Exponential attenuation of X-rays

As an X-ray beam passes through a material, some photons are absorbed or scattered out of the beam. The transmitted beam therefore has a lower intensity than the incident beam. This reduction in intensity is called attenuation.

For a homogeneous material and a beam whose photons all have the same energy, each equal thickness of material removes the same fraction of the intensity that reaches it. A later layer therefore removes less intensity than an earlier layer, because less remains to be removed. The decrease is exponential, not linear.

The transmitted intensity after a thickness $$$x$$$ is:

I = I_0\mathrm{e}^{-\mu x}

Here, $$$I_0$$$ is the incident intensity, $$$I$$$ is the transmitted intensity, and $\mu$ is the linear attenuation coefficient. The coefficient depends on the material and the photon energy. A larger $\mu$ means a greater fraction of the beam is removed in a given thickness.

The exponent $-\mu x$ is dimensionless, so $\mu$ has units of inverse length, such as $\mathrm{m}^{-1}$ . The units of $$$x$$$ must be consistent with those used for $\mu$ .

Mass attenuation coefficient

The mass attenuation coefficient, $\mu_m$ , relates the linear attenuation coefficient to the density $\rho$ of the material:

\mu_m = \frac{\mu}{\rho}

The quantity $\rho x$ is the mass of material traversed per unit area. Substituting $\mu = \mu_m\rho$ into the attenuation equation gives:

I = I_0\mathrm{e}^{-\mu_m\rho x}

This form relates attenuation to mass per unit area rather than thickness alone. For example, if density is measured in $\mathrm{kg,m^{-3}}$ and thickness in $\mathrm{m}$ , the mass attenuation coefficient has units of $\mathrm{m^2,kg^{-1}}$ .

Half-value thickness

The half-value thickness, $x_{1/2}$ , is the thickness that reduces the transmitted intensity to half its incident value. Setting $$$I/I_0 = 1/2$$$ in the exponential equation gives:

x_{1/2} = \frac{\ln 2}{\mu}

Each successive half-value thickness halves the intensity remaining after the previous one. Thus, after one half-value thickness the intensity is $$$I_0/2$$$ ; after two it is $$$I_0/4$$$ ; and after three it is $$$I_0/8$$$ .

Differential X-ray absorption by tissues

As an X-ray beam passes through tissue, some photons are absorbed or scattered out of the beam. This reduction in beam intensity is called attenuation. The intensity that reaches the detector depends on how strongly the tissue attenuates the X-rays and how much tissue the beam passes through.

Different tissues have different attenuation coefficients, which describe how strongly they attenuate a beam. The coefficient depends on tissue properties, including composition and density, and on the energy of the X-ray photons. For a given beam, a more strongly attenuating tissue transmits less intensity than a less strongly attenuating tissue of the same thickness. A thicker layer also transmits less intensity than a thinner layer of the same tissue.

Tissue differences and image contrast

The detector receives different intensities from X-rays that have passed through different tissues or thicknesses of tissue. These differences produce contrast in the X-ray image.

Bone generally attenuates X-rays more strongly than soft tissue. For comparable thicknesses, fewer X-rays pass through bone to reach the detector, allowing bone to be distinguished from surrounding soft tissue.