# Attenuation coefficient

For "attenuation coefficient" as it applies to electromagnetic theory and telecommunications see propagation constant. For the "mass attenuation coefficient", see the article mass attenuation coefficient.

The attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the medium, and a small attenuation coefficient means that the medium is relatively transparent to the beam. Attenuation coefficient is measured using units of reciprocal length.

The attenuation coefficient is also called linear attenuation coefficient, narrow beam attenuation coefficient, or absorption coefficient. Although all four terms are often used interchangeably, they can occasionally have a subtle distinction, as explained below.

## Overview

The attenuation coefficient describes the extent to which the intensity of an energy beam is reduced as it passes through a specific material. This might be a beam of electromagnetic radiation or sound.

• It is used in the context of X-rays or Gamma rays, where it is represented using the symbol $\mu$, and measured in cm−1.
• It is also used for modeling solar and infrared radiative transfer in the atmosphere, albeit usually denoted with another symbol (given the standard use of $\mu = \cos(\theta)$ for slant paths).
• In the case of ultrasound attenuation it is usually denoted as $\alpha$ and measured in dB/cm/MHz.[1][2]
• The attenuation coefficient is widely used in acoustics for characterizing particle size distribution.[1][2] A common unit in this contexts is inverse metres, and the most common symbol is the Greek letter $\alpha$.
• It is also used in acoustics for quantifying how well a wall in a building absorbs sound. Wallace Sabine was a pioneer of this concept. A unit named in his honor is the sabin: the absorption by a 1-square-metre (11 sq ft) slab of perfectly absorptive material (the same amount of sound loss as if there were a 1-square-metre window).[3] Note that the sabin is not a unit of attenuation coefficient; rather, it is the unit of a related quantity.

A small linear attenuation coefficient indicates that the material in question is relatively transparent, while a larger value indicates greater degrees of opacity. The linear attenuation coefficient is dependent upon the type of material and the energy of the radiation. Generally, for electromagnetic radiation, the higher the energy of the incident photons and the less dense the material in question, the lower the corresponding linear attenuation coefficient will be.

## Definitions and formulae

The measured intensity $I$ of transmitted through a layer of material with thickness $x$ is related to the incident intensity $I_0$ according to the inverse exponential power law that is usually referred to as Beer–Lambert law:

$I = I_{0} \, e^{-\alpha \, x},$

where $x$ denotes the path length. The attenuation coefficient (or linear attenuation coefficient) is $\alpha$.

The Half Value Layer (HVL) signifies the thickness of a material required to reduce the intensity of the emergent radiation to half its incident magnitude. It is from these equations that engineers decide how much protection is needed for "safety" from potentially harmful radiation. The attenuation factor of a material is obtained by the ratio of the emergent and incident radiation intensities $I/I_0$.

The linear attenuation coefficient and mass attenuation coefficient are related such that the mass attenuation coefficient is simply $\alpha/\rho$, where $\rho$ is the density in g/cm3. When this coefficient is used in the Beer-Lambert law, then "mass thickness" (defined as the mass per unit area) replaces the product of length times density.

The linear attenuation coefficient is also inversely related to mean free path. Moreover, it is very closely related to the absorption cross section.

## Attenuation versus absorption

The terms "attenuation coefficient" and "absorption coefficient" are generally used interchangeably. However, in certain situations they are distinguished, as follows.[4]

When a narrow (collimated) beam of light passes through a substance, the beam will lose intensity due to two processes: The light can be absorbed by the substance, or the light can be scattered (i.e., the photons can change direction) by the substance. Just looking at the narrow beam itself, the two processes cannot be distinguished. However, if a detector is set up to measure light leaving in different directions, or conversely using a non-narrow beam, one can measure how much of the lost intensity was scattered, and how much was absorbed.

In this context, the "absorption coefficient" measures how quickly the beam would lose intensity due to the absorption alone, while "attenuation coefficient" measures the total loss of narrow-beam intensity, including scattering as well. "Narrow-beam attenuation coefficient" always unambiguously refers to the latter. The attenuation coefficient is always larger than the absorption coefficient, although they are equal in the idealized case of no scattering.