Bruhat Order in the Full Symmetric sln\mathfrak{sl}_n Toda Lattice on Partial Flag Space

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations

Thermal equation of state and thermodynamic properties of iron carbide Fe 3 C to 31 GPa and 1473 K

Resent experimental and theoretical studies suggested preferential stability of Fe 3 C over Fe 7 C 3 at the condition of the Earth's inner core. Previous studies showed that Fe 3 C remains in an orthorhombic structure with the space group Pnma to 250 GPa, but it undergoes ferromagnetic (FM) to paramagnetic (PM) and PM to nonmagnetic (NM) phase transitions at 6–8 and 55–60 GPa, respectively. These transitions cause uncertainties in the calculation of the thermoelastic and thermodynamic parameters of Fe 3 C at core conditions. In this work we determined P‐V‐T equation of state of Fe 3 C using the multianvil technique and synchrotron radiation at pressures up to 31 GPa and temperatures up to 1473 K. A fit of our P‐V‐T data to a Mie‐Gruneisen‐Debye equation of state produce the following thermoelastic parameters for the PM‐phase of Fe 3 C: V 0  = 154.6 (1) Å 3 , K T 0 = 192 (3) GPa, K T ′ = 4.5 (1), γ 0 = 2.09 (4), θ 0  = 490 (120) К, and q  = −0.1 (3). Optimization of the P‐V‐T data for the PM phase along with existing reference data for thermal expansion and heat capacity using a Kunc‐Einstein equation of state yielded the following parameters: V 0  = 2.327 cm 3 /mol (154.56 Å 3 ), K T 0  = 190.8 GPa, K T ′ = 4.68, Θ E10  = 305 K (which corresponds to θ 0  = 407 K), γ 0  = 2.10, e 0  = 9.2 × 10 −5 K −1 , m  = 4.3, and g  = 0.66 with fixed parameters m E 1  = 3 n  = 12, γ ∞  = 0, β  = 0.3, and a 0  = 0. This formulation allows for calculations of any thermodynamic functions of Fe 3 C versus T and V or versus T and P . Assuming carbon as the sole light element in the inner core, extrapolation of our equation of state of the NM phase of Fe 3 C suggests that 3.3 ± 0.9 wt % С at 5000 К and 2.3 ± 0.8 wt % С at 7000 К matches the density at the inner core boundary. Key Points We present a P‐V‐T EOS for PM‐Fe 3 C with support from thermodynamic analyses We discuss uncertainties in magnetic transitions We applied EOS data for modeling carbon content in the corePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/101805/1/jgrb50396.pd

ALKALI-CONTAINING MINERALS WITHIN MELT INCLUSIONS IN OLIVINE OF MANTLE XENOLITHS FROM BULTFONTEIN KIMBERLITE PIPE (KAAPVAAL CRATON): EVIDENCE ON HIGH CONCENTRATIONS OF ALKALIS IN KIMBERLITE MELTS

The study reports the mineral assemblage of the crystallized secondary melt inclusions in the olivine of sheared peridotites xenoliths from Bultfontein kimberlite pipe (Kaapvaal Craton, South Africa). In this type of xenoliths, the inclusions may correspond in composition to primitive kimberlite melts related to the magmatism that formed the Bultfontein pipe. Among 32 daughter phases within the inclusions, there are both ordinary rock-forming and minor minerals for kimberlites (silicates, carbonates, oxides) and "exotic" (alkali carbonates, sulfates, and chlorides) for these rocks. In the inclusions, 20 alkali-containing minerals are present, 12 of which are Na-bearing and – 4 Na-K-bearing. For instance, the inclusions contain nyerereite, K-nyerereite, shortite, gregoryite, eitelite, bradleyite, northupite, tychite, burkeite, aphthitalite, arcanite, thenardite, sylvine, and halite. On the basis of these results, the kimberlite melt of the Bultfontein pipe had Na-specification rather than Ca or K ones. The carbonates, sulfates, and chlorides significantly prevail over silicates, which content (serpentine + micas) does not exceed 16 vol. %, in the inclusions.The obtained results pose fundamental questions regarding the petrogenesis of kimberlites: (i) initial sodium concentrations in kimberlite melts and rocks, which are "traditionally" considered as very low; (ii) composition and ratio of volatile components in kimberlite magmas, namely, the initial contents of both CO2 and the components such as Cl, SO3 and H2O; (iii) primary magmatic mineral association of kimberlite rocks, which loses diverse alkali-containing minerals, but mica, due to serpentinization process

Uakitite, VN, a new mononitride mineral from uakit iron meteorite (IIAB)

Uakitite was observed in small troilite–daubréelite (±schreibersite) inclusions (up to 100 µm) and in large troilite–daubréelite nodules (up to 1 cm) in Fe-Ni-metal (kamacite) of the Uakit iron meteorite (IIAB), Republic of Buryatia, Russia. Such associations in the Uakit meteorite seemed to form due to high-temperature (>1000 °C) separation of Fe-Cr-rich sulfide liquid from Fe-metal melt. Most inclusions represent alternation of layers of troilite and daubréelite, which may be a result of solid decay of an initial Fe-Cr-sulfide. These inclusions are partially resorbed and mainly located in fissures of the meteorite, which is now filled with magnetite, and rarely other secondary minerals. Phase relations indicate that uakitite is one of the early minerals in these associations. It forms isometric (cubic) crystals (in daubréelite) or rounded grains (in schreibersite). The size of uakitite grains is usually less than 5 µm. It is associated with sulfides (daubréelite, troilite, grokhovskyite), schreibersite and magnetite. Carlsbergite CrN, a more abundant nitride in the Uakit meteorite, was not found in any assemblages with uakitite. Physical and optical properties of uakitite are quite similar to synthetic VN: yellow and transparent phase with metallic luster; Mohs hardness: 9–10; light gray color with a pinky tint in reflected light; density (calc.) = 6.128 g/cm3. Uakitite is structurally related to the osbornite group minerals: carlsbergite CrN and osbornite TiN. Structural data were obtained for three uakitite crystals using the electron backscatter diffraction (EBSD) technique. Fitting of the EBSD patterns for a synthetic VN model (cubic, Fm-3m, a = 4.1328(3) Å; V = 70.588(9) Å3; Z = 4) resulted in the parameter MAD = 0.14–0.37° (best-good fit). Analytical data for uakitite (n = 54, in wt. %) are: V, 71.33; Cr, 5.58; Fe, 1.56; N, 21.41; Ti, below detection limit (<0.005). The empirical formula (V0.91Cr0.07Fe0.02)1.00N1.00 indicates that chromium incorporates in the structure according to the scheme V3+ → Cr3+ (up to 7 mol. % of the carlsbergite end-member). © 2020 by the authors. Licensee MDPI, Basel, Switzerland.Russian Foundation for Basic Research, RFBR: 17-05-00129, IGM SD 0330-2016-0005Government Council on Grants, Russian FederationMinistry of Science and Higher Education of the Russian FederationFunding: The investigations were partly supported by RFBR (grant 17-05-00129) and the State assignment project (IGM SD 0330-2016-0005). This work was also supported by the Initiative Project of Ministry of Science and Higher Education of the Russian Federation and by Act 211 of the Government of the Russian Federation, agreement no. 02.A03.21.0006