Clarifying the physical significance of the attenuation coefficient

I have seen several sources define the attenuation coefficient as the fraction of a beam's intensity which is attenuated per unit distance e.g.

This doesn't make sense to me.

Let $I_0=13Wm^{-2}$, $\mu=0.7cm^{-1}$, $x=1cm$

$I=I_0e^{-\mu x}=13e^{-0.7 \times 1}=6.4556$

$\frac{\Delta I}{I_0}=\frac{I_0-I}{I_0}=\frac{13-6.4556}{13}=0.50342\not=0.7$

The fraction of the initial intensity which has been attenuated is $0.50342$, not $0.7$ as the definitions would suggest.

However, I can sort of see where the definition has come from...

$-\frac{\Delta I}{\Delta x}=\mu I$

$-\frac{\Delta I}{I}\times\frac{1}{\Delta x}=\mu$

In words I think this says: (minus) the fraction of intensity which is attenuated ($-\frac{\Delta I}{I}$) per unit distance ($\frac{1}{\Delta x}$) is equal to the attenuation coefficient ($\mu$). This agrees with the definitions, but I suspect there might be some subtlety due to calculus which modifies the meaning of this equation.

Perhaps the definitions would be improved by saying that the attenuation coefficient is the instantaneous fraction of a beam's intensity which is being attenuated per unit distance, rather than the fraction of a beam's intensity which is attenuated per unit distance i.e. in one unit of distance.

• This also relates to radioactive decay, where I have seen the decay constant defined as the fraction of nuclei which decay per unit time. – Rational Function May 10 '18 at 19:17
• The problem is that your time interval is too large. per unit implies that you are calculating a derivative quantity, so the interval must be short. Retry your calculation usint $x=0.001$ cm. Note that this derivative quantity is applicable wherever you choose to apply it. As the radiation is attenuated, it loses intensity at the same fractional rate no matter where you look. Your calculation does not have that property. – garyp May 10 '18 at 19:35
• Aha, I had interpreted per unit as meaning per entire unit, not as a derivative quantity. I can see that as $x$ cm tends to zero the fraction of intensity being absorbed per unit length will tend to $0.7$, and that instantaneously the fraction of intensity being absorbed will be exactly $0.7$. Thank you! @garyp – Rational Function May 10 '18 at 19:43