When photons pass through matter, the intensity I of the...

When photons pass through matter, the intensity $I$ of the beam (measured in watts per square meter) decreases exponentially according to $$ I=I_{0} e^{-\mu x} $$ where $I$ is the intensity of the beam that just passed through a thickness $x$ of material and $I_{0}$ is the intensity of the incident beam. The constant $\mu$ is known as the linear absorption coefficient, and its value depends on the absorbing material and the wavelength of the photon beam. This wavelength (or energy) dependence allows us to filter out unwanted wavelengths from a broad-spectrum $\mathrm{x}-$ ray beam. (a) Two x-ray beams of wavelengths $\lambda_{1}$ and $\lambda_{2}$ and equal incident intensities pass through the same metal plate. Show that the ratio of the emergent beam intensities is $$ \frac{I_{2}}{I_{1}}=e^{-\left(\mu_{2}-\mu_{1}\right) x} $$ (b) Compute the ratio of intensities emerging from an aluminum plate 1.00 $\mathrm{mm}$ thick if the incident beam contains equal intensities of 50 $\mathrm{pm}$ and 100 $\mathrm{pm}$ x-rays. The values of $\mu$ for aluminum at these two wavelengths are $\mu_{1}=5.4 \mathrm{cm}^{-1}$ at 50 $\mathrm{pm}$ and $\mu_{2}=41.0 \mathrm{cm}^{-1}$ at $100 \mathrm{pm} .$ (c) Repeat part $(\mathrm{b})$ for an aluminum plate 10.0 $\mathrm{mm}$ thick.

When photons pass through matter, the intensity $I$ of the beam (measured in watts per square meter) decreases exponentially according to
$$
I=I_{0} e^{-\mu x}
$$
where $I$ is the intensity of the beam that just passed through a thickness $x$ of material and $I_{0}$ is the intensity of the incident beam. The constant $\mu$ is known as the linear absorption coefficient, and its value depends on the absorbing material and the wavelength of the photon beam. This wavelength (or energy) dependence allows us to filter out unwanted wavelengths from a broad-spectrum $\mathrm{x}-$ ray beam. (a) Two x-ray beams of wavelengths $\lambda_{1}$ and $\lambda_{2}$ and equal incident intensities pass through the same metal plate. Show that the ratio of the emergent beam intensities is
$$
\frac{I_{2}}{I_{1}}=e^{-\left(\mu_{2}-\mu_{1}\right) x}
$$
(b) Compute the ratio of intensities emerging from an aluminum plate 1.00 $\mathrm{mm}$ thick if the incident beam contains equal intensities of 50 $\mathrm{pm}$ and 100 $\mathrm{pm}$ x-rays. The values of $\mu$ for aluminum at these two wavelengths are $\mu_{1}=5.4 \mathrm{cm}^{-1}$ at 50 $\mathrm{pm}$ and $\mu_{2}=41.0 \mathrm{cm}^{-1}$ at $100 \mathrm{pm} .$ (c) Repeat part $(\mathrm{b})$ for an aluminum plate 10.0 $\mathrm{mm}$ thick.

Key Concepts

Intensity Ratio Analysis

Equating and comparing the attenuation expressions for different wavelengths leads to analyzing the ratio of the emergent intensities. This concept allows one to deduce the relative absorption differences through an exponential function of the difference in absorption coefficients and is important for practical applications in determining the filtering behavior of materials.

Wavelength Dependence of Absorption

The absorption characteristics of a material often change with the wavelength or energy of the incident photons. This wavelength dependence means that different wavelengths are absorbed at different rates, which is essential for applications like filtering and spectroscopy, where selectively removing certain wavelengths is required.

Linear Absorption Coefficient

The linear absorption coefficient is a material-specific parameter that quantifies the ability of a substance to absorb radiation. This coefficient depends on the composition of the material and the energy or wavelength of the photons, and it is a key factor in determining how quickly the beam's intensity decays with distance.

Exponential Attenuation

Exponential attenuation describes how the intensity of a beam, such as light or x-rays, decreases as it passes through a medium. This effect is governed by an exponential function that depends on the distance traveled through the material and the material's absorption properties, which is a fundamental concept in understanding photon interactions with matter.

Beer-Lambert Law

The Beer-Lambert Law provides the mathematical description of how light or x-rays are absorbed by a medium. It states that the transmitted intensity of the radiation decreases exponentially with the product of the linear absorption coefficient and the path length through the material, forming the basis for understanding the exponential attenuation model.

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