## 2 Materials and Methods
### RADTRAN 6 Code
3.1 Mathematical Equations Describing the Radiation Dose to Receptors from the Shipment during Normal (Incident-Free) Transport  
The mathematical equation for calculating single-point receptor's dose at a distance \(r\) from the traversing vehicle carrying radioactive consignment or package is as follows:  
\[D(x)=Q_{1}\frac{2\times k_{0}\times DR}{V}\int_{x}^{\infty}\frac{e^{-\mu r}B(r )\mathbf{dr}}{r\sqrt{r^{2}-x^{2}}} \tag{1}\]  
where \(D(x)\) is the dose (Sv) at a distance \(x\) from the traversing vehicle, \(Q_{1}=1000\) m/km, is the conversion factor needed because \(V\) is in km/h and \(k_{0}\) is in m, \(k_{0}\) is the package shape factor (m\({}^{2}\)), \(DR\) is the dose rate at 1 m from the shipping vehicle Sv/h, \(V\) is the speed of the shipping vehicle, \(x\) is the perpendicular distance of receptor from the shipment path (m), \(\mu\) is the attenuation coefficient (m\({}^{-1}\)), which depends on shielding factor and radiation energy, \(B(r)\) is the buildup factor expressed as a geometric progression, \(r\) is the distance between the receptor and shipment along the traveling route (m). Additionally, neutrons are rapidly attenuated in air and all gamma treatment provides a slightly conservative estimate of population dose for populations 10 m or more away from the vehicle [32]. Therefore, the maximum value of the product (\(e^{-\mu r}\)\(B(r)dr\)) is 1 for gamma radiation, which is nearly always the main source of radiation from a transport package. This value is mostly used in transportation analysis. Buildup factor (B(r)) is also the ratio of the total number of particles at a given point to the number of uncollided particles at that same site, and through a multiplicative change to the uncollided dose, buildup provides for the additional dose contributed by dispersed radiation. The mathematical equation for calculating population or collective dose at a distance r from the traversing vehicle carrying radioactive cargo is as follows:  
\[\mathbf{D}(\mathbf{p})=\mathbf{Q}_{2}\ \frac{\mathbf{2}\times\mathbf{k_{0}}\times\mathbf{D}\mathbf{R}\times \mathbf{P}\mathbf{D_{L}}\times\mathbf{L}}{\mathbf{V}}\int_{d}^{\infty}\int_{d}^{\infty}\frac{ \mathbf{e}^{-\mu r}\mathbf{B}(\mathbf{r})\mathbf{dx}\mathbf{dr}}{r\sqrt{\mathbf{r}^{2}-\mathbf{x}^{2}}} \tag{2}\]  
where \(D(p)\) is the collective dose of exposed population along the route, \(L\) is the distance of the route segment on which the responsible population dwells (km), \(PD_{L}\) is the population density (persons/km\({}^{2}\)), \(Q_{2}=(Q_{1})^{2}\) in which \(Q_{1}\) is a conversion factor equals 1000 m/km. In general terms, collective effective dose is the sum of all individual effective doses in a group of people over the time period, given \(X\) (man-Sv) is as follows:  
\[\mathbf{X}=\sum_{i}\mathbf{F_{i}}\mathbf{D_{i}} \tag{3}\]  
where \(F_{i}\) is the average effective dose for subgroup \(i\), \(D_{i}\) if the number of individuals in the subgroup \(i\). The basic mathematical equation used to calculate the radiation dose to the population receptors at a distance r from the shipment vehicle at the stop point is as follows:  
\[\mathbf{D_{stop}}=\mathbf{Q_{2}}\times\mathbf{D}\mathbf{R}\times\mathbf{P}\mathbf{D_{L}}\times\mathbf{t} \times\mathbf{k_{0}}\times\int_{d}^{\mathbf{r}}\frac{\mathbf{e}^{-\mu r}\mathbf{B}(\mathbf{r})\bm {dx}\mathbf{dr}}{r\sqrt{\mathbf{r}^{2}-\mathbf{x}^{2}}} \tag{4}\]  
where \(D_{stop}\) is the collective dose of exposed population at a stop point, \(t\) is the exposure time (h), \(d\) is the minimum distance of the receptor from the source (m).