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84891740664
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This is an average value over different samples estimated by fitting to X-ray data. See page 60 of ref 3 for more detail
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This is an average value over different samples estimated by fitting to X-ray data. See page 60 of ref 3 for more detail.
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5
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84891747139
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5b clearly indicating that the α to β transition temperature discussed in the literature is not the thermodynamic limit, in conflict with the interpretation of ref 9.
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5b clearly indicating that the α to β "transition temperature" discussed in the literature is not the thermodynamic limit, in conflict with the interpretation of ref 9.
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7
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Shang, S.1
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14
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84891739730
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Although the three-center bond of boron might seem unusual, the bonding scheme was first explained in terms of a simple linear combination of atomic orbitals description more than 40 years ago. For isolated trimer configurations, see ref 14. For more general cases, including extended systems, see ref 42
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Although the three-center bond of boron might seem unusual, the bonding scheme was first explained in terms of a simple linear combination of atomic orbitals description more than 40 years ago. For isolated trimer configurations, see ref 14. For more general cases, including extended systems, see ref 42.
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16
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84891736177
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Manuscript in preparation
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Ogitsu, T.; Gygi, F.; Reed, J.; Udagawa, M.; Motome, Y.; Schwegler, E.; Galli, G. Manuscript in preparation.
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Ogitsu, T.1
Gygi, F.2
Reed, J.3
Udagawa, M.4
Motome, Y.5
Schwegler, E.6
Galli, G.7
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17
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84891744363
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The replica Exchange Monte Carlo simulations have been performed down to 0.001 meV (or ∼10 mK) with no evidence of a phase transition.
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The replica Exchange Monte Carlo simulations have been performed down to 0.001 meV (or ∼10 mK) with no evidence of a phase transition.
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21
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33847033607
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A General Ab Initio Molecular Dynamics Program; Lawrence Livermore National Laboratory: Livermore, CA
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Gygi, F. GP 1.24.0, A General Ab Initio Molecular Dynamics Program; Lawrence Livermore National Laboratory: Livermore, CA, 2003.
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(2003)
GP 1.24.0
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41949118958
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Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Comput. Phys. Commun. 2008, 178, 685-699.
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Mostofi, A.A.1
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Lee, Y.-S.3
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Vanderbilt, D.5
Marzari, N.6
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29
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84891736800
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The number of possible configurations is calculated as follows: The 1280 atom supercell contains 12 rhombohedral cells. In the rhombo-hedral cell, the positions of 99 atoms correspond to fully occupied sites. This number is obtained by adding the total number of sites for Bl-B12, B14, and B15. The number of each site can be found in ref 3. Note that the numbers reported in the literature are for the hexagonal cell, so one should divide by 3 to obtain the numbers used here. In the 1280-atom supercell, 99 × 12, 1188 atoms have coordinates corresponding to fully occupied sites. Therefore, 1280-1188, 92 atoms have coordinates corresponding to POS. The total number of possible POS in the 1280-atom supercell is (6, 6, 6, 6, 6 +12) × 12, 504. Finally, the 92 POS atoms can be distributed over the 504 sites, which corresponds to [92 504] possible configurations
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504] possible configurations.
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33
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84891744657
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The deviation of the cell-parameters from rhombohedral symmetry was negligibly small in all of our simulations
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The deviation of the cell-parameters from rhombohedral symmetry was negligibly small in all of our simulations.
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34
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84891743881
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The B16 POS was first reported in ref 2, and the rest of the POS, B17-B20, were first reported in ref 3.
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The B16 POS was first reported in ref 2, and the rest of the POS, B17-B20, were first reported in ref 3.
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35
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84891744563
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In refs 8 and 9, the atomic density was fixed at 105 atoms/rhombohedral cell, while in ref 11, atomic densities corresponding to 104, 105, 106, and 111 atoms/cell were examined. However, only the occupations of B13 and B16 POS in their simplest possible configurations were considered. For example, with 106 atoms/cell, they considered only one POS configuration where all B13 sites and one B16 site were occupied
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In refs 8 and 9, the atomic density was fixed at 105 atoms/rhombohedral cell, while in ref 11, atomic densities corresponding to 104, 105, 106, and 111 atoms/cell were examined. However, only the occupations of B13 and B16 POS in their simplest possible configurations were considered. For example, with 106 atoms/cell, they considered only one POS configuration where all B13 sites and one B16 site were occupied.
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36
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84891747252
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12 considered only 22 possibilities out of over a million possible configurations within 214 atoms in the supercell consisting of a 2 × 1 x 1 rhombohedral cell. None of them systematically explored the POS configuration space within the hexagonal cell (or a larger supercell) at die experimental atomic density.
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12 considered only 22 possibilities out of over a million possible configurations within 214 atoms in the supercell consisting of a 2 × 1 x 1 rhombohedral cell. None of them systematically explored the POS configuration space within the hexagonal cell (or a larger supercell) at die experimental atomic density.
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The ZPE of both a- and β-boron were calculated from the full phonon dispersion by use of the linear response theory implemented in the PWSCF code. Additional details can be found in Supporting Information section 4. For β-boron, the perfect hR105 system was used, and its ZPE was 8.2 meV/atom lower than that of α-boron. The impact of POS occupation was estimated by the sums ofΓ-point phonon frequencies of the two most stable hR107 systems. This yielded a difference of 3 meV/atom, with respect to the energy of the perfect hR105. Therefore, we expect that the use of the hR105 structure to compute the ZPE (instead of hR107) has no impact on our conclusions regarding the relative stability of β- and a-boron. Note: The lower symmetry of hR107 would prohibitively increase the computational cost of the full phonon calculations
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The ZPE of both a- and β-boron were calculated from the full phonon dispersion by use of the linear response theory implemented in the PWSCF code. Additional details can be found in Supporting Information section 4. For β-boron, the perfect hR105 system was used, and its ZPE was 8.2 meV/atom lower than that of α-boron. The impact of POS occupation was estimated by the sums ofΓ-point phonon frequencies of the two most stable hR107 systems. This yielded a difference of 3 meV/atom, with respect to the energy of the perfect hR105. Therefore, we expect that the use of the hR105 structure to compute the ZPE (instead of hR107) has no impact on our conclusions regarding the relative stability of β- and a-boron. Note: The lower symmetry of hR107 would prohibitively increase the computational cost of the full phonon calculations.
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0034816901
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Jemmis, E. D.; Balakrishnarajan, M. M.; Pancharatna, P. D. J. Am. Chem. Soc. 2001, 123, 4324-4330.
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84891741848
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The perfect hR105 has 5 hole states per rhombohedral cell in the valence band. If one assumes that the number of valence bands does not change upon the introduction of POS atoms, the hole states would be perfectly filled at an atomic density of 106273 per rhombohedral cell (the experimental value). Our finding is that the stable hR1280 systems nearly satisfy this condition, except for the small deviation originating from the presence of B17-B18 pairs. See Supporting Information sections 5 and 6 for additional details.
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The perfect hR105 has 5 hole states per rhombohedral cell in the valence band. If one assumes that the number of valence bands does not change upon the introduction of POS atoms, the hole states would be perfectly filled at an atomic density of 106273 per rhombohedral cell (the experimental value). Our finding is that the stable hR1280 systems nearly satisfy this condition, except for the small deviation originating from the presence of B17-B18 pairs. See Supporting Information sections 5 and 6 for additional details.
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84891747109
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As the Kubo-Greenwood formula does not account for inelastic electron-phonon scattering, our estimate should be considered as an upper bound
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As the Kubo-Greenwood formula does not account for inelastic electron-phonon scattering, our estimate should be considered as an upper bound.
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Due to extremely slow dynamics at such a low temperature (sub-Kelvin, bringing a system into equilibrium is extremely challenging. Therefore, it is generally assumed that the spin ice systems freeze into a metastable state and do not exhibit the expected phase transition to an ordered phase at low temperature see ref 19
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Due to extremely slow dynamics at such a low temperature (sub-Kelvin), bringing a system into equilibrium is extremely challenging. Therefore, it is generally assumed that the spin ice systems freeze into a metastable state and do not exhibit the expected phase transition to an ordered phase at low temperature (see ref 19).
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