Updated:January 23, 2008
Generic Vibrational Mode Broadening
in a Linear Nanoparticle Cluster


Both molecular dynamics simulations and simple arguments show that the l = 2 spheroidal modes ("football modes") of a nanosphere are broadened into a continuous band when many nanospheres are placed side by side in a linear chain. The width of the band does not depend on the details of the forces that hold the nanospheres to each other.

   In each case, only the red sphere (the nanosphere at the center of the chain) is given an initial distortion, and only motion of atoms in the red sphere are Fourier transformed to yield the spectrum.
   For simplicity, the isotropic elastic properties of fused quartz are used.

Figure 1. Using Borland Turbo C++ program cc3bpow1.cpp
(a)
(b)
(c)
(d) In this case, the particles are not actually touching. As a result, the vibrational mode at η = 2.6 is not broadened.
(e)
(f)

   In each case, there always seems to be a mode with the frequency η = 2.6 which happens to be the free sphere mode frequency for the l = 2 spheroidal mode. This corresponds to the situation where adjacent spheres have vibrations that are exactly out of phase so that as they vibrate they exert no forces on each other, like so:
Figure 2.
md43a.cpp
43g.gif		 
The other vibrational frequencies are higher. Increasing the length of the chain simply increases the density of spacing of the frequencies. As for the highest frequency mode, close to η = 3.3, this corresponds to the situation where neighbouring nanospheres are vibrating in phase, like so:
Figure 3.
md43b.cpp
43h.gif		 
This is equivalent to an isolated nanosphere vibrating with hard walls fixed at its right and left edges. Therefore, as a reasonable approximation, the range of frequencies of the linear cluster does not depend on the details of the coupling between the nanoparticles! This seems like good news, since such details would be hard to determine in any case.

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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