Lecture 8: Case Study: UV - OH

1. Introduction

2. OH energy levels

Upper level

Lower level

3. Allowed radiative transitions

Transition notations

Allowed transitions

4. Working example - OH

Allowed rotational transitions from N"=13 in the A2Σ+←X2Π system

UV absorption of OH: A2Σ+ – X2Π (~300nm)

OH, a prominent flame emitter, absorber.Useful for T, XOH measurements.

2

1. Introduction

Selected region of A2Σ+←X2Π(0,0) band at 2000K

Steps in analysis to obtain spectral absorption coefficient

3

1. Introduction

1. Identify/calculate energy levels of upper + lower states

2. Establish allowed transitions

3. Introduce “transition notation”

4. Identify/characterize oscillator strengths using Hönl-London factors

5. Calculate Boltzmann fraction

6. Calculate lineshape function

7. Calculate absorption coefficient

Term energies

4

2. Energy levels

Separation of terms: Born-Oppenheimer approximation

G(v) = ωe(v + 1/2) – ωexe(v + 1/2)2

Sources of Te, ωe, ωexe Herzberg

Overall system : A2Σ+←X2Π

JFvGnTJvnE e ,,elec. q. no.

vib. q. no.ang. mom. q. no.

Electronic energy

Vibrationalenergy

Angular momentum energy (nuclei + electrons)

A2Σ+Te ωe ωexe X2Π

Te ωe ωexe

32682.0 3184.28 97.84 0.0 3735.21 82.21

in [cm-1]

Let’s first look at the upper state Hund’s case b!

Hund’s case b (Λ=0, S≠0) – more standard, especially for hydrides

5

2. Energy levels

Recall:

Σ, Ω not rigorously defined

N = angular momentum without spin

S = 1/2-integer values

J = N+S, N+S-1, …, |N-S|

i = 1, 2, …

Fi(N) = rotational term energy

Now, specifically, for OH?

The upper state is A2Σ+

6

2. Energy levels

For OH:

Λ = 0, ∴Σ not defined use Hund’s case b

N = 0, 1, 2, …

S = 1/2

J = N ± 1/2

F1 denotes J = N + 1/2

F2 denotes J = N – 1/2

Common to write either F1(N) or F1(J)

The upper state: A2Σ+

7

2. Energy levels

for pure case b

∴ the spin-splitting is γv(2N+1) function of v; increases with N

111

112

2

21

NNNDNNBNF

NNNDNNBNF

vvv

vvv

21 OHfor cm1.0constant splitting Av

Notes:

Progression for A2Σ+

“+” denotes positive “parity” for even N [wave function symmetry]

Importance? Selection rules require parity change in transition

γv(2N+1) ~ 0.1(5) ~ 0.5cm-1 for N2Compare with ∆νD(1800K) = 0.23cm-1

The ground state: X2Π (Λ=1, S=1/2)

8

2. Energy levels

Note:

1. Rules less strong for hydrides

2. OH behaves like Hund’s a @ low Nlike Hund’s b @ large N

at large N, couples more to N, Λ is less defined, S decouples from A-axis

3. Result? OH X2Π is termed “intermediate case”

L

Hund’s case a Hund’s case bΛ ≠ 0, S ≠ 0, Σ defined Λ = 0, S ≠ 0, Σ not defined

The ground state: X2Π

9

2. Energy levels

Notes:3. For “intermediate/transition cases”

where Yv ≡ A/Bv (< 0 for OH); A is effectively the moment of inertiaNote: F1(N) < F2(N)

22/122222

22/122221

14421

1414211

NNDYYNNBNF

NNDYYNNBNF

vvvv

vvvv

For large N

NNBF

NNBF

v

v

22

2

221 11

0121 2221 NNNBFF v

For small NBehaves like Hund’s a, i.e., symmetric top, with spin splitting ΛA

Behaves like Hund’s b, with small (declining) effect from spin

The ground state: X2Π

10

2. Energy levels

Notes:4. Some similarity to symmetric top

Showed earlier that F1 < F2

N

3

2

1

J

5/2

3/2

1/2

J

7/2

5/2

3/2F1: J = N + 1/2 F2: J = N – 1/2

Ω = 3/2 Ω = 1/2

Te = T0 + AΛΣFor OH, A = -140 cm-1

Te = T0 + (-140)(1)(1/2), Σ = 1/2+ (-140)(1)(-1/2), Σ = -1/2

∆Te = 140 cm-1

Not too far off the 130 cm-1 spacing for minimum J

130

Hund’s a → 2|(A-Bv)| Recall: Hund’s case a has constant difference of 2(A-Bv) for same J

F(J) = BJ(J+1) + (A-B)Ω2

(A–B)Ω2 ≈ -158.5Ω2

(A for OH~ -140, B ~ 18.5), Ω = 3/2, 1/2 Ω = 3/2 state lower by 316 cm-1

Actual spacing is only 188 cm-1, reflects that hydrides quickly go to Hund’s case b

The ground state: X2Π

11

2. Energy levels

Notes:5. Role of Λ-doubling

Showed earlier that F1 < F2

icid

diid

ciic FFJJJFFJJJFF

11

Fic(J) – Fid(J) ≈ 0.04 cm-1 for typical J in OH

c and d have different parity (p)

Splitting decreases with increasing N

N

3

2

1

J

5/2

3/2

1/2

J

7/2

5/2

3/2F1: J = N + 1/2 F2: J = N – 1/2

Ω = 3/2 Ω = 1/2

p

+–

–+

+–

p–+

+–

+–

Now let’s proceed to draw transitions, but first let’s give a primer on transition notation.

Transition notations

General selection rules Parity must change + → – or – → + ∆J = 0, 1 No Q (J = 0) transitions, J = 0 → J = 0 not allowed

12

3. Allowed radiative transitions

Full description: A2Σ+ (v')←X2Π (v") YXαβ(N" or J")

where Y – ∆N (O, P, Q, R, S for ∆N = -2 to +2)X – ∆J (P, Q, R for ∆J = -1, 0, +1)α = i in Fi'; i.e., 1 for F1, 2 for F2

β = i in Fi"; i.e., 1 for F1, 2 for F2

"or " JNXY

Notes:1. Y suppressed when ∆N = ∆J2. β suppressed when α = β3. Both N" and J" are used

Strongest trans.e.g., R1(7) or R17

Example: SR21: ∆J = +1, ∆N = +2F' = F2(N')F" = F1(N")

Allowed transitions

13

3. Allowed radiative transitions

Allowed rotational transitions from N"=13 in the A2Σ+←X2Π system

12 bands possible (3 originating from each lambda-doubled, spin-split X state) Main branches: α = β; Cross-branches: α ≠ β Cross-branches weaken as N increases

F1(13)F1c(13) F1d(13)

State or level

a specific v",J",N",andΛ-coupling

Allowed transitions

14

3. Allowed radiative transitions

Allowed rotational transitions from N"=13 in the A2Σ+←X2Π system

Notes: A given J" (or N") has12 branches (6 are strong; ∆J = ∆N) + ↔ – rule on parity F1c–F1d ≈ 0.04N(N+1) for OH for N~10, Λ-doubling is ~ 4cm-1, giving clear separation If upper state has Λ-doubling, we get twice as many lines!

Allowed transitions

15

3. Allowed radiative transitions

Allowed rotational transitions from N"=13 in the A2Σ+←X2Σ+ system

Note:1. The effect of the parity selection rule in reducing the number of

allowed main branches to 42. The simplification when Λ=0 in lower state, i.e., no Λ-doubling

Complete steps to calculate absorption coefficient

16

4. Working example - OH

1. Identify/characterize oscillator strengths using Hönl-London factors

2. Calculate Boltzmann fraction

3. Calculate lineshape function(narrow-band vs broad-band)

4. Calculate absorption coefficient

Absorption coefficient

0121

21 exp1cm,

kTh

e

fNcm

ek

0.0265 cm2/s

#/cm3 in state 1,a

a NNN 1

#/cm3 of species (OH), kTpA

Fractional pop. in state 1

ϕ[s] = (1/c) ϕ[cm]

To do: evaluate f12, N1/Na

Step 4 Step 5

Absorption oscillator strength

17

4.1. Oscillator strengths

1"2notation shorthandin or

1"2'"

'"'"'"

'"'"'"',',',',',",",","v,"

JSqff

JSqff

JJvvnnJJ

JJvvnnJvnJn

elec. vib. spin ang. mom. Λ-doubling

elec. osc. strength F-C factor H-L factor

'"vvf = band oscillator strength

(v',v") fv'v"

(0,0) 0.00096(1,0) 0.00028

Notes: qv"v' and SJ"J' are normalized

1'

'" v

vvq

2Xfor 4"

''" 121"2

elgJ

JJ SJS

this sum includes the S values for all states with J"

1 for Λ = 0 (Σ state), 2 otherwise

For OH A2Σ+–X2Π

Is SJ"J' = SJ'J"? Yes, for our normalization scheme! From g1f12 = g2f21, and recognizing that 2J+1 is the ultimate (non removable) degeneracy at

the state level, we can write, for a specific transition between single states

In this way, there are no remaining electronic degeneracy and we require, for detailed balance, that and

Do we always enforce for a state? No! But note we do enforce (14.17)

and (14.19)

where, for OH A2Σ←X2Π, (2S+1) = 2 and δ = 2.

When is there a problem? Everything is okay for Σ-Σ and Π-Π, where there are equal “elec. degeneracies”, i,e., g"el =

g'el. But for Σ-Π (as in OH), we have an issue. In the X2Π state, gel = 4 (2 for spin and 2 for Λ-doubling), meaning each J is split into 4 states. Inspection of our H-L tables for SJ"J' for OH A2Σ←X2Π (absorption) confirms ΣSJ"J' from each state is 2J"+1. All is well. But, in the upper state, 2Σ, we have a degeneracy g'el of 2 (for spin), not 4, and now we will find that the sum of is twice 2J'+1 for a single J' when we use the H-L values for SJ"J' for SJ'J". However, as there are 2 states with J', the overall sum as required by (14.19)

18

4.1. Oscillator strengths

1'2

'1'21"2

"1"2 "'"'

'"v'v"

J

SqfJJSqfJ JJ

vvelJJ

el

1"2'

'" JSJ

JJ

"''" JJJJ SS

"

"'J

JJS

v"v'v'v",'" qqff elel

121"2'

'" SJSJ

JJ

121'2"

'" SJSJ

JJ

41'2"

"' JSJ

JJ

Absorption oscillator strength for f00 in OH A2Σ+–X2Π

19

4.1. Oscillator strengths

Source f00

Oldenberg, et al. (1938) 0.00095 ± 0.00014

Dyne (1958) 0.00054 ± 0.0001

Carrington (1959) 0.00107 ± 0.00043

Lapp (1961) 0.00100 ± 0.0006

Bennett, et al. (1963) 0.00078 ± 0.00008

Golden, et al. (1963) 0.00071 ± 0.00011

Engleman, et al. (1973) 0.00096

Bennett, et al. (1964) 0.0008 ± 0.00008

Anketell, et al. (1967) 0.00148 ± 0.00013

Absorption oscillator strength

20

4.1. Oscillator strengths

Hönl-London factors for selected OH transitions

Transition SJ"J'/(2J"+1) ΣF1(J) ΣF2(J) Σ[F1(J)+F2(J)]Q12(0.5) 0.667 0 2 2Q2(0.5) 0.667R12(0.5) 0.333R2(0.5) 0.333P1(1.5) 0.588 2 2 4P12(1.5) 0.078P21(1.5) 0.392P2(1.5) 0.275Q1(1.5) 0.562Q12(1.5) 0.372Q21(1.5) 0.246Q2(1.5) 0.678R1(1.5) 0.165R12(1.5) 0.235R21(1.5) 0.047R2(1.5) 0.353P1(2.5) 0.530 2 2 4P12(2.5) 0.070P21(2.5) 0.242P2(2.5) 0.358Q1(2.5) 0.708Q12(2.5) 0.263Q21(2.5) 0.214Q2(2.5) 0.757R1(2.5) 0.256R12(2.5) 0.173R21(2.5) 0.050R2(2.5) 0.379

Transition SJ"J'/(2J"+1) ΣF1(J) ΣF2(J) Σ[F1(J)+F2(J)]P1(3.5) 0.515 2 2 4P12(3.5) 0.056P21(3.5) 0.167P2(3.5) 0.405Q1(3.5) 0.790Q12(3.5) 0.195Q21(3.5) 0.170Q2(3.5) 0.814R1(3.5) 0.316R12(3.5) 0.131R21(3.5) 0.044R2(3.5) 0.402P1(9.5) 0.511 2 2 4P12(9.5) 0.016P21(9.5) 0.038P2(9.5) 0.488Q1(9.5) 0.947Q12(9.5) 0.050Q21(9.5) 0.048Q2(9.5) 0.950R1(9.5) 0.441R12(9.5) 0.035R21(9.5) 0.014R2(9.5) 0.462

1. We seek the fraction of molecules in a single state for which

2. In general,

3. Electronic mode

21

4.2. Boltzmann fraction

1"2'

'" JSJ

JJ

re

kTii

QQQQQegNN i

v

/ //

OH 2

42

2

e

e

g

g

12 Sge 0,20,1

N"

J"=N"-1/2

J"=N“+1/2

c

dc

dA “state”

ΣSJ"J'=2J"+1 for each state

# of rot. levels produced by spin splitting & Λ-doubling = 4 for 2Π

nee

ee

kTnhcTSQ

QkTnhcTSN

nN

/exp12

//exp12

Elec. level

Note:hund’s (a) includes AΩ2

the sum of this over all levels is 1

4. Vibrational mode

5. Rotational mode (hund’s (b))

22

4.2. Boltzmann fraction

v

v

kThcGQ

QkThcGnN

nN

/vexp

//vexpv,

v Again, each of these →1 when summed

Nr

r

kTNhcFNQ

QkTNhcFNnN

NnN

/exp12

//exp12v,,v,

rr

r

vr

TQ

khcB

Tfor

/

Note: don’t use F1+F2(N) here; until we add spin splitting Now what about fraction of those with N in a given J?

1212

12,v,

,,v,

SNJ

NnNJNnN

≈ ½ for OH as expected

Since # of states in N is (2N+1)(2S+1)ϕ, while # of states in J is (2J+1) ϕ

1,,v,,,,v,

JNnN

pJNnN(fraction with spectral parity)

6. Combining

23

4.2. Boltzmann fraction

Note: 1. The fraction in a given state is 1/4 of that given by rigid rotor!2. Always know Σ(Ni/N) = 1, both in total and for each mode

separately.

re

ie

QQQ

NFGnTkThcJ

JNnNpJNnN

NnNJNnN

nNNnN

nNnN

NnN

NN

NpJNnN

v

1

vexp12

,,v,,,,v,

,v,,,v,

v,,v,v,

,,,v,

Proper Fi now!

We have 1 loose end to deal with:narrow-band and broadband absorption measurement.

(i.e., the Boltzmann fraction in state 1)

Narrow-band absorptionMeasured quantity

with

thus, if Tν (e.g. Tν0) is measured, and

if L, p, 2γ, T, f12 are known

then can solve for Nl

24

4.3. Narrow-band vs broad-band absorption measurement

LSIIT 120 exp/

kTh

e

fNcm

eS exp1121

2

12

Oscillator strength for transition

kTplN

NN l

ll

of dens. no. tot.1

Boltzmann fraction of species l = Fv",J",…(T)

species bd.

2i

ii X

Quantity usually sought

Let’s look at the classical (old-time) approach, pre 1975 Broadband absorption

25

4.3. Narrow-band vs broad-band absorption measurement

ν

Instrument broadening (no change in eq. width)

1 line

Eq. width WJ"J' (cm-1)

Integrated area is called: integrated absorbance, or eq. width

line JJ

JJline

dLK

dTdAWW

'"

'"

exp1

1

0 '"'" ,2ln2exp12ln

dxaxVLKWD

JJD

JJ

xD

2ln2

(for 1 line from 1 state)

Transform variables

iJJJJ PSK '"'"

Let’s look at the classical (old-time) approach, pre 1975 Broadband absorption Requires use of “curves of growth”

26

4.3. Narrow-band vs broad-band absorption measurement

ν

1 line

Eq. width WJ"J' (cm-1)

0 '"'" ,2ln2exp12ln

dxaxVLKWD

JJD

JJ

Procedure: measure WJ"J', calculate ∆νDand a, infer KJ"J', convert KJ"J' to Nspecies

Note:1. Simple interpretation only in optically

thin limit,

2. Measured eq. width is indep. of instrument broadening!

3. Before lasers, use of absorption spectroscopy for species measurements require use of Curves of Growth!

LfNmc

eLKW

dLKW

JJJJ

JJJJ

1212

2

'"'"

'"'" 11

Curve of growth

Consider spectral absorption coefficient of the (0,0)Q1(9) line in the OH A2Σ+–X2Π system, at line center.

27

4.4. Example calculation (narrow-band)

λ~309.6nm, ν~32300cm-1, T = 2000K, ∆νC = 0.05cm-1

Express kν as a function of OH partial pressure

4'"

v'v"9 1009.9947.000096.01"21

JSff JJ

Q

Oscillator strength (using tables)

Lineshape factor (narrow-band)

s1004.1or cm13.317.0cm05.0K2000

cm25.0K2000 1001

1

a

C

D

0'"",,,v,

221

scm10651.2cm JJ

a

Jna fN

NkTPk kTPN aa /

Consider spectral absorption coefficient of the (0,0)Q1(9) line in the OH A2Σ+–X2Π system, at line center.

28

4.4. Example calculation (narrow-band)

λ~309.6nm, ν~32300cm-1, T = 2000K, ∆νC = 0.05cm-1

Express kν as a function of OH partial pressure

0'"",,,v,

221

scm10651.2cm JJ

a

Jna fN

NkTPk kTPN aa /

Population fraction in the absorbing state

0.0193.08390 .9200 25.0

0.7529.6

287.0264.0

41

K66.26//K2313exp20

287.0/K2660exp

40exp

/5.9exp1"2/0exp/0exp 15.01

TTT

QkThcFJ

QkThcG

QkThcT

NN

rve

e

a

f c

Consider spectral absorption coefficient of the (0,0)Q1(9) line in the OH A2Σ+–X2Π system, at line center.

29

4.4. Example calculation (narrow-band)

λ~309.6nm, ν~32300cm-1, T = 2000K, ∆νC = 0.05cm-1

Express kν as a function of OH partial pressure

0'"",,,v,

221

scm10651.2cm JJ

a

Jna fN

NkTPk kTPN aa /

atmatmcm177

s1004.11009.9%93.1atmcm1066.3atm

scm10651.2cm

1

1043

182

21

a

a

P

Pk

LkII exp0Beer’s Law

59% absorptionfor L = 5cm, XOH = 1000ppm, T = 2000K, P = 1atm

Selected region of OH A2Σ+←X2Π (0,0) band at 2000K

30

4.4. Example calculation (narrow-band)

Notes:

Lines belonging to a specific branch are connected with dashed or dotted curve

Thicker dashed lines –main branches; thin dotted lines – cross branches

Bandhead in R branches if Bv'<Bv"; Bandhead in P branches if Bv'>Bv“

Note bandhead in RQ21branch

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Fundamentals Applications to Aeropropulsion