KAHIBARO
Table of Contents
Meaning of the coefficient
When a beam of radiation passes through matter, some photons are removed from the beam by interactions with the material. The quantity that describes how rapidly this happens is the linear attenuation coefficient, written as $\mu$.
If a narrow beam of photons enters a material, the number of photons remaining in the beam decreases with distance traveled inside the material. A larger value of $\mu$ means the beam is weakened more quickly. A smaller value of $\mu$ means the beam penetrates farther before being significantly reduced.
The word linear means that this coefficient is defined per unit length. Its SI unit is $\mathrm{m^{-1}}$, although in practice $\mathrm{cm^{-1}}$ is also very common.
The linear attenuation coefficient $\mu$ tells us the probability per unit path length that a photon is removed from the original beam by interaction in a material.
Its unit is
$$
[\mu] = \mathrm{length}^{-1}
$$
such as $\mathrm{m^{-1}}$ or $\mathrm{cm^{-1}}$.
Exponential decrease of beam intensity
Consider a thin slab of material of thickness $dx$. If the beam intensity at some point is $I$, then the small decrease in intensity through that thin slab is proportional to both $I$ and $dx$:
$$
dI = -\mu I \, dx
$$
The negative sign shows that the intensity decreases as the beam moves forward. Solving this differential equation gives the attenuation law:
$$
I(x) = I_0 e^{-\mu x}
$$
Here, $I_0$ is the initial intensity and $I(x)$ is the intensity after passing through thickness $x$ of the absorber.
This same form can also be written for photon number:
$$
N(x) = N_0 e^{-\mu x}
$$
where $N_0$ is the initial number of photons and $N(x)$ is the number remaining in the beam.
For a narrow monoenergetic photon beam passing through a uniform material,
$$
I(x) = I_0 e^{-\mu x}
$$
This is the basic attenuation formula.
Physical interpretation
The coefficient $\mu$ can be understood as a measure of how likely radiation is to interact as it travels through a material. If $\mu$ is large, interactions happen frequently, and the beam fades quickly. If $\mu$ is small, interactions are less frequent, and the beam travels farther.
A rough interpretation is that over a very small thickness $dx$, the fraction of the beam removed is
$$
\frac{-dI}{I} = \mu \, dx
$$
So $\mu dx$ is approximately the probability that a photon is removed while crossing that thin layer.
Dependence on material and photon energy
The linear attenuation coefficient is not a universal constant. Its value depends on the material and on the energy of the photons.
Dense materials usually have larger attenuation coefficients than light materials, because photons encounter more atoms per unit volume. Materials with high atomic number, such as lead, are often especially effective at attenuating X rays and gamma rays.
The photon energy also matters. At some energies, certain interaction processes are more likely, and this changes $\mu$. Because of this, the same material can attenuate low-energy and high-energy photons very differently.
Relation to thickness
The attenuation law shows that equal thicknesses do not remove equal amounts of intensity. Instead, each equal thickness removes the same fraction of what remains. This is why the decrease is exponential rather than linear.
For example, if one thickness reduces the intensity to half, then another equal thickness reduces it to half again, leaving one quarter of the original beam.
The fraction transmitted through thickness $x$ is
$$
\frac{I}{I_0} = e^{-\mu x}
$$
and the fraction removed is
$$
1 - e^{-\mu x}
$$
Simple numerical example
Suppose a material has linear attenuation coefficient
$$
\mu = 0.20 \, \mathrm{cm^{-1}}
$$
and a beam passes through thickness
$$
x = 5.0 \, \mathrm{cm}
$$
Then
$$
\frac{I}{I_0} = e^{-\mu x} = e^{-0.20 \times 5.0} = e^{-1} \approx 0.37
$$
So about $37\%$ of the original beam remains in the narrow transmitted beam, and about $63\%$ has been removed from it.
Comparison of different values of $\mu$
The effect of the coefficient becomes clearer when comparing materials or energies.
| Linear attenuation coefficient $\mu$ | Effect on beam |
|---|---|
| Small $\mu$ | Weak attenuation, strong penetration |
| Moderate $\mu$ | Noticeable reduction over moderate thickness |
| Large $\mu$ | Rapid attenuation over small thickness |
If two materials have the same thickness, the one with larger $\mu$ transmits less radiation.
Graphical view
A plot of transmitted intensity against thickness has the shape of a decreasing exponential curve. It starts at $I_0$ when $x=0$ and approaches zero as $x$ becomes large.
Microscopic picture
On a microscopic level, attenuation happens because photons interact with atoms and electrons in the material. Each interaction can remove a photon from the original beam direction or absorb it completely. The beam intensity falls because fewer photons continue forward unchanged.
This means attenuation is really a statistical process. Not every photon interacts at the same distance, but for a very large number of photons the average behavior follows the exponential law.
Important points to remember
The linear attenuation coefficient measures how strongly a material reduces a photon beam per unit length. It has units of inverse length, depends on both material and photon energy, and appears in the exponential attenuation law.
Key results:
$$
dI = -\mu I\,dx
$$
$$
I(x) = I_0 e^{-\mu x}
$$
$$
\frac{I}{I_0} = e^{-\mu x}
$$
A larger $\mu$ means stronger attenuation and less transmitted radiation for the same thickness.