Download Lecture Notes 14.5: Classical Non-Relativistic Theory of Scattering of EM Radiation, Thomson and Rayleigh Scattering

UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Classical, Non-Relativistic Theory of Scattering of Electromagnetic Radiation
We present here the theory of scattering of electromagnetic radiation from the classical, nonrelativistic physics approach. A quantum-mechanical, fully-relativistic of this subject matter can
also be obtained via the use of relativistic quantum electrodynamics {QED}, however, this more
sophisticated level of treatment is simply beyond the scope of this course.
“Generic” Theoretical Definition of a Scattering Cross Section:
There are all sorts of scattering processes that occur in nature; generically all of them can be
defined in terms of a scattering cross section  scat , which has SI units of area, i.e. m2.
In classical, non-relativistic physics the total scattering cross section  scat for a given EM
wave scattering process defined as the ratio of the total, time-averaged radiated power Prad  t 
( Watts ) radiated by a “target” object (e.g. a charged particle, an atom or molecule, etc.)
undergoing that scattering process, normalized to the incident EM wave intensity
 

I inc  r  0   Sinc  r  0, t  ( Watts m 2 ), evaluated at the location of the target/scattering

object (usually at the origin,   r  0  ):
Total Scattering Cross Section:  scat 
Prad  t 
Prad  t 
  

I inc  r  0 
Sinc  r  0, t 
(SI units of area, m2)
This relation is known as the total scattering cross section – because it contains no angular
 ,   information about the nature of the scattering process – these have been integrated out.
Physically speaking, the total scattering cross section can be thought of as an effective crosssectional area (hence the name cross section) per scattering object of the incident wave front that
is required to deliver the power that is scattered out of the incident wave front and into 4
steradians (i.e. into any/all angles  ,  ) – as shown schematically in the figure below.
The scattering object absorbs energy/power from the cross sectional area  scat in the incident EM
wave and then re-radiates (i.e. scatters) this absorbed energy.
Scattered
EM Wave
Incident EM
Plane Wave
Unscattered EM
Plane Wave
Scattering Object
(e.g. charge q, atom,
molecule, etc.)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Note also that the scattering cross section is explicitly defined using time-averaged (rather
than instantaneous) quantities in both the numerator and denominator in order to facilitate direct
comparison between theoretical prediction(s) vs. experimental measurement(s).
The time-averaged power radiated into a differential/infinitesimal element of solid angle
d   d cos  d  sin  d d (SI units: steradians; aka sterad, and/or sr) is:
d Prad  ,  , t 
d


 S rad  r , t  r 2 rˆ (SI units: Watts steradian  Watts sr ).

The infinitesimal vector area element dasph  dasph rˆ associated with
a sphere of radius r {centered on the scattering object - the “target” located at the origin  } is related to the differential solid angle

element d  via dasph  r 2 d  rˆ  r 2 d cos  d rˆ  r 2 sin  d d rˆ .
EM radiation scattered from the target object into a solid angle

element d passes through this area element dasph . The figure on the
right shows the relation between the differential solid angle element
d   d cos  d  sin  d d and the infinitesimal area element
dasph   rd cos   rd    r sin  d  rd   r 2 d  .
 Note also that d Prad  ,  , t  d  has no r-dependence!
The differential angular scattering cross section is defined as:
d Prad  ,  , t 
d 2 Prad  ,  , t 
d scat  ,   d 2 scat  ,  
1
1

  
  
d
d cos  d
d
d cos  d
Sinc  r  0, t 
Sinc  r  0, t 
SI units:
m 2 Sr
Note that the choice of {the usual} polar and azimuthal angles  ,   to describe the two
independent scattering angles means that the EM wave that is incident on the “target” object
(free charge q, atom or molecule, etc.) is propagating in the  ẑ direction.
It is also instructive to write out the differential scattering relation in more explicit detail:






Srad  r , t  r 2 rˆ
Erad  r , t   Brad  r , t  r 2 rˆ
d Prad  ,  , t 
d scat  ,  
1
  
  
  
 
d
d
Einc  r , t   Binc  r , t   zˆ
Sinc  r , t   zˆ
Sinc  r , t 
r 0
r 0
r 0
SI units:
m 2 Sr
As we derived in P436 Lecture Notes 14 (p. 1-5) from the Taylor series expansions of
 

tot  r , tr  and J tot  r , tr  to first order in r  , the only contribution to the EM power radiated is
associated with a non-zero value of {some kind of/“generic”} time-varying electric dipole
 

moment p  r , tr   p  r , tr  zˆ {n.b. oriented parallel to the ẑ -axis}, where, in the “far-zone” limit
 c  krmax
  2 rmax
   1 }, the instantaneous differential retarded EM
 r  1 and  rmax
{ rmax
power radiated by the time-varying E1 electric dipole, with retarded time tr  to  t  r c is:
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
dPrrad  ,  , t   rad 
 p 2  t 
Watts 
 S  r , t r 2 rˆ  o 2 o sin 2  

d
16 c
 steradian 
The accompanying retarded electric and magnetic fields, EM energy density, Poynting’s vector
associated with the radiating E1 electric dipole, in the “far-zone” limit are:
 
 p  t   sin 
Er  r , t   o o 
4  r
and:
o p  to   sin 
ˆ  
  , Br  r , t  

4 c  r

 
 
1
ˆ
ˆ

B
r
t

r

E
,


with
r
r r ,t 

c



i.e. Br  Er  rˆ  kˆ
 

 
 p 2  t   sin 2  
1  


Er  r , t   Br  r , t 
urrad  r , t   o 2 2o  2  and: S rrad  r , t  
16 c  r 
o
 rad 
o p 2  to   sin 2 
Sr  r , t  

16 2 c  r 2
o p 2  to   sin 2 
 ˆ
   ˆ 

16 2 c  r 2
 


 rˆ

c  crˆ

 rad 
 rˆ  cur  r , t  with
rˆ  kˆ

The total instantaneous retarded EM power radiated by a time-varying E1 electric dipole into

4 steradians, with vector area element da  r 2 sin  d d rˆ  r 2 d  rˆ in the “far-zone” limit is:
Pr
rad
 rad 
o p 2  to    2   2
o p 2  to 

sin  sin  d d 
 t   S Sr  r , t da 
(Watts)
16 2 c  0  0
6 2 c
We can then take the necessary time-averages of the instantaneous flux of incident EM energy
(i.e. Poynting’s vector) and the instantaneous differential radiated EM power and/or total
radiated EM power to use in the above cross section formulae.
Scattering of EM Radiation from a Free Electric Point Charge: Thomson Scattering
Suppose a free electric point charge q is located at the origin   x, y, z    0, 0, 0  with a plane
EM wave propagating in free space in the  ẑ direction, which is incident on the free charge q,

{which has mass m}. Noting that kinc  kkˆinc and that kˆinc  zˆ and using complex notation:
 
 
  1
 
  i kz t   
  i kz t 
1
ˆ

ˆ
Einc  r , t   Eo  r  e
, Binc  r , t   Bo  r  e
with Bo  r   kinc  Eo  r   z  E o  r 
c
c
 
The polarization of the incident EM wave is (by convention) taken parallel to the Eo  r  direction.

 

For definiteness’ sake, let us assume the incident EM plane wave to be linearly polarized in the
 
x̂ -direction, and for simplicity, let us also assume that Eo  r   Eo xˆ , and therefore:
 
 
  1
E
E
E
1
Bo  r   kˆinc  E o  r   zˆ  E o  r   o  zˆ  xˆ   o   yˆ    o yˆ .
c
c
c  c
c

 

  yˆ
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Poynting’s vector for the incident EM plane wave is:
 
 
1  
1 2 2i kz t 
1
2 i kz t 
Sinc  r , t  
Einc  r , t   Binc  r , t  
Eo e
 yˆ    o cEo2 e 
zˆ (using  o c 
)
xˆ

o
o c
o c
  zˆ
The EM plane wave incident on the free point electric charge q located at the origin  causes
the point charge q to accelerate/move, because two forces act on the point charge – an electric
force and a magnetic/Lorentz force. Noting that the resulting time-dependent position of the
point charge is at the source position r   0 (i.e. in the neighborhood/vicinity of the origin  ):





Ftot  r   0, t   qEinc  r   0, t   qv  r   0, t   Binc  r   0, t   ma  r   0, t 
The {transverse} electric field of the incident EM plane wave in the vicinity of r   0 is:

Einc  r   0, t   Eo e  it xˆ
The {transverse} magnetic field of the incident EM plane wave in the vicinity of r   0 is:



1 ˆ
1
1
1




ˆ
Binc  r  0, t   kinc  Einc  r  0, t   z  E inc  r   0, t   Eo e  it  zˆ  xˆ    Eo e  it yˆ



c
c
c
c

 

  yˆ
 
 

ˆ
 
 i t
 it v  r  0, t   y
 ma  r   0, t  .
Thus: Ftot  r   0, t   qEo e xˆ  qEo e
c


v c 1
However, for non-relativistic scattering of an EM plane wave by a point electric charge q , where
the motion of the charge is always such that v  c  or:   v c  1 , the 2nd (magnetic) Lorentz
force term will correspondingly always be small in comparison to the 1st (electric) force term,
hence we will neglect the magnetic Lorentz force term in our treatment here.

Physically, the transverse electric field of the incident EM plane wave Einc  0, t   Eo e  it xˆ



exerts a time-dependent force Ftot  0, t   qEo e it xˆ  ma  0, t   mx  0, t   m 2 xo e  it xˆ on the
free electric charge q, causing it to oscillate back and forth along the x̂ -axis in a time-dependent

manner: x  0, t   xo e it xˆ with (real) amplitude xo . Note that the wavelength of associated with
the incident EM plane wave for non-relativistic scattering is such that the variation of the electric
field strength in the vicinity of the free electric charge is negligible, i.e. that: xo    c f .


Note also that the instantaneous acceleration of the charge q is a  0, t   x  0, t    2 xo e  it xˆ .
The E-field induced oscillatory motion of the point electric charge q thus creates an induced


time-dependent electric dipole moment: p  0, t   qx  0, t   qxo e  it xˆ  po e  it xˆ oriented parallel

to the electric field Einc  0, t   Eo e  it xˆ of the incident EM plane wave.
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede

The induced oscillating electric dipole moment p  0, t  subsequently radiates electric dipole (E1)
EM radiation. Energy from the incident EM wave is {temporarily} absorbed by the point charge q
{this is necessary in order to get the charge q moving - i.e. to accelerate it}. The incident EM
energy absorbed by the charge q is radiated a short time later. Thus, this overall process is one type
of scattering of EM radiation! The geometrical setup for this situation is shown in the figure below:
r̂
Observer at
Field Point, r



S rad  1o Erad  Brad

S rad  kˆrad  rˆ


Einc  Eo xˆ



Sinc  1o Einc  Binc

Sinc  kˆinc  zˆ

 1c kˆinc  Einc  Bo yˆ


Binc




p  pxˆ

x̂


rad
ˆ
E
rad  Eo x


 ẑ
Brad  1c kˆrad  Erad
x


 Borad kˆrad  xˆ



x̂
ŷ
ŷ
Note also that the instantaneous mechanical power associated with the oscillating free charge q is


Pmech  t   Ftot  0, t v  0, t  , arising from the absorption of energy from the incident EM wave by the


free charge q. The velocity of the oscillating charge is v  0, t   dx  0, t  dt  i xo e it xˆ and hence
Pmech  t    m 2 xo e it xˆ  i xo e it xˆ   im 3 xo e 2it .
At the microscopic level, real photons of {angular} frequency  associated with the incident
EM wave (e.g. real photons in a laser beam that comprise the macroscopic EM wave output from
the laser) are {temporarily} absorbed by the electric point charge q, and then re-emitted {a short
time later} as {quantized} EM radiation {of the same frequency} – the emitted photons are
associated with the outgoing, or scattered EM wave! Two space-time/Feynman diagrams showing
examples of this QED scattering process for an electron are shown in the figure below:
ct
Incident /
Incoming
Photon
e*
Radiated /
Outgoing
Photon
ct
Incident /
Incoming
Photon
e*
Radiated /
Outgoing
Photon
Virtual /
Off-Shell e
Virtual /
Off-Shell e
x
x
The characteristic time interval t associated with the photon-free charge absorption-re-radiation
process is governed by the Heisenberg uncertainty principle E t  12  where   h 2 and h is
Planck’s constant,   1.0546 1034 Joule sec  6.582  1016 eV sec {n.b. 1 eV  1.602 1019 Joules }.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
For an incident photon e.g. with energy E  hf  hc   1 eV , since hc  1240 eV -nm this photon
has a corresponding wavelength of   1240 nm {which is in the infra-red portion of the EM
spectrum}. Then using E  E  1 eV in the Heisenberg uncertainty relation, we see that the
corresponding time interval t is:
t  12  E  12  E  12  6.582 1016 eV sec 1 eV   3.2611016 sec  0.32611015 sec  0.326 femto-sec
Note that the period of oscillation  associated with a E  hf  1 eV photon is:
   1 f   c  1240 109 m 3 108 m s  4.13 1015 sec  4.13 femto-sec
Thus, we see that the period of oscillation  associated with a 1 eV photon is ~ 10 longer than the
characteristic time interval t associated with the photon-free charge absorption-re-emission process.
 
Please also note that had we not neglected the magnetic qv  B Lorentz force term in the


original force equation, the B-field induced motion of the point charge q would have corresponded
to the creation of an induced, time-dependent magnetic dipole moment at r   0 :


m  0, t   I  0, t  A  I o A e it yˆ  mo e it yˆ
which in turn also would have subsequently radiated magnetic dipole (M1) EM radiation.
However, because Bo  Eo c  Eo we also see that mo  po c , and thus the power radiated by
this induced, time-dependent magnetic dipole would be  than the power radiated by the induced,
time-dependent electric dipole. As we have seen in P436 Lecture Notes 13.5 (p. 11), for “equal”
strength dipoles  i.e. mo  po c  , the amount of M1 radiation is far less than that for E1 radiation.
Note further that, formally mathematically speaking, M1 radiation appears at second-order in
 

the Taylor series expansion of tot  r , tr  and J tot  r , tr  , hence another reason why we neglected
this term, since we initially stated that we were only working to first order in this expansion.
In the “far-zone” limit, the instantaneous differential retarded EM power radiated by an
oscillating E1 electric dipole {situated at the origin, n.b. oriented along the ẑ -direction} into
solid angle element d is (see P436 Lecture Notes 14, p. 5):
dPrrad  ,  , t  
 p 2  t 

 Watts 
 Srad  r , t r 2 rˆ  o 2 o sin 2  

d
16 c
 steradian 
The angular distribution of the EM power radiated by an oscillating E1 electric dipole with

ˆ  it oriented parallel to an arbitrary â -axis
time- dependent electric dipole moment p  t   po ae
(e.g. where aˆ  xˆ, yˆ or zˆ axis) is shown in the figure below. The intensity {aka irradiance}



I rad  r   S rad  r , t  Watts m 2  is proportional to the distance from the origin    0, 0, 0 

to an arbitrary point r   r ,  ,   on the 3-D surface of the figure.
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
However, for the problem that we have at hand, our electric dipole is oriented in the x̂ -direction.
Thus, we must carry out a rotation of the above result such that it is appropriate for our situation:
dPrrad  ,  , t   rad
 p 2  t 
 Watts 
 S  r ,  ,  , t r 2 rˆ  o 2 o sin 2  x 

d
16 c
 steradian 
Here,  x is the opening angle between the observation/field point unit vector r̂ and the x̂ -axis.
Thus, cos  x is the direction cosine between the observation/field point unit vector r̂ and the
x̂ -axis: i.e. cos  x  rˆ xˆ  sin  cos  , in terms of the usual polar   and azimuthal   angles.
 
rx
   
Note also that since: r  x  r x sin  x or:    rˆ  xˆ  sin  x then:
r x
sin 2  x   rˆ  xˆ  rˆ  xˆ 
  sin  cos  xˆ  sin  sin  yˆ  cos  zˆ   xˆ    sin  cos  xˆ  sin  sin  yˆ  cos  zˆ   xˆ 
 sin 2  sin 2   cos 2 
which can also be obtained from:
sin 2  x  1  cos 2  x  1   rˆ xˆ   1   sin 2  cos 2    1  sin 2  1  sin 2  
2
 1  sin 2   sin 2  sin 2   cos 2   sin 2  sin 2   sin 2  sin 2   cos 2 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Thus, in terms of the usual polar   and azimuthal   angles, for the oscillating E1 electric
dipole oriented along the x̂ -axis:
dPrrad  ,  , t   rad
o p 2  to  2
 Watts 
2
sin  x 
 S  r ,  ,  , t r rˆ 

2
d
16 c
 steradian 
Becomes:
dPrrad  r , ,  , t   rad
 p 2  t 
 Watts 
 S  r ,  ,  , t r 2 rˆ  o 2 o  sin 2  sin 2   cos 2   

d
16 c
 steradian 
The angular distribution of the EM energy radiated from the oscillating E1 electric dipole oriented
along the x̂ -axis is thus similar to that shown in the figure below:

In the above formula, the second time-derivative of the electric dipole moment 
p  to  is to be

evaluated at the retarded time to  t  r c and also computed from the local origin,  { r   0 }:


p  0, to   p  0, t  r c  .


The instantaneous induced electric dipole moment is: p  0, t   qx  0, t   qx e  ito xˆ  p e ito xˆ .
o
o
o
o



Thus: p  0, to    2 po e  ito xˆ   2 qxo e  ito xˆ  q   2 xo  e ito xˆ  qx  0, to   qa  0, to  .
q 
q

From the above force equation, we see that: a  0, to   Einc  0, to   Eo e  ito xˆ
m
m
q

 q


Eo 
but: a  0, to   x  0, to    2 xo e ito xˆ and thus we see that:  2 xo  Eo or: xo   
2
m
 m

 q2 


and therefore: p  0, to   qa  0, to   q   2 xo  e it xˆ   Eo  e  ito xˆ
m 
q2
 q


E
Eo ,


n.b. here, p  0, to  is independent of frequency  , because: po  qxo  q 
o
2
m 2
 m

i.e. po  qxo 
8
1
2
2
 q 2  2ito




!!! Thus: p  0, to   p  0, to  p  0, to    Eo  e
.
m 
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
The time average of {the real part of !!!} this quantity, averaged over one complete cycle of
1 2
1  q2 
p 2  0, to   
p  0, to    Eo 
oscillation   1 f  2  is simply half of this value: 
2
2 m 
1
1
1
Re e 2ito dto 
cos 2 to dto  .
since Re e 2ito 


2
 one cycle
 one cycle



2

Note also that by carrying out the time-averaging process, this also helps us to completely sidestep/avoid the difficulty associated with experimentally dealing with the retarded time to  t  r c !
Thus, the time-averaged differential power radiated by the oscillating E1 electric dipole is:
d Prrad  ,  , to 
d
Or:
2
 rad
o p 2  to 
1 o  q 2 Eo 
2
2
2
sin  x  
 S  r ,  ,  , to  r rˆ 

 sin  x
16 2 c
2 16 2 c  m 
d Prrad  ,  , to 
 Watts 


 steradian 
2

1   q 2 Eo 
 Watts 
2
2
2
 S rad  r ,  ,  , to  r 2 rˆ   o2 
  sin  sin   cos   

2 16 c  m 
 steradian 
d
Likewise, the time average of the {magnitude of} the instantaneous flux of EM energy incident on


the free charge {located at the origin   r  0  }, Sinc  0, to    o Eo2 ce2ito is half of this value, i.e.:

1 
1
 Watts 
I inc  0   Sinc  0, to   Sinc  0, to    o Eo2 c 
2 
2
2
 m 
The scattering of EM plane waves by a free electric charge q is known as Thomson scattering
in honor of J.J. Thomson – the discoverer of the electron {in 1897}.
Thus, the differential Thomson scattering cross section {per scattering object of charge q} for an
incident plane EM wave propagating in the  ẑ -direction and linearly polarized in the x̂ -direction is:
d TLPx  ,  
d

d Prad  ,  , to 
1
1


2
d
1
Sinc  0, to 
 o E oc
2
2
 q2

sin 2  x

2 
 4 o mc 
using
 q4 E 2 
o


 m2 

 sin 2 
x
2
16 c
1
o
2
o  1  o c 2 
i.e.
d TLPx  ,    q 2

 q2

2


sin
sin 2  sin 2   cos 2  


x
2 
2  
d
 4 o mc 
 4 o mc 
2
2
m
2
sr 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
There are three interesting things associated with this result:
The differential Thomson scattering cross section is independent of both the frequency of the
incident EM radiation, and note also that it is independent of the strength (i.e. amplitude) of the
incident electric field Eo .
Note also that the differential Thomson scattering cross section varies as the square of the
electric charge, i.e. the Thomson scattering cross section is the same for +q vs. –q charged
particles, and note also that a scattering object with free electric charge q  2e (such as an
-particle) has a Thomson scattering cross section 4 greater than that associated with a
scattering object {of the same mass, m} that has free electric charge q   e .
Then since:
Pr
 to 
rad

d Prrad  ,  , to 
d
o  q 2 Eo 
d 


32 2 c  m 
Carrying out the azimuthal angle integrals first:
2
   sin
  2

0
  2
 
0
0
2
 sin 2   cos 2   sin  d d
sin 2  d   and
  2

0
d  2 .
Then carrying out the polar angle integration:
 

0
sin  sin  d  
2
 
 0
 
 
1
1 1 2
cos 2  sin  d   cos3 
  
0


3
3 3 3
And:

Thus:
   sin
0
  2
 
0
0
Then: Pr
rad
 
1
1
2 62 4


 1 
sin  d    cos   cos3  
 1     1    2  

3
3
3
3
3

  0  3  
3
 to 
2
2  4 4 8
 4 


 sin 2   cos 2   sin  d d        2   
3 3
3
3
 3 
2
o  q 2 Eo  8 
o  q 2 Eo 





2 
12 c  m 
m  3
4 32  c 
2
Thus, the total Thomson scattering cross section {per scattering object of charge q} for an
incident plane EM wave propagating in the  ẑ -direction and linearly polarized in the
x̂ -direction is:
 q 4 Eo2

6
2
12  c  m

1
 o Eo2 c
2
o

LPx
T
Prad  to 
d LPx  ,  
d  
 T
d
Sinc  to 
Thus: 
10
LPx
T
8

3
 q2


2 
 4 o mc 
2


 8

3
 q2


2 
 4 o mc 
2
using
o  1  o c 2 
m 
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
If the electrically-charged scattering object e.g. is an electron, with electric charge q = e, and
2
2
15
rest mass me then the quantity: re   e 4 o me c   2.82 10 m is known as the so-called
classical electron radius.
The classical electron radius re is defined as the radial distance from an electron where the
{magnitude of the} potential energy U e  re   eVe  re  associated with a unit test charge q = e
equals the rest mass energy of the electron Eerest
 me c 2 , i.e.

U e  re   eVe  re  
e2
4 o re
 me c 2 thus: re 
e2
 2.82 1015 m
2
4 o me c
The differential and the total Thomson scattering cross sections for free electron scattering of
a linear polarized plane EM wave, written in terms of the classical electron radius re are:
d TLPx
 ,  
e
d
 re2 sin 2  x  re2  sin 2  sin 2   cos 2  
m
2
sr per electron 
and:
 TLPx

e

8 2
re
3
 m2 per electron  where: re 
e2
 2.82 1015 m
4 o me c 2
Numerically, we see that:

 TLPx
e

2
8 2 8
2.82 1015   66.6 1030  0.67 1028
re 

3
3
m
2
per electron 
Physicists get tired of writing down {astronomically} small numbers all the time, so we have
defined a convenient unit of area for cross sections, known as a barn {originating from the phrase:
2
2
2
2
28
14
15
“It’s as big as a barn”}: 1 barn  10 m  10  m  10 10  m  10 fm   100 fm
2
2
2
where 1 fm = 1 Fermi = 1015 m (in honor of Enrico Fermi, nuclear physicist of mid-20th century).
Thus:

 TLPx
e

2
8 2 8
2.82 1015   0.67 1028 m 2 per electron  0.67 barns per electron
re 

3
3
In order to obtain a more physically intuitive understanding of the magnitude of this (and
various other) total scattering cross sections, note that the characteristic size of the nucleus of an
atom is typically rnucleus ~ 1  few  fm  1  few   1015 m whereas the characteristic size of an
atom is typically ratom ~ 1  few  Ångstroms  1  few  1010 m  0.11  few  nm .
Thus, the geometrical cross-sectional area of a typical nucleus in an atom is
2
Anucleus   rnucleus
~  0.03  1 1028 m   0.03  1 barns , whereas the geometrical cross-sectional
2
~  0.03  1 1018 m   0.03  1 1010 barns !!! .
area of a typical atom is huge: Aatom   ratom
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Hence, we see that the free electron Thomson scattering cross section is comparable to the
geometrical cross-sectional area for a typical nucleus, and is very much smaller than the
geometrical cross-sectional area for a typical atom, i.e.

LPx
T e

 8 re2 3  0.67 barns ~  Anucleus   0.03  1 barns    Aatom   0.03  1 1010 barns

Note that the Thomson scattering of a linearly-polarized EM plane wave propagating in the
ẑ -direction has rotational invariance about the ẑ -axis. In the original problem above, we
could have alternatively chosen the polarization of the EM plane wave to be parallel to e.g. the
 
 ŷ -axis instead of the x̂ -axis, i.e. Einc  r  0, t    Eo e  it yˆ , then the differential Thomson
scattering cross section for a free charge q would instead then be:
o p 2  to 
d Prad  ,  , to 
d TLPy  ,  
1
1
sin 2  y
 
 
2
d
d
16 c
Sinc  0, to 
Sinc  0, t 
where the direction cosine cos  y  rˆ  yˆ    sin  sin  and: rˆ    yˆ   sin  y thus:
sin 2  y   rˆ    yˆ   rˆ    yˆ     rˆ  yˆ  rˆ  yˆ 
  sin  cos  xˆ  sin  sin  yˆ  cos  zˆ   yˆ    sin  cos  xˆ  sin  sin  yˆ  cos  zˆ   yˆ 
 sin 2  cos 2   cos 2 
which can also be obtained from:
sin 2  y  1  cos 2  y  1   rˆ yˆ   1   sin 2  sin 2    1  sin 2  1  cos 2  
2
 1  sin 2   sin 2  cos 2   cos 2   sin 2  cos 2   sin 2  cos 2   cos 2 
Note that {importantly}, going through the same derivational steps as done originally, the
induced dipole moment associated with an incident EM plane wave propagating in the ẑ -direction


but linearly polarized in the  ŷ -direction is: p  0, t   qy  0, t   qyo e it yˆ   po e it yˆ , thus we
see that the induced dipole moment simply follows/tracks the polarization of the incident EM plane

wave, i.e. p  0, t  is parallel to the polarization/E-field vector of the incident EM plane wave!
The intensity maximum of the scattered radiation {here} would then be oriented perpendicular
to the ŷ -axis instead of the perpendicular to the x̂ -axis {n.b. please see/refer to the 3-D figure on
page 7 of these lecture notes}.
The differential Thomson scattering cross section for an incident EM plane wave propagating
in the ẑ -direction but linearly polarized in the  ŷ -direction is:
d TLPy  ,    q 2

 q2

2

sin
sin 2  cos 2   cos 2  



y
2 
2  
d
 4 o mc 
 4 o mc 
2
12
2
m
2
sr 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Integrating this expression over the polar and azimuthal angles  ,   , we obtain precisely the
same result for the total Thomson scattering cross section as above for the original x̂ -polarization
case:

LPy
T
8

3
2
 q2

=  TLPx

2 
 4 o mc 
m 
2
More generally, for an incident EM plane wave propagating in the ẑ -direction with arbitrary
 
ˆ as shown in the figure below
ˆ cos  xˆ  sin  yˆ and E inc  r  0, t   Eo e  it 
linear polarization 
{cf with/see also the 3-D figure shown on p. 4 of these lecture notes}:
ˆ cos  xˆ  sin  yˆ

 sin 

ẑ into page
cos 
x̂
ŷ
In this more general situation, the induced dipole moment is again parallel to/tracks the
polarization vector of the incident EM plane wave:


p  0, t   q 
 0, t   qro eit  cos  xˆ  sin  yˆ   po eit  cos  xˆ  sin  yˆ 
Thus, here the differential Thomson scattering cross section for a free charge q is:
o p 2  to 
d Prad  ,  , to 
d TLP  ,  
1
1
sin 2 
 
 
2
d
d
16 c
Sinc  0, to 
Sinc  0, to 
where {here} the direction cosine associated with an arbitrary linear polarization is:
ˆ  sin  cos  xˆ  sin  sin  yˆ  cos  zˆ  cos  xˆ  sin  yˆ   sin   cos 2   sin 2    sin  cos 2
cos   rˆ
ˆ  sin  thus:
and: rˆ 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
ˆ  rˆ 
ˆ
sin 2    rˆ 
Lect. Notes 14.5
Prof. Steven Errede
ˆ cos  xˆ  sin  yˆ

ˆ    sin  cos  xˆ  sin  sin  yˆ  cos  zˆ  
ˆ 
  sin  cos  xˆ  sin  sin  yˆ  cos  zˆ  
 4sin 2  cos 2  sin 2   cos 2   cos 2   sin 2    sin 2   4 cos 2  sin 2    cos 2 


1
sin 2 2
 sin 2  sin 2 2  cos 2 
which can also be obtained from:
ˆ   1  sin 2  cos 2 2  sin 2   cos 2   sin 2  cos 2 2
sin 2   1  cos 2   1   rˆ
2
 sin 2  1  cos 2 2   cos 2   sin 2  sin 2 2  cos 2 



 sin 2 2
This relation can also be obtained via a 3rd method – noting that since:
ˆ  rˆ cos  xˆ  sin  yˆ   cos   rˆ xˆ   sin   rˆ  yˆ    cos  cos  x  sin  cos  y
cos   rˆ
where: cos  x  rˆ xˆ  sin  cos  and: cos  y  rˆ  yˆ    rˆ yˆ   sin  sin  then:
ˆ   1   rˆ cos  xˆ  sin  yˆ    1   cos   rˆ xˆ   sin   rˆ  yˆ   
sin 2   1  cos 2   1   rˆ
2
2
 1  cos   sin  cos    sin    sin  sin     1  sin  cos 2   sin  sin 2  
2
2
 1  sin 2   cos 2   sin 2    1  sin 2  cos 2 2  cos 2   sin 2   sin 2  cos 2 2


2
 cos2 2
 cos 2   sin 2  1  cos 2 2   cos 2   sin 2  sin 2 2  sin 2  sin 2 2  cos 2 



 sin 2 2
Thus, we see that:
cos 2  x   rˆ xˆ   sin 2  cos 2 
2
cos 2  y   rˆ  yˆ     rˆ yˆ     sin  sin    sin 2  sin 2 
2
2
2
ˆ   1  sin 2   1  cos 2   sin 2  sin 2 2  sin 2   sin 2  sin 2 2
cos 2    rˆ
2
 sin 2  1  sin 2 2   sin 2  cos 2 2

 cos 2 2
ˆ cos  xˆ  sin  yˆ ,
with 
ˆ  xˆ and cos 2   cos 2  x  sin 2  .
Note that when   0 : 
ˆ   yˆ and cos 2   cos 2  y  sin 2  .
Note that when    2 : 
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
2
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
and:
sin 2  x   rˆ  xˆ  rˆ  xˆ   sin 2  sin 2   cos 2 
sin 2  y   rˆ    yˆ   rˆ    yˆ     rˆ  yˆ  rˆ  yˆ   sin 2  cos 2   cos 2 
ˆ  rˆ 
ˆ   sin 2  sin 2 2  cos 2 
sin 2    rˆ 
ˆ  xˆ and sin 2   sin 2  x  cos 2  .
Note that when   0 : 
ˆ   yˆ and sin 2   sin 2  y  cos 2  .
Note that when    2 : 
Thus, the differential Thomson scattering cross section for an incident EM plane wave
propagating in the ẑ -direction with arbitrary linear polarization in the
ˆ  cos  xˆ  sin  yˆ -direction is:

d TLP  ,    q 2

 q2

2



sin
sin 2  sin 2 2  cos 2  


2 
2  
d
4
mc
4
mc


o
o




2
2
m
2
sr 
 d  ,   
Then since  T    T
 d  , carrying out the angular integration over the polar   and
d


   sin
azimuthal   angles:
  2

0
 
 
0
0
2
 sin 2 2  cos 2   sin  d d , carrying out the  -integrals:
  2
2
  sin 4 

  and
sin 2 d   

8   0
2
2
2
 

0
d  2
Using sin 2   1  cos 2  and making the substitution u  cos  , hence du  d cos    sin  d ;
and when   0: u  cos 0  1 , when    : u  cos   1 , thus the  -integrals are:
 

0
 

0
sin 2  sin  d  
u 1
cos 2  sin  d  
u 1
u 1
u 1
1  u  du  u 
2
u 2 du  13 u 3
u 1
u 1
1
3
u 3 
u 1
u 1
 1  13   1  13   1  13   1  13   2 1  13  
 13    13   13  13 
2
3
Putting all of these results together:
   sin
 
 
0
0

LP
T
2
4
3
2
3
 sin 2 2  cos 2   sin  d d      2 
8

3
2
 q2

=  TLPx   TLPy   T

2 
 4 o mc 
4 4 8


!!!
3
3
3
m 
2
Thus we have explicitly shown that we obtain precisely the same result for the total Thomson
ˆ  cos  xˆ  sin  yˆ of an incident EM plane
scattering cross section for an arbitrary polarization 
wave as that obtained above for either the original x̂ - and/or the  ŷ - polarization cases.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15
4
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Due to the manifest rotational invariance/symmetry of this problem about the ẑ -axis {the
propagation direction of the incident EM plane wave}, intuitively we can understand why this
 
must be true, since the orientation of the induced electric dipole moment p  r   0, t  is such that
it is always parallel to/tracks the polarization vector ̂ of the incident EM plane wave.
ˆ of the incident EM plane wave certainly
Thus, we see that while the polarization vector 
matters greatly for the differential Thomson scattering cross section d T  ,   d  , the total
Thomson scattering cross section  T is unaffected/does not depend on the polarization vector
ˆ of the incident EM plane wave.

Since left- and right-circularly polarized EM plane waves are {complex, but} linear orthogonal
ˆ LCP  12  xˆ  iyˆ  and 
ˆ RCP  12  xˆ  iyˆ 
combinations of linearly-polarized EM plane waves: 
ˆ LCP 
ˆ *LCP 
ˆ RCP 
ˆ *RCP  1 },
{note that we have normalized these polarization vectors such that 
then we can also see that the total Thomson scattering cross section for LCP or RCP incident EM
plane waves is also:
8
T 
3
 q2


2 
 4 o mc 
2
m 
2
We leave it to the interested reader to determine the analytic form of the differential Thomson
scattering cross sections for LCP or RCP incident EM plane waves.
Using the same line of reasoning, we can additionally see that the total Thomson scattering
cross section for unpolarized EM plane waves incident on a free charge q is also
2

unpol
T

8  q 2

 T

2 
3  4 o mc 
m 
2
because an unpolarized EM plane wave is equivalent to a randomly-polarized EM plane wave,
ˆ  t  changes randomly from one moment to the next
whose time-dependent polarization vector 
within the azimuthal interval 0    2 . For a randomly-distributed  variable, note that the
probability distribution function dP   d is flat within the azimuthal interval 0    2 .
 
The instantaneous orientation of the induced electric dipole moment p  r   0, t  is parallel to
ˆ  t  of the randomly-polarized incident EM plane wave on a moment-tothe polarization vector 
moment basis; thus, while the differential angular distribution of the scattered radiation is
changing on a moment-to-moment basis, the total Thomson scattering cross section  T is
unchanged/time-independent for an unpolarized/randomly-polarized incident EM plane wave,
because the  ,   angular dependence has been integrated out. We explicitly prove this, below.
16
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
The probability is unity (i.e. 100%) for a randomly-polarized EM plane wave to have linear
ˆ  t  instantaneously oriented somewhere within the azimuthal interval 0    2 .
polarization 
Mathematically this means that the  -integral of the probability density
  2

0
 d P   

 d  1 .
 d 
 d P   
Since the azimuthal probability density 
 is flat {i.e. is constant} for a randomly d 
 d P   
distributed  distribution, then we can take 
 outside of this integral, and then since

d


d P   1

for an unpolarized/randomlyd
2
polarized EM plane wave, as shown in the figure below:
  2

0
d  2 , we see that the probability density
d P   1

2
d
 0
  2

ˆ  t  of the incident EM plane wave is changing randomly from
If the polarization state 
moment-to-moment, one might worry that the process of averaging the instantaneous differential
radiated power d Prad  ,  , to  d  over one period/one cycle of oscillation  would be
insufficient – i.e. it would be very “noisy” due to rapid fluctuations/random temporal changes in
ˆ  t  with time t.
the polarization state 
At the microscopic level, an unpolarized macroscopic EM plane wave consists of real photons,

ˆ  . Individual
each with a randomly oriented E -field/randomly oriented polarization vector 
photons which Thomson scatter off of a free charge q are first absorbed by the charge q and then
re-radiated a short time later, e.g. with characteristic time interval t  0.326 fs for a 1 eV photon,
compared to the period of oscillation for a 1 eV photon of    4.13 fs , ~ 10  longer than t .
Thus, for EM plane waves incident on a free charge q one might well worry that averaging
over a single period of oscillation/single cycle  would be likely to yield a noisy result due to
fluctuations. However, in actual/real-life scattering experiments, precisely because of such
concerns, the averaging time interval tavg is frequently orders of magnitude longer than either
of these two time scales, typically tavg is micro-seconds, to milli-seconds, seconds and/or even
longer in order to significantly reduce the level of such {statistical} fluctuations.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
17
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
ˆ  t   sin   t  , then with no loss of generality, we can very easily modify the
Since rˆ 
time-averaging of the differential power/differential scattering cross section formulae simply by
moving the sin 2   t  factor {n.b. previously assumed to time-independent in the above
examples} inside the time-averaging process, i.e.:
o p 2  to  sin 2   t 
d Prad  ,  , to 
d Tunpol  ,  
1
1
 
 
d
d
16 2 c
Sinc  0, to 
Sinc  0, to 

1

Sinc  0, to 
o p 2  to   sin 2  sin 2 2  t   cos 2  
16 2 c
Note that the polar angle  is fixed by the observer being at the field/observation point P.
The time-averaged value of the sin 2 2  to  factor is actually a probability-weighted integral
over all possible  -values that can/do occur during the time-averaging process over tavg :
sin 2 2  to   
  2
 0
1

2
  2  1 
 d P    2
2

 sin 2  to  d   0 
 sin 2  to  d
 2 
 d 
  2

0
  2
1   sin 4 
1 2 1




sin 2  to  d 


2
2  2
8   0
2 2
2
Thus, the differential Thomson scattering cross section for an unpolarized macroscopic EM
plane wave incident on a free electric charge q, averaged over a long time interval tavg is:
d Tunpol  ,    q 2

 q2

2


sin
sin 2  sin 2 2  to   cos 2 



2 
2 
d
 4 o mc 
 4 o mc 
2
2
2


2
2
 q2
 1 2

 1 q
2
sin
cos
sin 2   2 cos 2  









2
2  
 2  4 o mc 
 4 o mc   2
2
2

2


 
1  q2
1
q
2
2
2
sin   cos    cos    
1  cos 2  
 

2 
2  
 2  4 o mc 
2  4 o mc   


1
d Tunpol  ,   1  q 2

 
1  cos 2  
2  
d
2  4 o mc 
2
m
2
sr   n.b. has no  -dependence!
Hence, we see that for an unpolarized macroscopic EM plane wave incident on a free electric
charge q, averaged over a long time interval tavg , the differential Thomson scattering cross
section has no  -dependence, as we anticipated.
18
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
The total Thomson scattering cross section for an unpolarized macroscopic EM plane wave
incident on a free electric charge q, averaged over a long time interval tavg is:

unpol
T
 d  ,   

1  q2
  T
 d  
2 
d
2  4 o mc 


2
  1  cos   sin  d d
  2
 
0
0
2
But:
1   2  
2
1  cos 2   sin  d d 



2  0  0
2

 
 0
sin  d   
 
 0
 1  cos   sin  d
 
2
0
cos 2  sin  d  2   
2 6 2 8



3 3
3
3
Thus, here again we see that the Thomson scattering cross section for an unpolarized/randomlypolarized macroscopic EM plane wave incident on a free electric charge q, averaged over a long
time interval tavg is:
2

unpol
T

8  q 2
= T


2 
3  4 o mc 
m 
2
Even though an incident EM plane wave may be unpolarized, this does not mean that an

observer at the field/observation point P  r   P  r ,  ,   will observe unpolarized scattered
radiation – quite the contrary! The reason for this is simple – for a specific field/observation
point P  r ,  ,   the EM radiation that is scattered into that specific  ,   angular region depends
sensitively on the incident polarization state! We can easily see this from the following:
Consider an observer located in the x-z plane at P  r ,  ,   0  as shown in the figure below:
nˆscat  kˆrad  kˆinc
ˆ inc
ˆ

  y
ˆ inc  t 


kˆinc  zˆ

ˆ rad
ˆ

  y
ẑ

ˆ inc  xˆ

Incident EM
Plane Wave
(unpolarized)
x-z plane =
scattering
plane
ˆ rad  cos  xˆ  sin  zˆ

r̂
x̂
ˆ inc  t   cos   t  xˆ  sin   t  yˆ

ŷ
kˆrad  rˆ
Observer
Position
ˆ inc  sin   t  
ˆ inc
 cos   t  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
19
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
In the above figure, note that the x-z scattering plane is defined by the two wavevectors:
ˆ
kinc  zˆ and kˆrad  rˆ , where {here, with   0 } rˆ  sin  cos  xˆ  cos  zˆ  sin  xˆ  cos  zˆ .
The unit normal to the scattering plane nˆ is defined by: nˆ  kˆ  kˆ .
scat
scat
rad
inc
For an unpolarized/arbitrary/random polarization of the incident EM plane wave, from the
ˆ inc  t 
above figure, it can also be seen that the {instantaneous} polarization unit vector 
associated with the incident unpolarized EM plane wave can be decomposed into a component
ˆ inc and a component perpendicular to the x-z
parallel to/lying within the x-z scattering plane 
ˆ inc  t   cos   t  xˆ  sin   t  yˆ  cos   t  
ˆ inc  sin   t  
ˆ inc
ˆ inc

scattering plane 
 .
 :
ˆ rad  t  exists
Likewise, for the scattered EM radiation, whatever instantaneous polarization 
ˆ rad and a
can be decomposed into a component parallel to/lying within the x-z scattering plane 
ˆ rad
component perpendicular to the x-z scattering plane 
 .
However, as we have seen above for Thomson scattering with x̂ and  ŷ linearly-polarized
incident EM waves, the orientation of the incident vs. scattered polarization vectors is unchanged
by the scattering process in the following sense:
ˆ inc  xˆ parallel to
In the above figure, for an incident EM plane wave with linear polarization 
ˆ rad  cos  xˆ  sin  zˆ which
(i.e. lying in) the x-z scattering plane, the scattered polarization is 
also lies in the x-z scattering plane – notice the two blue polarization vectors in the above figure.
ˆ inc
ˆ
For an incident EM plane wave with linear polarization 
   y perpendicular to the x-z
ˆ rad
ˆ ˆ inc
scattering plane, the scattered polarization is 
   y  which is also perpendicular to the
x-z scattering plane – notice the two magenta polarization vectors in the above figure.
Thus, we can decompose the differential Thomson scattering cross section into that in which
the polarization vector of the scattered EM radiation lies in (i.e. parallel to) the x-z scattering
plane and that in which the polarization vector of the scattered radiation is perpendicular to the
x-z scattering plane, which we already have (!) from our above LPx and LPy results, namely:
d Tunpol   ,   0  1 d TLPx  ,   0  1  q 2


 
sin 2  x
2 
d
2
d
2  4 o mc 
2
2

m
2
sr 
2



1  q2
1  q2
2
2
2
 
cos 2 
sin  sin   cos   
2 
2 
2  4 o mc 
2  4 o mc 
2
2

d Tunpol   ,   0  1 d TLPy  ,   0  1  q 2

  2
1  q2
2
2
2







sin
sin
cos
cos
 





y


d
2
d
2  4 o mc 2 
2  4 o mc 2  
1

2


1  q2
1  q2
2
2


sin
cos
 





2 
2 
2  4 o mc   2  4 o mc 
2
m
1
2
sr 
ˆ inc
ˆ
ˆ inc  t  onto 
ˆ inc  xˆ and 
n.b. The 1/2 factor in the above arises from statistically projecting 
  y .
20
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Thus, for an observation point P  r ,  ,   0  lying in the x-z scattering plane:
d Tunpol   ,   0  1  q 2

 
cos 2 
2 
d
2  4 o mc 
2
m
2
sr 
And:
d Tunpol   ,   0  1  q 2

 
2 
d
2  4 o mc 
2
m
2
sr   n.b. has no  -dependence!
Note further that:
d Tunpol  ,   0 
d

d Tunpol   ,   0 
d

d Tunpol   ,   0 
d
2


1  q2
1  q2
2
 


cos

2 
2 
2  4 o mc 
2  4 o mc 
2

1  q2
 
1  cos 2  
2  
2  4 o mc 
m
2
2
sr 
Thus:
d Tunpol  ,   0  1  q 2
d Tunpol  ,  

2
 
 1  cos   
d
2  4 o mc 2 
d
2
m
2
sr 
We now introduce a quantity known as “the polarization” P  ,   0  which is formally a
specific type of asymmetry parameter A  x  – the normalized/fractional difference between two
related variables A  x  
a  x  b  x
. Thus, in general A  x  ranges between 1  A  x   1 .
a  x  b  x
(n.b. Sometimes A  x  is expressed in terms of a percentage).
Here, for our current Thomson scattering physics situation, the polarization {asymmetry}
P  ,   0  is defined as the normalized/fractional difference (i.e. asymmetry) between the
perpendicular    vs. parallel    differential Thomson scattering cross sections:
d Tunpol   ,   0  d Tunpol   ,   0 

1  cos 2 
sin 2 
d
d
PTunpol  ,   0  


d Tunpol   ,   0  d Tunpol   ,   0  1  cos 2  1  cos 2 

d
d
Here, we see that due to the physics associated with Thomson scattering of an unpolarized EM
plane wave from a free electric charge, PTunpol  ,   0  ranges between 0  PTunpol  ,   0   1 .
Due the geometrical constraints imposed in the overall scattering process, the instantaneous
orientation of the induced electric dipole yields useful non-zero time-averaged information!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
21
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
The angular dependence of the normalized differential Thomson scattering cross sections
 d Tunpol  ,   0  


d


 d Tunpol   ,   0  
 q2
 1
2



1
cos


,



2 
d
2
 4 o mc 


2
2
 q2
 1
 ,

2 
2
 4 o mc 
 d Tunpol   ,   0    q 2
 1
sin 2 
unpol
2
P




,
0


cos


and
the
polarization

 
T
2 
d
1  cos 2 
2

  4 o mc 
for unpolarized Thomson scattering as a function of  are shown together in the figure below:
2
Various things can be seen/learned from the four curves on the above graph:
a.) As mentioned above, the  differential Thomson scattering cross section is constant/flat,
independent of the scattering angle  . The  component is maximal at   0o and   180o .
b.) At   0o (forward scattering) and at   180o (backward scattering) the  vs.  differential
Thomson scattering cross sections are equal to each other, and because of this, the
polarization {asymmetry} vanishes, i.e. PTunpol   0,   0   PTunpol   180o ,   0   0 .
c.) At   90o the  differential Thomson scattering cross section vanishes, and because of this,
the polarization {asymmetry} is maximal, i.e. PTunpol   90o ,   0   1 , thus at   90o ,
the Thomson scattering of an unpolarized EM wave by a free charge q is purely/100%
due to the perpendicular    polarization component (only) of the incident EM wave!!!
22
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Scattering of EM Radiation by Neutral Atoms and/or Molecules
As discussed previously in P436 Lecture Notes 7.5 (Dispersion Phenomena in Linear Dielectric
Media) atoms and molecules are composite objects – consisting of {relatively light} electrons bound
to {relatively massive} nuclei. When a monochromatic (i.e. single-frequency) EM plane wave is
incident on a neutral atom (or molecule) – for simplicity’s sake, assumed to be spherical in shape –
 
the electric field of the incident EM plane wave Einc  r , t  induces electric dipole moment(s) in the
neutral atom/molecule {primarily} due to jiggling the light electrons at the angular frequency  of
 
the incident EM plane wave, arising from the driving force eEinc  r , t  acting on the bound electrons:
 


me r  t   me r  t   ke r  t   eEinc  r , t  ← inhomogeneous 2nd-order differential eqn.


 
 2 r  t 
r  t 

{again} neglected
me
 me
 ke r  t   eE  r , t  ← n.b. we have

2









t  
t
  B  eE term here...

ev
the


me a
Potential Force
(binding of atomic
electrons to atom)
Velocity-dependent
damping term
  damping constant

Driving Force
me  electron mass  9.11031 kg
For a driving force term sinusoidally varying in time with angular frequency  associated
with a monochromatic EM plane wave with linear polarization in the x̂ -direction incident on the
 

atom/molecule {located at the origin,   r  0  }: eEinc  r  0, t   eEo e  it xˆ



me x  me x  ke x  eEo e it xˆ .
the inhomogeneous force equation becomes:
 
The solution of this inhomogeneous second order differential equation is: x  r  0, t   xo e it xˆ
where: xo 
and:
eEo me
 2  02   i 


=
Atomic electron spatial
displacement amplitude
{n.b. complex!}
0  ke me (radians/sec) = characteristic/natural resonance {angular} frequency.
 
The corresponding {complex} induced electric dipole moment p  r  0, t  is:
 e2 Eo 
1
 

p  r  0, t   ex  t   exo e it xˆ   
e it xˆ

2
2
 me    0   i 
1
1 x  iy
x  iy

 2
with x   2  02  and y  
2
x  iy x  iy x  iy x  y
we can rationalize/rewrite this as:
Using the “standard trick” z 


2
2
 e 2 Eo     0   i   it
 
p  r  0, t    
 e xˆ

2
 me    2  02    2 2  
 
 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
23
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
 
The above expression describes the induced electric dipole moment p  r  0, t  of an atom /
molecule associated with a single QM transition/resonance at angular frequency 0  ke me
and natural linewidth    2  FWHM  sec1. However, real atoms/molecules are governed
by quantum mechanics and in general have many possible quantum mechanical bound states of
the atomic electrons, with transitions between them {resonances} with associated transition
energies E j  0 j and linewidths  j , as dictated by quantum mechanical selection rules.
Thus, a more realistic model of the atom/molecule that takes into account the various QM transitions
/ resonances present in an atom/molecule, properly weighted for each such resonance, gives:


2
2
 e 2 Eo   n osc   0 j  i j   it
 
ptot  r  0, t    
 e xˆ
  f j
  2   2 2   2 2  
 me   j 1
0j
j


 




where the angular frequency and natural linewidth associated with the jth resonance are:
0 j  ke j me rad/sec and  j   j 2  FWHM  sec1respectively, and the so-called
“oscillator strength” f
osc
j
n
th
associated with the j resonance is such that:
f
j 1
osc
j
1.
 
In the above expression for p tot  r  0, t  , note that the n individual contributions to the overall
/ total induced electric dipole moment of the atom/molecule are coherently added together, i.e.
added together at the amplitude level – the tacit assumption that has been made here is that the
wavelength  associated with the incident monochromatic EM plane wave/outgoing
scattered/radiated EM wave is much larger than the characteristic size of the atom/molecule, i.e.
  ratom , rmolecule and thus variation of phase(s) ei ~ eikr associated with the incoming/outgoing EM
waves, e.g. over the diameter of the atom/molecule are negligible and hence can be/are neglected.
Note that this approximation is certainly valid e.g. for EM radiation in the optical portion of the
EM spectrum (and below), since for visible light: violet ~ 400 nm , red ~ 650 nm whereas typically
ratom , rmolecule ~  0.1  few  nm .
 
Then evaluating p tot  r  0, t  at the retarded time to :


2
2
n
 02j  i j   it




e
E

 
osc
o
ptot  r  0, to    2 
 e o xˆ
  f j
2
m
2
2
2
2


 e   j 1
  0 j   j   

 





And thus:
2


2
2
2
n


i





 2ito


0
j
e
E


j
2

p tot
 0, to   p tot  0, to  p tot  0, to    4  o   f josc
 e
2
m
2
2
2
2


1

j
e

 
  0 j   j   


 
2

24



© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
From the discussion above (p. 3-9) for Thomson scattering of a monochromatic EM plane
wave linearly polarized in the x̂ -direction incident on a free electric charge q located at the
origin  , the instantaneous differential power radiated by an induced atomic/molecular electric
dipole moment, oriented along the x̂ -axis {due to the polarization of the incident EM radiation} is:
dPrrad  ,  , to   rad
 p 2  0, t 
 Watts 
 S  r , ,  , to r 2 rˆ  o tot 2 o sin 2  x 

d
16 c
 steradian 
where 
p 2  0, to  is given above; and: sin 2  x   sin 2  sin 2   cos 2   .
2
dPrrad  ,  , to   rad
o p tot
 0, to  sin 2  sin 2   cos 2   Watts 
2
 S  r , ,  , to r rˆ 
Thus:

  steradian 
d
16 2 c


Again, carrying out the time-averaging process on this quantity, and taking only the real part
of it for the physically-meaningful result for “far-zone” radiation associated with this scattering
2
process, however, because the above expression for 
p tot
 0, to  is extremely complicated, for the
purpose of discussing the salient physics features/behavior, we will assume for simplicity’s sake
that only a single resonance exists (with f1osc  1 ), rather than n of them. The full-blown
expression for n resonances can e.g. be coded up on a computer and results obtained numerically.
Thus, with the simplifying assumption of a single resonance in the atom/molecule:
d Prrad  ,  , to 
d


 2
 rad
1 o Re p  0, to 
2
 S  r ,  ,  , to  r rˆ 
sin 2  x
2
2
16 c
 4
2
2 2
2 2 
2         
2
0

1  Watts 
1   e Eo   
  2
  o2 
 sin  x and using: o 
 
2
2 16 c  me    2
 o c 2  sr 
2 2
2 2








0 
 
 
2


1
e2
4
sin 2  sin 2   cos 2  
  o Eo2 c 


2
2
2
 4 o me c   2  02    2 2 


The time average of the {magnitude of} the instantaneous flux of EM energy incident on the electric

1 
1
 Watts 
and thus:
dipole {located at the origin  } is: I inc  0   Sinc  0, to   Sinc  0, to    o Eo2 c 
2 
2
2
 m 
d Prad  ,  , to 
d atom  ,  
1
 
d
d
Sinc  0, to 
2
1
 o Eo2c
2

1
 o Eo2c
2
2


e2
4
sin 2  x

2 
2
2 2
2 2
4

m
c

o e

   0    
 m2 




 sr 


e2
4

 sin 2  sin 2   cos2  
2 
2
2 2
2 2
4
m
c


o e

   0    


© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
25
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
2
2
15
Since the classical electron radius is: re  e 4 o me c  2.82 10 m , we can write the
differential scattering cross section {per atom/molecule} for an incident EM plane wave
propagating in the  ẑ -direction and linearly polarized in the x̂ -direction in terms of re as:
LPx
d atom
 ,  
d
 re2
4
 2   2 2   2 2 
0


 sin
2
 sin 2   cos 2  
m
sr per atom/molecule 
2
Hence, we see that this result is very similar to that obtained for electron Thomson scattering for an
incident EM plane wave propagating in the  ẑ -direction and linearly polarized in the x̂ -direction:
d TLPx
 ,  
e
d
 re2 sin 2  x  re2  sin 2  sin 2   cos 2  
m
sr per electron 
2
For the atomic/molecular differential scattering cross section, we simply have the additional
2
dimensionless factor  4  2  02    2 2  arising from the physics associated with the


{quantum mechanical} internal structure of the atom/molecule!
Thus, since the total cross section for Thomson scattering of an EM plane wave incident on an
8 2
re  m 2 per electron  , the total cross section for atomic/molecular scattering
electron is:  T 
3
of an EM plane wave is:  atom 
8 2
4
re
3
 2   2 2   2 2 
0


m
2
per atom/molecule  .
Similarly, for each of the other polarizations of the incident monochromatic EM plane wave
discussed above for Thomson scattering, we obtain very similar results for atomic/molecular
scattering:
LPy
d atom
 ,  
d
LP
d atom
 ,  
d
 re2
 re2
4
  

2

2 2
0
 
2
2
 sin

 cos 2   cos 2  
m
2
sr per atom/molecule 
 sin 2 2  cos 2  
m
sr per atom/molecule 
2

4
 2   2 2   2 2 
0


 sin
2
unpol
d atom
 ,   0   1 r 2
4
1  cos2  
e
d
2  2   2 2   2 2 
0


26
m
2
2
sr per atom/molecule 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
with:
unpol 
d atom
 ,   0 
d
1 2
4
 re
2  2   2 2   2 2 
0


m
2
sr per atom/molecule 
and:
unpol 
d atom
 ,   0   1 r 2
4
cos 2 
e
2
d
2  2   2    2 2 
0


m
2
sr per atom/molecule 
Thus, we also see that the polarization {asymmetry} for atomic/molecular scattering by an
unpolarized EM plane wave is the same as that for Thomson scattering:
unpol 
unpol 
d atom
 ,   0 
 ,   0   d atom
1  cos 2 
sin 2 
unpol
d
Patom


 ,   0   unpol d 
unpol 
d atom  ,   0  d atom
 ,   0  1  cos 2  1  cos 2 

d
d
Hence the results on the graph shown on page 22 of these lecture notes for differential Thomson
scattering are in fact also valid for atomic/molecular scattering by an unpolarized EM plane wave.
Whereas the Thomson/free electron scattering cross section results are independent of the
frequency of the incident EM plane wave, from the above results, it is manifestly apparent that
the atomic/molecular bound electron scattering cross section results depend very sensitively on
the frequency of the incident EM plane wave.
4
Let us examine the frequency behavior of the resonance lineshape factor
 2   2 2   2 2 
in the above atomic/molecular scattering cross section results.
0


Noting that in atoms/molecules, the natural line widths associated with resonances/transitions
between distinct quantum states {typically in the UV portion of the EM spectrum for atoms} are
quite narrow, i.e. that:
  


2  FWHM   0  ke me .
x4
Defining x   0 and y   0 , a log-log plot of the resonance lineshape
 x 2  12  x 2 y 2 
vs. x is shown in the figure below.


Below the peak of the resonance   0  , note the linear 4 decade increase in the lineshape per
1 decade increase in x   0 which is due to the  4 dependence of the lineshape.
Above the peak of the resonance   0  , note that the lineshape is flat with frequency, i.e. it is
independent of frequency!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
27
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Referring to the above plot of the resonance lineshape, we see that there are three distinct
frequency regions to consider:
1.) Low frequencies:   0
When   0 , the factor   

2 2
0
4
  in the resonance lineshape  2
2
   02    2 2 
and additionally, since   0 , then for   0 :
2
4
0
4
4   
1 d atom  ,  


   2
.
4
2
4
2
2
re
d
 2   2    2 2  0     0  o 
0


4
4


Thus, at low frequencies   0  the differential and total atomic/molecular scattering cross
sections are strongly frequency-dependent, and behave as:
d atom  ,  
 
 re2    Angular Factor
d
 o 
4
m
2
sr per atom/molecule 
and:
 atom
28
8 2   

re  
3  o 
4
m
2
per atom/molecule 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
This is type of atomic/molecular scattering of EM plane waves at low frequencies   0 
is known as Rayleigh scattering, in honor or Lord Rayleigh, who carried out early theoretical
work associated with this topic in the latter part of the 19th century.
The strong  4 frequency dependence of the Rayleigh scattering cross section explains why
the sky and e.g. pure water (!) appears blue. Blue light is Rayleigh-scattered ~ 4-5 more than
red light! The following picture shows the gorgeous blue color arising from Rayleigh scattering
of light in the ultra-pure H2O tank of the Super-Kamiokande experiment (located in Japan):
unpol
The behavior of the polarization {asymmetry} Patom
 ,   0   sin 2  1  cos 2   for
unpolarized incident EM plane waves in the visible portion of the EM spectrum also explains
why the light from the sky is polarized, and especially so at   90o ,   0  ! Please go back/
refer to/look at P436 Lecture Notes 13, p. 17-18 where we discussed this originally.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
29
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
2.) Resonance:   0
On (or near) a resonance   0 (typically in the UV portion of the spectrum for atoms) the
factor  2  02   0 and thus for   0 :
2
04
02  0 
1 d atom  ,  
 2 2  2    2
2
re
d
 2   2    2 2   0 
  
0


2
4


Since   0    then 0    1 and thus we see that on (or near) a resonance the
2
differential and total atomic/molecular resonant scattering cross sections become extremely
large – incident EM radiation is absorbed/re-emitted/scattered prolifically on/near a resonance:
d atom  ,  
 
 re2  o   Angular Factor
d
  
2
m
2
sr per atom/molecule 
and:
 atom
8 2  o 
re  

3   
2
m
2
per atom/molecule 
3.) High frequencies:   0
For high frequencies   0  the bound atomic/molecular electrons behave as if they are free –
i.e. as in Thomson scattering of free electrons! When   0 the behavior of the factor  2  02 
in
4
  2  


2 2
0
is  2  02    4 and additionally, since   0 , then for   0 :
2
  2 2 

4
4
2
2
1 d atom  ,  
 4
 2 2
 2
 2 1 2
.
2
2
2
2
2
re
d
 2   2    2 2                
0


4


Thus, at high frequencies   0  the differential and total atomic/molecular scattering
cross sections are frequency-independent, and behave essentially identical to those associated
with Thomson scattering of free electrons:
d atom  ,  
 re2  Angular Factor
d
m
2
sr per atom/molecule 
and:
 atom 
30
8 2
re
3
m
2
per atom/molecule 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
2
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
4.) Extremely High frequencies:   me c 2 (and beyond).
At extremely high frequencies, when E    me c 2 the scattering of EM radiation by
atoms/molecules is no longer in the non-relativistic regime, and n.b. also violates the second
condition of our original Taylor series expansion for EM radiation in the “far-zone” limit
 r  1 and  rmax
 c  krmax
  2 rmax
   1 } because for E    me c 2 (and beyond),
{ rmax
 ,
the wavelength  is comparable to (and/or smaller than) the size of the atom/molecule rmax
   1 is not satisfied. Thus use of (any) of the above formulae
thus the requirement 2 rmax
would be extremely precarious in this regime.
This is the regime of hard x- and  -ray scattering by {essentially free} electrons bound to
atoms/molecules, and is known as Compton scattering, which is essentially “billiard-ball”
x- and/or  -ray photon-electron elastic scattering. In this high frequency/high energy regime,
the scattered EM radiation has a different (i.e. lower) frequency than the incident radiation.
The classical theory of the scattering of EM radiation is unable to explain, in terms of any
kind of macroscopic EM wave phenomena.
Only the relativistic quantum mechanical theory of photons interacting with electrons
{QED} succeeds in properly explaining Compton scattering of high-energy x- and  -rays
by electrons (and other charged particles).
For   me c 2 the frequency shift is small, but not precisely zero. The classical theory
works adequately well in this regime. We will discuss Compton scattering in more detail
when we get to the subject of relativistic kinematics (P436 Lect. Notes 17).
Scattering of EM Radiation by a Collection of Free Charges, Neutral Atoms and/or Molecules
Thus far, we have discussed scattering of EM radiation by a single free charge q and/or a
single neutral atom/molecule. What happens when a macroscopic EM plane wave scatters from a
collection/ensemble of many such objects?
Consider what happens when an incident EM plane wave scatters from just two such objects.

Suppose the first scattering object is located at the origin   r1  0  (as before), the other is

located at an arbitrary position r2  x2 xˆ  y2 yˆ  z2 zˆ .
Since the incident EM wave is a plane wave, the solutions of the two corresponding
inhomogeneous force equations are such that, at the common retarded time to the magnitudes of
 
 
the {complex} induced electric dipole moments are equal to each other: p  r , t   p  r , t  ,
1
1
o
2
2
o
 
 
i.e. p 2  r2 , to  can only differ from p1  r1 , to  by a relative phase
 
 
 
 
 
p 2  r2 , to   p1  r1 , to  ei 21  p1  r1 , to  eit21  p1  r1 , to  eikinc  r2 .
 
due to relative arrival time difference t21  kinc r2  of the incident EM wave at the 2nd


scattering object, at position r2 relative to the first, located at origin   r1  0  .


© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
31
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede



However, this is only half of the story. Because p1 is located at position r1  0 and p 2 is located

at position r2  x2 xˆ  y2 yˆ  z2 zˆ , the EM waves simultaneously radiated by each of the two electric
dipoles at the common retarded time to will arrive at the observation/field point position
P  r ,  ,  , t  such that the EM “far-zone” radiation fields associated with dipole # 2 have associated


with them an additional {relative} phase shift of ei 21  eit21  e  ikrad  r2 due to relative arrival time
 
   krad r2  of the scattered EM wave at the observation point P  r ,  ,  , t  from
difference t21


the 2nd scattering object, at position r2 relative to the first, located at origin   r1  0  .



Thus, there is an overall phase shift for scatterer # 2 at position r2 relative to scatterer # 1 located



 

 
 


i k k  r



at origin   r1  0  of: ei 21 ei 21  eit21 eit21  eikinc  r2 e  ikrad  r2  e inc rad 2  eik  r2 where: k  kinc  krad .
In general, the scattered EM wave(s) radiated from the two induced electric dipole moments
 
 
p1  r1 , t  and p 2  r2 , t  , since the radiation electric fields also obey the superposition principle,
will subsequently interfere with each other in the “far-zone” at the observation point P  r ,  ,  , t  .
We can generalize the above 2-scatterer result for the scattering of an incident EM wave to that
for a collection of N identical scattering objects, each of which is located at position

rn  xn xˆ  yn yˆ  zn zˆ for the nth scattering object, n = 1, 2,3,… N. Via use of the principle of linear


superposition, each such identical scattering object will contribute Erscat
 r , t  to the overall “farn
zone” EM radiation field at the observation point P  r ,  ,  , t  . Since the differential and total
scattering cross sections are both proportional to the modulus-squared of the overall/total EM

 2
radiation field Erscat
r
, t  , where:

tot
N 
N 
N
 
 
 

 scat 
 scat 
 scat 

ik  rn
ik  rn
ik  rn
scat 
scat 
Erscat
r
,
t

E
r
,
t

E
r
,
t
e

E
r
,
t
e
E
r
,
t

E
r
,
t
e
where












 rn
 r1

r1
rn
r1
tot
n 1
n 1
n 1


where Erscat
 r , t  is the scattered “far-zone” radiation field at the observation point P  r , ,  , t 
1

associated with a single scattering object located at the origin   r1  0  .

 2
Then we see that: Erscat
r,t 
tot
N
e
n 1
 
ik  rn
2

 2
Erscat
r,t 
1

Defining the so-called complex structure factor: F k  ,   

 e
N


ik  ,   rn
2
n 1
and neglecting multiple scattering effects1, the differential scattering cross section associated with
an incident macroscopic, monochromatic EM plane wave scattering off of a collection/ensemble
of N identical scattering objects (e.g. free electrons, atoms/molecules, etc.) can be written as:
1
i.e. the mean free path for scattering is large compared to the overall spatial dimensions of the collection of scatterers.
32
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015

d N  ,  
d 1  ,  
 F k  ,  

d
d


Lect. Notes 14.5
N
e


ik  ,   rn
n 1
2
Prof. Steven Errede
d 1  ,  
d
where d 1  ,   d  is the differential scattering cross section associated with a single

scattering object located at the origin   r1  0  .

It can be seen that the numerical effect of the structure factor F k  ,   

 e
N


ik  ,   rn
2
n 1
on the overall differential scattering cross section d N  ,   d  depends very sensitively on
the exact/precise details of the spatial distribution of the N identical scattering objects.
First, let us consider again the case of only two scattering objects for forward scattering,
when   0 . Then for two identical scattering objects, the structure factor is:

F k  ,   

 e
2
n 1


ik  ,   rn
2











 1  eik  ,   r2 1  e  ik  ,   r2  1  eik  ,   r2  e  ik  ,   r2  1








 2  eik  ,   r2  e ik  ,   r2  2  2 cos k  ,  r2  2 1  cos k  ,  r2 






However, for forward   0  scattering this means that: krad  kinc  zˆ i.e. krad  kinc  zˆ




and thus: k   0,    kinc  krad  0 and hence: F k   0,    0  2 1  cos  0    4 .








Thus we see that two identical scattering objects always constructively interfere with each

other for forward   0  scattering, independent of the location r2 of the 2nd scatterer relative to

that of the first, located at the origin   r1  0  . For forward   0  scattering the {relative} time
delay t2  z2 c in the arrival of the incident EM plane wave at the z-location of the 2nd scattering
object is exactly compensated by the {relative} decrease in the arrival time t2   z2 c of the
scattered/radiated wave in the “far-zone” at the observer’s position at  r ,   0,   .



Noting that since: k   0,    kinc  krad  0 for all N identical scattering objects for
forward   0  scattering, we see that for forward   0  scattering of an incident EM plane

wave by N identical scattering objects, the structure factor F k   0,    0 becomes:


F k   0,    0 


2
e
n 1


ik   0   rn
2

2
e
n 1
 
i 0  rn
2
 
 

 
 ei 0  r1  ei 0  r2  ...  ei 0  rN
2
2
 1  1  ...  1  N 2



N
2
and thus we see that N identical scattering objects always constructively interfere with each
other for forward   0  scattering, independent of the z-location of the nth scatterer relative to

that of the first {located at the origin   r1  0  }.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
33
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Thus, the forward   0  differential scattering cross section associated with N identical
scatterers is:
d N   0,  
d   0,  
 N2 1
d
d
It can also be seen that for backward     scattering the situation is not the same as for
forward   0  scattering. Again, we consider the two-scatterer case: when    then:






krad  kinc   zˆ and: k    ,    kinc  krad  2kinc  0 and:

 
F k    ,    2 1  cos 2kinc r2  .






 
When: 2kinc r2  2n , n  1, 2,3... {i.e. certain (angular) frequencies n  ckn  n c z2 }
 

Then: cos 2kinc r2  cos  2n   1 and: F k    ,    4 resulting in constructive






interference for backward     scattering.
 
When: 2 kinc r2   2n  1  , n  1, 2,3... {i.e. certain other (angular) frequencies
 

n  ckn  12  2n  1  c z2 } then: cos 2kinc r2  cos   2n  1    1 and: F k    ,    0






resulting in destructive interference for backward     scattering!
Thus, for backward     scattering with N identical scatterers, only if the scatterers are
arranged in some kind of highly-organized, regular array/3-D lattice (e.g. such as a crystal), will
coherent backward scattering effects (i.e. constructive/destructive interference effects) be observable.
In general, if the N identical scatterers are randomly organized, such as in a plasma, a gas, a
liquid or an amorphous solid, then it can be shown that the terms with m  n in the structure factor:

F k  ,   

 e
N


ik  ,   rn
n 1
2
N
N
  e

 
ik  ,    rn  rm 
n 1 m 1

contribute very little to the overall value of F k  ,   , due to stochastic cancellations


{n.b. the same principle is used to balance turbine blade assemblies in constructing jet engines}.
Thus, for a random collection of N identical scatterers, when m  n the double series associated

with F k  ,   becomes:


N
N
N



 
ik  ,  r  r
F k  ,     e    n n    eik  ,   0  1  N


n 1
n 1
n 1
and thus the differential scattering cross section associated with N randomly distributed
identical scatterers (except for the precisely   0 forward scattering) is:
d N   0,  
d   0,  
N 1
d
d
34
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
The overall scattering in this randomly-distributed situation is known as “incoherent scattering”.
For the situation associated with the scattering of a macroscopic, monochromatic EM plane
wave incident on a highly regular, cubical 3-D crystalline-type array/lattice consisting of N
identical scattering objects (e.g. atoms or molecules) of dimensions W  H  D  Lx  Ly  Lz and
lattice constant / lattice spacing a and N  N x N y N z   Lx a    Ly a    Lz a  where the N j are
the # of lattice sites in the jth direction, the structure factor for this highly regular array of N
identical scatterers is:

F k  ,   

 e
N


ik  ,   rn
n 1
2
 sin 2  1 N k a    sin 2  12 N y k y a    sin 2  1 N k a   
z
z
2

 N  2 22 1x x    2 2
 N sin  k a    N sin  1 k a    N 2 sin 2  1 k a   
x
z
2
2
y
2
 x
  y
 
  z
2
At low frequencies/long wavelengths    a  , only the forward-scattering peak at k j a  0

contributes to F k  ,   because {here} the maximum possible value of k j a is: 2ka  4 a   1 .


In this forward-scattering/small angle regime, using the small-angle Taylor series approximation
2
for sin 2  12 k x a    12 k x a  , the structure factor becomes:
 2 1
  sin 2  12 N y k y a    sin 2  12 N z k z a   

2  sin  2 N x k x a 



F k   0,    N 
2
2
2
sin x
1
1




1




N
k
a
N
k
a





N
k
a
 2 x x
  2 y y    2 z z
  where sinc  x   x


 N 2sinc 2  12 N x k x a   sinc 2  12 N y k y a   sinc 2  12 N z k z a 
A graph sinc(x) vs. x is shown in the figure below:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
35
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
In the low-frequency/long-wavelength regime, from the above graph of sinc(x) vs. x it can

be seen that the structure factor F k   0,   is only appreciable when 12 N j k j a   ,


which corresponds to an angular region of forward scattering   k j a  2 N j , which
becomes exceedingly small as N j   .
Note that for a typical lattice constant a  5Å  0.5 nm  5 1010 m and a typical macroscopic
sample size of L j  1 cm  0.01 m then: N j  L j a  102 5 1010  2 107 , corresponding to a
forward scattering angular region of:   2 2  107    107 radians  0.3 rad !!!
Thus, in regular crystalline arrays of scattering objects, e.g. single crystals of transparent
minerals such as diamond, emerald, quartz, rock salt, etc. there is essentially no scattering at long
wavelengths {except in the extreme forward direction}. The very small amount of large-angle
scattering that does occur in such samples is caused by/due to transitory thermal vibrations of the
atoms in the lattice away from the “perfect” configuration of the 3-D crystalline lattice.
At high frequencies/short wavelengths    2a ~ 1 nm  i.e. when ka   – typically in the

x-ray region of the EM spectrum – the structure factor F k  ,   has maxima when the


so-called Bragg scattering condition is satisfied: k j a  2n or:
k j  2kinc sin  j 
4

sin  j 
2n
, i.e. when n  2a sin  j .
a
Crystal planes spaced distance a apart
36
Typical X-ray diffraction pattern
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Experimental Aspects, Applications and Uses of
the Measurement of Differential and Total Scattering Cross Sections
At the beginning of these lecture notes, the theoretical/mathematical definitions of the
differential and total scattering cross sections were given:
Differential Scattering Cross Section {SI units: m 2 sr per scattering object}:
d Prad  ,  , t 
d scat  ,  
1
1

  

d
I inc  r  0 
d
Sinc  r , t 
d Prad  ,  , t 
d


S rad  r , t  r 2 rˆ
  
Sinc  r , t   zˆ
r 0
r 0
The differential scattering cross section {SI units: m2/sr} is the time-averaged differential/
angular EM power {SI units: Watts/sr} radiated by a scattering object (or a collection/ensemble
of scattering objects) into solid angle d   ,    d cos  d  sin  d d , normalized to the
time-averaged flux of EM energy (= EM intensity I inc , aka irradiance) {SI units: Watts}

incident on the scattering object/objects located at the origin   r  0  .
From the RHS of the above equation, we also see that:
d scat  ,  
d

Scattered Flux of EM Radiation Unit Solid Angle  Watts sr   m 2 



2 
Incident Flux of EM Radiation Unit Area
 Watts m   sr 
per
scattering
object
Total Scattering Cross Section {SI units: Area, i.e. m 2 per scattering object}:


d Prad  ,  , t  d   d Srad  r , t  r 2 rˆ d 
Prad  t 
Prad  t 
d  ,  

 scat 
  


  scat
d

 
 
I inc  r  0 
d
Sinc  r , t 
Sinc  r , t 
Sinc  r , t 
r 0
r 0
r 0
The total scattering cross section{SI units: m2} is the time-averaged total EM power {SI units:
Watts} radiated into all angles {i.e. 4 steradians (sr)} by a scattering object (or a collection /
ensemble of scattering objects) normalized to the time-averaged flux of EM energy (= EM
intensity I inc , aka irradiance) {SI units: Watts} incident on the scattering object/objects located

at the origin   r  0  .
From the RHS of the above equation, we also see that:
 scat 
Total Scattered Flux of EM Radiation into 4 sr  Watts 
  m2 

2 
Incident Flux of EM Radiation Unit Area
 Watts m 
per
scattering
object
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
37
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
In many common experimental situations associated with the scattering of a macroscopic EM
plane wave incident on a collection/ensemble of N identical scattering objects – e.g. free
electrons/ions in plasmas, or e.g. neutral atoms/molecules in solids, liquids and/or gases, the
{instantaneous} 3-D positions of the scattering objects are randomly distributed. For a collection
of N identical randomly distributed scattering objects, we have shown that the structure factor


F k  ,    N {except for precisely   0 forward scattering, where F k   0,    N 2 }




and thus the so-called incoherent differential scattering cross section associated with the
collective scattering of an incident EM plane wave by N identical randomly distributed
scatterers {except for precisely   0 forward scattering} is:
d N   0,  
d   0,  
N 1
d
d
Next, we consider a monochromatic macroscopic EM plane wave propagating in the
 ẑ -direction normally incident on a target consisting of a slab of “generic” matter (plasma, gas,
liquid or solid) of macroscopic volume V  H  W  D  m3  consisting of a total of N randomly
distributed identical scattering objects, characterized by a number density n  N V  # m3  and
mass volume density   M V  kg m3  . The incident EM plane wave uniformly illuminates the
cross sectional area A  H  W of the target slab of “generic” matter.
The macroscopic EM plane wave enters the front face of the target at normal incidence at
z = 0, and after propagating an infinitesimal longitudinal distance dz into the target, the
dN  N  dz D  randomly-distributed scattering objects contained within this infinitesimal
thickness dz of the target have collectively absorbed and then re-radiated into 4 steradians a
time-averaged amount of power dPo from the incident EM plane wave of {n.b. turning the
total cross section (per scattering object!) relation around: Prad  t    scat I o Watts  per
scattering object}:
dPo  dN I o  N  dz D   I o  N  dz A D   I o A   N V  dz I o A  n dzI o A Watts 
which corresponds to a decrease/reduction/loss in the incident intensity I o over the infinitesimal
distance dz of:
dI o 
dPo
dI
 n I o dz Watts m 2  with corresponding –ve slope: o   n I o Watts m3 
A
dz
Then for propagation of the macroscopic EM wave a longitudinal distance z into the target, this
latter relation becomes:
dI  z 
dI  z 
 n I  z   0
  n I  z  Watts m3  which can be rearranged as:
dz
dz
38
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
This relation is a simple homogeneous first-order differential equation with boundary conditions:
I  z  0   I o and: I  z     0 . The specific solution to this differential equation is: I  z   I o e  n z .
Thus we see that the incident EM plane wave is exponentially attenuated to 1 e  e 1  0.368
of its initial intensity I o in propagating a characteristic longitudinal distance known as the
attenuation length atten  1 n
m
z 
 n z
into the target. Then: I  z   I o e atten  I o e
.
The reciprocal of the attenuation length atten is known as the absorption coefficient:
  1 atten  n
 m  . Then: I  z   I e
1
o
 z
 I o e  z atten  I o e  n z .
Using an isothermal model of the earth’s atmosphere (i.e. the density air varies exponentially
with altitude), nair ~ 2.7  1025 molecules/m3 , typical value(s) of the attenuation length for
Rayleigh scattering of visible light by N2 and O2 molecules in the earth’s atmosphere are:
Red light (  = 650 nm): atten ~ 188 km
Green light (  = 520 nm): atten ~ 77 km
Violet light (  = 410 nm): atten ~ 30km
The % scattering of {direct} sunlight in the earth’s atmosphere as a function of wavelength
 is shown in the figure below:
Image: Copyright Robert A. Rohde, 2007
http://en.wikipedia.org/wiki/Image:Rayleigh_sunlight_scattering.png
Note also that Rayleigh scattering of the sun’s light by air molecules in the earth’s atmosphere
is also is responsible for giving the sky its apparent height above the ground.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
39
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Plots of the attenuation/absorption length atten and the absorption coefficient   1 atten for pure
water (& ice) vs. wavelength are shown in the figures below:
40
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
Typical Experimental Apparatus to Measure a Differential Scattering Cross Section
A typical experimental setup used to measure a differential scattering cross section is shown
in a plan view in the figure below, in the x-z scattering plane:
Incident EM
Plane Wave
Target

ẑ

d
R
Generic EM
Radiation
Detector
r̂
x̂
The “generic” EM radiation detector, located a distance R away from the center of the target
at polar angle  subtends a solid angle d from a point target. If the target has finite spatial
extent, then one obtains the solid angle subtended by the detector for a finite-volume target by
integrating over the volume of the target to obtain the target-weighted solid angle. In principle,
corrections also need to be made for a.) attenuation of the incident EM intensity in propagating
through the target and b.) multiple scattering of the incident EM radiation and also of the
scattered EM radiation in getting out the target; all of which become increasingly important for a
thick target.
Depending on the physics associated with the scattering of incident EM radiation in the target,
a specific choice must be made for the detector that is used for measuring the EM radiation
scattered into solid angle d   ,   0  . For example, for if one is interested in measuring the
differential scattering cross section associated with a continuous, high-intensity beam of incident
EM radiation in the infra-red (IR), visible or ultra-violet (UV) light region, use of an absolutelycalibrated {NIST-traceable} spectro-photometer with associated readout electronics {aka
radiometer} is frequently used, as shown in the figures below:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
41
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
International Light IL1700 Research Radiometer + Photometer:
Absolute calibration of photometer response (Ampere-cm2/Incident Watt) vs. wavelength :
 UV  | Visible  |  IR 
Wavelength,  (nm)
The spectrophotometer {of active area Asp } is frequently some kind of photodiode.
The photocurrent {in Amperes} produced in the junction of the photodiode by the EM radiation
incident on the spectrophotometer is accurately measured by the accompanying electronics of the
radiometer. Note that there is great flexibility in the experimental setup – it could use incident
EM radiation that is polarized in some manner (e.g. LPx, LPy, RCP, LCP etc) and/or for e.g.
unpolarized EM radiation incident on the target, polarizing filters can be placed in front of the
spectrophotometer to measure e.g. d unpol  ,   0  d  and d unpol  ,   0  d  and the
unpol
polarization {asymmetry} Patom
 ,   0  , etc.
42
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
In some physics situations, the illuminated target is e.g. a biological sample of some kind and
the detector of the scattered radiation is e.g. a microscope + CCD (or CMOS) camera, usually
oriented at fixed scattering angle, e.g.    2,   0  . Again the incident EM radiation could
{additionally} be polarized in some manner, or if unpolarized EM radiation is incident, then
polarizing filters placed upstream of the camera can be used to analyze the  vs.  polarization
states of the scattered EM radiation.
In other physics situations where the intensity of the incident EM radiation is extremely low,
the photodetector of choice often is a low-noise photomultiplier tube (PMT) or avalanche
photodiode (APD), often thermo-cooled {or even LN2 or LHe-cooled!} in order to reduce finite
temperature/thermal “dark noise”. These detectors then count single photons scattered from the
target. The {wavelength-dependent} quantum efficiency (QE) of the photon detector used in
such scattering experiments must therefore be accurately known; typical QE’s are on the order of
~ 10-20%.
In P436 Lecture Notes 5 (p. 18-26) we discussed the connection between intensity I and the
{time-averaged} number of photons n  t  e.g. associated with a laser beam with photon
energy E  hf  hc  :

I  S  t   c uEM  t   2 o Eo2rms  F   t  E  c n  t  E
 Watts 

2 
 m 
Thus, counting photons {quanta of the EM field} within solid angle d at fixed scattering angle
 ,   0  is equivalent to measuring intensity I scat within solid angle d at fixed scattering
angle  ,   0  .
Sometimes an EM wave scattering experiment involves e.g. microwaves or radio waves – e.g.
such as in RADAR applications. In engineering parlance, the type of scattering cross section(s)
that we have been discussing here in these lecture notes are all known as bi-static cross section
measurements, because the location of the source used to illuminate the target with EM radiation
is distinct from the location of the detector of the scattered EM radiation. A mono-static cross
section measurement is where the source and detector are co-located at the same point – i.e. the
detector only measures back-scattered     EM radiation. Additionally, engineers define the
RADAR differential scattering cross sections (RCS) differently than physicists, as: 4  d d   .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
43
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 14.5
Prof. Steven Errede
A typical differential RCS e.g. for a B-26 Invader
is shown as a polar plot in the figure on the right.
All airplanes {unless “stealthified”} have a
characteristic/distinct type of differential RCS that
can be useful e.g. in identifying the type of plane.
In plasma physics – e.g. fusion/tokamak
applications, the differential Thomson scattering
cross section {at small scattering angles} is
commonly used as a diagnostic tool e.g. using a
{fairly high-powered} laser to monitor the number
density and also the temperature of the plasma.
Thomson scattering of electrons in the tenuous
plasma surrounding our sun can be seen e.g. during a
total eclipse of the sun – this is the sun’s corona!
NASA’s STEREO mission generates 3-D images of
the sun by measuring the sun’s so-called K-corona using
two satellites, as shown in the RHS picture below:
The maximum possible EM luminosity of a star is
determined by the balance between the inward-directed
gravitational force Fgrav  GN M star m p r 2 and the
outward-directed force due to radiation pressure
associated with Thomson scattering of electrons

Frad  Lstar Te 4 cr 2 on the plasma surrounding the
star (assumed here to be 100% ionized hydrogen).
The condition that Fgrav  Frad constrains the maximum
luminosity of the star, known as the Eddington limit
(in honor of Sir Arthur S. Eddington):

Lstar  4 GN M star m p c  Te  1.3  1031 Watts
 3.3  104  M star M   L
where the sun’s mass: M   2  1030 kg and our
suns’ solar luminosity: L  3.85  1026 Watts
The cosmic microwave background (CMB)
– a relic of the Big Bang – is partially linearly
polarized due to Thomson scattering on electrons.
CMB experiments such as WMAP, the Planck
mission have recently measured the polarization
of the CMB radiation.
44
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.