# Between Harmonic Crystal and Glass: Solids with Dimpled Potential-Energy Surfaces Having Multiple Local Energy Minima

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## Abstract

**:**

## 1. Introduction

**Figure 3.**(color online) (

**a**,

**b**) Corrugated sombrero potentials with 4 (

**a**) and 8 (

**b**) equidistant LPEMs, given by Gaussians. (

**c**–

**f**) Networks of symmetry-equivalent displacements (black spheres) relative to $x=0$ (red dot). Two-dimensional basin with (

**c**) hexagonal 3-fold rotational symmetry (shaded is its inversion) and (

**d**) square 4-fold rotational symmetry. (

**e**) The [111] projection, with shading below [111] plane through 3 corners of the cube, and (

**f**) a view of the central part of a basin with 48 dimples, forming a 3D cubic lattice. LPEMs (black spheres) are displacements from a local maximum at the center (red dot). MEPs between pairs of LPEMs are represented by lines (light blue) that are not necessarily straight.

**Figure 4.**(color online) Potential energy E and force $F\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}dE/dx$ versus displacement x (in arbitrary units) for a 1D crystal, which is either (

**a**) harmonic for $E\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{E}_{H}$ with ${N}_{L}=1$ (as in Figure 1a) or (

**b**) anharmonic with ${N}_{L}=2$ (as in Figure 1b). For anharmonic case, forces at $-{x}_{L}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}x\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{x}_{L}$ (shaded) push an atom away from the unstable equilibrium at $x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$, as $F/x<0$. For $kT>{E}_{L}$, the average atomic position is $\u2329x\u232a=0$, and the effective linear force and harmonic potential at ${x}_{0}$ (filled circle) are shown (orange dashed lines).

**Figure 5.**(color online). NiTi potential energy vs. collective atomic displacement (which changes linearly from $x=0$ at unstable B2 (CsCl) to $\pm {x}_{L}$ at a LPEM) for a MEP from B2 to a representative austenitic structure (LPEM, $E\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$), from B2 to BCO (base-centered orthorhombic B33) ground state. The B2-to-BCO (B33) transition occurs via B19′ or R′ structures (thin dashed lines), see Figure 6 (or also Figure 4 in [8]). Additionally, there is a pathway for a (lower-barrier) transformation between two orientations of B19’ (deformed BCO B33) martensite via B19 (thick green dashed line), see Figure 6f,g or also Figure 1b,e,f,i in [8] and Figure 3 in [8] or Figure 3 in [5]. There are low-enthalpy barriers around each austenitic LPEM (thin solid black line near $x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$, $E\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$), see also Figure 4 in [5].

## 2. Models with Dimpled Potentials

#### 2.1. Networks of LPEM in MLM Solids

#### 2.2. Symmetry-Breaking Phase Transition

#### 2.3. Expected Thermal Atomic Motion

- $kT\le {E}_{h}\ll {E}_{l}$: harmonic vibration around a single LPEM. The small-displacement method can provide phonons at a LPEM (see Appendix B).
- ${E}_{h}<kT<{E}_{l}$: anharmonic vibration around a single LPEM.
- ${E}_{l}\le kT<{E}_{L}$: motion covers several LPEMs in the same basin. If such LPEMs are distributed symmetrically around $x=0$, then the time-averaged atomic position is $\u2329x\u232a=0$.
- ${E}_{L}\le kT<{E}_{B}$: motion covers a significant part of the PE basin, including neighborhoods of $x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$ and multiple LPEMs. If the PE surface has a negligible roughness (${E}_{L}\ll kT$) at the bottom of a nearly harmonic potential, then a finite atomic displacement method can be used to calculate phonons around $x=0$, see Appendix B.
- $kT\ge {E}_{B}$: atomic motion is no longer restricted by a PE basin; the solid has melted or sublimated.

#### 2.4. Atomic Forces in MLM Solids

#### 2.5. Examples for Corrugated-Sombrero Potentials

#### 2.6. Diffraction from MLM Solids

## 3. Example Applications

#### 3.1. NiTi Austenite

**Figure 6.**(color online). (

**a**) Ideal B2 (or CsCl) structure and (

**b**) its [111] projection, with Ni (yellow) and Ti (blue) atoms; length of NN (Ni–Ti) bonds is 2.6 Å, NNN (Ni–Ni or Ti–Ti) bonds are 3.0 Å. Stable atomic positions in cubic B2 [111] projection are shown in representative supercells (bounded by thin black line), containing (

**c**) 54 atoms and (

**d**) 108 atoms; for view of projections in 54-atom supercell, see Figure 1 in [20]. The NNN Ni–Ni and Ti–Ti bonds shorter than 2.75 Å (or 2.7 Å in B19) are shown. (

**e**) The kinetically limited unstable intermediate R structure (suggested from experiment [45]) is shown in cubic [111] (or hex [001]) projection (

**left**) and cubic [100] (or hex [111]) projection (

**right**). Shown in [001] (

**left**) and [100] (

**right**) projections are (

**f**) the unstable B19 and (

**g**) the BCO B33 ground state. The orthorhombic B19 and B33 structures can be viewed as monoclinic B19’ with a shear angle $\theta $ of ${90}^{\circ}$ and ≈107${}^{\circ}$, respectively.

**Figure 7.**(color online). Phonon density of states assessed from neutron diffraction [44] (black, shaded) and computed for NiTi austenite using several methods: small displacements from the stable LPEM austenitic representative structure [20] at $0$ K (thick blue) and unstable B2 at $0$ K (dashed red); large displacements from B2 in MD at $1586$ K using ThermoPhonon [46] (thin green line).

#### 3.2. Group 4 Metals: Ti, Zr, and Hf

#### 3.3. 1T-TaS_{2} Layered Crystal

_{2}layered crystal creates a dimple in the atomic potential and causes a collective atomic displacement resulting in a grouping of 13-atom Ta clusters (inset of Figure 9). As T is lowered, the crystal transforms from the high-symmetry hP3 to the lower-symmetry 1T-TaS

_{2}phase, as observed in bulk and quasi-2D samples [59,60].

_{2}was but one example.

#### 3.4. Ubiquity

## 4. Summary

_{2}(leading to the observed CDW). Similarly, we found no barriers on the transition pathways from ideal B2 (CsCl-type) to either a global (BCO B33 ground state) or a local energy minimum in NiTi, a well-known shape-memory alloy. For completeness, we discussed vibrations in a dimpled potential and we reviewed the applicability of phonon methods, which can address anharmonic vibrations at a fixed finite temperature (see Equation (4) and Appendix B), if those vibrations behave effectively as “harmonic” at sufficiently high temperatures. Our findings suggest that a possible MLM behavior should be checked as a part of the analysis in any polymorphic solid with competing structures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NASA | National Aeronautics and Space Administration |

DOE | U.S. Department of Energy |

USA | United States of America |

BCO | base-centered orthorhombic |

bcc | body-centered cubic |

hcp | hexagonal close-packed |

CDW | charge-density wave |

DOS | density of states |

LPEM | local potential-energy minimum |

MLM | multiple local minima |

MD | molecular dynamics |

MEP | minimal-enthalpy path |

NEB | nudged elastic band |

PE | potential energy |

QHA | quasiharmonic approximation |

T | temperature |

## Appendix A. Computational Details

## Appendix B. Phonon Calculations

#### Appendix B.1. Small-Displacement Method

#### Appendix B.2. Finite-Displacement Method

#### Appendix B.3. Phonons at Fixed Temperature

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**Figure 1.**(color online) Schematic 1D potentials with an average lattice constant a and barriers ${E}_{B}$ for (

**a**) harmonic PE (parabolic below ${E}_{H}$), (

**b**) dimpled case with 2 local minima per basin (here, ${E}_{l}\equiv {E}_{L}$), where at $kT<{E}_{L}$, each atom (filled circle) is displaced (arrow) from the high-symmetry unstable position (open circle), and (

**c**) glass-like amorphous case.

**Figure 8.**(color online) Minimal-enthalpy path from bcc to hcp has no barrier in Ti, Zr, and Hf metals.

**Figure 9.**(color online) Enthalpy versus collective atomic displacement in TaS

_{2}for the linear path between high-symmetry hP3 (at 0) and lower-symmetry 1T (at 1) phases and its continuation. Insets: Relaxed bulk 1T-TaS

_{2}in 110 (upper) and 001 (lower) projections. S (blue and yellow) is above and below the Ta (black) layer. Bonds shown are Ta–Ta (Ta–S) shorter than 3.28 Å (2.50 Å). The 13(TaS

_{2}) hexagonal supercell (thin red lines) with fixed a = 12.233 Å and c = 5.892 Å is composed of 13 hP3 primitive cells (a = 3.393 Å, c = 5.892 Å); internal atomic relaxation was at fixed supercell lattice constants.

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Zarkevich, N.A.; Johnson, D.D.
Between Harmonic Crystal and Glass: Solids with Dimpled Potential-Energy Surfaces Having Multiple Local Energy Minima. *Crystals* **2022**, *12*, 84.
https://doi.org/10.3390/cryst12010084

**AMA Style**

Zarkevich NA, Johnson DD.
Between Harmonic Crystal and Glass: Solids with Dimpled Potential-Energy Surfaces Having Multiple Local Energy Minima. *Crystals*. 2022; 12(1):84.
https://doi.org/10.3390/cryst12010084

**Chicago/Turabian Style**

Zarkevich, Nikolai A., and Duane D. Johnson.
2022. "Between Harmonic Crystal and Glass: Solids with Dimpled Potential-Energy Surfaces Having Multiple Local Energy Minima" *Crystals* 12, no. 1: 84.
https://doi.org/10.3390/cryst12010084