Inferring elastic properties of an fcc crystal from displacement correlations: Subspace projection and statistical artifacts

We compute the effective dispersion and vibrational density of states (DOS) of two-dimensional subregions of three-dimensional face-centered-cubic crystals using both a direct projection-inversion technique and a Monte Carlo simulation based on a common underlying Hamiltonian. We study both a (111) and (100) plane. We show that for any given direction of wave vector, both (111) and (100) show an anomalous ${\omega }^2\sim q$ regime at low $q$ where ${\omega }^2$ is the energy associated with the given mode and $q$ is its wave number. The ${\omega }^2\sim q$ scaling should be expected to give rise to an anomalous DOS, $D_{\omega }$, at low $\omega$: $D_{\omega }\sim {\omega }^3$ rather than the conventional Debye result: $D_{\omega }\sim {\omega }^2$. The DOS for (100) looks to be consistent with $D_{\omega }\sim {\omega }^3$, while (111) shows something closer to the conventional Debye result at the smallest frequencies. In addition to the direct projection-inversion calculation, we perform Monte Carlo simulations to study the effects of finite sampling statistics. We show that finite sampling artifacts act as an effective disorder and bias $D_{\omega }$, giving a behavior closer to $D_{\omega }\sim {\omega }^2$ than $D_{\omega }\sim {\omega }^3$. These results should have an important impact on the interpretation of recent studies of colloidal solids where the two-point displacement correlations can be obtained directly in real-space via microscopy.

Figure 1

Left: FCC crystal with $C=3$ cubic unit cells along each edge. Center: Rhombic $\left(111\right)$ FCC patch with $P=4$ atoms along each edge. Right: Square $\left(100\right)$ FCC patch with $P=4$ atoms along each edge.

Figure 2

Computed longitudinal and transverse dispersion curves scaled by wave number for plane waves traveling along the next-nearest-neighbor direction and lying in the first BZ for $\left(111\right)$ various patches.

Figure 3

Computed dispersion scaled by wave number, ${\omega }^2/K{a}^{2}q$, for a $\left(111\right)$ patch $\left(P=64,C=64\right)$ for waves in the first BZ. (a) Longitudinal; (b) transverse.

Figure 4

Computed longitudinal and transverse dispersion curves scaled by wave number for plane waves lying in the first BZ of various $\left(100\right)$ patches for plane waves traveling in the (a) nearest-neighbor $\left(\left[010\right]\right)$ direction and (b) the next-nearest-neighbor $\left(\left[011\right]\right)$ direction.

Figure 5

Computed dispersion scaled by wave number: ${\omega }^2/K{a}^{2}q$, for a $\left(100\right)$ patch $\left(P=64,C=64\right)$ for (a) longitudinal and (b) transverse waves in the first BZ.

Figure 6

Integrated DOS $N\left(\omega \right)={\int }^{\omega }{D}_{\omega }d\omega$. Frequencies, $\omega$, as a function of index, $N$, for $\left(100\right)$ and $\left(111\right)$ FCC patches $\left(P=32,C=32\right)$.

Figure 7

Computed DOS, $D_{\omega }$, normalized by the total number of normal modes, $N$, for various FCC crystal patches. (a) A patch from a $\left(100\right)$ crystal plane. (b) A patch from a $\left(111\right)$ crystal plane. Upper panels: $D_{\omega }/N$ scaled by the non-Debye scaling ${\omega }^3$. Lower panels: $D_{\omega }/N$ scaled by the Debye scaling ${\omega }^2$.

Figure 8

$S(\vec{q},\omega )$ for a $\left(111\right)$ FCC patch $\left(P=32,C=32\right)$ for various $\omega$. Upper row: Longitudinal. Lower row: Transverse. Left: $\omega /{\omega }_0=1.21$. Center: $\omega /{\omega }_0=1.65$. Right: $\omega /{\omega }_0=2.08$.

Figure 9

$S(\vec{q},\omega )$ for a $\left(100\right)$ FCC patch $\left(P=32,C=32\right)$ for various $\omega$. Upper row: Longitudinal. Lower row: Transverse. Left: $\omega /{\omega }_0=1.21$. Center: $\omega /{\omega }_0=1.65$. Right: $\omega /{\omega }_0=2.08$.

Figure 10

Isotropically averaged DSF $S(q,\omega )$, for a $\left(111\right)$ FCC patch $\left(P=32,C=32\right)$. The color bar is log scale. Left: Longitudinal. Right: Transverse.

Figure 11

Isotropically averaged DSF $S(q,\omega )$ for a $\left(100\right)$ FCC patch $\left(P=32,C=32\right)$. The color bar is log scale. Left: Longitudinal. Right: Transverse.

Figure 12

Evolution of the frequency spectrum of the Green's function of a $\left(111\right)$ FCC patch $\left(P=32,C=32\right)$ inferred from Monte Carlo simulations. $\tau$ is the number of MC sweeps per patch degree of freedom. Left: Short sampling times. Right: Long sampling times.

Figure 13

Scaled DOS, $D_{\omega }$, of a FCC crystal patch $\left(P=32,C=32\right)$ normalized by the number of normal modes $N=2P^2$, for different MC sampling times $\tau$. Left: Patch from (100) FCC plane. Right: Patch from (111) FCC plane. Upper panel: $D_{\omega }/N$ scaled by the non-Debye scaling ${\omega }^3$. Lower panel: $D_{\omega }/N$ scaled by the Debye scaling ${\omega }^2$.

Figure 14

DOS, $D_{\omega }$, of $\left(111\right)$ FCC crystal patches $\left(P=32,C=32\right)$ normalized by $N=2P^2$ for different MC sampling times $\tau$. Inset: $D_{\omega }/N$ as a function of frequency $\omega$. Main plot: $D_{\omega }/N$ and $\omega$, nondimensionalized by the maximum frequency ${\omega }_m$.