Interaction of Atoms and Electromagnetic Waves
Outline
- Review: Polarization and Dipoles
- Lorentz Oscillator Model of an Atom
- Dielectric constant and Refractive index
1
True or False?
-
+
+
+
p
+
++
-
-
E
2. The susceptibility relates the electric field to the
= o χ e E
polarization in this form: P
3. The refractive index can be written n
2
=
−q
-
1. The dipole moment of
= qy , and
this atom is p
points in the same
+q
direction as the
polarizing electric field.
y
o μo
μ
NI
2πr
Refractive Index: Waves in Materials
Index of refraction
How do we get from molecules/charges and fields to index of refraction ?
3
Index of Refraction
νλ = vp = c/n
frequency
wavelength
When propagating in a material,
c → c/n
λ → λo /n
k → ko /n
IR
UV
Microwaves
n
Vacuum
1
Air
1.000277
Water liquid
1.3330
Water ice
1.31
Diamond
2.419
Silicon
3.96
at 5 x 1014 Hz
λ (m) 103 102 101 1 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12
Radio waves
Material
“Hard” X-rays
˜0 ej(ωt−k0 nz) }
E(t, z) = Re{E
“Soft” X-rays
Gamma rays
Visible
4
˜0 ej(ωt−k z) }
E(t, z) = Re{E
Absorption
Photograph by Hey Paul on Flickr.
Why are these stained glass different colors?
˜0 e−αz/2 ej(ωt−k0 nz) }
E(t, z) = Re{E
Tomorrow: lump refractive index
and absorption into a complex
refractive index ñ
Absorption
coefficient
5
Refractive
index
Incident Solar Radiation
How do we introduce
propagation through a
medium (atmosphere)
into Maxwell’s
Equations?
Image created by Robert A. Rohde / Global Warming Art. Used with permission.
6
Microscopic Description of Dielectric Constant
y
-
+ +
+ +
+ +
-
+ +
+ +
+ +
E
-
-
-
NO external
small
amount
of charge
moved by
field E
field
E
Nucleus
Electron
“spring”
Density of dipoles…
P = N δx
Electric field polarizes molecules…
… equivalent to …
7
P = o χe E
Lorentz Oscillator
Lorentz was a late nineteenth century physicist, and quantum mechanics had not yet been
discovered. However, he did understand the results of classical mechanics and
electromagnetic theory. Therefore, he described the problem of atom-field interactions in
these terms. Lorentz thought of an atom as a mass ( the nucleus ) connected to another
smaller mass ( the electron ) by a spring. The spring would be set into motion by an
electric field interacting with the charge of the electron. The field would either repel or
attract the electron which would result in either compressing or stretching the spring.
Bx
Nucleus
Electron
“spring”
Hendrik Lorentz (1853-1928)
Nobel in 1902 for Zeeman Effect
Lorentz was not positing the existence of
a physical spring connecting the electron to an atom; however, he
did postulate that the force binding the two could be described
by Hooke's Law:
where y is the displacement from equilibrium. If Lorentz's system
comes into contact with an electric field, then the electron will
simply be displaced from equilibrium. The oscillating electric
field of the electromagnetic wave will set the electron into
harmonic motion. The effect of the magnetic field can be
omitted because it is miniscule compared to the electric field.
8
Image in public domain
Springs have a resonant frequency
Hooke’s Law
d2 y
m 2 = −ky
dt
d2 y
k
=− y
2
dt
m
Solution
y(t) = Asin(t
So we can write:
ωo2
k
) + Bcos(t
m
k
)
m
ω0
ω0
k
=
m
k = mωo2
9
Microscopic Description of Dielectric Constant
Nucleus
Bx
Electron
“spring”
Damping
Electron mass
field
E
Restoring force
force
(binding electron & nucleus)
10
Solution using complex variables
Lets plug-in the expressions for
and y
into the differential equation from slide 9:
Natural resonance
y
d2
d
q
2
y(t) + γ y(t) + ωo y(t) = Ey (t)
2
dt
m
dt
Ey (t) = Re{Ey ejωt }
+ +
+ +
+ +
2
ω y + jωγy +
p
+q
−q
ωo2 y
y(t) = Re{yejωt }
q
= Ey
m
q
1
y=
Ey
2
2
m (ωo − ω ) + jωγ
11
Oscillator Resonance
Driven harmonic oscillator: Amplitude and Phase depend on frequency
Low frequency
medium amplitude
Displacement, y
in phase with Ey
At resonance
large amplitude
Displacement,y
90º out of phase with Ey
12
High frequency
vanishing amplitude
Displacement y and Ey
in antiphase
Polarization
Since charge displacement, y, is directly related to polarization, P, of our material
we can then rewrite the differential equation:
= o E
+ P
D
Py = N qy
For linear polarization in
ŷ direction
2
d
d
N
q
( 2 + γ + ωo2 )Py (t) =
Ey (t) = o ωp2 Ey (t)
dt
m
dt
2
N
q
ωp2 =
o m
Py (t) = Re{Py ejωt }
ωp2
Py = 2
o Ey
2
(ωo − ω ) + jγω
13
Dielectric Constant from the Lorentz Model
= o E
+ P
D
ωp2
P = 2
E
o
(ωo − ω 2 ) + jγω
2
ωp
E
D = o 1 + 2
(ωo − ω 2 ) + jγω
= o 1 +
(ωo2 −
ωp2
ω2 )
14
+ jγω
Microscopic Lorentz Oscillator Model
ωp2
P = 2
E
o
(ωo − ω 2 ) + jγω
ωp2
N q2
=
o m
= r − ji
15
Real and imaginary parts
ωp2
= 2
(ωo − ω 2 ) + jωγ
ωp2 ωγ
ωp2 (ωo2 − ω 2 )
= 2
−j 2
2
2
2
2
(ωo − ω ) + ω γ
(ωo − ω 2 )2 + ω 2 γ 2
= r − ji
ωp2 (ωo2 − ω 2 )
i = 2
(ωo − ω 2 )2 + ω 2 γ 2
ωp2 (ωo2 − ω 2 )
r = 2
(ωo − ω 2 )2 + ω 2 γ 2
16
Dielectric constant of water
Magnitude
80
60
40
20
1
10
Frequency (GHz)
Microwave ovens usually operate at 2.45 GHz
17
Complex Refractive Index
18
Absorption Coefficient
1
n= √
r + 2r + 2i
2
1
κ= √
−r + 2r + 2i
2
˜0 e−αz/2 ej(ωt−k0 nz) }
E(t, z) = Re{E
Absorption
Refractive
index
2π
α = 2ko κ = 2 κ [cm-1]
λo
19
Absorption
˜0 e−αz/2 ej(ωt−k0 nz) }
E(t, z) = Re{E
Absorption
coefficient
I(z) = Io e−αz
Refractive
index
Beer-Lambert Law or Beer’s Law
20
Key Takeaways
Nucleus
Electron
Lorentz Oscillator Model
d2 y
dy
m 2 = −mωo2 y + qEy − mγ
dt
dt
ωp2
E
P = 2
o
(ωo − ω 2 ) + jγω
ñ = n − jκ
= r − ji
˜0 e−αz/2 ej(ωt−k0 nz) }
E(t, z) = Re{E
Absorption
coefficient
I(z) = Io e−αz
“spring”
Refractive
index
Beer’s Law
21
α = 2ko κ
Decay
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6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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