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