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INFRARED-MICROWAVE DOUBLE RESONANCE PROBING OF THE POPULATION-DEPOPULATION OF ROTATIONAL STATES IN THE NITROGEN DIOXIDE AND THE SULFUR DIOXIDE MOLECULES

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8307935
Khoobehi, Bahram
INFRARED-MICROWAVE DOUBLE RESONANCE PROBING OF THE
POPULATION-DEPOPULATION OF ROTATIONAL STATES IN THE
NITROGEN DIOXIDE A N D THE SULFUR DIOXIDE MOLECULES
North Texas State University
University
Microfilms
international
PH.D. 1982
300 N. Zeeb Road, Ann Arbor, MI 48106
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INFRARED-MICROWAVE DOUBLE RESONANCE PROBING OF THE
POPULATION-DEPOPULATION OF ROTATIONAL STATES
IN THE N 0 2 AND THE S0 2 MOLECULES
DISSERTATION
Presented to the Graduate Council of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Bahram Khoobehi, B .S ., M.S.
Denton, Texas
December,
1982
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INFRARED-MICROWAVE DOUBLE RESONANCE PROBING OF THE
POPULATION-DEPOPULATION OF ROTATIONAL STATES
IN THE N O 2 AND THE S0 2 MOLECULES
Approved:
Graduate Committee:
ajor Professor
Committee Member
Committee Member
B &-
Committee Member
Chairmgtfi of the Physics Department
Dean of ythe Graduate School
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Khoobehi, Bahram,
Infrared-Microwave Double Resonance
Probing of the Population-Depopulation of Rotational States
in the N O q and the SO? Mo l e c u l e s . Doctor of Philosophy
December, 198 2, 100 PP*/ H
graphy,
tables,
(Physics),
22 illustrations, biblio­
65 titles.
A 10.6 ym C 0 2 laser operating a power range S P 200
watts was used to pump some select vibrational transitions
in the N 0 2 molecule while monitoring the rotational transi­
tions
(9lrg
in the
*1 0 0/10),
(232/22
*242 ,23)
'
(4 0 2,38 — ”39 3 ,3 7 )
(0 , 0 , 0 ) vibrational level and the
rotational transition in the
(8 0 ,8 — ^ 1 7 )
(0 , 1 , 0 ) vibrational level.
These rotational transitions were monitored by microwave prob­
ing to determine how the population of states in the rotational
manifolds were being altered by the laser.
Coincidences
between some components of the V 3 -V 2 band of N0 2 and the C 0 2
infrared laser lines in the 10 ym region appeared to be
responsible for the strong interaction between the continuous
laser beams and the molecular states.
The same C 0 2 laser was used to establish the nature of
the coupling with the SO 2 .
There is a weak coupling of the
R-branch of the C 0 2 laser at 986 c m - -*- to the S0 2 molecule.
The
laser is made incident upon the S 0 2 while the rotational
transitions (4Q ^4
3i/3 )' (8l,7
163,13) ' (162,14
(215,17
224,18)f
-^l^'
(305,25
34? 2 7 ) an<3 (51g 42
72,6)f ^82,6
(172,16
l63,13),
296,24) ' (315,25
^°10 40^ ^n t^10
91,9}' (154,12
(213,19
306,24),
204,16)
(35,.
D fj U
vibrational
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level, the
(192 lg
transitions in the
18g 15) , (232 22>-- 2 2 g lg) rotational
(0 , 1 , 0 ) vibrational level and the
(26g 20---27g 2 g ) rotational transition in the
vibrational level are monitored.
(0, 0, 1)
There appear to be three
basic behaviour patterns we denote as being regular, phase
changing, and irregualr intensity changing in character.
Characteristic rate curves for population-depopulation
changes in the chosen rotational levels listed above were
found for various power-:levels from the laser pump and are
presented in this work.
Rate coefficients for intensity decay for various
laser powers were calculated from experimental data and are
presented.
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TA B LE OF C O N TENTS
Page
LIST OF TABLES ...........................................
LIST OF ILLUSTRATIONS....................................
Chapter
I.
II.
INTRODUCTION....................................
1
THEORY OF COLLISION-INDUCTED TRANSITIONS
BETWEEN MOLECULAR ROTATIONAL ENERGY STATES 10
Murphy-Boggs Theory of Transition Rate
Fractional Change in Intensity
III.
THEORY OF TWO PHOTON PROCESSES...............
23
Treatment by Schrodinger1s Equation
Interaction Diagrams
IV.
EXPERIMENTAL APPARATUS AND MEASUREMENT . . .
34
Infrared-Microwave D 6 uble Resonance
Spectrometer
General Technique of Data Acquisition
C O 2 Laser
V.
ANALYSIS OF EXPERIMENTAL R E S U L T S ...........
49
Results of The N0 2 and SC>2 Molecules
Phase Change in Tne S0 2 Molecule
VI.
C O N C L U S I O N ....................................
APPEN D I X ....................
A.
B.
75
81
Proof of Equation (2.38)...........
Evaluation of Tj j
(b,v) for
Common Inter- 1 2
molecular F o r c e s ..................
List of Tables (Experimental
Results) of Nitrogen Dioxide
Molec u l e ...........................
81
REFERENCES...........................................
97
C.
iv
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84
89
LIST OF TABLES
Table
Page
I.
Summary of Experimental Results
or Some Rotational States of the
Nitrogen Dioxide Molecule ........................ 61
II.
Summary of Laser Pumping of the
(232,22 *’24ji 2 3 ) Rotational
Component of the Nitorgen Dioxide
M o l e c u l e .......................................... 63
III.
IV.
V.
VI.
VII.
Summary of Experimental Results for
Some Rotational States of the Sulpher
Dioxide Molecule ...............................
70
Summary of Calculated Values of I/IQ
from Experimental Data for SO 2
M o l e c u l e .......................................... 73
List of Tables (Experimental Results)
of Nitrogen Dioxide Molecule ...................
89
Elapsed Time and Signal Amplitude for
Various Laser Power for the
(100 10 — *^1,9^ Rotational Transition
of tfte (0, 0^ 0) Vibrational Level
of the NC>2 Mdlec u l e ............................... 91
Elapsed Time Versus Amplitude for
Various Laser Power for the
(24i 2 3 — '232 22^ Rotational
Transition of the (0, 0 f 0)
Vibrational Level of the N O 2
M o l e c u l e .......................................... 92
VIII.
Elapsed Time and Signal Amplitude for
Various Laser Power for the (402 38— *■393 3 7 )
Rotational Transition of the
'
'
(0, 0, 0) Vibrational Level of N O 2
M o l e c u l e .......................................... 93
v
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LIST OF TABLES— Continued
Table
IX.
Page
Elapsed Time and Signal Amplitude
for Various Laser Power for the
(7-[^7 — ^^ 0 ,8 ^ Rotational Transition
of the (0, 1, 0) Vibrational
Level of the N O 2 M o l e c u l e ......................... 95
X.
Elapsed Time and Signal Amplitude
for Various Laser Power for the
(7 - ^ 7 -- ” 8 0 ,8 ) Rotational Transition
of the (0, 1, 0) Vibrational Level
of the N O 2 M o l e c u l e ................................ 94
XI.
Elapsed Time and Signal Amplitude
for a Fixed Laser Power of the
(241 , 2 3 ~ * ’232,22> Rotational
Component of the N O 2 M o l e c u l e ..................... 96
vi
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LIST OF ILLUSTRATIONS
Figure
1.
Energy Level Scheme for InfraredMicrowave Two-Ehoton Erocesses in NHg.
.
2.
The Relative Population of the Ground (a)
and Excited State (b) of Individual Msubstates of the K = 2, F = J + (5/2)
Levels of C F 3I . N® Represents the
Undistrubed Population, Nj the Population
in the Presence of the Laser Pump.
Populations are Plotted Relative to the
Ground State j = 4, K - 2, F = J + (5/2)
Level Assumed to have an Arbitrary Value
of 1 ........................................
3.
Diagram Used for the Steady-State Analysis
of the Lowest Four-Level Systems
(Ammonia)..................................
4.
Energy-Level Schemes for ir-Mw Two-Photon
Processes.
The Microwave Radiation (a)r )
is (a) Added to or (b) Subtracted from
the Infrared Laser Radiation (oi^).........
5.
Block Diagram of the Experimental Apparatus
Used in this Investigation ................
6.
Laser Schematic D i a g r a m ......................
7.
Energy-Level Diagram Showing Pertinent
Vibrational Levels of COg and N 2 . f =
Fast, s = Slow Radiation Decay ...........
8.
A Typical Recorder Output for Frequency
Measurement........... ......................
9.
Dual-Channel Trace of the Fourth Derivative
Profile and Frequency Marked Used for
Relative Intensity Measurement for the
S0 2 Molecule
10 . Semilogarithmic Plot of Elapsed Time Versus
Signal Amplitude for Various Laser Power
Levels for the (402 33 *— 39g 3 7 ) Transition
of N 0 o . . . . . . '........ ' ...............
vii
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LIST OF ILLUSTRATIONS— Continued
Figure
Page
11.
Semilogarithmic Plot of Elapsed Time Versus
Signal Amplitude for Various Laser Power
Levels for the (24.
— 239 00) transition
of n o 2 ............. : : . . . : ................... 52
12.
Semilogarithmic Plot of Elapsed Time Versus
Signal Amplitude for Various Laser Power
Levels for the (100 ig"— 9^ 9 );
(F,11.5-»— 10.5) transition of N 0 2 ................ 53
13.
Semilogarithmic Plot of Elapsed Time Versus
Signal Amplitude for Various Laser Power
Levels for the (10Q .g*— 9^ g) ; (F,9.5-<— 8.5)
Transition of N 0 2 - ' • • •
..................... 54
14.
Semilogarithmic Plot of Laser Power Versus
Change in Signal Amplitude at Fixed Time
Intervals for the (402 3 g-<— 39g 3 -7)
Transition of N 0 2 - .
. . . I ................ 56
15.
Semilogarithmic Plot of Laser Power
Versus Change in Signal Amplitude at
Fixed Time Intervals for the (24.
— 232 22)
Transition of N 0 2 .................................. 57
16.
Semilogarithmic Plot of Laser Power Versus
Change in Signal Amplitude at Fixed Time
Intervals for the (10g ng“*— 9-l 9 ); (F/ll.S"*—
10.5) Transition of N 0 2 . . .
58
17.
Semilogarithmic Plot of Laser Power Versus
Change in Signal Amplitude at Fixed Time
Intervals for the (10g n g ■*— 9^ 9 );
(F,9.5-*— 8.5) Transition of N 0 2 ...................59
18.
Systematic Decay Scheme for the 38495.28
M H 2 Rotational Component of the S 0 2
Molecule Undergoing a Phase Change Under
Laser Bombardment.
(15^ ^2 — 1 6 3 13) ;
(0, 0, 0) Component.
'
'
Laser power 126 W a t t s .............................. 65
viii
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LIST OF ILLUSTRATIONS— Continued
Figure
19.
Intensity Behavior for the 38230 MHz
Rotational Component of the SO 2
Molecule Under Laser Bombardment.
Probably the (72-^3
-- "^14 58^
Component.
Laser £ower 80 wAtts..........
20.
Systematic Signal Amplitude Decay and
Recovery for Laser On and Laser Off
Conditions for the S0 2 Molecule at
36834.87 MHz.
(192 1 8 ---183 1 5 ) ;
(0, 0 f 0) TransitioA.
The Upper
Trace is for Laser Power at 135 Watts
and the Lower Trace for Power of 105
Watts .........................................
21.
Su^tm^ry of
~
In
the Signal Amplitude Changes
(I/I0 ) with Power for Some Ground
and Vibrationally Excited Rotational Levels
of the S0 2 Molecule .........................
22.
Summary of the Signal Amplitude Changes
AI/Iq with Power for Some Ground and
Vibrationally Excited Rotational Levels
of the SC>2 Molecule
......................
ix
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CHAPTER I
INTRODUCTION
Microwave-infrared transitions are generally much weaker
than the normal single photon transitions but, by use of the
high sensitivity of laser spectroscopy, these transitions can
be observed with good signal to noise ratios.
Neither the
infrared nor the radiofrequency radiation might be directly
resonant with molecular transitions but, by using the nonlinearity
of molecular transition processes, we can observe
two-photon transitions for which the sum or the difference
of the frequencies of the two radiations correspond to mole­
cular transitions.
The idea is that, although the frequen­
cies of the infrared laser lines are not tunable and they
are not in resonance with molecular transitions in general,
we can take care of this discrepancy by tuning the frequency
of the "added" or "subtracted" rf frequency.
An alternative
way, of course, is to use some non-linear solid state element
for a mixing of the two radiations^, but since molecules
themselves are good non-linear elements the two radiations
can mix in the molecule.
The two-photon processes are well known processes since
the early years of quantum mechanics
moment
2-4
.
The transition
is expressed as
1
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2
=
where E
m
(1-1)
and E n are the electric field for the microwave and
I
the laser transition, respectively, y^ and yv are the perm­
anent dipole moment and the vibrational transition moment,
and Av is the difference between the laser frequency and the
frequency of the molecular transition
seen from Eq.
(see Fig. 1).
(1.1) that a two photon transition is weaker
than a normal vibrational transition by a factor of
2hAv|2 .
It is
| lly^E^^ /
If microwave radiation of sufficient power is used,
this factor is not very small even for relatively large Av.
Such two photon absorption has already been observed by using
two microwave radiations^'® .
The coincidence in frequency between various lines of
the CO 2 laser with the rotation-vibration bands of several
molecules opens the way for interesting spectroscopic m e a s ­
urements .
It is clear from experiments carried out in some
laboratories
7-20
that for a large number of molecular systems
many such coincidences exist.
The laser radiation selectively transfers the population
in some specific rotational ground states to vibrationally
excited states, or to upper rotationally excited groundvibrational states.
It is possible to determine the exact
transition by probing the participating rotational energy
levels by means of microwaves while the laser radiation is
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Fig. 1.
Energy level scheme for infraredmicrowave two-photon processes in N H 3 .
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4
transmitted through the cell.
It turns out that in such a
"double-resonance" experiment, not only the participating
levels but also other rotational levels, vibrational levels,
and even isotopic species are affected via collisional trans­
fer.
The rise time of the double-resonance signals depend on
the pressure of the absorbing gas and laser power level.
Collisional transfer of rotational energy was extensively
studied by Oka,
21-25
who used microwave-microwave double
resonance experiments.
We should mention here that in any real gas the rota­
tional energy levels of the molecules are coupled to each
other through transitions induced by molecular collisions.
Thus any non-Boltzman distribution caused by the absorption
of radiation in one transition is partially transferred to
the other nearby levels.
This process results in a corre­
sponding change in the absorption due to the adjacent transi­
tion.
This effect has been studied experimentally by using
the method of double resonance and can lead to valuable
information regarding the selection rules and the rates of
collision-induced transitions.
Oka
21-24
has conducted a
series of experiments and has measured the change in intensity
of a signal transition observed by saturating an adjacent
transition with high power radiation known as the pumping
radiation.
Gordon et al.
26
Microwave Double Resonance
used the method of Modulated
(MMDR) and measured the double
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5
resonance intensities and phase shifts in the 5,-type doub­
lets of HCN.
In both these investigations the measured
intensities were expressed in terms of a set of transition
ra t e s .
There is a method for calculating the transition rates
on the basis of the theory proposed by Murphy and Boggs
27
This theory has already been applied to explain the collision
broadening of microwave spectral lines in a wide variety of
cases
28-31
In general, the double-resonance technique provides
three types of information which are relevant to the under­
standing of the infrared spectrum.
From measurement of the
frequencies of rotational transitions which are affected by
the presence of laser radiation
(i. e., double-resonance
signals), and the subsequent assignment of these transitions,
the infrared transition which is accidentally coincidence
with a particular laser line can be accurately assigned.
Consequently, one can assign the precisely known laser fre­
quency to this transition and since the laser line must lie
within the Doppler width of the transition in order to
produce strong double-resonance signals, the accuracy of this
effective measurement is +1 Doppler width.
In addition to
producing this concrete starting point for subsequent analysis,
measurement of upper- and lower-state rotational transitions
in the form of double-resonance signals allow calculation of
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6
the constants for both these s t ates.
Provided the infrared
spectrum is free from strong perturbing influences, the
information so gained is usually sufficient to allow fairly
accurate calculations of the rovibrational structure of the
band concerned.
Finally, from rate equation
(AI/I fractional
change in intensity) which is main emphasis of this work, we
can get information about Boltzman distribution of different
gases and information about the energy hierarchy belonging
to some particular vibrational state.
Since there is not much
information about the third application of double-resonance
experiments, we propose to give a short account of the method
below.
The gas laser as the source of the infrared pumping
radiation in a double resonance experiment has the disadvantage
that output power is available only at quite widely separate
discrete frequencies.
Most ir-mw double resonance experiments
reported so far have usually relied on previously reported
coincidences, the source of such information being Stark laser
spectroscopy or laser line absorption studies, in conjunction
with molecules of large rotational constants.
Molecules with
large rotational constants have been selected for study, since
in such cases considerable information is available from con­
ventional infrared spectra.
However, for this class of mole­
cules, since the frequency separation between rotationalvibrational transitions is quite large, the spectral density
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7
is low and consequently the probability that an accidental
coincidence will occur is also low.
But for molecules with
small rotational constants, although little information is
available from infrared spectroscopy, the likelihood that
an accidental coincidence will occur is relatively high.
This is especially true if the spectrum is made more dense
because individual lines are split into hyperfine components
by effects such as nuclear quadrupole coupling.
In this
situation the problem becomes less one of finding a coinci­
dence between a laser line and a molecular transition and
more one of identifying the coincidence once it has been
found.
Jones and Kohler
32
used the symmetric top molecule,
CF^I to study population and depopulation effects of infrared
CC>2 laser lines on that molecule.
This molecule
was selec­
ted because it possesses the qualities enumerated above.
They reported extremely "large pumping effects due to using
6W of laser radiation, which caused relative changes in
intensity
(AI/I)
in ground state lines of up to twenty-five.
The population transfer into the excited state was so large
that many excited state lines, which had previously been
undetectable, produced signals up to thirty times more intense
than the corresponding undisturbed ground state lines
values of AI/I of ^6000 were achieved).
(i.e.
Population inversions
were produced by the laser pumping many of the K = 2 micro­
wave transitions, not only in those with levels directly
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13
13
12
12
11
11
EXCITED STATE
GROUND STATE
10
10
K = 2 ; F=(g+5/2)
9
9
8
8
7
7
6
6
5
4
5
POPULATION
0.90 0.92 0.94
Fig.
2.
K = 2 ; F = (j+5/2)
POPULATION
4
.96 .98 1.00
.02 .04
.06
,08
,10
12
The relative population of the ground (a) and excited
state (b) of individual M-substates of the K = 2, F =
J + (5/2)
levels of C F 3 I . Nj-represents the undisturb­
ed population, N- the population in the presence of the
laser pump.
Populations are plotted relative to the
ground state • = 4, K = 2, F =
J + (5/2)
level assum­
ed to have an arbitrary value of 1 .
9
pumped by the laser, but also in some connected only by
collisional transitions.
The result was that many of the
signals were observed as stimulated emissions rather than
absorptions".
The absolute population shifts produced by
the laser pump are estimated for symmetric top molecules,
CF^I.
Their experimental result is given in Fig. 2
32
To understand population shifts and the populationdepopulation mechanism of asymmetric rotors especially X Y 2
type of molecules which are subjected to strong electro­
magnetic fields such as those produced by laser, we selected
N 0 2 and S 0 2 for this study.
Before beginning a discussion of experimental data on
NC>2 and SO2 , in Chapter II we are going to show how the
experimental values of fraction change in intensity
relates to the transition rates.
(AI/I),
In Chapter III the theory
of double-photon processes is reviewed and in Chapter IV, the
C 0 2 laser and other experimental apparatus will be presented,
taken data presented in Chapter V and finally, conclusions
are given in Chapter V I .
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CHAPTER II
THEORY OF COLLISION-INDUCED TRANSITIONS BETWEEN
MOLECULAR ROTATIONAL ENERGY STATES
In any real gas the rotational energy levels of the
molecules are coupled to each other through transitions in­
duced by molecular collisions.
Thus any non-Boltzmann
distribution caused by the absorption of radiation in one
transition is partially transferred to the other nearby
levels.
This process results in a corresponding change in
the absorption due to the adjacent transition.
Theoretical
calculations of the transition rates have been done by Oka
using Anderson's Theory
35
.
33 34
'
Also, there is a method for cal­
culating the transition rates on the basis of the theory
proposed by Murphy and Boggs
Boggs
37
.
36
and modified by Prakash and
This theory has already been applied to explain the
collision broadening of microwave spectral lines in a wide
variety of cases"^-^ .
Here we introduce the Murphy and
Boggs Theory and then show how to find transition rates from
that theory.
The theory of Murphy-Boggs (M-B) is an impact theory.
M-B considered absorption and have assumed a Lorentz shape
for the absorption, referencing a paper by Weisskopf and
Wigner
42
.
There are two major assumptions commonly employed
10
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11
in impact-type theories, and considered by M-B.
The first
is the classical path approximation in which the relative
motion of the colliding molecules is treated classically
rather than quantum mechanically.
The second is the bi-
molecular collision approximation in which the effects of
collisions involving more than two molecules are assumed
negligible and ignored.
The Hamiltonian for the colliding molecules can be
written
H = H 0+ V ( W
(2.D
where HQ represents the Hamiltonian of an isolated molecule
and v(t)
represents the interaction between the two colliding
molecules.
The density matrix can be written
/%) = exp ^iHotA)Tli)/^(--;T"'(-)c)exp(iH.-t/fiJ
where T(t)
is unitary time development operator, and
(2-2)
satisfies
the following matrix differential equation.
A-fc
- exp (i.Hot/fc) v(i) exp
T(fc)
(2.3)
where
T(-*°) = Z-
<2 *4>
If the absorbing molecule is assumed to be initially in the
n,th energy level at t = - 00 then since vf-” ) is zero and H = HQ ,
we have:
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12
(2-5)
From equation
^
^
then
l f (— >1 h > < ^ l X ( . t ) t ^ > <-3 )exp
<^tnje x p
Since
and
(2.2)
(2 .6 )
\
(-i.
|exp ( i H o V ^ )
(2 .6 ) becomes
Iftt) l " > = H<«?1 T ( t ) \ i > < ? 1 7 ( - « ) l k > < k l f ' ' 0 0 \"> (2.7)
Using Eq.
(2.5)
< M
then, Eq.
=
s,„ SKn < « \ n - - ) i
">
(2.7) gives
eX
^ K l( + ° ° ) = l ' ^ n (+^)|
(2 .8 )
This is the probability that the molecule is still in the nth
level.
The probability that the collision has produced a
transition out of this nth level is given by
= l-jTvato ( + <>0)|51
From Eq.
(2.2), matrix element of T(t)
(2.9)
satisfies the
differential equation.
=
'dt
Z V^exrC-ito^t)
c-pb-
X Tc(, (2-10)
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13
where
^ckc~
(2 .11 )
^cxc 3=1 I
" '-a)
(
2 .12 )
and
Tcb=<c|T(i)\k,>.
(2.13)
By taking
T a k tt) = S a b ( t ) e x p [ - i / * j j.i' Va a (V)]
(2.14)
and it's derivative with respect to t
V - a t (Ta b r t ) ) = ( - % t
S a b ( - t ) j e x p [ - y t I < h V w (V)J
■'-CO
-‘
A A, IV)Sa(,(t)exp[-«/*j<JVVaatt')) (2.15)
By substituting Eqs.
in Eq.
(2.14) and (2.15) into Eq.
(2.10) can be eliminated.
(2.10) T's
The result is a differential
equation for Sab as
t
( % i
Sab ) exP [-i/*
J
Vaa IV)] » - 1/* £
_<*>
vac i t )
c^-a
xe'iW“^ Seklt) exp [-A ( Jt'v^tV)] •
The diagonal element Snn of Eq.
u.ui
(2.16) can be written as
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14
4 M IS„„) = -i/* 2Z yn[m{ i ) Sm„ exp f-i
- y*
W:£/l
*j
[ v u U ' ) - V MrtM 4 i ' ] - C 2 . n )
(Vo
Now let
t
'j [ v m „ U ' J - v „ n w ]
oik'
( 2 .18)
—oo
=-*■/* H
^
wt^rvl
W
s
The non-diagonal terms in Eq.
(Sw J = - Vk
wrt e x p r . i i o ^ i . - ^
L.
(2.16)
S MiaW -
(til (2 .1 9 )
satisfy the equation
21 \zj* (i) S
U)
(2*20)
k
where
fe+ ^
' £ l * > = V„,K l * ) e x p [-iu5m i k a i m K W j
Integration of Eq.
/t
sm„ w = - V *
/
(2.20) yields
^
A
v„,„U 0 S„„U')o i k - % . Z \
< # ' )
^+*"40
_«o
Substituting into Eq.
« / & (s„.it>
(2 .2 1 )
)=-^
SJtOdk'
(2.22)
(2.16) gives
n
f v ; „ h o O * ' ) o i r
vt\i=ko
-V-fc'il 21 1 v«'.Wv.',H') x
k^n
X
Equations
Eq.
similar to Eq.
^
4
(2.23)
(2.20) which can be substituted into
(2.23), may be used to eliminate the non-diagonal term
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15
S^Ct")
in E<3*
(2.23).
This process must be used repeatedly
to eliminate the non-diagonal elements in s(t), changing the
variable of integration and summation indicies as necessary.
This approach yields an expression for 9Snn/3t in the form of
an infinite series in powers of
(1/ifi) containing the only
diagonal element Sn n :
K^fcvio
*
(2.24)
Murphy and Boggs have obtained only the first term in this
expression and, they claimed, the higher order term in Eq.
(2.24)
can be retained from the pattern of the first two
terms given in Eq.
Snn(f')
(2.24) .
= Sn n (t) is made.
Moreover, the approximation
The reasoning is as follows.
Sn n (t') may be written as
(2.25)
Putting the first term of Eq.
(2.24)
into Eq.
(2.25) results
in
Now, substituting Sn n ( t1
from Eq.
(2.26)
into
(2.24), the
error in keeping only the Sn n (t) terms is less than the error
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16
in dropping the i/fi3 term in Eq.
4/m (s „ „ w )
= ->/*>£
(2.24).
Thus
1
<.<*>
+ ( iA ’ ) 2 T Z :
hlTjfcH
* VmnUO
Integration of Eq.
•
(2.27)
(2.27) yields
oo
/tn= (> a»; n
{oo
V„n (-fc*) + ...... J S nwU )
(+00)=
Where
\ dt' I d iv > )
K #1*1 '_ o o
5 1n^
l-*0 exP (_A„)
(2 .2 8 )
-U
U t I dt' v ;„ a ) c
**” L
L
x
c
n
(
(Y) - ( *a>)
t
it
l/M*n
r
,
( <*r U i ' »
i,
-o o
k =fci'i
x V ^ 0 : ) V * kli^
Using Eq.
(2.21), and ignoring rin
(2.29)
terms in that equation.
A n can be written as
Co
^
( J / V ) 2 1 jdfc J d i '
-Oo
'd) V ^ l i O e x ? [i*oBrrid-tOj.(2.3o)
__OO
This can also be written in terms of the Fourier transforms
+00
V (t)— */2TT ^ olco G(to) exp ( - i i o j c )
-c©
(2.31)
and
/v
+co
>
G(w) = l d d
\7(-tjexp(iwi:J
*-oo
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(2.32)
17
and the relationship
£>0
[
cJK exp ( I «*>'*) = TT £(u;) + A
(2.33)
'o
where (pis the principal value.
Then, as shown in Appendix A,
An becomes
< m' g
1-> i 2+4
^
-
f tOo* - i t i y
1« • “
>
Now let
r„ =
« . 35]
and
4^ ^ l/ 2TTh2 £ 1 (P j oltA3
(2.36)
-Oo
then
A „ = j/2T7, + i-
Now, if a molecule in state
cule in state j2 as ^ e
(2.37)
collides with another m o l e ­
perturber, Tn and (f>n are required
to be replaced by their expectation values O 2 M 21^nI 32M 2^ anc^
^j2M 2 1<()n |j2M 2^> averaged over all possible values of degenerate
states
and M 2 .
Therefore rn , <f>n in terms of expectation
values can be written as
Tji=!4jE
J,JZ
l < i ) , i j G ( w il j
vlul/ul(JZ
d
^2
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t2-38)
18
and
%
ykirV
=
-A
JZ
(P
a>3 '
(
" u))]^>'3
-®°
(2.39)
where
oo.
U ; o ; = K eK +e« H ES,+ e k ) j / *
= cou , + u v ; -
(2-4o»
is the Bohr frequency.
Since from Eq.
(2.14),
5
(-oo)
„[-*’)=
v / = T'nn
&/
the probability that any one of of the collisions will induce
a transition out of state
|Tn* (+ °°)| 2= exp
is given by Eq.
(2.9), and
{-f\n-f\*)= exp(-i^).
Therefore,
%
= 1 - e x P ^ r 3,32 ) '
(2'41)
The amount of collision-cross section between the impact
parameter b and b + db is 2irbdb.
And,
3/2
Vf[v)dV=
4 -TT JV (^/2TTkT) e x p (-•^\///2 k T >) V 3d V
(2.42)
is the number of molecules striking this area per unit time
at normal incidence and have speeds between v and v + dv.
Where N is the number of molecules per unit volume, F(v)
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43
is
19
the Maxwell-Boltzmann velocity distribution function, m
is reduced mass and v the relative speed in the center of
mass coordinates system.
The number of collisions per
unit time with impact parameters between
speeds
(v v + dv)
(b, b + d b ) , and
is then:
N2Trbv
F(\/) dbd- V.
Therefore, the number of transitions in state
(2.43)
per unit
time induced by collisions with molecules in state
then given by
Oo
(3 J
A
'
) =XTTN
1
00
) b d b \ V F ( v / R \ dv.
J0
(2.44)
J
o
Where the transition probability P^- i has been averaged
JiJ2
over all impact parameters and over all relative velocities,
and where the subscript i indicates the ith transition.
Since, for the first time we have used subscript i for ith
transition,
-follows
the P . . change to P^ . which are defined as
31^2
3l32
?M r c ^ ' ' exp{' r 4 (b' w^ '
(2'45)
Where C is called the normalization factor and is given by
C=['-exp{-E
J.
The expression
Ii
(2.44)
^
<2-46)
is then averaged over all the
rotational states of the perturbing molecules to give the
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20
average rate of the transition of state
A
k*,==£ * ‘
Do
where pB z
(2-47)
.
is the Boltzmann distribution.
In Appendix B we are going to show how to calculate
T.
.
for common intermolecular forces, but it is worth-
3132
while to mention here that the function
T.
.
depends on
3132
the type of interaction mechanisms involved, on the mole­
cular parameters, and upon the impact parameter b.
For
instance, taking dipole-dipole interaction between the
colliding molecules,
t^
,(
m
=-h
this function is expressed as
^ ; /
v
v
v
)
e
.
c
d
(m
;j
(2.48)
Where y-^ and y 2 are the dipole moments of the two molecules
and D (j jl) ,
d
(D2D2^ represent the dipole moment matrix
elements for the transition
One
may use Eq.
in intensity
(2.47)
3± +
and
^2 * ^2’ resPectiv e lY'
to calculate fractional change
(AI/I) due to a particular transition or, the
observed values of AI/I can be used to determine relative
values of the transition probabilities for various transitions by solving steady state rate equations.
Oka
34
used
ammonia for his study and found
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21
A I / j = | l « ((k„t - k , t ) / ( w
+ k,t + !
kj)
(2.49)
k , k„, k
, and k r are shown in Fig. 3 and are defined as
a
B
y
?
■
follows.
The transition rate k^ is for the dipole-type
(Aj = ±1 transitions), k^ for the dipole-type
(Aj = 0 transi­
tion) , and k ^ for the quadrupole-type transition.
other than a,
ties.
B, and
Transitions
y (Aj = ±2) have much smaller probabili­
They are summarized by £.
t, 4- denotes upward and
downward transitions, respectively.
+
i?
3
cx
3-1
Fig. 3.
3-i
T/3
_L
ex'
\
Diagram used for the steady-state
analysis of the lowest four-level
systems (ammonia).
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22
In
the Fig. 3 Vj = Vp is pumped frequency, and Vj
vs is the source frequency.
frequency region.
cules,
and
Both Vp, vs are in the radio
In our case for the N O 2 and SO 2 mole­
the pumped frequencies are in the infrared region,
the source one's are in the microwave region.
In
Chapter III details of a two photon process will
be discussed.
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CHAPTER III
THEORY OF TWO PHOTON PROCESSES
Like a solid state element, individual molecules can
work as a non-linear element and efficiently create sums and
difference of applied frequencies.
The only difference is
that, in molecules, the sum or difference has to satisfy
resonant conditions.
and Feyman
46
Kramers and Heisenberg
44..
45
, Goppert-Mayer
showed the transition probability of two-photon
processes can be derived as a result of second-order timedependent perturbation.
Here we will first derive the transi­
tion probability by using Schrodinger's time dependent equa­
tion and then more conveniently by using interaction diagrams.
3.1.
Treatment by Schrodinger's equation.
Now, in order to use the time dependent Schrodinger
equation and its results as applied to the infrared-microwave
two-photon processes, we use a diagram as shown in Fig. 4.
A microwave quantum is added or subtracted from an infrared
quantum in the energy-level schemes shown in Figs. 4 (a) and
4 (b), respectively.
The Hamiltonian of the molecule will be
given by
23
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24
where HQ is the vibrational-rotation Hamiltonian of the mole­
cule in free space and the second term represents the inter­
action energy between the molecule and the applied radiation
fields.
The molecular dipole moment y m can be written as
M vy\—
(3.1)
M p + Mi/ = M p +
where pip is the permanent dipole moment,
pv is the vibrational-
induced dipole moment and the Q s 's are normal coordinates.
The total radiation field E can be written m
its components
as
(3.2)
where E^ is the infrared radiation field with frequency to^
and Er is the radio frequency field with frequency tor .
3
3
o<
3
2
U3,
OO
wr
1
't
Fig. 4. Energy-level schemes for ir-Mw two-photon pro­
cesses.
The microwave radiation (cor ) is (a) added to or (b)
subtracted from the infrared laser radiation (to^) .
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25
In Eq.
(3.1) only one vibrational level nearly resonant with
the laser frequency
has been considered and other vibra­
tions and higher-order terms have been neglected.
neglect the effects of
yv .
->*
We also
,
->■
,
operating on
and E r operating on
These approximations do not introduce serious errors as
long as
|w0 - Wi|^o)r where w 0 = (E3 - E 2 )/h and ok is fre­
quency of a vibrational level other than u)0 .
The Hamiltonian, therefore, is
H=
H 4 -VVlt) = H 0 -
Mp£rCO«(tOri)-f»-v .^ C O S W Y f c ) .
(3.3)
We solve the time-dependent Schrodinger equation
(3.4)
To do this we apply a method analogous to the well-known
method of
varying constants to solve linear differential
equations
(P • Dirac 1926) .
Let
'VVI
N'<^
be the wave functions
of the stationary states of the unperturbed system.
Then
an arbitrary solution of the unperturbed wave equation can
be written as
(3.5)
To find the solution of the perturbed Eq.
expand
(3.5)
k.
(3.4), one can
in the form of
WT«""- a , i t ) a , (*)'%'+
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(3.6)
26
where the expansion coefficients are functions of time.
Substituting Eq.
functions
(3.6)
in Eq.
(3.4), and recalling that the
AV^°) satisfy the equation
tfi
= H.'tw
-at
yields
0 .7 )
K.
dT
Substituting v(t)
from Eq.
z.
it, t T k'0)
=
*
die
k
(3.3)
into Eq.
(3.7), we obtain
{jkrtrCOSll»r±hK-tx CO^,t))X'.
(3.8)
Multiplying both sides of this equation on the left by 'Y '
and integrating
a i/
iMZI
0) J o
X T X
OT
=
a
J ^^yn
K*
^
M
’f V 5 ^ 05^ * )
I
M v ■"e, c o s (u y k )}
yields
f t
t e r n .
=vmKe‘
/*(e"''E|-li:
&
Eq.
(3.9)
(3.9) leads us to the equation of motions for Q. (t)'s as
K
a i(^)= 2i.Xr C0S(u)rtj e x p ( - i t o ^ j
a z{i) ,
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27
d Iit)= 2 i ^ r C O S ( w r+ ) W ( * ‘°«ty 0 ,(i) + 2 A ^ C O S ^ t ) e x p ( - i w 0-):ja(-ty
d3 (-fc)= 2 a.9^ cos
exp (i^0t) &2Li) ,
(3 .io)
where id' = (E2 - E-^/ft, Xr = <^11 yp *Er |2>/2h and X^ 5 ^ 2 |UV *E^ |3)/
2h.
Contrary to a single-photon process where (i)^-~to0 , or
the right-hand sides of
plicit low frequency term.
(3.10) do not have any ex­
The low frequency terms are sup­
plied by high frequency parts of
into low frequency terms
.
We shall divide O^(t)'s
and high frequency terms CX^Ct) •
The magnitudes of 0 * ^ are much smaller than those of
but the magnitudes of
r
are tlie same order as those o f O ^ C t ) .
We can derive the high-frequency parts of O^(t) by
integration of Eqs.
0V
* )=
“ X r a M lt)
(3.10) to get the following results.
+
co0 -u)r
IK
lOo'4*
r
to0 4*cor
(•o - U)
=
0L.AH
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(3.11)
28
Since the frequencies have the magnitude relations:
' “ V/'l
^r'^o\ } }
|(u)l + lOr) - (^U30+
in deriving Eqs.
(3.11), terms with
U>'0)\ f
(3 .1 2 )
+ w 0 in denominators
were neglected.
Substituting
(3.11)
into
(3.10), yields the equations of
motion for the low frequency parts of Q.,. (t) as
n
0.l,Uk) = ASr a>»4tt)--gr AM exp [-A(Wo-LOs + O o ; - W r)i] a^j, \.i),
d j u W ^ - i U r - S i )
a 4 ,l i )y
d 3s w s r - L ^ a 3J ft) ~ U M * e x p [ ; ( w 0 - 4 +
w 0'- u;r)i](X^(3.i3)
where
(3.14)
^ = i 7 i y ( w» - ws-) ^
and M is the two-photon transition element,
(3.15)
and it is equal
to
(3.16)
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29
The 6r and the 6^ are the high frequency Stark shifts
47
due
to the R. F. radiation and the laser radiation, respectively.
When the transition moment for the two-photon process
the unit of angular frequency) Eq.
(in
(3.16), is compared with
the transition moment for the normal single-photon process,
(3.17)
we find that the transition probability of the two-photon
process is smaller than the corresponding single-photon pro­
cess by a factor
M.
(\ IMp-
A I-
t rl2)>
(3.18)
The
6 terms on the right-hand sides of the three equations
in (3.13) can be eliminated by transformations
t)-exp (-A-Srt)
c* (
/
W = e x p [ i ( ^ r -SJl)-k]
and
oc3 0 0 = e x p ( i £ i ) a 3fitk) ,
(3.19)
which shows that levels 1, 2 and 3 are Stark
-8rh,
Eq.
(<5r - S)h, and §h, respectively.
shifted by
Using E q s . (3.19)
(3.13) yields
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in
30
°<(( \ ) --- 4j- i.MCXp|^-A.
*s.0 :)
=
-V Sg •+ &r^ "fcj °<3 ("t),
- U^) - A ( ^ 0 -
0/
and
U 3 (i)
- --|- i H e x p [a («O0 -
-v (Oa - uDr + ^ + Sr)i]<*l tt).(3.20)
These equations of motion show that transitions occur
between levels 1 and 3 in the two-photon processes and level
2 is left out, although it provides intensity.
the two equations for aj and 03 in Eq.
photon case
Comparison of
(3.20) with the single-
48 49
'
indicates that the wave function oscillates
between levels 1 and 3 with a frequency of
r
- 0 - = j_( W 0 +
^
$
^ - * ° r -tSj t £ r ) + ^
a ”1^
J
(3.21)
Further comparison with the single-photon case indicates
that the line shape of the two-photon absorption should be
l ( w r) = c M / [ ( u j 0 * u < - 1oJ _ 1o r ) sv y s
+ H 5J
and that saturation is achieved when M
(322)
Hf, where'
pressure broadening of the two-photon line.
Using
y
is the
(3.16),
this condition is written more explicitly as
^
%
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(3.23)
31
The pumping efficiency <j> is calculated to be
S-
=
T
2Jh.
«*;
2/
/
= H / lw o ^ W o -
2
%
2
- L*-V) " V y + M
(3.24)
where n-^ and n 3 are populations of levels 1 and 3, and n° and
n° are their equilibrium values.
3.2.
Interaction diagrams.
Transition probabilities of multiphoton processes can
be greatly simplified by using the time-ordered Feynman
diagrams, as discussed by Peticolas
50
.
As an example, con­
sider diagram (a) for single-photon absorption, and diagram
(b) for single-photon emission.
These diagrams correspond to
terms in the perturbation expansion.
the initial state and
In absorption,
|m> is
|.n> is the final state, but in emission
(diagram b) this situation is reversed.
\v a >
•ft
I
(a)
(b)
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32
For two-photon processes and corresponding interaction,
diagrams are shown in Figures c and d.
In using these dia­
grams, we consider the number of infrared photons n^
and RF
photons n r together with the molecular states.
The transition
considered in the Schrodinger treatment section
(which is
shown in Fig. c) is expressed as
( ' V 1/
' ^ r - 1/
(V^r'H.))-
(3.25)
to
CO.
CO
(O
(d)
(Q
Figs. c. and d.
Energy diagrams and interaction diagrams
for IR-RF two-photon processes.
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33
Since
+ wr = ajQ +
=
(Eg - E-j^/ft, the total energy is
conserved between the initial and the final states,
However,
in the intermediate state, which is called the virtual state,
there is an energy deficiency of
(E2 - E-^) - 1iciJr = -hAoj.
The
transition moment for such a two-photon process is given by
=
|xrX^/Aw|, that is we multiply moments of the two transi­
tion and divide by the energy discrepancy at the virtual
level.
Since the energy is not conserved at the virtual level,
the system stays at the virtual level only for a time of
At~-^,
and the two transitions
simultaneously.
A similar method gives the transition moment
for the system given in Fig
M
(infrared and RF) occur almost
(d) as
< 2 l/,p-Erll><i|Mv E,|3> _
2 1)2&uo
In addition,
there are many other situations which may
occur other than a, b, c, and d which have been discussed in ref­
erence
(50) .
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CHAPTER IV
EXPERIMENTAL APPARATUS AND MEASUREMENT
A block diagram of the apparatus used in this experiment
is shown in Fig. 5.
The infrared radiation of about 5W to 200W, as required,
was generated by a CO 2 laser.
The block of C O 2 laser used
in this experiment is shown in Fig. 6 .
The laser consisted
of a water cooled plasma and an optical resonator.
The active
medium is a flowing mixture of nitrogen, helium, and carbon
dioxide.
This gas is admitted in the center of the plasma
tube and pumped through output at the e n d s .
The total pres­
sure in the tube was set at about 12 torr as this pressure
setting produced good laser stability.
The emission tun-
ability corresponds mainly to the Doppler profile of the ir
CC>2 emission lin e s .
When the active medium pressure is
increased, the emission line profile is no longer a Doppler
one but becomes the Lorentzian profile of a pressure-broadened
transition and offers a broad frequency profile.
Three electrodes are provided in the tube, an anode in
the center and cathodes at the e n d s .
Sufficient voltage is
applied to ionize the gas, and regulated current is passed
through the tube.
Energy is transferred from the discharge
electrons to the CC>2 molecules through collisional processes.
34
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
'VW
KLYS
CON UNIT
REG.
POW SUP
1~Z
MOD
-----
©
KLYS
r*
30 d b COUPLER
MOD
-► R EF TO
0 DET
ATT EN
ZIZQZ
■NaCI WINDOW
10 .b u m
C 0 2 LASER
C A R R IE R
GEN
XU BAND CELL
5mm
DIAMETER
v M XER
INTERP
RCVR
FREQ
STAND
CHOKE
MICA
WINDOW
MV MET
DETECTOR
DUAL PEN
RECORD ER
DUAL TR
OSCIL
0
'AT-
DET
PRE-AMP
T
R EF
Fig. 5.
Block diagram of the experimental apparatus used in this investigation.
u>
Ul
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
— Rear Mirror
Total Reflection
Concave.
Anode
(+6-10KV)
Gas Inlet
Cathode
Output Mirror
80% Reflection
20% Transmission
Flat
Cathode
Output
Beam
t M /W
Gas Outlet
Gas Outlet
Coolant Inlet
Figure 6.
Coolant Outlet
Laser Schematic Diagram.
U>
37
The plasma interactions involving the electrons are complex,
but the result is a population inversion of the CC>2 m o l e ­
cules in different excited vibrational states.
These excited
molecules give up energy to the field within the optical
resonator, thus intense coherent radiation builds up within
the resonator.
The optical resonator consisted of two mirrors supported
by a suitable rigid structure.
The rear mirror is coated
to produce maximum reflection at the operating wavelength.
The output mirror transmits about 20 percent of the incident
radiation.
The mirror substrate is germanium, which absorbs
less than 4 percent of the output radiation at 10.6 microns.
An anti-reflective coating is applied to the outside surface
of the output mirror to reduce Fresnel reflection losses.
Adjusting nuts are provided to facilitate alignment of the
mirrors.
Maximum output occurs when the two mirrors are
parallel and perpendicular to the axis of the plasma tube.
As we mentioned above, the active medium is a flowing
mixture of nitrogen, helium, and carbon dioxide.
Laser oscil­
lations may be obtained in pure CC>2 / but such oscillations
provide an output that is insignificant in comparison to the
outputs of CC>2 lasers containing additional gasses.
Nitrogen
serves as a carrier of excitation, He is a quenching gas,
which serves to remove excitation from terminal laser levels.
It gives us uniform discharge in the laser cavity and it
helps to reduce high voltage breakdown
51
.
Helium is required
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38
in comparatively large quantities and serves to remove mole­
cules from lower levels by cooling the discharge plasma.
With a mixture of 0.3 torr CO, 1.4 torr air, and 8.8 torr He,
a laser output of 20W was attained in a moderate size tube
cooled by liquid nitrogen
carbon dioxide,
52
.
With a mixture of 4.5 percent
13.5 percent nitrogen, and remaining mixture
of helium, we observed a maximum laser output of 200W for the
presently employed laser.
The vibrational levels of C O 2 involved in laser action
are shown in Fig. 7, with the first vibrational level of N 2
shown on the right side.
The 00°1 level is the level of the
lowest asymmetric-stretch vibrational mode and it is the
starting level of the C O 2 lasers.
either the 10°0 level
The terminal level is
(lowest symmetric vibration), or the
02°0 level (the second bending m o d e ) . The rotational levels
are very closely spaced and are not shown in the figure, how­
ever, transitions take place between the rotational sublevels
of these vibrational levels.
The 0 0 ° 1 — r 10°0 transitions
produce radiation around 10.6 ym, the 0 0 ° 1 — ►02°0 transition
around 9.6 pm.
These two transition types are in competition
with each other for molecules on their common starting level
and the 10.6 ym oscillations win out.
The 10.6 ym laser
transitions occur over a number of rotational levels.
In this
group— the 00°1 to 10°0 band— the P-branch transitions extend
from P(12)
to P(38), covering the wavelength range 10.5135 ym
to 10.7880 ym, the line with the highest gain being around
P(22) with a wavelength of 10.6118 ym5^.
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39
The 00°1 level is a good starting level for several
reasons, even in the absence of nitrogen.
First,
lowest one of a series of equally spaced l e vels.
it is the
Electron
collision in the discharge causes excitation of the C O 2
molecule to a number of levels of the type 00oV3.
a molecule excited to a v ^ l
cule in its ground
(00°0)
When such
state collides with a C 0 2 mole­
state.
An energy exchange is likely
to take place with the more energetic molecule losing one
quantum of vibrational energy to the other molecule.
In this
manner a (OC^Vg-l) and a (00°1) molecule are produced, and
ultimately most of the 00 ° V 3 type excitation is converted
into 00°1 type.
A similar degradation process takes place on
the other ladders, causing accumulation of molecules on the
10°0 and 01°0 levels.
The advantage of the 00°1 level are:
First, the 10°0 and the 0 2°0 levels drain relatively fast to
the 01°0 level; so they are suitable as terminal laser levels.
Second, it is the fact that direct electron collisions are
more likely to excite the asymmetric than the symmetric stretch
v i brations.
Finally,
the de-excitation is greatly enhanced by the
presence of He and
cool.
by keeping the walls of the laser tube
Most important is the role of nitrogen, whose molecules
accumulate in the v = 1 vibrational level, as a result of
excitation by electron collision to levels of different
v's
and subsequent collisions with ground-state N 2 molecules.
On
collision with ground-state C 0 2 molecules a resonant transfer
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40
3000r
Collision
v=l
00°1
2000
2331 cm
-1
1 0 .6 ym
9. 6 ym
Excitation
10°0
02 °0
1000
Rad
\
decay (f)
Rad. decay
(f)
Collisions and
rad. decay(s)
00°0
C0_ ground level
v =0
N_ ground level
Fig. 7.
Energy-level diagram showing pertinent
vibrational levels of CC>2 and N 2 . f = fast, s = slow
radiation decay.
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41
of excitation takes place at a relatively high rate because
the energy difference is only 18 cm- 1 , less than the mean
thermal energy of the molecules.
Transfer may also take
place at the v = 2 and 3 levels, but the contribution of
these transfers is not significant.
The fixed frequency CW CC>2 laser beam was passed through
a sodium chloride window through a 5m long x-band waveguide
cell made out of copper and coated with silver, or in some
transition through a 5 m long c-band waveguide cell made out
of Aluminum,to excite N O 2 and SO 2 molecules to the excited
vibrational or to the higher rotational state in the ground
vibrational state.
A shutter made of firebrick in front of the laser allows
us to send the laser beam for different amounts of time into
the absorption cell.
A timer gated by the window enables us
to determine the exposure time for the gas m o l e c u l e s .
Excess
laser power in the waveguide was adequately "dumped" by using
a 90 degree elbow and a taper section of waveguide in the
exit area.
A microwave spectrometer suitable for measuring frequency
and relative intensity in the spectrum of N O 2 and S O 2 was
employed for our measurements.
The basic conventional micro­
wave spectrometer utilized in this investigation was a source
modulated one and has been discussed elsewhere
54-57
.
Only
details pertaining to this set of measurements are given
below.
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42
Reflex klystrons provide a stable tunable source of
virtually monochromatic microwave radiation.
A variety of
klystron tubes manufactured by OKI and Varian were used,
which produced frequencies in the range of the S O 2 and N O 2
transitions under investigation.
In some cases it was nec­
essary to generate harmonics to reach higher frequency transi­
tions.
This was done using standard crossed-waveguide
multipliers.
The frequency range of the microwave source was varied
using the sawtooth sweep voltage from the time base of a model
561A Tektronics oscilloscope.
This method of sweeping enabled
a time varying signal, synchronized with the oscilloscope
trace, to be displayed visually on the oscilloscope screen.
The ramp pulse obtained from a Tektronics Model 561
oscilloscope as well as low-level sinusoidal signal were
put on a 133 KHz carrier as amplitude modulation, and this
signal was injected on the repeller via a modulator-demodulator
circuit.
Additionally,
a 31 KHz chopper signal was applied
to the repeller and this resulted in frequency modulation of
the source Klystron,
oscillator.
since the klystron is a voltage-controlled
The power absorption is thereby broken into com­
ponents, the fundamental oscillating at the modulating frequency.
Since the detector amplifier is tuned at 31 KHz, modulating
at 31 KHz gives the first derivative of the absorption sig­
nal, modulating at
etc.
(31/2) KHz gives the second derivative,
In some transitions,
since the signal was so weak, we
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43
used the derivative up to sixth order in order to see and
absorbed them on the trace of oscilloscope.
The absorption
signal was passed through a high-gain tuned amplifier cir­
cuit and onto one channel of the oscilloscope.
This addition­
al tuned circuit and chopping technique enable the power
levels to be kept quite small and signals not near the re­
sonance of the amplifier to be rejected by tuned circuits.
The calibrated frequency "comb" was generated from a
marker-mixer circuit as described below.
The marker-mixer generator
circuit consisted of a single
diode for both mixing and multiplication.
A General Radio
model 1112 A,B standard generator set was used to drive the
crystal multiplier and a Well-Gardner model BC-348Q radio
receiver was used to interpolate the beat signals between the
frequency standard's harmonics and the fundamental frequency
of the test klystron.
needed,
In order to assure the stability
the General Radio standard was slaved to a Hewlett-
Packard model 105 A/B quartz crystal oscillator, accurate to
one part in 10^-®.
The frequency standard described above was used to drive
a diode to generate a frequency near that for which the radio
receiver was tuned.
The remaining frequency interval was
spanned by using a variable tuned signal generator, Tektronics
model 191, tunable over a range of 0.05 to 100 MHZ.
This
scheme enabled the radio receiver to be set for a desired
frequency and the model 191 generator tuned so that the
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44
calibrated marker could be placed to coincide with any
point along the spectral signal profile.
It should be noted here that the receiver always gen­
erates at least two markers, one at vk - nvs another at
nvs - V}., where
is the klystron signal and n v s is the
proper harmonic of the standard signal source.
One of the
most satisfactory arrangements for generating standard mark­
ers consisted of using the "comb" of frequencies from the
General Radio signal generator and spanning the remaining
frequency intervals as needed with the auxiliary model 191
signal generator.
The use of the auxiliary generator made
it possible to span the coarse intervals of 100 MHz generated
by the General Radio signal generator to the degree of pre­
cision needed for accurate frequency determinations of each
spectral component.
The frequency marker was superimposed
upon the spectral component of interest and the difference
frequency counted with a Hewlett-Packard model 5383A elec­
tronic counter.
This technique enabled us to obtain frequenc­
ies accurate to a few Hertz,
marker width,
in theory, but due to finite
absolute frequencies were assumed to be not
better than 10 K H z .
Power levels incident on the detector were on the order
of 10
—8
watts, and resonant absorption by molecules may rep-
resent less than 10
of the ambient power level in the cell.
It is these small absorptions that the detection system must
follow in order to allow measurement of the absorption
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45
characteristic of a particular molecule.
To achieve this
degree of sensitivity requires precise matching of system
components and extreme care in implementation.
Crystal diodes were used for primary detection in the
range from ten gigahertz to forty gigahertz, germanium 1N26
diodes were used and above forty gigahertz, germanium 1N53
diodes were used.
Voltage levels of five to twenty milli­
volts were amplified by a set of matched high gain bandpass
pre-amplifiers constructed in the electronics shop.
These
amplifiers were tuned to thirty-two kilohertz to allow dif­
ferentiation by modulation as discussed earlier.
pre-amplifiers,
From the
the signal was sent to PAR-122 Lock-In
Amplifiers used as phase-sensitive detectors, whose phase ref­
erence came from the Heath EUW-27 Audio Frequency Generator.
The relative intensity was measured by using height of
transition with respect to the height of "laser-off" signal
on the chart recorder.
However,
at the same time, the method
of sweeping by a model 561A Tektronics oscilloscope enabled
a time varying signal synchronized with the oscilloscope
trace to be displayed visually on the oscilloscope screen.
On one channel of the scan a set of synchronized markers
appeared and on the other was displayed the line profile or
it's derivative.
A typical recorder output for frequency and
intensity measurement is shown in Fig. 8 and a recorder out­
put for repetitive scanning of the spectral components for
"laser on" and "laser off" conditions is given in Fig. 9.
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46
1MHz
Fig.
8.
A typical recorder output for frequency measurement.
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47
T I M E [secj
Fig. 9.
Dual-channel trace of the fourth derivative profile
and frequency marked used for relative intensity
measurement for the S02 molecule.
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48
Such scans were used to determine the intensity changes and
frequency location of each spectral component.
A summary
of the results for several components is given in the next
chapter.
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CHAPTER V
ANALYSIS OF EXPERIMENTAL RESULTS
In this chapter,
the experimental results of this
investigation are presented and discussed.
All measurement
reported here were made using the spectrometer described in
detail in Chapter IV.
All of the data were taken at
before the laser was turned on.
sent one standard deviation.
^27°C,
The errors indicated repre­
Rotational transitions of the
excited vibrational level of N O 2 molecule are weaker than
the ground vibrational transitions in both N O 2 and S O 2 mole­
cules being reported in this work, and standing waves accord­
ingly interfered with finding the transitions, posing, a
great
problem.
Three different absorption cells were tried,
but the standing wave problem persisted.
In order to observe
good signal to noise ratios and to reduce the effect of
standing waves,
in some part of the experiment we used and
observed higher derivative
absorption lines.
measurements
(i.e., fourth derivative)
of
There are no published relative intensity
due to the effect of IR radiation on these mole­
cules .
During measurement the pressure was maintained constant
as much as possible.
However,
there were pressure changes
within the absorption cell for laser-on and laser^off
49
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50
conditions.
It would appear that these pressure changes
could be attributed to impurity gases removed from the cell
walls by laser heating or perhaps a leak in the system.
The
pressure, however, returned to the original value for N 0 2
after a reasonable time and it is likely that kT effects
were responsible.
Similar experiments conducted on SO 2 in
the same absorption cell showed pressure changes with recovery
time much more rapid than for N 0 2 when the laser was turned
off.
There appeared to be two effects; one of a temperature
nature and the other possibly generation of transient mole­
cular species
(perhaps S O - or S02 of an excited vibrational
CQ
level
).
A worthwhile experiment would be a triple resonance
experiment in which one channel of S02 and one channel of SO~
are monitored while the laser beam is on in order to establish
whether SO- is being produced by the laser radiation.
This
experiment would require extensive modification of the present
equipment.
Figures 10, 11, 12 and 13 show the intensity decay scheme
for the four rotational components of the ground vibrational
level of the N 0 2 molecule studied for various laser power
levels.
It is clear from these figures that the rate of de­
population of the energy levels and hence the change of
intensity for each transition is laser power dependent.
More­
over, the highest laser power appears to produce the greatest
rate of decay.
This effect is expected to be reversed for
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51
40 r
• 30W
x 40W
n 80W
LOG SIGNAL AMPLITUDE (cm)
16,023.65MHz
ELAPSED TIME (sec)
Fig. 10.
Semilogarithraic plot of elapsed time versus
signal amplitude for various laser power levels for the
2,38*
393,37^ transition of NC>2 .
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52
LOG SIGNAL AMPLITUDE (cm)
26,633.83MHz
0
10
20
30
50
60
70
ELAPSED TIME (sec)
Fig. 11.
Semilogarithmic plot of elapsed time versus
signal amplitude for various laser power levels for the
(24i
2^222^ transition of N 0 2>
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53
▲18W
o58W
• 70W
x95W
®130W
LOG SIGNAL AMPLITUDE (cm]
661.38MHz
0
5
10
ELAPSED TIME (sec)
Fig. 12.
Semilogarithmic plot of elapsed time versus
signal amplitude for various laser power levels for the
(10q 10 ^
9^' (F,11.5<— 10.5) transition of NC^.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
54
18W
53W
75W
82W
LOG SIGNAL AMPLITUDE (cm)
40,671.06MHz
0
5
10
15
ELAPSED TIME (sec)
Fig. 13.
Semilogarithmic plot of elapsed time versus
signal amplitude for various laser power levels for the
(109. Q ) ; (F,9.5<—
8.5) transition of NO_.
u f J-U
J-/y
~
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55
some vibro-rotationally excited levels in accordance with
the results of Jones
7 59 - 6 I
'
.
However, we observed this effect
for some vibro-rotationally ground and excited levels on the
S02 molecule but not for the N 0 2 molecule.
can be seen that the
From Table I it
(0, 1, 0) rotational components of N 0 2
have a somewhat smaller decay rate than the
(0, 0, 0) compo­
nents .
We were unable to observe a systematic change in the
intensity for post-laser radiation in N 0 2 .
The relaxation
time for the system after laser bombardment appears to be
longer than ten minutes for all the N 0 2 transitions observed.
This is in contrast to the situation encountered with S02 in
which relaxation times for the transient behavior after the
laser was turned off was from a few seconds to several minutes
duration and even showed fluctuations in intensity for the
decay scheme^3 .
Although the curves given in Figures 10, 11, 12 and 13
do not in all instances produce ideal linear fits, relaxation
rates for exponential decay can be calculated^3 ' .
These
results are presented later in this thesis.
In order to compare the change in intensity versus laser
power, for each transition, curves like Figures 10-13 were
used to select laser beam exposure time for the NC>2 molecule.
A semi-logarithmic plot was then made of power versus relative
intensity change of the transition
intervals.
(Al/I)
for fixed time
These results are presented in Figures 14, 15, 16
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
x12sec
• 25sec
AlOOsec
C9
O
20
60
100
120
140
LASER POWER (watts]
Fig. 14. ^ S e m ilogarithmic p l o t of laser p o w e r v e r s u s c h ange in signal a m p l i t u d e
at fixed time i n t ervals for the (40
3^3 3 7 ^ t r a n s i t i o n of NC^.
2 , 36"~
Ul
<y\
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
<1
c
o
0.1
20
40
60
80
100
120
140
LASER POWER (watts)
Fig. 15. Semilogarithmic plot of laser power versus change in signal amplitude
at fixed time intervals for the (24^ 2 3 -«
232 22) transition of N02 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x 12sec
a 25sec
• lOOsec
40,661.38MHz
CD
o
0
20
40
60
80
100
120
140
U1
00
LASER POWER (watts)
Fig. 16. Semilogarithmic plot of laser power versus change in signal amplitude
at fixed time intervals for the (10„ n
9n 0) ; (F/ll.B-4--- 10.5) transition of N0o .
0,10
1 ,9 '
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1.
<J
cs
a 12sec
• 25sec
x lOOsec
40,671.38MHz
0.1
0
20
40
60
80
100
120
140
LASER POWER (watts!
Fig. 17. Semilogarithmic plot ot laser power versus change in signal amplitude
at fixed time intervals for the (10Q
91 g ) ; (F,9.5<--- 8.5) transition of NO? .
9 1 , 9 );
Ul
VO
60
and 17.
The figures are shown with log-linear plots indicated,
but careful observation will show that points representing
low laser power depart significantly, but in a systematic way,
from the extrapolated linear region of the figures.
This
effect indicates that the NC>2 molecule seems to have a differ­
ent decay branching scheme for low powers than it has for
high power laser radiation.
It will be necessary to make a
broader study of the rotational transitions of this molecule
before a complete picture of the branching scheme can be had.
A complete summary of the experimental results of this
investigation for the N O 2 molecule is given in Table I.
Additionally, we set the power of the laser at constant power
of about 40 W in an attempt to understand the nature of the laser
pumping, at fixed power, over the hyperfine components of a
given rotational transition.
components of the
The six rotational hyperfine
(0 , 0 , 0 ) vibrational level were investi­
gated for nearly constant power and pressure.
are summarized in Table II.
of relative intensity
These results
In both Table 1 and 2 logarithms
(I0 /I) over elapsed time have been
calculated for different laser power and transition,
from
experimental data given in Appendix C.
For SC>2 molecule,
some of the experimental results
appeared at first to be more confusing than enlightening.
One of the transitions observed near 30 GHz exhibited a
phase change of 180 degrees under continuous laser radiation,
the system upon relaxing went through phase reversal to the
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE I
SUMMARY OF EXPERIMENTAL RESULTS OR SOME ROTATIONAL
STATES OF THE NITROGEN DIOXIDE MOLECULE
J
Hk+i,k- 1T
+10
*---
F
Power
(watts)
p Ln ( l 0 /I)
Average
Pressure
sec " 1
(mtorr)
9 1,9
1 q
10.5 -— - 9 . 5
11.5 «— -10.5
18
58
70
95
130
0.00337±0.002
0.0285410.004
0.0377110.002
0.0483210.007
0.0757410.014
30
33
27
32
45
+ 1 0 0 ,1 0 ^ -
9 1,9
10.5 *-- - 9 , 5
9.5<-- - 8.5
18
53
75
82
0.0120410.007
0.0256210.003
0.0697610.007
0.0825410.017
26
27
32
30
+241 , 2 3 ^ “
2 3 2,22
23.5-*-- 22.5
23.5-----22.5
1
31
81
0.0136010.007
0.0455510.005
0.1123710.006
35
36
29
0,10
~
i
+402
^ /38**
oo
393 37
40.5 -*-- •39.51
41. 5 -- ■40.5
30
40
80
0.0071910.0008
0.0123810.0013
0.0436310.02
45
53
28
6.5 -*-- ■ 7.5
5.5 ■«-- - 6 . 5
23
43
64
0.0013710.00029
0.0006410.00023
0.0005510.00069
47
57
55
*
7 1,7 r
8 0 ,8
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE I— Continued
Vl,K-l
F
n k ; i ,k : x
*
?1
j.f7t<
-80 o
6 . 5 ^ --- 7.5
*
71
**•t71 <--
S 0,8
7.j<
8 .j
7.5 --- - 8.5
8 .5
---- 9.5
Power
(watts)
Average
Pressure
^•ln(I0/I) sec-1 (mtorr)
44
0.0002410.00014
0.0006310.00012
8.5
10.5
15
18
33
58
0.00099+0.00014
0.0011610.00007
0.0001025+0.00009
0.00386+0.00028
0.01035+0.00132
22
0
56
58
50
56
60
58
61
57
+ (o, 0 , 0 ) vibrational level
(0 , 1/ 0) excited vibrational level
O
to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE II
SUMMARY OF LASER PUMPING OF THE
(232 j2 2— *'2 4 l /23> ROTATIONAL
COMPONENT OF THE NITROGEN DIOXIDE MOLECULE
N K+1,K-1
n k+i,k : 1 ......... J .....
241,23*--
2 2 2 22
24.5 <-- 23.5
23.5 «-- 22.5
F
Power.
(watts)
£ln(I 0 /l)
Average
Pressure
sec -1
(mtorr)
24.5«-- 23.5
25.5 «-- 24.5
23.5 *-- 22.5
43
43
40
0.00830±0.00106
0.0081810.00045
0.0074710.00062
24.5 -- 23.5
23.5 ■«— 22.5
22 .5 -«— 21.5
38.5
42
42.5
0.0068510.00024
0.0083110.00019
0.0088410.00024
8
12
10
9
9.5
7
cr>
U>
64
original structure for the spectral component when the laser
was turned off.
We sought other transitions which exhibited
this characteristic in order to study the decay rates from
one connecting state to the next.
Examination of more than
ten transitions in the 20 - 35 GHz region produced only one
other candidate which showed the "phase change" character.
This component in S0 2 was found to be at 38495.28 MHz and
responded well for 126 watts of laser power.
Figure 18 shows
the time dependent detail of the character of this transition
for the laser on for 600 seconds and then the decay scheme for
relaxation when the laser is turned off.
The two regions A
and B are shown in rescan for a time scale expanded by 4 to
show the phase changing character of the spectral profile.
We interpret the change of phase by 180 degrees to represent
a change of absorption of microwave radiation in this frequency
channel to emission as the phase reverses from 0 to 180
degrees.
The molecular system can be treated as a multilevel
system with the two states E^ and E 2 being fed from all other
levels available for interaction with the laser beam.
states produce photons
These
(E^ - E 2)/h or absorb photons in a
manner dependent upon the energy hierarchy available to re­
plenish the population of levels E^ and E 2 ,
Oka et al. have predicted a phase reversal for spectral
components in a double resonance experiment wherein the laser
radiation shifts the candidate transition frequency from one
side to the other side of the Boltmann distribution of
the molecule.
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65
Fig. 18.
Systematic decay scheme for the 38495.28 MHz rota­
tional component of the SC>2 molecule undergoing a
phase change under laser bombardment. (154 1 2 --I 63 ^ 3 )/ (0,0,0) component.
Laser power 1^6 watts.
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66
Another set of spectral components in SC^, shown in
Figure 19, contained three components labeled A, B and C
which appeared to be of the same phase with and without the
power on.
The amplitude relationship can be readily seen as
the laser power is turned on for several hundred seco n d s .
The repetative scan shown in Figure 19 covers a total scan of
1000 seconds with the two conditions laser-on for 680 sec­
onds and then the laser is off 400 seconds.
The rates of
increase in intensity for the three components A, B, and C
are different, probably indicating that they are rotational
components from different vibrational manifolds.
It is also
interesting to note that the rates of decay when the laser
power is off are not the same as the rates of increase in
intensity for the laser on.
Since the SC>2 gas is outside the
main cavity of the laser it seems that coupling with the gain
profile: of the laser would be too weak to produce intercavity
coupling of the waveguide system to the laser.
It seems
that a more plausible explanation is that pumping of the inernal states of the S 0 2 molecule by the laser power is pro­
ducing some highly populated transitions by disturbing the
Boltzmann
distribution.
The two signals A and C of Figure 19,
appear with the same phase for laser off and laser on condi­
tions and are weakly sensitive to laser power.
This low
sensitivity may be an indication that little laser pumping is
taking place within these levels.
The coupling between the
laser and the microwave levels appeared to be dependent upon
i
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67
Fig. 19.
Intensity behavior for the 38230 MHz rotational compo­
nent of the SC>2 molecule under laser bombardment.
Probably the (72-j^ 59
^^14 58^ component.
Laser
power 80 watts.
'
'
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68
the microwave power level.
A microwave-microwave double
resonance experiment might be employed to further pursue
this effect.
We cannot explain the fact that the two signals B and
C of Figure 19 exhibit a shift in frequency with laser power
while the other signals observed seem relatively fixed in
frequency.
These two components may arise from spectral
manifolds wherein
the frequency shift is larger than for
other components, i.e.,the components which shift are compoents of highly excited states while those which have weak or
no shifts probably arise from ground vibrational levels and
probably low level rotational states within a given vibra­
tional level.
The decay scheme for S O 2 generally appeared as shown in
Figure 20 with a systematic decay in time with "laser on" and
a systematic return to equilibrium for the signals when the
laser was turned off.
Figure 20 shows the results of two
levels of laser power and the resultant decay schemes for laser
on and laser off conditions.
A complete summary of the experimental results of this
investigation for S 0 2 molecules is given in Table III.
It is
clear from this table that the rate of depopulation and the
rate of population of the energy levels and hence the change
of intensity for each transition is laser power dependent.
Where the table shows the value of i Jin I0 /I vs. laser power,
the
(+) sign indicates depopulation and the
(-) sign for
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69
Fig. 20.
Systematic signal amplitude decay and recovery
for laser on and laser off conditions for the SO 2 molecule at
36834.87 MHz.
(192.18--- 183,15^; (0,0,0) transition.
The
upper trace is for laser power at 135 watts and the lower trace
for power of 105 watts.
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70
TABLE III
SUMMARY OF EXPERIMENTAL RESULTS FOR SOME ROTATIONAL
STATES OF THE STJLPHER DIOXIDE MOLECULE
Frequency
Power
(MHz)
(watts)
24039.5
105
108
108
108
-0.00250
-0.00518
- 0.00210
-0.00273
24083.39
47
63
70
82
150
+0.00990
+0.00266
+0.00266
+ 0.0100
+0.0270
|ln(I 0 /I)
and
(Vi' V2 » V 3 1 )
21c
5,17
--- 2 2 .
4,18
(0 , 0, 0 )
° 2,6
°l,9
(0 , 0, 0 )
35
6,30
34
J*7,27
25049.43
130
(0 , 0, 0 )
Sl,7
7 2 ,6
(0 , 0, 0 )
172 ,16
163 ,13
(0 , 0, 0 )
4
--- 3
0,4
1,3
(0 , 0, 0 )
80
120
(sec- '*')
+ 0.0010
+0.0050
+ 0.0020
25392.0
83
+0.00230
28858.1
75
85
92
117
+0.00342
+0.00477
+0.00145
+0.00465
91
-0.00127
-0.00327
-0.00280
-0.00658
-0.00584
-0.00495
29321.46
120
120
140
140
158
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71
TABLE III— Continued
N-
K+1'K-1
K+1*K-1
Freguency
Power
(MHz)
(watts)
|ln(I0/I)
and
(vj_,
\>2 '
162,14
^
30205.52
45
’91
98
-0.00330
-0.00115
-0.00713
34376.02
100
110
-0.00044
-0.00155
34530.22
83
146
-0.00305
-0.00285
36338.05
63
120
136
165
+0.00141
+0.00330
+0.00540
+0.00613
22 .
36791.07
88
91
155
+0.00738
+0.00681
+0.0155
18.
3,15
36857.0
48
105
125
145
-0.00077
-0.00320
-0.00384
-0.00540
37335
42
95
105
105
125
-0.00253
-0.00319
-0.00377
-0.00419
-0.00538
37515.6
65
80
+0.00029
+0.00206
38518.2
100
138
-0.00201
•^■0.00575
171,17
(0 , 0 , 0 )
519 ,43
5 0 10,40
(0 , 0 , 0 )
315,2
306 ,24
(0 , 1, 0 )
305 ,25
296,24
(0 , 0 , 0 )
23
19
2,22
-3,19
(0, 1, 0)
2,18
(0 , 1, 0 )
213 ,19
204 ,16
(0 , 0 , 0)
266 ,20
275 ,23
(1 , 0 , 0)
*
154,12
(sec ■*■)
163,13
(0 , 0 , 0.)
Transitions that phase change take place.
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72
population increase.
The number of values of 7 - An
t
(I-./I)
U
for the S O 2 molecule is different from the number of
£ An
(I0 /I) values for the N O 2 molecule.
In the SO 2 case
we have oscillation of the relative intensity which is not
the case for NC>2 molecules.
As an example, we plotted the
logarithm of relative intensity versus elapsed time for a
typical transition,of the SO 2 molecule shown in Figure 21.
But in Table III only the first slope of such plots is
tabulated.
The combination of the frequency of the laser and the
frequency of the klystron in both sums and difference freq­
uencies opens channels such that the resultant frequencies
can match with a number of possible energy levels available
in the energy hierarchy of the molecules.
These frequency
components may cause oscillation of the intensity of some of
the spectral components as the photons are exchanged from one
state to another due to molecular collisions.
Finally, Table IV shows a summary of the results of
observed change
(AI/I0 ) values for several transitions of
the SC>2 molecule.
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TABLE IV
SUMMARY OF CALCULATED VALUES OF AI/IqFROM
EXPERIMENTAL DATA FOR S0 2 MOLECULE
Frequency
Vl.lL!
Vl.K-i
and
(\>1» v 2 ' v
3\
123 ,9
132 ,12
(0, 0, 0)
215,17
"•‘•4,18
(0, 0, 0)
8?
&ffoi-- Q j_^y
(0, 0, 0)
356,30
347 ,27
81,7
72 ,6
(0, 0, 0)
172 ,16
163,13
4n n
3, ,
0 f4
1/ j
(0, 0, 0)
^ 62,14
171,17
(0, 0, 0)
(MHz)
Power ^2- = Ipn - ^off
-'-off
(watts)
20335.4
62
-r0.412
24039.5
105
200
210
-0.368
-0.805
-0.762
24083.39
47
63
70
82
150
-0.613
-0.773
-0.481
-0.786
-0.815
25049.43
120
130
-0.518
-0.411
25392.0
83
-0.483
28858.1
92
92
117
+0.333
+0.421
+0.449
29321.46
91
125
140
158
+0.410
•i0.316
+0.593
+0.675
30205.52
45
91
98
-2.067
-2.083
-1.935
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74
TABLE IV— Continued
N.
K+1'K-1
N„
K+l/K.!
and
( V ^ , V
2>
V
Frequency
Power
(MHz)
(watts)
AI = Ion - Ioff
Loff
3)
51
__
3 9,43
501 0 ,40
(0, 0, 0)
34376.02
100
+0.282
31
34530
61
83
146
+1.802
+0.931
+2.371
36338.05
63
120
136
165
-0.398
-0.258
-0.598
-0.200
36791.07
88
91
155
-0.512
-0.556
-0.543
36857
48
105
105
125
145
-0.664
-0.758
-0.836
-0.782
-0.909
-3,19
284,16
(0, 0, 0)
37351.8
42
95
105
+2.768
+1.00
+1.039
27
5 ,23
266,20
(1, 0 , 0)
37515.6
65
80
-0.106
-0.585
68
100
120
120
138
-0.639
-1.636
-1.404
-1.459
-1.576
5,27
306 ,24
(0 , 1 , 0)
305 ,25
296 ,24
(0, 0 , 0)
23
19
21
-- -22
2,22
3,19
(0, 1, 0)
2,18
183,15
(0, 1, 0)
+
163 ,13
154 ,12
(0, 0, 0)
38518.225
A 1/1 = AA.
- A.
np
n
+Transitions that the phase change take place.
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C H A PTER V I
CONCLUSIONS
The intensity decay scheme for both SO 2 and N O 2 mole­
cules,
for fixed power,
followed a rate equation of the form
l W = i 0e x p l -/ ^ + )
where IQ is the original intensity of the radiation for a
given transition, and t the time elapsed for the laser beam
incident upon the molecules and Y
a rate coefficient.
We have designated the rate coefficient as
since
it does not appear as a constant but is dependent upon laser
power, choice of transition for study and seems to change
for long or short intervals of time for laser "beam on" con­
ditions.
The symbol i represents all of the states connect­
ing to the lower energy level and j represents the states
connecting to the upper energy level.
Measurements made upon the NC>2 molecule did not yield
the same results as were observed on SO 2 rotational states.
The decay schemes for "laser off" conditions showed no attempt
for restoration of the intensity to pre-laser bombardment
condition.
These results on the N 0 2 molecule indicated that
either permanent or very long time changes had been produced
in this molecule by laser bombardment.
75
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76
Data on the N 0 2 molecule showed that a very strong inter­
action between the C02 laser and the ground-vibrational
states takes place and the population statistics of the rota­
tional manifold in the ground-vibrational states are
altered from Boltzmanian distribution.
severely
Interaction between
the C 0 2 laser and the S02 molecule was not as strong as that
observed for the N 0 2 molecule.
The reason appears to be
that there is good coupling between some of the
and the
(0, 1/ 0)
(0, 0, 1) vibrational levels of the N 0 2 molecule with
the infrared laser.
Also, the energy "ladder" of NC>2 near
N = 10 begins to overlap the continuous region of the laser
spectrum.
However, we had to settle for the weaker coupling
case of the S02 molecule in that the R-branch of the C02
laser at 986 cm"1 couples less strongly than for the coupling
case of the C02 laser to the NC>2 molecules.
Besides, the
rotational energy of the ground-vibrational rotational states
begins to overlap the region of continuous C O 2 laser lines
for higher quantum numbers for S O 2 than for NC>2 .
An interesting use of the laser-microwave double photon
transitions for the study of collision-induced rotational
transitions is that we can establish this method a nonBoltzmanian rotational distribution which cannot be established
by the normal single-photon pumping. Previously the method
of double resonance was used for establishing a non-Boltzmanian
rotational distribution used for microwave double photon transi­
tions21" 2 4 '65- 6 7 .
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77
In Figure 2 1 a summary is given of the results observed
on the SC>2 molecule.
This figure is a plot of the data
given in Table III of Chapter V.
Figure 22 represents the
plot of value of AI/IQ versus the quantum number N after the
system had reach equilibrium behavior.
Such a plot could
not be made for the NC>2 results as each spectral component
became lost in the background noise rather than coming to
equilibrium.
The horizontal points plotted in Fig. 21 represent the
various laser power levels which we were able to establish.
For the quantum number N below 20 for most transitions the
power levels are increasing to the right,whereas above N = 20
the power levels are increasing toward the left.
It appears
that near N = 20 the effect of depopulation of the energy
levels by laser power is reversed.
It should be noted here
that Boggs and Murphy have shown the Boltzman distribution to
peak at N = 18 for SC^.
The two far left horizontal points plotted in Fig. 22,
which belong to the transitions
(154fi 2 ----^®3,13^
an<^
(1 6 2 , 1 4 --- 17 1 ,1 7 ) are the ones that show a phase change from
0° to 180° during the run.
Since N = 18 is the peak point of
the Boltzman distribution and since, as was mentioned in
Chapter V, that
Oka et a l . , have predicted a phase reversal
for spectral components in a double resonance experiment where
in the laser radiation shifts the candidate transition
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78
N
'
B-B.
50-
_^s=0— a
■-a-
20-
-M -
-1 60
-12 0
Fig. 21.
-80
-40
0
40
80
Summary of the signal amplitude changes
120
10s
In
(I/IQ ) with power for some ground and vibrationally
excited rotational levels of the SC>2 molecule.
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79
N
50
4 5-
4 0-
35
30
25-
20
-
15
10
0
-2.4
Fig.
-
1.6
22.
-
0.8
-r0
—
I—
0.8
—
1. 6
i—
2.4
3.2
Summary of the signal amplitude changes AI/I
with
power for some ground and vibrationally excited ro­
tational levels of the SC>2 molecule.
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80
frequency from one side to the other side of Boltzmann distri­
bution of the molecule", we predict that the Boltzmann distri­
bution shifted in such a way that the N = 15 component
shifted from one side of the Boltzmann distribution to the
other.
In other words, the laser interaction with the mole­
cules moves the Boltzmann distribution from a peak at N = 18
to some other quantum state N.
To understand the non-Boltzmanian rotational distribution
or the shifted Boltzmanian distribution for the N O 2 and the SO 2
molecules exactly we must construct an experiment like the one
done for different rotational component of the transition
(232 22
23)
t^ie N 0 2 m °lecule / f°r
quantum numbers which are available.
which is shown in Table II
fixed laser power
rotational
In that experiment
we ran the experiment with a
(P ^ 40 watts)
sure of approximately 10 mtorr.
and with a fixed average pres­
But this run with fixed power
and pressure is not available for all quantum numbers with
the set-up that we have
because in some cases transitions
are too weak to give good results.
Also, interaction between
the SC>2 molecule and the laser is so weak that we have to
maintain high laser power.
At higher powers the pressure
becomes quite unstable due to thermal effects and maintenance
of temperature around the inlet NaCl window and adjacent
waveguide becomes a problem.
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A P P E N D IX A
PROOF OF EQUATION
(2.38)
+ 00
<.00
00
0
m ± in■/-00
/r°°
+ !/vX
; (Ai)
J
a
^ rt
Using the Fourier transforms of
OO
Vl-fc) = y 2fT J oiw G(to) exp (-ifok:),
/ —00
s&>
dt v(t)ex?
G( U>) =
in Eq.
J- 00
(Al), one can get
/0
^ n = V t , !X
G »».( w *>")
vn^n
v* - > ( t i ) e x p ( - i w „ n - t J + > ^
jolt
vmtwki''-oo
-oo
-to
x f d i i^ f d u j G ^ M e x ? ( - - A U ) t ) A f f c l w ' G ^ l w ) e*p ( - * < ^ )
-°° * e*P [*“>*»« ( * - * » ) ] /
0
- y v l
^ ( ^ n ) !
— Oo
00
J
YV\-tn/ ~
c ( ^ V W n (|:,)e^P(-iwwrtiJ + y UTXV j
d ^ G nm(ou;Um '
00
J— 00
-00
X j cl-t, e x p p-A(ua- corirtri) t'l /
A**
<#»
)ex? H
^ w U
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*
82
j
+ /iTrVjT
^
vr\4*ft'- Oo
dt
e x p [-* (w-to*,,,)-}:]
J-oo
-V w m / 0 ‘t | J y Y ( u / +
X C X p [~ * {
ro
= y p , ZX
Gn
J
^
I
V«"» W
^
(-i W m ntl)+ / l W X
CO
CO
olto'^l^')
_ 00
'/-Uw + w ^ l
(alt exp(-iluo +
'-no
- 60
+ «o'Jf) -oli e x p f - i O - u ^ * ) * j y
Now letting t" = -t-^
+ J/(2TTfe/
m^= n
/"-t00
x Y
co
J u ) G „ „ ( i^ )
— 2TT
to -
7 7 T 7 — : --- r [ 2 T T ^ ( ^ + ^ j
/
00
Co
G**, (
=
vrntri
I G m „ ( ^ exp
^ 0
/_ o o
^
lYl
^
/n ^
e x p (Iu>mt ni ' ) d o
^
) I d
/l / _ oo
(1 w*';
G m ^ ( CO j j
/_ o o
^ "(w + W
J
fTT
'-
U3W m ) -i(to+ ^™„)j /
-oo
= w
£
y
Wifcw
-00
oo
e m „ ( - / d r e * p r ; ( ^ + “ m „ ) n + / 2 t t'
-{>
JduGnJv)] ct*>'Gnin(u')S(* + u')/-4i«>+ *>mJ -
**■£*{-00
fflVM
/*nV
J
'
A(tO+IA3wr,j
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83
-c°
/i
- */*"'*‘J_ p f ^ Gnnl i«; Gm„(-«)/(W- *>„„) - i/2
* P J 'd
CJH..)
w '+ w <"») '
= '/z**£
"•»'’>G„„ I-"»■«> + !/zn
m=£tn
G„„(“ m n) (?P“ G H
wigfefl
oo
/(co + ujm ^) - 0 i r 1 i 2 ^ j c l w
nvi^m
G„,„ (-*>)/( w - u j wrlj
_ CO
-
x / * n V ^ G M^ ( W W M )^)j d«o/GwwCwV(" + u;,«'")
n
y
Finally, the second term cancels the last term in above final
result, and the
^vn—
J
(^vrvn) + farrin'^^(P J^ 60 IG«noC^3)! / ( 10mm 0J)
m?tv\ -Ca
which can be written in the form of
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A P P E N D IX B
Evaluation of T. . (b, v) for common intermolecular forces,
^ 2
In Anderson's theory, it has been shown that T. . (b,
3132
is given by
r
( b , V ) = 2 K e S C b ) = ZS2(b)o ; ,
(Bl)
ald2
where S 2 (b)0/i is Anderson's collision efficiency function
with the term j£ =
neglected.
Taking the Fourier transforms of interaction energies
of molecules 1 and 2, which are
V(-fc)D D = (Mi M 2/ r 3)
IK
"z - 3 ('V
(B2)
’^ 0 - 2 ( ^ . % ) - t - 5 ( i a 1 n0 ){nr n0)] (B3)
vii) = ( 3 M 2l / LJr,4j[-^2^-^o-2
^,)(>a ,.^0)+ 5 C ^ 2 - ^ o ) ZJ
QD
'i'
(B4)
V W Q Q = ( ( 3 ^ 2)/>or5) j [ l ' 5 C V % ) - 5 ( ‘;>2- A 0)2+ 2(A,.A2)Z-20
* (A r ^o)(^2- ^ o ) 0 V ^ 2)+ 35 (Kf " o H " 1'"» I 1
V(i)OSp=(-l/4 ^ j [ f , « I/(«l+ ^ ) ] K ( 1) - K x (1) ] K ( 2) - " x ( 2)]
<2
x
-3
)J +
^o)(^2 -n o
3
[<*u (1) -<xx(l)J5(2; (i^z•n0f
84
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
(B5)
85
+ 3 [ * llU)-<*i lZ)]o<(l)(A2.lA0)%-3 [e<(l)0Cj.(2j + o<x o<^)3
(Bg)
where DD, DQ, QD, QQ and DSP stands dipole-dipole, dipolequadrupole, quadrupole-dipole, quadrupole-quadrupole and
dispersion interaction respectively, q is the molecular
quadrupole moment and in term of p the charge density of
the molecule is defined to be
(B7)
and fij, £12 denotes to unit vectors along either the dipole
moment,
the axis of the linear quadrupole moment, or the
symmetry axis of molecules 1 and 2, respectively;
r is the
intermolecular distance and ft0 represents a unit vector along
r;
a
is ionization potential;
a is the mean polarizability,
and is given by
o<=h | + 1^)/3 /
and finally, ot| |(i) , ^ ( i )
are the polarizabilities of the
ith molecule parallel and perpendicular to the symmetry a x i s .
Substituting the Fourier transforms of V ^ ,
v d Q'
V QD' V QQ anc^
Vgsp into equation
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86
yields to the following equations
r 2 U ( b ' w = ^ ( M' M* / W X
I
*!+3, K
1!
<B 8 >
*?r>Q(k16')
T>j>j(b,v^Q\)=7j7T4i^2^2A*
' Z
(B9)
^ P(il9'cJ2)Q (%i/&ij
^3, ^
x ^ dg ? ( ^ j O
l'jij) ^ I'/1 0 5 P " " ^ 0
(b1°)
■z..2i >0
^ ' ^ / < S|+ £ 2)^ [(°<||lt)-«i H)A'ti,v V
x
X
( q (i-i'D
^D
c-iX>
(B11)
3,vj, 3; l
+ f ^ i r « V l>- °vu);/<2u)]
in all of these equations
(B8-B11), the summations are over
all distinct energy levels which are represented by the j "
single quantum number.
fQQ(kjj'),
(B8-B11)
The function fD D (kjj'), fcQ(kjj^)
fpgpr l(kj j p
are normalized,
£ q D ^°) =
Q
=
and fD S p / 2(kjj'*) in the equations
therefore
^ ^D5p, 1 ^
= ^DS P,2
= ^
and in terms of the dummy variable.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
(B12)
87
•PD D W
= iy
K tl«.) K 0CK) + ^ 4 K 20 c^)]
[>*( 2 ^ + 5<k2+ 2)K* (*.) + 4<k3 (<*}+2 ) K, W
^DQ W
(B13)
K 0 (A)
+ 29L4 ( ^ + 1 ) K q2" (*.)]
(B14)
f
(9.J = (1/-l4t+)[<KZ( H ‘K-6+18,K.(++ %0 9tZ'4-'lLftf) K * w •+2 9C,(/i2<k4+62ftZ
QQ
+ 7 f ) K , ( * J K 0 (*) + * 4 ( 4 * q + 1 A » ? + 3 6 )
P
US V,1
k
£ (*)]
(B15)
=r (1/63) (29(.6+ U ^ 5+ 3 £}^4-v?H<JC3+A26<k.Z+126<K+63)e^p(-2Si;(B16)
£05p't= (l/50^,)(5if
216*7+ 66?**+ l7 4 0 * 5+ 3 n 6 A 4* 7 1 7 * 5
+
where K
n
402*1
10062 *
+
5031)
(SXp ( - 2
94)
(B 1 7 )
(%) is nth order of a modified Bessel function.
value of k in equations
(B8-B11)
The
is
h = ( b/ v ) \
JJ
(^/V)
||
I3
(B18)
Type of interaction allow one to use one of the B8-B11
equations, but in general case where the interaction energy
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
88
is a combination of many types of forces, V(t) would also
be the combination of all V.(t)'s, and can be written as:
<n
V ( 't)C 0 m =
And
^
V ^ D G > + V (‘t)QD + V W Q G ? + ^ D 5 P +
<B19>
4
(b, v)
can be written as:
-)l-’2
com
(b'^
+ V
b' v > n * V
b'
(B20)
+ 13,3I<b ''',a Q + 1il3.(b'W ^
4
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX C
LIST OF TABLES (EXPERIMENTAL RESULTS)
NITORGEN DIOXIDE MOLECULE
OF
TABLE V
ELAPSED TIME AND SIGNAL AMPLITUDES FOR VARIOUS LASER POW E R FOR THE
(10o 1 0 “— 9 1 ,9 ) ROTATIONAL transition of THE (0, 0, 0)
^ VIBRATIONAL LEVEL OF THE N 0 2 MOLECULE
p = 18 W
H k + 1-k -1
P = 50 W
P = 55 W
P = 58 W
N k + 1'k -1
l°n nn *--- 9 1 _
u ,j-0
i,y
t i m e (s )
h (cm)
t i m e (s )
h (cm)
t i m e (s )
h (cm)
t i m e (s )
h (cm)
off
off
3.56
5.42
5.92
6.24
7.55
8.01
12.26
13.4
13.0
15.1
13.5
11.0
14.5
13.5
10.1
11.2
11.5
10.5
off
2.84
4.23
4.17
4.51
5.36
4.51
5.36
7.46
10.5
7.0
7.4
6.1
5.6
4.4
5.6
4.4
3.2
off
5.36
5.94
5.42
6.05
6.13
3.43
6.06
15.3
9.3
8.0
6.3
5.9
3.0
3.1
3.3
off
2.68
3.36
2.69
1.82
2.06
5.75
11.02
14.16
11.5
11.0
7.9
9.5
9.0
7.3
6.5
4.9
3.2
89
90
g 00 rH
00
o • • • •
w rH H oo r- m
ifirlrl
&
o
n
rH
in
II
ft
:
cm
oo r~-
w mh 10 in ^ r~
d) m • • « *
g 0 H CM m cn
•rH
-p
in
in
»
6
4
2
00 O co in
e
• t • #
o
— 'CM a\oo lo in
X! H
TABLE
3
ft
8
in C\rH 00 oo co cm H cn
^ m
KDCO 00 roHcN
0) MH • • • • • • • •
e o H CM iN CM CM
•rH
+>
5
V--Continued
CTl
e *=* •CM• r-*cn• in• •
V •
00 O C
Tir- CO rH
XH
o
rII
ft
_
in G\ rH 00 00 CTi
MH LO CO in in
CO
a)m • • • t • •
CO KO00 r* o
e
o
"H
rH
-P
l"H
I
W
+
cr>
53
rH
I
o
rH
O
o
rH
N
su
g
oo
ro
K
KD
O
O
4*
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE V I
E LAP S ED T IM E AND S IG N A L A M P L IT U D E FOR V A R IO U S LA S E R POWER FOR THE
( 1 0 0 1 0 «— 9 1 9 ) R O T A T IO N A L T R A N S IT IO N OF THE ( 0 , 0 , 0)
'
V IB R A T IO N A L L E V E L OF THE N 0 2 MOLECULE
11 1 -
time(sec)
100,I0 *
40671.06
53 Watts
18 Watts
N K+ 1'K -1
9 1,9
h(cm)
time (sec)
off
7.4
off
1.89
7.5
3.22
1.47
6.2
5.59
75 Watts
h(cm)
14.9
time(sec)
82 Watts
h (cm)
time(sec)
h (cm)
off
8.3
off
9.6
2.39
9.1
4.75
8.2
4.49
9.2
2.89
6.2
2.64
6.2
7.0
6.50
5.6
2.93
5.7
3.76
6.8
3.13
6.5
7.00
7.1
5.63
3.4
5.14
3.9
3.19
5.8
5.41
6.1
7.23
2.4
4.32
2.0
9.31
5.8
5.68
4.4
• • • •
• • • •
• • • •
• • • •
12.37
4.5
8.32
4.0
• • • •
• • • •
t • • •
• • • •
12.18
2.5
• • • •
• • • •
• • • •
• • • •
10.0
MHz
• • * •
. . . .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE VII
ELAPSED TIME VERSUS AND AMPLITUDE FOR VARIOUS LAS E R POWER FOR THE
(2 4 1,23*— 232 2 2 } ROTATIONAL TRANSITION OF THE
(0,0'0) VIBRATIONAL LEVEL OF THE N 0 2 MOLECULES
P = 31 watts
NK+1,K_1
time(sec)
241,23
*
26633.83
232,22
MHz
off
2.91
2.33
3.20
3.63
3.54
5.28
6.49
3.20
7.13
11.08
11.27
14.02
15.41
h(cm)
17.4
16.4
16.0
16.0
16.5
14.8
13.7
13.0
12.7
10.1
9.1
7.8
7.2
5.0
52 watts
time(sec)
81 watts
h(cm)
time(sec)
120 watts
h(cm)
h(cm)
20.0
17.3
15.1
13.1
11.6
11.0
10.0
9.4
8.0
6.0
off
3.77
7.13
6.57
6.20
7.13
5.18
6.13
• • • •
• • •
•
• • • •
• • • •
• • • •
• • • «
• • • •
• • • •
• • • •
• • • •
• • • •
• • • •
• • * •
off
5.01
9.24
12.14
7.08
7.66
8.67
8.07
8.39
8.42
• • •
•
• • • •
• •
• •
• • • •
• • • •
• •
• •
off
2.38
2.49
3.38
3.07
•
17.1
10.2
7.2
5.0
3.7
• • •
• • • *
• • • •
• • • •
• • • •
•
• • • •
• • • •
second derivative of Lorenzian profile.
15.2
12.4
9.7
7.0
4.4
3.1
2.5
2.7
time(sec)
•
•••
• ••
• • • •
••••
••••
• • • •
• • • •
• • • •
• • • «
VO
to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE VIII
ELAPSED TIME AND SIGNAL AMPLITUDE FOR VARIOUS LASER POWER FOR THE
(40? 38'*— 393 3 7 ) ROTATIONAL TRANSITION OF THE (0, 0, 0)
'
VIBRATIONAL LEVEL OF N 0 2 MOLECULE
N»
T.
K+lK_l
Nt. , v*
K+lfK -1
40-p 0 0 ^ ; 3 9 ^ 07
P = 34W
P — 30W
time(sec)
off
h (cm)
14.0
12.8
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
4.0
6.0
6.0
7.0
10.0
6.0
6.0
7.0
13.4
10.7
12.0
10.5
10.5
10.5
12.0
10.5
10.1
9.9
10.0
7.9
9.5
9.7
9.1
9.3
8.0
6.2
7.4
7.5
6.3
time(sec)
off
1.0
2.0
h (cm)
13.9
13.8
11.7
3.0
3.0
4.0
4.0
4.0
10.5
9.5
9.0
8.7
10.8
P = 40W
time(sec)
off
4.0
4.0
4.0
4.0
4.0
4.0
h (cm)
13.6 '
12.5
10.7
8.0
8.2
8.6
4.0
4.0
4.0
10.1
9.5
9.5
9.2
7.3
6.8
P = 80W
time(sec)
off
4.0
4.0
3.0
h(cm)
7.2
5.3
3.5
4.0
94
s
CO
m
g in co in in cm ro in
0
• • • • • • •
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4
95
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rf; r» *-•
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+
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2
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p’
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11.2
11.7
11.2
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245
311
411
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g o
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cm i d
6 50 O r l O M O
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H i—I rH
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cm i n o'! cm i n
•P
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192
ft
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2
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2
•
212
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co
ft
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fCl rH rH rH rH rH rH
10.0
ft
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•
13.5
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CO>
n
P
< cw
2 oo o
CP f
H 1 *
a3 + H
p ft
ft O
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EinnriHiDci
177
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+
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cm co
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12.4
IX
TABLE
in
< P
> f t ft
EH >
ft
ft
o p p
f t (=5
2 P
ft O <
D H 2
P E jO
Eh
H
H Eh Eh
ft
2 ft
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cn r - h i d cm
cm i n <n CM o\
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CM<Dcn
96
w
w
gl
w
ft Hi
o D
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PS M
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£ O
O S
ft
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Ci
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r- CN CN 00 O
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