Download 3 Radiation processes 3.1 Atomic and molecular structure

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3.1
Radiation processes
Atomic and molecular structure
The binding energy of the hydrogen atom is
IH =
1 2
e4 me
2
2 = α me c = 13.6 eV,
2
2h̄
(3.1)
where α = e2 /h̄c = 1/137 is the fine structure constant. The energy levels are given by Bohr’s
formula
IH
(3.2)
Ei = 2 .
i
The Bohr radius is
h̄2
re
aH = 2
(3.3)
= 2 = 0.5 · 10−8 cm,
e me
α
where re = e2 /me c2 = 2.8 · 10−13 cm is the classical electron radius. In the Coulomb field, the
average kinetic energy is equal to the binding energy so that the electron velocity at the i-th level
could be estimated as v = αc/i. For hydrogen-like ions, IZ = Z 2 IH , aZ = aH /Z.
In all atoms, the energy of the first ionization is of the order of IH , from 4-5 eV for alkalis up
to 24 eV for He.
The binding energy of molecules is also of the order of IH , the distance between the atoms
being of the order of aH . The energy of the basic electron transitions is of the same order.
The atoms in the molecule can vibrate about the equilibrium position; transitions between
the vibrational energy levels may occur as a result of the emision/absorption of photons. Let
us consider a case of a diatomic molecule. The characteristic vibrational frequency may be estimated taking into account that if the vibration amplitude approaches the equilibrium interatomic
distance, the vibration energy approaches the binding energy. This condition is written as
1 2 2
µω a ∼ IH ,
2 v H
where µ is the reduced mass of the molecule. Then one finds
r
me
IH ∼ 0.1 eV,
εv = h̄ωv ∼
µ
(3.4)
(3.5)
so that this radiation is in the IR. The corresponding temperature is kT = εv ∼ 1000 K therefore
at the room temperature, the vibrational levels are not excited. But absorption/emission at
vibrational transitions is important even at the room temperature because the spectrum reaches
maximum at 3kT .
Dipole molecules could also emit/absorb photons at the transitions between rotation energy
levels. The characteristic energy of these photons is
εrot ∼
h̄2
me
∼
IH ∼ 10−3 eV,
2
µaH
µ
which corresponds to the mm band.
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(3.6)
3.2
Radiation in classic and quantum theory
The particle radiates if it is accelerated. Radiation power of a non-relativistic particle is given by
Larmor’s formula
2 e2 a2
.
(3.7)
P =
3 c3
In the simplest case circular motion with the velocity v and the period T , the particle radiates at
the frequency ν0 = 1/T , the power being
8π 2 e2 ν02 v02
P =
.
3
c3
(3.8)
If the particle performs periodic motion with the period T , it radiates at the frequencies ν = n/T ,
where n = 1, 2, . . . . The power of each harmonics is given by Eq. (3.8), where one has to
substitute v0 by the corresponding Fourier harmonics of the velocity.
In the case of non-periodic motion, the particle radiates continuous spectrum, which could be
found by performing the Fourier transform of the acceleration
Z ∞
aν =
a(t)e2πiνt dt.
(3.9)
−∞
The total radiated power may be presented making use of Parseval’s theorem as
Z
Z
Z ∞
4e2 ∞
2e2 ∞ 2
a dt = 3
|aν |2 dν.
E=
P dt = 3
3c
3c
−∞
0
−∞
(3.10)
This yields the spectral power (the energy radiated per unit time and per unit frequency interval)
Pν =
4 e2 |aν |2
.
3 c3
(3.11)
In the quantum theory, the radiation implies transition between the energy levels,
hνij = Ej − Ei .
(3.12)
The transition rate j → i due to spontaneous emission is determined by the Einstein coefficient
for spontaneous emission, Aji , as
dNi
= Aji Nj .
(3.13)
dt em
In the presence of the radiation field, the emission rate increases due to the induced emission.
Namely, the photon emission probability is proportional to 1 + n, where n is the occupation
number of photons already available in the state to which the new photon is emitted. Therefore
the total downward transition rate should be written as
h
i
dNi
= Aji 1 + n (νij ) Nj ,
(3.14)
dt em
where νij = Ej − Ei )/h and the overline denotes averaging over the angles.
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The upward transitions occur via absorption of photons with the frequency νij so that the rate
of the upward transition is proportional to the photon density at this frequency:
dNi
= −Aij n (νij )Ni .
(3.15)
dt abs
According to the principle of detailed balance, which in fact follows from the time reversibility of
quantum mechanics, gi Aij = gj Aji , where gi is the statistical weight of the i-th state. Finally the
evolution of atomic level population due to radiative transitions is described by the equation (the
master equation)
dNi
gj
= Aji Nj (1 + n) − nNi .
(3.16)
dt
gi
Now one sees that if the energy levels are populated according to the Boltzmann distribution,
Nj /Ni = (gj /gi ) exp[−(Ej − Ei )/kT ], and the photon occupation number is Plankian, n =
[exp(−hν/kT ) − 1]−1 , the system is in equilibrium, dN/dt = 0.
The correspondence principle implies that for highly excited levels, j, i 1, the transition
rate could be found from the classical theory. Making use of Eq. (3.8) for the classical rate of
spontaneous emission, one gets an estimate for the Einstein coefficient
A=
4π v 2
P
=
αν 2 .
hν
3
c
(3.17)
This relation provides a rough estimate even for the transitions between low levels, j, i ∼ 1. As an
example, let us estimate the probability of the transition 2 → 1 (Lyman-α line) in the hydrogen
atom. The line frequency is
ν21 =
3IH
3
c
= α2
= 2.5 · 1015 Hz,
4h
8 λC
(3.18)
where λC = h/me c = 2.4 · 10−10 cm is the electron Compton length. The electron velocity at the
second level is v = (1/2)αc. Now one gets
A21 =
π 3
π
c
α ν21 = α5
= 109 s−1 .
3
8 λC
(3.19)
A precise quantum calculation yields A21 = 4.7 · 108 s−1 .
3.3
Induced emission as negative absorption
When discussing the equation of radiation transfer (2.22), we have not mentioned the induced
scattering. According to the basic principles, the total rate of emission in the direction Ω should
be presented in the form jν (1 + n(ν, Ω)) so that the radiation transfer equation should have a form
dIν (Ω)
= jν [1 + n(ν, Ω)] − κ0ν Iν (Ω).
ds
(3.20)
Taking into account the relation (2.9) between n and I, one can reduce this equation to the
standard form (2.22), where
κν = κ0ν − (c3 /2hν 3 )jν .
(3.21)
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Therefore stimulated emission may be considered as negative absorption, which just modifies
the absorption coefficient. Note that Kirchhoff’s law (2.21) follows from the basic principles of
thermodynamics therefore it implicitly takes into account negative absorption. Therefore one can
safely find the absorption coefficient from the the spontaneous emission coefficient and Kirchhoff’s
law. However, if one finds the absorption cross-section straightforwardly (e.q., from quantum
mechanical consideration of the absorption process), one gets the absorption coefficient κ0ν = σν Nj
not corrected for stimulated emission. In this case, one could find κ making use eq. (3.21) and
Kirchhoff’s law:
0
κ0ν
−hν/kT
=
1
−
e
κnu .
(3.22)
κν =
(1 + (c3 /2hν 3 )Bν
One sees that the correction for stimulated emission is small at hν > kT .
3.4
Thomson scattering
In the radiation field, the free electron oscillates, me a = eE. For the monochromatic wave, one
finds a = (eE0 /me ) cos ωt. The oscillating electron radiates so one can say that the incident
radiation is reradiated by the electron; the process is called Thomson scattering. One finds from
Larmor’s formula that the average radiation power is
2 e4
1 e4
2
2
P =
E0 hcos ωti =
E02 .
2
3
2
3
3 me c
3 me c
(3.23)
The scattering cross-section is defined as the ratio of the scattered energy to the incident energy
flux, σ = P/S. Here S = hE 2 ic/4π = cE02 /8π is the Poynting flux. Now one finds the Thomson
cross-section as
8π 2
8π e4
=
r = 6.65 · 10−25 cm2 .
(3.24)
σT =
2
4
3 me c
3 e
3.5
Bremsstrahlung
Bremsstrahlung (free-free emission) occurs when an electron moving in the Coulomb field of an
ion emits photons. The electron motion could be considered in the scope of classical mechanics if
the deviation angle of the electron (we consider the hydrogen plasma, Z = 1),
θc ∼
∆v
1 F ∆t
1 e2 b
e2
∼
∼
=
,
v
v me
v b2 me v
me bv 2
(3.25)
where b is the impact parameter, exceeds the quantum diffraction angle, θq ∼ ň/b ∼ h̄/me vb. This
yields the condition on the electron velocity
v < αc.
(3.26)
For the thermal plasma this means that the classical treatment is valid at kT < IH .
Let us assume for simplicity that the trajectory remains nearly straight, which implies
mv 2 /2 ≥ e2 /b.
(3.27)
Since the particle motion is not periodic, the continuous spectrum is emitted. The spectral radiation power is given by Eq. (3.11). The acceleration is determined by the equations of the electron
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motion in the field of the ion. The function a(t) smoothly varies at the time-scale ∼ b/v therefore
the Fourier transform (3.9) is exponentially small at ν v/2πb. This yields the maximal impact
parameter at which the elctron with the velocity v could emit at the frequency ν:
bmax = v/2πν
(3.28)
R∞
In the opposite limit, ν v/2πb, one can substitute exp(2πiνt) by unity so that aν = −∞ adt =
∆v is independent of ν. Since the interaction mostly occurs at the minimal distance from the ion,
the velocity variation could be estimated as
∆v =
1 e2 2b
.
me b2 v
Then the spectral power of a single electron could be roughly presented in the form
(
16
e6
ν v/2πb;
2 c3 b 2 v 2 ;
3
m
e
Pν =
0;
ν v/2πb.
(3.29)
(3.30)
In order to obtain the emission coefficient (the energy radiated by the unit plasma volume per unit
time, unit solid angle and unit frequency interval), one has to multiply the obtained expression
by the electron flux, integrate Pν over the impact parameters, average the result over the electron
velocity distribution and multiply by the ion number density. For hydrogen plasma Ni = Ne = N ,
one can finds
Z
N2
v Pν · 2πbdb .
(3.31)
jν =
4π
The integral over the impact parameters, which is the radiation power from a unit flux of
electrons with the velocity v, could be presented as
Z
bmax
32π e6
ln
.
(3.32)
Qν = Pν 2πbdb =
3 m2e c3 v 2 bmin
The minimal impact parameter is determined by the condition (3.26) as bmin = e2 /me v 2 .
According to the obtained expression, the radiation spectrum is practically flat. The condition
bmin < bmax implies that this expression is valid at ν < me v 3 /4πe2 , which could be written as
hν
v
<
,
E
αc
(3.33)
where E = me v 2 /2 is the electron energy. Higher frequency photons are emitted when the condition (3.27) is violated so that when the electrons closely approach the ions. In this case, even
photons with hν > E could be emitted, so that the electron remains bound (recombination). This
process will be considered later on.
In the limit opposite to (3.26), the quantum treatment of the scattering process is necessary,
which yields in the Born approximation
√
√
32π e6
( E − E − hν)2
Qν =
ln
.
(3.34)
3 m2e c3 v 2
hν
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One sees that the quantum result coincides with the classical one to within a logarithmic factor,
but according to the quantum formula, Qν vanishes at hν = E automatically.
p Averaging Qν over the Maxwell distribution may be roughly performed by substituting vT =
kT /me for v and adding a multiplier exp(−hν/kT ), which takes into account that Qν = 0 at
me v 2 /2 > hν. The result of detailed calculations is presented in the form
jνff
8
=
3
2π
3kT me
1/2
hν
e6
N 2 e− kT g(kT /hν);
3
me c
(3.35)
where g is the Gaunt factor, which is only logarithmically dependent on the parameters. The total
emission power is
r
1/2
Z
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2πkT
e
32π
πkT IH
N2 = 2
σT cN 2 .
(3.36)
J ff = 4π jν dν =
3
3
3me
me c h
3
Having found the emission coefficient, one finds the absorption coefficient from Kirchhoff’s law
(2.21):
κffν
4
=
3
2π
3kT me
1/2
hν
e6
1
2
− kT
N 1−e
g = 3/2
3
hme cν
2π
IH
3kT
1/2
3
σT λ N
2
1−e
hν
− kT
g. (3.37)
In gaseous nebulae, the free-free absorption occurs in the radio band so that hν kT . In this
case the absorption coefficient is written as
κffν
1
=
(6π)1/2
me c2
kT
3/2
σT re λ2 N 2 g.
(3.38)
The free-free Rosseland mean (2.38) is estimated as
κffR = 7 · 1022
ρ2
cm−1 ,
T 3.5
(3.39)
where ρ is in g/cm3 , T in degrees.
3.6
Radiation properties of dust
The interstellar and interplanetary medium contains dust (roughly 1% by mass) composed of
small particles (a ∼ 0.1 µ and less), which absorb and emit light. At short wavelengths, ň < a,
the dust particles absorb a significant fraction of the radiation therefore one can roughly take
the absorption cross section σ ∼ πa2 (dust particles become transparent in X- and γ−ray bands
but here we interested only in optical-IR radiation). In the long wavelength limit, ň > a, the
absorption cross-section is roughly presented as
a
σ = πa2 .
ň
(3.40)
The dependence of the absorption cross-section on the wavelength results in the observed reddening
of distant stars.
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Close to hot, luminous stars, the dust is heated by the stellar light up to dozens K and therefore
emits in the IR band. The emission from the heated dust could be found from the Kirchhoff law:
if one puts a dust particle with the temperature T into a box filled by the blackbody radiation
with the same temperature, the energy emitted by the particle at any frequency should be equal
to the energy absorbed at this frequency. Therefore the spectral power emitted by the particle is
ν4
16π 3 a3 h
.
qν = 4πσBν =
c3
exp(hν/kT ) − 1
(3.41)
One sees that at small frequencies, the spectrum grows as ∝ ν 3 , faster than the Rayleigh-Jeans.
Dust clouds are typically transparent in the IR band therefore a cloud of the volume V heated to
the temperature T emits Lν = N V qν , where N is the density of particles.
The temperature of the dust is determined by heat balance of a single particle, q + = qR− , where
the heating is determined by the absorption of the light from external sources, q + = σFν dν,
whereas cooling is due to the thermal radiation of the particle itself,
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Z
16π 3 a3 h kT
−
A,
(3.42)
q = qν dν =
c3
h
R ∞ x4 dx
R∞
where A = 0 exp(x)−1
≈ 0 x4 e−x dx = 24 (one can neglect unity in the denominator because the
maximum of the integrand is at x ≈ 4).
3.7
Bound-bound transitions (line spectrum)
Radiative transitions between discrete energy levels produce the line spectrum. The transition
probabilities for dipole transitions are well described by the semiclassical formula (3.17).
Because the life-time of the excited state is finite, the width of the line is also finite, ∆ν ∼ Γ,
where Γ is the decay probability of the energy level. At the classical language, the line widening
occurs because oscillations are not strictly harmonic. Let us consider a harmonic oscillator with
the frequency ν0 and the energy E = mhv 2 i. If the oscillator is isolated, the lifetime is determined
by the radiation losses. The decay rate may be presented making use of Larmor’s formula (7.7) as
8π e2 ν02 2
dE
= hP i =
hv i = −γE;
dt
3 c3
(3.43)
where
8π 2 e2 ν02
8π 2 re
=
ν0
(3.44)
3mc3
3 λ0
is the decay constant. Therefore the actual motion is described as decaying oscillations, x =
x0 exp(−γt/2) cos 2πν0 t. Making Fourier transform, one finds the radiation spectrum from Eq.
(3.11) as (in the range |ν − ν0 | ν0 )
γ=
Pν = E0 φ(ν − ν0 );
φ(ν − ν0 ) =
γ
1
;
2
2
4π (ν − ν0 ) + (γ/4π)2
(3.45)
R∞
where E0 is the initial energy. The function φ represents the Lorentz profile. Note that −∞ φ(x)dx =
R∞
R∞
1 therefore 0 Pν dν = 0 P dt = E0 . The full width at half maximum, which is called the natural
line width, is
δν = γ/2π.
(3.46)
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In wavelength units, the natural line width is independent of the frequency, δλ = cδν/ν02 =
(4π/3)re .
In quantum systems, the upper layer could decay through different channels; then the profile
of each line is also described by the Lorentz profile with an appropriate ν0 but the line width γ is
determined
by the total decay probability for the level. If only radiation transitions are possible,
P
γi =
i0 Aii0 . If the level could be deactivated by collisions with other particles, γ should be
substituted by γ + 2/τ , where τ is the mean collisional deactivation time. Thermal motion of the
atoms results in the Doppler broadening of the lines, ν − ν0 = (v/c)ν0 . The Doppler profile is
gaussian,
( 2 )
ν − ν0
1
exp −
,
(3.47)
φ(ν − ν0 ) = √
∆νD
π∆νD
p
where the characteristic line width is ∆νD = ν0 2kT /mc2 .
The emission coefficient due to the radiative transitions i → i0 is presented as
hν
Aii0 φ(ν − ν0 )Ni .
(3.48)
4π
With account of the induced emission, the total emission rate is (cf. Eq. (3.14)) jν (1 + nν ). The
reverse transitions i0 → i are described by the absorption coefficient κν = σν Ni0 . The absorption
cross-section is found from the condition that in the thermodynamical equilibrium, the absorption
and emission are balanced,
jν =
hν
Aii0 φ(ν − ν0 )Ni (1 + nν ) = σν Ni0 Iν .
4π
Substituting nν = [exp(hν/kT ) − 1]−1 and Ni /Ni0 = (gi /gi0 ) exp(−hν0 /kT ) yields
σν =
λ20 gi
Aii0 φ(ν − ν0 ).
8π gi0
The cross section in the center of the line is now estimated as
λ2 gi Aii0
σmax ≈ 0
.
2π gi0 δν
(3.49)
(3.50)
(3.51)
Specifically in the case of natural broadening, γ ∼ Aii0 , therefore σmax ∼ λ20 . A relation independent of the line widening mechanism may be presented as:
Z
λ20 gi
σν dν =
Aii0 .
(3.52)
8π gi0
Substituting the estimate (3.17) for the Einstein coefficient, on can write
Z
2πgi
E
σν dν =
re c ,
3gi0
hν
where E is the electron energy at the upper level.
The radiation transfer equation in the line could be written as
dIν
2hν 3 gi0
gi0
= σν
Ni − Ni0 − Ni Iν .
ds
c2 g i
gi
(3.53)
(3.54)
One sees that in the case of a reverse level population, Ni > (gi /gi0 )Ni0 , the radiation intensity
grows exponentially along the ray - maser effect.
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