Primitivelatticevectors

Q:Howcanwedescribetheselatticevectors(thereareaninfinitenumberofthem)?A:Usingprimitivelatticevectors(thereareonlydoftheminad-dimensionalspace).

Fora3Dlattice,wecanfindthree primitivelatticevectors(primitivetranslationvectors),suchthatanytranslationvectorcanbewrittenas

𝑡 = 𝑛%�⃗�% + 𝑛(�⃗�( + 𝑛)�⃗�)where𝑛%, 𝑛( and𝑛) arethreeintegers.

Fora2Dlattice,wecanfindtwo primitivelatticevectors(primitivetranslationvectors),suchthatanytranslationvectorcanbewrittenas

𝑡 = 𝑛%�⃗�% + 𝑛(�⃗�(where𝑛% and𝑛( aretwointegers.

Fora1Dlattice,wecanfindoneprimitivelatticevector(primitivetranslationvector),suchthatanytranslationvectorcanbewrittenas

𝑡 = 𝑛%�⃗�%where𝑛% isaninteger.

Primitivelatticevectors

Red(shorter)vectors:�⃗�% and�⃗�(

Blue(longer)vectors:𝑏% and𝑏(

�⃗�% and�⃗�( areprimitivelatticevectors𝑏% and𝑏( areNOTprimitivelatticevectors

𝑏% = 2�⃗�% + 0�⃗�( �⃗�% =12𝑏% + 0𝑏(

Integercoefficients noninteger coefficients

Primitivelatticevectors

ThechoicesofprimitivelatticevectorsareNOTunique.

Primitivecellin2D

Theparallelogramdefinedbytwoprimitivelatticevectorsformaprimitivecell.

Ø theAreaofaprimitivecell:A = |�⃗�%×�⃗�(|Ø Eachprimitivecellcontains1site.

Primitivecellin3D

Theparallelepiped definedbythethreeprimitivelatticevectorsarecalledaprimitivecell.

Ø thevolumeofaprimitivecell:V = |�⃗�%. (�⃗�(×�⃗�))|Ø eachprimitivecellcontains1site.

Aspecialcase:acuboid

Wigner–Seitzcell

Ø thevolumeofaWigner-SeitzcellisthesameasaprimitivecellØ eachWigner-Seitzcellcontains1site(sameasaprimitivecell).

Rotationalsymmetries

Rotationalsymmetries:Ifasystemgoesbacktoitself whenwerotateitalongcertainaxesbysomeangle𝜃,wesaythatthissystemhasarotationalsymmetry.Ø Forthesmallest𝜃,2𝜋/𝜃 isaninteger,whichwewillcall𝑛.Ø Wesaythatthesystemhasa𝑛-foldrotationalsymmetryalongthisaxis.

ForBravaislattices,Ø Itcanbeprovedthat𝑛 canonlytakethefollowingvalues:1, 2, 3, 4 or6.

Mirrorplanes

MirrorPlanes:

2DBravaislattices

http://en.wikipedia.org/wiki/Bravais_lattice

3DBravaislattices

http://en.wikipedia.org/wiki/Bravais_lattice

Cubicsystem

Conventionalcells

Ø Forasimplecubiclattice,aconventionalcell=aprimitivecellØ NOTtrueforbody-centeredorface-centeredcubiclattices

Howcanweseeit?Ø sc:oneconventionalcellhasonesite(sameasaprimitivecell)Ø bcc:oneconventionalcellhastwosites(twiceaslargeasaprimitivecell)Ø fcc:oneconventionalcellhasfourcites(1conventionalcell=4primitivecells)

SimplecubicLatticesites:𝑎(𝑙𝑥? + 𝑚𝑦?+n�̂�)Latticepointperconventionalcell:1 = 8× %

EVolume(conventionalcell):𝑎)Volume(primitivecell):𝑎)Numberofnearestneighbors:6Nearestneighbordistance:𝑎Numberofsecondneighbors:12Secondneighbordistance: 2� 𝑎

Packingfraction:GH≈ 0.524

Coordinatesofthesites:(𝑙, 𝑛, 𝑚)Forthesite 0,0,0 ,6nearestneighbors: ±1,0,0 , 0, ±1,0 and 0,0, ±112nestnearestneighbors: ±1,±1,0 , 0, ±1, ±1 and(±1,0, ±1)

PackingfractionPackingfraction:WetrytopackNspheres(hard,cannotdeform).

Thetotalvolumeofthespheresis𝑁4𝜋 MN

)

ThevolumethesespheresoccupyV > 𝑁4𝜋 MN

)(therearespacing)

Packingfraction=totalvolumeofthespheres/totalvolumethesespheresoccupy

𝑃𝑎𝑐𝑘𝑖𝑛𝑔𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =𝑁4𝜋 𝑅

)

3

𝑉=4𝜋 𝑅

)

3

𝑉/𝑁=

4𝜋 𝑅)

3

𝑉𝑜𝑙𝑢𝑚𝑒𝑝𝑒𝑟𝑠𝑖𝑡𝑒

=4𝜋 𝑅

)

3

𝑉𝑜𝑙𝑢𝑚𝑒𝑜𝑓𝑎𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒𝑐𝑒𝑙𝑙

Highpackingfractionmeansthespaceisusedmoreefficiently

SimplecubicLatticesites:𝑎(𝑙𝑥? + 𝑚𝑦?+n�̂�)Latticepointperconventionalcell:1 = 8× %

EVolume(conventionalcell):𝑎)Volume(primitivecell):𝑎)Numberofnearestneighbors:6Nearestneighbordistance:𝑎Numberofsecondneighbors:12Secondneighbordistance: 2� 𝑎

Packingfraction:GH≈ 0.524

𝑃𝑎𝑐𝑘𝑖𝑛𝑔𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4𝜋 𝑅

)

3

𝑉𝑜𝑙𝑢𝑚𝑒𝑜𝑓𝑎𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒𝑐𝑒𝑙𝑙

=4𝜋 𝑅

)

3

𝑎)=4𝜋3(𝑅𝑎))=

4𝜋3(𝑎/2𝑎))=

𝜋6≈ 0.524

Ø Abouthalf(0.524=52.4%)ofthespaceisreallyusedbythesphere.Ø Theotherhalf(0.476=47.6%)isempty.

Nearestdistance=2RR= Nearestdistance/2=𝑎/2

bcc

Latticesites𝑎(𝑙𝑥? + 𝑚𝑦?+n�̂�)and𝑎[ 𝑙 + %

(𝑥? + 𝑚 + %

(𝑦? + 𝑛 + %

(�̂�]

Latticepointperconventionalcell:2 = 8× %E+ 1

Volume(conventionalcell):𝑎)Volume(primitivecell):𝑎)/2Numberofnearestneighbors:8

Nearestneighbordistance: (a()(+(a

()(+(a

()(� = )�

(𝑎 ≈ 0.866𝑎

Numberofsecondneighbors:6Secondneighbordistance:𝑎

Packingfraction: )� E𝜋 ≈ 0.680

Coordinatesofthesites:(𝑙, 𝑛, 𝑚)Forthesite 0,0,0 ,8nearestneighbors: ± %

(, ± %

(, ± %

(

6 nestnearestneighbors: ±1,0,0 , 0, ±1,0 and(0,0, ±1)

bccpackingfraction

Volume(primitivecell):𝑎)/2

Nearestneighbordistance: (a()(+(a

()(+(a

()(� = )�

(𝑎 ≈ 0.866𝑎

Packingfraction: )� E𝜋 ≈ 0.680

𝑃𝑎𝑐𝑘𝑖𝑛𝑔𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4𝜋 𝑅

)

3

𝑉𝑜𝑙𝑢𝑚𝑒𝑜𝑓𝑎𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒𝑐𝑒𝑙𝑙

=4𝜋 𝑅

)

3

𝑎)/2=8𝜋3(𝑅𝑎))=

8𝜋3(3�4 𝑎𝑎

))=3� 𝜋8

≈ 0.680

Ø About68.0%ofthespaceisreallyusedbythesphere.Ø About32.0%ofthespaceisempty.

Nearestdistance=2R

R= Nearestdistance/2= )�

b𝑎

fcc

Latticesites𝑎(𝑙𝑥? + 𝑚𝑦?+n�̂�) 𝑎[ 𝑙 + %

(𝑥? + 𝑚 + %

(𝑦? + 𝑛�̂�]

𝑎[ 𝑙 + %(𝑥? + 𝑚𝑦? + 𝑛 + %

(�̂�]

𝑎[𝑙𝑥? + 𝑚 + %(𝑦? + 𝑛 + %

(�̂�]

Latticepointperconventionalcell:4 = 8× %E+ 6× %

(= 1 + 3

Volume(conventionalcell):𝑎)Volume(primitivecell):𝑎)/4Numberofnearestneighbors:12

Nearestneighbordistance: (a()(+(a

()(+(0)(� = (�

(𝑎 ≈ 0.707𝑎

Numberofsecondneighbors:6Secondneighbordistance:𝑎

Forthesite 0,0,0 ,12nearestneighbors: ± %

(, ± %

(, 0 , ± %

(, 0, ± %

(and 0, ± %

(, ± %

(6 nestnearestneighbors: ±1,0,0 , 0, ±1,0 and(0,0, ±1)

fcc packingfraction

Volume(primitivecell):𝑎)/4

Nearestneighbordistance: (a()(+(a

()(+(0)(� = (�

(𝑎 ≈ 0.707𝑎

𝑃𝑎𝑐𝑘𝑖𝑛𝑔𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4𝜋 𝑅

)

3

𝑉𝑜𝑙𝑢𝑚𝑒𝑜𝑓𝑎𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒𝑐𝑒𝑙𝑙

=4𝜋 𝑅

)

3

𝑎)/4=16𝜋3

(𝑅𝑎))=

16𝜋3

(2�4 𝑎𝑎

))=2� 𝜋6

≈ 0.740

Ø About74.0%ofthespaceisreallyusedbythesphere.Ø About26.0%ofthespaceisempty.

Nearestdistance=2R

R= Nearestdistance/2= (�

b𝑎

0.740 isthehighestpackingfractiononecaneverreach.Thisstructureiscalled“closepacking”Thereareotherclosepackingstructures(samepackingfraction)

Cubicsystem

hexagonal

Primitivecell:arightprismbasedonarhombuswithanincludedangleof120degree.

Volume(primitivecell):𝑉𝑜𝑙 = 𝐴𝑟𝑒𝑎×ℎ𝑒𝑖𝑔ℎ𝑡

= 212×

3�

2𝑎%( 𝑎) =

3�

2𝑎(𝑐

2DplanesformedbyequilateraltrianglesStacktheseplanesontopofeachother

HexagonalClose-PackedStructure(hcp)

Ø Let’sstartfrom2D(packingdisks,insteadofspheres).Q:Whatistheclose-packstructurein2D?A:Thehexagonallattice(thesedisksformequilateraltriangles)

Ø Now3D:Q:Howtoget3Dclosepacking?A:Stack2Dclosepackingstructuresontopofeachother.

HexagonalClose-PackedStructure(hcp)

hcp:ABABAB… fcc:ABCABCABC…

Atomsinsideaunitcell

Ø WechoosethreelatticevectorsØ ThreelatticevectorsformaprimitiveoraconventionalunitcellØ Lengthofthesevectorsarecalled:thelatticeconstants

Wecanmarkanyunitcellbythreeintegers:𝑙𝑚𝑛𝑡 = 𝑙�⃗�% + 𝑚�⃗�( + 𝑛�⃗�)

Coordinatesofanatom:Wecanmarkanyatominaunitcellbythreerealnumbers:𝑥𝑦𝑧.Thelocationofthisatom:𝑥�⃗�% + 𝑦�⃗�( + 𝑧�⃗�)Noticethat0 ≤ 𝑥 < 1 and0 ≤ 𝑦 < 1 and0 ≤ 𝑧 < 1

Q:Whyxcannotbe1?A:Duetotheperiodicstructure.1isjust0inthenextunitcell

SodiumChloridestructure

SodiumChloridestructureFace-centeredcubiclatticeNa+ionsformaface-centeredcubiclatticeCl- ionsarelocatedbetweeneachtwoneighboringNa+ions

Equivalently,wecansaythatCl- ionsformaface-centeredcubiclatticeNa+ionsarelocatedbetweeneachtwoneighboringNa+ions

SodiumChloridestructure

Primitivecells

CesiumChloridestructure

Cesiumchloridestructure

SimplecubiclatticeCs+ionsformacubiclatticeCl- ionsarelocatedatthecenterofeachcube

Equivalently,wecansaythatCl- ionsformacubiclatticeCs+ ionsarelocatedatthecenterofeachcube

Coordinates:Cs:000Cl:%

(%(%(

NoticethatthisisasimplecubiclatticeNOTabodycenteredcubiclatticeØ Forabcclattice,thecentersiteisthe

sameasthecornersitesØ Here,centersitesandcornersitesare

different

Diamondstructure

Carbonatomscanform4differentcrystalsGraphene(NobelPrizecarbon)

Diamond(moneycarbon/lovecarbon)

Graphite(Pencilcarbon)

Nanotubes

NotalllatticesareBravaislattices:examplesthehoneycomblattice(graphene)

DiamondlatticeisNOTaBravaisLatticeeither

Samestoryasingraphene:Wecandistinguishtwodifferenttypeofcarbonsites(markedbydifferentcolor)Weneedtocombinetwocarbonsites(oneblackandonewhite)togetherasa(primitive)unitcellIfweonlylookattheblack(orwhite)sites,wefoundtheBravaislattice:fcc

CubicZincSulfideStructure

VerysimilartoDiamondlatticeNow,blackandwhitesitesaretwodifferentatomsfcc withtwoatomsineachprimitivecell

Goodchoicesforjunctions

Diode

Matter

Matter

gas/liquid:Atoms/moleculescanmovearound

solids:Atoms/moleculescannotmove

Crystals:Atoms/moleculesformaperiodic

structure

Randomsolids:Atoms/moleculesformarandom

structure