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