1. Symmetry-broken crystal structure of elemental boron at low temperature With Marek Mihalkovic (Slovakian Academy of Sciences) Outline: Cohesive energy puzzle (E  < E  ?) Optimization of partial occupancy in  Symmetry-restoring  phase transition

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

3. The structure of elemental Boron  -B.hR12McCarty (1958, powder, red)  -B.tP50Hoard (1958, 56 reflections, R=0.114)  -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 (relaxed DFT-GGA)  -B.hR12  E = 0.00 (meV/atom)  -B.tP50  E = +91.91  -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 3 rd law of thermodynamics!

4. Stability of  -Boron Possibility of Finite T phase transition (Runow, 1972; Werheit and Franz, 1986) Vibrational entropy can drive  transition (Masago, Shirai and Katayama-Yoshida, 2006) Quantum zero point energy can stabilize  (van Setten, Uijttewaal, de Wijs and de Groot, 2007) Symmetry-broken ground state , symmetric  phase restored by configurational entropy (Widom and Mihalkovic, 2008)

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

6. Structure and fluctuations Optimized structure Molecular dynamics T=2000K, duration 12ps

7. 2x1x1 Supercell Clock Model: “Time” shows occupancies Optimal times 02:20 and 10:00 Other times are low-lying excited states

8. Symmetry-restoring phase transition of clock model {  } = {all distinct clock configurations in 2x1x1 supercell}   = degeneracy of configuration  C TS U

9. Conclusions E  > E  conflicts with observation of  as stable Optimizing partial occupancy brings E  < E  Symmetry broken at low temperature (3 rd law) Symmetry restored through  phase transition  stabilized by entropy of partial occupation