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Communication: Designed diamond ground state via optimized isotropic monotonic pair potentials
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10.The forward approach specifies interparticle interactions for a many-particle system and then studies structural, thermodynamic, and kinetic features of the system.
11. M. Watzlawek, C. N. Likos, and H. Lowen, Phys. Rev. Lett. 82, 5289 (1999).
12. C. N. Likos, private communication (2006). Phonon spectra were not provided in Ref. 11.
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14.Since the nonconvexity of the potential is restricted to very small distances, one can add a short-range hard-core repulsion to Eq. (1) to obtain a convex potential, which will behave exactly like the original potential, provided that the pressure or temperature is not extremely large.
15.The diamond potential function is robust in that its parameters can be perturbed and still produce the diamond crystal as a ground state.
16.For example, we denote by D2 the potential that has the largest ratio of maximum to minimum stable pressures for the diamond ground state. Its parameters are a1 = −1.21189, and a2 = 0.788955. This potential possesses softer transverse phonon modes compared to the D1 potential.
17. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, New York, 1976).


Image of FIG. 1.

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FIG. 1.

(a) Optimized monotonic pair potential v(r) from Eq. (1) using the parameters from (2) : the D1 potential. (b) Second derivative d 2 v/dr 2 of the versus the radial distance r.

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FIG. 2.

Phonon spectrum in reduced units of the D1 potential for the diamond at dimensionless pressure p* = 0.078 and density ρ* = 0.271. Only a representative subset of wave vectors that lie on paths connecting high-symmetry points (Γ, K, W, X, and L) of the Brillouin zone 7 is shown. The D1 potential is chosen such that the lowest phonon frequency relative to the highest one at the X point is maximized.

Image of FIG. 3.

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FIG. 3.

Ground states of the D1 potential for a range of pressures obtained from steepest descent for a basis up to N = 16. The crystal phases indicated from zero pressure to higher pressures are the 12-coordinated face-centered cubic (gray), the 8-coordinated body-centered cubic (cyan), a 2-coordinated hexagonal (orange), a 3-coordinated buckled rhombohedral graphite (blue), the 4-coordinated diamond (red), a 5/6-coordinated deformed diamond (green), a 6-coordinated buckled hexagonal (violet), and a 8-coordinated flattened-hexagonal closed-packed (yellow). “Bonds” are indicated between nearest-neighbor particles for visualization purposes.

Image of FIG. 4.

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FIG. 4.

Schematics of three different types of energy landscapes as a function of the configurational coordinate. Boundaries of the basin of attraction associated with the global minima are indicated by dashed vertical lines. (a) Relatively rough energy landscape. (b) Energy landscape with a deep and narrow global minimum. (c) Energy landscape with a broad and smooth global minimum.


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Table I.

Frequency with which the ground-state diamond crystal is obtained from a steepest descent starting from a random configurations of N particles. For each N, the frequency is calculated using 10 000 trials, which results in standard deviations smaller than 0.5%. The D1 potential trials are carried out at p* = 0.078, while the star-polymer potential trials used (for which the ground state has a “packing fraction” η = 1.2) and an arm number f = 64 (see Ref. 11 for the definition of these parameters).


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We apply inverse statistical-mechanical methods to find a simple family of optimized isotropic, monotonic pair potentials (that may be experimentally realizable) whose classical ground state is the diamond crystal for the widest possible pressure range, subject to certain constraints (e.g., desirable phonon spectra). We also ascertain the ground-state phase diagram for a specific optimized potential to show that other crystal structures arise for pressures outside the diamond stability range. Cooling disordered configurations interacting with our optimized potential to absolute zero frequently leads to the desired diamond crystal ground state, revealing that the capture basin for the global energy minimum is large and broad relative to the local energy minima basins.


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Scitation: Communication: Designed diamond ground state via optimized isotropic monotonic pair potentials