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, non- relativistic 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 2 defined in terms of a scattering cross section  scat , which has SI units of area, i.e. m .

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 Ptrad  (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   2 IincrSrt00, inc    (Watts m ), evaluated at the location of the target/scattering  object (usually at the origin, r  0 ):

Ptrad Pt rad  Total Scattering Cross Section:    (SI units of area, m2) scat Ir 0  inc Srinc  0, t

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 cross- sectional 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 Unscattered EM Plane Wave Plane Wave

Scattering Object (e.g. charge q, atom, molecule, etc.)

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2015. All Rights Reserved. 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 ddcos d  sin dd (SI units: steradians; aka sterad, and/or sr) is:

dPrad ,, t    Srtrr,  2 ˆ (SI units: Watts steradian Watts sr ). d rad  The infinitesimal vector area element dasph da sph 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  22 2 element d via dasph  r d rˆˆ r dcos d r  rsin d d r ˆ. 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 ddd cos   sin dd and the infinitesimal area element 2 dasph  rdcos  rd r sin  d  rd  r d .

 Note also that dPrad ,, t d has no r-dependence!

The differential angular scattering cross section is defined as:

2 dP,, t d2 P  ,, t ddscat,, scat  11rad rad SI units:   dddcos d dd cos  2 Srinc0, t Sr inc  0, t mSr

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 zˆ direction.

It is also instructive to write out the differential scattering relation in more explicit detail:

   2 2 d, dP,, t Srtrrrad ,  ˆ ErtBrtrr,,  ˆ scat  1 rad  rad rad SI units:     dd ˆ 2 Srtinc , Srtzinc ,  ˆ ErtBrtzinc,, inc  mSr r0 r0 r0

As we derived in P436 Lecture Notes 14 (p. 1-5) from the Taylor series expansions of    tot rt, r and Jrttot , r 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rt,,rr prt zˆ {n.b. oriented parallel to the zˆ -axis}, where, in the “far-zone” limit

{ rrmax 1 and rckrmax max21 r max  }, the instantaneous differential retarded EM

power radiated by the time-varying E1 electric dipole, with retarded time tttrcr o 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

rad  2 dPr ,, trad  22 oo p t Watts  Srtrr,sinˆ    2  dc 16 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:

  pt   pt   oosin ˆ  oosin 1 Ertr ,   , Brtr ,  ˆ with Brrrt,, rˆ E rt 4 r 4cr c  ˆ ˆ i.e. BrrErk   2 2   rad  ooptsin  rad 1 and: urtr ,  22 2 and: SrtErtBrtrrr,,,  16 cr o

  pt22sin22 pt  sin ccr ˆ Srtrad ,, ooˆˆ oo rcurtˆ rad r  22 22 r  with 16cr 16 cr  rkˆ  ˆ rˆ

The total instantaneous retarded EM power radiated by a time-varying E1 electric dipole into  22 4 steradians, with vector area element da  rsin d d rˆˆ r d r in the “far-zone” limit is:

22  2 rad rad oop tpt2 oo   Ptrr Srtda ,sinsin   dd (Watts) S 1622cc00 6

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  xyz, , 0,0,0 with a plane EM wave propagating in free space in the zˆ direction, which is incident on the free charge q,  ˆ ˆ {which has mass m}. Noting that kkkinc inc and that kzinc  ˆ and using complex notation:

    ikz t ikz t  11 ErtEre,  , BrtBre,   with B rkErzErˆ  ˆ  inc o inc o oincoocc      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   xˆ -direction, and for simplicity, let us also assume that Eroo Exˆ , and therefore:

  11EEE B rkErzErzxyˆ  ˆˆ oooˆˆˆ y. oincooccccc      yˆ

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2015. All Rights Reserved. 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:

   11 2222ikzt ikzt 1 Srt,,, ErtBrt Ee xyˆˆ  cEe zˆ (using  c  ) inc inc inc  o o o o ooc oc  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 ):

     FrtqErtqvrtBrtmarttot0, inc 0, 0, inc 0, 0,

The {transverse} electric field of the incident EM plane wave in the vicinity of r  0 is:

  it Erinc  0, t Eex o ˆ

The {transverse} magnetic field of the incident EM plane wave in the vicinity of r  0 is:

 1111ˆ it it B rt0, kErt 0,  zErtEezxEeyˆˆ 0,  ˆˆ inccccc inc inc  inc o o  yˆ

  vr  0, t yˆ   it it   Thus: Ftot r  0, t qEe o xˆ qEe o  ma r  0, t . c  vc  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 vc or:   vc 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.

  it Physically, the transverse electric field of the incident EM plane wave EtEexinc0,  o ˆ   it 2 it exerts a time-dependent force Ftot0, t qEexmat oˆˆ  0, mxt  0, mxex o on the free electric charge q, causing it to oscillate back and forth along the xˆ -axis in a time-dependent  it manner: x 0,txex o ˆ 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    cf.  2 it Note also that the instantaneous acceleration of the charge q is atxt0,  0,  xexo ˆ .

The E-field induced oscillatory motion of the point electric charge q thus creates an induced  it it time-dependent electric dipole moment: p0,tqxtqxexpex  0, ooˆˆ  oriented parallel   it to the electric field EtEexinc0,  o ˆ 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:

rˆ Observer at  Field Point, r 1 SEBrad  rad rad o  Skrˆ ˆ    rad rad 1 rad SEBinc inc inc  EEx ˆ o   rad o  zˆ  ˆ  1 ˆ ˆ Skzˆ  x BkEradc rad rad EExinc o inc inc p  pxˆ   B 1 kEˆ Byˆ rad ˆ ˆ incc  inc inc o Boradkx xˆ xˆ yˆ yˆ

Note also that the instantaneous mechanical power associated with the oscillating free charge q is     PtFmech tot 0, tvt   0, , arising from the absorption of energy from the incident EM wave by the  it free charge q. The velocity of the oscillating charge is vtdxtdtixex0,  0,  o ˆ and hence  232it it  it Pmech t mxexixeximxe oˆˆ  o  o .

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 ct Radiated / Incident / Radiated / Incident / Incoming Outgoing Incoming Outgoing * Photon Photon Photon e Photon * e Virtual / Virtual /  Off-Shell e Off-Shell e

x x The characteristic time interval t associated with the photon-free charge absorption-re-radiation 1 process is governed by the Heisenberg uncertainty principle Et 2  where   h 2 and h is Planck’s constant,  1.0546 1034Joule 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 5 2005-2015. All Rights Reserved. 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 EhfhceV  1 , 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 EE 1 eV in the Heisenberg uncertainty relation, we see that the corresponding time interval t is: 111 16 16 15 t222  E  E 6.582  10 eV sec 1 eV  3.261  10 sec  0.3261  10 sec  0.326 femto - sec

Note that the period of oscillation  associated with a Ehf1 eV photon is:

98 15 1f c  1240 10 m 3  10 m s  4.13 10 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 :

   it it mt0, I  0, tAIAeymeyooˆˆ 

which in turn also would have subsequently radiated magnetic dipole (M1) EM radiation.

However, because Boo Ec E o we also see that mpcoo , 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 ie.. moo pc , 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 rt, r and Jrttot , r , 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 zˆ -direction} into solid angle element d is (see P436 Lecture Notes 14, p. 5):

dPrad ,, t   p2  t Watts r 22ˆ oo   Srtrrrad ,sin2   dc 16 steradian

The angular distribution of the EM power radiated by an oscillating E1 electric dipole with  it time- dependent electric dipole moment ptpae o ˆ oriented parallel to an arbitrary aˆ -axis (e.g. where axyˆˆˆ , or zˆ axis) is shown in the figure below. The intensity {aka irradiance}   2 IradrSrt rad ,  Watts m is proportional to the distance from the origin   0,0,0  to an arbitrary point rr ,, 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 xˆ -direction. Thus, we must carry out a rotation of the above result such that it is appropriate for our situation:

dPrad ,, t  p2  t Watts r rad 22ˆ oo  Sr,,, trr 2 sin x  dc 16 steradian

Here, x is the opening angle between the observation/field point unit vector rˆ and the xˆ -axis.

Thus, cosx is the direction cosine between the observation/field point unit vector rˆ and the

xˆ -axis: i.e. cosx rxˆˆ  sin cos , in terms of the usual polar  and azimuthal  angles.    rx Note also that since: rx rxsin  or: rxˆˆ sin  then: x rx x

2 sin x rxˆˆ  rx ˆˆ   sin cosxˆˆˆ sin  sinyzx cos ˆˆ sin  cos x ˆˆˆ sin  sin yzx cos   sin22 sin cos 2  which can also be obtained from:

222 2222 sinxx 1 cos  1rxˆˆ  1 sin cos  1 sin  1  sin 

 1 sin222222222 sin sin cos  sin  sin sin  sin cos 

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005-2015. All Rights Reserved. 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 xˆ -axis:

dPrad ,, t  p2  t Watts r rad 22ˆ oo  Sr,,, trr 2 sin x  dc 16 steradian

dPrad  r,,, t  p2  t Watts r rad 2222ˆ oo  Becomes: Sr,,,  trr 2  sinsin cos  dc 16 steradian

The angular distribution of the EM energy radiated from the oscillating E1 electric dipole oriented along the xˆ -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 ttrco  and also computed from the local origin,  { r  0 }:   p0,tptrco  0, .   itoo it The instantaneous induced electric dipole moment is: p0,tqxtqxexpexoooo  0, ˆˆ .

 22itoo it 2  it  o  Thus: p0,too pexqxexqˆˆ  o    xexqxt o ˆ 0, o  qat  0, o.

  qq it From the above force equation, we see that: at 0, E 0, t Eexo ˆ oincoomm

 it q q  2 o ˆ 2 but: at0,ooo xt 0,  xex and thus we see that:  xooE or: xoo2 E m m  q2  2 it it o and therefore: p0,tqatqoo  0,  xex oˆˆ Eex o m 2  qq n.b. here, p 0,to is independent of frequency , because: pooqx q22 E o  E o, mm

2 2 1   q 2it  2  o i.e. pqxoo2 !!! Thus: p0, toooo ptpt 0, 0,  Ee .  m 8 © 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 2 2 2211  q oscillation  12f  is simply half of this value: pt0,ooo pt 0,  E 22m

111 Reeedttdt22itoo Re it cos2  since  ooo. onecycle onecycle 2

Note also that by carrying out the time-averaging process, this also helps us to completely side- step/avoid the difficulty associated with experimentally dealing with the retarded time ttrco  !

Thus, the time-averaged differential power radiated by the oscillating E1 electric dipole is:

rad 2 2 2 dPr ,, tooo   p t 1  qE Watts rad 22ˆ oo 2 Sr,,, trrox22 sin sin x dccm 16 2 16  steradian

rad 2 2 dPr ,, to  1  qE Watts rad 2222ˆ oo  Or: Sr,,,  trro  2  sinsin  cos  dcm 216  steradian

Likewise, the time average of the {magnitude of} the instantaneous flux of EM energy incident on   2 2it o the free charge {located at the origin  r  0 }, Stinc0, o  o Ece o is half of this value, i.e.:

 112 Watts I 00,0,St St Ec inc inc  o  inc o o o 2 22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 zˆ -direction and linearly polarized in the xˆ -direction is:

4 2 1 qEo o  LPx dP,, t 2 m2 dT , 11rad o 2    2 sin x ddSt0, 1 2 16 c inc o  E c 2 o o 2 2 q 22 2 sinxoo  using  1 c 4 omc i.e. 22 dLPx , qq22  T  sin2222 sin sin  cos 2 22x  msr dmcmc 44oo  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005-2015. All Rights Reserved. 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 qe 2 (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 qe .

Then since:

rad 2 2 dP,, t 2 rad r o ooqE 22 2 Pt d sin sin  cos sin dd r o 2 00 dcm 32  2 2 Carrying out the azimuthal angle integrals first: sin2  d and d  2 .  0  0

Then carrying out the polar angle integration:

   1112624 sin23 sindd sin  cos  cos 3  1 1 2 00 333333 0

 1112 And: cos23 sind  cos      0 3333 0

2 22 2 42448  Thus: sin sin cos  sindd  2  00 33333

2 2  qE2 8   qE2 Ptrad  o o  oo Then: r o 2   4 32 c m 312cm

Thus, the total Thomson scattering cross section {per scattering object of charge q} for an incident plane EM wave propagating in the zˆ -direction and linearly polarized in the xˆ -direction is:

42  qEo o  6 2 2 LPx Pt 12 c m 2 LPx dT , rad o 8 q 2 T d   2  using oo 1 c  d 1 2 34 mc Stinc o  E c o 2 oo

2 8 q2  LPx  2 Thus: T 2 m 34 omc

10 © 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 re2242.8210 mc  15 m rest mass me then the quantity: eoe 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 Uree eVr ee associated with a unit test charge q = e equals the rest mass energy of the electron Emcrest  2 , i.e. e e

2 2 e 2 e 15 Uree eVr ee mc e thus: rme 2  2.82 10 4 oer 4 oemc

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LPx , Te  22 2 2 2 2 2  rrexesin sin sin  cos msr per electron d  and: 8 e2  LPx  r 2 2 rm 2.82 1015 Te  e m per electron where: e 2 3 4 oemc

Numerically, we see that:

2 LPx 88 2153028 2   re 2.82 10 66.6 10 0.67 10 m per electron Te 33 

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: 222 “It’s as big as a barn”}: 1 barn 1028mm 2 10 14 2   10 10  15  mfmfm 2 10  100 2 where 1 fm = 1 Fermi = 1015 m (in honor of Enrico Fermi, nuclear physicist of mid-20th century).

LPx 88 215282 2 Thus:   re 2.82 10 0.67  10 m per electron  0.67 barns per electron Te 33

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 15 atom is typically rfewfmfewmnucleus ~1 1 10 whereas the characteristic size of an 10 atom is typically rfewfewmfewnmatom ~ 1 Ångstroms  1 10 0.1  1 .

Thus, the geometrical cross-sectional area of a typical nucleus in an atom is 228 Anucleus r nucleus ~ 0.03 1 10 m  0.03 1 barns , whereas the geometrical cross-sectional 21810 area of a typical atom is huge: Aatom r atom ~ 0.03 1 10 m  0.03 1 10 barns !!!.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005-2015. All Rights Reserved. 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 8r2 3  0.67 barns ~ A  0.03 1 barns A  0.03 1 1010 barns Te  e nucleus  atom 

Note that the Thomson scattering of a linearly-polarized EM plane wave propagating in the zˆ -direction has rotational invariance about the zˆ -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    it yˆ -axis instead of the xˆ -axis, i.e. EincrtEey 0,  o ˆ , then the differential Thomson scattering cross section for a free charge q would instead then be:

LPy  2 d, 11dPrad,, t o oopt T  sin2 dd16 2 cy Stinc0, o St inc  0,

where the direction cosine cosy ryˆˆ   sin sin and: ryˆˆ sin y thus:

2 sin y ryryryryˆˆˆˆ    ˆˆˆˆ    sin cosxˆˆˆ sin  sinyzy cos ˆˆ sin  cos x ˆˆˆ sin  sin yzy cos   sin22 cos cos 2  which can also be obtained from:

222 2222 sinyy 1 cos  1ryˆˆ  1 sin sin  1 sin  1  cos 

 1 sin2 sin 22 cos cos 2  sin 22  cos sin 22  cos 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 zˆ -direction  it it but linearly polarized in the yˆ -direction is: p0,tqyt  0, qyeyooˆˆ pey, 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 yˆ -axis instead of the perpendicular to the xˆ -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 zˆ -direction but linearly polarized in the yˆ -direction is:

22 dLPy , qq22  T  sin2222 sin cos  cos 2 22y  msr dmcmc 44oo  

12 © 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 xˆ -polarization case: 2 8 q2  LPy =  LPx 2 TT2 m 34 omc

More generally, for an incident EM plane wave propagating in the zˆ -direction with arbitrary    it linear polarization ˆˆˆ cosx  sin y and Erinc 0, t Ee o ˆ as shown in the figure below {cf with/see also the 3-D figure shown on p. 4 of these lecture notes}:

ˆˆˆ cosx  sin y

sin  xˆ ˆ z into page cos

yˆ

In this more general situation, the induced dipole moment is again parallel to/tracks the polarization vector of the incident EM plane wave:

 it it p0,tq  0, tqre oo  cos xˆˆ  sin ype    cos x ˆˆ  sin y 

Thus, here the differential Thomson scattering cross section for a free charge q is:

LP  2 d, 11dPrad,, t o oopt T  sin2 dd16 2 c Stinco0, St inco 0,

where {here} the direction cosine associated with an arbitrary linear polarization is:

22 cos rxyzxyˆˆ  sin cos ˆ  sin sin ˆ  cosˆ cos  ˆ  sin ˆ  sin  cos  sin  sin cos 2

and: rˆˆ  sin  thus:

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 14.5 Prof. Steven Errede

2 sin rrˆˆ  ˆˆ   ˆ cos x ˆ  sin y ˆ  sin cosxyzˆˆˆ  sin  sin  cos ˆˆ sin  cos xyz ˆˆˆ  sin  sin  cos   4sin222 cos sin cos 2  cos 2 sin22222 sin 4cos sin  cos   1 sin2 2  sin22 sin 2 cos 2  which can also be obtained from:

2 22 22 2 2 22 sin 1 cos  1rˆˆ   1 sin cos 2  sin  cos  sin cos 2  sin22 1 cos 2  cos 2222 sin sin 2  cos   sin2 2

This relation can also be obtained via a 3rd method – noting that since:

cos rrˆˆ  ˆcos x ˆ  siny ˆ  cos  rx ˆˆ   sin  ry ˆ   ˆ  cos  cos x sin  cos y

where: cosx rxˆˆ  sin cos and: cosy ryˆˆ   ry ˆˆ  sin sin then:

2 2 sin22 1 cos  1rrxyˆˆ 2  1 ˆ cos ˆ  sin ˆ  1 cos  rxry ˆˆ   sin  ˆ   ˆ     

2 222  1 cos sin cos  sin   sin  sin  1 sin  cos   sin  sin 2  1 sin22 cos  sin 2   1 sin22 cos 2 cos 2  sin 2 sin 22 cos 2  cos2 2  cos2 sin 2  1  cos 2 2   cos 2  sin 22  sin 2  sin 22  sin 2  cos 2    sin2 2

Thus, we see that:

2222 cosx rxˆˆ  sin cos 

2 2 2222 cosy ryˆˆ   ry ˆˆ  sin sin   sin sin 222222222 cosrˆˆ   1 sin  1 cos  sin sin 2  sin   sin  sin 2  sin22 1 sin 2 sin 22 cos 2  cos2 2 with ˆˆˆ cosx  sin y , 222 Note that when   0 : ˆˆ x and cos cos x sin  .  222 Note that when  2 : ˆˆ y and cos cos y sin  .

14 © 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 and:

2222 sinx rxˆˆ  rx ˆˆ   sin sin  cos 2 22 2 siny ryryryryˆˆˆˆ    ˆˆˆˆ    sin cos  cos 2222 sin rrˆˆ  ˆˆ  sinsin 2  cos

22 2 Note that when   0 : ˆˆ x and sin sin x cos  .  22 2 Note that when  2 : ˆˆ y and sin sin y cos  .

Thus, the differential Thomson scattering cross section for an incident EM plane wave propagating in the zˆ -direction with arbitrary linear polarization in the ˆˆˆ cosx  sin y -direction is:

22 dLP , qq22  T  sin2222 sin sin 2  cos 2 22  msr dmcmc 44oo  

d, Then since  T d , carrying out the angular integration over the polar  and T    d   azimuthal   angles: sin22 sin 2 cos 2 sin dd , carrying out the -integrals: 00 2 2 sin 4 2   sin2 2d   and d  2  0   0 28 0 2 Using sin22  1 cos  and making the substitution u  cos , hence du dcos sin d ; and when  0: u cos 0  1 , when  : u cos  1 , thus the  -integrals are:

u 1 u1 2231111114 sin sinduduuu 1   3333333  1 1 1 1 2 1   01u u1 

u 1 u1 cos223 sin duduu111112  01u 333333u1

Putting all of these results together:

  42 448  sin22 sin 2 cos 2  sindd 2 !!! 00 33 333

2 8 q2  LP  =  LPx LPy 2 TTTT2 m 34 omc

Thus we have explicitly shown that we obtain precisely the same result for the total Thomson scattering cross section for an arbitrary polarization ˆˆˆ cosx  sin y of an incident EM plane wave as that obtained above for either the original xˆ - and/or the yˆ - polarization cases.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005-2015. All Rights Reserved. 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 zˆ -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 rt  0, is such that it is always parallel to/tracks the polarization vector ˆ of the incident EM plane wave.

Thus, we see that while the polarization vector ˆ of the incident EM plane wave certainly matters greatly for the differential Thomson scattering cross section ddT ,  , 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 ˆˆˆ1 x iy ˆˆˆ1 x iy combinations of linearly-polarized EM plane waves: LCP 2  and RCP 2  ** {note that we have normalized these polarization vectors such that ˆˆLCPLCPRCPRCP ˆˆ 1 }, then we can also see that the total Thomson scattering cross section for LCP or RCP incident EM plane waves is also:

2 8 q2   2 T 2 m 34 omc

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 8 q2  unpol   2 TT2 m 34 omc because an unpolarized EM plane wave is equivalent to a randomly-polarized EM plane wave, whose time-dependent polarization vector ˆ t changes randomly from one moment to the next within the azimuthal interval 02  . 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 rt  0, is parallel to the polarization vector ˆ t of the randomly-polarized incident EM plane wave on a moment-to- 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 polarization ˆ t instantaneously oriented somewhere within the azimuthal interval 02  .

2 dP  Mathematically this means that the -integral of the probability density d 1 .  0  d dP  Since the azimuthal probability densityis flat {i.e. is constant} for a randomly- d dP  distributed  distribution, then we can take  outside of this integral, and then since d 2 dP  1 d  2 , we see that the probability density  for an unpolarized/randomly-  0 d 2 polarized EM plane wave, as shown in the figure below:

dP  1  d 2    0   2

If the polarization state ˆ t of the incident EM plane wave is changing randomly from moment-to-moment, one might worry that the process of averaging the instantaneous differential radiated power dPrad,, t o 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 the polarization state ˆ t with time t.

At the microscopic level, an unpolarized macroscopic EM plane wave consists of real photons,  each with a randomly oriented E -field/randomly oriented polarization vector ˆ  . Individual 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tfs 0.326 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 17 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 14.5 Prof. Steven Errede

Since rtˆˆ  sin   t, then with no loss of generality, we can very easily modify the time-averaging of the differential power/differential scattering cross section formulae simply by 2 moving the sin  t factor {n.b. previously assumed to time-independent in the above examples} inside the time-averaging process, i.e.:

unpol  22 d, 11dPrad,, t o oop ttsin   T   dd 16 2 c Stinco0, St inco 0,

 222 2 1 ooptsin sin 2  t cos   16 2c Stinc0, o

Note that the polar angle  is fixed by the observer being at the field/observation point P. 2 The time-averaged value of the sin 2 to factor is actually a probability-weighted integral over all possible -values that can/do occur during the time-averaging process over tavg :

22dP  1 sin222 2ttdtd sin 2 sin 2  ooo00  d2 2 11sin412  2 1  sin2 2td    0 o  2228 0 2 22

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:

22 dunpol , qq22  T  sin2222 sin sin 2t  cos  22 o dmcmc 44oo   22 22 qq1122  2 2 22 sin cos  sin  2cos  42oomc 24   mc

22 2 2 1 q 22 21 q 2   2 sin  cos cos 2  1  cos  24 omc  24 omc 1

2 dunpol , 1 q2 T  1cos 2  2 2  msr  n.b. has no  -dependence! dmc 24 o

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:

2 2 d, 1 q 2  unpol T ddd1cossin2  T 2 00 dmc 24 o But: 122   1cossin2 dd 1cossin 2 d 2 00 2  0

  26 2 8 sindd cos2 sin  2 00 33 3 3

Thus, here again we see that the Thomson scattering cross section for an unpolarized/randomly- polarized macroscopic EM plane wave incident on a free electric charge q, averaged over a long time interval tavg is:

2 8 q2  unpol  =  2 TT2 m 34 omc

Even though an incident EM plane wave may be unpolarized, this does not mean that an  observer at the field/observation point Pr Pr ,,  will observe unpolarized scattered radiation – quite the contrary! The reason for this is simple – for a specific field/observation point Pr,,  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 Pr,, 0 as shown in the figure below:

inc ˆ ˆˆ ˆˆ y nkkscat rad inc ˆ inc t ˆˆrad y ˆ  kzinc  ˆ   zˆ 

x-z plane = ˆˆinc x scattering  rad plane ˆˆ cos x  sin zˆ Incident EM Plane Wave ˆ krrad  ˆ (unpolarized) ˆ xˆ r Observer ˆˆˆinc ttxtycos   sin  yˆ Position

inc inc  costtˆˆ sin  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005-2015. All Rights Reserved. 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 ˆ kzinc  ˆ and krrad  ˆ , where {here, with   0 } rxzxzˆˆsin cos cosˆˆ sin ˆ cos . ˆˆ The unit normal to the scattering plane nˆscat is defined by: nkkˆscat rad inc .

For an unpolarized/arbitrary/random polarization of the incident EM plane wave, from the above figure, it can also be seen that the {instantaneous} polarization unit vector ˆ inc t associated with the incident unpolarized EM plane wave can be decomposed into a component inc parallel to/lying within the x-z scattering plane ˆ  and a component perpendicular to the x-z inc inc inc inc scattering plane ˆ  : ˆˆˆˆˆttxtytcos   sin  cos    sin   t .

Likewise, for the scattered EM radiation, whatever instantaneous polarization ˆ rad t exists rad can be decomposed into a component parallel to/lying within the x-z scattering plane ˆ  and a rad component perpendicular to the x-z scattering plane ˆ  .

However, as we have seen above for Thomson scattering with xˆ and yˆ linearly-polarized incident EM waves, the orientation of the incident vs. scattered polarization vectors is unchanged by the scattering process in the following sense:

inc In the above figure, for an incident EM plane wave with linear polarization ˆˆ x parallel to rad (i.e. lying in) the x-z scattering plane, the scattered polarization is ˆˆ cos x  sin zˆ which also lies in the x-z scattering plane – notice the two blue polarization vectors in the above figure.

inc For an incident EM plane wave with linear polarization ˆˆ y perpendicular to the x-z rad inc scattering plane, the scattered polarization is ˆˆˆ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:

2 unpol  LPx 2 ddTT,011  ,0 q 22  2 sinx msr ddmc224 o

2 2 1 q2 1 q2  sin22 sin cos22 cos  2 2 24 omc 24 omc 22 unpol LPy 22 ddTT,011 ,0 qq2222 1   siny  sin cos cos ddmcmc2242422   oo 1 22 22 11qq22  2 22 sin cos  msr 24oomc 24   mc 1 inc inc inc n.b. The 1/2 factor in the above arises from statistically projecting ˆ t onto ˆˆ x and ˆˆ 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 Pr,, 0 lying in the x-z scattering plane:

2 dunpol  ,0 1 q2 T  cos2  2 2 msr dmc 24 o And: 2 dunpol ,0 1 q2 T  2 2 msr  n.b. has no  -dependence! dmc 24 o

Note further that:

ddunpol,0 unpol    ,0 d unpol   ,0 TT T ddd 22 22 11qq2  22 cos   24oomc 24   mc 2 2 1 q 22 2  1 cos msr 24 omc Thus: 2 dd unpol,0 1 q2  unpol   , T  1cos2  T 2 2  msr dmcd24 o

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 ax bx  related variables A x  . Thus, in general A x ranges between 11A x . ax bx  (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:

unpol unpol  ddTT,0 ,0  22 unpol dd1cos  sin P ,0   T ddunpol ,0 unpol   ,01cos22 1cos TT dd

Here, we see that due to the physics associated with Thomson scattering of an unpolarized EM unpol unpol plane wave from a free electric charge, PT ,0 ranges between 0,01PT  . 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 21 2005-2015. All Rights Reserved. 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 2 2 unpol 2 unpol 2 dT ,0 q 1 2 dT ,0 q 1 2 1cos , 2  , dmc 4 o 2 dmc 4 o 2 2 unpol  2 2 dT ,0 q 1 2 unpol sin  2  cos  and the polarization PT ,0 2 dmc 4 o 2 1cos  for unpolarized Thomson scattering as a function of  are shown together in the figure below:

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 unpol unpol o polarization {asymmetry} vanishes, i.e. PPTT0, 0   180 ,   0 0 . c.) At   90o the  differential Thomson scattering cross section vanishes, and because of this, unpol o o the polarization {asymmetry} is maximal, i.e. PT 90 , 0 1 , thus at   90 , 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 rt, 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:    nd mree t m r  t kr e  t eE inc  r, t  ← inhomogeneous 2 -order differential eqn. 2  rt rt  mmkrteErt  , n.b. we have {again} neglected eee2    ←   tt       the ev B  eE term here... ma   e

Velocity-dependent Potential Force Driving Force damping term (binding of atomic electrons to atom) 31   damping constant mkge electron mass 9.1 10

For a driving force term sinusoidally varying in time with angular frequency  associated with a monochromatic EM plane wave with linear polarization in the xˆ -direction incident on the     it atom/molecule {located at the origin,  r  0 }: eEinc r0, t eE o e xˆ   it the inhomogeneous force equation becomes: mxee m x kx e eEe o xˆ .

 it The solution of this inhomogeneous second order differential equation is: x rtxex0, o ˆ eE m Atomic electron spatial  oe where: xo  = displacement amplitude  22i 0 {n.b. complex!} and: 0  kmee (radians/sec) = characteristic/natural resonance {angular} frequency.

 The corresponding {complex} induced electric dipole moment p rt 0, is:

2   iteEo 1 it prtextexex0, o ˆˆ ex m 22i  e 0

11x iy x iy 22 Using the “standard trick” z  22 with x  0 and y   x iy x iy x iy x y we can rationalize/rewrite this as:

 2 22i   eEo 0 it prt0,  exˆ 222 22 me   0

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 23 2005-2015. All Rights Reserved. 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 rt 0, of an atom / molecule associated with a single QM transition/resonance at angular frequency 0  kmee 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  and linewidths  , as dictated by quantum mechanical selection rules. j 0 j j 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:

22 2 n i   eE 0 j j p rt0, o fosc ex i t ˆ tot j 2 m 22 22 e j1   0 j j where the angular frequency and natural linewidth associated with the jth resonance are:   km rad/sec and 2 FWHM sec1respectively, and the so-called 0 jjee jj  n osc th osc “oscillator strength” f j associated with the j resonance is such that:  f j 1. j1  In the above expression for ptot rt 0, , 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. iikr   rratom, molecule and thus variation of phase(s) ee~ 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

rratom,~0.1 molecule  fewnm.

 Then evaluating ptot rt 0, at the retarded time to :

22 2 n i   eE 0 j j it p rt0,  2 o fosc exo ˆ tot o j 2 m 22 22 e j1   0 j j And thus:

2 2 22 2 n i    eE 0 j j 2it ptptpt 240, 0, 0,  o fosc e  o tot o tot  o tot  o j 2 m 22 22 e j1   0 j j

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 xˆ -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 xˆ -axis {due to the polarization of the incident EM radiation} is:

dPrad ,, t  p 2  0, t Watts r oototorad 22ˆ  Sr,,, trrox 2 sin  dc 16 steradian

2 2222 where p 0,to is given above; and: sinx  sin sin  cos .

dPrad ,, t  p 2  0, t Watts r oototorad 2222ˆ  Thus: Sr,,,  trro 2  sinsin cos  dc 16 steradian

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 ptot0,t o is extremely complicated, for the purpose of discussing the salient physics features/behavior, we will assume for simplicity’s sake osc that only a single resonance exists (with f1 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:

rad  2 dPr ,, to  1 ooRept 0,  Srrad ,,, trr22ˆ sin dc ox216 2

4222 22 2   2 0 1 ooeE 2 1 Watts  2 sinxo and using:  2 2 2  216 cme 22 22  oc sr 0   2 24 1 2222e  ooEc2  sin sin cos 222 22 24 oemc   0

The time average of the {magnitude of} the instantaneous flux of EM energy incident on the electric  112 Watts dipole {located at the origin } is: Iinc00,0,St inc  o  St inc o o Ec o 2 and thus: 22m

1 2 2  ooEc 24 datom , 1 dPrad,, t o 2 e  2    2 sin x 222 22 ddSt0, 1 2 4 oemc 2 inc o 0   ooEc m 2  2 sr 24 e  22 2 2  sin sin cos  222 22 4 oemc   0

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 25 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 14.5 Prof. Steven Errede

22 15 Since the classical electron radius is: reeoe42.8210 mc m, we can write the differential scattering cross section {per atom/molecule} for an incident EM plane wave propagating in the zˆ -direction and linearly polarized in the xˆ -direction in terms of re as:

dLPx ,  4 atom  r 2222sin sin cos 2 e 2  msr per atom/molecule d 22  22 0

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 zˆ -direction and linearly polarized in the xˆ -direction:

dLPx , Te  22 2 2 2 2 2  rrexesin sin sin  cos msr per electron d 

For the atomic/molecular differential scattering cross section, we simply have the additional 2 dimensionless factor  42222  arising from the physics associated with the 0 {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 electron is:   r 2 m2 per electron , the total cross section for atomic/molecular scattering Te3  84   r 2 2 of an EM plane wave is: atom e 2 m per atom/molecule . 3  22 22 0

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:

dLPy ,  4 atom  r 2222sin cos cos 2 e 2  msr per atom/molecule d 22  22 0

dLP ,  4 atom  r 2222sin sin 2 cos 2 e 2  msr per atom/molecule d 22  22 0

dunpol ,0 1  4 atom  r 221cos  2 e 2  msr per atom/molecule d 2 22  22 0

26 © 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: dunpol ,0 1  4 atom  r 2 2 e 2 msr per atom/molecule d 2  22 22 0 and: dunpol  ,0 1  4 atom  r 22cos  2 e 2 msr per atom/molecule d 2 22  22 0

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  ddatom,0 atom  ,0  22 unpol dd1cos  sin P ,0   atom ddunpol ,0 unpol   ,01cos22 1cos atom atom dd

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  22 22 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  kee m  . x4 x    y    Defining 0 and 0 , a log-log plot of the resonance lineshape 2 x2221 xy 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 4 1 decade increase in x   0 which is due to the  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 27 2005-2015. All Rights Reserved. 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  4 222 4 When    , the factor   in the resonance lineshape 2 0 00  22 22 0 and additionally, since   0 , then for   0 :

4 444  1 d atom , 42 . 222422 22  rd   0  0 oe 0

Thus, at low frequencies   0 the differential and total atomic/molecular scattering cross sections are strongly frequency-dependent, and behave as:

4 d, atom 2  2 re Angular Factor msr per atom/molecule d o and: 4 8 2  2  atom r e  m per atom/molecule 3 o

28 © 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 22 The behavior of the polarization {asymmetry} Patom ,0sin1cos  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 29 2005-2015. All Rights Reserved. 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 222 factor 0 0 and thus for   0 :

4 42 2  000 1 d atom , 22 2 2 222 22   0   rde  0

2 Since   0  then 0 1and thus we see that on (or near) a resonance the 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, 2 atom 2 o 2 re Angular Factor msr per atom/molecule d  and: 2 8 2 o 2  atom r e  m per atom/molecule 3 

3.) High frequencies:   0

For high frequencies   0 the bound atomic/molecular electrons behave as if they are free –

222 i.e. as in Thomson scattering of free electrons! When  0 the behavior of the factor  0 4  222 4 in 2 is  0 and additionally, since   0 , then for   0 :  22 22 0

44422 1 d atom ,  221 . 22242222222 22  rd      e 0

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 r 2 Angular Factor msr2 per atom/molecule d e  and: 8   r 2 m2 per atom/molecule atom3 e 

30 © 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

2 4.) Extremely High frequencies:   mce (and beyond). 2 At extremely high frequencies, when Emc  e 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 2 { rrmax 1 and rckrmax max21 r max  } because for Emc  e (and beyond),

the wavelength  is comparable to (and/or smaller than) the size of the atom/molecule rmax ,

thus the requirement 21 rmax   is not satisfied. Thus use of (any) of the above formulae 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).

2 For  mce 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 rxxyyzz22ˆˆ 2 2ˆ .

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: p11rt,,oo p 2 rt 2 ,   i.e. p22rt, o can only differ from p11rt, o by a relative phase

      iit21  21 ikinc  r2 p2 rt 2,,oo p 11  rt e p 11  rt , o e  p 11  rt , o e .   nd due to relative arrival time difference tkr21inc  2  of the incident EM wave at the 2   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 31 2005-2015. All Rights Reserved. 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 is located at position r  0 and p is located  1 1 2 at position rxxyyzz22ˆˆ 2 2ˆ , 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 Pr,,, t such that the EM “far-zone” radiation fields associated with dipole # 2 have associated    with them an additional {relative} phase shift of eeiit21 21 eikrad  r2 due to relative arrival time    difference tkr21rad  2  of the scattered EM wave at the observation point Pr,,,  t from nd   the 2 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     ik k r     ii21 21 itit  21  21 ikinc r22 ik rad r  inc rad  2 ikr  2 at origin r1  0 of: ee e e e e  e e where: kkinc  k rad .

In general, the scattered EM wave(s) radiated from the two induced electric dipole moments   p11rt, and p22rt, , since the radiation electric fields also obey the superposition principle, will subsequently interfere with each other in the “far-zone” at the observation point Pr,,,  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  th rxxyyzznnˆˆ n nˆ for the n scattering object, n = 1, 2,3,… N. Via use of the principle of linear   superposition, each such identical scattering object will contribute Ertscat , to the overall “far- rn  zone” EM radiation field at the observation point Pr,,,  t. Since the differential and total scattering cross sections are both proportional to the modulus-squared of the overall/total EM   2 radiation field Ertscat , , where: rtot 

NN   N   ErtErtErteErtescat,,, scat scatikrnn  scat , ikr where ErtErtescat,, scat ikr  n rrrtot n 11  r   rrn 1  nn11 n  1  scat  where Ertr , is the scattered “far-zone” radiation field at the observation point Pr,,, t 1  associated with a single scattering object located at the origin  r1  0 .

2 22N   Then we see that: Ertscat,, eikr  n Ert scat rrtot   1  n1 2  N   ik ,  r Defining the so-called complex structure factor: F ke,  n 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 Lect. Notes 14.5 Prof. Steven Errede

2  N   ddd N ,, 11 ik ,  r   , F ke,  n dddn1

where dd1 ,  is the differential scattering cross section associated with a single  scattering object located at the origin r1  0 . 2  N   ik ,  r It can be seen that the numerical effect of the structure factor F ke,  n n1 on the overall differential scattering cross section ddN ,  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:

2  2  ikr,,,,, ikr  ikr  ikrikr    F ke,1111 n  ee22  ee 22  n1    2eeik,, r22  ik   r  2 2cos  kr ,  2 1  cos  kr  , 22   However, for forward   0 scattering this means that: kkzrad inc ˆ i.e. kkzrad inc ˆ   and thus: kkk0, inc rad 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  nd other for forward   0 scattering, independent of the location r2 of the 2 scatterer relative to  that of the first, located at the origin  r1  0 . For forward   0 scattering the {relative} time nd delay tzc22 in the arrival of the incident EM plane wave at the z-location of the 2 scattering object is exactly compensated by the {relative} decrease in the arrival time tzc22 of the scattered/radiated wave in the “far-zone” at the observer’s position at r,0, .  Noting that since: kkk0, inc rad 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:

22  22   2 ik 0  r ir00ir00 ir ir 2 2 F keeeeeN0,  0n nN 12   ...  1 1 ... 1 nn11    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 33 2005-2015. All Rights Reserved. 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: dd 0,  0, N  N 2 1 dd

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:   kkzrad inc ˆ and: kkkk,20 inc rad inc  and:  kkr,21cos2  . F inc 2

  When: 2krinc  2  2 n , n 1,2,3... {i.e. certain (angular) frequencies nnck n c z2 }    Then: cos 2krinc  2  cos 2 n 1 and: F k ,4  resulting in constructive interference for backward    scattering.

  When: 2krinc  2  2 n 1 , n  1,2,3... {i.e. certain other (angular) frequencies    1 nnck2 21 n  c z2 } then: cos 2krinc  2  cos 2 n 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 mn in the structure factor:

2  NNN    ik ,  r ik,  r r F ke, n  enm nnm111  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 mn the double series associated  with F k , becomes:

 NNN   ik,  r r ik ,0 F ke,1nn  e  N nnn111 and thus the differential scattering cross section associated with N randomly distributed identical scatterers (except for the precisely   0 forward scattering) is: dd 0,  0, N  N 1 dd 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 WHDLx  Lyz L and

lattice constant / lattice spacing a and NNNNLaLaLaxyz x y z 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:

2 222 1  N   11 ik ,  r sinNkasin 2 Nkayy  sin  Nka keN, n 2 22xx   zz F  2222  22  Nkasin11Nkasin 1   Nka sin n1 xx22yy2  zz

At low frequencies/long wavelengths  a , only the forward-scattering peak at kaj 0   contributes to F k , because {here} the maximum possible value of kaj is: 24ka  a  1. In this forward-scattering/small angle regime, using the small-angle Taylor series approximation 2 112 for sin 22kaxx  ka , the structure factor becomes:

22112 1   sinNkasin 2 Nkayy sin  Nka  2 22xx  zz F kN0, 222   sin x 111  22NkaxxNka  Nka zz where sincx  2 yy  x 2211 2 2 1 NNkaNkaNka sinc22xx sinc yy sinc  2 zz

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 35 2005-2015. All Rights Reserved. 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  1 be seen that the structure factor F k  0, is only appreciable when 2 Nkajj , which corresponds to an angular region of forward scattering kajj 2 N, which becomes exceedingly small as N j .

Note that for a typical lattice constant anmm 5Å 0.5 5 1010 and a typical macroscopic 2107 sample size of Lcmmj 1 0.01 then: NLajj10 5  10 2 10 , corresponding to a forward scattering angular region of: 2 2  1077   10 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   2~1anm 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: kaj 2 n or: 42 n kk2sin  sin  , i.e. when na  2sin . jincj ja j

Crystal planes spaced distance a apart Typical X-ray diffraction pattern

36 © 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: msr2 per scattering object}:

  Srtrr,  2 ˆ dscat , 11dPrad,, t dP rad   ,, t rad     dIrd0  d   inc  Srtinc , Srtzinc ,  ˆ 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 ddddd,cossin   , normalized to the

time-averaged flux of EM energy (= EM intensity Iinc , 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, Scattered Flux of EM Radiation Unit Solid Angle Watts sr m2 per scat  scattering  2  d Incident Flux of EM Radiation Unit Area Watts m sr object

Total Scattering Cross Section {SI units: Area, i.e. m2 per scattering object}:

  Pt Pt dP,, t d dS rt,  rrd2 ˆ  rad rad   rad   rad  dscat ,  scat      d  Ir 0   d inc Srtinc,, Srt inc Srt inc  , rr00 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 Iinc , 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:

Total Scattered Flux of EM Radiation into 4 sr Watts per 2 scattering  scat  2 m Incident Flux of EM Radiation Unit Area Watts m object

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 37 2005-2015. All Rights Reserved. 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 kN, {except for precisely   0 forward scattering, where F kN0, 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:

dd 0,  0, N  N 1 dd

Next, we consider a monochromatic macroscopic EM plane wave propagating in the zˆ -direction normally incident on a target consisting of a slab of “generic” matter (plasma, gas, liquid or solid) of macroscopic volume VHWDm3 consisting of a total of N randomly distributed identical scattering objects, characterized by a number density nNVm # 3 and mass volume density   M Vkgm3 . The incident EM plane wave uniformly illuminates the cross sectional area AHW 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: Ptrad   scat IWatts o  per scattering object}:

dPoo dN I 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 Io over the infinitesimal distance dz of:

dPo 2 dIo 3 dIoo n I dz Watts m with corresponding –ve slope: nI o  Wattsm 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 nIz  Wattsm3 which can be rearranged as: nIz  0 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: nz I zI0 o and: Iz 0 . The specific solution to this differential equation is: I zIe o .

Thus we see that the incident EM plane wave is exponentially attenuated to 1ee1 0.368 of its initial intensity Io in propagating a characteristic longitudinal distance known as the

z atten nz attenuation length atten 1 n m into the target. Then: I z Ieoo Ie .

The reciprocal of the attenuation length atten is known as the absorption coefficient:

1  znzz atten   1 atten n m . Then: I zIeIeoo  Ie o.

Using an isothermal model of the earth’s atmosphere (i.e. the density air varies exponentially 25 3 with altitude), nair ~2.7 10 molecules/m , 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 ~ 77km

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 39 2005-2015. All Rights Reserved. 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  zˆ  d Generic EM R Radiation Detector

rˆ

xˆ 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 absolutely- calibrated {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 41 2005-2015. All Rights Reserved. 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 unpol unpol spectrophotometer to measure e.g. dd ,0 and dd ,0 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 nt  e.g. associated with a laser beam with photon energy Ehfhc  :

 2 Watts I St cut 2 E tE  cntE  EM o orms F    2 m

Thus, counting photons {quanta of the EM field} within solid angle d at fixed scattering angle

,0 is equivalent to measuring intensity Iscat 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dd .

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 43 2005-2015. All Rights Reserved. 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 2 gravitational force FGMmrgrav N star p and the outward-directed force due to radiation pressure associated with Thomson scattering of electrons e 2 Fcrrad L star T 4 on the plasma surrounding the star (assumed here to be 100% ionized hydrogen).

The condition that FFgrav rad constrains the maximum luminosity of the star, known as the Eddington limit (in honor of Sir Arthur S. Eddington):

 L 41.310GM mce  31 Watts star N star p T 4  3.3 10 MMstar L

30 where the sun’s mass: M   210 kg and our 26 suns’ solar luminosity: L  3.85 10 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.