A THEORETICAL ASSESSMENT OF VIBRATICNAI To '
ROTATIONAL ENERGY EXCHANGE. IN THE HYDROGEN
ELMCRIDE CHEMICAL LASER

Dissertation for the Degree of Ph. D.
MICHIGAN STATEIUNIVERSITY
ROBERT CLINTON BROWN
1980

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ABSTRACT
A THEORETICAL ASSESSMENT OF VIBRATIONAL TO

ROTATIONAL ENERGY EXCHANGE IN THE
HYDROGEN FLUORIDE CHEMICAL LASER

BY

Robert Clinton Brown

Computer models for the H2+F2 chain reaction laser
oscillator and laser amplifier are deve10ped. Both models
include comprehensive formulations of chemical reaction
kinetics thought to be important to the HF laser. The models
employ vibrational to rotational energy transfer and
rotational nonequilibrium in an attempt to eXplain experi—
mental observations of rotational lasing and P-branch lasing
from high J states.

Vibrational to rotational energy exchange is found
important for the prediction of rotational lasing. The effect
of rotational lasing was found to be kinetically similar to
rotational relaxation. The effectiveness of the VR mechanism
in populating high rotational states is strongly dependent on
its relative contribution to the total kinetic rate.

The effect of different partitions between VT and VR
in the VR,T mechanism was considered. Increased VT produced
smaller predicted pulse duration and laser power. The major
effect of VB is on rotational levels above J=lZ. The prOper
choice of product rotational states for VR reactions was
suggested from simulations of laser fluorescence experiments
designed to detect VR. It was found that a VB reaction which

populated rotational states approximately 80% of a

Robert Clinton Brown
minimum energy defect VR reaction best simulated experimental
results.

Increased J-dependence for rotational relaxation rate
coefficients was effective in sustaining larger nonequili-
brium populations at high J levels but the effect was not
as pronounced as an overall decrease in rotational relaxation
rate. The prediction of excessive energy in the hot bands of
HF appears to be the result of insufficient vibrational deac-
tivation from these levels. Both increasing the endothermic
cold pumping reactions and the vibrational dependence of VR
rates were effective in decreasing hot band lasing. It was
found that near resonant VR reactions produce significant RV
rates and strong coupling between vibrational deactivation and
rotational relaxation was demonstrated. This coupling will
decrease the overall vibrational deactivation rates and VR

rate coefficients larger than v may be required to properly
describe vibrational deactivation.

Although the model compares favorably with experimental
studies for the lower three P-branch bands, excessive P-branch
lasing occurs for high J levels in the hot bands. Insuffi—
cient experimental data on the pulsed H2+F2 chemical laser
exists to reconcile theory with experiment.

The laser amplifier was used to assess the complex time
and space dependence of laser amplifier devices. It was
demonstrated that pulsed chemical amplifier performance is
complicated by the competing processes of chemical pumping

and saturation effects. Saturation of the amplifying medium

was demonstrated for sufficiently high input powers.

A THEORETICAL ASSESSMENT OF VIBRATIONAL TO
ROTATIONAL ENERGY EXCHANGE IN THE

HYDROGEN PLUORIDE CHEMICAL LASER

BY

Robert Clinton Brown

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1980

ACKNOWLEDGMENTS

My wife, Carolyn, performed the greatest number of roles
during the years of my graduate education and the writing of
this dissertation. She not only provided the inspiration and
enthusiasm frequently attributed to spouses of graduate
students but engineered the completed dissertation manuscript.
To her fell all the responsibilities of editing secretary and
draftsman. Her proficiency in these skills complemented my
lack of them.

I am also indebted to my thesis adviser, Dr. Ronald
Kerber, both friend and critic during four rewarding years
of study. In both capacities he was unfailing and I thank
him. I am extremely grateful for the time and interest of
Dr. John McGrath, Dr. Jes Asmussen, and Dr. Richard Schwendeman
who served on my faculty committee.

I acknowledge the valuable assistance of Pam Detine in
analyzing the quantities of data derived from my research.

To Warren Jaul goes the thanks reserved for a comrade—
in—arms.

I am grateful for the support of both my parents, Forest

and Beth Brown, and my parents-in—law, John and Dorothy Kay.

TABLE OF CONTENTS

Page
1. INTRODUCTION . . . . . . . . . . . . . . . . . . 1
1.1 History of the Hydrogen Fluoride Laser 1
1.2 Computational Studies of Chemical Lasers 4
1.3 Objectives of Present Research . . . . . 7
2. COMPUTER SIMULATION . . . . . . . . . . . . . . 19
2.1 General Model Formulation . . . . . . . . . 19
2.2 Comparison with VT Model . . . . . . . . . 29
2.3 Effect of Rotational Lasing . . . . . . . . 32
2.4 Effect of VR,T Energy Transfer . . . . . . 34
2.5 Effect of Rotational Relaxation . . . . . . 47
2.6 Vibrational Deactivation . . . . . . . . . 55
2.7 Comparison with Experiment . . . . . . . . 73
2.8 Effect of Initial Conditions on
Laser Performance . . . . . . . . . . . . . 93
3. AMPLIFIER SIMULATIONS: TEMPORAL AND SPATIAL
DEPENDENT MODEL . . . . . . . . . . . . . . . . 106
3.1 Introduction . . . . . . . . . . . . . . . 106
3.2 Chemically Pumped Amplifier Simulations . . 109
3.3 Simulation of Laser Fluorescence
Experiments Designed to Detect VR . . . . . 119
4. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . 126
APPENDIX A: RATE COEFFICIENTS FOR H2 + F2
CHEMICAL LASER . . . . . . . . . . . . 134
APPENDIX B: DERIVATION OF THE EQUATION FOR
CALCULATING EMPIRICAL QUENCHING
COEFFICIENTS . . . . . . . . . . . . . 137
REFERENCES . . . . . . . . . . . . . . . . . . . . . 140

iii

LIST OF TABLES

Table Page
1.1 Rate coefficients at 300 K for vibrational

to vibrational energy transfer for

HF(v) + HF(1)2HF(V+1) + HF(0) . . . . . . . . . 15

2.1 Relative rotational relaxation efficiencies
for several chaperon gases . . . . . . . . . . . 25

2.2 Single quantum VV rates of Wilkins (58) for

_. __ T I

HF(vl,Jl) + HF(v2,J2)._.HF(vl l,Jl ) + HF(v2+l,J2 ) 26
2.3 Possible VR relaxation distributions about J min

forHF(v,',J)+M2HF(vJ')+M......... 36
2.4 Abstraction reaction rate coefficients . . . . . 57
2.5 Effect of VR product rotational states on

predicted empirical quenching coefficients . . . 74
2.6 Comparison of model predictions with experiment . 79

iv

Figure

2.1

LIST OF FIGURES

Page

Vibrational to rotational energy transfer
channels. Multiquanta relaxation is
permitted . . . . . . . . . . . . . . . . . . . 28
Comparison of predictions of present VR
model with those of the VT model. Relative
lasing energy for every transition in the
vibrational bands is illustrated. This
format is used in all figures of relative
lasing energy that follow.
Gas Mixture:

0.02F:0.99F :1H :20He; Ti=300K, Pi=20 torr
Cavity Conditions:

Ro=0'7' RL=1.0, L=10cm, £=lOcm
(a) Standard VT model (Reference 24)
(b) VT model using Wilkins' (58) relaxation rates
(c) Present VR model . . . . . . . . . . . . . 31
Effect of rotational lasing on relative
P-branch lasing energy
Gas Mixture:

0.02F:O.99F :lH :ZOHe; Ti=3OOK, Pi=20 torr
Cavity Conditions:

RO=0.7, RL=1.0, L=10cm, £=10cm . . . . . . . 34
Effect of product rotational distributions on
relative band energy of P-branch lasing
Gas Mixture:

0.02F:0.99F :1H2:20He; Ti=3OOK, Pi=20 torr
Cavity Conditions:

Ro=0.7, RL=1.0, L=10cm, £=10cm . . . . . . . 37
Effect of VR,T and RR,T rates on laser energy
Gas Mixture:

0.02F:0.99F :lH :20He; Ti=3OOK’ Pi=20 torr
Cavity Conditions:

RO=0.7, RL=1.0, L=10cm, i=10cm
(a) Total P-branch lasing energy
(b) Total rotational lasing energy . . . . . . 40

Figure Page

2.6 Rotational pOpulation distributions predicted
by VR model at peak power of the laser pulse
Gas Mixture:
0.02F:0.99F :1H2:20He; Ti=300K, Pi=380 torr
Cavity Conditions:
RO=0.7, RL=1'O’ L=10cm, £=10cm . . . . . . . 42

2.7 Effect of increased VT contribution for VR,T
reactions on relative lasing energy
Gas Mixture:
0.02F:O.99F :lH :20He; T.=300K, P.=20 torr
Cavity Conditions: 1 l
Ro=0.7, RL=1.0, L=10cm, £=lOcm . . . . . . . 45

2.8 Effect of increased VT contribution for VR,T
reactions on rotational populations
Gas Mixture:
0.02F:0.99F :1H :20He; Ti=300K, Pi=20 torr
Cavity Conditions:
Ro=0'7’ RL=l.0, L=10cm, £=lOcm . . . . . . . 46

2.9 J-dependence of rotational relaxation rate
coeffiCients. O O I O O O O O O O I O O O O O O 50

2.10 Effect of rotational relaxation J-dependence
on rotational populations
Gas Mixture:
0.02F:0.99F :lH :20He; Ti=3OOK, Pi=20 torr
Cavity Conditions:
Ro=0'7’ RL=1.O, L=10cm, 2:10cm . . . . . . . 52

2.11 Effect of J-dependence of the rotational
relaxation rate coefficients on rotational lasing
Gas Mixture:
0.02F:0.99F :lH :20He; T.=300K, P.=20 torr
Cavity Conditions: 1 l
Ro=0'7' RL=1.0, L=10cm, £=10cm . . . . . . . 54

2.12 Effect of abstraction reaction on predicted
P-branch lasing energy
Gas Mixture:
0.02F:0.99F :1H2:20He; Ti=300K' Pi=20 torr
Cavity Conditions:
Ro=0’7’ RL=1.0, L=10cm, i=10cm . . . . . . . 58

2.13 Effect of pressure on band energy distribution
' Gas Mixture:
0.02F:0.99F :lH :20He; Ti=300K
Cavity Conditions:
Ro=0'7' RL=l.0, L=10cm, i=10cm . . . . . . . 62

vi

Figure Page

2.14

2.15

2.17

2.19

Dependence of vibrational deactivation on
vibrational quantum number . . . . . . . . . . . 65

Model calculations of empirical quenching
coefficients . . . . . . . . . . . . . . . . . . 69

Rotational populations during multiquanta

VR deactivation of vibrational level v=4

after 0.5usec

Gas Mixture:
1 torr HF vibrationally equilibrated except
for 1% distributed in v=4; all levels
initially rotationally equilibrated.

(a) v=0; maximum pOpulation in rotational
distribution is 1. 4x10 8 mole/cm3

(b) v=2; maximum population in rotational
distribution is 6. 4x10“ '12 mole/cm3

(c) v=4; maximum population 11D rotational
distribution is 8.3x10 mole/cm3 . . . . 72

Time resolved spectra of Parker and Stephens
experiment. The length of a line indicates
the duration of a transition with lower level
v,J and the dot indicates the time of
maximum power.
Gas Mixture:

0.05F:0.95F :lH :10He:0.2502;
Cavity Conditions:

Ro=0'8' RL=1.0, L=15cm, £=60cm . . . . . . . 81

T =300K, P =36 torr
1 1

Time resolved spectra of model predictions
Gas Mixture:

0.05F:O.95F :lH :10He:0.2SN2;
Cavity Conditions:

Ro=0'8' RL=1.0, L=15cm, £=60cm . . . . . . . 83

T.=300K, P =36 torr
1 1

Comparison with eXperiment: distribution of
band energy
Gas Mixture:
0.05F:0.95F :1H :10He:0.25N2;
Cavity Conditions:
RO=0.8, RL=1.0, L=15cm, l=60cm . . . . . . . 85

Ti=3OOK, Pi=36 torr

vii

Figure Page

2.20

2.21

2.

2.

22

.23

24

Effect of uncertainty in initial conditions

on distribution of band energy. Gas and
cavity conditions identical to those in Figure
2.19 except as noted.

(a) Initial gas mixture includes 1% HF

(b) Initial F/F2 ratio increased;
0.07F:0.97F2:1H2:10He:0.25N2

(c) Threshold gain increased 25% by decreasing
RL to 0.76 . . . . . . . . . . . . . . . . . 86

Effect of gas pressure on predicted laser
performance
Gas Mixture:
0.02F:0.99F :lH :20He; T.=300K
Cavity Conditions: 1
RO=0.7, RL=1.0, L=10cm, £=10cm

(a) Total P—branch lasing energy
(b) Maximum P-branch lasing power . . . . . . . 96

Effect of initial HF concentration on peak power
and total energy
Gas Mixture:
0.02F:(0.99-0.11x)F :(l-O.llX)H2:20He:O.22xHF
where x=SHF; T.=300 , Pi=20 torr

'Cavity Conditions:

Ro=0.7, RL=1.0, L=10cm, R=10cm . . . . . . . . 97

Effect of initial HF concentration on time to
maximum power and pulse duration
Gas Mixture:
0.02F:(0.99-0.1lx)F :(1-0.1lx)H2:20He:0.22xHF
where x=%HF; T.=300 , P.=20 torr
Cavity Conditions: 1
RO=0.7, RL=1.0, L=10cm, £=10cm . . . . . . . . 98

Effect of initial F=atom concentration on peak
power and total energy
Gas Mixture:
2x 2
§:§F:§:;F2:1H2:20He
where x=F/F2; Ti=300K’ Pi=20 torr
Cavity Conditions:
RO=0.7, RL=l.0, L=10cm, £=10cm . . . . . . . . 100

viii

Figure

2.25

3.2

3.3

Effect of initial F-atom concentration on time
to maximum power and pulse duration
Gas Mixture:
ééiF:§%;F2:1H2:20He
where x=F/F ; T.=300K, Pi=20 torr
Cavity Conditions:
RO=0.7, RL=1.0, L=10cm, £=10cm . . .

Effect of threshold gain on peak power and
total energy
Gas Mixture:

0.02F:0.99F :1H :ZOHe; T.=300K, P.=20 torr
Threshold gain is effected Ey changifig either
R0 or L; RL is assumed to be 1.0. . . . . . . .

Effect of initial Hz/F2 on peak power and
total energy
Gas Mixture:

0.02 l x -
1.02+xF:l.02+xF2:—_-1.02+xH2:20He; Where X'Hz/Fz
T.=300K, P.=20 torr

Cavity Conditions:
Ro=0'7’ RL=1.0, L=10cm, £=10cm .

 

Time and space dependent power for a pumping
dominated chemical laser amplifier
Gas Mixture:
0.02F:0.99F2:1H2:20He; Ti=3OOK’ Pi=380 torr
Input Pulse:
1.0W/cm2 of 2 nsec duration; no entrance time
delay . . . . . . . . . . . . .

Time and space dependent power for a saturation
dominated chemical laser amplifier
Gas Mixture:
0.02F:0.99F2:1H2:20He; Ti=300K, Pi=380 torr
Input Pulse:
1x105 W/cm2 of 2 nsec duration; 2 usec
entrance delay . . . . . . . . . . . .

Time dependence of power for saturation dominated

3cm chemical laser amplifier

Gas Mixture:
0.02F:0.99F2 2:

Input Pulse:
1x105 W/cm2 of 2 nsec duration; 2 nsec
entrance delay . . . . . . . . . . , . . , . .

:1H 20He; Ti=300K, Pi=380 torr

ix

Page

101

103

105

112

114

115

Figure

3.4

3.6

Dependence of saturation on input power
Gas Mixture:

0.02F:0.99F :lH :20He; T.=300K, P =380 torr
Length of amplifier = 0.1 cm, pulse duration
0.2 nsec; 0.5 nsec entrance time delay . . . .

Effect of entrance time delay on pulse
amplification
Gas Mixture:

0.02F:0.99F2:1H :20He; Ti=300K, P'1=380 torr
Input Pulse: 2 nsec duration . . . . . . . .

Simulation of laser fluorescence eXperiment to
detect VR

Gas Mixture: 0.1 torr HF at 300K

Input Pulse: 442 W/cm2 of 40 nsec duration .

Page

117

120

. . 124

l . INTRODUCTION

1.1 History of the Hydrogen Fluoride Laser

 

The first hydrogen fluoride (HF) chemical laser was
reported by Kompa and Pimentel (1) in 1967 using flash
photolysis of a SF6-H2 mixture. About the same time,
Deutsch (2) successfully produced lasing in a similar
mixture using electrical discharge. Both initiation methods
dissociate F-atoms from SF6 which then reacts with H2 to
form excited state HF. Although the resulting HF is in the
electronic ground state, the molecules are highly excited
vibrationally and rotationally and, under suitable condi-
tions will produce lasing in the infrared. Improvements
since that time have resulted in an efficient and powerful
laser pumped by the exothermic formation of HF.

The great advantage of the HF chemical laser is its
ability to sustain pumping through a chemical chain reaction.
A mixture of hydrogen and fluorine gas combine through the

chain reactions:

F + HZZHFWJ) + H (AH -3l.7 kcal/mole) (1.1)

H+F ZHF(V,J) +F (AH

2 -97.7 kcal/mole) (1.2)
where the vibrational energy level of HF is denoted by v
and the rotational energy level is denoted by J. The result—

ing laser power and spectral content are highly dependent on

2
the distribution of HF molecules over these vibrational
and rotational states. Although both reactions of the chain
are extremely exothermic, the exothermicity of reaction (1.2)
is three times that of reaction (1.1). Reactions (1.1) and
(1.2) shall be referred to as the cold and hot pumping reac—
tions, respectively, as is suggested by this difference. The
HF lasers of References l and 2 are not chain reaction lasers
and develop lasing energy only from the cold pumping reaction.
The first reports of H2 + F2 chemical lasers operating on
this chain were made in 1969. Batovskii (3), using electri-
cal discharge initiation, and Basov (4), using flash photo—
lysis, achieved chain reaction sustained lasing in HF.

The lasing characteristics of the HF chemical laser are
strongly dependent on the method of initiating the chain and
the gas mixtures employed. Initiation is the production of
a finite concentration of F-atoms to start the cold pumping
reaction. The amount and production rate of F-atoms strongly
effects laser energy and pulse duration. The existance of
ground state HF from prereaction in the initial gas mixture
can change the distribution of band energy. Optical losses
from mirrors and diffraction within the laser cavity deter—
mine the number of lasing transitions. Nevertheless, sever-
al quantitative features can be noted. Because of the
exothermicity of the pumping reactions, the purely chemical
contribution to the lasing energy can be very high. Unlike
Optically or electrically pumped lasers for which all lasing

energy is developed from external energy sources, the chemical

3
laser only uses Optical or electrical energy to initiate
chemical reactions which produce the balance of lasing
energy. Therefore, very high electrical efficiencies can
be obtained. For example, flash pumped solid state lasers
are typically less than 5% efficient and CO2 lasers may reach
25% efficiencies. Initiation efficiencies for chemical lasers
exceeding 170% have been reported by Greiner et a1. (5)
utilizing relativistic electron beams and by Kerber et a1. (6)
using electrical discharge initiation. Large volume initia-
tion has the potential of producing very high energy pulsed
systems. The ability of electron beam initiation to deposit
large, uniform quantities of energy over large cavities has
led to output energies of 2500J (5). Gain may reach thres-
hold simultaneously on a number of the many vibrational transi-
tions in HF resulting in output spectra rich in multiline
lasing. Lasing on all bands up to 6-5 have been observed,
peaking in the 2-1 band. Although a number of chemical lasers
have been developed, the diatomic hydrogen halide systems
are usually preferred. The reactions producing them generally
have the advantage of high exothermicity and the lasing species
have large electric dipole moments favorable to stimulated
emission (7,8). Of the hydrogen halide lasers, the HF laser
has the greatest exothermicity and is preferred for this
reason. For example, the cold pumping reaction of the hy—
drogen chloride chemical laser is actually endothermic.
Unfortunately, the atmospheric transmission characteristics

of the HF laser are poor, limiting its usefulness for a

4
number of applications. Despite this drawback, development
of HF lasers continues for several reasons. The kinetically
similar deuterium fluoride laser, although developing signi-
ficantly less power than the HF system, lases in a transmis-
sion window of the atmosphere. Because of the general avail-
ability of hydrogen gas and HF kinetic data compared to their
deuterated analogs, the largest effort is with the HF system.
An understanding of the HF system is important to the devel—
0pment of the DF system. In addition, the superiority of
the HF system in applications not dependent on atmospheric
transmission or for purposes of transferring energy to other
molecules is sufficient reason for its continued development.

1.2 Computational Studies of Chemical Lasers

 

The great expense of experimental investigations of
chemical lasers has lead to extensive computer modeling
studies of these systems. Modeling efforts also can easily
isolate a particular mechanism in the large number of reactions
occurring in a chemically reacting system and determine its
contribution to laser output characteristics. In addition,
the models develoPed for laser simulation can be used in
studies of specific reaction kinetics. Experimentally, HF
lasers have successfully been used to extract kinetic data
important for the HF system. Berry (9) and Pimentel and
coworkers (10) have deduced vibrational distributions for
the cold pumping reaction of HF by measuring the relative
gain coefficients of individual HF laser transitions. Sim-

ilarly, detailed computer models can determine the sensitivity

 

Vt

V«

as.
Q»

5
of predicted laser performance to changes in rate coefficients
and characterize the behavior of proposed mechanisms. Dif-
ficulties arise in choosing physically acceptable assumptions
for the models and determining appropriate kinetic rate co—
efficients.

The earliest theoretical models (8,ll-16) of the H + F

2 2
chemical laser made two prominent assumptions about the system.
The first was that lasing began when gain reached threshold
and that gain remained clamped at threshold for the duration
of lasing. It has been shown that this constant gain assump-
tion minimizes the effect of very fast nonlinear vibrational
to vibrational energy exchange reactions (13,15). Hough and
Kerber (l7) relaxed this assumption and allowed gain to rise
above threshold to allow more accurate modeling of nonlinear
deactivation mechanisms. They discovered that lasing did not
commence upon gain reaching threshold. Instead, lasing
effectively occurred only after gain was well above threshold.
Because of increased sensitivity to nonlinear deactivation
mechanisms, shorter pulses were predicted.

The second widely accepted assumption was that, although
vibrational nonequilibrium existed, rotational populations
remained in Boltzmann equilibrium distributions at the trans-
lational temperature. The chemiluminescense experiments of
Polanyi and coworkers (18,19) demonstrated that the nascent
rotational populations produced by the pumping reactions
(1.1) and (1.2) were not Boltzmann distributed. Nevertheless,
it was believed that rotational relaxation was fast enough to

equilibrate any departures from rotational equilibrium on a

6
time scale small compared to other kinetic mechanisms. This
assumption precluded simultaneous lasing within vibrational
bands and instead predicted rigid J—shifting patterns (13).
Experimental data indicate significant deviation from this
pattern and demonstrate definite multiline lasing (20-23).
The inclusion of rotational nonequilibrium in models by
several investigators (17,24-33) has shown that this equili—
brium assumption was a poor one. The effects of rotational
nonequilibrium are to increase the number of simultaneous
vibrational lasing transitions and lower the intensity and
increase the duration of each transition (17).

The inclusion of rotational relaxation mechanisms neces-
sarily complicates the models. The models given in References
(17,24-29) assume rotational nonequilibrium arises from chem-
ical pumping of rotational energy levels and P-branch lasing
transitions. Rotational levels are assumed to relax toward
equilibrium by simple rotational to translational (RT) energy
exchange reactions given by

HF(V,J) + MzHF(v,J-AJ) + M (1.3)
whereIAJis the change in rotational quantum number. In the
model of Kerber and Hough (17), the rate coefficient for
this reaction is

k

1/(TR[M]) (1.4)
where [M] is the concentration of the collisional species
and the rotational relaxation time, TR, is assumed to be of
the form suggested by the experimental results of Polanyi
et a1. (34). Polanyi et a1. observed two distinct peaks in

rotational distributions during rotational relaxation. After

Pr1

it!

7
considering several relaxation models, they concluded that
the bimodal relaxation was best described by an exponential
model of the form:

7k(v,J) = PRZHF_Me-B‘AE/kT (1'5)
where PR is the probability of rotational relaxation and
ZHF-M is the binary collision frequency for unit concentration
of HF with M. When rotational relaxation occurs the rotation-
al quantum number changes from J to J'lfiJ. The change in
energyIAE due to rotational relaxation is

AB = E(v,J) - E(v,J-AJ) (1.6)
where E(v,J) is the rotational energy of vibrational level v.
The parametric constant B approximates the decrease in tran-
sition probability with increasing J.

Despite the successes of these rotational nonequilibrium
models, there remain several discrepencies between theory and
experiment. In addition, there is growing evidence that vi-
brational-to-translational energy transfer does not adequately
explain vibrational relaxation in the hydrogen fluoride system.

These problems are addressed in this study.

1.3 Objectives of Present Research

 

The existing rotational nonequilibrium models predict
excessive P-branch lasing energy and do not give accurate
predictions of time resolved laser spectra despite qualita-
tive improvements in their capabilities.

Models to date have not included rotational energy levels
much above J=12 because of the apparent absence of mechanisms
to populate higher levels. The infrared chemiluminescence

(arrested relaxation) measurements by Polanyi et al. (18,19)

8
of nascent rotational distributions, although strongly non-
equilibrium, suggest rotational pumping is limited to rota-
tional levels less than 14 and typically peaking at J=7.
Polanyi and Woodall (18) found the mean fraction of avail-
able energy entering into vibration, rotation, and transla-
tion are 0.66, 0.08, and 0.26, respectively, for the cold
pumping reaction. Polanyi and Sloan (19) have investigated
the nascent distributions produced by the hot pumping reactions
and found the mean fraction of energy entering vibrational,
rotational, and translational modes to be 0.53, 0.03, and 0.44,
respectively. These results compare favorably with the theore-
tical calculations of several researchers (35-38). The models
also do not account for the possibility of pure rotational
band lasing. There is evidence that these omissions might
be important phenomena in the HF laser system. Several re-
searchers (39-41) have observed P-branch lasing from J levels
higher than are known to be pOpulated by pumping reactions.
Krogh and Pimentel (40) have observed emission from transi-
tions as high as Pl(20)' P2(16), and P3(18). Although the
emissions were achieved from flash photolysis of ClFx-Hz-Ar
(x=l,3,5) mixtures, excited state HF was produced by the
cold pumping reaction of equation (1.1). Krogh and Pimentel
noted that these high J transitions were characterized by
late threshold times and long lasing durations. Furthermore,
increasing total pressure decreased time to threshold and
brought high J transitions to threshold. Observations by
Sojka et a1. (41) of lasing on P4(l7), P5(15) and P6(12) are

the highest J levels for P-branch lasing from hot bands of HF

9
reported to date. Lasing on pure rotational bands of HF have
been observed by several researchers (42-49). Hinchen, et a1.
(49) using an infrared double resonance experiment with a
separate HF pump and probe laser measured rotational gains in
excess of l cm-l. Hinchen found it necessary to account for
rotational lasing in calculations of collisional relaxation
times. Sirkin et al. (43) observed rotational lasing from
levels as high as HF(v=0,J=3l) and HF(v=1,J=30) by photolysis
of halogenated olefins. Cuellar et al. (47) note that the
early threshold times for rotational lasing in HF formed
through photolysis of CH3CF2—Ar mixtures can be explained
from direct pumping of high rotational levels, but conclude
that additional mechanisms may play a role in sustaining the
laser emissions after photolytic pumping has ended.

The relative importance of rotational lasing is suggested
by the indirect method of Chen et al. (45) in detecting
rotational lasing in an atmospheric pulsed HF laser. They
placed black polyethylene which absorbs light with a wavelength
less than 15 microns over an energy meter. Assuming that the
P-branch lasing energy would be stopped by the polyethylene,
they observed that approximately 10% of the pulse energy was
transmitted. This observation suggests that rotational lasing
could be a significant fraction of the total laser power.

The occurence of lasing from very high rotational levels
indicates not only that these high rotational levels must be
included in qualitative models but that a nonequilibrium
mechanism pOpulating J levels well above that expected for

pumping must be identified. It is possible that these rotational

10
levels are populated directly by the chemical chain, but the
slow relaxation times expected by the model of Polanyi et al.
(34) make it unlikely that they would not be detected by
arrested relaxation methods. Krogh and Pimentel (40) also
rule out this possibility based on the pattern of P-branch
lasing observed from high J levels. They note that threshold
times for lasing from high J levels is substantially greater
than for low J levels. High J levels reach threshold and
continue to lase well after the cessation of flash pumping.
A potential mechanism to explain these phenomena is vibrational
to rotational, translational (VR,T) energy transfer.

Until recently, vibrational relaxation was assumed to
occur with the rotational levels in thermal equilibrium with
the translation temperature (50). This assumption results in
energy defects of several thousand cm"1 for typical vibrational
relaxation reactions. Vibrational to rotational energy transfer
assumes that part of the vibrational energy defect for the
reaction goes into rotational energy of the product states. If
the product rotational state is chosen to minimize the energy
defect, very high rotational states result with a very small
contribution to the translational energy mode. The near
resonance of the VR reactions might also explain the very fast
self-deactivation rates measured for HF. The fast vibrational
deactivation measured in fluorescence studies by Airey and
Fried (51) and the theoretical analysis of Shin (52) have
been ascribed to VR transfer rather than VT relaxation. The
role of rotational motion in the deactivation of vibrationally

excited diatomic molecules has been of increasing interest to

ll
investigators and a number of theoretical investigations have
been made (53-58). Kelly (53) found the importance of VR
energy transfer to be significant in heavy-atom light rotor
interactions suggesting its importance in HF. A dipole-
dipole and hydrogen bond interaction model by Shin (56,57)
predicted that VR energy transfer was more efficient than VT
energy transfer.

The three-dimensional classical trajectory calculations
of Wilkins (58) provide the first multiquanta VR,VV, and
rotational relaxation rate coefficients suggesting the impor-
tance of the VR mechanism in HF—HF interactions. Wilkins
employed a London-Eyring-Polanyi-Sato (LEPS) potential energy
function for short range interactions in combination with a
point—charge dipole-dipole potential energy function for long
range interactions between the four atoms comprising HF-HF
collisions. The energy surface is capable of supporting the
HF dimer concept suggested by other investigations (56,57)
but Wilkins concludes that dimers are not formed for typical
collisions at or above 300 K (58). Wilkins makes several
observations important to modeling VR kinetics in the HF
chemical laser. The trajectory calculations indicate that
the vibrational energy of the excited incident HF molecule
is transferred into rotational energy of the same HF molecule,
implying that little or no change occurs in the internal
energy of the target HF molecule. Furthermore, very high
rotational states of the incident HF molecules are pOpulated
by single and multiple quanta VR reactions, resulting in energy

defects much smaller than the corresponding VT reaction in

12
which the energy defect is given up to the thermal bath. The
results of Wilkins have successfully been applied by Wilkins
and Kwok (59) to predict the temperature dependence of self-
deactivation of HF(vzl).

The determination of vibrational to vibrational (VV) energy
exchange rates is dependent on the accepted VR,T deactivation
mechanism. Any experimental measurements of vibrational de-
activation for v>l include contributions from both VV and VR,T
processes. For a harmonic oscillator, the equally spaced
energy levels result in resonant VV energy transfer. Because
HF is anharmonic, some internal energy must be converted to
translational energy during the collision, but the reaction
is still near resonant compared to VT collisions. Because near
resonant reactions generally have faster reaction rates, it
was assumed that VV rate coefficients were much larger than
VT rates. In the past, this assumption suggested that VV
exchange could be made dominate in an experimental arrangement
for the purpose of determining these rates (60). The existance
of near resonant VR,T reactions would invalidate this assump—
tion. Wilkins (58) trajectory calculations suggest that VV
and VR,T transfer are approximately equally probable, making
experimental determinations of these contributions difficult.
Attempts to extract VV rate coefficients from experimental
measurement of vibrational deactivation rates have been
hampered by uncertainties in VR,T rate coefficients (61-64).
All of these experiments attempt to subtract the contribution
of VR,T from the total rate but because even the simplest of

these processes has not been measured directly, it is necessary

13
to estimate VR rate coefficients. It has regularly been
assumed that the VR,T rates increase linearly with vibrational
quantum number, resulting in very fast VV rate coefficients.
The laser absorption methods of Osgood, et al. (64) and the
fluorescence quenching experiments of Airey and Smith (65) have
resulted in VV rate coefficients exceeding the gas kinetic
collision frequency. Cohen and Bott (66) have shown that
such fast VV processes predict lasing from high vibrational
levels that is greater than is observed in experimental laser
performance. The importance of rotational states in VV energy
exchange has also been considered. Dillon and Stephenson (67)
found multiquanta vibrational exchange was most important in
systems with large transition moments, such as in HF. They
concluded that multiquanta rotational transitions play an
important role in diatomic-diatomic vibrational exchange
collisions by allowing the vibrational energy defect to be
absorbed by the rotational energy levels. Wilkins' (58) tra—
jectory study considered the role of rotational energy in VV
exchange processes. Although the anharmonicity of vibrational
levels in HF results in non-resonant VV reactions, the corre-
sponding energy defects are nevertheless relatively small.
These energy defects are usually computed assuming that the
net change in rotational energy is zero. If rotational states
are allowed to change during reactions, considerably smaller
energy defects are possible and can result in endothermic and
exothermic reaction rate coefficients of comparable magnitude.
Although it is not possible to experimentally measure these

energy defects, the trajectory studies of Wilkins calculated

14

both the endothermic and exothermic VV energy transfer rate
coefficients from which energy defects could be calculated.

The results for the reaction

HF (v) + HF(l).-=:HF(V+1) + HF(O)

are given in Table 1.1 and suggest that the energy defects are
smaller than is expected from the zero rotational energy assump-
tion. These small energy defects are the result of changes in
rotational quantum numbers for the reactant species. For ex-
ample, the reaction of two HF(v=1,J=3) molecules by VV exchange
with zero rotational energy change would be

HF(v=1,J=3) + HF(v=1,J=3)==:HF(v=0,J=3) + HF(v=2,J=3)

with energy defect of 175 cm-1. But the energy defect is
minimized if the J levels are allowed to change as given by

the reaction

HF(v=1,J=3) + HF(v=1,J=3)izzHF(v=0,J=2) + HF(v=2,J=4).

The resulting energy defect is only 85 cm-l. Based on Wilkins'

12 -1

exothermic reaction rate coefficient of 12 x 10 cm3mole-lsec ,

the endothermic reaction rate coefficients at 300K for the

12 and 8.0 x 1012 respectively:

above two equations are 5.2 x 10
the minimum energy defect reaction develops an endothermic
rate that is 1.5 times faster than the zero rotational energy
change reaction. Wilkins concludes that reagent and product
rotational states must be considered to calculate realistic
energy defects for the VV reaction. As the above example
illustrates, although the effect on the endothermic rate coef-
ficient is pronounced, the changes in energy defect produce

only a relatively small change in rotational states of the

colliding species. Wilkins (58) suggests that only single

15

Table 1.1 Rate coefficients at 300K for vibrational to
vibrational energy transfer in
HF(v) + HF(1)==HF(V+1) + HF(O)

 

 

 

Wilkins' Trajectory ' Zero Rotational
Calculations (58) Energy Change
v forward a back a AE back a AB 1
rate coeff rate coeff (cm’l) rate coeff (cm’ )
1 12.0 t 2.1 8.5 i 1.9 72 5.2 175
2 7.8 i 1.8 5.2 t 1.4 87 1.5 343
3 5.0 t 1.4 3.3 i 1.2 88 0.45 504
4 2.8 i 1.2 1.9 i 0.9 81 0.12 665
5 1.7 i 0.9 1.2 i 0.7 81 0.033 822
aRate coefficients should be multiplied by 1012 and have

units of cm3/mole/sec

l6
quantum processes are important in VV transfer and that v-
dependence in the exothermic direction decreases as 2(l-V).
Wilkins' analysis indicates both VR and VV processes have a
To'5 temperature dependence. This result implies that the
probability of energy transfer is independent of temperature.
The ability of the VR reaction to explain rotational

nonequilibrium phenomena is strongly dependent on the rate

of rotational relaxation. Rotational relaxation may be so
fast that the VR mechanism will not be able to effectively
populate high rotational levels. The measurements of Peterson
et al. (68) and Hinchen et al. (69,70) were among the first to
suggest that rotational relaxation rates were on the order of

1014

cm3/mole-sec. Nevertheless, there is experimental evidence
(34,69-71) that rotational relaxation decreases with increasing
rotational quantum number. Based on the fact that the spacing
of rotational levels increases with J, Polanyi et al.,(34)
proposed the relaxation model given by equation (1.5). The
parametric constant B approximates the decrease in transition
probability with increasing J. The experimental results of
Hinchen et al. (69,70) suggests this constant is very large.
Wilkins (58) determined that both rotational to translational
(RT) and rotational to rotational (RR) energy transfer were
important in rotational relaxation with single quantum tran—
sitions. The single quantum RT transition was found to be

the primary relaxation mechanism with single quantum RR tran-
sitions contributing to one-third the total rotational relax—

ation rate of a J level. The rate of rotational relaxation

from high rotational levels was found to be smaller than

l7

deexcitation from lower J levels, in concurence with the
observations of Polanyi (34). Because RR rate processes
have cross sections dependent on collision energy Wilkins
(58) found these rate coefficients to have complicated
temperature dependences.

The VR mechanism has been used qualitatively to explain
high rotational state phenomena observed by a number of
researchers (40,47,49). Krogh and Pimentel (40) note that
the action of VV reactions to enhance populations in high
vibrational states in combination with VR transfer can exPlain
the late threshold times and prolonged P-branch laser emissions
from high rotational states. Although Cuellar et al. (47)
believe the early threshold times of the rotational lasing
observed by them is the result of direct chemical pumping,
they note that VR transfer late in the pulse may be capable
of sustaining emissions. Clear experimental evidence of the
VR reaction does not yet exist. Hinchen (72) argues for indirect
evidence of VR from comparison of isotopic relaxation rates, but
Hinchen and Hobbs (49) were unsuccessful in their attempts to
measure VR transfer in HF. The occurence of this reaction in
HF systems would require reinterpretation of experimental rate
coefficients as well as complicate computational studies of
the HF chemical laser. Its potential ability to explain a
number of phenomena makes necessary careful study of its
behavior. A concurrent attempt to include the VR mechanism in
a constant temperature computer model of the F+H2 chemical
laser was made by Ben-Shaul and coworkers (30). The VR mech-

anism was assumed to convert two—thirds of the vibrational

18

energy lost by a colliding molecule into an increase in
rotational energy for the same molecule. They concluded that
the linear dependence of their VR energy transfer mechanism
was not effective in pOpulating high rotational levels. The
present work extends the preliminary formulation of a rotational
nonequilibrium model presented in Reference 33. Although this
preliminary model included vibrational to rotational energy
exchange, more extensive investigations are justified to
understand the performance of the hydrogen fluoride laser.
Because of the uncertainty in kinetic rates, careful consider-
ation will be given to the sensitivity of laser performance
to these potential mechanisms. In addition, other mechanisms
will be reassessed as they are effected by VR kinetics. The
major features to be incorporated and studied are:

1. Inclusion of vibrational to rotational energy transfer

2. Inclusion of rotational energy in vibrational to
vibrational energy exchange

3. Inclusion of rotational to rotational relaxation
processes as well as rotational to translational
mechanism

4. Effect of rotational lasing

5. Effect of pumping mechanisms in the VR model

6. The effect of gas composition and cavity conditions
on laser performance

7. Comparison with experimental results

8. Development of single pass amplifier model from
VR model

2. COMPUTER SIMULATION

2.1 General Model Formulation

 

The formulation of the computer model yields time histories
of the first seven vibrational levels of hydrogen fluoride and
their first thirty rotational levels. Vibrational levels v=7,8
are also included with their rotational levels assumed to be in
rotational equilibrium at the translational temperature. The
reactions represented in the model are:

l. The H2 + F2 chain

F + H23HF(V,J) + H
H + F2:HF(V,J) + F

2. Vibrational to rotational, translational (VR,T)
energy transfer HF(V,J) + M:=-’HF(v',J') + M

3. Vibrational to vibrational (VV) energy exchange
HF(vl,Jl) + HF(v2,J2);::HF(vl-1,Jl) + HF(v2+1,J2)

4. Rotational to rotational (RR) energy transfer
HF(V,Jl) + HF(v2,J2)=:!.HF(v1,Jl-AJ) + HF(v2,J2+AJ)

5. Rotational to translational (RT) energy transfer
HF(V,J) + Mz—’HF(V,J-AJ) + M

6. Dissociation - recombination

F2+MZF+F+M

H2+MS=2H+H+H

HF(v) + M::=H + F + M

The chemical reactions are written in shorthand as

k
r
21: ariNi f: )1: BriNi (2.1)

-r
where Ni is the molar concentration of species 1, ari and Bri

alie stoichiometric coefficients, and kr and k_r are forward

alixd backward rate coefficients. The rate of change of
19

20

concentrations for nonlasing molecules is
i
dE— - Xi (2.2)

o . 8 .
= _ r3 _ I]
where Xi gmri ori) (kr 2'le k__r EINJ. ) (2.3)
The rate of change of concentrations for the lasing HF

molecules is

dNHF (V,J) = + P(V J) + + R +
at " Xi ' RVRT RRT va

+ oRR(v,J)fRR(v,J) - aRR(v,J-l)fRR(v,J-l)
where NHF(V'J)IS the molar concentration of HF(V,J), and
P(v,J) = pumping rate into level v,J

RVRT = rate of concentration change resulting from
vibrational to rotational energy transfer

RRRT' 2 rate of concentration change from combined
rotational relaxation processes of RR and RT
transfer

va = rate of concentration change resulting from
vibrational to vibrational energy exchange

oVR(v,J) = gain on the P-branch transition with lower
level v,J

fVR(v,J) = photon flux on the P—branch transition with
lower level v,J

a (v,J) = gain on the rotational to rotational transition
RR .
With lower level v,J and

photon flux on the rotational to rotational

fRR(v,J)
transition with lower level v,J

The gain for lasing transitions is given by (27)

hN 9
J -. .— — -
“fliere N is Avogadro's number, w is the wavelength of the

A

transition, B is the Einstein isotropic absorption coefficient

bElsed on the intensity, and ¢ is the Voigt profile at line

21
center as given in Reference 73. The B coefficients for
P-branch transitions are from Emanuel and Herbelin (74) and
rotational transitions are calculated from the spontaneous
emission coefficients of Meredith and Smith (75). The molar
concentrations of the upper and lower levels are denoted as
Nu and N1 respectively, with corresponding level degeneracies

g and 91.

u

The rate equation for the lasing flux of both P-branch

and rotational transitions is

df(v,J) = c[h(v,J) ~ othr]f(v,J)L/£ (2.6)
E?

where c is the speed of light, and'
athr = -(l/2L)ln(RORL), (2.7)

where L is the length of the active medium, E is the mirror

spacing, and R and RL are the mirror reflectivities. The

0
gains and intensities of 72 P-branch transitions and 203 rota-
tional lasing transitions are computed by this method.

The energy equation for a constant density gas is

ZN c M = -P - dNi H (2 8)
i 1 vi 3? L 1 dt 1 ' '
where CVi and Hi are molar specific heat at constant volume

and molar enthalpy, respectively for species 1. The total

laser power per unit volume is given by P and T is the

L!
translational temperature. The output power of a given P-
branch or rotational lasing transition is given by

PLv,J = thAothrw(v,J)f(v,J) (2.9a)
Numerical integration of Equations (2.2), (2.4), (2.6), and

(2.8) by the fourth order Runge-Kutta method of Shampin and

22
Watts (76) yields the time evolution of species concentrations,
temperature, pressure, and gain on all transitions and the
intensity on all lasing transitions. Initiation is modeled
by the introduction of a finite F-atom concentration into the
cavity.
The laser energy extracted in each transition is then

determined by integration of the power
t
c

Ev,J =jf PLv,Jdt (2.9b)
0
where tC is the duration of the laser pulse. The energy of

a particular band is given by

Ev =ZJ:EV,J (2.9a)
and the total pulse energy is
E =;Ev (2.96!)

Although the exact formulations and rate coefficients for
pumping and deactivation mechanisms are parameters of this
study, general formulations can be presented before specific
mechanisms are considered in detail.

The chemical chain reactions given by Equation (1.1)
and (1.2) pump the transitions of the HF chemical laser and
represent major vibrational and rotational nonequilibrium
mechanisms in the system. The pumping rate into a particular
v,J level is given by

P(v,J) = R(v,J)P(v) (2.10)

where P(v) is the rate of chemical pumping by the chain into
level v and R(v,J) is the nascent rotational distribution

over a particular vibrational level.

23

The nonequilibrium distribution of rotational states
over the vibrational levels is taken from the infrared chemi—
luminescence measurements of Polanyi et al. (18,19).

Rotational relaxation is allowed to occur by the rotational
to translational (RT) reaction

HF(V,J) + Mz=HF(v,J — AJ) + M (2.11)

for all species M, and the rotational to rotational (RR) exchange

reaction

HF(vl,Jl) + HF(V2,J2):=HF(VI,Jl-.AJ) + HF(v2,J2+IAJ) (2.12)
The change in rotational quantum number,.AJ, is assumed to be
one or two for RT and is one for RR which is consistent with
the results of Wilkins (58). Polanyi's (34) model (Equation
1.5) for the calculation of rotational relaxation rates is
replaced with explicit values of relaxation rates. Because
rate coefficients for vibrational levels greater than one are
not presently available, the model assumes these unknown rates
are the same as that for Val. The complex temperature depend-
ence of rotational relaxation for all J levels is based on
the J=10 relaxation rates suggested by Wilkins (58):

16 -0.805e-2569/RT

k 1.023 x 10 T (2.13)

0,10-10 + AJ =

16 -0.8938-2436/RT

= 3.380 x 10 T (2.14)

k1,10» 10 + AJ
Although these relaxation rates will be shown to be too slow,
the form of their temperature dependences are useful. Specific
rate formulations shall be considered later.
Wilkins (58) observation that the RR mechanism contributed
to approximately one third of the combined relaxation rate is

uSed to compute separate RR and RT relaxation rates.

The chaperon M for RT reactions employs the relative

24
rotational relaxation efficiencies based upon the infrared
double resonance experiments of Hinchen (49) and are listed
in Table 2.1.

Vibrational to vibrational (VV) energy exchange occurs
for HF—HF collisions and HF-Hz collisions. The HF-HF col-
lisions are modeled by the equation
2'). (2.15)

The forward rate coefficients were obtained from the

HF(V,Jl) + HF(V2,J2)‘;2HF(Vl-1,Jl') + HF(V2+1,J

three dimensional trajectory calculations of Wilkins (58)

and are given in Table 2.2. Accurate modeling of VV exchange
including the effect of rotational energy changes would require
calculations of minimum energy defects in a manner similar to
the procedure for VR reactions. Because of the relatively
small changes in rotational states, the great complexity in
modeling these changes is not justified. In accordance with
the observation that no significant changes in rotational
states of the colliding species occurs, Jl equals J1' and J2
equals J2'. The difficulty with this procedure is the cal-
culation of endothermic VV reactions from detailed balance
considerations. As discussed previously, the larger energy
defects resulting will produce smaller endothermic reactions
than are recommended by Wilkins (58). This effect is partic-
ularly pronounced for large v (see Table 1.1). The endothermic
reaction has the ability to transfer vibrational quanta from
high vibrational levels to lower levels, hence the deacti-
Vation of high vibrational levels by VV exchange will not be

as pronounced as recommended by Wilkins. Wilkins' conclusion

that multiquanta VV reaction rates were found to be negligible

25

Table 2.1 Relative rotational relaxation efficiencies
for several chaperon gases

 

Species Relative Efficiency

HF 1.0

He 0.03
Ar 0.03
N2 0.03
SF6 0.03
F2 0.03
F 0.03
H2 0.10
H 0.03

26

m mo wocwpcmmwp Hmcoflumunfi> m mcflfismmm mump

N>IH>
.mCflxHflz Eouw Umumaommuuxm mums NAN> muons N>AH> How mmSHm> oxen
com
m.ol a macs

sh owflaaflufise on oasozm mans» much an mmumu cumzuom mzsm

 

 

mm.o n.o o
m.o o.o m.H m
am~.o msm.o mH.H m.m e
vv~.o mm¢.o msm.o mm.H m.m m
som.o mav.o mmm.o mo.a m.m o.m m
mq.o mm.o s.H m.~ o.m m.n o.~H H
e o m e m N H
H>
m1.~w.a+~>vmm + A.Hn.ana>vngnn.mn.~>vmm + .Hn.H>vmm

you Ammv mcwaflS mo mwumu >> Enucmsv mHOCflm m.~ manna

27
will be assumed here. Collisions of HF and H2 are modeled as
HF(V,J) + H2(v')::!HF(v+l,J) + H2(v'-l) (2.16)
where the forward rate is obtained from the recommended rate
coefficients of Cohen (77) (See Appendix A).

Vibrational to translational energy transfer is replaced
by the more complicated vibrational to rotational, translational
(VR,T) mechanism. The reaction is represented by the equation

HF(V,J) + MzzHFW'J') + M (2.17)
The reaction differs from the VT reaction in that instead of
vibrational energy being lost to the translational temperature
bath, it may appear in part as an increase in rotational ener-
gy of the collision species. The product rotational state,

J', takes on values ranging from J . -4 to J .
min m

+2 where J .
in m

in
is some specified fraction of the product rotational state
which gives the minimum energy defect for the VR relaxation
(see Figure 2.1). If this fraction is one, essentially all

of the vibrational energy goes into rotation resulting in
energy defects of only a few hundred inverse centimeters. It
is not clear from the results of Wilkins (58) what the relative
rate distributions over these seven rotational states should
be. Several distributions and choices of Jmin are examined.

It should be noted that whatever distribution is chosen, the
sum of these individual reaction channels is constant.

Because multiquanta vibrational transitions are expected to
exist, it is evident that very high J states can result if

near resonance VR occurs. The trajectory calculations of

Wilkins (58) provide a convenient point of reference from

28

HF (v,J) + M —>HF(V',J') + M

Energy of HF (v,J) Energy of HF(v',J')

# Jrn'm*2

 

 

‘Jnfin+4

 

 

#Jmin

 

 

__1 Jrnkl-‘

 

# “'t'n'm'2

 

J min '3

 

Jm’m'4
J.

Figure 2,1 Vibrational to rotational energy transfer

channels. Multiquanta relaxation is
permitted

29
which to study VR kinetics. Although these calculations are
applicable only to HF—HF collisions, they have been used to
estimate VR rate coefficients for other collisional species.
It has been assumed that the ratio of the calculated VR rate
to the previously accepted VT rate is roughly constant for

each collisional species.

2.2 Comparison with VT Model

 

The present model was developed from an existing rotational
nonequilibrium model (24) which simulated vibrational relaxation
by the simpler vibrational to translational energy exchange
mechanism. For comparison with the VT model, the VR model
incorporated the deactivation rate coefficients of Wilkins' (58)
trajectory calculations. The VT model used kinetic rate coef-
ficients suggested by Cohen (77). The distribution of energy
in each lasing band is illustrated in Figure 2.2. In this
comparison, there is a notable shift of P—branch lasing energy
to higher J levels in calculations with the VR model. This
shift may be the result of the VR mechanism populating high
rotational levels or the result of slower rotational relaxation
rates or a combination of both. To determine the origin of
this feature a second computation for the VT model was made
using the fast vibrational deactivation and slow rotational
relaxation rate coefficients recommended by Wilkins (58). The
results are also given in Figure 2.2. The use of Wilkins' data
in the VT model does not predict the correct band distribution
for P-branch lasing energy as well as it does in the VR model,
but it does shift lasing to higher J values. The first obser—

vation indicates the importance of the product rotational

30

Figure 2.2 Comparison of predictions of present VR
model with those of the VT model. Relative
lasing energy for every transition in the
vibrational bands is illustrated. This
format is used in all figures of relative
lasing energy that follow.

Gas Mixture:

0.02F:0.99F :1H :20He; T.=300K, P.=20 torr
Cavity Conditions: 1 l

R080.7, RL=1.O, L=10cm, 2210cm

(a) Standard VT model (Reference 24)
(b) VT model using Wilkins' (58) relaxation rates
(c) Present VR model

RElATIVE P-BRANCH USING ENERGY

31

Standard VT Male!

(0)

 

l0

 

LO" VT Model Using Wilkins Relaxation Rates
(b)

 

 

 

VR MODEL

 

 

 

Figure 2.2

32

distribution in VB energy exchange. The ability of the VR
mechanism to describe observed vibrational relaxation rates
is dependent on the subsequent rotational relaxation of the
high J states populated by the VR mechanism. The observation
that lasing shifts to higher J levels despite the absence of
the VR mechanism is the result of slower rotational relaxation
rates. The use of Cohen's (77) rate coefficients in the VT
model yields predictions of P-branch lasing energy distribution
closer to those of the VR model. Relative energy in the hot
bands for P-branch lasing is considerably higher than experi-
mental observations for all three model calculations, but it
is particularly excessive in the VR model. The difference
between the two models is suspected to be due to differences
in vibrational deactivation and is investigated in Section 2.6.

Several important results are not illustrated in Figure
2.2.- The VR model predicts the occurence of rotational lasing
on all vibrational bands whereas no rotational transition
reached threshold gain in either of the VT model computations.
The failure of even the slow rotational relaxation times of
Wilkins (58) to produce rotational lasing in the VT model
indicates the strength of the rotational nonequilibrium effect
of the VR mechanism and suggests its importance in explaining
rotational lasing. The effect of VR energy transfer on P-
branch lasing from high J levels cannot be determined from
this comparison. Although pulse lengths are similar for
P-branch lasing, the VR model develops considerably less
energy and peak power than the VT model. Subsequent analysis

will show this energy loss is not the result of rotational

33
lasing.

2.3 Effect of Rotational Lasing

 

Much of the interest in rotational lasing arises from
its potential parasitic effect in competing for energy avail-
able for P-branch lasing. Computer simulations have been
made with and without rotational lasing to assess this effect.
The effect of rotational lasing on P-branch lasing energy
is illustrated in Figure 2.3. Preliminary calculations employed
very small rotational relaxation rates and resulted in rota-
tional lasing that was 40% of the total lasing output. Although
this predicted lasing is greater than is to be expected, the
results indicate that although rotational lasing energy may
be a large fraction of P-branch lasing energy and can effect
the number of P-branch lasing transitions, the existence of
rotational lasing has only a small effect on P-branch lasing
energy. The presence of rotational lasing greatly attenuated
lasing from all transitions above J=5, but slightly increased
total P-branch lasing energy. This may be explained by noting
that rotational lasing and rotational to translational energy
exchange depOpulate rotational populations in similar fashion.
The presence of rotational lasing is almost kinetically equiv-
alent to increasing the rate of rotational relaxation. As has
been observed by Kerber and Hough (24), increased rotational
relaxation rates produce higher energies for the P-branch
transitions and spread band energy over fewer transitions.
Both effects are observed by the inclusion of rotational lasing.
The decrease in peak power and the increase in pulse

length produced by the inclusion of rotational lasing is not

Relative P-branch Losing Energy

34

No Rotational Losing

r 5+4
. 2+:

" 6+5

I-‘O

g)
01
*1

- 4+3
_ 3+2

 

 

O 5 l0
J

Rotational Losing

 

 

Figure 2.3 Effect of rotational lasing on relative
P-branch lasing energy
Gas Mixture:
0.02F:0.99F :lH2:20He; T.=300K, P.=20 torr
Cavity Conditigns: l l
Ro=0.7, RL81.0, L=10cm, £=lOcm

35

analogous to rotational relaxation effects. This behavior
can be attributed to the dissimilarities between these two
mechanisms. Unlike rotational relaxation, rotational lasing
relaxes rotational populations only for rotational gain above
threshold. Furthermore, depopulation by lasing is strongly.
dependent on the lasing flux time history which lags the gain
and population densities.
2.4 Effect of VR,T Energy Transfer

2.4.l Product Rotational State Distribution

In deactivating vibrational levels the VR mechanism
populates very high rotational states. Because of the statis-
tically small number of trajectories giving the necessary
product states, it is difficult to deduce the exact product
rotational states from the trajectory calculations of Wilkins
(58). Because of this uncertainty, several distributions
centered about the rotational state giving the minimum energy
defect for the reaction have been examined. These distributions
are listed in Table 2.3. The effect of these distributions on
P-branch lasing energy is illustrated in Figure 2.4. Distri-
bution 1 assumes the VR reaction populated only the rotational
level giving the minimum energy defect. Distribution 2 also
papulates one rotational level on either side of Jm. . The

in
third distribution assumes a smooth distribution about Jmin
incorporating all seven of the allowable rotational levels.
It should be noted that each succeeding distribution includes
more relaxation channels. Neither of the first distributions

correctly predict band energy distribution. Relatively good

results are obtained from the third distribution. It is

36

 

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37

Distribution

Effect of Product J-Distribution NO-l
:1 No.2

No.3

 
     
  

     
  

\\\\\\\\\\\\\\\‘

l/I/l/l/l/I/l/l/I/l/Il/II/I/ll/l/A

  

  

\\\\\\\\\\\\‘

  

\

  
 
     
 

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\\\\\\\

. /.///'///'/////////////1

 
 

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Relative Energy
0
on
I
\\\\\\\\\\

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[/[Z/[Z/////////////////fl//fl/////////A
\

 

  

 

 

 

 

 

 

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[Mi

140‘ 2—I ‘3—2‘ '4—3 5-4 6-5
Vibrational Band

‘ \\

 

Figure 2.4 Effect of product rotational distributions on
relative band energy of P-branch lasing
Gas Mixture:
0.02F:0.99F :lH :20He; Ti=3OOK' Pi=20 torr
Cavity Conditions:
RO=O.7, RL=1.0, L=10cm, £=10cm

38
evident that band distribution is a sensitive function of the
product rotational distribution produced by the VR mechanism.
It was also observed that total lasing energy decreased as
more relaxation channels were made available to the VR mecha-
nism. The ability of these rotational channels to influence
the effective vibrational deactivation rates indicates the
complexity of VR kinetics. This coupling will be considered
in detail later. Because Wilkins' (58) calculations indicate
that several product rotational states are produced by VR
reactions and because Distribution 3 best predicts P-branch
lasing band energy, it has been chosen as the standard distri-
bution for the remaining model computations.

2.4.2 Sensitivity of Laser Performance to VR Deactivation
Rate

The effectiveness of the VR mechanism in populating high
rotational states is dependent on its relative contribution
to the total kinetic rate. In particular, the phenomenon of
rotational lasing and P—branch lasing from very high rotational
levels will only appear if rotational relaxation is not too
fast compared to the VR rate. The effect of relative VR rate
on lasing energy is found in Figure 2.5. As VR rates are
increased, P-branch lasing rapidly decreases while rotational
lasing increases. Vibrational energy is converted to rotational
energy of high J states by the VR mechanism resulting in
increasing rotational lasing gains and decreasing P-branch
laSing gains. The P-branch bands shut off as VR deactivation
dominates the kinetics. When VR rates are made small, rota-
tional lasing becomes negligible. Under these circumstances,

VR is no longer the dominant collisional process; VV exchange

39

Figure 2.5 Effect of VR,T and RR,T rates on laser energy
Gas Mixture:
0.02F:0.99F :lH2
Cavity Conditions:
RC=O.7, Rle.0, L=10cm, i=10cm

(a) Total P-branch lasing energy

:20He; T.=300K, P.=20 torr
i l

(b) Total rotational lasing energy

P-BRANCH usmc
ENERGY lJ/l)

ENERGY RV"

ROIATIONM. lASlNG

loo.

‘0' -

 

102
no“

l0‘

l01

104+

I

10'5

 

noti
IO‘Z

 

40

(0)

/VR
/

10" 10° to 102 103
RATE/STANDARD RATE

/RRT

J J

l0" lO° lO‘ I02 103
RATE/STANDARD RATE

Figure 2.5

41

and rotational relaxation determine laser performance. Although
VV reactions now dominate vibrational exchange, the computations
assume that VV exchange produces no change in rotational quantum
numbers and the production of high rotational states rapidly
diminishes as VR rates are made small. When rotational relax-
ation depletes high J states faster than they are populated,
rotational lasing can no longer be sustained. The effect of
VR,T energy transfer on rotational populations is illustrated

in Figure 2.6. The rotational populations for vibrational
levels v=l,3, and 5 are predicted for a 380 torr F2:H2:He gas
mixture lasing at peak power. For this calculation it was

assumed that the VR product rotational states, Jmin

, were only
50% of the values predicted by assuming a minimum energy defect
for the reaction. Strong rotational relaxation rates at this
pressure establish equilibrium distributions for low J levels
while strong pumping into J=4-13 maintains these levels well
above equilibrium concentrations. The VR,T reaction is the
dominate mechanism populating J levels above thirteen. The
ability of VR to pOpulate high rotational levels is most
pronounced for low vibrational levels. This behavior is
expected from the additional multiquanta reactions which can
pOpulate these low vibrational levels from higher levels. In
addition, the larger energy defects associated with these
multiquanta reactions are converted into very high product
rotational states of low vibrational levels. The ability of
the VR reaction to form a hip in the rotational distribution

is strongly dependent on the choice of product rotational

states populated by the VR reaction as well as the rate of

Relative Population
0d: 0‘;

‘35

Figure 2.6

 

 

42

Rotational pOpulation distributions predicted
by VR model at peak power of the laser pulse
Gas Mixture:

0.02F:0.99F :lH :20He; T =300K, P.=38O torr
Cavity Conditions: 1 l

R080.7, RL81.0, L=10cm, £=lOcm

43

rotational relaxation.

2.4.3 Translational Contribution to the VR,T Mechanism

Earlier model assumptions that vibrational deactivation
occurred strictly by vibrational to translational energy
transfer produced large energy defects for the reaction which
appeared as a rise in the translational temperature of the gas.
The postulation of vibrational to rotational energy transfer
occurring with a minimum energy defect has several consequences.
The smaller energy defects result in population of very high
rotational states and suggest faster vibrational deactivation
rates than is expected for VT reactions. These effects are
consistent with experimentally observed phenomena of the HF
system. Because of the near resonant exchange of energy from
vibrational to rotational modes, however, the excited HF
molecules only very slowly relax toward equilibrium with the
translational temperature. The only effective relaxation
toward equilibrium is rotational to translational energy
transfer. The translational contribution to the VR,T mechanism
can be enhanced by allowing larger energy defects for the
reaction. There is little experimental or theoretical evidence
to suggest what the proper partitioning between VR and VT
reactions should be. The theoretical calculations of Shin (52)
suggest that the effect of VT energy transfer becomes important
as temperature increases, although the VT process in general is
muCh less efficient than the VR process. For example, Shin
calculates that the VR contribution decreases from 96% to 57%
as the temperature rises from 400 to 4000K. This temperature

dependence has not been simulated in the model since the

44

aggregate effect would be less than 5% over the typical range
of temperatures during laser energy extraction.

To test the effect of increasing VT contribution, the
product rotational state, Jmin’ predicted by the minimum
energy defect assumption was decreased by 50% for all VR,T
reactions. As is expected, an increased VT contribution to
vibrational deactivation resulted in smaller predicted pulse
duration and lasing energy. The effect on relative P-branch
and rotational lasing is shown in Figure 2.7. Because the
ability of the VR mechanism to pOpulate high rotational states
has been decreased by one-half, a change in spectral distri-
bution is expected. Although rotational lasing above J=10
disappears and the relative contribution to P-branch lasing
from high J levels slightly decreases, very little other effect
is predicted. To understand this behavior, the population
distributions after 2.05usec are plotted for the first excited
vibrational level as a function of J (Figure 2.8). It is
observed that the ability to pump very high rotational levels
is markedly decreased in the enhanced VT computations, but for
J=12 the populations for the two cases are nearly identical.
Because P-branch lasing is modeled only for rotational levels
less than this, the absence of spectral differences is imme-
diately understood. The small effect on populations below
J=l2 occurs because the lowest rotational level populated by
the standard VR reaction is J=13. Even a large decrease in
Jmin will have relatively small effect for the lowest rota-
tional levels. Although the endothermic back reactions of

VR,T will populate rotational levels below J=13 from very

45

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46

.040 Effect of VR on Population

  
  
     

P205455

55mm

 

m“ '
o k
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g '0 ‘ \ JMINIZ \
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Figure 2.8 Effect of increased VT contribution for VR,T
reactions on rotational populations
Gas Mixture:
0.02F:0.99F2:132:20He; Ti=300K, Pi=20 torr
Cavity Conditions:
Ro-0.7, Rle.0, L=10cm, £=lOcm

47

high rotational levels, this effect will be negligible for

the duration of the cold and hot pumping reactions. Hence,
the major effect of the VR reaction appears to be on P-branch
and rotational transitions not strongly pumped by the chemical
chain reactions.

2.5 Effect of Rotational Relaxation

 

The kinetic rate coefficients recommended by Wilkins
predict large and sustained rotational lasing. Gains of
rotational transitions are commonly of the order of l cm-l,
and rotational lasing energy is a large fraction of total
lasing energy. The persistance of rotational lasing long
after P-branch lasing has shut off suggests that the rotational
relaxation mechanism employed in the calculations is too slow.
Indeed, the rates calculated by Wilkins give probabilities of
rotational relaxation of approximately 0.01 at 300 K for J=10.
The effect of several different relaxation rates on lasing
energy were compared without changing the temperature depen-
dence of equations (2.13) and (2.14). The result, in which
Wilkins' recommendations are referred to as the standard
rate, are given in Figure 2.5. Decreasing the rotational
relaxation rates by a factor of ten has little effect on
rotational lasing, suggesting that rotational relaxation is
already so slow that it is no longer a competing mechanism
for high rotational states. Increasing the relaxation rates
of equations (2.13) and (2.14) so that for J=10 at 300 K they
are 20% of the gas kinetic rate decreases rotational lasing

energy by three orders of magnitude.

48

Only one rotational transition is above threshold gain even

at peak rotational lasing power. The slow rotational relaxation
times of Wilkins' effectively prevented depopulation of the
rotational manifolds except by rotational lasing, resulting in
strong and sustained rotational lasing. The very fast rotational
relaxation rate of HF(v=1,J=3) measured by Hinchen (70) was used
to scale rotational relaxation rates even higher. This increase
resulted in further reduction of rotational lasing and shorter
duration. These results indicate that rotational lasing drOps
approximately three orders of magnitude for probability of
rotational relaxation ranging between 0.01 and 0.10. In general,
as rotational relaxation rates increase, P-branch lasing energy
increased. It is clear that Wilkins' rates are too slow, never-
theless, the temperature and rotational quantum number depend-
encies will guide the present calculation. For this reason,

the rates have been retained by increasing them so that the
probability of rotational relaxation for J=10 is 0.20. These
shall be referred to as modified Wilkins' rates.

The effectiveness of the VR mechanism to produce popula-
tion inversions for high rotational levels is dependent on
relatively slow relaxation from these levels. Despite the
relative fast VR rates, however, present experimental results
indicate that rotational relaxation is very fast (68-70). It
has previously been noted that experimental studies indicate
that rotational relaxation decreases with increasing rotational
quantum number. The relaxation model of Polanyi (34) suggests

that a semilogarithmic plot of experimental rate coefficients

49

versus E(v,J) should present a linear curve with slope equal
to the parametric constant B described by Polanyi. The para-
metric constant B approximates the decrease in transition
probability with increasing J. For calculations with the VT
model, Kerber et al. (17,24) typically set this constant much
less than unity to produce relaxation rates that were approx—
imately independent of J. The resulting J-dependence for
B=0.0l is illustrated in Figure 2.9. Polanyi and coworkers
(71) study of rotational relaxation of HF by Ar suggest that
this constant is much larger. Assuming multiquanta relaxation
channels, they obtain B=0.9l7.

The classical trajectory calculations of Wilkins (58)
have resulted in J-dependent rotational relaxation rate coef-
ficients. They have been plotted in Figure 2.9 with the revision
discussed previously. These results suggest a much stronger
rotational quantum number dependence than was used in the VT
model. Nevertheless, the resulting B parameter of 0.52 cm is
still less than Polanyi's (71) value.

Hinchen and Hobbs (69,70) have used laser fluorescence
methods to study rotational relaxation in HF. They observed
unexpectedly fast rates of transfer between levels J and J-AGJ,
where :AJ was as large as five. Further analysis proved that
rotational lasing provided an additional transfer mechanism
and was altering the results. Because radiative transitions
do not substantially effect transfer between levels J and J+.AJ,
an earlier study by Hinchen and Hobbs (70) on transfer to
higher rotational levels was used to estimate the J-dependence

on rotational relaxation. Assuming single quanta relaxation

50

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51

and calculating the J to J-:AJ rates from detailed balance
considerations, the logarithmically plotted values yield a
parametric constant equal to 4.31; a large rotational quantum
number dependence which decreases relaxation rate coefficients
by ten orders of magnitude between J=l and 30. Rotational
relaxation measurements by Gur'ev (78) from J=8 is the highest
rotational level for which experimental data exists. This
relaxation probability of 0.10 fits the estimated J—dependence
of Hinchen's data.

The different J-dependence of Wilkins, Hinchen, and Polanyi
for rotational relaxation have been tested in the VR model.
Rotational relaxation rate coefficients for rotational levels
up to J=30 were extrapolated using the B parameters estimated
above. Rotational levels populated by the VR reaction were
assumed to be one-half those of a near resonant reaction. The
relaxation rate coefficients extrapolated from Hinchen's data
predicted maximum P-branch lasing energy in the v=l-O band.

This result is not consistent with experimental results and
suggests that a J-dependence which ranges over ten orders of
magnitude is unreasonably pronounced. The rate coefficients
extrapolated from the smaller J-dependence of Wilkins' and
Polanyi's data better predicted laser performance. Figure 2.10
gives the rotational pOpulations for the first excited vibra-
tional level at peak power for Wilkins' and Hinchen's relaxation
schemes. The results are compared to a rotationally equilibrated
pOpulation. The rotational nonequilibrium populations reveal
roughly three population regimes. For J less than about five,

rotational relaxation is fast enough that an approach toward

52

lo’iO

5
J.
u-

Fwamkn(nnhfimfii

 

 

'0‘” . 1 L

o 3 l0 IS 20
Rotational Level (J )

Figure 2.10 Effect of rotational relaxation J-dependence
on rotational populations
Gas Mixture:
0.02F:0.99F :lH :20He; T18300K, Pi=20 torr
Cavity Conditions:
RO=O.7, RL=1.0, L=10cm, £=lOcm

53

an equilibrium distribution can clearly be discerned. In the
range of J=6 to 13, the effect of chemical pumping is apparent.
Above J=13, the rotational population would rapidly drop off
except for the VR reaction. This reaction maintains the popu—
lation well above the equilibrium pOpulation distribution. It
is apparent that Polanyi's slow relaxation rates for high J
levels favors the formation of populations at high J levels
as compared to the relaxation scheme of Wilkins.

The effect of these schemes on rotational lasing power
can now be understood. The time history of rotational lasing
predicted from Wilkins' J-dependence and Polanyi's J-dependence
are illustrated in Figure 2.11 for 20 torr total pressure. The
slower relaxation of high rotational levels resulting from
Polanyi's relaxation scheme increases both peak power and
duration of rotational lasing. It was also observed that the
spectral distribution shifted to slightly higher J levels.
Increasing pressure to 380 torr quenched rotational lasing
except for an early, low energy spike due to the instantaneous
F—atom concentration introduced to initiate the pumping reactions.
Despite the increased lasing duration predicted by Polanyi's
rotational relaxation, rotational lasing existed only during a
fraction of the P-branch lasing pulse and appears to be primarily
a result of the initial F-atom concentration. The J—dependence
of rotational relaxation appears to have less influence on
predicted laser performance than do small changes in the

rotational relaxation rate coefficient.

Pmnrfwmfl

Figure 2.11

54

Effect of Rotational Relaxation J-Dependence
on Rotational Losing

0.2b

OJb

 

 

TMnQyud

Effect of J-dependence of the rotational

relaxation rate coefficients on rotational lasing
Gas Mixture:

0.02F:0.99F :1H2:ZOHe:
Cavity Conditions:
ROIO.7, RL=1.0, L=10cm, 2=lOcm

T.=300K, P.=20 torr
i l

55

2.6 Vibrational Deactivation

 

As has been previously noted, the present model formulation
predicts excessive energy from the hot bands of HF. The time
resolved spectra for a typical case at 20 torr pressure reveals
that the lasing persists in the upper two vibrational bands
well after the lower bands have shut off. This prediction
does not agree with experimental results in which lasing for
v>3 shuts off well before lasing for v<3. It is necessary to
identify the kinetic mechanism which is to provide the necessary
vibrational deactivation for the high vibrational bands. The
two candidates are the endothermic cold pumping reactions and
the vibrational dependence of the VR rate coefficients. These
mechanisms are considered below.

2.6.1 Effect of Cold Pumping Reaction

The cold pumping reaction is given by equation (1.1).

The kinetics of this reaction have been the subject of frequent
investigation and its rate coefficient and relative distribution
over vibrational levels has been reasonably well established (77).
However, the endothermicity of this reaction for vibrational
levels v>3 makes it important in investigations of lasing energy
from the hot bands of HF. It has previously been noted that
the model predicts excessive lasing energy in the hot bands of
HP. The back reaction for equation (1.1) provides a channel
by which HF(v>3) can be removed by H-atoms and hence is a
ENItential mechanism for decreasing hot band lasing. This
deactivation mechanism shall be refered to as abstraction (79) .
Bott and Heidner (80) have observed a sharp increase in

13162 removal for v=3 in reactions of HF with H but they note

56
that this effect can also be the result of vibrational deactivation
by H. Bartoszek et al. (79) concludes from chemiluminescence
measurements that the observed increase at v=3 can be attributed
primarily to the abstraction reaction. Hence, experimental
measurements (80-82) for HF(v=3) deactivation exceeding 1013cm3/
mole-sec suggest that abstraction may be important in the per-
formance of the HF chemical laser.

To determine the sensitivity of band energy distribution
to the abstraction reaction, several different rate coefficients
were studied for the endothermic reactions of equation (1.1).

These rates are compared to the recommendations of Cohen (77)
in Table 2.4. The predicted band energy distributions using
increased abstraction rates are found in Figure 2.12.

The standard rate coefficients of Cohen (77) only include
endothermic reactions v=3-6. These rate coefficients shall be
taken as standard for the following computations and their
predicted effect on laser performance at 20 torr pressure is
given in Figure 2.12. To determine the sensitivity of laser
performance to these reactions, the reaction rate coefficients
were increased by a factor of 100 (Run 123). At 20 torr
pressure, both P-branch and rotational lasing energies were
decreased by approximately 20%. Whereas the standard abstrac-
tion rates predict 34% of P-branch energy will appear in the
hot bands, increasing the rate of abstraction by a factor of
100 completely quenches lasing from bands above the 3—2 transition.

Also, time to peak power is considerably decreased.

Some of these increased rates exceed the binary collision

57

Table 2.4 Abstraction reaction rate coefficients

Rate Coefficient for HF(v) + H-*'F + H2

 

 

(1012 cm3/mole/sec)

v Cohen (77) Increased Gas Bartoszek (79)
by 100 kinetic

Run 122 Run 123 Run 116 Run 124

4 3.2 320 394 10.4

5 4.7 470 394 15.3

6 7.4 740 394 28.1

7 0 0 394 O

8 0 0 394 0

58

abstraction

   

 
  

7”’/”’/’/’/”//”’//’//’//”’

Bartoszek

a Cohen's Cold Pumping
I o

- s

C] Gas kinetic abstraction

R§BQ$§§SSNE\.

 

   

  

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IO-

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m.

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392 493 594 695

Vibrational Band

I90 2")

Effect of abstraction reaction on predicted

P-branch lasing energy

Gas Mixture:

. Figure 2.12

300K, Pi=20 torr

2:1H2:20He; Ti

Cavity Conditions:
RO=O.7, RL=1.O, L=10cm, £=lOcm

0.02F:0.99F

59

frequency by as much as a factor of two. Also, abstraction
reactions for vibrational levels above v=6 were not included
because no experimental data existed for them. Computations
for Run 116 included abstraction up to v=8 and set the reaction
rate coefficients equal to the binary collision frequency. The
effect on P-branch lasing is almost identical to Run 123.
Complete absence of hot band lasing is not to be expected.
Bartoszek et al. (79) have measured relative rates for the
abstraction reactions up to v=6. Despite the lack of measure-
ments for v>6, these values are incorporated into computations
for Run 124 by scaling absolute rates from Cohen's (77)
recommendations for the thermoneutral reaction v=3. The largest
of these rate coefficients (v=6) was 7% of the binary collision
frequency at 300 K. Recent experimental measurements of the
cold and hot pumping reactions have also been incorporated
into this computation. Huston (83) has measured pumping rates
for the temperature range of 190 to 375 K and Heidner (84)
has made measurements in the range of 295 to 765 K. Cohen (85)
has estimated the temperature dependence of the cold and hot
pumping rates for an eXpanded temperature range based on these
measurements:

11T0.8e-300/T

F + H : 4.9 x 10 (2.18)

2
H + F2: 3.0 x 109'1‘1'3e'480/RT

These new values result in faster hot pumping and slower

(2.19)

cold pumping. Computations using the revised pumping rate
coefficients resulted in a significant decrease in hot band
energy in comparison to the standard rates but did not completely

shut off hot band lasing. Only 18% of the total energy appeared

60

in the hot bands. Time to peak power was comparable to the
other results for increased abstraction.

Although improved kinetics for the abstraction reaction
has been shown successful in decreasing hot band lasing at
low pressure (20 torr), attempts to scale the model up to
higher pressures resulted in increasing contributions to
lasing from the hot bands (See Figure 2.13). An additional
mechanism to depOpulate high vibrational levels late in the
laser pulse is required. Because the pumping reactions rapidly
increase the concentration of HF with time, self deactivation
of HF will be of increasing importance as the laser pulse
proceeds. Hence, small changes in VR self deactivation rate
coefficients might produce significant depopulation of high
vibrational levels late in the pulse. An analysis was under-
taken to assess this mechanism.

2.6.2 The V-Dependence of Vibrational Deactivation

The v-dependence of vibrational deactivation may represent
the greatest uncertainty in HF kinetics. The anharmonicity of
the HF molecule renders the single quantum, linear v-dependence
of harmonic oscillator theory inappropriate. It is thought
that vibrational deactivation increases with vibrational
quantum number more than linearly because the decreasing
energy defects require less absorption of energy by the rota-
tional and translational modes (86). Poole and Smith (87)

prOVide strong experimental evidence for a v2 to vz'3

depend-
ence, lending support to Kwok's (81) suggested exponent of

2.3. A strong v-dependence will decrease lasing energy in the

61

Figure 2.13 Effect of pressure on band energy distribution
Gas Mixture:
0.02F:0.99F2:1H :20He; T18300K
Cavity Conditions:
RO=O.7, RL-l.0, L=10cm, £=lOcm

62

.. a
to. a set: 38.52.

a
is: .........

c . >
:2 8m "9...... .3252.

 

   

m..~ magmas

 

(c.-

 

. "l‘ 1" ‘

is «trial! .2. .

I
O/

I

 

:2 8 5:72 55...-..

O -I
.I

 

..~

 

 

l
5.

l
‘i

:
a?

 

.3

A083N3 MUTE?!

63

hot bands of HF but computational studies by Kerber et a1. (24)
with a VT model found Kwok's recommendation alone insufficient
to reconcile prediction with experiment.

The difficulty of distinguishing between VT transfer and
VV exchange in experimental studies has previously been described.
The proposed mechanism of vibrational to rotational energy
transfer might also complicate the interpretation of experimental
results. Because of the numerous energy transfer channels
available for relaxation, a more complicated model of vibrational
relaxation than is presently used may be necessary (59). The
ability of additional VR relaxation channels to decrease the
apparent relaxation rate has already been qualitatively noted.
Experimental verification of the VR,T mechanism would necessitate
reinterpretation of vibrational relaxation experiments. For
example, Green and Hancock (88) have pointed out that near
resonant VR,T reactions would have rate coefficients comparable
to rotational relaxation. Potential pooling of rotational
pOpulations could produce RV reactions which would decrease
the apparent vibrational relaxation rate. The relaxation
scheme of the Landau-Teller theory for the harmonic oscillator
would be inappropriate for extracting vibrational relaxation
rate coefficients from experimentally measured relaxation
times. For this reason, Wilkins (59) prefers to call these
experimental values "empirical quenching coefficients." These
coefficients represent the combined effects of VV, VRT, and
RRT kinetics in producing an overall deactivation rate for a
vibrational level.

The three-dimensional classical trajectory calculations

64

of Wilkins (58) provide the first detailed rate coefficients
for multiquanta VR reactions for collisions between HF molecules.
As Figure 2.14 reveals, Wilkins' rate coefficients do not
provide a strong v-dependence for the VR,T reactions. It is
much weaker than either Kwok's (81) recommendation or harmonic
oscillator theory. It was originally suspected that the remaining
v-dependence was the result of the coupling of the VR mechanism
and rotational relaxation. Because the VR mechanism papulates
high rotational levels, the rate of vibrational deactivation
is ultimately controlled by the rate at which the rotational
levels are relaxed. An additional v-dependence can be argued
qualitatively but does not seem capable of explaining the
disparity between Wilkins' data and experimental results. The
anharmonicity of the HF molecule results in smaller energy
spacing for vibrational levels with increasing v. The transfer
of vibrational energy to rotational energy by the VR mechanism
will thus give lower product rotational states with increasing
vibrational levels. Because the rate of rotational relaxation
is assumed to decrease for increasing J values, it can be
concluded that the VR,T mechanism coupled with rotational
relaxation will remove populations from high vibrational levels
more efficiently. The result will be an additional effective
v-dependence to the empirical quenching coefficient. The
effect of this process will depend on the anharmonicity of HF
vibrational levels and the J dependence of rotational relaxation.
Although this process provides additional v-dependence to

the VR deactivation mechanism, its actual effectiveness in HF

Rate (cm3/mole -sec)

65

 

 

Icfi4__
Wilkin's - ' °
C . /
. ,/ v2 dependence
Cohens |972
/
(0'3 i- -— J
/ Cohen's |978
2" ‘ ‘
/ .
/ . Harmonic
/ Oscillator
/ A
ICVZ ’ 1 l L 1 l l, l L‘J
I 2 3 4 5 6 7 8 9 IO
Vibrational Level (v)

Figure 2.14 Dependence of vibrational deactivation on
vibrational quantum number

66

appears to be limited. For example, the minimum energy defect
reaction v=l-0 for initial rotational state J=l predicts a
product rotational state J=15. The same initial rotational
state for v=6-5 results in a product rotational state of J=l4.
Even with the extreme J-dependence of Hinchen's rotational
relaxation data, the effect on v—dependence appears minimal.
Probably more important to an implicit v'dependence is the
additional multiquanta deactivation channels available with
increasing vibrational levels. It is assumed that seven
rotational levels participate in any VR relaxation from a
given vibrational level. Therefore the deactivation reaction
v=l—0 may relax into only seven rotational channels and the
six multiquanta reactions available to v=6 provides a total of
forty-two rotational channels for relaxation.

To quantify this analysis, the present model was modified
to simulate laser fluorescence experiments. In these experiments,
a vibrational level is populated by sequential photon absorption
from the ground state using a pump laser in a process that
selectively populates rotational levels within a vibrational
band. Nevertheless, it shares with other techniques of measuring
vibrational deactivation the usual assumption that rotational
equilibrium is attained before significant vibrational relaxation
occurs. Although recent rotational relaxation experiments
(68-70) may invalidate this assumption, deactivation studies
frequently omit information necessary for more detailed analysis.
Laser fluorescence studies which specify a unique pump tran-
sition usually employ low rotational levels (51,72). Despite

recent speculation about relaxation from high J levels,

67
relaxation from levels less than ten are thought to be relatively

fast and the rotational equilibrium assumption for these exper—
iments may be reasonable. Therefore, the initial gas composi-
tion for the computations consisted of l torr of hydrogen
fluoride with all but 1% vibrationally and rotationally equili-
brated. The remaining HF was Boltzmann rotationally distributed
in a single vibrational level. By excluding diluent gases,
pumping, dissociation and lasing, vibrational self deactivation
of HF could be simulated for the excited level.

By monitoring the populations, calculations of empirical
quenching coefficients are possible. It can be shown (Appendix
B) that the overall deactivation rate coefficient for vibrational

level v is

 

k = -1 l dIHF(vH

emp [HF] [HF(v)] -IHF(V)]eq dt

where [HF(v)]eq is the equilibrated population of vibrational

(2.20)

level v. Because these coefficients are the result of the
combined effect of VR and VV and the indirect contribution of
rotational relaxation, they are identified as the empirical
quenching coefficients, kemp' As such, they can be compared
directly to the overall deactivation rates (i.e., empirical
quenching coefficients) calculated from experimental measure-
ments. There exists the possibility that, because of the
coupling between rotational and vibrational levels, the
empirical quenching coefficient might be a function of time.
For example, as the VR reaction populated high J levels which
relax only slowly, RV reactions could become significant and
k would appear to decrease with time. It is therefore

emp
necessary to analyze the pOpulation data in a manner which

68
will reveal time dependence of the empirical quenching

coefficients. In Appendix B it is shown that for excursions
not far from equilibrium, equation (2.20) can be written

log [HF(v)] = -t/'r + A (2.21)
where t is time, A is an integration constant and T, the
characteristic relaxation time, is given by

T = (kmup[:M:I)-'l
with [M] the concentration of the collisional species. If
kemp is not a function of time, then the vibrational population
will decay exponentially as described by equation (2.21).
Linear regression analysis of log [HF(vfl as a function of
time provides an average kemp for the deactivation of a vibra-
tion level. The resulting correlation coefficient reveals
deviation from a linear relationship and hence provides a
measure of the time dependence of kemp' All calculations by
this method gave correlation coefficients of one to five sig-
nificant figures and it was concluded that kemp was constant
with time. Unless otherwise specified, the VR reaction is
defined for model computations such that the product rotational
states are 50% of the values eXpected from a minimum energy
defect VR reaction.
The first determination was whether Wilkins' (58) VRT,

VV and RRT rates predicted the prOper v-dependence for the
kamp coefficients. Figure 2.15 illustrates the resulting
coefficients. If vibrational deactivation rates are independent
of rotational relaxation, the deactivation rate from a given

vibrational level is expected to be kVR+gkvvnwhere-kVRiis;the

total VR rate coefficient and kVV is the total VV rate coefficient

Rate Coefficient (cm3/mole-sec)

 

69

 

 

 

 

Io'4lr-
- A
e O
I I
(03- .
r O kVV-l-kVRusing Wilkins"VR
. [5 kemp predicted from Wilkins' VR
0 kVV+kVR using v2 dependent VR
(- ‘ kemp predicted from v2
dependent VR
7
I012" A L l 1 1 1 1 1 J
I 2 3 4 5 6 7 8 9 IO
Vibrational Level (v)

Figure 2.15 Model calculations of empirical quenching

coefficients

70

for that vibrational level. For both reactions, ground state
HF is the major collisional species and the only contribution
to the VV reaction is assumed to be the endothermic rate coef-
ficients given in column 5 of Table 1.1. The kVR+ kVV values
obtained from Wilkins' data are also given in Figure 2.15. The
fact that the kemp values are slightly greater than the cor-
responding kVR+ kvv values and because this divergence increases
with v indicates that there is an implicit v-dependence to the
deactivation rate coefficients, but it is very small. Accom-
panying the above plotted values are error bars giving the
range of deactivation rate coefficients observed experimentally
as a function of vibrational level (61,81,87,89). It is
apparent that Wilkins' data cannot predict experimentally
observed deactivation rates; they produce deactivation rates
that are too large for low vibrational levels. The result
suggests-the use of VR,T rate coefficients with much larger
explicit v-dependence. Cohen's 1972 review of HF rate coef-

12cm3/mole-sec

ficients (77) recommends a value of 1.04 x 10
for vibrational deactivation of the first excited state of HF

at 300 K. This value has been used to scale VR rate coefficients
with a v2 dependence. The resulting kemp and their corresponding
kVR + kvv values are illustrated in Figure 2.15. Because kemp
values are nearly identical to their corresponding kVR + kVV
values, coupling of rotational relaxation and vibrational
deactivation appears to be minimal in these simulations. These
predicted deactivation rates correspond more closely to the
experimental data than do the kemp predicted from Wilkins' rate

coefficients. Nevertheless, greater deactivation from the

higher vibration levels is desirable; a v-dependence as high

71

as 2.3 is suggested.

To ascertain the role of rotational relaxation on
vibrational deactivation, the rotational relaxation rate
was decreased by a factor of 100. The formation of large
pOpulations in high rotational levels did not decrease the
rate of vibrational deactivation as would be expected. Figure
2.16 illustrates rotational population distributions for
simulated deactivation from the fourth excited vibrational
level of HF. Single and multiquanta VR reactions produce
large nonequilibrium humps in the rotational populations for
vibrational levels v<4 and create reactant and product popu-
lations that are approximately equal. That these large non-
equilibrium humps do not produce significant RV reactions
capable of decreasing the overall deactivation rate indicates
that the endothermic reaction rate coefficient is not large.
This feature is understood by recalling that presently VR has
been chosen such that the product rotational states populated
are only one-half those of a minimum energy defect reaction.
The larger energy defects associated with these reactions result
in larger equilibrium constants and the RV reactions are small.
For example, vibrational relaxation of HF(v=4,J=2) by a minimum
energy defect reaction will populate HF(v=3,J=13) and has an
equilibrium constant of 11.97. The present formulation of the
VR reaction will only populate HF(v=3,J=7), but has an equili-
brium constant of 6.123 x 105. In this case, even if reactant
and product state populations are approximately equal the RV

reaction is negligible and rotational relaxation does not

effect the resultant deactivation. Although the minimum energy

Relative Population

Figure 2.16

72

93

‘25

(i

5.
0)

ES
a:

 

 

Rotational populations during multiquanta
VR deactivation of vibrational level v=4

after 0.5usec
Gas Mixture:

l torr HF vibrationally equilibrated except
for 1% distributed in v=4; all levels
initially rotationally equilibrated.

(a) v=0; maximum
distribution

(b) v82; maximum
distribution

(c) v=4; maximum
distribution

population in rotational
is 1.4x10‘3 mole/cm3

population in rotational
is 6. 4x10‘12 mole/cm3

pOpulation lin rotational
is 8.3x10’1mole/cm3

73

defect reaction has VR and RV rate coefficients which are

both large, the smaller population associated with this higher
product rotational state produces an RV rate which is generally
smaller than the VR rate. Nevertheless, the minimum energy
defect reaction is more sensitive to rotational relaxation.
Table 2.5 compares the predictions of deactivation simulations
assuming minimum energy defect VR reactions with the previous
results. Although the VR rate coefficients were unchanged,
corresponding RV rate coefficients calculated from detailed
balance considerations were larger because of the smaller
energy defects. The resulting RV rates are pronounced enough
to significantly decrease the empirical quenching coefficients.

2.7 Comparison with Experiment

 

Although a number of experimental studies of the HF
pulsed chemical laser have been reported in the literature
(20-23,45), considerable discrepancy exists between their
results. These differences reflect the extreme sensitivity
of laser performance to initial gas composition and cavity
conditions. Parameters such as initial HF concentration from
prereaction, degree of FZ-dissociation and optical losses
from the cavity effect laser performance and hence must be
carefully defined for comparisons with computational predictions.
The degree of prereaction strongly effects the vibrational
band energy distribution from the HF laser. Because this HF
is primarily in the ground state, it will greatly decrease
lasing from \7= l-O band. Suchard et al. (20) found that by

raising total gas pressure to 100 torr about 10% prereaction

 

Table 2.5 Effect of VR product rotational states on predicted
empirical quenching coefficients
V kViz-“‘vv kemp kemp
for JMIN/2 for JMIN

2 0.94 0.73 -

3 1.09 1.03 0.99
4 1.71 1.71 1.22
5 2.61 2.68 2.56
6 3.75 3.90 3.58

75

occurred. They observed lasing on the 1-0 band to be greatly
decreased under these conditions and found lasing from this
band to completely disappear at higher pressures. Both Suchard
et al. (20,21) and Parker and Stephens (22) include oxygen in
their gas mixtures to suppress prereaction. Oxygen effectively
scavenges free hydrogen atoms which produce prereaction but was
not observed to effect laser output for F2:02 ratios as high as
1:1. The oxygen inhibitor is employed in low enough concen-
trations so that it is not able to quench the chain reaction
after initiation by the main electrical discharge or flashlamp
photolysis. Suchard et al. (20) estimated prereaction by
monitoring the UV absorption of F2 in the laser tube. They
report an ability to infer F2 pressure to an accuracy of better
than 5%. Because they were able to measure no F2 dissociation
below 75 torr, the upper limit on initial HF was 0.1%. Nichols
et al. (23) note that UV absorption methods may not be adequate
for determining minimal HF concentrations capable of adversely
influencing HF laser performance. Instead, they employed a
small HF probe laser operating on the P1(2) transition to
directly detect ground state HF with an effective sensitivity
of 10"3 torr HF. They found that a standard mixture H2:F2:N2 =
7.5:7.5:75 torr had a steady state HF concentration of 0.017
torr at room temperature for a residence time of 1.8 seconds.
This concentration doubled with a doubling of the residence
time. Neither the results of Parker and Stephens nor the
atmospheric pressure experiments of Chen et a1. (45) included
measurements of prereaction.

The method of initiating the chemical chain reaction also

76
significantly effects the spectral output of the laser. Ideally,

initiation instantaneously dissociates molecular fluorine into
atomic fluorine which react with molecular hydrogen via the
cold pumping reaction. The method of flash photolysis of
Suchard et al., Nichols et a1. and Chen et al. suffers from
uncertainties in Optical coupling and the extended duration of
F-atom generation. The photolysis of F-atoms well into and
possibly beyond the laser pulse duration may be responsible

for the irregular pattern of spectral output observed by these
researchers. The use of electrical discharge by Parker and
Stephens greatly improves the time definition of F2 dissociation
but adds the possibility of ion deactivation. They did not
have a method for directly determining F-atom yield but instead
estimated the yield by comparing measured peak power and pulse
length with a computer model of the H2 - F2 laser similar to
that of Reference 9.

Chen et al. (45) have attempted to measure directly
initial F atoms produced from photolysis by substituting HCl
for H2 in the gas mixture. The F atoms produced react with
HCl via the very fast reaction

F + HClq-‘ZHF + Cl
Since no H2 is used, the quantity of HF measured should equal
the F atoms produced by the flash. At a total pressure of 1.1
atm, they found a dissociation of F2 of about 0.4 + 0.1% inde-
pendent of F2 partial pressure. This constant fraction was
assumed to be the result of the gas being optically thin for
their experimental conditions.

None of these experimental results estimate the uncertainty

77
in optical losses for their systems. Chen et al. found that

density gradients exist within the laser cavity during laser
operation and speculate that resulting photon scattering may
represent a significant loss mechanism.

The carefully defined initial conditions of Nichols et al.
(23) make their results attractive for comparison with model
predictions. Unfortunately their spectral results appear to
be incomplete. Lasing on the v=6~=5 and 5J-4 bands are not
reported and their apparatus apparently lacked sufficient
sensitivity to detect lasing from several J levels reported
by other researchers. Uncertainties inherent in flash photo-
lysis of F2 suggest that the results of Parker and Stephens
are more appropriate than those of Suchard et al. for a detailed
comparison of time resolved spectra.

Parker and Stephens' experiments were performed with an
initial gas composition of F2:H2:He:02 = l:l:lO:0.25 at 35 torr
pressure with an active medium length of 15 cm. Mirror
separation was 60 cm with an output coupling mirror of 80%
reflectivity. Initial F2 dissociation was estimated at 2.5%.
Rate coefficients for the following comparisons were chosen
from the results of previous sections. Rotational relaxation
was based on the modified relaxation rate coefficients of
Wilkins (58). The recommendations of Bartoszek et al. (79) were
used for the endothermic cold pumping reactions. Because the
deactivation results of Section 2.6.2 suggest that v2 dependent
VR rates do not provide sufficient deactivation from high
vibrational levels, the most recent recommendations of Cohen (91)

2.3

are used. These VR rates increase approximately as v The

78
product rotational state distribution was chosen to be 50% of
the value expected for a minimum energy defect reaction.

The results of comparing the prediction to the experimental
results of Parker and Stephens are found in Table 2.6 and Figures
2.17-2.19. Although the model predicts smaller peak power than
was experimentally observed, predicted energy was greater than
the experimental value. The comparison of band energies in
Figure 2.19 demonstrates that although the model compares
favorably for the lower three bands, excessive P-branch lasing
occurs for high J levels in the hot bands. The time resolved
spectra of Figures 2.17 and 2.18 reveal that these high J levels
may also be responsible for the extended pulse duration predicted
by the model. Parker and Stephens' results suggest that the
hot bands shut-off before the cold bands. This behavior is not
predicted by the model.

Because of the considerable uncertainty in initial condi-
tions for the experimental results, it is useful to consider
model sensitivity to these parameters before assessing the
origins of these discrepancies. Table 2.6 and Figure 2.20
summarize the effect of increased HF, increased F—atom con-
centration and increased threshold on model predictions.

Adding 1% HF to the initial gas mixture decreased laser
energy by nearly 60% as well as decreased peak power and pulse
duration. Although energy for all transitions decreased, the
effect was especially pronounced for the v=lI—0 band and the
lower J transitions for the v=2 -1 band, as would be expected
from the presence of ground state HF.

Increasing the initial F2 dissociation will increase

79

Table 2.6 Comparison of model predictions with experiment

 

Peak Power Energy Time to Pulse
Peak power Duration
(W/CC) (J/R) (nsec) (nsec)
Parker & Stephens
Experimenta 185 1.77 1.0 6.0
Simulation 271 2.33 2.9 11.8
Simulation with
increased initial HF 113 0.94 5.2 10.0
Simulation with
increased F-atom
dissociation 385 2.71 4.7 9.0
Simulation with
increased threshold 230 1.80 2.3 11.0
Simulation with
80% JMIN 239 1.86 3.01 11.3

aGas Mixture: 0.05F:0.95F2:1H2:10He:0.2502;

Cavity Conditions: RO=0.8, RL=1.0, L=15cm, £=60cm

T.=300K, P.=36 torr
1 l

80

Figure 2.17 Time resolved spectra of Parker and Stephens
experiment. The length of a line indicates
the duration of a transition with lower level
v,J and the dot indicates the time of
maximum power.
Gas Mixture:
0.05F:0.95F :lH :1OHe:0.2502: Ti=300K, Pi=36 torr
Cavity Conditions:
Roa0.8, RL-l.0, L=15cm, R-60cm

81

 

 

 

 

 

 

 

 

v J
IO
:39"
5
0
IO
_.'°_'_
-o——-
4 +-
0
l0
_..:_°_—
-qr—————
3 "'
0
IO .___
_-:——*———
2 .7:—
.2"—
O
' ——..":!:“
K) £~5*____-
.____.—A;‘
i _.._""'"—
:3:-
0
.______.______
I0 i
__.—._—_
t
O +-
t
O l 1 4 1 L l
O 5 IO
Time (usec)

Figure 2.17

82

Figure 2.18 Time resolved spectra of model predictions
Gas Mixture:
0.05F:0.95F :lH lOHe:0.25N2;
Cavity Conditi ns:
Ro=0.8, RL=1.O, L=15cm, £=60cm

Ti=300K, Pi=36 torr

83

 

 

 

;_b-b-#pbb-—Lh——br-bb-p_ -——-.b—prb-b .bb-np-pp

 

 

 

 

 

 

 

 

 

 

 

 

..
.
. _

 

 

 

 

 

 

 

 

 

 

r:

 

 

 

 

 

 

p .p-b-

P

+

 

.-

 

 

 

 

 

phthnb

7.

 

 

tom

om

om

om

nv

m

rt.
nu

m

nu

l0.0

5.0
Time (nsec)

Figure 2.18

84

Figure 2.19 Comparison with experiment: distribution of
band energy

Gas Mixture:

0.0SF:0.95F :12 :1OHe:0.25N2; Ti=300K’ P.=36 torr
Cavity Conditions: 1

R080.8, Rle.0, L=15cm, £=60cm

Figure 2.20 Effect of uncertainty in initial conditions
on distribution of band energy. Gas and

cavity conditions identical to those in Figure
2.19 except as noted.

(a) Initial gas mixture includes 1% HF

(b) Initial F/F2 ratio increased;
0.07F:0.97F2:1H2:10He:0.25N2

(c) Threshold gain increased 25% by decreasing
RL to 0.76

Energy (J/ l)

O.|5

 

 

1'"
33

Parker 8 mStephens
Experlme

 

 

   
 

 

 

 

P
C
O.iO __—
0.05 :
TI 5:4
0 b /5/.:-\ 4 +3
0 l0
J
' Model Pmdictions
0J5:- Standard Case 2“"
0.10 :-
.
0'05 T 3+2 4+3
0+: __/5,‘
O to
J
0J5 C. Model Prediction 2.”
: 80% JMIN
O.lO 1
0.05 1
o .

 

 

 

 

86

(a)

2+l

 

 

 

0

Increased F/FZ

P

0J5 *-

 

l0

 

 

pP-pn O
m 0
Q

.3. .92.”.

(C)

m

+FP..EL.....-

nv

0.

 

:u
Au

0

 

Figure 2.20

87

reaction rates, hence decrease pulse duration and increase
peak power. Based on the model predictions of Hough and
Kerber (17), initial F2 dissociation was increased from 2.5%
to 3.5%. Although the above desired effects were observed,
energy also decreased. Very little effect on band energy is
produced.

Threshold was increased by 25% by increasing the Optical
losses through mirror RL. The result was to decrease laser
energy and pulse duration. The relative contribution Of
lasing from high J levels in all bands was decreased. Increased
Optical losses decrease lasing energy by two methods. Not
only does decreasing the reflectivity Of RL decrease the
fractional output from the output coupling mirror, but the
resulting higher threshold gain makes it more difficult for
lasing to occur.

It would appear that uncertainty in a single initial
condition cannot account for the discrepancy between the
experiment and theory, although the uncertainties taken
together might explain much of the behavior.

The prediction of excessive lasing and pulse duration is
traced to P-branch lasing from high J levels (J=8-12) in the
hot bands. Although the VR mechanism is capable of populating
these high J levels, it has previously been noted that the VR
mechanism does not have a strong effect on pOpulations below
J=13. Increasing the v dependence Of the VR mechanism has a
pronounced effect on lasing from the hot bands, but this
decreases lasing from all transitions within these bands. A

selective decrease for high J levels is desired. Because of

88
the strong evidence that rotational relaxation decreases for

rotational quantum number, it does not appear that this
mechanism can explain the Observed lasing.

The ability Of VV energy exchange to populate high
vibrational levels must be considered as a mechanism to explain
the prediction Of excessive hot band lasing. As has previously
been described, the model assumption that rotational levels do
not change during VV reactions produces larger equilibrium
constants for the VV reactions than are suggested by the
trajectory calculations Of Wilkins (58). Because the exo-
thermic reactions have the ability to transfer vibrational
quanta from low vibrational levels to higher levels, the
present formulation may promote excessive lasing from the hot
bands. To test this hypothesis, VV rates were decreased by
a factor Of ten. If the exothermic reactions were effecting
high vibrational level populations, this decrease should
significantly reduce hot band lasing. The results showed no
change in hot band lasing and it can be concluded that other
mechanisms are responsible for this behavior.

The hot pumping reaction may be responsible for excessive
lasing from high J levels. The total hot pumping rate coef-
ficient as given by equation 2.19 is discussed in Section
2.7.1. The distribution over vibrational levels v=0 to 8 is
recommended by Cohen's (77) 1972 review of HF kinetics. The
nascent rotational distribution comes from the chemiluminscence
measurements Of Polanyi and Sloan (19). The hot pumping
reaction has not been studied as extensively as the cold pumping

reaction and as a result, is not known with as much certainty

89
(66). Cummings et a1. (89) have found good agreement between
theoretical and experimental laser performance using the cold
pumping reaction and suggested that the hot reaction kinetics
are more suspect than those for the cold reaction. They
conclude that experimental and theoretical predictions Of the
H2 + F2 reaction could be resolved if the rate Of the hot
pumping reaction was slower thanthe value suggested by Cohen.
Another possibility is that the relative vibrational pumping
distribution for the hot reaction used in the models is not
correct. Classical Monte Carlo calculations by Wilkins (38)
indicate that no molecules are formed with v<2 or v>7. Low
vibrational levels Observed in experiments are attributed to
VV energy exchange and pumping by the cold reaction. High
vibrational levels observed were presumably the results of VV
processes.

It is not clear that changes in the nascent vibrational
distribution can explain the time history for hot band lasing
reported by Parker and Stephens (22). They Observed hot band
lasing to reach threshold after cold band lasing and to quench
well before cessation of lasing from v<4. The threshold
behavior is expected from the initial F-atom concentration
initiating the cold reaction and is correctly predicted. The
experimental observation Of early quenching for the hot bands
is not consistent with accepted rate data for the pumping
reactions. In a typical lasing mixture, quasi-equilibrium
between F and H atoms is quickly reached and the temperature
steadily increases with time. For example, an F:H ratio of

1:6 and temperatures in excess of 1000K can be expected late

90
in the laser pulse. The equilibrium constant for the cold
and hot pumping reactions at 700K are approximately 2.8 and
1.7 x 108, respectively. It is evident that the hot pumping
reaction is expected to promote lasing from the hot bands
long after the cold pumping reaction is no longer pOpulating
the lower vibrational bands. Parker and Stephens' results
are inconsistent with this result. Even very strong vibrational
deactivation from high v levels cannot eXplain this time history.
The only other reported time resolved spectra for the hot bands
is from Suchard (21). His observation of hot band lasing late
in the pulse does not agree with Parker and Stephens.

In these comparisons with experiment, model computations
assumed that the rotational states populated by VR were only
one-half the values for minimum energy defect reactions.
Subsequent calculations in Section 3.5 suggest that higher
rotational levels should be populated. Simulations in which
rotational states populated are 80% the values for minimum
energy defect reactions produce rotational populations in
agreement with laser fluorescence experiments. Although the
results of Section 2.4.3 indicate this change in product
rotational states will not significantly influence the distri-
bution of P-branch energy, a simulation Of Parker and Stephens'
experiment was performed with this change to assess the
effect. The results are found in Table 2.6 and Figure 2.19.

.As was expected, this change in VR product rotational states did
not significantly effect band energy distribution. Although

higher rotational states are produced, the predominate effect

91
is on levels well above those involved in P-branch lasing.

Little eXperimental work has been done to compare rota-
tional and P-branch lasing in a chemically pumped laser. Chen
et al. (45) suggest rotational lasing may be as much as 10%
of the pulse energy of an atmospheric H2 + F2 laser, although
they were not able to determine whether it occurred before,
during or after P-branch lasing. Cuellar et al. (47) noted
time histories for both rotational and P-branch lasing in HF
elimination reactions at 62 torr pressure. They noted that
threshold times were earlier for rotational lasing but rota-
tional transitions had durations comparable to P-branch lasing
transitions. The first rotational line to lase was v=0,

J=13-12 at 0.2 usec and was quenched at 21 nsec. In comparison,
P-branch transitions for the lowest two bands reached threshold
at about 2.8 usec and had a duration Of 18 usec. Upon raising
argon partial pressure to 242 torr, all but the v=l, J=l4-13
rotational transition were quenched. These time histories

have not been confirmed by other researchers.

As has been noted previously, model predictions of rota-
tional lasing are especially sensitive to rotational relaxation
rates. Model predictions based on the modified rotational
relaxation rate coefficients of Wilkins' (58) were used to
make the following observations. Even at 20 torr pressure,
the model predicts only limited rotational lasing. Typically,
.rotational lasing energy is less than 0.5% of total pulse energy
and is quenched even before P-branch lasing reaches maximum

power. Energy and duration for rotational lasing can be

92
increased by decreasing the overall rotational relaxation
rate but little experimental evidence exists to justify such

a change.

93

2.8 The Effect of Initial Conditions on Laser Performance

The sensitivity of laser output characteristics to
variations in gas composition and cavity conditions is of
importance in Opthmizing laser performance for experimental
systems and in testing the validity of model predictions.
In the previous section it was pointed out that considerable
uncertainty exists in the initial conditions of experimental
results. Because the prediction of laser performance is
strongly dependent on specification Of these conditions, an
investigation of important laser parameters is justified.
Total gas pressure, initial HF concentration, initial F—atom
concentration, initial hydrogen concentration and threshold
gain are among the most important parameters and their effect
on laser performance is considered here.

For the model computations, the standard reaction rate
coefficients are defined as follows. Rotational relaxation
is given by the modified rate coefficients Of Wilkins (58).
Endothermic rates for the cold pumping reaction are determined
from the rate coefficients measured experimentally by Bartoszek
et a1. (79). The product rotational states for VR deactivation
are distributed about the J level which is one-half of the
value predicted for a minimum energy defect reaction. The
rate coefficients for VR reactions are from Reference 91 and
increase approximately as v2'3.
.2.8.1 Effect of Total Pressure
The effect Of total gas pressure on laser performance

was assessed for an initial gas composition Of 0.02F:0.99F2:

94

1H2:20He at 300 K. The predicted pressure dependence of
maximum lasing power and total energy are illustrated in
Figure 2.21. As expected, power increases approximately as
the square of pressure. This relationship reflects the fact
that binary reaction rates scale proportional to the square
of pressure. Predicted total lasing energy increases linearly
with pressure because the energy available to lasing is pro—
portional to the chemical energy Of the gas. At very low
pressures fewer transitions reach threshold and the lasing
energy is less than would be expected. The effect of three
body recombination reactions and pressure line broadening
are not observed for pressures less than one atmosphere.
2.8.2 Effect of Initial HF Concentration

The effect of initial HF concentrations on laser perfor-
mance was studied for a total pressure of 20 torr with initial
composition identical to the previous computations. Initial
HF concentration up to 0.5% of the total gas pressure were
tested. Total hydrogen and fluorine atoms for the system
were held constant as initial HF was changed. It is expected
that lasing energy will decrease as the degree of prereaction
is increased. The eXperimental studies of Suchard, et al. (20)
showed that lasing from the 1-0 band is especially suppressed
in the presence of ground state HF.

Results for the computations are found in Figures 2.22
_ and 2.23. Both peak power and total energy for the system
drOp rapidly as initial HF concentration increases. Time

to peak power decreased and pulse duration decreased as initial

95

Figure 2.21 Effect of gas pressure on predicted laser
performance
Gas Mixture:
0.02F:0.99F2:132:208e; Ti=300K
Cavity Conditions:
Ros0.7, RL=1.0, L=10cm, £=lOcm

(a) Total P-branch lasing energy
(b) Maximum P-branch lasing power

Energy (J/l)

P am (VI/cc)

96

 

 

 

 

Iozr
(a)
to' .
to°-
lO'"
'02 1
to1 not (0°
Prensa (otrn)
m5r
lo‘- (b)
m3”
mar
m‘r
ld’z ‘ i
n D’ p°
Pnuunimm)

Figure 2.21

97

 

 

 

6 .. _l
l— -‘ OJO

4 — .. g
’5 >
o O
E A 5
C
&. DJ
4) " 7 c»
g .s
O. .1 m
E 0.05 3
:5 .c:
E 2 - - 8
g _ E
.o
2 - a.

-— A

J

O 1' J J l O
0 0.) 0.2 0.3 0.4 0.5

Initial HF (°/e)

piqure 2,22 Effect of initial HF concentration on peak power
' and total energy
Gas Mixture:
0.02F:(0.99-0.llx)F :(1-0.llx)H :20He:0.22xHF
where =%HF; Ti=300 , P1820 torr
Cavity Conditions:
RO=O.7, RL=1.O, L=10cm, £=10cm

Time to Maximum Power (nsec)

Figure 2.23

98

 

 

 

"' " 60
5 F. “5C)
8
4»F' - 4C)
3
C
.9
8
3 e- r 30 5
O
a)
.59.
3
D.
2'.. 4 2C)
- I0
0 I l i I O
O O.) 0.2 0.3 0.4 0.5

- initial HF (°/o)

Effect Of initial HF concentration on time to

maximum power and pulse duration

Gas Mixture:
0.02F:(0.99-0.11X)F7:(1-0.11X)H2:20He:0.22XHF
where x=%HF; Ti=300K, Pi=20 torr

Cavity Conditions:
RO=O.7, RL=1.0, L=10cm, £=10cm

99

HF increased. For low levels of prereaction, the greatest
decrease in power is observed in the 1-0 band but at high
HF concentrations, collisional self-deactivation begins to
decrease lasing on higher bands. Lasing in the 1-0 band
and all bands above 3-2 is quenched when initial HF concen-
tration reaches 0.5%. This corresponds to 5.5% prereaction.
2.8.3 Effect of Initial F-Atom Concentration

The effect of initial F-atom concentration on laser pulse
characteristics was examined. With 20 torr total pressure, the
F/F2 ratio was changed while maintaining the total F-atom con-
centration. The results are given in Figures 2.24 and 2.25.

As the figures illustrate, increasing the F/F2 ratio

increases peak power while decreasing the pulse duration.
The higher F-atom concentrations increase pumping reaction
rates and result in higher powers. These faster reaction
rates consume the reactants more quickly and shorter pulse
durations result.

Kerber and coworkers (17,24) predict that a maximum in
the energy should occur at P/F2=0.2 but this value is expected
to be a function of initial gas and cavity conditions. The
results reported here do not establish a maximum at such low
ratios. Attempts to establish a maximum at higher ratios
were unsuccessful. The trends in peak power and pulse duration
do not suggest a maximum is to be expected. The lasing energy
is the integrated power over time, hence a maximum in the
energy is expected when peak power increases more slowly

than pulse duration decreases. These trends do not deve10p

100

300 - '4 LS

200

Maximum Power (W/cc)
l
'5
P-branch Losing Energy (J/I)

 

 

 

 

I00 - 0.5
o ' o
o 0.5 L0

F/ F2

Figure 2.24 Effect of initial P-atom concentration on peak
power and total energy
Gas Mixture:

éé;F:§§§F2:lHZ:20He

where x=F/F2; Ti=300K, Pi=20 torr
Cavity Conditions:

Ro=0.7, RL=1.0, L=10cm, £=lOcm

101

 

 

 

 

 

- so

3 .—
33
‘3 420 r
.. 2 — 8
0) (D
3 £3
0— I:
g e
.g g
g o
5 $
.9 - IO as
a) l _
.E
..-

I
o 0.5 LO

F/F2

Figure 2.25 Effect of initial F-atom concentration on time
to maximum power and pulse duration
Gas Mixture:
2X F- 2 F °lH '20He
747:? ”5:33? 2'
where x=F/F ; T.=300K, Pi=20 torr
Cavity Conditions:
Ro=0.7, RL=l.O, L=10cm, £=lOcm

102

in the present model computations.

This lack of a maximum in energy is the result of the
increased vibrational deactivation rates incorporated in
the present model. In Section 2.4 it was discovered that
excessive hot band lasing could be reduced by incorporating
abstraction rate coefficients measured by Bartoszek et al.
(79) for the endothermic cold pumping reactions and using the
v?”3 dependent VR rates of Cohen (90). This reduced lasing
from the hot bands to approximately 3% of total lasing energy.
In comparison, hot band lasing predicted by the models of
Kerber and coworkers (17,24) was as high as 25% of total
lasing energy. As the F/F2 ratio is increased, the cold
pumping reactions begin to dominate the pumping reactions
and eventually hot band lasing is completely quenched. The
loss of hot band lasing decreases total lasing energy and
is responsible for the maximum in energy reported in References
17 and 24. Because the present rate coefficient mechanisms
yield model predictions with very little energy in the hot
bands, quenching of these bands does not significantly effect
total energy and no maximum is predicted.
2.8.4 The Effect of Threshold Gain

Initial gas and cavity conditions were maintained at
the same values as the previous computations except that the
reflectivity of the output coupling mirror was changed to
change the threshold gain. As threshold gain is increased,
the number and duration of transitions which reach threshold
is expected to decrease and peak power and total energy should

be reduced. The sensitivity of laser performance to threshold

 

’4
C)
L)

 

 

 

 

30 r —
S
3
(\9 s.
Q)
5 L5
35 2*
5 '35
G- 20 — —- 3
E .r:
3 0
.§ 5
x h-
E '9
CL
”‘ -‘ 0J0
no ‘ o
0.0| 0.02 0.03
aihr (cm'I)
It 1
0.8 0.7 0.6
L (cm)
I J 1 l A 1 1 J
IO 8 6 4
R0
Figure 2.26 Effect of threshold gain on peak power and

total energy
Gas Mixture:

O. 02?: 0. 99? :20He; =300K, P. =20 torr
Threshold gain is Heffected Thy changing either
R0 or L; RL is assumed to be 1.0

104
gain is illustrated in Figure 2.26. As expected, peak power
rapidly decreases as threshold gain is increased. It should
be noted that, although power generated within the laser is
maximum for threshold gain equal to zero, this requires the
mirrors to be 100% reflecting and power through the output
coupling mirror will be zero. The results of Figure 2.26.
are consistent with the predictions of Kerber et al. (17).
2.8.5 Effect of H2/F2 Ratio

The effect of initial Hz concentration on laser perfor-
mance was evaluated while maintaining total gas pressure at
20 torr and holding F/F2 constant at 0.0202. The effect on
maximum power and total lasing energy are illustrated in
Figure 2.27. Laser energy is quite sensitive to initial H2
concentration for H2 lean mixtures while variations were
small for H2 rich mixtures. This observation is in concurrence
with the predictions of Kerber et a1. (17).

Although peak power does not strongly decrease until
Hz/F2>2, the large decrease in energy for H2/F2>l can be
attributed to a rapid decrease in pulse duration for H2 rich
mixtures. The decrease in pulse duration corresponds to the
decrease in hot pumping reaction rates for large Hz/F2 ratios.
As explained in Section 2.7, the hot pumping reactions
normally persist after cold pumping has shut off and hence
the hot pumping reaction determines pulse duration. The
quenching of hot band lasing predicted above is attributed
to the relative decrease in F2 molecules required by the hot

pumping reactions.

Maximum Power (W/cc)

 

105

P-branch Losing Energy (J/I)

 

 

30"
-O.2
20-
-O.|
IO-
l l -
O 2 3 4 5

Figure 2.27

Hz/FZ

Effect of initial HZ/FZ on peak power and
total energy
Gas Mixture:

0.02 g. l r . x . .
l.02+x‘ '1.02+x* 2'“1.02+xH2'2°He'
T.=300K, P.=20 torr,where.x = 112/I"2

Cavity Conditions:
RO=0.7, RL=l.O, L=10cm, £=lOcm

 

3. AMPLIFIER SIMULATION: TEMPORAL
AND SPATIAL DEPENDENT MODEL

3.1 Introduction

 

The rate of change of flux for an optical beam through
an absorbing or emitting gas is given by

8f 3f
“ems—>2

3 = co(x,t)f (3.1)

where f = flux
c = speed of propagation through the medium
a(x,t) = absorption or emission coefficient for the
the medium (gain factor)

Closed form solutions for this partial differential
equation cannot be obtained. Complex numerical schemes are
normally applied to obtain solutions.

In calculations for a pulsed laser device, space averaged
fluxes and gains are assumed and the spatial dependence of
equation (3.1) is removed. For cw lasers, an assumption of
steady state Operation results in time independent quantities
and the partial derivative with respect to time is removed.

In both cases, the partial differential equation reduces to

an ordinary differential equation to which relatively simple
numerical integration techniques can be applied. For a
chemically driven amplifier, however, both spatial and temporal
behavior must be included and solutions are more difficult to

obtain. Although the reaction mixture may be initially

assumed homogeneous, the transit of a laser beam through the

106

107
cavity results in spatially dependent absorption while rapid
chemical reactions produce time dependent gain for the medium.
A simplified solution may be obtained by choosing a
Lagrangian description for the system in which the path of a
flux element is considered. Hence, the flux equation becomes

2f.
dt

= caf (3.2)
and the input signal which is to be amplified by the medium

is divided into incremental flux elements. Equation (3.2)

must be solved for each of these flux elements as it passes
through the gain medium. Each integration time step,.At, moves
the flux element a distance.Ax = c.At through the gain medium,
hence the space dependence of the flux can be obtained in
increments of c.At through the medium.

In practice, solutions are obtained by dividing the
amplifier into sections of lengthmAx and choosing the step
size, A¢.=.Ax/c, such that a flux element is integrated
across the section in one time step. Hence, across a given
section of lengtthx the gain seen by a flux element is constant.
The gain can change after each time step and is the result of
the combined action of chemical reactions and radiative inter—
actions with the previous flux element which passed through
the section. For each section, an input signal is entered at

At time intervals until the pulse has passed through the
section. The output from the section is the result of succes-
sive integrations of equation (3.2) using successive flux
elements for the initial condition. The resulting values

represent the time dependent flux after amplification through

108
the section and must be retained for subsequent integration
across the next section. Hence, a cavity of N sections requires
N computer runs. Because a flux element takes time At = Ax/c
to pass through a section, computations for subsequent sections
must include an appropriate delay after initiating the chemical
reactions before the pulse enters the section. The delay is
equal to the time required for the leading flux element to
pass through all previous sections. Integration across a
section continues until the train of flux elements for the
pulse has exited the section.

The choice of section length determines integration step
size and ultimately sets the limitation on pulse width that
can be reasonably integrated through the amplifier sections.
Because gain is assumed constant through a section, absorption
must not be so great that gain significantly changes and section
lengthqiox, must be chosen accordingly. The pulse width is
then constrained by the number of integration steps appropriate
to the problem.

The amplifier model was developed from the laser oscillator
model described in Section 2. P-branch lasing for the first
six vibrational bands with twelve transitions per band is
allowed to occur. The major revision to the oscillator model
was the substitution of space averaged fluxes (calculated in
the previous time step) with single pass fluxes (entered from
the previous section) in the flux equation (3.2). Optical
losses are neglected in the amplifier hence threshold gain
has not been included in the flux equations.

The amplifier model is useful for two types of simulations.

109
It can be used to assess the performance of chemically driven

amplifiers. The complex time dependence of gain for a chem—
ically reacting and optically active gas makes necessary de—
tailed calculations to determine the extractable power. The
amplifier model also provides an opportunity to simulate laser
fluorescence experiments. Although Section 2.6.2 considered
simulation of such experiments to the extent that vibrational
deactivation could be calculated for an initially excited state,
pumping into this state via absorption was not simulated. With
the amplifier model, absorption of pump radiation by a quiescent
gas and subsequent relaxation can be simulated. The simulation
is especially useful for comparison with experiments designed to
detect vibrational to rotational energy exchange reactions.

Both applications of the amplifier model are considered below.

3.2 Chemically Pumped Amplifier Simulations

 

For high power applications it is often desirable to in-
crease beam power by passing the laser pulse through an ampli-
fying medium. The complex interaction of chemical pumping
and saturation effects which determine amplifier spatial and
temporal prOperties justifies detailed simulation of amplifier
chemical kinetics. Several properties of the chemical ampli-
fier are characterized in the discussion that follows. P-branch
lasing fluxes predicted for a 380 torr mixture of 0.02F:0.99F2:
1H2:20He at peak power are used for input fluxes.

3.2.l Sensitivity of Model Predictions to Integration
Step Size

As with other numerical problems, integration step size,

.At, is chosen as large as is consistent with accurate numerical

110

results. In the amplifier simulations there is an additional
constraint because the length of a cavity section,AAx is
related to step size by Ax = c At. Because gain is assumed
constant for an integration step, it is necessary that.Ax
be chosen small enough to satisfy this condition.
Several calculations were carried out to determine the sensi-
tivity of model predictions to the length of cavity section.
Lengths of 0.5, 1.0, and 3.0 cm were tested for a 380 torr
mixture of 0.02F:0.99F2:1H2:203e which was initiated when a
1.0 W/cm2 pulse entered the cavity. Pulse duration for these
calculations was limited to 2.0 nsec. No significant varia-
tions in predicted output were obtained. Gains for individual
transitions were typically of the order of 0.02 cm-l. Although
the results suggest that even larger cavity sections could
successfully be employed, it should be recalled that the
gain can increase rapidly in a chemically pumped amplifier.
Smaller section lengths are required for large gain systems
to satisfy the condition that gain be constant across a section.
3.2.2 Spatial and Temporal Dependence of Power

As a flux element passes through an amplifier, the power
associated with it is expected to increase exponentially with
distance as given by equation (3.1). If chemical pumping is
large, the gain subsequently encountered by the flux element
will be greater than in the previous cavity sections and the
change in power across the amplifier may be greater than

simple exponential. In a pulsed chemical amplifier the gain

is dependent on the rates of pumping, collisional deactivations,

induced emission, and absorption. Beam power can increase or

lll

decrease depending on which of these processes is dominate. If
pumping into a level is greater than the rate of induced emission
and collisional deactivation out of the level, the gain and power
will increase with time. If induced emisSion depopulates a level
faster than pumping replenishes it, gain and power will decrease.

An example in which pumping predominates is illustrated
in Figure 3.1. The initial gas mixture for the amplifier was
0.02F:0.99F2:1H2:20He at 380 torr and was initiated when a
1.0 W/cm2 laser pulse entered the amplifier. The illustration
shows the relative increase in power across the amplifier
with increasing time. Gain for all transitions start at zero
but pumping is so rapid that saturation is not possible at
these low input powers,hence the output power increases with
time. For early times, the power across the amplifier appears
to increase almost linearly because the gain is small, but as
the gain increases with time, a much greater increase in
power occurs. As was described earlier, power is expected
to increase at least exponentially across the amplifier and,
if gain is increasing rapidly with respect to the pulse transit
time, power will increase even faster than exponentially.
This is demonstrated in Figure 3.1. At l.6ns, the ratio of
output power to input power is 1.0023 for an effective gain
of 0.0023 cm-1. A simple exponential increase across a 3 cm
cavity is expected to increase power by a factor of 1.0069
but in fact the power output power (which requires 0.067
nsec to transverse the additional 2 cm) is increased by a
factor of 1.0077. Therefore, for systems in which saturation

effects are not important, a lower bound in power amplification

112

Power (Output/input)

 

 

 

 

l.8 ns
IIID8i-
l.6ns
L4ns
ILXNSr'
LZns
LOns
ifK)4-
0.8ns
0.6ns
iLXIZ-
O.4ns
e—OCIZnS
LOOO ' 1 1 ‘
C) I 2 3

Position (cm)

Figure 3.1 Time and space dependent power for a pumping
dominated chemical laser amplifier
Gas Mixture:
0.02F:0.99F2:1H2:20He; Ti=300K, Pi=380 torr
Input Pulse:
l.0W/cm2 of 2 nsec duration; no entrance time
delay

113

for very long gain lengths can be established by extrapolating
from a unit length assuming an exponential increase in power.

Amplification effects in which induced emission is large in
the medium may be observed by entering a very large flux into
the amplifier or delaying the input pulse until chemical
pumping can no longer replenish the gain. The result for a
combination of these two methods is illustrated in Figure 3.2.
The cavity and gas conditions are identical to the previous

5 2 and

example except that the input power is l x lO W/cm
entrance time is delayed for 2.0 usec. Again, the exponential
increase in power across the amplifier is evident. It is
observed that output power decreases monotonically with time
because the rate of chemical pumping is no longer greater
than the rate of induced emission. To better illustrate this
effect the output power from the 3 cm amplifier is plotted
as a function of time in Figure 3.3. The output power decreases
with time until pumping equals the rate of induced emission.
The output power becomes constant when the gain reaches a
steady state value. This effect is different than that for
a simple saturable medium for which input flux causes the
gain to drOp to zero and the output power approaches the
input power. If the input flux is not too large, pumping
can sustain gain at some positive steady state value.

As input power increases, this steady state value for
gain approaches zero. For large enough input power, the

ratio of output power to input power approaches one and the

amplifier resembles a simple saturable medium. This phenomena

Power (Output/ Input)
5
l

 

114

/

 

IC) J l l
C) I 2 3
Position (cm)
Figure 3.2 Time and space dependent power for a saturation

dominated chemical laser amplifier
Gas Mixture:
0.02F:0.99F2:1H2:ZOHe; Ti=300K, Pi=380 torr
Input Pulse:
1x10S W/cm2 of 2 nsec duration; 2 usec
entrance delay

115

2.0—

Power (Output/ Input)
{.11
l

 

 

LO 1 l I
0 0.5 LO L5

Time (nsec)

Figure 3.3 Time dependence of power for saturation dominated
3cm chemical laser amplifier
Gas Mixture:
0.02F:0.99F2:1H2:20He; Ti=300K, Pi=380 torr
Input Pulse:
1x105 W/cm2 of 2 nsec duration; 2 nsec
entrance delay

116

is illustrated in Figure 3.4 for a 0.1 cm amplifier with
initial gas mixture of 0.02F:0.99F2:1H2:20He at 380 torr
pressure. Pulse entrance time into the amplifier was delayed
for 0.5 nsec. For very low levels of input power, gain
remains approximately constant and equation (3.2) can be
integrated to give

f/fo = ecuct (3.3)
where fo is the input flux. Therefore the ratio of output
power to input power is constant at nonsaturating power
levels. This feature occurs in Figure 3.4 for input power
less than 1 x 105 W/cmz. For greater input powers, the ratio
drops rapidly as the medium is saturated. At about 1 x 109
W/cm2 saturation is complete and the output power is approx-

imately equal to the input power.

Analytical expressions for gain saturation exist for
sufficiently simple systems. It is useful to compare the
predictions of these equations with the results of the amplifier
model. For steady state pumping, gain saturation for a three
level system with a homogeneously broadened transition is

given by (93)
ao(w)

(1(a)) = ——-7—T-T- (3.4)
l + Pm PS m
where ao(w) is the zero signal gain of the medium, Pm is the

input power per unit area for the transition with wavenumber

w and
2
4nhc w3
P (w) = (3.5)
S (R7tspont)¢(w)
where h is Planck's constant, c is the speed of light and

o(w) is the Voigt line width, t is the spontaneous

spont

117

 

 

 

1.015 i-
I.Ol0 '-
E
L.
Q)
5
O.
‘\
"5
O
L.
Q)
5 Laos--
CL
IKKXD l I l J J
I03 I05 107 :09
Input Power (W/cmz)
Figure 3.4 Dependence of saturation on input power

Gas Mixture:

0.02F:0.99F2:1H2:208e; Ti=300K, P =380 torr
Length of amplifier = 0.1 cm, pulse duration
0.2 nsec; 0.5 nsec entrance time delay

118
emission lifetime of the transition and r is the total lifetime
of the upper state of the transition.
Saturation of the P2(7) transition is representative of
the model calculations. The gain on this transition dropped

1 5 -1

from a zero signal value of 0.23 cm- to 7.6 x 10- cm for

an input power of 7.5 x 107 W/cmz. Lifetime for the upper
state is estimated to be 0.39 usec based on collisional
deactivation rates. Spontaneous emission lifetime is 5.2 msec
and the line width is 293 cm. From equation (3.4) the input
power required to decrease gain on the P2(7) as prescribed
above is 4.34 x 104 W/cmz. This value is considerably less
than predicted by the model calculations. The assumption of
homogeneous broadening is very good at 380 torr pressure and
is not the source of the large discrepancy. The error is
attributed to two other assumptions employed in deriving
equation (3.4). The assumption of constant pumping rates is
not satisfied for a pulsed chemical device and the large
number of vibrational and rotational levels participating in
lasing and chemistry are not well represented by a simple
three level system.
3.2.3 Effect of Entrance Time on Output Energy

The gain of the lasing medium is strongly dependent on
time even in the absence of saturating flux. Gain increases
rapidly after the chemical chain is initiated and continues
to rise until collisional deactivation and induced emission
exceed the chemical pumping rate. For sufficiently long

times chemical equilibrium is approached and the gain for

119
all transitions becomes negative since the medium can no
longer sustain an inversion. The dependence of output energy
on entrance time is examined in Figure 3.5 for a 380 torr gas
mixture of 0.02F:0.99F2:1H2:20He. The percent increase in
pulse energy for a 1cm amplifier is illustrated for input
powers of 1 W/cm2 and 1x105 W/cmz. Percent increase in output
energy for longer amplifier lengths can be estimated by
assuming an exponential increase in energy across the amplifier
as discussed in Section 3.2.2. For example, a pulse delay of
1.0 usec produced a 21% increase in energy for a 1.0 W/cm2
input pulse of 10 nsec duration. This represents an effective

gain of 0.194 cm-1

and a 10 cm gain length will amplify input
energy by at least 590%.

It should be noted that in Figure 3.5 amplification did
not increase significantly for time delays greater than 1 nsec
for a 380 torr gas mixture. After this time, pumping no
longer exceeds collisional deactivation and amplification will

eventually decrease as the gas approaches chemical equilibrium.

3.3 Simulation of Laser Fluorescence Experiments Designed
to Detect VR

Very simple simulations of laser fluorescence experiments
were described in Section 2.6.2 and were designed to predict
vibrational deactivation rate coefficients. Hinchen and Hobbs
have applied the method of laser fluorescence to measurements
of rotational relaxation (49,69) and more recently have
completed preliminary experiments designed to detect vibrational
to rotational energy transfer (92). They attempted to directly

verify the occurence of VR transfer by detecting the presence

120

 

 

O
O

20 - A
>~
2’ A
o
C
LIJ
a) O
..‘L’
a A
.55 o
8 0 1.0 W/cc
g A Ixi05 W/cc
£5 ”DP-
‘5
a:
i

‘OAL l _J

O LO 2.0

Time Delay (nsec)

Figure 3.5 Effect of entrance time delay on pulse amplification
Gas Mixture:
0.02F:0.99F2:1H2:203e; Ti=300K, Pi=380 torr
Input Pulse: 2 nsec duration

121
of nonequilibrium populations in high rotational states.
They measured the rotational distribution in v=0 of HF by
probing this manifold after pumping population into v=l
with pulsed laser radiation. The pump laser operated on
the P1(4) transition with pulse energy of 0.05 mJ which could
be Q-switched down to pulse durations of 40 nsec (FWHM).
The probe laser consisted of cw radiation on a single v=1—0
wavelength which could be tuned to resonance with P-branch
transitions in the manifold. Both probe and pump beams were
3 mm in diameter and were passed colinearly through the axis
of a 40 cm absorption cell. Although the uncertainties in
their data were large, Hinchen and Hobbs (92) found rotational
populations exceeding equilibrium values for J>10.

Laser fluorescence experiments are able to deduce
rotational populations by measuring gain of P—branch transi—
tions. The gain for a probed transition is proportional to
the degeneracy-weighted population difference between the
upper and lower transition states (see Equation 2.5). In
probing very high rotational levels with weak signals, the
upper state population will be much smaller than the lower
state population and eXperimental studies assume that the
gain is prOportional to the lower state population. It is
well known that nonequilibrium pOpulations decay approximately
exponentially with time (see Appendix B). Hence, laser
fluorescence experiments can determine rotational relaxation
rate coefficients from the rate of absorption of probe

radiation as well as relative rotational populations from

122
instantaneous gain measurements.

The amplifier model allows simulation of Hinchen and
Hobbs' experiment. Computations were performed for a pump
laser pulse passing through a 1 cm amplifier section.
Although Hinchen and Hobbs' experiment was performed with a
40 cm absorption cell, the qualitative nature of their pre-
liminary results do not justify a more detailed and costly
simulation using a 40 cm cell versus a 1 cm cell at this time.

The experimental limitations on detector sensitivity
which make necessary such long absorption lengths are not
encountered in numerical simulations and the VR effect should
be discernable for a smaller cell. It is evident from the
discussion of Section 2.4.3 that the results of simulations
will be strongly dependent on the definition of product
rotational states. The choice of a minimum energy defect
VR reaction will result in rotational populations centered
about J=15. Because greater VT contribution to VR,T was
expected, the product rotational states were distributed
about the J level which was one-half the value expected for
a minimum energy defect reaction. This VR scheme was used
in a simulation employing a 0.04 nsec square wave pump pulse.
No perturbation from a Boltzmann distribution was detected
in the v=0 manifold except for a small dip at J=4 where the
pump laser removed population. The absence of a VR hump
about J=8 is understood by estimating the expected rate of
change of HF(v=0,J=8). Even after 0.1 usec, the largest

population to be expected in HF(v=0,J=8) is only 1% greater

123
than the Boltzmann population. Rotational relaxation out of
this level will further decrease the expected population.
A second simulation was performed using a minimum energy
defect VR reaction. The resulting rotational populations for
v=0 after 0.04 nsec are plotted with the data of Hinchen and
Hobbs in Figure 3.6. Both sets of data show clearly defined
humps in the populations at high rotational numbers. It is
evident that the use of a minimum energy defect VR reaction
in the simulation causes the VR bump to occur at higher
rotational levels than is suggested by the experimental data;
the humps peak at J=ll and J=l4 for the experiment and simu—
lation respectively. Unfortunately, the experimental data
does not extend above J=13 but clearly a nonresonant VR reac-
tion is suggested by comparing the available data with the
simulation. A‘VR reaction for which the product rotational
states are approximately 80% those of a minimum energy defect
reaction might produce rotational distributions in better
agreement with the experimental data. Such a choice of
product rotational states was employed in the model and the
effect on rotational populations after 0.04 nsec appears in
Figure 3.6. The perturbation in the Boltzmann distribution
is not as pronounced as for the minimum energy defect reaction
because of larger pOpulations in the lower rotational levels.
Better resolution is obtained by allowing the system additional
relaxation time. After 0.5 msec, the VR hump is clearly
discernable (Figure 3.6). The hump has shifted to lower

rotational levels and compares favorable with the experimental

Rehiivo Population

Figure 3.6

124

 

 

 

 

 

0
IO r
1’Bmuhmu
llamumuusnMadJmN
«CEMuuc
IO’1 - I Simiaflm: 80% o: JMIN
inOD4uuc
iiflmme:&N6MJMW
caOSuun
Io’z -
A
A
'03 i.
g A
IO" -
a
O
10'5 -
I“ JVLL 1 1 L 1 _1_ 1
9 to ll )2 i3 i4 l5 )6
FMMWmmiLmel
Simulation of laser fluorescence experiment

to detect VR
Gas Mixture: 0.1 torr HF at 300K
Input Pulse: 442 W/cm2 of 40 nsec duration

125
results. The differences in relative magnitudes between the
simulation and the experimental data simply reflects differences
in absorption lengths and times allowed for relaxation. There
is also considerable uncertainty in the eXperimental values.
The relative distribution over the product states is the

important result shown by the model.

126

4. SUMMARY AND CONCLUSIONS

Computer models for the H2 + F2 chain reaction laser
oscillator and laser amplifier have been developed. Both
models include comprehensive formulations of chemical reaction
kinetics thought to be important to the HF laser, including
rotational nonequilibrium. There is clear experimental and
theoretical evidence that rotational nonequilibrium is
important in the Operation of the HF chemical laser. Models
that attempt to simulate rotational nonequilibrium with
pumping, lasing, and RT energy transfer reactions have improved
predictive capability but have not fully resolved laser
performance characteristics. In particular, the experimental
observations of rotational lasing and P-branch lasing from
high J states cannot be explained with existing nonequilibrium
mechanisms. Vibrational to rotational energy exchange has
the ability to populate high rotational states and has been
used to qualitatively explain high J state phenomena in HF.
Although conclusive experimental evidence of VR in HF does
not yet exist, its potential ability to eXplain rotational
nonequilibrium behavior in HF motivated the inclusion of it
in the present model formulations to assess its effect on

laser performance.

127

Vibrational to rotational energy exchange is found
important for the prediction of rotational lasing. A simpler
VT model for vibrational deactivation predicted no rotational
lasing transitions to reach threshold. The effect of rota-
tional lasing was found to be kinetically similar to rotational
relaxation. Like rotational relaxation, rotational lasing
produced slightly higher energies for P-branch transitions
and spread band energy over fewer transitions.

The effectiveness of the VR mechanism in populating high
rotational states is strongly dependent on its relative
contribution to the total kinetic rate. As VR rates are
increased, P-branch lasing rapidly decreases while rotational
lasing increases. Conversely, increasing rotational relaxa-
tion rates decreased rotational lasing sharply while P-branch
lasing increased.

There is little experimental or theoretical evidence to
suggest the proper translational contribution to VR,T energy
transfer. The effect of different partitions between VT and
VR on predicted laser performance was considered. It is
known that increasing the VT contribution produced lower
product rotational populations and slower back reactions for
VR,T. This results in smaller predicted pulse duration and
laser power. Because the major effect of VR is on rotational
levels above J=12, the relative distribution of P-branch
lasing energy changes very little. An acceptable choice of
product rotational states for VR reactions was suggested
from computations with the amplifier model. An amplifier

model was develOped that simulated the absorption of the

128
pump transition and subsequent VR deactivation. The amplifier
model was used to simulate the laser fluorescence experiments
of Hinchen and Hobbs (92). In their experiments, a pump laser
on the v=1-0 band of HF produced excited vibrational pOpula-
tions which then relaxed to the vibrational ground state. It
was found that a VR reaction which populated rotational states
approximately 80% those of a minimum energy defect VR reaction
best simulated the experimental results of Hinchen and Hobbs,
but the preliminary nature of their results made further
conclusions impossible. The occurence of product rotational
states with the above distribution would produce rotational
nonequilibrium at relatively high rotational levels and would
result in RV rate coefficients of comparable magnitude to the
VR rate coefficients.

It was found that decreasing rotational relaxation rates
with respect to VR rates increased rotational lasing. Several
researchers (68-70) have measured rotational relaxation rates
at low J levels which suggest these rates are very fast.
However, measurement of relaxation rates which decrease
rapidly with rotational quantum number (34,58,69-71) might
result in relaxation rates which are apprOpriately small at
high J levels. It was found from model calculations that
although increased J-dependence for rotational relaxation
rate coefficients was effective in sustaining larger nonequi-
librium populations at high J levels and increasing rotational
lasing, the effect was not as pronounced as an overall
decrease in rotational relaxation rate.

The prediction of excessive energy in the hot bands of

129

HP appears to be the result of insufficient vibrational
deactivation from these levels. The two mechanisms suspected
to be responsible for this behavior are the endothermic cold
pumping reactions and the vibrational dependence of the VR
rate coefficients.

The endothermicity of the cold pumping reaction for
v>3 provides a channel by which HF(v>3) can be removed by
H-atoms. The rate coefficients for this reaction measured
by Bartoszek et al. (79) are faster than previously accepted
values. Model computations proved these rates were effective
in substantially reducing P-branch lasing from the hot bands
and showed hot band lasing to be sensitive to small changes
in the endothermic cold pumping reactions. Increasing gas
pressure resulted in increasing contributions to lasing from
the hot bands and it was necessary to identify a mechanism
capable of depopulating high vibrational levels late in the
laser pulse. The vibrational quantum number dependence cf
VR self deactivation rate coefficients was investigated
because of the uncertainty in these fast rate coefficients.

The v-dependence of vibrational deactivation may represent
the greatest uncertainty in HF kinetics because of the great
difficulty of distinguishing it from other collisional
mechanisms. Not only do experimental measurements of vibrational
deactivation for v>l include contributions from both VV and
VR,T processes, but vibrational relaxation may be coupled to
rotational relaxation. An attempt to deduce the correct

vibrational deactivation rates was made by simulating

130
vibrational deactivation experiments in which the exponential
decay of excited vibrational level populations is used to
measure vibrational deactivation rate coefficients. The
empirical quenching coefficients were predicted to be nearly
equal to the VR rate coefficients if it was assumed that the
VR reactions populated rotational levels one-half those of
near resonant reactions. A v-dependence as large as v2‘3
is suggested by these results. It was discovered that decreas-
ing the energy defect for the VR reactions resulted in diver-
gence of predicted empirical quenching coefficients and the
VR rate coefficients used to predict them. The large energy
defects associated with strongly nonresonant VR reactions
result in corresponding RV reaction rate coefficients which
are very small. Even the formation of very large nonequili-‘
brium populations at high rotational levels will not produce
significant RV reactions and vibrational deactivation will
be independent of rotational relaxation. It was shown,
however, that near resonant VR reactions produce significant
RV rate coefficients and a strong coupling between vibrational
deactivation and rotational relaxation was demonstrated. If
rotational relaxation is not fast enough to remove rotational
populations formed by VR, RV reactions will decrease the
overall vibrational deactivation rates and VR rate coefficients

larger than v2'3

will be required to properly describe
vibrational deactivation. If the simulations of Hinchen and
Hobbs' (92) laser fluorescence experiments are correct in

predicting VR reactions which populate rotational levels 80%

131
those of near resonant reactions, then significant coupling
between vibrational deactivation and rotational relaxation
is to be expected.

The ultimate goal of laser modeling is to correctly
predict laser performance. Although several experimental
studies of the HF laser exist, considerable discrepancy exists
between their results and reflect the extreme sensitivity of
laser performance to gas composition and cavity conditions.
The relatively well-defined results of Parker and Stephens
(22) were chosen for comparison with the model. Although the
model compares favorably for the lower three bands, excessive
P-branch lasing occurs for high J levels in the hot bands.

It does not appear that uncertainties in a single initial
condition can account for this discrepancy. Faster deacti—
vation of high rotational levels in the hot bands of HF could
produce the desired effect but cannot presently be justified.
Although changes in the nascent vibrational distribution could
significantly decrease hot band lasing energy, it has been
shown that Parker and Stephens' observation of early quenching
for the hot bands is not consistent with accepted rate data
for the pumping reactions.

Unfortunately, little experimental work has been done to
compare rotational and P-branch lasing in a chemically pumped
laser. Although Chen et a1. (45) suggest rotational lasing
may be as much as 10% of total pulse energy, presently accepted
rotational relaxation rates (68-70) do not predict rotational

lasing energy much above 0.5% of.total pulse energy.

132

A parametric study of the effect of total gas pressure,
initial HF concentration, initial F-atom concentration,
initial H2 concentration and threshold gain on laser perfor-
mance was completed. The results compared favorably with
other theoretical studies.

The development of a laser amplifier model from the
laser oscillator model allowed simulation of the laser
fluorescence experiments of Hinchen and Hobbs and provided
information important to understanding the mechanism of VR
energy transfer. It was also valuable in assessing the complex
time and space dependence of laser amplifier devices. Ampli-
fier performance is complicated by the competing processes of
chemical pumping and saturation effects. When input powers
are sufficiently small, chemical pumping exceeds the rate of
laser induced emission and gain can be maintained or even
increased with time. Saturation of the medium, in which
output power approaches input power, was observed by either
raising the input power or decreasing the rate of chemical
pumping. Under these conditions, induced emission depletes
gain faster than pumping restores it. It was discovered
that saturation of a chemically pumped amplifier is different
from a simple saturable medium. Although gain begins to
decrease as a high power beam passes through the amplifier,

a point will be reached where pumping equals the rate of
induced emission. If this occurs at some positive, nonzero
value of gain, the ratio of power out to power in will

asymptotically approach a value greater than one. For

133
sufficiently large input powers, the amplifier will resemble
a simple saturable medium and the output power will equal
the input power. For a given chemical system it was also
demonstrated that amplification of a laser pulse was strongly
dependent on the entrance time of the pulse into the amplifier,
as expected.

The study was successful in assessing the effect of VR
energy transfer on the HF chemical laser. Nevertheless,
confirmation of several predicted phenomena and a complete
understanding of some mechanisms awaits more comprehensive
experimental studies. Insufficient experimental data on the
pulsed H2 + F2 chemical laser exists to resolve theory with
experiment. Of special interest is time resolved spectra from
the hot bands of HF. The number and duration of these transi-
tions must be known before the prediction of excessive energy
from these bands can be resolved. Although rotational lasing
has been experimentally studied in some detail, an understanding
of its relationship to P-branch lasing in the 32 + F2 laser is
imcomplete. Presently accepted rotational relaxation data do
not predict significant rotational lasing, but this result
also awaits experimental confirmation. Simulation of the
preliminary laser fluorescence experiments of Hinchen and
Hobbs (92) was successful in suggesting the distribution of
product rotational states for VR deactivation. More detailed
simulations are possible when further experimental results

are reported.

APPENDICES

APPENDIX A

RATE COEFFICIENTS FOR THE H2 + F2 CHEMICAL LASER

134

APPENDIX A

RATE COEFFICIENTS FOR THE H2 + F2 CHEMICAL LASER

The rate coefficients for the H2 + F2 chemical laser
are from a variety of sources. Dissociation, pumping,HF-H2
VV reactions and Hz-H2 VT reactions were originally formu-
lated from the recommendations of Cohen (77) (see Table A.l).
The pumping reactions are parameters of this study and
changes in these rate coefficients are discussed in the text.

Rate coefficients for rotational relaxation, VR,T and
VV reactions were originally formulated from the trajectory
calculations of Wilkins (58). These rate coefficients were
also parameters of the study and specific formulations are

presented in the text.

Table A.l Rate Coefficients for the H2 + F2 Chemical Laser

Excluding VR Mechanisms

Reaction Collision Partner M Forward Ratea

Dissociation Recombination

F2 + M i'F + F + M all species KfFZ - 5.*1013¢—35.3e
H + H + "12 32(0) + M all species, Kfuz - 1.*1019T-1

2.932, 203
HF(O) + M.3 a + r + M all species foF(o) _ 1_2*1018T-1e-135.se
HF(l) + MI: E + r + M all species foF(1) . 1.2*10181'1e'12‘°5°
HF(Z) + M 3 H + F + M all species foF(2) . 1.2*1018T-le-113.66
HF(3) + M1: H + F + M all species KfHF(3) _ 1-2*1018T-1e-103'3e
HF(A) + M.I a + F + M all species foF(4) + 1.2,..1018,1.-1e-93.39a
HF(S) + M 2'H + F + M all species KfaF(5) + 1.2a10181’1e’83-969
HF(6) + M 2 H + F + M all species KfHF(6) . 1,2a10181’le'74‘979
HF(7) + M iiu + p + M all species KfHF(7) _ 1.2*1018T-1e--66.438

Cold Pumping

F + H2(0) z:l-Il"(l) + H K§(1) . 2.6*1013e'1‘66
F + 32(0) 241112) + H x§(2) - 8.8*1ol3e'1'66
F + H2(O) z:HI“(3) + H K§(3) - 4,4a1013e’1°69
r + M2(O) I’gp(4) + H K:(4) _ 1.46*1013T-0.1O7e-ll.340
F + 32(0) 2 HF(S) + H x§(5) - 2.17*1013T'°'1°7e'2°°786
F+ 32(0) 2 HF(6) + H K;(6) , 3.76*1013T-0.107e-29.826

aAll rate coefficients in this table are from Cohen (77).
Other rate coefficients considered in this study are discussed
in the text. The value of 8 is 1000/RT where R is the gas
constant and is equal to 1.98725 kcal/mol-K

135

 

Table A.l (continued)
Reaction Collision Partner M Forward Rate

Hot Pumping
u + F2 2 HF(O) + r x:(0) - 1. 1*1012e 4 2°
H + F2 1 HF(l) + F K:(1) =H5*1012 2 49
a + F2 1 33(2) + r x?(2) - 3. 5*1012e 2 26
a + F2 : HF(3) + p x?(3) - 3. 6*1012e 2 ‘9
H + 22 I HF(A) + F K?(4) - 1. 61"1013e'2 ‘9
a + 32 I HF(S) + F K:(5) - 3. 6*1013e 2 ‘9
a + F2 2 HF(6) + F K2(6) . 4. 8*1013e 4 29
H + F2 2 HF(7) + F K2(7) a 5. 5*1012e 2 49
H + F2 I'HF(8) + F K2(8) - 2. 5*1012e 2 “6

HF-Hz vv
HF(O) + 32(1) 1 HF(l) + H2(0) KfHF'H2(o,0) - 9*1010
um) + 112(1) I mm + 112(0) K'f“ “2a 0) - 2.9m12
HF(Z) + H2(l) I'HF(3) + 32(0) fHF'Hz(2,0) . 9*1012
HF(3) + 32(1) 2 HF(4) + H2(0) foF'“2(3,0) - 2*1013
HF(O) + “2(2) 1 HF(l) + H2(1) KfHF'“2(o.1) = 9*1011
HF(l) + H2(2) I HF(Z) + u2(1) KfHF H2(1, 1) a 2.9*1012

HZ-HZ VT
H2(1) + M 2 H2(O) + M all species KfHZ-H2(l) - 2.S"‘10'-4T2°3

an ,an
2
H2<2) + M I H2(1) + M all species KfHZ-H2(2) = 5.o*1o“‘1~‘"3
4H2,4H

136

APPENDIX B

DERIVATION OF THE EQUATION FOR CALCULATING EMPIRICAL
QUENCHING COEFFICIENTS

137

APPENDIX B

DERIVATION OF THE EQUATION FOR CALCULATING EMPIRICAL
QUENCHING COEFFICIENTS

The experimentally observed rate at which a vibrational
level is collisionally deactivated can be described by an
empirical quenching coefficient. Derivation of the relation
giving this coefficient is as follows. Vibrational deacti-
vation of HF(v) is given by the reaction

2:92”

HF(v) + M :: HF(V') + M (3.1)
kb(v')

where M is the collisional species. In deactivation studies,
a single collisional species is usually made dominate. In
the self deactivation studies of interest here, it is assumed
that the gas consists of HF in Boltzmann equilibrium at the
translational temperature with a small perturbation in the
vibrational level of interest. Therefore, deactivation of
HF(v) occurs primarily by collisions with ground state HF.
Both VV and multiquanta VR,T reactions must be included in

the rate equation for vibrational deactivation HF(v):

igéfle- g'ka') [HF(v)] [M] + ‘é'kbww [HF(v')] [n] (3.2)

138

where kf(v') combined VV and VR forward rate coefficient

combined VV and VR backward rate coefficient

kb(V')
An expression for the empirical quenching coefficient
can be obtained by assuming that the product state HF(v') is
never far from its equilibrium concentration, HF*(v')
Substituting this value for HF(v') and recalling that the

forward and backward rate coefficients are related by

k
f _ HF*(v‘)
r; "EHF‘TVTJJ (3'3)

equation B.2 can be written as

d . :

IEEWH = -[HF(v)] [M123 ka ) + [@252ka ) [11sz (v)] (8.4)
or
d[HF(v)]

__ . a -1 1
"empz "f‘V’ W W at ”3'”

where the asterisk indicates equilibrium concentrations.

 

Therefore, the empirical quenching coefficient, kemp'
includes the effect of VV and multiquanta VR reactions. This
formulation does not allow for the formation of large nonequi-
librium product concentrations and will underestimate kemp if
back reactions become significant.

Equation (B.5) can be modified to a form useful for
analysis of time dependent population data. Because

[HF(v)] >> HF*(v)] equation (_B.5) can be written as

d HF(v) _ _
W - kemp [M]dt (B . 6)

which can be integrated to give

log[HF(v)]=-1T3+ A (13.7)

139
_ '1
where '1' - (k pEMJ)

A integration constant

Applying linear regression analysis to the natural
logarithm of pOpulation versus time effectively produces a

time averaged empirical quenching coefficient.

LI ST OF REFERENCES

11.

12.

13.

14.

140

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