Crystalline Lattice

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.

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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.

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( , )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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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.png


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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.svg


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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

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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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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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Zincblende(ZnS)structure:

identicalto

diamond,

except

thattwospeciesofatoms

alternatebetweensites.

Perovskite(CaTiO3)structure:

Ca: asimplecubiclattice.

Ti:atthebodycenter

O:onthefacecenters.

Ca

O

Ti

Crystalline Lattice

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