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Spare me the math: Raman scattering

August 12, 2013

For most of my (still nascent) scientific career,  I have worked on the physics of materials. This likely sounds pretty humdrum to you.  To me, at least, the terms “materials” or “materials science” conjure up images of stodgy old nerds meticulously optimizing the chemical composition of some slurry to be used in one or another mind-numblingly specific manufacturing process.

“Hmm, perhaps we should add another 250 ppm of Vanadium…”

[The likely source of this prejudice is my stodgy old materials science professor from college, who made us spend 4 months memorizing the iron-carbon phase diagram.]

But for a physicist, the study of materials can be something significantly more dramatic and imaginative.  In short, every new material is like a new, synthetic universe.  It has its own quantum fields that arise from the motions and interactions of the atoms in the material.  And these fields give rise to their own kinds of particles, which may look a lot like the electrons, atoms, and photons we’re used to, or which may have completely different rules of engagement.

For example, the recently-discovered graphene is essentially a two-dimensional universe where electrons have no mass and the speed of light is 300 times smaller than its normal value.  For a physicist, that’s a dramatic thing.

The downside to working in materials is that it’s often hard to see what’s going on, and to know whether the material you have is the same as the material you think you have.  For this reason materials scientists become dependent on a barrage of characterization methods, each of which probes some slightly different aspect of the material’s properties.

In this second edition of Spare Me the Math, I want to describe one of the most crucial and ubiquitous of the material characterization tools: Raman scattering.

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In Raman scattering, you shine light on a material and see some of it get reflected back with a different frequency (that is, a different color).   Since light is made of individual photons, and the energy of these photons is proportional to their frequency, this shift in frequency means that the light is gaining or losing some of its energy inside the material.

As it turns out, that lost energy goes into exciting a vibration in the material.  (Or, conversely, the light can gain energy by stealing some existing vibrational energy from the material).  Thus, the shift of the light tells you something about the way the material vibrates, and therefore about what the material is made of.

But how, exactly, is the light frequency getting shifted?  How does light energy get mixed up with vibrational energy?

Usually the process of Raman scattering is explained only in terms of some “electron-phonon scattering” and accompanied by opaque diagrams like this one:

Raman_vertex

or this one.  But what is really going on?

In this post I want to try and demystify this process a little bit, by explaining how, exactly, the light frequency gets shifted.  It turns out that pretty much everything can be understood by imagining that your material is made of stretchy metal balls.

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Imagine, for a moment, a metal ball.  Like this:

ball

This metal ball will stand for a molecule; you should imagine that there are bazillions of little metal balls making up your material.

A metal ball is like a molecule in the sense that it can rearrange its electrons a bit to adapt to the presence of an electric field: the ball is polarizable.  So when an electric field gets applied, the ball moves some positive charge to one side and some negative charge to the other side.  Like this:

ball-E

In polarizing this way, the ball creates its own electric field that partially counteracts some of the applied electric field.

When an oscillating electric field (a beam of light) is applied to the ball, the electric charge on the ball redistributes to point in the direction of the light’s electric field.  Like this:

polarized_ball

In this picture, the arrow above shows the direction and magnitude of the electric field coming from the light, and the colors show the induced charge density (red for positive, blue for negative).

It is common to say that by this process of polarizing back and forth, the metal ball “scatters” some of the incident light.  But one can just as well say that in sloshing its charge back and forth, the ball creates is own light waves that emanate outward.

(I like this language a little better.  When the sky is a beautiful vibrant blue, I like to think how all the little molecules in the air are getting excited by the sun and glowing bright blue light in my direction, like bioluminescent algae.)

In the picture above, however, the ball’s electric charge is oscillating in lock-step with the applied electric field, which means that its own radiated light is at exactly the same frequency \omega as the incoming light.  To get the frequency shift implied by Raman scattering, we have to make the ball a bit stretchy.

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So imagine now that the metal ball is a bit squishy, and can get stretched out in different directions the same way that a real molecule can.  Say, like this:

stretched_ball

Crucially, you should notice that in its stretched-out state, the ball responds differently to an applied electric field.  Basically, it puts positive and negative charge further apart, and in doing so it does a better job of screening out the applied field.  Like this:

stretched_ball-E

Since the ball is elastic, however, it can easily start wobbling back and forth when it gets stretched out.  Like this:

wobbling_ball

The frequency of this wobbling, which I call \Omega, depends on how stiff the ball is.  Every molecule has its own characteristic wobbling frequency (or rather, its own set of frequencies, one for each different way it can be excited.)

Now, the essence of Raman scattering can be understood by thinking about what happens if you try to apply an oscillating electric field while the ball is wobbling.  Basically, the electric polarization frequency and the wobbling frequency both get involved in determining how light is radiated by the ball.  The picture is something like this:

wp_ball

Here you can see how the light frequency and wobbling frequency get mixed up.  The ball is sometimes at its fattest while the electric field is strongest (making ah enhanced dipole) and sometimes at its thinnest while the electric field is strong (making a weakened dipole).  As a result, the ball radiates some light at the original frequency \omega and radiates some light at the shifted frequencies \omega + \Omega and \omega - \Omega.  Its just like beating in sound waves: when something is the outcome of two frequencies simultaneously, you see (hear) the sum and the difference of the two frequencies also.

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And that’s how you can shine light on a material and see some of it come back to you at a different frequency.  The point is that the molecules inside the material are like squishy metal balls: they polarize, and they wobble. And so the light that they radiate has information about the frequencies of both processes.  By applying light with a known frequency, you can figure out how quickly the material’s constituent molecules wobble, and thereby say something about what they are made of.

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Footnote

You may well ask “what gets the ball wobbling in the first place?”  It turns out there are two possible answers.  First, the ball can start wobbling just by random kicks that it gets from its thermal environment.  In this case the intensity of the scattered light is proportional to the square of the temperature multiplied by the square of the incident electric field.

On the other hand, if the temperature is small or the intensity of the applied light is large, then the ball can start wobbling due to the stretching forces it feels from the light itself.  (Briefly, when a molecule gets electrically polarized, it feels a force pushing it apart that is proportional to the strength of the applied field.)  This is called “stimulated Raman scattering,” and it produces a scattered light intensity that is proportional to the fourth power of the incident electric field.

3 Comments leave one →
  1. October 9, 2015 1:33 pm

    Great post, thanks. Regarding the classic explanation of Raman scattering, I still don’t understand why one would introduce ‘virtual energy level’ concept.

    • Brian permalink*
      October 9, 2015 1:43 pm

      Thanks, I’m glad you liked it.

      I guess that the energy level is “virtual” in the sense that it is not stable: any energy put into the system will be radiated away again. And the language of virtual energy levels might be the more appropriate description when you think quantum mechanically about individual photons being absorbed. But I guess my point here is that the shift in frequency can be understood at a completely classical level.

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