Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 8 pdf

3.225 21Dispersion • Dispersion can be defined a couple of ways same, just different way – when the group velocity ceases to be equal to the phase velocity – when the dielectric constant

3.225 17

• Usually Clausius-Mosotti necessary due to high density of dipoles

Ionic Polarizability

(  





− + +

=

= +

−

−

2

3

1 3

2

1

ω

ω α

α ε

ε

α

ε

ε

oi o

o tot

r

r

M

e

v

N

By convention, things are abbreviated by using εs and ε∞:

(  



 + +

= +

−

<< + − 2 2

3

1

2

1

,

oi o

s

s

oi

M

e

v α α ω ε

ε

ε

ω

ω

[ + −

∞

+

−

= +

−

>> α α

ε

ε

ε

ω

ω

v

n

n

o

oi

3

1

2

1

2

1

, 2

2







 +

+

=

−

− +

=

∞

2

2 ,

1

2

2

2

2

s oi

T

T

s

r

ε

ε ω

ω

ω

ω

ε ε

ε

ε

εr

ω

ωT

n2=ε∞

εs

© E Fitzgerald-1999

)

)

]

Orientational Polarizability

• No restoring force: analogous to conductivity

H

H

O

p +

-C

p=0

+q -q

θ

For a group of many molecules at some temperature:

T

k

pE T

k

U b

b e e

f

θ cos

=

=

−

After averaging over the polarization of the ensemble molecules (valid for low E-fields):

T

k

p

b

DC

3

~

2

α

Analogous to conductivity, the

molecules collide after a certain

time t, giving:

ωτ

α

α

i

DC

o = 1 −

Dielectric Loss

© E Fitzgerald-1999

• For convenience, imagine a low density of molecules in the gas phase

• There will be only electronic and orientational polarizability

ωτ

ε

ε

ε

α ε

ε

ωτ

ωτ ε α χ

χ

ε

i

n

n

N n i N n

so

r

o DC so

r

o DC o

e

r

−

−

+

=

∴

+

=

=

<<

− +

=

+

+

=

1

3

,

,

1

) 1 ( 3 1

2

2

2 2

We can write this in terms of a

real and imaginary dielectric

constant if we choose:

ωτ τ ω

ε ε

τ

ω

ε

ε

ε

ε

ε

2 2

2 so 2

2

2

2

1 ''

;

1

'

''

'

+

−

= +

−

+

=

+

=

n n

n

i

so

r

Water molecule: τ=9.5x10-11sec, ω~1010

microwave oven, transmission of E-M waves

log ωτ

ε ’ , ε ’’

n2

εso αe+ αi

αe

Dielectric Constant vs Frequency

• Completely general ε due to the localized charge in materials

ω

ε

1

n2

αo

αi

αe

molecules

ions

electrons

Dispersion-free regions, vg=vp

© E Fitzgerald-1999

3.225 21

Dispersion

• Dispersion can be defined a couple of ways (same, just different way)

– when the group velocity ceases to be equal to the phase velocity

– when the dielectric constant has a frequency dependence (i.e when dε/dωnot 0)

k

Dispersion

k c

r

ε

ω =

g r

k

c k

∂

∂

=

=

ε ω

g r

k

c k

∂

∂

≠

=

ω ε

ω

) (

© E Fitzgerald-1999

Spontaneous Polarization

Remember form of orientational polarization:

kT

C kT

p

or= = 3

2 α

With C ≡Curie constant Define a critical temperature Tcby

k

NC

Tc

0

3ε

=

Noting further

Thus

© H.L Tuller, 2001

or

0 3ε

αor

c N T

T =

Fig 1 The Curie-Weiss law illustrated for (Ba,Sr)TiO3 From L.L Hench and J.K West, Principles of Electronic Ceramics, Wiley, 1990, p 243.

1 3

3 0 = 0 =

c

kT C N N

ε ε α

c

c

T T

T

−

χ

• Each unit cell a dipole!

• Large PR(remnant polarization, P(E=0)

• Coercive Field EC, electric field required to bring P back to zero.

Ferroelectrics

E

R

Ro

large enough reverse E-field to get over barrier

E

P

‘normal’ dielectric

Ps

Ec

PR

© E Fitzgerald-1999

3.225 3

Ferroelectrics

• ‘Confused’ atom structure creates metastable relative positions of

positive and negative ions

© E Fitzgerald-1999

Ferroelectrics

Applications

• Capacitors

• Non-volatile memories

• Photorefractive materials

Characteristics of Optical Fiber

• Snell’s Law

n1

n2

θ1

θ2 Refraction

Boundary conditions for E-M wave gives Snell’s Law:

2 2 1

1sinθ n sinθ

n2

n1

θ1

θ2

Internal Reflection: θ 1 =90°

2 1 1

n

n

c

−

=

= θ θ

Glass/air, θc=42°

© E Fitzgerald-1999

• Attenuation

– Absorption

• OH- dominant, SiO2tetrahedral mode

– Scattering

• Raleigh scattering (density fluctuations) αR~const./ λ4(<0.8 µ m

not very useful!)

• Dispersion

– material dispersion (see slide i13)

– modal dispersion

Characteristics of Optical Fiber

x

• Light source always has ∆λ

• parts of pulse with different l propagate

at different speeds Black wave arrives later than red wave

Solution: grade index

y

n

n2

n1

© E Fitzgerald-1999

3.225 7

Characteristics of Optical Fiber

© E Fitzgerald-1999

Characteristics of Optical Fiber

3.225 9

Colors Produced by Chromium

Above: alexandrite, emerald, and ruby

Center: carbonate, chloride, oxide

Below: potassium chromate and ammonium dichromate

© H.L Tuller, 2001

Electron distribution in the ground state of a chromium atom (A) and a trivalent chromium ion (B)

Chromium Electronic Structure

© H.L Tuller, 2001

3.225 11

Interaction of the d orbitals of a central ion with six ligands in

an octahedral arrangement

Octahedral Environment of Transition Metal Ion

© H.L Tuller, 2001

The splitting of the five 3d orbitals in a tetrahedral and an octahedral ligand field

Note: hen the element is a mid-gap dopant, transitions within this element lead to absorption and/or emission via luminescence

Crystal Field Splitting

W

3.225 13

Optical Transitions in Ruby

Optical absorption spectrum tied to Cr transitions in ruby

© H.L Tuller, 2001

Optical Transitions in Emerald

Optical absorption spectrum tied to Cr transitions in emerald

© H.L Tuller, 2001