Line group theory of commensurate and incommensurate modulations

Kirschner, I.; Mészáros, Cs.; Laiho, R.

doi:10.1007/s100510050240

Cited by 7 publications

(18 citation statements)

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“…(Obviously, the formula represents a special simple particular case of ). In some of our earlier studies we had illustrated the relevance of the projective representations of line groups in Fourier analysis necessary for interpreting results emanating from various types of scattering experiments realized with incommensurately modulated condensed matter (sub)systems of different types. We have also proposed that in the most general case, the structure modulation of crystals should be described by scalar products given in the form

(δ q, z) = (q, v),

where the z ‐coordinate axis denotes the modulation direction, and which may therefore be identified as a main axis of the relevant line group, too.…”

Section: Relevance Of the Projective Representationsmentioning

confidence: 99%

“…486–491). Then, we will apply here the relation from Kirschner et al represented concisely as

\overset{falseˆ}{D} [(R | {boldv}_{R} + T)] e^{i (Φ_{0} + 2 π \frac{p}{m})} = e^{i (Φ_{0} + 2 π \frac{p}{m})} \cdot e^{- 2 π i \frac{k}{p}}

where R denotes now a point group element, to which the fractional translation

{boldv}_{R}

is directly coupled, and

T

is a translation vector generated by purely integer elementary translation vectors along the main axis of the Q1D system being described by the adequate line group (i.e. the transformation is necessary to be realized by the element belonging to the subgroup of generalized translations of the actual line group).…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

“…the transformation is necessary to be realized by the element belonging to the subgroup of generalized translations of the actual line group). Finally, we will also recall here some of the relevant formulae for demonstrating the equivalence of our originally introduced projective representation method of line groups with other well‐known and frequently applied Fourier‐analytical methods in evaluation of experimentally obtained results in the structural investigation of incommensurately modulated crystal by X‐ray diffraction in kinematic approximation . Accordingly, we have applied the following expression for describing the modulation of the electron‐density functions in such types of 3D crystal structures

ρ (r - {boldu}_{n γ}) \Rightarrow ρ (r - {boldR}_{n γ} - {boldu}_{n γ}) \equiv ρ_{1} (r - {boldu}_{n γ})

where the vectors

{boldR}_{n γ}

give the original positions of the atomic‐type scattering centers in the ideal 3D crystals (where the index “ n ” counts the elementary cells and the index “γ” counts the scattering centers inside the unit cell), and the vectors

{boldu}_{n γ}

are the displacements from their initial places in the elementary cell of the ideal crystal.…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

“…Accordingly, we have applied the following expression for describing the modulation of the electron‐density functions in such types of 3D crystal structures

ρ (r - {boldu}_{n γ}) \Rightarrow ρ (r - {boldR}_{n γ} - {boldu}_{n γ}) \equiv ρ_{1} (r - {boldu}_{n γ})

where the vectors

{boldR}_{n γ}

{boldu}_{n γ}

are the displacements from their initial places in the elementary cell of the ideal crystal. Then, in the case of the modulated electron density function the next‐type Fourier expansion method is to be applied

ρ_{1} (r) = \sum_{boldk} ρ_{1 k} e^{2 π i k \cdot r} e^{- 2 π i k \cdot {boldu}_{n γ}} \approx \sum_{boldk} ρ_{1 k} e^{2 π i k \cdot r} - 2 π i \sum_{boldk} k \cdot {boldu}_{n γ} ρ_{1 k} e^{2 π i k \cdot r}

and which is equivalent to the frequently applied form of the modulated electron density function usually given as

ρ (x) = ρ + ρ_{2 k_{F}} \cos (2 k_{F} x + φ)

relevant for longitudinal displacements related to the periodic lattice distortion induced by a Peierls‐type instability. (In Equation the X ‐axis of a rectangular Cartesian coordinate system is assumed to be the main axis of the actual co...…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

“…Therefore, we will apply here the relation in a developed form and compare it to . We have

\overset{falseˆ}{D} (Q | {boldv}_{R} + T) ρ_{1} (r - {boldu}_{n γ}) = \sum_{{k}} ρ_{1 Q k} (T) \cdot e^{2 π i (Q k, r - {boldu}_{n γ})}

where the relevant Fourier coefficients (i.e., structure factors) have the explicit form

ρ_{1 Q k} (T) \equiv ρ_{1 k} e^{- 2 π i (Q s, {boldv}_{R} + T)} .

Finally, if we apply the following relation

Q k = k + δ k

expressing the basic orthogonal symmetry relation between the point group of the crystal structure before the modulation and the isogonal point group of the actual line group, describing the modulation appearing in a structural phase transition. Then, as it has been mentioned, by the reciprocal space vector difference

δ k

, the modulated structures of both commensurate‐, and incommensurate characters can be described .…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

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Symmetry Analysis of Some Basic Structural Properties of Incommensurately Modulated Crystals by Projective Representations of Unimodular Groups

Mészáros

Nikolényi

Bálint

2019

Physica Status Solidi (b)

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Projective representations of the unimodular groups are applied in the general symmetry theory of incommensurately modulated crystals. It is demonstrated that the algebraic condition relevant for the existence of line groups can be directly connected to those related to linear projective groups. As a concrete application example, the formalisms of the thermodynamic modelling of the structural phase transitions and the Fourier‐analysis formulae of the scattering processes in kinematic approximation are discussed by use of this group‐theoretical technique, introduced newly in this study for symmetry analyses of the general structural features of incommensurate systems.

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(δ q, z) = (q, v),

where the z ‐coordinate axis denotes the modulation direction, and which may therefore be identified as a main axis of the relevant line group, too.…”

Section: Relevance Of the Projective Representationsmentioning

confidence: 99%

“…486–491). Then, we will apply here the relation from Kirschner et al represented concisely as

\overset{falseˆ}{D} [(R | {boldv}_{R} + T)] e^{i (Φ_{0} + 2 π \frac{p}{m})} = e^{i (Φ_{0} + 2 π \frac{p}{m})} \cdot e^{- 2 π i \frac{k}{p}}

where R denotes now a point group element, to which the fractional translation

{boldv}_{R}

is directly coupled, and

T

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

ρ (r - {boldu}_{n γ}) \Rightarrow ρ (r - {boldR}_{n γ} - {boldu}_{n γ}) \equiv ρ_{1} (r - {boldu}_{n γ})

where the vectors

{boldR}_{n γ}

{boldu}_{n γ}

are the displacements from their initial places in the elementary cell of the ideal crystal.…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

“…Accordingly, we have applied the following expression for describing the modulation of the electron‐density functions in such types of 3D crystal structures

ρ (r - {boldu}_{n γ}) \Rightarrow ρ (r - {boldR}_{n γ} - {boldu}_{n γ}) \equiv ρ_{1} (r - {boldu}_{n γ})

where the vectors

{boldR}_{n γ}

{boldu}_{n γ}

ρ_{1} (r) = \sum_{boldk} ρ_{1 k} e^{2 π i k \cdot r} e^{- 2 π i k \cdot {boldu}_{n γ}} \approx \sum_{boldk} ρ_{1 k} e^{2 π i k \cdot r} - 2 π i \sum_{boldk} k \cdot {boldu}_{n γ} ρ_{1 k} e^{2 π i k \cdot r}

and which is equivalent to the frequently applied form of the modulated electron density function usually given as

ρ (x) = ρ + ρ_{2 k_{F}} \cos (2 k_{F} x + φ)

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

“…Therefore, we will apply here the relation in a developed form and compare it to . We have

\overset{falseˆ}{D} (Q | {boldv}_{R} + T) ρ_{1} (r - {boldu}_{n γ}) = \sum_{{k}} ρ_{1 Q k} (T) \cdot e^{2 π i (Q k, r - {boldu}_{n γ})}

where the relevant Fourier coefficients (i.e., structure factors) have the explicit form

ρ_{1 Q k} (T) \equiv ρ_{1 k} e^{- 2 π i (Q s, {boldv}_{R} + T)} .

Finally, if we apply the following relation

Q k = k + δ k

δ k

, the modulated structures of both commensurate‐, and incommensurate characters can be described .…”

Section: Generalizations Of the Kinematic X‐ray Scattering Formalismmentioning

confidence: 99%

See 3 more Smart Citations

Symmetry Analysis of Some Basic Structural Properties of Incommensurately Modulated Crystals by Projective Representations of Unimodular Groups

Mészáros

Nikolényi

Bálint

2019

Physica Status Solidi (b)

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Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory of coupled transport processes

Mészáros

Kirschner

Bálint

2013

Continuum Mech. Thermodyn.

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Symmetry based properties of the transition metal dichalcogenide nanotubes

et al. 2000

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The full geometrical symmetry groups (the line groups) of the monolayered, 2Hb and 3R polytypes of the inorganic MoS2 and WS2 micro-and nanotubes of arbitrary chirality are found. This is used to find the coordinates of the representative atoms sufficient to determine completely geometrical structure of tubes. Then some physical properties which can be deduced from the symmetry are discussed: electron band degeneracies, selection rules, general forms of the second rank tensors and potentials, phonon spectra.PACS. 61.46.+w Clusters, nanoparticles, and nanocrystalline materials -78.66.-w Optical properties of specific thin films, surfaces, and low-dimensional structures -63.22.+m Phonons in low-dimensional structures and small particles

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Line group theory of commensurate and incommensurate modulations

Abstract: PACS. 02.20.-a Group theory - 64.70.Rh Commensurate-incommensurate transitions - 75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.),

Cited by 7 publications

References 0 publications

Symmetry Analysis of Some Basic Structural Properties of Incommensurately Modulated Crystals by Projective Representations of Unimodular Groups

Symmetry Analysis of Some Basic Structural Properties of Incommensurately Modulated Crystals by Projective Representations of Unimodular Groups

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory of coupled transport processes

Symmetry based properties of the transition metal dichalcogenide nanotubes

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