Basic Physics of Nuclear Medicine/Print version

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Relationship between the Linear Attenuation Coefficient and the Half Value Layer

As was the case with the Radioactive Decay Law, where we explored the relationship between the Half Life and the Decay Constant, a relationship can be derived between the Half Value Layer and the Linear Attenuation Coefficient. We can do this by using the definition of the Half Value Layer:

$I_x = \frac{I_0}{2}$

when

$x = x_{\frac{1}{2}}$

and inserting it in the exponential attenuation equation, that is:

$I_x = I_0\ \text{exp}\ (-\mu x)\,\!$

to give

$\frac{I_0}{2} = I\ \text{exp}\ (-\mu x_{\frac{1}{2}})$

Therefore

$\frac{1}{2} = \text{exp}\ (-\mu x_{\frac{1}{2}})$

$\therefore 2^{-1} = \text{exp}\ (-\mu x_{\frac{1}{2}})$

$\therefore \ln 2^{-1} = -\mu x_{\frac{1}{2}}$

$\therefore \ln 2 = \mu x_{\frac{1}{2}}$

$\therefore 0.693 = \mu x_{\frac{1}{2}}$

$\mu = \frac{0.693}{x_{\frac{1}{2}}}$

and

$x_{\frac{1}{2}} = \frac{0.693}{\mu}$

These last two equations express the relationship between the Linear Attenuation Coefficient and the Half Value Layer. They are very useful as you will see when solving numerical questions relating to attenuation and frequently form the first step in solving a numerical problem.