Radiation Dose Rate Calculation w/ Gamma Ray Sources

[pone]

Hi, I am having much trouble on this particular question. Here it is:

Four “point” gamma ray sources are permanently implanted in tissue so that the sources are at the corners of a 1 cm X 1 cm square. Each source has an initial activity of 15 MBq (1 Bq = one decay per second), every decay produces a 30 keV gamma ray, and the half-life is 60 days. The dose rate from each source falls off with distance, r, according to

Dose Rate = (Constant)(exp -ur)(r-2)

where u = 0.25 cm-1. Calculate the total dose delivered by the implanted sources to a point at the centre of the square. Sketch the isodose distribution in the plane that contains all of the sources.

Hint: Use conservation of energy to evaluate the constant in the equation above.

Attempt at a solution:
This question confuses me because there is no real time frame given. There is a half life, but I am unsure how to use it. Also, when evaluating the Constant, I do not get proper units. The way I have been looking at it is:
constant = (15 x 10^6 decay/s)(30,000 eV)(1.602 x 10^-19 J/eV) which gives me J/s.
I know this must be wrong, but do not know where to go from here. Any help?

 

[Astronuc]

In determing an accumulated dose, one must account for geometric (spatial) effects, shielding (absorption) and temporal effects where the source strength is decreasing with time. Sheilding and spatial effects are somewhat dependent because the absorption reduces the dose rate at the receiver, but are independent of time unless the source is moving.

One can calculate the spatial and absorption time dependent dose rate at the receiver, then integrate over time.

 

[sir-pinski]

Hello,

Thank you for reaching out for help with this question. I understand that it can be confusing and overwhelming at first glance, but let's break it down step by step.

Firstly, let's define some variables for easier understanding:
- A = initial activity of each source (15 MBq)
- t = time (in seconds)
- λ = decay constant (calculated using the half-life, which is 60 days or 5,184,000 seconds)
- r = distance from the source (in cm)
- u = 0.25 cm^-1 (given in the question)

Now, let's look at the equation for dose rate:
Dose Rate = (Constant)(exp -ur)(r^-2)

As you correctly stated, the constant is found using conservation of energy. This means that the total energy emitted by the source must be equal to the energy absorbed by the surrounding tissue. In other words, the total activity of all four sources must be equal to the dose rate at the centre of the square.

Therefore, we can write the equation as:
A = (Constant)(exp -u0)(0^-2)

Since the dose rate at the centre of the square (r = 0) is equal to the total activity of all four sources, we can solve for the constant:
Constant = A/exp -u0

Substituting the values:
Constant = (15 x 10^6 decay/s)/(exp -0.25 cm^-1 x 0 cm)(0^-2)
= (15 x 10^6 decay/s)/1
= 15 x 10^6 decay/s

Note that the units for the constant are in decay/s, which makes sense since we are dealing with decay rates.

Now, to calculate the total dose delivered by the implanted sources to the centre of the square, we can use the equation for dose rate:
Dose Rate = (15 x 10^6 decay/s)(exp -0.25 cm^-1r)(r^-2)

Since the sources are at the corners of a 1 cm x 1 cm square, the distance from the centre to each source is 0.5 cm. Therefore, we can calculate the dose rate at the centre by substituting r = 0.5 cm:
Dose Rate = (15 x 10^6 decay/s)(exp -0.25 cm^-1 x 0.5 cm)(0