \right
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)^2 \left [ \frac{\kappa'}{\kappa} + \frac{\kappa}{\kappa'} + \mu^2 - 1
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\right ]
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\frac{d\sigma_{KN}}{d\mu} = \pi r_e^2 \left ( \frac{k'}{k} \right)^2 \left
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[ \frac{k'}{k} + \frac{k}{k'} + \mu^2 - 1 \right ]
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where :math:`\kappa` and :math:`\kappa'` are the ratios of the incoming and
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exiting photon energies to the electron rest mass energy equivalent (0.511 MeV),
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where :math:`k` and :math:`k'` are the ratios of the incoming and exiting
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photon energies to the electron rest mass energy equivalent (0.511 MeV),
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respectively. Although it appears that the outgoing energy and angle are
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separate, there is actually a one-to-one relationship between them such that
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only one needs to be sampled:
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@ -181,32 +180,31 @@ only one needs to be sampled:
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.. math::
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:label: compton-energy-angle
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\kappa' = \frac{\kappa}{1 + \kappa(1 - \mu)}.
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k' = \frac{k}{1 + k(1 - \mu)}.
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Note that when :math:`\kappa'/\kappa` goes to one, i.e., scattering is elastic,
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the Klein-Nishina cross section becomes identical to the Thomson cross
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section. In general though, the scattering is inelastic and is known as Compton
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scattering. When a photon interacts with a bound electron in an atom, the
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Klein-Nishina formula must be modified to account for the binding effects. As in
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the case of coherent scattering, this is done by means of a form factor. The
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differential cross section for incoherent scattering is given by
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Note that when :math:`k'/k` goes to one, i.e., scattering is elastic, the
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Klein-Nishina cross section becomes identical to the Thomson cross section. In
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general though, the scattering is inelastic and is known as Compton scattering.
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When a photon interacts with a bound electron in an atom, the Klein-Nishina
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formula must be modified to account for the binding effects. As in the case of
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coherent scattering, this is done by means of a form factor. The differential
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cross section for incoherent scattering is given by
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.. math::
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:label: incoherent-xs
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\frac{d\sigma}{d\mu} = \frac{d\sigma_{KN}}{d\mu} S(x,Z) = \pi r_e^2 \left (
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\frac{\kappa'}{\kappa} \right )^2 \left [ \frac{\kappa'}{\kappa} +
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\frac{\kappa}{\kappa'} + \mu^2 - 1 \right ] S(x,Z)
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\frac{k'}{k} \right )^2 \left [ \frac{k'}{k} + \frac{k}{k'} + \mu^2 - 1
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\right ] S(x,Z)
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where :math:`S(x,Z)` is the form factor. The approach in OpenMC is to first
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sample the Klein-Nishina cross section and then perform rejection sampling on
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the form factor. As in other codes, `Kahn's rejection method`_ is used for
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:math:`\kappa < 3` and a direct method by Koblinger_ is used for :math:`\kappa
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\ge 3`. The complete algorithm is as follows:
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:math:`k < 3` and a direct method by Koblinger_ is used for :math:`k \ge 3`.
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The complete algorithm is as follows:
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1. If :math:`\kappa < 3`, sample :math:`\mu` from the Klein-Nishina cross
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section using Kahn's rejection method. Otherwise, use Koblinger's direct
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method.
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1. If :math:`k < 3`, sample :math:`\mu` from the Klein-Nishina cross section
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using Kahn's rejection method. Otherwise, use Koblinger's direct method.
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2. Calculate :math:`x` and :math:`\bar{x}` using :eq:`momentum-transfer` and
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:eq:`xmax`, respectively.
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@ -243,7 +241,8 @@ scattering angle:
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.. math::
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:label: pz
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p_z = \frac{E - E' - EE'(1 - \mu)/(m_e c^2)}{-\alpha \sqrt{E^2 + E'^2 - 2EE'\mu}},
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p_z = \frac{E - E' - EE'(1 - \mu)/(m_e c^2)}{-\alpha \sqrt{E^2 + E'^2 -
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2EE'\mu}},
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where :math:`\alpha` is the fine structure constant. The maximum momentum
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transferred, :math:`p_{z,\text{max}}`, can be calculated from :eq:`pz` using
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@ -269,7 +268,7 @@ The sampling algorithm is summarized below:
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:ref:`incoherent-sampling`.
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2. Sample the electron subshell :math:`i` using the number of electrons per
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shell as the PDF.
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shell as the probability mass function.
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3. Sample :math:`p_z` using :math:`J_i(p_z)` as the PDF.
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@ -325,7 +324,7 @@ heavier elements.
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When simulating the photoelectric effect, the first step is to sample the
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electron shell. The shell :math:`i` where the ionization occurs can be
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considered a discrete random variable with probability density function
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considered a discrete random variable with probability mass function
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.. math::
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:label: photoelectron-shell-pdf
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@ -339,7 +338,7 @@ shell has been sampled, the energy of the photoelectron is calculated using
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:eq:`photoelectron-kinetic-energy`.
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To determine the direction of the photoelectron, we implement the method
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described in [Kaltiaisenaho]_, which models the angular distribution of the
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described in Kaltiaisenaho_, which models the angular distribution of the
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photoelectrons using the K-shell cross section derived by Sauter (K-shell
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electrons are the most tightly bound, and they contribute the most to
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:math:`\sigma_{\text{pe}}`). The non-relativistic Sauter distribution for
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@ -372,10 +371,11 @@ where
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.. math::
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:label: mu-pdf-factors
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\psi(\mu_{-}) &= \frac{(1 - \beta_{-}^2)(1 - \mu_{-}^2)}{(1 - \beta_{-}\mu_{-})^2}, \\
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\psi(\mu_{-}) &= \frac{(1 - \beta_{-}^2)(1 - \mu_{-}^2)}{(1 -
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\beta_{-}\mu_{-})^2}, \\
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g(\mu_{-}) &= \frac{1 - \beta_{-}^2}{2 (1 - \beta_{-}\mu_{-})^2}.
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In the interval :math:`(-1, 1)`, :math:`g(\mu_{-})` is a normalized PDF and
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In the interval :math:`[-1, 1]`, :math:`g(\mu_{-})` is a normalized PDF and
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:math:`\psi(\mu_{-})` satisfies the condition :math:`0 < \psi(\mu_{-}) < 1`.
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The following algorithm can now be used to sample :math:`\mu_{-}`:
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@ -426,7 +426,7 @@ Accurately modeling the creation of electron-positron pair is important because
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the charged particles can go on to lose much of their energy as bremsstrahlung
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radiation, and the subsequent annihilation of the positron with an electron
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produces two additional photons. We sample the energy and direction of the
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charged particles using a semiempirical model described in [Salvat]_. The
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charged particles using a semiempirical model described in Salvat_. The
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Bethe-Heitler differential cross section, given by
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.. math::
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@ -441,9 +441,9 @@ constant, :math:`f_C` is the Coulomb correction function, :math:`\Phi_1` and
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:math:`\Phi_2` are screening functions, and :math:`\epsilon = (E_{-} + m_e
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c^2)/E` is the electron reduced energy (i.e., the fraction of the photon energy
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given to the electron). :math:`\epsilon` can take values between
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:math:`\epsilon_{\text{min}} = \kappa^{-1}` (when the kinetic energy of the
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electron is zero) and :math:`\epsilon_{\text{max}} = 1 - \kappa^{-1}` (when the
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kinetic energy of the positron is zero).
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:math:`\epsilon_{\text{min}} = k^{-1}` (when the kinetic energy of the electron
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is zero) and :math:`\epsilon_{\text{max}} = 1 - k^{-1}` (when the kinetic
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energy of the positron is zero).
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The Coulomb correction, given by
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@ -478,7 +478,7 @@ where
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.. math::
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:label: b
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b = \frac{Rm_{e}c}{2\kappa\epsilon(1 - \epsilon)\hbar}.
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b = \frac{Rm_{e}c}{2k\epsilon(1 - \epsilon)\hbar}.
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and :math:`R` is the screening radius.
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@ -487,15 +487,15 @@ described above will not be accurate at low energies: the lower boundary of
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:math:`\epsilon` will be shifted above :math:`\epsilon_{\text{min}}` and the
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upper boundary of :math:`\epsilon` will be shifted below
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:math:`\epsilon_{\text{max}}`. To offset this behavior, a correcting factor
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:math:`F_0(\kappa, Z)` is used:
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:math:`F_0(k, Z)` is used:
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.. math::
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:label: correcting-factor
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F_0(\kappa, Z) =~& (0.1774 + 12.10\alpha Z - 11.18\alpha^{2}Z^{2})(2/\kappa)^{1/2} \\
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&+ (8.523 + 73.26\alpha Z - 44.41\alpha^{2}Z^{2})(2/\kappa) \\
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&- (13.52 + 121.1\alpha Z - 96.41\alpha^{2}Z^{2})(2/\kappa)^{3/2} \\
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&+ (8.946 + 62.05\alpha Z - 63.41\alpha^{2}Z^{2})(2/\kappa)^{2}.
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F_0(k, Z) =~& (0.1774 + 12.10\alpha Z - 11.18\alpha^{2}Z^{2})(2/k)^{1/2} \\
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&+ (8.523 + 73.26\alpha Z - 44.41\alpha^{2}Z^{2})(2/k) \\
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&- (13.52 + 121.1\alpha Z - 96.41\alpha^{2}Z^{2})(2/k)^{3/2} \\
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&+ (8.946 + 62.05\alpha Z - 63.41\alpha^{2}Z^{2})(2/k)^{2}.
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To aid sampling, the differential cross section used to sample :math:`\epsilon`
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(minus the normalization constant) can now be expressed in the form
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@ -512,23 +512,23 @@ where
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.. math::
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:label: u
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u_1 &= \frac{2}{3} \left(\frac{1}{2} - \frac{1}{\kappa}\right)^2 \phi_1(1/2), \\
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u_1 &= \frac{2}{3} \left(\frac{1}{2} - \frac{1}{k}\right)^2 \phi_1(1/2), \\
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u_2 &= \phi_2(1/2),
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.. math::
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:label: phi
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\phi_1(\epsilon) &= \frac{1}{2}(3\Phi_1 - \Phi_2) - 4f_{C}(Z) + F_0(\kappa, Z), \\
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\phi_2(\epsilon) &= \frac{1}{4}(3\Phi_1 + \Phi_2) - 4f_{C}(Z) + F_0(\kappa, Z),
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\phi_1(\epsilon) &= \frac{1}{2}(3\Phi_1 - \Phi_2) - 4f_{C}(Z) + F_0(k, Z), \\
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\phi_2(\epsilon) &= \frac{1}{4}(3\Phi_1 + \Phi_2) - 4f_{C}(Z) + F_0(k, Z),
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and
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.. math::
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:label: pi
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\pi_1(\epsilon) &= \frac{3}{2} \left(\frac{1}{2} - \frac{1}{\kappa}\right)^{-3}
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\pi_1(\epsilon) &= \frac{3}{2} \left(\frac{1}{2} - \frac{1}{k}\right)^{-3}
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\left(\frac{1}{2} - \epsilon\right)^2, \\
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\pi_2(\epsilon) &= \frac{1}{2} \left(\frac{1}{2} - \frac{1}{\kappa}\right)^{-1}.
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\pi_2(\epsilon) &= \frac{1}{2} \left(\frac{1}{2} - \frac{1}{k}\right)^{-1}.
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The functions in :eq:`phi` are non-negative and maximum at :math:`\epsilon =
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1/2`. In the interval :math:`(\epsilon_{\text{min}}, \epsilon_{\text{max}})`,
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@ -545,10 +545,10 @@ sample the reduced electron energy :math:`\epsilon`:
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.. math::
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\epsilon &= \frac{1}{2} + \left(\frac{1}{2} - \frac{1}{\kappa}\right)
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\epsilon &= \frac{1}{2} + \left(\frac{1}{2} - \frac{1}{k}\right)
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(2\xi_1 - 1)^{1/3} ~~~~&\text{if}~~ i = 1 \\
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\epsilon &= \frac{1}{\kappa} + \left(\frac{1}{2} -
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\frac{1}{\kappa}\right) 2\xi_1 ~~~~&\text{if}~~ i = 2.
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\epsilon &= \frac{1}{k} + \left(\frac{1}{2} -
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\frac{1}{k}\right) 2\xi_1 ~~~~&\text{if}~~ i = 2.
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3. If :math:`\xi_2 \le \phi_i(\epsilon)/\phi_i(1/2)`, accept
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:math:`\epsilon`. Otherwise, repeat the sampling from step 1.
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@ -563,13 +563,14 @@ distribution,
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.. math::
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:label: sauter–gluckstern–hull
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\frac{d\sigma_{pp}}{d\Omega_{\pm}} = C(1 - \beta_{\pm}\mu_{\pm})^{-2},
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p(\mu_{\pm}) = C(1 - \beta_{\pm}\mu_{\pm})^{-2},
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where :math:`C` is a normalization constant and :math:`\beta_{\pm}` is the ratio of
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the velocity of the charged particle to the speed of light given in :eq:`beta-2`.
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where :math:`C` is a normalization constant and :math:`\beta_{\pm}` is the
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ratio of the velocity of the charged particle to the speed of light given in
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:eq:`beta-2`.
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The inverse transform method is used to sample :math:`\mu_{-}` and :math:`\mu_{+}`
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from :eq:`sauter–gluckstern–hull`, using the sampling formula
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The inverse transform method is used to sample :math:`\mu_{-}` and
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:math:`\mu_{+}` from :eq:`sauter–gluckstern–hull`, using the sampling formula
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.. math::
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:label: sample-mu
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@ -638,7 +639,7 @@ and Auger electrons:
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photon assuming it is from a captured free electron and terminate.
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2. Sample a transition using the transition probabilities for the vacancy
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shell as the PDF.
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shell as the probability mass function.
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3. Create either a fluorescence photon or Auger electron, sampling the
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direction of the particle isotropically.
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@ -688,18 +689,14 @@ expressed in the form
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\chi(Z, T, \kappa),
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where :math:`\kappa = E/T` is the reduced photon energy and :math:`\chi(Z, T,
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\kappa)` is the scaled bremsstrahlung cross section,
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.. math::
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:label: scaled-bremsstrahlung-dcs
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\chi(Z, T, \kappa) = \frac{\beta^2}{Z^2} E \frac{d\sigma_{\text{br}}}{dE}.
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\kappa)` is the scaled bremsstrahlung cross section, which is experimentally
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measured.
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Because electrons are attracted to atomic nuclei whereas positrons are
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repulsed, the cross section for positrons is smaller, though it approaches that
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of electrons in the high energy limit. To obtain the positron cross section, we
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multiply :eq:`bremsstrahlung-dcs` by the :math:`\kappa`-independent factor used
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in [Salvat]_,
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in Salvat_,
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.. math::
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:label: positron-factor
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@ -777,9 +774,9 @@ transport electrons. However, the bremsstrahlung emitted from high energy
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electrons and positrons can travel far from the interaction site. Thus, even
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without a full electron transport mode it is necessary to model bremsstrahlung.
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We use a thick-target bremsstrahlung (TTB) approximation based on the models in
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[Salvat]_ and [Kaltiaisenaho]_ for generating bremsstrahlung photons, which
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assumes the charged particle loses all its energy in a single homogeneous
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material region.
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Salvat_ and Kaltiaisenaho_ for generating bremsstrahlung photons, which assumes
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the charged particle loses all its energy in a single homogeneous material
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region.
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To model bremsstrahlung using the TTB approximation, we need to know the number
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of photons emitted by the charged particle and the energy distribution of the
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@ -906,11 +903,6 @@ angles.
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.. _LA-UR-04-0488: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-ur-04-0488.pdf
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.. rubric:: References
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.. _Kaltiaisenaho: https://aaltodoc.aalto.fi/bitstream/handle/123456789/21004/master_Kaltiaisenaho_Toni_2016.pdf
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.. [Kaltiaisenaho] T. Kaltiaisenaho, "Implementing a photon physics model in
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Serpent 2." M.Sc. Thesis, Aalto University, 2016.
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.. [Salvat] F. Salvat, J. M. Fernández-Varea, and J. Sempau, "PENELOPE-2011:
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A Code System for Monte Carlo Simulation of Electron and Photon Transport,"
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OECD-NEA, Issy-les-Moulineaux, France (2011).
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.. _Salvat: http://www.oecd-nea.org/globalsearch/download.php?doc=77434
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