Cluster modes, 8th of June 2012.
I have first read about viruses as physical systems sometime in year 2003 or 2004. This subject appeared with another system I studied
at that time: icosahedral clusters. It is known, namely, that the clusters containing several hundreds of atoms are often non-crystalline
and have central, icosahedral symmetry. The same kind of symmetry is characteristic of viruses.
Clusters with closed shells are particularly symmetric. These appear for "magic" number of atoms: 13, 55, 147, 309, 561, 923, 1415 ... We can imagine
these clusters as "Russian dolls" (babushkas) in which the icosahedral shells of atoms are one within the other, all the way down to the atom
in the center of the cluster. For example, in the image above, the left icosahedral cluster (upper row) is within the one in the middle, and
the one in the middle is within the right one. We could say that the magic clusters are built in a layered manner as "icosahedral onion".
It is even more interesting that for the same array of magic numbers we can also build shelled clusters, but of different geometry and symmetry.
The clusters that I talk about are pieces of FCC (face-centered-cubic) crystal and they have a form of cuboctahedra (octahedra with 6 pyramids
around the vertices cut off, see the lower row o clusters in the image above). Since we know that the crystal is a favorable energy state (at
T=0 K) of a large number of atoms, we expect that the cuboctahedral clusters are favorable energywise after some critical and large enough
number of atoms i.e. shells.
The answer to this question was known within the classical-mechanical approach to the problem, and I wanted to see how the situation changes
in the quantum-mechanical description. To calculate the quantum-mechanical energy, I had to calculate the vibrational modes of the clusters and
their frequencies. This is where the connection of this research and Construction of Reality begins.
Namely, the calculation of vibrational modes, besides their frequencies, also gives the so-called eigen-vectors of vibration. Each of
these vector
contains an array of displacements of each atom in the cluster and it shows the pattern of vibration of all the atoms in the cluster for a
particular mode. I, of course, wanted to visualize the results I got and I spent quite some time thinking about how to do it in the
best and the simplest way.
In the end, I decided to use the Graphics3D representation of Wolfram's Mathematica. Although such 3D representation is relatively poor and
really basic, its advantage is that it can be easily written down. So, my FORTRAN code produced the output which was interpreted and represented
by Mathematica. It all became even better, since at that time (2003/2004) a very stable version of Java applet for representation of
Graphics3D objects appeared. It was a sort of a Java interpreter of the Mathematica syntax for graphic objects. The applet was coded by
Martin Krauss, and you can see it in action in the window above (by the way, my Firefox 11.0 has some problems in showing the applet
correctly (interestingly, it works nicely in IE9) - it is obvious that a lot has changed in the web world since year 2003).
Logo of LiveGraphics3D application with >> a link can be seen
in the image below
If you move the mouse over the window which shows the vibrational mode of the cluster, you will be able to rotate the cluster by clicking
the mouse and moving it. If you combine SHIFT + click + mouse movement you will be able to zoom in/out of the object.
You should easily note that each atom in the cluster has the corresponding arrow. When the cluster vibrates, its atoms move along the directions
denoted by arrows. The arrows thus denote a collective nature of cluster vibration (mode).
For the cluster in the window above, one can see that the vibrating atoms "move in" the cluster from the two parallel planes, and "move out"
of it across two parallel planes, perpendicular to the "entry" planes (do not forget that, during the vibration, atoms also return back along
the same lines, so that the oscillate along the lines denoted by arrows).
The number of cluster vibrational modes is 3N - 6, where N is the total number of atoms in the cluster. The clusters shown in the two windows
above have thus 3*55 - 6 = 159 modes of vibration. I will, of course, not show all of them here. I have chosen another one of them and
shown it in the window above. This vibrational mode is similar to the pattern of lines of force of a rod magnet. The atoms in some
"middle line" of the cluster move along the line, and all the surrounding atom "return back", closing thus the "flow" of vibration.
Somewhat more detailed story about the patterns and energies of vibration can be found in
>> my paper
(A. Šiber, Phys. Rev. B 70, 075407 (2004)). In this work, I made some of the images by exporting Graphics3D syntax to postscript (I intend
to devote several posts to postscript format in the future).
I will say nothing about the mode in the window above. I leave it to you to try to grasp its three-dimensional nature (hint: it has something to do with triangle).
In the end, in the window above I show a mode with a really high frequency (energy). In it, the whole icosahedral shell of 12 atoms moves
toward the central atom and away from it. Such way of vibration requests a low of energy, so that the corresponding frequency is high.
Download the paper >> A. Šiber, Phys. Rev. B 70, 075407 (2004) for more details.
Of course, the clusters can also be seen in the desert, as is shown in the image below. >>
Click on the image (panorama) to see it in double resolution.
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Last updated on 8th of June 2012.