Einstein Coefficients
Plan:
More fun with thermal equilibrium!
We’ve previously discussed the Einstein A
coefficient:
Einstein Rate Coefficients
• Definitions
• Relations among each other
• Relation to absorption/emission
coefficients
• Relation to “oscillator strengths”
Why? We’re building up machinery needed
to think about atoms other than Hydrogen
Einstein Coefficients
You can use the Einstein A coefficient to
calculate the rate of spontaneous downward
transitions
_
_
in previous notation. We’re switching
to Rybicki & Lightman, Ch. 1.6
You might also see
A21 as well.
Einstein Coefficients
There is a similar Einstein Coefficient for
absorption:
_
_
_
= number of
downward radiative
transitions per
second, per volume.
Probability per unit time
of an electron decaying
radiatively from an upper
state u to a lower state l
_
Probability per unit time
of an atom absorbing a
photon, causing an
electron to be excited
radiatively from a lower
state l to an upper state u
You might also see
B12 as well.
Einstein Coefficients
You can use the first Einstein B coefficient to
calculate the rate of upward transitions, in the
presence of a radiation field of intensity J!.
Line Widths
Caveat: In reality, transitions respond to a small
range in frequency, rather than a single frequency
Line
profile:
_
_
= number of upward
radiative transitions
per second, per
volume.
(where J! is evaluated at ! corresponding to the
frequency of the transition)
Einstein Coefficients
The final, least intuitive Einstein coefficient
is the B coefficient for “stimulated emission”
So more properly:
But, because lines are narrow, and J! usually varies slowly, we won’t bother with writing the bar!
Einstein Coefficients
You can use the second Einstein B coefficient
to calculate the rate of stimulated downward
transitions
_
_
You might also see
B21 as well.
Probability per unit time
of an electron decaying
radiatively from an upper
state u to a lower state l,
in response to light of
intensity J!.
_
_
= number of downward
stimulated radiative
transitions per second,
per volume.
Note: sometimes B’s are defined in terms of a response to a photon energy density u!, not
an intensity. This leads to differences in units, and factors of (4"/c)
Einstein Coefficients
All 3 Einstein Coefficients are Related
Emission
Emission
Rate
depends on
radiation
field
All 3 Einstein Coefficients are Related
This can be rearranged to solve for the
intensity:
Rate
depends on
radiation
field
Assume Detailed Balance & Thermal Eq:
All 3 Einstein Coefficients are Related
Why Bother? Because at Thermal Equilibrium:
Thus, Aul, Bul, and Blu must be related in
such a way that the top expression looks
like a Planck spectrum!
All 3 Einstein Coefficients are Related
All 3 Einstein Coefficients are Related
Let’s compare to a Planck function:
First, we can assume a Boltzmann occupation
of the upper and lower levels:
Only works under thermal equilibrium assumption!
All 3 Einstein Coefficients are Related
All 3 Einstein Coefficients are Related
Let’s compare to a Planck function:
The only way to have J!=B!(T) for any T, ! if:
All 3 Einstein Coefficients are Related
Rate
depends on
radiation
field
Emission
All 3 Einstein Coefficients are Related
The only way to have J!=B!(T) at all T, ! if:
No dependence on T!
Holds out of ThEq!
If B’s are defined in terms of a response to a photon energy density
u! instead an intensity.
A Diversion:
Notation:
Einstein coefficients are related to the
standard coefficients used for emission
and absorption in radiative transfer....
j! & #!
$! & %!
We’ll use the specific intensity I!, rather
than the angle averaged mean intensity J!.
dEν = Iν (k̂, x, t) k̂ · n̂ dA dΩ dν dt
1
Jν =
4π
�
Iν dΩ
Why? Because radiative transfer usually
involves paths (i.e., has direction)
j! in units of ergs/s/cm3/ster/Hz
Very Basic Radiative Transfer:
Monochromatic Emission Coefficient j!
Along a path S, energy can be added (emission)
or subtracted (absorption), so I! is not
necessarily constant in space.
j! & #!
j! & #!
dE! = j! dV dt d& d!
dropping k direction
for notational
convenience...
For isotropic emission (or randomly oriented emitters):
$! & %!
j! = P! / 4"
where P! is radiated power per volume per Hz
Neglecting scattering for now...
21
Shu’s “Radiation” Ch. 3 shows how to calculate with scattering
22
#! in units of ergs/s/gm/Hz
j! in units of ergs/s/cm3/ster/Hz
I! in units of ergs/s/cm2/ster/Hz
Emissivity "! (angle integrated, tied to mass)
Monochromatic Emission Coefficient j!
j! & #!
j! & #!
For isotropic emission (or randomly oriented
emitters), assuming a density '
ds
dA
dE! = (#! ') dV dt (d&/4") d!
And comparing to the definition for the emission coeff j!
dEν = Iν (k̂, x, t) k̂ · n̂ dA dΩ dν dt
dE! = j! dV d& d! dt
The intensity added to the
beam is therefore:
j! = #! ' / 4"
23
Shu’s “Radiation” uses j! where Rybicki & Lightman use #!.
Travelling along a path of
length ds a beam of cross
section dA travels through a
volume dV = dA ds
24
dI! = j! ds
$! in units of 1/cm
I! in units of ergs/s/cm2/ster/Hz
%! in units of cm2/g
I! in units of ergs/s/cm2/ster/Hz
Monochromatic Absorption Coefficient #!
Monochromatic Mass Absorption Coefficient "!
dI! = -$! I! ds
$! = '%!
$! & %!
$! & %!
Density n of
absorbers
Cross section for
absorption $!
More commonly, %! is known as the “opacity”
$! = n(!
25
$! in units of 1/cm
I! in units of ergs/s/cm2/ster/Hz
Monochromatic Absorption Coefficient #!
dI! = -$! I! ds
A wrinkle:
Although $! is called an “absorption
coefficient” sometimes “stimulated
emission” (which is also proportional to
incident intensity) gets wrapped into the
term for convenience. Thus, although $! is
positive when dominated by absorption, it
can sometimes be negative!
Radiative Transfer:
$! %!
&
j! #!
&
dI! = -$! I! ds
dI! = j! ds
Along a path S, the change in I! is:
dIν
= −αν Iν + jν
ds
Scattering makes the solution to this differential equation
extremely complicated, since you have to keep track of d$
Optical Depth:
Along a path S, the
change in I! is:
Optical Depth:
dIν
= −αν Iν + jν
ds
We define the “optical depth” )! as:
d)! = $! ds
dIν
jν
= −Iν +
dτ
αν
“Source Function”
dIν
jν
= −Iν +
dτ
αν
$! in units of 1/cm
)! is dimensionless
d)! = $! ds
A differential equation that can be solved!
If no
emission Iν (τν ) = Iν (0)e−τν
(j!=0)
dIν
= −Iν
dτ
At an optical depth of 1, intensity down by 1/e
Optical Depth:
dIν
jν
= −Iν +
dτ
αν
More general solution (using exponential
integrating factor):
� τν
�
Iν (τν ) = Iν (0)e−τν +
Sν (τν� )e−τν dτν�
The Einstein Coefficients can be related to
the emission and absorption coefficients.
j!
Emission coefficient
#!
Absorption coefficient
0
S! is the “source function” of the material that is
at the position that corresponds to )!’
dIν
= −αν Iν + jν
ds
The Einstein Coefficients can be related to
the emission and absorption coefficients.
_
_
j!
Emission coefficient
_
_
#!
Absorption coefficient
The Einstein Coefficients can be related to
the emission and absorption coefficients.
_
_
j!
Emission coefficient
_
_
Absorption coefficient
Depends on intensity, so
acts more like negative
absorption
The Einstein Coefficients can be
related to the emission coefficient.
Definition
_
_
The Einstein Coefficients can be
related to the emission coefficient.
Definition
Spontaneous decay produces
nuAul photons of energy h! per
sec, per volume, distributed in
frequency with some line
profile &(!)d! over 4' sterrad
For emission in one particular transition
#!
The Einstein Coefficients can be
related to the absorption coefficient.
Along a line of sight, the energy lost per
volume, per second, per frequency, due to
absorption is:
Adding in a “per sterradian”:
The Einstein Coefficients can be
related to the absorption coefficient.
Along a line of sight, the energy lost per area,
per second, per frequency, per sterradian is
just the change in intensity dI!.
dEν = Iν (k̂, x, t) k̂ · n̂ dA dΩ dν dt Def’n
The Einstein Coefficients can be
related to the absorption coefficient.
Along a line of sight, the energy lost per
volume, per second, per frequency, per
sterradian due to absorption is:
Changing the volume element to a path length:
The Einstein Coefficients can be related to the
absorption coefficient.
What about emission due
to stimulated emission?
_
“Correction for Stimulated Emission”
_
_
Emission
_
Rate
depends on
radiation
field
Mathematically, acts more
like “negative absorption”!
“Correction for Stimulated Emission”
But, Bul and Blu are related through detailed
balance:
Factoring:
“Correction for Stimulated Emission”
Substituting:
This is true in general, even out of Th.Eq.
“Correction for Stimulated Emission”
In thermal equilibrium, can assume
Boltzmann populations:
In Thermal Equilibrium:
“Correction for Stimulated Emission”
We can also try expressing the “Source
Function” in terms of the Einstein Coefficients
Radiation Transfer:
In terms of
the optical
depth:
Define the source function:
“Correction for Stimulated Emission”
Outside Thermal Equilibrium:
Use Definition of the
Excitation Temperature:
True in General!
“Correction for Stimulated Emission”
We can also try expressing the “Source
Function” in terms of the Einstein Coefficients!
We can also try expressing the “Source
Function” in terms of the Einstein Coefficients!
If in Thermal
Equilibrium,
Boltzmann:
Factoring out nuBul:
Using guBul = glBlu:
This Looks Awfully Familiar!
For a path length element ds passing
through a material in thermal equilibrium
Question: Why isn’t this nuts?
Note: you can still use this for radiative transfer through a medium of
varying temperature, as long as each slab is in thermal equilibrium!
Kirchoff’s Law:
There is a general relation between the
emission and absorption coefficient for
material in thermal equilibrium
Outside of thermal equilibrium:
Very generally:
Def’n of
Excitation
Temperature:
For a path length element ds passing
through any material!
What We’ve Done:
• Einstein A and B coeff’s.
• Relations among eachother.
• Intro to radiative transfer
• Relation to emissivity and absorption coeff’s.
Immediate Plan:
• Einstein A and B coeff’s in terms of
“oscillator strengths”
• Bound-free transitions
Next up:
• Rate coefficients for different kinds of
collisions
What are the values of the Einstein
Coefficients?
Values are frequently defined in terms of
“oscillator strengths”:
Cross section for absorption:
Classical
Prediction
QM
Correction
Oscillator
strengths are of
order 1
Line
Profile
What are the values of the Einstein
Coefficients?
What are the values of the Einstein
Coefficients?
In terms of the oscillator strength, the
Einstein B coefficient for absorption is:
Cross section for absorption:
Line profile has
units of 1/Hz
Of order
unity
What are the values of the Einstein
Coefficients?
Where !ul is the (positive) frequency of the
transition
Note: Emission oscillator strengths
are negative.
One can define equivalent quantities for
emission:
Where !lu is the (negative) frequency of the
transition
Where !lu is the
(negative) frequency
of the transition
Bul>0 !lu<0
ful<0
Easier to use flu:
Because:
What are the values of the Einstein
Coefficients?
Using the relations between A and B:
Where the frequency and oscillator
strengths are both postive.
You can also define oscillator strengths for
“bound-free” transitions to the continuum
_
Define in terms
of transition to an
energy range dE
for the electron
_
Constraint:
This is how you can put in the “stimulated recombination”
term into the Milne Relation
There are relationships among the oscillator
strengths for a given state n
“The Sum Rule”
If there are closed shells, and
only q “active” electrons:
= zero for ground state, or
negligible for meta-stable states
Number of
electrons in
the atom
An example for Hydrogen:
All bound transitions
upwards:
Most of which are
n=1->2 or 3:
Continuum transitions
(ionization):
Continuum and Bound are comparable, so
you ionize at about the same time you excite!