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Chapter 2. Electromagnetic Radiation and its Interaction
with Atoms and Molecules
2.1 Electromagnetic Radiation
Ey = A sin(2πνt - kx)
Eq. (2.1)
Hz = A sin(2πνt - kx)
where, k = 2π/λ
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2.2 Absorption and Emission of Radiation
ΔE = En - Em = hν = hcν~
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Eq. (2.2)
2
There are 3 processes which occur in this two-state system:
Eq. (2.3)
1. induced absorption M + hν Æ M*
2. spontaneous emission
Eq. (2.4)
M* Æ M* + hν
3. induced emission
M* + hν Æ M + 2 hν Eq. (2.5)
For an induced absorption process (from state m to state n)
Rate of change of population Nn of state n
dN n
dt
=
N m Bmn ρ (ν~ )
Eq. (2.6)
where Bmn is Einstein coefficient for induced absorption
ρ (ν~ ) is spectral radiation density
3
~
8πhcν
~
ρ (ν ) =
exp(hcν~ / kT ) − 1
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Eq. (2.7)
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For an induced emission process (from state n to state m)
dN n
~)
Eq. (2.8)
−
N
B
ρ
(
ν
=
n nm
dt
Note: Einstein coefficient for induced emission Bnm = Bmn
For a spontaneous emission process (from state n to various lower
states)
dN n
dt
=
− N n Anm
Eq. (2.9)
where Amn is Einstein coefficient for spontaneous emission
In the presence of radiation, all 3 processes occur
dN n
dt
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~)
(
N
N
)
B
ρ
(
ν
=0=
− N n Anm
m
n
nm
Eq. (2.10)
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At equilibrium, Nn and Nm are related through the Boltzmann
distribution law
Nn
gn
∆E
∆E
=
exp(−
) = exp(−
)
Nm gm
kT
kT
Eq. (2.11)
gn and gm are the degrees of degeneracy.
Recall that E = En – Em = hν = hc ν~
Assuming gn = gm, putting Eqs (2.11) and (2.7) into Eq. (2.10),
one gets
Anm = 8πhcν~ 3 Bnm
Eq. (2.12)
This indicates that as ν~ increases,
spontaneous emission increases rapidly relative to induced
emission
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Sample calculation 2.1
For temperatures of 25 and 1000ºC, calculate the ratio of
molecules in a typical excited rotational, vibrational, and
electronic energy level to that in the lowest energy level,
assuming that the levels are 30, 1000, and 40000 cm-1,
respectively, above the lowest energy level. (Three significant
figures are sufficient. Assume that, for the excited rotational
level, the rotational quantum number J is 4 and remember that
each level is (2J+1)-fold degenerate. Assume that the
vibrational and electronic energy levels are non-degenerate.)
Possible solutions:
Form Eq. (2.11), we consider rotational level J =4,
Nn/Nm = NJ/N0 = [(2J+1)/1]•exp(-hcν~/kT)
= 9•exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x
30 cm-1)/(1.382 x 10-23 J•K-1 x 298 K)]
= 7.79 at 25ºC (298 K);
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Nn/Nm = 8.70 at 1000ºC (1273 K).
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For vibrational level v,
Nv/N0 = exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x 1000
cm-1)/(1.382 x 10-23 J•K-1 x 298 K)]
= 8.01 x 10-3 at 25ºC (298 K);
Nv/N0 = 0.323 at 1000ºC (1273 K).
For electronic level e,
Ne/N0 = exp[-(6.626 x 10-34J•s x 2.998 x 1010 cm•s-1 x 40000
cm-1)/(1.382 x 10-23 J•K-1 x 298 K)]
= 1.40 x 10-84 at 25ºC (298 K);
Ne/N0 = 2.35 x 10-20 at 1000ºC (1273 K).
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Bnm (Einstein coefficient for induced processes) can be
related to the wavefunctionsΨm and Ψn through the
transition moment Rnm
Rnm =
∗
ψ
∫ n µψ m d τ
Eq. (2.13)
µ is the electric dipole moment operator
µ = ∑ qi ri
Eq. (2.14)
i
where qi and ri are the charge and position vector of
the ith particle (electron or nucleus)
R
nm 2
Bnm =
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is the transition probability and is related to Bnm by
8π 3
nm
R
(4πε 0 )3h 2
2
Eq. (2.15)
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2.3 Line Width
An absorption line is not infinitely narrow. A typical line exhibits
a half-width at half maximum (HWHM) of ∆ν.
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2.3.1 Natural Line Broadening
If the species M* in state n decays to the lower state by a
first-order process
dN n
= kNn
Eq. (2.22)
dt
where k is the 1st-order rate constant
τ
1
k
=τ
Eq. (2.23)
is the life time of state n
If spontaneous emission is the only process by which M*
decays, then k = Anm
Eq. (2.24)
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Heisenberg uncertainty principle:
τ∆Ε ≥ ħ, where ħ = h / 2π
Eq. (2.25)
Since τ is finite, all energy levels are smeared out to some
Extent with resulting line broadening.
To relate Anm (Einstein coefficient for spontaneous emission)
2
to the transition probability R nm , one combines Eqs.
(2.12) and (2.15) and gets
64π 4ν 3
nm 2
Anm = ( 4πε )3hc 3 R
0
Thus, Eq. (2.25) becomes
32π 3ν 3
nm 2
∆ν ≥
R
3
( 4πε 0 )3hc
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Eq. (2.26)
Eq. (2.27)
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Since ∆ν is dependent on ν3, ∆ν is much larger for an
excited electronic state than an excited rotational state.
∆ν is ~ 30 MHz for an excited electronic state
~ 10-4
– 10-5 Hz for an excited rotational state
Eq. (2.27) illustrates the natural homogeneous line broadening,
which results in a Lorentzian line shape.
In general, the natural line broadening is very small compared
to other causes of spectral line broadening.
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Gaussian and Lorentzian lineshapes of the same FWHM
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2.3.2 Doppler Broadening
If a molecule is traveling towards the detector with
a velocity υa, then the observed frequency νa is related
to the actual frequency ν in a stationary molecule by
νa =
⎛ υa ⎞ -1
ν ⎜1 − ⎟
c⎠
⎝
Eq. (2.28)
Because the velocities of molecules exhibit a Maxwell
distribution, it gives rise to an inhomogeneous
broadening with a line width Δν
Δν =
ν
c
1/2
⎛ 2kT ln 2 ⎞
⎜
⎟
m
⎝
⎠
Eq. (2.29)
This Doppler broadening results in a Gaussian line shape.
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2.3.3 Pressure Broadening
If a gas-phase spectroscopic experiment is performed
with a cell, molecular collisions may take place, leading
to so-called pressure broadening with a line width
Assume that τ is the mean time between collisions and
each collision results in a transition, the Heisenberg
uncertainty principle (Eq. 2.25) gives
Δν = (2πτ)-1 Eq. (2.29)
This pressure broadening causes a Lorentzian line shape,
similar to that of homogeneous natural line broadening.
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2.3.4 Removal of Line Broadening
2.3.4.1 Effusive Molecular Beams
An effusive molecular beam is produced by pass the gas at a
high pressure (1-3 atm) through a tiny hole into a vacuum (~10-6
Torr) system. This isentropic expansion process quenches the
internal rotations and vibrations of molecules. It results in a
narrow spectral line. Since the rotational cooling is more
efficient than the vibrational cooling, scientists often referred to
as a rationally cooled process.
Pressure broadening of spectral lines is removed in an effusive
beam. If observations are made perpendicular to the direction of
the beam, Doppler broadening is considerably reduced because
the velocity component in the direction of observation is very
small.
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Fluorescence
excitation spectrum of
NO2: (a) conventional
gas cell, room temp.
0.04 Torr, (b)
molecular beam, (c)
seeded molecular
beam, 5% NO2 in Ar.
Ref.: Smalley, et al.
J. Chem. Phys. 61,
4363 (1974).
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Ref.: Smalley, et al. J. Chem. Phys. 64, 3266 (1976).
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Ref.: Smalley, et al. J. Chem. Phys. 64, 3266 (1976).
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2.3.4.2 Lamb Dip Spectroscopy
If a narrow light source (e.g. laser) is tuned to a frequency νa
which is higher than the resonance frequency νres at the line
center, only molecules like 1 and 2 which have a velocity
component va away from the source in Fig. 2.6 absorb radiation.
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On the return journey of the radiation back to the detector,
molecules like 6 and 7 which have a velocity component –va
away from the source absorbed at –va. When the radiation is
tuned to νa , the number of molecules in the lower state m of the
transition with velocity component va away from the source is
depleted. This is referred to as hole burning.
Molecules like 3, 4, and 5 which have zero velocity component
away from the source absorb radiation at vres regardless the
radiation travels towards or aways from R, resulting in
“saturation”. The result is a dip in the absorbance curve as seen
in Fig. 2.5 and is known as a Lamb dip which marks an accurate
value of vres.
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