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The Harmonic Approximation
Figure 5:
Probability distribution P(x) for inplane displacements
about spheres' mean positions for singleparticle trajectories
averaged over 1024 video frames (33 seconds).
Statistics were acquired along an arbitrary direction,
taking advantage of the separability
of Eq. (7).
The solid lines are fits to Eq. (8).
(a)
.
(b)
.

Each sphere in a colloidal crystal diffuses in a potential well
created by interactions with its neighbors.
Regardless of the form of the underlying pair potential,
the potential of mean force for a stable solid may be approximated
by a spherically symmetric harmonic well,

(6) 
centered about each sphere's equilibrium position.
The adequacy of this model can be judged by comparing
the measured probability distribution for sphere displacements
from their mean positions in d dimensions with

(7) 
where
is the thermal
energy scale at temperature T.
Figure 5 shows
for the crystals in this study
binned along
one Cartesian coordinate for a period of 33 seconds.
The choice of projection is arbitrary and quantitatively equivalent
results are obtained with other projections.
The solid lines in Fig. 5 are twoparameter fits to the form

(8) 
where
is an arbitrary additive offset.
We expect Eq. (7) to fail for
displacements larger than the Lindemann limit,
.
Consequently, the data in Fig. 5 are fit over the range
but plotted
over a wider range to illustrate limitations of the harmonic
approximation for large displacements.
The resulting estimates for the dimensionless curvature of the
potential of mean force,
and
for the
dilute and dense crystals respectively,
will be used in Sections 4 and 5
to estimate the crystals' elastic parameters.
Next: Diffusion in the Potential
Up: The Potential of Mean
Previous: The Potential of Mean
David G. Grier
19980608