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Incommensurate Crystals and Quasicrystals
Incommensurate Crystals and Quasicrystals Uwe Grimm
Max Scheffer
The Open University
Chemnitz University of Technology
I. II. III. IV.
Aperiodic Crystals The Structure of Quasicrystals Physical Properties of Quasicrystals Concluding Remarks
GLOSSARY Aperiodic crystal Crystalline structure without threedimensional lattice periodicity. Approximant phase Periodic crystalline phase that approximates a quasicrystalline phase. Commensurate Having a rational ratio. Crystal Ordered structure with essentially pure Bragg diffraction pattern; the current definition includes both periodic and aperiodic crystals. Decagonal phase Quasicrystal with one periodic direction and a 10-fold rotational symmetry. Dodecagonal phase Quasicrystal with one periodic direction and a 12-fold rotational symmetry. Icosahedral phase Quasicrystal with icosahedral symmetry and no periodic direction. Icosahedral symmetry The symmetry of the regular icosahedron. It is the largest symmetry group of a threedimensional regular polyhedron and comprises two-, three-, and fivefold rotational symmetry axes. Incommensurate Having an irrational ratio.
Incommensurate phase Aperiodic crystal whose structure is based on periodic lattices. The aperiodicity may be due to an incommensurate modulation of the periodic lattice or an incommensurate combination of several periodic lattices. Lock-in transformation An incommensurate-to-commensurate phase transition. Modulated structure Result of a (small) periodic distortion of a periodic pattern. It is periodic when the two periods are commensurate; an incommensurate modulation results in an aperiodic structure. Octagonal phase Quasicrystal with one periodic direction and an eightfold rotational symmetry. Phason Degree of freedom associated with relative phases of incommensurate waves. Quasicrystal Aperiodic crystal without an underlying periodic lattice structure, usually showing noncrystallographic symmetries that cannot occur in periodic crystals. Quasiperiodic A particular type of aperiodicity where the Fourier transform is pure point, i.e., consists only of
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Max Scheffer
The Open University
Chemnitz University of Technology
I. II. III. IV.
Aperiodic Crystals The Structure of Quasicrystals Physical Properties of Quasicrystals Concluding Remarks
GLOSSARY Aperiodic crystal Crystalline structure without threedimensional lattice periodicity. Approximant phase Periodic crystalline phase that approximates a quasicrystalline phase. Commensurate Having a rational ratio. Crystal Ordered structure with essentially pure Bragg diffraction pattern; the current definition includes both periodic and aperiodic crystals. Decagonal phase Quasicrystal with one periodic direction and a 10-fold rotational symmetry. Dodecagonal phase Quasicrystal with one periodic direction and a 12-fold rotational symmetry. Icosahedral phase Quasicrystal with icosahedral symmetry and no periodic direction. Icosahedral symmetry The symmetry of the regular icosahedron. It is the largest symmetry group of a threedimensional regular polyhedron and comprises two-, three-, and fivefold rotational symmetry axes. Incommensurate Having an irrational ratio.
Incommensurate phase Aperiodic crystal whose structure is based on periodic lattices. The aperiodicity may be due to an incommensurate modulation of the periodic lattice or an incommensurate combination of several periodic lattices. Lock-in transformation An incommensurate-to-commensurate phase transition. Modulated structure Result of a (small) periodic distortion of a periodic pattern. It is periodic when the two periods are commensurate; an incommensurate modulation results in an aperiodic structure. Octagonal phase Quasicrystal with one periodic direction and an eightfold rotational symmetry. Phason Degree of freedom associated with relative phases of incommensurate waves. Quasicrystal Aperiodic crystal without an underlying periodic lattice structure, usually showing noncrystallographic symmetries that cannot occur in periodic crystals. Quasiperiodic A particular type of aperiodicity where the Fourier transform is pure point, i.e., consists only of
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