It is well known that a triangle can be divided by mid-point refinement into four sub-triangles with the same shape. Similarly, a tetrahedron can be parted into eight subtetrahedra, which are generally not uniform in ...It is well known that a triangle can be divided by mid-point refinement into four sub-triangles with the same shape. Similarly, a tetrahedron can be parted into eight subtetrahedra, which are generally not uniform in shape. This paper proves that there exist a set of tetrahedra, which is called isometrically subdivisible tetrahedra(IST) and can be divided into eight isometric subtetrahedra, including identical and reflection ones. And a new classification of tetrahedra is put forward, based on which all tetrahedra can be categorized into 26 classes according to both the number of maximum equal edges and topological relations. The IST belongs only to three of the classes. That result provides a new viewpoint of spatial structure and may be used to tile or subdivide space uniformly or isometrically.展开更多
We report the detailed physical properties of quaternary compound Ba2BiFeS5 with the key structural ingredient of isolated FeS4 tetrahedra.Magnetization and heat capacity measurements clearly indicate that Ba2BiFeS5 h...We report the detailed physical properties of quaternary compound Ba2BiFeS5 with the key structural ingredient of isolated FeS4 tetrahedra.Magnetization and heat capacity measurements clearly indicate that Ba2BiFeS5 has a paramagnetic to antiferromagnetic transition at about 30 K.The calculated magnetic entropy above ordering temperature is much smaller than theoretical value for high-spin Fe^3+ion with S=5/2,implying the possible short-range antiferromagnetic fluctuation in Ba2BiFeS5.展开更多
Compounds containing(pseudo-)tetrahedral main group(semi-)metal units are fundamentally important in three aspects.First,they provide new strategies for the stabilization of tetrahedral units,such as Pn_(4)(Pn=P,As)an...Compounds containing(pseudo-)tetrahedral main group(semi-)metal units are fundamentally important in three aspects.First,they provide new strategies for the stabilization of tetrahedral units,such as Pn_(4)(Pn=P,As)and their isoelectronic analogs,in the coordination sphere of d-/f-block metal atoms.Second,they serve as first-step models for the activation of tetrahedral molecules.Third,they are starting materials for subsequent transformations toward larger clusters or materials.Partial or full isoelectronic replacement of the pnictogen atoms in P4 or As_(4) results in a variety of(pseudo-)tetrahedral anions,Tt_(4)^(4−)(Tt=Si,Ge,Sn,Si/Ge),(Ge_(3)Pn)^(3−)(Pn=P,As;formed in situ),(Tt_(2)Pn_(2))^(2−)(Tt/Pn=Ge/P,Ge/As,Sn/Sb,Sn/Bi,Pb/Sb,Pb/Bi),or(TrBi_(3))^(2−)(Tr=Ga,In,Tl),and also cations,(P3Ch)+(Ch=S,Se,Te).The anions have proven to be excellent starting materials and building blocks for multinary clusters in the context of Zintl chemistry.Such compounds,which push forward the boundary of intriguing P4 chemistry,have never been thoroughly summarized before.In this minireview,we introduce all known intact(pseudo-)tetrahedral group 13–15 units,as well as the coordination compounds based on them,along with two new missing links,[Au(η^(2-)Tt_(2)Bi_(2))_(2)]^(3−)(Tt=Sn,Pb).展开更多
A general local C-m(m greater than or equal to 0) tetrahedral interpolation scheme by polynomials of degree 4m + 1 plus low order rational functions from the given data is proposed. The scheme can have either 4m + 1 o...A general local C-m(m greater than or equal to 0) tetrahedral interpolation scheme by polynomials of degree 4m + 1 plus low order rational functions from the given data is proposed. The scheme can have either 4m + 1 order algebraic precision if C-2m data at vertices and C-m data on faces are given or k + E[k/3] + 1 order algebraic precision if C-k (k less than or equal to 2m) data are given at vertices. The resulted interpolant and its partial derivatives of up to order m are polynomials on the boundaries of the tetrahedra.展开更多
Two special tetrahedra packing clusters were analytically calculated and discussed.Diamond and double-diamond,periodic superstructures emerge.Both structures exemplify an efficient,congruent tetrahedra space tiling st...Two special tetrahedra packing clusters were analytically calculated and discussed.Diamond and double-diamond,periodic superstructures emerge.Both structures exemplify an efficient,congruent tetrahedra space tiling strategy.The diamond superstructure exhibits a remarkable packing density of φ∼0.76 exceeding maximal density for the packing of equal spheres.The presented architectures display long range order with specific,repetitive structural motifs.Due to their simple tetrahedral symmetry,the investigated clusters have the potential to become interesting,unique sub-units both in construction technology and in self-assembled systems design.The modified structure of presented cluster can entirely tile three-dimensional Euclidean space R^(3) through periodic tessellations.Moreover,detailed analysis of the cluster enables definition of a new plesiohedron.展开更多
The Raman spectra of synthetic hydroxyapatite (HAP), natural silicate, sulfate bearing apatite (NAP) and ellestadite (ELLE) were compared. Raman characteristics of [SO 4],[SiO 4] tetrahedra in apatite structure were d...The Raman spectra of synthetic hydroxyapatite (HAP), natural silicate, sulfate bearing apatite (NAP) and ellestadite (ELLE) were compared. Raman characteristics of [SO 4],[SiO 4] tetrahedra in apatite structure were discussed. As [SiO 4] and [SO 4] enter the structure, their Raman intensity increases and that of [PO 4] decreases. The ν 1 peaks of [PO 4],[SiO 4],[SO 4] shift to lower wave number region as tetrahedra T P bong length and volume of unit cell increase.展开更多
The distribution of Al （j） and the structural units distribution of Qi T in calcium aluminosilicate melts were studied by means of molecular dynamics simulation. The results show that provided there exists lower-fie...The distribution of Al （j） and the structural units distribution of Qi T in calcium aluminosilicate melts were studied by means of molecular dynamics simulation. The results show that provided there exists lower-field strength cation relative to Al3+, such as alkaline and alkaline earth metals, Al will be four-coordinated but not six-coordinated. Meanwhile, if there exist a large number of higher-field strength cations such as Si4+ and little lower-field strength cation, six-coordinated aluminum will be formed. The relation of structural units distribution of Qi T with chemical composition shift was also extracted, showing that as Ca2+ exists, the distributions of Qi Si, Qi Al or Qi T have the similar changing trend with the variation of component. Because of high-temperature effect, the Al-tetrahedral units in melts are greatly active and unstable and there exist dynamic transforming equilibria of Al（3）Al（4） and （Al（5））Al（4）. The three-coordinated oxygen and charge-compensated bridging oxygen are proposed to explain phenomena of the negative charge redundancy of AlO4 and location of network modifier with charge-compensated function in aluminosilicate melts.展开更多
A brilliant purple octahedral single crystal is hydrothermally synthesized by the reaction of CoCl26H2O, H3BO3 and H3PO4 in NaOH aqueous solution of CH3(CH2)15N(CH3)3Br, and its crystal structure has been characterize...A brilliant purple octahedral single crystal is hydrothermally synthesized by the reaction of CoCl26H2O, H3BO3 and H3PO4 in NaOH aqueous solution of CH3(CH2)15N(CH3)3Br, and its crystal structure has been characterized by single-crystal X-ray diffraction. The compound, NaCo(H2O)2BP2O8稨2O (Mr = 336.72), belongs to hexagonal, space group P6122 with a = 9.447(5), c = 15.83(1) , V = 1223(1) 3, Dc = 2.742 g/cm3, Z = 6, F(000) = 1002 and m = 2.606 mm-1. The three-dimensional framework in the compound is built up from the linkage tetrahedral ribbons, in which the BO4 and PO4 tetrahedra alternate with CoO6 octahedra. The sodium ions and water molecules are located within the free thread of the helical ribbons.展开更多
基金Supported by National Key Basic Research Program(2004CB318000)National Natural Science Foundation of China(60573154,61227802 and 61379082)
文摘It is well known that a triangle can be divided by mid-point refinement into four sub-triangles with the same shape. Similarly, a tetrahedron can be parted into eight subtetrahedra, which are generally not uniform in shape. This paper proves that there exist a set of tetrahedra, which is called isometrically subdivisible tetrahedra(IST) and can be divided into eight isometric subtetrahedra, including identical and reflection ones. And a new classification of tetrahedra is put forward, based on which all tetrahedra can be categorized into 26 classes according to both the number of maximum equal edges and topological relations. The IST belongs only to three of the classes. That result provides a new viewpoint of spatial structure and may be used to tile or subdivide space uniformly or isometrically.
基金Project supported by the National Key Research and Development Program of China(Grant No.2016YFA0300504)the National Natural Science Foundation of China(Grant Nos.11574394,11774423,and 11822412)+1 种基金the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(RUC)(Grant Nos.15XNLQ07,18XNLG14,and 19XNLG17)
文摘We report the detailed physical properties of quaternary compound Ba2BiFeS5 with the key structural ingredient of isolated FeS4 tetrahedra.Magnetization and heat capacity measurements clearly indicate that Ba2BiFeS5 has a paramagnetic to antiferromagnetic transition at about 30 K.The calculated magnetic entropy above ordering temperature is much smaller than theoretical value for high-spin Fe^3+ion with S=5/2,implying the possible short-range antiferromagnetic fluctuation in Ba2BiFeS5.
基金supported by the German Research foundation(Deutsche Forschungsgemeinschaft,DFG)F.P.gratefully acknowledges a research fellowship by the Alexander von Humboldt foundation.
文摘Compounds containing(pseudo-)tetrahedral main group(semi-)metal units are fundamentally important in three aspects.First,they provide new strategies for the stabilization of tetrahedral units,such as Pn_(4)(Pn=P,As)and their isoelectronic analogs,in the coordination sphere of d-/f-block metal atoms.Second,they serve as first-step models for the activation of tetrahedral molecules.Third,they are starting materials for subsequent transformations toward larger clusters or materials.Partial or full isoelectronic replacement of the pnictogen atoms in P4 or As_(4) results in a variety of(pseudo-)tetrahedral anions,Tt_(4)^(4−)(Tt=Si,Ge,Sn,Si/Ge),(Ge_(3)Pn)^(3−)(Pn=P,As;formed in situ),(Tt_(2)Pn_(2))^(2−)(Tt/Pn=Ge/P,Ge/As,Sn/Sb,Sn/Bi,Pb/Sb,Pb/Bi),or(TrBi_(3))^(2−)(Tr=Ga,In,Tl),and also cations,(P3Ch)+(Ch=S,Se,Te).The anions have proven to be excellent starting materials and building blocks for multinary clusters in the context of Zintl chemistry.Such compounds,which push forward the boundary of intriguing P4 chemistry,have never been thoroughly summarized before.In this minireview,we introduce all known intact(pseudo-)tetrahedral group 13–15 units,as well as the coordination compounds based on them,along with two new missing links,[Au(η^(2-)Tt_(2)Bi_(2))_(2)]^(3−)(Tt=Sn,Pb).
基金Supported partially by NSFC under Project 1967108, Croucher Foundation of Hong Kong and FRG ofHong Kong Baptist University.
文摘A general local C-m(m greater than or equal to 0) tetrahedral interpolation scheme by polynomials of degree 4m + 1 plus low order rational functions from the given data is proposed. The scheme can have either 4m + 1 order algebraic precision if C-2m data at vertices and C-m data on faces are given or k + E[k/3] + 1 order algebraic precision if C-k (k less than or equal to 2m) data are given at vertices. The resulted interpolant and its partial derivatives of up to order m are polynomials on the boundaries of the tetrahedra.
基金supported by the H2020-INFRAIA-2016-2017 project under research grant"EUSMI-European infrastructure for spectroscopy,scattering and imaging of soft matter",contract number GA731019,funded under H2020-EU.1.4.1.2.-RIA.
文摘Two special tetrahedra packing clusters were analytically calculated and discussed.Diamond and double-diamond,periodic superstructures emerge.Both structures exemplify an efficient,congruent tetrahedra space tiling strategy.The diamond superstructure exhibits a remarkable packing density of φ∼0.76 exceeding maximal density for the packing of equal spheres.The presented architectures display long range order with specific,repetitive structural motifs.Due to their simple tetrahedral symmetry,the investigated clusters have the potential to become interesting,unique sub-units both in construction technology and in self-assembled systems design.The modified structure of presented cluster can entirely tile three-dimensional Euclidean space R^(3) through periodic tessellations.Moreover,detailed analysis of the cluster enables definition of a new plesiohedron.
基金Key Research and Development Program of Shandong Province,China (No. 2021ZLGX01)the National Natural Science Foundation of China (No. 52071190)the Shandong Provincial Natural Science Foundation,China (No. ZR2021YQ34)。
文摘The Raman spectra of synthetic hydroxyapatite (HAP), natural silicate, sulfate bearing apatite (NAP) and ellestadite (ELLE) were compared. Raman characteristics of [SO 4],[SiO 4] tetrahedra in apatite structure were discussed. As [SiO 4] and [SO 4] enter the structure, their Raman intensity increases and that of [PO 4] decreases. The ν 1 peaks of [PO 4],[SiO 4],[SO 4] shift to lower wave number region as tetrahedra T P bong length and volume of unit cell increase.
文摘The distribution of Al （j） and the structural units distribution of Qi T in calcium aluminosilicate melts were studied by means of molecular dynamics simulation. The results show that provided there exists lower-field strength cation relative to Al3+, such as alkaline and alkaline earth metals, Al will be four-coordinated but not six-coordinated. Meanwhile, if there exist a large number of higher-field strength cations such as Si4+ and little lower-field strength cation, six-coordinated aluminum will be formed. The relation of structural units distribution of Qi T with chemical composition shift was also extracted, showing that as Ca2+ exists, the distributions of Qi Si, Qi Al or Qi T have the similar changing trend with the variation of component. Because of high-temperature effect, the Al-tetrahedral units in melts are greatly active and unstable and there exist dynamic transforming equilibria of Al（3）Al（4） and （Al（5））Al（4）. The three-coordinated oxygen and charge-compensated bridging oxygen are proposed to explain phenomena of the negative charge redundancy of AlO4 and location of network modifier with charge-compensated function in aluminosilicate melts.
基金NNSFC (No. 20173017) and State Key Basic Program (No. G2000048)
文摘A brilliant purple octahedral single crystal is hydrothermally synthesized by the reaction of CoCl26H2O, H3BO3 and H3PO4 in NaOH aqueous solution of CH3(CH2)15N(CH3)3Br, and its crystal structure has been characterized by single-crystal X-ray diffraction. The compound, NaCo(H2O)2BP2O8稨2O (Mr = 336.72), belongs to hexagonal, space group P6122 with a = 9.447(5), c = 15.83(1) , V = 1223(1) 3, Dc = 2.742 g/cm3, Z = 6, F(000) = 1002 and m = 2.606 mm-1. The three-dimensional framework in the compound is built up from the linkage tetrahedral ribbons, in which the BO4 and PO4 tetrahedra alternate with CoO6 octahedra. The sodium ions and water molecules are located within the free thread of the helical ribbons.
