What is a crystal?

0.13 MB·English

by  Ron Lifshitz

What is a crystal?
Ron Lifshitz
Condensed Matter Physics 114-36, California Institute of Technology, Pasadena CA 91125, U.S.A.∗
(Dated: January 1, 2007)
Almost 25 years have passed since Shechtman discovered quasicrystals, and 15 years since the
Commission on Aperiodic Crystals of the International Union of Crystallography put forth a pro-
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visional definition of the term crystal to mean “any solid having an essentially discrete diffraction
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diagram.” Havewelearnedenoughaboutcrystallinityinthelast25years,ordoweneedmoretime
0
toexploreadditionalphysicalsystems? Thereismuchconfusion andcontradictionintheliterature
2
in using the term crystal. Are we ready now to propose a permanent definition for crystal to be
n used by all? I argue that time has come to put a sense of order in all theconfusion.
a
J

2 Confusion is a word we have invented for an Unfortunately, by the end of the 18th century, when
order which is not understood. Hau¨ybeganformulatingthemathematicaltheoryofcrys-
]
i tallography [4], Kepler’s insightful drawings of aperi-
c
s Henry Miller odic tilings with decagonal symmetry were long forgot-
- ten. Mathematicalcrystallographywasfoundeduponthe
l
r premisethattheinternalorderofcrystalswasnecessarily
t achieved through a periodic filling of space. Thus, crys-
m I. PERIODICITY OR ORDER?
tallography treated order and periodicity synonymously,
.
t usingeitherpropertyinterchangeablytodefinethenotion
a Foralmosttwomillenniacrystallographywaspredom-
of a crystal. The periodic nature of crystals was “con-
m
inantly concerned with the external morphology of crys-
firmed” with the discovery of x-ray crystallography and
- tals. Crystallographers studied the naturally-occurring
numerous other experimental techniques throughout the
d
n facets of crystals, which always intersect at precise and 20th century. Being more “elaborate and intricate” and
characteristicangles. Itwasonlyinthe17thcenturythat
o less commonly found in Nature, aperiodic crystals were
c moderncrystallographywasborn,thankstothebrilliant completely overlooked. Periodicity became the under-
[ idea—attributed to such great scientists as Kepler and
lying paradigm, not only for crystallography itself, but
Hooke—that crystal shapes were the result of internal
1 also for other disciplines such as materials science and
order of “atomic” units.
v condensed matter physics, whose most basic tools, like
9 In his study of tilings of the plane by polygons, the Brillouin zone, rely on its existence.
2 Kepler [1] was quick to realize that very few regu- This historical oversight was corrected with Shecht-
0 lar polygons—namely the triangle, the square, and the
man’s dramatic discovery of quasicrystals in 1982 [5]—a
1
hexagon—can tile the plane without introducing over- discoverythatsparkedabonafide Kuhnianscientificrev-
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laps or leaving holes. Yet, this observationdid not deter
7 olution[6,7]. Inhisdescriptionoftheevent,Hargittai[8]
0 him from constructing a well-orderedtiling of the plane, comments that
/ consisting ofpentagons and decagonsthat requiressome
at of the decagons to overlap, and leaves holes in the form “...the discovery was a kind of legalistic dis-
m of 5-fold stars—a tiling that was to be rediscovered by covery. This happens when the human clas-
Penrosemorethan350yearslater[2,3]. Without realiz- sification system is more restrictive than the
-
d ingit,Keplerhaddiscoveredsomeofthebasicproperties laws ofnatureanddiscoveries appear tobreak
n of aperiodic order. In his words: the laws that had been artificially constructed
o
in the first place.”
c
“If you really wish to continue the pattern,
:
v certain irregularities must be admitted, two Nature had found a way of achieving order without pe-
i decagons must be combined, two sides being riodicity, and Shechtman was careful enough not to dis-
X
removed from each of them. As the pattern miss it as an experimental artifact, as many must have
ar is continuedoutwards five-cornered forms ap- done before him. He confronted a skeptical scientific
pear repeatedly: ...So as it progresses this community that was unwilling to relinquish its most ba-
five-cornered pattern continually introduces sic paradigm that order stems from periodicity. In an
something new. The structure is very elab- article, suggesting a description of Shechtman’s icosahe-
orate and intricate.” dral quasicrystal as multiple twinning of periodic cubic
crystals—adescriptionthat waslater shownto be incor-
rect [9]—Pauling [10] concluded by saying that
“Crystallographers can now cease to worry
∗Permanent address: School of Physics & Astronomy, Raymond
that the validity of one of the accepted bases
andBeverlySacklerFacultyofExactSciences,TelAvivUniversity,
69978TelAviv,Israel;Electronicaddress: [email protected] of their science has been questioned”.
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Today, thousands of diffraction diagrams later, com- The 1992 definition is consistent with the notion of
pounded by high-quality experimental data—such as long-rangeorder—oneofthe basicnotions ofcondensed-
images from high-resolution transmission electron mi- matter physics [14, 15]—dating back to ideas of Lan-
croscopes and atomic-resolution scanning tunneling dau in which the symmetry-breaking transition from a
microscopes—the existence of order without periodicity disordered (high-symmetry) phase to an ordered (low-
has been unequivocally established. Not only has the symmetry) phase is quantified by the appearance of a
periodicity paradigm been questioned, as Pauling wor- non-zero order parameter. Stated in plain words, long-
ried, it has been completely invalidated. It is time now, range order—orin the context of our current discussion,
as Ben-Abraham [11] argues in this volume, to make a long-rangepositional order—is a measure of the correla-
clear statement associating the notion of a crystal with tionsbetweenthepositionsofatomsindistantregionsof
its more fundamental and underlying property of order, the material. It measuresthe extent of our ability to de-
ratherthanthepropertyofperiodicity, whichisonlyone scribethepositionsofatomsinfarawayregionsofspace,
way of achieving order (even if it is the more common based on our knowledge of their positions nearby.
wayinwhichNaturechoosestodoso). Accordingtothis, Howisthisrelatedtothe requirementofhavinganes-
crystals that are periodic should explicitly be called pe- sentiallydiscretediffractiondiagram? Ifρ(r)isadensity
riodic crystals, allothersshouldbe calledaperiodic crys- function,describingacertainmaterial,thenthe function
tals. thatmeasurescorrelationsbetweentwopoints,separated
Some may object by saying that there are no right or in space by a vector R—the two-point autocorrelation
wrong definitions, and that a definition is merely a mat- function—is known in crystallography as the Patterson
ter of mutual agreement. They would further claim that function,
the notion of a crystal should strictly be associatedwith
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periodicity, because overa century ofliterature in math- P(R)= lim drρ(r)ρ(r−R). (1)
ematical crystallographycontains statements and proofs V→∞V Z
about crystals that are, in fact, true only for periodic
Its Fourier transform is exactly the intensity I(k) =
crystals. A change of the definition would cause great
|ρ(k)|2 thatis measuredinmostdiffractionexperiments.
confusionwhenreadingthisliterature. Thistypeofargu-
It was realized long ago (e.g. Ref. [16]), that the or-
mentistypicaloftheparadigm-shiftingstagewearecur-
derparameterthatsignalsatransitionfromadisordered
rently undergoing in this scientific revolution [12]. Nev-
liquid to an ordered solid, indicating the emergence of
ertheless, the discovery of quasicrystals demands such a
non-trivial correlations as measured by the Patterson
paradigm shift and along with it a new definition of the
function (1), appears in the form of delta functions, or
notionofacrystal. Onesimplyneedstobeaware,when-
Bragg peaks, at non-zero wave vectors in the diffraction
ever reading old literature, that the term crystalmay be
diagram. Thus, the observation of essentially discrete
used in its old restricted meaning.
pointsinthediffractiondiagramis theindicationofhav-
ing long-range positional order, and therefore of crys-
tallinity. From the point of view of condensed-matter
II. BUT, WHAT IS ORDER?
physics, long-range positional order is characterized by
having Bragg peaks in the Fourier transform of the den-
In 1992, the International Union of Crystallography, sityfunctionρ(r). Inthelanguageofspectraltheory,the
throughitsCommissiononAperiodicCrystals[13],pub- diffraction spectrum of ρ(r) must contain a pure-point
lished a provisional definition of the term crystal that component, for ρ(r) to be considered crystalline.
abolishes periodicity, but does not go as far as naming
order asits replacement. The commissionwasnotready
to give precise microscopic descriptions of all the ways III. PERIODIC, QUASIPERIODIC, AND
in which order can be achieved. Clearly, periodicity is ALMOST-PERIODIC CRYSTALS
one way of achieving order, quasiperiodicity as in the
Penrose-Kepler tiling is another, but can we be certain Following H. Bohr’s theory of almost periodic func-
that there are no other ways that are yet to be discov- tions [17], solids whose density functions ρ(r) may be
ered? The Commissionoptedtoshiftthe definitionfrom expanded as a superposition of a countable number of
a microscopic description of the crystal to a property of plane waves
the datacollectedinadiffractionexperiment. Itdecided
on a temporary working-definition whereby a crystal is ρ(r)= ρ(k)eik·r, (2)
kX∈L
“any solid having an essentially discrete
diffraction diagram.” are to be called almost periodic crystals. In particular,
if taking integral linear combinations of a finite num-
The new definition was left sufficiently vague so as not ber D of wave vectors in this expansion can span all the
to impose unnecessary constraints until a better under- rest, then the crystal is to be called quasiperiodic. The
standing of crystallinity emerges. diffraction pattern of a quasiperiodic crystal, therefore,
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contains Bragg peaks, each of which can be indexed by IV. WHAT ELSE IS CRYSTALLINE?
D integers. If D is the smallest number of wave vec-
tors that can span the whole set L using integral linear
The discovery of quasicrystals and the possibility for
combinations, then D is called the rank, or the indexing
aperiodic order has made a great impact on mathemat-
dimension of the crystal. Periodic crystals form a spe-
ics[21,22,23,24],assummarizedbyLagarias[25]. Much
cial subset of all quasiperiodic crystals, whose rank D is
efforthasbeeninvestedinstudyingthecharacteristicsof
equal to the actual physical dimension d (the number of
order, as well as in the development of diffraction the-
components in the vectors r and k).
ory[26, 27],once itwasrealizedthatperiodicity wasnot
anecessaryconditionfororderandfortheappearanceof
ForperiodiccrystalsthesetofBraggpeaksistrulydis-
crete, because the set of wave vectors k in their Fourier Bragg peaks in the diffraction spectrum.
One interesting line of study [25] is an attempt to
expansion (2) is discrete. For quasiperiodic crystals
characterize the local mathematical properties of peri-
whose rank D is greater than the physical dimension d,
odic crystals and quasicrystals that are responsible for
the set L of wavevectorsin the expansion(2) is dense—
there are k’s in L that cannot be surrounded by a finite the emergence of long-range order. For periodic crystals
d-dimensional ball that contains no other k’s. Neverthe- it is well-understood that any of a number of localprop-
ertiesissufficientforensuringperiodicity,andthusorder.
less, in actual experiments, where the total integrated
Thesenotionscanbegeneralizedtocharacterizethelocal
diffraction intensity is finite, Bragg peaks are not ob-
served at wave vectors k for which the intensity |ρ(k)|2 properties of quasicrystals,yet it is still unclear whether
anyofthesenotionscanserveasasufficientconditionfor
is below a certain threshold. In this sense the observed
ensuring quasiperiodicity, or more generally, long-range
diffraction pattern is essentially discrete, abiding by the
order.
1992definition,eventhoughthesetLisnotdiscrete. As
the resolution of the diffraction experiment is improved, Rather than trying to characterize local structural
additional peaks will appear in the diffraction diagram. properties that ensure Bragg peaks in the diffraction, a
Ifthecrystalisperiodictheaddedpeakswillappearfur- number of authors have considered the diffraction prop-
therawayfromtheorigin. Ifthecrystalisquasiperiodic, erties of structural models that are constructed by par-
new peaks can appear in between already-existing ones, ticular methods. Could there be a constructive proce-
yetallthenewpeakswillstillbeindexablebyDintegers. dure that perfectly encompasses all structures that pos-
If the crystal is almost periodic, new vectors may be re- sesslong-rangeorder? Theanswertothisquestionisstill
quiredinordertogeneratetheadditionalpeaks,thusthe unknown,althoughanumberofunintuitiveandevensur-
rank D will seem to grow as the resolution is improved. prising results have already been obtained.
It is well-understood that structure models that are
It should be noted that all experimentally observed constructed by the cut-and-projecttechnique from finite
crystalstodatearequasiperiodic. Thetermquasicrystal, dimensional spaces [21] are all quasiperiodic, and there-
whichwasfirstintroducedbyLevineandSteinhardt[18], fore have a Bragg spectrum and possess long-range or-
is simply short for quasiperiodic crystal, but is used to der. The same holds for structures that are generated
refertothosequasiperiodiccrystalsthatarestrictlyape- using the dual grid method and its various generaliza-
riodic(i.e.withD >d). Someauthorsrequirecrystalsto tions [28, 29]. These construction methods are therefore
possess so-called “forbidden symmetries” in order to be too limited to encompass all ordered structures. On the
regardedasquasicrystals. Ihaveexplainedelsewhere[19] other hand, it turns out that other approaches that are
why such a requirement is inappropriate. commonly used to generate quasicrystal models are too
general, often producing structures that do not possess
Oneshouldalsorecallthatcertainclassesofquasiperi-
Bragg peaks in their diffraction spectrum, and therefore
odiccrystalswereknownlongbeforeShechtman’sdiscov-
are not crystalline.
ery. Thesearetheso-calledincommensurately-modulated
Atypicalexampleofaconstructionmethodthatistoo
crystals and incommensurate composite crystals, (or in-
general to be used to define crystallinity is that of sub-
tergrowth compounds). These crystals did not pose any
stitution rules—either of letter sequences in one dimen-
serious challenge to the periodicity paradigm because
sion, or of tiles in two or three dimensions. The famous
they could all be viewed as periodic structures that had
Fibonacci sequence, produced by the substitution rules
been slightly modified. Order was still obtained through
L → LS and S → L, produces a rank-2 quasicrystal.
periodicity, leaving the paradigm intact.
What may seem as a minor change to these substitution
Almost periodic crystals have not (yet) been found rules may generate sequences that do not have a Bragg
in Nature, but can be constructed artificially. A pos- spectrum. Models based on such sequences do not pos-
sible way of constructing such crystals is to consider the sesslong-rangeorderandarethereforenotcrystals,even
limit of an infinite sequence of periodic or quasiperiodic thoughtheyareconstructedusingthesamedeterministic
structures. If chosen properly, such a limit can yield procedurethatisusedtogeneratemanyquasicrystals. It
so-called limit-periodic or limit-quasiperiodic structures, has been understood for some time [30] that what deter-
whose spectrum was shown to consist of Bragg peaks, mineswhetherastructure,generatedbysubstitution,has
even though their rank is infinite [20]. long-range order is an algebraic property of the matrix,
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which describes the substitution. The generated struc- Finite-sizeissuescanbeformulatedasacriterionofmin-
ture contains Bragg peaks in its Fourier spectrum only imumsizeinrealspace,orasaconditiononthewidthof
if the largest eigenvalue of the substitution matrix is a Bragg peaks or their scaling with the number of atoms.
so-called Pisot-Vijayaraghavannumber. In some of these cases it is a matter of taste to decide
Thefactthattherearenocorrelationsbetweenthelet- what constitutes a crystal, and one may need to rely on
ters of a sequence that is constructed by a deterministic common sense. Questions such as “how many atoms are
procedure may seem counter-intuitive, but readers who neededtoformacrystal?” or“whenisanatomiccluster
are familiar with the notion of deterministic chaos may sufficientlylargetobeconsideredananocrystal?” should
notbesoalarmed. Ben-Abraham[11]hassuggestedcon- notbeofanyrealconcern. Onthe otherhand,questions
sideringallstructuresthataregeneratedbydeterministic of short-range versus long-range order in a given homo-
proceduresascrystals. Alongthesamelines,Mackay[31] geneous material should be considered with care.
has argued that More interesting are structures that by design possess
both long-range order and some degree of disorder. An
“a crystal is a structure the description of important example is that of random-tiling models of
which is much smaller than the structure it- quasicrystals[34, 37]. These models are believedto have
self.” an underlying Bragg spectrum, indicative of quasiperi-
odic long-rangeorder,superimposedwitha diffuse back-
I objectto suchdefinitions thatconfusethe notionofor- ground stemming from phason disorder, which is inher-
der with determinism. As counter-intuitive as it may ent to the randomness of the models. In this case, the
seem, deterministically-generated structures might be underlying Bragg spectrum is often viewed as being the
disordered. If this were not the case it would not have essential component, and therefore the structure is con-
been possible to design computer algorithms that gen- sidered to be crystalline.
erate random numbers. Random number generators use An opposite example is the one-dimensional Thue-
deterministic procedures to generate uncorrelated num- Morse sequence, generated from two letters by the sub-
bers that look as if they were obtained by the random stitution rule L → LS and S → SL. If represented
flipping of a coin. When considering the actual struc- as a sequence of short (S) and long (L) segments with
tures, deterministic disorder is practically the same as atoms at the joining points, then 1/2 of the atoms form
randomdisorder,andshouldnotbeassociatedwithcrys- aperiodiclatticewithaunitcelloflengthS+L. There-
tallinity. Noteverydeterministicstructurethatissimple maining atoms divide the unit cell into an L followedby
to generate is necessarily a crystal. an S, or vice versa, in a completely disordered fashion.
Baake and several co-workers [32, 33, 34, 35, 36] are In this case, many authors tend to view the underlying
currently performing a systematic study whose purpose periodic backbone as an unimportant artifact, and the
istocharacterizewhichdistributionsofmatterdiffractto structureitselfasbeingdisordered. Infact,onecancare-
produce a pure point component in their spectrum, and fully choose to represent the Thue-Morse sequence in a
thus can qualify as possessing long-range order. In some non-generic way in which all of the Bragg peaks become
casesitisevendifficulttodeterminewhethertheFourier extinguished, leaving only the diffuse part of the diffrac-
transform of a structure exists, in the sense that it has tion spectrum [30]. If viewed as being essentially disor-
a unique infinite volume limit. When it is clear that the dered, the Thue-Morse sequence should not be regarded
Fourier transform exists it may still be difficult to rigor- as a crystal, although one should always remember that
ously know whether it contains Bragg peaks. This is a it does contain an underlying periodic lattice.
difficultstudywhichmighttakealongtimetocomplete,
yetitis animportantstudy thatmayeventually provide
a proper characterizationof long-range order. VI. SO, WHAT IS A CRYSTAL?
Having said all this, I would like to conclude with a
V. THE GRAY ZONE definition:
Therearemanystructuresthatarepartially-ordered— 1. A crystal is a solid that has long-range posi-
tional order.
eitherbydesign,ordueto“practical”circumstancessuch
as thermal fluctuations, or the existence of defects and 2. Long-range positional order can be inferred
grain boundaries. In some of these cases it might be dif- from the existence of Bragg peaks in the
ficult to decide at what point the structure ceases to be Fourier spectrum of the solid.
crystalline. Often, details regarding the shape of Bragg
peaks offer a convenient way of doing so. Melting of a It is common to add a descriptive prefix to the term
crystaldue to thermal fluctuations, for example, may be crystal to refer to soft materials, metamaterials, or any
formulatedinterms ofa real-spaceconditionsuchasthe other form of matter that is crystalline but is not an
Lindemann criterion, or as a condition on the reduction atomic solid. Examples are photonic crystals and liq-
ofBragg-peakintensitiesduetotheDebye-Wallerfactor. uidorsoftcrystals—moreappropriatelycalledcrystalline
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liquids [38] to distinguish them from the more com- VII. ACKNOWLEDGMENTS
mon liquid crystals that have long-range orientational
rather than positional order. These should now be un-
derstood in the context of the new definition of crys-
tal to include periodic soft and photonic crystals as well I thank S.I. Ben-Abraham for insisting that we re-
as soft quasicrystals [39, 40] and photonic quasicrys- consider now the definition of crystal, rather than wait
tals [41, 42, 43, 44] that are not periodic. for another 10 or 15 years. I also thank the Editor of
I would happily welcome a revision of the second part Zeit. Krist., Walter Steurer, for suggesting to me that
of the definition once a more complete characterization I put my thoughts on this matter into writing. Finally,
oforderisestablished. Inthemeantime,onemustfollow I thank David Mermin for his thoughtful comments on
the only reliable characterizationin terms of the Fourier this manuscript.
spectrum. I would be even happier to welcome a second
scientific revolution in materials science that would re-
quire yet another paradigm shift and a completely new My research in crystallography is currently supported
definition. Until that happens, I suggest sticking to or- by the Israel Science Foundation through grant number
der. 684/06.
[1] J.Kepler,HarmonicesMundi,(Frankfurt,1619)BookII. 1947).
English translation by E. J. Aiton, A. M. Duncan, and [18] D. Levineand P.J. Steinhardt,“Quasicrystals I:Defini-
J. V. Field, “The harmony of the world,” (Am. Philos. tion and structure,” Phys. Rev. B34, 596 (1986).
Society,1997). [19] R. Lifshitz, “Quasicrystals: A matter of definition,”
[2] R. Penrose, “The role of aesthetics in pure and ap- Foundations of Physics 33 (2003) 1703.
pliedmathematicalresearch,”Bull.Inst.Math.Appl.10 [20] M. Baake, R. V. Moody, and M. Schlottmann, “Limit-
(1974) 266. (quasi)periodicpointsetsasquasicrystalswithp-adicin-
[3] M. Gardner, Sci. Am. 236 (1977) 110. ternal spaces,” J. Phys. A: Math. Gen. 31 (1998) 5755.
[4] R.-J. Hau¨y, Essai d’une Th´eorie sur la Structure des [21] M. Senechal, Quasicrystals and Geometry, (Cambridge
Crystaux (Gogu´e & N´ee, Paris, 1784). Univ.press, Cambridge, 1996).
[5] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, [22] R. V. Moody (Ed.), The Mathematics of Long-Range
“Metallicphasewithlong-rangedorientationalorderand Aperiodic Order, (Kluwer, Dordrecht,1997).
no translational symmetry,” Phys. Rev. Lett. 53, 1951 [23] J. Patera (Ed.), Quasicrystals and discrete geometry,
(1984). (American Mathematical Society,Providence, 1998).
[6] J.W. Cahn,“Epilogue,” in Proceedings of the 5th Inter- [24] M.BaakeandR.V.Moody(Eds.),Directions inMathe-
national Conference on Quasicrystals, Ed. C. Janot and matical Quasicrystals, (American Mathematical Society,
R.Mosseri (World Scientific, Singapore, 1996) 807. Providence, 2000).
[7] R.Lifshitz,“Therebirthofcrystallography,”Zeit.Krist. [25] J. C.Lagarias, “Theimpact ofaperiodic orderon math-
217 (2002) 432. ematics,” Materials Science and Engineering 294-296
[8] I.Hargittai,“Symmetryincrystallography,”ActaCryst. (2000) 186.
A54 (1998) 697. [26] A.Hof,“Diffractionbyaperiodicstructures,”inRef.[22],
[9] See,forexample,P.A.Bancel,P.A.Heiney,P.M.Horn, p. 239.
and P. J. Steinhardt, “Comment on a Paper by Linus [27] J. C. Lagarias, “Mathematical quasicrystals and the
Pauling,” Proc. Natl. Acad. Sci. USA 86 (1989) 8600. problem of diffraction,” in Ref. [24], p.61.
[10] L. Pauling, “Apparent icosahedral symmetry is due to [28] N.G.deBruijn,Proc.K.Ned.Akad.Wet.,Ser.A:Math.
directedmultipletwinningofcubiccrystals,”Nature317 Sci. 84 (1981) 39.
(1985) 512. [29] F. G¨ahler and J. Rhyner, “Equivalence of the gener-
[11] S. I. Ben-Abraham, “What is a crystal?” Zeit. Krist. alised grid and projection methods for the construction
Thisvolume(2007)[EDITOR:Pleaseinsertcorrectref- of quasiperiodic tilings,” J. Phys. A 19 (1986) 267.
erence]. [30] F. G¨ahler and R. Klitzing, “The diffraction pattern of
[12] T. S. Kuhn, The Structure of Scientific Revolutions, self-similar tilings,” in Ref. [22], p. 141.
(Universityof Chicago Press, 1962). [31] A. L. Mackay, “Generalized crystallography,” Structural
[13] International Union of Crystallography, “Report of the Chemistry 13 (2002) 215.
executive committee for 1991,” Acta Cryst. A 48, 922 [32] M. Baake and R.V.Moody, “Diffractivepoint sets with
(1992); see p. 928. entropy,”J. Phys. A 31 (1998) 9023.
[14] P. W. Anderson, Basic notions of condensed matter [33] M.Baake,R.V.Moody,andP.A.B.Pleasants,“Diffrac-
physics (Addison Wesley,Reading MA, 1997), Chap. 2. tion from visible lattice points and kth power free inte-
[15] J. P. Sethna, Entropy, order parameters, and complexity gers,” Disc. Math. 221 (2000) 3.
(Clarendon Press, Oxford, 2006), Chap. 9. [34] M. Baake and M. H¨offe, “Diffraction of random tilings:
[16] N. D. Mermin, “Crystalline order in two dimensions,” Some rigorous results,” J. Stat. Phys. 99 (2000) 219.
Phys. Rev. 176 (1968) 250. [35] M.H¨offeandM.Baake,“SurprisesinDiffuseScattering,”
[17] H.Bohr,Almost periodicfunctions,(Chelsea,NewYork, Zeit. Krist. 215 (2000) 441.
6
[36] M. Baake, R. V. Moody, C. Richard, and B. Sing, [41] C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Ban, and
“Which distribution of matter diffracts?” In Quasicrys- B.Sun,“Bandgapandwaveguidingeffectinaquasiperi-
tals: Structure and Physical Properties ed.H.-R.Trebin, odicphotoniccrystal,”Appl.Phys. Lett.75(1999)1848.
(Wiley-VCH,Berlin, 2003) 188. [42] M. E. Zoorob, M. D. B. Charlton, G. J. Parker,
[37] C. L. Henley, “Random tiling models,” in J. J. Baumberg, M. C. Netti, “Complete photonic
Quasicrystals—The State of the Art, D. P. DiVin- bandgaps in 12-fold symmetric quasicrystals,” Nature
cenzo and P. J. Steinhardt, Eds. (World Scientific, 404 (2000) 740.
Singapore, 1991), p.429. [43] R. Lifshitz, A. Arie, and A. Bahabad, “Photonic qua-
[38] D. C. Wright and N. D. Mermin, “Crystalline liquids: sicrystals for nonlinear optical frequency conversion,”
thebluephases,” Rev. Mod. Phys. 61 (1989) 385. Phys. Rev. Lett. 95 (2005) 133901.
[39] X.Zeng,G.Ungar,Y.Liu,V.Percec,A.E.Dulcey,and [44] B. Freedman, G. Bartal, M. Segev, R. Lifshitz,
J.K.Hobbs,“Supramoleculardendriticliquidquasicrys- D. N. Christodoulides, and J. W. Fleischer, “Wave and
tals,” Nature 428 (2004) 157. defect dynamics in nonlinear photonic quasicrystals,”
[40] R. Lifshitz and H. Diamant, “Soft quasicrystals - Why Nature 440 (2006) 1166.
are they stable?” To appear in Phil. Mag. (preprint
cond-mat/0611115).