## 22 November 2015

### Do nano-objects have color?

I've been reading Jim Pivarski's blog Coffeeshop Physics for some time, and I always find the topics interesting and the perspective refreshing. However, I think that his latest post "Viruses have no color" contains a number of fundamental errors, beyond the imprecisions inherent in a simplified account.

Pivarski's stated point is that objects smaller than the wavelength of light have no color, and he explains this by the uncertainty principle. Instead, he illustrates that small objects scatter less light than large ones, using a "geometrical" point of view that ignores the composition of the objects and sees them simply as opaque to the incoming light. Of course, in this approximation even large objects are colorless, since their scattering properties will not change much over the visible spectrum1.

The relevant parameter when discussing the color of an object is not the wavelength but the frequency of the incoming light. For instance, gold nanoparticles a few tens of nanometers in diameter both absorb and scatter green light more effectively than at other visible frequencies because in this range the electromagnetic field couples very effectively with the oscillation modes (plasmons) of the conduction electrons in the particle. Dispersions of such particles are therefore green when seen in reflection and red in transmission, as illustrated by the Lycurgus cup. Even atoms can be said to "have color" if we think of their characteristic transition lines (for instance, sodium lamps glow yellow).

The uncertainty principle2 only tells us that the image of the nanoparticles cannot be sharper than the wavelength used to look at them, not that this image is colorless (see such colored images here and here).

1. I neglect here the λ4 dependence in Thompson scattering, leading to the "blue-sky effect".
2. I preserve here the author's terminology, although "the uncertainty principle" is generally associated with quantum mechanics. Here the reasoning is completely classical, so we might as well call the result "the Abbe resolution limit".

1. This is Jim Pivarski. Hi!

Yes, if you get a bunch of viruses together and crystallize them into a solid block or something, that block has a color. It might have different colors depending on the geometry of that crystal (structural color, like the colors in a peacock's tail: see Wikipedia), but there would probably be some consistency among them because the material of the virus's shell absorbs some frequencies of light better than others.

My point was to get the reader to think about the unusual world of objects whose size is at or smaller than the frequency of light. We live way at the other end of the scale: we use light as a probe of most everyday objects and most everyday objects are much larger than the wavelength of light.

This micro world represents a more general case and provides a good lead-in to the bandwidth theorem. The uncertainty principle in quantum mechanics is a special case of the bandwidth theorem, and I wanted to get there by thinking about classical waves, colors, and wifi data throughput, rather than the spookiness that is usually associated with quantum mechanics.

The title of my article, "Viruses have no color," was deliberately provocative and has attracted rebuttals from others as well, especially experts who crystallize viruses for study every day. However, I still stand by it. I feel that the concept of "color" is laden with size >> wavelength assumptions that break down in interesting ways.

For instance, I also refuse to admit that my eyes are blue. They're all sorts of colors. And then I learn that "blue" eyes are also products of structural coloration: put my eyes in a blender, whip them into a milkshake, and the blue is gone. That's not supposed to happen when we assert that X is blue!

1. Reference: https://medium.com/@ptvan/structural-eye-color-is-amazing-24f47723bf9a

2. Hi Jim, thank you for replying!

I believe the main difference between our perspectives is that in your post you only consider structural color, resulting from an ordering of the building blocks of the material over length scales comparable to the wavelength of visible light. By definition, then, the individual components do not have structural color, but they can still have an "intrinsic" color, as you mention in the first paragraph of your reply.

There are two problems with this separation:
- First of all, structural colors are not very common in Nature, most substances getting their color from microscopic-scale phenomena.
- Second, there is not always a clear separation between structure-related and "intrinsic" color: gold nanoparticles are green not in spite of their small size, but because of it (macroscopic gold is not green!) and the shade of green varies with their size. If you were to bring them together in a crystal, their color would again shift due to plasmon coupling between neighbours.

I understand that you intended the original article to be provocative, but you achieve this at the expense of silently redefining a very common and well-established concept, that of "having color" in a very restrictive way: "having structural color". You may have to choose between provocative and pedagogical...

3. The opposite: I consider pigments to be real colors, structural color to not be real. The reason is because the common (macroscopic) understanding of color is as a property of a substance, not a side-effect of shape and size.

For instance, people would rarely say that the bottom of a CD is "rainbow colored," even though the grating causes you to see rainbows when you hold it at a certain angle to the light. In the ordindary, everyday understanding of color, that rainbow effect would be considered a "sheen" of "trick of the light," not an intrinsic color like the redness of an apple.

But this everyday concept of intrinsic color breaks down when you get to the more general case of objects at the scale of the waves that are bouncing off of them. The reason my title was provocative was to confront the ordinary understanding of color as an intrinsic property of a substance. In a more fundamental sense, there is no intrinsic color. It's all a "trick of the light." At an extreme, there's absolutely no color that could be associated with an electron.

4. If I understand you correctly, leaves are green but chlorophyll has no color?