Where does this formula for absorption cross section come from?

In summary, my professor gave us a formula for absorption cross section, but he said he did not remember where he found it. He speculated that it might be related to the electrostatic approximation for the absorption cross section of a dielectric sphere.
  • #1
Haorong Wu
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TL;DR Summary
Where does this formula of absorption cross section come from?
My professor gave us a formula for absorption cross section, but he said he did not remember where he found it.

The formula is given by

$$\sigma_a =\frac {1} {| \mathbf E_i|^2} \int_V k \epsilon^{''}_r | \mathbf {E} |^2 dV = k \epsilon^{''}_r |\frac 3 {\epsilon_r +2}|^2 V.$$

Where does this formula come from? I want to know what those variables stand for.

He gave this formula just after the formula of Rayleigh scattering cross section. It seems these two equations are related somehow, because he also gave the counterpart-formulars in Mie scattering theory.
 
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  • #2
Haorong Wu said:
My professor gave us a formula for absorption cross section, but he said he did not remember where he found it.

This made me giggle and made my soul hurt at the same time.

This looks like what you'd get if you took the imaginary part of the Clausius-Mossotti formula for electrostatic fields in a dielectric sphere (see Jackson, 3rd edition equation 4.56 and the derivation leading up to it). I'm going to go out on a limb and guess that this formula you gave is an electrostatic approximation for the absorption cross section of a dielectric sphere. The electrostatic approximation is valid when the wavelength is much larger than the radius of the sphere (Rayleigh scattering). In that scenario, the electric field due to the wave is approximately constant over the whole volume of the sphere, so you can use the Clausius-Mossotti formula (which assumes uniform fields).

As far as the symbols go, the standard conventions say:
##\epsilon_r''## ought to be the imaginary part of the relative permeability (the real part would be ##\epsilon_r'##) (phyiscally, the imaginary part corresponds to absorption of the EM wave by the material. If the permeability was entirely real, it'd just reflect or refract the light per the Fresnel equations)
V here is just the volume of the sphere
k is the wavenumber of the light
##E_i## is just the amplitude of the incident electric field
 

Related to Where does this formula for absorption cross section come from?

1. What is absorption cross section?

Absorption cross section is a measure of the probability that a particle or photon will interact with a target object and be absorbed by it.

2. How is absorption cross section calculated?

The absorption cross section is calculated by taking the ratio of the number of particles or photons absorbed by the target object to the number of particles or photons incident upon the object.

3. Why is absorption cross section important in scientific research?

Absorption cross section is important because it helps us understand how particles and photons interact with matter, which is crucial in fields such as atmospheric science, astrophysics, and materials science.

4. Where does the formula for absorption cross section come from?

The formula for absorption cross section comes from theoretical models and experimental data. It is based on the physical properties of the target object and the properties of the incident particles or photons.

5. Can absorption cross section vary for different types of particles or photons?

Yes, absorption cross section can vary for different types of particles or photons. It depends on factors such as the energy and wavelength of the incident particles or photons, as well as the composition and structure of the target object.

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