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The Point Kernel Method

Overview

The point-kernel method calculates the uncollided radiation flux from a point source through layers of shielding material. It is the simplest analytical method for shielding calculations and forms the basis for engineering dose estimates.

The formula

For a point isotropic source of strength \(S\) (particles/sec) at the origin, the uncollided flux at distance \(R\) through layers of material is:

\[\Phi(R) = \frac{S}{4\pi R^2} \cdot \exp\left(-\sum_i \Sigma_{r,i} \cdot t_i\right) \cdot B\]

Where:

  • \(\frac{S}{4\pi R^2}\) is the geometric attenuation (inverse square law)
  • \(\exp(-\sum_i \Sigma_{r,i} \cdot t_i)\) is the material attenuation (exponential decay through each layer)
  • \(\Sigma_{r,i}\) is the macroscopic removal cross section of layer \(i\) (cm\(^{-1}\))
  • \(t_i\) is the thickness of layer \(i\) (cm)
  • \(B\) is the build-up factor (correction for scattered radiation, \(B \geq 1\))

Transmission fraction

The transmission fraction is the pure material attenuation without geometric spreading:

\[T = \exp\left(-\sum_i \Sigma_{r,i} \cdot t_i\right)\]

This is a dimensionless number between 0 and 1. It answers: "what fraction of particles pass through the shield without interacting?"

Dose rate

To convert flux to dose rate, multiply by the ICRP-116 dose conversion coefficient \(h(E)\):

\[\dot{D} = \Phi \cdot h(E) \cdot 3600 \cdot 10^{-12} \quad \text{[Sv/hr]}\]

Where \(h(E)\) is in pSv\(\cdot\)cm\(^2\) and depends on the particle energy and irradiation geometry (AP, PA, etc.).

Removal cross sections

The removal cross section \(\sigma_r\) is not the total cross section. It accounts for the fact that forward-scattered particles effectively continue in the beam direction and so shouldn't be counted as "removed":

\[\sigma_r(E) = \sigma_t(E) - f_{\text{fwd}}(E) \cdot \sigma_s(E)\]

where \(\sigma_s\) is the scattering channel that dominates the forward peak (elastic for neutrons, coherent/Rayleigh for photons), and \(f_{\text{fwd}}(E) \in [0, 1]\) is the fraction of that channel scattering into the forward cone \(\mu \in [\mu_0, 1]\), with \(\mu_0 = \cos\theta_0\). We use \(\mu_0 = 0\), i.e. the forward hemisphere (\(\theta < 90^\circ\)) is treated as "not removed".

Neutrons

\[\sigma_r(E) = \sigma_t(E) - f_{\text{fwd}}(E) \cdot \sigma_{\text{el}}(E)\]

\(f_{\text{fwd}}\) is obtained by integrating the ENDF elastic angular distribution \(p(\mu, E)\) from \(\mu_0\) to 1:

\[f_{\text{fwd}}(E) = \int_{\mu_0}^{1} p(\mu, E)\, d\mu\]

If no angular distribution is available the scattering is treated as isotropic, giving \(f_{\text{fwd}} = (1 - \mu_0)/2 = 0.5\) at \(\mu_0 = 0\).

Photons

\[\sigma_r(E) = \sigma_t(E) - f_{\text{fwd}}(E) \cdot \sigma_{\text{coh}}(E)\]

For photons the forward peak is coherent (Rayleigh) scattering, and \(f_{\text{fwd}}\) is computed from the Thomson-weighted atomic form factor \(F(x, Z)\):

\[f_{\text{fwd}}(E) = \frac{\displaystyle\int_{\mu_0}^{1} (1 + \mu^2)\,[F(x, Z)]^2\, d\mu}{\displaystyle\int_{-1}^{1} (1 + \mu^2)\,[F(x, Z)]^2\, d\mu}, \qquad x(E, \mu) = \frac{E \sqrt{(1 - \mu)/2}}{hc}\]

Incoherent (Compton) and photoelectric contributions are not subtracted because they genuinely remove the photon from the beam.

Why not simply \(\sigma_t - \sigma_s\)?

Subtracting all of the elastic (or coherent) cross section would assume every scattered particle keeps going in the forward direction; this overestimates transmission. Subtracting nothing (just \(\sigma_t\)) assumes every scatter removes the particle; this underestimates it. Weighting by the angular distribution is between the two and, for anisotropic scatterers at high energy, much closer to reality.

Compound materials

For a compound the macroscopic removal cross section is:

\[\Sigma_r = \rho \cdot N_A \cdot \sum_i \frac{w_i \cdot \sigma_{r,i}(E)}{A_i}\]

where \(w_i\) is the mass fraction, \(A_i\) the atomic mass, and \(\rho\) the density. The microscopic removal cross sections \(\sigma_{r,i}\) are pre-computed per nuclide from ENDF/B-VIII.0 using the endf package (see examples/removal_xs_to_json.py in that repo for the extraction script).

Limitations

  • Uncollided only: Without build-up factors, the method only counts particles that haven't interacted. For thick shields, this severely underestimates the dose.
  • Spherical geometry: Layers are concentric spheres. Real geometries with ducts, penetrations, or non-spherical shapes need Monte Carlo.
  • No energy degradation: Particles that scatter and lose energy are not tracked; the build-up factor corrects for this empirically.