Table Of Content21
Crystal Packing
by Angelo Gavezzotti and Howard Flack
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http://www.iucr.org/iucr-top/comm/cteach/pamphlets/21/21.html
InternationalUnionofCrystallography
CommissiononCrystallographicTeaching
CRYSTAL PACKING
AngeloGavezzottia andHowardFlackb
a DipartimentodiChimicaStrutturaleeStereochimicaInorganica,
Universita` diMilano,Milan,Italy,andb Laboratoirede
Cristallographie,Universite´ deGene`ve,Geneva,Switzerland
1 Introduction
Weallknowbyeverydayexperiencethatmatterhasmanydifferentstatesofag-
gregation. Chemistsalsoknowthatmatterismadeofatoms,ionsandmolecules,
andthatthemacroscopicpropertiesofanyobjectdependonthesize, shapeand
energiesofthesemicroscopicconstituents.
Onemoleofgaseoussubstanceoccupiesabout24litresatroomtemperature,
while the volume of the same amount of substance in the liquid or solid state is
afewtenstoafewhundredmillilitres. Itfollowsthatthemolecules1 aremuch,
muchclosertoeachotherinaliquidandasolidthaninagas.Aneasycalculation
shows that in condensed phases the average volume per molecule is about one
andahalftimesthevolumeofthemoleculeitself. Moleculesaretightlypacked
inspace,andthereforethecompressibilityofcondensedmediaisverysmall. You
cansitonarocksimplybecauseitsatomsandmoleculesaresoclosetoeachother
thattheycannotgivewayunderexternalpressure.
Agaswilldiffuseveryquicklyoutofanopenbottle,whileasolidcanusually
be left in the open air almost indefinitely without apparent change in size and
shape(thereareexceptions,likemothballs). Besidesrepellingeachotheratshort
distances,moleculesinasolidarereluctanttoleavetheirneighbours;thismeans
that some sort of attraction is holding them together. Temperature has a much
moredramaticeffectonallthisthanpressure: ordinaryliquidsboilwhenheated
mildly,andevensolidrockmeltsandvaporizesinvolcanicdepths.
1Fromnowon,thetermmoleculedenotesamoleculeproper,oranyotherchemicalentityalso
recognizableinthegasphase(aheliumatom, anNa+ orSO24− ion, anFe2(CO)9 complex). In
general,itcanbesaidthatamoleculeisadistinguishableentitywhentheforcesactingwithinitare
muchstrongerthantheforcesactingonitinthecrystal. Difficultiesarisewithinfinitestringsor
layers; diamondandNaClcrystalsareexamplesofthree-dimensionallyinfinitesystemswherethe
termmoleculeismeaningless. Also, wheneverorganiccompoundsarementionedinthetext, one
shouldreadorganicandorganometalliccompounds.
1
Throughsimplereasoningonelementaryevidence,weareledtothefollowing
conclusions: upon cooling or with increasing pressure, molecules stick together
and form liquid and solid bodies, in which the distance between them is of the
same order of magnitude as the molecular dimensions; and an increasing repul-
sionarisesiftheyareforcedintoclosercontact. Thereverseoccursuponheating
orloweringtheexternalpressure.
While a layman may be more than satisfied at this point, a scientist must
ask him- or herself at least two further questions: (1) What is the nature and
magnitudeoftheforcesholdingmoleculestogether? (2)Whatisthegeometrical
arrangement of molecules at close contact? Restricting the scope, as we do in
this pamphlet, to crystalline solids, these questions define the subject of crystal
packing.Sincecrystalsareendowedwiththebeautifulgiftoforderandsymmetry,
thespatialpart(2)isnottrivial. Packingforcesandcrystalsymmetrydetermine
thechemicalandphysicalpropertiesofcrystallinematerials.
2 Thermodynamics and kinetics
Now put yourself in the place of a molecule within a pure and perfect crystal,
beingheatedbyanexternalsource. Atsomesharplydefinedtemperature,abell
rings, you must leave your neighbours, and the complicated architecture of the
crystalcollapsestothatofaliquid. Textbookthermodynamicssaysthatmelting
occurs because the entropy gain in your system by spatial randomization of the
molecules has overcome the enthalpy loss due to breaking the crystal packing
forces:
T[S(liquid)–S(solid)]>H(liquid)–H(solid)
G(liquid)<G(solid)
This rule suffers no exceptions when the temperature is rising. By the same
token,oncoolingthemelt,attheverysametemperaturethebellshouldringagain,
andmoleculesshouldclickbackintotheverysamecrystallineform. Theentropy
decreaseduetotheorderingofmoleculeswithinthesystemisovercompensated
bythethermalrandomizationofthesurroundings, duetothereleaseoftheheat
offusion;theentropyoftheuniverseincreases.
Butliquidsthatbehaveinthiswayoncoolingaretheexceptionratherthanthe
rule; in spite of the second principle of thermodynamics, crystallization usually
occurs at lower temperatures (supercooling). This can only mean that a crystal
ismoreeasilydestroyedthanitisformed. Similarly,itisusuallymucheasierto
dissolveaperfectcrystalinasolventthantogrowagainagoodcrystalfromthe
resultingsolution.Thenucleationandgrowthofacrystalareunderkinetic,rather
thanthermodynamic,control.
2
3 Forces
Amoleculeconsistsofacollectionofpositivelychargedatomicnucleisurrounded
by an electron cloud. Even if the molecule has no net charge, such an object
can hardly be considered as electrically neutral. Its electrostatic potential is a
superpositionofthefieldsofallnucleiandelectrons. Anapproachingchargecan
alter, by its own electrostatic field, the electron distribution in a molecule; this
phenomenoniscalledpolarization.
Theattractiveforcesholdingmoleculestogetherareaconsequenceofmolec-
ularelectrostaticpotentials. Forpurelyioniccrystals,onecanjustuseCoulomb’s
lawwithintegercharges; fororganicmolecules,ittakesamorecomplicatedex-
pression, involving an integration over continuous electron densities. Alterna-
tively,thechargedistributioncanberepresentedbyaseriesexpansionusingmul-
tipoles, and the interaction energy can be calculated as a function of multipole
moments.
Different atoms have different electronegativities. Larger charge separations
within the molecule – in the jargon of the trade, more polar molecules – build
up stronger intermolecular forces. Ionic crystals are very hard and stable, while
naphthalene or camphor (two common ingredients of mothballs) sublime rather
easily. Thesenon-polarhydrocarbonmoleculesmustrelyonmutualpolarization
toproduceattraction;theresultingforcesarefeeble,andarecalleddispersionor
vanderWaals’forces; theyareusuallydescribedbyempiricalformulae. Inthis
way,evenargonmanagestoformasolidatverylowtemperature.
Ubiquitous in crystals is the hydrogen bond, a polar interaction which is the
mosteffectivemeansofrecognitionandattractionbetweenmolecules; soeffec-
tive,thatmoleculeswithdonorandacceptorgroupsformhydrogenbondswithout
exception.Thereisnocase(atleast,totheauthors’knowledge)whereamolecule
thatcanformhydrogenbondsdoesnotdosointhecrystal.
Therepulsionatshortintermoleculardistancearisesfromaquantummechan-
icaleffect. AccordingtoPauli’sprinciple,electronswiththesamequantumnum-
bers,nomatterifbelongingtodifferentmolecules,cannotoccupythesameregion
ofspace. Thus,Pauli‘forces’–althoughtheyarenotforcesinthesenseofNew-
tonian mechanics – steer electrons to mutual avoidance, and the total energy of
theelectronsystemrisesifpairedelectronsarepulledtogether.
Table1collectsthesimplepotentialsmentionedsofar.Directbutnon-specific
measures of the strength of crystal forces are the melting temperature and the
sublimationenthalpy.
4 Crystal symmetry
Intermolecular attraction brings molecules together, but there is a priori no im-
plication of order and symmetry. Glasses, in which molecules are oriented at
3
Table1: Formulaeforpotentialenergiesincrystals
Electrostatic i.e. ions or point charges; q , q are the charges and R is the distance
i j ij
betweenthetwo:
X
E = (q q )/R .
i j ij
i,j
Electrostatic(moleculesAandBwithelectrondistributionsσ andσ ):
A B
ZZ
E = σ (r )σ (r )|r −r |−1dr dr .
A 1 B 2 2 1 1 2
Dispersion-repulsion(A,B,C,D,m,...,Qareempiricalparameters;R isthedistance
ij
betweentwosites–usually,atomicnuclei–ondifferentmolecules):
X
E = Aexp(−BR )−CR−6+DR−m+···+QR−1.
ij ij ij ij
i,j
Hydrogenbond:empiricalpotentialsinvolvinglocalcharges,localdipoles,etc.(thereisa
varietyofapproachesintheliterature).
random, are sometimes as stable as crystals, in which molecules are arranged
in an ordered fashion. The ordering of irregularly shaped, electrically charged
molecules does however imply anisotropy; for mechanical properties, it results
inpreferentialcleavageplanes, whiletheconsequencesofoptical, electricaland
magneticanisotropyleadtoavarietyoftechnologicalapplicationsofcrystalline
materials. Butwhatisthelinkbetweenorder,symmetryandcrystalstability?
Crystalsymmetry2 hastwofacets. Ononeside,inamilestonemathematical
development, it was demonstrated that the possible arrangements of symmetry
operations(inversionthroughapoint,rotation,mirrorreflection,translation,etc.)
give rise to no less and no more than 230 independent three-dimensional space
groups. After the advent of X-ray crystallography, space-group symmetry was
determinedfromthesystematicabsencesindiffractionpatternsandusedtohelp
inthecalculationofstructurefactorsandelectron-densitysyntheses.
The other side of crystal symmetry has to do with the crystal structure, as
resulting from mutual recognition of molecules to form a stable solid. This is a
fascinating and essentially chemical subject, which requires an evaluation and a
comparisonoftheattractiveforcesatworkinthecrystal. Space-groupsymmetry
is needed here to construct a geometrical model of the crystal packing, and it
comesintoplayinjudgingrelativestabilities.
It should be clear that the necessary arrangements of symmetry operations
2The term crystal symmetry refers to microscopic relationships between molecules or parts of
molecules,andnottomacroscopicmorphology.
4
Table2: Space-groupfrequenciesfrom[1]forasampleoforganiccrystals
Rank Group No. of Moleculesin Point-group
crystals generalposition symmetry
1 P2 /c 9056 8032(89%) 1
1
2 P2 2 2 4415 4415(100%) 1
1 1 1
3 P1 3285 2779(85%) 1
4 P2 2477 2477(100%) 1
1
5 C2/c 1371 802(58%) 2,1
6 Pbca 1180 1064(90%) 1
7 Pna2 445 445(100%) 1
1
8 P1 370 370(100%) 1
9 C2 275 225(82%) 2
10 Pnma 266 33(12%) m,1
12 Pbcn 205 94(46%) 1,2
14 P2 /m 127 40(31%) 1,m
1
16 P2 2 2 92 46(50%) 2
1 1
17 Fdd2 88 51(58%) 2
in space bear no immediate relationship to crystal chemistry. The fact that 230
spacegroupsexistdoesnotmeanthatmoleculescanfreelychooseamongthem
whenpackinginacrystal. Farfromit, thereareratherstrictpackingconditions
that must be met, and this can be accomplished only by a limited number of
arrangements of very few symmetry operations; for organic compounds, these
are inversion through a point (1), the twofold screw rotation (2) and the glide
reflection(g). Somespacegroupsaremathematicallylegitimate,butchemically
impossible, and the crystal structures of organic compounds so far determined
belongtoaratherrestrictednumberofspacegroups[1,2,3](Table2).
Whenchargeisevenlydistributedinamolecule,andthereisnopossibilityof
forminghydrogenbonds, no specialanchoringpointsexist. Every region ofthe
molecule has nearly the same potential for intermolecular attraction, and hence
it is reasonable to expect that each molecule be surrounded by as many neigh-
boursaspossible,formingasmanycontactsaspossible. Emptyspaceisawaste,
and molecules will try to interlock and to find good space-filling arrangements.
Thisclose-packingideaappearedveryearlyinitsprimitiveform[4],butwascon-
sciouslyputforwardbyKitaigorodski[5].
Order and symmetry now come to the fore, since for an array of identical
objects a periodic, ordered and symmetrical structure is a necessary (although
notsufficient)conditionforanefficientclosepacking. Whenspecialinteractions
(likehydrogenbonds)arepresent,theclose-packingrequirementmaybealittle
5
Figure1: Amoleculewithoutstronglypolarsitesorhydrogen-bondingcapabil-
itychoosestoclose-packinthecrystal,bumpintohollows,inordertomaximize
dispersiveinteractions.Whenstrongforcesarepresent,theclose-packingrequire-
mentmaybelesscompelling(waterisanextremeandalmostuniqueexample).
less stringent (Figure 1), but it turns out that all stable crystals have a packing
coefficient3between0.65and0.80.
5 Symmetry operations
Inacrystal,somesymmetryoperationscanbeclassifiedasintramolecular,mean-
ing that they relate different parts of the same molecule and thus belong to the
molecular point-group symmetry. The other symmetry operations, which act as
true crystal-packing operators, may be called intermolecular, and these are the
ones which relate different molecules in the crystal. This classification implies
thatmoleculesbedistinguishableincrystals.
The simplest intuitive way of viewing a symmetry operation is that it repro-
ducesinspaceone,ormoreifappliedrepetitively,congruentorenantiomorphic
3Thepackingcoefficientistheratioofvolumeoccupiedbythemoleculesinthecelltothevolume
ofthecell.Molecularvolumescanbecalculatedinanumberofways;thesimplestonesaredescribed
byKitaigorodski[5],andothersbyGavezzotti[7].
6
Figure2:Sketchesoftheeffectofsymmetryoperations.Top:inversionthrougha
point. Below,left:twofoldscrewrotation;belowright,glidereflection. Thelatter
twooperationsgiverisetostringsintheydirection.
copiesofagivenobject,accordingtoawell-definedconvention(Figure2). The
spatialrelationshipbetweentheparentandthereproducedmoleculesisstrict,so
amoment’sponderingwillconvincethereaderthatsomeoperatorsaremoreef-
fectivethanotherstowardsclosepacking. Forobjectsofirregularshape, mirror
reflectionandtwofoldrotationproducebump-to-bumpconfrontation,whileinver-
sion through a point, screw rotation and glide reflection favour bump-to-hollow,
moreclose-packedarrangements(Figure3). Onemustnotforgetthatpuretrans-
lation(t)isalwayspresentinacrystal. Exceptwheninfinitestringsorlayersare
present,itisanintrinsicallyintermolecularoperator,whoseroleinclose-packing
isprobablyintermediate(Figure4);spacegroupP1istheeighthmostpopulated
onefororganicsubstances.
The clearest proof of the leading role of 1, 2 and g in close packing comes
1
fromastatisticalanalysisofthespace-groupfrequenciesoforganiccompounds,
care being taken to distinguish between inter- and intramolecular symmetry op-
7
Figure3: Amirrorreflection(mirrorplaneperpendiculartothepage,tracealong
thesolidline)cannotproduceclose-packing. Translationalongsomedirectionis
requiredtoallowinterlockingofmolecularshapes.
Figure4: Atwo-dimensionalpatternobtainedbypuretranslation: notsobadfor
interlocking and close packing. For a complete set of two-dimensional space-
fillingdrawingsinallthe17planegroups,see[6].
8
