Annotated Bibliography of the
Fourier-space approach to crystallography
(Bienenstock-Ewald/Rokhsar-Wright-Mermin approach)
John F. Huesman
December 2004
University of South Florida
revisions
Charles Hemphill
April 2008
This plain-text file is linked from
http://ewald.cas.usf.edu/research/quasi.html .
See there for additional links. Please e-mail additions and
corrections to David Rabson at davidra (at) ewald.cas.usf.edu .
47 papers as of April 2008
A. Bienenstock and P.P. Ewald
Symmetry in Fourier Space
Acta Cryst. 15 (Dec), 1253 (1962)
The beginning and basis of it all. The authors show, for the first time, that
all of crystallography can be done in reciprocal space. Their Fourier-space
formulation proves to be more instructive and more intuitive than
crystallography done in real space. This is due to the fact that all symmetries
of the crystal are observed in the diffraction pattern, symmetries which can
easily be missed in real space. The authors give a detailed correspondence
between the two methods of crystallography, and this author recommends this
paper most highly.
---------
D.S. Rokhsar, D.C. Wright, and N.D. Mermin
Rudimentary Quasicrystallography
Phys. Rev. B 35(11), 5487-5495 (15APR1987)
The first major application of the approach described by Bienenstock and Ewald.
Until the discovery of quasicrystals, the Fourier-space approach to
crystallography was nothing more than an interesting alternative means to show
previously-established results. With the realization that the distinction
between periodic crystals and quasicrystals essentially disappears in Fourier
space, this method now became a powerful way to classify quasi-periodic
structures. It is in this paper that the Rokhsar-Wright-Mermin (RWM) method of
Fourier-space crystallography is formally introduced and derived. The concepts
of "reciprocal lattice", "equivalent densities", and "phase functions" are
defined in a manner consistent with including a wide array of non-periodic
structures while keeping periodic crystals as a special case. A lattice is now
defined as a finitely-indexed set of vectors that is closed under addition.
Using this definition and the super-space projection method, it is shown that
there are only three icosahedral, 3-dimensional, real-space lattices and that
the dodecahedral lattice is unique.
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N.D. Mermin, D.S. Rokhsar, and D.C. Wright
Beware of 46-Fold Symmetry: The Classification of Two-Dimensional
Quasicrystallographic Lattices
Phys. Rev. Lett. 58(20), 2099-2101 (18MAY1987)
This paper relates the problem of classifying 2-dimensional lattices with N-fold
symmetry to the cyclotomic fields, \mathbb{Q}_N, and cyclotomic integers,
\mathbb{Z}_N, of algebraic topology. It is then easily shown that the number
of equivalent classes of N-fold lattices is equal to the number of equivalent
ideals in \mathbb{Z}_N, the first problematic case, other than the rather
trivial case N=2, occurring when N=46. For 46100 to show how difficult and how
quickly the problem gets "out of hand".
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D.S. Rokhsar, D.C. Wright, and N.D. Mermin
The Two-dimensional Quasicrystallographic Space Groups with Rotational Symmetry
less than 23-fold
Acta Cryst. A 44, 197-211 (1988)
Applying the RWM method of Fourier-space crystallography and the equivalence of
these N-fold rotational lattices and the cyclotomic integers, this paper shows
a full derivation and classification of the 2-dimensional space groups for
rotational order less than 23. This paper also uses the gauge-invariant phase
functions to derived the criteria for necessary extinctions in the diffraction
pattern, i.e. the Type-I invariant. Also of great importance is the method
(shown in Section 4.A) for constructing the appropriate gauges needed in the RWM
method to check that extinctions are not accidental results of the initial
choice of gauge.
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D.S. Rokhsar, D.C. Wright, and N.D. Mermin
Scale equivalence of quasicrystallographic space groups
Phys. Rev. B 37(14), 8145-8149 (15MAY1988)
This paper's main focus is the calculation of the icosahedral space groups
without using the super-space projection methods. It is shown that in addition
to space groups being equivalent if they differ by a gauge, there is also an
equivalence under scale invariance. If two densities \rho(k) and
rho(\lambda k) differ only in this scale factor \lambda, they certainly
belong to the same class of space groups. This is equivalent to saying that
\Phi_g(k) and \Phi_g(k / \lambda) characterize the same space group on
two lattices differing only by a rescaling. Or one can think of them as
\emph{different} phase functions on the same lattice. Since they may not, in
general, be gauge equivalent, but must represent the same space group, this
provides an equivalence on space groups. This paper also contains one of the
first allusions to the relationship between group invariants and physical
aspects of the crystal (in this case, extinctions in the diffraction pattern).
This is a fundamental step toward the realization of reciprocal-space
crystallography as group cohomology.
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D.A. Rabson, Tin-Lun Ho, and N.D. Mermin
Aperiodic Tilings with Non-Symmorphic Space Groups p2^jgm
Acta Cryst. A 44, 678-688 (01SEP1988)
Shows an explicit method for constructing tilings having the trivial space group
p2^jmm as well as the not-at-all trivial space group p2^jgm.
Although this paper's domain is a 2-dimensional grid space (and associated 2-
dimensional tiling space), the generalization to 3-dimensional grids and tilings
should not prove exceedingly difficult. This paper also shows the explicit
equivalence of this method with that of super-space projections.
---------
D.A. Rabson, Tin-Lun Ho, and N.D. Mermin
Space Groups of Quasicrystallographic Tilings
Acta Cryst. A 45, 538-547 (01AUG1989)
Applying the methods derived in their earlier paper, Rabson, Ho, and Mermin look
at a larger set of quasicrystalline space groups and show that their method can
again be used to construct a tiling for each of them.
---------
A.P. Smith and D.A. Rabson
Comment On "Icosahedral Quasiperiodic Ground-States?"
Phys. Rev. Lett. 63 (25), 2768-2768 (18DEC1989)
Smith and Rabson take on the assertion that near-neighbor correlations act as
the stabilizing mechanism for certain quasicrystals. A crystalline structure
having a higher near-neighbor correlation than the proposed quasicrystal is
given, thus showing that this cannot be the sole means of stabilization.
---------
N.D. Mermin
Reinventing crystallography: the forbidden lattices and space groups
XVIIth Intl. Colloq. on Group-Theoretical Methods in Physics, Y. Saint-Aubin and
L. Vinet, eds., World Scientific, Singapore (1989).
The author describes how the Fourier-space methods of crystallography both
preserve the physics of periodic crystals while extending these ideas to classes
of aperiodic structures, specifically the "forbidden" rotational symmetries
exhibited by quasicrystals.
---------
N.D. Mermin, D.A. Rabson, D.S. Rokhsar, and D.C. Wright
Stacking quasicrystallographic lattices
Phys. Rev. B 41(15), 10498-10502 (15MAY1990)
Shows the classification of 3-dimensional space groups in the special case where
the quasicrystal consists of 2-dimensional, N-fold crystals that are "stacked".
The way in which these 2D, N-fold lattices can be stacked is highly dependent
on the n-fold axial symmetry. When n is a power of 2, both vertical and
alternating stackings are allowed, but only vertical stackings are permitted
when n is even but not a power of 2. In the cases of odd n, the only
permissible stackings occur when n is the power of a prime p and N=2n;
these stackings repeat after p layers. It is also briefly shown that the 3D,
crystallographic lattices are included in this derivation.
---------
D.A. Rabson, N.D. Mermin, D.S. Rokhsar, and D.C. Wright
The Space-Groups of Axial Crystals and Quasicrystals
Rev. Mod. Phys. 63(3), 699-733 (JUL1991)
Computation of the 3-dimensional space groups for those crystals and
quasicrystals having n-fold, axial point groups. This is done without appeal to
super-space and projection, but is done in the 3-dimensional reciprocal space.
Along with a review of the RWM method and the extinction criteria, many
calculations are done in great detail showing, among other important things, the
process of calculating gauges which set certain phase functions to zero (a
requirement of the RWM method). The authors consider the full range of point-
group generators and both vertical and staggered stackings of the 2D
sublattices. Finally tables are presented containing the full classification of
the groups.
---------
N.D. Mermin
The Space-Groups of Icosahedral Quasi-crystals and Cubic,
Orthorhombic, Monoclinic, and Triclinic Crystals
Rev. Mod. Phys. 64(1), 3-49 (JAN1992)
This paper finishes the work of the previous paper by calculating the space
groups of the non-axial crystals and quasicrystals. Again, this is done using
the reciprocal-space formulation without super-space projection. This paper
again stresses the fundamental idea of indistinguishability rather than identity
as being the important association of structures. From a review of the RWM
method to detailed calculations of numerous space groups, this paper serves as
an excellent resource and reference for those interested in the topic at any
level.
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N.D. Mermin and R. Lifshitz
Bravais Classes for the Simplest Incommensurate Crystal Phases
Acta Cryst. A 48, 515-532 (01JUL1992)
It is shown that incommensurately modulated crystals can be treated on an equal
footing with periodic crystals and quasicrystals in Fourier space. Calculations
are done of the Bravais classes for the (3+3) modulated cubic crystals and (3+1)
modulated crystals in the remaining six systems. In this notation, (3+d), d is
the number of incommensurate modulations of the 3-dimensional crystal. The
integer d also signifies the number of additional vectors, beyond the
dimensionality of the real space, required to form a basis of the reciprocal
lattice. Perhaps as significant as the enumeration of the 16 Bravais classes is
the demonstration that using the super-space approach can lead one astray, as it
did to Janner, Janssen, and de Wolf who found 24 Bravais classes. This error is
because the super-space formulation fails to distinguish classes that differ by
trivial aspects such as scale equivalence.
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N.D. Mermin
Copernican Crystallography
Phys. Rev. Lett. V68 No. 2, 1172-1175 (1992)
This paper details the advantages of the reciprocal space formulation of
crystallography over the derivation in real space. With the discovery of
quasicrystals, either the idea of periodicity must be abandoned or periodic
structures must be found in higher, non-physical dimensions and then projected
onto real space. Mermin argues, and convincingly demonstrates, that it is best
to abandon the periodicity requirement and do crystallography in reciprocal
space, where the distinction between periodic crystals, modulated crystals, and
quasicrystals virtually disappears.
---------
J. Dr\"{a}ger and N.D. Mermin
Superspace Groups without the Embedding: The Link between Superspace and
Fourier-Space Crystallography
Phys. Rev. Lett. 76(9), 1489-1492 (1992)
Gives an exact formulation of the equivalence of reciprocal-space
crystallography and super-space crystallography.
---------
N. David Mermin
Crystallography Without Periodicity
Proceedings of the XIX International Colloquium on Group Theoretical Methods in
Physics (1993)
Gives a broad review of Fourier-space crystallography, also introducing new
topics (at that time) of the usefulness of the technique for application to a
wide range of periodic and aperiodic structures. This paper, perhaps more than
any other, gives a lead-in to the cohomology approach later described by Rabson
and Fisher. The author not only describes phase functions as belonging to
disjoint classes, but describes the equivalence relation in a way that will
"clearly" be seen as an element of a cohomology theory.
---------
R. Lifshitz and N.D. Mermin
Space Groups of Trigonal and Hexagonal Quasiperiodic Crystals of Rank 4
Acta Cryst. A 50, 72-85 (1994)
This paper shows how the Fourier-space (RWM) method of crystallography can be
extended to include the case of modulated crystals. In fact, it is shown that
this case is an extension of the periodic case when done in the Fourier-space
point-of- view. This eliminates the need to view the diffraction pattern as sets
of primary, secondary, tertiary, etc. patterns, as required for a real-space
treatment. The rank of a lattice is the smallest number of linearly-independent
(over \mathbb{Z}) vectors required to generate the lattice, and the number of
"lower-order" patterns depends on the rank of the crystal. This paper
focuses on (3+1) modulated crystals where the modulation can be either
commensurate (a rational ratio between the set of generating vectors) or
incommensurate (an irrational ratio between the set of generating vectors).
---------
R. Lifshitz and N.D. Mermin
Bravais Classes and Space Groups for Trigonal and Hexagonal Quasiperiodic
Crystals of Arbitrary Finite Rank
Acta Cryst. A 50, 85-97 (1994)
This paper extends the findings of "Space Groups of Trigonal and Hexagonal
Quasiperiodic Crystals of Rank 4" to the general (n+d) case. This now includes
modulated quasicrystals (n>3) and periodic crystals with more than a single
modulation (d>1), while retaining singly-modulated periodic crystals as a
special case. The focus of the paper is on the (3+d) case, but the method is
given that applies to the general case. Calculations are described in detail,
and full lists of classes and space groups are listed in numerous tables.
---------
J. Draeger, R. Lifshitz, and N.D. Mermin
Tetrahedral Quasicrystals
Proc. 5th Intl. Conf. on Quasicrystals, C. Janot and R. Mosseri, eds., World
Scientific, Singapore (1995), 72-75.
This paper derives the rank-6 space groups of tetrahedral quasicrystals. Their
relationship to the space groups of rank-6 icosahedral is explored, as well as
their relationship to rank-3 tetrahedral approximants and tetrahedrally
modulated crystals.
---------
R. Lifshitz and N.D. Mermin
Color Symmetry of Aperiodic Structures
Aperiodic '94, an Intl. Conf. on Aperiodic Crystals, G. Chapuis and W. Paciorek,
eds., World Scientific, Singapore, 77-81 (1995).
The Rokhsar-Wright-Mermin formulation is extended to any quasiperiodic field.
This has applications to magnetically-ordered structures and liquid crystals.
Examples are done for quasiperiodic colorations of a 10-fold lattice.
---------
R. Lifshitz and N. D. Mermin
Symmetry Changes in Rank-Lowering Phase Transitions
Aperiodic '94, Ed. G. Chapuis and W. Paciorek
(World Scientific, Singapore 1995) 267-271.
Available at: http://www.cmp.caltech.edu/%7Elifshitz/pub5.html
The authors examine the possible changes in space-groups when the rank of a
lattice is lowered. Lowering the rank of the lattice results in (1) new linear
relationships between the phase functions that describe the point group, and (2)
reduces the gauge freedom existing in the higher-rank lattice.
---------
R. Lifshitz
Quasiperiodic Spin Space Groups
Proc. 5th Intl. Conf. on Quasicrystals, C. Janot and R. Mosseri, eds., World
Scientific, Singapore (1995), 43-46.
An example of the field extension of the RWM method is shown for spin density
fields. In this case, the phase functions of RWM are now indexed not only by a
point-group element, but also by component transformations, \gamma, which are
proper rotations possibly combined with a time-reversal operation. The group-
compatibility condition in this field is \Phi_{gh}^{\gamma \delta}(k)=
\Phi_g^\gamma (h k)+\Phi_h^\delta (k). The criterion on the phase functions
leading to extinctions in the neutron- diffraction pattern are derived.
---------
R. Lifshitz and N.D. Mermin
The Symmetry of Composite Crystals
Aperiodic '94, an Intl. Conf. on Aperiodic Crystals, G. Chapuis and W. Paciorek,
eds., World Scientific, Singapore, 82-86 (1995).
An application of the RWM method to composite crystals is given in this paper.
These are normally thought of as two, or more, periodic crystals, having lattice
spacings that are either rational ratios (commensurate modulation) or irrational
ratios (incommensurate modulations) that are embedded in each other.
---------
R. Lifshitz
Introduction to Fourier-Space Crystallography: Lecture notes for the
International School on Quasicrystals
Available on the web at http://www.cmp.caltech.edu/~lifshitz/pub7.html.
Lecture notes, given at the 1995 International School on Quasicrystals, to
provide a tutorial overview of the state of Fourier-space crystallography. One
of the most distinctive, and appealing, aspects of this paper is the inclusion
of several Exercises, allowing the reader to gain first-hand, detailed insight
by actually applying the concepts discussed.
---------
N.D. Mermin
Extinctions in Scattering from Periodic or Aperiodic Crystals
Physica Status Solidi A 151(2), 275-279 (16OCT1995)
Mermin deftly shows that the criticism by some authors, that the RWM method
cannot be as useful for finding extinctions in diffraction patterns as the
super-space method, is totally unfounded. It is, in fact, easier and more
natural to look for these extinctions in Fourier-space since that is where the
diffraction pattern is found. (Diffraction patterns are rarely found in 6-
dimensional hyper-spaces.) In order to totally allay the fears of his
detractors, Mermin shows the entire calculation in a few lines.
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R. Lifshitz
Extinctions in Scattering from Magnetically Ordered Quasiperiodic Crystals
Lecture notes for the International Workshop on Application of symmetry analysis
to diffraction investigations, July 6-9, 1996, Krakow, Poland.
Available at: http://www.cmp.caltech.edu/%7Elifshitz/pub11.html
The spin-space classification of Litvin and Opechowski is extended to a Fourier-
space treatment of extinctions in neutron scattering from magnetically-ordered
crystals. This shows another example where a super-space geometric treatment can
be made more accessible though Fourier-space crystallography.
---------
R. Lifshitz
The symmetry of quasiperiodic crystals
Physica A 232(3-4), 633-647 (01NOV1996)
This paper describes how the Fourier-space method of crystallography is
applicable to all currently-known crystallographic structures: periodic
crystals, incommensurately modulated crystals, composite crystals,
quasicrystals, and even modulated quasicrystals. It is shown how the idea of
indistinguishability naturally extends the long-held ideas of symmetry, point
group, space group, etc. to all these types of aperiodic structures.
---------
R. Lifshitz
Theory of color symmetry for periodic and quasiperiodic crystals
Rev. Mod. Phys. 69(4), 1181-1218 (OCT1997)
This paper extends the idea of color symmetry to non-periodic structures. Using
the idea of indistinguishability to replace that of identity, and removing the
requirement of a translational symmetry, allows this extension in Fourier space.
While a general element of a color group is of the form (g, \gamma), where g
is in the spatial point group and \gamma is a permutation of the colors, this
paper concentrates on the lattice color group, having elements (e, \gamma).
---------
A. K\"{o}nig and N.D. Mermin
Electronic level degeneracy in nonsymmorphic periodic or aperiodic crystals
Phys. Rev. B 56(21), 13607-13610 (01DEC1997)
The authors describe the demonstration that orbital electronic energy levels at
certain wave vectors, in nonsymmorphic crystals, are necessarily degenerate as
"subtle". This is an understatement of epic proportions. Because the Rokhsar-
Wright-Mermin method recasts the problem of crystallography in the reciprocal
space, it is natural, and thus much easier, to determine such subtle aspects of
the electronic band structure. This paper shows this much simplified calculation
for all cubic and icosahedral, nonsymmorphic space groups, as well as
I2_12_12_1 and I2_13 which have no systematic extinctions.
---------
R. Lifshitz
Lattice Color Groups of Quasicrystals
Proc. 6th Intl. Conf. on Quasicrystals, S. Takeuchi and T. Fujiwara, eds., World
Scientific, Singapore (1998).
This paper uses the Fourier-space methods of crystallography to study the
partitioning of a periodic or aperiodic set of points into n symmetry-related
subsets. This method has applications ranging from anti-ferromagnetism in
tetragonal crystals to decorations of a lattice by multiple types of atoms.
---------
R. Lifshitz
Symmetry of magnetically ordered quasicrystals
Phys. Rev. Lett. 80(12), 2717-2720 (23MAR1998)
Fourier-space methods are used to investigate the observation of long-range,
magnetic ordering in icosahedral quasicrystals. Diffraction patterns for this
class of quasicrystal are derived and listed in numerous tables.
---------
A. K\"{o}nig and N.D. Mermin
Screw rotations and glide mirrors: Crystallography in Fourier space
Proc. Natl. Acad. Sci. USF 96, 3502-3506 (MAR1999)
This paper shows a distinction in screw axes and glide planes, in that some can
be described as "essential", while others are "removable". These are examined
from both the real and reciprocal spaces, leading to a geometric criterion on
the Bravais classes to distinguish between essential and removable screw axes
and glide planes. It is further shown that the two exceptional groups,
I2_12_12_1 and I2_13, owe their nonsymmorphic nature not to an absence of
Bragg peaks, but to the presence of electronic level degeneracies.
---------
A. K\"{o}nig and N.D. Mermin
Symmetry, extinctions, and band sticking
Am. J. Phys. 68(6), 525-530 (JUN2000)
Summarizing and restating the language of earlier papers, the authors here show
the absolute link, in Fourier-space, between symmetry and its physical
consequences. For periodic crystals, this link is often difficult to
demonstrate, but in Fourier space, this connection is natural and straight-
forward. It is shown how symmetries naturally describe necessary extinctions
(missing Bragg points in the diffraction pattern) and necessary degeneracies
(electronic energy band sticking).
---------
R. Lifshitz
Magnetic quasicrystals: What can we expect to see in their neutron diffraction
data?
Mat. Sci. Eng. A-Struct. 294, 508-511 Special Issue (15DEC2000)
The concept of indistinguishability is extended to include vector fields,
S(k). This allows for investigation of long-range, magnetic ordering in
icosahedral quasicrystals. It is shown that only two of the three icosahedral
Bravais classes admit nontrivial lattice spin groups; F^*, face-centered in
reciprocal space, being the exception. The author then predicts and lists
expected neutron-diffraction data.
---------
D. Rabson and B. Fisher
Fourier-Space Crystallography as Group Cohomology
Phys. Rev. B 65(2) (2001).
This paper shows how the phase functions, requirement of gauge invariance, and
other aspects of the Rokhsar-Wright-Mermin formulation are equivalent to the
elements of a cohomology theory. This allows very powerful, established
mathematics to be applied to Fourier-space crystallography. The formulation of
the method in terms of the cohomology of groups allows for a much easier
classification of space groups, the need to construct a gauge is removed, but
the duality of cohomology and homology allows the necessary extinctions and
necessary degeneracies to be easily determined.
---------
R. Lifshitz
The Rebirth of Crystallography
Zeit. Krist. 217, 342-343 (2002).
The author summarized the Fourier-space approach to crystallography, with a
concentration on how this method places large classes of aperiodic structures on
an equal footing with periodic crystals. Despite the redefinition of
"crystal" by the International Union of Crystallography, the author sees a
large resistance to adapting the easier, more natural, and more powerful methods
of Fourier-space crystallography. By showing how the method both preserves and
extends the ideas of crystal, symmetry, order, space group, point group, etc.
the author argues for wider use of this method and against the more popular, but
much more difficult, "super-space" methods.
---------
B.N. Fisher and D.A. Rabson
Applications of group cohomology to the classification of quasicrystal
symmetries
J. Phys. A 36(40), 10195-10214 (10OCT2003)
The authors demonstrate the usefulness of using cohomology to classify the space
groups of quasicrystals. Not only is the necessity of constructing a gauge
removed in this formulation, but it is shown that entire classes of
quasilattices can be evaluated with one calculation, rather than a case-by-case
calculation as was done previously. This paper demonstrates not only the
usefulness of the Fourier-space methods of crystallography, but also the
simplicity brought by the recasting of the method in the language of cohomology
of groups. By exploring, in depth, the duality of homology and cohomology, this
paper shows how this method easily describes the physical implications of the
algebraic classes.
---------
R. Lifshitz
Quasicrystals: a Matter of Definition
Found. Phys. 33, 1703-1711 (2003).
This paper discusses the unofficial adoption of the idea by much of the
crystallography world that a quasicrystal must possess forbidden symmetries.
Along with a discussion of incommensurately-modulated crystals and
incommensurate composite crystals, both of which have the forbidden symmetries,
the paper presents several examples of structures that have no forbidden
symmetries, yet should be classified as quasicrystals.
---------
D.A. Rabson, J.F. Huesman, and B.N. Fisher
Cohomology for Anyone
Found. Phys. 33(12), 1769-1796 (DEC2003)
This paper attempts to give an easy-to-follow overview of the use of group
cohomology in crystallography. Elements of group theory, algebraic topology,
group cohomology and crystallographic invariants are introduced at an elementary
level. Detailed examples then used to give the reader the full flavor of this
approach. This paper also includes the first computer code used in the attempt
to automate the process of classifying space groups using this method.
---------
Benji N. Fisher and David A. Rabson
Group Cohomology and Quasicrystals I: Classification of Two-Dimensional Space
Groups
Ferroelectrics 305, 37-40 (2004)
This paper shows how, using integer representation theory, all lattices, of all
ranks, symmetric under a point group G can be classified with a single
calculation. This "G-first" point-of-view is much more efficient than choosing a
lattice then finding the point group and doing the calculation, the "L-first"
point-of-view. Also removed is the assumption, used by many, that the lattice
must have minimal rank consistent with the point group. This paper deals with 2-
dimensional lattices, for which the point group is either C_N (cyclic of order
N) or D_N (dihedral of order 2N).
---------
Benji N. Fisher and David A. Rabson
Group Cohomology and Quasicrystals II: the Three Crystallographic Invariants in
Two and Three Dimensions
Ferroelectrics 305, 25-28 (2004)
This paper discusses the fundamental invariants from the cohomological approach
to Fourier-space crystallography and their physical implications. Along with a
review of the Type-I invariant being the condition for missing Bragg peaks and
the Type-II invariant being the condition for necessary degeneracy, a third type
of invariant is derived and discussed. It is shown that all classes of the first
homology group can be expressed as sums of these three types of invariants, and
thus these three invariants are the only ones.
---------
R. Lifshitz and S. Evan-Dar Mandel
Magnetically-ordered quasicrystals: Enumeration of spin groups and calculation
of magnetic selection rules
Acta Cryst. A 60 (2004) 167-178
This paper begins by filling in many of the details, only previously outlined
by the first author, on magnetic symmetries in quasicrystals. A process is
then given by which these can be classified by spin space group, spin point
group, and the diffraction selection rules in neutron scattering can be
calculated. This process is demonstrated using an octagonal quasicrystal in
two dimensions. Separate attention is given to geometric selection rules and
magnetic selection rules.
---------
S. Even-Dar Mandel and R. Lifshitz
Symmetry of Magnetically Ordered Three Dimensional Octagonal Quasicrystals
Acta Cryst. A 60 (2004) 179-194
The authors use the formalism derived in a companion paper (above) to classify
all three-dimensional octagonal spin point groups and spin space groups. These
results are then used to calculate the selection rules for neutron diffraction.
All point groups, space groups, and selection rules are listed in numerous tables.
---------
R. Lifshitz,
Quasicrystals: From Kepler to Shechtman
PhysicaPlus 3 (2004)
[Article is in Hebrew]
The theory of crystallography had developed for centuries under the premise that
crystals are necessarily periodic. No one had ever imagined that long-range
order could be achieved by any other means. All this changed overnight, some 20
years ago, when the Israeli physicist Danny Shechtman discovered the first
quasicrystal. Ever since Shechtman's discovery, crystallography has been in the
midst of a scientific revolution, in which we are revising basic notions in
condensed matter physics, such as the definition of "crystal" as well as the
notions of "order" and "symmetry" in crystals and their implication on the
physical properties of crystals.
In this article we shall try to explain, without going into too much detail,
where we stand today on a number of these issues. We shall give the current
definition of "crystal", demonstrate what aperiodic order might look like, and
explain what it means to say that an aperiodic crystal has certain symmetry.
---------
R. Lifshitz
Magnetic point groups and space groups
Encyclopedia of Condensed Matter Physics
(Elsevier Science, Oxford, 2005); In press.
Also on cond-mat: http://xxx.tau.ac.il/abs/cond-mat/0406675
This paper gives a broad overview of the theory of magnetic ( a.k.a.
antisymmetry, Shubnikov, black-and-white, Heesch, or Opechowski-Guccione)
groups. Derivations are shown for periodic and quasiperiodic crystals, along
with a generalization to multi- color groups and freely-rotating axial
vectors.
---------
Ron Lifshitz and Haim Diamant
Soft quasicrystals - Why are they stable?
arXiv:cond-mat/0611115v1 5 Nov2006
The paper first reviews the field of soft quasicrystals, highlighting
the useful ness of indistinguishability and guage functions in this
arena. Using a Lifshit z-Petrich extension of the Swift-Hohenberg
equation for describing different pat tern forming systems (Lyapunov
"effective free energy"), they arrive at a formlation that shows
stabilization for dodecagonal QC, but not octagonal or decagonal QC.
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G. Barak and R. Lifshitz
Dislocation dynamics in a dodecagonal quasiperiodic structure
"Philosophical Magazine", Vol. 86, Nos. 6-8, 21 February 11 March 2006, 1059-106
4
Starting with fournier analysis and specific numerical tools
(Lyapunov functiona l, Lifshitz-Petrich equation), the paper explores
how dislocations propagate uniquely in quasiperiodic structures. In
quasicrystals, weak diffusions under stress show irregular pinning
compared to regular pinning in periodic structures.
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This material is based on work supported by the National Science Foundation
under grant DMS-0204845. Any opinions, findings, or conclusions or
recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of the National Science Foundation.