On the Lattice of Convex Sublattices of a Lattice

1 Introduction Let L be a lattice and CS(L) be the family of all convex sublattices of L, includingthe empty set.Then CS(L), partially ordered by inclusion, forms an atomic algebraic lat-tice. K. M. Koh investigated some basic concepts and results of CS(L), and raised the following Problem.

Light-modulated graphene-based φ_(0) Josephson junction and -φ_(0) to φ_(0) transition

We investigate the chiral edge states-induced Josephson current–phase relation in a graphene-based Josephson junction modulated by the off-resonant circularly polarized light and the staggered sublattice potential.By solving the Bogoliubov–de Gennes equation,a φ_(0) Josephson junction is induced in the coaction of the off-resonant circularly polarized light and the staggered sublattice potential,which arises from the fact that the center of-mass wave vector of Cooper pair becomes finite and the opposite center of-mass wave vector to compensate is lacking in the nonsuperconducting region.Interestingly,when the direction of polarization of light is changed,-φ_(0) to φ_(0) transition generates,which generalizes the concept of traditional 0–πtransition.Our findings provide a purely optical way to manipulate a phase-controllable Josephson device and guidelines for future experiments to confirm the presence of graphene-based φ_(0)Josephson junction.

Alloy gene Gibbs energy partition function and equilibrium holographic network phase diagrams of AuCu_3-type sublattice system

Taking AuCu3-type sublattice system as an example, three discoveries have been presented: First, the third barrier hindering the progress in metal materials science is that researchers have got used to recognizing experimental phenomena of alloy phase transitions during extremely slow variation in temperature by equilibrium thinking mode and then taking erroneous knowledge of experimental phenomena as selected information for establishing Gibbs energy function and so-called equilibrium phase diagram. Second, the equilibrium holographic network phase diagrams of AuCu3-type sublattice system may be used to describe systematic correlativity of the composition?temperature-dependent alloy gene arranging structures and complete thermodynamic properties, and to be a standard for studying experimental subequilibrium order-disorder transition. Third, the equilibrium transition of each alloy is a homogeneous single-phase rather than a heterogeneous two-phase, and there exists a single-phase boundary curve without two-phase region of the ordered and disordered phases; the composition and temperature of the top point on the phase-boundary curve are far away from the ones of the critical point of the AuCu3 compound.

Alloy gene Gibbs energy partition function and equilibrium holographic network phase diagrams of Au_3Cu-type sublattice system

Taking Au3Cu-type sublattice system as an example, three discoveries have been presented. First, the fourth barrier to hinder the progress of metal materials science is that today’s researchers do not understand that the Gibbs energy function of an alloy phase should be derived from Gibbs energy partition function constructed of alloy gene sequence and their Gibbs energy sequence. Second, the six rules for establishing alloy gene Gibbs energy partition function have been discovered, and it has been specially proved that the probabilities of structure units occupied at the Gibbs energy levels in the degeneracy factor for calculating configuration entropy should be degenerated as ones of component atoms occupied at the lattice points. Third, the main characteristics unexpected by today’s researchers are as follows. There exists a single-phase boundary curve without two-phase region coexisting by the ordered and disordered phases. The composition and temperature of the top point on the phase-boundary curve are far away from those of the critical point of the Au3Cu compound; At 0 K, the composition of the lowest point on the composition-dependent Gibbs energy curve is notably deviated from that of the Au3Cu compounds. The theoretical limit composition range of long range ordered Au3Cu-type alloys is determined by the first jumping order degree.

Magnetizations of Multisublattice Bilayer System

Magnetizations of four sublattice bilayer system have been systematically investigated by the use of effectivefield theory with self-spin correlations and differential operator technique. The effects of the crystal field, longitudinal magnetic field, and intra- and inter-monolayer interaction constants on magnetizations axe examined in detail. Some interesting results are obtained, which may potentially be related to experimental work.

Luminescence and Energy Transfer of Gd_xY_（1－x）P_5O_（14）：Ce， Tb──Ⅱ． Energy Transfer of Ce→Gd→Tb

The energy transfer phenomenon of Ce→Gd→Tb via Gd sublattice and its depandence has been investigated in GdxY-1-xP5O14:Ce,Tb.The fluorescent and excitation spectra of Gdp5O14,Gdp5O14:Ce,Gdp5O14:Tb and GdxY-1xP5O14:Ce,Tb and absorption spectrum of Gdp5O14 have been studied.The results show that as x is larger than 0.7.the energy transfer from Ce3+ via Gd3+to Tb3+ is obvious.The main reason for the energy transfer of Ce→Gd→Tb being efficient in the region x>0.7 is that the spectral overlap between Ce3+ emission spectrum and Gd3+ absorption spectrum increases and the structure changes from monoclinic Ⅱ(C2/c) layer structure(x<0.7) to monoclinic I(P21/c) ribbon structure.

Luminescence and Energy Transfer of Gd_xY_（1－x）P_5O_（14）：Ce，Tb──I．Preparation，Structure and Spectral Properties

Preparation, structure and spectral properties of rare earth pentaphosphates Gd_xY_(1-x)P_5O_(14): Ce, Tb have been investigated. When x>0. 7, the pentaphosphates belong to monoclinic crystal system Ⅰ with space group P2_1/c (C). When x≤0. 7. they belong to monoclinic crystal system Ⅱ with C2/c (C). The fluorescent and excitation spectra of Gdp_5O_(14), GdP_5O_(14): Ce. GdP_5O_(14) : Tb and Gd_xY_(1-x)P_5O_(14) : Ce, Tb have been studied and the energy transfer phenomenon from Ce(3+)→Gd(3+)→Tb(3+) by the medium of Gd(3+) sublattice has been determined.

Mean-Field Solution of the Mixed Spin-2 and Spin-5/2 Ising Ferrimagnetic System with Different Single-Ion Anisotropies

The mixed spin-2 and spin-5/2 Ising ferrimagnetic system with different anisotropies (DA/z|J|) for the spin-2 and (DB/z|J|) for the spin-5/2 is studied by the use of the mean-field theory based on the Bogoliubov inequality for the free energy. First, the ground state phase diagram of the system at zero temperature is obtained on the (DA/z|J|,DB/z|J|) plane. Topologically, different kinds of phase diagrams are achieved by changing the temperature and the values of the single ion anisotropies DA/z|J| and DB/z|J|. Besides second-order transition lines, first order phase transition lines terminating at tricritical points, are found. The existence and dependence of a compensation temperature on single-ion anisotropies is also investigated.

Single-Parameter Quantum Pumping in Graphene Nanoribbons with Staggered Sublattice Potential

We present a theoretical study of quantum charge pumping in metallic armchair graphene nanoribbons using the Floquet Green function method. A central part of the ribbon acting as the scattering region is supposed to have staggered sublattiee potential to open a finite band gap. A single ae gate is asymmetrically applied to a part of the scattering region to drive the pumping. Corresponding to the gap edges, there are two pumped current peaks with opposite current directions, which can be reversed by changing the position of the ac gate relative to the scattering region. The effects of the parameters, such as the staggered sublattice potential, the driving frequency and the geometric parameters of the structure, on the pumping are discussed.

Copper chalcogenide thermoelectric materials

Cu-based chalcogenides have received increasing attention as promising thermoelectric materials due to their high efficiency,tunable transport properties,high elemental abundance and low toxicity.In this review,we summarize the recent research progress on this large family compounds covering diamond-like chalcogenides and liquid-like Cu2X (X=S,Se,Te)binary compounds as well as their multinary derivatives.These materials have the general features of two sublattices to decouple electron and phonon transport properties.On the one hand,the complex crystal structure and the disordered or even liquid-like sublattice bring about an intrinsically low lattice thermal conductivity.On the other hand, the rigid sublattice constitutes the charge-transport network, maintaining a decent electrical performance.For specific material systems,we demonstrate their unique structural features and outline the structure-performance correlation. Various design strategies including doping,alloying,band engineering and nanostructure architecture,covering nearly all the material scale,are also presented.Finally,the potential of the application of Cu-based chalcogenides as high-performance thermoelectric materials is briefly discussed from material design to device development.

DIRECT PRODUCT DECOMPOSITION OF MOLECULAR LATTICES AND STRUCTURE OF GENERALIZED ORDER HOMOMORPHISMS

In this note, we shall give the direct product decomposition of a molecular lattice using the concepts of molecules, then we shall discuss the structure of a generalized order homomorphism. About the concepts and notations in this note, refer to [1—7]. L(M) denotes a molecular lattice (i. e. a completely distributive lattice) with M as the set of molecules in L. Let {L_i:i∈I} be a family of mo-

Magneto-electric behaviors of La_(0.67-x)Nd_(x)Sr_(0.33)MnO_(3) system

The polycrystalline samples La_(0.67-x)Nd_(x)Sr_(0.33)MnO_(3)(x=0.1,0.2,0.3,0.4,0.5)were prepared by standard solid-state reaction method,and the M–T curves,electron spin resonance(ESR)curves,infrared spectra,Raman spectra,ρ–T curves and MR–T curves of the compounds above were measured.The results show that single phase can be obtained at different Nd substitution ratios,and with the increase in Nd doping,the magnetic structure of the system transforms from long-range ferro-magnetic order to spin cluster glass state and antiferro-magnetic state;and the phase separation appears in the samples with x=0.3 and 0.4.The transport property exhibits variation with the increase in Nd doping,and the electric behavior of the sample transforms(from metal state to insulator–metal transition,and then insulator state)is accompanied by colossal magnetoresistance(CMR)effect,which can be attributed to the different couplings of sublattices induced by Nd doping and the interface tun-neling effect related to the spin.

RELATIVELY COMPLEMENTED DISTRIBUTIVE LATTICE

In this paper, we have proved that for a relatively complemented distributive lattice L,there exists one and only one completion ? of L such that for every ξ∈?, there exists afamily {S_α}_(α∈△) of non-vacuous subsets of L satisfying ξ= ∨[∧f(S_α)]. Such a completion? is called the entire completion of L. We have in this paper extended the homomorphic extension theorem of a generalizedBoolean lattice to that of a relatively complemented distributive lattice and proved that the lat-tice of the congruences on a relatively complemented distributive lattice is isomorphicwith the lattice of the convex sublattices containing a fixed element and that the entire comple-tion of a relatively complemented distributive lattice L, the lattice of the completable con-gruences on L and the lattice of the completable convex sublattices of L containing a fixed ele-ment are isomorphic.

Two-Sublattice Thermodynamic Model on Carbonitride Precipitation in Microalloyed Steels Bearing Nb-V-Ti

Based on the two sublattice model of the regular solution,one being metal atom sublattice and another being interstitial atom sublattice,a thermodynamic model for the precipitates of niobium carbonitride,vanadium carbonitride and titanium carbonitride was established to study the starting-temperature of precipitates and the austenite compositions at given temperature in a low carbon steel.The calculation results show that starting-temperature of the precipitation of niobium carbonitride,vanadium carbonitride and titanium carbonitride are 1100℃,920℃ and 1340℃,respectively,the mole fraction of carbonitride precipitates is 8.65×10-4 in the 0.053C-0.0028N-1.28Mn-0.008S-0.031Al-0.046Nb-0.008Ti0.029V-Fe steel.When the N content is from 0.0028% to 0.0056%,the starting-temperature of the precipitation of the titanium carbonitride changes from 1340℃ to 1430℃.And the C content is from 0.053% to 0.07%,the startingtemperature of the precipitation of the titanium carbonitride hardly changes,but the atomic fraction of niobium in the carbonitride obviously increases.

“Phonon” scattering beyond perturbation theory

Searching and designing materials with intrinsically low lattice thermal conductivity （LTC） have attracted extensive considera- tion in thermoelectrics and thermal management community. The concept of part-crystalline part-liquid state, or even part-crystalline part-amorphous state, has recently been proposed to describe the exotic structure of materials with chemical-bond hierarchy, in which a set of atoms is weakly bonded to the rest species while the other sublattices retain relatively strong rigidity. The whole system inherently manifests the coexistence of rigid crystalline sublattices and fluctuating noncrystalline substructures. Representative materials in the unusual state can be classified into two categories, i.e., caged and non-caged ones. LTCs in both systems deviate from the traditional 7-1 relationship （T, the absolute temperature）, which can hardly be described by small-parameter-based perturbation approaches. Beyond the classical perturbation theory, an extra rattling-like scattering should be considered to interpret the liquid-like and sublattice-amorphization-induced heat transport. Such a kind of compounds could be promising high-performance thermoelectric materials, due to the extremely low LTCs. Other physical properties for these part-crystalline substances should also exhibit certain novelty and deserve further exploration.

Spectrum of L-fuzzy Prime Ideals of a Distributive Lattice

A topological space denoted by Fspec(D),called L-fuzzy prime spectrum of a bounded distributive lattice D is introduced.This space Fspec(D) is compact and it contains a subspace homeomorphic with the prime spectrum of D which is dense in it.The correspondence associating D to the topological space Fspec(D) is shown to define a contravarient functor from the category of bounded distributive lattices into the category of compact topological spaces.