# The accumulated radiation dose when moving up to a source

Source: Dr Chris Robbins, Grallator

## The accumulated dose when moving up to a source

### Dr Chris Robbins, Grallator

Consider the situation where you move up to a radioactive source, perform some operations for a period of time, and then move away. What total dose will be received?

The accumulated dose when moving up to a source

Let's introduce a few definitions and assumptions.

$$\qquad R_r \qquad$$ is the reference dose rate measured at a reference distance $$x_r$$ from the source.

$$\qquad T_m \qquad$$ is the time it takes to move up to the source.

$$\qquad T_w \qquad$$ is the working time spent stationary near the source.

$$\qquad x_s \qquad$$ is the starting distance away from the source (m).

$$\qquad x_w \qquad$$ is the working distance from the source. (m)

It will be assumed that:

$$\qquad$$ $$x_r = 1$$ m.

$$\qquad$$ $$x_w < x_s$$.

$$\qquad$$ You move up to the source at constant speed.

$$\qquad$$ You move away from the source at the same constant speed.

The easy bit is the dose received at a fixed distance near the source while working.

The dose rate at a distance $$x$$ from the source is obtained from the reference dose rate at a distance $$x_r$$ by

$R(x)=R_r \left (\frac{x_r}{x} \right )^2.$

With the assumption $$x_r = 1$$ this simplifies to

$R(x)=\frac{R_r}{{x}^2}.$

The total dose at this fixed distance is the simply given by $D = RT_w = \frac{R_r T_w}{{x_w}^2}$ (which is simply the dose received per unit time multiplied by the time).

The harder bit is the dose received when moving as $$x$$ is constantly changing. The first thing to do is to find an expression that links the distance from the source to time. At time $$t = 0, x = x_s$$, and at time $$t = T_m, x = x_w$$. The [constant] speed of movement is

$s=\frac{x_s-x_w}{T_m}$

and at any point in time $$0 \leq t \leq T_m, x$$ is given by

$x=x_s-st$

The dose received, $$\delta D$$ in a small amount of time $$\delta t$$ at time $$t$$ is given in a similar way to that found while working by

multiplying dose rate by time as

$\delta D=R \cdot \delta t = \frac{R_r \cdot \delta t}{{x}^2}=\frac{R_r \cdot \delta t}{\left ( x_s-st \right )^2}$

The total dose is found by letting $$\delta t \rightarrow 0$$ and integrating.

$D = \int_{0}^{T_m} \frac{R_r}{\left ( x_s-st \right )^2}dt = R_r \int_{0}^{T_m} \frac{1}{\left ( x_s-st \right )^2}dt$

To solve this use the substitution $$u=x_s-st$$. This gives \begin{align} \frac{du}{dt}&=-s \Rightarrow \frac{-1}{s}du=dt \\ \\ u(t=0) &= x_s \\ \\ u(t=T_m) &= x_s - sT_m = x_w \\ \\ \end{align}

So that

\begin{align} D &= \frac{-R_r}{s} \int_{x_s}^{x_w} \frac{1}{u ^2}du \\ \\ &=\frac{R_r}{s} {\left [ \frac{1}{u} \right]_{x_s}^{x_w}} = \frac{R_r T_m}{x_s-x_w} {\left [ \frac{1}{u} \right]_{x_s}^{x_w}} \\ \\ &=\frac{R_r T_m}{x_s-x_w} {\left ( \frac{1}{x_w} - \frac{1}{x_s} \right )} \\ \\ &=\frac{R_r T_m}{x_s-x_w} {\left ( \frac{x_s-x_w}{x_s x_w} \right )} \\ \\ &=\frac{R_r T_m}{x_s x_w} \end{align}

We can now answer the original question:

Consider the situation where you move up to a radioactive source, perform some operations for a period of time, and then move away. What total dose will be received?

$\text{Total dose} = \underbrace {\frac{R_r T_m}{x_s x_w} }_{\text{moving to}} + \underbrace {\frac{R_r T_w}{{x_w}^2} }_{\text{working}} + \underbrace {\frac{R_r T_m}{x_s x_w} }_{\text{moving away}}$

As a last note, if $$x_r \neq 1$$ then all the doses need scaling by $$x_r^2$$ to give

$\text{Total dose} = \underbrace {\frac{R_r T_m x_r^2}{x_s x_w} }_{\text{moving to}} + \underbrace {\frac{R_r T_w x_r^2}{{x_w}^2} }_{\text{working}} + \underbrace {\frac{R_r T_m x_r^2}{x_s x_w} }_{\text{moving away}}$

Or, tidying things up

\begin{align} \text{Total } &=\frac{R_r x_r^2}{x_w} \left ( \underbrace {\frac{T_m}{x_s} }_{\text{moving to}} + \underbrace {\frac{T_w}{x_w} }_{\text{working}} + \underbrace {\frac{T_m}{x_s} }_{\text{moving away}} \right ) \\ \\ &=\frac{R_r x_r^2}{x_w} \left ( \frac{2 T_m}{x_s} + \frac{T_w}{x_w} \right ) \end{align}

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Find below a neat interactive graphic illustrating the above mathematical concepts. Have a play by moving the coloured sliders!

Reality is merely an illusion, albeit a very persistent one

– Albert Einstein -