Confinement-deconfinement transition and spin correlations in a generalized Kitaev model

We present a spin model, namely, the Kitaev model augmented by a loop term and perturbed by an Ising Hamiltonian, and show that it exhibits both confinement-deconfinement transitions from spin liquid to antiferromagnetic/spin-chain/ferromagnetic phases and topological quantum phase transitions between gapped and gapless spin-liquid phases. We develop a fermionic resonating-valence-bonds (RVB) mean-field theory to chart out the phase diagram of the model and estimate the stability of its spin-liquid phases, which might be relevant for attempts to realize the model in optical lattices and other spin systems. We present an analytical mean-field theory to study the confinement-deconfinement transition for large coefficient of the loop term and show that this transition is first order within such mean-field analysis in this limit. We also conjecture that in some other regimes, the confinement-deconfinement transitions in the model, predicted to be first order within the mean-field theory, may become second order via a defect condensation mechanism. Finally, we present a general classification of the perturbations to the Kitaev model on the basis of their effect on it's spin correlation functions and derive a necessary and sufficient condition, within the regime of validity of perturbation theory, for the spin correlators to exhibit a long-ranged power-law behavior in the presence of such perturbations. Our results reproduce those of Tikhonov et al. [Phys. Rev. Lett. 106, 067203 (2011)] as a special case.

Evolving Random Geometric Graph Models for Mobile Wireless Networks

We consider evolving exponential RGGs in one dimension and characterize the time dependent behavior of some of their topological properties. We consider two evolution models and study one of them detail while providing a summary of the results for the other. In the first model, the inter-nodal gaps evolve according to an exponential AR(1) process that makes the stationary distribution of the node locations exponential. For this model we obtain the one-step conditional connectivity probabilities and extend it to the k-step case. Finite and asymptotic analysis are given. We then obtain the k-step connectivity probability conditioned on the network being disconnected. We also derive the pmf of the first passage time for a connected network to become disconnected. We then describe a random birth-death model where at each instant, the node locations evolve according to an AR(1) process. In addition, a random node is allowed to die while giving birth to a node at another location. We derive properties similar to those above.

Thermodynamic study of oxygen in liquid lead tin alloys

The activity coefficients of oxygen in liquid lead-tin alloys have been measured between 550 and 1100°C by use of solid oxide galvanic cells Pt, Ni-NiO I Zr02 Solid electrolyte I 0 (Pb + Sn), Cermet, Pt Pt, Fe-FeO I Zr02 Solid electrolyte I O(Pb + Sn), Cermet, Pt Alcock and Richardson's quasi-chemical equation, with the coordination number of atoms set to 2, is found to predict successfully the activity coefficients of oxygen in these alloys.The relative partial molar enthalpy and entropy of oxygen ?t 1 atom per cent in the alloys have been calculated from ttva variation of the activity coefficient with temperature. The addition of tin to an unsaturated solution of oxygen in lead is shown to decrease significantly both the partial molar enthalpy and entropy of oxygen. As the measurements were restricted to a narrow range between 750-1100'C in lead-rich alloys, however, the pronounced variation of the partial molar enthalpy of oxygen with temperature at constant alloy composition predicted by the quasi-chemical model could not be verified.

The oxygen potential of the systems Fe+FeCr2O4+ Cr2O3 and Fe+FeV2O4+ V2O3 in the temperature range 750 1600o C

From electromotive force (emf) measurements using solid oxide galvanic cells incorporating ZrOz-CaO and ThOz-YO~.s electrolytes, the chemical potentials of oxygen over the systems Fe + FeCrzO 4 + Cr20 ~ and Fe + FeV204 + V203 were calculated. The values may be represented by the equations: 2Fe(s, I) + Oz(g) + 2Cr2Oa(s) -- 2FeCr204 (s)Akto2 = - 151,400 + 34.7T (• cal= -633,400 + 145.5T(• J (750 to 1536~ A~tO2 = -158,000 + 38.4T(• cal= -661,000 + 160.5T(*1250) J (1536 to 1700~2Fe (s, I) + O2 (g) + 2V203 (s) -- 2FeV204 (s) A/~Oz = - 138,000 + 29.8T(+300) cal= - 577,500 + 124.7T (• J (750 to 1536~A/IO2 = -144,600 + 33.45T(-300) cal = -605,100 + 140.0T(~-1250) J (1536 to 1700~At the oxygen potentials corresponding to Fe + FeCrzO a + Cr203 equilibria, the electronic contribution to the conductivity of ZrO2-CaO electrolyte was found to affect the measured emf. Application of a small 60 cycle A.C. voltage with an amplitude of 50 mv across the cell terminals reduced the time required to attain equilibrium at temperatures between 750 to 9500C by approximately a factor of two. The second law entropy of iron chromite obtained in this study is in good agreement with that calculated from thermal data. The entropies of formation of these spinel phases from the component oxides can be correlated to cation distribution and crystal field theory.

Thermodynamics of CuAlO2 and CuAl2O4 and phase equilibria in the system Cu2O CuO-Al2O3

The standard Gibbs free energies of formation of CuAlO2 and CuAl2O4 were determined in the range 700° to 1100°C, using emf measurements on the galvanic cells (1) Pt,CuO +] Cu2O/CaO-ZrO2/O2,Pt; (2) Pt,Cu +] CuAlO2+] Al2O3/CaO-ZrO2/ Cu +] Cu2O,Pt; and (3) Pt,CuAl2O4+] CuAlO2+]Al2O3/CaO-ZrO2/O2,Pt. The results are compared with published information on the stability of these compounds. The entropy of transformation of CuO from tenorite to the rock-salt structure is evaluated from the present results and from earlier studies on the entropy of formation of spinels from oxides of the rock-salt and corundum structures. The temperatures corresponding to 3-phase equilibria in the system Cu2O-CuO-Al2O3 at specified O2 pressures calculated from the present results are discussed in reference to available phase diagrams.

Gibbs free energies of formation of ZnAl2O4 and ZnCr2O4

The standard free energies of formation of zinc aluminate and chromite were determined by measuring the oxygen potential over a solid CuZn alloy, containing 10 at.−% Zn, in equilibrium with ZnO, ZnAl2O4+Al2O3(χ) and ZnCr2O4+Cr2O3, in the temperature range 700–900°C. The oxygen potential was monitored by means of a solid oxide galvanic cell in which a Y2O3 ThO2 pellet was sandwiched between a CaOZrO2 crucible and tube. The temperature dependence of the free energies of formation of the interoxidic compounds can be represented by the equations, The heat of formation of the spinels calculated from the measurements by the “Second Law method” is found to be in good agreement with calorimetrically determined values. Using an empirical correlation for the entropy of formation of cubic spinel phases from oxides with rock-salt and corundum structures and the measured high temperature cation distribution in ZnAl2O4, the entropy of transformation of ZnO from wurtzite to rock-salt structure is evaluated.

Thermodynamics and phase equilibria in the system Cu2O CuO ()Ga2O3

The standard Gibbs free energies of formation of CuAlO2 and CuAl2O4 were determined in the range 700° to 1100°C, using emf measurements on the galvanic cells (1) Pt,CuO +] Cu2O/CaO-ZrO2/O2,Pt; (2) Pt,Cu +] CuAlO2+] Al2O3/CaO-ZrO2/ Cu +] Cu2O,Pt; and (3) Pt,CuAl2O4+] CuAlO2+]Al2O3/CaO-ZrO2/O2,Pt. The results are compared with published information on the stability of these compounds. The entropy of transformation of CuO from tenorite to the rock-salt structure is evaluated from the present results and from earlier studies on the entropy of formation of spinels from oxides of the rock-salt and corundum structures. The temperatures corresponding to 3-phase equilibria in the system Cu2O-CuO-Al2O3 at specified O2 pressures calculated from the present results are discussed in reference to available phase diagrams.

Gibbs free energy of formation of magnesium stannate

he thermodynamic properties of the spinel Mg2SnO4 have been determined by emf measurements on the solid oxide galvanic cell,View the MathML source in the temperature range 600 to 1000°C. The Gibbs' free energy of formation of Mg2SnO4 from the component oxides can be expressed as View the MathML source,View the MathML source These values are in good agreement with the information obtained by Jackson et al. [Earth Planet. Sci. Lett.24, 203 (1974)] on the high pressure decomposition of magnesium stannate into component oxides at different temperatures. The thermodynamic data suggest that the spinel phase is entropy stabilized, and would be unstable below 207 (±25)°C at atmospheric pressure. Based on the information obtained in this study and trends in the stability of aluminate and chromite spinels, it can be deduced that the stannates of nickel and copper(II) are unstable.

Electrochemical determination of activities in Cr2O3 Al2O3 solid solutions

The activity of Cr20~ in Cr20~-A12Oa solid solution has been determined in the temperature range 800~176 from electromotive force measurements on the solid oxide galvanic cell Pt,Cr + Cr2OJY~O~-ThO2/Cr + Cr~A12-xO~,Pt The activities of Cr203 and A120~ in the solid solution show both positive and negative deviations from Raoult's law. The heat and entropy of mixing of the solid Solution obtained from the temperature dependence of the emf can be expressed as AH = XCr203XA1203 [31,700Xcrzo3 -}- 37,470XA1203] J mole -I hS = -- 1.8R [Xcr2o3 In Xcr2o3 + XA12o3 In XAaos]The entropy of mixing is 10% lower than that predicted by the Temkin model.The large positive heat of mixing in the Cr2Os-A12Oa solid solution, however, suggests that this apparent: entropy discrepancy originates with the clustering of positive ions on the cation sublattice. The asymmetric miscibility gap exhibited in the CrzOa-A12Oa system below 900~ is consistent with the thermodynamic data trends recorded at the more elevated temperatures.

Vapor pressure and thermodynamic properties of MgIn2O4

The vapor pressure of pure indium, and the sum of the pressures of (In) and (In2O) species over the condensed phase mixture {In} + 〈MgIn2O4〉 + 〈MgO〉, have been measured by the Knudsen effusion technique in the temperature range 1095–1350 K. The materials under study were contained in a zirconia crucible, which had a Knudsen orifice along the vertical wall. The major vapor species over the condensed phase mixture were identified as (In) and (In2O) using a mass-spectrometer. The vapor pressure of (In2O) corresponding to the reaction,View the MathML source was deduced from the experimental results;View the MathML source The standard free energy of formation of the inverse spinel 〈MgIn2O4〉 from its component oxides, is given by,View the MathML source View the MathML source The entropy of transformation of 〈In2O3〉 from the C rare-earth structure to the corundum structure is evaluated from the measured entropy of formation of (MgIn2O4) and a semi-empirical correlation for the entropy of formation of spinel phases from component oxides with rock-salt and corundum structures.

Solubility and activity of oxygen in liquid antimony

The solubility limit of oxygen in liquid antimony has been measured by a novel isopiestic technique in the temperature range 995--1175 deg K. The results can be expressed by the equation log c = --5500/T + 3.754 ( plus/minus 0.04) with c in at.% O and T in deg K. The oxygen potential over Sb + O alloys equilibrium with Sb2O3 has been measured by a solid state cell using a fully stabilized CaO.ZrO2 electrolyte. The cell was designed to contain the Sb + Sb2O3 mixture in a closed volume, that the vaporization of the oxide can be minimized and true equilibrium attained. The Gibbs free energy of the reaction 2 Sb(s) + 3/2 O2 = Sb2O3(s) is Delta G deg = --719560 + 274.51 T( plus/minus 500) and Sb(l) + 3/2 O2 = Sb2O3(l), Delta G deg = --704711 + ( plus/minus 500) ( Delta G deg in J/mole, T in deg K). The combination of these results with Sieverts' law yields the standard free energy of solution of oxygen in liquid antimony according to the reaction 1/2 O2 = \O\Sb,at.% as Delta G deg = --129620 + 14.23 T ( plus/minus 950). The standard enthalopy and entropy of the solution of oxygen in Sb are compared with values for other metal- oyxgen systems, and with the standard enthalpies of formation of corresponding oxides. The resulting correlations permit the estimation of the standard free energy of solution of oxygen in pure metals for which experimental information is lacking. 24 ref.--AA

A comparison of the structure of liquid alloys derived from thermodynamic and diffraction studies

A significant amount of research on the thermodynamic properties of molten alloys is undertaken for obtaining insights into their structure . The partial and integral molar enthalpies, entropies and volumes of mixing provide some general information on the nature and strength of atomic bonds and the distribution of atoms. However, until recently it has been difficult to derive specific quantitative information because the excess entropy of mixing contains configurational , vibrational , electronic , and sometimes magnetic contributions which cannot be easily separated.

Correlations between structure and thermodynamic properties of spinels and their applications in technology

Measurements a/the Gibbs' energy enthalpy and entrupy vffarmation oj chromites, vanadites and alumlnat.:s 0/ F", Ni. Co'. Mn, Zn Mg and Cd, using solid oxide galvanic cells over a ternperature range extending approximately lOOO°C, have shown that the '~'Ilir"!,,, J'JrIl/iJ~ tion 0/ cubic 2-3 oxide spinel phases (MX!O,), from component oxide (MO) with rock-salt and X.Os whir c(1f'l/!ldwn st!'llt'lw,·. call b,' represented by a semi-empirical correlalion, ~S~ = --LiS + L'i,SM +~S~:"d(±O.3) cal.deg-1 mol-1 where /',.SM Is the entropy 0/calian mixing oillhe tetrahedral alld octahedral sites o/the spinel and Sr:~ is tlie enfropy associaf,'d Wifh Ih,' randomization a/the lahn-Telier distortions. A review a/the methods/or evaluating the cation distriblltion lfl spille!s suggeJ{j' l/r,l! Ihe most promising scheme is based Oil octahedral site preference energies from the crystal field theory for the Iral1silioll IIIl'f"! IlIIL';. For I/""-Irallsifioll melal cal ions site preference energies are derived relative /0 thol'lt fLI, [ransilion metal ions from measured high tClllP('ftJi ure Cal iUlI disll iiJuriol1 in spine! phases thar contail! one IransilioJl metal and another non-transition metal carion. For 2-3 srinds compulatiorrs b,IS"J Oil i.!c[J;' Temkin mixing on each catioll subialtice predici JistributionJ that are In fair agreement with X-ray and 1I1'IIIrOll ditTraction, /IIdg""!ic dll.! electrical propcrries, and spectroscopic measurements. In 2-4 spineis mixing vI ions do not foliow strictly ideal slllIistli:al Jaws, Th,' OIl/up) associated with the randomizalion 0/the Jllhn-Teller dislOriioll" appear to be significant, only ill spinels witll 3d'. 3d', 3d' (ifld~UI' iOtls in tetrahedral and 3d' and 3d9 ions in octahedral positions. Application 0/this structural model for predicting the thermodynamic proputies ofspinel solid .,olutiofl5 or,' illustrated. F,lr complex systems additional contributions arising from strain fields, redox equilibria and off-center ions have to be qllalllififti. The entropy correlation for spinels provides a method for evaluating structure tran:.jormafiofl entropies in silllple o.\id.-s, ["founlllion on the relative stabilities ofoxides in different crystallCtructures is USe/III for computer ea/culaliof! a/phase dfugrullls ofIlIrer,',,1 III (N.lll1ie5 by method, similar to thost: used by Kaufman and Bernstein for refractory alloy systems. Examples oftechnoiogical appliCation tnclude the predictioll ofdeoxidation equilibria in Fe-Mn-AI-O s),slelll at 1600°C duj ,'Ulllpltfalion 0/phase relutions in Fe-Ni-Cr-S system,

Solubility and activity of oxygen in liquid manganese

The oxygen concentration of liquid manganese in equilibrium with MnAl2+2xO4+3x and α−Al2O3 has been determined in the temperature range 1520 to 1875 K. The oxygen content of quenched samples, wrapped in oxygen-free nickel foil, was determined by an inert gas fusion technique. The results are combined with accurate data now available on the Gibbs energies of formation of MnO and Al2O3−saturated MnAl2+2xO4+3x to derive the oxygen content of liquid manganese in equilibrium with MnO and the Gibbs energy of solution of diatomic oxygen gas in liquid manganese. The enthalpy and entropy of solution of oxygen in manganese are compared with similar data on other metal-oxygen systems.

Inter and intra crystalline ion exchange equilibria in the system Fe Cr Al O

The tie lines delineating ion-exchange equilibria between FeCr2O4FeAl2O4 spinel solid solution and Cr2O3Al2O3 solid solution with corundum structure have been determined at 1373 K by electron microprobe and EDAX point count analysis of oxide phases equilibrated with metallic iron. Activities in the spinel solid solution are derived from the tie lines and the thermodynamic data on Cr2O3Al2O3 solid solution available in the literature. The oxygen potentials corresponding to the tie-line composition of oxide phases in equilibrium with metallic iron were measured using solid oxide galvanic cells with CaOZrO2 and Y2O3ThO2 electrolytes. These electrochemical measurements also yield activities in the spinel solid solution, in good agreement with those obtained from tie lines. The activity-composition relationship in the spinel solid solution is analysed in terms of the intra-crystalline ion exchange between the tetrahedral and octahedral sites of the spinel structures. The ion exchange is governed by site-preference energies of the cations and the entropy of cations mixing on each site.