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Chapter 7. Emission and Absorption and Rate Equations
7.1 IntroductionFor most considerations (total relaxation
rate is much faster than the rate at which external forces cause electron to jump between atomicenergy levels.
)A(2
112121
(6.5.18)
The result of the external force,
F=-eE is only to produce a gradual
increase or decrease in probability.
)A(2
112121
(6.5.19)
Therefore, such a fast phenomena can often be
treated with sufficient accuracy in an average sense. Absorption rate / Stimulated emission rate
(Rates of increase and decrease in probability)
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7.2 Stimulated Absorption and Emission Rates
When )A(2
12121
quasisteady-state approximation, 02112
)(2
A 21*
12222111111 i
)(2
)A( 21*
122221222 i
)(2
)( 1122
*
1212 ii
)(2
)( 11222121 ii
(6.5.14)
(6.5.17)
In Chapter 6, density matrix equations considering the relaxation effects are given by
is possible ;
(6.5.17) => )(2/
1122
*
12
i
i
)(2/
112221
i
i: Adiabatic solution(7.2.1)
(condition for adiabatic following to occur),
2112 , adiabatically follow the inversion 1122,
※
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)( 112222
2
21*
12
i
(6.5.14) =>)(
2/112222
2
222111111
A
)(2/
)( 112222
2
2221222
A
: Population rate equation(7.2.2)
1122 , are coupled only to each other
※
<Stimulated Absorption and Emission Rates>
Stimulated absorption or emission rates
2221
2
0
)(
2//ˆrateAbsorption
Ee 21r
2221
2
0
)(
2//ˆrateemissionStimulated
Eεe 21r
1) For nondegenerated transitions,
(7.2.3)
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Calculation of ||2
In many cases (unpolarized radiation, rotational or collisional disorientation, etc), orientational average of |is simpler and useful
2
022
012
2
0norientatio
2
0
3
1
3
1
ˆ
EDEe
EEe
221
21
rr
r
DεrD 21 ˆand e
)(2
12
2
12
2
122
1222 zyxeeD 221 rrD
(7.2.4)
where, : Complex dipole moment and its projection on ε̂
In terms of cartesian components,
Homework : Problem 7.1
22
2
02
2221
2
012 6)(
2//
EDE
R
Induced transition rates (abbreviation), : 12R
(orientation-averaged)(7.2.7)
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2) For degenerated transitions (Homework : Refer to Appen. 7.A)
1
2 g2=5
g1=3
Absorption cross section, : [Refer to (7.4.2)]
12
200
12abs
2/fluxenergy radiation incident
1 levelin atomper rate absorptionenergy R
Ec
R
abs12 R
where,
(7.2.8)
: photon flux
<Example of degenerated transition>
21 ,
212
21 ),(1
mm
mmAg
A
21 ,
211
12 ),(1
mm
mmRg
B
21 ,
212
21 ),(1
mm
mmRg
B
: In the case of natural excitation (the # of atoms in each of the different degenerated states of the same level are equal) ;
22221111 /,/ gPgP
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7.3 Population Rate Equations
Densities of atoms in levels 1 and 2 ;
222111 , NNNN (7.3.1)
where, N : total density of atoms
(7.2.2) => )( 12221111 NNNANN
)( 12221222 NNNANN (7.3.2)
※ ??0221121 NNNN
: This indicates that ineleastic collisions will takes all of the atoms out of levels 1 and 2 into other atomic levels. Nevertheless, are practically small relative to , and the intermediate time behavior is of the most interest. => We can ignore the
21, ,21A
21,
2
1
2
1
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constant21 NNN
NNA
NNNAN
221
22212
)2(
)2(
22)0()(
21
)2(
2122
21
A
Ne
A
NNtN tA
(No inelastic collision), then 21 NNN
Sol)
(7.3.7)
)( 122211 NNNAN
)( 122212 NNNAN (7.3.4)
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Examples)
1) No radiation field ; 0
)decaysspontaneou()0()( 2122
tAeNtN
2) Weak radiation field ;21A
, 0)0(2 N
, 0)0(2 N
)excitaionweak(1)( 21
212
tAeA
NtN
3) Strong radiation field ;21A , 0)0(2 N
NA
NNt
212,
: Lorentz classical theory is valid.※
)statesteadysaturated(222
)0()( 222
NNe
NNtN t
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NN )0(20)0(2 N
[Fig. 7.1]
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Power Broadening
In the limit, 1)2( 21 tA
(7.3.7) =>
2)(
212 A
NN
)statesteady(/
2/
/)(21
/)()(
21222
212
21
212
A
A
A
A
N
N
2/1
21
2/1
21
22
2/1
)0(21
A
A
Half width ;
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7.4 Absorption Cross Section and the Einstein B Coefficient
abs12 R (7.2.8), )(2/200 IEc
(7.2.7) => 22210
2
)(3
c
D
)(6
)()2()()2(
2
3
0
2
221
2221
221
0
2
Lc
D
c
D
212Put,
221
221
21
)()(
/)(
Lwhere, : Lorentzian line shape function
)(S : generaliztion for arbitrary line shape function
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Examples)
1) For descrete radiation frequencies
)(S
21
0
hI
h
IS
c
DR
)(
6 0
2
12
For a single frequency, :00
00
0
2
12
)(
6
h
IS
c
DR
(narrow band limit)2) For continuous band radiation
)(S
)(I
dh
Sc
DR
00
2
12
)()(
6
cI
21
If 021 )()(, 21 S
)()(6 21212
0
2
12
B
DR
: Einstein’s empirical definition
20
2
6 D
B (broad band limit)
0012 )(1
IBSc
R
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7.5 Strong Fields and Saturation
What is the criterion for “strong” field ? => Saturation the population ; !2/2 NN
This criterion is satisfied in (7.3.13), if 2221
221 /or 2/ AA
Define,
intensity saturation : 2/ I
flux (photon) saturation : 2/
21sat
sat
21sat
A
A
ex) 1821
210 s10~ ,resonance)(on cm10~ A2
sat218sat W/cm0.3I and s,photons/cm 10
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7.6 Spontaneous Emission and the Einstein’s A Coefficient
An atom in an excited state will eventually drop to a state of lower energy, even in the absence of any field or other atoms. => Spontaneous emission(※ Spontaneous emission would occur even for a single excited atom in a perfect vacuum !)
ex) Luminescence, Fluorescence, Phosphorescence
tAeNtN 21)0()( 22
(7.3.8) =>
212 /1 A : characteristic time constant (excited state “life time’)
- In the case that there are multi-channels for radiative transition,
,22 m
mAAmm AA 22
2
11
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Quantum mechanical expression ;
222
31
3
32
0
2
pol3
32
0
4224
1
sinˆ24
1
nmnmnmnm
kkkknmnm
nm
zyxc
e
ddc
eA
εr
30
32
3
32
0 33
4
4
1
c
D
c
DA nmnmnmnmnm
(7.6.4)
(7.6.5)
2222nmmnnmnm eeD rrr where,
Line shape for the spontaneous emission : Lorentzian
2rad
20
rad
)(
/)(
S
where, )(4
112rad AA
: Natural linewidth
(7.6.7)
Homework : Problem 7.3