Symmetry properties of the normal modes of vibration of calcite and α-corundum

1969 ◽  
Vol 47 (13) ◽  
pp. 1381-1391 ◽  
Author(s):  
E. R. Cowley
Keyword(s):  
Theoretical Analysis ◽  
Normal Modes ◽  
Dispersion Curves ◽  
Dynamical Matrix ◽  
Space Group ◽  
Symmetry Properties ◽  
Modes Of Vibration ◽  
Symmetry Species ◽  
Group Theoretical Analysis

A group theoretical analysis is carried out on the symmetry properties of the normal modes of vibration of crystals with the calcite and α-corundum structures. Both of these structures have the space group [Formula: see text]. The symmetry species present in the dispersion curves are determined for important wave vectors, and a transformation to block-diagonalize the dynamical matrix at the zone center is given.

Space Group Theoretical Analysis of Grain Boundaries in Ordered Alloys

1986 ◽  
Vol 93 (1) ◽  
pp. 45-55 ◽  
Author(s):  
D. Farkas ◽  
A. Ran
Keyword(s):  
Theoretical Analysis ◽  
Grain Boundaries ◽  
Space Group ◽  
Ordered Alloys ◽  
Group Theoretical Analysis

scholarly journals Group-theoretical analysis of 1:3 A-site-ordered perovskite formation

2019 ◽  
Vol 75 (2) ◽  
pp. 379-397 ◽  
Author(s):  
Mikhail V. Talanov
Keyword(s):  
Phase Transitions ◽  
Theoretical Analysis ◽  
Perovskite Structure ◽  
Functional Materials ◽  
Order Parameters ◽  
Space Group ◽  
Symmetry Phase ◽  
Structural Mechanisms ◽  
Low Symmetry ◽  
Group Theoretical Analysis

The quadruple perovskites AA′3 B 4 X 12 are characterized by an extremely wide variety of intriguing physical properties, which makes them attractive candidates for various applications. Using group-theoretical analysis, possible 1:3 A-site-ordered low-symmetry phases have been found. They can be formed from a parent Pm{\bar 3}m perovskite structure (archetype) as a result of real or hypothetical (virtual) phase transitions due to different structural mechanisms (orderings and displacements of atoms, tilts of octahedra). For each type of low-symmetry phase, the full set of order parameters (proper and improper order parameters), the calculated structure, including the space group, the primitive cell multiplication, splitting of the Wyckoff positions and the structural formula were determined. All ordered phases were classified according to the irreducible representations of the space group of the parent phase (archetype) and systematized according to the types of structural mechanisms responsible for their formation. Special attention is paid to the structural mechanisms of formation of the low-symmetry phase of the compounds known from experimental data, such as: CaCu3Ti4O12, CaCu3Ga2Sn2O12, CaMn3Mn4O12, Ce1/2Cu3Ti4O12, LaMn3Mn4O12, BiMn3Mn4O12 and others. For the first time, the phenomenon of variability in the choice of the proper order parameters, which allows one to obtain the same structure by different group-theoretical paths, is established. This phenomenon emphasizes the fundamental importance of considering the full set of order parameters in describing phase transitions. Possible transition paths from the archetype with space group Pm{\bar 3}m to all 1:3 A-site-ordered perovskites are illustrated using the Bärnighausen tree formalism. These results may be used to identify new phases and interpret experimental results, determine the structural mechanisms responsible for the formation of low-symmetry phases as well as to understand the structural genesis of the perovskite-like phases. The obtained non-model group-theoretical results in combination with crystal chemical data and first-principles calculations may be a starting point for the design of new functional materials with a perovskite structure.

Kristallstrukturen, Schwingungsspektren und Normalkoordinatenanalysen von trans-[PtX2(acac)2], X = CI, Br, I, SCN / Crystal Structures, Vibrational Spectra and Normal Coordinate Analyses of trans-[PtX2(acac)2], X = Cl, Br, I, SCN

1996 ◽  
Vol 51 (10) ◽  
pp. 1400-1406 ◽  
Author(s):  
D. Rickert ◽  
W. Preetz
Keyword(s):  
Crystal Structures ◽  
Normal Modes ◽  
Force Constants ◽  
Monoclinic Space Group ◽  
Normal Coordinate ◽  
Triclinic Space Group ◽  
Valence Force ◽  
Space Group ◽  
X Ray ◽  
Modes Of Vibration

The crystal structures of trans-[PtCl2(acac)2](monoclinic, space group P21/c, a - 7.616(5), b = 12.759(5), c = 7.892(5) Å, β = 118.459(5)°, Z = 2), trans-[PtBr2(acac)2] (triclinic, space group P1̅, a = 7.502(5), b = 7.665(5), c = 8.155(5) Å, α = 114~508(5), β = 94.537(5), γ = 117.669(5)°. Z = 1) and trans-[Pt(SCN)2(acac)2] (triclinic, space group P1̅ , a = 7.9095(10), b = 7.9393( 10), c = 7.9631 Å, a = 114.051 (10), β = 100.955(10), γ = 100.573(10)°, Z = 1) have been determined by single crystal X-ray diffraction analysis. The crystal structure of trans- [Ptl2(acac)2] is known from the literature. To enhance the spectroscopic resolution, the IR and Raman spectra of the four complexes have been measured at low temperature (10 K). Using the X-ray data, normal coordinate analyses based on a modified valence force field have been performed and the normal modes of vibration for the octahedral skeleton [PtX2O4] have been assigned. With a set of 19 or 23 force constants taking into account the inner-ligand vibrations a good agreement between observed and calculated frequencies has been achieved. The valence force constants are e.g.fd (PtCl) = 2.16, fd (PtBr) = 1.45, fd (PtI) = 1.01, fd (PtS) = 1.80 mdyn/Å, and fd (PtO) ranges from 1.89 to 1.91 mdyn/Å.

Kristallstrukturen, Schwingungsspektren und Normalkoordinatenanalysen von Tetrahalogenooxalatoosmaten (IV), [OsX4(ox)]2-, X = Cl, Br, I / Crystal Structures, Vibrational Spectra and Normal Coordinate Analyses of Tetrahalogenooxalatoosmates (IV), [OsX4(ox)]2-, X = Cl, Br, I

1997 ◽  
Vol 52 (3) ◽  
pp. 315-322 ◽  
Author(s):  
W. Preetz ◽  
A. Krull
Keyword(s):  
Crystal Structures ◽  
Normal Modes ◽  
Force Constants ◽  
Monoclinic Space Group ◽  
X Ray Diffraction ◽  
Normal Coordinate ◽  
Valence Force ◽  
Space Group ◽  
X Ray ◽  
Modes Of Vibration

Abstract The crystal structures of [(C5H5N)2CH2][OsCl4(ox)] (monoclinic, space group I2/m, a = 10.260(5), b = 13.841(5), c = 12.273(5) Å, β = 92.050(5)°, Z = 4), [(C5H5N)2CH2][OsBr4(ox)]·H2O(monoclinic, space group P21/n, a = 11.666(3),b = 11.591(5), c = 14.926(2) Å, β = 102.45(2)°, Z = 4) and [P(C6H5)4]2 [OsI4(ox)]·2CH2Cl2 (triclinic, space group P1̄, a = 14.597(2), b = 11.9185(9), c = 22.5624(14) Å, α = 80.284(8), β = 78.903(8), γ = 69.432(8)°, Z = 2) have been determined by single crystal X-ray diffraction analysis. The IR and Raman spectra of the three complexes were measured at room temperature. Using the molecular parameters of the X-ray determinations normal coordinate analyses based on a modified valence force field have been performed and the normal modes of vibration are assigned. With a set of 25 force constants and taking into account the innerligand vibrations a good agreement between observed and calculated frequencies has been achieved. The valence force constants of the X-Os-X axis are fd(OsCl) = 1.77, fd(OsBr) = 1.48, fd(OsI) = 1.0 mdyn/Å, of the X′-Os-O• axes are fd(OsCl′) = 1.88, fd(OsBr′) = 1.6, fd(OsI′) =1.1 mdyn/Å and fd(OsO•) ranges from 2.7 to 2.8 mdyn/Å.

Kristallstrukturen, Schwingungsspektren und Normalkoordinaten- analysen von cis- und trans-[OsX2(acac)2], X = Cl, Br, I / Crystal Structures, Vibrational Spectra and Normal Coordinate Analyses of cis- and trans-[OsX2(acac)2], X = Cl, Br, I

1997 ◽  
Vol 52 (8) ◽  
pp. 965-974 ◽  
Author(s):  
K Dallmann ◽  
W Preetz
Keyword(s):  
Crystal Structures ◽  
Normal Modes ◽  
Force Constants ◽  
Monoclinic Space Group ◽  
Normal Coordinate ◽  
Triclinic Space Group ◽  
Valence Force ◽  
Space Group ◽  
X Ray ◽  
Modes Of Vibration

The crystal structures of trans-[OsCl2(acac)2] (triclinic, space group P1̄, a = 7.4114(5), b = 7.6419(9), c = 7.9944(6) Å, α = 62.692(7), β = 87.687(6), γ = 60.667(6)°, Z = 1), trans-[OsBr2(acac)2] (triclinic, space group P1̄, a = 7.467(3), b = 7.621(3), c = 8.260(3) Å, α = 115.03(3), β = 92.78(3), γ = 117.91(3)°, Z = 1), cis-[OsCl2(acac)2] (monoclinic, space group C2/c, a = 13.8532(13), b = 7.7990(8), c = 13.6202(12) Å, β = 108.375(10)°, Z = 4) and cis-[OsBr2(acac)2] (monoclinic, space group C2/c, a = 13.944(2), b = 8.0347(13), c = 13.743(2) Å, β = 106.757(12)°, Z = 4) have been determined by single crystal X-ray diffraction analysis. To enhance the spectroscopic resolution, the IR and Raman spectra of the six complexes have been measured at low temperature (10 K). Using the molecular parameters of the X-ray determinations normal coordinate analyses based on a modified valence force field have been performed for trans-[OsCl2(acac)2], trans-[OsBr2(acac)2] and cis-[OsCl2(acac)2], and the normal modes of vibration assigned. With sets of 31 or 32 force constants, taking into account the innerligand vibrations, a good agreement between observed and calculated frequencies has been achieved. The valence force constants for the X-Os-X axes are fd(OsCl) = 1.81, fd(OsBr) =1.61 mdyn/Å, and for the Cl′-Os-O* axis are fd(OsCl’) = 1.94, fd(OsO*) = 2.81 mdyn/Å and fd(OsO) ranges from 3.27 to 3.31 mdyn/Å.

Structures of the ordered double perovskites Sr2YTaO6 and Sr2YNbO6

2005 ◽  
Vol 61 (3) ◽  
pp. 258-262 ◽  
Author(s):  
Christopher J. Howard ◽  
Paris W. Barnes ◽  
Brendan J. Kennedy ◽  
Patrick M. Woodward
Keyword(s):  
Theoretical Analysis ◽  
Group Symmetry ◽  
Double Perovskites ◽  
Monoclinic Structure ◽  
Space Group Symmetry ◽  
Space Group ◽  
Recent Group ◽  
Triclinic Structure ◽  
Group Theoretical Analysis

The ordered perovskite Sr2YTaO6, distrontium yttrium tantalum hexaoxide, has been reported as showing an unusual triclinic structure, at odds with the results from a recent group-theoretical analysis. A new investigation establishes that Sr2YTaO6 and the isostructural Sr2YNbO6, distrontium yttrium niobium hexaoxide, in fact both adopt the commonly occurring monoclinic structure, with the space-group symmetry P21/n.

Group-theoretical analysis of octahedral tilting in ferroelectric perovskites

2002 ◽  
Vol 58 (6) ◽  
pp. 934-938 ◽  
Author(s):  
Harold T. Stokes ◽  
Erich H. Kisi ◽  
Dorian M. Hatch ◽  
Christopher J. Howard
Keyword(s):  
Phase Transitions ◽  
Theoretical Analysis ◽  
Landau Theory ◽  
Group Symmetry ◽  
Theoretical Methods ◽  
Space Group Symmetry ◽  
Space Group ◽  
First Order ◽  
Octahedral Tilting ◽  
Group Theoretical Analysis

Group-theoretical methods are used to analyze perovskite structures where both ferroelectric cation displacements and simple tilting of octahedral units are present. This results in a list of 40 different structures, each with a unique space-group symmetry. The list is compared with that of Aleksandrov & Bartolomé [Phase Transit. (2001), 74, 255–335] and a number of differences are found. The group–subgroup relationships between the structures are also determined, along with an indication of those phase transitions that must be first order by Landau theory.

VIBRATE!A program to compute irreducible representations for atomic vibrations in crystals

2009 ◽  
Vol 42 (6) ◽  
pp. 1194-1196 ◽  
Author(s):  
A. M. Glazer
Keyword(s):  
Crystal Structure ◽  
Group Theory ◽  
Group Analysis ◽  
Normal Modes ◽  
Factor Group ◽  
Irreducible Representations ◽  
Information File ◽  
Atomic Vibrations ◽  
Modes Of Vibration ◽  
Symmetry Species

VIBRATE!is a computer program that uses group theory to carry out factor group analysis of a crystal structure. The symmetry species of the normal modes of vibration are derived, together with information relating to the symmetry-adapted vectors. The program is simple to use, relying on input mainly from a crystallographic information file. The output is presented in a form that should be familiar not only to crystallographers but also to others such as chemical spectroscopists.

The symmetry properties of the normal modes of vibration of crystalline carbon disulphide and chlorine

1975 ◽  
Vol 8 (11) ◽  
pp. 1620-1632 ◽  
Author(s):  
P J Grout ◽  
J W Leech ◽  
P S English
Keyword(s):  
Carbon Disulphide ◽  
Normal Modes ◽  
Symmetry Properties ◽  
Modes Of Vibration

Ordered double perovskites – a group-theoretical analysis

2003 ◽  
Vol 59 (4) ◽  
pp. 463-471 ◽  
Author(s):  
Christopher J. Howard ◽  
Brendan J. Kennedy ◽  
Patrick M. Woodward
Keyword(s):  
Irreducible Representation ◽  
Theoretical Analysis ◽  
Irreducible Representations ◽  
Cation Ordering ◽  
Double Perovskites ◽  
Cation Site ◽  
Theoretical Methods ◽  
Space Group ◽  
Octahedral Tilting ◽  
Group Theoretical Analysis

Group-theoretical methods are used to enumerate the structures of ordered double perovskites, A 2 BB′X 6, in which the ordering of cations B and B′ into alternate octahedra is considered in combination with the ubiquitous BX 6 (or B′X 6) octahedral tilting. The cation ordering on the B-cation site is described by the irreducible representation R_1^+ of the Pm \overline 3 m space group of the cubic aristotype, while the octahedral tilting is mediated by irreducible representations M_3^+ and R_4^+. There are 12 different structures identified, and the corresponding group–subgroup relationships are displayed. Known structures are briefly reviewed.

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