lecture 5: quantitative emission/absorption lecture... · lecture 5: quantitative...
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Lecture 5: Quantitative Emission/Absorption
Light transmission through a slab of gas
1. Eqn. of radiative transfer / Beer’s Law
2. Einstein theory of radiation
3. Spectral absorption coefficient
4. Radiative lifetime
5. Line strengths
L
Io(ν) I(ν)
Collimated light @ ν
Gas
Energy balance
2
1. Eqn. of radiative transfer / Beer’s Law
T
ontransmissiscatteringreflectionabsorption100
1 Tspectral absorptivity, or absorbance spectral transmissivity
units no
I
dIdxk
Spectral absorption coefficient (the fraction of incident light Iν over frequency range ν→ν+dν which is absorbed per unit length dx
spectral intensity in or
1
2
cmW/cm
Hz
W/cm2
1-cm/
I
dxdIk
dII 2W/cmIntegrate over ν
for total I
dx
Iν Iν +dIνThin sample of emitting/absorbing gas
Collimated light @ ν
Also applies to I @ ν
Energy balance
3
1. Eqn. of radiative transfer / Beer’s Law
Blackbody spectral radiancy
dx
Iν Iν +dIνThin sample of emitting/absorbing gas
Collimated light @ ν
Consider emission from the gas slab
units nounits no
bb
em
bb
em
II
II
Kirchhoff’s Law – “emissivity equals absorptivity”
Spectral emissivity
II bb
absorptionemission
IIII
dIbbbb
absorptionemission
Differential form of the eqn. of radiative transfer
IIdxkdI bb
Energy balance
4
1. Eqn. of radiative transfer / Beer’s Law
L
Io(ν) I(ν)
Collimated light @ ν
Differential form of the eqn. of radiative transfer
IIdxkdI bb
Integrate over L
LkILkII bb exp1exp0
Integrated form of the eqn. of radiative transfer
Optical depth
Consider two interesting cases: Emission, Absorption
Gas
Case I: Emission experiment (no external radiation source)
5
1. Eqn. of radiative transfer / Beer’s Law
Spectral radiancy:
Spectral emissivity:
Integrate over ν
00 I
LkILILk
LkILI
bb
bb
exp1,
exp1
04
00
exp11
exp1
dLkITI
LIL
dLkIdLILI
bbbb
bb
Note:
Stefan-Boltzmann constant
4
0TdII bbbb
-4-1-25 Kscmerg1067.5
Single/multiple lineSingle/multiple bandsContinuum
Emission types:
Optical depth: Optically thick:
Optically thin:
LkεILkLILk
ILILkbb
bb
, ,1
,1
LkILkII bb exp1exp0
Case II: Absorption experiment
6
1. Eqn. of radiative transfer / Beer’s Law
Alternate form:
bbII 0
LkILI exp0
LkILkII bb exp1exp0
Beer’s Law / Beer-Lambert Law
00
expIILk
IIT
Observations:1. The same equation would apply to the transmission of a pulse of
laser excitation, with energy Eν [J/cm2/cm-1], i.e., Tν=Eν/Eν0
2. The fundamental parameter controlling absorption over length L is the spectral absorption coefficient, k.
How is k related to fundamental molecular parameters?
= I0v exp(-v)
= absorbance
Simplified theory (Milne Theory)
7
2. Einstein theory of radiation
Einstein coefficients of radiation
State 2
State 1
Transition probability/s of process per atom in state 1 or 2
Spontaneous Emission
Induced Absorption
Induced Emission
A21 B12ρ(ν) B21ρ(ν)Energy
hEE 12
Total transition rate [molec/s] N2A21 N1B12ρ(ν) N2B21ρ(ν)
B12ρ(ν)
B21ρ(ν)
A21
The probability/s that a molecule in state 1 exposed to radiation of spectral density ρ(ν) [J/(cm3Hz)] will absorb a quantum hν and pass to state 2. The Einstein B-coefficient thus carries units of cm3Hz(J s).
The probability/s that a molecule in state 2 exposed to radiation of spectral density ρ(ν) will emit a quantum hν and pass to state 1.
The probability/s of spontaneous transfer from state 2 to 1 with release of photon of energy hν (without regard to the presence of ρ(ν)).
Simplified theory (Milne Theory)
8
2. Einstein theory of radiation
Equilibrium
Detailed balance 02 state leaving molec/s
21212
2 state entering molec/s
1212
BANBNN rad
equil. lstatistica
1
2
equil. rad.
2121
12
1
2 exp
kTh
gg
BAB
NN
eq
eq
1/exp
/8 33
kThch
eq Planck’s blackbody
distribution
1/exp
/8
1/exp
/
3321
12
2
1
2121
kThch
kThBB
gg
BAeq
21213
3
21
212121
/18
BchA
BgBg
Radiative lifetime Where is the link to kν?
Note: for collimated light
1-2
-13
sW/cm
sJ/cm
chnI
hn
p
peq
cI /
Find kν for a structureless absorption line of width δν
9
2. Einstein theory of radiation
Recall Beer’s Law: LkIIT
exp0 dxI
dIk
Now, let’s find fraction absorbed using Einstein coefficients
dx
Iνδν Iνδν+(dIν)δν
LkI
TI
Pabs
exp1
1
W/cm absorbedfraction over power incident
0
0
2Absorbed power
Optically thin limit
W/cm2s-1 s-1
1dxk
dxkIP
dxkIP
abs
abs
absorbedfraction 0
0
Gas
Find kν for a structureless absorption line of width δν
10
2. Einstein theory of radiation
Energy balance
absorption inducedemission sspontaneouemission induced0
dI
dx
Iνδν Iνδν+(dIν)δνGas
for collimated light
Induced emission =
Induced absorption =
photonper energy emission of prob/s
12
1 statein molec/cm
1
photonper energy emission of prob/s
21
2 statein molec/cm
2
2
2
hBdxn
hBdxn
cI / Recall:
kThBnchk
BnBnchk
dxIdI
dxIchBnBndI
/exp11cm
1
1211
121212
121212
Since kν is a function of δν, we conclude depends on linewidths + hence shape; next, repeat with realistic lineshape
Where are we headed next?Improved Einstein Theory, Radiative Lifetime, Line Strength
Water vapor absorption spectrum simulated from HITRAN
3. Spectral absorption coefficient
with proper lineshape
4. Radiative lifetime
5. Line strengths
Temperature dependence
Band strength
11
= ab
sorb
ance
Wavenumber [cm-1]
Eqn. of radiative transfer
12
3. Spectral absorption coefficient
Recall:
LkILkII bb exp1exp0
Next: use realistic lineshape
L
Io(ν) I(ν)
Collimated light @ ν
Gas
1, cmdxIdIk
• For structureless absorption line of width δν (Hz), we found
kThBnchk /exp11cm 121
1
∝n1, B12, and 1/δν
Independent of lineshape!
Repeat derivation of kν using an improved lineshape model
13
3. Spectral absorption coefficient
Structureless absorption line of width δν
A typical absorption line with typical structure
Replace with realistic lineshape
1 ,sor cm lineline
ddkk
kν
14
3. Spectral absorption coefficient
Recall Beer’s Law:
Define:
LkIIT
exp0 T
Lk ln1
Normalized lineshape function
Inverse frequency Note: max,kdk
Average width
1max, dkk
pk
Relevant transition probabilities have the same spectral dependence (shape) as kν and φ(ν)
And we can anticipate that 1/v will be replaced by in kv equation
Modified model
15
3. Spectral absorption coefficient
Einstein coefficients of radiation
State 2
State 1
Transition probability/s/molec (in level 2 or 1) for range ν to ν+dν
Spontaneous Emission
Induced Absorption
Induced Emission
A21φ(ν)dν B12φ(ν)dνρ(ν) B21φ(ν)dνρ(ν)Energy
hEE 12
A21φ(ν)dν
B12φ(ν)dνρ(ν)
B21φ(ν)dνρ(ν)
Recall:
The probability/s of a molecule undergoing spontaneous emission, in the range ν→ν+dν.[Note that the integral of this quantity over the range of allowed is just A21 [s-1], i.e., .]
The probability/s of a molecule undergoing a transition from 1→2, in the range ν→ν+dν.
The probability/s of a molecule undergoing a transition from 2→1, in the range ν→ν+dν.
2121 AdA
cI /
Energy balance
16
3. Spectral absorption coefficient
[W/cm2 in ν to ν+dν]dx
Iνdν Iνdν+(dIν)dνGas
0121
tonenergy/pho0
dfor molec prob/s
21
molec/cm
/cc#
2 //
in absorptionin emission
2
hcIdBdxnhcIdBdxn
ddddI
212121 BnBn
chk
dxIdI
kThBn
chk /exp1121
Integrated absorption / Line strength 11
12 scm --
line
dkS kThBn
chS /exp112112
Line strength – alternate forms
17
3. Spectral absorption coefficient
Line strength does not depend on lineshape, but is a function of n1, T, B12
11121
2
12
11
1
2211
2
12
scm /exp1
scm /exp18
kThfncmeS
kThggAnS
e
kThfnS actual /exp1Hzcm0265.0 1212
,12 kTpn 1
1
@ STP, n1=n=2.7x1019cm-3, exp(–hν12/kT)<<1 1272
12 10380.2atm/cm fS
Oscillator strength kThSS
fclassical
actual
/exp1,12
,1212
Hzcm0265.0, 22
1
2
,12
cmen
cmeS
eeclassical
where
122
121 f
ggf
Important observations
18
3. Spectral absorption coefficient
1. From the original definition of kν and S12 we have
2. When
12Sk
as is common for electronic state transitions
1
2211
2121
2
121
21
12
8
Hzcm0265.0
Hzcm
ggAn
fn
fncmeSe
1/ kTh
21
221121212 cm51.1/
ggfAf
Radiative lifetime of the 2→1 transition 2121 /1 A
1/exp1 kTh
Aside:@λ=1440nm, hν/k=104K@λ=720nm, hν/k=2x104K@λ=360nm, hν/k=4x104K
Example: “Resonance Transition”
19
3. Spectral absorption coefficient
Resonance transition – one that couples the ground state to the first excited state
Electronic transition of a sodium atom
s1024.5cm51.1 92
1
2nm58912
ggf
Conventions:atoms: (L-U)molecules:(U↔L), arrow denotes absorption or emissionfij: i denotes initial state, j denotes final
Measured: 189 s1062.0s101.16 A
cm1089.5nm589 ,1 5
1
2 gg
Strong atomic transition: single electronMuch smaller for molecular transitions: ~ 10-2-10-4
325.0f
(U)upper
2/12
(L)lower
2/12 33 PSNa
Oscillator strength
20
3. Spectral absorption coefficient
Oscillator strengths of selected sodium transitions
Transitions f21 λ [nm]
32S1/2 – 32P1/2 0.33 589.6
32S1/2 – 32P3/2 0.67 589.0
32S – 42P 0.04 330.2
Absorption oscillator strengths of selected vibrational and vibronicbands of a few molecules
Molecule v'←v" Electronic Transition
Band center [cm-1] f12
CO1←0 - 2143 1.09x10-5
2←0 - 4260 7.5x10-8
OH1←0 - 3568 4.0x10-6
0←0 2Σ←2Π 32600 1.2x10-3
CN 0←0 2Π←2Σ 9117 2.0x10-2
Radiative and non-radiative lifetimes
21
4. Radiative lifetime
Rate equation for radiative decay
Rate equation for non-radiative decay
Simultaneous presence of radiative and non-radiative transitions
onlyemission sspontaneou l
luuu Andtdn
Upper level u Lower level l
lluuu Atntn exp0
Initial number density
llu
r A1Radiative lifetime
(zero-pressure lifetime)
nr
uunr
nr
u nnkdtdn
Rate parameter [s-1] Non-radiative decay time, depends on the transition considered and on the surrounding molecules
111 , nrru
nr
u
r
uu nnndtdn
←Life time of level u
u
l
Alternate forms – Line strengths
22
5. Line strengths
1.
2.
3.
4.
cmcmcm 212
1 Sk
11
1
s/1cmcm/1cm
scm/scm
dcd
c
/scm/1cm 112
212
ScS
kThggAc
PnPSS
ii /exp1
8atmatm/cmatm/cm
1
2122
1212
212
Number density of absorbing species i in state 1
Ideal gas law
kTSS atmcmdynes/1013250cmmolec/cmatm/cm
221*2
12
erg/K1038054.1 16k 1221*
atmcm1034.7
TSS
1219* atmcm104797.2 SS@ T=296K
atm
cc/moleccmmolec/cmatm/cm21*
212 P
nSS
HITRAN unit
Alternate forms – Beer’s Law n = number density of the absorbing species
[molecules/cm3] σν = absorption cross-section [cm2/molec] S = line strength [cm-2atm-1] or [cm-1sec-1/atm] βν = frequency-dependent absorption coefficient [cm-1/atm] Pi = partial pressure of species i [atm] φν = frequency-dependent lineshape function [cm] or [s] v = kvL = absorbance
23
5. Line strengths
LPS
LPLn
LkII
i
i
expexpexp
exp,,
0
Common to use atmosphere and wavenumber units in IR
kThggA
Pnc
kThfPn
cPS
dS
i
i
i
/exp18
/exp1atm
1082.8
atmscm
/atmcm
1
221
12
12113
1112
212
iPk / absorption coefficient per atmosphere of pressure
Temperature dependence
24
5. Line strengths
1
0
,0,0
0
"00
0
exp1exp1
11exp
kThc
kThc
TTkhcE
TT
TQTQTSTS
ii
iii
Line strength in units of [cm-2atm-1]
Line strength in units of [cm-1/(molecule·cm-2]
1
0
,0,0
0
"0
0**
exp1exp1
11exp
kThc
kThc
TTkhcE
TQTQTSTS
ii
iii
T
TTSTS
TSTS
i
i 0
0*
*
0
Band strength
25
5. Line strengths
Example: Heteronuclear Diatomic Band Strength
0"1'
"
01"'
01"'
01vv
JJJJJ RSPSS
band
linesband SS
"
"102
610
"
"102
610
1"2"
810013.1
1"21"
810013.1
J i
J
J i
J
JJ
nnA
kTPS
JJ
nnA
kTRS
"
" 1/J
iJ nn
kTcATS 2
10610
810013.1
kThAJJA
JJ
gg
kTnncRS R
J
J
i
JJJ /exp1
1"21"
1"21'2
10013.1/81010
"
'6
"2
10"'
atm ,"
i
JPn
1 1010
1"2" A
JJAP
Based on normalized Hönl-London factor
Band strength
26
5. Line strengths
Example: Heteronuclear Diatomic Band Strength
band
linesband SS
/atmcm280102.3K273 22
102810
ASCO
1cm2150 113s104.6
110 s36 A
kTcATS 2
10610
810013.1
Band strength of CO:
s028.010
Compare with previous example of τNa≈16nsIR transitions have much lower values of A and longer radiative lifetime than UV/Visible transitions due to their smaller changes in dipole moment