Latest revision as of 22:54, 24 July 2022

Claimed by Zhaoxian Zhang (Summer 2022)

Light scattering is the change in momentum and energy of light when it passes through an object. Light scattering occurs because light, as a form of electromagnetic wave[link], interacts with charge. This often happens with light interacting with dipoles[link]. dusts in the air, molecules in the air, particles in solutions, water droplets, and many other common things can form dipoles. These interactions lead to everyday phenomena like whiteness of cloud or the redness of sunset, and even governs the color of the sky.

Light scattering can be characterized into elastic or inelastic scattering, depending on how energy is exchanged during the interaction. In elastic scattering, the scattered particle conserves its original kinetic energy.

Mathematical Models of Scattering

In modeling scattering effects, a dimensionless parameter [math]\displaystyle{ x }[/math] is often used to characterize the size of the scattering particle in relation to the scattered wavelength. It is calculated with the following relation: [math]\displaystyle{ x = \frac{2 \pi r}{\lambda} }[/math] When [math]\displaystyle{ x\lt \lt 1 }[/math], or when the scattering particle is much smaller than the wavelength of scattered light, the scattering event can be characterized by Rayleigh Scattering[link]. When the particle size is larger, the scattering event is characterized by Mie theory[link] or discrete dipole approximation[link] by solving Maxwell equations[link].

Computational Models of Scattering

Examples of Scattering

Compton Scattering Compton scattering is an example of an inelastic scattering event, in which a photon gives a part of its momentum to an electron when it collides with it and scatters off at an angle. Because the transfer in momentum, the photon changes its wavelength after collision. The amount of change in wavelength can be derived from conservation of energy and momentum. Recall the energy equation: [math]\displaystyle{ E^2 = (\rho c)^2 + (mc)^2 }[/math] Since the energy lost by the photon is gained by the electron, [math]\displaystyle{ \delta E_{photon} = -\delta E_{\electron} }[/math] Consider the change in momentum of the photon, as it flies off at a different angle, is simply the vector difference of the momentums. When squared in energy calculation, this difference can be simplified using scalar product: [math]\displaystyle{ \delta E_{\gamma} = (vec{P_{gamma}}- vec{P_{gamma’}})*(vec{P_{gamma}}- vec{P_{gamma’}})c^2 \delta E_{\gamma} = (P_{gamma}^2 + P_{gamma’}^2 - 2 P_{gamma} P_{gamma’} cos \theta )c^2 }[/math] For momentum of a photon [math]\displaystyle{ P = hf/c }[/math], the above expression becomes: [math]\displaystyle{ \delta E_{\gamma} = ((\frac{hf}{c})^2 + (\frac{hf’}{c})^2 - \frac{hf}{c} \frac{hf’}{c} cos \theta )c^2 }[/math] Alternatively, the square of the change in energy of the electron is: [math]\displaystyle{ \delta E_{\e} = (mc^2)^2 +(\delta P_{gamma} c)^2 \delta E_{\e} = (mc^2)^2 =(\frac{hf}{c} - \frac{hf’}{c})^2 c^2 }[/math] Equate these expressions in energy and simplifying, we eventually get to the equation describing Compton scattering: [math]\displaystyle{ (\lambda - \lambda’) = \frac{h}{m_{e}c} (1-cos \theta) }[/math] Which describes how the frequency of the photon changes when scattering from an electron.

The Blue Sky While the derivation of original Rayleigh scattering equations is much more complicated, a simple dimensional analysis, as Lord Rayleigh did in his first analysis, is much simpler and reveals how the sky appears blue. Say the scattered light has electric field [math]\displaystyle{ E_{s} }[/math] . The intensity of the scattered light ==Connectedness

History

John William Strutt (Lord Rayleigh) (1842-1919) was born in the United Kingdom. He made fundamental discoveries in the fields of acoustics and optics that are basic to the theory of wave propagation in fluids. He was also responsible for much of the early discoveries in scattering.

His work on scattering began with the question of the color of the sky. John Tyndall[link] first discovered that light scattered from small particles appeared blue from the famous Tyndall effect. But it was Lord Rayleigh who worked out the math and solved the mystery of the color of the sky.

@@ Line 1: / Line 1: @@
-by Irene Hammel
+'''Claimed by Zhaoxian Zhang (Summer 2022)'''
-==Visible Light Spectrum==
+Light scattering is the change in momentum and energy of light when it passes through an object. Light scattering occurs because light, as a form of electromagnetic wave[link], interacts with charge. This often happens with light interacting with dipoles[link]. dusts in the air, molecules in the air, particles in solutions, water droplets, and many other common things can form dipoles. These interactions lead to everyday phenomena like whiteness of cloud or the redness of sunset, and even governs the color of the sky.
-Electromagnetic Radiation is light, but only a small part is visible to the human eye.  This portion is known as the visible light spectrum, the rest os the spectrum is either too large or too small for our eyes to receive. Visible light falls between infrared (IR) and ultraviolet (UV). It has wavelengths of about 740 nanometers (nm) to 380 nm. The image belows shows how small the portion of visible light is in comparison to all the wavelengths that exist.  The most important characcteristic of visible light is color.  Our eyes percieve each wavelength as a different color. The following section will go more in detail of how wavelengths relate to colors and how humans perceive them.
-[[File:Visible_light_spectrum.PNG]]
-===Wavelengths===
-The wavelengths of reflected light determine what color you see.  Light at the lower end of the visible spectrum is perceived as red.  Green light is found in the middle of the spectrum. While light at the upper end of the spectrum is seen as violet. All other colors that we perceive are mixtures of these colors. White light contains all colors and black is the absence of light.  Our eyes are able to percieve these colors because of cones our eyes contains.  Cones are photoreceptors, in other words tiny cells in the retina that respond to light. There are around 6 to 7 million cones in the light-sensitive retina at the back of your eye. In the image below you can see which wavelength represents each colors and also the relation of color and amplitude.
-[[File:Wavelength.PNG]]
-===Prism===
-A prism is is a transparent optical element with flat, polished surfaces that refracts light, its traditional shape is a triangle either way a prism will have two surfaces with an angle between them.  A prism breaks a white light into its individual colors, with their unique wavelengths.  This type of prism was created by Isaac Newton.
-[[File:Prism.PNG]]
-==Lightscatering==
-Light scattering can be thought of as the deflection from a straight path of a ray of light.  Objects are visible because of the light scattering from their surfaces. Scattering of light depends on the wavelength and/or frequency of the light.
-===Rayleigh Scattering===
-Rayleigh scattering is the scattering of light by molecules that are much smaller than the wavelength of the light. It occurs when light penetrates any of the three main phases of matter, gas, liquid, and solid. Rayleigh scattering intensity depends strongly on the size of the particles. This kind of scattering can be considered to be elastic scattering because the photon energies of the scattered photons do not change.
-=Why is the Sky Blue?=
-The sky is blue because of the light scattering phenomenon.  The rays of light from the sun travel through the atmosphere which contains air particles.  The sun rays appear white because they contain all the colors and wavelengths.  When the light goes through the air it hits the light molecules and bounces off into different directions.  Due to Rayleigh scattering blue is more scattered because it has a shorter wavelength and Rayleigh scattering states that shorter wavelengths scatter more strongly.  This scattered light that gives the surrounding sky its brightness and its color. Longer wavelengths pass through and are not difracted which is why they do not affect the color of the sky.
 [[File:Blue_sky.PNG]]
+Light scattering can be characterized into elastic or inelastic scattering, depending on how energy is exchanged during the interaction. In elastic scattering, the scattered particle conserves its original kinetic energy.
+==Mathematical Models of Scattering==
+In modeling scattering effects, a dimensionless parameter <math>x</math> is often used to characterize the size of the scattering particle in relation to the scattered wavelength. It is calculated with the following relation:
+<math>
+x = \frac{2 \pi r}{\lambda}
+</math>
+When <math>x<<1</math>, or when the scattering particle is much smaller than the wavelength of scattered light, the scattering event can be characterized by Rayleigh Scattering[link]. When the particle size is larger, the scattering event is characterized by Mie theory[link] or discrete dipole approximation[link] by solving Maxwell equations[link].
+==Computational Models of Scattering==
+==Examples of Scattering==
+Compton Scattering
+Compton scattering is an example of an inelastic scattering event, in which a photon gives a part of its momentum to an electron when it collides with it and scatters off at an angle. Because the transfer in momentum, the photon changes its wavelength after collision. The amount of change in wavelength can be derived from conservation of energy and momentum.
+Recall the energy equation:
+<math>E^2 = (\rho c)^2 + (mc)^2</math>
+Since the energy lost by the photon is gained by the electron,
+<math>\delta E_{photon} = -\delta E_{\electron}</math>
+Consider the change in momentum of the photon, as it flies off at a different angle, is simply the vector difference of the momentums. When squared in energy calculation, this difference can be simplified using scalar product:
+<math>
+\delta E_{\gamma} = (vec{P_{gamma}}- vec{P_{gamma’}})*(vec{P_{gamma}}- vec{P_{gamma’}})c^2
+\delta E_{\gamma} = (P_{gamma}^2 + P_{gamma’}^2 - 2 P_{gamma} P_{gamma’} cos \theta )c^2
+</math>
+For momentum of a photon <math> P = hf/c </math>, the above expression becomes:
+<math>
+\delta E_{\gamma} = ((\frac{hf}{c})^2 + (\frac{hf’}{c})^2 - \frac{hf}{c}  \frac{hf’}{c} cos \theta )c^2
+</math>
+Alternatively, the square of the change in energy of the electron is:
+<math>
+\delta E_{\e} = (mc^2)^2 +(\delta P_{gamma} c)^2
+\delta E_{\e} = (mc^2)^2 =(\frac{hf}{c} - \frac{hf’}{c})^2 c^2
+</math>
+Equate these expressions in energy and simplifying, we eventually get to the equation describing Compton scattering:
+<math>
+(\lambda - \lambda’) = \frac{h}{m_{e}c} (1-cos \theta)
+</math>
+Which describes how the frequency of the photon changes when scattering from an electron.
-=Why is the Sunset Red?=
+'''The Blue Sky'''
-As the sun begins to set, the rays of sun must travel further through the atmosphere before it gets to you. The longer it takes the more wavelengths are already scattered, so the color of the sun itself appears to change, first to orange and then to red. Since blues and greens have shorter waveengths by the time they reach you hey are already scattered.  Only the longer wavelengths are left which is why you are able to see a red sunset.
+While the derivation of original Rayleigh scattering equations is much more complicated, a simple dimensional analysis, as Lord Rayleigh did in his first analysis, is much simpler and reveals how the sky appears blue.
+Say the scattered light has electric field <math>E_{s}</math> . The intensity of the scattered light
-[[File:Red_sunset.PNG]]
+==Connectedness
-=Why is the Universe Black=
-Since there is no air in the atmosphere no light wavelength is difracted so it appears to be black.
-[[File:Space.PNG]]
-=History=
-Isaac Newton (1643-1727) was born in England.  Although studied law, he was an established physicist and mathematician, and is credited as one of the great minds of the 17th century Scientific Revolution.  Between his biggest discoveries, are the theory of gravity, the three laws of motion, calculus and the refraction of light.
-[[File:Isaac_newton.PNG]]
-John William Strutt (Lord Rayleigh) (1842-1919) was born in the United Kingdom.  He made fundamental discoveries in the fields of acoustics and optics that are basic to the theory of wave propagation in fluids.
+==History==
+John William Strutt (Lord Rayleigh) (1842-1919) was born in the United Kingdom.  He made fundamental discoveries in the fields of acoustics and optics that are basic to the theory of wave propagation in fluids. He was also responsible for much of the early discoveries in scattering.
 [[File:Rayleigh.PNG]]
+His work on scattering began with the question of the color of the sky. John Tyndall[link] first discovered that light scattered from small particles appeared blue from the famous Tyndall effect. But it was Lord Rayleigh who worked out the math and solved the mystery of the color of the sky.
 = See also =

Light Scattering: Why is the Sky Blue: Difference between revisions

Latest revision as of 22:54, 24 July 2022

Contents

Mathematical Models of Scattering

Computational Models of Scattering

Examples of Scattering

History

See also

Further reading

External links

References

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