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7/29/2019 Crystalline Lattice
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CrystallineLattice
(Ch1&
2of
CMP)
BravaisLatticeandPrimitiveVectors
UnitCells
CrystalStructures
Symmetries: spacegroupandpointgroup
ClassificationofLatticesbySymmetry
1
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2
Crystallinesolids
Foralmostalltheelementsand
foravastarrayofcompounds,
thelowestenergystateis
crystalline,exceptforhelium
whichremainsliquidatzero
temperatureand
standard
pressure.
naturally occurringcrystals of iron pyrite
1st
publishedpicture of a
crystal structure (minerals consisting ofiron disulfide)
Whyare
low
energy
arrangements
of
some
atoms
so
often
periodic?
Noonereallyknows.
Crystallineorderisthesimplestwaythatatomscouldbepossiblybe
arrangedto
form
amacroscopic
solid.
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3
A
fundamental
concept
in
the
description
of
any
crystalline
solid
is
theBravaislattice, acollectionofpointsinwhichtheneighborhood
ofeachpointisthesameastheneighborhoodofeveryotherpoint
undersometranslation.
CrystallineLattice:repeatingoverandover
, i.e.,arepeatedarrayofpointswithan
arrangementand
orientation
that
appears
exactly
the
same,
from
whicheverthepointsofthearrayisviewed.
1 1 2 2 3 3,R n a n a n a= + +
where
arecalled
primitive
vectors,
andmustbelinearlyindependent.la
Thelocationofverypointinsuch
alatticecandescribedintheform
also called a triangular lattice,
symmetric under reflection about x
& y axes, and 60-rotation
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Intwodimensions,thereare5Bravaislattices.
4
symmetric underreflection about x & y
axes, and 90-rotation
a square lattice losing
the 90rotational
symmetry.
a compression of thehexagonal lattice without the
60rotational symmetry.
arbitrary choice of a1 and a2with no special symmetry,
still possessing inversion
symmetry.
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Areprimitivevectorsunique?
5
No,forhexagonallattice.
However,onecouldequally
choose
Cana latticeofafivefoldrotationalsymmetryexist?
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6
ConsideratwodimensionalBravais latticethatisinvariant
underarotation aroundtheorigin.
isaBravais lattice
point
with
abeing
alattice
constant,andthereisarotationsymmetryof,mustbeaBravais latticepoint.
( cos , sin )a a
( cos , sin )a a
Hence
isaBravais lattice
point,
(0, 2 sin )a
0, 2sin ) (cos , sin ) (cos , sin( ) =
On
other
hand,
if
we
choose
the
primitive
vectors
as1 ( ,0)a a=
2 ( cos , sin )a a a =
1 2(0, 2 sin ) ( ,0) ( cos , sin )a n a n a a = + 1 22cos 0; 2nn + ==
cos integer / 2 =
ItisimpossibleforaBravais latticetoafivefoldrotationaxis.
2 2 2 2 20, , , , ,
2 3 4 5 6
= X
( ,0)a
( cos , sin )a a
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Because
lattices
are
created
by
repeating
basic
units
over
and
over
throughoutspace,thefullinformationofacrystalcanbeobtainedin
asmallregionofspace.Sucharegion,chosentobeassmallasitcan
be,iscalledprimitiveunitcell.
UnitCells
7
Primitivecell:avolumeofspacethat,whentranslatedthroughall
thevectors inaBravaislattice,justfillsallofspacewithouteither
overlappingitselforleavingvoids.
Eachprimitivecellcontains
onlyonlatticepoint.
Primitivecells
are
not
unique.
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Onecanalsofillspaceupwithnonprimitivecells(knownas
conventionalcells).Theconventionalunitcell isaregionthat
justfillsspacewithoutanyoverlappingwhentranslated
throughsomesubsetofthevectorsofaBravaislattice.
8
Theconventionalunitcellisgenerallychosentobebiggerthan
aprimitive
cell
and
to
have
the
required
symmetry.
WignerSeitzcell: aprimitivecellwithfull
symmetryoftheBravaislattice.Thecellabouta
lattice
point
is
the
region
of
space
that
is
closer
to
anyotherlatticepoint.
TheWignerSeitzcellcanbe
constructedby
drawing
the
perpendicularbisectorofallthelines
betweenalatticepointtoeachofits
neighbor.
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CrystalStructure:latticewithabasis
Acrystalstructureconsistsofidenticalcopiesofthesamephysical
unit,called
the
basis,
located
at
all
the
points
of
aBravais
lattice.
9
Forexample,a2Dhoneycomb,thoughnot aBravaislattice,canbe
constructedbyatriangular(hexagonal)Bravaislattice.
triangularlatticenotinvariantunder
reflection
1
3 1( , )
2 2a a=
2 3 1( , )2 2
a a=
Primitivevectors
1 1( , 0)2 3
v a=
1
1( , 0)2 3
v a
=
basisparticles
atTheleftandrightparticlesineach
cellfindtheirneighborsoffat
differentsets
of
angles.
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Onceonedecoratesalatticewithabasis,itssymmetriesmight
change.
10
In
general,
theses
symmetries
of
a
lattice
can
be
destroyed
by
addingbasiselements.
Inatriangularlatticedecoratedwith
chiral
molecules,
the
rotational
symmetries oftheoriginallatticeare
preserved,butnotthereflection
symmetry.
Anobjectorasystemiscalledchiral if
itdiffersfromitsmirrorimage,andits
mirror
image
cannot
superimpose
on
theoriginalobject. Anonchiralobject
iscalledachiral andcanbe
superimposedonitsmirrorimage.
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Introductionto
group
theory
Agroup
is
set
of
elementsA, B, C..
such
that
aform
of
group
multiplicationsatisfiestherequirements:
11
Ref:M.Tinkham,GroupTheoryandQuantumMechanics
Theproductofanytwoelementsisintheset.
The
associative
law
holds;
e.g.,
A(BC)=(AB)C. ThereisaunitelementEsuchthatEA=AE=A. ThereisaninverseA-1 ofeachelementA suchthatAA-1=A-1A=E.
Crystal
symmetry
operations
such
as
translation,
rotation,
reflection,
inversion,canformagroup.
Forexample,allrotationsbyn/3 (n=0,1,2,5) aboutsomeaxis form
a
group
termedC
6.
In
such
a
group,
(1) themultiplicationAB meanstherotationfirstbyB,thenA.
(2) theunitelementisnooperationatall.
(3) Theinverseoperationisarotationthesameangleinreversesense.
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Thespacegroup oftheBravaislattice:acompletesetofallrigid
bodyoperationcomposedofatranslationandarotation.
12
( , )a n = +G
R There are 230 space groups and 32point groups in three dimensions.
Thepointgroup oftheBravaislattice:asubsetofthefullsymmetry
group,leaving
aparticular
point
fixed,
e.g.,
C6.
Notethatthepointgroupofalatticecannotdefinethelattice,
becausedifferentlattices,suchasrectangularandcentered
rectangular,can
be
invariant
under
precisely
the
same
set
of
point
symmetryoperations.
Althoughrectangularandcenteredrectangularlatticesshare
pointgroup
symmetries,
they
have
different
space
groups
and
aredifferentlattices.
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13
Notonlyforclassifyingcrystalstructures,thesymmetryofcrystal
canefficientlyreducethecomputationeffort insolvingthe
Schrodingers
equation
in
periodic
potentials,
and
lead
to
significantpredictionsofthesolutions.Symmetriesalsoprovide
uswithqualitativeinformation.
Forexample,withpracticallynoworkatall,onecanuse
symmetriestoidentifyplacesinkspacewhereenergybandswillbedegenerate.
Inthree
dimensions,
There
are
only
7distinct
point
groups
that
a
Bravaislaticecanhave.Outofthese,thereare14differentkindsof
Bravaislattice.
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14
bcc
fcc
b
ac
bct
(trigonal)
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15
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SchnfliesnonationC4
Cn: Cyclic,groupshavingonlyone
nfold
rotation
axis
Dn: Dihedral,groupshavingn 2fold
rotationaxis totheprincipalCn
axis.
Additionalsymbolsformirrorplanes:
h, v, dh: horizontal= totherotationaxis
v: vertical=// therotationaxis
d: diagonal=//the
main
rotation
axis
intheplaneandbisectingtheangle
betweenthe2foldaxes tothe
principalaxis.
http://upload.wikimedia.org/wikipedia/commons/3/3e/Uniaxial.pnghttp://upload.wikimedia.org/wikipedia/commons/3/3e/Uniaxial.pnghttp://upload.wikimedia.org/wikipedia/commons/3/3e/Uniaxial.pnghttp://upload.wikimedia.org/wikipedia/commons/3/3e/Uniaxial.png7/29/2019 Crystalline Lattice
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17
Sn
: Spiegel (German for mirror), containing nfold
axisfor
improper
rotation
(rotoinversion),
i.e.,
acombinationofarotationandaninversion
inapointontheaxis
S4
O: Octahedral,thegroupof24proper
rotationswhichtakeanoctahedron
(orcube)intoitself. 8C3
6C4, 3C2
6C2T: Tetrahedral,
the
group
of
12
proper
rotationswhichtakearectangular
tetrahedronintoitself.
http://upload.wikimedia.org/wikipedia/commons/e/e7/Dual_Cube-Octahedron.svghttp://upload.wikimedia.org/wikipedia/commons/e/e7/Dual_Cube-Octahedron.svghttp://upload.wikimedia.org/wikipedia/commons/e/e7/Dual_Cube-Octahedron.svghttp://upload.wikimedia.org/wikipedia/commons/e/e7/Dual_Cube-Octahedron.svg7/29/2019 Crystalline Lattice
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18
Usingaprojection torepresenta3Dcrystal
on2Dplane:
Stereographicprojections
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Stereographicprojections
ofthe32crystallographic
pointgroups
l l
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SomeImportantandPopularCrystalStructures
HexagonalClosePackedLattice:
Astackingof2Dtriangularlattices,
ahexagonallatticewithtwopoint
basis:(0,0,0)and ( , , )2 22 3
a a c
20
Thestructureisclosepackif
(Homework:Problem2.4)
8
3
c
a=
Diamondstructure:
Afcc
lattice
with
two
pointbasis:(0,0,0)and
(1/4,1/,4,1/4)a
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21
Rocksalt(SodiumChloride)
AfcclatticewithNaat(0,0,0)and
Clat(1/2,0,0)a.
CsClstructure:
Abcc
lattice
with
Cs
at
(0,
0,0)andClat(1/2,1/2,
1/2)a.
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22
Zincblende(ZnS)structure:
identicalto
diamond,
except
thattwospeciesofatoms
alternatebetweensites.
Perovskite(CaTiO3)structure:
Ca: asimplecubiclattice.
Ti:atthebodycenter
O:onthefacecenters.
Ca
O
Ti