Polyhedra of three quasilattices associated with the icosahedral group

Haase, R W; Krämer, Ludwig; Krämer, Peter; Lalvani, Haresh

doi:10.1107/s0108767387098970

Cited by 13 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In passing we note that this six-dimensional representation of the full cubic group could be obtained from the non-trivial one-dimensional representation of the subgroup D2h by induction (Haase, Kramer, Kramer & Lalvani, 1987).…”

Section: The Transformed Representation D6(g)= M(/3)d6(g)[m([3)] -T Smentioning

confidence: 99%

Continuous rotation from cubic to icosahedral order

Krämer¹

1987

Acta Cryst Sect A

Self Cite

View full text Add to dashboard Cite

A1-Mn alloys display the competition of condensed matter phases with periodic cubic and non-periodic icosahedral order respectively. Both types of order are connected by a single continuous rotation M(/3) in the hypercubic lattice of 6-space projected to 3space. The rotation M(/3) results from Schur's lemma [Schur (1905). Sitzungsber. Preuss. Akad. applied to the cubic and icosahedral point groups. For 0 <-/3 < 7)" it preserves tetrahedral symmetry. For /3 ~= 0 ° one finds cubic symmetry, for /3i= 13.28 ° icosahedral symmetry. Implications for the AI-Mn structure are presented.

show abstract

Section: The Transformed Representation D6(g)= M(/3)d6(g)[m([3)] -T Smentioning

confidence: 99%

Continuous rotation from cubic to icosahedral order

Krämer¹

1987

Acta Cryst Sect A

Self Cite

View full text Add to dashboard Cite

show abstract

“…These polytopes in E 3 are the 'shadows' of the p-dimensional boundary h(p; g) obtained by projection. As shown by Haase et al (1987) h2(p; g), p=6, 5, 4, 3 are zonohedra with p(p -1) rhombic faces. For p = 6 we obtain the Kepler triacontahedron with 30 faces and for p = 3 the two different rhombohedra with six faces.…”

Section: Kinematical Factors and Their Computationmentioning

confidence: 99%

“…1 to enumerate the axes. The action of I on E 3 used so far corresponds in representation theory to one of the two threedimensional irreducible representations of/, denoted by [312] by Haase, Kramer, Kramer & Lalvani (1987). The two three-dimensional representations appear in the explicitly reduced form of a six-dimensional representation.…”

Section: The Icosahedral and Hyperoctahedral Point Groupsmentioning

confidence: 99%

“…The two three-dimensional representations appear in the explicitly reduced form of a six-dimensional representation. The matrices of the generators for these representations are given by Kramer (1987). The sixdimensional representation is technically obtained as a representation of I induced from a one-dimensional representation of D5 (Haase et al, 1987).…”

Section: The Icosahedral and Hyperoctahedral Point Groupsmentioning

confidence: 99%

“…Its explicit reduction into the representations [312 ] and [312 ] is obtained by transforming with the 6 x 6 matrix M given in Table 3 which is adapted to the present notation from Kramer (1987). The matrix (2.7)…”

Section: Propositionmentioning

confidence: 99%

See 2 more Smart Citations

Structure factors for icosahedral quasicrystals

Krämer¹,

Zeidler²

1989

Acta Cryst Sect A

Self Cite

View full text Add to dashboard Cite

The restriction of a hypercubic lattice in six dimensions to a subspace of three dimensions yields a well known quasiperiodic description of quasicrystals with non-crystallographic icosahedral point symmetry. A quasicrystal model is considered where this description is further reduced to a non-periodic quasilattice formed from two types of rhombohedra. Given the density on two representative rhombohedral cells, the full Fourier transform is expressed in closed form through structure factors, quasilattice factors and kinematical factors. The diffraction from point scatterers in the quasilattice is computed as an example.

show abstract

1-15 Introduction

Smith¹

Landolt-Börnstein - Group IV Physical Chemistry

View full text Add to dashboard Cite

Polyhedra of three quasilattices associated with the icosahedral group

Cited by 13 publications

References 3 publications

Continuous rotation from cubic to icosahedral order

Continuous rotation from cubic to icosahedral order

Structure factors for icosahedral quasicrystals

1-15 Introduction

Contact Info

Product

Resources

About