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
() 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
(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
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 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