Rayleigh ScatteringOutline

1. Electric fields by charges2. Electric potential of charges3. Dipoles4. The electric potential of a dipole far away5. The “retarded potential” (yes, that’s right) and the speed of light in a vacuum.6. Polarization (induced dipole) of neutral atoms by an external electric field7. Oscillating polarization of matter by a polarized E+M wave – the Rayleigh approximation8. Polarization of bulk matter9. The radiation component of of an oscillating dipole using the retarded potential 10. The definition of polarized Rayleigh scattering11. The intensity and phase function of polarized Rayleigh scattering12. The polarization of the scattered wave13. The intensity and phase function of unpolarized Rayeligh scattering14. Examples from the blue sky15. Examples from radar meteorology16. Back-scatter of a radar signal by a particle17. The importance of -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.

• Static (not moving) charges fill the space around them with an electric field that follows Gauss’ Law

• 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…)

rE ˆ4 2

0rq

q2

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

• 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.

rqV

V

V

041

'

E

drEr

Dipoles• A “dipole” is simply two opposite charges

separated by some distance s

• Both E and V are additive

• 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.

rrqV 114 0

r+r-

s

The Electric Potential far from a Dipole

• We simplify things by defining a “center” of the dipole for our coordinate system

• Far away from the particle, we can do a Taylor expansion to get

• Putting this into our dipole potential and doing another Taylor expansion yields

• We define a “dipole moment”, which captures the relevant properties of the dipole far from the charges

r+r-

s

r

22

2cos

21

srsrr

cos24 2

0

srqV

cos

41

rsrr

• 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.

• 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

sp q 204ˆr

Vrp

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

• The main point here is that if the two sides of a dipole are moving (oscillating, in our case), then:

)/(41

0 crtrqV

v

V(t)

r(t)

r(t-r/c)

V(t)

200 4

ˆ)/()/(

1)/(

14

)(rcrt

crtrcrtrqtVd

rp

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.

• The potential created by the new dipole, Vd, is governed by its moment, and added to the potential by the external field:

+

No electric field

ElectronShell (-)

Nucleus (+)

+-

Induced Dipole

Eexts

extq Esp

20

20 4

ˆ4

ˆrr

V extd

rErp

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

• This field instantaneously causes a dipole moment in the atom

• This implies that the dipole potential, Vd, will also oscillate, creating, in effect, a wave of its own.

• This is Rayleigh scattering.

Induced Dipole

)/~2cos(

)~2cos()(

0

0

ct

ttext

E

EE

)/~2cos()( 0 ctt Ep

Particulate Rayleigh Scattering by E+M Waves• A bulk medium has an induced dipole

moment per unit volume P.

• 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.

• The total dipole moment of a polarized sphere is

• The degree to which they shift is quantified by the electrical susceptibility of the medium, 0, 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.

E0

-

+

-

+

exteEP 0

extpep RR EPp 30

3

34

34

NN

Particulate Rayleigh Scattering by E+M Waves• Because of this polarization, the electric

field is reduced below that of the external field

• 1 + e 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.

• We often don’t know the polarizability of an individual atom, , and instead compute it from that of the bulk medium.

exte

EE

11

int

NN

N

NR

RNR

NR

R

e

ext

ext

i

exti

0

0

3

0

3

3

0int

int

333

1

34

41

343441

Ep

pEp

pp

pE

EEp

3

3

extnEE 2int

1

33

21

0

e

e

N

KK

Radiation of an oscillating dipole• Now let’s go back to the retarded

potential of an atomic dipole

• We simulate the dipole oscillation as

• Where, you recall,

• 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?

r+r-

s

r

)/(1

)/(1

4),(

0 crtrcrtrqtVd

r

cos

4)(1)(rtsrtr

)()/2cos()( 0

tqctt

sEp

)/2cos()( 0 ctq

t Es

Radiation of an oscillating dipole• 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.

• A small term does show up with 1/r dependence…

• Remember,

r+r-

s

r

)/(2sin2cos

)/(2sin2cos),(

0

0

0

0

crtcr

E

crtcr

qstVd

r

VE 242

0

20

24

20

02

4sin),(

ˆ)/(2cos4

sin),(

rIctI

crtcr

Ect

r

θrE