.. _methods_photon_physics: ============== Photon Physics ============== Photons, being neutral particles, behave much in the same manner as neutrons, traveling in straight lines and experiencing occassional collisions which change their energy and direction. Photons undergo four basic interactions as they pass through matter: coherent (Rayleigh) scattering, incoherent (Compton) scattering, photoelectric effect, and pair/triplet production. Photons with energy in the MeV range may also undergo photonuclear reactions with an atomic nucleus. In addition to these primary interaction mechanisms, all processes other than coherent scattering can result in the excitation/ionization of atoms. The de-excitation of these atoms can result in the emission of electrons and photons. Electrons themselves also can produce photons by means of bremsstrahlung radiation. ------------------------------ Coherent (Rayleigh) Scattering ------------------------------ The elastic scattering of a photon off a free charged particle is known as Thomson scattering. The differential cross section is independent of the energy of the incident photon. For scattering off a free electron, the differential cross section is .. math:: :label: thomson \frac{d\sigma}{d\mu} = \pi r_0^2 ( 1 + \mu^2 ) where :math:`\mu` is the cosine of the scattering angle and :math:`r_0` is the classical radius of the electron. Thomson scattering can generally occur when the photon energy is much less than rest mass energy of the particle. In practice, most elastic scattering of photons off electrons happens not with free electrons but those bound in atoms. This process is known as Rayleigh scattering. The radiation scattered off of individual bound electrons combines coherently, and thus Rayleigh scattering is also known as coherent scattering. Even though conceptually we think of the photon interacting with a single electron, because the wave functions combine constructively it is really as though the photon is interacting with the entire atom. The differential cross section for Rayleigh scattering is given by .. math:: :label: coherent-xs \frac{d\sigma(E,E',\mu)}{d\mu} = \pi r_0^2 ( 1 + \mu^2 ) \left [ ( F(x, Z) + F'(E) )^2 + F''(E)^2 \right ] where :math:`F(x,Z)` is a form factor as a function of the momentum transfer :math:`x` and the atomic number :math:`Z` and :math:`F' + iF''` is a factor that accounts for `anomalous scattering`_ which can occur near absorption edges. In a Monte Carlo simulation, when coherent scattering occurs, we only need to sample the scattering angle using the differential cross section in :eq:`coherent-xs` since the energy of the photon does not change. In OpenMC, anomalous scattering is ignored such that differential cross section comes .. math:: :label: coherent-xs-openmc \frac{d\sigma(E,E',\mu)}{d\mu} = \pi r_0^2 ( 1 + \mu^2 ) F(x, Z)^2 To construct a proper probability density, we need to normalize the differential cross section in :eq:`coherent-xs-openmc` by the integrated coherent scattering cross section: .. math:: :label: coherent-pdf-1 p(\mu) d\mu = \frac{\pi r_0^2}{\sigma(E)} ( 1 + \mu^2 ) F(x, Z)^2 d\mu. Since the form factor is given in terms of the momentum transfer, it is more convenient to change variables of the probability density to :math:`x^2`. The momentum transfer is traditionally expressed as .. math:: :label: momentum-transfer x = \kappa \alpha \sqrt{1 - \mu} where the coefficient :math:`\kappa` can be shown to be .. math:: :label: kappa \kappa = \frac{m_e c^2}{\sqrt{2}hc} \approx 29.14329, :math:`m_e` is the mass of the electron, :math:`c` is the speed of light in a vacuum, and :math:`h` is Planck's constant. Using :eq:`momentum-transfer`, we have that :math:`\mu = 1 - [x/(\kappa\alpha)]^2` and :math:`d\mu/dx^2 = -1/(\kappa\alpha)^2`. The probability density in :math:`x^2` is .. math:: :label: coherent-pdf-x2 p(x^2) dx^2 = p(\mu) \left | \frac{d\mu}{dx^2} \right | dx^2 = \frac{2\pi r_0^2 A(\bar{x}^2,Z)}{(\kappa\alpha)^2 \sigma(E)} \left ( \frac{1 + \mu^2}{2} \right ) \left ( \frac{F(x, Z)^2}{A(\bar{x}^2, Z)} \right ) dx^2 where :math:`\bar{x}` is the maximum value of :math:`x` that occurs for :math:`\mu=-1`, .. math:: :label: xmax \bar{x} = \kappa \alpha \sqrt{2} = \frac{m_e c^2}{hc} \alpha, and :math:`A(x^2, Z)` is the integral of the square of the form factor: .. math:: :label: coherent-int-ff A(x^2, Z) = \int_0^{x^2} F(\chi, Z)^2 d\chi^2. As you see, we have multiplied and divided the probability density by the integral of the squared form factor so that the density in :eq:`coherent-pdf-x2` is expressed as the product of two separate densities in parentheses. In OpenMC, a table of :math:`A(x^2, Z)` versus :math:`x^2` is pre-generated and used at run-time to do a table search on the cumulative distribution function: .. math:: :label: coherent-form-factor-cdf \frac{\int_0^{x^2} F(\chi,Z)^2 d\chi^2}{\int_0^{\bar{x}^2} F(x,Z)^2 dx^2} Once a trial :math:`x^2` value has been selected, we can calculate :math:`\mu` and perform rejection sampling using the Thomson scattering differential cross section. The complete algorithm is as follows: 1. Determine :math:`\bar{x}^2` using :eq:`xmax`. 2. Determine :math:`A_{max} = A(\bar{x}^2, Z)` using the pre-generated tabulated data. 3. Sample the cumulative density by calculating :math:`A' = \xi_1 A_{max}` where :math:`\xi_1` is a uniformly distributed random number. 4. Perform a binary search to determine the value of :math:`x^2` which satisfies :math:`A(x^2, Z) = A'`. 5. By combining :eq:`momentum-transfer` and :eq:`xmax`, calculate :math:`\mu = 1 - 2x^2/\bar{x}^2`. 6. If :math:`\xi_2 < (1 + \mu^2)/2`, accept :math:`\mu`. Otherwise, repeat the sampling at step 3. ------------------------------- Incoherent (Compton) Scattering ------------------------------- Before we noted that the Thomson cross section gives the behavior for photons scattering off of free electrons valid at low energies. The formula for photon scattering off of free electrons that is valid for all energies can be found using quantum electrodynamics and is known as the Klein-Nishina_ formula after the two authors who discovered it: .. math:: :label: klein-nishina \frac{d\sigma_{KN}}{d\mu} = \pi r_0^2 \left ( \frac{\alpha'}{\alpha} \right ) \left [ \frac{\alpha'}{\alpha} + \frac{\alpha}{\alpha'} + \mu^2 - 1 \right ] where :math:`\alpha` and :math:`\alpha'` are the ratios of the incoming and exiting photon energies to the electron rest mass energy equivalent (0.511 MeV), respectively. Although it appears that the outgoing energy and angle are separate, there is actually a one-to-one relationship between them such that only one needs to be sampled: .. math:: :label: compton-energy-angle \alpha' = \frac{\alpha}{1 + \alpha(1 - \mu)}. Note that when :math:`\alpha'/\alpha` goes to one, i.e., scattering is elastic, the Klein-Nishina cross section becomes identical to the Thomson cross section. In general though, the scattering is inelastic and is known as Compton scattering. When a photon interacts with a bound electron in an atom, the Klein-Nishina formula must be modified to account for the binding effects. As in the case of coherent scattering, this is done by means of a form factor. The differential cross section for incoherent scattering is given by .. math:: :label: incoherent-xs \frac{d\sigma}{d\mu} = \frac{d\sigma_{KN}}{d\mu} S(x,Z) = \pi r_0^2 \left ( \frac{\alpha'}{\alpha} \right )^2 \left [ \frac{\alpha'}{\alpha} + \frac{\alpha}{\alpha'} + \mu^2 - 1 \right ] S(x,Z) where :math:`S(x,Z)` is the form factor. The approach in OpenMC is to first sample the Klein-Nishina cross section and then perform rejection sampling on the form factor. As in other codes, `Kahn's rejection method`_ is used for :math:`\alpha < 3` and a direct method by Koblinger_ is used for :math:`\alpha \ge 3`. The complete algorithm is as follows: 1. If :math:`\alpha < 3`, sample :math:`\mu` from the Klein-Nishina cross section using Kahn's rejection method. Otherwise, use Koblinger's direct method. 2. Calculate :math:`x` and :math:`\bar{x}` using :eq:`momentum-transfer` and :eq:`xmax`, respectively. 3. If :math:`\xi < S(x, Z)/S(\bar{x}, Z)`, accept :math:`\mu`. Otherwise repeat from step 1. Doppler Energy Broadening ------------------------- LA-UR-04-0487_ and LA-UR-04-0488_ -------------------- Photoelectric Effect -------------------- Atomic Relaxation ----------------- --------------- Pair Production --------------- --------------------------- Thick-target Bremsstrahlung --------------------------- .. _Koblinger: http://www.tandfonline.com/doi/abs/10.13182/NSE75-A26646 .. _anomalous scattering: http://pd.chem.ucl.ac.uk/pdnn/diff1/anomscat.htm .. _Kahn's rejection method: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/aecu-3259_kahn.pdf .. _Klein-Nishina: https://en.wikipedia.org/wiki/Klein%E2%80%93Nishina_formula .. _LA-UR-04-0487: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-ur-04-0487.pdf .. _LA-UR-04-0488: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-ur-04-0488.pdf