# Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry

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## Abstract

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## 1. Symmetry in Crystal Structures—The Past and the Present

## 2. Atomistic Crystal Growth Process

## 3. Pair Potentials and Structural Stability

#### 3.1. Stability of Structural Motifs

#### 3.2. Stability of Unary Structures

## 4. Methods

## 5. Conclusions

“It is the harmony of the diverse parts, their symmetry, their happy balance [...] that permits us to see clearly and to comprehend at once both the ensemble and the details”[39].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Platonic Solids with increasing symmetry, i.e., number of symmetry elements: the self-dual tetrahedron, the octahedron with the dual cube, as well as the dodecahedron with the dual icosahedron.

**Figure 2.**Common structural symmetries of unary phases: body-centered cubic (bcc), face-centered cubic (fcc), simple cubic (sc), and hexagonal close-packed (hcp). Space-filling is realized with distorted octahedra (solid blue) for bcc, a combination of octahedra and tetrahedra (opaque green) for fcc and hcp, and cubes (solid violet) for sc.

**Figure 3.**Atomistic crystal growth process for the first few atoms with structural motifs (upper) and derived crystal symmetries (lower part). Due to their high symmetry, the platonic solids present, with respect to total energy, the most stable cluster motifs with a maximum number of pair potential equilibrium distances (beige sticks), and a minimum of energetically less stable distances (orange, red sticks). After the first atomic pair (orange), the third atom (green) and the forth atom (dark blue) may satisfy equilibrium distances without restrictions forming the tetrahedron motif (opaque green). Limitations of 3D space arise for the fifth atom (light blue), which selects the pyramid or the ditetrahedron according to the specific pair potential. Likewise, the sixth atom (violet) may favor the regular octahedron (blue), its bcc variant, or the prism (opaque yellow). Further structural transformations may occur to the cube (opaque violet) at the 8th atom (yellow), the icosahedron (beige) at the 12th atom (dark red) and the dodecahedron (orange) at the 20th atom (gray). For the last two, the high volume favors centered motifs as well (not shown here). The high symmetric crystal symmetries for unary phases: hcp, fcc, bcc, hp, and sc evolve directly or as combination of the presented motifs.

**Figure 4.**Short-range and long-range flexibility of the Modified Morse type potential for four sets of parameters in comparison to the Standard Morse Potential and the Lennard-Jones Potential.

**Figure 5.**Phase diagram depicting the equilibrium lattice symmetries in dependence on the Modified Morse Potential $MM[a,\phi ]$ (uncalculated region in gray). The Standard Morse Potential (blue line) and the Lennard-Jones Potential (point $MM[6,0]$ on red line) lack the flexibility in independent short- and long-range interaction to represent the stable bcc region and to describe the large parameter space for phase transformations (orange and green boundaries).

**Figure 6.**Equilibrium relaxation states of hcp, bcc, and fcc lattices for exemplary pair potentials, namely the Standard Morse $MM[6,1]$, a broad-well potential $MM[4,2.5]$ and an extremely long-range potential $MM[3,0.5]$. The weights of the shell (ball size) within the total energy sum are given up to a contribution of ${10}^{-3}$ energy units per atom. For $MM[3,0.5]$, the significant influence of higher coordination shells forces the first shell to contract even to anti-binding distances.

**Table 1.**Total energies ${E}_{\mathrm{tot}}$ and relaxation $\Delta {R}_{\mathrm{min}}$ for optimized cluster motifs of high structural symmetry in dependence on chosen pair potentials. The order of stability of the motifs (most to least stable: dark to light green, yellow, orange) changes in dependence on soft- and long-range interaction, in addition to the significance of relaxation.

Motifs | N atom | Standard Morse MM[6,1] | Lennard Jones MM[6,0] | Modified Morse MM[3.5,2.5] | Modified Morse MM[10,−5] | ||||
---|---|---|---|---|---|---|---|---|---|

ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | ||

Ditetrahedron | 5 | −0.07 | −1.809 | −0.09 | −1.821 | −0.37 | −1.814 | −0.01 | −1.861 |

Pyramid | 5 | −0.45 | −1.665 | −0.45 | −1.695 | −2.21 | −1.717 | −0.07 | −1.743 |

Octahedron | 6 | −0.45 | −2.081 | −0.45 | −2.119 | −2.21 | −2.146 | −0.07 | −2.179 |

Octahedron-bcc | 6 | −1.13 | −1.475 | −1.14 | −1.546 | −6.03 | −1.615 | −0.18 | −1.724 |

Prism | 6 | −1.18 | −1.668 | −1.15 | −1.742 | −6.30 | −1.839 | −0.18 | −1.858 |

Cube | 8 | −1.87 | −1.772 | −1.86 | −1.910 | −11.13 | −2.133 | −0.29 | −2.184 |

Icosahedron | 12 | −0.68 | −2.630 | −0.91 | −2.800 | −5.00 | −2.749 | −0.13 | −3.410 |

Icosahedron+C | 13 | −3.53 | −3.265 | −3.62 | −3.419 | −6.12 | −3.383 | −2.31 | −3.872 |

Dodecahedron | 20 | −1.34 | −1.658 | −1.79 | −1.875 | −16.62 | −1.963 | −0.26 | −3.755 |

Dodecahedron+C | 21 | −2.57 | −1.771 | −2.83 | −2.053 | −22.94 | −2.703 | −0.44 | −3.924 |

**Table 2.**Total energies ${E}_{\mathrm{tot}}$ and relaxation $\Delta {R}_{\mathrm{min}}$ for optimized structures of high symmetry in dependence on chosen pair potentials. Again, the order of stability of the structures (most to least stable: dark to light green, yellow, orange) changes in dependence on soft- and long-range interaction, in addition to the significance of relaxation.

Structure | Standard Morse MM[6,1] | Lennard Jones MM[6,0] | Modified Morse MM[4,2.5] | Modified Morse MM[3,0.5] | ||||
---|---|---|---|---|---|---|---|---|

ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | ΔR_{min}/% | E_{tot}/atom | |

sc | −4.12 | −8.613 | −4.87 | −11.364 | −22.37 | −12.588 | −36.99 | −55.871 |

bcc | −4.00 | −13.225 | −4.75 | −16.451 | −10.29 | −14.534 | −33.82 | −64.299 |

fcc | −1.75 | −13.784 | −2.87 | −17.196 | −7.42 | −14.404 | −31.97 | −64.518 |

hcp | −1.75 | −13.787 | −2.84 | −17.197 | −7.39 | −14.406 | −31.94 | −64.451 |

hp | −3.17 | −10.523 | −3.17 | −13.495 | −16.15 | −13.010 | −34.94 | −59.250 |

diamond | −0.94 | −6.391 | −2.35 | −7.960 | −2.54 | −6.276 | −33.10 | −31.726 |

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**MDPI and ACS Style**

Zschornak, M.; Leisegang, T.; Meutzner, F.; Stöcker, H.; Lemser, T.; Tauscher, T.; Funke, C.; Cherkouk, C.; Meyer, D.C.
Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry. *Symmetry* **2018**, *10*, 228.
https://doi.org/10.3390/sym10060228

**AMA Style**

Zschornak M, Leisegang T, Meutzner F, Stöcker H, Lemser T, Tauscher T, Funke C, Cherkouk C, Meyer DC.
Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry. *Symmetry*. 2018; 10(6):228.
https://doi.org/10.3390/sym10060228

**Chicago/Turabian Style**

Zschornak, Matthias, Tilmann Leisegang, Falk Meutzner, Hartmut Stöcker, Theresa Lemser, Tobias Tauscher, Claudia Funke, Charaf Cherkouk, and Dirk C. Meyer.
2018. "Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry" *Symmetry* 10, no. 6: 228.
https://doi.org/10.3390/sym10060228