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Address #1023 comments
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@ -5,7 +5,7 @@ Photon Physics
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==============
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Photons, being neutral particles, behave much in the same manner as neutrons,
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traveling in straight lines and experiencing occassional collisions which change
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traveling in straight lines and experiencing occasional collisions which change
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their energy and direction. Photons undergo four basic interactions as they pass
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through matter: coherent (Rayleigh) scattering, incoherent (Compton) scattering,
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photoelectric effect, and pair/triplet production. Photons with energy in the
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@ -31,11 +31,11 @@ cross section is
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.. math::
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:label: thomson
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\frac{d\sigma}{d\mu} = \pi r_0^2 ( 1 + \mu^2 )
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\frac{d\sigma}{d\mu} = \pi r_e^2 ( 1 + \mu^2 )
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where :math:`\mu` is the cosine of the scattering angle and :math:`r_0` is the
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classical radius of the electron. Thomson scattering can generally occur when
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the photon energy is much less than rest mass energy of the particle.
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where :math:`\mu` is the cosine of the scattering angle and :math:`r_e` is the
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classical electron radius. Thomson scattering can generally occur when the
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photon energy is much less than the rest mass energy of the particle.
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In practice, most elastic scattering of photons off electrons happens not with
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free electrons but those bound in atoms. This process is known as Rayleigh
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@ -50,30 +50,33 @@ The differential cross section for Rayleigh scattering is given by
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.. math::
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:label: coherent-xs
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\frac{d\sigma(E,E',\mu)}{d\mu} = \pi r_0^2 ( 1 + \mu^2 ) \left [ ( F(x, Z) +
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F'(E) )^2 + F''(E)^2 \right ]
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\frac{d\sigma(E,E',\mu)}{d\mu} &= \pi r_e^2 ( 1 + \mu^2 )~\left| F(x,Z)
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+ F' + iF'' \right|^2 \\
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&= \pi r_e^2 ( 1 + \mu^2 ) \left [ ( F(x,Z)
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+ F'(E) )^2 + F''(E)^2 \right ]
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where :math:`F(x,Z)` is a form factor as a function of the momentum transfer
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:math:`x` and the atomic number :math:`Z` and :math:`F' + iF''` is a factor that
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accounts for `anomalous scattering`_ which can occur near absorption edges. In a
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Monte Carlo simulation, when coherent scattering occurs, we only need to sample
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the scattering angle using the differential cross section in :eq:`coherent-xs`
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since the energy of the photon does not change. In OpenMC, anomalous scattering
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is ignored such that differential cross section becomes
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:math:`x` and the atomic number :math:`Z` and the term :math:`F' + iF''`
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accounts for `anomalous scattering`_ which can occur near absorption edges. In
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a Monte Carlo simulation, when coherent scattering occurs, we only need to
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sample the scattering angle using the differential cross section in
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:eq:`coherent-xs` since the energy of the photon does not change. In OpenMC,
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anomalous scattering is ignored such that the differential cross section
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becomes
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.. math::
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:label: coherent-xs-openmc
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\frac{d\sigma(E,E',\mu)}{d\mu} = \pi r_0^2 ( 1 + \mu^2 ) F(x, Z)^2
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\frac{d\sigma(E,E',\mu)}{d\mu} = \pi r_e^2 ( 1 + \mu^2 ) F(x, Z)^2
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To construct a proper probability density, we need to normalize the differential
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cross section in :eq:`coherent-xs-openmc` by the integrated coherent scattering
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cross section:
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To construct a proper probability density, we need to normalize the
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differential cross section in :eq:`coherent-xs-openmc` by the integrated
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coherent scattering cross section:
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.. math::
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:label: coherent-pdf-1
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p(\mu) d\mu = \frac{\pi r_0^2}{\sigma(E)} ( 1 + \mu^2 ) F(x, Z)^2 d\mu.
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p(\mu) d\mu = \frac{\pi r_e^2}{\sigma(E)} ( 1 + \mu^2 ) F(x, Z)^2 d\mu.
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Since the form factor is given in terms of the momentum transfer, it is more
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convenient to change variables of the probability density to :math:`x^2`. The
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@ -84,23 +87,24 @@ momentum transfer is traditionally expressed as
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x = \kappa \alpha \sqrt{1 - \mu}
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where the coefficient :math:`\kappa` can be shown to be
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where :math:`\alpha` is the ratio of the photon energy to the electron rest
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mass, and the coefficient :math:`\kappa` can be shown to be
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.. math::
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:label: kappa
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\kappa = \frac{m_e c^2}{\sqrt{2}hc} \approx 29.14329,
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:math:`m_e` is the mass of the electron, :math:`c` is the speed of light
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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 that :math:`\mu = 1 - [x/(\kappa\alpha)]^2` and :math:`d\mu/dx^2 =
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we have :math:`\mu = 1 - [x/(\kappa\alpha)]^2` and :math:`d\mu/dx^2 =
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-1/(\kappa\alpha)^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_0^2 A(\bar{x}^2,Z)}{(\kappa\alpha)^2 \sigma(E)} \left (
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r_e^2 A(\bar{x}^2,Z)}{(\kappa\alpha)^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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@ -116,7 +120,7 @@ and :math:`A(x^2, Z)` is the integral of the square of the form factor:
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.. math::
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:label: coherent-int-ff
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A(x^2, Z) = \int_0^{x^2} F(\chi, Z)^2 d\chi^2.
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A(x^2, Z) = \int_0^{x^2} F(x,Z)^2 dx^2.
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As you see, we have multiplied and divided the probability density by the
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integral of the squared form factor so that the density in :eq:`coherent-pdf-x2`
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@ -127,7 +131,7 @@ run-time to do a table search on the cumulative distribution function:
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.. math::
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:label: coherent-form-factor-cdf
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\frac{\int_0^{x^2} F(\chi,Z)^2 d\chi^2}{\int_0^{\bar{x}^2} F(x,Z)^2 dx^2}
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\frac{\int_0^{x^2} F(x,Z)^2 dx^2}{\int_0^{\bar{x}^2} F(x,Z)^2 dx^2}
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Once a trial :math:`x^2` value has been selected, we can calculate :math:`\mu`
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and perform rejection sampling using the Thomson scattering differential cross
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@ -162,7 +166,7 @@ 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_0^2 \left ( \frac{\alpha'}{\alpha} \right
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\frac{d\sigma_{KN}}{d\mu} = \pi r_e^2 \left ( \frac{\alpha'}{\alpha} \right
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) \left [ \frac{\alpha'}{\alpha} + \frac{\alpha}{\alpha'} + \mu^2 - 1 \right
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]
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@ -188,7 +192,7 @@ differential 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_0^2 \left (
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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{\alpha'}{\alpha} \right )^2 \left [ \frac{\alpha'}{\alpha} +
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\frac{\alpha}{\alpha'} + \mu^2 - 1 \right ] S(x,Z)
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@ -213,6 +217,10 @@ Doppler Energy Broadening
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LA-UR-04-0487_ and LA-UR-04-0488_
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Compton Electrons
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+++++++++++++++++
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Photoelectric Effect
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--------------------
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@ -253,7 +261,7 @@ Thick-Target Bremsstrahlung Approximation
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+++++++++++++++++++++++++++++++++++++++++
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.. _Koblinger: http://www.tandfonline.com/doi/abs/10.13182/NSE75-A26646
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.. _Koblinger: https://doi.org/10.13182/NSE75-A26663
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.. _anomalous scattering: http://pd.chem.ucl.ac.uk/pdnn/diff1/anomscat.htm
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