=.:'AEDC.-TR-80-47 - . - . . . , .~ o ° ~

, , - ~ ~ ' . '

SEP 4 1981

t~0V 0 4 198B

JUL 7 1988

• • • Computer Modeling of Laser-Excited Molecular ~ ~ and Atomic-Molecular Systems

D. H. Campbell ARO, Inc.

August 1981

Final Report for Period October 1, 1979 - September 1, 1980

Approved for public release; distribution unlimited.

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, AEuC . . . . . ~' L.;J ...:i #' "' F40@~0-3 " ' : , .

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AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE

NOTICES

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Approved for publication:

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UNCLASSIFIED REPORTDOCUMENTATION PAGE

1 R E P O R T N U M B E R 12 G O V T A C C E S $ | O N N O

I AEDC-TR-80-47 L

4 T I T L E ( ~ d Sub!l~le)

COMPUTER MODELING OF LASER-EXCITED MOLEC- ULAR AND ATOMIC-MOLECULAR SYSTEMS

? A U T H O R ( s )

D. H. Campbell, ARO. Inc., a Sverdrup Corporation Company

9 P E R F O R M I N G O R G A N I Z A T I O N NAME AND ADDRESS

Arnold Engineering Development Center/DOT Air Force Systems Command Arnold Air Force Station, TN 37389

I I C O N T R O L L I N G O F F I C E NAME AND ADDRESS

Arnold Engineering Development Center~DOS Air Force Systems Command Arnold Air Force Station, TN 37389

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Final Report - Oct. i, 1979 - Sept. I, 1980

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Available in Defense Technical Information Center (DTIC).

19 K EY WORDS (ContJnue on reverae cede i f n e c e a e e ~ ~ d I den t i f y ~ bJock number)

laser induced fluorescence computerized simulation Raman spectroscopy electronic states gas analysis diatomic molecules combustion carbon monoxide

20 A B S T R A C T (ContJnue on reverse srde I t n e c e e e e ~ ~ d Iden t i f y ~ bJock numbe~

The potential application of the laser-induced fluorescence method for determining number densities and temperatures of Kas species is investigated through computer modeling of the detailed relaxation of laser-excited molecular and atomic-molecular systems. The rate equation models for a sodium-diatomic molecular system and a single diatomic molecule system are explained and applied to the case of sodium-nitrogen and pure carbon monoxide.

F O R . 1 4 7 3 EO, T,ON O r ' NOV 6S IS OBSOLETE D D , .AN 73

UNCLASSIFIED

A E D C-TR-80-47

PREFACE

The research reported herein was performed by the Arnold Engineering Development Center (AEDC), Air Force Systems Command (AFSC). Work and analysis for this research were done by personnel of ARO, Inc., AEDC Group (a Sverdrup Corporation Company), operating contractor for the AEDC, AFSC, Arnold Air Force Station, Tenneseee. The work covered the period from October l, 1979 to September 1, 1980 and was done under Project Number P32M-01A. Capt. K. H. Leners was the Air Force project manager. The manuscript was submitted for publication on September I l, 1980.

Mr. D. H. Campbell is currently employed at the Air Force Rocket Propulsion Laboratory, Edwards Air Force Base, CA.

A E DC-TR -80-47

CONTENTS

Page

1.0 INTRODUCTION ........................................................ 5

2.0 REVIEW OF LASER-INDUCED-FLUORESCENCE (LIF)

2.1 Two-Level Model ..................................................... 6

2.2 Three-Level Model ............................................ . . . . . . . . 8

3.0 SODIUM-DIATOMIC MOLECULE SYSTEM RELAXATION

3.1 Delails of Qt,enching Mechanism ........................................ 9

3.2 Sodium-Nitrogen Relaxation Model ...................................... 10

3.3 Results of Sodium-Nitrogen Modeling .................................... 15

4.0 SINGLE DIATOMIC SPECIES RELAXATION

4.1 Breakdown of Two-Level Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Single Diatomic Species Relaxation Model ................................ 23

4.3 Carbon Monoxide Modeliqg Resulls ..................................... 26

5.0 SUMMARY ............................................................. 34

REFERENCES ........................................................... 35

ILLUSTRATIONS

Figure

I. Two-Level Model Schematic ................................................ 7

2. Three-Level Model Schematic ............................................... 9

3. Sodium-Nitrogen Collisional Quenching Probability Distribution . . . . . . . . . . . . . . . . 13

4. Quenching and Reverse Quenchiqg Rate Constants at T = 2,000°K . . . . . . . . . . . . . . . 14

5. Sodium 3P Level Mixing Exchange Rate Constants and Total Cross Sections . . . . . . . 15

6. Time History of Population in Sodium Levels for High Density, T = 1,000°K . . . . . . 16

7. Time History o f Population in Nitrogen Vibrational Levels for High Density,

T = 1,000°K ............................................................. 17

8. Time History of Population in Sodium Levels for High Density, T = 2,000°K . . . . . . 18

9. Time History of Population in Nitrogen Vibrational Levels for High Density,

T = 2,000°K ............................................................. 19

10. Time History of Population in Sodium Levels for Low Density, T = 2,000°K . . . . . . 20

I I. Time History of Population in Sodit, m Levels for Low Density, T = 300°K . . . . . . . . 21

12. Time History of Population in Nitrogen Vibrational Levels for Low Density,

T = 300°K ............................................................... 22

13. Time History of Rotational Population Distribution in Lower Electronic State

Laser-Coupled Vibrational Level ............................................ 26

3

AE DC-TR-80-47

Figure Page

14. Time History of Rotational Popttlation Distribulion in Upper Electronic State Laser-Coupled Vibrational Level ............................................ 27

15. Time History of Population in Upper Electronic Stale Laser-Coupled

Vibrational Level for Three Values of Quenching Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 28

16. Time History of Vibralional Population Distribution in Upper Electronic

State, T = 300°K ......................................................... 29

17. Time Hislory of Vibrational Population Distribution in Lower Electronic State, T = 300°K ......................................................... 30

18. Time History of Population in Upper Electronic State Laser-Coupled Vibrational Level, T = 2,000OK ............................................. 31

19. Laser Radiation Densily Effects on Ratio of Calculated Peak Number

Density and Two-Level Model Steady-State Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 32

TABLES

1. Constants for Sodium-Nitrogen Model ....................................... 12

2. Maximunl and Steady-State Number Densities and Times . . . . . . . . . . . . . . . . . . . . . . . 33

APPENDIX

A. Molecular Exchange Rate Constants ......................................... 39

NOMENCLATURE .......................................................... 37

A E DC-T R i80147

1,0 INI"ROI)LJC'I'ION

Aunong the requirements encoulHered in groundibased simulation test facilitic.,, i,, the

need to determine the ga,~eous conlposit ion, temperatures, and flow velocily in both simulation flow fields and the exhaust plumes of various engines. A variety of techniques are

being developed and applied to such problenas, but the measurement of the propertic,, of

minor species has bee.1 .no,~t difficuh. Thi,, capability is de,~ired to pro,..idc both spatially and

temporally resol,..ed nleasurcullerHs of the number densilic,~ of ~.arious low concerti rat ion, or

minor , species in cornbuslinlg flov,, fields and engine exhaust, plumes, with OH, H, and, in

sornc cases, H2 being exemplary species.

It is knlowrl thal for ,..cry low concentrat ions (mole fractions OIl the onden of 10- : or less)

of such species, local measuremcnls using Raman scattering are impractical because of low

signal level,~. However, fluorescence techniques do po,~ses,, the necessary requi,,itcs of

molectnlar specificity, and spatial and temporal resolution. Also, since fluorescence excitation cross sections exceed Raman scattering cro,~s sections by over eight orders of magni tude, the sen,,itivity of fluore,,cence is exceptionally good. Further, unllikc charged-

particle fltnorcscence excitation, the laser-induced fhnorescence [I.IF} lechniquc is energy

eigcnstate specific, which simplifies con,~iderably the rc,,uh ing fluorescence spectra. These

advantage,, of LIF are lessened considerably, hov,.ever, by recognition of the exi,,tcnce of

quenching processes of the excited, radiatnng species, for now the radiation intensity is no

longer a linear function of the local density. Furthcn'naore, the dcgrcc of nonlincanity is a

function of the ntLmber density of each ,~pccies in the saunr~le.

However, it is known thai for sul'ficieruly intcn,,e laser beam source,, inlcr:tcling with

,,iml~le two- and three-energy level sy,,tcm,,, the laser-induced ,,timulatcd emis,,ion

predominates over the 1o,,,~ of cxcilcd slate species by colli.~ional quenching. Con,,cquently,

for such saturation condition,,, collisional quenching is uniml~ortant, and the laser-induced

J'luore,,ce]lCe pro,,.idcs a direct measurcmcxu of the local ,,petit,, dcnsity. Un fort unatly, mo',t

rnolecular specie,, of interest are ilot accuralcl',, dc,,cribcd by .,,imple Iwo- anld thr~:c-lcvcl

syslem,~ but rather arc mt.hi-.,,talc systems. To evaluate tile fca,,ibility and adv.amagc.,, of the

use of laser-induced fluon'cscetlce for such ,,pccics, it is nccc.,,sar.v to tlcscribe analylically the

response of such systems to i~tlen,~e la,,er bean1 sources and IO determine Ihc effects and

imporlancc of collisional qtncnching proce,,,,e.~. This report dc,,cribcs the t c.,,ull,, of a .,,I tidy of

the lillle-dcr~crldenl fluonenccnt .,,ignal of a common]ly encOtmlercd I'~httllc ga', ',pccic,, CO.

During this study it became clear tha! even if saturation conditiorls were not achieved, a

second rnethod of application of LIF was possible if the fluorescent species ~',.as a trace gas

species in a binaary mixture. For a situation in which collisional quenching of the

5

A E DC-T R -80-47

fluorescence of the trace species exists, the fluorescence signal is a dirccl function of the local

density of the ho,,t gas, or major specie% of the mixture; i.e., for a specified excitation

source the fluorescence respon,,e of the trace species is dominated and determined by the

density of the quenching species, and thi., dcpctldcnce should yield a direct measure of the major species density. To investigate the possibility of measurement of the density of the

major species using tracer gas impurity fluorescence, it was decided to focus attention on a

problem area for which such a technique would be useful - - namely, the study of the

boundary-layer properties of a low density N2 nozzle expansion to x~.hich Na is added in vcr).

low concentrations as the tracer gas. The feasibility of applying this technique to such a

problem was investigated by developing a theoretical description of the time-resolved

fluorescence of Na in a N2 host gas enx.ironment, and the results of this ~.ork arc described in this report.

2.0 REVIEW OF LASER-INDUCED FLUORESCENCE (LIF) MODELS

The basic method of measuring gas species parameters by the laser-induced fluorescence (LIIO method is to excite an atom or molecule from the ground electronic state to an excited

electronic state by absorption of visible or ultraviolet laser radiation. The subsequent

fluorescence produced by spontaneous emission back to the ground electronic state is then

measured. The intensity of the fluorescence is proportional to the total number density of

atoms (or molecules) in the upper excited state from which the radiation originates. The

total number density in the original excited ground state is then related to the upper state

number density by an appropriate excitation-emission model. Finally, the total species number density is determined from an appropriate original distribution function over all the

possible energy states which may be populated at the time of laser excitation. For molecular

species, the temperature can be obtained by measuring the relative fluorescence intensities resulting from excitation of two or more ground state rotational-vibrational le,,.els. For

either temperature or number density measurer bnts, the important step in obtaining values of those parameters with LIF is the excitation-emission model used to relate the measured

fluorescent intensities to the original state of the gas species being investigated. The next two

sections present the two most popular models used in LIF analysis, the two-level and three- level models.

2.1 TWO-I,EVEL MODEL

The simplest excitation-emission model is the two-level model shown in Fig. i. In this

model it is assumed that the only important energy exchange processes are those which

transfer energy between the two laser-coupled states. The rate equation for the population of level 2 can be written:

A E DC-TR-80-47

tIN 2 ,l~ - = NI [}12Pv - [Q2 I A2I " [ }21 5 ] N 2 ( i )

The notat ion for this and subsequent equations is described in the Nomenclature .

, 2

orB12 PUB21 A21 Q21

Figure 1. Two-level model schematic.

If it is assurned that at t ime = 0 all of the populat ion is in level 1, then

N. I. = N I+N 2 = N'~ (2)

For steady-state condit ions the rate of change of populat ion in level 2 is zero, and Eq. (I)

can then be rearranged:

Nn N2[ Q21 + '~21 4 (" Bt2 ~ B21 ) pv] (3) I = 13 P 12 t,.'

which gives the ground state number density in terms of known (or measurable) constants

and the number density in level 2, which is proport ional to the fluorescent signal.

If the laser radiation density (Q,,) is large so that (BI2 + Bzt) 0,' ~" Q21 + A21, saturation condit ions exist, and Eq. (2) reduces to

131 , B 2 ~) N o 2 ' 1 N 2 (4)

I = BI k 2 j

which allows deterrnination of the gas number density without the necessity of knowing the

quenching rate Qzl.

Taking Eq. (2) and solving for N2 and recognizing that the fluorescent signal (S) is

related to the upper state number density by

S = C A21 N 2 (5)

7

A E D C-T R -80-47

then

s'~ B12 (6) S = C A21 (021 + A21)/pt, ' + ~Bt2 + B21)

If a binomial expansion about 1/0~ is performed, the following expression is obtained:

2 I o 1 S --- C A21 N~ - N1 (Q21 A21) (7) 1312- 1"121 p v ( B i 2 ... 1321) t- . . .

A plot of S versus I/o,, will ~ive N? as a function of the intercept, and O~ as a function of the slope (for known N~). This "near" saluration technique provides another method for

determining the number density without knowledge of Q21. The two-level near-saturation technique has been used to measure number densities of C2 (Ref. 1), MgO (Ref. 2), CH and

CN (Ref. 3), and Na (Ref. 2) in flames. Additionally, OH number densities have been measured using the two-level model at low-pressure short pulse conditions to eliminate quenching effects (Ref. 4). Finally, Bechtal (Ref. 5) has used the two-level model with quenching effects included for measuring OH concentrations in a flame.

2.2 THREE-LEVEL MODEL

The three-level model, as shown schematically in Fig. 2, is particularly useful when two closely lying electronic states occur as the first excited levels above the ground state, as in

sodium and potassium. For the case when level 2 is excited, the rate equation is

dN 2

Assuming that all of the population is in level 1 before laser excitation, then

N, r : N j + N 2 ~- N~ = N~

At steady-state, Eq. (6)can be rearranged:

(9)

-n ~ (P '21 - 1"312) Pv -"~2 + O21 I - Q I I 2 t Q23]-~2 - I 'Q32 - Q12 - 13 ]2 I°vJ N3 I~] rl12Pv + Qj 2 (10)

If all quenching and coilisional exchange rates are known, then N i ° can be determined from

the total fluorescent signal from levels I and 2. Under saturated conditions Eq. (10) reduces tO

A EDC-TR-80-47

/ "B]2 + B21 i N 2 (!1) N~ = N 3 + '~2

which would allow calculation of N~' from LIF measurements without the necessity of knowing the various collisional quenching and exchange rates. The three-level saturated LIF

technique has been used to measure Na number densities in a flame (Ref. 6).

A31 Q31

QI3

i

PvB12 Q21 A21

orB21 Q12

' r

Figure 2. Three-level model schematic.

3.0 SODIUM-DIATOMIC MOLECULE SYSTEM RELAXATION

3.1 DI'TAILS OF QUENCHING MECHANISM

Experimental (Refs. 7 through 9) and theoretical (Ref. 10) analysis has shown that the

collisional energy exchange collisions between sodium and diatomic molecules involve

energy transfer between the electronic states of sodium and the vibrational state of the molecule. The quenching (Q21, Q31) and reverse quenching (QI2, Qi3) rates in the rate equations for the sodium levels [Eq. (8)] are therefore dependent on the vibrational

population distribution of the quenching molecule. Specifically,

Q21 = N d~ kv~ , V W ~,

012 = ~ N w ~ k(~.,, %%' 't,' ' ~ %V

(12)

where k,.,,, is the rate constant (cnP sec-t) for an upward vibrational transition causing a

downward sodium transilion (quenching), and k',.,, is the rate constant for a dow.nward vibrational transition causing an upward sodium transition (reverse quenching). The upward

and downward rate constants are related by the principle of detailed balancing:

9

A E DC-TR -80-47

~ ~ n [ - ( A E 2 t - . . , ] k,~.~ = k . , , , - aE~)i'k'r g[ (13)

where g2 and g! and the statistical weight of sodium levels 2 and 1 (Fig. 2), AE21 is the sodium level energy difference, and AE~. is the vibrational transition energy.

During an excitation-relaxation process in a sodium-diatomic mixture, the population in the vibrational levels of the quenching molecule may not maintain a Boitzmann distribution because of the relatively slow V-V exchange rates (Appendix A) and unequal sodium- molecule rate constants k,., for different vibrational transitions. Consequently, the values of the total quenching and reverse quenching rates in Eqs. (8) and (10) may vary with time.

3.2 S O D I U M - N I T R O G E N R E L A X A T I O N M O D E L

To check the effects of the possible variation in quenching rates, a computer code was developed to calculate the time-dependent populations in all sodium electronic and molecular vibrational levels during laser excitation of one of the sodium 3S-3P transitions for nitrogen-sodium mixtures.

The rate equations for the ground electronic state (32S1.,.2) and doublet first excited states (32PI/2 and 32p3/2) of sodium are, for excitation of the 32Ph.2 state (level 2):

,I---/- = Ni Ili2 ,, i%, ~,, .~. iV

- N 2 IH2IP~., - A.21 i-N3k23 +Ni.k.~3-1-~N~,, i~_>~ v "vii" ] +N3 [N2k32+ I-X.l" k ,~ 2 ]

d Iltl1,1

d-"-~" = I :~ ,v ~<w

- N . t [ A31 i-N2 k-t2 + ~1. i k~2 + I~.,. N ~w ~,. k ] (14)

10

A E DC-T R -80-47

where k32 and k23 are the P-level exchange rate constants for collisions with sodium atoms in the ground state (32Si/2-1evel I) and k.~2 and k~3 are the P-level exchange rate constants for collisions with nitrogen molecules (total nitrogen number density = N-r). It is assumed that the quenching rate constants (k',.,. and k((.,, are equal for both P-levels, and the reverse quenching rate constants from detailed balancing are calculated [(Eq. (13)] so that for v _< w It k','~. #: k,,,,,.

For a nitrogen vibrational level the rate equation broken into its component parts is

" = -N Ilv depleting VV tit v Nw , v,; v4-1, w-I t- [iv. w, v-l, wa-1

+ NT ~w Ilvw I depleting VT

popular mg VV

l • 4-N T £ Rwv populatingVT

-N" IN1 w~<v(kv'w+kv~v) depleting reverse Q

t-(N 2 t-N 3 ) w>_v~ kwv I depletingQ

w>v populating reverse Q

+ (N 2 + N a ) £ N~ kwv populating Q (15) w< v

The V-V and V-T rate constants are discussed in Appendix A.

if the complete system of sodium electronic level and nitrogen vibrational level rate equations and initial conditions of a Boltzmann distribution in the nitrogen vibrational

I1

A E DC-TR-80-47

levels and all sodium population in the ground electronic state is used, the time-dependent

populations are found by numerically solving the system of equations by the Gear linear

muhistep method (Ref. l I).

Table 1 lists the values of the energy exchange cross sections, along with some of the

other atomic and molecular constants used in this analysis. The rate constantsfor use in the

rate equations are calculated by multiplying the cross section by the relative speed of the

colliding species, which is dependent on the gas temperature. The quenching rate constants

are based on the change in vibrational level (Av) during the quenching collision. A Gaussian-

like distribution of the quenching probability is used, as suggested by the experimental

scattering experiments of Ref. 12. Figure 3 shows the probability distribution used in the

present work. Notice that the peak is placed at v = 3 and v = 4, not at the resonance

condition ( - v = 8). It is assumed that there is no dependence on the initial (or final)

vibrational level. The actual quenching rate constants are calculated by taking the relative

probabilities and, using a Bohzmann distribution at the gas temperature, summing the

products of number density and quenching probability for each vibrational level and then normalizing so that

Y. x, = Q j (16) V %%' > %'

Table 1. Constants for Sodium-Nitrogen Model

Parameter Value Source

A21 = A31

q32

q23

q32

~ ( 3 2 p 1 / 2 -

\ ( 3 2 P 3 / 2 -

32Si/2 )

32S I/2 )

6.13 x

283 A 2

552 A 2

144 A 2

73.8 ~2

5895.9 A 2

5889.9 A

107 sec -1 Ref. 17

Ref. 18

Detailed Balance

Ref. 14

Detailed Balance

Ref. 19

Ref. 19

at 400°K

12

A E DC-TR-80 -47

1 . 0

0 . 9

0 . 8 ;,,*o

r "

2 :-;,. 7

C r... 0 . 6

- 0 . 5 g -3

0 . 4

0 . 3

~" 0 . 2

0 1

0 1 2 3 4 5 6 1 8 9 .tO . ',v

Figure 3. Sodium-nitrogen collisional quenching probability distribution.

where Qr is the total quenching rate (see- I). This rate is calculaled, in turn, from a chosen

total quenching cross section and the relative speed of the colliding species. The quenching

rate constants for the sodium 32p3.,2 and 32p~,,2 levels are assumed to be equivalent. The

reverse quenching rate constants are calculated from the quenching rate constants by

detailed balancing Eq. (13). The reverse quenching-rate constants for the two sodium P

levels w.ill be similar but somewhat different in value because of the 17.3-cm - t energy

difference between the two P levels, and to the different statistical weights of the levels. The I .

sodium-nitrogen total quenching cross sections (q3t = q21) are taken from the review article

by Lijnse and Elsenaar (Ref. 8), which shows an inverse temperature dependence of the total

quenching cross section below about 1,000°K.

Figure 4 presents the rale constants for quenching and reverse quenching collisions

betv,.een the sodium 3P levels and nitrogen for vibrational transitions originating (for

quenching collisions) in the 0th vibrational level, at a temperature of 2,000°K. From the

figure it is apparent that reverse quenchil~g rate,~ are small cotnpared to quenching rates Ibr

the lower v levels but that at the higher v levels reverse quenching collision rates are

comparable to quenchillg rates. As shall be seen in the next section, tiffs relationship between

the quenchilag and reverse quenching rates for different vibrational levels has a ,,Iiong

influence on the time hi,,tory of the laser-excited sodium-nitrogen sy,,tcm.

13

A E D C - T R - 8 0 - 4 7

i0-I0

i0-II

10"12

E

c - o

L 3

¢ 0 P s 10-13

10-14

10-15

m

P I

E ~J ~ .o'"

111 II

I I

t

// I // /

I !

! I

D

o 0-vQuenching

o 3Pl/2, v -*0 Reverse Quenching

z~ 3p312 , v - 0 Reverse Quenching

I I I I I I I

2 3 4 5 6 7 8

V

Figure 4. Quenching and reverse quenching rate constants at T = 2,000=K.

14

A EDC-TR-80-47

The 3P level mixing exchange cross sections for collisions with molecules (q'~, q~2) are

temperature adjusted using an exponential fit to the velocity-dependent cross section data of

Ret'. 13 along with the cro,~s section at 400°K.from Rcf. 14. Figure 5 ~how.~ the cross section.~

and rate constants resulting from this procedure. Also shown in the figure is the rate

constants which would result if a constant 3P mixing cross seclion x~,ere used. The

application of a ternperature correction on the 3P mixing cross section reduces the

temperature dependence of the rate constant so that the rate constants al different

temperature are more nearly equal. The 3P level mixing exchange cross sections for

collisions with 3S sodium atom,~ (q23, q32) are not temperature-corrected because of a lack of

information on the velocity dependence of these cross ~cctions. However, the contribution

of the 3S level collisional 3P mixing exchange rates on the relaxalion in a sodium-nitrogen

mixture i~ small because of the much lower .sodium number dcnsilies compared Io the

nitrogen concentration for the conditions of into, est.

30

25

,~ 20

tO

/ o k - Temperature-Adjusted -'l 240

o k (Based on 144 A 2 at 400°KI ~ , ~ 230

tx Total Cross-Section Iq) / ~ 1 . 3 , 220 /

"\ 1N

o / " ' - . 7 / . . A -J 130

/ --~-- ~ ....... "~ I 120 I I I I I I I I I

100 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000

T (°K)

Figure 5. Sodium 3P level mixing exchange rate constants and total cross sections.

3.3 RI"SUI,TS OF SODIUM-NITROGEN MODELING

For relatively high number densities of nitrogen and sodium (2 x 10 I~) cm --~ and

1.0 x 1013-cm -3. respectively) and a temperature of 1,000°K, the time history of the

15

A E DC-T R-80-47

populations in the three sodium levels and in the nitrogen vibrational levels is shown in Figs. 6 and 7 for a constant laser radiation density (step function on at t = 0) of 1.0 x 10- l0 ergs

cm -3 H z - i exciting the 3P]/2 level. At this laser excitation energy, which is well below the saturation energy, the sodium level populations reach an equilibrium value after about 0.1 nsec, and maintain this constant level until approximately 0.1 msec, when a transition to a

new steady-state level occurs. The vibrational population distribution continually changes as

more and more molecules are put into higher vibrational levels, because of quenching collisions with sodium. The relatively slow V-V rates prevent the vibrational state from maintaining an equilibrium distribution, and eventually there is sufficient change from the original Boltzmann distribution to alter the total quenching cross section and to permit significant contributions from reverse quenching processes. For lower temperatures and/or higher laser intensities the results at these high number densities are qualitatively the same. For higher temperatures this is not the case, as illustrated in Figs. 8 and 9 for identical conditions to those in Figs. 6 and 7 except for a temperature of 2,000°K. At this temperature there is no transfer to a second steady-state condition. Despite the fact that there is a significant change in the vibrational population distribution, this change is not large enough to cause the transfer to a second steady-state condition. The vibrational population distribution attains a steady state after approximately 1 msec, so that there is no possibility of a change at times longer than those for which the calculation was performed.

.J

¢-

¢ r,

-i

q~ ® N -,d

¢d

E 0

Z

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Nitrogen

Run Number 58

Mol. N.D. = 2.000 x 1019 cm -3

Sodium N.D. = 1.000 x 1013 cm -3

Temp. = l,O00°K

Num. Vib. Levels = 11

3Pl/2 Excited

Laser Rad. Den. = 1.0 x I0 -I0

Total Q = 20 Sq. AnE.

\

j -12.0 -11.0 -10.0 -9.0

Figure 6.

J

- 8 . 0 - 7 . 0 - 6 . 0 - 5 . 0 - 4 . 0 - 3 . 0 - 2 . 0

Log T i m e , s e c

Time history of population in sodium levels for high density, T = 1,000°K.

3S

3P3/2

'3Pl/2

1

-1.0

16

A E D C - T R - 8 0 - 4 7

0

- 3 . 0

- 6 . 0

"~ t = 1.118 x 10 -6 s e c

Nitrogen

Run Number 58

Mol. N.D. = 2.000 x 1019 cm -3

Sodium N.D. = 1.000 x 1013 cn -3

Temp. = 1,000°K

Num. Vzb. Levels = 11

3Pl/2 Excited

Laser Rad. Den. = 1.0 x i0 -I0

Total Q = 20 Sq. Ant.

4~

.j

z

t.

O

q

0

-4,0

-8.0

O

-5.5

-11.0

0

- 7 . 0

- I d .0

0 = o

-i.0

4~

,-4

-2.0

°

~ - 1 . 0

~ - 2 . 0 0

= . _ o~ 0

: -2.0

-d.0 0 2 4 6 8 10

t = 1 . 0 8 7 s e c

1: = 1. 104 x 10 - 2 s e c

0 2 4 6 8 10

V

Figure 7. Time history of population in nitrogen vibrational levels for high density, T = 1,000°K.

i

At lower number densities a still different picture emerges. Figure 10 shows the time

history results for nitrogen and sodium number densities of 2.0 x 10 ~7 and 1 x 1010 era-a , a

temperature of 2,000°K and laser radiation density of 1.0 x 10-10 ergs cm -3 H z - I . The

steady-state condition is not attained until after about 7 nsec with the 3PI/2 level showing a

pronounced peak at about 0.2 nsec. This behavior occurs because o f the dependence of the

3P level mixing exchange rates on the nitrogen number density. For these lower number

densities the 3P3/2 populating collisions occur at a reduced rate so that the 3P3/2 level

population process will lag the laser-excited 3Pi/2 level population. At these conditions the

3P1/2 level attains a number density higher than its steady-state value before collisions with

nitrogen, causing a transfer to the 3P3/2 level to become effective. After about 0.2 nsec this

transfer process begins to depopulate the 3P1,,2 level until the steady-state condition is

obtained. At lower temperatures (Fig. ! l) the same picture emerges with the exception that

at very long times ( - 0. I sec) a transition to a second steady-state condition occurs, because

17

A E DC -T R -80-47

4J

f-

e"

E

0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0. i

0 -12.0 -11.0 -10.0

N i t r o g e n

Run Number 59

H o l . N.D. = 2 . 0 0 0 x 1019 cam - 3

Sodium N.D. = 1.000 x 1013 ~ - 3

Temp. = 2 , 0 0 0 ° K

Num. V~b. L e v e l s = 11

3 P l / 2 E x c i t e d

L a s e r Rad. Den. = 1 . 0 x 10 - 1 0

T o t a l Q = 20 Sq. Ang.

- 9 . 0 - 8 . 0 - 7 . 0 - 6 . 0 - 5 . 0 - 4 . 0 - 3 . 0 - 2 . 0 - 1 . 0 O

Log Txmo, sec

3S

3P3/2

'3Pl /2

Figure 8. Time history of population in sodium levels for high density, T = 2,000°K.

of the redistribution of populat ion in the vibrational level,, of nitrogen (Fig. 12). It takes a

longer time at t he,~e condit ions to attain sufficient populat ion in the higher vibrat tonal levels

to cause the transfer to a second steady state than for the higher number density condit ions.

It should also be noted Ihat at the same la,~er radiation density the ,~ystem is,c'loser to

saturation (Ni = N2) at the lower nitrogen densities because of a reduction in the total quenching rate.

In summary , this analysis has shown that there can, in fact, be a ,~ignificant effect on the relaxation of a sodium-ni t rogen laser-excited system attributable to redistribution o!"

vibrational populat ion by quenching collisions. However, this effect occurs at sufficiently

long times so Ihal, for short laser pulse nncasurennenl,, (_< I #~ec) the ,,ready-state relation,,

[Eq. (10)] for a three-level system are applicable. On the other hand, extremely short pul,,e.,,

may not be long enough for ,~teady ,~tate to be obtained uulder .,,onne condition.,, becau,,c of the finite 3P level exchange collision rate.

18

0

-2.5

-5.0

t = 1 . 0 7 0 x 1 0 - 6 s e c

Nitrogen

Run Number 59

Mol. N,D. = 2.000 x 1019 cm -3

Sodimn N.D. = 1.000 x 1013 cm -3

Temp. = 2,000°K

Num. Vlb. Levels ~ ii

3 P l / 2 Excited

Lauor Rad. Den. = 1.0 x i0 -I0

Total Q - 20 Sq. Ang.

¢~ - 3 . 5 - 1 . 5 = • "

- 7 . 0 ~ -3.0

~' 0 ~ 0

-7.0 ~ -3.0

o ° = • -- i,

-3.5 -2.0

-7.0 -4.0

0 2 4 6 8 I0 0 2 4 6 8 I0

V V

Figure 9. Time history of population in nitrogen vibrational levels for high density, T = 2,000°K.

~> m

c)

-11

do O J~ -,J

b.3

4~

0)

0;

"O 0; N -,4

E O

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Nitrogen

Run Number 50

Mol. N.D. = 2.000 x 10 -17 cm -3

Sodium N.D. ffi 1.000 x I0 I0 cm -3

Temp. = 2,000°K

Num. Vib. Levels ffi ii

3 P l / 2 E x c i t e d

L a s e r Rgd. Den. ffi 1 . 0 x 10 - 1 0

T o t a l Q = 20 Sq . Ang.

3 P 3 / 2

,3S '3Pl 12

~> m tD O

"11

do O

~j

0.1

0

-12.0 -11.0 -i0.0

Figure 10.

-9.0 -8.0 -7.0 -6.0 -5,0 -4.0 -3.0 -2,0 -I.0

Log Time, sec

Time history of population in sodium levels for low density, T -- 2,000°K.

t ,J

1 . 0

0.9

0 .8

0 .7 ,,,H tO

.E

N ,F,I r - I

o z

0.6

0.5

0.4

0.3

0.2

0.1

3P l / 2

Nitrogen

Run Number 48

Mol. N.D. = 2.000 x 1017 cm -3

Sodium N.D. = 1.000 x i0 I0 cm -3

Temp. = 300°K

Num. Vib. Levels = 11

3P1/2 ExciCed

Laser Rad. Den. = 1.0 x 10 -10

Total Q - 45 Sq. Ang.

'2

0 -12.0 -11.0 -10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0

Log Time, sec

- 1 . 0

Figure 11. Time history of population in sodium levels for low density, T = 300°K.

~> m

c~

~0

O

~J

t ~ t,J

0

- 6 . 0

- 1 2 . 0 t - 1 . 0 1 1 x 1 0 - 6 s c c

, 4

-7.0

t = 1 . 0 4 9 x I0 -8 sec -14.0

0

o z - 8 . 0

° \ x I0 -I0 s e c -16.0 ..-

0

1 0 . 5

21.0 t - 1 . 1 3 1 x 1 0 - 1 2 s e c

0 2 4 6 i0

4J

g

0 z

~0 o

- 6 . 0

- 3 . 0

0

0

- 3 , 5

- 7 . 0

0

- 4 . 5

- 9 . 0

Nitrogen

Run Number 48

Mol. N.D. = 2.000 x i0 J7 cm -3

Sodlum N.D. = 1.000 x i0 I0 cm -3

Temp. = 300°K

Num. V1b. Levels = ii

3 P i / 2 E x c i t e d

L a s e r R a d . D e n . = 1 . 0 x 1 0 - 1 0

T o t a l Q = 4 5 S q . A n g .

\ t = 1 . 0 2 8 s e c

t - 1 . 0 4 6 x 1 0 - 2 s e c

0 2 4 6 8 10

V

Figure 12. Time history of population in nitrogen vibrational levels for low density, T = 300°K,

~> m o c)

-n & 0

~4

A EDC-TR-80 -47

4.0 SINGLE DIATOMIC SPECIES RELAXATION

4.1 BREAKDOWN OF TWO-LEVEL MODEL ASSUMPTIONS

Real molecules contain complex rotational and vibrational structure which may cause

significant breakdown of the assumptions in the tv,.o-level LIF model. Collisional energy

exchange collisions during the laser excitation pulse will transfer energy between the laser-

coupled vibrational-rotational states and the other vibrational-rotational state~ of the upper

and lower electronic states at a finite rate. Additionally, spontaneous emission from the

laser-populated upper level will occur not only to the laser-excited lower level, but also to

other vibrational levels of the ground electronic state. Quenching collisions will also produce

transitions to lower electronic vibrational-rotational states other than the laser-excited one.

The possible effects of these processes will be:

I. Time-dependent populations and fluorescent intensities

2. Non-Bohzmann vibrational and rotational population distributions

3. Time dependent total number density in laser-coupled states [breakdown of Eq.

(2)1.

Lucht and Laurendeau (Ref. 15) have shown that when the ratio of the rotational relaxation

rates of the upper and lower electronic states to the quenching rate are 100 or less, the

steady-state rotational state population distributions are significantly non-Boitzluann. To

investigate the steady-state vibrational population distribution and the time history of the

rotational and vibrational distributions, a computer code has been constructed which

follows the detailed relaxation of a diatomic molecule during laser excitation. The next

section discusses the model used in the computer code.

4.2 SINGLE DIATOMIC SPECIES REI,AXATION MOI)EL

For the case of a single diato,nic species in which one particular vibrational-rotational

(V-R) level in the ground electronic state of the rnolecule is pumped by laser radiation to one

particular vibrational-rotational level in the first excited electronic state of the rnolecule, the

following processes need to be considered:

1. Laser excitation and stimulated emission

2. Fluorescence from all upper eleclronic V-R states

23

d

A E D C-TF1-80-47

3. Collisionai exchange between rotation and translation (R-T)

4. Collisionai exchange between the vibrational modes of collision partners (V-V)

5. Collisional exchange between vibration and translation (V-T)

6. Collisional quenching, i.e., nonradiative transitions from upper to lower

electronic state

7. Radiative transitions between V-R levels of the same electronic state

(heteronuclear molecules only)

8. Self-absorption of fluorescence

It is assumed that the gas is optically thin so that process 8 can be neglected. Additionally,

the radiative lifetime for transitions between V-R levels of the same electronic state is

typically much greater than the time scales for processes l through 6, so process 7 will also be

neglected. Including processes I through 6, the rate equation for the lower electronic level, laser-excited V-R state is then (with its component parts specified):

dN 1 (~", .l") F

_ _ ( v . ; .,-, L ' ' dl

+ ~..j NI (v){l:l~.. ,, v.._j., v_t+Rv.., v:,.._], v+1 } , , depletingV'i,;

J

stimulated emission

N 2 (v', J') B21 a~,

V J

I N I I v ' " ' ] " ) I~F~/Nl(V) IRv" . - I .v ; v " , v - ] ,

i | lv"-I, ~, v", vLL I

laser absorption

radiation and quenching from upper electronic level

populating VV

24

A E D C - T F~ - 8 0 - 4 7

-,- { ,,,.,., J

populating VT & RT

07)

Where v" and J" denote the laser-excited lower electronic state vibrational and rotational

level, v' and J ' denote the laser-excited upper electronic state vibrational and rotational

level, N~ and N2 are the lower and upper electronic state number densities respectively, NT is

the total gas number density, B~2 and B2j are the absorption and simulated emission

coefficients, 0,, is the laser radiation energy density, A the radiation rate (Einstein

coefficient), Q the quenching rate, and the R's are the various collisional exchange rate

constants (cm-a/sec) which are identified by their subscripts. R,.w is the V-T rate constant

for transfer fv'om vibrational state v to vibrational state w ' . Ru; is the R-T rate constant tbr

for transfer from rotational state J to state J ' , and Rv+ ~.w': v, w+l is the rate constant for

V-V exchange for molecules in vibrational states v + l and w' colliding and exchanging

vibrational energy to end up in states v and w + I. For V-T and V-V processes, lay1 = i

transitions only are considered, but for R-T processes all AJ's are included in the analysis.

The sum over J for the A and Q terms is limited to allowed dipple transitions only, i.e., P,

Q, and R branch transitions in the case of CO. The quenching rates are based on the value of

the Einstein A coefficients for the various vibrational-rotational transitions.

The rate equation for the laser-excited upper electronic V-R state is

,INT. r,v" .I')

dt v, J)-.' Q (v', .I ", v, .1!)

~. N 2 (v ) (R , , , . ,,, , , '4 I . , , -] - R v ' . v ; , "-t . , , + ] ) V

J P

+ N 2 (,,'. J ") I,~. N2 (,') (ll, . ,+l" ,.; ,.'. ,._]

• n.,.,,} 1 .

- ' Rv,_1 " ,,, ~ ". ~ t l )

(]8)

Tile rate equations for the other V-R levels in the upper and lower electronic states are the

sa*ne as those for the laser-coupled states withoul the absorplion and stimulated emission ICrlllS.

25

A E D C-TR -80-47

4.3 CARBON MONOXIDE MODELING RESULTS

The total number o f vibrational-rotational levels used in the relaxation model was

restricted by computer time and core usage limitations; it was not possible to use a large

number o f rotational and vibrational levels simultaneously. The analysis was therefore

divided into two parts. The first part used 31 rotational levels and 3 vibrational levels to

investigate the rotational relaxation, and the second part neglected rotational relaxation and

used l0 vibrational levels in each electronic state. For both test cases a step function laser

pulse was applied at time t = 0 and a constant laser radiation density was maintained for the

duration of the run time. Excitation is from a rotational level in the 0 th vibrational level of

the X ' [ ; + ground electronic state o f CO up to one particular vibrational-rotational state in

the A ' H first excited electronic state. The absolute transition probabilities (Einstein A

values) of Ref. 16 are used for the various vibrational transitions from the upper to the lower electronic state.

The time history of the rotational population distributions in the upper and lower laser-

coupled vibrational levels is shown in Figs. 13 and 14 for atmospheric pressure conditions at

300°K and a laser radiation density o f i.0 x l0 -8 ergs cm -~ H z - i, which is well under the

saturation radiation density. The results agree qualitatively with those in Ref. 17. The

distributions are highly non-Bohzmann with an overpopulation in the upper electronic state

laser-pumped rotational level and an underpopulation in the lower electronic state laser- pumped rotational level.

C a r e e n ~ o n o x l d e

T o t a l G a s N , O . ~ 2 . 4 4 8 x 1 0 1 9 cm - 3

Gas " l ~ m p . = 3 0 0 ° K

~ ¢ , R o t , L e v e l s - 3 1

~ u m . V l b . L e v o l s = 3

J = 6 V - 0 t o J - 5 V ffi 1 E x c t t a t x o s

L a s e r R a d . D u n . = 1 . 0 x 1 0 - 8

Q - 1 0 0 x A

Uppor H l o c . L o v o l

0.5

0 6 12 1 8 24

t = 1 . 0 3 1 x 10 - 4 s e e

t ffi 1 . 0 2 2 x 1 0 - 6 s e e

t = 1 . 0 0 7 x 1 0 - 8 s e c

t = 1 . 0 2 5 x 1 0 - 1 0 s e c

, t - 1 . 0 7 5 x 10 - 1 2 s e c

3 0

B o L a t t o n a m Q ' Ja .~ tum N u m . ~ o r , J

Figure 13. Time history of rotational population distribution in lower electronic state laser-coupled vibrational level.

26

A E D C - T R - 8 0 - 4 7

C a r b o n M o n o x i d e

T o t a l Gas N . D . - 2 . 4 4 8 x 1 0 1 9 cm - 9

O~s Temp. = 3 0 0 ° K

Num. R o t . L e v e l s = 31

Num. Vzb. Levels = 3

J = 6 V = 0 t o J = 5 V = i E x c ~ t a t l o n

L a s e r R a d . D e n . = 1 . 0 x 1 0 - 8

q = 1 0 0 x A

L o w e r E l u c . L e v e l A A

A 0.5

o 0 0 6 12 18 24 30

Rotational Quant~ Number, J

A t - 1 . 0 3 1 x 1 0 - 4 s o c

t - 1.022 x 10 -6 sec

t = 1 . 0 0 7 x 10 - 8 s e c

t = 1 . 0 2 5 x l 0 - 1 0 s e e

t = 1.075 x 10 -12 sec

Figure 14. Time history of rotational population distribution in upper electronic state laser-coupled vibrational level.

For pure vibrational relaxation, Fig. 15 shows the time history of the number density of

the upper electronic state laser-excited vibrational level for room temperature, atmospheric

pressure conditions with three values of the quenching rate (Q). For all cases the population does not come to steady state until very long times ( - 0 . 1 see). The initial drop in number

density (around 0.1 nsec for Q = 100 x A) depends strongly on the quenching rate and is

attributable to the depletion of population from the upper laser-excited state by quenching

and radiative relaxation to the various ground electronic state vibrational levels where the

finite V-V rates prevent fast enough collisional exchange to refill the depleted laser-excited

vibrational level. Figures 16 and 17 show the time history of the vibrational population

distributions in the upper and lower electronic states for Q = 100 x A. Highly non-

Boltzmann distributions exist at all times during the relaxation. V-V coilisional exchange

tends to smooth out the vibrational population distribution in the ground electronic state,

with the most pronounced effects occurring after about I msec. At a temperature of 2,000°K, (Fig. 18) the time history is qualitatively similar to that at 300°K except for a higher

absolute steady-state number density at an earlier time. This effect is attributable to the

much higher V-V and V-T rates at the higher temperature.

27

Oo

? == o

.El

S

o ,.3

20.0

18.5 / / ~

17.0

15.5

14.0

12.5

11 ,0

9 . 5

8 . 0

6 . 5

5 . 0 -12.o - l i . o - lo.o -9.o

C a r b o n Monoxi d e

Tot=L1 N.D. = 2 . 4 4 8 x 1 0 1 9 cm - 3

Gas Temp. = 3 0 0 ° K

Num. V t b . L e v e l s = 10

V - 0 t o V = 1 E x c i t a t i o n .0 10_8

~ Q = tO x A

q = 1 0 0 x A

- 8 . o - ~ . o -8".o 5;0 -4;0 -3.o -2".o -1".o o 1~o Log Tzme, sc,~.

Figure 15. Time history of population in upper electronic state laser-coupled vibrational level for three values of quenching rate.

~>

m O C~

-n

~o o J= ~4

I'J

~J ,d e- r~

F. -j

0

0 ,-1

0

-2.0

- 4 . 0 t/ /

I

t = 1 . 0 3 6 x 1 0 - 6 s e c

0

- 2 . 0

- 4 . 0

0

- 2 . 0

- 4 . 0

~ C .

C = o -4.0 ~ e

-8,0

0 2 4 6 8 10

Carbon Monoxide llun Number 57 Total N.D. - 2.448 x 10 ]9"~n Gas Temp, = 300°K Num. Vtb, Levels = 10 V = 0 to V - 1 Excitation

Laser Had. Den. = 1.0 x 10 -8

Q ffi 100 x A

Lower Elec. Level

- 6 . 0

- 1 2 . 0

t = 1 , 0 4 5 s e c

o_

~ O ~ c ~ - 6 . 0

- 1 2 . 0 i . . . . 0

-5 .0

- 1 0 . 0 4 . . . .

0 2 4 6 8

V V

Figure 16. Time history of vibrational population distribution in upper electronic state, T -- 300°K.

10

~> m

o

io d0 0 j~ ~4

0

-7 .0

-14.0

0 4~ ,r4 m

= -7.0

-14.0

z 0

0 z -7 .0

0 ,-.I -14.0

-7.0

-14.0

~ 1 . 0 3 6 x 10 - 6 soc

= ~ = 1.054 x 10 -8 sec

- ~ x , 10 -10 s e c

. ~ 1 . 0 3 9 x 10 - 1 2 s~c

0 2 4 6 8

Figure 17.

I0

Carbon Monoxzde

Run Number 57

Total N.D. = 2.448 x 1019 cm -3

Gas Temp. = 300°K

Num. Vzb. Levels = i0

V = 0 to V ~ 1 Excitation

l,aser Rad. Den. = 1.0 x 10 -8

q = I00 x A Uppor Elec. Level

=~ -7.0 ~

'~ -14.0

e~

~ -7 .0 Z

sk -14.0 0 Z

3

-7 .0

-14.0

0 2

t = 1.045 s e c

t = 1 . 0 3 6 x 10 - 2 sec

t = 1 . 0 6 8 x 10 - 4 sec

.1 6 8 10

V V

Time history of vibrational population distribution in lower electronic state, T -- 300°K.

]> m

o o

~0

o & -4

t ~

2 0 . 0

1 8 . 5

1 7 . 0

~uE 1 5 . 5

14.0

,IQ E , j

o ,...1

1 2 . 5

1 1 . 0

9 . 5

8 . 0

6 . 5

5 . 0 - 1 2 . 0 - 1 1 . 0 - 1 0 . 0

Ca rbon Monoxlde

Run Ntunber 64

Total N.D. = 2.448 x 1019 cm -3

Gas Temp. = 2,000°K

Max. N.D. ffi 7.720 x 1018 cm -3

Num. V1b. Levels = I0

V = 0 to V ffi i Excltation

l,aser l{Rd. Den. = 1.0 x 10 -8

Q ffi 100 x A

- 9 . 0 - 8 . 0 - 7 . 0 - 6 . 0 - 5 . 0 - 4 . 0 - 3 . 0 - 2 . 0 - 1 . 0

Log T i m e , s e c

Figure 18. Time history of population in upper electronic state laser-coupled vibrational level, T -- 2,000°K.

1 . 0

m

-i1

o & ~J

A E DC-TR-80-47

The effects of laser radiation density on the upper vibrational level number density are

presented in Fig. 19 and Table 2. In the figure is plolted the peak number density in the

upper vibrational level normalized by the steady-state number density predicted by the two-

level model for both 300°K and 2,000°K. At the lower temperature the peak number density

is equal to the two-level model prediction for laser radiation densities greater than l0 -6 erg

cm 3 Hz l . The ratio falls quickly around 10 ~ erg cm --~ Hz -I and again levels out at a lower

lower ratio for radiation densities less than 10 -10 erg cm -3 Hz -I For the higher

temperature case the same pattern emerges, except that at the saturated conditions the ratio

is below a value of one. In Table 2 the steady-state time and number density are essentially

constant for all the laser radiation densities tested at 300°K. At 2,000°K the time of steady

state is virtually constant but the number density does show a reduced value at the lower radiation densities.

in summary, for saturated laser cxcitation of carbon monoxide the peak of the number

density in the upper laser-excited vibrational level is comparable to the two-level prediction

at room temperature. For lower radiation densities, for high temperatures, or for prediction

of the time history and steady-stale conditions, the two-level model fails completely because

of the finite collisional exchange rates and large contribution of multipath radiative and

collisional relaxation from the upper electronic state vibrational manifold to lhe ground

electronic state vibrational levels.

.~ 0 . 9

~ 0 . 8

I f . f " ~ , 0 . 7 , . o ~

¢ O. 6 s SS

~z 0 . 5

I- ," I

N 0 . 3

~ 0 . 2

• ~ O. 1

" ' 0

_O - E P " " " Z f ' 0 300°K Q 2, O00°K

I I I I I I I I - 1 2 - 1 1 - 1 0 - 9 - 8 - 7 - 6 - 5

Log ( 0 ~ )

Figure 19. Laser radiation density effects on ratio of calculated peak number density and two-level model steady-state prediction.

I - 4

32

Table 2. Maximum and Steady-State Number Densities and Times

t~J

v Maximum Steady State Maximum Steady State

10 -4

10 -6

10 -8

i0 -I0

10 -12

I. 625 x 1019 -12

1.0x I0

1.610 x 1019

-12 2.0x I0

9.879 x 1018

7.0 x I0 -II

3. 809 x 1017

5.0 x I0 -I0

4.044 x 1015

] .0 x 10 -9

8.447 x 108

0.2

8.447 x 108

0.2

8.447 x 108

0.2

8.446 x 108

0.2

8.405 x 10 8

0.2-~ 0.3

1.275 x 1019

1.0 x 10 -12

I. 264 x 1019

-12 2.0x I0

7. 720 x 1018

7.0 x 10 -12

2.987 x 1017

5;0 x I0 -I0

3. 172 x 1015

1.0 x 10 -9

3. 160 x 1014

5.0 x 10 -6

3. 160 x 1014

5.0 x 10 -6

3. 158 x 1014

5.0 x 10 -5

3.023 x 1014

5.0 x 10 -6

2.069 x 1014

7.0 x 10 -6

Number Density (cm -3) Time After Laser Radiation Initiation (seconds)

m

o o

o

A E DC-T R-80-47

5.0 SUMMARY

The objective of this work v,,as to assess the feasibility of using the laser-induced

fluorescence technique to measure space-and time-resolved species number densities in

gaseous systems. Specifically, computer codes were developed to model the detailed kinelic

rate relaxation of Na atom--N2 molecule and CO molecule systems to evaluate the

usefulness of the simple two- and three-energy level models in deriving number densities from fluorescence intensities.

For sodium-nitrogen mixtures the time-depcndence of the fluorescence intensity was

shown to be dependent upon both the total N2 number density and the N2 ,,.ibrational mode number density distribution function. Specifically, it was shown that a metaslable steady-

state fluorescence intensity of Na was attained within 0.1 - 10 nsec after initiation of the

laser source radiation. Further, the metastable steady-state fluorescence intensity for the three-level Na model was dependent upon the N2 number density, thereby providing a means

of measurement of the local N2 number density using Na as a fluorescent tracer. As time

increases, the effects of vibration-vibration and vibration-translation energy transfer

processes in the N2 assume added importance, and a transition occurs in the fluorescence signal from the metastable state to a final steady state. For the parameters used in this study

it was seen that this transition occurred approximately ! msec after initiation of the laser

source radiation. Consequently, for laser pulses sufficiently long in duration to attain the

metastable steady-state condition, and shorter than the time of transition to the final steady- state condition, the laser-induced fluorescence technique using the three-level model appears

to be feasible for measuring nilrogen, or other similar diatomic molecule, number densities

with sodium as a tracer. Experimental measurements under controlled conditions are recommended to validate further the applicability of this technique.

For molecular systems the preliminary resulls using pure carbon monoxide have shown

the vital importance of multipath relaxation from the upper laser-excited level. For carbon

monoxide the two-level model is totally inadequate for predicting the sleady-stale

populations. Consequently, the laser-induced fluorescence technique based on the two-level model cannot be used with any degree of reliability for carbon monoxide number density

measurements. Furthermore, these results make it clear that it is highly desirable to extend

the computer modeling techniques developed here to more realistic muhispecies molecular

systems and to species of interest, such as OH, in order to evaluate the applicability of laser- induced fluorescence to these systems.

34

A E OC-TR -80 -47

R E F E R E N C E S

. Baronavski, A. P. and McDonald, J. R. "Application of Saturation Spectroscopy to

the Measurement of C2 3Hu Concentrations in Oxy-acetylene Flames." Applied Optics, Vol. 16, No. 7, July 1977, pp. 1897-1901.

. Pasternack, L., Baronavski, A. P., and McDonald, J. R. "Application of Saturation Spectroscopy for Measurement of Atomic Na and MgO in Acetylene

Flames." Journal of Chemical Physics, Vol. 69, No. 11, December 1978, pp.

4830-4837.

. Bonczyk, P. A. and Shirley, J. A. "Measurement of CH and CN Concentrations in

Flames by Laser-Induced Saturated Fluorescence." Combustion and Flame, Vol.

34, 1979, pp. 253-264.

. Stepowski, D. and Cottereau, M. J. "Direct Measurement of OH Local Concentration in a Flame from the Fluorescence Induced by a Single Pulse."

Applied Optics, Vol. 18, No. 3, February 1979, pp. 354-356.

5. Bechtel, J. H. and Teets, R. E. "Hydroxl and Its Concentration Profile in Methane- Air Flames." Applied Optics, Vol. 18, No. 24, December 1979, pp. 4138-4144.

6. Daily, J. W., and Chan, C. "Laser-Induced Fluorescence Measurement of Sodium in Flames." Combustion and Flame, Vol. 33, 1978, pp. 47-53.

7. Starr, W. L. "Excitation of Electronic Levels of Sodium by Vibrationaily Excited Nitrogen." Journal of Chemical Physics, Vol. 43, No. 1, July 1965, pp. 73-75.

. Lijnsc, P. L. and Elsenaar, R. J. "The Temperature Dependence of the Quenching

of Na-D-Doublet by N2 and H20 in Flames of 1500-2500 K." Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 12, 1972, pp. 1 i 15-1128.

. Krause, H. F., Fricke, J., and Fire, W. L. "Excitation of Na D-Line Radiation in

Collisions of Sodium Atoms with Internally Excited H2, D2, and N2." Journal of Chemical Physics, Vol. 56, No. 9, May 1972, pp. 4593-4605.

10. Bauer, E., Fisher, E.R., and Gihnore, F. R. "De-excitation of Electronically Excited

Sodium by Nitrogen." Journal of Chemical Physics, Vol. 51, No. 10, November 1969, pp. 4173-4181.

35

A E DC-TR -B0-47

11. Gear, C. W. Numerical Initial Value Problems in Ordinary D([ferential Equations. Prentice Hall, Englewood Cliffs, N. J., 1971.

12. Hertel, i. V., Hofmann, H., and Rost, K. J. "Electronic to Vibrational Energy Transfer in the Differential Scattering of Na* by N2 Molecules." Physical Review Letters, Vol. 36, No. 15, April 1976, pp. 861-863.

13. Apt, J. and Pritchard, D. E. "Velocit}. Dependence of Fine-Structure Changing of

Na with Ne, Ar, Xe, N2 and CO2." Journal o[Physics B: Atomic and Molecular Physics, Vol. 12, No. 1, 1979, pp. 83-98.

14. Stupauski, M. and Krause, L. "Inelastic Collisions Betx~.een Excited Alkali Atoms and Molecules V. Sensitized Fluorescence in Mixtures of Sodium with N2, H2, HD,

and D2." Canadian Journal o f Physics, Vol. 46, No. 19, October 1968, pp. 2127-2131.

15. Lucht, R. P. and Laurendeau, N. M. "Two-Level Model for Near Saturated Fluorescence in Diatomic Molecules." Applied Optics, Vol. 18, No. 6, March 1979, pp. 856-861.

16. Mumma, M. J., Stone, E. J., and Zipf, E. C. "Excitation of the CO Fourth Positive Band System by Electron Impact of Carbon Monoxide and Carbon Dioxide."

Journal of Chemical Physics, Vol. 54, No. 6, March 1971, pp. 2627-2634.

17. Kibble, B. P., Copley, G., and Krause, L. "Effect of Imprisonment of Radiation in Sodium Vapor o11 Measured Lifetime of the 32p States." Physical Review, Vol. 153, No. 1, January 1967, pp. 9-12.

18. Pitre, J. and Krause, L. "Sensitized Fluorescence in Vapors of Alkali Metals, X. Energy Transfer in Sodium-Sodium Collisiorts." Canadian Journal o f Physics, Vol. 46, 1968, pp. 125-128.

19. Kuhn, H. G. Atomic Spectra. Academic Press, New York, 1961.

36

A U

Bu

C

E

gi

k

k U

Ni

NT

Qu

QT

qlj

R~.l;~..w

Rij

S

T

t

V-V

A EDC-TR-80-47

N O M E N C L A T U R E

Einstein spontaneous emission coefficient rate for i-j transition

Radiative absorpt ion coefficient (i < j), st imulated emission coefficient

(i > j )

Constant proportional to viewing volume and optical parameters [(Eq. (,5)]

Energy

Statistical weight of level i

Bol tzmann's constant

Rate constant involving energy states i and j

Number density of level i

Total diatomic gas species number density

Quenching (i > j) or reverse quenching (i < j) rate

Total quenching rate

Cross section for i-j collisional exchange

V-V rate constant for transition from states i, j to states v, w

R-T or V-T rate cortstant for i-j transition

Fluorescent signal

Kinetic temperature

Time

Vibral ional-,,.ibrat ional t raqnit ion,,

37

A E DC-TR-80-47

V-R

V-T

V~W

Q~

Vibrational-rotational transition

Vibrational-translational transition

Vibrational quantum level

Wavelength

Excitation radiation energy density

38

A E D C - T R - 8 0 - 4 7

A P P E N D I X A

M O L E C U L A R E X C H A N G E RATE C O N S T A N T S

1.0 I N T R O D U C T I O N

The theoretical formulat ion of Herzfeld and Litovitz (HL) (Ref. A-I) for collisionai

energy exchange attributable to short range interactions is employed to calculate molecular

V-V and V-T exchange rate constants. Their expressions yield values for the V-T and V-V

transition probabilities (P) based on the kinetic (translational energy exchange) rate constant

(R t) at a given temperature, so that the V-T or V-V rate constant will be

the kinetic collision rate (Rt) is

I1 -- pri~ (A-l)

lit I - (2rr k ' l ' /~ = 7,_ t~ c r = d ~" \ ~ / ( A - 2 )

where d is the coilisional interaction radius (molectflar diameter), ~, is the average relative

velocity of Ihe colliding molecules, o the collision cross section, and ~ is ttle reduced mass.

2.0 V I B R A T I O N A L E X C H A N G E

2.1 V-T PROBABII . IT IFS

Following tile HL analysis, tile V-T transition probability is separated into three componen ts

] ] I ( A - 3 ) P V I ' ~ Z . Z.,~,_- Z tr"

Zo is a geometrical factor estirnated by HL to be 3. Single quan tum vibrational transitions only are considered in the HL formulat ion. For the general case of vibration exchange in

molecule B-C attribulable to a V-T collision with molecule A,

z M c ) M., I ( M n t- rr 2 0__ (A-4) Z,,...c: M B M C (M~ .. M H ,-- M C I O"

v, here M.x, MB, and Mc are the molecular weighls of molecule A, a tom B, and a tom C respectively, 0 is the characteristic temperature of the vibrational transition,

__ IF:] - F'21

( A - 5 )

39

A E D C-T R -80-47

for vibrational energies El and E2, and finally

16rr4 : ~ 2 v2 O" =

k

where m is the reduced mass of the system

(A-6)

M A M B (, = - (A-7)

M A t- .MHC

p is the frequency corresponding to the vibrational transition (v = ErE2 /h ) , k is

Boltzmann's constant and I'is lhe range of repulsive forces between the colliding molecules. Next

' .,-' E' (-:-)" 'J exp - - + - - (A-8~ Ztr"

for translational temperature T.

The final form of the HL V-T probability is

J ",. _J I J Y ( 2 , 2 ) e "~/'kr l - e "0/T -J (A-9 ) PV'I ' I 017 Zo 7t, sc Ztr"

where rc/ro is the ratio of the stopping point (re) for a molecule coming from infinity and

undergoing an inelastic collision and the Lennard-Jones constant (to), Y(2,2) is the

"Suther land correct ion" factor

,.,.,..,>: 0.,6 (, .,., # ) (A-IO)

and ~ is the Lennard-Jones potential well depth. Table A-1 lists the values for the constants

used in the sodium-nitrogen and carbon monoxide relaxation codes.

2 .2 V - V P R O B A B I L I T I E S

The V-V probability' is similarly divided into three components:

I l I Pvv = P~-. ,,-t - (A-I I)

Zo Z,)sc Zt~ w = ~ w + I

40

A E DC-TR -80-47

Table A-1. Constants for V-T, V-V and R-T Calculations

Par ame ter Va lu e Source

o

rc/r o

--/k £

F

C

3.75

1.0

i 91 .5

2.117 X 10 -9

0.15

0.10

Ref. A-4

Approximation

Ref. A-I

Ref. A-I

w

See Appendix Section 3.0

where

z,, "2 "~ "3- l ? kl" 2':i 7 (A-12)

with

0 ~ 2 -: 16~r* ~l'-' k P'~ (0j -02) 2

h 2 (A-13)

Note that the third term in the exponential, because of the anharmonicity of vibrational

motion, accounts for the difference between endothermic and exothermic collisional transfer, i.e., detailed balance effects. In the actual calculation of l /Ztr for both V-T and V-V probabilities, exothermic probabilities only are calculated using the full equations, with

the corresponding reverse probabilities calculated using the detailed balance relation

Pcn = Pc,, e-aE/kT where AE is the energy given up to translation during the collision because of the mismatch in vibrational energies during the V-V exchange collision. Now,

Substituting for O'

I -- "1,,74 ( A - 1 4 ) Zasc 4MI'I Mr: I 4MH MC 2 O! O;

and O and simplifying yields

(2 2) %t B + s,1 C

Z.sc I

, , ,2 • 4h4 (A-15)

~ '~-~r~ :'~ / ,6 ~r 4 ~2 "' (t!:, - L 2 , (I.:, - r. 2 )2

41

A E DC-T R -80-47

Zo, sometimes referred to as the steric factor, is the term which adjusts for the fact that the

theoretical analysi~ ~.hich producc~ tile mathematical expressions for Z~, and Zo," neglecls the rotational motion of the colliding molecules. HL suggest a value of Zo = 9.

Figure A-I shows the V-V and V-T transition probabilities at various translational

temperatures for nitrogen. For temperatures below approximately 1,000°K the V-T

probabilities for the lower vibrational levels are negligible compared to the V-V

probabilities. Despite the smallness of the V-T probabilities it was necessary to include V-T

transfer in the relaxation code to obtain steady-state conditions at a finite time.

For carbon monoxide it is known that long range dipole interactions play a significant

role in V-V exchange processes for the lower vibrational levels (Refs. A-3 and A-4). This

effect tends to increase the V-V probabilities calculated by the HL method by approximately

two orders of magnitude at a temperature of 300OK for collisions involving the lower

vibrational levels (v < 10), although the shape of the probability distribution is also altered

so that the deviation from the HL probabilities is not constant for each transition.

Nonetheless, for the purposes herein, which are primarily qualitative, the HL V-V

probabilities multiplied by 100 are used as representative of the true V-V probabilities for carbon monoxide.

3.0 R-T RATE CONSTANTS

The exponential gap law as proposed by Polanyi and Woodall (Ref. A-2) is used to

calculate the R-T probabilities. For a downward transition from J to J ' the transition probability is

P.l.,, = F ' e x p [ - C ( E . , - F ; j , ) / k T J (A-16)

where F is the normalization constant, and C is the parameter governing the exponential decrease in the probability with increasing energy gap AE = Ej - E j,.

Table A-I gives the values o f the constants in Eq. (A-16) which were used in the present

analysis. These values are somewhat arbitrary because of the lack of any experimental values

for the R-T exchange rates in CO. They are chosen so that the probability distribution variation with AJ and J approximates that found for HC1 in Ref. A-2. The value of P j j, for

J = J ' (AJ = 0) is adjusted so that the total probability is equal to one.

42

A E D C - T R - 8 0 - 4 7

Note All Temperatures in DeErees Kelvin

V - V I

1 0 . 2 =- V - T . . . . i

I

,o_,

,o, ~---- \ \ ~ i-~ ,ooooj

10_5

. \ " ~c~ = ~oo° 7 ~o-6 == ..-" \ \ /

10-7

10 -8 ///// //// ~, / //

I /

zo -9 ,',/,,Ivtn,l,,,,l~'~,,I,,,,lll==lvl=l I 0 5 10 15 20 25 30 35

V

Figure A-1. Transition probability diagram.

43

A E D C -T F1-80-47

REFERENCES

A-I. Herzfeld, K. F. and Litovitz, T. A. Absorption and Dispersion of Ultrasonic Waves. Academic Press, New York, 1959.

A-2. Polanyi, J. C. and Woodall, K. B. "Mechanism of Rotational Relaxation." Journal of Chemical Physics, Vol. 56, No. 4, February 1972, pp. 1563-1572.

A-3. Jeffers, W. Q. and Kelley, J. D. "Calculations of V-V Transfer Probabilities in CO-CO Collisions." Journal of Chemical Physics, Vol. 55, No. 9, November 1971, pp. 4433-4437.

A-4. Lordi, J. A., Falk, R. J., and Rich, J. W. "Analytical Studies of the Kinetics of Electrically Excited, Continuously Operating CO Flow Lasers." AIAA Paper No. 74-563, June 1974.

NOMENCLATURE

C Exponential parameter [(Eq. (A-16)]

c'r Average relative velocity

d Collision interaction radius (molecular diameter)

Ei Energy of level i

F Normalization constant [Eq. (A-16)]

h Planck's constant

k Boltzmann's constant

e Repulsive intermolecular force range

M, Molecular weight of species i

6a Reduced mass

P Transition probability

44

Rt

r¢

ro

T

Y(2,2)

Zo,Zosc,Ztr

O,

012'

O"

Collision rate defined by Eq. (A-2)

Stopping point in molecular collision

Lennard-Jones constant

Temperature

Sutherland correction factor

Components of vibrational transition probabilities

Lennard-Jones potential well depth

Characteristic temperature

Term defined by Eq. (A-6)

Term defined by Eq. (A-13)

Frequency

Collisional cross section

A E D C-TR -80-47

45