Transcript Rayeligh_Scattering

Rayleigh Scattering
Outline
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Electric fields by charges
Electric potential of charges
Dipoles
The electric potential of a dipole far away
The “retarded potential” (yes, that’s right) and the speed of light in a vacuum.
Polarization (induced dipole) of neutral atoms by an external electric field
Oscillating polarization of matter by a polarized E+M wave – the Rayleigh approximation
Polarization of bulk matter
The radiation component of of an oscillating dipole using the retarded potential
The definition of polarized Rayleigh scattering
The intensity and phase function of polarized Rayleigh scattering
The polarization of the scattered wave
The intensity and phase function of unpolarized Rayeligh scattering
Examples from the blue sky
Examples from radar meteorology
Back-scatter of a radar signal by a particle
The importance of l-4 and R6.
Electric Fields by Charges
•
•
All molecules are made of protons and
electrons, which are positively and
negatively charged particles having the
fundamental unit of charge e.
E
q
40 r
2
rˆ
Static (not moving) charges fill the
space around them with an electric
field that follows Gauss’ Law
q2
•
If there is another charge, q2, at
location r2, it will feel a force equal to
q’s electric field times its own charge.
F = E(r2)q2
•
(This second charge will also create its
own electric field that will exert a force
on the first one…)
r2
q
Electric Potentials by Charges
•
Because E depends ONLY on distance
from the particle, we can simplify its
representation by defining an electric
potenial, V

V   E  dr '
r
•
We just made the math way simpler by
describing all the vector information in
E within a scalar V. E has the
magnitude and direction of the
downhill gradient in V.
•
If you take a 2-D slice through space
centered on a point charge, and then
plot V as a third dimension, you end
up with “volcanos” around positive
charges, and funnels below negative
ones.
•
You can visualize the downhill gradient
as being the force on a positive
charge. A negative charge will “fall”
uphill.
E  V
1
q
V
40 r
Dipoles
•
A “dipole” is simply two opposite
charges separated by some distance s
•
Both E and V are additive
V
q 1 1
  
40  r r 
•
Very far from two oppositely charged
particles, their fields tend to cancel
(same magnitude, opposite direction)
•
Between the particles, the fields
reinforce one another
•
In the plane perpendicular to s, the
part of the field directed away from the
particles cancels. What’s left behind is
a field that’s parallel to their separation
vector, s.
r+
r-
s
The Electric Potential far from a
Dipole
•
We simplify things by defining a “center” of
the dipole for our coordinate system
1
s
r  r 2  rs cos    
2
2
•
Far away from the particle, we can do a
Taylor expansion to get
•
Note V also falls off like s/r2 , which is
faster than for the 1/r dependence of
the point charges alone, due to the
cancellation
r+
r-
Putting this into our dipole potential and
doing another Taylor expansion yields
V
•
Note that the dipole potential is zero in
the plane z = 0 (i.e.  = 90º).This does
NOT mean the electric field is zero,
since there is still a gradient.
2
s


r  r 1  cos  
 4r

•
•
q
s
cos 
40 r 2 2
 r
s
We define a “dipole moment”, which
captures the relevant properties of the dipole
far from the charges
p  qs
V
p  rˆ
40 r 2
The Retarded Potential
•
This simply means that if a charge is
moving (i.e. r = r(t)), the potential far
from the particle does not
instantaneously reflect the new
position of the particle
•
Instead, the “information” travels at the
speed of light (through a vacuum).
•
Thus the electric potential (NOT field)
at a location far from the particle is
given by
V
•
q
40 r (t  r / c)
r(t-r/c)
r(t)
v
1
V(t)
V(t)
The main point here is that if the two
sides of a dipole are moving
(oscillating, in our case), then:
 ! p(t  r / c)  rˆ
q 
1
1

 
Vd (t ) 

40  r (t  r / c) r (t  r / c) 
40 r 2
The point “sees” the + and –
sides of the dipole at different
times in its history! This matters!
Polarization of Neutral Atoms by an Electric field
•
Atoms are normally electrically neutral,
because the centers of charge of the
positive nucleus and negative electron
shells are collocated.
•
When a uniform external electric field
Eext surrounds an atom, the nucleus is
pulled in the direction of Eext and the
electron shell is pulled oppositely.
•
Now that the charges are not
collocated, a new electric field is set up
such that the attraction between the
nucleus and the electrons balances the
effect of Eext.
•
Far from the atom, this is perceived as
an induced dipole moment, p. That is,
the atom has become polarized by Eext.
No electric field
Nucleus (+)
+
Electron
Shell (-)
Induced Dipole
Eext
s
+
-
p  qs  Eext
•
The potential created by the new
dipole, Vd, is governed by its moment,
and added to the potential by the
external field:
Vd 
E ext  rˆ
p  rˆ

40 r 2 40 r 2
Atomic Rayleigh Scattering by E+M Waves
•
E+M waves are simply self-propagating
electric (and magnetic) fields
•
If an atom is small compared to the
wavelength of radiation passing by it, it
experiences an effectively uniform field
around it
Eext (t )  E0 cos( 2~t )
 E cos( 2~ct / l )
0
•
This field instantaneously causes a
dipole moment in the atom
p(t )  E0 cos( 2~ct / l )
•
This implies that the dipole potential,
Vd, will also oscillate, creating, in effect,
a wave of its own.
•
This is Rayleigh scattering.
Induced Dipole
Particulate Rayleigh Scattering by E+M Waves
•
A bulk medium has an induced dipole
moment per unit volume P.
P   0 c e Eext
•
E0
Instead of thinking about each individual
atom becoming a dipole, it’s more
convenient to think of two continuous
clouds of charge – one positive, and one
negative that get shifted by an external
field.
N
•
-
+
The total dipole moment of a polarized
sphere is
4
4
p  R 3p P  0 c e R 3p Eext
3
3
•
+
The degree to which they shift is
quantified by the electrical susceptibility
of the medium, c0, which will be related
to the polarizability, , of the atoms
within, and how tightly packed together
these atoms are, N. The field within the
medium feeds back on the atoms, and so
you can’t just sum up all the Vps if the
atoms were alone.
-
Particulate Rayleigh Scattering by E+M Waves
•
Because of this polarization, the electric
field is reduced below that of the external
field
E int 
•
1
E ext
1  ce
1 + ce is also called the dielectric
constant, K, where K = n2 and n is the
index of refraction we’ve already
introduced. These are all different sides
of the same die.
E int 
•
1
E ext
n2
We often don’t know the polarizability of
an individual atom, , and instead
compute it from that of the bulk medium.

K 1
K 2
3 c e
 0
N ce  3
p i   E ext  Eint 
Eint
1
p

40 R 3
4
p  R 3p i N
3
4 3 
1 p 

p  R N  E ext 
3 
3
40 R 

4 3
R NE ext
p 3
N
1
3 0
ce 
3N
3 0  N
Radiation of an oscillating dipole
•
Now let’s go back to the retarded
potential of an atomic dipole
Vd (r, t ) 
•

q 
1
1



40  r (t  r / c) r (t  r / c) 
•
The particle now sees the potential
from the far side of the dipole later
than it sees the near side. We won’t
work out the details (because it gets
cumbersome), but how does this work
out?
We simulate the dipole oscillation as
p(t )  E0 cos( 2ct / l )
 qs(t )
s(t ) 
•
E0
q
r+
r-
cos( 2ct / l )
Where, you recall,
 s (t )

r (t )  r 1 
cos  
4r


 r
s
Radiation of an oscillating dipole
•
r+
In the end, there’s a large term that
looks simply like the 1/r2 dipole
potential. We don’t care about this,
because for it to radiate energy
indefinitely, we need a 1/r term.
r r
s
•
A small term does show up with 1/r
dependence…
qs0 cos 
 2c

sin 
(t  r / c) 
2 0 lr
 l

E cos   2c

 0
sin 
(t  r / c) 
2 0 lr
 l

Vd (r, t ) 
•
Remember,
E  V
c 2E0 sin 
 2c
ˆ
E(r, t ) 
cos
(
t

r
/
c
)

θ
40 l2 r
l


c 4 2 I 0 sin 2 
I (r, t ) 
40 2 l4 r 2