Friday, March 14, 2014

Light Scattering

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I often discuss the scattering of light. We mention four types of scattering, each differentiated by the name of the brilliant scientist who first studied it: Compton scattering, Thomson scattering, Rayleigh scattering, and Raman scattering. Let’s see if we can get these all straight.

Compton Scattering

In Chapter 15 (Interaction of Photons and Charged Particles with Matter) of IPMB, Russ and I analyze Compton scattering. This is a particularly simple case: a photon interacts with a free electron, resulting in a scattered photon of lower energy and a recoiling electron. This type of scattering is particularly important for x-rays. You might be wondering how often do we encounter a free electron? Aren’t most electrons bound to atoms? If the incident photon has an energy much greater than the binding energy, then the electron is to a first approximation free and Compton scattering occurs. In the interaction of x-rays with biological tissue, Compton scattering is the dominant mechanism contributing to the interaction cross-section at intermediate energies; say, one tenth to a few MeV. Since the electrons act almost as if they were free, the atomic number of the target atom is unimportant and scattering depends only on how many electrons are present (meaning the mass attenuation coefficient is nearly independent of atomic number). You don’t really want to do imaging of tissue when Compton scattering is the dominate interaction because you don’t get much discrimination between different tissues (the weak dependence on atomic number) and, well, you get a lot of scattering that blurs the image.

Compton scattering is named after Arthur Holly Compton (1892–1962), an American physicist who played a key role in the Manhattan Project. Compton scattering was important in the development of quantum mechanics. The light quanta hypothesis had been developed by Planck and Einstein, but was not widely embraced until 1923, when Compton analyzed his x-ray scattering data by treating the x-ray photon as a particle with energy and momentum, interacting with another particle, the electron. Compton won the 1927 Nobel Prize in Physics for his discovery.

Thomson Scattering

When Compton scattering occurs at such a low energy that we can ignore the difference in energy between the incident and scattered photons, the process is called Thomson scattering. We can analyze Thomson scattering by treating the incident light as an electromagnetic wave rather than a photon. The electric field accelerates the electron, causing it to radiate an electromagnetic wave at the same frequency. The direction of the electric field is important for determining the distribution of the outgoing dipole radiation, so Thomson scattering depends on the polarization of the incident light. This type of scattering is particularly important in plasma physics, where many free charged particles are present. It is not too important in biology and medicine, because usually either the photon energy is so high that Compton scattering occurs, or else the photon energy is so low that one cannot treat the electron as being free. Because the frequency of the light (and therefore the energy of the photons) does not change, Thomson scattering is a type of elastic scattering.

Thomson scattering was first analyzed by, and was named after, J. J. Thomson (1856–1940), the British physicist who discovered the electron, for which he received the Nobel Prize in Physics in 1906. I have my own connection to Thomson: academically speaking, he is my great-great-great-great-great-grandfather.

Rayleigh Scattering

Rather than scattering from a single electron, light can also scatter from an entire atom or molecule, and even larger particles. When the wavelength of the light is much larger than the size of the particle, we get Rayleigh scattering. Like for Thomson scattering, in Rayleigh scattering the light is treated as an electromagnetic wave. However, unlike Thomson scattering, in Rayleigh scattering the scatterer is not a single particle, but instead can be represented by a continuous, polarizable medium. The electric field of the light causes the induced charge distribution to oscillate at the same frequency as the incident light, resulting in the scattered light having the same frequency as the incident light. In IPMB, Russ and I refer to Rayleigh scattering as coherent scattering, because the atom responds coherently as a whole, rather than as individual charged particles. In tissue, coherent scattering dominates Compton scattering at low energies (say, below 1 keV), but such low energy photons also interact by the more important photoelectric effect, so Rayleigh scattering is often not very important. It is crucial for understanding how sunlight scatters off the molecules of the air, causing the blue color of the sky.

When I was an undergraduate at the University of Kansas, I had my first research experience in Professor Wes Unruh’s laboratory studying light scattering off of colloidal impurities in crystals. We were able to determine the size of the impurities by measuring the scattered light as a function of angle. However, these colloids tended to be large, so that you could not ignore interference between light scattered from different parts of the particle. In that case, you must use a more advanced theory, called Mie theory, to calculate the distribution of scattered light. I recall struggling to learn Mie theory from Milton Kerker’s book The Scattering of Light and Other Electromagnetic Radiation. I didn’t work much with Unruh himself, but rather was mentored by then-graduate student Robert Bunch. The first item in my CV is an abstract resulting from that research (Bunch, Roth, and Unruh, 1983, “Size Distributions of Ni and Co Colloids Within MgO,” March Meeting of the American Physical Society).

Rayleigh scattering is named after English physicist John William Strutt (1842-1919), also known as Lord Rayleigh. He was awarded the Nobel Prize for Physics in 1904 for the discovery of argon. Because one of Rayleigh’s students was J. J. Thomson, Rayleigh is my academic great-great-great-great-great-great-grandfather. Rayleigh was the second Cavendish Professor of Physics at the University of Cambridge, following Maxwell and succeeded by J. J. Thomson, Ernest Rutherford, and William Bragg; quite an impressive bunch.

Raman Scattering

In IPMB, Russ and I discuss Raman scattering in Chapter 14 (Atoms and Light). The mechanism of Raman scattering is similar to Rayleigh scattering, in that the scattering occurs off an entire molecule. However, it is unlike Rayleigh scattering in that the scattered light does not have the same frequency as the incident light (inelastic scattering). Instead, some of the energy induces transitions between different vibrational energy levels. These transitions result in the scattered light having a lower energy (Stokes) or a higher energy (Anti-Stokes). Also, because the vibrational energy levels are quantized, the spectrum of Raman scattered light consists of a series of discrete lines. This spectrum contains information about the vibrations within the molecule, and therefore about the chemical bonds.

The description of Raman scattering given above (and in IPMB) is a quantum view that depends on the presence of discrete energy levels. However, one can also develop a classical model of Raman scattering. For instance, treat a simple diatomic molecule as two atoms attached by a spring, so that the molecule has its own natural frequency of oscillation, fo. If an electric field of frequency f is incident on the atom, it will respond by not only oscillating both at frequency f (Rayleigh scattering) but also at frequencies f+fo and f-fo (Raman scattering). The frequency difference between adjacent lines is fo, which is the same frequency as one would expect in the infrared absorption spectrum. (For those who have read Appendix F of IPMB and are wondering why the the scattered light oscillates with a component at the natural frequency, realize that the charge induced by polarization depends on the electric field, so the force on the charge--charge times electric field--depends on the square of the electric field and the problem is nonlinear.)

Raman scattering was named after Indian physicist C. V. Raman (1888–1970), whose discovery led to the 1930 Nobel Prize for Physics.

Four types of scattering, named after four Nobel Prize winners. Here are some ways to keep them straight: Compton and Thomson scattering is off a single charged particle (usually an electron), whereas Rayleigh and Raman scattering is off an entire atom or molecule or particle. Thomson and Rayleigh scattering are elastic, whereas Compton and Raman scattering are inelastic. Thomson and Rayleigh scattering are most commonly described using the classical wave theory of light, whereas Compton and Raman scattering are typically analyzed using quantum mechanics (although Raman scattering is sometimes analyzed with classical theory).

I admire all four scientists: Compton, Thomson, Rayleigh, and Raman. Who is my favorite? I like Rayleigh best. Love those Victorians.

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