MOLECULAR CONSTANTS. 0F N02 FROM
HIGH-RESOLUTION ABSORPTION SPEB'I'RA
AND
MAGNETIC CIRCULAR DICHROISM AND MAGNETIC
CIRCULAR BIREFRIIIGENCE 0F I‘IO: AND N02"

Thesis for the Degree of Ph. ’D.
MICHIGAN STATE UNIVERSITY ’ '
RICHARD E. BLARK
1970

i

LIBRARY

Michigan Sta to
University

 

This is to certify that the

thesis entitled

MOLECULAR CONSTANTS OF N02 FROM HIGH-
RESOLUTION ABSORPTION SPECTRA AND

MAGNETIC CIRCULAR DICHROISM AND MAGNETIC I
CIRCULAR BIREFRINGENCE OF NO AND N02

presented by

RICHARD E. BLANK

has been accepted towards fulfillment
of the requirements for

__Ph_'P_L_ degree in

3, & %zz¢i\

Major professor

PHYSICS

Date Aprll 17, 1970

 

0-169

ABSTRACT

MOLECULAR CONSTANTS OF NO2 FROM

HIGH-RESOLUTION ABSORPTION SPECTRA
AND
MAGNETIC CIRCULAR DICHROISM AND MAGNETIC

CIRCULAR BIREFRINGENCE OF NO AND NO2

by Richard E. Blank

High-resolution, near-infrared absorption spectra of
the (0,0,3), (1,0,3), and (3,0,1) vibration-rotation bands

l4N1602 and 15N1602 have been analyzed. Ground state

of
constants have been given by Olman and Hause as well as
vibration-rotation interaction constants from which initial
values for A, B, and C in the upper states were predicted.
Improved values for the upper state constants including
centrifugal distortion and empirical P6 terms have been
obtained. Prediction of improved ground state constants
showed no significant difference from those previously
reported.

Spin splitting was observed and analyzed for a number
of transitions in the (0,0,3) band of both isotopes since
the splitting was large and resolution favorable in that

region (~0.03 cm-l

). Effective spin-rotation coupling
constants were obtained for the excited state. No spin
splitting was observed in either of the (1,0,3) or (3,0,1)

bands due to weakness of absorption and reduced resolution

(No.05 cm-l) in that region.

Richard E. Blank

The (1,0,1) and (2,0,1) bands previously analyzed in
this laboratory combined with the results from the (0,0,3),
(1,0,3), and (3,0,1) bands have been used to calculate
vibrational and vibration-rotation interaction constants.

Finally methods based on those described by Stalder
and Eberhardt for measurement of magnetic circular
dichroism and magnetic circular birefringence were used to
obtain these Spectra for the (0,0,3) band of N02. Magnetic
circular birefringence spectra were also observed for the
3-0 band of NO.

The Jones-Mueller matrix mathematics of polarized light
is discussed, and a general matrix describing the effect of
a dichroic, birefringent sample on any incident state of
polarization has been derived. The Jones-Mueller calculus
was then used to derive intensity expressions for magnetic

circular dichroism and magnetic circular birefringence.

MOLECULAR CONSTANTS OF NO2 FROM
HIGH—RESOLUTION ABSORPTION SPECTRA
AND
MAGNETIC CIRCULAR DICHROISM AND MAGNETIC

CIRCULAR BIREFRINGENCE OF NO AND No2
By

Richard E. Blank

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1970

TO KAREN

ACKNOWLEDGMENTS

It has been a privilege to work and study under the
patient guidance of Professor C. D. Hause. His dedication

to teaching and research is evident to all who know him.

 

I am grateful to Professor T. H. Edwards who has given

freely of his time and who has provided many good ideas.

If '31:... “(fig 3—. U"
V

I appreciate the aid of Professor P. M. Parker who has taken
the time to answer many questions. My thanks also to
Professor R. D. Spence for the help that he has given.

I would like to thank Dr. Melvin Olman who contributed
much to the development of this work. I appreciate the
assistance received from Dr. Donald Keck and Dr. Lamar
Bullock who contributed computer programs which were very
useful in reducing the data. I wish also to thank Dr. Peter
Willson, Dr. Richard Peterson, Dr. Lewis Snyder, Dr. Tom
Barnett, and Dr. Kent Moncur for their stimulating discussions
and various contributions. .

I am grateful to the Department of Physics and the
National Science Foundation for their support of this work
through assistantships, fellowships and grants. My thanks
to the Michigan State University Computer Center staff for
their assistance and consideration.

I am deeply grateful to my parents, Mr. and Mrs. E. H.

Blank, for their encouragement. My father contributed

iii

several ideas for experimental apparatus.
A special thanks to my wife, Karen, for her help in
the preparation of this thesis and for making it all

worthwhile.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS. . . . . . . . . .

LIST OF FIGURES

LIST OF TABLES

LIST OF APPENDICES . . . . . . . . .
INTRODUCTION . . . . . . . . . . .
CHAPTER. . . . . . . . . . . . .

I ENERGY EXPRESSIONS. . . . . .

II

III

GEOMETRY . . . . . . .
SYMMETRY CONSIDERATIONS . .
SELECTION RULES . . . . .
ASYMMETRIC ENERGY EXPRESSIONS

SPIN-ROTATION INTERACTION. .

EXPERIMENTAL METHODS AND ANALYSIS.

IDENTIFICATION OF TRANSITIONS
ANALYSIS OF ASYMMETRIC ENERGY
UPPER STATE CONSTANTS . . .
SPIN SPLITTING ANALYSIS . .
GENERAL FEATURES OF THE BANDS

VIBRATIONAL CONSTANTS . .

LEVELS

Page
. iii
. vii
.viii
. ix
. l
. 3
. 3
. 5
. 6
. 7
. 10
. 16
. l7
. l7
. l8
. 22
. 22
. 32

VIBRATION-ROTATION INTERACTION CONSTANTS. 39

THE MATHEMATICS OF POLARIZED LIGHT

JONES-MUELLER CALCULUS. . .

42

 

IT'S-3:64:331'ffis“ ~_L_ . -.
1 -

IV PHASE SENSITIVE DETECTION OF MAGNETIC
CIRCULAR DICHROISM (MCD) AND MAGNETIC
CIRCULAR BIREFRINGENCE (MCB) IN NO AND
NO .

SUMMARY. .

REFERENCES.

APPENDICES.

2

THE JONES INSTRUMENT MATRIX OF THE
SAMPLE O O O O O I O O O

CONVENTIONAL MAGNETIC ROTATION .

MAGNETIC CIRCULAR DICHROISM . .

MAGNETIC CIRCULAR BIREFRINGENCE.

DICHROISM AND BIREFRINGENCE SPECTRA

SUGGESTIONS FOR CONTINUED WORK .

Vi

48

49

50

52

57

62

66

68

69

71

10.

11.

12.

l3.

14.

15.

16.

LIST OF FIGURES

Geometry of the N02

The spin splitting scheme.

molecule.

Spin splitting in the P, Q, and R branches

The complete (0,0,3) band of 14N02. . .

The K_l = o subband of the (0,0,3) of 14No

Typical spin doublets in the (0,0,3) of 14

The complete (1,0,3) band of 14N02. . .

The K_1 = O and l subbands of the (1,0,3)
of 14NO . .

2. O O O O
14

The (3,0,1) band of NO

2"

Experimental apparatus for magnetic circular

dichroism . . . .

Chopper construction .

Schematic diagram of the difference signal

The balancing scheme for magnetic circular

dichroism . . . .

Experimental apparatus for magnetic circular

-birefringence . . .

Magnetic circular birefringence in the 3-0

of NO. . . . . .

Magnetic circular dichroism and birefringence

for the (0,0,3) of NO

vii

2

Page

l3
14
33
34
35

36

37

38

52

S3

54

56

58

63

64

Win-cu ”“177.“ L ' VFW—1
‘ i

Table

II

III

IV

VI

VII

VIII

IX

LIST OF TABLES

Ground State Constants for N0 .
Upper State Constants for the (0,0,3) of N0
Upper State Constants for the (1,0,3) of NO
Upper State Constants for the (3,0,1) of N0

Spin Split Lines in the (0,0,3) of N0

2

2

Effective Spin-Rotation Coupling Constants.

Vibrational Constants. .

Vibration-Rotation Interaction Constants

Jones Vectors . . . .

Jones Instrument Matrices

viii

2

2

2

Page

19
23
24
25
26
29
40
41
46

47

.3. man.- “it?

 

Appendix

II

III

IV

VI

VII

LIST OF APPENDICES

Computer Programs.

VIBCON

ROTCON

Frequencies for

Frequencies
Frequencies
Frequencies
Frequencies

Frequencies

for
for
for
for

for

the
the
the
the
the

the

ix

of

of

of

of

of

of

14NO

15NO

14N0

ISNO

l4No

lSNo

Page

71
72
80
88
94
99
103
107

110

‘ . In—r—Iu-nwm”
l ; !

INTRODUCTION

Nitrogen dioxide and nitric oxide have been of interest

in molecular spectroscopy because they are among the very

few stable, paramagnetic molecules. The single unpaired

electron possessed by these molecules produces unique spin

 

coupling and magnetic effects.

This work is primarily concerned with studies of
vibration-rotation bands and magnetic effects in N02(l)-(6).
Also included are experimental magnetic circular birefring-
ence spectra of NO which are of interest in themselves.

The NO spectra, because of their intensity, were useful in
refining the experimental techniques for the weaker magnetic
circular birefringence and dichroism spectra of N02.

The first section of this work details the analysis of
the (0,0,3), (1,0,3), and (3,0,1) vibration-rotation spectra

14N16 15N16

of 0 and 02 in the near-infrared. The spectra

2
are analyzed on the basis of the slightly asymmetric rotor
with the spin-rotation interaction as a perturbation. From
these results combined with the previously analyzed (1,0,1)
and (2,0,1) bands are derived the upper state constants,
spin-rotation coupling constants, vibrational and vibration—

rotation interaction constants.

The second part of this work details the experimental

results of recent techniques (7) (8) for obtaining separa-
tion of magnetic circular birefringence and magnetic

circular dichroism effects which have generally been measured
in a combined form in magnetic rotation studies (9) (10).

An introduction to the Jones-Mueller mathematics of polari-
zation is given which is used to derive intensity expressions
for the two experimental arrangements. Experimental spectra
are presented for N02 and NO as well as an indication of

the methods by which predicted spectra may eventually be

generated and compared to the observed.

Whowhnmu‘m 7:? . . .- L A.
‘ 7 I
,.

CHAPTER I

THEORY

ENERGY EXPRESSIONS

This part of the work is a continuation of the invest-

igations begun by Olman and Hause on the absorption spectra

14Nl60 and 15N160

of 2 2. A discussion of the theory of

energy levels, selection rules, and spin-rotation interaction

is presented.

GEOMETRY

Nitrogen dioxide is a slightly asymmetric, non-linear,
triatomic molecule whose geometry is illustrated in Figure
l. The assignment of axes a, b, and c is as shown. Each
axis has associated with it a moment of inertia Ia, Ib,
and IC respectively. It is conventional to define recipro-

cal quantities:

h
Rz—T—

8n cI
r

where R = A,B,C and r a a,b,c respectively. Molecules are
classified according to the relative value of the reciprocal
moments:

A = B = C Spherical Top

A > B = C Prolate Symmetric Top

4:0-N-O = 134"4', rs

o
= 1.1934 A

 

 

 

Figure 1

Geometry of the N02

molecule.

 

fif—w..7_i-.' ._.
a -_

A = B > C Oblate Symmetric Top

A > B > C Asymmetric Top
In each case the axes have been assigned to the molecule
such that A 1.3 1 C.

Nitrogen dioxide closely approaches the prolate

symmetric top since B and C are nearly equal. One method
of denoting the degree of asymmetry is by the value of Ray's
asymmetry parameter

K = 2B - A - C
A - C

 

where K = +1 (-1) for an oblate ( prolate ) symmetric top.
Nitrogen dioxide is found to have K = -0.994 so that it is
a close approximation to the prolate symmetric top. This
is an aid in analysis since energy levels for symmetric
tops can be expressed in closed form, and perturbation
theory yields excellent values for levels of slightly asy-
mmetric tops.

SYMMETRY CONSIDERATIONS

 

As can be seen from Figure l, nitrogen dioxide belongs
to the C2V group with the b-axis corresponding to the
symmetry axis. This symmetry has an important consequence
in that if the two identical nuclei have zero or integral
spin the total eigenfunction for the state must be symmetric
and the molecule follow Bose statistics. This means that
even though the symmetric top energy levels are split by
the asymmetry of the molecule, only the totally symmetric

levels are populated i.e. half of the levels are missing.

The missing levels can be determined as follows. All
transitions discussed here are within the 22 ground state
which is symmetric. If the vibrational state is denoted by
(vl,v2,v3) then the vibrational eigenfunction is symmetric
(antisymmetric) as v3 is even (odd), and all upper states
observed here have v3 odd. Finally the nature of a rotation
about the b-axis is such that the rotational eigenfunction
is symmetric under a 1800 rotation about the b-axis if K_1
and K+1 are both odd or both even, and antisymmetric if one
is odd and one is even (11). Thus the rotational wave—
function is symmetric if K_l + K+1 is even and antisymmetric

if K_1 + K+1 is odd. Since the total eigenfunction

must be symmetric, we have for the ground state
(Symmetric)x(Symmetric)x(Symmetric) + Symmetric

and for the upper state
(Symmetric)x(Antisymmetric)x(Antisymmetric) + Symmetric
Thus YR must be symmetric in the ground state and anti-
symmetric in the upper state. As a consequence of the

A

missing levels symmetric rotor notation, KANK(N), may be

used without ambiguity where K is K—l'

SELECTION RULES

 

All observed bands were of Type A which correspond to
parallel bands in the near-by prolate limit. For such bands

the selection rules are

AK__l = 0
AN = 0,11 if K_l # O (1)
AN = :1 if K = 0

AK_l = :2 transitions which would be allowed by the finite
asymmetry of the molecule were not observed presumably due r“

to the very low intensity of such lines.

ASYMMETRIC ENERGY EXPRESSIONS

 

. ‘d't "M'fi-.- . r ' .

The energy levels have been determined from the

 

perturbation expansion:

E = E0 + El + E2 + ESR

ESR represents the spin-rotation interaction energy and
will be given in detail in the next section.
E0 is the rigid rotor energy for the slightly asymmetric

molecule based on the Wang reduced energies(12).

E0 = (gamma) + (A-(B+C))w (2)
2 ‘7—

A,B, and C are the molecular rotational constants, and w is

the Wang reduced energy. The method used to obtain the

Wang energies is based on the diagonalization of the Wang

reduced energy matrices by numerical techniques(l3).

El includes P4 or centrifugal distortion terms of the

4
E = Z T.X. = + T

1 Taaaaxl bbbbx2 + Taabbx3 + Tababx4 (3)

The T's are treated as parameters and account is taken of

possible vibrational dependence by allowing the T'S to be
determined independently in the upper and lower states. It

was found that there is a strong correlation between Taabb

and Tabab in each of the bands studied. In order to deter-

mine a Significant estimate of Taabb the value of Tabab was

calculated from the expression of Chung and Parker (27).

= -16 A B C

Tabab e e

 

LONG)

(A)

Explicit expressions for the x's are given by Hill (l4)(21).

x1 = 1 r2[N2(N+l)2-61-2N(N+l)62] - (r-2)[2rN(N+1)<P2>
I6 2
4
+ 2rd3 - (r-2)<Pz>]}
X2 = l {(l+S)2[N2(N+l)2—2N(N+1)<P2> + <P4>]
1'6 Z Z

— (l-s)261 + 2(1-SZ)[N(N+1)62 + 531} (4)
x3 = l{(l+s)[rN2(N+l)2 - 2(r-l)N(N+1)<P2> + (r-2)<P4>]
5 Z Z

+ r[(1-s)61 - 25N(N+1)62] - 2[1+s(1-r)]s3}

r 2 4
X = 1 N(N+1)<P > - <P > - 6 }
4 21 z z 3
_ 2 _ 4 _ 2 2
61 — (2w<Pz> <Pz> w )/bp
5 = (<p2> - w)/b
2 z p
_ 2 _ 4
63 — (w<PZ> <Pz>)/bp
and
_ 2 2
r = C2/A2 S _ Ce/Be
e e

Since the equilibrium rotational constants Ae’ Be’ and Ce

are not known it is necessary to use the ground state values

for those constants in the calculation of r and s.

P6 TERMS

 

E2 consists of P6 terms as described by Pierce, {I

DiCianni, and Jackson (15). These terms are of the form I

E = H N3(N+1)3 + H

2 2 2
2 N NKN (N+l) <Pz> + H

4 6
NN(N+l)<Pz> + H <Pz> (5)

K K

Where the H's are empirical constants and the <P2> are
expectation values of the operator P2 5 Pa' <Pz> is very
nearly K_l since in the prolate limit the wavefunctions are

eigenfunctions of Pz‘
THE COMPLETE ENERGY EXPRESSION

Thus combining E0, E1, and E2 the complete energy
expression exclusive of the spin-rotation interaction can

be written

E = (gimme) + [A - (signw
2 2

4
+ E Tixi (6)

3 3 2 2 2 4 6
+ HNN (N+l) + HNKN (N+l) <Pz> + HKNN(N+1)<PZ> + HK<PZ>

10

SPIN-ROTATION INTERACTION

 

The only angular momentum of an electronic character
in non-linear molecules is a spin momentum since the orbital
motion of electrons is generally "quenched" or suppressed.
Thus the primary perturbation on the energy levels of the

non-linear NO2 molecule is an Mfg interaction where N’is the

‘ 'nur‘rg‘T-“ifl m1
F

rotational and S the spin angular momentum. The spin

magnetic moment interacts with the magnetic field due to

 

molecular rotation which gives an interaction energy prop-
ortional to N-S. The field arises partly from the motion
of charges in the molecule due to simple rotation, and
partly from the transfer of a small amount of angular
momentum from end-over-end rotational motion to electronic
orbital motion(16). The spin Splitting is small compared
to the differences in rotational energy levels. The
result is a doubling of the spectral lines for K_1 # 0.

A detailed treatment is discussed by Lin (17) in
which the spin-rotation Hamiltonian can be represented in
the form:

_ N-g Z EijNiNj i,j = a,b,c (7)

HSR " i , j
N“‘(“TTN+ '

The spin splitting observed in this work is at most
0.2 cm-1 so the symmetric rotor approximation may be used.

In this approx1mation eij = 0 for i # j and Ebb = Sec“ So

HSR can be written in the fOrm

11

_ . 2 2 2 _ 2
H — N S [e aNa + ebb(Nb + NC)] — N‘S [eaaNa

2 2
SR _ _ a + ebb(N -Na)]

 

(8)

Since the symmetric top eigenfunctions are eigenfunctions

 

of N: and N2 and Since

<JIN°§IJ> = J(J+1) - N(§+1) - S(S+l) (9)
we have

ESR = [Ebb + (Eaa'ebb)§:fi¥II<J|§f§IJ> (10)

Then neglecting Ebb in comparison to the second term in

Eq. (10) and defining an effective 6 and a K such that

 

SR
6 = Eaa — Ebb (11)
and
2
KSR = K e (12)
NIN+I)

we can write

ESR = KSR[J(J+1) - N(N+l) - S(S+l)] (13)

2

Since there is one electron and one unit of spin angular

momentum, S = 1/2, J can assume only the values
J+ = N + 1/2
(14)
J' = N - 1/2
then

 

'1
'1
'1
‘i
i.

12

+
ESR — KSRN/Z

(15)
ESR - -KSR(N+1)/2

thus the magnitude of the splitting of a given level is

 

 

P3

+ - 2 . S

ESR ' ESR = KSR(N+l/2) = K é§;i{§)€ (16) i

The only transitions observed have AJ = AN so that the I
spin splitting is not directly measurable as seen in Figure 2. p”
Denoting the J+" to J+' and J-"to J-' transitions by 0+ and P:

v- reSpectively we have for AK = 0

 

 

0+ — v- = K2(N'+l/2)e' - K2(N"+l/2)e" (17)
N'(N‘+l) N"(N"¥1)
It is important to note that even if 5' = e" splitting

will occur in the P and R branches due to the dependence on
N. In the more typical case the effective spin‘ splitting
constants differ so that splitting also arises in the Q
branch. Then as can be seen in Figure 3, it is possible for
the spin splitting to be increased in the R(P) branch and
decreased in the P(R) branch. In addition the spin splitting
is not symmetrical about the asymmetric energy levels so
one need not expect symmetrical splitting in the observed
transitions.

If one makes the assumption that the electronic
structure is unchanged by isot0pic substitution it follows

(18) that

13

 

 

 

 

 

Figure 2
J‘I'" E’+ E+I
‘ s.r
E, /
_’ -
\i E! + Egic .
/\ ' ' *
v+ 1’.
+II
J E"+ EII’

I!

_ _..II
fJ

 

II _
FE+%£

A schematic diagram of the observed transitions giving rise

to spin doubling.

14

Figure 3

 

 

 

- N +1
N ~ UPPER
N —'I/2 I STATE

 

 

 

 

 

 

 

 

 

A A
N +1 I
N GROUND
N -—I/ L. STATE '
P Q R
A schematic diagram of spin splitting in the F, 0, ind R

branches .

15

 

(18)

where R assumes the values A,B,and C as r assumes a,b, and
c. Since Baa is so nearly equal to the effective 5 in N02,

it is a valid approximation to write

= ——— (19)

This makes it possible to combine the spin data from isotopes
of the molecule for a particular band. Constants determined
from the combined data agree very closely with those deter-
mined from each isotope separately. Conversely one can use
this as a verification of the premise that the electronic

structure is unchanged by isotopic substitution.

CHAPTER II
EXPERIMENTAL METHODS AND ANALYSIS

The Michigan State University high-resolution, near-
infrared spectrometer as previously described (9)(10)(l9),
was employed to produce absorption, Zeeman, and magnetic
rotation spectra of the (l,0,3),(3,0,l) and (0,0,3) bands

l4N1602 and 15N1602. The spectra were recorded on

of
strip charts utilizing a dual pen recorder in which one

pen traced the spectrum of NO and the other recorded

2:
calibration fringes for the purpose of accurate interpolation
between calibration lines of known frequency. The speCtra
were photographed and measured on the Hydel system (30),

and each chart was measured twice to reduce measuring error.

Two runs were made of the (1,0,3) of 15NO and the (3,0,1)

2

and (0,0,3) bands of both isotopes. The (1,0,3) of 14NO2
required additional runs because the only available calib-
ration lines were situated directly above the band.

The (0,0,3) band was run at a pressure of 15-18 Torr
and an absorbing pathlength of 6.3 meters. The (1,0,3)
and (3,0,1) bands were run at 50 Torr and 40 Torr respectively
with a 15.75 meter pathlength. The method of calibration
has been described by Rao et. al. (20). The spectra were

calibrated using the precise near-infrared absorption

frequencies measured by Rank and co—workers (20).

16

 

‘17

The standard errors of the least squares fits to
obtain calibration constants averaged 0.002 cm.1 for the
(0,0,3) band and 0.003 cm‘1 for the (1,0,3) and (3,0,1)
bands. The effective resolution obtained was approximately

1 1

0.05 cm‘ for the (1,0,3) and (3,0,1) bands and 0.03 cm_ for

the (0,0,3) band where conditions were favorable.
IDENTIFICATION OF TRANSITIONS

The individual transitions were identified through the
use of ground state combination differences. These differ-
ences have been determined by microwave authors and Olman
and Hause (l) (22). For a given band all possible differ-
ences of the frequencies were formed and compared to the
known ground state combination differences. ThoSe which
agreed within a given tolerance were tabulated and were
extremely useful in identifying series of lines and
consequently subbands. Once a few transitions had been
identified the predicted frequencies from the least squares
fits could be used in conjunction with the ground state
combination differences in order to identify additional

lines.
ANALYSIS OF ASYMMETRIC ENERGY LEVELS
GROUND STATE

The ground state energy levels and molecular constants

are assumed known from previous work. Attempts to refine

18

the ground state constants employing the data from the
(1,0,3), (3,0,1), and (0,0,3) bands yielded no significant
changes in the values reported by Olman and Hause (1).
Thus these ground state values are accepted and for

completeness are listed in Table I.

UPPER STATE CONSTANTS

 

Analysis of the upper state rotational, centrifugal
distortion, empirical P6 terms, and spin splitting constants
proceeded as follows:

1. A sufficient number of unperturbed lines were
identified to enable prediction of the entire unperturbed
spectrum of the band.

2. Spin split lines were identified and analyzed
yielding effective spin splitting constants.

3. A single unperturbed frequency was determined
for each spin split line and these values, in addition to
the unsplit lines, were entered in the least squares fits
of the unperturbed spectrum.

The upper state constants may be determined by forming upper
state combination differences or by using ground state
constants plus observed transitions to determine the set of
upper state vibrational and rotational energy levels. The
latter method was used because it is difficult to determine
A and H from combination differences, and furthermore,

K

series deviating from the predicted Spectrum are clearly

19

 

 

 

Table I
Ground State Constants for N02(cm-l)
14 15
N02a NOZa r:
A 8.00251 1 0.00010b 7.63062 1 0.00029 1
B 0.433665 1 0.000004 0.433717 1 0.000016 I
c 0.410493 1 0.000014 0.409492 1 0.000030 1
1 (-11.37 1 0.28) x 10‘3 (-9.15 1 0.5) x 10'3 i,
aaaa .
—6 -6
Tbbbb (-1.418 1 0.023) x 10 ( 1.40 1 0.04) x 10
1 (6.91 1 0.24) x 10""5 (5.87 1 0.4) x 10"5
aabb
x -8.215 x 10‘6 -8.l75 x 10‘6
abab
—7
HKN (-1.1 1 0.6) x 10 0.0
”K (2.8 1 1.1) x 10"5 0.0
a Ref. (1)

b 95% Simultaneous confidence intervals throughout

The 95% confidence interval is approximately 4.5
times as large as the standard deviation. The ground
state constants were obtained by forming ground state

combination differences.

20

defined.

The complete asymmetric energy level expression
given by Eq.(6) is not in a form directly suitable for '
least squares and iterative procedures. The expression

was linearized by Hill (21) by writing the approximation

E = CA + E(B+C) + n(B-C)
0 T T ‘20)
After some manipulation (22) one finds the expressions
_ 2
g _ <pz>
2
g = N(N+l) - <Pz> (21)
n = (<PZ> - w)/b
Z P
where
b = C-B
p 2A-B-C (22)

So the complete linearized energy expression can be written

E = cA + £(B+C) + n(B-C) + Xxiri + HNN3(N+1)3 + HNKN2(N+1)2<P:>
2 2 i

4 6
+ HKNN(N+1)<PZ> + HK<PZ> (23)

Corrections to the energy levels in a given vibrational state

can be expressed as:

21

AB = cAA + £A(B+C) + nA(B-C) + inAri + N3(N+1)3AHN +
2 2 i (24)

2 2 2 4 6
N (N+1) <PZ>AHNK + N(N+l)<Pz>AHKN + <PZ>AHK

Experimentally one observes not energy levels but
transitions between energy levels so we can write the freq-

uency of an observed line as

v = 00 + E - E (25)

where V0 is the band origin, and a prime indicates the
upper state while a double prime indicates the lower state.
One finds corrections in the molecular parameters by using
least squares to minimize the differences between the

observed and predicted frequencies. This can be written

Av + AE' - AB" (26)

vobs - vpred = o

If the ground state is known AE" vanishes and

vobs - vpred = Avo + AE' (27)

In order to employ iterative methods it was necessary to
provide initial values for A, B, and C in the upper state
from which c, E, n, and <P2> could be calculated. The
least squares fits were done utilizing program Specfit
contributed by Olman (22). The resulting upper state

constants are given in the following tables.

(0,0,3) Table II

22

(1,0,3) Table III

(3,0,1) Table IV

SPIN SPLITTING ANALYSIS

 

Least squares methods were used to analyze the spin
splitting data. Since 6' and 8" occur linearly in Eq. (17)
least squares methods can be applied directly to determine I
values for these constants. The data was analyzed utilizing

program SPINFIT contributed by Olman (22). The values of

 

5 derived from the various fits are given in Table VI.
The combined fits agree very closely with the separate
fits which may also be used as a confirmation that the elect-

ronic structure is unchanged by isotopic substution.

GENERAL FEATURES OF THE BANDS

 

(0,0,3) of N02

The (0,0,3) is a strongly absorbing band located in

the region from 4695 cm"1 to 4775 cm_1 for 14NO and from

2
4611 cm"1 to 4676 cm-1 for 15N02. A previous analysis

ascribed the observed structure to the overlapping of a

type A and a type B band, but with the exception of one

weak series underlying the (0,0,3) of 15NO2 we have found

that at room temperature the structure can be accounted
for entirely by the type A band. Analysis has produced

fits of nearly all observed lines with K_ as high as 7

l
and N as high as 48. The branches of differing subbands

23

 

 

 

 

Table II
Upper State Constants for N02 (cm-1)
003
14 15
N02 N02
00 4754.20910.006 4655.22810.009
A 7.342710.0012 7.021710.0019
B 0.42545710.000032 0.42578110.000046
C 0.40291610.000024 0.402214:0.000036
-2 -2
Taaaa (—0.992:0.025)x10 (—0.8361 0.044)x10
-5 -6
Tbbbb (-l.5110.14) x 10 (-l.5810.l9) x 10
-5 -5
Taabb (9.0610.38) x 10 (7.2610.81) x 10
-6 -6
Tababb —8.215 x 10 -8.l75 x 10
-7
HKN (-l.7720.31) x 10 0.0
HK (2.96110.093)xlo"5 (0.2910.17) x 10‘5
No. Lines
Identified 550 466
No. Lines 416 308
Used
SD of Fit 0.0082 0.0092
a 95% simultaneous confidence intervals throughout

b Calculated values

The value of Tabab is in all cases fixed at the theoretical

value given by Chung and Parker (27).

24

 

 

 

Table III
Upper State Constants for N02(cm-l)
103
14N02 15N02

“o 5984.70510.004a 5874.95110.010
A 7.414010.0087 7.079110.0057
B 0.42321110.000039 0.4235510.00016
c 0.39935810.000017 0.39872810.000065
Iaaaa (-1.2010.19) x 10‘2, (—0.9910.15) x 10"2
Tbbbb (-1.54010.076)xlo‘6 (-1.3910.40) x 10"6
Taabb (8.612.1) x 10'5 (6.811.0) x 10'5
Tababb —8.215 x 10'6 -8.175 x 10'6
HKN 0.0 0.0
HK 0.0 (1.0710.71) x 10"5
No. Lines
Identified 301 323
No. Lines 214 233

Used
s0 of Fit 0.0065 0.0199

 

a 95% simultaneous confidence intervals throughout
b Calculated values

25

 

 

 

Table IV
Upper State Constants for N02 (cm-l)
301 F:
14N02 15N02 2

00 5437.54010.012a 5367.31610.026 g
A 8.014010.0056 7.532810.0090 i
B 0.4238410.00011 0.4245210.00027 y
c 0.39951910.000079 0.3991110.00019
Taaaa (-3.9710.31)xio"2 (4.8610.895)x10"2
Tbbbb 0.0 0.0
raabb (1.1610.37)xio'4 0.0
Tababb -8.215xio'6 -8.175xio'6
HN 0.0 0.0
HNK 0.0 (4.613.6)xio‘8
HKN 0.0 (—9.413.3)xio'6
HK (2.2210.24)xio'4 (-2.70010.097)xio"3
No. Lines
Identified 228 152
No. Lines

Used 171 121
so of Fit 0.009 0.016

 

 

a 95% simultaneous confidence intervals throughout

b Calculated values

Branch

P

FUIOIOIOOIOIOOO'U'U

I! 50 9U 50 5U 5U

50

9U

K‘

2

6

Table V

(0,0,3) Band Spin-Splittings (cm-1)

11

12

14

OBS
+0.0515

+0.0642

-0.0440

-0.0995

_000571

-0.063O
-0.0428
-O.1008
-0.0755
-0.0552

-0.0440

-001080

-0.0873

NO

2
PRED
+0.0594

+0.0595

-0.0356
-0.0865

-0.0592

-0.0654
-0.0441
-0.0992
-0.0730
-0.0568

-0.0460

-001010

-000818

15

OBS

+0.049l
-000652

-000459

-0.0626

-0.1105

-0.0569
-0.0593

-0.1049

-0.0431
-001845
-000982

-000673

NO

2

PRED

+0.0425
-000465

-000359

-0.0523

-0.0987

-0.0511

-0.0651

-0.0942

-0.0436
-001674
‘000957

-000785

Branch

R

W W W W W W W W W W W

W W W W

W

(0,0,3) Band Spin-Splittings (cm-
14

9
10
11
12

13

10
11
12
13
14
15
16
10
12
15
16
17

18

Table V

OBS

-0.0652
-0.0444
-0.0286
-0.0260
-0.0224

-0.0876
-0.0636
-0.0729
-0.0592
-0.0384
-0.0407

-0.0422

-001036
-0.0626
-0.0440

-0.0524

2

NO

7

2
PRED

-0.0582
-0.0505
-0.0445
-0.0397
-0.0357
-0.1578

-0.0789
-0.0695
-0.0620
-0.0559
-0.0507
-0.0464

-0.0427

-0.0983
-0.0668
-0.0615

-0.0570

(con't)

1)

15

OBS
-0.0542
-0.0534
-0.0461

-0.0835

-0.0556

-0.0491

-0.04l9

-000269

-O.1027

—0.0461

NO

2
PRED

-000642
-0.0551

-0.0860

-0.0747

-0.0528

-0.0479

-0.0404

-O.1093

«0.0501

Branch

R

W W W W

K

-1

(0,0,3)

N
19
20
21
22

23

2

8

Table V

14

OBS

-000416

-0.0403

NO

Band Spin-Splitting

2
PRED

-000437

-0.0413

1)

15

(cm-

OBS
“000437
-000403

-000354

-000447

PRED
-0.0468
-0.0439
-0.0413

-0.0369

29

Table VI

Effective Spin-Rotation Coupling Constants

For the (0,0,3) Band of NO

 

 

 

14 15
N02 N02
a
1000 0.181644 0.173210
1201b 0.1782 1 0.0014 0.1693 1 0.0016
b
(1000 - 6201) 0.0036 1 0.0010 0.0038 1 0.0010
1003C 0.1629 1 0.0013 0.1555 1 0.0010
C
(8000 - 6003) 0.0187 1 0.0013 0.0177 1 0.0010
EOOBd 0.1627 1 0.0012 0.1556 1 0.0019
d
(€000 - 1003) 0.0189 1 0.0012 0.0176 1 0.0019
a. From microwave values
b. From Ref. (1)
c. Fitting 14NO2 and 15N02 separately
d. 14NO2 and 15N02 data combined

0.”!!! vnmnum

30

overlap considerably, and the Q branches of the various

subbands are widely separated. In the (0,0,3) of 14NO2

the Q lines for Kel = l are grouped about 4752 cm"1 while

for K—l = 6 they occur about 4730 cmnl. Figure 4 displays

the complete picture of the (0,0,3) band of 14NO2 and

Figure 5 shows a detailed picture of a portion of the

(0,0,3) showing the K_1 = 0 subband.

A perturbation was found affecting the K_1 = 5
subband in the (0,0,3) of 14N02. The deviation of observed
from predicted frequencies increased with N and was as

large as 0.16 cm.1 at N = 42. A similar perturbation was

noted in the K_l = 4 subband of 15NO2 and of approximately

the same deviation.
Spin splitting has been observed and analyzed in
the (0,0,3) band. The high resolution, strong absorption,

and large splittings permitted observation of 32 resolved

or nearly resolved spin doublets in 14NO and 27 such

2
doublets in 15N02. Splitting is small in the P branch,
intermediate in the Q branch, and large in the R branch.

The possibility of this was considered previously. The

maximum observed splitting was 0.135 cm-1

14NO 1 for R4(4) in 15N02. Very few doublets

for R5(6) in

2 and 0.185 cm—

were observed in the P branch due to the small splitting,
overlapping, and weakness of the spin split lines. A list

of the observed doublets is given in Table V. A small

region of the (0,0,3) of 14N02 is shown in Figure 6 with

 

31

several resolved spin doublets. Effective spin rotation

splitting constants are listed in Table VI.

(1,0,3) of N02

The (1,0,3) is a weakly absorbing band located in

the region from 5923 to 6000 cm"1 for 14N02 and from

1 15

5825 to 5890 cm’ for NO The complete (1,0,3) band

2.
is shown in Figure 7, and a portion of the (1,0,3) is

seen in Figure 8 where the K_1 = 0 and the K_l = l tran—
sitions are indicated. Note the missing lines in the

K_l = 0 series. Q branches of the subbands were present
but too weak to be analyzed. Transitions have been ident-
ified with K__l as high as 5 and N as high as 46. Good fits
were obtained for series through K_l = 3. The K_ = 5

1

series was very weak, and the K_1 = 4 series was perturbed.
Spin splitting was not observed due to overlapping,
low resolution (~0.05 cm-l) in the region, and weak absorp-

tion of the low N lines.

(3,0,1) of NO

2
The (3,0,1) is a Type A band extending from 5380 cm-1
to 5456 cm"1 for 14N02 and from 5330 to 5387 cm-1 for 15N02

P, Q, and R branches were observed in both bands. P and R
lines were identified and analyzed by least squares methods
to yield molecular constants. This also led to correct

predicted positions for the O subbranches wh.ch are present.

 

'9?
I
E
r;

If

. .

32

The weakness and overlapping of the Q subbranches prevented

identification of individual lines. Figure 9 Shows the K_

1
= 0 subband of the (3,0,1) of 14N02. Lines were identified

14 15

through K_ = S for NO and K_ = 4 for NO The K_ = 4

1 l

subband was perturbed.

2 1 2'

Due to the weak absorption and overlapping of band

lines spin splitting was not observed in the (3,0,1) bands.

VIBRATIONAL CONSTANTS

 

 

The vibrational energy levels can be expressed as a

sum of harmonic oscillator energy levels linear in the vi

quantum numbers and quadratic anharmonic terms in the form

3 i
G(v v ,v ) = )30).(v.+l) + '23 Z X..(v.+ 1)(v.+ l) (28)
1' 2 3 1 1 1 2 i=1 j=l 13 1 2 3 2

with a zero point energy of

i
z x.. (29)

G(0,0,0) = 1
lj=l 3

"saw

Ami + I
i

1
2 41

Thus the band origins of the measured bands are given by
vo(vl,v2,v3) = G(vl,v2,v3) - G(0,0,0) (30)

This equation is in a form adaptable to least squares
analysis.

In addition to the (1,0,3), (3,0,1), and (0,0,3)

bands the (1,0,1) and (2,0,1) bands of 14N02 and 15N02

analyzed by Olman and Hause (l) were included in the least

33

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02

ea

we seen Am.o.ov mueaasoo mes

 

it

an? IJ—hlt‘

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02

4F , L._.)

«cased...
.=__::_____.C____.%.L _ _ _ _ _ .1.

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35

 

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02:

Va

we came Am.o.HC wumeasoo one

 

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Jar

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37

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.mcflmmflfi wocHH z co>w Sufl3 wco ppo cm can mcfimmfifi mmcfla z poo

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38

 

 

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3“

 

«Jen omen . WES oo_¢m 818 .

1:3; e I s;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5:: :Z:::::...

3 Eon e 2 cu as $96..

 

.2... to .n

39

squares analysis. Note that in each of the bands v2 = 0.
Thus it was necessary to supply values for X12 and X23 from
the work of Arakawa and Nielsen (5). The accuracy of the
results thus depends on the accuracy of that work.

Program VIBCON was written to perform the fits utiliz-
ing the Efroymson least squares subroutine. The resulting
vibrational constants as well as all the constants from
Arakawa and Nielsen are given in Table VII and a listing of

VIBCON is given in Appendix I.

h

VIBRATION-ROTATION INTERACTION CONSTANTS

 

The rotational constants A, B, and C depend on the

vibrational state as follows:
3
R. = R - Z a.v. (31)

where R = A, B, or C and the vi are the vibrational quantum
numbers. This equation is easily adapted to least squares
analysis and program ROTCON listed in Appendix I, was
written to perform the desired fits. Values of a? and a?

for both isotopes of NO2 are given in Table VIII.

40

TABLE VII

 

 

VIBRATIONAL CONSTANTS (cm'l)
Constant 14No a 14No b'c 15N0 a 15N0 b'c
2 2 2 2
61 1357.8 1355.9 1342.5 1338.5
02 756.8 756.8 747.1 747.1
03 1665.8 1663.5 1628.0 1628.3
xll -9.0 -8.1 -8.8 -7.4
x12 -9.7 -9.7 -9.5 -9.5
x13 -28.7 -29.8 -27.7 -28.4
X22 -005 -005 -004 -0c4
X23 “-207 "207 —206 .2.6
x33 -16.4 -15.6 -15.6 -15.3

 

Arakawa and Nielsen's values

Present work

w2, X12, X22, and X23 are fixed at the values given by
Arakawa and Nielsen

41

TABLE VIII

VIBRATION-ROTATION INTERACTION CONSTANTS

 

 

141102 15N02
0? —0.075310.0083 -0.04210.038
a? 0.00235410.000094 0.0022210.00019
6E 0.0028210.00050 0.0026910.00058
a? 0.220810.0070 0.19910.032
63 0.00271810.000079 0.0026510.00015
cg 0.0026410.00042 ' 0.0025610.00049

 

CHAPTER III

THE MATHEMATICS OF POLARIZED LIGHT

In the discussion which follows it is interesting and
economical to apply the powerful methods of matrix algebra
which have been developed to describe the polarization state
of electromagnetic radiation and the changes introduced by
various optical elements such as polarizers and retarders.
Since these methods are not generally common knowledge it
is thought desirable to include a brief description of the

mathematics and some useful matrices. More complete infor-

mation is available in POLARIZED LIGHT by Shurcliff (23)

 

and the RCA Institute Lecture Notes entitled MODERN OPTICS

 

by Sutton and Panati (24). Also references to several

papers are found within these references.

THE JONES-MUELLER CALCULUS

Two forms of matrix algebra have been developed to
aid computation in dealing with phase relationships,
intensity, and polarization of light. One form, the Mueller
calculus utilizes 4x4 matrices and the Stokes vector as its
basic units. It is most useful in dealing with incoherent

beams and partially polarized light. The matrices in this

42

raw .1 J" _ .‘7.'.-?.-" 1

gm. fie—.5 - .- M '5." ;-

43

theory have been derived empirically, and thus have a firm
experimental basis.

Jones calculus is based on 2x2 matrices and the Jones
vector which is simply a vector of the amplitudes and phases
of the x and y vibrations for a beam propagating along the
z axis. Jones calculus is suited to completely polarized
light and addition of coherent beams. Thus Jones calculus
is best suited for this work and will be discussed briefly.

The general Jones vector can be written

J = X . (31)

for some particular amplitudes of the electric vector EX
and Ey with some definite phase angle ex and sy. An
harmonic time variation is assumed. The type of polarization
depends on Ex' Ey’ and the relative phase difference. There
is ng_Jones vector for unpolarized light or partially
polarized light.

Where absolute phases and intensities are unnecessary
one uses standard normalized Jones vectors. For example
the complete vector for light plane polarized along the x

axis is

J = X (32)

The normalized Jones vector is

 

44

eiex
J

1/2 = (33)

 

(J1J)

where J+ is the Hermitian conjugate of J.
Since ex is arbitrary one writes the standard normal-
ized Jones vector
1
J' = (34)
0
A Jones matrix can be associated with each optical element
which alters the polarization state but leaves the direction
of prOpagation unaltered. These matrices when operating on
the Jones vector of the incident light yield the Jones
vector of the emergent radiation. A list of useful Jones
vectors is found in Table IX and Jones instrument matrices
are found in Table X.
As an example consider the case of a polarizer-
analyzer assembly with the axis of the analyzer making an

angle 6 with the polarizer axis. Suppose the polarizer

produces linearly polarized light with a Jones vector

1
(35)
0
The Jones matrix of the analyzer (Table X) operating on
the incident Jones vector can be written
2 . 2
Cos 0 COSOSInB l Cos 0
. . 2 = (36)
CosGSine Sin 0 0 CosOSine
.__

 

45

so the amplitudes are

2

Ex = Cos 0
(37)
E = SinGCose
Y
and they are in phase. The intensities are found from
I * * 2
I = J J = E E + E E = C03 6 (38)
x x y y

in agreement with the Law of Malus.

In this way the theory of polarization is reduced to
a series of matrix multiplications often with great
simplification of the problem. Note that if light described
by a Jones vector, [J] , is incident on a series of elements
in the order A,B,C, to yield a Jones vector, [J'] , then the

matrix multiplication takes the form
[J'] = CBA [J] (39)

The Mueller calculus is similar in application since
the Stokes vectors and Mueller matrices are tabulated. The
Mueller calculus deals directly with intensities and is

useful only where phase information is not required.

 

46

 

 

 

 

 

TABLE IX JONES VECTORS
POLARIZATION FORM JONES VECTOR
FULL STANDARD
NORMALIZED
LINEAR POLARIZATION
P - - r1
0 Axele 1
Horizontal 0 = 0 ———
0 0
' u) (hi
r — F‘
o 0 0
Vertical 0 = 90 is
A e l
I Y y
h- .1» 1.4)
fl
0 is 1 11 l
Oblique 0 = :45 X Axe x —
11-l filed

 

CIRCULAR POLARIZATION

i q
Axe 8x -i
Right Circular

hue
H

Axe1(sx+w/2)

 

‘

is
Axe X 1

ei(sX-‘n'/2) /2

Left Circular
A
x

F——_-_'1
H P-

ELLIPTICAL POLARIZATION

Axelex (CosR)e-1Y/2
General Elliptical
Ayeley (SinR)e1Y/2

 

 

 

 

TanR = Ay/AX y = e + e

 

47

TABLE X JONES INSTRUMENT MATRICES

 

 

LINEAR POLARIZERS

 

 

AZIMUTH OF TRANSMISSION AXIS, MATRIX
0
P cl
0 l 0
0
0 0
.L -
F q
0 0 0
90
0 l
J- ‘I
V 2 .
Cos 0 Cosesine
General 0 2
CosBSine Sin G‘J

 

IDEAL LINEAR RETARDERS
or RETARDANCE 6y= 90o

AZIMUTH OF FAST AXIS, e

 

 

- WI
0 CIR/4 0
0 .
0 e-lfl/4
b. -.
F.-.
o e 10/4 0.1
90 11/4
0 e
r
1 1 ii
0 = 1450 —
/2 ii 1

CIRCULAR POLARIZERS

Right Circular

HWIH

Left Circular

NIH
Fri
I
I—‘ P_]' I—‘ I-“|

 

64H
F7
P.

5.1- d, n 'l

 

 

CHAPTER IV

PHASE SENSITIVE DETECTION OF MAGNETIC CIRCULAR DICHROISM (MCD)
AND MAGNETIC CIRCULAR BIREFRINGENCE (MCB) IN NO AND NO2

Conventional magnetic rotation spectra are obtained
by inserting a sample of gas between a crossed polarizer-
analyzer pair, applying a magnetic field, and recording the
intensity of the radiation emerging from the analyzer. These
spectra are a combination of contributions from magnetic
circular dichroism and birefringence.

In this series of experiments adaptations of the
techniques described by Eberhardt and Stalder of the Georgia
Institute of Technology (7)(8) have been applied to separate
and record magnetic circular dichroism (MCD) and magnetic
circular birefringence (MCB).

The techniques for the measurement of MCB were applied

to the 3-0 band of NO and the (0,0,3) band of NO Measure-

2.
ment of MCD requires production of right and left circularly
polarized light. This can be accomplished with quarter-

wave plates and properly oriented polarizers or by electro-
Optical techniques such as Pockel's effect. In these initial
experiments quartz quarter-wave plates were selected as

being the simplest method to verify that MCD signals could

be detected. The quarter-wave plates were purchased from

48

 

49

Carl Lambrecht Co. and were centered at 2.11 microns which

is the center of the (0,0,3) band of N02. The radiation

at 2.11 microns was essentially circularly polarized and

MCD could be measured, however, the radiation at the 3-0

of NO at 1.8 microns showed such ellipticity that measurement

was not feasible.
THE JONES INSTRUMENT MATRIX OF THE SAMPLE

A general Jones matrix to describe a sample with

 

circular dichroism and circular birefringence may be
determined as follows.

Assume that there exists a matrix which describes
the way in which the sample operates on any incident state
of polarization. Further assume that the matrix has the
circular polarization states as eigenvectors. Thus we
have an eigenvalue problem. For right circularly polarized
light
_;_D = (T+)l/2ei6/2_i_ ’1

(40)
CD/2'1 /21

where T+ is the right circular transmission fraction and
0 is the relative phase shift between the right and left
circularly polarized components. Likewise for left

circularly polarized light

1/2e-1s/2 1 1

/§ 1 (41)

__ = (T-)
C D /2 1

50

where T- is the left circular transmission fraction. Solution

yields the matrix

 

 

 

A B
(42)
-B A
r:
where
A = (T+)l/2ei6/2.+ (Té)l/2 e—iG/Z (43)
2
and v,
r~
21

This matrix operating on the incident Jones vector gives the
emergent Jones vector. This permits the following work to

be described completely in terms of Jones calculus.

CONVENTIONAL MAGNETIC ROTATION (MR)

 

The signal observed when a sample in a magnetic field
is placed between crossed polarizers may be readily derived.

The first polarizer generates a horizontal beam described by

then the sample and final polarizer operate to give

('01
J I; 0 A B l

 

 

51

which simplifies to

J; 0
= (46)
J' -B
Y
and
f 1
I s J J = B B (47)

B is evaluated from Eq.(44) so the intensity becomes

+ 2

I a 1[T + T" -2(T+T-)l/

4

C056] (48)

Assume an exponential absorption law

where 0+ (0”) is the absorption coefficient for right
(left) circularly polarized light and 0 is the relative
phase shift between right and left circularly polarized
components. This is usually expressed in terms of the
Faraday angle

6 = 0/2 (49)
where
(n- - n+)L (50)
Then Eq.(48) may be rewritten

I « [(T+)1/2— (T')1/2]2 + (T+T')1/251n28 (51)

4

 

which separates into a magnetic circular dichroism term

52

 

 

AzaV

 

 

 

 

 

 

 

 

 

 

 

 

 

+ l 2 - 2 2
MCD = [(T ) / - (T )1/ 1 (52)
4
and a magnetic circular birefringence term
MCB = (T+T’)1/251n28 (53)
Magnetic rotation spectra are composed of the combined
effects of dichroism and birefringence.
MAGNETIC CIRCULAR DICHROISM (MCD)
The MCD of the (0,0,3) band of NO2 was measured
using the experimental setup of Figure 10.
Figure 10
ANALYZER
MAGNET I
BEAM -
////////// DETECTOR!
AMPLIFIER
-——+ SAMPLE SPECTROMETER L“ AND '
RECORDER
//////////
BALANCING
(J QUARTER-
CHOPPER WAVE

PLATE

53

The construction of the chopper is shown if Figure 11

Figure 11

axis of rotation

 

where R stands for the right circular polarizer and

L stands for the left circular polarizer.

The right and left circular polarizers were constructed
by mounting a quartz quarter—wave plate preceeded by an HR
polarizer in a teflon assembly. The axis of the HR polarizer
was mounted at an angle of :450 to the fast axis of the
quarter-wave plates which generates left and right circularly
polarized light respectively.

The relative orientation of the circular polarizers
was not important since circular polarization has no
unique axis. As thechOpper rotates the incident beam is
alternately converted to right and left circularly polarized
pulses. It is therefore possible to look for a difference
in the right and left signals directly. This is accomplished

by observing that the primary signal occurs at a frequency

54

20 where v is the frequency of rotation. Also present is a
difference signal at 0 which is proportional to the difference
in intensities of the right and left pulses after passing
through the sample. The origin of the signal can be

schematically visualized by reference to Figure 12
Figure 12

R represents
right circular

”"‘~\\\__‘(/,w pulses

L represents
R L R L R L left circular
7 pulses

 

 

 

 

 

 

 

 

 

 

 

This difference signal is a direct measure of the circular

dichroism and can be expressed as

D = (I - IL)Sin(wt + y) (54)

R

where w = 200 and y is an arbitrary phase angle.

The signal was detected using a Princeton HR-8 lock-
in amplifier which is a sensitive tuned amplifier plus a
phase sensitive detector. The phase sensitive detector
requires an externally generated reference Signal which
has a definite phase relationship to the actual signal.
The reference for convenience was generated by reflecting
a laser beam from a half blackened disk mechanically locked
to the chopper disk. The beam was detected by a photo

detector, amplified, and used to provide the reference

55

required by the HR-8. The lock-in amplifier effectively
selects a frequency, converts the output of the detector
to a sinusoidal wave form at the desired frequency whose
amplitude is proportional to the signal, rectifies the
resultant AC signal utilizing the reference signal as a
timer or gate, and filters the output to generate a DC.
voltage. Appropriate time constants can be selected for
the filtering in order to smooth out the high frequency
noise while retaining the signal.

In practice the right and left circular polarizers
do not pass precisely the same intensity of radiation, and
the grating is polarization sensitive. Any imbalance in
the right and left signals produces an absorption spectrum
which could mask the desired circular dichroism. Thus a
compensating scheme was necessary. Balancing was accomp-
lished by inserting a quarter-wave plate and a linear polar-
izer into the optical train after the sample as shown in
Figure 13. The quarter-wave plate converts the right and
left circularly polarized light to linearly polarized light
at :450 to the fast axis of the plate.

In practice the final polarizer is aligned with the
grating to give maximum sensitivity and the fast axis of
the quarter-wave plate is rotated until balancing is achieved
as shown is Figure 13.

The initial imbalance was small but compensation was

required and this scheme allowed effective balancing. The

56

F .
A Figure 13

The linearly polarized pulses
emerging from the balancing
quarter-wave plate are shown
resolved along the axis of
the analyzer.

R is the result of conversion
of the right circular pulse to
linear polarization.

 

 

L is the result of conversion
UNBALANCED of the left circular pulse to
linear polarization.

F is the fast axis of the
F A balancing quarter-wave plate.

A is the axis of the analyzer.

Both cases are for zero
-applied field.

Balancing is achieved when
the components along the
analyzer axis are equal in
the absence of a field.

 

 

BALANCED

method is also useful because small rotations of the
quarter-wave plate produce a difference which can be used
as a test signal when tuning and phasing the electronics
even in the absence of a sample. The net result of the
balancing scheme is to reduce the signal slightly but to
retain the same dependence of the signal on (IR-IL).

The overall result of a change in the relative
values of (IR-IL) is to introduce a negative sign -(IR-IL)
which is equivalent to a 1800 phase shift.

D = -(IL-IR) Sin(wt + y) = (IL-I ) Sin(wt + y + w) (56)

R

57

In a lock-in amplifier a 1800 phase shift inverts the
output voltage which makes it possible to record positive

and negative deflections.
MAGNETIC CIRCULAR BIREFRINGENCE (MCB)

Magnetic circular birefringence is usually measured
by the degree of rotation of a linearly polarized beam of
radiation in passing through a sample subjected to a long—
itudinal magnetic field. A linearly polarized beam can be
thought of as composed of a right and a left circularly
polarized component of equal magnitudes. Thus mathematically
the Jones vector of a vertical linearly polarized beam can

be represented as

l
5' (57)

ll
MI I-‘

From a classical point of View the right and left components

can have different indices of refraction denoted by nR = 11+

and nL = n- which yields a difference in relative phase as
the components traverse the sample.

The following is a description of an experimental
arrangement which can measure the relative difference

n - n . The experimental setup for measurement of MCB is

R L
shown in Figure 14. The polarizer and analyzer were crossed
and the rotating polarizer was mounted on a cylinder set

in a 1.75 inch I.D. ball bearing. The assembly was rotated

58

 

 

 

 

 

 

Figure 14
MAGNET ANALYZER
BEAM
——————9» SAMPLE O SPECTROMETER AMPLIFIER
——’ 7777777777 / RECORDER
I ROTATING
POLARIZER POLARIZER

using pulleys and a belt drive powered by a 1/20 H.P.
synchronous motor. The reference required for the lock-in
was derived by aligning a laser beam with a hole bored in
the cylinder which produced two brief pulses per revolution.
The pulses were detected with a photo detector and amplified
in a tuned circuit then sent to the tuned reference amplifier
of the HR—8 lockein amplifier. The tuned circuits converted
the brief pulses into sinusoidal waves suitable as a refer-
ence.

A mathematical analysis of the system is required to.
find the form of the radiation emerging from the analyzer,
but a brief physical picture may be useful. If one neglects

circular dichroism then the result of magnetic circular

birefringence on a linearly polarized beam is to produce a

59

rotation of the plane of polarization. This occurs because
the relative phase of the two equal amplitude circular compo-
nents into which the linearly polarized beam can be decomposed
is changed due to their differing indices of refraction. Thus
neglecting dichroism we have a linearly polarized beam
emergent from the sample and incident on the rotating polarw
izer but at different angles depending on the MCB.

In the experimental arrangement of Figure 14 and in
the absence of a sample one has a primary signal at 40 where
v is the rotational frequency of the polarizer. The maxima

O

of the signal occur at 0 = 45°, 135 , 225O

, and 3150 where
0 is the angle of rotation of the polarizer measured from the
horizontal. When the sample is introduced and the beam is
rotated by +8 degrees alternate pulses increase and the
remaining pulses decrease, and for -e the relative pulse
difference is reversed. This means that going from +5 to
-e changes the phase of the signal impressed at 20 by 180°.
This is the reason that positive and negative deflections
can be observed.

The following calculation with Jones calculus shows
how the observed signal is related to the angle of rotation.
Assume that the first polarizer produces a horizontal beam
(1
L0

which is incident on the sample. Then we have the following

multiplication of Jones matrices for the experimental

60

arrangement of Figure 14.

(58)

where Cl and S1 are C050 and Sin¢ and the angle 0 is the
azimuthal angle of the rotating polarizer measured from
the horizontal. A and B were defined previously. This

reduces to

 

 

r- q
0
2
ClslA f SIB
:— .I

The emergent intensity is found from

' I « J1J (60)

which after appropriate simplification gives

2COS(2¢ + 0)) (61)

I « Sin2¢(T+ + T" + 2(T+T-)1/
where o = mot is the rotating polarizer angle. Recall

0 is measured from the horizontal. If we wish to measure

from the vertical

0 + ¢ + fl/Z

and the expression becomes

I « C052¢(T+ + T- - 2(T+T~)1/2C052(¢ + 6)) (62)

 

‘I
(y.

u- _._. - _._.._.__._._ -A_‘_-

61

where 6 = 0/2 which reduces in the appropriate limits to the
conventional magnetic rotation signal (10). This intensity
is converted to an equivalent electrical signal at the
detector.

The remaining problem is to Show that if the lock-in
amplifier is tuned to twice the rotational frequency one gets
a signal proportional to the angle of rotation of the beam
or the Faraday angle. For this to be true two conditions
must be met:

(a) The MCD must be very small compared to the MCB and

(b) The rotation of the beam in the sample must be a

few degrees or less.
These conditions have been experimentally verified for N02
and theoretically for N0 (10).)
The terms dependent on 20 may be found as follows.

First expand
065(2¢ + 0) = CosZ(¢ + 0) = 1 - 28in2(¢ + e) (63)

SO

+)1/2 —(T-)
4

1/2]2

I a [(T Cosz¢ + (T+T')1/20652651n2(¢+0) (64)

 

The first term is the MCD which is assumed small. Then
complex exponentials can be used to reduce Coszosin2(¢+0)
into simple trigonometric functions. The only term in 20
which remains is Sin0Sin(2¢+0). The observed intensity

then assumes the form

62

12¢ « (T+T-)l/zsinesin(2¢ + e) (65)

or in terms of the absorption coefficients

12¢ = Ioe[‘(‘1++ a-)L]/28in63in(2¢ + e) (66)
The signal depends on an amplitude factor and a phase
shift. The Sin(2¢ + 6) term introduces a small phase shift
but the dominant signal arises from the Sine term. As 0
goes from + to -, 12¢ changes Sign which corresponds to a
1800 phase shift in the lock-in amplifier. This makes it
possible to observe positive and negative deflections of

the recorder about some relative base line.

MCD AND MCB SPECTRA

 

NO 3-0 BAND

Magnetic circular birefringence spectra were observed
in the 3-0 band of l4N160 utilizing the Michigan State
University near-infrared spectrometer. The spectra are
shown in Figure 15. The absorption and magnetic rotation
spectra have been observed previously and are included
in Figure 15 for comparison. The MCB spectra were run
at a pressure of 10 cmHg and with an absorbing pathlength
of 18.9 meters at fields of 2000 and 4000 Gauss. The

signal was detected at a frequency of 47 Hz 11th a lead

sulfide detector.

ALIA. ____..-...‘-

 

. .3 U
. . bu
1 skin’s), , !

 

 

 

 

.mmsmu ooov mo caoflm m cufl3 omazz «0 pawn cum
on» cw mnuommm mu: .m: .coHuQHOmnm "ma madman
menu in . Tic . hi
_ _ _
1§,%CE%E
“81353!-
3
6
.b.o IIBEI
gigs/L
0.
.3. am.
sIEISI¢

a use)»

64

 

.|¢~ I ..1I
. .. n a.”
. a . I. . x....Plutwivil01-52ulflllk

 

N

.mmsm 0 ma 0 m o
w ooov m pH .w u omHZvH m

mnuommm no: can .moz .mz .coflumnomnd "ma madman

mfih‘ . .LZU 09!.

331x12 8..

 

akf>PS%Xé<$33§$§€322>Eé§/H _, .3.

«a... «6.0

 

65

The spectra at once prove that for the P and R branches
the 2H1/2 and 2H3/2 lines are rotated in opposite directions.

The direction of rotation is positive for the 2H P and R

1/2
lines and negative for the 2H3/2 lines. The Q branch shows a
very strong signal which is predominantly rotated in the

negative direction. The directions of rotation are in agree-

ment with the work of Aubel (9) and Keck (10).. A complete

explaination of the spectrum awaits a detailed analysis.

(0,0,3) of NO2

It was possible to observe both MCB and MCD signals
’ 14 16

in the (0,0,3) band of N 02.. Absorption and magnetic
rotation spectra were also run for camparison. Typical
absorption, MR, MCB, and MCD spectra are shown in Figure 16.

MCB and MCD spectra were run at a pressure of 3 cm Hg
and a pathlength of'18.9 meters at fields of 2000 and 4000
Gauss. The MCD signals were much weaker than the MCB and
required that the entrance and exit slits of the spectrometer
be opened from 100 microns to 200 microns. Furthermore it
was necessary to increase the gain by a factor of 2.5. This
supports the premise that the MCD signals are much smaller
than the MCB signals.

The strongest MR, MCB, and MCD signals occur around
the Q branch lines which are closely spaced in N and widely

separated according to the value of K-l' The MR lines from

right to left in Figure 16 arise from K_l = 2,3,4,5,6,7 and
8.

 

 

 

Hm.

66

The theory required for prediction of Zeeman patterns

2
that a detailed study of these patterns will lead to further

for NOV has been detailed by Olman (22), and it is hoped

meaningful interpretation of these spectra.

SUGGESTIONS FOR CONTINUED WORK

 

Much work has already been done on the magnetic rotation
spectra of paramagnetic molecules. Mann and Hause (25) (26),
Aubel (9), and Keck (10) have develOped a considerable body
of information about the MR of NO and N02. The calculation
of MR spectra requires prediction of both magnetic circular
birefringence and dichroism terms. In principle calculation
of either birefringence or dichroism signals should be simpler
than the MR signals where they are combined.

A semiclassical approach to the calculation of hire-
fringence and dichroism effects has been outlined by Aubel
(9) and continued by Keck (10) with application to the NO
molecule. Briefly, in this approach one calculates the
absorption coefficients, 0+ and 0-, from expressions which
apply to DOppler, Lorentz, or intermediate line shapes. Then
the indices of refraction, n+ and n-, are related to the
absorption coefficients through the Kramers-Kronig relations.

Once n+ and n- are known 0 is determined and can be
inserted into Eqs. (66) and (56) which are the intensity
expressions for birefringence and dichroism. This enables

computation of an ideal predicted spectrum. To compare this

67

with the observed spectrum the effect of the spectrometer slit
function must be included. Semi-quantative agreement between
predicted and observed signals provides strong support for
the validity of the Zeeman and dispersion theories.

It is hoped that this approach will be fruitful for
interpretation of the observed MCB and MCD spectra of NO

and N02.

";~'. 25‘

 

SUMMARY

The (0,0,3), (1,0,3), and (3,0,1) vibration-rotation
bands of 14N1602 and 15N1602 have been analyzed on the basis 1
of the slightly asymmetric top molecule. Least squares

methods were used to compare the observed to predicted trans-

‘

itions and by iterative procedures yielded the upper state
constants for those bands. This information in conjunction
with the previously analyzed (1,0,1) and (2,0,1) bands was
used to predict improved vibrational and vibration-rotation
interaction constants.

Spin splitting was observed in the (0,0,3) band and
was analyzed to give spin-rotation interaction constants.

Magnetic circular birefringence and dichroism studies
were undertaken for NO and N02. Experimental spectra were
obtained which are qualitatively in agreement with previous
experimental work and theoretical predictions. Finally a
method was suggested whereby theexperimental spectra may

be compared to theoretical predictions.

68

 

10.

ll.

12.

13.

14.

15.

REFERENCES

M. D. Olman and C. D. Hause, J. Mol. Spectry. £6.241

(1968).

 

M. D. Olman and C. D. Hause, J. Chem. Phys. 49, 4575
(1968). a

G. R. Bird et 31., J. Chem. Phys. 49, 3378 (1964).

R. M. Lees, R. F. Curl, Jr., and J. G. Baker, J. Chem.
Phys. 45, 2037 (1966).

E. T. Arakawa and A. H. Nielsen, J. Mol. Spectry. a,
413 (1958).

G. E. Moore, J. Opt. Soc. Amer. 43, 1045 (1953).

A. F. Stalder, Thesis, Georgia Institute of Technology
(1967).

A. F. Stalder and W. H. Eberhardt, J. Chem. Phys. 41,
1445 (1967).

J. L. Aubel, Thesis, Michigan State University (1964).

D. B. Keck, Thesis, Michigan State University (1967).

C. H. Townes and A. L. Schawlow, Microwave Spectroscopy

 

(McGraw-Hill Book Co., New York, 1955), p. 93.
Ibid., p. 84.
Swalen and Pierce, Journal of Math. Phys. 3' 736 (1961).
R. A. Hill and T. H. Edwards, J. Mol. Spectry. 9, 494
(1962).
L. Pierce, N. DiCianni, and R. H. Jacksvn, J. Chem. Phys.

38, 730 (1963).

(5 (a

a.“

l6.

17.

18.

19.

20.

21.

22.

23.

24.

27.

28.

29.

3‘).

70

Townes and Schawlow, 92. giE., p. 175.

C. C. Lin, Phys. Rev. 116, 903 (1959).

w. T. Raynes, J. Chem. Phys. 41, 3020 (1964).

D. B. Keck EE.E£" "Symposium on Molecular Structure and
Spectroscopy," Columbus, Ohio, 1966, Paper H-Z.

K. Narahari Rao, C. J. Humphreys, and D. H. Rank, E2237

length Standards in the Infrared, Appendix III and

 

pp. 160, 161, 171. Academic Press, New York, 1966.
R. A. Hill, Thesis, Michigan State University (1963).
M. D. Olman, Thesis, Michigan State University (1967).

W. A. Shurcliff, Polarized Light (Harvard University

 

Press, Cambridge, Mass., 1962).

A. M. Sutton and C. F. Panati, Modern Optics (RCA

 

Institute, Inc. Lecture Notes, 1969).

G. Mann, Thesis, Michigan State University (1959).

G. A. Mann and C. D. Hause, J. Chem. Phys. 31, 1117
(1960).

K. T. Chung and P. M. Parker, J. Chem. Phys. 43,3869
(1965).

G. Herzberg, Infrared and Raman Spectra 9£_Polyatomic

 

 

Molecules, Molecular Spectra and Molecular Structure

 

II (D. Van Nostrand Company Ltd., New York, 1945).

M. A. Efroymson, Mathematical Methods for Digital

 

Computers (John Wiley and Sons, Inc., New York, 1960),

 

Ch. 17, pp. 191-203.

T.L. Barnett, Thesis, Michigan State University (1967).

 

 

APPENDIX I
COMPUTER PROGRAMS

Several computer programs were utilized in the course
of this work. Several programs were contributed as previously
acknowledged. The following programs were written to perform
least squares fits based on a least squares routine by M. A.
Efroymson (29). Typical data decks are listed as well as the

programs.
VIBCON

This program performs a least squares fit on Eq. (30)
using the experimentally determined band origins as data
points to determine vibrational constants. It is necessary
to input approximate values of the vibrational constants and
any or all of the constants may be fixed or varied as indicated

in the data deck.
ROTCON

This program performs a least squares fit on Eq. (31)
using the experimentally determined rotational constants A,
B, and C as data points to determine vibration-rotation
interaction constants. The program is otherwise similar in

operation to VIBCON.

71

 

v

 

k?
D

CJN
L) H!

'4

,4

fl

3.;
C")

M“

(U

12

104

l:

PRQGFAM

7?

VIBCON
COMfiOA/123/NK(30).
COMMON/lZ/ IHFC1.INF02.
10EmT<2>

NFELMAX, YDFV“AX

COMMOA/A/ NAVE(39)

DINEMSION

NUlfFD). ~02<5r). ”U3(5U)

INF33.NU“CCN.JUNP.COVSI(301:5?IN;EF0UT}TOL

DIMENSION COW(30:50).F038(60).FCfiLC(5u)oWT(50))NUSE(503

READ 20a

III

F09MAT(12)

PRINT 210

III

FORMATti THE FCLLUNING VIBRATIONAL CONSTANTS APE FOR &I2‘N02t)

REAUII
PRINT 1.
FORMAIt21

READ5JINFC1JIVFOEIINEOE

PRINT 5 .
FORMAT!
N = 0

N : N

READ 20

PRINT12 ,

FOEHAI(1XJ

IF(CO\8T(

NUMCUNaJUMpv

315:

i 1
CONST(N)J
F09MAI¢F19.4o

YDFVVAV; NPELMAX
KUICOH.JUHP, XDFVMAX. NVELMAX
5»F13.4)I5)

:TOL0:FIV0EFOUT

19501. INFDZ. INFDS. TCL:EVINoEF”UT

6F1§.5)

NKlw)

I1)

C0M3T(N)o ~K(N)
F190402X0I1)
N).NE.3) G” TO 102

NCOUNT = W - 1

NN = C

“N = NM + 1

READ 4: HU1(NN).N02(NV).NUK(MN)p F088(NN).WT(V”)

pRINT 4,NU1(NN),HU?(MM):VU3(NN):

FORMATt3I

IF(FOBS(NN).NE.O)

INDATA =
DRINTéi I
FORMAT!
NAME(1)
NAHE(2)
NAMEIS)
NAME(4)
MAVE(5)
NAVE(6)
NAMEI7)
NAME(8)
NAME(9)
NCONPl =
00 110.NN
COV(19MW)
CON(2aNN)
CON(3oNN)
CON(4.HN)
CON‘SoNN)
CUNI515UJ)
CON(7.MH)
CONI8.NM)
COM(9.NN)

IIIINIIIIflIlflflfi

FCBS(NN),%T(0”)
3: 2F10.4)
GO TO 104
NM - 1

WDATA

INDATA ISt
6HOHEGA1
6HOMEGL2
3HX11

SHXZE

3HX33

SHXLE

3HX13

3HX21
NUMCDN + 1
' 1, {NBATA
”01(HN)
NU?(NN)
NU3(NN)
((KUI(NN)
((WUZ(NV)
((NU3(NN)
((hUiCNN)
((“01(NN)
((NUE(NN)

I5)

F,5)*(901(“N)
0.5)*(“U3(1N)
n.5)*(NU?(VN)
0.5)*(UU3(VN)
n.5)*(MU3(”N)

"- "0 "- I. II- "5 fl '0 II II
+4-+«++ +
1-+-+4»+1+
U‘“lh‘lfl a

 

0.25
0.95
0.75
0.25
0.95
0.25

110
Can

001

10
200

165
166

9100

9110

175

73

CONTINUE

PRINT 1800. C0N

FORMAT(F10.4)

PRINT 1001. NAVE

FOGMATCAB)

DO 200. NV = 1. INDATA

FCALC£NN) é COKST(1)*CON(1.NN) + CONSTI2)tCOH(?.NN) + CONST(3)6
100NI3.NN) + CONST(4)oCCN(4.NM) + CONST(5)tCON(5.NN) + CONST(6)*
2000(6.NN) + ConsT(7)wCCN<7,NM) + CONST(B)tCON(8oNN) + CovST(9)*
3COM(9.NN)

PquT 10. 001(IN).N02(MN).N03(NN).rCALC<wN)

FORMAT(313. F10.4)

CONTINUE

CALL STEPFIT (00*.F0DS.F0ALC.NUSF. NT. NCONP1. INDAT!)

END

SURROUTINE PRINTCUT(0ATA.F038.FCALC.MUSE.wT.N30NP1.IMDATA)

COMMON/1237 NK(31).N“ELMAX,X0EVMAX

COMMOk/zs/hQDATA.NOVMI,AVEwHT.ST0Y.NQSTtD, N. NTN, FPEOPQD, YPRED,
1 DEV

CQqMQN/lgU/JJ(935), KK(900)1 JHEL(QCQ), KDEL(900). NUDFL¢900I

019EMSIONDATA(NCOMP1.INDATA).F089(INDATA),FCALC(INDATA).HT(INDATA)
1. MUSEtleATA)

ENTRY PRINT 2

PRINTléé

RETURN

ENTRY PRIVT 3

PRINT 165. N. KDEL(N). JFELtN). KMN). JJ(N). WIN. F088(N). FCALC(
10), FRFQPRP. DATAIMCONPl. N). YPCEfl. DFV

-‘ em;
up 6

 

rfi—v—s
‘uai’fi, m

REYURN

FORMAT (15.x. 2R1. 12. 1H,. I2. 3x, sz2. 6F11i4)

'FORMAT (*1 NO. NT CBS FREQ CALC FPEC PRED FREQ
loBS-CALC PRED-CALC CBS-PRED3)

END

SUPROUTINE STEPFITIDATA:FOPS:FPALCaNUSFp NT: NCONPloINFATA)
COMMON/1237 NK(3?)aN”ELMAX.XUEVMAX
COMMON/lZ/IN501aINc02aIN7031NUNCCNaJUMPaCOMST(30):FFIN3EF0UTaTOL.
llDENT(2)

COMMON/23/NODATA3NOVVI.AVEWHT18T9Y:NOSTEP. V, NTN, FPEQPPD. YPREU.
1 DEV

COMMON/A/ NAME(30)

DI”EVSICN “1136)9VECTOR(31131)9IHDFX(3O):ICEXIBOI1SICMA(30)1
ICOEN(35)aSIGMCC(30):MOTI~(30),XCCNST(30)
DIMENSIONDATA(NC0N91.INDATA):FOBS(INDATA)oFCALC(INPATAIoWTCINDATA)
1: NUSE‘INDATA)

TYPE INTEGER S€.VV

TYPE DOUBLE VECTOR.SIGNA:CRENpQICMC0aSIGYoPEFR.VAR

IFIIUENTI1I) 9109:9110

IDENT(1)88HINPUT DA

IDENT(2) =2HTA

CONTINUE

IF(.N0T.EFIN) FFIN=EFDUT=1.E-8

IFCTOL.EO.C.0) TGL=0.001

MOVAR a 1

9591

74

F0 157 I=leUMCON
IFINK(I))158.1‘7
N1(NOVAR)=WAMEII)
IFtJUHP) 8158,9159
XCQNSTINCVLR)=CO\ST(I)
NUVAP=NOVAR+1
COHTINUE

KODATAze

00254 N=1.IMDATA
IFINT(N))252.2“3
NUSE<N>S0

GO TO 254
NUSEIRI=1
NODATA:NODATA+1

CONTINUE

NDEL=0
NDIN=K=WSENT=NCMIsznMAx=vaR=FLEVEL=a,6
L03P=o

NOVHI : NOVAP - 1

NOVPL = NOVAR + 1

00 176 I = 1.NOVDL

00 176 J = 1.N0VPL

VECTOR‘I'J’ = C.Q

IF(NDEL)501.500

SUVNHI=C.O

IDEXINOVAR)=NCCNP1

00525N=1.INDAIA

NU“=0

00 512 1:1.NUMCON

IF!NK(I))511.512

NUM=MUM+1

IDEXIHUM)=I

CONTINUE

DATA(NCONP1.N):FOBS(M)-FCALC(N)
SUMNHTsSUMW4T+WT(M)

AVEwHTasuwwHT/NODATA

00510N=1.INDATA

IF(NUSF(N))146.510

WHT=WT(N)/AVENHT

DO 540 I = 1, NOVAR
VECT0R(I.NOV8L):VECTOR<I.N0VPL)+3ATA(IDfiX<I).H).HHT
DO 540 J = I.NOVAR
VECT0R(I.JI=VECTOR(I.J)+PATA(IDEX(I).N)*5ATA(IDEX(J),N)thT
VECTORINOVPL.NCVPL) = VECTPR(N0V9L.NOVPL) + WHT
COWTINUE

IF(INF01)580.6‘0

PRINT 1115

ropMAT(.1su4 or vAo./)

NOMAX:MOVMI

IF(NOVMI.GT.7) NOMAX = 7

PRINT 15. TI.I=1.NQMAX)

PRINT 17. VECTOR<NOVAR.NPVPL).(VFCTOR([.movpL),1:1.N0MAX)
IFcNOVMI.E0.NOMAX) 00 T0 581

WOMAX = NOVAX + 1 T NOMAX = MQMAX + 7

 

'81

~82
'99

$03

650

794

910
800

895

1001

1002

1010

75

IFINOMAX.GT.UQVMI) NCWAX = Nfivvl

PRINT 1115. (1.I=M0MAX.N0MAX)

PRINT 17. (VECTORII.POVPL). I = “OMAX.HCMA¥)
60 T0 9581

CONTINUE

NOVIN = 1 5 Iowax = NDVMI 5 NGO = 5
IF(NOVMI.GT.NGP) NOMAX = N80

PRINT 152.II.I = many». NOVA!)

DO 180 J = NONIN. mnvvl

JJJ = J
IFIJJJ.GT.NOMAX> JJJ = MOMAX IN;
PRINT 153. J, (VECTOPII.J). I = “OVIN. JJJ) ‘

PRINT 154. (VECTCRII,JCVAR). 1 = NUMIN, woMAx)
IFINOMAX.EC.VJVMI) 60 T0 603

NOWIN = NOMAX + 1 1 N60 = N60 + 5 I HOMAX 2 NOVHI
GO T0 602

COVTINUE i
DRINTISS. VECTORINfiVAR.NUVAd) g
MOSTFP = -1 E4
ASSIGN 1320 To NUMRED h
DEFR = VECTOP(NOVPI:KUVPL) ' 10G

00 800 I = 1.NOVAR

IFIVECTORII.I)) 792.794,919

PRINT 7939 I

 

GOT0172

PRINT 795. I

SIGWAII) = 1.3

00 T0 800

SIGMAII) =FSGRT (VECTOR (1.1))
VECTORII.I) = 1.0

DO 830 I = 1.NCVMI

1P1 = I + 1

00 830 J = 1P1. NOVA:

VECTOR(1.JI = VECTOR(1.J) /( SIG”A<I)~ SIGVAIJI)
VECTORIJ.I) a VECTORII.J)

IFIINFO2) 870.1001

NOMIM s 1 s NOMAK = MOVMI 5 new 2 15 w NOMINP a 2
IFINOVMI,GT.NGC) NOMAX = Nno

PRINT 159. (1.! = NONI“. NOMAXI

00 885 I = NOMINP. NOV”!

III = I - 1

IFIIIIIGT.NOMA¥) II! = NOMAX

PRINT 1600 10(VECTCR(IoJ)o -J =1: III)

PRINT 161.IVECTOP(I.NOVAR). I = “O“IN. AOMAX)
IFINOMAX.E0,NQVMI) so To 1051

NOMIN : NOMAX + 1 g HONINP = NCMIN + 1

N00 = NGO + 15 T Nomnx s NOVMI

50 TO 1602

NOSTFP = NCSTEP + 1

IF (VECTORI NJVA8.NOVAR)) 1002.100?.1013

MSTPH1 = NOSTEP . 1

PRINT 1004. NsTPMl

GO T0 1381

SIGY = SIGaA(NOVAR) *PSCRT (VECTORINOVAR.VOVAR)/ DFFR)

1017

1020

1042

1860
1080

1100

1301

8302
8303
8305

8306
8304
1320

76

DEFR =DEFR-1.3

IF (UEFR ) 1017.1517. 1020

PRINT 1019 .ICSTEP

GO ID 1381

VMIN=VWAX=NOIn=3

00 1053 I = 1.NOVMI

IF <VECTORII.I)) 1042. 1¢50. 10 c
PRINT 1044. I. NOSTEP

GO TO 1381

IFIVECTOR(I.I)9T0L) 1350. 1080. 1080
VAR=VECIORII.NCVAR)~VECTPR(NOVAR.II/VECT0R(I.I)
IFIVARI1110. 1050. 1110

NOIN = RUIN 1

INDEX(NUIN) I

COFNINOIL) VECTORII.N0VAR) * SIGMAINOVAP) / SIGMA (1)
SIGMCCILOIN) = (slay / SIGMAIII) *nsowr (VFCTDQIIoII)
1r <vm11> 1160.1179.QJ4

PRINT 906

GOT0172

VMIN = VAR

MOMIN : 1

GO TO 1053 U

IF(VAR - VMIN)1050.1050.1173

IF (VAR - V4AXI1150.1350.121P

VHAX : VAR

NOMAX 2 I

CONTINUE

IF (“OIN) 933.1240.1330

PRINT 907

GO TO 172

STDY=SIGY

GO TO 1350

IFIINF03)1312.1320

IF INCENT) 1311.1311.131?

PRINTLOBO NOSTFPo K

GO TO 1314

PRINT169. NUSTFP. K

IFILOCP) FLEVEL=FL

PRINT 162.K.FLEVEL.SIGY
D01331J=10NOIV

N=INDEX(J)
PRINT1630N3H1(N)QCOEM(J)OSIGMCO(J)
IFIJUMP) 8332.8304

IFILOOP) 8303.8304

PRINT 8305

rooMAT¢.2FINAL CONSTANTS AFTFR CORRECTIONS MERE ADDEDtl)
DO 8306 J=1.NOIN

M=Ih0EXIJI

XWENCON=COENIJI+XC0NST(N)

PRINT 163.N.N1(N).XNFNCUN.SIGMCO(J)
GO TO NUMBER.(1320.1580)

FL=FLEVEL
FLEVEszMIN*0EFR/VECTOR(IOVAR.NOVAR)
IFIEFOUT + FLEVEL) 1350. 1350. 1‘40

I! “4-

W‘ ' A' .1'1Wm

134m
134?

1350

1361
137a

1391
1392

1400
143R
1460

144a
141a

1506
1481

154a
1520

1389
1381

157a

1586

1630

1651

1652

77

K = NCMIV

NOCNT : 3

GO TO 1391

FLFVEL = VV1X * PEFR / (VEnTUR(W7VAR.H0VAR)- VMAX)
IF (EFIN - FLEVEL) 13731136111383
IF (FFIN) 1383,1389p1370

K = NOVAX

NOFET = K

IF(K) 1392.139?.14F3

PRINT 1395: KOSTFP

8070172

F0 1410 I = 1,10VAQ
IF (I-K) 1438.1410.1450
DD 1440 J = j, NPVAR

IF (J-K) 1460.1440.1463
VECTOR(I.J)=VECTOQ(I.J)-VEFTOR(I.K)tVEFTOR(<.J)/VECTOR(K.K)
CONTINUt

CONTINUE

DO 146J I = 1. NOVA?

IF (I‘K) 1500:14R301?30

VECTOR (I.K) = - VECT?R (I.K) / VECTUP ((.K)
CONTINUE

no 1520 J t 1. NOVAR

IF (J-K) 1540.1520.1R40

VECTORCK.JS = VECTOR (¥.J) / VECTOR (K.K)

COYTINUE
VECTCR(K.K) = 1.0 / VFCTDP(K.V)
80701031

PRINT 167.IUE~T(1).IPEHT(2).NOUATA.NOVNI.HPELMAX.XDEVMAX.AVENH7o

lSTDY.NOSTEP

COHTIVUE

ASSIGN 1583 TO NHWSEQ

L00P=1

GO TO 1513

CONTIAHE

IF(INF03) PRINT 1596.<L.VECT0R(L.L).L=1aNuV%I>
CALL PRINT 2

VFII=DEVMAX=SwT=XIP=C.O
001660N=1.I1DATA

NTM=NUSE(VStHT(N)

MHT=NTN/AVEAHT

YPQFD = 0.0

no 1633 I = 1.N01~

LASSIE = IPEX(INDEX(I))

YPQED = VPPED + COEVCI)*DATA(LASSIE.N)
DEV=FORS(N1-FCALC(M)-YPPFD
IF(1E.-WT(N))1652.1662.1651
VFIT=VFIT+HTN«PEV**2

XI?=XIR*1.C

SNT=SNT¢NT?1)

FRCDPRfi=FCALC(N)+YPRFD

NHTszTiAVEdHT

CALL PRINT 3(OkTA.FORS.FCALC. NT. MUSE. NCDJPl. INDAIA)
ADEV=ARSF(PEV)*SORTF(NHT)*NUSE(N)

A 1______.__.._ .—~

7R

IFCDEVHAX-ADEV)1615.1613o1‘6?
1613 NMAX=N

DEVMAX=ADEV
1660 CONTINUE

VFIT:VF1T~X1P/t(XIP-NOIN)*S~T)

STDFIT=SQRTF(VFIT)

PRIOTISC. VFIT. STOFTT

IF(NDELMAX§NDEL)17?.172.16?4
1624 IF(DEV”AX-XDEVVAXJ172.179.1690
1620 NUQE(NMAX)=3

NDEL=NDEL+1

HODATA=NODATA-1

SUMWHT:SUMkHT-kT(NMAX)

DRINT 9201.NPAX

GOTO495
172 RETURN
15 pquAT(w Yt7!17o/)

17 FOQMAT(X1P7E17.6.Eib.6.//)
1116 FORhAT(113.6117./)
150 FORMAT(*CSTAT FROM LINES qu0 LT 10.0*/¢CV¢R. =*710.6§ STD. 0E
1V =tF754/i7t)
151 FORMAT <10AO)
152 FORMATttRAw sscp MaTnlvt/5126)
153 FURMATCXI2olpSF26.10)
154 FOQMATtt Y~1P5E26.1§)
155 FORMAT(* Y vs Y*F15.4)
156 FORMAT (1311.35x12.10x15.F5.QXA6>
159 FOQMAT (i-PAPTXAL c092. COfiFF.i//1618)
160 FODFAT (13.16F8.4)
161 FOQMAT (* Y¢16F8.4)
162 FORMAT (. F LEVEL OF X-*11.E10.2.9XFSTO rev OF (O-P)5F7.4//10X*V
1AR1ARLE COEFFICIFuT STD ERROR*/)
163 FORMAT (111.A9.2F17.12)
167 FORMAT(*1LEAST SQUARFS FIT OF $268 /3 FIT wldt DATA POINTS TO .
112* VARIABLES*7t OELFTFS up T0 «I2* POTMTS TF (O-PT 18 GT *F6.3(
2. HHT NORw w F6.2/* STO PEV 0F (C-C)*F8.4/* COMPLETED i1?» STEPSw>
160 FOQMAT (*GSTEP NO*13/* VAR. REMOVEOwTO)
169 FOQMAT (*OSTFF NOwt3/r VAR. ENTEDEO*13T
701 FOQMAT (4F15)
702 FORMAT (*ABDVE NCNvEIGENVALUF NOT DIAS; AFTER 1o CYCLE9*I
793 FORMAT (* ERROR 9&8!“ SO VAR*I3* IS MEG"
795 FORMAT (*UVARtIS* IS CONSTfi)
850 FORMAT (X613.F20.F10.23X18>
906 FOQMAT (*OFRROF. VMIM pus.)
907 FORMAT (OBFRROP NOIN NFG*)
1000 FORMAT (*OLINE 00.14. DELETEDtSX613.2F10.4)
1004 FOPHAT (*SY SQUARE NF3 STEDFIS)
1019 FORMAT (rOZERO DEG FQEEDON STEPaTsa
1044 FORMAY (*SOUARE x-il?* NFGATIVF STEP*13)
1395 FORMAT (*K=D STEPwtsa
1586 FORMAT (*0 DIAG ELEMENTSv/r VAR 00 VALUE*//(I4.F12I6))
9000 FORMAT (215.?10)
9201 FOQMAT (*+LINE 00:14: REINC DELETED FROM FITt)
END

79

7/9 RUN CAWH 4F SECUVJS

TYPICAL DATA DFCK

C IDENTIFY THE ISDTUDE
15
c THE MAXIMJM NUMBER OF COTSTAVTS AND TYPICAL TOLERANCFS
9 1 1.8 2
1 1 1 c.00001 1.0 0.0

C VALUES OF CWEGA 1.2.3 AND A PAnAVETEQ 0 IF VOT VARIED 1 IF VARIED
1342.5 1
747.1 0
1078.0 1
C THE PRELIMINARY X SUP IJ VALUES WITH 1 (8) IF VARIFD (NOT VAQIFD)
-8.8 1
.004 0
.1506 1
-9.5 0
.2707 1
-2.5 0
C A BLANK CARD IS INSEPTFD HFRF
C THE FOLLOWING ARF DATA CLRHS HIT” THE FIRST THPEE INTEGEQS IDENTIFYING
5 THE UPPER STATE FOLLQAED BY TAF nAND ORIGIN AVD THF WEIGHT

1 0 1 2558.7071 1.0

2 0 1 4123.3673 1.0

O 0 3 4655.2?53 1.0

3 0 1 536703149 103

1 0 3 5874.9510 1.9

1 n 0 1336.0 C.01

C A BLANK CARD IS INSEQIFD HERE

‘ r 1.!

 

P? 6..

,80 ,

MPRQGRAM ROTCON
COHMON/lZS/NK(3O). NDELNAX. xDEVNAx

CQUMON/12/ INFO1 INEOZ INFO3; NUMCON JUMP C0DSY13011EF1N EFOUT. TOL

1c IDENT‘2)
icOMMON/A/ NAME(30)

DIMENSION CON(5 30), F0831301oFCAL613oiaflT130).NUSE(30)

A DIMENSION NU1£301.ND2(302.NUS(30)
READ 20, A, 11
20 FORMAT( A8, 12)
PRINT 21, A, 111

21 FORMAT1t THE FOLLON1N3 NOTAYIONAL CONSTANTS ARE FOR THE tABt 0F c!

12tN02t)
.100 NUMcoN a 4
NCONP1 a NUMCON i 1

NAME¢1) = 4HR000

NAMEcZ) g 6HALPHA1
NAME‘S) = 6HALPHA2
NAMEc4) . 6HALPHA3

READ 1, JUMP
PRINY 1. JUMP
FORMAT115)

1“

READ 2. co~sr¢1).coNsr¢2), CONST(3),CONST(4)
PRINT 2, c0N51(1),cDNsr(2>; CONSTt3).C0NST(4)

2 FORNAT(‘F15,6)
7 ,1 N s U
101 N I Nt1

READ 3: NU11N)aNU2(N)nNU3(N>.FOBSQN).NY(N),

(d

,FORHAT‘ 3131P10.63F10.1’
CON(1DN’ ' 1'0
CON12an EHNU1(N) W
CON‘SgN) =-NU2(N)
CON‘4,N) =-NU3(N)
IF¢FOBS(N),NE,0) GO YO 101
,,XNDATA 9 N a 1

PRINT 7. INDATA
7 FORMAT(* INDATA IS . 14)

READ 4. NK(1).NKtZ),NK13).NK(4).XDEVMAX.NDELFAX, INF01, INPDZ.

llNPQS. 70L: EFINI EFOUT
‘ F0RH‘T1‘11,5X.F10.5,41292X.3F10.5)

PRINT 10, NK‘1),NK(2).NK(3;.NK(4),XDEVMAX,NDELNAX,INF01,!NP02,

' ieros, 70L. EFIN, EFOUT
10 FORMATg1X,4l1 5x r10 5 412 2x. 3r1o. 5,
DD 120. N n 1,1NDATX

FCALC(N) I CONST(1)GCON(1. N) o CONSI‘2)tCON(2.N) ¢ CONST(3)0

160N(3,N) . CONsr(4)-DDN(4
. , PRth_a. NUi¢N2.Nu2(N).Nu5<N:. FCALCtN)
a FORMAT! 3:3 Fin. 4)
120 CONTINUE
CALL STEPF!T(CON.EDNS.FDALC NUSE. HY. NCDNR1.
READ 5. x106 NT
5 ronnnrtas)
PRINT_ 11.XIDENT
11 FDRNAT¢1x. A8)

PRINT 3. NU1<N).NU2(N).NU3(N).FDBscN).wT(N)

lNDAfA)

1mm- 81
__LPQXIDENIJEQABHNEN DATA) GO TO 109 _
END
‘___SNBRDUTINE PRINTOUT1DATAafoesAFCALCANUSEANT_.NCQNP1A1NDATA)
COMMON/125/ NK(30A.NDELNAX. XDEVNAX
coNNONIZS/NQDATA.NDVN1.AvENNT STDY.NOSTEP, _N. NTN, FREQPRD, VPRED.
1 DEV
__GQHNDN11QQIJJ1900AA KK1900Ag-JDEL19002; KDEL19OOA..NUDEL(900A
DIMENSIONDATA(NcoNP1;:NoATAA;FOBS(lNDATAA.PCALC(1NDATAA.HT(INDATAA
-1 NUSEAINDATAA
ENTRY PRINT 2
N_“PRINT166
RETURN
ENTRY EBJNT -5 _
PRINT 165. Ne KDEL¢NA. JDELCNAA KK1N). JJIN). NTN FOBS<NA. FCALCA
.1NA. FREDERD; DATAtNgONP1._ NA. YPRED. DEV .

RETURN
115 FORMAT 115.x. 2R1. 12.11H.. 12. 3x, :6. 2. 6F11. 7A
166 FORMAT ('1 No. HT 033 FREQ CALC FRED PRED FREQ
1gas.QALC 93E0.9ALC_-_065.PRED.)__ __ __“__~ H
ND

_SUBRDUTINE STEPFITIDATAAPDBSAFCALGANUSE. NT. NCONP1.TNDATA)
COMMON/123/ NK(30A.NDELNAX. anVNAx
V1E0NHON£1ZIINF01AINFOZ,lNF03,NUHCON;JUMP,_CQNST(30A. .EFAN;Erou1,AoL.
IDENT( A
M_ COMMON/za/NODATA,Nova AVEHHT. srnv. NOSTEP, N. NTN, FREOPRD, YPRED,
1 DEV
QDHMDN/A/ NANE1301
DIMENSION N1(30) VECTDR(31. 31).!NDEXt3DA IDEX(30) SIGHA(30A,
1CQEN¢30A.sxaMcoxsnA.N011N¢30A.XCONSTt30A
DIMENSIONDATlcNCONP1 ANuATAA FOBS(!NDATA). FCALccanATAA.NT¢1NDATAA
111 NuSE¢1NDAYAA _m__ _ -H
TYPE INTEGER ss vv
TYPE DOUBLE VEGTOR.S!ENA.COEN. SIGNED. SIGY.DEFR2 VAR
lrcanNfclAA 9100. 911a
”3100 IDENT11IEEHINPUT DA.
tDENTcZA a2NTA
9110 GONTTNUE m u..
!F(.NOT, EFouTA Eroutéi.E.a
1Fg.N01 EFINA EEIN=EPDUT51 E.a
IFCTDL.éG.o .0) TOLIO 001
17_5 _NOVAR . 1
DO 157 131 NUNCON
IE1NNLJA1158 15?-
158 N1¢NovAnAaNANE<AA
. 1F¢JUMPA 8155. Q159_
8158 NCONsrcNOVARAsCONsT(!A
__§122_MDXAB'NQ¥AB$1_#_N .
157 CONTANUE
NDDATA11__
00254 No1éINDATA
1H11N11___L22§_
2 3 NUSE!N)ID
80 T0 254
252 NUS§¢N3I1

“—- _ —— u—rn

 

82

_“N0DATAaNODATA91
254 CONTINUE
_ NDEL‘O
495 NOINsKaNOENT=NOMIN= NonAx=VAR=FLEVEL=o. o
LOOPeO m, _ “W
NOVH! : NOVAR . 1
NOyPL 9 NQVAR o 11
no 176 1 = 1,NDVPL
no 176 4,: 1,NOVPL
176 VECTOR¢:.J> = o. o
, IF(NDEL25011500_
500 SUMHHTsO. 0
_!DEX(NOVAR)§NCONP1
00525N=1.1NDATA
1_~uM.o
DO 512 1:1.NUMCON
___HIF(NK(1))5111512
511 NUMzNUHt1
-1 -XDEX(NUM)=LN.
512 CONYINUE
__DATA(NC0NP11N)=FOBS1N)~ECALC(N)
525 SUMNHTtSUMNHY*NT(N)
501 AVEHHTISUMNHT/NODATA
00510N31,!NDATA
.IF(NU$E(N))146.510
146 HHY:HT(N)/AVENHT
__ Do 540 I a 1& NOVAR
VECTOR(I.NOV L):VECTOR<I.NOVPL;+DAIA<105x<x;;N:.uH1
no 540 4 a I,NOVAR I
546 VECTOR(!. Jasvecroa¢x. JyooATA¢105x11).N)cDAvAcztex(J;.Ny.uH7
, “ VECTOR1N0VPL1NOVPL) a VECTORQNOVPL,NOVPL) . EH?
51o CONTINUE
-IF(INF01)5501650
580 PRINT 1115
1115 FORMAT(*1SUM 0F VARtl)
NOHAX=NOVMI
_IF(NOVMI.GT.7).NOMAX.5«Z“
PRINT 15, (I.I=1.NonAX) .
1_PR!NT 17. VECTORtuovna.NOVPL>.(vecroa<1.Nov91121s1;~cMAx>
9581 IF1NOVMI.E0,NOMAX) GO to 581
MoMAx a NOMAX * 1 5‘ VOHAX a MOMAX o 7
IF<NOMAX.GT,NOVMI) Nowax : NOVMI
PRINT 1116. (I IzMOnAx.~onAx2 _
PRINT 17. (VECTOR(I,NOVPL). x - MOMAX.NOMAX)
1-60 To 9581
581 CONTINUE
. -1- NOMXN a 1 s NOHAX - NOVM! 3 N60 3 5
602 IP¢NOVMI.GT.NGO) NOHAX s NGO
599_‘PRINT-1521‘101 ' NOHIV. NOMAX)
DO 180 J = NOMIN. NOV”!
_‘_JJJ - J _,-
IF¢JJJ. GT.NOMAX) JJJ-B-NOMAX
180__PRINTL153,_Jp,(VEcTUR(1,J), 1 . NOMIN, JJJ) f
PRINT 154, (vecroacx;vov1R). 1 s NOMIN. NOHAX)

 

—-‘

,» ____ _ W" I _. 83,
_ IFINOMAX. E0 NOVMI) so T0 605 _ _ . U
NOMIN I NOMAX 1 1 1 N00 I N00 1 5 s NOMAx-n NOVMI
Go 10 602 , I _, _ ,
603 CONTINUE
PRINT155. VECTORINONAR.NOVARI
650 NOSTEP a -1
I - I- ASSIGN 1320 To NUMBER _
DEFR - VECTORINovPL. NOVPL) - 1.0
00 800 I = 1 NOV AR
IFIVECTORII.I)) 792, 794,010
792 PRINT 793,
0070172
794 PRINT 795. I
SIGHAII) : 1.0
Go To 800
010 SIGMAII) =DSORT (VECTOR II.I))
000 VECTORII. I) = 1 0
Do 830 I = 1, NOVMI

 

IP1 a I t 1 ‘
DO 830 J = IP1. NOVAR
VECTDRI I.J) 5 VECTORII.JI II SIGMAIIIt SIGNAIJI)
830 VECTORI IJ.I) 3 VECTDHII.JI
IFIINFOZ) 8

70.1001
870 NOMIN a 1 s NOMAX : NOVHI 3 ~00 a 15 s NGMINP a 2
11602 IFINOVMI.GT.N00) NDMAX : NGO
PRINT 159. (1.! = NUMIVo NOHAXI
Do 085 I = NOMINP. Nova
III I I P 1
IFIIII.GT.N0MAX) III = NOMAX
885 PRINT 160. I.IVECTOHII.JI. J = 1. III)
PRINT 161.IVECTOR(I,NOVARI. I = NOMIN. NOMAXI
IFINOHAX.EO.NOVMI) 00 TO 1001
NOMIN a NOMAx 1 1 s NOMINP s NOMIN + 1
N60 a NGO + 15 s ~051x = NOVMI

, GO TO 1602
1001 NOSTeP n NOSTEP 1 1
._ If. IVECTDRI NOVAR. NDVARII 1002.1002. 1010
1002 NSTPM1 5 NOSTEP n 1
PRINT 1004. NSTPM1 ,
GO TO1318 I
-1010 SIGY RibegéGMAINDVARI tDSORT' IVECTORINOVARaNCVARI/ DEFR)
' I
, I: F(DEER 1 1017010171 1020 .._0,1 1. IM_ 74 _
1017 PRINT 1019 ,NO STEP
£0 10 1301
1020 VMINIVMAXINOINIO
/ DO 1050 I I 1. NOVM
IF (VECTORII. II) 1042. 1050, 1060
I 1042 ?RINT.10141 In NOSTEP _MV ,, 0_,W
GO To 1381
1060 IFIVEGTORII. IIPTOLI 1050. 1080.1080
1080 VAnavECTORII. NOVAR):VECTORINOVAR. IIIVECTORII. II
IFIVARIIIOO. 1050. 1110
1100 NOIN a NOIN ‘

.-

,1..—

_904"

1170.

'1160
__1110
1210

‘1050
'06:
1240
”1300
_1310
1311

”1313

.1311"

1301

E_8302

8303
.8305

8306
8304
1320

‘1340
1345
1350

1361
1370

”1391
f 1392

1400

84d

INDEXINOINII I.w 0,, . § 3 , _ -_m¥w __ .

COENINOIN) VECTOHII.NOVARI - SIGMAINOVAR) / SIGMA (I)

SIGMCOINOINI = ($101 I SIGMAIIII *DSORT (VECTORII.I))

IF (VMIN) 1160,1170,9ua

PRINT 906mm

GOT0172

VNIN a VAR

NOMIN I I

60 TO 1050 m_ _

IFIVAR - VMINI1050.1050.1170

IF_IVAR - VHAXI1050.1050.1210 _

VHAX a VAR um
NOHAX a I_1_ _ , _ 1 If ,,u I
CONTINUE 1
IF INOIN) 903.1240.1300 .
PRINT 907

GO TO 172_‘

STDY-SIGY 1
GO TO 1350 ,
IFIINF03)1310.1320
IF INOENT) 1311,1311;1313 1
PRINT168, NOSTEP, K

00 TO 1314

PRINT169, NOSTEP, K

IFILOOP} FLEVELsPL

PRINT 162,K,FLEVEL,SISY

DO1301J§1,NOIN

NaINOEXIJ)

PRIN116$.N.N11NI}00:NIJI.SIGMCOIJ)

IFIJUMP) 6302,8304

IFILOOPI 8303.8304 _

PRINT 8305

FORMATItZFINAL CONSTANTS AFTER CORRECTIONS HERE A0050:/)

00 8306 J81.NOIN ‘

NIINDEXIJI

XNENCONSCOENIJI¢XCONSTINI

PRINT 163.”.N1IN),XNENCON.SIGMC0IJI

GO To NUMBERII1320.15802

FL=FLEVEL

FLEVELIVHINtDEFR/VECTORIN0VAR.N0VARI

IFIEFOUT o FLEVEL) 1350, 1350. 1340

K - NOMIN

NOENT q 0 .

GO TO 1391

FLEVEL - VMAX . DEPR / IVECTORINOVAR.NOVARI- VMAX)

IF IEFIN - FLEVEL) 1320.1361;1300

IF IEFIN) 1380.1380,1370

K a NOMAX

NOENT . K _ ‘

IFIK) 1392.1392.1400

PRINT 1395, NOSTEP

0010172

00 1410 I I 1.NOVAR

IF II-KI 1430.1410.1430

1"

NY? 'I I. .' 7'

 

l

143fl_

.1460

1440
141D

'1900

11570

1

7 1480

1540
1520
138 so

1
1381

1560

__ ,. ,7 7 7 85 ,_ 7
D0 1440 J = 1. NOVAR 1
IF (J-KI 1460.1440.1460
VECTORII J’IVECTOR(10J"VECTOR(I K1*VECTOR¢K. JI/VECTORIK1K1
CONTINUE
CONTINUE
D0 1480 I = 1. No
IF (1-K1 1500,14ao,1500 ‘

VECTOR (I K1 I - VECTOR (I.K1 / VECTOR IK. K1
CONTINUE

DO 1520 J = 1. NO OVA

IF (J-KI 1540 1520 1540

VECTDRIK J1 I VECTOR (K. J1 / VECTOR (K, K1
CONTINUE I, ,,_,,
VECTORIK K) = 1.0 / VECTORIK,K1

GOT0100 01

PRINT 167,IDENTI11.IOENT(21.NOOATA.NOVMI.NDELMAx.anVMAx.AVEwHT.
STDY NOSTEP

CONTINUE

EASSIGN 1550 .70 NUMBER,H,H, I, , ,, ,VW ,11 E,,,

LOOP-1
GO.TO 1310
CONTINUE

VIFIXNFOSI PglNT 15BO}(LIVECTORIL1L11L=11NOVMI)

CALL PRINT

"VFlfaDEVHAX!SHTaXIRa0.0, , W,n,w,, ,, , ,, 11,

001660N31.INDATA
HTN!NUSE(N)'HT(N)
HHTQHTN/AVEHHT

DO 1630 I u loNOIN

__LA§§1E,a.IOEX(INOEXI111 7 ,,-E, , ,, E

686
1121

1652

J1610

_M1660

1624

, 1620

YPRED = YPRED ¢ OOEN(I1~OATA¢LASSIE.N1
DEV-EOBSIN1uPCALC(N1-¥PRED

IVT1O.-HT(N111652.1052 1551

VELI-VFIT~HTN-DEv-~2 ,

XIRIXIRt1. 0

8H 7:5 HTtHT‘N

FREQPRDEFCALCIN)OYPRED

HHTauHTtAVENHT

CALL PRINT SIDATA,FUBS,ECALC} NT, NUSE, NCONF1; INDATA)
ADEV:ABSF(DEV)tSDRTFINHT10NUSEIN)
IFTOEVMAx-ADEV11610;1610,1660

NMAX3N
DEVMAXEADEV
CONTINUE
VFIT-VFIT-XIR/Tcx
STDFITBSGRTFIVFIT
PR1NT150, VFIT, s
IF(NDELMAX- -NDEL)1
IFIDEVMAX- -XDEVMAX
NUSEINMAXI= 0
NDELaNDELol
NOOATA-NODATA-1
SUMNHT: SUMHHT-NTINMAX)

R-NOINI'SHT)

624

I
1

T

7 1
172 1620

DFIY
2117?:
172117

 

86

__PRINY 9001,NHAX

OOT0495
I172,RETURN __n_ _
15 FORMATIO v.7117./1

._"17_.FORMATIx1R7E17.6.E18.6.1/1
11116 FORMA7I11016117111
_150 FORMATIAOSTAT_ FROM LIVES HHTD LT 10 0./.0VAR _..F1o,e. sTO. DE
1V 30F7, 4/t7t)
151 FORMAT I10A81
152 FORMATIARAN SSCP MATRIx./5120)
_‘153__FORMATIX12,1PSEZO.101
154 FORMATIO 1.195E26.101
__155-FORHATIR_ Y VS YRF1>.4III
150 FORMAT 11311.35x12.10x151F5.EXA81
“159 FORHAT It-PARTIAL CORR. COEFF.A//16181
160 FORMAT (l3 16F8 .41
“.161 FORMAT_IR_ 1'16F8.41
162 FORMAT (A F LEVEL OF x-t11.E10.2,9XRSTD DEV OF (o.R1AF7 4//1OXRV
_ _ IARIABLE COEFFICIENT "m_mmm-STD-ERR0Rt/1_pumwnwmmmm "I
163 FORMAT I111.A9 2’17. 121
_167 FORMATItlLEAST SOUARES FIT OF t2A8 It FIT .14. DATA POINTS To .
112. VARIABLESA/v DELEYE§ UP To .12. POINTS IF IO-PI 13 GT th. 3/
2A HHT NORM i F6 Z/A STD DEV OF Io- -C)aF8, 4/o COMPLETED «12: STEPs.)
16B FORMAT (ODSTEP No.15]. VAR.REMOVEDa13)
169 FORMAT (ROSTER NOR!3/*__ VAR, ENTERED-I31
701 FORMAT (4F151
702 FORMAT_I.ABOVE NON-tIsENVALUE NOT OIAO. AFTER 10 CVCLESA1
793 FORMAT I. ERROR RESIO SO VARRISA Is NEG.)
795 FORHAT_(¢0VAR*I§. 15 CONST.)
050 FORMAT Ix613.F20;F10;23xA81
__906 FORMAT_(A0ERR0R,_VMIN Ros¢1mm
907 FORMAT (.0ERROR NOIN NEGRI
_1000 FORMAT (OOLINE No.14. DELETEDO5X613,2F10.4)
1004 FORMAT (.01 SQUARE NEO STERA151
1019 FORMAT I.DlERO DEG FREEDOM STEPOISI
1044 FORMAT («SQUARE x..;z. NEGATIVE STEPAISI
_ 1395 FORMAT (8K: 0 STEPAISI __ .
1586 FORMAT (.0 OIAO ELEMENISEIR VAR NO VALUEo/III4LF12.611
9000 FORMAT 1215. F101
9001 FORMAT IvtLINE No.14. BEING DELETED FROM FIT-1
END

7/9 RUN CARD 45 SECONDS

 

87

c TYPICAL DATA DECK
c THIS CARD IDENTIFIEs THE ROTATIONAchONSTANT AAD ISOTOPE
A H 14
C THIS Is A SNITCHING FARAHETER CALLED JUMP
1

c INROT THE VALUE OF THE-ROTATIONAL CONSTANT_AND ALPHA 1.2;3
c THE FOLLOHING ARE DATA.ROINTs NITH IDENTIFICATION OF THE OFFER
c STATE AND THE HEIGHT ASSIGNED To THE POINT

1 '0 1 7.85404 1.0 ' '

2 -0 1 7092649 100

3 o 1 8.01300 1.0

O 0 3 7.3427 1.0

1 0 3 7.4140 1.0
C “INSERT'A BLANK CARD HERE
0 THE FIRST FOUR COLUMNS CAUSE VARIATION OF YHE'CONSTANTS A 500 zERO,
c MALPHA 1.2. AND 3 RESPECTIVELV THE REMAINING PARAMETERS ARE USED
0 INTERNALLY AND ARE sET TO TYPICAL VALUES
0101 "'0.0002 ”1 2 1 1 1 0.0001 ' 0.0 0.0
c

INSERT A BLANK CARD HEREI

a: ‘41.. v Fla-Rn."

 

 

X

APPENDIX 1!

FREQUENCIES FOR THE (0.0.31 BAND OF 14N02

nun‘on‘onflnnnnnuv‘onvunnnuunufiuvv‘ouunvnn‘onnnv
MNNNMNNHHHJ—l-HHHHHHHHHHHHHHHHHHHOOOODO00000?)

063 O-P I. K N 085 O-P NT
4696.466 ~0.004 0.3 P 0 46 4699.567 -0.002 0.3
4702.626_ 0.000 0.3 P 0 42 4705.616 0.011 0.0
4708.529 0.005 0.5 P 0 36 4711.365 0.004 0.8
4714.184 0.004 0.6 p 0 34 4716.913 .0.004 0.6
4719.597 0.003 1.0 P 0 30 4722.212 0.000 1.0
4724.769_ 0.001 1.0 P70 26 4727.266_ 0.003 1.0.
4729.699 -0.002 0.5 P 0 22 4732.064 0.007 1.0
4734.396 -0.004 1.0 P 0 16 4736.654 0.005 0.5
4736.643 -0.001 0.3 P 0 14 4740.960 0.002 1.0
4743.055 0.002 1.0 P 0 10 4745.067 .0.001 1.0
4747.020 0.001 0.6 P 0 6 4746.913 0.004 0.5
4750.737 .0.001 0.6 _ P 0 2 4752.505 .0.000_0.6
4694.569 0.018 0.3 P 1 47 4697.915 -0.000 0.3
4697.752 0.011 0.3 P 1 45 4700.997 0.026 0.0
4700.693 0.026 0.0 P 1 43 4703.946 .0.012 0.3
4703.946 _0.019 0.3 P 1 41 4706.669 .0.003 0.3
_4706.933 0.006 0.5 P 1 39 4709.761 -0.001 0.6

_4712.576 _0.006 0.3 p 1 36 4712.736__0.004 0.5
4715.316 -0.001 1.0 P 1 34 4715.527 -0.011 0.0
4716.009 -0.004 0.5 P 1 32 4716.294 _0.013 0.5
4720.630 -0.002 0.6 P 1 30 4720.964 0.003 1.0
4723.202 0.004 1.0 P 1 26 4723.574_-0.004 1.0
4725.705 0.002 1.0 P 1 26 4726.134 0.004 0.6
4726.146 0.001 1.0 P 1 24 4726.619 -0.001 1.0
4730.539 0.009 0.6 P 1 22 4731.043 -0.004 0.6
4732.661 0.006 0.5 P 1 20 4733.411 _0.000 0.6
4735.121 0.006 0.5 P 1 16 4735.713 0.002 1.0
4737.307 -0.009 0.3 P 1 16 4737.947 .0.001 1.0
4739.456 0.000 0.6 P 1 14 4740.123 0.002 1.0

_4741.534 -0.001 1.0 R 1 12 4742.230 .0.002_1.0
4743.554 0.002 0.5 P 1 10 4744.260 0.001 1.0
4745.510 0.000 1.0 P 1 6 4746.265 -0.003 0.5
4747.406 -0.000 0.6 P 1 6 4746.162 .0.000 0.6
4749.236 -0.002 1.0 P 1 4 4750.040 _0.002 1.0
4751.009 -0.004 1.0 P 2 46 4692.965 0.003 0.0
4696.166_ 0.011 0.3 P 2 45 4696.195 -0.003 0.5
4699.301 0.016 0.3 P 2 43 4701.230 -0.012 0.3
4702.352 0.005 0.0 P 2 41 4704.231 0.006 0.0
4705.337 -0.005 0.3 P 2 39 4707.137 -0.005 0.5
4706.267 -0.003 0.5 P 2 37 4709.996 -0.001 0.6
4711.135 0.002 0.5 P 2 35 4712.781 -0.009 0.6

_4713.925 -0.005 0.6 p 2 33 4715.527 0.006 0.5

 

'UTJU'UYJU‘UT)U‘UVJU‘UTlv‘DTIW'DYJV'UTin‘U1JU'UTJW‘UTJ”‘07)U‘DTJU‘DTJU'DT)U‘DTIU'UYJU‘DT3U

32
30
28
26
24
22
20
18

14
12
10

46
46
44
42
40
36
36
34

30
28
26
24
22
20

16
14
12
10

45
43
41
39
37
35
33
31
29
27
25
22
20
18
16
14

2Ath.§h-A.bALblshhb¢bhmbJSGCJOJM(AOIMCdOHNOiHCADI“CdO‘GL‘OLquOIMPONDNPUNHUR)NFONMUflJNTU

18‘

4716.660
4719.336
4721.927
4724.469
4726.942
4729.352
4731.694
4733.974
4736.165
4736.340
4740.427
4742.450
4744.410
4746.299
4746.140
4690.169
4693.324
4696.403
4699.434
4702.393
4705.290
4706.122
4710.692
4713.599
4716.236
4716.621
4721.337
4723.794
4726.165
4728.514
4730.762
4732.963
4735.121
4737.207
4739.223
4741.176
4743.055
4744.663
4690.605
4693.633

'4696.590

4699.511

4702.352‘

4705.141
4707.667
4710.522

4713.119'
4715.656_

4716.135
4721.731

4724.056'
4726.311_

4726.514
52391644

“00003

00006
.00006
~0.002
'00003
'00001
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f0p003
-0.007
-o.003
.0.003
70.001

0.000
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0,006
’0'016
“00013
.01021

'-0.013

“0.013
.09012
-0.011
'00009
-0.007
'00010
-0,006
'00007
‘0.004
'0.004
-0.004

‘-0.001

.09004
.0.008
0.000

-0.001’

'09000
“00015
“00016
0.034
_0.026
0.010
0.021
0.013
0.017
0,020
_0.013
0.011
0.010
0.012
0.009
0.012
0.007
0.011
0.004

UCGCQCGGC

H

DHHDOOPOODCOCOOCODHHQOOHHO1DHHPHODCDODDC‘OQOHHHOHHHDHHQH

-h£bb<béhbd>b-§£>Ame>AHbJ>ACdOHH(HOHH(ROHUCdOJUCNOHNCNOHNCHOKNOJNFURJNWORHUflJNHURJNFOfl)N

‘01)?'DTJV'U1JTVUWJV7013U"013"UTJU‘UTJV'UWJU'D1JV‘OTJU‘U1317C133'CYJU'U1TU'DTJTIU‘DTJV‘UWJ

4716.164”

4720.791
4723.338
4725.819

’4726.237

4730.598

‘4732.691

4735.121
4737.307
4739.415
4741.460
4743.440
4745.366
4747.226
4749.042
4691.600
4694.913
4697.969

'4700.944

4703.672
4706.734
4709.530
4712.267
4714.934
4717.546
4720.095
4722.579
4724.996
4727.362
4729.656
4731.694
4734.060
4736.165
4736.222
4740.205
4742.132
4743.965
4669.063

4692.124

4695.130

4696.062"

4700.944

”4703.757

4706.509
4709.196
4711.631
4714.396
4716.913
4719.336
4722.695

4725.192'

4727.416
4729.566
4731.694

“00005
'09005
“00002
701003
n0.005
u0,002
-0.004
-0.009
0.005
0.003
0.001
un,004_

:0.001

.0.002
0.016
200012
-0.012
01°l4_

'60.016

-0.012
.0.009
30.010
-0.006
-0.010
$0,006
90.003
00.003
-0.007-
00.001
.0.001
0.001
.0.006
0.009
90,001
.00004
0.000
“00007
-0.034
0.026
0.030
0.020
_0.022
0,019
-0.016
06012
0.015
0.012
_01921
'0.002
_0.004
0.010
_0.005
0.009
0.006

 

0Acratawracaumacrocac30c::M3c39c1rtuwfihuchh-o(aame:c>ozacrocycma+4HJAwMom3HbohaherAwsa!
.. - C O C C I. O i. O p C . C . C O C D O D C . I b O . O . . - O . O 1. . O . . . O 1. . C b . . O O O C O Q .

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OU‘DU‘O'UTITJTJ'U‘OU'D'UTJ’O'U‘JUUU‘O'UU‘U‘U‘J‘U‘JU'O‘UTJ'U‘J‘U‘UDW'U'DU‘UTJUU’J‘JU‘D‘JU'D’J
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12
10

42
40

36
36
34
_32

30
26
26
24
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20
16
16
14
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41
39
37
35
33
31
29
27
25
23
21
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17
15
13
10

(I)

37
35
33
31
29
27
.25
23
21
19
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11
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4732.721
4734.735
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4710.324
4712.602
4714.627
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4722.119
4724.101
4662.044
4664.793
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4692.661
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4700.014
4702.352
4704.596
4706.630
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-4754.215

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37
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21
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9
42
40
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36
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32
30
28
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9

7
36
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32
30
28
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‘4696.590

4733:737
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‘4737.635

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4699.369
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4712.380
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4717.153
4719.446
4721.689
47239851
4725.965
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4682.738
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4691.289
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4706.794
4709.174
4711.468
4713.718
4715.918
4718.041
4721.123
4723.109
4725.046
4683.419
4686.145
4688.773
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UHm\flm\nUHflUL535AJbb.thbJ>AJ>QMNOdUCAUCHQKNOKAOHMCdOJQMKOHURJNIU

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5992.669
5991.931
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5984.819
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5990.326

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5987.093
5986.166
5985.190
5984.117
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5984.621
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40
36
32
28
24
20
16
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39

FREQUENCIES FOR THE (10003)

088

5824.847
5831.387
5837.575
5843.508
5848.972
5854.137
5858.977
5863.464
5867.629
5871.464
5826.289
5830.626
5833.768
5836.794
5839.789
5842.658
5845.469
5848.203
5850.847
5853.403
5855.867
5858.264
5860.569
5862.776
5864.929
5866.990
5868.963
5870.867
5872.676
5829.327
5832.414
5835.457
5838.414
5841.263
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5853.197
5855.617
5857.944
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5862.359

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APPENDIX V

2:
A
7:

1.50.1...O..’-...1..1.'.i.”.C. .i

t.
otDUanflOWfiOJQCDOWMLd0H3ouOCDCMDCDCDO(DUHDCMUCDQHwOHGCDUHNOHW\NUHD(3

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IURHURHURJNhONFONHUhJNF‘H+‘HE‘PF‘H+‘P+‘PF*HWJH14FHDC)OCDCCDO(DOW:

BAND OF 15N02

30

CBS

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58341548
5840.612
58463257
58511610

. 58560605

58610267
5865.601
58693595
58731255
5829.543
58321710
58353794
58380799
5841.723
58441540
58473287
58490957

5852.530'

58551006
58573416
58593714
58613956
5864;101
58661163
58680161
58703037
58711864
5827;699
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5833.962
5836.946
5839;865
5842;696
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5867.402
5869.314
5828.272
5831.308
5834.326
5837.221
5840.047
5842.771
5845.469
5848.012

5850.506-

5852.912

5855.244-

58570476
58590655
58610733
58630728
58650645
58240454
58290003
58330370
58360189
58380946
58410600
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58470908
58500284
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5856.903

5858.977-

5860.924

5824.291”

5827.208
5830.069
5834.206
5836.845
5839.434
5841.935
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5831.885_

5833.640
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5838.521
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5874.287
5873.970
5869.954

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34
32
30
28
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14
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33
31
28
26
24
22
20
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13
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33
31
29
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24
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24
22
20
18
16

V001!“

5866.429
5868.348
5826.713

58290813_

58321844
5835.794
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5841.426
5844.112

5846;739_

58490265
5851.714
58543100
58560372
5858.573
5860;698
58621776
58641693
58224897
5827.488
58300455
5834.794
58371575
58400285
58423897
5845.369
58496105
58513436
58533690
58551867
58573944
58590940
58220779
58250732
58280643

58320844_

58350537

5838;146_

58400690
58431120
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5849.979
58520139
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GOGGOOQCGAOOOQOOOOOCIGIOODGCGDOQOOGOOOGQOCH0000000000009O
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oououuuuoouoouuouuJluoaouoaammo-mmomuuomomnmormounmaouomm

5890.448
5890.313
5888.314
5886.867
5885.035
5882.830
5880.276
5877.358
5890.448
5890.064
5888.477
5887.912
5887.288
5886.555
5885.713
5884.777
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5882.725
5881.578
5879.816
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5883.795

5882.592”

5881.700

5880.754—

5879.164
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34
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29
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36
34
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31
29
27
25
23
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13
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27
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58900448
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5886;693
5885.773
58843731
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FREQUENCIES FOR THE (3.0.1)

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