We report on the results of a systematic ab initio study of the magnetic structure of Fe rich fcc FeNi binary alloys for Ni concentrations up to 50 at. %. Calculations are carried out within density-functional theory using two complementary techniques, one based on the exact muffin-tin orbital theory within the coherent potential approximation and another one based on the projector augmented-wave method. We observe that the evolution of the magnetic structure of the alloy with increasing Ni concentration is determined by a competition between a large number of magnetic states, collinear as well as noncollinear, all close in energy. We emphasize a series of transitions between these magnetic structures, in particular we have investigated a competition between disordered local moment configurations, spin spiral states, the double layer antiferromagnetic state, and the ferromagnetic phase, as well as the ferrimagnetic phase with a single spin flipped with respect to all others. We show that the latter should be particularly important for the understanding of the magnetic structure of the Invar alloys.
First-principles calculations are employed to investigate the stability and properties of cubic rock-salt like (Cr_{1−x}Al_{x})_{2}O_{3} solid solutions, stabilized by metal site vacancies as recently reported experimentally. It is demonstrated that the metal site vacancies can indeed be ordered in a way that gives rise to a favorable coordination of all O atoms in the lattice. B1-like structures with ordered and disordered metal site vacancies are studied for (Cr_{0.5}Al_{0.5})_{2}O_{3} and found to a have cubic lattice spacing close to the values reported experimentally, in contrast to fluorite-like and perovskite structures. The obtained B1-like structures are higher in energy than corundum solid solutions for all compositions, but with an energy offset per atom similar to other metastable systems possible to synthesize with physical vapor deposition techniques. The obtained electronic structures show that the B1-like systems are semiconducting although with smaller band gaps than the corundum structure.
First-principles calculations are employed to investigate the stability and properties of cubic rock-salt-like (Cr_{1−}_{x}Al_{x})_{2}O_{3} solid solutions, stabilized by metal site vacancies as recently reported experimentally. It is demonstrated that the metal site vacancies can indeed be ordered in a way that gives rise to a suitable fourfold coordination of all O atoms in the lattice. B1-like structures with ordered and disordered metal site vacancies are studied for (Cr_{0.5}Al_{0.5})_{2}O_{3} and found to have a cubic lattice spacing close to the values reported experimentally, in contrast to fluorite-like and perovskite structures. The obtained B1-like structures are higher in energy than corundum solid solutions for all compositions, but with an energy offset per atom similar to other metastable systems possible to synthesize with physical vapor deposition techniques. The obtained electronic structures show that the B1-like systems are semiconducting although with smaller band gaps than the corundum structure.
Wedescribe an efficient first-principles method that can be used tocalculate mixing enthalpies of transition metal nitrides with B1 structureand substitutional disorder at the metal sublattice. The technique isbased on the density functional theory. The independent sublattice modelis suggested for the treatment of disorder-induced local lattice relaxationeffects. It supplements the description of the substitutional disorder withinthe coherent potential approximation. We demonstrate the excellent accuracy ofthe method by comparison with calculations performed by means ofthe projector augumented wave method on supercells constructed as specialquasirandom structures. At the same time, the efficiency of thetechnique allows for total energy calculations on a very finemesh of concentrations which enables a reliable calculation of thesecond concentration derivative of the alloy total energy. This isa first step towards first-principles predictions of concentrations and temperatureintervals where the alloy decomposition proceeds via the spinodal mechanism.We thus calculate electronic structure, lattice parameter, and mixing enthalpiesof the quasibinary alloy c-Ti_{1−x}Al_{x}N. The lattice parameter follows Vegard'slaw at low fractions of AlN but deviates increasingly withincreasing Al content. We show that the asymmetry of themixing enthalpy and its second concentration derivative is associated withsubstantial variations of the electronic structure with alloy composition. Thephase diagram is constructed within the mean-field approximation.
We have performed a numerical study of the process of electron localization in reduced ceria. Our results show that different localized charge distributions can be attained in a bulk system by varying the lattice parameter. We demonstrate that the effect of electron localization is mainly determined by lattice relaxation and an accurate account for the effects of electronic correlation is necessary to achieve localized charge distribution.
The new high-temperature (T), high-pressure (P), body-centered cubic (bcc) phase of iron has probably already been synthesized in recent diamond anvil cell (DAC) experiments (Mikhaylushkin et al 2007 Phys. Rev. Lett. 99 165505). These DAC experiments on iron revealed that the high-PT phase on quenching transforms into a mixture of close-packed phases. Our molecular dynamics simulation and structural analysis allow us to provide a probable interpretation of the experiments. We show that quenching of the high-PT bcc phase simulated with the embedded-atom model also leads to the formation of the mixture of close-packed phases. Therefore, the assumption of the stability of the high-PT bcc iron phase is consistent with experimental observation.
The Gibbs free energies of bcc and fcc Mo are calculated from first principles in the quasiharmonic approximation in the pressure range from 350 to 850 GPa at room temperatures up to 7500 K. It is found that Mo, stable in the bcc phase at low temperatures, has lower free energy in the fcc structure than in the bcc phase at elevated temperatures. Our density-functional-theory-based molecular dynamics simulations demonstrate that fcc melts at higher than bcc temperatures above 1.5 Mbar. Our calculated melting temperatures and bcc-fcc boundary are consistent with the Mo Hugoniot sound speed measurements. We find that melting occurs at temperatures significantly above the bcc-fcc boundary. This suggests an explanation of the recent diamond anvil cell experiments, which find a phase boundary in the vicinity of our extrapolated bcc-fcc boundary.
Phase diagrams of refractory metals remain essentially unknown. Moreover, there is an ongoing controversy over the high-pressure melting temperatures of these metals: results of diamond anvil cell (DAC) and shock wave experiments differ by at least a factor of 2. From an extensive ab initio study on tantalum we discovered that the body-centered cubic phase, its physical phase at ambient conditions, transforms to another solid phase, possibly hexagonal omega phase, at high temperature. Hence the sample motion observed in DAC experiments is very likely not due to melting but internal stresses accompanying a solid-solid transformation, and thermal stresses associated with laser heating.
The magnetic anisotropy energy (MAE) of Fe, Co, and Ni is presented for tetragonal and trigonal structures along two paths of structural distortion connecting the bcc and the fcc structure. The MAE was calculated from first principles with the full-potential linear muffin-tin orbital method and the force theorem. As is expected from symmetry considerations, the MAE increases by orders of magnitude when the cubic symmetry is broken. For tetragonal structures of Co and Ni a regular behavior of the MAE is observed, i.e., only the symmetry dictated nodes at the cubic structures appear along this path of distortion. In the case of tetragonal Fe, additional reorientations of the easy axis occur that are attributed to a topological change of the Fermi surface upon distortion. For the trigonal structures of all three elements the strain dependence of the MAE is more complicated, with additional reorientations of the easy axis and an unexpectedly large MAE for certain distortions of Ni, and a strongly nonlinear behavior for trigonal structures of Co close to fcc. Furthermore, the linear magnetoelastic coupling coefficients are calculated from the MAE at small distortions from the cubic equilibrium structure of the three elements. Two different Brillouin-zone integration techniques were used to calculate the MAE. Since the Gaussian broadening method smears out details of the Fermi surface, it results in a different MAE as compared to the tetrahedron method in some cases.
The uniaxial magnetic anisotropy energy (MAE) of L 10 FePt and Fe1-x Mnx Pt, x=0-0.25, was studied from first principles using two fully relativistic computational methods, the full-potential linear muffin-tin orbitals method and the exact muffin-tin orbitals method. It was found that the large MAE of 2.8 meV/f.u. is caused by a delicate interaction between the Fe and Pt atoms, where the large spin-orbit coupling of the Pt site and the hybridization between Fe 3d and Pt 5d states is crucial. The effect of random order on the MAE was modeled by mutual alloying of the sublattices within the coherent potential approximation (CPA), and a strong dependence of the MAE on the degree of chemical long-range order was found. The alloying of FePt with Mn was investigated with the virtual crystal approximation and the CPA as well as supercell calculations. The MAE increases up to 33% within the concentration range studied here, an effect that is attributed to band filling. Furthermore, the dependence of the MAE on the structural properties was studied. © 2005 The American Physical Society.
By means of ab-initio calculations, we perform an analysis of the configurational thermodynamics, effects of disorder, and structural energy differences in Fe-Ni alloys at the pressure and temperature conditions of the Earths core. We show from ab-initio calculations that the ordering energies of fcc and hcp-structured Fe-Ni solid solutions at these conditions depend sensitively on the alloy configuration, i.e., on the degree of chemical disorder, and are on a scale comparable with the structural energy differences. From configurational thermodynamic simulations we find that a distribution of Fe and Ni atoms in the solutions should be very close to completely disordered at these conditions. Using this model of the Fe-Ni system, we have calculated the fcc-hcp structural free energy difference in a wide pressure-temperature range of 120-360 GPa and 1000-6600K. Our calculations show that alloying of Fe with Ni below 3000 K favours stabilisation of the fcc phase over the hcp, in agreement with experiments. However, above 3000 K the effect is reversed, and at conditions corresponding to those of the Earths inner core, Ni acts as an agent to stabilise the hcp phase.
Due to a large discrepancy between theory and experiment, the electronic character of crystalline boron carbide B13C2 has been a controversial topic in the field of icosahedral boron-rich solids. We demonstrate that this discrepancy is removed when configurational disorder is accurately considered in the theoretical calculations. We find that while the ordered ground state B13C2 is metallic, the configurationally disordered B13C2, modeled with a superatom-special quasirandom structure method, goes through a metal to nonmetal transition as the degree of disorder is increased with increasing temperature. Specifically, one of the chain-end carbon atoms in the CBC chains substitutes a neighboring equatorial boron atom in a B-12 icosahedron bonded to it, giving rise to a B11Ce(BBC) unit. The atomic configuration of the substitutionally disordered B13C2 thus tends to be dominated by a mixture between B-12(CBC) and B11Ce(BBC). Due to splitting of valence states in B11Ce(BBC), the electron deficiency in B-12(CBC) is gradually compensated.
Configurationally disordered crystalline boron carbide, with the composition B4C, is studied using first-principles calculations. We investigate both dilute and high concentrations of carbon-boron substitutional defects. For the latter purpose, we suggest a superatoms picture of the complex structure and combine it with a special quasirandom structure approach for disorder. In this way, we model a random distribution of high concentrations of the identified low-energy defects: (1) bipolar defects and (2) rotation of icosahedral carbon among the three polar-up sites. Additionally, the substitutional disorder of the icosahedral carbon at all six polar sites, as previously discussed in the literature, is also considered. Two configurational phase transitions from the ordered to the disordered configurations are predicted to take place upon an increase in temperature using a mean-field approximation for the entropy. The first transition, at 870 K, induces substitutional disorder of the icosahedral carbon atoms among the three polar-up sites; meanwhile the second transition, at 2325 K, reveals the random substitution of the icosahedral carbon atoms at all six polar sites coexisting with bipolar defects. Already the first transition removes the monoclinic distortion existing in the ordered ground-state configuration and restore the rhombohedral system (R3m). The restoration of inversion symmetry yielding the full rhombohedral symmetry (R (3) over barm) on average, corresponding to what is reported in the literature, is achieved after the second transition. Investigating the effects of high pressure on the configurational stability of the disordered B4C phases reveals a tendency to stabilize the ordered ground-state configuration as the configurationally ordering/disordering transition temperature increases with pressure exerted on B4C. The electronic density of states, obtained from the disordered phases, indicates a sensitivity of the band gap to the degree of configurational disorder in B4C.
The elastic properties of alloys between boron suboxide (B_{6}O) and boron carbide (B_{13}C_{2}), denoted by (B_{6}O)_{1−x}(B_{13}C_{2})_{x}, as well as boron carbide with variable carbon content, ranging from B_{13}C_{2} to B_{4}C are calculated from first-principles. Furthermore, the mixing thermodynamics of (B_{6}O)_{1−x}(B_{13}C_{2})x is studied. A superatom-special quasirandom structure approach is used for modeling different atomic configurations, in which effects of configurational disorder between the carbide and suboxide structural units, as well as between boron and carbon atoms within the units, are taken into account. Elastic properties calculations demonstrate that configurational disorder in B_{13}C_{2}, where a part of the C atoms in the CBC chains substitute for B atoms in the B12 icosahedra, drastically increase the Young’s and shear modulus, as compared to an atomically ordered state, B_{12}(CBC). These calculated elastic moduli of the disordered state are in excellent agreement with experiments. Configurational disorder between boron and carbon can also explain the experimentally observed almost constant elastic moduli of boron carbide as the carbon content is changed from B_{4}C to B_{13}C_{2}. The elastic moduli of the (B_{6}O)_{1−x}(B_{13}C_{2})_{x} system are also practically unchanged with composition if boron-carbon disorder is taken into account. By investigating the mixing thermodynamics of the alloys, in which the Gibbs free energy is determined within the mean-field approximation for the configurational entropy, we outline the pseudo-binary phase diagram of (B_{6}O)_{1−x}(B_{13}C_{2})_{x}. The phase diagram reveals the existence of a miscibility gap at all temperatures up to the melting point. Also, the coexistence of B_{6}O-rich as well as ordered or disordered B_{13}C_{2}-rich domains in the material prepared through equilibrium routes is predicted.
An accurate and easily extendable method to deal with lattice dynamics of solids is offered. It is based on first-principles molecular dynamics simulations and provides a consistent way to extract the best possible harmonic-or higher order-potential energy surface at finite temperatures. It is designed to work even for strongly anharmonic systems where the traditional quasiharmonic approximation fails. The accuracy and convergence of the method are controlled in a straightforward way. Excellent agreement of the calculated phonon dispersion relations at finite temperature with experimental results for bcc Li and bcc Zr is demonstrated.
Charge redistribution at low oxygen vacancy concentrations in ceria have been studied in the framework of the density functional theory. We propose a model to approach the dilute limit using the results of supercell calculations. It allows one to reproduce the characteristic experimentally observed behavior of composition versus oxygen pressure dependency. We show that in the dilute limit the charge redistribution is likely to be driven by a mechanism different from the one involving electron localization on cerium atoms. We demonstrate that it can involve charge localization on light element impurities.
Effect of volume change on the ionic conductivity in ceria has been studied in the framework of the density functional theory. We show that properly controlling external conditions one can treat the lattice constant of ceria as an adjustable parameter and change the topology of the energy landscape for the oxygen ion diffusion. We reveal the existence of the narrow range of lattice parameters, which optimize the ionic conductivity in ceria.
We have developed a thorough and accurate method of determining anharmonic free energies. The technique is based in ab initio molecular dynamics and map a model Hamiltonian to the fully anharmonic system. We can accurately deal with low-symmetry systems, such as random alloys. The formalism and the numerics are described in great detail. A number of practical examples are given, and favourable results are presented, both with respect to experiment and established techniques.
We have developed a thorough and accurate method of determining anharmonic free energies, the temperature dependent effective potential technique (TDEP). It is based on ab initio molecular dynamics followed by a mapping onto a model Hamiltonian that describes the lattice dynamics. The formalism and the numerical aspects of the technique are described in detail. A number of practical examples are given, and results are presented, which confirm the usefulness of TDEP within ab initio and classical molecular dynamics frameworks. In particular, we examine from first principles the behavior of force constants upon the dynamical stabilization of the body centered phase of Zr, and show that they become more localized. We also calculate the phase diagram for ^{4}He modeled with the Aziz et al. potential and obtain results which are in favorable agreement both with respect to experiment and established techniques.
The lattice dynamics for the monocarbides and mononitrides of the early transition metals has been studied using first-principles density functional perturbation theory. It is shown that the superconductivity of transition metal carbides is directly related to anomalies in the phonon spectra which in their turn are connected to the number of valence electrons. The calculated electron-phonon interaction constants are in excellent agreement with experimental data. Superconductivity is considerably enhanced for substitutional M Cx N1-x alloys. We also predict that perfect VC in the sodium chloride structure (B1) is a superconductor with a transition temperature Tc =11.5 K. The experimental failure to sythesize some transition metal carbides and nitrides within the B1 structure is connected to their dynamical instability. © 2005 The American Physical Society.
An accurate equation of state (EOS) is determined for the high-pressure orthorhombic phase of boron, B(28), experimentally as well as from ab initio calculations. The unique feature of our experiment is that it is carried out on the single crystal of B(28). In theory, we take into consideration the lattice vibrations, often neglected in first-principles simulations. We show that the phonon contribution has a profound effect on the EOS of B(28), giving rise to anomalously low values of the pressure derivative of the bulk modulus and greatly improving the agreement between theory and experiment.
The knowledge of lattice thermal conductivity of materials under realistic conditions is vitally important since many modern technologies require either high or low thermal conductivity. Here, we propose a theoretical model for determining lattice thermal conductivity, which takes into account the effect of microstructure. It is based on ab initio description that includes the temperature dependence of the interatomic force constants and treats anharmonic lattice vibrations. We choose ScN as a model system, comparing the computational predictions to the experimental data by time-domain thermoreflectance. Our experimental results show a trend of reduction in lattice thermal conductivity with decreasing domain size predicted by the theoretical model. These results suggest a possibility to control thermal conductivity by microstructural tailoring and provide a predictive tool for the effect of the microstructure on the lattice thermal conductivity of materials based on ab initio calculations.
The knowledge of lattice thermal conductivity of materials under realistic conditions is vitally important since most technologies either require either high or low thermal conductivity. Here, we propose a theoretical model for determining lattice thermal conductivity with the effect of microstructure. This is based on ab initio description that includes the temperature dependence of the interatomic force constants, and treats anharmonic lattice vibrations. We choose ScN as a model system, comparing the computational predictions with the experimental data by Time Domain Thermoreflectance (TDTR). Our results show a trend of reduction in lattice thermal conductivity with decreasing grain size, with good agreement between the theoretical model and experimental data. There results suggest a possibility to control thermal conductivity by tailoring the microstructure of ScN. More importantly, we provide a predictive tool for the effect of the microstructure on the lattice thermal conductivity of materials based on first-principles calculations.
The accuracy of the exact muffin-tin orbitals method combined with the coherent potential approximation (EMTO-CPA) for total energy calculations for systems with magnetic and chemical disorder, which is present simultaneously, is investigated. The mixing enthalpy of ordered, as well as disordered FeCo, FeNi, and FeCu equiatomic ferromagnetic alloys is calculated with the EMTO-CPA method and with the full-potential projector augmented wave (PAW) method. The results are compared and found to be in excellent agreement with each other. The EMTO-CPA method, in combination with disordered local moment model, is then applied to calculate the mixing enthalpy of the random paramagnetic face-centered cubic (fcc) FeCo alloy, as well as body-centered cubic (bcc) FeCr and FeV alloys over the whole concentration range. The results are compared with experimental data and a very good agreement is found again.
Herein, we use first principles calculations to study the energy of the (11 (2) over bar1) twin boundary in Zr, Zn, Mg, Ti, and Be. This boundary is important for understanding the microyielding and damping of hexagonal close-packed metals. The (11 (2) over bar1) twin boundary is unique in that it is composed of-and can form by the glide of-basal dislocations nucleating at every c lattice parameter. The effect of the number of atoms between boundaries on the boundary energy, and the resulting lattice strains of the relaxed structures are quantified. It is shown that the energies obtained converge within 32-64 atoms/supercell. The structures with a higher second-order elastic constant term, c(44), also have higher boundary energies. It is further shown that the critical resolved shear stresses of the basal dislocations at 0 K, which make up the (11 (2) over bar1) twin, are so low as to be below the threshold of the first principles calculations.
Static local displacements of ions in disordered face-centered cubic Fe50Ni50 alloy are studied from first principles in the framework of the density functional theory. The disordered alloy is modeled using a 64 atom supercell constructed as a special quasirandom structure. Fully relaxed atomic positions inside the supercell are calculated by means of projected augmented wave method as implemented in Vienna ab initio simulation package. According to our calculation, the relative changes of mean nearest neighbor interatomic distances due to local lattice relaxations are relatively small (0.6%), in agreement with experiment. At the same time, we predict that for all types of pairs, Fe–Fe, Fe–Ni, and Ni–Ni, the dispersion of the nearest neighbor interatomic distances is rather large, and the individual changes of distances between certain pairs of atoms due to local lattice relaxations can be one order of magnitude larger than the mean values for the corresponding pair of atoms
The sound velocity of Mo along the Hugoniot adiabat is calculated from first principles using density-functional theory based molecular dynamics. These data are compared to the sound velocity as measured in recent experiments. The theoretical and experimental Hugoniot and sound velocities are in very good agreement up to pressures of 210 GPa and temperatures of 3700 K on the Hugoniot. However, above that point the experiment and theory diverge. This implies that Mo undergoes a phase transition at about the same point. Considering that the melting point of Mo is likely much higher at that pressure, the related change in the sound velocity in experiment can be ascribed to a solid-solid transition.
A first-principles study of the effect of local environment on the electronic structure of random face-centered cubic Ag-Pd and Cu-Pd alloys is presented. The core-level shift for each atom in the equiatomic alloys is calculated and compared to experimental data. It is shown how the initial-state and final-state distributions contribute to the total broadening. We find that the initial-state and the final-state contributions together increase the broadening for the investigated core levels in Cu and Ag, whereas they cancel each other to a large degree for Pd. We also demonstrated how local lattice relaxations influence the binding energy shift. We find that relaxation does not influence the average shift, though it is able to affect the broadening of the simulated x-ray photoelectron spectroscopy spectra. © 2005 The American Physical Society.
It has been shown that the Earths inner core has an axisymmetric anisotropic structure with seismic waves traveling similar to 3% faster along polar paths than along equatorial directions. Hemispherical anisotropic patterns of the solid Earths core are rather complex, and the commonly used hexagonal-close-packed iron phase might be insufficient to account for seismological observations. We show that the data we collected are in good agreement with the presence of two anisotropically specular east and west core hemispheres. The detected travel-time anomalies can only be disclosed by a lattice-preferred orientation of a body-centered-cubic iron aggregate, having a fraction of their [111] crystal axes parallel to the Earths rotation axis. This is compelling evidence for the presence of a body-centered-cubic Fe phase at the top of the Earths inner core.
The influence of the tetragonal and orthorhombic axial distortions on the body-centered cubic (bcc) phase of Fe at extreme conditions has been studied by means of first-principles calculations. We unambigiously demonstrate that the energy minimum corresponding to the body-centered tetragonal (bct) (c/a approximate to 0.9) structure, previously found in Fe upon the axial tetragonal distortion of the bcc phase along the Bains path under compression at zero temperature, is an artifact of the structural constraint. When the bcc structure is examined using the orthorhombic distortion involving the tetragonal distortion as a particular case, the bct (c/a approximate to 0.9) structural framework represents a saddle point between two mirrored face-centered cubic minima rather than a local minimum. Therefore we conclude that there is no ground to emphasize on possible thermal stabilization of the bct structure with a particular c/a ratio apart from the whole family of structures obtained by tetragonal, orthorhombic, or another type of axial distortions.
Mechanical and thermodynamic stability of the isoelectronic ternary inverse perovskites Sc3EN (E=B,Al,Ga,In) has been studied from first principles. We confirm stability of recently synthesized cubic phases Sc3AlN and Sc3InN, and predict the stability of cubic Sc3GaN and a triclinic phase aP20-Sc3BN. Substantial phonon softening in Sc3AlN and Sc3GaN is observed indicating a possibility that structural defects could form readily. In accord, our experiments show that magnetron sputter deposited films contain regions with high density of nonperiodic stacking faults along the < 111 > growth direction. We suggest that defect-free crystals may exhibit anomalies in the carrier properties, promising for electronic applications.
The high-pressure structural behavior of the binary intermetallic compounds CoSn, FeSn, and NiIn with the peculiar void containing CoSn (B35)-type structure has been studied by means of room-temperature diamond anvil cell and high-temperature multianvil experiments, as well as by first-principles calculations. All three compounds remain structurally stable at pressures up to at least 25 GPa, whereas first-principles calculations predict high-pressure structural changes below 20 GPa. A plausible explanation for the discrepancy is that at room temperature, a sizable activation barrier inhibits kinetically the transformation into more close-packed polymorphs. It is supported by our experiments at temperatures around 1000 °C and a pressure of 10 GPa. At these conditions, NiIn transforms into the temperature-quenchable stoichiometric CsCl-type high-pressure phase, which has been predicted in our first-principles calculations. However, CoSn and FeSn decompose into a mixture of compounds richer and poorer in tin, respectively. Nevertheless, it might be possible that lower temperatures and higher pressures may afford theoretically predicted polymorphs. In particular, a phase transformation to the FeSi-type structure predicted for CoSn is of interest as materials with the FeSi-type structure are known for unusual thermal and transport properties.
A Reply to the Comment by C. Cazorla, D. Alfé, and M. J. Gillan.
The peculiar bonding situation in gamma boron is characterized on the basis of an experimental electron-density distribution which is obtained by multipole refinement against low-temperature single-crystal x-ray diffraction data. A topological analysis of the electron-density distribution reveals one-electron-two-center bonds connecting neighboring icosahedral B-12 clusters. A unique polar-covalent two-electron-three-center bond between a pair of atoms of an icosahedral cluster and one atom of the interstitial B-2 dumbbell explains the observed charge separation in this high-pressure high-temperature polymorph of boron.