ep 364 solid state physics · introduction aim of solid state physics ... elementary...
TRANSCRIPT
EP
36
4 S
OL
ID S
TA
TE
E
P 3
64
SO
LID
ST
AT
E
EP
36
4 S
OL
ID S
TA
TE
E
P 3
64
SO
LID
ST
AT
E
PH
YS
ICS
PH
YS
ICS
PH
YS
ICS
PH
YS
ICS
Cou
rse
Coord
inato
r
Pro
f. D
r. B
eşir
e G
ön
ül
INTRODUCTIO
NIN
TRODUCTIO
N
�A
IM O
F S
OLID
ST
AT
E P
HY
SIC
S
�W
HA
T IS
SO
LID
ST
AT
E P
HY
SIC
S
AN
D W
HY
DO
IT
?
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
AN
D W
HY
DO
IT
?
�C
ON
TE
NT
�R
EF
ER
EN
CE
S
Aim
of Solid S
tate
Physics
�S
olid
state
phys
ics
(SS
P)
exp
lain
sth
epro
pert
ies
of
solid
mate
rials
as
found
on
eart
h.
�T
he
pro
pert
ies
are
exp
ect
ed
tofo
llow
from
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�T
he
pro
pert
ies
are
exp
ect
ed
tofo
llow
from
Sch
röd
ing
er’s
eqn.
for
aco
llect
ion
of
ato
mic
nucl
ei
and
ele
ctro
ns
inte
ract
ing
with
ele
ctro
static
forc
es.
�T
he
funda
menta
lla
ws
gove
rnin
gth
ebeha
vio
ur
of
solid
sare
know
nand
well
test
ed.
Cry
sta
lline S
olids
�W
ew
illde
al
with
crys
talli
ne
solid
s,th
at
isso
lids
with
an
ato
mic
stru
cture
base
don
are
gula
rre
peate
dpatt
ern
.
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�M
any
import
antso
lids
are
crys
talli
ne.
�M
ore
pro
gre
sshas
be
en
mad
ein
unders
tan
din
gth
ebeh
avi
our
of
crys
talli
ne
solid
sth
an
that
of
non-
crys
talli
ne
mate
rials
since
the
calc
ula
tion
are
easi
er
incr
ysta
lline
mate
rials
.
What is solid sta
te p
hysics?
�S
olid
state
phys
ics,
als
okn
ow
nas
conden
sed
matter
phys
ics,
isth
est
udy
of
the
beha
vio
ur
of
ato
ms
when
they
are
pla
ced
incl
ose
pro
xim
ityto
one
anoth
er.
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
they
are
pla
ced
incl
ose
pro
xim
ityto
one
anoth
er.
�In
fact
,co
nde
nse
dm
att
er
phys
ics
isa
much
better
nam
e,
since
many
of
the
conce
pts
rele
vant
toso
lids
are
als
oapplie
dto
liquid
s,fo
rexa
mple
.
What is the p
oint?
�U
nders
tan
din
gth
ee
lect
rica
lpro
pert
ies
of
solid
sis
rightatth
eheart
ofm
odern
soci
ety
and
tech
nolo
gy.
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�T
he
entir
eco
mpute
rand
ele
ctro
nic
sin
dust
ryre
lies
on
tunin
gof
asp
eci
al
class
of
mate
ria
l,th
ese
mic
ond
uct
or,
whic
hlie
srig
ht
at
the
meta
l-in
sula
tor
boun
dary
.S
olid
state
phys
ics
pro
vide
aback
gro
un
dto
unders
tand
what
goes
on
inse
mic
onduct
ors
.
Solid sta
te p
hysics(S
SP)
is the a
pplied p
hysics
�N
ew
tech
nolo
gy
for
the
futu
rew
illin
evi
tably
invo
lve
deve
lop
ing
and
un
ders
tand
ing
ne
wcl
ass
es
of
mate
rials
.B
yth
eend
of
this
cours
ew
ew
illse
ew
hy
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
this
isa
non-t
rivi
alt
ask
.
�S
o,
SS
Pis
the
ap
plie
dphys
ics
ass
oci
ate
dw
ithte
chnolo
gy
rath
er
than
inte
rest
ing
fundam
enta
ls.
Electrical re
sistivity o
f th
ree
sta
tes o
f solid m
atter
�H
ow
can
this
be
?A
fte
ra
ll,th
ey
ea
chco
nta
ina
syst
em
of
ato
ms
an
de
spe
cia
llye
lect
ron
so
fsi
mila
rd
en
sity
.A
nd
the
plo
tth
icke
ns:
gra
ph
iteis
am
eta
l,d
iam
on
dis
an
insu
lato
ra
nd
bu
ckm
inst
er-
fulle
ren
eis
asu
pe
rco
nd
uct
or.
Th
ey
are
all
just
ca
rbo
n!
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�A
mong o
ur
aim
s -
unders
tand w
hy
one is
am
eta
land
one a
n in
sula
tor,
and t
hen t
he p
hys
ical o
rigin
of th
em
ark
ed featu
res.
�A
lso t
hin
k about
therm
al p
ropert
ies
etc
. etc
.
CONTENT
�C
hap
ter
1.C
ryst
al S
tru
ctu
re
�C
hap
ter
2.X
-ra
y C
ryst
allo
gra
ph
y
�C
hap
ter
3.In
tera
tom
ic F
orc
es
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�C
hap
ter
3.In
tera
tom
ic F
orc
es
�C
hap
ter
4.C
ryst
al D
yna
mic
s
�C
hap
ter
5.F
ree
Ele
ctro
nT
he
ory
CHAPTER 1
.CRYSTAL S
TRUCTURE
�E
lem
enta
ry C
ryst
allo
gra
phy
�S
olid
ma
teri
als
(cr
ysta
llin
e,
po
lycr
ysta
llin
e,
am
orp
ho
us)
�C
ryst
allo
gra
ph
y�
Cry
sta
l La
ttic
e
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�C
ryst
al L
att
ice
�C
ryst
al S
tru
ctu
re�
Typ
es
of
La
ttic
es
�U
nit
Ce
ll�
Dir
ect
ion
s-P
lan
es-
Mill
er
Ind
ice
s in
Cu
bic
Un
it C
ell
�T
ypic
al C
ryst
al S
truct
ure
s
(3D
–1
4 B
rava
is L
att
ice
s a
nd
th
e S
eve
n C
ryst
al S
yste
m)
�E
lem
ents
of S
ymm
etr
y
CHAPTER 2
. X-R
AY C
RYSTALLOGRAPHY
�X
-ray
�D
iffra
ctio
n
�B
ragg e
quatio
n
X-r
ay
diff
ract
ion m
eth
ods
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�X
-ray
diff
ract
ion m
eth
ods
�Laue M
eth
od
�R
ota
ting C
ryst
al M
eth
od
�P
ow
der
Meth
od
�N
eutr
on &
ele
ctro
n d
iffra
ctio
n
CHAPTER 3
. IN
TERATOMIC
FORCES
�E
nerg
ies
of In
tera
ctio
ns
Betw
een A
tom
s
�Io
nic
bondin
g�
Na
Cl
�C
ova
lent
bondin
g
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�C
ova
lent
bondin
g�
Co
mp
ari
sio
n o
f io
nic
an
d c
ova
len
t b
on
din
g
�M
eta
llic
bondin
g
�V
an d
er
waals
bondin
g
�H
ydro
gen b
ondin
g
CHAPTER 4
. CRYSTAL D
YNAMIC
S
�S
ou
nd
wa
ve�
La
ttic
e v
ibra
tion
s o
f 1
D c
ysta
l�
Ch
ain
of
ide
ntic
al a
tom
s�
Ch
ain
of
two
typ
es
of
ato
ms
�P
ho
no
ns
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�P
ho
no
ns
�H
ea
t C
ap
aci
ty�
De
nsi
ty o
f S
tate
s�
Th
erm
al C
on
du
ctio
n�
En
erg
y o
f h
arm
on
ic o
scill
ato
r�
Th
erm
al e
ne
rgy
& L
att
ice
Vib
ratio
ns
�H
ea
t C
ap
aci
ty o
f L
att
ice
vib
ratio
ns
CHAPTER 5
. FREE E
LECTRON T
HEORY
�F
ree e
lect
ron m
odel
�H
eat ca
paci
ty o
f fr
ee e
lect
ron g
as
�F
erm
i funct
ion, F
erm
i energ
y
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
�F
erm
i funct
ion, F
erm
i energ
y
�F
erm
i dirac
dis
trib
utio
n f
unct
ion
�T
ransp
ort
pro
pert
ies
of co
nduct
ion e
lect
rons
REFERENCES
�C
ore
book:
Solid
state
phys
ics,
J.R
.Ho
ok
an
dH
.E.H
all,
Seco
nd
editi
on
(Wile
y)�
Oth
er
books
ata
sim
ilar
leve
l:S
olid
state
phys
ics,
Kitt
el(
Wile
y)
EP
364 S
OLID
ST
AT
E P
HY
SIC
S
IN
TR
OD
UC
TIO
N
Solid
state
phys
ics,
Kitt
el(
Wile
y)S
olid
state
phys
ics,
Bla
kem
ore
(Cam
bridge)
Fund
am
enta
lsof
solid
state
phys
ics,
Chri
stm
an
(Wile
y)
�M
ore
adva
nce
d:
So
lidst
ate
phys
ics,
Ash
croft
and
Merm
in
CH
AP
TE
R 1
CR
YS
TA
L S
TR
UC
TU
RE
Ele
menta
ry C
ryst
allo
gra
phy
Typ
ical C
ryst
al S
truct
ure
sE
lem
ents
Of S
ymm
etr
y
Obje
ctives
By
the
en
d o
f th
is s
ect
ion
yo
u s
ho
uld
:
�b
e a
ble
to
ide
ntif
y a
un
it ce
llin
a s
ymm
etr
ica
l
Cry
stal S
truct
ure
18
�b
e a
ble
to
ide
ntif
y a
un
it ce
llin
a s
ymm
etr
ica
l p
att
ern
�kn
ow
th
at
the
re a
re 7
po
ssib
le u
nit
cell
sha
pe
s
�b
e a
ble
to
de
fine
cu
bic
, te
tra
go
na
l, o
rth
orh
om
bic
an
d h
exa
go
na
l un
it ce
ll sh
ap
es
Ma
tt
er
ma
tt
er
ma
tt
er
Cry
stal S
truct
ure
19
GA
SE
SL
IQU
IDS
L
IQU
IDS
L
IQU
IDS
L
IQU
IDS
an
da
nd
an
da
nd
LIQ
UID
L
IQU
ID
LIQ
UID
L
IQU
ID
CR
YS
TA
LS
CR
YS
TA
LS
CR
YS
TA
LS
CR
YS
TA
LS
SO
LID
SS
OL
IDS
SO
LID
SS
OL
IDS
Gases
�G
ase
s h
ave
ato
ms
or
mo
lecu
les
tha
t d
o n
ot
bo
nd
to
on
e
an
oth
er
in a
ra
ng
e o
f p
ress
ure
, te
mp
era
ture
an
d v
olu
me
.
�T
he
se m
ole
cule
s h
ave
n’t
an
y p
art
icu
lar
ord
er
Cry
stal S
truct
ure
20
�T
he
se m
ole
cule
s h
ave
n’t
an
y p
art
icu
lar
ord
er
an
d m
ove
fre
ely
with
in a
co
nta
ine
r.
Liquids a
nd Liquid C
rysta
ls
�S
imila
rto
gase
s,liq
uid
shave
n’t
any
ato
mic
/mole
cula
rord
er
and
they
ass
um
eth
esh
ape
ofth
eco
nta
iners
.
�A
pply
ing
low
leve
lsof
therm
al
en
erg
yca
neasi
lybre
ak
the
exi
stin
gw
eak
bonds.
Cry
stal S
truct
ure
21
bre
ak
the
exi
stin
gw
eak
bonds.
Liq
uid
crys
tals
have
mob
ilem
ole
cule
s,but
aty
pe
of
lon
gra
nge
ord
er
can
exi
st;
the
mole
cule
shave
aperm
ane
nt
dip
ole
.A
pply
ing
an
ele
ctric
field
rota
tes
the
dip
ole
and
est
ablis
he
sord
er
with
inth
eco
llect
ion
of
mole
cule
s.
Cry
tals
�S
olid
sco
nsi
sto
fa
tom
so
rm
ole
cule
sexecuting
thermal
motion
ab
ou
ta
ne
qu
ilib
riu
mp
osi
tionfixedatapoint
insp
ace
.
�S
olid
sca
nta
keth
efo
rmo
fcr
ysta
llin
e,
Cry
stal S
truct
ure
22
�S
olid
sca
nta
keth
efo
rmo
fcr
ysta
llin
e,
po
lycr
sta
llin
e,o
ra
mo
rph
ou
sm
ate
ria
ls.
�S
olid
s(a
ta
giv
en
tem
pe
ratu
re,
pre
ssu
re,
an
dvo
lum
e)
ha
vest
ron
ge
rb
on
ds
be
twe
en
mo
lecu
les
an
da
tom
sth
an
liqu
ids.
�S
olid
sre
qu
ire
mo
ree
ne
rgy
tobreakthe
bonds.
SO
LID
MA
TE
RIA
LS
CR
YS
TA
LL
INE
CR
YS
TA
LL
INE
CR
YS
TA
LL
INE
CR
YS
TA
LL
INE
PO
LY
CR
YS
TA
LL
INE
PO
LY
CR
YS
TA
LL
INE
PO
LY
CR
YS
TA
LL
INE
PO
LY
CR
YS
TA
LL
INE
AM
OR
PH
OU
SA
MO
RP
HO
US
AM
OR
PH
OU
SA
MO
RP
HO
US
(No
n(N
on
(No
n(N
on
- ---cr
ys
ta
ll
ine
)c
ry
st
al
lin
e)
cr
ys
ta
ll
ine
)c
ry
st
al
lin
e)
EL
EM
EN
TA
RY
CR
YS
TA
LL
OG
RA
PH
YE
LE
ME
NT
AR
Y C
RY
ST
AL
LO
GR
AP
HY
Cry
stal S
truct
ure
23
Sin
gle
Cry
sta
l
Types o
f Solids
�S
ingle
crs
ytal,
poly
crys
talli
ne,
and a
morp
hous,
are
the
thre
e g
enera
l typ
es
of so
lids.
�E
ach
typ
e is
chara
cterize
d b
y th
e s
ize o
f ord
ere
d
regio
n w
ithin
the m
ate
rial.
Cry
stal S
truct
ure
24
regio
n w
ithin
the m
ate
rial.
�A
n o
rdere
d r
egio
n is
a s
patia
l volu
me in
whic
h a
tom
s or
mole
cule
s have
a r
egula
r geom
etr
ic a
rrangem
ent
or
periodic
ity.
Cry
sta
lline S
olid
�C
ryst
alli
ne
So
lidis
the
solid
form
of
asu
bst
ance
inw
hic
hth
eatomsormolecules
are
arr
ang
ed
ina
defin
ite,re
peatin
gpatt
ern
inth
ree
dim
ensi
on.
�S
ing
lecr
ysta
ls,
idea
llyhave
ahig
hdegre
eof
ord
er,
or
regu
lar
ge
om
etr
icpe
riod
icity
,th
rough
out
theentire
Cry
stal S
truct
ure
25
regu
lar
ge
om
etr
icpe
riod
icity
,th
rough
out
theentire
volumeofthematerial.
Cry
sta
lline S
olid
�S
ing
lecr
ysta
lhas
an
ato
mic
stru
cture
that
repeats
peri
od
ically
acr
oss
itsw
hole
volu
me.
Eve
nat
infin
itele
ngth
scale
s,each
ato
mis
rela
ted
toeve
ryoth
er
equ
iva
lent
ato
min
the
stru
cture
by
transl
atio
nals
ymm
etr
y
Cry
stal S
truct
ure
26
Sin
gle
Cry
sta
l
Sin
gle
Pyri
te
Cry
sta
l
Am
orp
ho
us
So
lid
Polycry
sta
lline S
olid
�P
oly
crys
tal
isa
mate
ria
lm
ad
eup
of
an
agg
regate
ofmanysmallsinglecrystals
(als
oca
lled
crys
talli
tes
or
gra
ins)
.
�P
oly
crys
talli
ne
mate
rialh
ave
ahig
hd
egre
eof
ord
er
ove
rm
an
yato
mic
or
mole
cula
rdim
ensi
ons.
�T
hese
orderedregions,
or
single
cryt
al
regio
ns,
vary
insi
zeand
orienta
tion
wrt
one
anoth
er.
Cry
stal S
truct
ure
27
Po
lycry
sta
llin
e
Pyri
te
form
(Gra
in)
one
anoth
er.
�T
hese
regio
ns
are
calle
dasgrains(domain)
and
are
sep
ara
ted
from
on
eanoth
er
bygrainboundaries.Theatomicordercanvaryfromonedomaintothenext.
�T
he
gra
ins
are
usu
ally100nm-100micronsindiameter.
Poly
crys
tals
with
gra
ins
thatare
<10
nm
india
mete
rare
calle
dnanocr
ysta
lline
Am
orp
hous S
olid
�A
mo
rph
ous
(No
n-c
ryst
alli
ne
)S
olid
isco
mp
ose
do
fra
nd
om
lyo
rie
nta
ted
ato
ms
,io
ns,
or
mo
lecu
les
tha
tdo
not
form
de
fined
pa
tte
rns
or
lattic
est
ruct
ure
s.�
Am
orp
ho
us
ma
teri
als
have
ord
er
only
with
ina
few
ato
mic
or
mo
lecu
lar
dim
en
sio
ns.
�A
mo
rph
ous
ma
teri
als
do
no
th
ave
an
ylo
ng
-ran
ge
ord
er,
bu
tth
ey
Cry
stal S
truct
ure
28
�A
mo
rph
ous
ma
teri
als
do
no
th
ave
an
ylo
ng
-ran
ge
ord
er,
bu
tth
ey
ha
veva
ryin
gd
eg
ree
so
fsh
ort
-ra
nge
ord
er.
�E
xam
ple
sto
am
orp
hou
sm
ate
ria
lsin
clu
de
am
orp
ho
us
silic
on,
pla
stic
s,a
nd
gla
sse
s.�
Am
orp
ho
us
silic
on
can
be
use
din
sola
rce
llsa
nd
thin
film
tra
nsi
sto
rs.
Departure
Fro
m P
erfect Cry
sta
l
�S
tric
tlysp
ea
kin
g,
on
eca
nn
ot
pre
pa
rea
pe
rfe
ctcr
ysta
l.F
or
exa
mp
le,
eve
nth
esu
rfa
ceo
fa
crys
tal
isa
kin
do
fim
pe
rfe
ctio
nb
eca
use
the
pe
rio
dic
ityis
inte
rru
pte
dth
ere
.
�A
no
the
re
xam
ple
con
cern
sth
eth
erm
alv
ibra
tion
so
fth
ea
tom
s
Cry
stal S
truct
ure
29
�A
no
the
re
xam
ple
con
cern
sth
eth
erm
alv
ibra
tion
so
fth
ea
tom
sa
rou
nd
the
ire
qu
ilib
riu
mp
osi
tion
sfo
ra
ny
tem
pe
ratu
reT
>0
°K.
�A
sa
thir
de
xam
ple
,a
ctu
al
crys
tal
alw
ays
con
tain
sso
me
fore
ign
ato
ms,
i.e.,
imp
uri
ties.
Th
ese
imp
uri
ties
spo
ilsth
ep
erf
ect
crys
tals
tru
ctu
re.
CRYSTALLOGRAPHY
Wh
at
is c
rysta
llo
gra
ph
y?
The b
ranch
of sc
ience
that deals
with
the g
eom
etr
ic
desc
riptio
nof cr
ysta
ls a
nd t
heir in
tern
al a
rrangem
ent.
Cry
stal S
truct
ure
30
desc
riptio
nof cr
ysta
ls a
nd t
heir in
tern
al a
rrangem
ent.
Cry
stallo
gra
phy
is e
ssentia
l for
solid
sta
te p
hys
ics
�S
ymm
etr
yof
acr
ysta
lca
nhave
apro
foun
din
fluence
on
itspro
pert
ies.
Cry
sta
llogra
phy Cry
stal S
truct
ure
31
on
itspro
pert
ies.
�A
ny
crys
tal
stru
cture
shou
ldbe
speci
fied
com
ple
tely
,co
nci
sely
and
unam
big
uousl
y.
�S
truct
ure
ssh
ould
be
class
ifie
din
todiff
ere
nt
types
acc
ord
ing
toth
esy
mm
etr
ies
they
poss
ess
.
�A
basi
ckn
ow
ledg
eof
crys
tallo
gra
phy
isess
entia
lfo
rso
lidst
ate
phys
icis
ts;
�to
spe
cify
an
ycr
ysta
lstr
uct
ure
an
d�
tocl
ass
ifyth
eso
lids
into
diff
ere
nt
typ
es
acc
ord
ing
toth
esy
mm
etr
ies
the
yp
oss
ess
.
EL
EM
EN
TA
RY
E
LE
ME
NT
AR
Y
EL
EM
EN
TA
RY
E
LE
ME
NT
AR
Y
CR
YS
TA
LL
OG
RA
PH
YC
RY
ST
AL
LO
GR
AP
HY
CR
YS
TA
LL
OG
RA
PH
YC
RY
ST
AL
LO
GR
AP
HY
Cry
stal S
truct
ure
32
the
sym
me
trie
sth
ey
po
sse
ss.
�S
ymm
etr
yof
acr
ysta
lca
nhave
apro
found
influ
ence
on
itspro
pert
ies.
�W
ew
illco
nce
rnin
this
cours
ew
ithso
lids
with
sim
ple
stru
cture
s.
CRYSTAL LATTIC
E
Wh
at
iscry
sta
l(s
pace)
latt
ice?
Incr
ysta
llogra
ph
y,only
the
geom
etr
ical
pro
pert
ies
of
the
crys
tala
reof
inte
rest
,th
ere
fore
one
repla
ces
each
ato
mby
age
om
etr
ical
po
int
loca
ted
at
the
equ
ilib
rium
posi
tion
of
thatato
m.
Cry
stal S
truct
ure
33
thatato
m.
Pla
tinum
Pla
tinum
surf
ace
Cry
stal l
att
ice a
nd
stru
cture
of
Pla
tinum
(scan
nin
gtu
nn
eli
ng
mic
rosco
pe)
�A
nin
finite
arr
ay
of
poin
tsin
space
,
�E
ach
poin
tha
s
Cry
sta
l Lattice
α
a
bC
BE
D
OA
y
x
Cry
stal S
truct
ure
34
�E
ach
poin
tha
sid
entic
al
surr
oun
din
gs
toall
oth
ers
.
�A
rrays
are
arr
ang
ed
exa
ctly
ina
period
icm
anner.
aO
Ax
Cry
sta
l Structu
re
�C
ryst
al
stru
cture
can
be
obta
ined
by
att
ach
ing
ato
ms,
gro
ups
of
ato
ms
or
mole
cule
sw
hic
hare
calle
db
asi
s(m
otif
)to
the
latt
ice
sides
ofth
ela
ttic
epoin
t.
Cry
sta
lS
tru
ctu
re=
Cry
sta
lL
att
ice
+B
asis
Cry
stal S
truct
ure
35
A two-dimensional Bravais lattice with
different choices for the basis
EC
BD
y
αb
CB
ED
y
Basis
��A
gro
up
of
ato
ms
wh
ich
de
sc
rib
e c
rys
tal
str
uc
ture
A g
rou
p o
f a
tom
s w
hic
h d
es
cri
be
cry
sta
l s
tru
ctu
re
Cry
stal S
truct
ure
37
E HO
A
Fb
Gx
a
α
a
b
OA
x
b)
Cry
sta
lla
ttic
eo
bta
ine
db
y
ide
nti
fyin
ga
llth
ea
tom
sin
(a)
a)
Sit
uati
on
of
ato
ms
at
the
co
rne
rso
fre
gu
lar
he
xa
go
ns
Cry
sta
l structu
re
�D
on't
mix
up
ato
ms
with
latt
ice
poin
ts
�Latt
ice
poin
tsare
infin
itesi
ma
lpo
ints
in Cry
stal S
truct
ure
38
infin
itesi
ma
lpo
ints
insp
ace
�Latt
ice
poin
tsdo
not
nece
ssari
lylie
at
the
centr
eofato
ms
Cry
sta
lS
tru
ctu
re=
Cry
sta
lL
att
ice
+B
asis
Cry
sta
l L
att
ice
Bra
va
is L
att
ice
(B
L)
No
n-B
rava
is L
att
ice
(n
on
-BL
)
�A
llato
ms
are
of
the
sam
eki
nd
�A
llla
ttic
epoin
tsare
equiv
ale
nt
�A
tom
sca
nbe
of
diff
ere
nt
kind
�S
om
ela
ttic
epoin
tsare
not
equiv
ale
nt
�A
com
bin
atio
noftw
oor
more
BL
Cry
stal S
truct
ure
39
�A
com
bin
atio
noftw
oor
more
BL
Types O
f Cry
sta
l Lattices
1)
Bra
vais
latt
ice
isan
infin
itearr
ay
of
dis
crete
poin
tsw
ithan
arr
an
ge
ment
and
orie
nta
tion
that
appea
rsexa
ctly
the
sam
e,
from
wh
ich
eve
rof
the
po
ints
the
arr
ay
isvi
ew
ed.
Latt
ice
isin
variant
under
atr
ansl
atio
n.
Cry
stal S
truct
ure
40
Latt
ice
isin
variant
under
atr
ansl
atio
n.
Nb
fil
m
Types O
f Cry
sta
l Lattices
�T
he
red
sid
eh
as
an
eig
hb
ou
rto
its
2)
2)
Non
Non--B
rava
is L
att
ice
Bra
vais
Latt
ice
Not only
the a
rrangem
ent
but als
o t
he o
rienta
tion
must
appear
exa
ctly
the s
am
e fro
m e
very
poin
t in
a b
rava
is la
ttic
e.
Cry
stal S
truct
ure
41
�T
he
red
sid
eh
as
an
eig
hb
ou
rto
itsim
me
dia
tele
ft,
the
blu
eo
ne
inst
ea
dh
as
an
eig
hb
ou
rto
itsri
gh
t.�
Re
d(a
nd
blu
e)
sid
es
are
eq
uiv
ale
nt
an
dh
ave
the
sam
ea
pp
ea
ran
ce�
Re
da
nd
blu
esi
de
sa
ren
ot
eq
uiv
ale
nt.
Sa
me
ap
pe
ara
nce
can
be
ob
tain
ed
rota
ting
blu
esi
de
18
0º.
Ho
neyco
mb
Tra
nslational Lattice V
ecto
rs –
2D
Asp
ace
latt
ice
isa
set
of
poin
tssu
chth
at
atr
ansl
atio
nfr
om
any
po
int
inth
ela
ttic
eby
ave
ctor;
Rn
=n
1a
+n
2b
P
Cry
stal S
truct
ure
42
Rn
=n
1a
+n
2b
loca
tes
an
exa
ctlyequivalent
po
int,
i.e.
apoin
tw
ithth
esa
me
envi
ron
ment
asP
.T
his
istr
ansl
atio
na
lsy
mm
etr
y.T
he
vect
orsa,b
are
kno
wn
as
latt
ice
vect
ors
and
(n1,
n2)
isa
pair
of
inte
gers
whose
valu
es
depe
nd
on
the
latt
ice
poin
t.P
oin
t D
(n1, n
2)
= (
0,2
)
Po
int
F (
n1, n
2)
= (
0,-
1)
�T
he
two
vect
ors
aand
bfo
rma
set
of
lattic
eve
ctors
for
the
latt
ice.
Lattice V
ecto
rs –
2D
Cry
stal S
truct
ure
43
�The
choice
oflattice
vectors
isnotunique.
Thus
one
could
eq
ua
llyw
ell
take
the
vect
ors
aand
b’a
sa
latt
ice
vect
ors
.
Lattice V
ecto
rs –
3D
An
idea
lth
ree
dim
ensi
on
al
crys
tal
isdesc
rib
ed
by
3fu
ndam
enta
ltr
ansl
atio
nve
ctors
a,
band
c.If
there
isa
latt
ice
poin
tre
pre
sente
dby
the
posi
tion
vect
or
r,th
ere
isth
en
als
oa
latt
ice
poin
tre
pre
sen
ted
by
the
posi
tion
vect
or
whereu,v
andw
are
arb
itrary
inte
gers
.
Cry
stal S
truct
ure
44
r’ =
r +
ua
+ v
b +
wc
(1)
Five B
ravais Lattices in 2
D
Cry
stal S
truct
ure
45
Unit C
ell in 2
D
�T
he
smalle
stco
mponent
of
the
crys
tal
(gro
up
of
ato
ms,
ions
or
mole
cule
s),
whic
hw
he
nst
ack
ed
togeth
er
with
pure
transl
atio
nal
repetit
ion
repro
duce
sth
ew
ho
lecr
ysta
l.
Cry
stal S
truct
ure
46
S a
b
SS
S
SS
SSS
SS
SS
S
S
Unit C
ell in 2
D
�T
he
smalle
stco
mponent
of
the
crys
tal
(gro
up
of
ato
ms,
ions
or
mole
cule
s),
whic
hw
he
nst
ack
ed
togeth
er
with
pure
transl
atio
nal
repetit
ion
repro
duce
sth
ew
ho
lecr
ysta
l.
Cry
stal S
truct
ure
47
S
S
Th
e c
ho
ice o
f
un
it c
ell
is n
ot
un
iqu
e.
a
Sb
S
2D U
nit C
ell e
xam
ple -(N
aCl)
Cry
stal S
truct
ure
48
We d
efin
e l
att
ice p
oin
ts; th
ese
are
poin
ts w
ith identical
environments
Choic
e o
f origin
is a
rbitr
ary
-la
ttic
e p
oin
ts n
eed n
ot be
ato
ms
-b
ut
unit
cell
size
should
alw
ays
be t
he s
am
e.
Cry
stal S
truct
ure
49
This
is a
lso a
unit
cell
-it
doesn
’t m
att
er
if yo
u s
tart
fro
m N
a o
r C
l
Cry
stal S
truct
ure
50
-or
if yo
u d
on’t
start
fro
m a
n a
tom
Cry
stal S
truct
ure
51
This
is N
OT
a u
nit
cell
eve
n t
hough t
hey
are
all
the
sam
e -
em
pty
space
is n
ot
allo
wed
!
Cry
stal S
truct
ure
52
In 2
D,
this
IS
a u
nit
cell
In 3
D,
it is
NO
T
Cry
stal S
truct
ure
53
Why can't the b
lue triangle
be a
unit cell?
Cry
stal S
truct
ure
54
Unit
Cell
in 3
D
Cry
stal S
truct
ure
55
Unit
Cell
in 3
D
Cry
stal S
truct
ure
56
Thre
e c
om
mon U
nit
Cell
in 3
D
Cry
stal S
truct
ure
57
UN
IT C
EL
L
Pri
mit
ive
Co
nve
nti
on
al &
No
n-p
rim
itiv
e
�S
ingle
latt
ice
poin
tper
cell
�S
malle
stare
ain
2D
,or
�S
malle
stvo
lum
ein
3D
�M
ore
than
one
latt
ice
poin
tper
cell
�In
tegra
lmulti
ble
sof
the
are
aof
prim
itive
cell
Cry
stal S
truct
ure
58
prim
itive
cell
Bo
dy c
en
tere
d c
ub
ic(b
cc
)B
od
y c
en
tere
d c
ub
ic(b
cc
)
Co
nve
nti
on
al
Co
nve
nti
on
al ≠ P
rim
itiv
e c
ell
≠ P
rim
itiv
e c
ell
Sim
ple
cu
bic
(sc
)S
imp
le c
ub
ic(s
c)
Co
nve
nti
on
al
Co
nve
nti
on
al=
Pri
mit
ive
ce
ll
= P
rim
itiv
e c
ell
The C
onventional Unit C
ell
�A
un
itce
llju
stfil
lssp
ace
whe
ntr
ansl
ate
dth
rough
asu
bse
tof
Bra
vais
latt
ice
vect
ors
.
�T
he
conve
ntio
na
lunit
cell
is
Cry
stal S
truct
ure
59
�T
he
conve
ntio
na
lunit
cell
isch
ose
nto
be
larg
er
than
the
prim
itive
cell,
but
with
the
full
sym
metr
yofth
eB
rava
isla
ttic
e.
�T
he
size
of
the
conve
ntio
na
lcell
isgiv
en
by
the
latt
ice
const
ant
a.
Prim
itive a
nd conventional cellsof FCC
Cry
stal S
truct
ure
60
1
1ˆ
ˆˆ
()
2a
xyz
=+
−r
Prim
itive a
nd conventional cellsof BCC
Primitive Translation Vectors:
2 3
1ˆ
ˆˆ
()
2 1ˆ
ˆˆ
()
2
axyz
axyz
=−
++
=−
+
r r
Pri
miti
ve a
nd
co
nve
ntio
na
l ce
lls
Bo
dy
cen
tere
d c
ub
ic (
bcc
):
con
ven
tion
al ≠
pri
miti
ve c
ell
a
bc
Fra
ctio
na
l co
ord
ina
tes
of la
ttic
e p
oin
ts in
co
nve
ntio
na
l ce
ll:
Cry
stal S
truct
ure
62
a
bc
Sim
ple
cu
bic
(sc
):
pri
miti
ve c
ell=
con
ven
tion
al c
ell
Fra
ctio
na
lco
ord
ina
tes
ofla
ttic
ep
oin
ts:
00
0, 1
00
, 0
10
, 00
1, 11
0,1
01
, 011
, 111
aco
nve
ntio
na
l ce
ll:
00
0,1
00
, 01
0, 0
01
, 11
0,1
01
, 011
, 111
, ½ ½
½
Bo
dy
cen
tere
d c
ub
ic (
bcc
):
pri
miti
ve(r
om
bo
he
dro
n)
≠co
nve
ntio
na
l ce
ll
a
b
c
Fra
ctio
na
l co
ord
ina
tes:
Pri
miti
ve a
nd
co
nve
ntio
na
l ce
lls
Cry
stal S
truct
ure
63
aF
ract
ion
al c
oo
rdin
ate
s:
000, 100,
101,
110,
110,1
01,
011
, 211
, 200
Fa
ce c
en
tere
d c
ub
ic (
fcc)
: co
nve
ntio
na
l≠ p
rim
itive
ce
ll
a
bc
Fra
ctio
na
l co
ord
ina
tes:
000,1
00,
010,
001,
110,1
01,
011
,111
, ½
½ 0
, ½
0 ½
, 0 ½
½ ,
½1 ½
, 1
½ ½
, ½
½ 1
He
xag
on
al c
lose
pa
cke
d c
ell
(hcp
):
con
ven
tion
al=
pri
miti
ve c
ell
poin
ts o
f prim
itive
cell
Pri
miti
ve a
nd
co
nve
ntio
na
l ce
lls-h
cp
Cry
stal S
truct
ure
64
con
ven
tion
al=
pri
miti
ve c
ell
Fra
ctio
na
l co
ord
ina
tes:
1
00
, 01
0, 11
0, 1
01
,011
, 111
,00
0, 0
01
ab
c
�T
he
un
itce
llan
d,
conse
que
ntly
,th
eentir
ela
ttic
e,
isuniquely
dete
rmin
ed
by
the
six
latt
ice
const
ants
:a,b,c,
α,β
and
γ.
Unit C
ell
Unit C
ell
Cry
stal S
truct
ure
65
const
ants
:a,b,c,
α,β
and
γ.�
Only
1/8
of
each
latt
ice
poin
tin
aun
itce
llca
nact
ually
be
ass
igne
dto
thatce
ll.�
Each
un
itce
llin
the
figure
can
be
ass
oci
ate
dw
ith8
x1/8
=1
lattic
epoin
t.
�A
prim
itive
unit
cell
ism
ade
of
prim
itive
transl
atio
nve
ctors
a1
,a2,
and
a3
such
that
there
isno
cell
of
smalle
rvo
lum
eth
at
can
be
use
das
abuild
ing
blo
ckfo
rcr
ysta
lstr
uct
ure
s.
Prim
itive U
nit C
ell a
nd v
ecto
rs
Cry
stal S
truct
ure
66
�A
prim
itive
unit
cell
will
fill
space
by
repetit
ion
of
suita
ble
crys
tal
transl
atio
nve
ctors
.T
his
defin
ed
by
the
para
llelp
iped
a1,
a2
and
a3.
The
volu
me
of
aprim
itive
unit
cell
can
be
found
by
�V
=a
1.(
a2
xa
3)
(vect
or
pro
duct
s)C
ub
ic c
ell v
olu
me
= a
3
�T
he p
rim
itive
unit
cell
must
have
only
one la
ttic
e p
oin
t.
�T
here
can b
e d
iffere
nt
choic
es
for
latt
ice v
ect
ors
, but th
e
volu
mes
of th
ese
prim
itive
cells
are
all
the s
am
e.
Prim
itive
Unit
Cell
a
Cry
stal S
truct
ure
67
P =
Pri
mit
ive
Un
it C
ell
NP
= N
on
-Pri
mit
ive
Un
it C
ell
1a
Wigner-Seitz M
eth
od
Asi
mply
way
tofin
dth
eprim
itive
cell
whic
his
calle
dW
igner-
Seitz
cell
can
be
done
as
follo
ws;
1.
Choose
ala
ttic
epoin
t.
Cry
stal S
truct
ure
68
1.
Choose
ala
ttic
epoin
t.2.
Dra
wlin
es
toco
nnect
these
latt
ice
poin
tto
itsneig
hbours
.3.
At
the
mid
-poin
tand
norm
al
toth
ese
lines
dra
wn
ew
lines.
Th
e v
olu
me e
nclo
sed
is c
all
ed
as a
Wig
ner-
Seit
z c
ell
.
Wigner-Seitz C
ell -
3D
Cry
stal S
truct
ure
69
Lattice S
ites in C
ubic U
nit C
ell
Cry
stal S
truct
ure
70
Cry
sta
l Directions
�W
ec
ho
ose
on
ela
ttic
ep
oin
to
nth
elin
ea
sa
no
rig
in,
sa
yth
ep
oin
tO
.C
ho
ice
of
ori
gin
isco
mp
lete
lya
rbit
rary
,s
inc
ee
ve
ryla
ttic
ep
oin
tis
ide
nti
ca
l.
�T
he
nw
ec
ho
os
eth
ela
ttic
eve
cto
rjo
inin
gO
toa
ny
po
int
on
the
lin
e,
sa
y
Cry
stal S
truct
ure
71
Fig
. S
ho
ws
[111
] d
ire
cti
on
join
ing
Oto
an
yp
oin
to
nth
elin
e,
sa
yp
oin
tT
.T
his
ve
cto
rc
an
be
wri
tte
na
s;
R =
n1
a +
n2
b +
n3c
�T
od
isti
ng
uis
ha
latt
ice
dir
ecti
on
fro
ma
latt
ice
po
int,
the
trip
leis
en
clo
se
din
sq
ua
reb
rac
ke
ts[
...]
isu
se
d.[
n1n
2n
3]
�[n
1n
2n
3]
isth
es
ma
llest
inte
ger
of
the
samerelativeratios
.
210Exam
ples
Cry
stal S
truct
ure
72
X =
1 ,
Y =
½ , Z
= 0
[1 ½
0]
[2 1
0]
X =
½ , Y
= ½
, Z
= 1
[½ ½
1]
[1 1
2]
Negative d
irections
�W
he
nw
ew
rite
the
dir
ec
tio
n[n
1n
2n
3]
de
pe
nd
on
the
ori
gin
,
ne
ga
tiv
ed
ire
cti
on
s
Z d
irecti
on
Cry
stal S
truct
ure
73
ca
nb
ew
ritt
en
as
�R
=n
1a
+n
2b
+n
3c
Dir
ec
tio
n m
us
t b
e
sm
all
es
t in
teg
ers
.
Y d
irecti
on
(ori
gin
) O
-Y
dir
ecti
on
X d
irecti
on
-X
dir
ecti
on
-Z
dir
ecti
on
][
32
1n
nn
][
32
1n
nn
Exam
ples o
f cry
sta
l directions
Cry
stal S
truct
ure
74
X =
-1 , Y
= -
1 ,
Z =
0
[110]
X =
1 ,
Y =
0 ,
Z =
0
[1 0
0]
Exam
ples
Cry
stal S
truct
ure
75
X =
-1 , Y
= 1
, Z
= -
1/6
[-1 1
-1/6
]
[
6 6
1]
We c
an
mo
ve v
ecto
r to
th
e o
rig
in.
Cry
sta
l Planes
�W
ithin
acr
ysta
lla
ttic
eit
isp
oss
ible
toid
en
tify
sets
of
eq
ua
llysp
ace
dp
ara
llelp
lan
es.
Th
ese
are
calle
dla
ttic
ep
lan
es.
�In
the
figu
red
en
sity
of
latt
ice
po
ints
on
ea
chp
lan
eo
fa
set
isth
esa
me
an
da
llla
ttic
ep
oin
tsa
reco
nta
ine
do
ne
ach
set
of
pla
ne
s.
Cry
stal S
truct
ure
76
pla
ne
s.
b
a
b
a
Th
e s
et
of
pla
nes in
2D
latt
ice.
Miller In
dices
Mill
er
Ind
ices
are
asy
mbolic
vect
or
repre
sen
tatio
nfo
rth
eori
enta
tion
of
an
ato
mic
pla
ne
ina
crys
tall
attic
ea
nd
are
de
fine
da
sth
ere
cip
roca
lso
fth
efr
act
ion
ali
nte
rce
pts
wh
ich
the
pla
ne
ma
kes
with
the
crys
tallo
gra
ph
ica
xes.
To
de
term
ine
Mill
er
ind
ice
s o
f a
pla
ne
, ta
ke th
e fo
llow
ing
ste
ps;
Cry
stal S
truct
ure
77
To
de
term
ine
Mill
er
ind
ice
s o
f a
pla
ne
, ta
ke th
e fo
llow
ing
ste
ps;
1)
De
term
ine
the
inte
rce
pts
of th
e p
lan
e a
lon
g e
ach
of th
e th
ree
cr
ysta
llog
rap
hic
dir
ect
ion
s
2)
Ta
ke th
e r
eci
pro
cals
of th
e in
terc
ep
ts
3)
If fra
ctio
ns
resu
lt, m
ulti
ply
ea
ch b
y th
e d
en
om
ina
tor
of th
e s
ma
llest
fr
act
ion
Axi
sX
YZ
Inte
rcep
t
po
ints
1∞
∞
Exam
ple-1
Cry
stal S
truct
ure
78
po
ints
1∞
∞
Recip
rocals
1/1
1/
∞1
/ ∞
Sm
all
est
Rati
o1
00
Mill
er
Đnd
ice
s (
10
0)
(1,0
,0)
Axi
sX
YZ
Inte
rcep
t
po
ints
11
∞
Exam
ple-2
Cry
stal S
truct
ure
79
po
ints
11
∞
Recip
rocals
1/1
1/
11
/ ∞
Sm
all
est
Rati
o1
10
Mill
er
Đnd
ice
s (
110
)(1
,0,0
)
(0,1
,0)
Axi
sX
YZ
Inte
rcep
t
po
ints
11
1(0
,0,1
)
Exam
ple-3
Cry
stal S
truct
ure
80
po
ints
11
1
Recip
rocals
1/1
1/
11
/ 1
Sm
all
est
Rati
o1
11
Mill
er
Đnd
ice
s (
111
)(1
,0,0
)
(0,1
,0)
Axi
sX
YZ
Inte
rcep
t
po
ints
1/2
1∞
Exam
ple-4
Cry
stal S
truct
ure
81
po
ints
1/2
1∞
Recip
rocals
1/(
½)
1/
11
/ ∞
Sm
all
est
Rati
o2
10
Mill
er
Đnd
ice
s (
21
0)
(1/2
, 0
, 0
)
(0,1
,0)
Axi
sa
bc
Inte
rcep
t
po
ints
1∞
½
Exam
ple-5
Cry
stal S
truct
ure
82
po
ints
1∞
½
Recip
rocals
1/1
1/
∞1
/(½
)
Sm
all
est
Rati
o1
02
Mill
er
Đnd
ice
s (
10
2)
Axi
sa
bc
Inte
rcep
t
po
ints
-1∞
½
Exam
ple-6
Cry
stal S
truct
ure
83
po
ints
-1∞
½
Recip
rocals
1/-
11
/ ∞
1/(
½)
Sm
all
est
Rati
o-1
02
Mill
er
Đnd
ice
s (
10
2)
Miller In
dices R
eci
pro
cal n
um
bers
are
: 21 ,
21 ,31
Pla
ne in
terc
epts
axe
s at
c
ba
2 ,2 ,
3
Indic
es
of th
e p
lane (
Mill
er)
: (2
,3,3
)
a2
2
bc
[2,3
,3]
Cry
stal S
truct
ure
84
(100)
(200)
(110)
(111
)(1
00)
Indic
es
of th
e d
irect
ion:
[2,3
,3]
a
3
Cry
stal S
truct
ure
85
Exam
ple-7
Cry
stal S
truct
ure
86
Indices o
f a F
am
ily o
r Form
�S
om
etim
es
wh
en
the
un
itce
llh
as
rota
tion
al
sym
me
try,
seve
raln
on
pa
ralle
lp
lane
sm
ay
be
eq
uiv
ale
nt
by
virt
ue
of
this
sym
me
try,
inw
hic
hca
seit
isco
nve
nie
nt
tolu
mp
all
the
sep
lan
es
inth
esa
me
Mill
er
Ind
ice
s,b
ut
with
curl
yb
rack
ets
.
Cry
stal S
truct
ure
87
pla
ne
sin
the
sam
eM
ille
rIn
dic
es,
bu
tw
ithcu
rly
bra
cke
ts.
Th
us
ind
ice
s{h
,k,l}
rep
rese
nt
all
the
pla
ne
se
qu
iva
len
tto
the
pla
ne
(hkl
)th
rou
gh
rota
tion
als
ymm
etr
y.
)1
11(
),1
11(
),11
1(),
11
1(),
11
1(),1
11(
),1
11
(),
111
(}
111
{
)00
1(),
100
(),
01
0(),
001
(),
010
(),
100
(}
100
{
≡≡
�T
he
rea
reo
nly
seve
nd
iffe
ren
tsh
ap
es
of
un
itce
llw
hic
hca
nb
est
ack
ed
tog
eth
er
toco
mp
lete
lyfil
lall
spa
ce(i
n3
dim
en
sio
ns)
with
ou
to
verl
ap
pin
g.
Th
isg
ive
sth
ese
ven
crys
tal
syst
em
s,in
wh
ich
all
crys
tal
stru
ctu
res
can
be
cla
ssifi
ed
.
3D
3
D
3D
3
D – –––
14
BR
AV
AIS
LA
TT
ICE
S A
ND
TH
E S
EV
EN
CR
YS
TA
L S
YS
TE
M1
4 B
RA
VA
IS L
AT
TIC
ES
AN
D T
HE
SE
VE
N C
RY
ST
AL
SY
ST
EM
14
BR
AV
AIS
LA
TT
ICE
S A
ND
TH
E S
EV
EN
CR
YS
TA
L S
YS
TE
M1
4 B
RA
VA
IS L
AT
TIC
ES
AN
D T
HE
SE
VE
N C
RY
ST
AL
SY
ST
EM
TY
PIC
AL
CR
YS
TA
L S
TR
UC
TU
RE
ST
YP
ICA
L C
RY
ST
AL
ST
RU
CT
UR
ES
TY
PIC
AL
CR
YS
TA
L S
TR
UC
TU
RE
ST
YP
ICA
L C
RY
ST
AL
ST
RU
CT
UR
ES
Cry
stal S
truct
ure
88
cla
ssifi
ed
.
�C
ub
icC
ryst
alS
yste
m(S
C,
BC
C,F
CC
)�
He
xag
on
alC
ryst
alS
yste
m(S
)�
Tri
clin
icC
ryst
alS
yste
m(S
)�
Mo
no
clin
icC
ryst
alS
yste
m(S
,B
ase
-C)
�O
rth
orh
om
bic
Cry
sta
lSys
tem
(S,
Ba
se-C
,B
C,
FC
)�
Te
tra
go
na
lCry
sta
lSys
tem
(S,
BC
)�
Tri
go
na
l(R
ho
mb
oh
ed
ral)
Cry
sta
lSys
tem
(S)
Cry
stal S
truct
ure
89
Coord
inatıon N
um
ber
�C
oord
inatıon
Num
be
r(C
N)
:T
he
Bra
vais
latt
ice
poin
tscl
ose
stto
agiv
en
poin
tare
the
neare
stneig
hbours
.
�B
eca
use
the
Bra
vais
latt
ice
isperi
od
ic,
all
po
ints
ha
ve
Cry
stal S
truct
ure
90
�B
eca
use
the
Bra
vais
latt
ice
isperi
od
ic,
all
po
ints
ha
veth
esa
me
num
ber
of
neare
stneig
hb
ours
or
coord
inatio
nnum
ber.
Itis
apro
pert
yofth
ela
ttic
e.
�A
sim
ple
cub
ichas
coord
inatio
nnum
be
r6;
abody-
cente
red
cub
icla
ttic
e,
8;
and
afa
ce-c
ente
red
cub
icla
ttic
e,1
2.
Ato
mic P
acking F
acto
r
�A
tom
icP
ack
ing
Fa
cto
r(A
PF
)is
de
fine
da
sth
evo
lum
eo
fato
ms
with
inth
eu
nit
cell
div
ide
db
yth
evo
lum
eo
fth
eu
nit
cell.
div
ide
db
yth
evo
lum
eo
fth
eu
nit
cell.
1-C
UBIC
CRYSTAL S
YSTEM
�S
imp
leC
ub
ich
as
on
ela
ttic
ep
oin
tso
itsp
rim
itive
cell.
�In
the
un
itce
llo
nth
ele
ft,
the
ato
ms
at
the
corn
ers
are
cut
be
cau
seo
nly
ap
ort
ion
(in
this
case
1/8
)b
elo
ng
sto
tha
tce
ll.
a-Sim
ple C
ubic (SC)
Cry
stal S
truct
ure
92
be
cau
seo
nly
ap
ort
ion
(in
this
case
1/8
)b
elo
ng
sto
tha
tce
ll.T
he
rest
of
the
ato
mb
elo
ng
sto
ne
igh
bo
rin
gce
lls.
�C
oo
rdin
atin
atio
nn
um
be
ro
fsi
mp
lecu
bic
is6
.
a
bc
a-Sim
ple C
ubic (SC)
Cry
stal S
truct
ure
93
Ato
mic P
acking F
acto
r of SC
Cry
stal S
truct
ure
94
b-B
ody C
ente
red C
ubic (BCC)
�B
CC
ha
stw
ola
ttic
ep
oin
tsso
BC
Cis
an
on
-pri
miti
vece
ll.
�B
CC
ha
se
igh
tn
ea
rest
ne
igh
bo
rs.
Ea
cha
tom
isin
con
tact
with
its
Cry
stal S
truct
ure
95
Ea
cha
tom
isin
con
tact
with
itsn
eig
hb
ors
on
lya
lon
gth
eb
od
y-d
iag
on
ald
ire
ctio
ns.
�M
an
ym
eta
ls(F
e,L
i,Na
..e
tc),
incl
ud
ing
the
alk
alis
an
dse
vera
ltr
an
sitio
ne
lem
en
tsch
oo
seth
eB
CC
stru
ctu
re.
a
bc
0.68
=
VV =
APF
cell
unit
atoms
BCC
Ato
mic P
acking Facto
r of BCC
Cry
stal S
truct
ure
96
2(0
,433a)
c-Face C
ente
red C
ubic (FCC)
�T
he
rea
rea
tom
sa
tth
eco
rne
rso
fth
eu
nit
cell
an
da
tth
ece
nte
ro
fe
ach
face
.�
Fa
cece
nte
red
cub
ich
as
4a
tom
sso
itsn
on
pri
miti
vece
ll.�
Ma
ny
of
com
mo
nm
eta
ls(C
u,N
i,Pb
..e
tc)
crys
talli
zein
FC
Cst
ruct
ure
.
Cry
stal S
truct
ure
97
stru
ctu
re.
3 -
Fa
ce
Ce
nt
er
ed
Cu
bıc
Cry
stal S
truct
ure
98
Ato
ms a
re a
ll s
am
e.
0.68
=
VV =
APF
cell
unit
atoms
BCC
FC
C0,7
4
Ato
mic P
acking Facto
r of FCC
Cry
stal S
truct
ure
99
4(0
,353a)
Ato
ms
Sh
are
d B
etw
ee
n:
Ea
ch
ato
m c
ou
nts
:
corn
er
8 c
ells
1/8
face
ce
ntr
e2
ce
lls1
/2b
od
y ce
ntr
e1
ce
ll1
Unit cell conte
nts
Countin
g t
he n
um
ber
of ato
ms
with
inth
e u
nit
cell
Cry
stal S
truct
ure
100
bo
dy
cen
tre
1 c
ell
1e
dg
e c
en
tre
2ce
lls1
/2
latt
ice
typ
ec
ell c
on
ten
ts
P1
[=8
x 1
/8]
I2
[=(8
x 1
/8)
+ (
1 x
1)]
F4
[=
(8 x
1/8
) +
(6
x 1
/2)]
C
2
[=(8
x 1
/8)
+ (
2 x
1/2
)]
Exam
ple; Ato
mic P
acking F
acto
r
Cry
stal S
truct
ure
101
2 -
HEXAGONAL S
YSTEM
�A
crys
tal
syst
em
inw
hic
hth
ree
equ
al
copla
nar
axe
sin
ters
ect
at
an
ang
leof
60
,and
aperp
en
dic
ula
rto
the
oth
ers
,is
ofa
diff
ere
nt
length
.
Cry
stal S
truct
ure
102
2 -
HEXAGONAL S
YSTEM
Cry
stal S
truct
ure
103
Ato
ms a
re a
ll s
am
e.
Cry
stal S
truct
ure
104
3
3 --
TRIC
LIN
IC
TRIC
LIN
IC
4
4 --
MONOCLIN
IC C
RYSTAL S
YSTEM
MONOCLIN
IC C
RYSTAL S
YSTEM
�T
ricl
inic
min
era
lsa
reth
ele
ast
sym
me
tric
al.
Th
eir
thre
ea
xes
are
all
diff
ere
nt
len
gth
sa
nd
no
ne
of
the
ma
rep
erp
en
dic
ula
rto
ea
cho
the
r.T
he
sem
ine
rals
are
the
mo
std
iffic
ult
tore
cog
niz
e.
Cry
stal S
truct
ure
105
Tri
clin
ic (
Sim
ple
)
α ≠
α
≠
α ≠
α
≠ ß
≠ γ
≠≠
γ ≠
≠ γ
≠≠
γ ≠
90
oa
≠
≠
≠
≠ b
≠
≠
≠
≠ c
Mon
ocli
nic
(S
imp
le)
α ααα=
γ γγγ=
90
o, ß
≠
≠
≠
≠ 9
0o
a ≠
≠
≠
≠
b ≠ ≠≠≠
c
Mo
no
clin
ic (
Ba
seC
ente
red
)
α ααα=
γ γγγ=
90
o, ß
≠
≠
≠
≠ 9
0o
a ≠
≠
≠
≠
b ≠
≠
≠
≠
c,
5 -
ORTHORHOMBIC
SYSTEM
Cry
stal S
truct
ure
106
Ort
ho
rho
mb
ic (
Sim
ple
)
α ααα=
ß =
γ γγγ=
90
o
a ≠
≠
≠
≠
b ≠
≠
≠
≠
c
Ort
ho
rho
mb
ic (
Ba
se-
cen
tred
)
α ααα=
ß =
γ γγγ=
90
o
a ≠
≠
≠
≠
b ≠
≠
≠
≠
c
Ort
ho
rho
mb
ic (
BC
)
α ααα=
ß =
γ γγγ=
90
o
a ≠
≠
≠
≠
b ≠
≠
≠
≠
c
Ort
ho
rho
mb
ic (
FC
)
α ααα=
ß =
γ γγγ=
90
o
a ≠
≠
≠
≠
b ≠
≠
≠
≠
c
6 –
TETRAGONAL S
YSTEM
Cry
stal S
truct
ure
107
Tet
ragonal
(P
)
α=
ß =
γ=
90
o
a =
b ≠
c
Tet
ragonal
(B
C)
α=
ß =
γ=
90
o
a =
b ≠
c
7 -
Rhom
bohedra
l (R
) or Trigonal
Cry
stal S
truct
ure
108
Rhom
bohed
ral
(R)
or
Tri
gonal
(S)
a =
b =
c, α
= ß
= γ
≠ 9
0o
THE M
OST IMPORTANT
CRYSTAL S
TRUCTURES
�S
od
ium
Ch
lori
de
Str
uct
ure
Na
+C
l-
�C
esi
um
Ch
lori
de
Str
uct
ure
Cs+
Cl-
Cry
stal S
truct
ure
109
�C
esi
um
Ch
lori
de
Str
uct
ure
Cs
Cl
�H
exa
go
na
l Clo
sed
-Pa
cke
d S
tru
ctu
re
�D
iam
on
d S
tru
ctu
re
�Z
inc
Ble
nd
e
1 –
Sodium
Chloride S
tructu
re
�S
od
ium
ch
lori
de
als
o
cry
sta
lliz
es
ina
cu
bic
latt
ice,
bu
tw
ith
ad
iffe
ren
tu
nit
ce
ll.
So
diu
mc
hlo
rid
estr
uc
ture
Cry
stal S
truct
ure
110
�S
od
ium
ch
lori
de
str
uc
ture
co
ns
ists
of
eq
ua
ln
um
be
rso
f
so
diu
ma
nd
ch
lori
ne
ion
s
pla
ce
dat
alt
ern
ate
po
ints
of
a
sim
ple
cu
bic
latt
ice
.
�E
ac
hio
nh
as
six
of
the
oth
er
kin
do
fio
ns
as
its
ne
are
st
ne
igh
bo
urs
.
Sodium
Chloride S
tructu
re
�If
we
tak
eth
eN
aC
lu
nit
ce
lla
nd
rem
ove
all
the
red
Cl
ion
s,
we
are
left
wit
ho
nly
the
blu
eN
a.
Ifw
eco
mp
are
this
wit
hth
efc
c/
cc
pu
nit
ce
ll,
itis
cle
ar
tha
tth
ey
are
ide
nti
ca
l.T
hu
s,
the
Na
is
ina
fcc
su
bla
ttic
e.
Cry
stal S
truct
ure
112
Sodium
Chloride S
tructu
re
�T
his
stru
cture
can
be
consi
dere
das
afa
ce-
cente
red-c
ub
icB
rava
isla
ttic
ew
itha
basi
sco
nsi
stin
gof
aso
diu
mio
nat
0and
ach
lori
ne
sod
ium
ion
at
0and
ach
lori
ne
ion
at
the
cente
rof
the
conve
ntio
nalc
ell,
�LiF
,NaB
r,K
Cl,L
iI,etc
�T
he la
ttic
e c
onst
ants
are
in
the o
rder
of 4-7
angst
rom
s.)(
2/
→→
→
++
zy
xa
2-C
esium
Chloride S
tructu
re
Cs+Cl-
�C
esiu
mch
lori
de
cry
sta
lliz
es
ina
cu
bic
latt
ice.
Th
eu
nit
cell
may
be
dep
icte
das
sh
ow
n.
(Cs+
iste
al,
Cl-
isg
old
).
Cry
stal S
truct
ure
114
Cl-
isg
old
).
�Cesiumchlorideconsistsofequal
numbers
ofcesium
andchlorine
ions,placedatthepoints
ofa
body-centeredcubiclatticesothat
eachionhaseightoftheotherkind
asitsnearestneighbors.
Cesium
Chloride S
tructu
re
Cs+Cl-
�T
he
tra
nsl
atio
na
lsy
mm
etr
yo
fth
isst
ruct
ure
isth
at
of
the
sim
ple
cub
icB
rava
isla
ttic
e,
an
dis
desc
rib
ed
as
asi
mp
lecu
bic
latt
ice
with
aba
sis
sim
ple
cub
icla
ttic
ew
itha
ba
sis
con
sist
ing
of
ace
siu
mio
na
tth
eo
rig
in0
an
da
chlo
rin
eio
na
tth
ecu
be
cen
ter
�C
sBr,
CsI
crys
talli
zein
this
stru
ctu
re.T
he
latt
ice
con
sta
nts
are
inth
eo
rde
ro
f4
an
gst
rom
s.
)(
2/→
→→
++
zy
xa
Cesium
Chloride C
s+Cl-
8ce
ll
3–Hexagonal Close-P
acked S
tr.
�T
his
isa
no
the
rstr
uc
ture
tha
tis
co
mm
on
,p
art
icu
larl
yin
me
tals
.In
ad
dit
ion
toth
etw
ola
ye
rso
fa
tom
sw
hic
hfo
rmth
eb
as
ea
nd
the
up
pe
rfa
ce
of
the
he
xa
go
n,
Cry
stal S
truct
ure
117
the
up
pe
rfa
ce
of
the
he
xa
go
n,
the
reis
als
oa
nin
terv
en
ing
laye
ro
fato
ms
arr
an
ge
ds
uc
hth
at
ea
ch
of
the
se
ato
ms
rest
ove
ra
de
pre
ssio
nb
etw
ee
nth
ree
ato
ms
inth
eb
as
e.
Bra
vais
La
ttic
e : H
exa
go
na
l La
ttic
eH
e, B
e, M
g, H
f, R
e (G
rou
p II
ele
me
nts
)A
BA
BA
B T
ype
of S
tack
ing
Hexagonal Close-p
acked S
tructu
re
a=
b a
=1
20
, c=
1.6
33
a,
ba
sis
: (
0,0
,0)
(2/3
a ,1
/3a
,1/2
c)
Cry
stal S
truct
ure
118
A
AA
AA
A
AA
A
BB
BB
CC
C
Clo
se p
ack
B
AA
AA
AA
B
BB
Packing
Cry
stal S
truct
ure
119
AA
AA
AA
AA
A
B
B
BB
B
B
BC
CC
CC
C
C
Se
qu
en
ce A
BA
BA
B..
-he
xag
on
al c
lose
pa
ck
Se
qu
en
ce A
BC
AB
CA
B..
-fa
ce c
en
tere
d c
ub
ic c
lose
pa
ck
AA
A
BB
Se
qu
en
ce A
AA
A…
-si
mp
le c
ub
ic
Se
qu
en
ce A
BA
B…
-b
od
y ce
nte
red
cub
ic
Cry
stal S
truct
ure
120
4 -
Diam
ond S
tructu
re
�T
he
dia
mo
nd
latt
ice
isc
on
sis
to
ftw
oin
terp
en
etr
ati
ng
fac
ec
en
tere
db
rava
isla
ttic
es
.
�T
he
rea
ree
igh
ta
tom
inth
es
tru
ctu
reo
fd
iam
on
d.
�E
ac
h a
tom
bo
nd
s c
ova
len
tly t
o 4
oth
ers
eq
ua
lly s
pre
ad
ab
ou
t a
tom
in
3d
.
Cry
stal S
truct
ure
121
ab
ou
t a
tom
in
3d
.
4 -
Diam
ond S
tructu
re
�T
he c
oord
inatio
n n
um
ber
of dia
mond
stru
cture
is 4
.
�T
he d
iam
ond la
ttic
e is
not a B
rava
is
�T
he d
iam
ond la
ttic
e is
not a B
rava
is
lattic
e.
�S
i, G
e a
nd C
cry
stalli
zes
in d
iam
ond
stru
cture
.
Cry
stal S
truct
ure
123
5-Zinc B
lende
�Z
incb
len
de
ha
se
qu
al
nu
mb
ers
of
zin
ca
nd
sulfu
rio
ns
dis
trib
ute
do
na
dia
mo
nd
latt
ice
soth
at
ea
chh
as
fou
ro
fth
eo
pp
osi
teki
nd
as
ne
are
stn
eig
hb
ors
.T
his
stru
ctu
reis
an
ne
are
stn
eig
hb
ors
.T
his
stru
ctu
reis
an
exa
mp
leo
fa
latt
ice
with
ab
asi
s,w
hic
hm
ust
sod
esc
rib
ed
bo
thb
eca
use
of
the
ge
om
etr
ica
lp
osi
tion
of
the
ion
sa
nd
be
cau
setw
oty
pe
so
fio
ns
occ
ur.
�A
gI,
Ga
As,
Ga
Sb
,In
As,
5-Zinc B
lende
5-Zinc B
lende
Zin
cB
len
de
isth
en
am
egiv
en
toth
em
ine
ral
Zn
S.
Ith
as
acu
bic
clo
sep
ack
ed
(face
cen
tre
d)
arr
ay
of
Sa
nd
the
Zn
(II)
sit
inte
tra
he
dra
l(1
/2o
ccu
pie
d)
site
sin
the
lattic
e.
Cry
stal S
truct
ure
126
�E
ach
of
the
unit
cells
of
the
14
Bra
vais
latt
ices
has
one
or
more
types
of
sym
metr
ypro
pert
ies,
such
as
inve
rsio
n,
refle
ctio
nor
rota
tion,e
tc.
EL
EM
EN
TS
OF
SY
MM
ET
RY
EL
EM
EN
TS
OF
SY
MM
ET
RY
EL
EM
EN
TS
OF
SY
MM
ET
RY
EL
EM
EN
TS
OF
SY
MM
ET
RY
Cry
stal S
truct
ure
127
SY
MM
ET
RY
INV
ER
SIO
NR
EF
LE
CT
ION
RO
TA
TIO
N
Lattic
e g
oes
into
itse
lf thro
ugh
Sym
metr
y w
ithout tr
ansl
atio
n
Op
era
tion
Ele
me
nt
Inve
rsio
nP
oin
t
Cry
stal S
truct
ure
128
Re
flect
ion
Pla
ne
Ro
tatio
nA
xis
Ro
toin
vers
ion
Axe
s
Invers
ion C
ente
r
�A
cen
ter
of
sym
me
try:
Ap
oin
ta
tth
ece
nte
ro
fth
em
ole
cule
.(x
,y,z
)--
>(-
x,-y
,-z)
�C
en
ter
of
inve
rsio
nca
no
nly
be
ina
mo
lecu
le.
Itis
no
tn
ece
ssa
ryto
ha
vea
na
tom
inth
ece
nte
r(b
en
zen
e,
eth
an
e).
Te
tra
he
dra
l,tr
ian
gle
s,p
en
tag
on
sd
on
'th
ave
ace
nte
ro
fin
vers
ion
sym
me
try.
All
Bra
vais
latt
ice
sa
rein
vers
ion
Cry
stal S
truct
ure
129
inve
rsio
nsy
mm
etr
y.A
llB
rava
isla
ttic
es
are
inve
rsio
nsy
mm
etr
ic.
Mo
(CO
)6
Reflection P
lane
Cry
stal S
truct
ure
130
�A
pla
ne
ina
cell
such
that,
when
am
irro
rre
flect
ion
inth
ispla
ne
isperf
orm
ed,th
ece
llre
main
sin
variant.
Exam
ples
Cry
stal S
truct
ure
131
�T
ricl
inic
has
no
refle
ctio
npla
ne.
�M
onocl
inic
has
one
pla
ne
mid
way
betw
een
and
para
llelt
oth
ebase
s,and
sofo
rth.
We
ca
n n
ot fin
d a
lattic
e th
at g
oe
s in
to it
self
un
de
r o
the
r ro
tatio
ns
Rota
tion S
ym
metry
Cry
stal S
truct
ure
132
•A
sin
gle
mole
cule
can h
ave
any
degre
e
of ro
tatio
nal s
ymm
etr
y, b
ut an in
finite
periodic
lattic
e –
can n
ot.
Rota
tion A
xis
90
°
Cry
stal S
truct
ure
133
�T
his
isan
axi
ssu
chth
at,
ifth
ece
llis
rota
ted
aro
un
dit
thro
ugh
som
eangle
s,th
ece
llre
main
sin
variant.
�T
he
axi
sis
calle
dn-f
old
ifth
eangle
ofro
tatio
nis
2π
/n.
12
0°
18
0°
Axis o
f Rota
tion
Cry
stal S
truct
ure
134
Axi
s of
Rota
tion C
ryst
al S
truct
ure
135
5-f
old
sym
metr
y Cry
stal S
truct
ure
136
Can n
ot be c
om
bin
ed w
ith t
ransl
atio
nal p
eriodic
ity!
Gro
up d
iscu
ssio
n
�K
ep
ler
wo
ndere
dw
hy
snow
flake
shave
6co
rners
,neve
r5
or
7.B
yco
nsi
deri
ng
the
pack
ing
of
po
lyg
ons
in2
dim
en
sio
ns,
dem
onst
rate
why
penta
gons
an
dhepta
gons
should
n’t
occ
ur.
Cry
stal S
truct
ure
137
Em
pty
sp
ace
not
allo
wed
90°
Exam
ples
�T
ricl
inic
has
no
axi
sofro
tatio
n.
�M
onocl
inic
has
2-f
old
axi
s(θ
=2π
/2=
π)
norm
alt
oth
ebase
.
Cry
stal S
truct
ure
138
Cry
stal S
truct
ure
139