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Symmetry-broken crystal structure of elemental boron at low temperature With Marek Mihalkovič (Slovakian Academy of Sciences) Outline: Cohesive energy puzzle (E < E ?) Optimization of partial occupancy in (predicted new low temperature phase!) Symmetry-restoring phase transition
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Bond lengths: Occupancy: 100% 75% 9% 7% 27% 4%
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Total Energy Calculations Electronic density functional theory Generalized gradient approximation (GGA) All-electron projector augmented wave (PAW) potential Full relaxation of lattice parameters and coordinates VASP program
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The structure of rhombohedral Boron -B.hR12McCarty (1958, powder sample) -B.hR105Geist (1970, 350 reflections, R=0.074) -B.hR111Callmer (1977, 920 reflections, partial occ. R=0.053) -B.hR141Slack (1988, 1775 reflections, partial occ. R=0.041) The energies of elemental Boron -B.hR12 E = 0.00 (meV/atom) -B.hR105 E = +25.87 105 atoms/105 sites -B.hR111 E = +0.15 106 atoms/111 sites -B.hR141 E = 0.86 107 atoms/141 sites -B.aP214 E = 1.75 214 atoms/214 sites
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Stability of -Boron Possibile structural transition in (Werheit and Franz, 1986) E < E (Mihalkovič and Widom, 2004) Vibrational entropy could drive transition (Masago, Shirai and Katayama-Yoshida, 2006) Quantum zero point energy could stabilize (van Setten, Uijttewaal, de Wijs and de Groot, 2007) Symmetry-broken ground state , symmetric phase restored by configurational entropy (Widom and Mihalkovič, 2008) 3 rd law/Landau Theory requires ( ) phase transition Symmetry breaking and superlattice (extra diffraction peaks)
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Occupancy: 100%75% 9%7% 27% 4% 100% cell center, partial occupancy All sites Optimal sites Clock model
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Structure and fluctuations Optimized structureMolecular dynamics T=2000K, duration 12ps Create B17B18 pair from adjacent B13 atoms (12x) Place two non-adjacent B16 atoms (9x) =108 configurations/cell TS = k B T log( )/107 atoms 4 meV/atom at T=1000K.
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2x1x1 Superlattice Clock Model: “Time” shows occupancies Optimal times 02:20 and 10:00 Other times are low-lying excited states
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Superlattice clock ordering energies CellClock+Clock´ E (meV/atom) 2x1x1.......... 1x1x102:200 2x1x106:20+04:30-0.11 2x1x1.......... 2x1x102:20+02:00-0.75 2x1x106:20+10:00-0.82 2x1x102:20+10:00-0.89 ´ 02:00 02:2004:30 06:20 10:00
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Symmetry-restoring phase transition of clock model { } = {all distinct clock configurations in 2x1x1 superlattice} = degeneracy of configuration C TS U T m =2365K
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Vibrational modes Modes around 60 meV localized on B13 and B16 sites, reduce free energy of by ~ 15 meV/atom at high T.
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Conclusions E < E conflicts with observation of as stable Optimizing partial occupancy brings E < E Symmetry broken at low temperature (3 rd law) Superlattice ordering (extra diffraction peaks?) Symmetry restored through transition stabilized by entropy of partial occupation
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