Kap

. 5C

ryst

allo

grap

hyan

d cr

ysta

lstru

ctur

es

Con

dens

eph

ases

:Li

quid

sS

olid

mat

eria

lsA

mor

phou

sm

ater

ials

Gla

ssC

ryst

allin

em

ater

ials

1 D

im. p

hase

sC

rO3,

carb

onna

notu

bes

2 D

im. p

hase

sV

2O5,

grap

hite

3 D

im. p

hase

sTi

O2,

diam

ond

Inor

gani

cm

ater

ials

/ un

itsS

epar

ate

units

Ele

men

tsM

olec

ules

Ions

Com

plex

esA

r(g)

CO

2(g)

SO

42-(a

q)P

tCl 42-

(aq)

CH

4(g)

CO

32-(a

q)C

u(N

H3)

42+(a

q)H

2O(g

)(l)

Ag(

NH

3)2+

(aq)

XeO

F 4(l)

Fe(C

N) 6

3-(a

q)

Stru

ctur

frag

men

ts in

sol

id s

tate

SiO

44-te

traed

ralb

uild

ing

units

in s

ilicat

esC

N =

4 te

traed

ral

’CuO

2’-la

yers

in h

igh

Tc-m

ater

ials

CN

= 4

pla

nequ

adra

tic’M

nO6’-

octa

hedr

a in

oxi

des

CN

= 6

oct

ahed

ra

Uni

tcel

l

Uni

tcel

l

x

z

y

Uni

tcel

lTr

anst

altio

nal

ong

x, y

, z

x

y

z

() c

ba

V×

⋅=

Cou

ntin

gof

atom

s

Cou

ntin

g of

ato

ms

in 2

D

Ato

ms

in a

cor

ner=

¼A

tom

s on

an e

dge

= 1 / 2

Ato

ms

insi

deth

ece

ll=

1

Cou

ntin

gof

atom

s in

3D

Aco

rner

-ato

mis

sha

red

betw

een

8ce

lls⇒

1 / 8at

oms

pr. c

ell

An

edge

-ato

mis

sha

red

betw

een

mel

lom

4ce

lls⇒

1 / 4at

om p

r cel

lA

surfa

ce-a

tom

is s

hare

dbe

twee

n2

cells

⇒1 / 2

atom

pr c

ell

A a

tom

insi

deon

ece

ll⇒

1at

om p

r cel

l

Poss

ition

ing

ofat

oms

(0.5

,0.5

,0.5

)

(1.5

,0.5

,0.5

)

(2.5

,0.5

,0.5

)

Cry

stal

syst

em

Cry

stal

syst

ems

A c

olle

ctio

nof

poin

tgro

ups

that

in c

omm

ongi

veca

ract

eris

ticsy

mm

etry

oper

atio

ns

The

unit

cell

is c

hose

nso

that

the

men

tion

sym

met

ryel

emen

ts a

reea

sily

obse

rved

.B

y de

scrib

ing

the

sym

met

ryof

the

unitc

ellt

hesy

mm

etry

ofth

eco

nden

sed

mat

eria

l is

desc

ribed

fully

.

Bra

vais

Lat

tices

Hex

agon

alC

ubic

Tetra

gona

l

Trig

onal

Orth

orom

bic M

onoc

linic

Tric

linic

Sym

met

ry-

oper

atio

ns

Sym

met

yop

erat

ions

Mirr

orpl

ane

mR

otat

iona

xis

n(2

,3,4

,6)

Inve

rtion

axis

n(1

,2…

)S

entro

sym

met

ry1

Glid

emirr

orpl

ane

n, d

, a, b

, cS

crew

axis

2 1, 3

1, ..6

3

Poi

ntgr

oup

sym

met

ry

Spe

chia

lsy

mm

etry

oper

atio

ns

Mirr

orpl

ane

m

Rot

atio

naxi

sn

4-fo

ld ro

tatio

nax

is

α=

360/

n

Inve

rsio

naxi

sn

Rot

atio

n+

inve

rsio

n

α=

360o /n

Scre

wax

is, X

yTr

ansl

atio

n: y

/x

Rot

atio

n: 3

60º/x

ordi

nary

6-fo

ld ro

tatio

nax

issc

rew

rota

tion

axis

6 5

Glid

e pl

ane

012

Scr

ewax

is2 1

a

012

Glid

e m

irror

plan

e4 1

(x,y

,z)

(-y,x

,z+¼

)

(-x,-y

,z+½

)

(y,x

,z+¾

)

{}

()

tR

rt

r3

3t|4

+=

+×

=r

r

{}

{} t|1

t|44

rr

=

Poin

tgro

ups

Poin

tgro

ups

•A c

hrac

teris

ticco

llect

ion

ofsy

mm

etry

elem

ents

•The

sym

met

ryel

emen

ts h

as o

rigo

as c

omm

onpo

int

•Sym

met

ryel

emen

ts a

nd p

oint

grou

psy

mbo

l:Tw

osc

hem

es:

Sch

önfli

esH

erm

ann

Mau

guin

•Illu

stra

ted

in fo

rm o

fste

reog

raph

icpr

ojec

tion

(x,y

,z) r

(x’,y

’,z’)

r’

r’=

R. r

R is

a s

ymm

etry

oper

ator

=

zyx

aa

aa

aa

aa

a

'z'y'x

3332

31

2322

21

1312

11

Sym

met

ryop

erat

ions

:

Iden

tity:

Ia i

j= 0

a ii=

1

10

00

10

00

1[

]{

}()

−−=

−

−=

zyx

zyx

10

00

10

00

1xy

z00

12R

otat

ion:

n, C

n

+ ro

tatio

nax

is[u

vw]

n[uv

w] o

r Cn[u

vw]

23

46

[]

{}(

)

−=

−

=zxy

zyx

10

00

01

01

0xy

z00

14

42

{} (

)(

) z,y

,xz,y,x

)i(1−

−−

=

Inve

rsio

n/ c

entro

sym

met

ry: 1

i

i

Rig

htha

nd b

ecom

esle

ftha

nd

[]

{}(

)(

) z,y,x

z,y,x01

0m

−=

Mirr

orpl

ane:

m

m

m is

def

ined

from

the

norm

al o

fthe

plan

e

zH

2O

Rot

atio

nax

isO

ther

sym

met

ryel

emen

ts

C2z

2V

ertic

alm

irror

plan

e

yB

F 3

xC

3z3

Hor

izon

talm

irror

plan

eV

ertic

alm

irror

plan

e

yX

6

Rot

atio

nax

isO

ther

sym

met

ryel

emen

ts

C6z

6Tw

o-fo

ldax

isM

irror

plan

esIn

vers

ion

cent

er

yP

tCl 4-

xC

4z4

C2

axis

Mirr

orpl

anes

Inve

rsio

nce

nter

x

yC

5H5

Rot

atio

nax

isO

ther

sym

met

ryel

emen

ts

C5z

55

C2

axis

Mirr

orpl

anes

x

Left

hand

rule

for x

,y,z

Hig

hest

rota

tion

axis

|| z

Tric

linsy

stem

1an

d 1

does

not i

mpl

yan

yre

stric

tions

for a

,b,c

or α

,β,γ

Poi

ntgr

oup

with

elem

ents

:1

I1

I,i

Ifa

2-fo

ld a

xis

is a

dded

, the

nth

esy

stem

bec

omes

:

Mon

oclin

icsy

stem

with

the

sym

met

ryop

erat

ions

2 (C

2) o

r 2 =

m (σ

h)2

I, C

2m

I, σ h

Pre

senc

eof

furth

er2

or m

is a

crit

eria

for a

n or

thor

ombi

csy

stem

Wha

tabo

utm

nor

mal

to 2

?It

does

not c

hang

eth

ecr

iteria

for a

,b,c

or α

,β,γ

and

is h

ence

poss

ible

.

[]

{}

[]

{}

{} 11

00

01

00

01

10

00

10

00

1

10

00

10

00

100

12

001

m=

−−

−=

−

−

−=

2/m

sent

rosy

mm

etric

, with

the

oper

atio

ns: I

,C2,σ

h,i

Ste

reog

raph

icpr

ojec

tion

•The

crys

tali

s su

rroun

ded

by a

sph

ere

•We

are

inte

rest

edin

the

proj

ecte

dsu

rface

in a

xy-

plan

eth

roug

hth

esp

here

•The

proj

ecte

dpo

inti

s de

term

ined

by th

ein

ters

ectio

nof

a co

nnec

tion

line

from

the

poin

tofi

nter

rest

to th

epo

le o

fthe

oppo

site

side

.•T

hetw

oha

lves

oft

hesp

here

sar

eno

ted

by a

ssig

ning

+ an

d –

and

to u

sefil

led

and

open

sum

bols

Pol

e

Pol

e

xy-p

lane

- +

2

HH

Cl

Cl

HH

Cl

Cl

m

HH

Cl

Cl

m

mm

2

Gen

eral

ly4

pos.

Spe

chia

lly2

pos.

2 po

s.1

pos.

Sym

met

yop

erat

ions

Mirr

orpl

ane

mR

otat

iona

xis

n(2

,3,4

,6)

Inve

rtion

axis

n(1

,2…

)S

entro

sym

met

ry1

Glid

emirr

orpl

ane

n, d

, a, b

, cS

crew

axis

2 1, 3

1, ..6

3

Poi

ntgr

oup

sym

met

ry

Spe

chia

lsy

mm

etry

oper

atio

ns

2(C

2)

2/m

(C2h

)

mm

m(D

2h)

(fullt

sym

bol 2

/m 2

/m 2

/m)

4mm

(C4v

)

Exa

mpl

esof

ster

eogr

ams

ofpo

intg

roup

sP

oint

sS

ymm

etry

oper

atio

ns

Poin

tgro

ups:

A c

ryst

allo

grap

hic

poin

tgro

upis

a s

elec

tion

ofsy

mm

etry

elem

ents

that

can

oper

ate

ona

thre

edi

men

sion

alla

ttice

.Th

isis

onl

ym

et b

y 32

poi

ntgr

oups

.

Cry

stal

syst

emC

ryst

allo

grap

hic

poin

tgro

upTr

iklin

ic1,

-1M

onok

linin

c2,

m, 2

/mO

rthor

ombi

c22

2, m

m2,

mm

mTe

trago

nal

4, -4

, 4/m

, 422

, 4m

m, -

42m

, 4/m

mm

Trig

onal

3, -3

, 32,

3m

, -3m

Hex

agon

al6,

-6, 6

/m, 6

22, 6

mm

, -6m

2, 6

/mm

mC

ubic

23, m

3, 4

32, -

43m

, m3m

Oft

hese

are

11 c

entro

sym

met

rican

d 21

non

-cen

trosy

mm

etric

.

Poin

tgro

ups:

A c

ryst

allo

grap

hic

poin

tgro

upis

a s

elec

tion

ofsy

mm

etry

elem

ents

that

can

oper

ate

ona

thre

edi

men

sion

alla

ttice

.Th

isis

onl

ym

et b

y 32

poi

ntgr

oups

.

Cry

stal

syst

emC

ryst

allo

grap

hic

poin

tgro

up, f

ull s

ymbo

lTr

iklin

ic1,

-1M

onok

linin

c2,

m, 2

/mO

rthor

ombi

c22

2, m

m2,

2/m

2/m

2/m

Tetra

gona

l4,

-4, 4

/m, 4

22, 4

mm

, -42

m, 4

/m2/

m2/

mTr

igon

al3,

-3, 3

2, 3

m, -

32/m

Hex

agon

al6,

-6, 6

/m, 6

22, 6

mm

, -6m

2, 6

/m2/

m2/

mC

ubic

23, 2

/m-3

, 432

, -43

m, 4

/m-3

2/m

Oft

hese

are

11 c

entro

sym

met

rican

d 21

non

-cen

trosy

mm

etric

.

Rom

grup

pe

Ther

eTh

ere

are

are

230

230

spac

esp

ace --

grou

psgr

oups

!!

Spa

cegr

oup

sym

bol:

Xefg

Bra

vais

latti

ce:

P (R

)F,

IA

,B,C

Sym

met

ry fo

r cha

ract

eris

tic d

irect

ions

(dep

ende

nt o

n cr

ysta

l sys

tem

)

Sym

met

ryop

erat

ions

with

outt

rans

latio

n:In

vers

ion

-1,1

Rot

atio

nn

Mirr

orm

Rot

atio

n-in

vers

ion

n

Sym

mor

fesp

ace

grou

ps(7

3 gr

oups

)

Sym

met

ryop

erat

ions

with

trans

latio

n:S

crew

-axi

sn m

,2 1

,63,

etc.

Glid

epla

nea,

b,c,

n,d

Non

-sym

mor

fesp

ace

grou

ps(1

57 g

roup

s)

Sym

met

rypl

anes

Sym

bol

Tran

slat

ion

Mirr

orm

none

Axi

algl

ide

aa/

2b

b/2

cc/

2

Dia

gona

l glid

en

(a+b

)/2, (

b+c)

/2, (

a+c)

/2(a

+b+c

)/2 fo

r cub

ican

d te

trago

nal o

nly

Dia

mon

dgl

ide

d(a

±b)/4

, (b±

c)/4

, (a±

c)/4

(a±b

±c)/4

for c

ubic

and

tetra

gona

l onl

y

7C

ryst

alsy

stem

s

14 B

rava

is la

ttice

s

32 P

oint

grou

ps

230

Spa

cegr

oups

In o

rder

to id

entif

yth

epo

intg

roup

ofa

spac

egr

oup

one

mus

t:C

hang

esy

mbo

ls fo

r sym

met

ryel

emen

ts w

ithtra

nsla

tion

with

corre

spon

ding

sym

bol f

or s

ymm

etry

elem

ents

with

outt

rans

latio

n

Viz

.n m

-> n

for a

scr

ewax

isa,

b,c,

dor

n ->

mfo

r glid

e pl

ane

Exa

mpl

e: P6 3

/mm

c6 3

/mm

c6/

mm

mP

nma

nma

mm

m

CuO

a =

465

pm, b

= 3

41 p

m, c

= 5

11 p

m, β

= 99

,5°

Spa

cegr

oup

C2/

cC

uin

4c

O in

4e

y =

0.41

6

Cry

stal

syst

em: a

= b

= c

, α =

γ=

90°,

β=

90°

Bra

vais

-latti

ce: C

mon

oclin

ic, s

ide

cent

ered

Cor

resp

ondi

ngcr

ysta

llogr

aphi

cpo

intg

roup

: 2/m

O Cu

CuO

a =

465

pm, b

= 3

41 p

m, c

= 5

11 p

m, β

= 99

,5°

Spa

cegr

oup

C2/

cC

uin

4c

O in

4e

y =

0.41

6

Cry

stal

syst

em: a

= b

= c

, α =

γ=

90°,

β=

90°

Bra

vais

-latti

ce: C

mon

oclin

ic, s

ide

cent

ered

Cor

resp

ondi

ngcr

ysta

llogr

aphi

cpo

intg

roup

: 2/m

Ato

m c

oord

inat

es:

Cu

¼, ¼

, 0O

0, 0

.416

, ¼

Sym

met

ryop

erat

ions

ona

gene

ral p

ositi

on:

(x,y

,z)

(-x,

y,½

-z)

(-x,

-y,-z

) (x

,-y, ½

+z)

(½+x

, ½+y

,z)

(½-x

, ½+y

,½-z

)(½

-x, ½

-y,-z

) (½

+x, ½

-y, ½

+z)

CuO

Mill

erpl

an

http

://fo

lk.u

io.n

o/ol

nils

en/C

D_M

EF.

zip

Cry

stal

plan

e an

d cr

ysta

ldire

ctio

ns

A p

lane

(h k

l)

A s

etof

equi

vale

ntpl

anes

{h k

l}

A d

irect

ion

[h k

l]

A s

etof

equi

vale

ntdi

rect

ions

<h k

l>

The

equi

vale

ntpl

anes

and

dire

ctio

nsar

ea

resu

ltof

the

syst

ems

sym

met

ry

e.g.

fcc

<111

>

[111

] [1

11]

[111

] [1

11]

[111

] [1

11]

[111

] [1

11]

Mill

er in

dice

s, 2

D

PI

(100

)(2

00)

(100

)(2

00)

(200

)

(111

)

[112

]

(1,1

,2)

(½, ½

,1)

(0,0

,0)

(1,0

,1)

(2,0

,2)

[101

] [½0

½] =

[101

] = [2

02] =

n[1

01]

Par

alle

ldire

ctio

nsha

ve s

ame

inde

x

Dire

ctio

ns

FC

I

hkl;

h+k

= 2n

k+l=

2n

l+h

= 2n

all o

ddal

l eve

n

h+k

= 2n

h+k+

l= 2

nC

ondi

tions

for B

ragg

refle

ctio

ns.

(100

)(2

00)

In a

dditi

onto

this

, the

rew

illbe

effe

cts

from

: scr

ewax

isan

d gl

ide

plan

es.

P la

ttice

has

noco

nditi

ons

Bra

ggs

lov θ

λsi

n2

hkl

d=

BD

= d

sin

θD

C =

d si

n θ

⇒B

DC

= 2

d si

n θ

= nλ

Røn

tgen

tetth

et

Den

sity

•Exp

erim

enta

l(py

knom

etric

)

•Cal

cula

ted;

X-ra

yde

nsity

base

don

the

assu

mpt

ion

that

the

unit

cell

is k

now

nor

th

ata

mod

elex

ists

•Wet

ting

•Por

es in

the

mat

eria

l

ρ x-ra

y>

ρ exp

.

Aun

itcel

lra

yX

Nvo

lum

eU

nitc

ell

cell

/un

itsof

num

ber

wei

ght

Form

ula

Vm⋅

⋅=

=ρ

−

() c

ba

V×

⋅=

Den

sity

and

defe

cts

ρ obs

ρ cal

c

•V u

nitc

ell;

is d

eter

min

edex

perim

enta

lly•F

orm

ula

wei

ght

Mod

elas

sum

ptio

ns:

A/B

< 1

AB

1+y

A1-

xB

ρ(in

ters

titia

lB) >

ρ(p

erfe

ct) >

ρ(v

acan

tspa

ceA

)

Endr

ing

av

enhe

tsce

lle

NaC

lC

aC2

Cub

icz

= 4

Non

-cub

ic

NaC

lC

aC2

Cub

icz

= 4

Non

-cub

ic

Tetra

gona

lz

= 2

CaC

2(lt

)C

aC2

(ht)

fcc

(F)

Z=4

bcc

(I)Z=

2

Dis

orde

red

Cu 0

.75A

u 0.2

5H

igh

tem

p

Low

tem

pO

rder

edC

u 3A

u

Dis

orde

red

Cu 0

.50A

u 0.5

0H

igh

tem

p

Low

tem

pO

rder

edC

uAu

AB

O3

Per

ovsk

itea=

b=c,

α=β

=γ=9

0°cu

bic

c

a

b

a

b

ac-p

lane

bc-p

lane

Iden

tical

Pro

ject

ion

onth

eab

-pla

ne:

Cel

ldim

ensi

ons

are

dete

rmin

edby

:A

-O-A

a

b

A3B

3O6

Per

ovsk

itea=

b=c,

α=β

=γ=9

0°Te

trago

nal

c

a

bA

ssum

eth

atth

eab

-pla

ne is

unc

hang

edvi

z. a

=b. A

ssum

ech

ange

dc-

axis

YBa 2

Cu 3

O6

a

b

Pro

ject

ion

onth

eab

-pla

ne:

a=b

tetra

gona

la=

bor

thor

ombi

c

Kul

epak

king

Ato

ms

as s

pher

es:

-ion

s-m

etal

atom

s-m

olec

ules

Ioni

cbo

ndin

g

Van

der

Waa

ls b

ondi

ng

Met

albo

ndin

g

Cov

alen

tbon

ding

Sphe

repa

ckin

g

The

entit

ies

have

to b

e:

•Sph

eric

al•O

fsam

e ty

pe (s

ize)

•Non

-com

pres

sibl

e•N

on-r

epul

sive

/ con

tract

ive

Idea

l sph

ere

pack

ing

mod

el

Any

obse

rved

devi

atio

nfro

m th

eid

eal m

odel

will

be e

xpla

ined

by th

atth

ere

quire

men

tsar

eno

t ful

lym

et.

Clo

sest

(den

sest

) pac

king

ofsp

here

s:

74%

oft

hevo

lum

eis

fille

dby

the

sphe

res

26%

voi

ds/ v

acan

tspa

ce

The

void

s/ho

les

will

have

diff

eren

tapp

eara

nce:

•Oct

ahed

rals

hape

•Tet

rahe

dral

shap

e•(

Trig

onal

pris

mat

icho

les)

•(Tr

igon

al b

ipyr

amid

ale

hole

s)

The

void

s/ho

les

may

be fi

lled

with

atom

s•o

fthe

sam

e ty

pe a

s th

epa

ckin

gsp

here

s•o

fdiff

eren

ttyp

e

AB

……

hcp

hexa

gona

lclo

sepa

cked

AB

C…

..cc

pcu

bic

clos

epa

cked

AA

…..

prim

itive

hex

agon

alpa

ckin

g

Den

sesp

here

pack

ing A

B

AB

AA

BC

hcp

ccp

A

Tetra

eder

hol

e +

Tetra

eder

hol

e -

Oct

aede

rhol

e

Oct

ahed

ra h

oles

CN

= 6

Tetra

hedr

a ho

les

CN

= 4

Hex

agon

alpa

ckin

g(A

A..)

Trig

onal

pris

mat

icho

les

CN

= 6

Hex

agon

alcl

osep

acke

d(A

B..)

Trig

onal

bip

yram

idal

CN

= 5

A B A

Type

ofh

ole

Num

ber

Max

. rad

ius

Trig

onal

pris

mat

ic2N

0.52

8Te

trago

nal

2N0.

225

Oct

ahed

ral

N0.

414

Plas

serin

gbc

c, fc

c, c

cp

bcc,

cu

cic,

I-c

ente

red

(0,0

,0) +

(1/2

,1/2

,1/2

)C

N =

8

CsC

l-typ

est

ruct

ure,

CN

= 8

M in

(0,0

,0)

X in

(1/2

,1/2

,1/2

)N

ot I-

cent

ered

, ->

P

fcc,

Cub

icF-

cent

ered

latti

ce

Stru

ctur

e=

latti

ce+

basi

s (m

otif)

F-ce

nter

edla

ttice

with

met

alin

(0,0

,0)

NaC

l-typ

est

ruct

ure

= cu

bic

+ ba

sis

F-ce

nter

edla

ttice

:N

a in

(0,0

,0)

Cl i

n (1

/2,0

,0)

hcp

(hex

agon

al c

lose

pac

ked)

Z =

22

atom

s in

unitc

ell:

(0, 0

, 0),

(2 / 3, 1 / 3,

1 / 2)

ccp

(cub

ic c

lose

pac

ked)

Z=4

4 at

oms i

n un

itcel

l:

(0, 0

, 0) (

0, 1

/ 2, 1 / 2)

(1 / 2,

0, 1

/ 2) (1

/ 2, 1 / 2,

0)

bcc

Z=2

2 at

oms i

n un

itcel

l:

(0, 0

, 0) (

1 / 2,

1 / 2,

1 / 2)

Hul

l

(111

)

Dia

gona

l = 4

r,V

olum

eof

cube

= (2√2

r)3

Vol

ume

of4

sphe

res

= 4*

π*4/

3 r3

Den

sity

= 16

π/3

/ (2√

2)3

= 0.

7405

60°

120°

Den

sity

ofpa

ckin

g

Coo

rdin

atio

nN

ame

Den

sity

num

ber(

CN

)

6S

impl

e cu

bic

0.52

368

Sim

ple

hexa

gona

l0.

6046

8B

ody-

cent

red

cubi

c0.

6802

10B

ody-

cent

red

tetra

gona

l0.

6981

12C

lose

stpa

ckin

g0.

7405

Stru

ktur

er fr

aku

lepa

kkin

g

Stru

ctur

e(ty

pes)

der

ived

from

den

secl

osep

acki

ngof

sphe

res

Prin

cipe

: Clo

sepa

cked

laye

rsof

diffe

rent

type

s of

sphe

res

Fillin

g of

hole

s w

ithsm

alle

rsph

eres

(oct

ahed

ra-,

tetra

hedr

a-, t

rig. b

ipyr

amid

al.-

hole

s)

Com

bina

tions

of

and

A B CA

B

A

TiA

l 3W

Al 5

= W

Al 3

+ A

l 2

BFi

lling

ofh

oles

(int

erst

itial

poss

ition

s)

AB

nM

mX

X =

Pack

ing

sphe

re

Filli

ng d

egre

eA

Bn

Mm

XSp

here

pack

ing

ccp

hcp

All

octa

eder

hole

sA

BM

XN

aCl

NiA

sA

ll te

trae

derh

oles

AB

2M

2XC

aF2

½te

raed

erho

les

AB

MX

ZnS(

bl.)

ZnS(

wu.

)½

octa

eder

hole

sA

B2

M1/

2XC

dCl 2

CdI

2[C

d(O

H) 2

]1 / 3

octa

eerh

oles

AB

3M

1/3X

CrC

l 3B

iI 3, β

-ZrC

l 3

CM

ixed

sphe

res

in d

ense

pack

edla

yers

+ fil

ling

ofin

ters

titia

lhol

es

A, B

catio

nsX

anio

n

Aan

d X

ofsi

mila

rsiz

eB

is s

o sm

all t

hati

t fits

into

octa

eder

hole

s

AX 3

dens

epac

ked

laye

rs

Thos

eoc

tahe

dra

hole

s w

ith6

neig

hbou

rsof

Xty

pe is

fille

dw

ithB

AB

X 3pe

rovs

kite

type

stru

ctur

e

Per

ovsk

ite

A

Per

ovsk

ite

OA

Per

ovsk

ite

OA

Per

ovsk

ite

OA B

Per

ovsk

ite

OA B

Per

ovsk

ite

OA B

Per

ovsk

ite

OA B

AB

O3

Spa

ce g

roup

: Pm

-3m

(No.

221

)H

igh

sym

met

ry

Larg

e M

ande

lung

cons

tant

Larg

e cr

ysta

l ene

rgy

Per

ovsk

ite

OA B

Ena

bles

cos

tly e

lect

ron

conf

igur

atio

ns

Inte

rest

ing

prop

ertie

s

Prop

erty

Prop

erty

Com

poun

d ex

ampl

esC

ompo

und

exam

ples

Insu

lato

rIn

sula

tor

LaG

aOLa

GaO

33, L

aAlO

, LaA

lO33,

LaC

rO, L

aCrO

33, L

aFeO

, LaF

eO33

Hig

hH

igh --

K d

iele

ctric

K d

iele

ctric

BaTi

OBa

TiO

33, B

a, B

a 22E

uZrO

EuZ

rO5.

55.

5, C

aCu

, CaC

u 33TiTi

44OO1212

Sem

icon

duct

ivity

Sem

icon

duct

ivity

LaM

nOLa

MnO

33, P

bCrO

, PbC

rO33,

RTi

O, R

TiO

33(R

= L

a...T

m)

(R =

La.

..Tm

)H

alf

Hal

f met

allic

itym

etal

licity

LaB

aMn

LaB

aMn 22

OO5.

55.

5, Y

BaM

n, Y

BaM

n 22OO

5.5

5.5

Sr

Sr 22

FeM

oOFe

MoO

66, B

a, B

a 22Fe

MoO

FeM

oO66,

Ca

, Ca 22

FeM

oOFe

MoO

66, ,

Ca

Ca 22

FeR

eOFe

ReO

66M

etal

lic c

ondu

ctiv

ityM

etal

lic c

ondu

ctiv

ityLa

NiO

LaN

iO33

Sup

erco

nduc

tivity

Sup

erco

nduc

tivity

YBa

YBa 22

Cu

Cu 33

OO77,

HgB

a, H

gBa 22

CuO

CuO

44, L

a, L

a 1.51.5N

dN

d 0.50.5C

aBa

CaB

a 22C

uC

u 55OO

zz, ,

Bi

Bi 22S

rS

r 22C

aC

a 22C

uC

u 33OO

1010–– dd

, HgB

a, H

gBa 22

Ca

Ca 22

Cu

Cu 33

OO8+

d8+

dC

olos

sal

Col

ossa

l m

agne

tore

sist

ance

mag

neto

resi

stan

ceAA

0.3

0.3L

aLa0.

70.

7MnO

MnO

33(A

= C

a,

(A =

Ca,

Sr

Sr ,

Pr,

, Pr,

Pb

Pb ))

Mul

ti M

ulti

ferr

oics

ferr

oics

BiM

nOB

iMnO

33, B

iFeO

, BiF

eO33,,

Ferr

oela

ctic

ityFe

rroe

lact

icity

LaC

oOLa

CoO

33Fe

rrom

agne

tic

Ferro

mag

netic

S

rRuO

SrR

uO33,

LaM

nO, L

aMnO

3.15

3.15

, La

, La 11

–– xxC

aC

a xxM

nOM

nO33,

Sr

, Sr 11

–– xxLa

MnO

LaM

nO33

Anti

Anti

ferr

ofe

rro

BiM

nOB

iMnO

3,

3, L

aFeO

LaFe

O3,

3,

LaM

nOLa

MnO

33P

iezo

elec

trici

tyP

iezo

elec

trici

tyPb

ZrPb

Zr0.

470.

47TiTi

0.53

0.53

OO33

Spin

gla

ssSp

in g

lass

CaR

uOC

aRuO

33M

ulti

vale

nce

Mul

ti va

lenc

e m

ater

ials

mat

eria

lsC

aC

a 33C

oC

o 22OO

66, S

r, S

r 44FeFe

44OO1111

, YBa

Mn

, YBa

Mn 22

OO5.

55.

5

Per

ovsk

ite

Poly

eder

pakk

ing

Spac

efil

ling

ofpo

lyhe

dra

Stru

ctur

esca

nbe

des

crib

edas

con

nect

ions

ofpo

lyhe

dra

that

shar

e: Cor

ners

Edge

sFa

ces

The

poly

hedr

a ar

esi

mpl

ified

for v

isua

lcla

rity.

Type

ofp

olyh

edra

:

Tetr

ahed

raO

ctah

edra

Trig

onal

pris

mat

ic… B

asic

ally

the

sam

e ty

pes

ofpo

lyhe

dra

as m

entio

nfo

r sph

ere

pack

ing

Lim

ited

units

, Oct

ahed

ra

Isol

ated

octa

hedr

aM

X 6

Dim

erM

2X11

(Nb 2

F 11- )

Dim

erM

2X10

(Nb 2

Cl 10

)(U

2Cl 10

)

Dim

erM

2X9

(Fe 2

(CO

) 9)

(I 2O

94-)

Oct

ahed

raTe

trah

edra

Con

nect

edby

:C

orne

rsE

dges

(Fac

es)

Con

nect

edby

:C

orne

rs(E

dges

)

How

thes

eun

itsco

nnec

twill

affe

ctth

ech

emic

alco

mpo

sitio

n, a

nd v

ice

vers

a.

Poly

mer

izat

ion

ofM

X 6oc

tahe

dra

Cor

ner s

harin

g:d(

M-M

) = 2

*d(M

-X)

Edg

esh

arin

g:d(

M-M

) = √

2*d(

M-X

)

Face

shar

ing:

d(M

-M) =

1.1

6*d(

M-X

)

CC

P C

l-w

ith N

a+ in

all

Oct

ahed

ral h

oles

La

ttice

: fcc

Mot

if: C

lat (

0,0,

0); N

a at

(1/2

,0,0

) 4N

aCl i

n un

it ce

ll C

oord

inat

ion:

6:6

(oct

ahed

ral)

Cat

ion

and

anio

n si

tes

are

topo

logi

cally

iden

tical

NaC

lN

aCl 6,

ClN

a 6

CdI

2

CdC

l 2/ C

dI2

type

str

uctu

res

B B BA A A

C A BB C A

With

inW

ithin

the

the

laye

rsla

yers

: CdX

: CdX

66--oc

tahe

dra

octa

hedr

aB

etw

een

Bet

wee

nth

eth

ela

yers

laye

rs: : o

nly

only

van

der W

aals

va

n de

r Waa

ls in

tera

ctio

nsin

tera

ctio

ns

Pol

ytyp

es:

in 2

-dim

ensi

ons

-> s

ame

stru

ctur

e w

ith s

trong

bon

dsdi

ffere

nt re

petit

ion

in th

e 3r

d di

rect

ion,

can

hav

e w

eek

bond

s

CdC

l 2C

dI2

Infin

ite s

yste

ms;

oct

ahed

ra b

y co

rner

shar

ing

Num

bero

fcor

ners

sha

red

in a

giv

en o

ctah

edra

:2,

(3),

4, (5

), 6

AX

5ch

ains

: (ci

s-, t

rans

-)ci

s-VF

5

tran

s-B

iF5

SnF 4

K2N

iF4

AX

33D

net

wor

k:

AX

4la

yers

:

ReO

3Fe

F 3A

BX 3

pero

vski

te

ReO

3K

2NiF

4

Poly

mer

izat

ion

ofM

X 4te

trah

edra

Cor

ner s

harin

g:d(

M-M

) = 2

*d(M

-X)

Edg

esh

arin

g:d(

M-M

) = 1

.16*

d(M

-X)

Face

shar

ing:

d(M

-M) =

0.6

7*d(

M-X

)

2.0

only

obse

rved

for S

iO4

Si4+

-Si4+

repu

lsio

ns

Stru

ctur

esba

sed

onte

trah

edas

1A

2X7

Fini

tem

olec

ule

or p

yro

ion

Cl 2O

7, S

2O72-

, etc

.

2(A

X3)

nC

yclic

mol

ecul

e,S

3O9,

Se 4

O12

, (P

NC

l 2)n

or m

eta-

ion

(P4O

12)4

- , (S

i 3O9)

6-,

infin

itech

ain

(SO

3)n,

(PO

3)nn-

3(A

2X5)

nFi

nite

poly

hedr

al,

P4O

10do

uble

cha

in,

Al[A

lSiO

5]la

yero

rP

2O5,

Li2S

i 2O5

3D s

truct

ure

P2O

5, La

2[Be 2

O5]

4(A

X2)

nLa

yer,

HgI

2(re

d)do

uble

laye

r, or

CaS

i 2Al 2O

8(h

exag

.)3D

stru

ctur

eS

iO2

stru

ctur

es, G

eS2

Ver

tices

com

mon

to th

ree

tetra

hedr

a

3(A

X2)

nIn

finite

laye

rA

lOC

l, G

aOC

l

Ver

tices

only

shar

ed, V

ertic

esco

mm

onto

two

tetra

hedr

aN

o.sh

ared

Form

ula

Type

ofc

ompl

exE

xam

ples

verti

ces

No.

shar

edFo

rmul

aTy

pe o

fcom

plex

Exa

mpl

esed

ges

Stru

ctur

esba

sed

onte

trah

edas

1A

2X6

Fini

tedi

mer

Al 2C

l 6, F

e 2C

l 62

(AX

2)n

Infin

itech

ain

BeC

l 2, S

iS2,

Be(

CH

3)2

3(A

2X3)

nIn

finite

doub

le c

hain

Cs(

Cu 2

Cl 3)

4(A

X)

nIn

finite

laye

rLi

OH

, PbO

6(A

2X) n

3D s

truct

ures

Li2O

, F2C

a

Ver

tices

and

edge

ssh

ared

(AX

) nD

oubl

e la

yer

La2O

3, C

e 2O

2S, U

2N2S

b

(AX

) n3D

stru

ctur

eβ-

BeO

Edg

eson

lysh

ared

, Edg

esco

mm

onto

two

tetra

hedr

aC

aF2

FCa 4

-tet

rahe

daN

a 2O

NaO

4-t

etra

heda

Sink

ble

nde

ZnS 4

, SZn

4

Wur

tsitt

ZnS 4

, SZn

4

A4X

11

AX

3

A6X

17

A2X

5

Silic

ates

:

SiO

4te

trahe

das

Cor

ner (

verti

ce) s

harin

g, n

ever

edg

eor

face

(too

stro

ngS

i4+-S

i4+re

puls

ions

)

Onl

ytw

oS

iO4

tetra

hedr

a sh

are

a co

mm

onco

rner

Brid

ging

oxyg

ens

coun

t½N

on-b

ridgi

ngco

unt1

1:4

1:3.

5

1:3

Rin

gs1:

3D

oubl

e rin

gs1:

2.5

Laye

r1:

2.5

Dou

ble

laye

r…

3D1:

2

ZnS

ZnS

Stru

ctur

alpo

lym

orph

s:

Zink

blen

deSt

able

at n

orm

al P

,TW

urst

ittSt

able

at T

> 1

020

°C a

t P =

1 a

tm

Met

asta

ble

at R

T, b

uttra

nsfo

rms

by c

rush

ing

Ther

mod

inam

ics

Kin

etic

s

Zink

blen

decc

p½

tetra

edra

hol

es fi

lled

Wur

stitt

hcp

½te

traed

ra h

oles

fille

d

ZnS

Zinc

Ble

nde

(Sph

aler

ite)

CC

P S

2-w

ith Z

n2+

in h

alf T

etra

hedr

al h

oles

(onl

y T+

{or T

-} fi

lled)

La

ttice

: fcc

4ZnS

in u

nit c

ell

Mot

if: S

at (

0,0,

0); Z

n at

(1/4

,1/4

,1/4

) C

oord

inat

ion:

4:4

(tet

rahe

dral

) C

atio

n an

d an

ion

site

s ar

e to

polo

gica

lly id

entic

al

ZnS

Wur

tzite

HC

P S

2-w

ith Z

n2+

in h

alf T

etra

hedr

al h

oles

(onl

y T+

{or T

-} fi

lled)

La

ttice

: Hex

agon

al -

P

a=

b, c

ÅÃ

(8/3

)aM

otif:

2S

at (

0,0,

0) &

(2/3

,1/3

,1/2

); 2Z

n at

(2/3

,1/3

,1/8

) & (0

,0,5

/8)

2ZnS

in u

nit c

ell

Coo

rdin

atio

n: 4

:4 (t

etra

hedr

al)

Zink

-ble

nde

Wur

tsitt

ZnS

4–t

etra

hedr

aD

iam

ond

type

stru

ctur

eif

Zn =

SN

on-c

entro

sym

met

ric

ZnS

4–t

etra

hedr

a of

+ ty

pe

Poin

tsym

met

ry

S in

(0,0

,0) ½ ½

½½

¼

¼¾

¾

x

y

Che

ckfo

r:In

vers

ion

Mirr

orpl

anes

Rot

atio

nax

isR

otat

ion

inve

rsio

nax

is

Cub

iccr

ysta

lsys

tem

: X

abc

a =

alon

g<1

00>

b =

alon

g<1

11>

c =

alon

g<1

10>

a)2,

-2, 4

, -4

?b)

3, -3

?c)

2, -2

(=m

) ?

-43m

ZnS

–w

ürts

itt

Z =

2 pr

. hex

agon

alun

itce

ll

Oth

erre

late

dst

ruct

ures

:

MX

ZnO

MM

’X2

LiG

aO2

M2M

’M’’X

4Li

2BeS

iO4

LiPO

4…

Pol

ymor

phZn

S(b

lend

e), Z

nS(w

urts

itt)

Allo

trope

mod

ifica

tions

Dia

mon

d, g

raph

ite, C

60)

Pol

ytyp

esC

dCl 2,

CdI

2

…A

BC

…vs

.…

AB

…

…A

BC

A…

vs.

…A

BA

C…

vs.

…A

VA

CB

…

?

CaF

2

CC

P C

a2+

with

F-i

n al

l Tet

rahe

dral

hol

es

Latti

ce: f

ccM

otif:

Ca2

+ at

(0,0

,0);

2F-a

t (1/

4,1/

4,1/

4) &

(3/4

,3/4

,3/4

) 4C

aF2

in u

nit c

ell

Coo

rdin

atio

n: C

a2+

8 (c

ubic

) : F

-4 (t

etra

hedr

al)

In th

e re

late

d A

nti-F

luor

ite s

truct

ure

Cat

ion

and

Ani

on p

ositi

ons

are

reve

rsed

CaF

2

NiA

s N

icke

l Ars

enid

e

HC

P A

s w

ith N

i in

all O

ctah

edra

l ho

les

Latti

ce: H

exag

onal

-P

a

= b,

c Å

Ã(8

/3)a

Mot

if: 2

Ni a

t (0,

0,0)

& (0

,0,1

/2)

2As

at (2

/3,1

/3,1

/4) &

(1

/3,2

/3,3

/4)

2NiA

s in

uni

t cel

l C

oord

inat

ion:

Ni 6

(oct

ahed

ral)

: A

s 6

(trig

onal

pris

mat

ic)

CdI

2 C

adm

ium

Iodi

de

Latti

ce: H

exag

onal

-P

M

otif:

Cd

at (0

,0,0

); 2I

at (

2/3,

1/3,

1/4)

& (1

/3,2

/3,3

/4)

1CdI

2 in

uni

t cel

l C

oord

inat

ion:

Cd

-6 (O

ctah

edra

l) : I

-3

(bas

e py

ram

id)

WC

WC

6, C

W6

trig

onal

pris

mat

icN

iAs

NiA

s 6oc

tahe

dra

AsN

i 6tr

igon

al p

rism

atic

AlB

2A

lB12

hexa

gona

lpris

mat

icB

Al 6

trig

onal

pris

mat

ic

MoS

2SM

o 3tr

igon

al p

yram

idM

oS6

trig

onal

pris

mat

ic

CsC

l Ces

ium

Chl

orid

e

•Lat

tice:

Cub

ic -

P (N

.B.P

rimiti

ve!)

•Mot

if: C

lat (

0,0,

0); C

s at

(1 / 2,1 /2,1 / 2)

•1C

sCl i

n un

it ce

ll•C

oord

inat

ion:

8:8

(cub

ic)

•Ado

ptio

n by

chl

orid

es, b

rom

ides

and

iodi

des

of la

rger

cat

ions

,e.

g.C

s+, T

l+ , N

H4+

MoS

2 M

olyb

deni

te

Not

e:H

exag

onal

laye

rs o

f S a

tom

s ar

e N

OT

Clo

se-p

acke

d in

3D

La

ttice

: Hex

agon

al -

P

Mot

if: 2

Mo

at (2

/3,1

/3,3

/4) &

(1/3

,2/3

,1/4

) 4I

at (

2/3,

1/3,

1/8)

, (2/

3,1/

3,3/

8), (

1/3,

2/3,

5/8)

& (1

/3,2

/3,7

/8)

2MoS

2 in

uni

t cel

l C

oord

inat

ion:

Mo

6 (T

rigon

al P

rism

atic

) : S

3 (b

ase

pyra

mid

)

MoS

2 M

olyb

deni

te