100 Nano-Stories: Mie Scattering & Rayleigh Scattering (Mathematics)!
Episode #78: Scattering Coefficient Geometry & Optical Theory (Radar Cross Section x Atmosphere)!
Preface! ✨
It’s your favorite material science & nanotechnology enthusiast! Today, we will cover a new theory in the series of nanotechnology and aerogel: Mie Scattering Equations!
If you need a reminder on one main application of the Mie theory in aerogels, check out this article below!
Don’t worry, it’s only a 4-minute read! 😁
TL;DR → Mie Scattering & Mie Theory! 🔑
- Mie Scattering → A form of scattering that can occur in a spherical particle that either has the same diameter or larger than a certain wavelength of light that will enter the molecule/spherical particle (aka incident light).
- Mie Scattering will mostly occur where the sky (atmosphere) is less monochromatic. This will result in the sky blending in with white/ less saturation. Mie Scattering is the white halo that you see around the Sun (makes The Sun look white in the sky).
- Saturation → A color with no mixture of white (pure red, pure purple, etc.).
- Monochromatic → Contains only one wavelength of light and/or one color.
Now that we have the main definitions done, let’s introduce some warm equations to kick us going! Don’t worry, I won’t confuse you that fast! 😉
Mie & Rayleigh Scattering Equations! 💡
Authors Note! 🔑
When I think of aerogels being exposed to light, the obvious source of light comes from the Sun and the atmosphere. So for this article, I will use the atmosphere, particles/molecules in the atmosphere, and the photons (light) from The Sun.
While this demo is not 100% accurate for aerogels (because the aerogel network of particles is inside the aerogel, not exposed to the atmosphere), I want to explain it abstractly before going specific! 😄
Concepts! 🔑
Rayleigh Scattering is a form of scattering of photons that occurs in the atmosphere if the particle is smaller than 1/10 of the wavelength of light in the size of a certain color in the visible light spectrum. (Particle Size << 1/10 (λ)).
Mie Scattering is a form of scattering of photons that occurs in the atmosphere if the particle is bigger than 1/10 of a wavelength of light in size in the visible light spectrum but is smaller than a full wavelength of light in size of a certain color in the visible light spectrum. (1/10 (λ) << Particle Size << (λ)).
Visible Light Spectrum Wavelengths are between 380–700 Nanometers (nm). A nanometer is one-billionth of a meter.
Wavelength Symbol → λ.
Equations! 🔑
To begin the explanation, I want to show you this photo that describes the direction of the light when it hits a certain particle that has the size that will obey either Rayleigh or Mie Scattering.
Incident Light is the light that will enter the molecule/spherical particle/material.
The reason the Rayleigh Scattering prefers to be scattered in all directions is that the size of the particles (mostly oxygen and nitrogen from the atmosphere) is less than 50 nanometers.
But notice that some light is reflected in the direction that the incident light came from. That light is interfering with the incident light that is trying to pass/scatter with that air molecule/particle. Sometimes, that light will either get absorbed by other molecules or the amplitudes of the wavelengths could extinguish each other/reflected light goes extinct.
Amplitude → The amplitude starts at the middle/origin to the crest/hilltop of a wave.
For those confused, here’s a photo of a possible scenario of the extinction of the light that is reflected towards the incident light (aka back-scattering):
When this happens, we need to bring out our old friend: The Extinction Coefficient! 😄
The extinction coefficient is the sum of an absorption coefficient (the light that could get absorbed by other particles) and a scattering coefficient/efficiency (the light that gets scattered by the air molecules or the particles/material themselves).
However, we want to use the scattering coefficient to find the scattering cross-section of the particles. The Scattering Cross-Section is where the radiation–target interaction occurs (radiation is the wavelength of light, the target is the particle). If the light passes through this cross-section of the particles and the incident light, the light will be deflected.
The equation to describe the scattering cross-section due to the wavelength and the particle would be:
x≡2πr/λ
≡ → Identical to (DOES NOT mean equal to).
x → The size of the particle.
r → Radius.
λ → Wavelength.
The reason why we only use radius instead of diameter (aka the length of the particle) is that there will be a region of the particle where the light will not be able to pass because it will be deflected at an angle before it crosses this area. In this case, the area that the particle can’t cross is half the diameter of the particle (aka radius.)
To make the equation easier to understand, we can divide 2π into both sides to get this equation:
x/2π = r/λ
Our extinction coefficient is now a function of the scattering cross-section of r/λ!
Scattering Cross-Section Explained via Geometry! 🔑
Scattering Coefficient → σ
σ=πr²
Geometrical Scattering Cross Section is:
σ/πr² (πr²/πr²)
As the particle gets bigger (x), so does the radius of the particle divide by the wavelength (r/λ).
Once the particle size (x) is roughly equal to 2π or 1, the scattering will start moving in the forward direction, or the same direction as the incident light.
This happens because the scattering cross-section has reached its peak and now begins to decrease and oscillates around 2π or 1, which makes the waves smaller and closer to each other. This results in the waves of light having a higher frequency, and therefore, the wavelengths of light have been closer to the blue and ultraviolet wavelengths of light.
There is a reason for this case of higher frequency for the (scattering cross-section) coefficient of the extinction coefficient to be around 2π or 1.
Optical Theory! 🔑
- As the scattering efficiency/cross-section increases to 2π or 1, the frequency will increase, and the power/direction of the scattering will move in the forward direction. This makes the wavelengths very small, at the same size as UV Light, which makes the particle look like a disc to the light approaching the particle. This causes a shadow to form on the forward scattering, which has to occur to cancel the incident wave of light to create the shadow (shown below).
Note that we are only talking about the wavelength of light, not the particle itself. Rayleigh Scattering has long wavelengths of blue light, while Mie Scattering can have shorter wavelengths (higher frequencies) that are in the UV Light, which results in this forward scattering!
Closing Thoughts! 💭
This was a tough article to write, readers!
To summarize this article:
- The Incident Wavelength of Light and the particle size can play a huge role if the scattering will obey Rayleigh Scattering or Mie Scattering.
- Instead of using the extinction coefficient to find out the scattering efficiency of the particles, we used The Scattering Cross-Section which is where the radiation–target interaction occurs (radiation is the wavelength of light, the target is the particle).
- The Geometry of Mie Scattering says that once we reach a particle size of 2π or 1 (which is a constant number), we begin to oscillate and the frequency of the wavelengths get smaller and smaller, which pushes more towards blue, violet, and Ultraviolet (UV) Wavelengths of light, which is backed by The Optical Theory.
If I didn’t clarify well enough in this article, please throw a comment, and I will edit this to help you understand more about Mie Scattering itself!
See you tomorrow for what happens if the particles are bigger than a certain wavelength of light! ✌🏽
Vocabulary! 📓
Mie Scattering → A form of scattering that can occur in a spherical particle that either has the same diameter or larger than a certain wavelength of light that will enter the molecule/spherical particle (aka incident light).
Monochromatic → Contains only one wavelength of light and/or one color.
Incident Light → The light that will enter the molecule/spherical particle/material.
Wavelength Symbol → λ
Rayleigh Scattering → The wavelength of a certain ray of light is short, then the wavelength of light will scatter more than a ray of light where the wavelength is larger, and Rayleigh Scattering tends to scatter blue light because of its wavelength (450–485 nanometers).
The Scattering Cross-Section → The radiation–target interaction occurs (radiation is the wavelength of light, the target is the particle).
Amplitude → The amplitude starts at the middle/origin to the crest/hilltop of a wave.
Optical Theory → The higher the frequency of a wave (smaller wavelength of light like blue light or UV Light), the more scattering of light will go in the forward direction.
UV → Ultraviolet Light. The wavelength of UV Light is between 100–400 nanometers.
Bonus Resources! 📓
Previous “100 Nano-Stories!” 🔖
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