1. 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

2. Bond lengths:   Occupancy: 100% 75% 9% 7% 27% 4%

3. 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

4. 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

5. 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)

6. Occupancy: 100%75% 9%7% 27% 4% 100%  cell center, partial occupancy All sites Optimal sites Clock model

7. 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.

8. 2x1x1 Superlattice Clock Model: “Time” shows occupancies Optimal times 02:20 and 10:00 Other times are low-lying excited states

9. 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

10. Symmetry-restoring phase transition of clock model {  } = {all distinct clock configurations in 2x1x1 superlattice}   = degeneracy of configuration  C TS U T m =2365K

11. Vibrational modes Modes around 60 meV localized on B13 and B16 sites, reduce free energy of  by ~ 15 meV/atom at high T.

12. 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