Download Thomson scattering: - Ira-Inaf

 Thomson scattering:
''It is the scattering of electromagnetic radiation by a free non­relativistic charged particle. The electric and magnetic components of the incident wave accelerate the particle. As it accelerates, it, in turn, emits radiation and thus, the wave is scattered. Thomson scattering is an important phenomenon in
plasma physics and was first explained by the physicist
J. J. Thomson 1856­1940
Nobel Prize 1906
The main cause of the acceleration of the particle will be due to the electric field component of the incident wave. The particle will move in
the direction of the oscillating electric field, resulting in electromagnetic
dipole radiation. The moving particle radiates most strongly in a
direction perpendicular to its motion and that radiation will be
polarized along the direction of its motion. Therefore, depending on where an observer is located, the light scattered from a small volume
element may appear to be more or less polarized.''
Daniele Dallacasa – Radiative Processes and MHD Section ­4 : Thomson Scattering, Compton scattering & Inverse Compton
Thomson scattering
photon­electron interaction ­1­
hv ≪ mec2
– for a non relativistic particle, the incoming (low­energy) photons can be expressed as a continuous e­m wave and the magnetic field (Ho=Eo) can be “neglected” (i.e. studied as a consequence of the E field behaviour). – the average incoming Poynting flux is <|S|> = cEo2/8π
– the charge feels a force due to a linearly polarized incoming wave:
– the energy of the scattered radiation is the same as the incoming wave
Thomson scattering (2)
– the force is due to a linearly polarizesd wave:
 = e  E sin t
F
o
o
2
 = e
̈ = e ̈r =
d
r

d
– the dipole moment is
d = −
e Eo
m
 sino t
 
2
e Eo
2
o
m
 t 
 sin o t = a
– which describes a dipole with amplitude
 
2
do = −
e Eo
m
2
o

– taking into account the Larmor's formula, the time averaged emitted W is
4 2
¨2

e
Eo
dW
2d
d
2
〈
〉 = 〈
〉
=
〈
sin
〉
=
〈S
〉
3 c3
d
d
8 m 2 c 3
– where
4
d
e
2
2
2
= 2 4 sin =r o sin 
d
m c
is the differential cross section for the scattering
Particle acceleration
(due to incoming radiation)
the emitted power by the scattering particle is:
from this equation we define the electron Thomson cross section by integrating over all the angles:
2
Back to the ''geometric problem'' the angular distribution of the radiation is given by and this means that each individual scatter produces polarized radiation
Cross section independent from frequency (breaks down at high frequencies when the classical physics need quantum mechanics)
Compton scattering ''In physics, Compton scattering or the Compton effect, is the decrease in energy (increase in wavelength) of an X­ray or gamma ray photon, when it interacts with matter.
Inverse Compton scattering also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength increases by is called the Compton shift. Althoughnuclear compton scattering exists, what is meant by Compton scattering usually is the interaction involving only the electrons of an atom. A.H. Compton 1892­1962
Compton effect was observed by Arthur Holly Compton in 1923, for which he earned the 1927 Nobel Prize in Physics.
The effect is important because it demonstrates that light cannot be explained
purely as a wave phenomenon.
Thomson scattering, the classical theory of charged particles scattered by an electromagnetic wave, cannot explain any shift in wavelength. Light must behave
as if it consists of particles in order to explain the Compton scattering.''
Compton scattering photon­electron interaction ­1­
Compton scattering photon­electron interaction ­2­
energy conservation:
h i m e c
2
=

2
h f  p e c
2
2
m e c
4
momentum conservation: pi = pf  pe  p=E /c =h  / c
pe = pi − pf

photon momentum

2
2
2
p e =  pi − pf ⋅ pi − pf  = p i p f −2 p i p f cos 
p 2e = h 2  i2  h 2 f2 − 2 h 2 i f cos 
2
multiply by c then take the square of energy conservation
p 2e c 2m 2e c 4 = h 2 i2h 2  f2m 2e c 42 h  i m e c 2−2hf m e c 2−2 h 2  i f
2
compare the two by highlighting
2
pec
2
we get
2
2
2
−2h i f cos  = 2 h i m e c −2 h f m e c −2h i f
2
2
2 h i  f 1−cos  = 2 h m e c i − f 

1−cos
me c
f
i
1
1
h
−
=
−
=
1−cos
2
f
i
c
c
me c
f −i =
h
1−cos  = o 1−cos 
me c
2
=
 i − f
h i  f
=
1
h

1
1
−
f
i

Compton wavelength for electrons
o =
h
= 0.02426 A
me c
Compton scattering a Java applet showing the geometry of the scattering can be found at
http://www.student.nada.kth.se/~f93­jhu/phys_sim/compton/Compton.htm
Compton scattering a Java applet showing the geometry of the scattering can be found at
http://www.student.nada.kth.se/~f93­jhu/phys_sim/compton/Compton.htm
h  e=
h i
1h i 1−cos / me c
2
h
⋅1−cos  = o 1−cos  = 0.02426⋅1−cos  [A]
me c
Final energy of the scattered photon is a function of:
e−i =
– initial energy of the photon
– scattering angle
The photon always looses energy, unless θ = 0, and the scattering is closely elastic when ≫c (i.e. hv≪ mec2 )
When the photons involved in the collision have large energies, the scattering become less efficient and quantum electrodynamics effects reduce the cross section:
the Thomson cross section becomes the Klein­Nishina cross section
Cross – section: Thomson and Klein – Nishina In case hv ~ mec2 , the probability of interaction decreases, also as a function of the scattering angle; additional constraints come from relativistic quantum mechanics; h
if we use x ≡
me c 2
 = T
[ 

3 1x 2 x 1x 
1
13 x
−ln12
x


ln12
x
−
2
4 x3
12 x
2x
12 x 
known as the Klein – Nishina cross section;
if x ≫1
  = T
it originates from a differential cross section based on the scattering angle
d
1 2
2
2
3
= r e [ P h ,  − P h  ,sin   P h  ,  ]
2
d
where the probability is defined as
P h  , =
1
1
h
me c
2
1−cos
and the Klein – Nishina cross section becomes
the Thomson cross section at low photon energies
]

3 1
1
ln 2 x 
8 x
2

Cross – section Klein – Nishina The Klein – Nishina cross section is relevant at high photon energies (Hard X­Rays and beyond)
x = Inverse Compton scattering photon­electron interaction ­3­
Moving, energetic (relativistic), electrons “cool down” by increasing the energy of seed photons.
To make computations simpler, it is possible to consider the centre of momentum frame where the photon energy is much less than mec2 and then T can be used.
In the electron rest frame we have Thomson scattering
1. Lab system ⇨ electron rest frame:
h e = h L  1 −  cos 
2. Thomson scattering in electron rest frame
h e ' = h e
3. Electron rest frame ⇨ Lab system
h L ' = h e '  1   cos 1 ' 
i.e. ⇨
2
h L ' ≈  h L
Very efficient energy transfer from electron to photon!
Inverse Compton scattering The scattering angle is important for the energy of the outgoing electron:
1 ­ maximum energy gained by the electron if =and 1' = 0
⇨ in the electron rest frame the photon is blue – shifted (face on collision)
2 – minimum energy gained by the electron if = 0and 1' = 
⇨ in the electron rest frame the photon is red – shifted (end on collision)
the maximum energy that can be gained is (averaged over all angles):
or, in terms of frequency
4 2
' ≈  
3
4 2
' ≈  
3
In case it is possible to measure both the initial and final photon energy/frequency,
then  2 can be deduced
 ≈

3 '
4 
Inverse Compton scattering emitted energy
It is a Lorentz invariant, i.e. the same in the laboratory and electron rest frame
E ' = h  ' = h  1   cos  = E  1   cos 
Let's consider a region where there is a plasma of relativistic electrons and a radiation field:
in the electron framework, the photon energy is
2
〈 E ' rad 〉
d
d '
=
= c T
dt
dt '
8
2
2
since the emitted energy is a Lorentz invariant. It is then possible to write
2
〈E rad 〉
= U ph
1 2
〈1   cos  〉 = 1 
3
2
and, finally, the energy coming out the scattering region is
d out
dt
2
= c  T  〈1   cos  〉 〈E rad 〉

1 2
2
= c  T U ph  1 
3

The energy associated with the photons, prior of the scattering is
d in
= −c  T U ph
dt
and then the net effect of the Inverse Compton scattering is
d out
d in
d
1 2
2
−
=
= c  T U ph  1   − 1
3
dt
dt
dt ic
2
2
2
 −1 =  
 
 
then
d
dt
ic
[
4
2 2
= c  T   U ph
3
and if we recall the synchrotron emission
we get  
 
d
dt
d
dt
syn
ic
 ]
=
UH
U ph
 
d
dt
syn
=
4
2 2
c T   U H
3
The “Local Radiation Field”
Evaluation of Urad: as taken in a number of selected locations
X­rays radio
Spatial variation of the total radiation field throughout the Galaxy. Black line : total radiation field for GC. Magenta line : total radiation field for R=0, z=5 kpc,
Blue line : total radiation field for R= 4, z=0 kpc,
Red line : total radiation field for R=12, z=0 kpc,
Green line : total radiation field for R=20, z=0 kpc.
This diagram will be discussed a bit further when considering the ISM
Synchrotron & Inverse Compton scattering radiative losses
The same relativistic electrons radiate via synchrotron and inverse Compton; their contributions add up:
 
d
−
dt
sy ic
[
= C s 2 H 2  8 U rad
]
The increased cooling rate implies that the electron radiative lifetime is consequently reduced:
t
∗
sy ic
≈
1

∗
8
1
3⋅10
≈
2
2
H 8 U rad
H /8U rad 
1
−3
[eV cm ]

Therefore.... hard times for relativistic electrons!
Example: radiative lifetime for a relativistic electrons in a radio lobe.
Heq , Urad , strong dependence on z!
BCMB ~ 3.28 (1+z)2 G
∗
[GeV ]
yr
Synchrotron Self­Compton Catastrophe
Let's consider a spherical cloud with radius R filled with magnetized relativistic plasma located at a distance d from the observer. The low­energy synchrotron photons may be up­scattered by the relativistic electrons via IC:
2
Ls = 4 d
∫ S d 
= U rad
R
2
≈ 4 d S max c f  =
4c 2
R
3
the energy density of the radiation field can then be derived
U rad =
3 Ls
2
4 R c
finally the comparison between the IC and Synchrotron losses gives:
LIC
Ls
=
U rad
2
H /8 
=
6L s
2
2
R H c
≈
S max c
2
2
 H c
f  =
5
  
T Bmax
12
10 K
c
GHz
f 
For brightness temperatures above 1012 K, the radiation field will undergo to a dramatic amplification; IC would become the most efficient and dominant cooling process (X­Rays) and a source would radiate its energy in a very short time [Compton catastrophe]. Brightness temperature limit in radio source is  1012 K
Comptonization:
We want to evaluate if there is a substantial change in the spectrum of the seed photons in case of multiple scattering with hot (but non­relativistic) electrons in thermal equilibrium. Let's consider the process from the point of view of the photons
There are interactions where energy is transferred from photons to electrons (C scattering)
h 
h
〈
〉
≈ −
h  phot
me c 2
There are interactions where energy is transferred from electrons to photons (IC scattering)
(positive since we are considering the process from the point of view of the photons)

4 v

〈
〉 ≈
 el
3 c
2
4 3kT
4kT
=
=
2
2
3c m e
me c
The net energy transfer is:
E
4kT − h 
〈
〉 =
2
E
me c
Thermal equilibrium
(1/2)mev2 = (3/2)kT
Comptonization: ­2­
There is not a net energy transfer when photons loose energy and electrons (gas) heats up
photons gain energy and electrons (gas) cools down
Let's consider the latter case in a region of size D:
D
The opacity to the scattering is
and the mean free path is
Comptonization: ­3­
The total number of collisions is:
the total energy gain is
the mean photon energy as a function of time becomes
where y is known as ''Compton y­parameter''
The spectrum of the radiation is substantially modified when vf >~ vo
and the final equilibrium is achieved when
h f = 4kT
i.e.
y =
 
1
4kT
ln
4
h o
namely
e =
2 1/2
[   ]
4kT
ln
h o
me c
4kT
If this condition is met (thermal equilibrium), the modified photon
spectrum must follow the Bose­Einstein distribution
Modified spectrum
At equilibrium, the photons will follow a Bose ­ Einstein distribution:
3
8 h 
1
u  d  =
d
3
h /kT 
c
e
−1
where  is known as chemical potential
(which is =0 for Planck's BB spectrum)
and measures the rate at which photons
are produced
h
in case   ≫ 1
kT
we have
3
8 h  −h /kT −
u d  =
e
e
3
c
i.e. the Wien's law modified by e −
Kompaneets' equation:
The exact description of the photon scattering (i.e. find the observed spectrum originated by various values of the y­parameter) is better obtained
in the phase­space through the Kompaneets' equation:
where Recoil effect
Induced/stimulated emission
Doppler motion
3
9 1

Emission spectra come out as power­law with index 3+m where m=− ±
2
4 y

Examples: Accretion disks in AGNs , and Q; S­Z effect
Thermal bremsstrahlung spectrum with comptonization
(accretion disks)
Comptonization: the spectrum is modified as a function of the opacity to this process
Comptonization: Astrophysical examples
possible models of comptonization
Comptonization: Astrophysical examples (2)
courtesy of G. Ghisellini
Sunyaev­Zeldovich effect
Rashid Sunyaev
Jakov Zeldovich 1914 – 1987
­ thermal: high energy electrons interact with CMB photons via IC
scattering
­ kinematic: bulk motions modify the spectrum, second order effect, also known as Ostriker–Vishniac effect
Where does it take place?
Sunyaev­Zeldovich effect
Galaxy clusters have hot electrons
capable of IC scattering of CMB photons. A CMB photon have a 1% probability
of interacting with a high energy ICM
electron.
v3+m
fixed cross­over freq
Sunyaev­Zeldovich effect: the Planck's view
The same galaxy cluster as a function of frequency
as seen by the Planck satellite
.
A full sky temperature map is at
http://www.astro.cardiff.ac.uk/~spxcen/CMB_Sims/Planck_comb_rbcol_scaled.png