# Point-Kernel Code Development for Gamma-Ray Shielding Applications
## 3 Gamma Flux Solution for Simple Source Geometries
### Line Gamma Source  
This section presents the application of the PK method to an isotropic gamma line source emitting \(S_{l}\) particles/cm/s. If we take a differential element of line length \(dl\) (cm), then the product \(Sl\)-\(dl\) represents point gamma source (particles/s). The detector response at distance \(R\) is then given by PK definition as integration over the total source length \(l\)  
\[\phi(R)=\int\limits_{l}S_{l}G(R)dl. \tag{14}\]  
This result is completely general, so an actual problem setup is needed to perform specific integration. One should bear in mind how effects of problem boundaries are not included, and gamma rays, which may be scattered multiple times in the shield before reaching the detector, are modeled by means of \(B\)-factors. We are considering a long line representing a linear isotropic radiator of photons in front of a slab shield with thickness \(t\) (see Figure 2). Figure 2: Geometry setup for gamma line and slab shield [2]. The exponential kernel of Equation (2) is substituted in Equation (14) for a general line source. The distance \(R\) for any point source to the detector is expressed in perpendicular distance from the line source to the point detector P (variable \(z\)) and angle \(\theta\), i.e., \(R=z\cdot\sec\theta\). The attenuation kernel for the uncollided flux is written as a function of variables \(z\), \(t\), and \(\theta\), giving  
\[G(z,t,\theta)=\frac{\exp(-\mu t\cdot\sec\theta)}{4\pi R^{2}}=\frac{\exp(-\mu t \cdot\sec\theta)}{4\pi(z\cdot\sec\theta)^{2}}. \tag{15}\]  
To perform integration, we express the differential line element \(dl\) as a function of variables \(z\) and \(\theta\). This is first performed by a derivation of expression \(R^{2}=z^{2}+l^{2}\), giving \(l\cdot dl=R\cdot dR\) in turn. The same is repeated for the differential element \(dR\), i.e., the derivation of \(z=R\cos\theta\) is giving \(dR=(l\cdot d\theta)/\cos\theta\). Combining obtained expressions, we eliminate the \(l\) variable and finally obtain a connection between differentials as \(dl=z\cdot(\sec\theta)^{2}d\theta\), which will permit two-part integration over the whole line length \(l\) in terms of angle \(\theta\) as  
\[\phi(R)=\frac{S_{l}}{4\pi z}\int\limits_{l}\exp(-\mu t\cdot\sec\theta)d\theta= \frac{S_{l}}{4\pi z}\left[\int\limits_{0}^{\theta_{1}}e^{-\mu t\cdot\sec \theta}d\theta+\int\limits_{0}^{\theta_{2}}e^{-\mu t\cdot\sec\theta}d\theta \right]. \tag{16}\]  
It is customary to introduce the secant or Sievert integral \(F(\theta,x)\), so the uncollided flux solution can be finally expressed as  
\[F(\theta,x)=\int\limits_{0}^{\theta}e^{-x\cdot\sec\theta}d\theta\to\phi_{l}(z, t)=\frac{S_{l}}{4\pi z}[F(\theta_{1},\mu t)+F(\theta_{2},\mu t)] \tag{17}\]  
Sievert integral plots can be found in radiation shielding books [1; 23] for various values of \((\theta,x)\) or in tabulated values of auxiliary function \(\overline{F}(\theta,x)\) as \(F(\theta,x)=\theta e^{-x}\overline{F}(\theta,x)\) or in form of series expansion over ranges of \((\theta,x)\). For the purposes of programming, the tabulated \(F\)-function approach was selected in this work [24]. To obtain the buildup flux, one must include the \(B\)-factor in exponential kernel and perform integration. The Taylor form becomes  
\[B=A\exp(-\alpha_{1}\mu t\cdot\sec\theta)+(1-A)\exp(-\alpha_{2}\mu t\cdot\sec \theta), \tag{18}\]  
and the corresponding kernel for buildup flux is  
\[G(z,t,\theta)=\frac{1}{4\pi(z\cdot\sec\theta)^{2}}\Big{[}Ae^{-\mu t\cdot\sec \theta(1+\alpha_{1})}+(1-A)e^{-\mu t\cdot\sec\theta(1+\alpha_{2})}\Big{]}. \tag{19}\]  
The total or buildup flux from the isotropic gamma line emitter becomes  
\[\phi_{l}(z,t)=\frac{S_{l}}{4\pi z}\Bigg{[} \begin{array}{c}A\{F(\theta_{1}, \mu t(1+\alpha_{1}))+F(\theta_{2},\mu t(1+\alpha_{1}))\}+\\ (1-A)\{F(\theta_{1},\mu t(1+\alpha_{2}))+F(\theta_{2},\mu t(1+\alpha_{2}))\} \end{array} \Bigg{]}.