How Does Self-Absorption Affect Intensity in a Spherical Gamma-Ray Source?

[physfan01]

Encountered the following bonus question in a hwk assignment and neither myself nor my other class-mates have a clue on how to approach it. Or, if someone has ran across this or a similar problem in a text or online, please point me in that direction.
Any help is greatly appreciated.

Given: A gamma-ray source in the shape of a homogeneous sphere of radius R, throughout which there is a homogeneous temperature distribution.

a. Calculate the factor by which the intensity is reduced through self-absorption. Express this as a function of:

1) source radius &
2) source temperature

b. What is the magnitude of these reductions in the case of a 198-Au source (μ = 2.9 cm-1) of radius R = 0.15 cm when the temperature is changed by ΔT = 1000°C (linear coefficient of expansion α = 17 x 10-6 per °C).

 

[Astronuc]

I'd suggest the ring method, but IIRC it will be a complicated integral. I think I did a similar problem about 25+ years ago.

The intensity is simply a function of distance d and the radius, r which will determine the attenuation.

The thermal expansion will simply change the electron density, but I don't think by much.


ΔT = 1000°C (linear coefficient of expansion α = 17 x 10-6 per °C) means a difference of about 2% based on 1000 * 17 x 10-6 = 0.017

 

[sir-pinski]

Unfortunately, I am not familiar with this specific problem and cannot provide a direct solution. However, I can offer some general guidance and suggest some resources that may be helpful in finding a solution.

Firstly, it is important to understand the concept of self-absorption in a gamma-ray source. Self-absorption occurs when gamma-ray photons emitted from the source are absorbed by the material of the source itself, leading to a decrease in the intensity of the emitted radiation. This phenomenon is dependent on the material properties of the source, such as its density and atomic number, as well as its size and temperature.

To approach this problem, you can start by looking at the equations and concepts related to self-absorption in a homogeneous sphere. One important equation to consider is the Beer-Lambert law, which describes the relationship between the intensity of radiation and the material properties of the source. You can also look into the concept of mass attenuation coefficient, which takes into account the material properties of the source as well as the energy of the gamma-ray photons.

In terms of resources, you can refer to textbooks or online sources that cover the topic of gamma-ray sources and self-absorption. Some recommended textbooks include "Introduction to Nuclear Engineering" by John R. Lamarsh and "Radiation Detection and Measurement" by Glenn F. Knoll. You can also search for online lectures or tutorials on this topic.

Additionally, you can try breaking down the problem into smaller parts and solving them individually, using the relevant equations and concepts. This may help in understanding the problem and finding a solution.

I hope this helps and good luck with your assignment!