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ChemistryChemistry-251
Questions for Monday the 14th
March 8, 2004
• Describe the d-levels in a tetrahedrally coordinated
transition metal compound
I
Cr3+
[Cr(I)6]3-
Matter and Light
• Optical properties of matter
• Properties of waves in matter
• Some applications
• Dielectric function
Figure 21.26 The d Orbitals in a Tetrahedral Arrangement of Point Charges
C M S M a s t e r s : Electronic Structure of Solids: Light
Optical Properties
© B.Dam2005
The spectrum
Wavelength mm
Energy eV
Frequency Hz
• The optical phenomena reflect the electronic
properties of a material
Visible spectrum
wavelength
0.4 µm
• Human perception of solids depends on their
interaction with visible light (1.5-3 eV):
– why are metals shiny and 'metallic' ?
– why are diamonds transparent ?
– why is white paint white ?
1 Angstrom
1 nanometer
Blue
0.5 µm
Green
0.6 µm
Orange
Yellow
1 micrometer
1 millimeter
1 meter
C M S M a s t e r s : Electronic Structure of Solids: Light
Violet
0.7 µm
Red
1 kilometer
© B.Dam2005
1
Optical Properties
Translucency
• Reflection
– Scattering at interfaces of materials with different
index of refraction n
– The larger ∆n, the larger the scattering
• Scattering may occur
at:
• grain boundaries in
poly-crystalline
materials
• fine pores in
ceramics
• The boundaries of
a phase mixture
– Translucency
• Light scattered at interfaces within a transparant
material
Privacy window with polymer
dispersed liquid crystals
C M S M a s t e r s : Electronic Structure of Solids: Light
Sol-gel mixture
© B.Dam2005
Absorption
Absorption by electronic excitation
• Transition between well-defined electronic levels
http://www.nhn.ou.edu/~kieran/reuhome/vizqm/figs/hydrogen.gif
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
Energy scales
Absorption in a gas or liquid
• 26 meV ~ 300 K
• 1eV/atom~ 96 kJ/mol
k = 2π/λ = 2πν
k = 2π/λ = 2πν = E/hc
• Vibration spectra of weakly interacting molecules
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
•
λ = 1 µm
ν = ??? cm-1
•
λ = 1 µm
Ε = ??? eV
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
B.Dam2004
2
Impurities give large bandgapbandgap-crystals color
Impurity absorption in sapphire
• Corundum
• Al2O3
• Ruby, sapphire, topaz,
amethyst
http://home.achilles.net/~jtalbot/glossary/photopumping.gif
http://www.valleydesign.com/images/sapp.jpg
– Colorless in pure form
– Impurities result in
different colors
• A similar technique is used to colour glasses or
pottery glaze by adding impurities into the molten
state:
The Cr levels give strong
absorptions at 400 nm (green) and
600 nm (blue) leaving only red to
be transmitted
– Cu2+: blue-green, Cr3+: green
– Co2+: blue-violet, Mn2+: yellow
Small particles show size dependent absorption
Absorption
• Transition between well-defined electronic levels
• Electron excitation across the band gap
• Electron excitation to defect levels in the band gap
• Electronic Polarization
C M S M a s t e r s : Electronic Structure of Solids: Light
Absorption by electronic excitation
Band Gap Energy and Color
optical spectroscopy can
give an idea of band-gap
Color that
corresponds to
band gap energy
Apparent color
of material
(unabsorbed light)
ultraviolet
colorless
Absorption
Bandgap energy (eV)
4
3 violet
© B.Dam2005
yellow
blue
green
2
1
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
orange
yellow
red
infrared
red
.
shiny
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
3
Absorption
Absorption by electronic excitation
• Transition between well-defined electronic levels
optical spectroscopy can
give an idea of band-gap
• Electron excitation across the band gap
• Electron excitation to defect levels in the band gap
• Electronic Polarization
Conduction
Band
Valence
Band
insulator = gap in VUV
semicon.
semicon. = gap at
border IR/vis
IR/vis..
C M S M a s t e r s : Electronic Structure of Solids: Light
Absorption by electronic excitation
Band gaps & excitons (non(non-metals)
excitons
peaks above threshold: exciton
Eg
electron:
-ve
Eopt
hole: +ve
insulator = gap in VUV
© B.Dam2005
Frenkel or Wannier
excitons....
......details come in
Ch. 7
semicon.
semicon. = gap at
border IR/vis
IR/vis..
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
Band gaps & excitons (non(non-metals)
Absorption
excitons
• Transition between well-defined electronic levels
mean that the optical spectra don't give
the true band-gap
Eopt
• Electron excitation to defect levels in the band gap
photoconductivity
or
• Electronic Polarization
PES - IPES:
N -1
N+1
• Electron excitation across the band gap
PES-IPES
EgPES= 3.5 eV
4
Absorption by polarization
Matter and Light
•
The electronic polarizability of large atoms leads
to absorption
• Optical properties of matter
•
This also leads to a reduction of the speed of
light!
• Properties of waves in matter
• Some applications
• Dielectric function
The electronic polarization increases for larger atoms
•The addition of Pb to glass causes
•a slightly darker color (absorption)
•an enhanced refractivity
Linked?
Electromagnetic Wave in vacuum
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
Electromagnetic Wave in a medium
ε = permittivity
µ = permeability
o - in a vacuum
r - relative
Speed of light is related to
electric permittivity and
magnetic permeability
c=
1
ε o µo
C M S M a s t e r s : Electronic Structure of Solids: Light
Electromagnetic Waves
εµ
=
ε οµ o
n=c =
v
= 3 x 108 m / s
E ( x ) = E0e − iωt eikx
E ( x ) = E0e − iωt eikx
In a medium:
In a medium:
n ≅ εr
© B.Dam2005
Electromagnetic Waves
In vacuo:
ω 2π
k= =
λ
c
nω 2nπ
=
c
λ
n = n'+in' '
n ≅ εr
εr µ r
C M S M a s t e r s : Electronic Structure of Solids: Light
© B.Dam2005
In vacuo:
ω 2π
k= =
λ
c
k=
1
v= 1
ε οµ o
εµ
c=
nω 2nπ
=
c
λ
n = n'+in' '
k=
n ≅ εr
E ( x ) = E0e −iωt e
iω ( n ' + in '') x
c
⎛ iωn ' x
⎞
−iωt ⎟ − ωn '' x
⎜
c
⎠
c
E ( x ) = E0 e ⎝
E ( x ) = E0e
e
⎛ i 2πn ' x
⎞
−iωt ⎟ −2πn '' x
⎜
⎝ λ
⎠
λ
e
5
Absorption coefficient
Electromagnetic Waves
⎛ i 2πn ' x
⎞
−iωt ⎟ −2πn '' x
⎜
λ
⎠
λ
In vacuo:
ω 2π
k= =
c
λ
E ( x ) = E0 e ⎝
E ( x ) = E0e − iωt eikx
−4πn '' x
I ( x ) = EE* = E02e
In a medium:
Decay of wave amplitude
nω 2nπ
k=
=
c
λ
n = n'+in' '
e
α=
λ
= I 0e −αx
4πn(ω )' '
λ
Phase velocity
⎛ i 2πn ' x
⎞
−iωt ⎟ −2πn '' x
⎜
λ
⎠
λ
n ≅ εr
E ( x ) = E0e⎝
e
Absorption coefficients of various semiconductors
Complex index of refraction
Photon energy (eV)
5 4
3
1
2
0.9
0.8
0.7
108
Ge
In0.7Ga0.3As0.64P0.36
In0.53Ga0.47As
Si
106
GaAs
InP
m
α((m-1-1))
107
⎛ I ( x) ⎞
⎟⎟
αx = − ln⎜⎜
⎝ I0 ⎠
105
Drastic increase in
α at energies larger
than the bandgap
except for Si!
a-Si:H
⎛
iσ ⎞
⎟ = ε ' + iε ' '
ε r = ⎜⎜ ε 'r +
ωε ⎟
n = µ rε r ≈ ε r
⎝
0
⎠
Note the removal of the subscript
If εr has imaginary parts, the refractive index is:
complex:
n=n’-in”
and ω-dependent: n(ω )
104
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
n(ω ) = n'+in"
n ≅ εr
103
1.8
Wavelength (µm)
n'2 −n"2 = ε '
2n' n" = ε "
Fig. 9.19: Absorption coefficient (α) vs. wavelength (λ) for
various semiconductors (Data selectively collected and combined
from various sources.)
Vacuum/dielectric interface
Vacuum/dielectric interface
T+A+R = 1
ER
ET
EI
I0 = e-αz
α = (4πn”)/λ
n2
n1 = n1’ + in1”
ET
n2
n1 = n1’ + in1”
n = n1 /n2 = n’ + in’’
n = n1 /n2 = n’ + in’’
C M S M a s t e r s : Electronic Structure of Solids: Light
ER
From matching the
tangential E and Hcomponents across the
EI
boundary one finds for
the reflection coefficient
R = ER/EI
⎛ 1 − n ⎞ (n'−1) + n"
R=⎜
⎟ =
2
2
⎝ 1 + n ⎠ (n'+1) + n"
2
© B.Dam2005
2
2
6
Absorption and reflectivity
Why are Metals Shiny?
proportion of light reflected by a solid, R :
2
⎛ 1 − n ⎞ (n'−1) + n"
R=⎜
⎟ =
2
2
⎝ 1 + n ⎠ (n'+1) + n"
2
2
R is large when:
n' >> 1 or n' << 1
n" >> 1
n ' = normal refraction
n '' = absorption
• Metals Eg = 0 eV
• All light (with λ above X-ray wavelengths) absorbed by
continuous number of unoccupied states above Ef.
• Light is reemitted with exact energy of absorption as
electrons fall back into lowest state. Metals appear
reflective as the light we see is re-emitted.
Si:
Si: Eg = 1.1 eV
in vis.
vis. light it's
reflective and
'metallic'.....
Ef
occurs for E > Eg (non(non-metal)
absorption
emission
Switchable Absorbing Mirrors: 3 of a kind
Matter and Light
• Optical properties of matter
1st generation:
• Properties of waves in matter
RE-Hx (YHX)
Huiberts et al, VU
• Some applications
2nd generation:
• Dielectric function
GdMgHX
Vd Sluis Philips
3rd generation:
Black
state
Mg2NiHX
Giebels + Lohstroh VU
The MetalMetal-Insulator transition in YHx
Hcp
fcc
hcp
10
1 .0
YH0
1
YH3
YH2
0 .8
0 .6
0 .4
0 .1
0 .2
0
500
1000
1500
Transmission (a.u.)
© B.Dam2004
Resistivity (mΩ cm)
C M S M a s t e r s : Electronic Structure of Solids: Light
2000
T im e (s )
Huiberts et al. Nature (1996)
7
Visualisation of H diffusion in Y
Den Broeder, van der Molen et al, Nature 394 (1998) 656
H
Y2O3
Pd
Y
3
H/Y
2
hcp- γ
hcp- γ
β
metal
insulator
1
α
0
The MI transition depends on the HH-concentration only
10
Van Gogh et al. PRL
(1999)
10
YH3
2.90
2.86
1
101
0.1
10-1
0.01
-2
0.1
ρ
0.2
0.08
0
500
1.9
1.0
0.8
0.6
0.4
YH2
0.2
1000
1500
Transmission (a.u.)
1
2.93
102
Resistivity (mΩ cm)
10
ρ (mΩcm)
x=H/Y
YHx
YDx
ρ (mΩcm)
100
Switchable mirror applications
•
•
•
•
3
Switchable mirrors for office building windows
Switchable absorbers for solar collectors
Optical hydrogen sensors
Search for new hydrogen storage materials
– Complex, light-weigth metal hydrides appear to be
insulating.
– Their bandgap can be used for identification
2000
Time (s)
1.8 2.0 2.2 2.4 2.6 2.8 3.0
10
x = H(D)/Y
1
10 T (K) 100
The biggest problem in storage is weight!
45
40
MJ/kg
MJ/ltr
35
System energy
densities
30
25
C M S M a s t e r s : Electronic Structure of Solids: Light
Hoekstra, Rosenbaum et al (2001)
The energy density of
hydrogen in a MgH2 based
hydrogen tank (6 wt % of
hydrogen) is 6x less than
that of gasoline
© B.Dam2004
The AJA sputter system
3”
Turbo
20
15
Argon box
TSP
10
5
0
Li battery
MgH2
LH2 in
tank
700atm gasoline
H2 gas
P < 10-8 mbar
6 DC/RF
Turbo
8
Optical screening for hydrogen storage properties
Mg rich
Ni-rich
Matter and Light
• Optical properties of matter
• Properties of waves in matter
Mg rich
MgH2
Ni-rich
• Some applications
3 CCD
Camera
• Dielectric function
Mg2NiH4
?
Loading cell
300 oC / 10 bar H2
Light source
C M S M a s t e r s : Electronic Structure of Solids: Light
Absorption: simple model
Absorption: Lorentz model
absorbing mode:
electron on a spring
Now put the oscillators in the solid,
with N per unit volume
force constant:
vibrational frequency ω0
interaction with surroundings:
decay time τ
ω0
ω0
ε’
© B.Dam2004
ε’
ε”
ε”
damping,
= plasma frequency
Single absorbtion line at ω0
ε (ω) has real (ε ') and imaginary (ε ") parts
Width ~1/τ
Absorption: Lorentz model
Absorption: Lorentz model
strongest absorption
Now put the oscillators in the solid,
with N per unit volume
ε ' (ω ) = 1 +
ω0
ε’
ε(ω)
ε”
ε ' ' (ω ) =
ω p 2 (ω0 2 − ω 2 )
strongest absorption
(ω0 − ω 2 ) 2 + ω 2 / τ 2
2
ω p 2ω / τ
(ω0 − ω 2 ) 2 + ω 2 / τ 2
2
ε’
ω0
ε”
ε (ω ) = 1 +
ωp2
ω 0 2 − ω 2 + iω / τ
Shift is small if τ >> ω0
ω p ω /τ
2
(ω0 − ω ) + ω / τ
2
2 2
2
Width of ε’’ ~ 1/τ
2
ω p (ω0 − ω 2 )
2
ε '= 1+
n’
n”
Neglect local fields
2
ε ''=
(ω0 − ω 2 ) 2 + ω 2 / τ 2
2
9
Absorption: Lorentz model
ε ' (ω ) = 1 +
ε ' ' (ω ) =
ω p 2 (ω0 2 − ω 2 )
strongest absorption
(ω0 − ω ) + ω / τ
2
2 2
Absorption: Lorentz model
2
ω p 2ω / τ
(ω0 − ω ) + ω / τ
2
2 2
2
2
ε ' ' (ω ) =
ωp
ε’
ε”
ω
2
n' (ω ) ≈ 1 −
2
ω p2 /τ
ω3
ε ' ' (ω ) =
ω0
ω > ω0
ε ' (ω ) = 1 −
ε ' (ω ) = 1 +
2
n' ' (ω ) ≈
ωp
2
2ω
2
ε ' (ω ) = 1 +
ε ' ' (ω ) =
3
ε ' ' (ω ) =
ω p 2 (ω0 2 − ω 2 )
strongest absorption
ω p 2ω / τ
(ω0 − ω ) + ω / τ
2
2
n(ω ) = n'+in"
n'2 −n"2 = ε '
2n' n" = ε "
ωp
2
ω0
2
ω0
n' ' (ω ) <
4
ε ' ' (ω ) =
ω0
ε’
ε”
ε(ω)
ωω p 2 / τ
2ω0
4
ω p 2 (ω0 2 − ω 2 )
strongest absorption
(ω0 − ω 2 ) 2 + ω 2 / τ 2
ω0
ω p 2ω / τ
(ω0 − ω ) + ω / τ
2
2 2
2
ε’
2
n'2 −n"2 = ε '
2n' n" = ε "
ε”
ε(ω)
Trm. Abs. Refl. Trm
n(ω ) = n'+in"
n(ω)
Gross structure
(for solid with a single absorption mode):
increasing ω or E
2ω 2
1
Absorption and reflectivity
n(ω)
ω
⎛ 1 − n ⎞ (n'−1) + n"
R=⎜
⎟ =
2
2
⎝ 1 + n ⎠ (n'+1) + n"
2
2
2
T+A+R=1
I. The best white pigment....?
White:
no absorption: gap > 3 eV
strongest absorption
high transmission at low E
High reflectivity in visible
ω0
n' (thus e’(ω)) as big as possible
ε’
absorption edge just above 3eV
ε”
high reflectivity
transmission again
ωp
1
2
2
ω
absorption band
ω0
ε’
ε”
n' (ω ) ≈ 1 +
ωω p 2 / τ
ε ' (ω ) = 1 +
(ω0 − ω 2 ) 2 + ω 2 / τ 2
2 2
(ω0 − ω 2 ) 2 + ω 2 / τ 2
Absorption: Lorentz model
2
2
ω p 2ω / τ
2
ω < ω0
Absorption: Lorentz model
ε ' (ω ) = 1 +
strongest absorption
(ω0 − ω 2 ) 2 + ω 2 / τ 2
1
ω p2 /τ
2ω
ω p 2 (ω0 2 − ω 2 )
2
Transm. Abs. Refl. Transm
n’
n”
strong scatterer
e.g powder
rutile: TiO2
!
also cheap and non-toxic
10
MgO
QM and the oscillator model?
Oscillator frequency equals
energy between quantum levels
The optical behaviour of MgO
(continuous lines) can roughly
be understood on the basis of R
single oscillater model (broken
line)
Decay time τ (inverse of the
damping γ) is related to the
transition probabilities to all
available quantum states
ω0 ~ 14 eV
2
γ= 1/τ = 8 eV
n
strongest absorption
MgO
ω0
ε’
ε”
The gap is 7.6 eV, the peak
just below is due to an exciton
1
Transm. Abs. Refl. Transm
k
n’
20
10
Absorption and reflectivity: metals
Absorption and reflectivity: metals
reflects
A metal has no gap: absorption frequency ω0=0
ωp2
ω 2 + iω / τ
ωp
2
ωp
ω2
⎛ Ne 2 ⎞
⎟⎟
= ⎜⎜
⎝ ε 0m ⎠
n purely
imaginary
ε”
2
ε (ω ) = 1 −
photon
frequency (ω)
ω0
ε’
transmits
dielectric const. ε(ω)
negative
Also the damping can be neglected
ε (ω ) = 1 −
n”
⎛ Ne 2 ⎞
⎟⎟
ω p 2 = ⎜⎜
⎝ ε 0m ⎠
n'=0
N is density of
conduction
electrons
n’
n”
n = n ' - i n ''
for E < plasma frequency, metals reflect
High reflection already at ω=0
Plasma frequency
depends on electron density, N
Absorption and reflectivity: metals
From optical measurements one deduces the plasma
frequency
for most metals, ωp is in the far UV
→ high reflectivity in visible
= 'shiny'
l
Metal
Li
Na
K
Al
Ag
E (calculated)
8.04
6.05
4.40
15.8
8.98
E (measured)
6.64
5.94
4.15
14.7
3.93
Due to ‘bound’ charges
11
Silver
Aluminium
In the dielectric function of
silver you find an anomaly at
ωp which is due to interband
transitions
This metal has only
little bound charges at
2 eV, resulting in a
slightly reduced
reflectivity
Accordingly, a dip in the
reflectance is found at ~4 eV
The reflectanc remains low
for E > ~9eV
Thus the spectrum is
composed of a free electron
part εf and a bound electron
part εb
Plasma frequency
Plasma frequency Sn1-xSbxO2 (x=3%)
depends on electron density, N
So the carrier density is a
new parameter to play with!
for most metals, ωp is in the far UV
→ high reflectivity in visible
= 'shiny'
ωp in IR: transparent
low N metal:
e.g. doped metal oxide
Sn1-xSbxO2 (x=3%)
EELS
Reflection in the IR (due to the low electron density):
good IR-mirror.
It is a good conductor too
The bandgap is >3eV, transparant in the visible
II. Switchable Absorbing Mirrors
The switching of complex Mg2NiH4
c
1st generation:
a
RE-Hx (YHX)
z
Huiberts et al, VU
x
z
x
y
y
2nd generation:
b
GdMgHX
Vd Sluis Philips
3rd generation:
Mg2NiHX
Giebels + Lohstroh VU
c
b
a
Black
state
Spacegroup P6222
g = 120 o
a = 0.52 nm
b = 0.52 nm
c = 1.32 nm
Mg2Ni
Symmetry
Stacking
Bonding
30 % volume
Spacegroup C2/c
Angle b = 113.5o
a = 1.43 nm
b = 0.64 nm
c = 0.65 nm
Mg2NiH4
12
Reflection of Mg2NiHx
The ‘impossibility of the black state
3.0
2.5
Photon
2.0
1.5
energy
Lo
1.0
(eV)
3nm Pd
RT / up to 1bar H2
Lambert-Beer
For a 220 nm film
we need n’’ > 2
to explain
T < 0.0001
T ≅e
4.0
0.6
(n'−n0 )2 + n' '2
R≅
(n'+ n0 )2 + n' '2
Reflectance
Reflectance
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
1000
2000
3000
s)
e ( Isidorsson et al.
4000
im
APL80 (2002) 2350
5000
gt
n
i
220nm Mg2.17Ni
6000
ad
3.5
3.0
ω
− 2 n ''d
c
2.5
2.0
1.5
0.4
0.4
0.3
0.2
0.2
0.1
0.0
4.0
0.1
T
3.5
Lohstroh PRL 89 (2004) 197404
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
Photon energy (eV)
Film Surface
Mg2Ni columns
20 x 200nm
Mg2NiH4
substrate palladium
Interface
Substrate
Mg2Ni
0.5
then R > 0.4 !
Mg
22NiH
Mg
NiH0.30.3
Pd
Mg2NiH0.3
0.0
0.6
0.3
Pd caplayer
palladium substrate
Pd
0.5
Thin film grain size and loading behaviour of Pd/Mg2Ni
The black state viewed from two sides
Pd
1.0
R
0.5
Mg2NiH4
Mg<2εNiH
> 4
Substrate
• Smallest grains near interface
• Smallest grains have lowest activation energy for transition
• Surface energy confines the phases in two layers
Westerwaal MH2004
Black state solar absorption Mg1.7NiHx
Area Solar Spectrum Area absorption
The emissivity at around
100 oC is less than 16%
1238.1
959.4
0,9
700
0,8
600
0,7
500
Total
absorption
78 %
400
300
200
Black body power output
5 < λ < 50 µm
Solar power input within
0.25 < λ < 2 µm
0,6
0,5
0,4
0,3
Atmospheric transmittance
around 10 µm
0,2
100
0
0,0
Absorption
Solar spectrum
SS x Absorption
800
1,0
Power input/output
0,1
0,5
1,0
1,5
2,0
2,5
3,0
Energy [eV]
3,5
4,0
0,0
4,5
Sensitivity of the
human eye
0.4 > λ < 0.7 µm
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Project goal
Solar contrast: reflective/absorbing state
Find a layer which is adapting its optical properties according to solar
input and heat output
Solar absorbance
Reflecting state ~ 40%
VAREM
Solar collector
Solar absorbance
Black state
~ 80%
H2O
VAREM: Variable absorbing and reflecting mirror which switches
between two states:
1)Absorb the solar spectrum, without emitting or absorbing heat
2)Reflect the solar spectrum
Multiple absorptions
In many structures both electronic
and vibronic absorption is seen
Multiple absorptions
strongest absorption
Added contribution due to slow lattice vibrations
ε ' (ω ) = 1 +
εopt is related to the bandgap!!
Ge (0.7 eV) => ε = 16
NaCl (8.5 eV)=> ε = 2.3
ωe average
excitation
energy
ω0
ε’
ε”
ω <<n’ω0
strongest absorption
Added contribution due to slow lattice vibrations
ω0
ε’
In many structures both electronic
and vibronic absorption is seen
ω << ωvn’
n”
ωp2
ωe 2
ε”
εs related to lattice vibrations
Ge (0.7 eV) => ε = 16
NaCl (8.5 eV)=> ε = 2.3
ε 's (ω ) = ε opt (ω ) + A
n”
NeT
Mε 0ωv 2
ωv vibrational frequency
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