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

Bond lengths:

Occupancy:

100% 75%

9%

7% 7%

27%

4%

The structure of elemental Boron

-B.hR12 McCarty (1958, powder, red)-B.tP50 Hoard (1958, 56 reflections, R=0.114)-B.hR105 Geist (1970, 350 reflections, R=0.074)-B.hR111 Callmer (1977, 920 reflections, partial occ. R=0.053)-B.hR141 Slack (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

3rd law of thermodynamics!

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)

Occupancy: 100% 75%

9% 7% 7%

27%

4%

100%

cell center, partial occupancy

All sites Optimal sites

Clock model

Structure and fluctuations

Optimized structure Molecular dynamics

T=2000K, duration 12ps

2x1x1 Supercell

Clock Model:

“Time” shows occupancies

Optimal times02:20 and 10:00

Other times are low-lying excited states

Symmetry-restoring phase transition of clock model

ZTkF

Z

B

TkE B

ln

e /

{} = {all distinctclock configurationsin 2x1x1 supercell}

= degeneracy ofconfiguration

C

TS

U

Conclusions

• E > E conflicts with observation of as stable

• Optimizing partial occupancy brings E < E

• Symmetry broken at low temperature (3rd law)

• Symmetry restored through phase transition

• stabilized by entropy of partial occupation