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This is an authorized facsimile
and was produced by microfilm-xerography
in 1980 by
UNIVERSITY MICROFILMS INTERNATIONAL
Ann Arbor, Michigan, U.S.A.
London, England

A THEORETICAL INVESTIGATION
OF ROTATIONAL NONEQUILIBRIUM
PHENOMENA IN THE PULSED HYDROGEN

FLUORIDE LASER SYSTEM

BY

Keith Emery

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the degree of

MASTER OF SCIENCE
Department of Electrical Engineering

1979

ABSTRACT

A THEORETICAL INVESTIGATION OF ROTATIONAL NONEQUILIBRIUM
PHENOMENA IN THE PULSED HYDROGEN FLUORIDE LASER SYSTEM

BY

Keith Emery

A rate equation model of a pulsed H2 + F2 chemical laser is
used to examine the relative effect of rotational nonequilibrium'
mechanisms on laser performance. This computer model yields the
population time histories for the first seven vibrational and thirty
rotational level of HF. The model also yields the time resolved
spectra for the first twelve vibrational - rotational P branch
transitions. The major thrust of the present work was to evaluate the
relative importance of vibrational to rotational (VR), vibrational to
vibrational (VV), rotational to rotational (RR) and rotational to
translational (RI) energy transfer on laser performance. The character
of the spectra is significantly different from that of other F + H

2
and H2 + F2 models which assume that rotational equilibrium is present
in VR,T and VV reactions. The effect of single and multi vibrational
quanta VR are assessed. The effect of increasing the multiquanta VR

- rate was found to populate higher rotational levels of HF, increase P

branch pulse energy and duration.

ii

ACKNOWLEDGEMENTS " '

I would like to thank the many people who have helped bring this
thesis to completion. My adviser, Dr. Ron Kerber has provided much
needed technical advice and assistance. I would like to thank Captain
Larry Rapognani for providing information on the CYBER 176 operating
system at Kirkland Air Force Base in Albuquerque, New Mexico where the
model was executed.

I would also like to thank the Division of Engineering Research
at Michigan State University for assistance in preparing the manuscript.
Finally, I would like to thank Betty Hutcheson at Colorado State

University for assistance in typing this thesis.

iii

TABLE OF CONTENTS

PREVIOUS STUDIES OF ROTATIONAL
NONEQU ILIBRIUM PHENOMENA o o o o o o e o o o o o o o 1

1.1 Nonequilibrium Pumping. . . . . . . . . . . . . 2
1.1.1 Cold Pumping . . . . . . . . . . . . . . 3
1.1.2 Hot Pumping. . . . . . . . . . . . . . . 6

1.2 Rotational Population Transfer. . . . . . . . . 10
1.2.1 Experimental Studies . . . . . . . . . . 10
1.2.2 Rotational Lasing. . . . . . . . . . . . 12
1.2.3 Theoretical Studies. . . . . . . . . . . 15

1.3 Vibrational Relaxation of HF . . . . . . . . . 18
1.3.1 Theoretical Investigations of VR . . . . 19
1.3.2 Theoretical Investigations of VV . . . . 26

STUDIES OF ROTATIONAL NONEQUILIBRIUM IN
THE PULSED H2 + F2 SYSTEMS . . . . . . . . . . . . 32
2.1 Time Resolved Spectroscopy. . . . . . . . . . . 32
2.2 Computer Modeling of Rotational

Nonequilibrium Phenomena; . . . . . . . . . . . 36

MODEL FORMULATION. . . . . . . . . . . . . . . . . . 42
RESULTS AND DISCUSSIONS. . . . . . . . . . . . . . . 55

4.1 Effect of Vibrational to Rotational

Relaxation on HF Population . . . . . . . . . . 55
4.2 Effect of Vibrational to Rotational

Relaxation on the Power and Energy. . . . . . . 61

SUMMARYOOOOOOOOOOOOOOOOO00.000.00.00... ..... 69

APPENDIX A: RATE COEFFICIENTS FOR THE

H2 +F2 CHEMICAL LASER. . . . . . . . . 70

APPENDIX 3:" DERIVATION or xrad . . . . . . . . . . 33

APPENDIX C: DESCRIPTION OF COMPUTER

SIMULATIONS FOR THE PULSED

HF CHEMICAL LASER . . . . . . . . . . . 93
APPENDIX D: MINIMUM ENERGY DEFECTS FOR

VIBRATIONAL ROTATIONAL

RELAXATION. . . . . . . . . . . . . . . 109

REMENC ES 0 O O O O O O O O O O O O O O O O O O O O 11 S

Table

1.1

1.2

1.3

1.4

1.5

1.6

1.7

2.1

3.1

4.1

A.1

C01

13.1

iv

LIST OF TABLES

Relative Pumping Efficiencies for
115+}!2 (0',J') K§(V) I-IF(v)+H. . . . . . . . . . .

#

RelativehPumping Efficiencies for
H + F Rf(v) HF(v) +»F . . . . . . . . . .

Q

Observed HF Rotational Laser Transitions . . .

Multiquanta VR Rates of Wilkins [51]

HF(v,J) +'M I HF(v',J') + M. . . . . . .
Vibrational to Vibrational Rate Coefficients

in 1012 cc/mol-sec at 300°K for HF(v-l) +

HF(v'-1) +HF(v) +HF(v'-O). . . . . . . . . . . . .
Single Quanta VV Rates of Wilkins [51]

HF(v,J1) + HF(v*,J2) I HF(v-1,Ji) +

HF(V*+1,J£)O o o o o o o e o o o o e o o o o

Vibrational Dependence of the Total VR, T
Rate coefficient 0 O O O O O I O 0 O I O I O 0 O 0

Comparison of Computer Models Incorporating
Rotational Nonequilibrium Phenomena in HP
or DF 0 O O O O O O O O O O O O O O O O O O O O O O 0

Relative Rotational Relaxation Efficiencies. . . . .

Possible VR Relaxation Distributions About
Jmax for HF(v,J) + M I HF(v',J') +-M. . . . .

Rate Coefficients for the H2 + F2 Chemical
Laser. . . . . . . . . . . . . . .

Nomenclature . . . . . . . . . . . . . . . . . .

Most Probable Multiquanta VR Transitions
and Their Energy Defect AB in cm.1 . . . . . . . .

Page

13

14

27

29

30

33

49

56

72

109

LIST OF FIGURES

Figure

1.1 Rotational distribution for the cold
pumping reaction. The solid line is the
distribution of Polany;et a1. [11] and the
dashed line is a 300°K equilibrium rotational
distribution . . . . . . . . . . . . . . . . . .

1.2 Rotational distribution for the hot pumping
reaction. The solid line is the distribution
of Polany; et a1. [26] and the dashed line
is a 300°R equilibrium rotational distri-
bution ... . . . . . . . . . . . . . . . .

1.3 Calculated rotational transition probabilities
versus rotational level [51]. The equation for
line A is 0.788 exp( 0.1357 J). The equation
for line B is 0.0406 exp(-0.1376 J). . . . . .

1.4 A typical VR relaxation path. Multiquanta
vibrational relaxation is allowed. The values
of J' range from J -4 to J +2 where

max max
J is the value of J' with smallest
max
energy defect. . . . . . . . . . . . . . . . . .

1.5 Schematic of a typical VV reaction path. The
specific reaction shown is HF(2,10) + HF(0,4)

zar(1,10) + HF(1,4) + 102.8 cm’l. If zero

rotational energy is assumed then AE -

171.2 cm’1 . . . . . . . . . . . . . . . . . .

2.1 Comparison of time-resolved spectral output.
(a) Experimental data reported in Ref. 69
for a 1 32:1 F2:60 He mixture at 50 Torr. .

2.2 Time resolved spectra of Nichols et a1. [69]

for a 1:1:10 mixture of H2:F2:N2 at 90 Torr

with F/F2 - 0.232 and 0.002 HP. The active

medium length is 30 cm and the mirror spacing
is 113 cm, R0~. 0.7 and RL - 1.0 . . . . . .

vi

LIST OF FIGURES - CONTINUED

Figure

2.3 Time resolved spectra of Parker and
Stephens [70] for a 1:1:10:0.25 mixture

of F2:H2He:02 at 35 Torr with F/Fz-SZ.

The active medium length is 15 cm and
the mirror spacing is 60 cm, Ro - 0.8

and RL - 1.0. The lower level of a P

branch transition is denoted by v and

J. The length of the lines correspond

to the duration of lasing on a given line.

The dot on the line corresponds to the

time at which the maximum power was

achieved on that line. . . . . . . . . . . . .

A.l Pumping rates for the cold reaction. The
pumping rates employed in the present
formulation have been suggested by
Cohen [24] . . . . . . . . . . . . . . . . . .

A.2 Rate coefficients for the hot pumping~
reaction. The present formulation uses the
rate coefficients that have been suggested
by Cohen [24]. . . . . . . . . . . . . . . . .

A.3 The temperature dependence of selected
vibrational to vibrational exchange rate
coefficients. The VV rates calculated by
Wilkins [51] and given in Table 1.5 are
employed in the present formulation. . . . . .

A.4 Vibrational to rotational, translational rates.
the that for HF+HZ, H, F2, H, Wilkins [51]

rates are scaled according to Cohen's [22]
rates and are employed in the present
formulation. . . . . . . . . . . . . . . . . .

C.1 ' A warnier-Orr [90] diagram of the Fortran
program MSUHFVR, a comprehensive computer

model of the pulsed Hz + F2 laser system . . .

l. PREVIOUS STUDIES OF ROTATIONAL NONEQUILIBRIUM PHENOMENA

Laser action from the hydrogen fluoride (HF) molecule initiated
by flash photolysis of UF6-H2 mixtures was reported by Kompa and
Pimentel [1] in 1967 shortly after the first HCl chemical laser was
demonstrated in 1965 by Kasper and Pimentel [2]. At about the same
time, Deutsch [3] reported similar laser action resulting from initia-
tion by pulsed electrical discharge from Freon-H2 mixtures. Since the
advent of this first laser important advances in the development and
understanding of the HF laser have occurred.

With the discovery of the chemical laser, it was recognized that
one essential advantage of this laser was its potential for high
efficiency, and that this efficiency could be realized through the

use of a chain reaction to achieve a population inversion. The chain

consists of a cold pumping reaction,

F + H2 (v,J') : HF(v,J) + H AH - 31.9 kcal/mole (1.1)

and a hot pumping reaction,

H + F2 : HP(v,J) + F AH - 97.9 kcal/mole (1.2)

where cold and hot refers to the relative exothermicity of the reactions.
The vibrational energy level of HP is denoted by v, the rotational
energy levels of HFare given by J, and the net change in molar enthapy
of the reaction is denoted as AB. The vibrational and rotational
levels of Hz are denoted as v', J'. We note that the nascent vibra—

tional and rotational distribution of HP is significantly different

from the thermal equilibrium population.

Many initiation techniques for the H2+F chain reaction laser,

2
e.g., electron-beam-irradiated discharge, electrical discharge and laser
photolysis, with a large assortment of experimental apparatus and a
variety of gas mixing procedures have been developed yielding a wide
range of results [4]. A review of the literature by Gross and Bott
[4] indicate that electrical efficiencies of 1602 have been obtained in
converting electrical energy to laser output energy with electrical
initiation [5]. Output energies of 2500 J [6] have been demonstrated
-using electron-beam initiation. Current efforts in high-energy pulsed
systems are directed toward achieving successful large—volume initiation
of the laser medium. The electron beam is considered a prime candidate
for accomplishing this objective because of its ability to deposit
large amounts of energy uniformly over a large volume of the gas mixture.
Several theoretical models for the analysis of the H2+F2 chemical
laser have been developed. The assumption that lasing begins when
gain reaches threshold and that gain equals loss during the lasing
period have been used in the models developed by references [7,8,9,10].
The assumption of thermal equilibrium in the rotational population yields
a rigid J-shifting pattern and multiline lasing in a given branch is

minimal. The study of the role of rotational nonequilibrium.mechanisms

in HF lasers is a major part of this study.

1.1 Nonequilibrium Pumping

The nascent vibrational and rotational distributions for the cold
pumping reaction Equation (1.1) and the hot pumping reaction Equation
(1.2) have been measured by several workers [11,12,13,14,26,27].
Theoretically predicted distributions have also been determined [15,16,

17,18,19,28,29].

1.1.1 Cold Pumping
The nascent vibrational and rotational distribution for the

cold pumping reaction has been measured by Polanyi and coworkers [11.12,
13,14]. Theoretically predicted distributions have also been determined
using classical trajectory [15,16,17,18,19] and quantum mechanical
methods [20]. The measured and calculated relative vibrational distri-
butions of HF produced by the cold reaction are presented in Table 1.1.

The results of Polanyi and Woodall [11] using the arrested relaxa-
tion variation [21] of the infrared chemiluminescence (IRC) technique
indicated that the mean fraction of available energy entering into
vibration, <fv>, rotation, <fr> and translation, <ft> are 0.66,
0.08 and 0.26 respectively. The arrested relaxation method allows the
nascent product distribution‘to be measured before relaxation becomes
significant. The fraction of energy entering into rotation and vibra-
tion is far from the thermal equilibrium distribution at the transla-
tional temperature. The rotational energy distributions observed by
Polanyi, et a1. [11] for the vibrational levels pumped by the cold
pumping reaction are compared with a 300°K thermal equilibrium rotation-
al distribution in Figure 1.1.

Douglas and Polanyi [12] investigated the effect of rotationally
excited H2 on the vibrational and rotational distribution of HF.
Their work showed that rotationally excited Hz has a small but
noticeable effect on the nascent vibrational distribution and no
observable effect on the nascent rotational distribution of HF. Perry
and Polanyi [13] using the infrared chemiluminescence technique obtained
the vibrational and rotational distribution of HP from 77K to 1315K.

Berry [14] has obtained vibrational distributions for the cold pumping

Table 1.1 Relative Pumping Efficiencies for

F + H2(0',J') K;(v) arm + Ha

#

 

 

 

 

Q
J fv(0) fv(l) fv(2) fv(3) Technique Reference
erimental
0 - 0.6 0.4 - Laser Experiment Konpa [22]
0 - 0.07 0.40 <0.53 Laser ’ periment (534K) Parker [22]
0 - 0.23 0.7 - Laser Experiment Krough [22]
0 0.03 0.15 0.50 0.32 Laser Experiment Berry [14}
0 - 0.16 0.57 XJ.27 IRC Polanyi [22]
0 - 0.17 0.56 0.27 IRC Anlauf [22]
0 - 0.14 0.49 0.37 130 Jonathan [22]
0 - 0.17 0.56 0.27 IRC Jonathan [22]
0 - 0.17 0.56 0.27 IRC Polanyi [11]
0 - 0.14 0.55 0.41 IRC Douglas [12]
l - 0.17 0.56 0.27 IRC Douglas [12]
2 - 0.11 0.53 0.33 IRC Douglas [12}
0 0.02 0.15 0.53 0.30 IRC Perry [13]
Theoretical
0 - 0.09 0.61 0 . 30 Semi-empirical Monte Carlo Wilkins [16]
0 - 0.15 0.67 0.18 Semi-classical Monte Carlo Polanyi [17]
1 - 0.31 0.58 0.11 Semi-classical Monte Carlo Jaffe [18)
0 - 0.561 0.438 0.001 Variation theory Monte Carlo Jaffe [22]
0 - 0.016 0.665 0.319 Information Theoretic Suprisal Connor [20]
0 0.001 0.006 0.483 0.510 Information Theoretic Suprisal Connor [20)
Iaaeeeus

0 0 0.17 0.55 0.28 - - Cohen [22,23]
0 0.07 0.15 0.52 0.26 Used in this article Cohen [24]

 

fv(V) cc/nol-sec.

DThe distribution was calculated assuming that the total available energy is 35.40 k cal/mol.
c
The distribution was calculated assuming that the total available energy is 37.40 R cal/mol.

.. a.

‘ .
The forward vibrational pumping rate K;(v) is given by K;(v) - 1.6x10

14

exp ('2400/RT)

.. ,_. -0 -

 

Fraction of available energy entering rotation

 

<f>

I

0.3

<f>

r
<f >

r
Figure 1.1.

 

 

r+n 3 HF(1.J) + H

2

 

n+3 I HF(2.J) + a

2

 

2 4 6 8 10 12 14

  

P+H 3 HF(3.J) + a

2

2 4 6 8 10 12 14

HF Rotational level J

Rotational distrubution for the cold pumping reaction.
The solid line is the distrigution of Polanyi et a1. [11}
and the dashed line is a 300 K equilibrium rotational
distribution.

reaction usingzachemical laser technique which measures the time to
threshold while minimizing the effects of vibrational deactivation. His
results were in agreement with those of Polanyi, et a1. [13].

Muckerman [17], Jaffe, et a1. [18] and Polanyi, et a1. [19] have
calculated the nascent vibrational and rotational distribution of HP
from three dimensional classical trajectory calculations. Connor, et a1.
[20] applied information theory to extract vibrational and rotational
distributions of HF populated by the hot and cold pumping reaction. The
product distribution obtained by Polanyi, et a1. [19] and Connor, et
al. [20] compared favorably with infrared chemiluminescence measurements.
The classical trajectory and quantum mechanical calculations predict
larger rotational energies than have been observed experimentally.

The vibrational distribution recommended by Cohen [22,23,23] will

be used in a comprehensive model of the H +F2 pulsed chain reaction

2
laser system. The reaction rate recommended by Cohen [22,23,24] for the

cold reaction is:
K§(v) = 1.6 1014 exp (-l600/RT) fv(v) cc/mol-sec (1.3)

where R is in units of Kcal/mol-K. This recomendation concurs with
the subsequent analysis of Cummings, et a1. [25]. .

1.1.2 Hot Pumping

The relative vibrational distribution for the hot pumping reaction
has been measured and calculated using various methods. Results of
these calculations are listed in Table 1.2.

Polanyi and Sloan [26] have investigated the product vibrational

and rotational distribution for the hot reaction (1.2) and found that

the mean fraction of energy entering into vibration, rotation and

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translation were 0.53, 0.03 and 0.44 respectively. The rotational
distribution of HF for the hot reaction is presented in Figure 1.2.

The vibrational distribution of the constants measured by Jonathan, et
a1. [27] compared favorably with those obtained by Polanyi, et a1. [26]
using the arrested relaxation method.

The three dimensional Monte Carlo classical trajectory calculations
of Polanyi [28] and Wilkins [29] and the quantum mechanical results of
Connor [20] yield vibrational and rotational energy distributions
which are compared in Table 1.2 to observations of Polanyi [26],
Jonathon [27] and others [22].

In a comprehensive review of the literature, Cohen [22] suggests
that HP is not formed by the hot pumping reaction for vibrational levels
less than three or vibrational levels greater than five. This conclu-
sion is based upon the classical Monte Carlo trajectory calculations of
Wilkins [29]. The experimental results of Polanyi and Sloan [26] were
renormalized so that vibrational levels outside of the v - 3 to 6
manifold are neglected,

Z <fv(v)> .. 1 (1.4)
V

giving the values listed in Table 1.2 which are recommended by Cohen
[22,23]. Subsequent analysis by Cummings, et al. [25] supports the
recommendation of Cohen [22,23]. It is assumed that the forward rate
coefficient for the hot pumping reaction measured by Albright [30] and

recommended by Cohen [22.23.24] and Cummings, et a1. [25] is:

K:(v) 8 1.2 x 1014 exp (-2400/RT) fv(v) cc/mole-sec (1.5)

 

 

 

    
    
 

0.3-
1"
0.2_’
<f > I
r I
g 7
fl 0.1-
u
M
u
0
u
E 0
C 0 2 4 6
O
u
C
9 0.3_
>:
U‘
3
IR
5 I! \
<fr>
0
A
.Q
G
n:
.4
G
>
U
ea
0
8
d 0.3-
u
0
8
h /\
I \
0.2.]! ‘\
<fr> ;
0.1,!
I
0.
0

 

Figure 1.2.

0.3

:HF(1,J)+F 3ar(2,J)+r

/‘ n+r2

H+F2

 

 

 

 

 

0.2
<fr>
0.1
o,
a 10 12 14 o 2 4 6 J 8 10 12 14
0.3 r
a+rziar(3,J)+r ,"\ H+F2:HF(4,J)+F
0.2 I ‘
<fr>
0.1
o,
14
001-
* . ¢
H+F2*HF(5,J) I"\ n+r2+nr(6,a)
0.2
‘3’

0.1

o.

a 10 12 14 o

 

HP Rotational Level J J
Rotational distribution for the hot pumping reaction.
The solid line is the distributio of Polanyi,et. a1.
[26] and the dashed line is a 300 K equilibrium
rotational distribution.

10

1.2 Rotational Population Transfer

1.2.1 Experimental Studies

Once the HF molecule is formed in a vibrationally and rotationally
excited state, rotational relaxation may occur. The rate of rotational
population transfer from a given vibrational and rotational level has
been measured by Polanyi, et al. [31,32], Hinchen and Hobbs [33,34],
Peterson, et a1. [35], Vasilev, et a1. [36], Gur'ev [37] and Emanuel
[38]. Polanyi and Woodall [31] used the infrared chemiluminescence
method to observe the rotational distribution of HCl pumped by the hot
reaction (1.2) with Cl

replacing F Polanyi, et a1. [31] determined

2 2'
that the empirical formula for the rate of rotational relaxation out

of a given rotational level is of the form

P z -BAE/kT _

KJ,J-AJ ' r HF~M 9 (1'6)

.1.
rrP
where Pr is the probability of rotational relaxation, and zHF-M is
the binary collision frequency for unit concentration of HF with M. The
gas pressure is P and Tr is the rotational relaxation time constant.
When rotational relaxation occurs the rotational quantum number J

undergoes a change from J to J-AJ. The change in energy AE due

to rotational relaxation is

AE I E(v,J) - E(v,J-AJ) . (1.7)

The rotational energy in vibrational level v. is denoted as E(v,J).
The parametric constant B approximates the decrease in transition
probability with increasing AJ. For AJ - r1 Polanyi [31] determined
that B was 0.016. Ding and Polanyi [32], using a heated supersonic

primary beam of HCl or HF and a target beam containing He, Ar, Kr,

11

HX(XEF, Cl, Br, I), H25 or propane, examined the infrared emission
from colliding molecular beams. Ding, et a1. [32] observed that the
parametric constant B was also a function of collision energy.
Emanuel [38] investigated the sensitivity of temperature and gas non-
uniformities in chemiluminescent experiments and concluded that these
nonuniformities can have a significant impact on the observed rotation-
al distributions. Vasil'ev, et a1. [36] studied rotational relaxation
of HF with H

D He and Xe. Their results indicated that H

2’ 2’ 2
and D2 depleted the rotational levels of HF, through a resonant
transfer of rotational energy an order of magnitude more rapidly than
He or Xe.

Gur'ev, et a1. [37] obtained rate coefficients for rotational
relaxation of HF(v'0, J-8) by several chaperon gases by monitoring
the relaxation losses that occur when light is passed through a
resonantly absorbing medium. Their rate coefficients for rotational
relaxation of HF(v-0, J-8) by a chaperon M correspond to relative
rotational relaxation efficiencies of 1.0 for MtHF, 0.03 for MPH

2

or D2 and less than 0.005 for M-He, Ar, Kr or Xe.

Peterson, et a1. [35] obtained rotational relaxation rates of HF
from specific vibrational and rotational levels by using a pump laser
to populate a particular level of HF and then observing an exponential
decay out of that level with a probe laser. They assumed that the
rotational relaxation rate is related to the collisional linewidth y

L
of HF by

1

.- ' my; (LB)

‘1'

12

where c is the velocity of light in vacuum and YL is in cm . The
relaxation time Tr was obtained by Peterson, et a1. [39] by conducting
high resolution linewidth measurements of HF.

Hinchen and Hobbs [33,34] observed rotational relaxation effects
in HF using infrared double resonance. This technique employed a
pulsed pump laser to populate a particular vibrational and rotational
level of HF and a continuous wave probe laser to monitor the rotational
relaxation of HF to adjacent HF levels. The relative rotational

relaxation efficiencies for HF, H and He were found by Hinchen,

2
et a1. [33] to be 1.0, 0.1 and 0.03 respectively. Specific relaxation
rates for HF indicated that a mechanism, in addition to rotational
relaxation, was preferentially transferring population to a lower level
(gJ-l,2,3,etc.). This additional mechanism was observed by Hinchen
[33,34] to be lasing between adjacent rotational levels.

1.2.2 Rotational Lasing

Deutsch [40] observed laser action on pure rotational bands of HF
produced by a pulsed electrical discharge in a mixture containing CF

4
and H2. Table 1.3 contains a list of rotational lasing transitions
that have been observed by Deutsch [40], Sirken, et al. [41], Krogh, et
al. [42], Skribanowitz, et a1. [43], Chen, et al. [44], Hinchen [33,34],
Akitt, et a1. [45], Cueller, et a1. [46] and Rice [47]. Rotational

lasing occurs by the following mechanism

Hf(v,J+l) + 1r;i + HF(v,J) + 2 h% (1.9)

where observed values of 1 range from 12 microns to more than 100

microns. Overtone rotational lasing has not been observed.

Table 1.3.

Observed HF rotational laser transitions

 

Wavelength chEV’J) (v:J+l+J) Reference
Micron cm v J

9.381 1065.0 0 31 41

9.960 1004.0 1 30 44
12.262 815.53 1 22 40,45,47
12.767 783.27 2 22 47
13.303 751.71 3 22 47
13.726 728.54 2 20 40,47
13.730 728.33 1 19 40,47
13.774 726.01 3 21 47
13.785 725.43 0 18 40,47
14.336 697.54 1 18 47
14.441 692.47 0 l7 40,46,47
15.017 665.91 1 17 40,47
15.624 660.04 2 17 47
15.785 633.51 1 16 47
16.021 624.18 0 15 40,47
16.422 608.94 2 16 45,47
16.657 603.61 1 15 45,47
16.981 588.89 0 14 45,47
17.328 577.10 2 15 45,47
17.654 566.44 1 14 45,47
18.086 552.91 0 13 45,46,47
18.801 531.89 1 l3 40,46,47
19.369 516.29 0 12 46,47
36.475 274.16 1 6 33,43
42.432 235.67 1 5 33,43
50.800 196.85 1 4 33,43
63.383 157.77 1 3 33,43
84.388 118.50 1 2 33,43
126.41 79.107 1 1 33,43

14

Sirken, et a1. [41] observed laser emission between high rotational

2, CHE-CH2

etc.). Rotational lasing from levels as high as HF(v-0,J-31) and

states of HF by the photolysis of halogenated olefins (CHz-CF

HF(v'l,Je30) were observed. Similar results were observed by Krogh

[42] using ClFx-F Skribanowitz, et a1. [43] and Hinchen, et a1.

2.
[33,34] observed rotational lasing by using a pump laser to populate
the upper level of the rotational lasing transition. Skribanowitz

[43] used a helical transverse discharge HF pin laser with a mixture

of SF6 and H2 as a pump laser. The sample cell used a ring laser
cavity configuration allowing the gain to be monitored in the same or
opposite direction that the light from the pump laser entered. Pres-
sures in the sample cell ranged from 50 millitorr for the P1(3) and
P1(4) pump lines to 6 torr for the P1(8) pump line. The variation
in cell pressure compensates for the changing absorption coefficient

of HF. Skribanowitz [43] estimates that under certain conditions the
incremental gain for rotational lasing to be in excess of 1 cm-l. Using
an infrared double resonance experiment utilizing a separate HF pump
and probe laser Hinchen, et al. [33,34] observed the same transitions
as Skribanowitz [43]. The gain at pressures from 0.007 torr to 0.85
torr ranged from 0.02 cm”1 to a maximum of 1.15 cm“1 for the HF(1,3)
to HF(1,2) transition at 0.17 torr. The gain expression for rotational
lasing is given in Appendix B. The large gain for single quanta
rotational lasing effectively blocks rotational lasing for AJ larger
than 1. Chen, et a1. [44], in an atmospheric pulsed HF laser, observed
indirect evidence of rotational lasing. Their method of observation

involved placing black polyetheylene which absorbs light with a wave-

length less than 15 microns over an energy meter. Chen, et a1. [44],

15

observed that approximately 10% of the total energy of 1.5 joules was
transmitted through the polyetheylene. Akitt, et a1. [45] observed
laser emission at 16 wavelengths between 11 and 18 microns in a pulsed
3. N2 and H20 were found to decrease the
lasing energy while He increased the total emission energy by a

discharge containing FE

factor of 4. Cuellar, et a1. [46] observed laser emission from
rotational transitions of HF formed through chemically activated
CH3CF3 and from photo-excited CHZCFZ.
rotational lasing by exploding wires in F

Rice, et a1. [47] observed
2 and an inert gas mixture
at pressures up to 500 torr.

There is clear experimental evidence that rotational lasing can
and does occur. Lasing can occur from very high rotational levels
indicating a nonequilibrium rotational distribution that is well
above what would be expected from pumping. One mechanism that can
populate high rotational levels is vibrational to rotational energy
transfer. .

1.2.3 Theoretical Studies

Sentman [48] developed a computer model to determine PrzHF-M and

B in Equation (1.3) and found that for rotational relaxation of HF by HF to

be PrzHF-HF.2°45 and B-0.359. Sentman [48] found that PrzHF-HF and
B determined from Hinchen's [34] experiment were comparable to those
Obtained from Polanyi and WOodall's [31] data.

The rotational relaxation of polar gases such as HF was investi-
gated by Zeleznik [49] using two dimensional classical dynamics to
investigate the rotational relaxation of polar molecules. Zeleznik

[49] found that the change in rotational energy per collision averaged

over the number of collisions per unit time can be expressed as

16

d HF(v,J) ‘ HF(v) Boltz(v,J) - HF(v,J)

dt 1
r

(1.10)

where HF(v,J) represents the hydrogen fluoride and Tr corresponds

to the mean time for rotational relaxation, and HF(v) represents the
total rotational population of a specific vibrational level of HF.

Boltz (v,J) is the Boltzman distribution for the rotational populations

at the translational temperature T and is given by

2J+l e-th(v,J)/kT
(1:0)

where the value of the rotational partition function 0:(T) may be

Boltz (v,J) - (1.11)

taken from the data given in reference [7] and E(v,J) is the rotation—
al energy calculated from Dunham coefficients [50].

From three dimensional trajectory calculations Wilkins [51,52]
obtained detailed rate coefficients for rotational to rotational (RR)
and rotational to translational (RT) energy transfer. Wilkins [51,
52] found that the rate coefficients for de-excitation from the upper
rotational levels were much smaller than rate coefficients for de-
excitation from the lower rotational levels in agreement with the
exponential dependence of the energy defect in Equation (1.6), the
anharmicity of the energy levels of HF, and observations of Polanyi and
coworkers [31,32]. The rotational transition probabilities calculated
by Wilkins [51] AJ-l,2, and 3 are presented in Figure 1.3. The
probabilities for rotational relaxation with J greater than 15 were
extrapolated from the probabilities between J - 10 and Jl-lS. The
extrapolation technique was a mean square regression fit to an exponen-

tial curve.

Rotational relaxation probablllty 91.1”)

17

    

 

 

1.0»
- O nglh)
o D E’_‘..J-2(J)
_ c)<3<3
<3 c)<) A: P.., 3(J0
C)
C) ' v
.. 00 ar- pwflm + ng3(ol
<3 C)<3
Dell. 0
a-
_ C1
0 it
_ . A
[J * ”_
[1 l3
- [543'45 ‘3 e ,.
AAD *
£1 a
0.011- a
C1
- D *
A D
A
OchIl .L,1 a. 1 1 .111 1 1 .1 1.411h.1 1 1, 1 l I 1 11,4 1 l 1 1
10 15 Z) 25 30
Rotational level 1
Figure 1.3. Calculated rotational transition probabilities versus

rotational level [51]. The equation for line A
is 0.788 exp(-0.1357 J). The equation for line B is
0.0406 exp(-0.l376 J).

18

A small rotational relaxation probability at high rotational
levels is consistent with observations of rotational lasing at high J

levels.

1.3 Vibrational Relaxation of HF
The vibrational relaxation of HF may be expressed by vibrational
to rotational, translational energy transfer (VR,T) and vibrational to

vibrational energy transfer (VV) reaction
HF(v,J) + M 3 HF(v',J') + M + AB (1.12)

* _ r * r
HF(v, J1) + HF(v,JZ)‘:~HF(v 1,J1)-+HF(v +1,J2)4-AE (1.13)

where M is a collision partner and AB is the energy defect. Until
very recently, the interpretation of experiments reported in the litera-
ture usually assumed that rotational levels J and J' are in thermal -
equilibrium at the translational temperature with the energy defect

being released to the thermal bath [4]. This interpretation results in
energy defects of several thousand cm.1 for Reaction (1.12). The

recent results of Wilkins [51,52] and others [34] suggest that Reaction
(1.12) occurs by a near resonant multiquanta VR process where the
resultant HF(v',J') loses one or more quanta of vibrational energy
with the excess energy going into rotation. The "optimum" value of J'

is obtained by minimizing the energy defect

AE =‘Minimum.[E (v,J) - E (v',J')] (1.14)

total total

I 7
where Etotal(v’J) and Etotal(v ,J ) are the sum of the vibrational

and rotational energy of HF(v,J) and HF(v',J'), respectively. A

l9

typical multiquanta VR reaction path is shown in Figure 1.4. The
values of v', J' and AE are given in Table D.l for the range of v
and J of.interest in HF chemical lasers. The first column contains
the rotational level J associated with the upper vibrational level
v. The rotational level Jmax that yields a minimum energy defect
given) v, J and v' is found in columns 3, 5, 7, 9, 11, and 13. The
energy defect associated with a given multiquanta VR reaction is
found in columns 4, 6, 8, 10, 12, and 14. The energy defect is quite
small even when there are large changes in vibrational quanta, indica-
tive that the multiquanta VR rates are near resonant..

It has usually been assumed that Reaction (1.13) occurs through
intermolecular VV energy exchange while neglecting rotational energy

1 [7]. The recent

resulting in an energy defect of several hundred cm-
results of Wilkins [51, 52] suggests that rotational energy levels
should be included in Reaction (1.13) and that J1 and Ji are
equal and J2 and J5 are equal. The energy defect for Reaction
(1.13) is typically one hundred cm.1 indicating a near resonant VV
process. A typical VV reaction path is shown in Figure 1.5.

1.3.1 Theoretical Investigations of VR Relaxation

The role of vibrational to rotational energy transfer in diatomic-
diatomic or diatomic-atomic collisions has been investigated by many
researchers. The role of rotational motion in the deactivation of
vibrationally excited diatomic molecules by collision with an atom has
been investigated by Kelly [53]. His theory is based upon the solution
of the two dimensional classical equations of motion, for an initially

vibrating rotationless oscillator bound by a Morse potential, colliding

with a particle of mass m. The importance of VR energy transfer was

20

HF(v,J) +11: HF(v',J') + M

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

 

 

 

 

Figure 1.4. A typical VR relaxation path.
Multiquanta vibrational relaxation
is allowed. The values of J'

range from Jmax-4 to Jmax+2 where

J is the value of J' with
max

smallest energy defect.

HF(v,J) Population Energies cm-1

21

* .
HF(v,Jl) + HF(v ,J2) HF(v-1,Ji) + HF(v*+1,Jé)

10

(v,JI)

(2,10)

l I l l l I l I If T 1

10
(V‘l sJi)

(1'10) (v*+1,1§)

 

IAT

(1.4)

*
_ . (v .Jz)

- . (0,4)

 

10 .__

Figure 1.5 Schematic of a typical VV reaction path.
The specific reaction shown is

ar<2,1o1 + ar(0,4) 3 ar(1,10) + ar(1.4) + 102.8cm’1

If zero rotational energy is assumed then AE-l7l.Z cm- .

1

22

found by Kelly [53] to increase with m, indicating the intramolecular
VR processes should be important in HF. The role of near resonant

VR energy transfer in BCl diatomic collisions have been investigated

3
by Poulsen et a1. [54] and Frankel et a1. [55]. The model of Poulsen
et a1. [54] accounted for near resonant transitions between a quantized
2, 02). Further
refinement of the theory of Poulsen et a1. [54] by Frankel et a1. [55]

oscillator (BC13) and a quantized rotor (HCl, DCl, H

indicated that attractive long range forces were necessary for BCl3
deactivation rates to compare reasonably with infrared double resonance
experiments [54] at temperatures above 300°K. Poulsen [54] noted that
his theory was most useful for systems in which a small quantum of
vibrational energy is relaxed by a hydrogen containing species where the
vibrational and rotational quanta are comparable. A fully quantum
mechanical 'coupled states' approximation was used by.McGuire and

Toennies [56] to be a dominant process in He-H relaxation even though

2
the rotational coupling for this system is weak. Multiquanta rotational
and vibrational relaxation processes were found by Dillon and Stephenson
[57] to be important when transition moments are large as in the case
for collisions involving HF. Infrared chemiluminescence experiments
conducted by Nazar et a1. [67] with highly excited rotational levels

in hydrides suggest that relaxation occurs through a resonant VR process.
This cascading process involves near resonant vibrational exchange in
which the vibrationally excited hydride is vibrationally deactivated
with retention of its initial rotational quantum number. The quantum

mechanical theory of Dillon and Stephenson [57] indicates that multi-

quantum rotational transitions play an important role in vibrational

23

energy exchange involving diatomic-diatomic collisions by allowing the
vibrational energy defect to be absorbed by the rotational energy
levels.

A one-dimensional analytical model which includes the effects of
dipole-dipole and hydrogen—bond interactions has been formulated by
Shin [58,59] to explain the efficient vibrational deactivation of HF
at low temperatures. For HF-HF collisions Shin [58,59] predicted that
VR energy transfer was more efficient than VT energy transfer. The
results of Shin [58,59] compare favorably with experimental data over a
wide range of temperatures. Shin noted that as a result of the strong
dipole-dipole attractive forces the HF molecules attract each other at
large distances even when each molecule is rotating rapidly and this
attractive force between HF molecules would tend to form an HF dimer
intermediary at low temperatures.

The 3-dimensional classical trajectory calculations of Wilkins
[51] provide for the first time detailed rate coefficients for multi-
.quantum VR, VV and rotational relaxation for reactions involving HF
with HF. The potential that Wilkins [51] employed for HF-HF inter-
actions is a combination of two functions, a London-Eyring-Polanyi-
Sato (LEPS) potential energy function for short range interactions and
a point-charge dipole-dipole potential energy function for long range
interactions between the four atoms. The energy surface is capable of
supporting the HF dimer concept which has been suggested as being
responsible for the efficient VV energy transfer in the experiment of
Airey and Fried [60]. The fast self-relaxation rates of HF measured
by Airey and Fried [60] in their opinion are not the result of dimer

formation. At the pressures and temperature of their experiment less

24

Table 1.4. Multiquanta VR Rates of Wilkins [51]
HF(v,J) +gn 3 HF(v',J') + Ma

 

v Total 0 1 2 3 4 s 6 7
1 3023.0 30.
2 3923.1 15. 24.
3 4333.2 11. 17. 15.
4 49.0:3.2 9.9 3.9 14. 17.
s 53.633.2 9.4 7.1 11. 7.1 19.
6 sa.0:3.0 8.5 5.6 8.5. 7.5 8.9 13.0
7 60.6b 7.6 7.6 7.6 7.6 7.6 7.6 15.
b

8 63.6 7.0 7.0 7.0 7.0 7.0 7.0 7.0 14.6

aThe forward VR rates in this table should be multiplied by
1012(3T0)0.5

bThe values for v=7,8 were obtained from a least squares power

fit to the total rates from val to 6 of the form 29.89v0'363

25

than 0.001% of the HF exists as dimers. Thompson [67] points out that
this view is correct if it is assumed that vibrational relaxation occurs
between an excited HF molecule and the dimer, but it is not necessarily
true if the effective intermediate is a long-lived complex. If a long—
lived complex is formed then the vibrational relaxation rates are
critically dependent on the rate of formation and dissociation of the
dimer.

The following qualitative results were noted by Wilkins [51]: (1)
HF dimers are not formed in the typical HF-HF collision in the tempera-
ture range at or above 300°K. (2) The vibrational energy of the
vibrationally excited incident HF molecule is transferred into rotation-
al energy of the same HF molecule implying that there is little or no
change in the internal energy of the target HF molecule. This result
allows the HF-M rates to be inferred from Wilkins [51] HF-HF rates
based upon previously measured or calculated HF-M VT rates. Similar
results were obtained for HF-DF trajectory calculations of Wilkins [52].
(3) Very high rotational states of the incident HF molecules are popu-
lated by single and multiple quanta VR reactions listed in Table D.l.
(4) The energy defect for these multiquantum VR reactions are much less
than the energy defect for the corresponding multiquantum VT reaction
where the energy defect is given up to the thermal bath.

The rate coefficients for vibrational to rotational energy transfer

process
HF(v,Jl) + HF(0,J2) I HF(v',Ji) + HF(O,J£) (1.15)

calculated by Wilkins [51] are presented in Table 1.4. For vibrational

energy levels greater than 6 a least squares fit of the form

26

Avn - 29.89v0‘36 (1.16)

was obtained for the vibrational dependence of this rate based upon the
overall rate coefficient. The resultant total rate was then converted
to a multiquanta rate with the single quantum rate twice the multi-
quantum rates.

1.3.2 [Theoretical Investigations of VV Relaxation

Theoretical calculatiOns of VV exchange rates for HF-HF collisions
were conducted by Shin and Kim [61] and Wilkins [51]. The one 1
dimensional arrested rotational relaxation model used by Shin and Kim
[61] gives VV rates for the temperature range of 200°K to 2000°K. The
transfer of energy was assumed by Shin and Kim [61] to occur through the
formation of non-rigid dimers at low temperatures (T<300°K) and through
the rotational motion of the colliding molecules at temperatures above
300°K. In the non-rigid dimer model the energy defect in Reaction (1.13)
is transferred to the restricted translational motion of individual
molecules of the dimer. At temperatures above 300°K the energy defect
is transferred to the rapid rotational motion of the colliding mole-
cules. Shin and Kim's model confirmed previous predictions that two
quantum VV processes HF(v) + HF(O) +-HF(v-2) + HF(Z) have very low
probability. Their theory does not indicate the mechanism by which
vibrational energy is converted to rotational energy and it does not
include multiquantum VR processes. The VV rate coefficients for
v - 2 to 5 and v' - 0 have been calculated by Wilkins [51] and
measured by Kwok and Wilkins [62], Airey and Smith [63], and Osgood
et a1. [64] are compared in Table 1.5. In the analysis of vibrational

2.3

relaxation data a v dependence on the VR,T is assumed.

Table 1.5. Vibrational to Vibrational rate coefficients in

1012 cc/mol—sec at 300°K for HF(v-l) + HF(v'=1) -
HF(v) + HF(v'=0)

Wilkins [51] Shin. Kim Kwok, Wi kins Aire . S ith Osgood et a1.

 

Owl-FUN

 

.forwgzg :gvgrgg 63] L°uJ
8.5:1.9 12:2.1 8.5 5.0 6.1 11.0
5.2:1.“ 7.8il.8 9.8 3.2 b 18.4
3.311.2 5.011.“ 10. >7.7 3.7
1.9:0.9 2.8:1.2 2.3 >7.5

1.2:O.7 1.7:O.9

aThe VV rate was extracted from the total vibrational relaxation rate

by assuming a v2'3 dependence for the contribution due to VR,T

bSubtracting the v2'3'contribution due to VR,T resulted in a

negative rate.

28

The three dimensional trajectory calculations of Wilkins [51]
predict that the formation of a dimer is not necessary to account for
VV processes. The VV rate coefficients calculated by Wilkins [51] and
presented in Table 1.6 indicate that the VR rates in Table 1.4 are
larger than VV rates.

In order to interpret the experimentally determined rate coeffi-
cients for the deactivation of HF by HF the effects of VR,T (Reaction
1.12) and VV (Reaction 1.3) must be uncoupled. There is very little
data available except for systems involving v - 1 single quantum tran-
sitions and only a linear v dependence of VR,T rates. A vibration-
al dependence of vz'3 has been suggested by Kwok and Wilkins [62]
based upon flow tube studies. The vibrational dependence due to VR,T
is summarized in Table 1.7. The experimental data of Kwok [62],

Osgood [64], and Airey [63] are compared with the theoretical data of
Shin [59] and Wilkins [51] in Table 1.5 assuming a v2°3 dependence. A
linear v dependence for VR,T in interpreting experimental data to
deduce VV rate coefficients for HF(v) + HF(l) + HF(v+l) + HF(O)
results in a rate coefficient larger than gas kinetic.

The lack of direct experimental confirmation of the findings of
Wilkins [51] indicates a need for experimental study of multiquanta
VR reactions, VV reactions and rotational relaxation processes under
well defined conditions..

One of the goals of the present effort is to assess the effect of
VR processes on laser performance. This goal may be achieved in part
through the introduction of mechanisms for VR, VV and rotational relaxa-
tion with rotational and P branch lasing in a comprehensive computer

model. Time resolved spectroscopy of pulsed H2+F2 lasers and computer

Table 1.6 Single Quanta VV Rates of Wilkins [51]
HF(v,Jl) + 1111qu : HF(v-1,Ji) + arcvs+1,15)a

29

 

 

4 5 6 7
1 12.0 7.8 5.0 2.8 1.7 0.85 0.42
2 6.6 3.3 1.65 0.825 0.413 0.207
3 3.9 1.95 0.975 0.488 0.244
4 2.3 1.15 0.575 0.289
5 1.2 0.6 0.3
6 0.7 0.35
a 12 T 0.5
The forward rates in this table should be multiplied by 10 (3009 .

b

e a
The values for v =6 and *v =7 were obtained by assuming a vibra-
v-v

tional dependence of 2

30

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a

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_m~_ aweaaeao - . m.a n n a a s a and:-
e O O O O O a

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as A

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_n~_ :aeoo - - - am an a as a a a a a
_-_ assoc - n n n n «N aN a a cameo.

scumuuuumo

cancers: 5 a A a n e n N 3 cause.
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no eqmcum
yoquuuuuou uumu H.z> amuOU ecu mo uocuveomov floccuumubu> s.~ edema

31

models incorporating rotational nonequilibrium effects are discussed

in the next section.

32

2. STUDIES OF ROTATIONAL NONEQUILIBRIUM IN THE PULSED

H2+F2 SYSTEM

2.1 Time Resolved Spectrosc0py

Experimental measurements of the time resolved spectra of pulsed
H2+F2 lasers have been made by Suchard, Gross and Whittier [68],
Suchard [71], Nichols, et a1. [69] and Parker and Stephens [70]. The
duration of transitions and initial conditions are shown in Figure 2.1,
2.2 and 2.3 for the data reported by Suchard, et a1. [68], Nichols,
et al. [69] and Parker, et a1. [70]; respectively. In the figures, the
duration of the P-branch spectra is denoted by a line at an elevation
associated with the v,J state corresponding to the lower level of the
P-branch transition. The dot corresponds to the point where power
peaked on the given transition. Theoretical models that assume rota-
tional equilibrium do not permit simultaneous lasing on adjacent
transitions in the same band. This phenomena can be seen by comparing
the spectra predicted by the comprehensive model reference [7] in
Figure 2.1 with the spectra of Suchard, et a1. [68].

In laser systems initiated by flashlamps [68], [71], the sequence
of appearance of individual lines is irregular within a given vibration-
al band. One possible explanation of this phenomena is nonuniform
absorption of the light from the flashlamps. The initial F atom con-
centration is difficult to determine in flashlamp initiated systems,
and F atom generation by photolysis occurs well into and often beyond
the duration of the laser pulse. The time resolved spectra observed by
Nichols, et a1. [69] and shown in Figure 2.2 was obtained with carefully
defined initial conditions. The second harmonic emission (3471 A) from

a 30n sec variable amplitude Q—switched ruby laser initiated the

133

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34

 

 

 

 

 

 

 

 

 

 

v J
1° ; l l l r
3 E
or: '
l0“;-
L: .2:
2 E ......:::E—’p3'5'
0-5: .31.
'0 E ___ 1.39217) 3
l : ’2— " 3.
o_E_ :'
WE"
o ;--—__.___._.r 9,16) lol
Oji—_l l I l
'0’ I l l —\ .F
5 _____-—.~—.-"'" 96110] E
0]: Ps(10l\_€
lO' - __ _;
4 E .—-—-"""""‘-._ §
‘31:. P4llll. _-
r: __—_ :
3 E ._:'~ 5
A. 5
‘ l
0 lb) 3
' .
30 l00

Figure 2.1.

 

TIME AFTER lNiTlATlO-‘J, uscc

Comparison of time-resolved
spectral output. (a) Experi-
mental data reported in Ref-
erence 69 for a 1H2:1F2:60 He
mixture at 50 Torr: (b) cal-
culated spectrum for the some
mixture from Reference7

who re rotational equilibrium
was assumed.

10

35

P3(5)

\

 

10

P2(6)

 

10

 

IIIIIIIIII [llllllllll ll llllrll]

Time After Initiation (usec)

Figure 2.2.

Time resolved spectra of Nichols et al. [69]
for a 1:1:10 mixture of HZ:F2*N2 at 90 torr

with F/P2 = 0.23% and 0.002 HP. The active
medium length is 30 cm and the mirror spacing

is 113 cm, R0 = 0.7 and RL = 1.0

36

H2+F2 chain reaction laser by the photodissociation of F The level

2.
of initiation was obtained by considering the mirror and window
transmissities, the photodissociation cross section of F2 and the
energy of the initiating ruby pulse with the assumption that the energy
is uniformly absorbed.

Small amounts of initial HF can have a significant effect on laser
performance. The initial HF concentration was measured by Nichols,
et a1. [69] by utilizing an HF probe laser operating on the P1(2)
transition to measure the ground state concentration of HP. The experi-
mental apparatus of Nichols, et a1. [69] is limited to measuring the
time resolved power on 9 transitions and the total power.

The time resolved spectroscopy of Parker and Stephens [70] shown
in Figure 2.3 was obtained utilizing transverse electric discharge
initiation. A preionizer consisting of a linear array of spark
discharges was used to provide a background of electrons insuring a
uniform glow discharge. The sequence of appearance and disappearance
of transitions within a band is quite regular with positive J-shift as '
observed by Nichols, et a1. [69] and Parker and Stephens [70]. However,
there is strong simultaneous lasing on adjacent transitions within a
given band which is not possible with rotational equilibrium population
distributions. The initial F atom concentration was obtained by come

paring the peak power and pulse duration with the predictions of a

computer model of the 32+?2 laser [71].

2.2 Computer Modeling of Rotational Nonequilibrium Phenomena
The effects of ratational nonequilibrium in pulsed HF lasers have

been modeled by Kerber and coworkers [72,73], Ben-Shaul, et al. [75],

Figure 2.3

37

Time resolved spectra of Parker and Stephens

[70] for a l:l:lO:0.25 mixture of Fzzfizfle:02

at 36 Torr with F/F2=5%. The active medium

length is 15 cm and the mirror spacing is
60 cm,Ro=O.8 and RL=l.0. The lower level of

a P branch transition is denoted by v and J.
The length of the lines correspond to the
duration of lasing on a given line. The

dot on the line corresponds to the time at
which the maximum power was achieved on that
line.

38

P&(9)

(8)

5

/.
i l

P6(6)
P

(9)

 

 

 

ii

 

i

9.,(13)

P1(13)

 

 

 

 

 

_ i

 

a

 

 

 

 

f

 

::_::_::_:.L::::__ 5755.: :LE:
0 0 o o

0 o o
.I 1 1 l

P____ b___LP_—__
0 Jq

1

I; _:_:::_:
a m

10

 

Time After Initiation (usec)

Figure 2.3

39

Moreno [76] and Creighton [77]. Rotational nonequilibrium phenomena
have been modeled by Sentman [78,79] and Skifstad, et a1. [80] for the
continuous wave (cw) HF laser system and by Hall [81] for the cw DF
laser. These models vary significantly in their ability to model
rotational nonequilibrium phenomena in a manner that approximates the
actual physical mechanisms. The major characteristics and assumptions
of these models are outlined in Table 2.1.

The cold pumping reaction is included in every model listed in the
table. The rotational distribution suggested by Polanyi, et al. [11]
(see Figure 1.1) was used by references [73-75 and 77-79] and this work.
The Boltzmann rotational distribution is used by reference [80] and the
effect on the pumping distribution is examined in references [73-75].
The rotational distribution for references [73,74] and this work is a
given fraction of a Boltzmann distribution and the distribution of
Polanyi [l]. The distribution computed by Wilkens [16] is used by
references [72,76].

The hot pumping reaction is included in the models of Kerber and

coworkers [72,74] and Moreno [76] with the rotational distribution
measured by Polanyi, et al. [26]. The rotational distribution for
references [73,74] and this work is a given fraction of a Boltzmann
distribution and the distribution of Polanyi [76] (see Figure 1.2).
The inclusion of the hot pumping reaction is essential in modeling the
112+}?2 chain reaction, but greatly increases the complexity of a model
by increasing the number of vibrational levels that must be considered
from 3 to 8.

The vibrational to vibrational (VV) energy transfer process is one

of the energy transfer mechanisms that is responsible for the excitation

40

and de-excitation of vibrationally excited HF. Rotational levels are at
their equilibrium values during the intermolecular VV energy exchange
process in all models considered with the exception of the present work
and that of Ben-Shaul, et a1. [75] The present work and that of refer-
ence [75] require that the rotational levels are not perturbed during
VV relaxation.

Vibrational relaxation can also occur by the following VR,T

reaction:

HF(v,J) + M I HF(v',J') + M (2.1)

This reaction is a VT reaction if J and J' are given by their
thermal equilibrium values. With the exception of the present work and
reference [75], reaction (2.1) is treated as a VT reaction. The

effects of multiquantum VT were considered by Kerber, et al [73] who
found as one would expect that multiquanta deactivation tended to de-
crease the pulse energy, peak power and shorten the pulse duration over
single quantum (v' -v-l) cases. Ben-Shaul, et a1. [75] incorporated the
effects of vibrational to rotational energy transfer (VR) in his model
for reactions of HF(v,J) with HF and F. The rates for this process
increase linearly with vibrational energy. The change in rotational

energy due to the VR process was determined by the relationship [87],[88]

~ 2
<AE > 3 <AEv (2.2)

rot ib>

Ben-Shaul concluded that VR cannot serve as an efficient pumping

mechanism of high rotational levels since his mechanisms for depopulating

41

high rotational levels were more efficient than his VR mechanism for
populating high rotational levels. Although these results are contrary
to the results of the present efforts this is the first attempt to model
VR processes.

Rotational relaxation for all models with the exception of the
present work is assumed to consist of AJ r 1 RT relaxations of the

form
HF(v,J) + M I HF(v,J-l) + M . (2.3)

These models, with the exception of reference [80], employ the rota-
tional relaxation model of Polanyi, et a1. [31] as discussed in Section
1.2. Two approaches to modeling RT relaxation are discussed by Kerber
[73]. These methods consist of the rotational relaxation time constant
approach where rotational levels relax towards their equilibrium values
(Equation 1.11) and the detailed kinetic approach where explicit RT
reactions are modeled. The results of Kerber indicate that (1) the
major temporal characteristics are the same including the sequential
J-shift pattern, (2) the detailed kinetic model predicts a smaller
fraction of energy in the vibrational levels pumped by the hot reaction
than the time constant model predicts, (3) the detailed kinetic model
predicts higher pulse energies and less power over the time constant
model, and (4) the detailed kinetic model predicts a sharper distribu-
tion of energy as a function of rotational quantum number for each band.
The detailed kinetic model also predicts that the rotational transitions
undergoing lasing will appear and terminate sooner than what is pre-

dicted by the time constant model.

42

The line shape in references [75,77,79] is assumed to be Doppler
broadened. The Doppler broadening profile is valid only at pressures
under a few torr. In the model of the HZ+F2 superradient laser of
reference [76], the line shape is approximated by a rectangle. The
line shape used by Kerber and coworkers [72,74] is a Voigt profile which
includes the effects of Doppler boradening and Lorentz broadening [7,
86].

All_of the models listed in Table 2.1 include P branch lasing.

The model of Skifstad [80] restricts lasing to only single lines.

The effects of R branch lasing were found by Sentman [78] to decrease
with increasing rotational relaxation rates. The present study includes
the effects of P branch lasing.

The only models which adjust the gas temperature according to
the energy balance equation are those of Kerber and coworkers [72,74],
Moreno [76], and Hall [81] and the present model. The other models
maintain a fixed temperature throughout the simulation.

The effect of rotational nonequilibrium on the time resolved
spectra was found by Kerber and coworkers [72,74] (1) to increase the
number of transitions that lase simultaneously, (2) to lower the inten-
sity of each transition, (3) to extend the duration of lasing on each
transition, and (4) to shift the laser energy to higher rotational
levels. These conclusions appear to be consistent with the other
models in Table 2.1. The rotational distribution was noted by referen-
ces [35,37] to approximate the pumping rotational distribution for
early times and approach but not reach an equilibrium rotational distri-

bution towards the end of the pulse.

43

3. MODEL FORMULATION

The computer model of the pulsed hydrogen fluoride laser system
is described. The mechanisms for a rate equation model of the pulsed
H2+F2 chain reaction laser with P branch and RR lasing are pre-
sented.

The present formulation yields the time history of the first seven
vibrational levels of hydrogen fluoride and their first thirty rota-
tional levels. In addition vibrational levels (v-7,8) are included
and their rotational levels are assumed to be in rotational equilibrium
at the translational temperature.

The cold pumping reaction is

K:(v) P (v,J)
r + 32(0) 1 ° ~ HF(v,J) + a (3.1)

kficv.»

 

 

where K§(v) is the forward rate coefficient recommended by Cohen [24].
The normalized rotational pumping distribution is given by Pc(v,J).
The present formulation assumes this to be a variable fraction of the
distribution suggested by Polanyi [11] and the Boltzmann distribution.
The backward pumping rate is denoted as K§(v,J) and is obtained from
the forward rate K§(v) Pc(v,J) by considering detailed balancing.
The cold reaction is exothermic for v-O to 3 and endothermic for v-4
to 6. Pumping of v-O has not been observed experimentally. Pc(v,J)
is shown in Figure 1.1 and K§(v) is given in Appendix A.

The hot pumping reaction is

K:(v) Ph(v,J)
“‘ HF(v,J) + F (3.2)

 

+
H erh

Kf(v,J)

44

where K:(v) is the forward rate coefficient recommended by Cohen [24].
The normalized pumping distribution‘ Ph(v,J) for the hot pumping
reaction is also assumed to be a fraction of the distribution suggested
by Polanyi [26] and the Boltzmann distribution. The backward rate
coefficient, K:(v,J), is computed from the equilibrium constant for
the cold reaction. The hot reaction forms vibrationally excited HF
from v=0 to v=8. The forward rate K:(v) is given in Appendix A and
Pb(v,J) is illustrated in Figure 1.2.
The HF vibrational to rotational (VR) quantum exchange mechanism

is given by

K¥R(v,v')

HF(v,J) +lM “‘ HF(v',J') + M (3.3)
E‘f’R(v,v' ,J,J')

 

where K¥R(v,v') is the forward rate coefficient. The forward rates,
K¥R(v,v'), are obtained from the overall rate coefficients recommended
by Wilkins [51] for a given v by making the rate for single quantum
tranisitions two times the rate for multiple quantum transitions. Thus
if only single quantum transitions are to be considered K¥R(v,v-1)
would be the total rate fora given vibrational level. The forward rate
constants KgR(v,v') for a change in vibrational quanta up to 7 are given
in Table 3.1. The backward rate K:R(v,v',J,J') is obtained from de-
tailed balancing considerations. A typical multiquanta VR reaction
path is illustrated in Figure 1.4.

The rotational level yielding a minimum energy defect between
HF(v,J) and HF(v',J') is given in Table 1.4. Equal probability is
assigned to each of the allowed vibrational to rotational transitions

from HF(v,J) to HF(v',J'). In the results given by Wilkins [51],

45

only self relaxation of HF is considered. Collision partners other
than HF must be considered for VR relaxation. Single quantum VR
processes are assumed for all other species with the exception of H
atoms. A temperature dependent scale parameter was formulated based
upon Cohen's [23] recommended HF-M VT rates. For H2 this parameter

is the M-H rate divided by Cohen's [23] HF+HF VT rate KHF

2
6 x 107 T
W (3")
For F2, Ar, N2 and SP6 this parameter is,
7 7 x 10‘7 T
' (3.5)

 

50KHF

where the scale factor for He is twice that of Ar. The factor for

 

 

F is
1.5 x 1012 e-700/RT (3.6)
10KHF
and for H2 this factor is
1.6 x 1013 e-2700/RT (3.7)

200KHF

where T is the temperature in degrees Kelvin and R is the gas con-
stant in calories/mole-K. The scale factor for MhHF is l. The RF-

HF VT rate KHF is given by

14 T-l 4 T2.26

3 x 10 + 3.5 x 10

Vibrational to vibrational (VV) quantum exchange for HF-HF

collisions are modeled as

46

 

 

Ky(v,v*)
HF(v,Jl) + HF(v*,J2) ‘- HF(v-1.Ji) +
VVV<v v* J J )
Kb ’ ' 1’ 2 HF(v*+l,Jé) (3.8)

where the forward rate coefficient K¥v(v,v*) is obtained from the
three-dimensional trajectory calculations of Wilkins [51] (Table 1.7).

The rotational levels J and J are held fixed and as before

1 2
K:v(v,v*,Jl,J2) is obtained from detailed balance considerations. To

prevent duplications in the HF-HF VV reactions considered” v,v*,Jl

and J2 are given by

63y*3y31 and 293323; 39 . (3.9)

l

Vibrational to vibrational energy transfer of HF-H2 is modeled as

KEF-H2(v,v*)
\ HF(v+1,J) + H2(v*-l) (3.10)
‘HF

HF(v,J) + H2(v*)
Kb -HZ(v,v*)

where the forward rate REF-H2(v,v*) is obtained from the recommended
rate coefficients of Cohen [24]. For v*-1, the first 4 (v-O to 3)
vibrational levels of HF are considered. For v*=2, the first 2
vibrational levels of HF are considered. Vibrational to translational

energy transfer in H may be.modeled as

 

2
3:2‘32(V)
H (v) +IM H (v-l) +iM (3.11)
2 H -H 2
Kb2 2(v)
where ng-H2(v) is the forward rate coefficient, M represents all

collision partners and v is l or 2 as recommended by Cohen [24].

47

The effects of rotational relaxation excluding lasing are modeled

by the following mechanisms; rotational to translational (RT)

K§T(V.J.AJ)
HF(v,J) + M r HF(v,J-AJ) + M (3.12)

EEka,J,AJ)

 

and rotational to rotational (RR) quantum exchange

K§R(v.J1.AJ)
HF(v,Jl) + HF(v*,J2) "‘- HF(v,Ji-AJ) +

@(vmlszJ) HF(v*,Ji+AJ) (3.13)

 

 

where AJ, the change in rotational quantum number, is 1 or 2 for
RT and AJ is l for RR. To prevent duplications in the rotational

relaxation reactions considered, we require
73y*3yzp and (29 - AJ) 3_J2 3_J1 3_AJ. (3.14)

The forward rates for RT and RR processes are based upon the total

rate coefficients for the following two reactions:

KETV'O)
HF(0,10) + HF(v,J)'---“"HF(0,10-AJ) + HF(v,J) (3.15)

where the forward rate KECv-O) is

16T-0.805 e-2569/RT

KEN-0) - 1.023x10 (3.16)

and

KEEN-1)
HF(1,10) + HF(v,J) ---‘*- HF(l,10-AJ) + HF(v,J) (3.17)

where the forward rate K§(v-l) is given by

48

K¥(v-l) = 3.38x1016'r'0'893 {2436”I (3.18)

where KECV) is in units of cc/mole-sec.

The total rotational relaxation rate K:(v) was calculated for
rotational levels 5, 10 and 15 by Wilkins [51]. The intermediate value
J-lO was chosen as a reference. The rate coefficient K§(v) is scaled
according to the transition probabilities presented in Figure 1.4 about
J-lO.

The present formulation assumes that the rate coeffiCient for
vibrational levels greater than 1 is the same as the K§(l) rate.
Wilkins [51] noted that the contribution due to RR.mechanisms contri-
buted to approximately one third of the total rate coefficient. Assume
ing that indeed one third of the total rate is due to RR and two thirds
of the total rate is due to RT mechanisms then the following rates for

K§R(V,J,AJ) and K§I(V,J,AJ) may be obtained:

 

 

 

K§R(v, J ,1)- :f(V) PAJ'1(J) (3 10)
PAJ- -1(1°) . I
PICR(v) P (J)
KRT 2 AJ- -1
K30?) P (J)
KRT 2P AJ- -2

The chaperon M for RT reactions employs the relative rotational
relaxation efficiencies based upon the infrared double resonance experi-

ments of Hinchen [33,34] and are listed in Table 3.1.

49

Table 3.1 Relative Rotational Relaxation Efficiencies.

 

 

Species Relative Efficiency

HF 1.0

He 0.03
Ar 0.03
N2 0.03
SF6 0.03
F2 0.03
F ' 0.03
Hz 0.1

H 0.03

 

50

The dissociation of HF is given by
HF
Kf (V)

K’f‘chJ)

 

HF(v,J) +'M H +'F + M (3.22)

where the rate coefficients are of the form suggested by Cohen [24] and
listed in Appendix A.

The dissociation of H is given by

2
K122

H2(v=0) +M -—-‘- H+H+M (3.23)
H
KbZ
where the rate coefficient KHZ has been suggested by Cohen [24].

f
The dissociation of F is represented by

2
15:2
F +M—-=-F+F+M (3.24)
2 ‘F

KbZ

F

KfZ is the sum of the rate suggested by Cohen [24].

Initiation may be modeled by the instantaneous introduction of F

atoms or by a continuous dissociation of F simulating a flashlamp.

2

The rate of photodissociation of F due to a flashlamp is of the

2
form presented in Nest [89].

dF z
”53' - —pl- sin (Flt—9 (1 - e'zze) . (3.25)
flash

The quantity 21 is the maximum value of the flashlamp output in moles
of photons relative to a unit volume of the gas mixture, p is the

density of the mixture, t is the time in seconds from the start of

51

photodissociation, tflash is the length of time of the photodissociation
l

of, F2 in moles/cc. The absorption coefficient av in cm- of F2

and the active medium length L in cm, is related to 22 by
a L
z--v

. (3.26)
2 F2

 

The chemical reactions are written in the general form

K
r
{a N— is N (3.27)
i ti 1 EF_ 1 ri i
-r
where N is the concentration of species 1 in mole/cc, a and

i ri

8:1 are stoichometric coefficients, Kr and K-r are the forward and
backward rate coefficients. For the specific reactions introduced

earlier the chemical reactions yield a concentration rate of change of

the form
dN a B
i rj ri
3-2 (8:1 - ari)(Kr | I Nj - K_r l | Nj ) . (3.28)

r 3 J

The general term for the rate of change of HF(v,J) can be

written as

dHF v J
dt x1 4'

{Xrad(v’J) - Xrad(v,J-1)} +

{o(v,J) f(v,J) - o(v-1,J+1) f(v-1,J+l)} (3.29)

where X1 is the rate of change of HF(v,J) due to kinetics and
a(v,J) is the gain in cm-1 on the P branch v+14v transition with
lower level J and f(v,J) is the photon flux in mole/cmz-sec.

The model can be madified to include ratational lasing where the

52

rate of lasing from rotational level J+1+J is denoted as Xrad(v,J).
The rate equation for the lasing flux if [72]

df v J

dt ] (3.30)

c L
l [0(V’J) - athr

where c is the speed of light, L is the active medium length and

2 is the spacing between the mirrors in cm. The threshold gain is

“cm: =- - "2‘11' ln(R0RL) (3.31)

where RO is the reflectivity of the output mirror and RL includes

the loss at the second mirror and other cavity losses. The gain for

a P branch transition at line center is [72]

h N '
A 2J+1
4n wc(v,J) ¢(v,J) B(v,J) [2J-1 HF(v+1,J-1) -

 

o(v,J) -

HF(v,J)] (3.32)

where NA is Avagadro's number, h is Planck's constant, mc(v,J)

is the wavenumber of the P branch transition, B(v,J) is the
Einstein isotropic absorption coefficient based on the intensity, and
¢(v,J) is the Voigt profile at line center with the lorentzian line-
width based upon the data of Meredith [27] or Hough [86]. Lasing
transitions are assumed to be homogeneously broadened and do not

overlap.

The power on a given P branch transition is [72]

PVR(v,J) = h c N a

A thr wc(v,J) f(v,J) . (3.33)

53

 

The output power through R0 is [27]
' l - R
O t
2V; (v,J) - PVR(v,J) o . (3.34)

1+\/:i-L;- 1"me

The rate of photon emission due to RR lasing for a transition from

J+1 to J given by Xrade,J) can be expressed in terms of a rate
equation for the RR lasing flux. An alternate form for Xrad(v’J)

that does not require the addition of up to 210 flux terms to the
integration list is discussed in Appendix B. The power due to rotation-
al lasing is

PRR(v,J) - h c NA mcR(v,J) Xrad(v,J) (3.35)

where mcR(v,J) is the wavenumber of the rotational lasing transition
from 'J+l to J. Research is currently underway assessing the effect
of Equation (3.35) on laser performance and will not be presented here.
The laser energy extracted in each band is then determined by integra-

ting the power

t
Ev.R(v,J) - f° PVR(v,J) dt (3.36)
O

t
ERR(v,J) - gc PRR(v,J) dt

there tc is the length of time in seconds of the laser pulse.
The time rate of change of temperature for a constant density gas

based on conservation of energy is [72]

54

 

dNi
- [PVR + PR + 2 —dt 31(1)]
dT i
a: " (3.37)
X Nini(T)
i
where PVR is the total power due to P branch lasing, PRR is the
total power due to rotational lasing, Cvi(T) is the molar specific

heat of species i for constant volume at a given temperature and
Hi(T) is the molar enthalpy.
A Wernier-Orr [80] diagram outlining the major processes in the

model is.given in Appendix C.

55

4. RESULTS AND DISCUSSION
A comprehensive computer model has been developed and presented
that simulates the performance of the pulsed H2

laser. Rate equations are used to represent the chemical kinetic and

+ F2 pulsed chemical

stimulated emissions processes occurring in a representative unit
volume within a Fabry-Perot cavity. All processes are assumed uniform
throughout the cavity. Lasing is permitted on all P' branches in the
vibrational - rotational bands at or below v I 6 and J - 12, and may
respond to gain fluctuations during the lasing period. This model per-
mits the examination of the effect of vibrational to rotational, rota-
tional to rotational, rotational to translational and vibrational to
vibrational energy transfer on the spectral distribution. The effect
of nonequilibrium pumping distributions and line selection on the
spectral content of the laser pulse can also be examined. The major
thrust of the present work has been to examine the effects of Wilkins

[51] VR, VV and RR,T rate coefficients on laser performance.

4.1 Effect of Vibrational to Rotational Relaxation on HF POpulation
The present formulation supports the predictions of Wilkins [51,

52] and others [23,34] that VR is the primary mechanism for popu-

1ating high rotational states of HF. The effect of various distri-

butions of J' about Jmax for the multiquanta VR reaction

HF(v,J) + M; HF(v',J') + M + AE (4.1)

where Jmax is the value of J' that minimizes the energy defect.
Several distributions both multi quanta and single quanta have been

studied and are listed in Table 4.1. The total forward reaction rate

56

 

 

Table 4.1 Possible VR relaxation distributions about Jmax for
HF(v,J) + M I HF(v',J') + M
Distri-
bution max 4 max 3 Jmax.-2 Jmax 1 max Jmax+1 max+2
1 1/7 1/7 1/7 1/7 1/7 1/7 1/7
2 o o o ' 1/3 1/3 1/3 0
3 0 0 0 0 1 -0 0
4 0.05 0.05 0.05 0.25 0.3 0.25 0.05

 

57

is obtained by multiplying K¥R(v,v') by a particular distribution.

The rate K¥R(v,v') was obtained by multiplying a temperature dependent
scale parameter based upon Cohen's [22,23] recommended HF +1M single
quanta VT ‘rate and Wilkins [51] recommended VR rates for HF+HF
collisions. These rates are illustrated in Figure A.4.

To assist in understanding the numerous coupled processes occurring
in the model,plots were made of the population time history of HF(v,J)
and time resolved spectra for all cases. Integration difficulty was -
encountered for the first distribution where J' is 2 above and 4
below Jmax and all seven levels are equally weighted. The difficulty
arose from the integration step size being too large. Prohibitively
large computation times would have been necessary to eliminate this
problem. The concentration of HF(v,J) versus J at 1 sec, 10 sec,
50 sec and at the end of the lasing pulse when the total power has
dropped below 1% of the maximum are illustrated in Figure 4.1 for run
18a. At early times in the simulation the rotational population
matches the pumping rotational distribution. As time progresses the
population of rotational levels above the levels pumped increases
dramatically. The VR relaxation processes were found to be primarily
responsible for populating these upper levels. The VV relaxation
mechanism will tend to redistribute the specific upper rotational level
pumped by VR about the 7 vibrational levels considered. The effect
of rotational relaxation on the derivative of HF(v,J) was found to
be important even at high rotational levels.

The relative contributions to the derivative of HF(V‘1,J) are
compared in Figure 4.2. For a typical HF(v,J) there are several

hundred contributions to the derivative. A total of 92,398 reactions

log HF(v,J)) mol/cc

log (HF(v,J)) mol/cc

58

RUNXBR VR,VV,RR: P LRSING

 

 

 

 

 

 
 

-18

-11

0

3‘1 .

'2

h»

3. -13 I

it

B

E --14

- _15 11 1 1 1 1

s 12 18 24 a a s 12 1a 24 so
Rotatlonal level (I) Raggggonal level (I)

0
/12

 

 

 

 

 

. s A 5
’ ‘1
-11 . ‘~VQ§ ‘11 r
P V, \ 8
.‘.
-12 F 8 -12
’ 2
' H
'13 ,- nre - .m—ea 3‘ '13
i E:
L U
-14 8’ '14
-15 I 1 l l l _15 .J l l l 1
s 6 12 1a 24 a a 6 12 1s 24 35
Rotational level (J) Rotational level (I)

Figure 4.1 Concentration of HF(v,J) versus rotational
level at selected times.

59

 

 

15 20 25 30

Rotational level J

10

 

 

)
J

n ’

O l.

.1 (\

t .r

.a H

a. :1

n c at

.1 o o l

D. s e

m u v s

D. .d .1 _m

.. u

o m v u

h 9 fl 1

a a. ..... .. I -...... IIII

.w “u m1 o. .I-v.I. I!

'I".
A A O I
\
\a\\..
x \\ IIIIIIII 'l"-
\.\\
II
I,
. AI-
i \r i
.AhV\)II I .z II.I 13.1 I. I I..I I. I.Iu In .I I. I I. I. .I l.l
\‘Ii‘ 3. .1
In hlIHnu Lu-
I. .I.fl.....II-I..I......I..IIII--I....
nu.l.du I I. l.l I. I n..l I
A IIIIIIIIIIII IIIIII.
4.

.l‘ lllllll llllllll llllllllll llllll 1
A. .1 .NrI.LH ...I .. I I. I .I I .I I IIIIIIIIIIIIIIIIIIIIIII
a. - fill—lb? - n n p —b. a p

a q . a
nu nu Av Au
1 1 I. .1
AuoquoexoUV u a
dines

Contribution to the derivative of HF(1,J) at peak power for

run 18a.

Figure 4.2a.

A dashed line denotes a negative contribution to

the total derivative.

60

Population of HF(1,J)
in units of nol/cc

Rotational relaxation
* 10‘3

 

 

 

Auenlno§\00v

 

~H ~wh= v

30

15

Rotational level J

Figure 4.2b.

Contribution to the derivative of HF(1,J) at peak power for

run 18a.

61

contribute to the derivative of all 210 vibrational and rotational
levels of HF considered. ~The major contribution to the derivative

of HF(1,J) is from VR relaxation. The pumping and P branch lasing
processes compete with VR relaxation at rotational levels less than 13.
The peak in the derivative of HF(1,J) due to VR at J - 15 corres-
ponds to the transition from HF(v-Z, 3.0-4) to HF(1,15). This peak
near J - 15 has been observed for all vibrational levels that are
considered. The ability of VR to transfer energy from several of the
lowest rotational levels to a single rotational level near J - 15
provides an efficient mechanism for populating this level. The effect
of rotational relaxation is to redistribute the rotational distribution
towards thermal equilibrium.counteracting the effects of VR at higher
rotational levels. The derivative of HF(v,J) due to VV has the
effect of redistributing the rotational population among all 7 vibra-
tional levels.

The total derivative of HF(v,J) versus J is illustrated in
Figure 4.3. For v-O-Z the derivatives are nearly equal. The large
peak in the derivative of HF(0,29) is an artifact of the model and
is caused by the VR mechanism. There are 9 separate VR reaction paths
which contribute to this derivative [HF(1,J-25,26), HF(2,J-21,22)
HF(3,16,17), and HF(4,J-8-10)]. Examination of the derivatives due to
VR indicate that the multiquanta VR relaxation path from HF(4,J-

9,10) is responsible.

4.2 Effect of Vibrational to Rotational Relaxation on the Power and
Energy
The effect of distributions 2, 3 and 4 are examined in Table 4.2

which compares the relative band energies, pulse energy and duration.

62

 

 

7
.
0
l

AooaIAoI\90v

 

30

20

15

10

Rotational level J

Total contribution to the derivative of HF(v,J) at
peak power for run 18a.

Figure 4.3a.

d "F v J

d t

(cc/nol-scc)

63

10'5

I IIIIII

10

 

I»
é"

2

 

 

10’7 . .

: v-3

L.

_ 5

we

10‘8 .

C

b
10'9 1 n .1 J j

0 5 10 15 25 30

20

Rotational level J

Figure 4.3b.

peak power for run 18a.-

Total contribution to the derivative of HF(v,J) at

64

The relative band energies in Table 4.2 indicate that the maximum band
energy for cases 12a and 12b is found in the 5-4 band which is inconsis-
tent with experimental observations. The maximum band energy for run
12c and 18a are on the 2-1 band which is consistent with experimental
observations. The excess energy in the hot bands could be reduced by
incorporating the recent recommendation for the vibrational pumping
distribution of Cohen [22] or Cummings et a1 [25]. From Table 1.2 it
can be seen that a significant reduction in the direct population

of vibrational levels 4 to 8 due to pumping would result if Cohen's

[22] or Cummings et a1 [25] pumping distribution were adopted over the
distribution recommended by Cohen [24] and employed in the present
formulation. The effect of VR and VV relaxation mechanisms also are
believed to contribute to the excess energies observed in the hot

bands. The decrease in pulse energy, maximum power and time to maximum
power for multiquanta VR cases over single quanta VR cases is consistent
with the predictions of the comprehensive VT, RT model of Kerber and
co-workers [72-74].

The total power for cases 12a, 12b, 12c, and 18a are compared in
Figure 4.4. Comparing cases 12a and 12b in Figure 4.4 illustrates the
effect of single and multiquanta VR on pulse energy, maximum power and
time to maximum power. As more VR channels about Jmax become
available the pulse energy decreases.

The time resolved spectra of Run 18a for all 72 P branch transitions
considered are illustrated in Figure 4.5. The initial spike in the
power is a result of the instantaneous introduction of a finite amount
of F atoms. This effect would be less pronounced if F atoms were

introduced over a longer period of time by the dissociation of F2

65

.maamma cosmos a .EU ow a a .EU ow a a .m.c u as ..H n as

.namauasHva\nHmsfimva mg m coauouneou no common 0:80

.H mo soaummeou
Hm=o«umun«> mo mufidaomnoue m an“: H.e manna s“ moumfia mum mowumxmaou m> How nonfiuonwhumfio ooh

A

"msoaufionou huq>mu

 

 

o

“to“ cm a am .x.oom u as . =o~“~=au~maa.oum~c.c "annexes mace

m.o m.wm H.9H N¢.n mH.c Hm.o mm.o aa.c Nu.o o.H ec.o c «mama mad
o.o c.- H.HH cH.n mN.c a~.o om.o Ho.c mm.o o.H Hm.o N. onsum UNH
n.o m.moH q.oa mo.¢ ~m.o Ha.o o.~ mH.c mc.o mq.o ed.o n amass ANH
m.o ~.noa o.ce «c.w Nn.o ca.o cud hm.o om.o mm.o -.o m mawsfim mud
on Hanan? away uo\3 a\H mic «In mlw Nln HIN 01H a.umao m> mmmu

mem m>u unfiwuosw comm o>fiumdom mucosa «wane
. no odwsfim

 

me> no summon ~.e wanes

(watts/cc)

Power

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.60

1.0

0.0

 

 

 

 

66

 

run 12a
run 12c
1.1
I
run 12b
.. : I
.1 I
l ; n. i”
-w ‘ . run 18a
t: I
» ||
' l
1 1 1n 1 n
0.0 20. 40. 60. 80. 100.
Time (uses)

Figure 4.4.

Comparison of the total power as a function of time

«oomzw meme
.0: .oo .o: .o~

   

\\
Z'O

fl'O

UL: .m> a him. g

l
9'0
(ao/siiem) Jamod

1
8'0

 

0‘1

Anomav mafia
.o: .om
_

c
6
0

son .00
m _

0

\\\

Z'O

n.~

fi‘O
(oo/siiem) JaMOd

9'0

8'0

mt.» .m> A ~.~. «mica

O'I

.oau can new muuuomm oo>~0mou weds
Anomaw mafia

 

uzuh .m> A h.vv mUaom

Aommav made
.8 .oo .o: .o~
_ _

 

mime..m> . n._. muse;

 

I
9‘0 fi'O 2'0

(so/siiem) :euod

8'0

 

z'p

fl'O

9'0
(so/3412M) Janna

8'0

 

 

... apnoea
Anomav wads
.om .oo .o. .ow

Ha: .m> A him. 5.5L

Anomav osua
.om .oo .o: .om

 

— _

.A

 

ULuh .m> . Hflcv KHiVm

o.oO

fl‘O

1
9'0
(ao/siiem) JOMOJ

J
8'0

 

h

0.00

0

8'0 9'0 fl'O Z'O
(oo/siiem) :amod

0'1

 

68

given by Equation 3.25. The rapid oscillations in the power is much
more pronounced than the VR,T model of Kerber et al. [73,74]. This
phenomena is believed to be caused in part by the rapid redistribution
of energy levels of HF(v,J) by multiquanta VR. The late time powers
observed in the hot bands is quite pronounced. The mechanism contri-
buting to the spectra of P4(12) is believed to be caused in part

from the suppression of P branch lasing from rotational levels greater
than 12. The mechanism resulting in lasing from P5(4), P5(5), P5(4),
P6(7) and P6(8) is not clear although examination of the population

of HF(v,J) at 50 usec in Figure 4.1 indicates that the population

of HF(é,J) and HF(5,J) are nearly equal from J equals 0 to 12.

69

5. SUMMARY

A comprehensive model of the pulsed H2+F2 chemical laser system
has been presented. The assumption of rotational equilibrium has been
relaxed for all mechanisms contributing to the population of HF(v,J).
This model permits examination of the effect of VR, VV, RR and RT
relaxation mechanisms on the spectral content of the laser pulse. The
model is also capable of examining the effects of various nonequilibrium
pumping distributions and line selection on the time resolved spectra.

This comprehensive model supports the conclusion of Wilkins [511
that VR relaxation processes are responsible for populating very high
rotational levels of HF. The effect of multiquanta VR relaxation was
found to increase the population of the upper rotational levels (J>lS)
of HF over the entire pulse duration while RR and RT relaxation
mechanisms slowly depleted these levels. The VV relaxation mechanism
permits the energy of a given HF(v,J) to be transferred to HF(v-1,J)
or visa versa which has the effect of redistributing a given rotational
level over all vibrational levels.

Recent experimental results indicate that rotational lasing may be
significant. The present formulation is being extended to assess the
effects of rotational lasing on the population time histories and time

resolved spectra.

70

APPENDIX A

RATE COEFFICIENTS FOR THE HZ + F2 CHEMICAL LASER

The rate coefficients that have been recommended by Cohen [22] and
incorporated in the present formulation for dissociation, recombination,
cold and hot pumping, HF-H2 VV and HZ-H2 VT energy transfer are
presented in Table A.l.

The cold pumping rates that have been suggested by Cohen [22,24]
and Cummings et a1. [25] are compared in Figure A.l. The specific cold
pumping reactions listed in Figure A.l are plotted as a function of
temperature for the listed forward rate. Reactions for all figures in
this appendix are shown in the exothermic direction. The cold pumping
reaction is exothermic in Figures A.l(a) to A.l(c) and endothermic in
Figures A.l(d) to A.l(j). Both the forward and reverse rate coeffi-
cients are included in the model. The hot pumping rates are compared
in Figure A.2. In the present study the rate coefficients recommended
by Cohen [24] listed in Table A.l are combined with the rate coeffi-
cients calculated by Wilkins [51] for VV, VR, and rotational relaxa-
tion of HF to form a chemical kinetic model of the pulsed H2 + F2
laser system.

The VV rates calculated by Wilkins [51] are compared with the VV
exchange rates suggested by Cohen [22,24] and Cummings et a1. [25] in
Figure A.3. The multiquanta VR rates calculated by Wilkins [51] are
only valid for collisions of HF with HF. For collision partners other
than HF, Wilkins rate coefficients are scaled according to the re-
commended VR,T rates of Cohen [22]. These rate coefficients are

compared with the suggested rate coefficients of Cohen [22,24] and

Cummings et a1. [25] in Figure A.4. The recommended rate coefficients

71

of Cohen [24] assume that the VR-T rates are linear in v. and a

single quanta, while those rates recommended by Cohen [22] and Cummings

et al. [25] have a v2'3 vibrational dependence for multiquanta VR,T

reactions.

72

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

Reaction Collision Partner M Forward Ratea

 

Dissociation Recombination

F2 + M : F + F + M all species KfFZ = S.*lol3e’35°3e
a + a + M 3 32(0) + M all species. xffiz = 1.*1019T‘1

2.932, 208
HF(O) + M 3 a + F + M all species xf“F(0) = 1.2*1018r’1e'135°33
HF(l) + M 3 H + F + M all species KfHF(l) = 1.2*1018T'1e'124'56
HF(Z) + M I H + F + M all species . KfHP(2) = 1.21r1ol8-r'le‘ll3'6E
HF(3) + M 2 H + F + M all species foF(3) e l-2*1018T'le'1°3‘3e
HF(4) + M I H + F + M all species KfHF(4) + 1,2«1018T'le'93-395
HF(S) + M 3 H + F + M all species KfHF(5) + 1_2,1018T-le-83.966
HF(G) + M 2 H + F + M all species KfHF(5) a 1 2,lolaT-le-V‘i.97E?
HF(7) + M I H + F + M all species KfHP(7) = l.2*1018T-1e.66’436

Cold Pumping

F * "2‘°’ 3 HF(I) + H K§(1) = 2.6*1ol3e‘1°6e
F + 32(0) : HF(Z) + H K§(2) g 3.3.10133'1-56
F + 32(0) 3 HF(3) + a x§(3) a 4.4.1013e-l.66
r , 32(0) : HF(4) + a x§(4) . 1.46.1013T-0.107e-ll.34e
a +.32(0) : HF(S) + a x§(5) a 2.17*1013T'°'l°7e'2°'786
r+ fi2(0) 3 HF(6) + a x§(6) a 3.76'1013T-°'l°7e-29'826

aWith the exception of VV and VR,T reactions this table contains the
recommended rate coefficients of Cohen [24]. The value of 6 is lOOO/RT.
The gas constant R is 1.98725 Kcal/mol-‘R.

73

 

Table A.l (continued)
Reaction Collision Partner M Forward Rate
Hot Pumping
a + F 3 HF(O) + F x2(0) = 1.1*1012e'2'46
a + r 3 HF(l) + F x2<1> - 2.5*1012e'2°46
u + r 3 ar(2) + r K§(2) = 3.5.10123-2-46
a + r 3 HF(3) + r x2(3) . 3.6*1012e'2'4e
n + r 3 HF(4) + F x2(4) - 1.6*1013e‘2'4e
a + r 3 HF(S) + r x2(5) . 3.6*1ol3e'2'49‘
a + r :HF(6) + :- x’f‘m - 4.8.10133—249
a + r :nrm + 2‘ K2”) 2 .=..s*1012e'2"‘6
a + r :uma) + r x’f‘m = 2.5'1012e-2'4e
ar-az vv
arm) 4» 32(1) :Hru) 112(0) Rf" 2(0.0) =- 9*1010
nr-a - 12
am) + 32(1) :arm 32(0) xf 2(1.0) . 2.9*1o
ar-a 12
33(2) 4' 32(1) 3317(3) 82(0) Kf 2(2o0) " 9'10
nr(3) + 32(1) sear(4) 32(0) xfxf'“2(3.0) - 2*1013
arm) + 32(2) =urm 32(1) foF'“2(o.1) - 9'10“
arm + 112(2) :arm 32(1) x,”’32(1.1) =- 2.921012
32-32 VT
3 (1) + M‘;ZH.(0) + M all species x H2’“2(1) - 2.5«10’424‘3
2 2 f
432.43
H2(2) + MfiH (l) + M ‘ all species Ksz-H2(2) - S.0*10-4T4'3

482,43

74

 

  
     
 

L00 RI"! (CC/flOLE-SECI

 

’ - 1 1 - 1 1, : A: A: : 1:
2 r * - i ' : ' ‘ ' '

‘ used in present formulation
3‘- :
“ : (a)
3‘,

’ a Cohen [24]

L
a L" F 9 112101: NF!“ 0 H A Cohen [22]

p

. e Cummings [25]
o E'

p

1 1 1, 1 1 1 1, 1 1 11 1 1 1111, L 1 1 1 11

 

F 100 300 S 900 1100 1300 1500 1700 1000

00 700
TERPERRTURE (DEGREES KELVIN)

L00 BRIE 1 CC/flOLE-SECI

 

 

a:
‘ used in present formulation
2
a (b)
‘
3
. F v 11210) s 111121 c 11
<0
7 1 1 1 1, 1 1 1 1 1 1 1 1 1 1 11 11 1 1 1
r-100 300 1100 1300 1800 1700 1000

500 700 000
TEMERRTURE 1 OEDREES KELV 1111

Figure A.l. Pumping rates for the cold reaction.

The pumping rates employed in the present formulation
have been suggested by Cohen [24].

75

used in present formulation

13

12

(c)

LOQORRIE (CC/HOLE-SECI

 

F 0 H2101 = HF(3) e H

1L111_14111111111111

c.100 300 500 700 900 1100 1300 1500 1700 1000
TEHPERRTURE (DEGREES KELVIN)

18

12

11

used in present formulation

L000801E (CC/HOLE-BECI

(d)
HF(41 o H: F 9 H2101

 

'JIJILIL91411LIIJJ111]

9'100 300 800 700 000 1100 1300 1800 1700 000
TEHPERHTURE (DEDHEEB KELVIN) t

Figure A.l. (continued)

76

13

(e)

L00 BRIE lCC/HOLE-OECI
10 11

HF141 o H 3 H2111 o F

 

llJJJLlJlLllLLIJ

600 700 000 1100 1300 1800 1700 1900
TEHPERRTURE (DEOQEEB KELVINI

13
I
ll
1|
11
0
II
(I
1 ll
1 h
71.
1 h
0
1 1
1
1
1 I
1 l
1 1
1 l
1 1

 

L00 R01! 1CCINOLE-0E01

 

N
C!
used in present formulation
d
(f)
6
‘
arts) 1 H s F o "2101

O
O

7 1 ,11 1 1, ,L, 1 11 1 11 1, 11 1 1 1 1 1, 11 1 ,1
r-100 300 700 000 1100 1300 1600 1700 1000

500
TEHPERHTURE (DEGREEfl KELVIN)
Pigure A.l. (continued)

77

Y '1'!

V ‘0‘"

7

<9)

I

11

Logonntt (CC/HOLE-SECI
varv III.

I

HF(S! o H = H211) 0 F

VIII 0 '0‘.

Y

 

llLlllLlilllillLllJ

l‘100 300 500 700 000 1100 1300 1500 1700 1000
YEBPERHTURE (DIOREEB KELVIN)

10

12

used in present formulation

11

(h)

Logolfllt (CC/NOLE-OECI

HF(B) 9 H = F 9 H2101

 

JIJJIJJJJLJJIIJLILJ

9‘100 000 500 700 000 1100 1300 1500 1700 1000
TEMPERRTURE (DEGREES KELVIN)

Figure A.l. (continued)

78

10

12

(i)

11

LogolfltE (CC/HOLE-SECI

HF(B) o H = H2111 o F

 

JilllLllJJJlllLLIJJ

f‘100 300 500 700 000 1100 1300 1500 1700 1000
TEHPERHTURE (0EOREEO KELYINI

10
1
1
P
1
1
r
1

(j)

11

L00bRRIE (CC/HOLE-IECD

HF(B) e H 2 H2121 e F

l l L 1 l J J 1 1 1 LL! 11 1 1 1111

9'100 300 500 700 000 1100 1300 1500 1700 1000
TEHPERBIURE (DEGREE! KELVIN)

Figure A.l. (continued)

L00 HHIE 100lfl0LE-0EC)

79

 

L00 RHIE (CC/flOLE-0ECI

 

 

     
   
       
 

“ LW
2
used in present formulation
2
(g)
(S
110 F2:=HF10141F

1m
0

7 1 ,1 1 1 1 1,11, 1 1 1 1 1 1 1 1, 1 1 1 1
r-100 300 800 700 000 1100 1300 1500 1700 900

IEHPERHTURE (OEDREEB KELVIHl ‘

n:-
“D

1 —:‘::::::::"—
an r “K\\\_—-
“D

- used in present formulation
::'

: (h)
53 r

p

_ H 0 F2 = HF(7) o f
I»:

1

 

1 1 1 ,1 1 1 1, 1 1 1 ,11,1 1, 1 1 1 1 1 1
r-100 300 800 700 000 1100 1300 1500 1700 1900
IEHFEHHIUHE (DEOREES HELVIHI

Figure A.2. (continued)

80

m

0

used in present formulation

11 12
77" rYVVIF'T III. 1

II

I

1

 

I

(c)

LogbRflIE (CC/HOLE-BEC)

H 6 F2 3 HF(Z) o F

07" U 0".

 

l J 1 l l 1 I l l l J l J J 1 L L l J

3‘100 300 500 700 000 1100 1300 15
IEflPERHIURE 10EOREEO KELVIN) 00 ‘700 1000

 

 

 

 

 

2E 77,;:,1_
:- _-W
U ___
7.1:-
uv‘ -
g .
‘ - used in present formulation
5...-
8"
“P
.-
:1.
e53 F (d)
ab
.1

. H 9 F2 3 HF(31 O F
a»
O
1 1 1 1 ,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f'100 300 1100 1300 1500 1700 1000

500 700 000
IEHPERHIURE (DEDREES KELVIN)

Figure A.2. (continued)

 

 

81

 

 

 

 

    
  
     
 

 

'7 L'
d I
8 .
u:
‘5‘ .- - - - : : :—
u" ’ : : — - -
.1 .
O
33 1
E.
..3 : sed in present formulation
3' _
1‘;
e3 r (a)
S 1 11: Cohen 1241
11 9 F2 2 HF(010 F A COhen-[ZZ]
o e Cummings [25]
C
1 1 1 1 1 1 1 1 1 1 1 1 1, 1 1 1 1 1 11
'— 7011 son 1100 1300 1500 1700 1900
l~100 30° sIgHPERHIURE 10EOREEO KELVIH)
o :-
"‘ 111
z: . _ _ _ _ : __7_
U - - 3 I 3 — v- - '
‘?:g r ,_ — __ _
3 : X
2
3 . used in present formulation
3: E'
u D
E . (b)
‘9 :'
e— .
o b
J
’ 11 . n s 111111). 1'
a F
O
1 1 1 1 1, 1 1 1 1 1 1 1 1 000‘ 1000' 1000J
700 000 1100 1300 1
r.‘o° 30° sIgflPERHIURE (DEDREEO KELVIH)
Figure A.2.

Rate coefficients for the hot pumping reaction.

The present formulation uses the rate coefficients
that have been suggested by Cohen [24].

10

U Y1 0"

‘V"

LogbRflIE (CC/HOLE-CECI
vvv' I‘vrvrv

70"

82

00"

V

_ ‘-
‘v

- L._
*- — v _ —

“\\\_—_used in present formulation

(1)

     

U

H ‘ F2 8 HE‘S) e F

  

 

J l I L l l J l I l J l J l L J l j J
500 700 son 1100 1300
rtnrtnntunt (ornate: HELVIH) xsoo ‘7°° "°°

Figure A.2. (continued)

83

 

  
    

L00 RHIE (CC/flOLE-SEC)

:3 F _ .. — : : : 3‘ 3 3 3 ‘2 f: 3: '
r
r
e: r .
a 1 used in present formulation
p
:3 F (e)
r
0 '.'
d D
p
r
H 0 F2 = HF(4) o F
on

 

LiLlLlliillLLJLllJJ

p100 soo son 700 son 1100 1300 1500 1700 moo
ttnrtanruns (ornate: KELVIH)

_—
“——
_————-—'—'-—
.o—
-——"'_‘

1:
\

./I used in present formulation

(f)

L00000IE (CC/HOLE-0EC)

! H 9 F2 3 HF(E) O F

LILLLILIIJIJLJILLLJ

1-1oo soo 500 700 can 1100 1300 1500 1700 1900
rearranrunt (asserts xtvau)

Figure A.2. (continued)

L00 RHIE (CC/HOLE-BECI

84

(1
h n

 

U I)
(I (I
11 ll
U n

A
v

A—‘
v

1) I)
I) (I
0 u
(r (I

(I I)
I

I1 I)
1 (I
1 (I

 

 

 

 

 

   

 

 

 

 

 

 

L00 IHIE (CC/HOLE-OECI

2! “\\
A:X:AE::3‘V::A
:: \\\\_—- e__
used in present formulation
‘2
(a)
S cfiohen [24)
.Cohen [22]
‘Wilkins [51]
C
1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1, 1 1
r-100 300 500 700 000 1100 1300 1500 1700 1000
IEHPERHTUHE (DEGREE! KELV1H)
22
00
CI.
2 used in present formulation
a
d
HF(S) e HF(S) = HF(4) o HF(O)
1.
.

 

IJLJIILILJLLLJJ‘LIALJ

1-160 :00 too 700 900 1100 1300 1500 1700 1000
TEHPERHTURE (DEDREEB KELVIH)

Figure A.3. The temperature dependence of selected vibrational
to vibrational exchange rate coefficients. The VV
rates calculated by Wilkins [511 and given in
Table 1.5 are employed in the present formulation.

85

 

 

1 CCMOLE-OEC 1

 

 

L00 [HIE

2‘.

1'. ,,,,,,11-1,1,-1,

as - _ .1 _ - w‘ L L : j:—

ue ' _ _ — w — — - _ - +_

3 , used in present formulation lCohen [2“

(6) .Cohen [22]

a - eCuumings (25]
. W11) 0 112 s HF101 o 112 xWilkins [51)

- Cohen [22]

JIJJLJLJJILJIIJLJLJ

Moo soo son 7011 son 1100 use woo was man
tau-mm: manners 11mm

1
I111
111)
II II

 

 

1.00 IHIE (CE/HOLE-OEN

2 r7 — 2 :
u __,,,,,.__.....——-————————.—.——_____._..____..
d
- used in present formulation
d
9 ()3)
fl
HF“) c1 H s 1010). u
D
.

llllliJJlLlJljllllJ

Moo soo too no son use 1:110 1500 ma 1900
1mm (crusts: 112mm

Figure A.4. Vibrational to rotational. translational rates.

~ Note that for HF+H ,H,F ,H. Wilkins [Sl] rates are
scaled according t6 Cohzn's [22] rates and are
employed in the present formulation.

86

U “"

(c)

Hf!!! 9 F! 8 ”FIG! 0 F2

L00 RITE fist/“OLE-OECI

'10

  
    

fiv'j" V V V". j V 'U' V V 'V' V V 7" V V '1'

sed in present formulation

 

 

IJILJLLIIILILIJJ

I~t00 son " 500 700 can xxoo :300 1:00 t700 1000
ttnrzxarun: (occurs: xevaua

 

 

 

L00 INTI (CC/HOLl-IEBD

 

 

E

’, (d)
a :- - - - A - A
d, - _ - r

b

P - : 3 : ; ‘ ~~

, sed in present formulation
:7
o:- ‘232—H—
"r-
.r fiftuofsflflmof

b

h},
e’
' _ 1r,1 ,4, 1 1 1 4 1, 1, 1 1,,1 1 ,4i 1, 1, 1, n J
9:00 sou 11W $300 3600 1700 mo

sob won one
trnrcnutusr throats: nszxua

Figure A.4. (continued)

87

I!
w

A.

    

‘2

HI
hi!

vva v Viv] v vvv‘
L/‘
In
C.

D-
4
4
‘
4
U
l}

in present formulation

1!

LoguPflTE (CC/flOLE-SEC)

flftii 0 ”F s ”7(0) 9 B}

V VV' V

VVV' V

j

 

[LilLLlllllllLllllJ

0 500 700 900 ixoo 1300 1500 1700 ‘800
t~100 30 TEflPERflTURE (DEGREE! KELVIN)

f}-

   

   
 

ll

(CC/"OLE-IEC’
vvv
1
n
n
H
1
4
1
b
b
i
>
D
>
?
F

I!
VVV' V

‘\\—;—used in present formulation

(f)
HF!!! 0 H? s flFtS) o “F

L00 nut:
10

V, T-VT' V VVV' 1 VVV'

 

lllllLJlgLLillklllJ

r-lOO 300 $00 700 900 1100 1300 1500 1700 I900
TEflPERflTURE (OBOE!!! KELVI“)

 

Figure A.4. (continued)

88

APPENDIX B

DERIVATION 0F Xra FOR ROTATIONAL LASING

d
The rate equation for HF(v,J). is

dHF v J

dt - X(V9J) + Xrad(V.J+l) " Xrad(v,J) + A(V,J) (3.1).

where X(v,J) is the net change in HF(v,J) resulting from chemical
reactions and P branch lasing. The rate of change in HF(v,J) due

to rotational-lasing can be expressed in terms of the gain, c(v,J),

and xrad(v’J)' The Einstein coefficient for spontaneous rotational

decaying A(v,J) has been tabulated by Meredith [91] and can be ex-

pressed in terms of the Einstein isotropic absorption coefficient

B(v,J) as shown in Equation 3.2.

A(v,J) - 2hcch(v,J)BB(v,J) molauleml-secnl . (3.2)

The gain coefficient for RR lasing is derived in terms of the
Einstein B coefficient, the Voigt profile lineshape function

¢(v,J), and the wave number ch‘v,J) of the transition (v,J+l)-*(v,J).
This gain coefficient can be written as [72]

hNchR

4n

(v,J)B(v,J)[2£ig-HF(v,J+l) - HF(v,J)] . (8.3)

“R(V’J) ' 2J+3

The gain coefficient has been estimated by Hinchen [34] to be on the
order of 1 cm.1 for a population difference of 10-4 torr and cross
section of 2.2x1013 cmz. Similar gains have been measured by
Skirbanowitz et al. [43] with the same range of wavelength and pressure.

The rate of photon emission Xrad(v,J) can be written in terms of the

photon flux and the gain [72].

89

Xrad(v’J) ' cR(v,J)fR(v,J) (3.4)

The present formulation is capable of modeling rotational lasing
based upon Equation (3.4) which requires up to 210 additional rotation-
al lasing flux variables to be integrated. However, the model currently
requires over 30 minutes on a CDC 7600 for a typical run and the
additional variables may significantly increase computation time. An
alternate expression for Xrad(v,J) will be derived which does not
require additional integration variables. If a population inversion
exists then the following condition holds

2J+l

2J+3 HF(v,J+l) - HF(v,J):Q . (3.5)

With the large gains that have been observed it may be reasonable to

expect that during lasing

doR(v,J) ~
T. 0 (3.6)

hence Equation (3.3) reduces to

 

 

2J+1 dHFLv.J+l) _ 9.1911211 ; o , (3.7)
2J+3 dt dt

If equality is assumed in Equation (3.7) then Equations (3.1) and

(3.7) may be solved for the rotational lasing photon emission rate

2J+l 2J+1
[2J+3 + l] xrad(v’J) 2J+3 X(v,J+l) - X(v,J) (3.8)

For two adjacent lasing transitions, Xrad(v,J) and Xrad(v,J+l) can

be written as

90

2J+l 2.1+___1_x 2_J__+l
[2J+3 + 1] xrad(v’J)- 2.1+3x ed“ M”) 2J+3 x“ ML) ”x“ J)
2J+3 2J+3

 

 

- X ad(v,J) + [ + l] Xrad(v’J+l) - X(v,J+3) a X(v,J+1)

2J+5 2J+5

(3.9)
With inductive reasoning Equation (3.9) can be generalized to a system

of n adjacent lasing transitions with n unknowns.

 

 

 

A a B xrad , (3.10)
_ __1 .- 1 .. ._
-l 1+G2 -G2 . . . 0 3(2) xrad(2)
0 -l 1+Gk --Gk 0 - B(k) xrad(k) (3.11)
o . . . g -1 “Ga-1 -Gn_1 B(n-l) Xradon-l)
0 . . . 0 -1 14GIn B(n) L-de(n) .—

 

 

 

where 3(k) is given by

30:) . G dHF v k-l-l _ dHF v k (3.12)
k dt dt
and the degeneracy factor Gk is given by
G --2——-k+1 (13.13)

k. 2k+3

\

91

The matrix A defined by Equations (3.10) and (3.11) is a continuant

matrix. The matrix A can be readily transformed to the upper triangular

 

 

 

 

 

 

 

 

 

matrix A'.
V . ' n
A B xrad (p.14)
r “ - r ' " ‘
Eli-G1 G ... 0 3(1) :xrad(l)
0 [l-I-G2 1+6]. -G2 0 3(2) + l+Gl :xrad(‘)
i
le MkD
o [1+c - " H: o - B(k)+—-:-— x (k)
k 1+Gk-l k 1+Gk_1 rad
0 [1+6 - 611-2 ] G B(n-l) +£53.32- X (n-l)
°'° n-l 1+G n-l 1+G rad
n-2 G n—2
n-1 BSn-l)
0 ... 0 [1+Gn - 1+Gn-1] B(n) + 1+Gn-1 xrad(n)
__ _ L. _J L _3
(3.15)

Since there are non zero terms only on the main diagonal and the dia-
gonal above it, a simple algorithm can be developed to obtain xrad

by noting that

 

G
_ _ n-l 3(n-l)
xrad(n) [HGn 1 +Gn-1] / [B(n) + 1 +Gn-1] (3.16)

and through repeated substitution Xrad(k) is obtained

 

6
[1+6 - k'l ] (3.17)

. - .. .B_<k_-1).
xrad(k) [ karad(k+l) B(k) + 1+G ] / k 1+kal

k-l

92

The photon emission rate for the kth transition Xrad(k) can be
obtained in terms of known quantities through Equation (3.16). This
derivation assumes that all elements in xrad are positive which means

that if a negative Xra k) is obtained the system of n adjacent

dC
lasing transitions is re-evaluated for k-l adjacent transitions.

Once the solution vector Xr is obtained with all elements positive

ad
the matrix A' is reformulated for the subgroup of transitions from
k+l to n. A new group of n transitions is then reformulated based
upon the criterion of Equation (3.5). This process is repeated until
all positive Xrad have been computed.

The rotational lasing power PRR(v,J) may be expressed in terms

of xrad(v’J) as

PRR(v,J) 8 thchR(v,J) xrad(v’J) . (3.18)

where the rate of photon emission Xrad(v,J) can be written in terms

of Equation (3.4) or (3.17).

93

APPENDIX C

DESCRIPTION OF COMPUTER SIMULATIONS
FOR THE PULSED HF CHEMICAL LASER

The simulation of the pulsed HF laser system is accomplished
through the numerical integration of Equations (3.27), (3.30), and
(3.38). The basic structure of the computer model consists of a
controlling main program designated by MSUHFVR, and five major sub-
routines designated PRINT, HFRATE, GAIN, RKF45, and DIFFUN. The
purpose of MSUHFVR is to initialize computational variables such as
species concentration, photon flux, temperature and time; define the
cavity conditions; obtain the necessary spectroscopic and thermo-
dynamic data from data files; and provide control parameters that
direct the laser simulation. Subroutine HFRATE obtains the forward
reaction rate constants and the detailed balance factors. Subroutine
GAIN is used to compute the medium gain of the laser for any given
allowed P branch transition by means of Equation (3.32). The popula-
tion difference in Equation (3.32) is computed in DIFFUN to reduce
discontinuities in computing Equation (3.30).

Time progression within the model proceeds along user definable
integration steps from time T to Tout' Numerical integration is
accomplished through subroutine DIFFUN, which utilizes a fourth order
Runge-Kutta technique develOped by Shampine and Watts [92]. The
derivatives required for the integration are computed from rate Equa-
tions (3.27), (3.30), and (3.38) and are obtained by means of sub-
routine DIFFUN. Laser initiation may be accomplished by the introduc-
tion of a finite amount of F,F and H2 at a given temperature.

2

Other gases such as HF, He, Ar, N2 and SF6 may also be present.

94

Fluorine atoms may also be produced by photodissociation of F2 in
Equation (3.25).

A warnier-Orr diagram describing the computer model is presented
in Figure C.1. The important variables and symbols used in the computer
program, listed in Figure C.1 and discussed in the text are defined
in Table C.1. This diagram is an overview that illustrates the
hierarchy and structure that produces a chain of commands and decisions
that are followed to obtain the time evolution of species concentra-
tions within the laser medium as well as temperature, and gain on all
transitions and the intensity on all lasing transitions. TheOWarnier-
Orr diagram illustrates the hierarchy of the processes involved in the
simulation of the pulsed HF laser by observing that the leftmost brace
denotes the main program MSUHFVR. The series of braces to the right
of MSUHFVR brace. The computer simulation proceeds from the first
process in the first level INPUT through the tree structure to the last
process in the first level TERMINATE RUN. The initial conditions are
printed by subroutine PRINT prior to the INTEGRATION LOOP. The
INTEGRATION LOOP is a recursive structure that calls subroutine RKFAS
at T based upon the values

1 out

of Y1 at T and the derivatives of Y1. The variables Y1 consti-

tute a 298 variable one-dimensional array and contain the species

which computes the solution vector Y

-concentration, photon flux and temperature. The INTEGRATION LOOP calls
subroutines RKF45, GAIN, HFRATE, and PRINT. The computer simulation is
terminated when the total power Ptot drops below 1% of the maximum

power PMax and the time is greater than the expected pulse duration

T For computer simulations where P branch lasing is suppressed

Final'

the simulation is terminated when T is greater than T Due to

final'

95

program execution times exceeding 30 minutes on a CDC 7600 the Fortran
Program MSUHFVR produces a file that is capable of restarting the

simulation at a later date if desired.

_ “_uw__—__—_—-——————-———1 '

 

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an
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AN I u- C Alla—ugh” c... fit I MIN
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An_.aaav
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>> :

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II“

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SI

 

 

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5 > mam ~H_>mh=v

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w .+.a..a.a..o .I . . . o
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105

Table C.1. Nomenclature

Ar

B(v,J)
Cvi(T)

B(v,J)
ERR(V.J)I ERR

f(v,J)

fr(v,J)

PB(J,T)

Flux

hc/k

81(T)

82(V')

He

HF(v,J)

Print

IRestart

Concentration of argon, mol/cc

Einstein isotropic intensity absorption coefficient,
cm2/molecule-J-sec

Molar specific heat at constant volume of species
1’ CBl/DO]. - 9K

Rotational energy of state (v,J), cm-l

Energy density of laser output due to rotational
lasing, J/cc

Photon flux for the transition (v,J+l)+(v,J)
mol/cmZ-sec

Photon flux for the transition (v,J+l)+(v,J)
mol/cmz-sec

Concentration of fluorine atoms, mol/cc
Concentration Of fluorine molecules, mol/cc

HF self broadening coefficients recommended by
Bough [86]

14

Background photon flux. Typically 10- mol/cmz-sec

The gas constant is k, h is Plancks constant,
c is the speed of light in units of 1.4387886
cm-°K

Concentration of atomic hydrogen, mol/cc

Molar enthalpy of species i, Kcal/mol

Concentration of hydrogen in vibrational level
v', mol/cc

Concentration of helium, mol/cc

Concentration of hydrogen fluoride in vibrational
level v and rotational level J, mol/cc

A flag which controls the output from the laser

simulation and the rotational lasing option included
in the simulation

A flag which specifies whether or not the simulation
is started from previous conditions

ssg

J,J'

KbRT(v.J.AJ)
%m(VDV*9JDA‘-T)
wa<v.v*.J.J')

K; M (1:26))

R
Kf (V)
KfRR(v,J,AJ-=l)
KfRT(v,J,AJ)
VR

Kf (v,v')

Kfvv(v,v*)

106
A flag which specifies whether or not there is P
branch lasing
Rotational energy level

Rotational energy level that minimizes the energy
defect in the VR reaction HF(v,J)+EF(v',Jmax)

Rotational energy level on a given vibrational band
of maximum gain

F atom recomination rate coefficient

H atom recombination rate coefficient

HF recombination rate coefficient

Detailed balance factor for multiquanta VR, where

J' ranges from -4 to +2 of Jmax' The most probe

able VR path is denotes as J .
max

Detailed balance factor for RT with AJ 8 l or 2

Detailed balance factor for RR with AJ - 1
Detailed balance factor for HF—HF VV

Forward rate coefficient of cold (hot) pumping from
reference [24], cc/mol-sec

Forward rate coefficient for F dissociation
recommended by Cohen [24], cc/mol-sec

Forward rate coefficient for H2 dissociation
recommended by Cohen [24], cc/mol-sec

Forward rate coefficient for HF dissociation
recommended by Cohen [24], cc/mol-sec

Forward rate coefficient for rotational relaxation
recommended by Wilkins [51], cc/mol-sec

Forward rate coefficient for rotational to rotational
quantum exchange, cc/mol-sec

Forward rate coefficient for rotational to transla-
tional relaxation, cc/mol-sec

Forward rate coefficient for mulitquanta VR,
cc/mol-sec

Forward HF VV rate coefficients recommended by
Wilkins [51], cc/mol—sec

Stepsize, Hi

I

TFinal

TFlash

ILeft

Out

107

HF-HF VT rate recommended by Cohen [23], cc/mol-sec
Spacing between R0 and RL, cm
Concentration of nitrogen, mol/cc

Normalized cold pumping distribution of Polanyi
and Wbodall [ll]

Normalized hot pumping distribution of Polanyi and
Sloan [26]

Power density cf laser output from.rotational lasing
for the transition (v,J+1)»(v,J), w/cc

Total power due to rotational lasing, w/cc

Power density of laser output from P branch lasing
for the transition (v+1,J-l)+(v,J), w/cc

Total power from P branch lasing, w/cc

Power density of laser output through R0 for the
transition (v+1,J-l)+(v,J), w/cc

Total power from P branch lasing through mirror
R0, w/cc

HF rotational relaxation probability for the transi-
tion J+J~AJ

Rotational partition function for level v
Universal gas constant, 1.98725 cal/mol-°K
Reflectivity of the back mirror

Reflectivity of the output mirror

Concentration of sulfurhexafluoride in mol/cc

The integration stepsize is Tout—T, sec

The gas temperature, °K

Expected duration of the simulation

Length of time of the photodissociation of F2, sec

The amount of remaining program execution time in-
sec

The integrator RKF45 integrates dY/dt from Time

to Tom:

TStart’ TSkip

Time (t)

Av
My.”

xrad(v’J)

wc(v,J)

ch(v,J)

108

Controls the time at which the output is printed
in MSUHFVR

Time, sec
Vibrational energy level

The variable list Y contains 298 variables, HF(v),
HF(VIJ). H2(V'). H. F2. F. 1‘. and f(V.J)

The derivitive of Y1 evaluated in DIFFUN

Maximum value of flashlamp output in moles of
photons per unit volume, mol/cc-sec

Photon absorption parameter of F2, cc/mol
Gain of transition (v+l,J-1)+(v,J), cm"1
Lorentz half—half width, cm'l
Doppler half—half width, and

l

Gain of transition (v,J+1)+(v,J), cm-
Threshold gain, cm-l

Density of the gas mixture, mol/cc
Number of vibrational quanta lost in VR
Normalized line profile, cm

Rate of photon emission of transition (v,J+l)+(v,J)
mol/cc-sec

Wave number of transition.(v+l,J-1)+(v,J), cm-1

Wave number of transition (v,J+l)+(v,J), cm.-1

109

APPENDIX D

MINIMUM ENERGY DEFECTS FOR
VIBRATIONAL-ROTATIONAL RELAXATION

The J' levels associated with the vibrational-rotational relaxa-
tion processes given in Equation (1.12) with minimum energy defect
for the relaxation paths of interest are tabulated in Table D.l. The

corresponding energy defect is also noted.

Table 0.1
VIiIV'IO

J J' A6
1 i5 '263
2 15 -223
3 i5 .166
6 i5 -26
5 i5 131
6 16 -260
7 16 ~20
6 i6 269
9 17 -62
10 i7 267
11 i6 i6
12 i9 ~251
i3 i9 209
16 20 -19
i5 21 '299
16 21 321
17 22 133
16 23 -52
19 26 -235
20 25 -913
21 25 31k
22 26 166
23 27 25
26 26 -115
25 29 -252
26 30 -366
27 30 507
26 31 600
29 32 295
36 33 192

Most Probable Multiquanta VR Transitions
Defect AB in cm-

V82.V'I0
J' A6
20 236
20 276
20 350
21 -292
21 '191
21 96
21 27k
22 -250°
22 50
22 367
23 ~59
23 399
26 ~56
26 ~21
25 56
26 -302
26 276
27 ~67
26 ~36:
26 306
29 26
30 -266
30 507
31 265
32 29
33 -202
3B ~926
36 952
35 25k
36 61

1

110

v-2.I'-i 7-3.v--I

J.

16
15
15
15
15
16
16
16
17
17
16
19
19
26
21
21
22
23
26
25
25
26
27
26
29
30
30
31
32
33

I66

256
'237
-161

'97

106
“273

-66

217

'62

256

-6
-266

179

~66
'257

266

105

-72
‘267
‘920

279

139

1
'133
'265
-396

M63

360

259

151

J6
25
25
25
25
25
25
26
26
26
27
27
27
26
26
29
29
36
30
31
32
32
33
36
3h
35
36
36’
37
36
39

66

~23:
~19~
~12:
~12
133
315
~37I
~116
171
~635
~76
315
'216
266
~2~6
276
~159
616
s
~397
275
~92
~656
zoo
~23
~auo
662
196
-65
~357

733,9‘81
J' A6
26 162
20 219
20 292
21 ~326
21 -160
21 1
21 219
22 '265
22 3
22 327
23 '101
23 290
26 '96
26 360
25 6
26 ~336
26 217
27 '91
26 '39“
26 296
29 '20
30 -269
30 661
31 209
32 '16
33 -260
36 -657
36 367
35 196
36 10

and Their Energy

V'SQU'32 V‘9IU'39

J.

16
15
15
15
15
15
16
16
17
17
16
19
19
20
21
21
22
23
29
26
25
26
27
26
29
30
30
31
32
33

AB

223
~25|
~177

~66

77

259

~66

166
-ili

222

~26
~276

1&9

~61
~27I

252

76
~92
'260

362

265

110

~21
~15:
~277
~66:

621

322

225

131

J.

26
26
29
29
29
29
29
29
36
30
30
31
31
32
32
32
33
33
3b
35
35
36
36
37
36
36
39
90
60
61

A5

636
671
-636
-332
'192
'17
191
636
-290
20
366
-263
125
~660
'6
693
066
521
27
'556
166
'256
637
25
-375
391
21
-337
691
161

Table 0.1 (continued)

96 0"! 6'66 6"! 60" 6-

00'» III“ ROI“ fill“ fill» 86" r. fl’O‘ fl>69 flit. H'I‘
“'66 C 96" ”I19 06" F'GD‘D Cl'i 6“” 6'6! N I. G

V‘6.'.'1 V3“,V'.z

J6

25
25
25

25
25
26
26
26
26
27
27
26
26
29
29
36
36
31
32
32
33
36
36
35
36
36
37

36'

AE

~29o
~255
~166

-79'
60‘

235
~626
'166

96

667
~162

236
'276

166
~362

199
~231

329

~66
~656

193
~15?
'561

216

~39!
369
113
~15»
~615

J6

26
26
26
26
21
21
21

'22
-22

22
23
23
26
26
25
26
26
27
26
26
29
30
3o
31
32
33
36
36
35
36

AE

136
165

235
'361

"216

'63
165
~319
"61

:‘269

-162
233
.166
366
~37
~369
162
~133
~626
192
~65
~319
376
153
~66

~277'

'666
322
166
~37

111

V'6'V'U3 '85".8.

J6

19
16
15

15'

15
15
16
16
17
17
16
19
19
20
21
21
22
23
26
26
25
26‘
27
26
29
3o
36
31
32
33

AZ

192
227
~193
~67
52
227
~66
156
~126
196
~69
~267
121
~61
~262
220
52
~111

°~273

366
212
62
~66
~166
~269
~666
379
266

- 191
. 161

J'.

632

32
32
32
32
32
32
33
33
33
33
36
36

.36

35
35
36

'36

37
37
36
36

66
66
61

.61

62
63
63

AE

~316
~266
~217
-116
16
166
367
~666
~176
119
669
~272
119
562
-166
375
~231
366
~266
331
~196
666
~65
~523
169
~255
696
16

~312

697

725.7021
4' AB
26 326
26 361
26 629
'29 ~610
29 ~276
29 ~106
29 92
29 326
36 ~370
36 ~71
30 256
31 ~366
31 26
31 656
32 ~160
32 366
33 ~136
33 667
36 ~67
36 526
35 66
36 ~363
36 323
37 ~71
3I ~656
36 277
39 ~77
6o ~621
60 371
61 ' 56

V8557‘82

J6

25
25
25'
25
25
25
25
26
26
26
27
27
26
26
29
29
36
36
31
31
32
33
33
36
35
36

'36

37
36
39

A6

~367
~313
~266
~165
~16
157
356
~261
26
322
~265
156
~332
96
$366
121
~292
265
~132
659
116
~222
663
136
~161
~656
299
36
~222
~672

v~5.v'-3
~J5 A2

26 79
20 113
26 161
26 262
21 ~255
21 ~67
21 ‘113
21 367
22 ~65
22 212
23 -~163
23 - 176
26 ~166
26 261
25 ~62
25 -399
26 169
27 ~175
27 396
26 137
29 ~110
36 ~35:
36 312
31 - 96
32 ~11I
33 ~315
33 636
36 256
35 63
36 ~67

Table 0.1 (continued)

I.

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u “I!“ NIhOlV ADI“ hi!“ «1!. P r. p r. p»po p y. p
9 131. N 06‘” t at!» H 6610 fll'fl U'\fl C on!” fi»:: 0

V85.V'86
J‘ A5
16 162
16 196
15 -20!
15 '10?
15 27
15 195
16 -106
16 127
17 -136
17 160
16 -70
16 290
19 93
26 -101
21 -293
21 167
22 27
23 -130
26 -265
26 306
25 179
26 55
27 -65
26 -165
29 -301
30 -616
30 337
31 266
32 157
33 7L

V=S.V'30

J.

35
35
35
35
35
35
35
35
36
36
36
36
37
37
36
36
36
39
39
60
60
61
61
62
62
63
66
66
65
66

AE

-626
-396
-329
-232
~103
57
250
676
-367
-101
216
561
-195
269
-502
'60
667
-196
366
-255
335
-227
609
-116
566
73
-603
335
-109
-539

V86.V'81
J‘ AE
32 -621
32 -366
32 -326
32 ~22?
32 -96
32 62
32 255
32 679
33 -267
33 -1
33 315
36 -376
36 -2
36 602
35 -219
35 261
36 -361
36 173
37 ~370
37 196
36 ~309
36 306
39 -166
39 69 2
60 55
61 ~370
61 369
62 -65
63 -626
63 366

112

v=e,v-=2
3' AS
23 222
23 256
23 319
23 615
29 -357
29 -19s
29 -3
29 223
3a -666
3a -162
3a 153
31 -663
31 ~51
31 337
32 -191
32 270
33 -221
33 29a
36 -151
36 666
35 ~15
3s ~626
33 21a
37 -166
37 512
33 153
39 -175
an osoa
no 239

61

'5‘

V86,V'83
J' AE
26 376
25 -370
25 ~365
25 ~20.
25 -79
25 61
25 276
26 ~301
26 -66
26 239
27 -267
27 79
26 -365
26 15
29 ~616
29 65
30 -352
30 163
31 -199
31 366
32 35
33 ~26?
33 369
36 55
35 -231
35 669
36 206
37 -66
36 ~291
36 661

V‘ip"'5
J. A5
20 3o
20 52
20 127
20 22“
21 .291
21 -130
21. 62
21 2.6
22 -129
22 157
23 .222
23 123
26 -220
2“ 155
25 '126
25 335
26 55
27 -215
27 325
2. 52
29 ‘155
30 '36.
30 239
31 B3
32 ‘15?
33 .353
33 366
36 19“
35 25
36 .137

986.0'85
J' AE
16 132
16 166
15 -222
15 -125
15 3
15 166
16 -125
16 99
17 -155
17 130
13 '91
16 256
19 65
20 ~121
21 -305
21 155
22 2
23 -169
26 -297
26 '266
25 167
26 26
27 -66
26 -202
29 -316
29 366
30 296
31 209
32 123
33 60

Table 0.1 (continued)

OONU‘WCNNVQ

u 0"“ N'fiilv «306 A3!» A... n on h»r- n F. » yo 9
GODflOmcuNPGOONOWJ‘UNF‘O

'379V'30
J' 65
37 311
37 362
37 606
37 697
36 '523
36 -369
36 '165
36 29
36 273
36 567
39 -307
39 26
39 363
60 '396
66 1o
60 659
61 '251
61 261
62 ~627
62 113
63 '513
63 73
66 .516
66 116
65 -636
65 231
66 -265
66 616
67 -62
66 -526

V87. v.81

JD

36
35
35
35
35
35
35
35
36
36
36
36
37
37
37
36
36
39
39
60
60
61
61
62
62
63
66
a.
65
65

AE

505
~519
.657
~366
-261

-67

97

311
-515
-261

60

392

-332
56

669
-166

260
-335

161
-396

169
-370

236
-266

363

-66
-566

159
-265

669

V379v732

J.

31
31
32
32
32
32
32
32
33
33
33
31.
36
3a.
35
35
36
36
3 7
37
3e
3e
39
39
60
61
61
62.
62
63

62

639

_ 670

.629
'336
'212
'56
125
360
-395
'121
161
-663
.126
263
'332
106
'650
62
.679
61
'623
163
‘267
361
-77
-666
199
-176

190

113

V87.V'!3
3' A5
26 117
26 166
26 210
26 303
26 627
29' -263
29 -99
29 115
29 359
30 -252
36 5c
30 332
31 -162
31 225
32 -231
32 159
33 ~311
33 161
36 -255
36 265
35 -117
35 659
36 97
37 -255
37 336
36 69
39 -277
39 626
60 127
61 -163

'379V'86
4' AE
26 267
26 316
25 -366
25 -271
25 -167
25 6
25 190
26 -361
26 ~116
26 156
27 ~326
27 3
27 363
26 -56
26 356
29 -31
30 ~612
30 60
31 -267
31 276
32 -63
33 -353
33 255
36 -26
35 -301
35 366
36 117
37 -125
36 -361
36 373

V879V'85
J' 65
20 -16
20 12
26 76
29 167
20 299
21 '172
21 11
21 226
22 .171
22 161
23 -251
23 69
26 .250
26 127
25 '170
25 270
26 6
27 '256
27 260
26 27
29 0200
29 356
30 166
31 '11
32 .206
33 .391
'33 296
36 125
35 -31
36 .166

v;7.v'=6
J' 62

16 103
16 136
16 196
15 '166
15 -21
15 133
16 ~163
16 70
17 -173
17 160
19 '112
16 219
19 36
20 -166
20 273
21 126
22 -23
23 -166
26 -310
26 236
25 116
26 9
27 '116
26 '226
29 -326
29 361
36 255
31 171
32 90
33 9

(—

ONOUIJ'unp-o

O
u an!» ~>n9|6 hit» All» hil‘ » r- n r- n p. p p. p
a «11. N ah\n t on!» n»¢5co cn-fl a-\n : 1» N r- a «3

Table 0.1

V369 V .35

J.

60
60
60
60
60
60
60
60
61
61
61
61
62
62
62
63
63
63
66
66
65
65
66
66
67
67
66
66
69
69

AE

~323
~293
~236
~165
~27
119
296
501
~663
~161
107
626
~617
~67
366
~622
23
693
~209
306
~356
203
~619
179
~606
231
~315
352
~160
537

(continued)
V86.V'=1
J' AE
37 126
37 157
37 216
37 305
37 623
36 ~525
36 ~369
36 ~166
36 69
36 351
39 ~666
39 ~151
39 191
60 ~555
60 ~159
60 261
61 ~619
61 50
61 563
62 ~73
62 666
63 ~115
63 666
66 ~79
66 536
65 26
66 ~665
66 203
67 ~257
67 660

v=3.v'-z
3' A2
36 336
36 ' 355
36 625
35 ~696
35 ~37:
35 ~z31
35 ~56
35 150
35 313
36 ~3ao
35 ~91
35 225
37 ~669
37 ~99
37 295
33 ~332
3a 113
39 ~676
39 17
39 533
60 z
61 ~516
61 55
62 ~617
62 zoo
63 ~zso
63 603
66 ~19
65 ~625
65 272

116

9'699'83
J' A!
31 296
31 326
31 367
32 ~665
32 ~32?
32 ~160
32 ’3
32 201
32 636
33 ~261
33 67
33 366
36 ~265
36 126
35 ~667
35 ~26
35 619
36 ~91
36 602
37 ~73
37 666
36 21
39' ~611
39 167
60 ~216
60 621
61 69
62 ~316
62 370
63 33

986,V'86
J' 55
26 16
26 66
26 103
26 192
26 310
29 ~371
29 ~196
29 10
29 266
30 ~362
30 ~52
30 263
31 ~257
31 112
32 ~373
32 66
33 ~602
33 67
36 ~350
36 165
35 ~220
35 336
36 ~17
37 ~365
37 253
36 ~67
39 ~376
39 269
60 2
61 ~276

V 86” '85
J' AS
26 200
26 230
26 269
25 ~336
25 ~216
25 ~66
25 107
25 312
26 ~167
26 76
26 363
27 ~73
27 270
26 ~133
26 262
29 ~109
29 336
30 ~3
31 ~335
31 160
32 ~123
33 ~619
33 150
36 ~110
35 ~373
35 262
36 23
37 ~209
36 ~636
36 263

V359

J.

20

20

20
20

20
21

21

21
22
22
23
23
26
26

25
25
26
27
27

26
29
29
30
31

32
32
33

36
35
36

V'=6
AC

'67
~37
21
109
226
~215
~35
166
~215
66
~361
15
~365
59
~215
206
~69
~29?
19~
'20
~266
313
120
~66
~252
366
221
61
~91
~261

10.

11.

12.

13.

14.

115

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