scholarly journals Normally supportive sublattices of crystallographic space groups

2020 ◽  
Vol 76 (1) ◽  
pp. 7-23
Author(s):  
Miles A. Clemens ◽  
Branton J. Campbell ◽  
Stephen P. Humphries
Keyword(s):  
Normal Subgroup ◽  
Point Group ◽  
Group Extension ◽  
Normal Subgroups ◽  
Symbolic Form ◽  
Space Group ◽  
Space Groups ◽  
Significant Step ◽  
Crystal Class ◽  
Crystallographic Space Groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

scholarly journals Finite-index normal subgroups of crystallographic space groups

2017 ◽  
Vol 73 (a1) ◽  
pp. a373-a373
Author(s):  
Miles A. Clemens ◽  
Branton J. Campbell ◽  
Stephen P. Humphries ◽  
H. T. Stokes
Keyword(s):  
Finite Index ◽  
Normal Subgroups ◽  
Space Groups ◽  
Crystallographic Space Groups

Systematic prediction of new inorganic ferroelectrics in point group 4

1999 ◽  
Vol 55 (4) ◽  
pp. 494-506 ◽  
Author(s):  
S. C. Abrahams
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Inorganic Crystal Structure Database ◽  
Space Group ◽  
Space Groups ◽  
Group 4 ◽  
Structure Database ◽  
Inorganic Crystal ◽  
Crystal Structure Database

The latest release of the Inorganic Crystal Structure Database contains a total of 87 entries corresponding to 70 different materials in point group 4. The structures reported for 11 materials in space group P4 satisfy the criteria for ferroelectricity, as do four in P41, one each in P42 and P43, 12 in I4, including seven that form three families, and another three in I41. Three previously known ferroelectrics were also listed in I4 and one in I41. In addition, the listing for point group 4 contains 22 entries for nonferroelectric materials and three with misassigned space groups. Among the newly predicted ferroelectrics in point group 4, assuming the validity of the underlying structural reports, are Ce5B2C6, modulated NbTe4, Na3Nb12O31F, Ca2FeO3Cl, K4CuV5O15Cl, TlBO2, CrOF3, PbTeO3, VO(HPO3)(H2O).3H2O, MgB2O(OH)6, β-tetragonal boron, CuBi2O4, WOBr4, Na8PtO6, SbF2Cl3, Ba1.2Ti8O16, Ni[SC(NH2)2]4Cl2, Ca2SiO3Cl2, the mineral caratiite, NbAs, β-NbO2 and Ag3BiO3.

Orthorhombic sphere packings. V. Trivariant lattice complexes of space groups belonging to crystal class 222

2014 ◽  
Vol 70 (6) ◽  
pp. 591-604 ◽  
Author(s):  
Heidrun Sowa
Keyword(s):  
Sphere Packing ◽  
Tetragonal Symmetry ◽  
Orthorhombic Symmetry ◽  
Sphere Packings ◽  
Homogeneous Sphere ◽  
Space Group ◽  
Space Groups ◽  
Characteristic Space ◽  
Different Types ◽  
Crystal Class

This paper completes the derivation of all types of homogeneous sphere packing with orthorhombic symmetry. The nine orthorhombic trivariant lattice complexes belonging to the space groups of crystal class 222 were examined in regard to the existence of homogeneous sphere packings and of interpenetrating sets of layers of spheres. Altogether, sphere packings of 84 different types have been found; the maximal inherent symmetry is orthorhombic for only 36 of these types. In addition, interpenetrating sets of 63nets occur once. All lattice complexes with orthorhombic characteristic space group give rise to 260 different types of sphere packing in total. The maximal inherent symmetry is orthorhombic for 160 of these types. Sphere packings of 13 types can also be generated with cubic, those of seven types with hexagonal and those of 80 types with tetragonal symmetry. In addition, ten types of interpenetrating sphere packing and two types of sets of interpenetrating sphere layers are obtained. Most of the sphere packings can be subdivided into layer-like subunits perpendicular to one of the orthorhombic main axes.

Procedures for Point and Space Group Analysis

1989 ◽  
Vol 47 ◽  
pp. 488-489
Author(s):  
M. Tanaka
Keyword(s):  
Point Group ◽  
Group Analysis ◽  
Convergent Beam Electron Diffraction ◽  
Special Issue ◽  
Zeroth Order ◽  
Space Group ◽  
Space Groups ◽  
Flow Charts ◽  
Crystal Space ◽  
Convergent Beam

Description of general procedures for point- and space-group determination by convergent-beam electron diffraction(CBED) soon appears in a special issue of J. Electron Microsc. Tech. The essence of them was already given in the flow charts in ref.2. We here demonstrate the procedures using an example of La2 CuO4 - 6, which is the end member of a 40K class superconductor La2-x Mx CuO4 - 6 (M = Ba, Sr and Ca). The CBED method has utilized to date only dynamical extinction(GM lines) appearing in zeroth-order Laue-zone(ZOLZ) reflections for the space-group determination. We emphasize that the use of GM lines appearing in higher-order Laue-zone(HOLZ) reflections makes it easy to determine crystal space-groups.The substance was already known to belong to the orthorhombic system with the lattice parameters of a=5.3548Å, b=5.4006Å and c=13.1592Å. Fig. 1(a) shows a CBED pattern taken at the [001] incidence, Fig.l(b) being the central part of Fig.l(a). The whole pattern has a symmetry 2mm. This indicates that the crystal has two mirror symmetries perpendicular to each other and that the point group is mmm or mm2.

scholarly journals The space groups of point group C 3: some corrections, some comments

2002 ◽  
Vol 58 (5) ◽  
pp. 893-899 ◽  
Author(s):  
Richard E. Marsh
Keyword(s):  
Point Group ◽  
Crystallographic Data ◽  
Cambridge Structural Database ◽  
Cambridge Crystallographic Data Centre ◽  
Space Group ◽  
Space Groups ◽  
Data Centre ◽  
Group Assignment ◽  
Structural Database

A survey of the October 2001 release of the Cambridge Structural Database [Cambridge Structural Database (1992). Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge, England] has uncovered approximately 675 separate apparently reliable entries under space groups P3, P3_1, P3_2 and R3; in approximately 100 of these entries, the space-group assignment appears to be incorrect. Other features of these space groups are also discussed.

Algorithms for deriving crystallographic space-group information

1999 ◽  
Vol 55 (2) ◽  
pp. 383-395 ◽  
Author(s):  
R. W. Grosse-Kunstleve
Keyword(s):  
Three Dimensional ◽  
Basis Matrix ◽  
Space Group ◽  
Space Groups ◽  
Group Type ◽  
Group Information ◽  
Commercial Applications ◽  
Crystallographic Space Groups ◽  
Symmetry Operations

Algorithms are presented for three-dimensional crystallographic space groups, handling tasks such as the generation of symmetry operations, the characterization of symmetry operations (determination of rotation-part type, axis direction, sense of rotation, screw or glide part and location part), the determination of space-group type [identified by the space-group number of the International Tables for Crystallography (Dordrecht: Kluwer Academic Publishers)] and the generation of structure-seminvariant vectors and moduli. The latter are an algebraic description of allowed origin shifts, which are important in crystal structure determination methods or for comparing crystal structures. The space-group type determination produces a change-of-basis matrix which transforms a given space-group representation to the standard one according to the International Tables for Crystallography. The algorithms were implemented and tested using the SgInfo library. The source code is free for non-commercial applications.

scholarly journals Space groups and crystallographic symmetry: writing a multi-featured tutorial in a new style

2021 ◽  
Vol 77 (9) ◽  
Author(s):  
Bruce M. Foxman
Keyword(s):  
Chemical Properties ◽  
Lattice Points ◽  
Group Symmetry ◽  
Space Group ◽  
Space Groups ◽  
Crystallographic Symmetry ◽  
Additional Section ◽  
Crystal Class ◽  
Physical And Chemical ◽  
And Chemical Properties

Knowledge of space groups and the implications of space group symmetry on the physical and chemical properties of solids are pivotal factors in all areas of crystalline solids. As Jerry Jasinski and I met to bring our ideas in teaching this subject to life, we both felt that `early and often' – teaching the concepts with textual and visual reinforcement, is the key to providing a sound basis for students in this subject. The tutorial contains > 200 PowerPoint `slides', in five modules, arranged by crystal class; a sixth module covers special topics. A `credits' module gives the direct addresses of all embedded links. Space-group diagrams appear in International Tables format. The triclinic and monoclinic groups (2 + 13) are built from `scratch', and are derived from the Hermann–Mauguin symbol. An additional section provides practice on many (but not all) of the orthorhombic groups in crystal class 222. Finally, a `Special Topics' section on enantiomorphous space groups features space groups P41 and P43. In the tutorial, lattice points build iteratively and interactively via keyclick, and the coordinates of points `pop up' as the unit cell is filled. We trust that the elements of guidance, inquiry and occasional humor will make the learning process enjoyable.

Symmetry breaking in convergent-beam diffraction

1989 ◽  
Vol 47 ◽  
pp. 480-481
Author(s):  
D.J. Eaglesham
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
X Ray Diffraction ◽  
Multiple Diffraction ◽  
Space Groups ◽  
Atomic Displacements ◽  
Small Probe ◽  
Phase Sensitive ◽  
Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

Packing Effect on the Transfer Integrals and Mobility in α,α′-bis(dithieno[3,2-b:2′,3′-d]thiophene) (BDT) and its Heteroatom-Substituted Analogues

2011 ◽  
Vol 64 (12) ◽  
pp. 1587 ◽  
Author(s):  
Ahmad Irfan ◽  
Abdullah G. Al-Sehemi ◽  
Shabbir Muhammad ◽  
Jingping Zhang
Keyword(s):  
Electron Transfer ◽  
Electron Mobility ◽  
Hole Mobility ◽  
Experimental Investigations ◽  
Hole Transfer ◽  
Space Group ◽  
Space Groups ◽  
Transfer Integrals ◽  
Packing Effect ◽  
Good Agreement

Theoretically calculated mobility has revealed that BDT is a hole transfer material, which is in good agreement with experimental investigations. The BDT, NHBDT, and OBDT are predicted to be hole transfer materials in the C2/c space group. Comparatively, hole mobility of BHBDT is 7 times while electron mobility is 20 times higher than the BDT. The packing effect for BDT and designed crystals was investigated by various space groups. Generally, mobility increases in BDT and its analogues by changing the packing from space group C2/c to space groups P1 or . In the designed ambipolar material, BHBDT hole mobility has been predicted 0.774 and 3.460 cm2 Vs–1 in space groups P1 and , which is 10 times and 48 times higher than BDT (0.075 and 0.072 cm2 Vs–1 in space groups P1 and ), respectively. Moreover, the BDT behaves as an electron transfer material by changing the packing from the C2/c space group to P1 and .

scholarly journals Multiplicity, Parity and Angular Momentum of a Cooper Pair in Unconventional Superconductors of D4h Symmetry: Sr2RuO4 and Fe-Pnictide Materials

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1435
Author(s):  
Victor G. Yarzhemsky
Keyword(s):  
Angular Momentum ◽  
Cooper Pair ◽  
Point Group ◽  
Group Symmetry ◽  
Cooper Pairs ◽  
Group Approach ◽  
Angular Momentum Projection ◽  
Space Group ◽  
Unconventional Superconductors ◽  
Even Pairs

Sr2RuO4 and Fe-pnictide superconductors belong to the same point group symmetry D4h. Many experimental data confirm odd pairs in Sr2RuO4 and even pairs in Fe-pnictides, but opposite conclusions also exist. Recent NMR results of Pustogow et al., which revealed even Cooper pairs in Sr2RuO4, require reconsideration of symmetry treatment of its SOP (superconducting order parameter). In the present work making use of the Mackey–Bradley theorem on symmetrized squares, a group theoretical investigation of possible pairing states in D4h symmetry is performed. It is obtained for I4/mmm , i.e., space group of Sr2RuO4, that triplet pairs with even spatial parts are possible in kz direction and in points M and Y. For the two latter cases pairing of equivalent electrons with nonzero total momentum is proposed. In P4/nmm space group of Fe- pnictides in point M, even and odd pairs are possible for singlet and triplet cases. It it shown that even and odd chiral states with angular momentum projection m=±1 have nodes in vertical planes, but Eg is nodal , whereas Eu is nodeless in the basal plane. It is also shown that the widely accepted assertion that the parity of angular momentum value is directly connected with the spatial parity of a pair is not valid in a space-group approach to the wavefunction of a Cooper pair.

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