Viliani’s group in Trento has sent us coordinates for three samples of a Lennard-Jones glass, with 500, 2048, and 6980 atoms. Bill Garber and I have examined the vibrational normal modes in harmonic approximation. Some results are shown below.

First a word about the structure. The Lennard-Jones (LJ) potential V(r
)=4e((s/r)^{12}-(s/r)^{6})
contains a long-range Van der Waals attraction and a short range, nearly
hard-core repulsion. As is shown in textbooks, like Kittel’s, this interaction
gives a nice description of the rare gas solids like Ar, which crystallize
in the fcc structure. In this structure, each sphere is surrounded by 12
nearest neighbors whose position is just a little inside the minimum of
the Lennard-Jones potential at r=2^{1/6}s.
There is a shell of second neighbors more distant by 2^{1/2}, and
then increasingly large shells at increasingly smaller separations.

For the Lennard-Jones glass, we have looked at the mean distance to
the N’th neighbor, and also at the variance <dr^{2}>^{1/2}
of this quantity, which are plotted to the left. The units were chosen
to correspond to the Lennard-Jones potential which fits best crystalline
Ar. The closest 10 neighbors lie in narrow intervals near the distance
3.7 Angstroms found in crystalline Ar. The 11^{th} to 13^{th}
neighbors start to have less well-defined positions. The 15^{th}
to 18^{th} atoms have the least well-defined positions, lying near
the second neighbor shell of the crystal. More distant neighbors start
to be packed as in a random gas, with distance starting to scale like N^{1/3}.
A shell of radius r contains on average (4p/3)rr^{3}
atoms, where r is the number density. The mean
distance from an atom to its N’th neighbor is thus (3N/4pr)^{1/3}.

The structure of the glass seems somewhat like that of a smeared-out
crystal. The density is about 5% smaller. There may be icosahedral aspects
in the local packing, as has been studied for randomly packed hard-spheres.

The vibrational density of states is shown to the right. All three
models gave essentially the same result, which is shown here for the 6980-atom
model. The results for the glass are compared with the corresponding
density of states of the crystal. The crystalline results closely resemble
the experimental result for crystalline argon (Y. Fujii, N. A. Lurie, R.
Pynn, and G. Shirane, Phys. Rev. B**10**, 3647 (1974)) except for a
factor of two that came from a value of e that
was too large by 4. However, the LJ glass has a surprisingly
different density of states, being essentially featureless, with no remaining
indication of a distinction between longitudinal and transverse modes.

The
failure of glassy vibrational modes to exhibit diverse longitudinal or
transverse character is clearly illustrated in the next picture (to the
left), which shows the bond-stretching character of each mode in black,
contrasted with the same quantity for the crystal in red. The bond-stretching
parameter is defined by looking at one vibrational eigenvector, and calculating
what fractional component of the displacement vectors (of the atoms on
either side of a first-neighbor bond) are directed oppositely and along
the bond. This number can vary from 0 to 1 in a simple cubic crystal,
but is more tightly constrained in the close packed fcc structure.
The crystal has diverse modes. At a given frequency there are many
modes with different wavevectors, some of which may be primarily transverse,
and others primarily longitudinal, leading to the broad and colorful range
of normal modes. In the glass, the absence of symmetry means that
all purely sinusoidal modes of a given frequency are coupled to all other
purely sinusoidal modes of the same frequency (and different wavelength.
When the coupling terms are eliminated by unitary rotation to the normal
mode basis, the mixing effect of the coupling terms appears to rob the
resulting eigenvectors of all traces of individuality. The modes
appear to have a property of "local indistinguishability." This
means you could play the following game. Pick any two modes |1> and
|2> of nearly equal frequency and any two separated local regions R_{1}
and R_{2}. Then show your opponent the two eigendisplacements
|1> and |2> on the atoms in region R_{1}, explaining which is mode
1 and which is mode 2. Finally show your opponent the two eigendisplacments
in region R_{2}, without identifying which is mode 1 and which
is mode 2. Your opponent has to figure this out. There is no
way to do it short of obtaining the eigenvectors in between the two regions
by diagonalizing the huge dynamical matrix.

Finally we look at the localization
properties. At the highest frequencies, each normal modes is Anderson-localized
on an atom and a small number of adjacent sites. At very low frequencies,
most of the normal modes remember their propagating plane-wave character.
In between, they have neither property. We call them "diffusons"
because if a wave-packet is built from such modes ( in a narrow window
of frequency), and then allowed to evolve in time, there is no way to adjust
the amplitudes and phases of the component normal modes so that the resulting
wavepacket will propagate ballistically, but on the other hand, there is
also no way to prevent the energy in the normal mode from spreading diffusively
to the boundaries of the sample, in a time proportional to the square of
the distance to the boundary.

The
degree of localization is estimated from the participation ratio (inverse
of the sum of the fourth power of the eigendisplacement on an atom, where
the sum is over all atoms.) This quantity is equal to the number
of atoms if the mode has the same amplitude on each atom, as in a simple
plane wave, but if the mode is localized on N atoms, the participation
ratio is approximately N. We see that most of the modes in the 500
atom sample are delocalized over about 250, or half the atoms, and most
of the modes in the 2048 and 6980-atom samples are also delocalized over
half the atoms. These are modes which are definitely **not** Anderson-localized,
because the participation ratio is proportional to the size of the sample.
At frequencies higher than 17meV, the participation ratio scales less rapidly
than the number of atoms, and at frequencies higher than 18meV, the participation
ratio seems independent of the number of atoms. These last modes
are certainly Anderson-localized. Where exactly is the mobility edge
which separates localized from delocalized modes? It is not possible
to answer with high certainty from these three samples, but the answer
seems to be about 16.5 to 17meV. As the mobility edge is approached
from the delocalized (low frequency) side, modes have significant amplitudes
on smaller and smaller fractions of the atoms. For example, at 16meV,
in all samples, the modes seem to have significant weight on 28% of the
atoms. At 16.5meV, the proportion has gone down closer to 22% and
may be slightly sample-size dependent. Still larger samples
are required to fix the mobility edge definitely by this method, but we
can say with reasonable confidence that no more than about 7% of the modes
(those with energies higher than 16.5 meV) are localized. At the
very upper end, near 20meV, states are localized on very few atoms.
No doubt a careful examination of eigenvectors would show that these vibrations
localize in locally unusual regions of the glassy lattice.

Finally there is another puzzle, the occurrence of small participation ratios for a few of the low-frequency modes. A careful analysis has not yet been done. These modes are in the region where they should remember their propagating plane-wave character, at least over distances of a few wavelengths. However, it is well known that glasses have various low-energy anomalies: (1) "2-level systems" which dominate the thermodynamics at low frequencies and scatter sound waves rather efficiently, (2) the "boson peak" or excess low-frequency vibrations detected in various probes such as Raman scattering, and (3) anomalously large, and sample-dependent, low temperature thermal expansion. Are the low frequency modes with low participation ratios are connected with some of these anomalous properties? Are they intrinsic properties? The data shown above suggest that in ideal, homogeneous glasses (such as the three samples studied here), the anomalous participation ratio is sample-size dependent. This is perhaps consistent with the idea that the modes are "resonant" (i.e., not truly localized.) This means that they are temporarily trapped in locally peculiar regions which have a vibrational resonance at the frequency of the mode, but they also escape and propagate. As the sample size gets bigger, more and more of the modes should perhaps find a small region where they can resonate, but they also spend a correspondingly smaller fraction of their time in the resonant regions because these have a small, fixed size, while the rest of the sample grows with sample size.

last modified 10 November, 2001.

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