Kap
. 2K
ap. 2
Che
mic
alC
hem
ical
bond
ing
bond
ing
Ioni
cbo
ndin
g
Van
der
Waa
ls b
ondi
ng
Met
albo
ndin
g
Cov
alen
tbon
ding
Cry
stal
line
mat
eria
ls s
how
a ra
nge
ofdi
ffere
ntbo
ndty
pes:
Blen
ded
Blende
d
Ioni
cbo
ndin
gIo
ns a
rech
arge
dsp
ecie
s
Cl
Na
OC
O
O
2
Ebo
nd= E
elec
tro-E
rep
Cl
Na
Na
Mad
elun
gen
ergy
r
()(
)r
ere
eE e
02
04
4πε−
=πε−
+=
Like
cha
rges
repe
llea
chot
her
()(
) eioniccharg
Geometry
re
E e×
×
πε−=
02
4
2
02
A4
Zr
eN
E eα
πε−
=
Ioni
cst
ruct
ures
–La
ttice
ener
gy-N
aCl
Na+
is s
urro
unde
dby
6 C
l-
re)
Z)(
Z(6V
2−
+−=
r2e)
Z)(
Z(12
V2
−+
+=
Na+
is s
urro
unde
dby
12
next
neig
hbou
rNa+
at √
2r
Na+
is s
urro
unde
dag
ain
by 8
Cl- a
t √3r
r3e)
Z)(
Z(8V
2−
+−=
+−
+−
−=−
+....
4638
2126
re)
Z)(
Z(V
2
Mad
elun
gen
ergy
Ions
als
osh
ow re
puls
ive
forc
esw
hen
they
are
too
clos
e!
()
24
2X
M02
An
mZ
Zr
eN
E e+
α
πε=
Ioni
cbo
ndin
g
Non
-dire
ctio
nal
Hig
hco
ordi
natio
nnu
mbe
ras
poss
ible
Cha
rged
, non
-com
pres
sibl
e, n
on-p
olar
izab
lesp
here
s
Pure
lyio
nic
bind
ing
rare
lyoc
curs
,a
netc
harg
eof
mor
e th
an+1
are
unlik
ely
Use
fula
s st
artin
g po
intt
o vi
sual
ize
stru
ctur
es
For
visu
aliz
atio
n, o
nene
eds
a se
tof
ioni
calr
adiis
Ioni
cra
diis
The
ioni
calr
adiis
from
Pau
ling
and
Gol
dsch
mid
t hav
e be
enre
vise
ddu
e to
info
rmat
ion
from
pre
sent
hig
h-qu
ality
X-ra
ydi
ffrac
tion
wor
k.
a)Io
ns a
rees
sent
ially
sphe
rical
b)Io
ns a
reco
mpo
sed
ofa
cent
ralc
ore
with
mos
t oft
heel
ectr
ons
and
anou
ters
pher
ew
ithve
rylit
tleel
ectr
onde
nsity
.c)
Ass
ignm
ento
frad
iiis
diff
icul
t
Ioni
cra
diis
The
size
ofan
ion
is d
epen
anto
nho
wth
eou
ter
orbi
tals
shi
eld
the
char
gefr
om th
enu
cleu
s.
s p d
Poorer shielding
s-an
d p-
bloc
k, ra
diin
crea
sew
ithat
omic
num
berf
or a
nyve
rtica
lgro
up.
For i
soel
ectro
nic
serie
s of
catio
ns, r
adii
decr
ease
with
incr
ease
ing
char
ge, N
a+,
Mg2
+ , A
l3+an
d S
i4+
Cat
ion
radi
us d
ecre
ase
with
incr
ease
ing
oxid
atio
nst
ate,
V2+
, V3+
, V4+
, V5+
Cat
ion
radi
us in
crea
sew
ithco
ordi
natio
nnu
mbe
r, C
N =
4 v
s. C
N =
6
Lant
hani
deco
ntra
ctio
nan
d tra
nsiti
onco
ntra
ctio
ndu
e to
poo
rshi
eldi
ng
Ioni
cra
diis
Bas
edon
empi
rical
valu
esa
= 2(
r M+r
X)
Con
sist
ents
ett o
fval
ues
The
valu
esar
ede
pend
ent o
n:
Coo
rdin
atio
nnu
mbe
rTy
pe o
fcoo
rdin
atio
npo
lyhe
dra
Oxi
datio
nnu
mbe
r
Met
allic
oxid
esSe
tt of
radi
isde
pend
ent o
nty
pe o
forb
itals
use
d??
Ioni
cst
ruct
ures
–ge
nera
l prin
cipl
es
1.Io
ns a
rein
fact
char
ged,
ela
stic
and
pola
rizab
lesp
here
s
2.G
eom
etric
alan
nran
gem
ent
cont
ract
ion:
an
ion
–ca
tion
repu
lsio
n:
anio
n –
anio
nca
tion
–ca
tion
3.A
s hi
ghco
ordi
natio
nnu
mbe
ras
poss
ible
to m
axim
ize
nete
lect
rost
atic
attra
ctio
n. D
epen
dson
the
ratio
ofi
on ra
dii.
4.N
extn
eigh
bour
ions
are
as fa
r as
poss
ible
. →M
axim
izat
ion
ofvo
lum
e!
5.Lo
cale
lect
rone
utra
lity.
Val
ence
= Σ
elec
trost
atic
bond
stre
ngth
Ioni
cst
ruct
ures
–bo
ndst
reng
th
2 r)e
Z)(e
Z(F
−+
=Fo
rce
betw
een
two
ions
:
Ele
ctro
stat
icbo
ndst
reng
th:
The
char
geis
div
ided
amon
gth
enu
mbe
rofb
onds
.
The
sum
ofe
bson
an io
n m
ust b
alla
nce
the
char
ge
nmebs=
Spi
nel:
MgA
l 2O4
Oct
ahed
ralA
l3+→
ebs
= 3/
6 =
½Te
trahe
dral
Mg2
+→
ebs
= 2/
4 =
½
→O
xyge
nte
trahe
dral
ysu
rroun
ded
by 3
Al a
nd 1
Mg:
Σebs
(3A
l3+
+ 1M
g2+)
= 2
xnm=
∑
SiO
2 ?
mx)X(
CN
)M(
CN
XM
xm
=
Ioni
cst
ruct
ures
–bo
ndst
reng
th
Exa
mpl
e:
NaC
l typ
e st
ruct
ure,
Na+
in o
ctah
edra
hol
es →
CN
(Na+
) = 6
CN
(Cl- )
?
CN
(Cl- )
= 1
/1*6
= 6
CaF
2ty
pe s
truct
ure,
F-in
tetra
hedr
a ho
les →
CN
(F- )
= 4
CN
(Ca2
+ )?
CN
(Ca2
+ ) =
2/1
*4 =
8
Rut
il ty
pe s
truct
ure
(TiO
2);C
N(T
i4+) =
6
CN
(O2-
)?
CN
(O2-
) = 1
/2*6
= 3
Ioni
cst
ruct
ures
–bo
ndst
reng
th
Ioni
cst
ruct
ures
–R
adiu
s ra
tio
Idea
llyio
ns s
urro
und
them
selv
esw
ithas
man
yio
ns o
p th
eop
osite
char
geas
pos
sibl
e. T
heio
ns m
ust b
e in
con
tact
→C
N d
epen
dson
the
radi
us ra
tios.
Con
side
ran
octa
hedr
a:
Oct
aede
rhol
e
2 r x
2 r x
2 (r M
+r x)
()
()
()
[]
() 414
.01
2r
r
rr
22
r2
rr
2r2
r2
x
M
xM
x
2x
M2
x2
x
=−
=
+=
+=
+
Ioni
cst
ruct
ures
–R
adiu
s ra
tio
Ioni
cst
ruct
ures
–R
adiu
s ra
tio, b
orde
rline
GeO
2ha
s po
lym
orph
sw
here
Ge
may
have
CN
= 4
or 6
Al3+
may
be fo
und
in o
ctah
edra
lenv
ironm
ents
for s
pine
lMgA
l 2O4,
buta
lso
in te
trah
edra
sfo
r sili
cate
s.
Ti in
PbT
iO3
is s
lighl
tlyto
osm
all a
nd m
ayra
ttle.
Dis
plac
edby
0.2
Å.
Ioni
cst
ruct
ures
–R
adiu
s ra
tio
Ioni
cst
ruct
ures
–La
ttice
ener
gy
NaC
l(s) →
Na+ (
g) +
Cl- (g
), ∆
H =
US
ublim
atio
nen
ergy
2
2
re)
Z)(
Z(F
−+
=Fo
rce
betw
een
two
ions
Col
umbi
cpo
tent
iale
nerg
yre)
Z)(
Z(Fdr
V2
r−
+
∞−=
=∫
12...5
nrB
Vn
==
Sho
rt-ra
nge
repu
lsiv
e fo
rces
Ioni
cst
ruct
ures
–La
ttice
ener
gy
max
O6
r2
Nh
25.2CNr
BNe
rN
ZZ
Ae
Uν
+−
+−=
−ρ
−−
+
Attr
activ
eel
ectr
osta
ticfo
rces
Bor
nre
puls
ive
term
Zero
poi
nten
ergy
Van
der W
aals
attr
activ
efo
rces
Kap
ustin
skii
foun
d th
at if
the
Mad
elun
gco
nsta
nt fo
r a
give
n st
ruct
ure
is d
ivid
ed b
y th
e nu
mbe
r of i
ons
in o
ne
form
ula
unit
(n) t
he re
sulti
ng v
alue
s ar
e al
mos
t con
stan
t:
The
Kap
ustin
skii
equa
tion
gene
ral l
attic
e en
ergy
equ
atio
nth
at c
an b
e ap
plie
d to
any
crys
tal r
egar
dles
s of
the
crys
tal s
truct
ure
1
ac
ac
mol
kJr
r345
.01
rr
ZVZ
5.1200
U−
−+
+−
+−=
Mos
t im
porta
nt a
dvan
tage
of t
he K
apus
tinsk
iequ
atio
n:
It is
pos
sibl
e to
app
ly th
e eq
uatio
n fo
r lat
tice
calc
ulat
ions
ofc
ryst
als
with
pol
yato
mic
ions
(e.g
. KN
O3,
(NH
4)2S
O4
...).
A s
et o
f „th
erm
oche
mic
alra
dii“
was
der
ived
for f
urth
er c
alcu
latio
ns
of la
ttice
ent
halp
ies
The
Kap
ustin
skii
equa
tion
The
latti
ce e
nerg
y ca
n be
cal
cula
ted
by u
sing
Hes
s la
w a
nd
the
follo
win
g sc
hem
e:
−∆H
f°(N
aCl,s
)
∆Hf°(
Na,
g)
∆Hf°(
Cl,g
)
I 1(N
a)E
(Cl)
Latti
ce e
nerg
y
H
NaC
l(s)
Na(
s) +
½C
l 2(g)N
a+(g
) + C
l- (g)
Na(
g) +
Cl(g
)
Na(
g) +
½C
l 2(g)
Na+
(g) +
e-+
Cl(g
)
Bor
n H
aber
cycl
e
Cov
alen
tbon
ding
Cry
stal
line
mat
eria
ls s
how
a ra
nge
ofdi
ffere
ntbo
ndty
pes:
Cov
alen
tbon
ding
Cov
alen
tbon
ding
is a
resu
ltof
orbi
tal o
verla
psSp
ins
in o
verla
ppin
g or
bita
ls m
ust b
e an
tipar
alle
l
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
LiLi22
[He 2
](σ2s
)2
Bon
d le
ngth
= 0.
267
nmB
ond
ener
gy=
101
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
Be
Be 22
[He 2
](σ2s
)2(σ
∗ 2s)
2
Bon
d le
ngth
= -n
mB
ond
ener
gy=
-kJ/
mol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
BB22
[Be 2
](π2p
)2
Bon
d le
ngth
= 0.
159
nmB
ond
ener
gy=
289
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
CC22
[Be 2
](π2p
)4
Bon
d le
ngth
= 0.
124
nmB
ond
ener
gy=
599
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
NN22
[Be 2
](π2p
)4(σ
2px)
2
Bon
d le
ngth
= 0.
110
nmB
ond
ener
gy=
941
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
OO22
[Be 2
](π2p
)4(σ
2px)
2 (π*
2p)2
Bon
d le
ngth
= 0.
121
nmB
ond
ener
gy=
494
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
FF 22
[Be 2
](π2p
)4(σ
2px)
2 (π*
2p)4
Bon
d le
ngth
= 0.
142
nmB
ond
ener
gy=
254
kJ/m
ol
σ1s
σ∗1s
σ2s
σ∗2s
π2p
π∗2p
σ2p x
σ∗2p
x
Ne
Ne 22
[Be 2
](π2p
)4(σ
2px)
2 (π*
2p)4
(σ2p
x)2
Bon
d le
ngth
= 0.
142
nmB
ond
ener
gy=
254
kJ/m
ol
Bon
ding
betw
een
unlik
eat
oms
Orb
ital h
ybrid
isat
ion
Mul
tiple
bon
ds
Res
onan
cePa
rtia
l cov
alen
t bon
ding
Cov
alen
tbon
ding
occu
rsw
hen
the
oute
rele
ctro
nic
char
gede
nsity
onan
ani
on is
pol
ariz
edto
war
dsan
d by
a n
eigh
bour
ing
catio
n.
The
nete
ffect
is e
lect
ron
dens
itybe
twee
nth
eat
oms.
SrO
NaC
l-typ
eB
aON
aCl-t
ype
HgO
linea
r O-H
g-O
segm
ents
AlF
3R
eO3
rela
ted
ioni
cso
lidA
lCl 3
Laye
red
poly
mer
icst
ruct
ure
AlB
r 3M
olec
ular
Al2B
r 6di
mer
AlI 3
Mol
ecul
arAl
2I 6di
mer
Incr
easi
ngdi
ffere
nse
inel
ectro
nega
tivity
Sand
erso
nsm
odel
The
prop
ertie
sof
solid
s is
muc
hde
pend
ent o
nho
wva
lenc
eel
ectro
ns’fe
elth
esi
tuat
ion’
.
s p d
Poorer shielding
Cor
rela
tions
betw
een
effe
ctiv
enu
clea
rcha
rge
and:
-Ioni
zatio
npo
tent
ial
-Ele
ctro
naf
finity
-Ele
ctro
nega
tivity
χ-A
tom
icra
dius
Effe
ctiv
enu
clea
rcha
rge
Sand
erso
nsm
odel
–A
tom
icra
dii
Ato
mic
radi
ivar
yco
nsid
erab
lyde
pend
ing
onC
N a
nd p
olar
ityof
bond
.
Non
-pol
arco
vale
ntra
dii(
r c)ca
nbe
mea
sure
dac
cura
tely
(C-C
)
δ−
=B
rr
cS
ande
rson
sat
omic
radi
i
Non
-pol
arco
vale
ntra
dii
Ato
m s
peci
ficco
nsta
ntPar
tialc
harg
e
The
parti
alch
arge
sca
nno
t be
mea
sure
ddi
rect
lybu
test
imat
edfro
m S
ande
rson
sel
ectro
nega
tivity
scal
e.
Sand
erso
nsm
odel
–El
ectr
oneg
ativ
ity
San
ders
onde
velo
ped
a ne
wel
evtro
nega
tivity
tabl
e(S
) whe
re:
aDD
S=
Ele
ctro
nde
nsity
ofth
eat
om
Ele
ctro
nde
nsity
deriv
edfro
m
linea
r int
erpo
latio
nof
iner
t gas
el
emen
ts.
Prin
cipl
eof
elec
trone
gativ
ityeq
ualiz
atio
n 006
.2S
SS
FNa
b=
=
The
elec
trone
gativ
ityis
div
ided
betw
een
both
atom
s in
the
bond
Sand
erso
nsm
odel
–Pa
rtia
lcha
rge
Par
tialc
harg
eis
the
ratio
ofc
hang
ein
ele
ctro
nega
tivity
unde
rgon
eby
an
atom
on
bond
form
atio
nto
the
chan
geit
wou
ldha
ve u
nder
gone
onbe
com
ing
com
plet
ely
ioni
cw
ithch
arge
+ or
-1
A p
oint
ofre
fere
nce
is n
eces
sary
.Th
ebo
nds
in N
aFis
75%
ioni
c.
The
chan
gein
ele
ctro
nega
tivity
onac
quiri
nga
+ or
-1 c
harg
eis
∆S
c:
S08.2
S c=
∆
Par
tialc
harg
eca
nth
enbe
def
ined
as:
bc
SS
SSS
−=
∆∆∆
=δ
Exam
ple:
BaI
2
SB
a=
0.78
∆S
cBa
= 1.
93S
I=
3.84
∆S
cI=
4.08
Sb
= 3 √
(0.7
8*3.
84*3
.84)
= 2
.26
∆S
Ba=
2.26
–0.
78 =
1.4
8∆
SI=
-3.8
4 +
2.26
= -1
.58
78.093.148.1
SS BacBa
==
∆∆=
δ
39.008.458.1
SS I cI−
=−
=∆∆
=δ
Ba+
0.78
I-0.3
9
Ato
mic
radi
us c
anbe
cal
cula
ted
from
:δ
−=
Br
rc
Ba
Ir c
1.98
Å1.
33 Å
B0.
348
1.38
4
r Ba
= 1.
98 Å
–0.
348*
0.78
Å=
1.71
År I
= 1.
33 Å
+ 1.
384*
0.39
Å=
1.87
Å
Ba-
I= 3
.58
Å, O
bser
ved:
3.5
9 Å
!
Moo
ser–
Pear
son
plot
s an
d io
nici
ties
The
radi
us ra
tio is
uns
atis
fact
ory
in p
redi
ctio
nof
stru
ctur
ety
pe.
Alte
rnat
ive:
Plo
t the
-A
vera
gepr
inci
palq
uant
umnu
mbe
rvs.
Diff
eren
cein
ele
ctro
neva
tivity
Hug
er io
ns w
ithla
rge
Dx
give
sio
nic
stru
ctur
es.
Sm
alle
rato
ms
tend
to h
ave
dire
ctio
nalb
ondi
ng.
Bon
d va
lenc
ean
d bo
ndle
ngth
Vale
nce
bond
theo
ryis
eas
ilyad
apta
ble
to m
olec
ules
.
An
appr
oach
for c
ryst
allin
eso
lids
is to
intr
oduc
ebo
ndva
lenc
e(b
v).
The
sum
ofb
vfo
r all
bond
sto
an
atom
mus
t equ
alth
eva
lenc
eof
that
atom
. ->
the
vale
nce
sum
rule
. ∑=
jij
ibv
V
The
bvis
foun
dem
piric
ally
from
oth
erst
ruct
ures
and
vary
with
bond
leng
th
Each
bond
is
trea
ted
as a
n in
divi
dual
1
2 3
4
N0
ijRR
b
=
Bon
d va
lenc
ean
d bo
ndle
ngth
The
bond
vale
nce
sche
me
van
be u
sed
to:
a)C
heck
the
corr
ectn
ess
ofa
prop
osed
stru
ctur
e.
b)Lo
cate
hydr
ogen
ato
ms
for s
truc
ture
sde
term
ined
by X
-ray
diffr
actio
n
c)D
istin
guis
hbe
twee
nA
l3+an
d Si
4+in
alu
min
osoi
licat
es.
Bon
d va
lenc
ean
d bo
ndle
ngth
, exa
mpl
eYB
CO
Wha
tis
the
oxid
atio
nst
ate
for C
u?
1.91
1
1.94
3
1.96
92.
206
1.96
51.
912
∑∑
==
==
6 1j
6 1J
N
j,Cu0
j,Cu
Cu
RRc
bV Fo
r Cu:
c =
0.33
3; R
0=20
6.8
pm ;
N=5
.4
VC
u(1)
= 2*
0.51
0+2*
0.46
6+2*
0.43
4 =
2.82
VC
u(2)
= 2*
0.50
9+2*
0.43
9+1*
0.23
5 =
2.13
Y3+
Ba2
+ 2Cu?
3O7
lead
s to
a C
uav
erag
eof
2.33
Non
-bon
ding
elec
tron
effe
cts,
Cry
stal
field
theo
ryN
on-b
ondi
ngel
ectr
onef
fect
s, C
ryst
alfie
ldth
eory
Non
-bon
ding
elec
tron
effe
cts,
Cry
stal
field
theo
ryIo
nic
Rad
ii. F
or a
giv
en o
xida
tion
stat
e, th
e io
nic
radi
us d
ecre
ases
ste
adily
on
goin
g fro
m le
ft to
righ
t in
a tra
nsiti
on s
erie
s.
Pop
ulat
ing
antib
odin
gor
bita
ls(i.
e. fi
lling
the
e gle
vels
in a
n oc
tahe
dron
) lea
ds to
an
incr
ease
in io
nic
radi
us a
nd to
wea
ker b
onds
. The
refo
re, t
he io
nic
radi
us d
epen
ds
on th
e sp
in s
tate
of t
he m
etal
(i.e
. hig
h sp
in o
r low
spi
n).
M2+
M3+
Hig
hsp
in
Low
spin
Hig
hsp
in
Low
spin
Non
-bon
ding
elec
tron
effe
cts,
Jah
n-Te
ller
d
e g t 2g
dx2 -
y2
dz2
dxy dx
zdy
z
d4(H
S)
Mn3
+ (H
S)
Non
-bon
ding
elec
tron
effe
cts,
Jah
n-Te
ller
d
e g t 2g
dx2 -
y2
dz2
dxy dx
zdy
z
d7 (L
S)
Ni3
+ (L
S)
Non
-bon
ding
elec
tron
effe
cts,
Jah
n-Te
ller
d
e g t 2g
dx2 -
y2
dz2
dxy dx
zdy
z
d9
CuC
l2
Non
-bon
ding
elec
tron
effe
cts,
d8
d
e g t 2g
dx2 -
y2
dz2
dxy dx
zdy
z
d8
PdO
, PtO
Cry
stal
field
stab
iliza
tion
Low
–sp
in∆
crys
t.fie
ld>
PH
igh
–sp
in∆
crys
t.fie
ld<
P
∆(5
d) >
∆(4
d) >
∆(3
d)lo
wsp
inhi
ghsp
in
Oct
ahed
ricfie
ld: C
ryst
alfie
ldst
abiliz
atio
n
[]
egg2t
n6n4
10−
∆=
Exa
mpl
e:V
2+C
FSE
= ∆
/10[
4*3-
0] =
1.2∆
Mn2
+ (h
s)C
FSE
= ∆
/10[
4*3-
6*2]
= 0
∆∆
Non
-bon
ding
elec
tron
effe
cts,
Iner
t pai
r effe
ct
Hea
vyp-
grou
pel
emen
ts (T
l, Sn
, Pb,
Sb)
com
mon
lysh
ow
vale
nce
two
less
than
grou
pva
lenc
e.
PbO
, red
Fact
ors
dete
rmin
ing
the
stru
ctur
e
Siz
er M
/r XC
N =
2,4
,6,8
…
Bon
ding
type
Ioni
cC
oval
ent
Met
allic
CN
MX
1D
irect
iona
ldep
enda
ntde
nse
pack
edM
X2
2bo
nds,
hyb
ridiz
atio
nst
ruct
ures
MX
33
....
sp3
tetra
hedr
ate
tr. h
oles
....
d2sp
3oc
tahe
dra
oct.
hole
sM
Xn
ntri
g. h
oles
trig.
bip
yr.
poly
hedr
a
ener
getic
alor
derin
g
Ele
ctro
nica
leffe
cts
such
as ’l
one
pairs
’, lig
and
field
stab
ilizat
ions
Met
albo
ndin
g
Cry
stal
line
mat
eria
ls s
how
a ra
nge
ofdi
ffere
ntbo
ndty
pes:
Met
allic
bond
ing
Bro
aden
ing
ofat
omic
orbi
tal i
nto
ener
gyba
nds
The
broa
deni
ngis
dep
ende
nt o
nth
eor
bita
l ove
rlap
Will
Mg
([Ne]
3s2 )
be
a m
etal
?
Free
elec
tron
gas
The
elec
trons
are
free
to m
ove
thro
ugh
the
solid
as
if it
wer
e an
ele
ctro
n ga
s in
a c
onta
iner
det
erm
ined
by
the
oute
r per
imet
er o
f the
sol
id.
The
Ferm
iene
rgy
Bril
loui
nzo
nes
Mov
ing
elec
trons
do
inte
ract
with
the
atom
ic n
ucle
us!
k2sin
2n
dn
π=
θ=
λ
K,4
,3
,2
,a
aa
aπ
±π
±π
±π
±=
k
Ban
ds in
ioni
can
d co
vale
ntso
lids