Electronic Spectroscopy I:Line Intensities and CN

Peter BernathDepartment of Chemistry

University of York

“Quantitative” means (to me) line positions and line intensities.

Second Edition-April 2005

Second edition includes line strength formulas and their derivation for microwave (JPL), infrared (HITRAN) and electronic transitions plus light scattering.

Most of the equations, etc. and many of the figures from my three lectures are from this book.

CN, For Example

“The CN radical has been the subject of extensive studies because of its importance in chemical kinetics, flame diagnostics, and astrophysics. The CN radical is found in many extraterrestrial sources including the Sun, stellar atmospheres, comets, dark interstellar clouds, and diffuse interstellar clouds.” from Ram et al. JMS, 237, 247 (2006).

Interstellar CN Violet 0-0

Diffuse interstellar cloud absorption towards the star HD169454

Gredel et al. AA, 251, 625 (1991)

See also, A. McKellar, PASP, 52, 187 (1940)

(Bob McKellar’sfather!)

Trot = 3 K (same temperature as the cosmic background radiation)[CN]/[13CN]=38

Circumstellar CN Violet 0-0

Circumstellarcloud absorption towards the star HD56126

Bakker & Lambert, ApJ502, 417 (1998)

Trot = 12 K

[CN]/[13CN]=38

Lines are saturated.

Solar CN Violet (Fraunhofer Lines)

Absorption in photosphere of the Sun

L. Wallace,

Kitt Peak Solar Atlas

Trot = 6000 K in the solar photosphere[CN]/[13CN]=90

Same as on Earth

Comet Linear CN Violet Emission

Notice the “Swings effect”

Arpigny et al. Science 301, 1522 (2003)

[CN]/[13CN]=115,

[CN]/[C15N]=140

(269 on Earth)

The Beginning: The Two Level System

dN1/dt = B1<-0 ρN0 – B1->0 ρN1 – A1->0N1

Absorption, Stimulated Emission and Spontaneous Emission

Einstein Relationships (SI units!)

Absorption, stimulated emission and spontaneous emission “rate constants”are all related to each other and to the transition dipole moment.

B1<-0 = B1->0

A1->0 = (8πhν3/c3) B1<-0

or

g(ν-ν10) is a line shape function (e.g., Gaussian, Lorentzian, Voigt, etc.) (Subscript nuis usually not included!)

These “familiar”, famous equations are all for transitions between non-degenerate states

μ10= <ψf|μ|ψi> is transition dipole moment

but most applications involve transitions between degenerate energy levels.

Beer’s Law

Again equations for cross section, Beer’s law, etc., assume no degeneracy.

What is a line?Transition between two levels J′ <-> J″ with MJ-degeneracy of

2J+1. True for both atoms and molecules, regardless of whether the line is resolved. N.B. with this definition the degeneracy is always 2J+1, there are no extra factors or 2 or 2S+1, etc., to consider.

J′; N1/(2J′+1) per M state

J″; N0/(2J″+1) per M state

SJ′J″ is defined as the line strength.

Einstein Relations (Revised) Absorption Cross section

Beer’s Law

Beer’s Law (Thermodynamic Equil.)

Oscillator Strength DefinitionCross sections to A values

Radiation Relationships (SI)

Integrated cross section to get rid of line shape function g. HITRAN intensity units are an example of an integrated cross section.

“Practical” Units

debye2

HITRAN units

SI for B

dimensionless

e. g., “wavenumber”, cm-1 for frequency, ν

My advice is to work in SI units for all of the basic equations and then convert to “practical” units only at the end. For example, don’t switch to cgs units or atomic units or measure concentrations in atmuntil the end.

What are HITRAN units?

HITRAN Units

Start with Beer’s Law:

Assume thermodynamic equilibrium to eliminate N0and N1 in favour of N:

Beer’s law again:

q is the partition functionHITRAN form of Beer’s law with N (total conc.) in molecules/cm3, l in cm, g in cm (wavenumbers used in line shape function) so S′ in cm/molecule.

S′=∫ln(I0/I)dν/(Nl)

Diatomic Electronic Transitions

Transition Dipole MomentIgnore rotation for the moment

CN Transition Dipole Moments

Calculated ab initio by Schwenke (unpublished) for the A2Π-X2Σ+ and B2Σ+-X2Σ+ transitions.

Franck-Condon Factors (CN radical)

Electronic Line Intensities

In which q=|<v′|v″>|2 is a Franck-Condon factor, Re the electronic transition moment and S a Hönl-London factor for rotation. N.B. S is used for 3 quantities (line strength, HITRAN-like line strength and Hönl-London factor). Often |<v′|R(r)|v″>|2 is used instead of q|Re|2.

Singlet states only!

Line Intensity Equations

is substituted into basic equations given before to get:

Best to use Einstein A’s!

Astronomers prefer f values (oscillator strengths).

Fourier Transform Emission Spectroscopy of the B2Σ+-X2Σ+

(Violet) System of CN

R.S. RamDepartment of Chemistry, University of Arizona, Tucson, AZ 85721

S. P. Davis, L. WallaceNational Optical Astronomy Observatory, Tucson, AZ 85726

R. EnglemanDepartment of Chemistry, University of New Mexico, Albuquerque,

NM 87131D. R. T. Appadoo

Canadian Light Source, 101 Perimeter Road, Saskatoon, Sask. Canada S7N OX4

P. F. BernathDepartment of Chemistry, University of Waterloo, Waterloo, Ont.,

Canada N2L 3G1

J. Mol. Spectrosc. 237, 247 (2006)

McMath-Pierce Solar Observatory

McMath-Pierce FTS

CN Violet Δv=+2 Sequence

CN was made by S. Davis using a microwave discharge of nitrogen with a trace of methane. The discharge tube was cooled by a flow of N2vapor from liquid nitrogen boil-off.

CN Violet 10-10 Band

CN Violet B2Σ+-X2Σ+

From Herzberg’s Diatomics, notice now N replaces K and F1 has e parity and F2 has f.

Each rotational level N (except N=0) is split into 2 by the electron spin with J=N+1/2 (F1, e) and J=N-1/2 (F2, f). There are 4 main branches (R1, R2, P1, P2) and 2 weak satellite Q branches (RQ21and PQ12 “R-form” and “P-form”). It is important to include all branches for work in astronomy because main lines are often saturated so satellite lines are needed to get the “opacity”correct.

CN Data Used

Starting point was previous analysis of Prasad & Bernath 1991 (jet-cooled emission data, plus IR and microwave)

• New microwave: Klisch et al. v=0-7• New IR: Horka et al. 1-0 to 8-7 bands• Electronic: Our new B-X data for v’ & v”

up to 14 and 15-15 band plus Douglas & Routly, Ito et al. for v’s > 15

CN X2Σ+ Constantsv

Tv Bv 106 Dv γv γDv

0 0.0 1.891089596(96) 6.39726(64) 7.25514(52) H 10-3 -1.91(11) H 10-7

1 2042.42143(24) 1.873665288(90) 6.40576(60) 7.17376(74) H 10-3 -1.83(11) H 10-7

2 4058.54933(29) 1.856186883(85) 6.41672(60) 7.0850(12) H 10-3 -1.47(11) H 10-7

3 6048.34329(35) 1.83865221(11) 6.42731(56) 6.9814(12) H 10-3 -1.40(11) H 10-7

4 8011.76637(42) 1.82105914(21) 6.44121(73) 6.8631(14) H 10-3 --

5 9948.77554(56) 1.80340409(27) 6.4530(38) 6.7198(14) H 10-3 --

6 11859.32721(61) 1.78568472(29) 6.4651(44) 6.5417(15) H 10-3 --

7 13743.37442(66) 1.76789824(29) 6.4812(46) 6.3136(14) H 10-3 --

8 15600.86884(71) 1.75004020(28) 6.4835(64) 6.0121(15) H 10-3 --

9 17431.75410(77) 1.73210149(27) 6.5334(85) 5.6133(22) H 10-3 --

10 19235.95846(76) 1.71405029(27) 6.6424(73) 5.2004(74) H 10-3 1.17(25) H 10-6

11 21013.2936(11) 1.694997(24) 1.30(15) c 1.424(21) H 10-2 -8.156(75) H

12 22765.7297(11) 1.677358(16) 8.954(75) 1.3336(21) H 10-1 -2.339(15) H

13 24488.7281(13) 1.659510(16) 6.645(40) 1.750(17) H 10-2 --

14 26185.6934(15) 1.641291(26) 6.58(12) 1.153(21) H 10-2 --

15 27856.2000a 1.622749(62) 6.38(25) 3.6(1.7) H 10-3 2.83(59) H 10-5

16 29500.37(71) 1.6004(23) 4.3(1.8) -- --

17 31115.064(25)b 1.58528(44) 4.2(1.8) -- --

18 32703.724(20)b 1.56699(18) 7.85(36) -- --

CN B2Σ+ State Constants

v Tv Bv 105 Dv γv γDv

0 25797.86825(43) 1.9587413(13) 0.660855(81) 1.7154(52) H 10-2 -8.58(29) H 10-7

1 27921.46650(55) 1.9380444(45) 0.67324(29) 1.8162(82) H 10-2 -1.044(81) H 10-6

2 30004.90632(77) 1.916503(10) 0.7021(27) 1.840(13) H 10-2 -2.51(59) H 10-6

3 32045.94678(73) 1.894180(15) 0.7105(60) 2.453(16) H 10-2 -7.4(1.1) H 10-6

4 34041.97036(68) 1.8704809(66) 0.7448(15) 2.1169(97) H 10-2 -5.18(35) H 10-6

5 35990.0970(21) 1.847108(24) 0.9132(54) 4.31(83) H 10-3 1.691(39) H 10-4, c

6 37887.42418(74) 1.8193429(54) 0.8092(11) 2.5237(87) H 10-2 -8.50(27) H 10-6

7 39730.53401(80) 1.790761(12) 1.1054(58) 6.126(58) H 10-3 B

8 41516.64296(84) 1.7621417(59) 0.9040(13) 3.4942(98) 10-2 -2.000(32) H 10-5

9 43242.98350(93) 1.730285(12) 0.9243(58) b 1.567(11) H 10-2 -1.897(36) H 10-5

10 44908.7905(14) 1.697091(90) 4.73(20) b 3.1270(75) H 10-1 -4.287(33) H 10-3, c

11 46511.39508(97) 1.6649929(84) 1.0311(21) 2.138(14) H 10-2 -1.690(50) H 10-5

12 48053.7300(11) 1.629723(27) 1.798(19) -9.15(23) H 10-3 -7.81(22) H 10-5

13 49537.3389(13) 1.598042(16) 1.0877(37) 3.325((22) H 10-2 -2.613(59) H 10-5

14 50964.6127(38) 1.56437(49) 9.5(1.4) 1.159(60) H 10-2 B

15 52340.0303(17) a 1.532490(75) 1.223(34) 9.90(18) H 10-2 -4.39(11) H 10-4, c

16 53664.4700(98) 1.49988(12) 1.289(22) 8.4(2.1) H 10-3 5.45(51) H 10-5

17 54944.838(690) 1.4656(21) 0.91(16) 7.2(2.1) H 10-3 B

18 56178.130(22) a 1.43704(30) 0.83(10) 0.0 B

19 57371.297(14) a 1.40806(13) 1.212(26) 0.0 B

CN Energy Levels

• Levels up to high v (v′=19, v″=18) and high J (100) are calculated using spectroscopic constants on previous slides.

• Experimental term values are used to replace all of the theoretical ones if they are available.

• All possible lines are computed for all possible branches from the combined experimental-theoretical line list. Goal is to have a linelist that can be used at high and low resolution and at high (6000 K) and low (3 K) temperatures.

CN Line Intensities• Calculate RKR potential curves for B and X states using Bob LeRoy’s

program (http://leroy.uwaterloo.ca/). Need G(v)=ωe(v+1/2)+… and Bv=Be– αe(v+1/2)+… polynomials for each state (perturbations are a problem!).

• Use LeRoy’s LEVEL program to calculate the Einstein A for each possible transition (v′,N′-N″,v″) using Schwenke’s B-X ab initio electronic transition dipole moment points (Re(r) function)

• Convert N to J for each P, Q and R line and then get the correct Hönl-London factors for a 2Σ-2Σ transition

SI eq.

“Practical” equation

CN Linelist for AstrophysicsM. Rusilowicz (York)

# Column 1: v', upper state vibrational level# Column 2: v", lower state vibrational level# Column 3: Upper state Omega (always 0.5)# Column 4: Lower state Omega (always 0.5)# Column 5: J", lower state J# Column 6: Parity: e or f# Column 7: Branch type: P=1, Q=2, R=3# Column 8: Transition wavenumbers in cm-1# Column 9: Einstein A value for the transition in s-1# Column 10: Lower state energy level in cm-1#0 0 0.5 0.5 1.5 e 1 25794.082 9901821.73425 3.78578090 0 0.5 0.5 1.5 f 1 25790.433 9897619.61284 11.3354260 0 0.5 0.5 0.5 f 2 25794.093 4952108.82250 3.77489880 0 0.5 0.5 2.5 e 1 25790.440 8907865.51100 11.3535610 0 0.5 0.5 2.5 f 1 25786.917 8903142.72253 22.6776480 0 0.5 0.5 1.5 f 2 25790.458 990003.159574 11.3354260 0 0.5 0.5 1.5 e 2 25805.808 991533.255510 3.78578090 0 0.5 0.5 0.5 e 3 25801.794 4955948.81000 00 0 0.5 0.5 0.5 f 3 25805.819 4958865.86465 3.7748988

Red Rectangle CN Violet

Emission and absorption!

Hobbs et al. ApJ 615, 947 (2004)