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. --------- 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". --------- 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. --------- 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. --------- 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. --------- 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. --------- 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. --------- 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. --------- 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. ---------------------------------------------------------------------- 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.