Lecture 5: Quantitative Emission/Absorption Lecture... · Lecture 5: Quantitative Emission/Absorption Light transmission through a slab of gas 1. Eqn. of radiative transfer / Beer’s

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


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


Also applies to I @ ν

Energy balance

3


Blackbody spectral radiancy

dx

Iν Iν +dIνThin sample of emitting/absorbing gas


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


L

Io(ν) I(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


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


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


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


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


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


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


with proper lineshape


5. Line strengths

Temperature dependence

Band strength

11

= ab

sorb

ance

Wavenumber [cm-1]

Eqn. of radiative transfer

12


Recall:

LkILkII bb exp1exp0

Next: use realistic lineshape

L

Io(ν) I(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


Structureless absorption line of width δν

A typical absorption line with typical structure

Replace with realistic lineshape

1 ,sor cm lineline

ddkk

kν

14


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


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


[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


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


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


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


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


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

Next: Spectral Lineshapes

Doppler, Natural, Collisional and Stark Broadening Voigt Profiles

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Lecture 5: Quantitative Emission/Absorption Lecture... · Lecture 5: Quantitative Emission/Absorption Light transmission through a slab of gas 1. Eqn. of radiative transfer / Beer’s