Occupation modulation
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.5,
p.
[ doi:10.1107/97809553602060000624 ]
is the sum of δ functions over the reciprocal lattice of the basic structure: Consequently, the diffraction peaks occur at positions H given by (
9.8.1.7) . For a simple sinusoidal modulation [ m = ±1 in (
9.8.1.29) ], there are only main ...
Basic symmetry considerations
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.4,
p.
[ doi:10.1107/97809553602060000624 ]
*).
An element R of K then transforms the basic vectors,,, q into ones of the form (
9.8.1.15) . If one denotes, as in (
9.8.1.2), q by, this implies with Γ*(R) a 4 × 4 matrix with integral entries. In the case ...
Four-dimensional space groups
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.4.5,
p.
[ doi:10.1107/97809553602060000624 ]
The groups were determined on the basis of algorithms developed by Zassenhaus (1948), Janssen,
Janner & Ascher (1969 a, b), Brown (1969), and Fast & Janssen (1971). In the book by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus ...
The diffraction pattern
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.3.1,
p.
[ doi:10.1107/97809553602060000624 ]
in the structure factor, which is given by the expression where is the atomic scattering factor (which still, in general, depends on H). Using the Jacobi–Anger relation, one can rewrite (
9.8.1.5) as where is the m th-order Bessel ...
Description in four dimensions
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.4.2,
p.
[ doi:10.1107/97809553602060000624 ]
leaving a lattice with basis vectors (
9.8.1.14) invariant. Indeed, one can consider the vectors (
9.8.1.15) as projections of four-dimensional lattice vectors, which can be written as where [ cf. (
9.8.1.14) ] m has now been ...
The symmetry
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.3.2,
p.
[ doi:10.1107/97809553602060000624 ]
transformations all elements ({ R | v }, {ɛ|Δ}) satisfying (
9.8.1.13) . These form a space group in four dimensions.
The reciprocal to the basis (
9.8.1.11) is A general reciprocal-lattice vector is now The projection of this reciprocal ...
Introduction
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1,
p.
[ doi:10.1107/97809553602060000624 ]
cases, cover essentially the one-dimensional modulated case [ d = 1 in equation (
9.8.1.1) ].
9.8.1.2. The basic ideas of higher-dimensional crystallography
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Incommensurate modulated crystals are systems ...
The basic ideas of higher-dimensional crystallography
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.2,
p.
[ doi:10.1107/97809553602060000624 ]
as crystalline phases and generalize for that reason the concept of a crystal. The positions of the Bragg diffraction peaks given in (
9.8.1.1) are a special case. In general, they are elements of a vector module M * and can be written as This leads ...
Generalized nomenclature
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.4.4,
p.
[ doi:10.1107/97809553602060000624 ]
...
The simple case of a displacively modulated crystal
Janssen, T.,
Janner, A.,
Looijenga-Vos, A. and
Wolff, P. M. de,
International Tables for Crystallography
(2006).
Vol. C,
Section 9.8.1.3,
p.
[ doi:10.1107/97809553602060000624 ]
there is an integer N such that N q belongs to the reciprocal lattice, one may restrict the values of m in (
9.8.1.7) to the range from 0 to N − 1.
9.8.1.3.2. The symmetry
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There is more than one way ...