Snow Crystals

A snow crystal is a single crystal of snow, whereas a snow flake is a clump of snow crystals. Snow crystals show a six-sided symmetry, and often lie in a single plane. There is a book by W. A. Bentley, who photographed snow crystals for about five decades; his 2,400-some photos are in a 1931 book which has been republished by Dover. See the figure for an example, and see the end of this post for pointers to the book and some online resources.

Why are snow crystals symmetrical? There are a couple of hypotheses. (A) One, from Ken Libbrecht at Caltech Physics, is this: “Branches begin to sprout from the six corners of the hexagon… Since the atmospheric conditions (eg, temperature and humidity) are nearly constant across the small crystal, the six budding arms all grow out at roughly the same rate.” Secondly, Libbrecht notes, symmetrical crystals are rare — irregular crystals are much more common.

Another hypothesis (B) is that snow crystals grow upon a charged core, and the charge promotes growth which fills in gaps in the structure. That is, symmetrical growth is a lower energy state, hence encouraged.

These two alternative hypotheses might be tested, by a statistical examination of snow crystal photographs. Consider two arms separated by 60 degrees; call that type 60 symmetry. Or consider two arms separated by 120 degrees; that is type 120 symmetry. Or, opposite arms, type 180 symmetry. If hypothesis A (external conditions) holds, we would expect the context to be roughly constant across the crystal; hence type 60, vs 120, vs 180 symmetry should be about the same across a population of crystals. If hypothesis B holds (charge migration, energy minimum) we would expect type 60 symmetry to be stronger than type 120 symmetry, and type 120 to be stronger than type 180.

So there is interesting work to be done. Opportunity beckons.

Here are some pointers to get you started…

Book: Snow Crystals, by W. A. Bentley and W. J. Humphreys, 1931, Dover reprint 1962. A beautiful book to browse.

Website: http://snowcrystals.com — this website was written by Ken Libbrecht of Caltech Physics, and links to his academic pages at Caltech.

Website: http://www.its.caltech.edu/~atomic/snowcrystals/faqs/faqs.htm — this web page has the hypothesis A which was quoted above, as well as other interesting info, and links to other comparable pages.

Best wishes,
Ken Roberts
01-Dec-2015

Nice Integral for Zeta

Reading Feynman’s Statistical Mechanics book is a pleasure. There are gems to be found everywhere. Here is one little item, from page 37. You probably know this already, but it was nice to see the following explicit link between Fermi-Dirac integrals and the Riemann zeta function.

Consider the integral I = int(from 0 to +infinity) of (x / (exp(x) + 1)) dx. Here it is in Latex, which will give a nice picture at the top of this post: However, I will stick with plain text for the math in the remainder of this post. It is simpler to write plaintext math fast, and you can make the necessary translations to math symbolisms.

Write x/(exp(x)+1)=(x*exp(-x))/(1+exp(-x)) and expand that as a power series in w=exp(-x) as x*w-x*w^2+x*w^3-… which is, because w=exp(-x), simply an alternating sum of terms of the form x*exp(-n*x) for some n=1,2,3,… Those terms are to be integrated from 0 to +infinity to determine the value of I.

The integral of x*exp(-n*x)*dx over [0,+infinity] can be determined by integration by parts, differentiating u=x and integrating dv=exp(-n*x)*dx, to obtain 1/n^2. Thus I equals the alternating sum 1-1/2^2+1/3^2-1/4^2+…

You probably already recognize that. But if not, then let J=1+1/2^2+1/3^2+… which is the Riemann zeta function zeta(2) and (proof we owe to Euler in the first instance) equals pi^2/6. J/4=J/2^s equals 1/2^2+1/4^2+1/6^2+… so I=J-J/2. Hence I equals pi^2/12.

Very tidy. Familiar to you very likely. I suppose in some sense, previously familiar to me. But it was nice to encounter the Riemann zeta function in a discussion of a physical topic, ie the statistical mechanics of a Fermi gas. There are interesting connections between number theory and physics. This is only one indication of links.

I have read that Euler worked for ten years on the proof that zeta(2) equals pi^2/6. It was a famous problem, and there was numerical evidence, but a proof was not found — though some of the best minds (eg, the Bernoullis) were working on the challenge. Euler made his attempt, and probably revisited the problem from time to time, and ten years after he first encountered it, came up with his method of solution. Moral of this anecdote: keep working away at a problem, revisit it occasionally, and don’t be discouraged if you are not getting a solution rapidly.

Best wishes,
Ken Roberts
07-Jun-2014