THE MICROWAVE ROTATIONAL SPECTRUM OF
METHANE IN THE GROUND VIBRONIC STATE
by
CRAIG WARD HOLT
B.A., Northwestern University, 1963
M.Sc, University of California at Berkeley, 1966
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in the Department
of
Chemistry
We accept this thesis as conforming
to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
April, 1976 In presenting this thesis in partial fulfilment of the requirements for
an advanced degree at the University of British Columbia, I agree that
the Library shall make it freely available for reference and study.
I further agree that permission for extensive copying of this thesis
for scholarly purposes may be granted by the Head of my Department or
by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of
The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5
Date I 3 ABSTRACT
The microwave rotational spectrum of ground state methane has been observed for the first time. Seven AJ=0 Q-branch transitions have been observed between 7.8 and 20 GHz with peak absorption coefficients < 6 X 10"11 cm-1: these transitions are the weakest ever observed with a Stark-modulated waveguide absorption microwave spectrometer.
The spectrometer employed bolometer detectors in a source- noise cancellation scheme. The spectrometer signal was integrated by a signal averager with integration times ranging up to one week. Special experimental techniques are discussed in detail. The bolometer time constant is calculated from hot-wire anemometer heat loss data.
The theory required to perform the experiment is presented using the octahedral group 0^ and its rotational subgroup 0 along with the tetrahedral group T^. An understanding of the groups employed by various workers in the field removes many of the contradictions between them. The symmetric top rotational wavefunction parity introduced by Wang is used to obtain the correct total wavefunction parity from the electronic, nuclear spin, vibrational and rotational wavefunctions.
The one quartic tensor distortion constant D,p = 132943.41
+ .71 Hz, the two sextic constants H4T =-16.9839 +.0076 Hz
and H6T = 11.0342 +.0086 Hz, and for the first time the three
octic distortion constants L^T = (20.07 +.24) X 10"" Hz,
1 h L.T = (-26.77 +.35) X lO" * Hz and LRT = (-30.0 +1.8) X I0~ Hz iii have been determined using the seven transition frequencies measured here, the J=2 ortho-para splitting known for methane and the two J=7 Q-branch transitions measured earlier in an
infra-red laser - microwave double resonance experiment. The
above errors are standard deviations given by weighted linear
least squares analysis. The estimated absolute errors are
also given. Term values are presented which allow the
accurate calculation of all ground state splittings of methane
up to J=21. iv
TABLE OF CONTENTS
CHAPTER Page
1. INTRODUCTION 1
2. THEORY OF THE MICROWAVE ROTATIONAL SPECTRUM OF METHANE • 4
2.1 Introduction and Historical Development ... 4
2.2 Symmetry of the Rotational Wavefunctions
of Methane 8
2.3 The Rotational Hamiltonian of Methane .... 17
2.3.1 Fourth Degree Tensor Hamiltonian ... 22
2.3.2 Sixth Degree Tensor Hamiltonian ... 23
2.3.3 Eighth Degree Tensor Hamiltonian ... 24 2.3.4 Exact Evaluation of the Tensor Hamiltonian Eigenvalues 26
2.4 Dipole Moment 27
2.5 Selection Rules 29
2.6 Stark Effect 30
2.7 Physical Interpretation 31
2.8 Intensities of Transitions 33
3. EXPERIMENTAL APPARATUS 35
3.1 Introduction 35
3.2 Stark Modulation 36
3.3 Stark Cell 40
3.3.1 X-band 41
3.3.2 Cell above X-band 41
3.4 Frequency Stabilization and Control 43
3.4.1 Microwave Reference Frequency .... 44
3.4.2 Stabilized VHF Reference Oscillator . 44 V
CHAPTER Page
3.4.3 Stabilized Microwave Oscillators ... 46
3.4.4 Frequency Determination 47
3.5 Time Averaging 48
3.6 Noise Minimization 55
3.6.1 Stray Pick-up 55
3.6.2 Detector to Preamplifier Impedence Match 56
3.6.3 Minimization of Microwave Oscillator Noise 57
3.6.4 Noise Cancellation 59
3.6.5 Cell Noise 61
3.7 Bolometer Detectors 64
3.7.1 Bolometer Noise 64
3.7.2 Power Saturation 66
3.7.3 Conversion Gain 70
3.7.4 Bolometer Time Constant 74
3.7.5 Bolometer Matching Impedence 76
4. EXPERIMENTAL CONDITIONS AND RESULTS 78
4.1 Introduction 78
4.2 Initial Search 79
4.3 Second Experimental Phase 80
4.4 Final Experimental Phase 88
4.5 Experimental Conditions 90
4.5.1 Line Strengths 96
4.5.2 Sweep Times 96
4.5.3 Stark Fields 97
4.6 Line Identification . 98 vi
CHAPTER Page
4.7 Determination of Distortion Constants .... 100
4.8 Influence of Dectic and Higher Order Distortion Constants . 105
5. CONCLUSIONS 115
APPENDIX A BOLOMETER TIME CONSTANT 119
APPENDIX B THE SYMMETRIC TOP ROTATIONAL WAVEFUNCTION PARITIES 126
BIBLIOGRAPHY 129 vii
LIST OF TABLES
TABLE Page
I Character Table for the Full Molecular Symmetry
Group of Methane 11
II Reduction of the Irreducible Representations of
0^ into the Subgroup T^ and the Rotational
Subgroup 0 : 12
III Reduction of the Representations and of r g u the Full Rotation-Inversion Group onto T^ .... 14
IV Reduction of the Representations and of r g u
the Full Rotation-Inversion Group onto 0^ .... 15
V The Allowed Symmetry Species of the Wavefunctions
of Methane in the Ground Vibronic State 18
VI N610B/38B7 Bolometer Conversion Gain 71
VII The Product G-Prf 72
VIII Relative Bolometer Signal Output Power Given by
the Product G«Prf 73
IX Predicted Bolometer Behavior with Increased
Bias Current 74
X Bolometer Operating Values near 200 ohms 77
XI Experimental Frequencies of X- and P-band AJ=0
Transitions of 12CHit in the Ground Vibronic State
together with Predictions based on the Distortion
Constants DT, H^T, and HgT of Curl and Tarrago . . 82
XII Experimental Conditions 95
XIII Centrifugal Distortion Constants of 12CIU in the
Ground State 101 XIV The AJ=0 Transitions of Ground State ^CH* . . . . 103 viii
TABLE Page
XV Frequency Contributions of the Terms Associated
with the Distortion Constants Listed in Table XIII
to the Transition Frequency 106
XVI Predicted Transition Frequencies based on the
Distortion Constants of Tarrago et al and the
Distortion Constants determined here 108
12 XVII Tensor Distortion Energy ET for C1U in the
Ground State Ill
XVIII Bolometer Heat Dissipation 121 ix
LIST OF FIGURES
FIGURE Page
3.1 Spectrometer used between 7.8 and 18 GHz with
BWO as Source and Single Ended Detection Scheme . 37
3.2 Spectrometer used between 18 and 24 GHz with
Klystron Source and Balanced Noise Cancelling
Detection Scheme 38
3.3 HP-K15-8400B Microwave Spectroscopy Source
Configuration used at X- and P-bands 45
3.4 Improvement in Signal to Noise Ratio Resulting
from the Summation of Two Separate Three Day
Runs to give a Six Day Run 50
3.5 Frequency Analog Voltage Conditioning Amplifier . 53
3.6 Bolometer Equivalent Circuit given by Cohn ... 76
4.1 512 Sample minus 512 Baseline 51.2 second
Scans to give the J=12 E1 -»• J=12 E2 Transition
of Ground State 12CR* 83
12 4.2 The J=12 Rotational Energy Levels for CHH ... 84
4.3 512 Sample minus lk 1024 Baseline 51.2 second
Scans to give the J=13 E1 -> J=13 E2 Transition
of Ground State 12CH-t 85
4.4 1024 Sample minus 1024 Baseline 51.2 second
Scans to give the J=14 E2 -»• J=14 E3 Transition
of Ground State 12CH-» 86
4.5 512 Sample minus 512 Baseline 51.2 second
Scans to give the J=15 E1 + J=15 E2 Transition
of Ground State 12GHi» ." 87 X
FIGURE Page
4.6 384 Sample minus 384 Baseline 51.2 second
Scans to give the J=14 E1 J=14 E2 Transition
of Ground State .^CHi, 91
4.7 The J=14 Rotational Energy Levels for .12CHi» ... 92
4.8 3328 Sample minus 3328 Baseline 51.2 second
Scans to give the J=16 E2 -»- J=16 E3 Transition
LZ of Ground State CHK 93
4.9 3072 Sample minus % 4096 Baseline 51.2 second
Scans to give the J=18 E2 •> J=18 E3 Transition
of Ground State 12C1U 94 xi
ACKNOWLEDGEMENTS
The work reported in this thesis was completed in the
Department of Chemistry at the University of British Columbia under the patient and able direction of Dr.M.C.L. Gerry.
I would like to thank Dr. Gerry for his emphasis on fair-play, his pleasant manner, his optimism (especially throughout the long "initial search period"), his readiness to listen to new ideas, and the large degree of independence allowed all my work.
Dr. Irving Ozier is gratefully acknowledged for presenting the problem treated in this thesis as well as for the inexhaustible enthusiasm which he gave to the experiment.
The authors of various disciplines cited in the
Bibliography are to be credited for the indispensable work they performed laying the foundation upon which this experiment rests.
For the use of critical equipment during various stages of the experiment Dr. M. Bloom, Dr. A.V. Bree, Dr. J.E. Eldridge,
Dr. J.B. Farmer, Dr. D.C. Frost, Dr. W.N. Hardy, and
Dr. C.A. McDowell are given thanks. The many discussions and constant encouragement given by Dr. Walter N. Hardy have been quite useful.
I would like to express my gratitude to the University of
British Columbia for extending a Graduate Fellowship to me for several years and the National Research Council of Canada for a bursary.
My wife and three children receive special appreciation xii for accepting a Spartan existence, forgoing many conveniences and pleasures for a considerable period of time.
Dr. and Mrs. F. Clark White are thanked for their liberal financial support over the years which became indispensable during the last.
Thanks are given Dr. Robert F. Curl for the communication of his experimental results prior to publication, a post-doctoral fellowhip and much patience during the completion of this thesis. 1
CHAPTER 1
INTRODUCTION
Microwave spectroscopy is a well established field of research [25,26,91,93,97,105] which allows the determination of, amongst other things, bond distances and angles, molecular dipole moments, nuclear quadrupole couplings, barriers to internal rotation, nuclear magnetic moments and nuclear quadrupole moments [25,26]. The field is characterized by very precise frequency measurements. The frequency of a molecular transition is typically determined with a precision of one part in a million. This precision allows the above determinations to be done accurately.
The transitions observed in microwave spectroscopy occur between rotational levels of a molecule. The pattern of the levels is dictated by the molecular shape. These allow classification of the molecules into various types: asymmetric rotors, in which all three principal moments are unequal; symmetric rotors, which have two equal principal moments; linear molecules, which are actually a subset of symmetric rotors; and spherical tops, in which all three principal moments are equal.
The observation of a molecular pure rotational spectrum requires the molecule to have a permanent electric or magnetic dipole moment [26], Most molecules possess a permanent electric dipole moment; these include linear molecules without 2 a center of symmetry, non-planar symmetric tops, and virtually all asymmetric tops (without a center of symmetry).
A few molecules, such as 02 , NO, N02, C102 and NF2, possess large magnetic moments which permit rotational transitions to be observed. All diamagnetic spherical tops, on the other hand, possess zero electric dipole moment in their equilibrium configurations, and until recently were believed to possess no pure rotational spectrum.
This thesis will deal with the rotational spectrum of the spherical top methane, CHi*. It was shown recently by Watson
[100] that, contrary to popular belief, spherical tops without a center of symmetry, and in particular those having tetrahedral symmetry like CHi* or CCTit, can develop a very small dipole moment in its ground state. Such a moment is induced by centrifugal distortion, and may be thought of in simple terms as arising by rotation about one of the chemical bonds, such as the C-H bond in methane. On the other hand molecules like SF6, which do have a center of symmetry, cannot develop such a dipole moment. The existence of a small dipole moment in the tetrahedral molecules led to the possibility of observing pure rotational transitions. Watson [100] considered in particular the case of methane, and predicted that certain transitions which occur at microwave frequencies should have peak absorption coefficients of up to 4.4X10"11 cm"1 (ie. that 4.4 X10"11 of incident radiation should be absorbed per centimeter).
Sugden and Kenney [93] give the practical minimum absorption coefficient which can be detected with a 3 conventional Stark modulated microwave spectrometer to be in the region of 10~9 cm-1. Townes and Schawlow [97] state the limit of sensitivity for such a spectrometer to be from
10"9 to 10~8 cm-1 in typical cases. Long [54], however, had reported the observation of a line with a peak absorption coefficient of 1.33X10"10 cm"1 at dry-ice temperature. This was the weakest line ever reported prior to this experiment using techniques similar to those used here. Since the initial methane line had a calculated intensity only three times weaker than that of the line observed by Long et al [54], the methane experiment seemed to have a reasonable chance of success and was thus undertaken. The result was the detection
of lines with a peak absorption coefficient, ym , of r r max
2X10"11 cm"1 at a signal to noise ratio of three. These were clearly the weakest ever observed using a Stark modulated spectrometer.
In the work to follow, chapter 2 contains a discussion of the theory needed to perform the experiment. Chapter 3 contains a detailed account of the apparatus used in this experiment. Chapter 4 lists the experimental conditions used and discusses the results obtained. The concluding chapter 5 lists possible future uses of the results obtained here.
Appendix A contains interesting and useful bolometer detector calculations. Appendix B deals with some aspects of parity and its application to methane. 4
CHAPTER 2
THEORY OF THE MICROWAVE ROTATIONAL SPECTRUM OF METHANE
2.1 Introduction and Historical Development
A rotational energy level of a tetrahedral spherical top
such as methane, characterized by the total angular momentum
quantum number J in the ground state, has a field free
degeneracy of (2J+1)2 in the rigid rotor approximation if nuclear spin statistics are ignored. This degeneracy is the
product of a (2J+1) K-degeneracy corresponding to the number
of ways the total angular momentum, J, can be oriented in
the molecular framework and a (2J+1) m-degeneracy corresponding
to the number of ways J is oriented in space upon application
of an external field. The rotational energy of a rigid
methane molecule is given by [31]:
Ej = BoJQJ+1) (2.1)
where Bo = h2/8iT2I is the rotational constant in Hertz, h is
Planck's constant and I = ?mir2 is the moment of inertia [26]
where mi is the mass of the i-th particle and ri is the
perpendicular distance from the axis of concern to the i-th
particle.
The rigid rotor degeneracy is partially lifted by
centrifugal distortion. For a given J level of methane in the
ground state, the components of different symmetry species
(sec 2.2), K, under the molecular point group are affected
differently by centrifugal distortion, resulting in a splitting 5
of the rotational levels. This ground state rotational
splitting for a tetrahedral molecule is due to spherical
tensor operators, tt^, in the Hamiltonian where Z gives the tensor rank [84]. This ground state splitting was first
considered in detail by Hecht [30] in 1960. The tensor
splitting operator, Qn, considered by Hecht was fourth
degree in J. With the inclusion of a scalar operator fourth
degree in J he showed that the energy levels become [30]:
2 2 Ej k = BoJ(J+l) - DSJ (J+1) + DTf(J,K) (2.2) where Dg and D^ are quartic scalar and tensor distortion
constants respectively in Hz and the eigenvalues of Slit,
f(J,<), are splitting functions which vary with J and the
symmetry species K. The term in Dg collectively shifts all
the components of a given J the same amount. The term in Drj,
splits a rotational level characterized by J into components
with a total statistical weight of (2J+1) if the components
of symmetry species (sec 2.2) A, E and F are assigned the
weights 1, 2 and 3 respectively. This splitting corresponds
to a removal of the K-degeneracy described above. The splitting
function, f(J,<), which cannot be given in closed form, was
first tabulated by Hecht [30] who also gave the value
Drr, = .12 MHz.
It was assumed methane displayed no pure rotational
spectrum in the ground vibronic state because its high 1
degree of symmetry in the equilibrium configuration ruled out
a permanent dipole moment. In 1971, Watson [100], Fox [21]
and Aliev [1,2] proposed a mechanism by which a molecule
with no permanent dipole moment, but without a center of 6 symmetry, acquired one in the ground state by centrifugal distortion. This dipole moment, represented by the constant
Qxy, together with the tensor splittings of Hecht, paved the z way for the observation of pure rotational spectra of tetrahedral molecules described in this thesis. Also in
1971, Ozier [71] determined the value of Watson's dipole moment factor 0X^ to be 2.41X10"5 Debye by means of a molecular beam experiment.
Transitions due to the ground state centrifugal distortion dipole moment of methane appear in two regions: the AJ=1,
R-branch transitions appear in the far infrared and the AJ=0,
Q-branch transitions appear in the radio frequency-microwave region. Far IR, R-branch pure rotational transitions of methane have been observed by Rosenberg et al [89], Rosenberg and
Ozier [88] and Cole and Honey [14]. The term values of equation (2.2) lead to transition frequencies:
3 Vj = 2B0(J+1) -4Dg(J+l) (2.3)
This spectrum [88] has given the rotational constant
Bp = (5.245±.004) cm"1 and the scalar distortion constant
Dg = (1.19±.09) X10~" cm-1 for methane. Similar R-branch
transitions have been observed for the tetrahedral molecules
silane [85,87] and germane [75,86] by Rosenberg and Ozier.
One AJ=0, Q-branch J=2 splitting was determined by Ozier in
1971 [70].
In 1972 Dorney and Watson [17] extended Hecht's splitting
function f(J,<) to J=20 and calculated the line strengths and
first-order Stark coefficients for the E rotational components 7 of methane. With accurate splitting functions, line strengths and Stark coefficients [17] and a good value of D^, derived
from the J=2 splitting of methane by Ozier [35,70], the experiment which is reported here was begun.
The accurate measurement of two J=7 Q-branch transitions of ground state methane in a microwave-infrared laser double resonance experiment by Curl et al [16] and Curl [15] demonstrated the need for further refinement of the ground state theory of tetrahedral molecules. This was done to sixth degree in J by Kirschner and Watson [45] and Moret-Bailly
[63,65]. Using the notation of Kirschner and Watson [45],
the rotational energies to sixth degree in J are:
E = B J J+1 2 2 3 3 J K ° ( ) - DSJ (J+1.). . + HgJ (J+l)
+ [DT + H4TJ(J+l)]f(J,K) + H6Tg(J,K) (2.4) where Bo, Dg, D,j, and f(J,<) are as previously defined, Hg
is a sextic scalar distortion constant in Hz, is a sextic
tensor distortion constant in Hz associated with the fourth rank spherical tensor operator oli» which gives the eigenvalue f(J,K), and H^ is a sextic tensor distortion constant in Hz
associated with the sixth rank spherical tensor operator fi6, which gives the eigenvalue g(J,<) by first-order perturbation
theory [45]. Values of the sextic tensor distortion constants
introduced by Kirschner and Watson, H^T and Hg^,, along with a new value of D,p, were determined by Curl [15] ; these were of
considerable value in defining the search regions employed in
the experiment reported here.
The first four AJ=0, Q-branch microwave transitions of 8 methane observed in this work showed the need to carry the ground state theory of tetrahedral molecules to yet higher order. Accordingly the theory was extended to eighth degree in J by Ozier [72], Michelot, Moret-Bailly and Fox [59],
Hilico and Dang-Nhu [33] and Watson [102]. The eighth degree theory combined with the seven microwave transitions [35] reported in this thesis, the J=2 splitting determined by
Ozier [71] and the two J=7 transitions observed by Curl [15] permitted the determination of the six distortion constants
Drj,, H^, Hgrp, L^, Lgrj, and LgT reported here (table XIII) and allowed an accurate prediction of all AJ=0 ground state methane rotational transitions to J=21 (table XVII).
The discussion thus far has concerned a centrifugally induced dipole moment which is the principal contribution to the dipole moment in the ground state of a tetrahedral molecule. A similar mechanism described by Watson [100] gives rise to a dipole moment perpendicular to the figure axis of
symmetric tops such as PH3 and AsH3 in addition to the strong dipole moment along the figure axis. Transitions due to this centrifugally induced dipole moment perpendicular to the figure axis have been observed by Chu and Oka [12].
2.2 Symmetry of the Rotational Wavefunctions of Methane
The parity assigned to a rotational wavefunction together with the molecular point group used governs the symmetry species designations used to classify the rotational levels. In addition to being a basis of the nomenclature used, parity is important here because it helps to explain the 9 selection rules and the first-order Stark shift of the
E levels upon which the success of this experiment largely depended.
The parity of a state is given by the quantum number ±1 which results from the application of the inversion operator
[104], i, to the total wavefunction:
^total - ^total <2'5>
i i i The inversion operator is given by X •+ —X, Y -»• —Y, Z •*• — Z.
Parity is important because the Hamiltonian of an isolated system commutes with the inversion operator so that parity is conserved. Moreover, since parity is a property of field-free space, it is always a good quantum number under this condition
[58].
It is universally accepted that methane has tetrahedral symmetry, and thus belongs to the point group T^ [96]. The operations of this group, besides the identity, consist of eight three-fold rotations, three two-fold rotations, six plane reflections and six rotation-reflections [96]. Various groups have been used to classify the rotational wavefunctions of methane. For a time [31,103] the rotational wavefunctions of a molecule of point group T^ were classified by only the species A, E and F of the rotational subgroup T of T^.
Anderson and Ramsey [4] used the full T^ point group and gave the rotational wavefunctions of methane the species A2, E and
Fi. Herzberg [31] pointed out that if inversion of a tetrahedral molecule were considered, the proper group to use was 0, with the rotational subgroup 0. Under 0 the rotational 10
wavefunctions assume the species Ai, A2, E, Fi, and F2.
Hougen [36] has introduced a full permutation-inversion group for methane based on the ideas of Longuet-Higgins [55]. The character table for these groups is given in table I. The operations to the left of the dividing line all consist of rotations and are "feasible", whereas those to the right of the dividing line invert the molecule and are thus
"non-feasible". It is assumed throughout the thesis that at equilibrium the full permutation-inversion group [36] is identical to the point group 0^ and that the subgroup of all
"feasible" operations is identical to the rotational subgroup
0 of 0^ [31]. Hougen [37] has made the identification between the "feasible" operations and rotations without inversion: this supports the use here of the rotational subgroup 0 of 0^ as an alternate to the somewhat more confusing designation "isomorphic T^". The identification of the group 0 with permutation-inversion T^ resolves the
"contradictions" between the work of Hougen [38] and those of
Anderson and Ramsey [4] and Moret-Bailly [64]. This resolution is aided by use of the correlation table, table II.
Two different systems of parity have been assigned to the rotational wavefunctions of methane thus far. Jahn [40] assigned the spherical surface harmonic parity (—1)^.
Hougen [36] concluded by means of trigonometric identities among various rotational wavefunctions that the parity was always even. Because of the parity assumed, Jahn assigned
the J=l rotational wavefunction the symmetry species F2, but Hougen assigned the symmetry species Fi (both symmetries Table I Character Table for the Full Molecular Symmetry Group of Methanea
T E 8C3 3C2 d 6S-, 6ad
0 E 8C3 3C2 6C- 6C2
E 8C3 3C2 6C-* 6C2 i 8S6 3a 6S-» 6a h d °h * P-I E 8(123) 3(12)(34) 6(1423)* 6(L2)* E 8(123)* 3(12) (34* 6(1423) 6(12)
+ Ai Ai Aie Ai i 1 1 1 1 1 1 1 1 1 O + n- A2 A2 A2 A2 1 1 1 -1 -1 1 1 1 -1 -1 o + E E E E 2 -1 2 0 0 2 -1 2 0 0 go Fi Fi Fi+ 3 0 -1 1 -1 3 0 -1 1 -1 6 + F2 F2 F 2 3 0 -1 -1 - 1 3 0 -1 -1 1 *g F
A2 Ai Aiu Af 1 1 1 1 1 -1 -1 -1 -1 -1
Ax A2 A2u A2" 1 1 1 -1 -1 -1 -1 -1 1 1 E E E ~ 2 -1 2 0 0 -2 1 -2 0 0 Eu 1 F2 Fi Fiu Fi" 3 0 -1 1 -1 -3 0 -1 1
Fi F2 F2u F2~ 3 0 -1 -1 1 -3 0 1 1 -1
Columns are labelled with the symmetry operations of T^, 0 and 0^ together with the permutation-inversion (P-I) operations of Longuet-Higgins [37,55]. All operations in a column are identical when applied to a molecule of tetrahedral symmetry in an equilibrium configuration. 12
Table II Reduction of the Irreducible Representations of 0^ into
the Subgroup T, and the Rotational Subgroup 0 [104]. 13 in T table III) . d" The total wavefunction of methane, has been total' shown by various authors [22,68,74,106] to be of either or both parities for a given J value. Assuming the rotational wavefunction parity of either Jahn [40] or Hougen [36], when wavefunction is written in the Born-Oppenheimer [6] approximation:
= (2.6) ^elec'WVLb^rot *total the left side of the equation yields only one parity for a given J, in contrast to the two parities given on the right side. Attempts have been made by Yi, Ozier and Anderson [106] and Ozier and Fox [74] to add a pseudoscalar of unit modulus, co, or an inversion function, u(p) , to the left side of the above equation (2.6) to secure the required two parities.
These attempts have been helpful and have led the author to conclude that the symmetric top basis function parity given by Wang [98] is appropriate to the methane problem.
The ground state electronic wavefunction of methane has
even parity and is AXg in 0^, corresponding to Ai in 0 and T^.
The nuclear spin wavefunctions of methane are of even parity [37,58]. These wavefunctions are not all of symmetry
Aj, so they must be considered further. The four symmetrically
1 placed protons of spin /z combine to give 2'*=16 wavefunctions.
The correct linear combinations of these wavefunctions which transform as irreducible representations of T [103] and
Td [4] have been determined: the five 1=2 (MI=2,1,0,-1,-2)
spin functions of meta-methane transform as Ai, the one Table III Reduction of the Representations D and D of r g u the Full Rotation - Inversion Group onto T,
J J T D D d g u
J, I = 0 Ax A2
1 Fi F2
2 E + F2 E + Fi
3 A2 + Fi + F2 Ai + Fi + F2
4 Ai + E + Fi + F2 A2 + E + Fi + F2
5 E + 2Fi + F2 E + Fi + 2F2
6 Ai + A2 + E + Fi + 2F2 Ai + A2 + E + 2Fj + F2
7 A2 + E + 2Fi + 2F2 Ai + E + 2Fi + 2F2
8 Ax + 2E + 2Fi + 2F2 A2 + 2E + 2Fi + 2F2
9 Ai + A2 + E + 3Fi + 2F2 Ai + A2 + E + 2Fj + 3F2
10 Ai + A2 + 2E + 2Fi + 3F2 Ai + A2 + 2E + 3Fi + 2F2
11 A2 + 2E + 3Fi + 3F2 Ai + 2E + 3Fi + 3F2
12 2Ai + A2 + 2E + 3F! + 3F2 Ai + 2A2 + 2E + 3Fi + 3F2 Table IV Reduction of the Representations D and D of
the Full Rotation - Inversion Group onto 0,
DJ DJ °h u g
J,£ = 0 Alg Aiu
1 F >g Fiu
2 E + F + g ^g Eu F2u
F 3 A2g + + F + + F2u >g ^g A2u Fiu
4 Alg + £ + Fl + F2g Aiu + + Fiu + F2u g g Eu
E F 5 + 2Flg + + 2Fiu + F2u g ^g Eu
F 6 Alg + A2g + E + + 2F2g Aiu + A2U + + Fiu + 2F2U g >g Eu
E 7 A2g + + 2Flg + 2F2g A2U + + 2F1U + 2F2u g Eu
8 Alg + 2Eg + 2Flg + 2F2g Aiu + 2EU + 2Fiu + 2F2U
9 Aig + A2g + + 3Flg + 2F2g + A2U + + 3Fau + 2F2U Eg Aiu Eu
10 Alg + A2g + 2Eg + 2Flg + 3F2g Aiu + A2U + 2EU + 2Fm + 3F2U
11 A2g + 2Eg + 3Flg + 3F2g A2U + 2EU + 3FiU + 3F2u
12 2Alg + A2g + 2Eg + 3Flg + 3F2g 2AiU + A2U + 2EU + 3Fiu + 3F2U 16 doubly degenerate 1=0 (Mj=0) spin function of para-methane transforms as E, and the three triply degenerate 1=1
(Mj=l,0,-1) spin functions of ortho-methane transform as F2.
Since the spin functions are of even parity, the species under
[4] give the species under 0 and 0^, where a g is to be added to the species designation in 0^. These spin functions are conveniently listed by Yi, Ozier and Anderson [106].
The ground state vibrational wavefunction of methane has even parity [27] and is Aig in 0^, corresponding to Ai in 0
and Td.
If the rotational wavefunctions of methane are constructed using a basis set of Wang functions [98]:
iJmK*)- (2)"1/ 2 (| JmK) ± |Jm-K)), (2.7) wavefunctions of either parity can be constructed for a given
J. For J even, | JmK+^> has even and |JmK~^> has odd parity.
For J odd, |jmK+^> has odd and | JmK~^> has even parity. The rotational wavefunction | Jm0^> has the parity (—1)^. The Wang functions are discussed in greater detail in Appendix B.
The symmetry of the total wavefunction must now be considered [55]; it can be deduced in the following way.
Fermi-Dirac statistics dictate that the total wavefunction must be anti-symmetric to an odd exchange of protons. Under
T^, the operations E, C3 and C2 give an even number of exchanges requiring the character +1, while Si* and give an odd number of exchanges requiring the character -1; thus
the total wavefunction must be of species A2 under T^.
Under 0,, the operations not in T-, more than merely interchange identical particles [55]: they take protons from even octants into odd octants (the molecule fixed frame defined by the
Si* axes) and thus the Pauli principle is non-committal with respect to these operations. Under 0^, therefore, the total
wavefunction can be either Ai or A2 , and under 0 either o
Ai or A2 .
The only possible symmetry species of the methane wavefunctions are given in table V constructed using the irreducible representations of 0^. The A and F levels have a nuclear spin degeneracy of 5 and 3 respectively. The E levels on the other hand exhibit a two-fold degeneracy where the two levels have opposite parities.
If the rotational subgroup 0 of 0^ is used to classify the rotational wavefunctions, the subscripts g and u are to be dropped. In this case the symmetry species of the rotational wavefunctions obtained by reducing and into 0 are the same as obtained from the reduction of
into T, [17,32,37] giving a compatible system.
2.3 The Rotational Hamiltonian of Methane
The rotational Hamiltonian of methane can be written in terms of various spherical tensor operators, fi^, where I gives the rank of the tensor [84]. The number of independent tensors [30,40,62,102] of rank £ is given by the number of times the irreducible representation A (the Hamiltonian is totally symmetric and has even parity) appears upon the reduction of the representations of the full rotation- Table V The Allowed Symmetry Species of the Wavefunctions
of Methane in the Ground Vibronic State
Spin Degeneracy ^elec Deg. *vib *rot *total
A Alg Aig 5 Alg A2g *g 5
Alg Aig 5 Aig Aiu Aiu 5
Alg 1 Aig Eg A2g 2
Alg E 1 Alg Eu Aiu 8 *
F Alg F2g 3 Aig >g A2g 3
Alg F Alg 2u 3 *g 3 F Aiu 19 inversion group [96] into the irreducible representations of
0^ with J assuming the role of £. This reduction is given in table IV. The irreducible representation Aig appears once for each of J = £ = 0,4,6,8,9,10 and twice for J = £ = 12. The spherical tensor operator of rank £ = 0, &Q , is a scalar
associated with the scalar distortion constants, ft2 does not appear and fig has to be excluded because of time reversal symmetry [37].
The rotational Hamiltonian, W t> can thus be written
[30,40,45,72] in terms of various even [37] degrees of the angular momentum operator, which when taken to eighth degree is :
2 2 2 2 3 2 Wr£Jt = [B0J -DS(J ) +HS(J ) +Ls(J )-]^o (2.8)
2 2 2 2 + [DT+H4TJ +L4T(J ) ]filt + [H6T+L6TJ ]fi6 + LgTfi8
The quartic, sextic and octic distortion constants D, H and L respectively correspond to terms of fourth, sixth and eighth degree in J. The subscripts S and T differentiate scalar and tensor constants. The numerical subscripts 4, 6 and 8 of the distortion constants refer to the rank of the associated spherical tensor operator. In this expression the scalar fio has been shown explicitly, but normally it is assimilated into the scalar distortion constants Bo, Dg, Hg and Lg. The effects of tenth and higher degree terms are assimilated into all the constants given above and will change them slightly from their true values when a fit to experimental data is made.
The Hamiltonian can be divided into a scalar part (all terms associated with fio) and a tensor part (all other terms): 20
W (2.9) rot
The scalar Hamiltonian Wg affects all components of a given value of J equally. To eighth degree in J its eigenvalues are given by:
2 2 3 3 k Eg = B0J(J+1) -DgJ (j+l) +HgJ (J+l) + Lg J (J+l) * (2.10)
The energy Eg would give that of a rotational level if all the tensor distortion terms D^,, H^, . . , Lg^, assumed a value of zero. In the experiment described in this thesis, only
Q-branch (AJ=0) transitions were observed; thus the scalar terms need not be considered further.
The tensor Hamiltonian to eighth degree in J is thus:
2 2 2 2 WT = [DT +H4TJ +L4T(J ) ]SU + [H6T +L6TJ ]fi6 +LgTfie (2.11) where the terms are as defined earlier. It splits the levels of a given J value into various A,E and F components which are conveniently listed in table XVII to J=21. It is these splittings which were observed in this work. The degree of splitting is given by the values of the tensor distortion constants, and the spherical tensor operators , and
They can be constructed from general symmetry arguments [30,72]
and are: r (2.12)
(2.13)
C8 Ve/53 T8,o + ^/^"(Te,, + TB,_k )
+ Vs/^Te.s + T8j-e ) (2.14) where the C's are constants and the T, are obtained from J ,m the spherical harmonics by replacing x ± iy and z with
J± and Jo:and symmetrizing any resulting products.
In terms of the Cartesian components of the total
rotational angular momentum, J, oriented along the molecule
fixed Sit axes of methane, J , J , and J , tin [17,106] and X y Z ft6 [45] assume the form: f 2 2 5 -72 (35Jz+25j;) + (15J;+3)J' - 72(J ) - A(j;+Jl) (2.15)
(
2 2 15 ftG = Vi e (1UZ + 35JZ + 14JZ) - /i e (21JZ + 35JZ + 4) J
2 2 5 + 7i6(21JZ+8)(J ) - /ie(J )
21 21 2 - /32[J+(HJZ -J +6)J+] - /32[J!(HJZ -J +6)J ] (2.16)
where
J J T iJ (2.17) ± - x y
The operator ft8 expressed in Cartesian components is
quite lengthy. The expressions for T8i0, T^i*, and T8j±8 have
been given by Ozier [72], Watson [102] has given an
expression for the tensor operator ft8 in terms of spherical
tensor operators of lower rank:
2 3465/1430 ft8 = 3003 ft* - 2080(4J -183) ft<
+ 252(12j't -489J2 + 2645)ft-
- 572J2(J2-2) (4J2-3)(4J2-15) (2.18)
Eigenvalues for the tensor Hamiltonian can be obtained
by diagonalizing it. The procedure used in the present work
will be described below. However, the method is quite complex 22 and depends on the values of the distortion constants. Because of this, and because, furthermore, this degree of accuracy is not required in some circumstances, various approximate methods have been developed. They were of use at various stages of the present work, and it is appropriate to describe them here. We describe first all the approximate methods and then the procedure for exact diagonalization.
2.3.1 Fourth Degree Tensor Hamiltonian
This approximation was used in the initial search period of the work. It considered W^, to contain only terms fourth degree in J, making it: Wm = Drpftlt (2.19)
The eigenfunctions and eigenvalues have been determined to
J=13 by Hecht [30] and extended to J=20 by Dorney and Watson
[17], using Cartesian axes x,y, and z oriented in the molecule in two different ways: 1) along the S•+ axes and 2) with the
z-axis along a C3 axis and the x-axis in a plane. The eigenfunctions which diagonalize n-* are the zeroth order wavefunctions used in the following sections. The eigenvalues are: (2.20) where D^, is an experimentally determined distortion constant and
(2.21) is a tabulated spherical tensor splitting function which 23 depends on J, the symmetry species K, and a running index, t, which labels components of a given J and K in order of increasing energy.
2.3.2 Sixth Degree Tensor Hamiltonian
A sixth degree perturbation treatment was used to calculate energy levels during the period in which the first four lines of methane were observed. The Hamiltonian was assumed to be:
2 WT = [DT + H^J ]^ + H6Tft6 (2.22)
The fourth degree tensor Hamiltonian D^fiif was diagonalized yielding the zeroth order wavefunctions. Upon going to sixth
As degree, the added terms in the tensor Hamiltonian H^J2^ and
Hg,pft6 can be treated using first-order perturbation theory with the zeroth order wavefunctions. As
Although the term H^J2^ could be treated using first order perturbation theory, this is not necessary because it As can be treated exactly. The operator J2 is diagonal in the zeroth order basis set; thus
H^jJ2^!^ ^ = H^^J2^
= H4TJ(J+1)^Q
= H4TJ(J+l)f(J,K,t) (2.23)
First order perturbation treatment of Hg^,ft6 requires evaluation of the matrix elements
fi6) = g(J,K,t) (2.24) o 24
These have been tabulated by Kirschner and Watson [45].
The eigenvalues of the tensor splitting Hamiltonian taken to sixth degree in J are thus determined using first order perturbation theory:
E(J,K,t) = [DT + H4TJ(J+l)]f(J,K,t) + H6Tg(J,K,t) (2.25) where Drj,, H^ and Hg^ are experimentally determined centrifugal distortion constants and f(J,K,t) and g(J,K,t) have been tabulated by Kirschner and Watson [45].
2.3.3 Eighth Degree Tensor Hamiltonian
This of course is the full tensor Hamiltonian required in the present work. It has the form: (2.26)
2 Z 2 2 WT = [DT + H4TJ + L4T(J ) ]SA + [H6T + L6TJ ]fi6 + LgTfi8 where all the parameters are as described earlier.
Two methods have been used to obtain its eigenvalues.
One, an approximate method using second order perturbation theory was developed by Ozier [72]. In this case, fourth degree terms are treated exactly, sixth degree terms require a second order perturbation treatment, but eighth degree terms require only a first order perturbation treatment.
As before, since J2 is diagonal in the zeroth order wavefunctions, a second order treatment is not required for
2 H4^,J fii, since it can be treated exactly. Likewise a first
2 2 order treatment is not required for L4^,(J ) fiit because it, too, can be treated exactly. The exact solution of the term associated with fl*. is:
2 2 [DT + H4TJ(J+1) + L4TJ (J+l) ]f(J,K,t) (2.27) 25
Using the above argument, the first order treatment of
T2 L/-6rrr J fi6 becomes
2 2 L6TJ fie) = L6T(j ftf
= L6TJ.(J+D(n./o
= L6TJ(J+l)g(J,K,t) (2.28)
The second order perturbation treatment of HgTfi6 gives [72]:
2 (H6Tfi6^ = H6Tg(J,K,t) + [H6T /DT]g(J,K,t) (2.29) where
2 (j,K,t) = y i
The first order perturbation treatment of Lg^fts requires the matrix elements
^fi8^)o = h(J,K,t) (2.31)
These have been tabulated by Ozier [72] along with the above g(J,K,t).
The eigenvalues of the tensor splitting Hamiltonian taken to eighth degree using appropriate perturbation theory are thus:
2 2 ET = [DT + H4TJ(J+1) + L4TJ (J+l) ]f(J,K,t)
+ [H6T + L6TJ(J+l)]g(J,K,t) + L8Th(J,K,t)
2 + [H6T /DT]£(J,K,t) (2.32)
where DT, H4T, HgT, L4T, LgT and LgT are experimentally determined constants, f(J,K,t) and g(J,K,t) have been tabulated by Kirschner and Watson [45], and g(J,K,t) and h(J,K,t) have been tabulated by Ozier [72].
2.3.4 Exact Evaluation of the Tensor Hamiltonian
Eigenvalues
This method was the one ultimately used in evaluating the distortion constants presented in this thesis. Because the wavefunctions resulting from the diagonalization depend on the
distortion constants [72], the method was an iterative one.
It was developed by Fox and Ozier [22] and is described here.
The procedure first assumes that all distortion constants
are zero except for DT> The matrix <^JK' K | WT| JKK^> is
constructed in the so-called "primitive" basis |JKK^>, which
is a suitably defined set of Wang functions [22,98], It is
then transformed to a new, so-called "preliminary", basis
iJKt) [75] by the method of Fox and Ozier [22]. This basis has no particular physical significance [72] and has matrix
1 elements ^Jict |WT| J The matrix in the "preliminary" basis is diagonalized to give the eigenfunctions which are called the "second- order" basis. For the first iteration, since only Drj, was assumed non-zero, the eigenfunctions are those determined by Jahn [40], Hecht [30] and Dorney and Watson [17]. The diagonal elements of the spherical tensor operators P^, fi6 * Fox and Ozier [22] use yp instead of K. The species of ^rot in T is given by y. The value {1,2} given p indicates the species {Ai,A2) in T^ of the pseudoscalar or inversion function [22,74,106] required to give the correct parity [68] assuming the ^rot parity of Jahn [40]: (-1)^ and even parity for ^elec*^NS*^vib. and fie are next calculated in the "second-order" basis which for the first iteration are f(J,K,t) [17], g(J,K,t) [45] and h(J,K,t) [72] respectively so that the energies can be expressed as a linear function of the six distortion constants: DT, H4T, HgT, L^T, LgT and LgT. A new set of distortion constants is obtained by a linear least squares fit to the experimental frequencies. This new set of constants is used to set up a new matrix in the "primitive" basis at which point the second iteration is begun. The iterations are continued until the distortion constants converge to the desired accuracy. 2.4 Dipole Moment Using permutation-inversion arguments Hougen [37] has shown that each of the space-fixed components of the electric dipole moment operator in isomorphic T^ (our 0) is of species A2. The transformation between molecule-fixed axes x,y,z and space-fixed axes X,Y,Z is given by equation (B.2) in appendix B. Hougen [38] has shown that the molecule u fixed components x>Vy,Vz of the dipole moment are related to the space-fixed components by a similar transformation, and has determined as a result that the species of these components are F2 F2 ,F2 respectively. X y WW In order for a molecule to possess a permanent dipole moment at least one translation must be totally symmetric. Because this is not so [96], as is well known, methane has no permanent dipole moment in its equilibrium configuration. However, rotation about one of the C3 axes of methane can distort the molecule in such a manner that a dipole moment 28 is created [17]. The distortion of the molecule can be described in terms of the normal coordinates, Q. Of the 3N — 6 = 9 normal coordinates, only the six corresponding to the infra-red active vibrations v3 and Vi» (of symmetry species F2) need be considered. The centrifugally induced dipole moment is given by [100]: 6^a - £(3V9VfiQk (2.33) where the distortion of the individual normal coordinates, Q^, produced by centrifugal distortion is given by [100] : 60, = /9Q J J (2.34) 8TT2X 3Y k> 6 Y k By .-1 with an element of the I" tensor and the force constant of the associated normal vibration [104]. The centrifugally induced dipole moment can be written in the equivalent alternate form [100]: 6y (2.35) a / i a 3 y BY where the Ja are the components of J in units of h/2iT and 31 fay 03Y = 0Y6 = 2 3Y a a a (2.36) 3Ql 3Qi is a constant third rank tensor of symmetry Ai with B^ the rotational constant about the a axis, 1^ the product of inertia about the B,Y axes and the frequency of the normal vibration Q^. The dipole moment is an even.function 29 of J because of time-reversal symmetry. There is only one independent dipole moment constant 0^ for methane because the irreducible representation Aj appears only once in the direct product [32,100]: 2 2 T(T) X [r(R)] = F2 X L [Fi]1 \ / sym sym = F2 X [Ai + E + F2] = Ai + E + 2Fi + 3F2 (2.37) Watson [100] finds for a tetrahedral molecule such as CEU six non-zero values of but only one is independent: 0yz = 0zy = 0xz = 0zx = 9xy = 0yx (2 3g) xxyyzz N/ where the x,y,z axes are coincident with the Si* axes of the molecule. The value of ©x^ has been calculated by Watson [100] to be 2.6 X 10"5 Debye, calculated by Fox [20] to be 1.8 X 10"5 D, and measured by Ozier [71] as 2.41 X 10"5 D. 2•5 Selection Rules Selection rules are determined by matrix elements of the form: ^a|y|b^ where the dipole moment operator, u, is in the space fixed frame. Allowed transitions, a«-+-b, are those with non-zero matrix elements. These are non-zero only if the direct product r& X X contains the totally symmetric representation Ax [96]. The point group 0^ will be used to derive the selection rules because parity is an important consideration here. In the point group 0^, = A2u in the space fixed frame. The above direct product will contain Aig only if r& X contains A2u. The only 30 allowed transitions in 0^ are: Alu •«-*• A2g, Eg «-»• Eu, and Fig •<-»• F2u. The selection rules in the subgroup 0 are the same as those in 0^ except that the subscripts g and u are dropped. 2.6 Stark Effect If an electric field is applied to a molecular s system, a perturbation term Wgt ^ ^ added to the Hamiltonian. Every term in the Hamiltonian of methane is totally symmetric, Ai, under the operations of the molecular point group T^ because these operations merely interchange identical particles and thus have no effect on the energy with or without an applied external field. The field free terms of the Hamiltonian are of even parity because parity is conserved for an isolated system [58]. With the application of an external field, the system is no longer isolated. The term Wgtar^ mixes states of opposite parity [58] and thus Wgtark is of odd parity. The Stark Hamiltonian of methane is Ai under T^, but the odd parity leads to the species A2u under 0^ and A2 under 0 (see table II). The first-order perturbation Stark Hamiltonian is: WStark --^Z (2"39) with the electric field, e, defined along the space fixed Z axis. The Stark Hamiltonian is evaluated by taking the average over the unperturbed wavefunctions [26], thus = £ a a 2 40 EStark - ( l^zl ) < ' > s oc Since i- °f *d parity (A2u under 0^) , the first-order 31 Stark energy, Egtar^, is zero unless a given level contains both parities. As shown in table V, only the E rotational components have wavefunctions of both parities simultaneously and thus only the E rotational components exhibit a first-order Stark shift. An accidental degeneracy of certain levels can give rise to a first-order Stark shift. The first-order Stark matrices are seen to be very similar to those governing the selection rules: a first-order Stark shift will arise only with an accidental degeneracy of Aiu and A2g levels or Fig and F2u levels. Nearly degenerate levels of the proper symmetry may become pseudolinear at high fields [17]. The signed Stark coefficients C(J,E,t) required to calculate the Stark energy shifts: AE(J,E,t) = C(J,E,t)m0xye (2.41) z where AE(J,E,t) is the Stark energy shift in Debye«kV/cm, m is the magnetic quantum number, ©xy is the-dipole moment factor in Debye [100] and e is the electric field strength in kV/cm, have been calculated by Dorney and Watson and are listed in table III of reference [17]. The Stark shift for a given transition is calculated using the difference between the two appropriate signed Stark coefficients. 2.7 Physical Interpretation Dorney and Watson [17] present a physical model to explain the centrifugally induced dipole moment and Stark effect which is valid for high J and maximum positive and negative values of the tensor splitting function f(J,<,t). 32 They note that levels pushed highest in energy by the tensor operator ft* occur in nearly degenerate sets with a spin-free degeneracy of 8, which is attributed to two directions of rotation about each of the four C3 axes of methane. Rotation about a C3 axis tends to pivot the off-axis hydrogens away from the C3 axis creating a dipole moment along that axis and lowering the molecular symmetry from T^ to C^v- The largest distortion dipole moment gives rise to the largest first-order E level Stark shifts. On the other hand, the energy levels pushed lowest in energy by the tensor operator occur in nearly degenerate sets with a spin-free degeneracy of 6, attributed to the two directions of rotation about each of the three S^ axes. Rotation about an Sn axis tends to lower the molecular symmetry from T^ to T>2^ but not to induce a dipole moment. Since a first-order Stark shift is proportional to the molecule fixed dipole moment through the direction cosines, rotation about an Si, axis will give no (or a quite small) first-order Stark shift. The direction cosines are the nine cosines relating the three molecule fixed axes to the three space fixed axes [91] and are the transformation (B.2) in appendix B. In the particular limiting cases discussed above it is to be remembered that the dipole moment along the C3 axis: X y,. [17] or alon0g the S4 axis: y = 0 ^J J is not the same £ zzxy as the dipole moment factor 0*^ = 2.41 X 10"5 D, which is a constant. 33 2.8 Intensities of Transitions The line strengths for isotropic radiation have been calculated by Dorney and Watson [17] using the transition moment: \ 1 (2.42) fa3 summed over the three directions of space, f, and all components a and 3 of a and b respectively. The calculation can be performed for the spin-free problem; nuclear spin and inversion doubling is then taken into account in the following manner: the effect of internal degeneracy is removed by dividing the intensities of A, E, and F transitions by 1, 2, and 3 respectively, then the intensities of the A and F transitions are increased by the factors 5 and 3 respectively to take into account nuclear spin degeneracy and the E level intensities are multiplied by 2 to take into account the inversion doubling. Dorney and Watson have tabulated the reduced line strength XY 2 XY [S^B/(0 ) ] for a distortion dipole factor 0 = 1 Debye in table IV of reference [17], where S'-^ is the spin free line strength divided by the A,E,F internal degeneracy of 1,2,3 respectively. In microwave spectroscopy the peak absorption coefficient 8TT2P -E, /kT Y. v0Sabe »b (2.43) max 3cQ(kT)2Av is of interest [17]. In this expression Q is the partition function, P/Av is the reciprocal of the pressure broadening parameter which is nearly independent of pressure, Vo is the 34 absorption center frequency, S ^ is the transition moment -E /kT eq. (2.42), and e b' is the Boltzmann factor. Dorney and Watson [17] have tabulated y (cm-1) assuming a pressure 111 3.X broadening parameter of 1 MHz/torr, T = 300°K, B-= 5.241 cm-1, y 5 DT = .12 MHz and 0* = 2.41 X 10" Debye. The peak absorptior n coefficients, Y , for the transitions observed max' in this experiment are listed in table XII calculated on the basis of a pressure broadening parameter of 2.7 MHz/torr [76] and DT = .133 MHz. Calculated values of YMAX range from 2 X 10"11 to 6 X 10"11 cm"1. Consideration of higher order terms in the dipole moment expansion [78] suggests that the absorption coefficients of methane may be smaller by a factor of two. The pressure broadening parameter observed in this experiment [73] was approximately 5 MHz/torr which further reduces the value of Y_ 1 max 35 CHAPTER 3 EXPERIMENTAL APPARATUS 3.1 Introduction This experiment involved observation of extremely weak absorptions readily masked by noise. To improve the signal to noise ratio, efforts were taken to enhance the signal as much as possible while minimizing the noise. The conventional method of Stark modulation combined with phase sensitive detection described in section 3.2 reduced noise in three ways: 1) the detection frequency was increased from near zero to 1 kHz., drastically reducing the effectiveness of 1/f noise sources; 2) the noise bandwidth was reduced by means of a simple post detection time constant; and 3) the phase detector did not respond to noise in quadrature with the signal. The signal was increased approximately four-fold by constructing a 13 meter cell to replace the 3 meter cells normally used in the laboratory. The difficulties created by higher-order modes [82] in the cell are discussed in section 3.3. The use of a signal averaging computer (see section 3.5) to accumulate many scans also enhanced the signal. The long scan times used required precise frequency control which is described in section 3.4. In addition to the Stark modulation mentioned above, noise was reduced by a) minimizing stray pickup, b) matching the 36 detectors to the preamplifier, c) minimizing microwave oscillator noise, d) subtracting microwave oscillator noise from the cell detector output, e) minimizing noise generation in the cell, and f) using low noise detectors. Steps a) through e) are discussed in section 3.6. The subject of low noise bolometer detectors is treated in section 3.7 and appendix A. In X-band (8+12.4 GHz.) and P-band (12.4+18 GHz.) the microwave generators were backward wave oscillators contained in a Hewlett-Packard 8690A microwave sweeper of a HP-K15-8400B microwave spectroscopy source (figures 3.1 and 3.3). Between 18 and 24 GHz. (referred to here as K-band) OKI 20V10 and 24V10 air cooled klystrons were used (figure 3.2). Schematic diagrams of the configurations of the spectro• meters used between 8 and 18 GHz. and between 18 and 24 GHz. are given in figures 3.1 and 3.2 respectively. 3.2 Stark Modulation Microwave spectroscopy in its simplest form consists of passing the tunable, virtually monochromatic power from a microwave generator through a gas filled cell and determining the amount of power absorbed as a function of frequency by means of some detector. Since the cross-sectional dimensions of the cells normally used are quite small in relation to the wavelength of the microwave radiation which passes through them, small irregularities in the transmission properties of the cell result. These transmission irregularities, although small, can be very much larger than the lines under observation because of the characteristic weakness of microwave absorptions. Microwave I3m. CELL Spectroscopy tt STARK Source SEPTUM N6IOB/38B7 5mfd Bolometer HP-KI5-8400B PHASE DET 1kHz PAR 128 "He-f Synchronizer I 2K WW Reference L HP 8708A Oscillator /77 ""TRIAD rh G-IO — 12V. | DC- FM MOD SIGNAL RAMP AVERAGER -OSCILLOSCOPE T NICOLET -X-Y RECORDER 1072 Figure 3.1 Spectrometer used between 7.8 and 18 GHz with BWO as Source and Single Ended Detection Scheme. OKI IkHzJ~L 20VI0 GENERATOR SEPTUM 20db Xlodb l3m.CELL SLIDE SCREW TUNER UJUU Power Supply Narda 62 Al 2KWW MA460DR Synchronizer ] Sage 244 25MHz IF Counter HP5246L N 5252A Prescaler X N6I0B/38B7 N6I0B/38B7 Bolometer sample o reference in 20db PHASE DEI ro PAR 128 1kHz Reference RF^ Oscillator 500mv RF HP8467B DC-FM jf RAMP SIGNAL OSCILLOSCOPE ERROR Synchronizer <— Figure 3.2 Spectrometer used between 18 and 24 GHz with Klystron Source and Balanced Noise Cancelling Detection Scheme. 39 If the transmission characteristic of the waveguide cell has a small 1% dip, for instance, it will make the observation of a .001% molecular absorption difficult because the dip is one-thousand times larger than the absorption due to the molecule. The importance of transmission irregularities can be significantly reduced by employing Stark modulation combined with phase sensitive detection [39]. Here the microwave absorption is turned on and off at the modulation frequency whereas the transmission irregularities are in principle unaffected; since the phase sensitive detector responds only to the modulation frequency, the effect of! the transmission irregularities is greatly reduced (in the foregoing example, from 1000X to .01). It is assumed the Stark cell is perfectly rigid so that the Stark modulation does not affect transmission irregularities. If a microwave spectrometer is operated in its simplest form, the detected bandwidth can be restricted to a small range about the repetitive sweep frequency to minimize noise. If the same repetitive sweep frequency is used with a Stark modulated spectrometer, the same bandwidth is required, but it is centered at the modulation frequency, not the sweep frequency. This is of great advantage because most power supplies, microwave generators and detectors exhibit much less noise per unit bandwidth as the center frequency is raised. In this experiment, the modulation frequency was set as high as possible consistent with the bolometer detector response and was chosen to be 1 kHz. This low modulation frequency 40 enabled the 13 meter cell employed to be driven to 2 kV/cm with ease since the power the Stark generator [9] must supply to the septum varies directly as the frequency [91]. The Stark modulation was supplied as a zero based square wave with a peak voltage just sufficient to give complete (99+7o) modulation of the line. An unnecessarily high Stark field was not used because this would aggravate a non-zero baseline slope which resulted in some cases from Stark field induced vibrations of the non-rigid cell. The Stark modulation was obtained from a solid-state square wave generator which has been described in detail by Britt [9]. The square wave generator with seven series connected transistors to charge the cell and seven additional transistors to discharge the cell was designed to operate at 25 kHz; operation at 1 kHz was accomplished without modification. An external audio oscillator operating at approximately 1 kHz (992 Hz) drove the square wave generator and simultaneously supplied a reference signal to the PAR-128 phase sensitive detector. 3.3 Stark Cell The cell consisted of one 12 ft. and three 10 ft. lengths of X-band copper waveguide connected in series to give a total length of 12.8 meters; each length contained a Stark septum held midway between and parallel to the broad faces of the waveguide by Teflon strips, to give a septum spacing of 0.47 cm. In order to fit the cell into available laboratory space and to allow the cell ends to be adjacent for a noise cancellation scheme, five 90° elbows were used to fold the 41 cell onto itself in the shape of a paper clip opened 90°. Mica windows sealed the ends of the cell and a longitudinal slit in the center of the broad face of the waveguide served as the gas entry port. 3.3.1 X-band At X-band (8-<->12.4 GHz) the cell attenuation coefficient 3 was «c = 1.4 X 10~ /cm so that 60 mw of input power yielded 10 mw of detector power. In this case the optimum cell = 1 length [26] (LQpt 2a" ) was 14.3 meters which was close to the 12.8 meters used. At higher frequencies, it was found that microwave generator powers of greater than 100 mw gave less than 10 mw of power at the detector as a result of increased cell attenuation. The cell had greater than optimum length at higher frequencies, but it was considered impractical to vary the length of the cell with the operating frequency. 3.3.2 Cell above X-band For a given frequency of operation, the waveguide size is normally chosen so that it can transmit energy in only one mode: the dominant mode. If it is desired to double the frequency of operation, for instance, a waveguide is chosen twice as small to insure that only the dominant mode is propagated. The dominant mode is advantageous because it is transmitted with the least loss and because it gives the best performance with most microwave components. Since this experiment was originally conceived as an X-band experiment, the X-band cell was constructed for low attenuation and predictable performance. If the dominant mode could have been maintained as the sole means of energy 42 transmission as the frequency of operation was increased, the cell would have exhibited no more attenuation at 40 GHz than at 8 GHz [82]. But above 12 GHz, X-band waveguide transmits higher modes characterized by higher losses, which at 16 GHz are at least 2X greater than that of the dominant mode. Above 12 GHz, any discontinuity in the cell which distorts the microwave field of the dominant mode (a mis• aligned flange or a Stark septum end, for instance) creates higher modes. The Stark septa will have no effect on the propagation of a mode provided it is parallel to the microwave magnetic field and perpendicular to the microwave electric field at every point [81]. The septa thus do not interfere with the dominant TEio mode or any of the related higher TEnQ modes. But the higher TEai and TMXa modes [81] are totally reflected by the septa. The Stark septr a r pass all TE modes but reflect others. no The tapered waveguide transitions used at each end of the cell in frequencies above X-band, however, reflect all higher modes. All TEnQ modes not reconverted to the dominant TEi0 mode are reflected back and forth over the Stark septa in the cell until dissipated. Other higher order modes not reconverted to the dominant TEio mode are restricted to Stark septum free waveguide regions and are also reflected back and forth until dissipated. In an effort to cope with higher modes, the waveguide was not tapered at the detector end. Instead an X-band slide screw tuner and X-band detector mount with tunable stub were used. In this way the higher modes which did reach the detector end together with the dominant mode could be coupled to the detector element in the best possible manner. The application of the 1 kHz square wave Stark voltage to the septa produced an audible vibration in the Stark septa- waveguide system [91]. The vibrating septa gave rise to reflections and higher modes which varied at the modulation frequency in phase with the molecular absorption and could not be discriminated against. Fortunately the "signal" due to the vibrating septa was broad in character in comparison with a methane line and thus appeared as a sloping baseline which was easily compensated for. 3.4 Frequency Stabilization and Control The sharpness of the microwave transitions observed and the long periods of time required to obtain them required close frequency control. This could not have been obtained with free running (unstabilized) X- and P-band backward wave oscillators contained in the HP-8690A sweeper or with free running OKI 20V10 or 24V10 klystrons. This control was achieved by means of commercially available synchronizers which not only stabilized the frequency, but were responsive enough to keep the controlled oscillator in phase step with a well controlled microwave reference frequency. This was important because the FM noise [67] (or spectral line width) of the microwave oscillators was reduced to that of the purer reference oscillator. The microwave reference frequency was obtained by harmonic multiplication of a signal from a stabilized very 44 high frequency (VHF) reference oscillator. This had the benefit of allowing the microwave frequency to be determined and swept by performing these same operations on a VHF oscillator much lower in frequency. 3.4.1 Microwave Reference Frequency The microwave reference frequency was a harmonic of the output of an HP-8466A (200^-450 MHz) VHF solid state variable frequency oscillator (VFO), see figures 3.2 and 3.3. At X- and P-bands the output of this VHF reference oscillator was fed directly to a harmonic mixer. At K-band the output of the VHF reference oscillator was fed through a HP-8767B broad band power amplifier (which gave a maximum of 500 mw power at 350 MHz) to a MA-460DR varactor harmonic multiplier in an X-band non-tunable detector mount. 3.4.2 Stabilized VHF Reference Oscillator The HP-8466A (200++450 MHz) VHF reference oscillator was not particularly stable, so it had to be stabilized with a HP-8708A synchronizer, see figures 3.2 and 3.3. This VHF synchronizer was employed in the usual manner except that the sampling rate (a front panel control) was set at 50 kHz. This spaced the microwave lock points approximately 3 MHz apart and simplified instrument operation when searching regions 3 MHz at a time; The HP-8708A VHF synchronizer had a d-c coupled FM input, a desirable feature because it permitted very slow, linear frequency sweeps. The deviation from linearity of the entire system was checked using frequency markers and was found to be within 1/2%. 45 HP-5246L Counter with 5252A Prescaler freq 0 HP-8466A VHF monitor Reference (-L Oscillator vT out error frequency vW analog fm HP-8708A voltage in Synchronizer directional coupler vhf 20 sample db HP-8690A Microwave Sweeper out O * * helix m + 4- o o CM HP-8709A Synchronizer Harmonic ln Mixer Irror °" HP-934A X-band 0) HP-932A > Cti CD P-band O PJ u o •H to S 20 db x to cell Figure 3.3 HP-K15-8400B Microwave Spectroscopy Source Configuration used at X- and P-bands 46 The use of the FM input to sweep the frequency produced a disadvantage. In this mode, the overall frequency stability and purity was ultimately determined by an L-C oscillator. This was so because the FM input varied the frequency of a .3 to .4 MHz VFO in the HP-8708A synchronizer. By means of a phase lock loop this variation was transferred in an additive manner to a 19.95 to 20.05 MHz VFO in the same synchronizer. This frequency transfer reduced the stability demands on the .3 to .4 MHz VFO by a factor of 50, but the overall stability was still worse than that obtained from a crystal oscillator. A long term frequency stability of 1 part in 106 (.02 MHz at 20 GHz) was maintained by "trimming" the frequency of the spectrometer every few hours to compensate for the effects of slow temperature drifts. The frequency of the 19.95 to 20.05 MHz VFO in the HP-8708A synchronizer was then divided by 400 to drive a sampling gate at 50 kHz. The output frequency of the HP-8466A VHF reference oscillator was sampled at a 50 kHz rate (which meant only a fraction of a cycle was sampled every 7000 cycles at a controlled frequency of 350 MHz) to give a corrective - error signal back to the sampled VHF oscillator. The stability of the 19.95 to 20.05 VFO was thus transferred to the 200-<~*-450 MHz VFO, but in a multiplicative manner so that there was no gain in relative stability. 3.4.3 Stabilized Microwave Oscillators The frequency of the microwave oscillators was stabilized by means of servo loops (figures 3.2 and 3.3). In all cases, one percent of the output power was mixed with the microwave 47 reference frequency to give a beat frequency. This beat frequency was in turn compared to a crystal controlled offset frequency by means of a phase sensitive detector, giving a corrective error voltage to the microwave oscillator, closing the loop. At X- or P-band (figure 3.3) the mixer was either a HP-934A or HP-932A harmonic mixer which also served to generate the microwave reference frequency. The synchronizer was a HP-8709A with an offset frequency of 20 MHz. At K-band (figure 3.2) the mixer was a 1N26 crystal mounted in a tunable K-band mount. The microwave reference frequency was supplied by a varactor in an X-band mount combined with an X- to K-band transition. Both the mixer and the varactor harmonic multiplier were mounted on opposite arms of a 20-db cross guide dual directional coupler, which was placed between the klystron and the isolator to minimize interference from cell reflections. The synchronizer was an LFE 244 (less the harmonic generator) with an offset of 25 MHz. Phase lock was ascertained by observing the corrective error signal on an oscilloscope. 3.4.4 Frequency Determination The frequency of the microwave oscillator was always maintained ±20 or ±25 MHz with respect to the microwave reference frequency at X- and P-bands or K-band, respectively. If the foregoing sign was known, along with the VHF reference oscillator frequency and the correct harmonic number, the microwave frequency was easily determined. At X- and P-bands (figure 313) this determination was 48 exceptionally easy. Here the frequency was correctly determined by a factory modified HP-5246L counter with a 5252A prescaler if the system was in lock and the HP-8690A sweeper dial setting was in approximate agreement with the frequency displayed on the counter. The line at 7861 MHz was below the normal range of the HP-8690A sweeper with an X-band plug-in, but could be reached in frequency by the application of an appropriate d-c voltage from the front panel FM input. Since in this case the front panel frequency calibration was no longer valid, the frequency had to be determined by establishing phase lock with no FM input at 8 GHz and then simultaneously increasing the FM input and decreasing the reference frequency while maintaining phase lock until 7861 MHz was reached. This frequency was in agreement with an independent determination made with a HP-5245L counter utilizing a HP-5255A heterodyne converter plug-in. At K-band (figure 3.2) the klystron frequency was determined to within 10 MHz with a Narda 12K1 cavity wavemeter. This was sufficient to determine unambiguously the harmonic number and the correct offset frequency sign, which when combined with the precisely determined reference oscillator frequency gave an accurate klystron frequency. 3.5 Time Averaging The observed signal to noise ratio in an experiment can be improved by restricting the noise bandwidth of the instrument employed. This can be done simply by increasing the output time constant. In order for the output to follow 49 a spectral feature reasonably, however, the sweep rate must be slowed simultaneously. But as the sweep rate is lowered, the time rate of change of any spectral feature becomes less, increasing,the relative importance of any slow variation in experimental conditions which might be due, for example, to temperature or power supply voltage fluctuations. On the other hand, the signal to noise ratio can be improved by additively superimposing many scans of the identical frequency region, while retaining the original sweep rate. In this case the signal increases linearly with time, but the noise increases only as the square root. The influence of low frequency noise is significantly reduced because it is divided into many time segments and summed. This is the familiar process of "signal averaging" [29,47]. Figure 3.4 shows the improvement in the signal to noise ratio resulting from the summation of two separate three day runs to give a six day run. A signal averaging computer allows the sweep time to be varied almost at will, with the number of sweeps adjusted to give the same total sweep time. In this experiment, the ability to determine the frequency with precision set the upper limit to the sweep rate. If, for instance, the counting rate of an electronic counter is 1 MHz, the counter must have a dwell time of 1 second to determine a frequency with a precision of 1 part in 106. If the precision need be only 1 part in 105, the required dwell time in this example can be reduced to .1 second and the sweep rate may be correspondingly increased. 18562 18563 FREQUENCY (MHz) Figure 3.4 Improvement in Signal to Noise Ratio Resulting from the Summation of Two Separate Three Day Runs (top two traces) to give a Six Day Run (bottom trace). o The signal averager was a Nicolet #1072 with a memory of 1024 channels divided into quarters with provision to transfer data in various ways between quarters. This feature allowed baseline subtraction. The baseline slope in some instances exceeded the line height by a factor of from 16 to 32 within a 3 MHz range. The sloping baseline was compensated for to a considerable extent by alternately accumulating sample and baseline runs of several hours each in separate quarters and taking the difference to give a methane absorption on a relatively flat baseline. The sample and baseline runs were identical in all respects except for the presence of methane in the cell. The baseline subtraction method worked well because the baseline slope remained reasonably constant with time. The signal to noise ratio in principle was degraded by 1.414 using this method, but if the baseline slope was considered noise instead of signal, there was a great improvement. On occasion, the instrument broke down during a run destroying the information being accumulated. The seriousness of any breakdown was diminished by periodically transferring the contents of the working quarter to an appropriate storage quarter; in this manner only a few hours work at most would be ruined per interruption. Because some runs were unusable on occasion, the baseline subtraction could not always be performed directly. In these cases, the data was multiplied by a binary constant during transfer, which when done the correct number of times gave equal weight to sample and baseline. 52 The Nicolet #1072 offered two modes of operation: "flyback" in which the memory channels were swept always in one direction, or "sweepback" in which the memory channels were first swept up and then down in number. The sweepback mode was chosen because the "time constant distortion" [91] suffered by a line was symmetrical, broadening the line but not affecting the peak absorption frequency. The sweepback mode was chosen also because it lacked the large discontinuity at flyback of the other mode. This discontinuity in the flyback frequency analog voltage caused the synchronizer servo loops to "drop out of lock" frequently. In addition, a sloping baseline yielded a discontinuous baseline on flyback, which when acted upon by the phase sensitive detector output filter caused a very curved baseline at the beginning of each scan resulting in an appreciable loss of information. Both these disadvantages were avoided with the sweepback mode. The signal averager provided an output voltage proportional to the memory channel being swept. This was a 0 to 4 volt staircase ramp and was used as a frequency analog voltage to drive both the HP-8708A synchronizer (figures 3.2 and 3.3) and a noise tracking electromagnet (figure 3.2). Since neither the required synchronizer FM input voltage nor the required electromagnet voltage was 0 to 4 volts, two variable gain, variable offset precision d-c amplifiers (figure 3.5) were constructed. The "starting points" were set with the offset controls. The gain controls had no effect on the "starting point" settings because here the ramp voltage uA741 and yA777 10KQ operational amplifiers and uA7805 - 5 volt 30Kfi voltage regulator lOKft -wv manufactured by Fairchild Semiconductor A/W lOKft uA777 AAA/ + © SYNCHRONIZER unity gain inverting 5KQ< amplifier 3X gain inverting amplifier with variable offset +3Vo A/W lOKfi OFFSET 30Kft m out AAA/ RELAY lOKfi MAGNET WV common 1 amp voltage follower 3X gain inverting amplifier with variable offset +4V o VVV lOKfi OFFSET Figure 3.5 Frequency Analog Voltage Conditioning Amplifier 54 was 0.0 volts. The settings corresponding to a ramp voltage of 4.0 volts were then made "on the rim" with the gain controls. The current going to the noise tracking electro- magnet had to be adjusted on occasion midway through a run, for this reason a voltage follower was put into the amplifier to prevent the gain adjustment of the electromagnet amplifier from upsetting the end-point frequencies. The staircase ramp of the Nicolet 1072 introduced a noise spike into the system each time the frequency was discontinuously increased from one value to the next. These noise spikes were observed in the signal channel of the PAR 128 phase sensitive detector. To smooth out these discontinuities the ramp was fed through an R-C filter with a time constant of 1/2 second. This filter affected the triangular ramp voltage in two ways: it rounded the sharp peaks and troughs and it retarded the wave in time by an amount equal to the time constant [61]. The effect was to add hysteresis to the frequency sweep system. In the center of the sweep, the frequency was shifted alternately 1% low and 17o high as the frequency analog voltage was swept up and down. Sharp frequency markers appeared doubled due to this hysteresis. With a 3 MHz sweep range, each marker was alternately shifted down and up in frequency 30 kHz. This was estimated to have broadened the methane absorptions approximately 45 kHz, which was small compared with the usual total width at half-height of 450 kHz. To test the overall frequency linearity of the system and the accuracy of the method of frequency determination, several 55 runs were made using marker doublets spaced 1 MHz apart to give 3 doublets within a 3 MHz sweep range. An extrapolation employing doublet centers gave end-point frequencies to within .01 MHz (counter accuracy) of those determined by observing the counter "on the run". The total system deviation from linearity was experimentally determined to be less than l/27o which was close to the minimum which could be expected: a 1 channel error in 256 gives a .4% deviation. 3.6 Noise Minimization In addition to the noise minimization affected by using Stark modulation combined with phase sensitive detection, other steps were taken to minimize noise. These are discussed in detail below. 3.6.1 Stray Pick-up Many components critical to this experiment exhibited microphonism: these were the microwave and stabilizing oscillators, the cell, and the detectors. The laboratory was situated in a basement below grade which probably helped minimize the pick-up of building vibrations. In addition, care was exercised to insure that the vibration of vacuum pumps did not couple into the system. Troublesome cooling fans were supported in such a manner that they didn't affect the system. To minimize electrical pick-up, much attention was given to the detector input system which operated at the lowest signal level. The ground loop formed by the cell, detector mounts and input circuitry, was broken by covering the detector mount flanges with polyester tape and securing them 56 with plastic screws. The Triad G-10 input transformer was very well shielded: three u-metal and two copper shields together with humbucking coils were specified to give up to a 135 db (6 X 106 voltage ratio) reduction in pick-up. In addition, the leads from the detectors and to the phase sensitive detector were double shielded and short. The bolometer bias battery (Eveready 732) was shielded and all connections to it and the input transformer were made within a steel box. The 2000 ohm load resistors in the bolometer bias circuit were non-inductively wound for low pick-up. In spite of well regulated power supplies, large noise spikes were introduced into the system when the room lights were switched on or off. This was remedied by leaving the lights on in the laboratory around the clock. Clearly, however, since the luxury of an electromagnetically shielded room [66, 69] was not to be had, transients originating in other parts of the building had to be tolerated (the overall effect of these transients could not be assessed). 3.6.2 Detector to Preamplifier Impedence Match The bolometer detectors used were low impedence devices operated at a d-c resistance of about 200 ohms. The operator's manual [79] for a PAR 128 lock-in amplifier showed that for this resistance with a modulation frequency of 1 kHz, the noise figure was approximately: 11 db; that is, the signal to noise ratio obtained from the detector would have been degraded by a factor of 3.V2 upon passing through the first few stages of the phase sensitive detector. However, the use of an input transformer increased the impedence seen by 57 the lock-in by the transformer turns ratio squared. If the source impedence seen by the PAR 128 was between 250 kilo-ohms and 7 megohms at 1000 Hz, a noise figure of better than .05 db was achieved, that is the signal to noise ratio is only degraded 1%. Assuming a bolometer detector dynamic resistance of 275 ohms (section 3.7.5) and a turns ratio of 1:37.7,: the impedence seen by the lock-in amplifier is 400 kilo-ohms, giving a noise figure of better than .05 db. A 5 microfarad polyester capacitor was placed in series with the primary of the transformer to prevent a deleterious d-c magnetizing current from flowing. 3.6.3 Minimization of Microwave Oscillator Noise The microwave oscillators produced both AM and FM noise. Both types of noise were important for this experiment. AM noise was important because the detectors responded directly to it. FM noise was important because it gave the microwave oscillators spectral width (similar to the effect of optical slit width) which in principle broadened the absorption lines. FM noise was also important because it could be converted to AM noise in various ways and thus also be detected by the bolometers. The phase stabilization of the microwave oscillators reduced the FM noise to the point that it approached that of the microwave reference frequency. The AM noise was unaffected by the synchronization. At X- and P-bands the only noise reduction of the backward wave oscillators was brought about by the synchroni• zation process. The power levelling option increased the noise 58 seven fold at 1 kHz and was not used. At K-band, the noise of the klystrons, although perhaps not improved, was kept from getting worse. The output power of a klystron appears as a "mode"; the plot of output power versus reflector voltage (frequency) is parabolic. A klystron which is stationary in frequency can be placed on the top of the mode, which has zero power versus frequency slope, by suitably adjusting the reflector voltage and the mechanical size of the cavity. At this point, the minimum noise is obtained because a frequency variation gives the smallest power variation, that is, FM to AM noise conversion is least efficient. FM to AM noise conversion [69, 92] was important because the FM noise of a microwave oscillator is usually much greater than the AM noise [7, 41]. Any mechanism which affected FM to AM noise conversion did not need to be very efficient to produce a large increase in AM noise. The amount of extra AM noise produced by a klystron upon tuning off the top of the mode depends on the FM noise present and the power-versus-frequency slope. Mueller [66] found the AM noise (as an average .3 to 5 kHz off carrier) of a 2K39 increased a mere IV2 db upon going from the top of the mode to the half power point, but that of an X-13 klystron increased 19 db (nearly 100X). In the case of the OKI klystrons used here, the AM noise at a modulation frequency of 1 kHz was found to increase typically 8X within 3 MHz at 19 GHz. The AM noise generated by a klystron can be minimized across a frequency range by varying the size of the klystron 59 cavity in synchronism with the frequency sweep. The klystron cavity size affects only the noise in this case because the frequency is determined by the microwave reference frequency through the synchronizer servo loop and the reflector voltage. This noise reduction across a frequency range was achieved with an OKI 20V10 klystron by using an electromagnet to vary the size of the cavity. The electromagnet was secured to the tuning knob bracket; this acted upon a hinged iron plate which was linked by a rod to the large hollow lever adjustment screw on top of the klystron. In the region of magnetic saturation, it was found the required electromagnet current was a linear function of the frequency and to be without significant hysteresis. A variable gain, variable offset d-c amplifier (figure 3.5) was constructed. It allowed the frequency analog voltage of the signal averager to drive the electromagnet so that the noise minimum precisely tracked the frequency. The AM noise of the klystron was thus minimized across the sweep range. 3.6.4 Noise Cancellation Even with the klystron maintained on the top of the mode for minimum AM noise, the noise was still several times greater than that of the backward wave oscillators used at X- and P-band; for this reason a noise cancellation scheme was tried and successfully employed with the K-band klystrons. In this scheme the "klystron noise" signal detected before the cell was subtracted from the "methane absorption plus klystron noise" signal detected after the cell leaving only a "methane absorption" signal. If the detectors contributed no noise of 60 their own and in addition, the cell contributed no noise, then in principle the system should have worked very well. In practice, it did work very well, too. Each bolometer detector required approximately 10 mw of microwave power for optimum operation. This was supplied to the "noise reference" bolometer by suitably attenuating a ten percent fraction of the microwave power going to the cell. The remaining 90% of the power, which was severely attenuated by the four section - 13 meter cell, gave approximately 10% of the total power to the "sample" bolometer with associated attenuator. The subtraction was performed by bridging the output of the two bolometers across the primary of the Triad G-10 input transformer (figure 3.2) where the noise common to both detectors was rejected as a common mode voltage. The power levels going to the bolometers were adjusted with attenuators for best noise cancellation consistent with optimum bolometer bias power. The noise cancellation scheme worked so well that some• times the microwave power could be turned completely off with no detectable difference in the noise. Yet with the microwave power back on, when the microwave power going to either detector individually was removed, the noise voltage would increase by a factor of twenty or more. The noise cancellation scheme was so effective that the elaborate noise tracking scheme described in the preceding section made only a marginal difference. The detector noise, of course, was 1.4 times greater than 61 if only one bolometer were used, but this was no penalty compared with the benefits of noise cancellation. This simple noise cancellation scheme worked extremely well because it worked at audio frequencies and not at microwave frequencies where the exact equality of sample and reference arms necessary for noise cancellation over a frequency span is many times more difficult to achieve. A phase change of 180° in one arm of a microwave bridge turns noise cancellation into noise reinforcement. The same microwave frequency phase change with audio frequency balancing would have caused only a negligible change (1 kHz/10 GHz = 10"7) because the detectors were envelope detectors and didn't respond to individual microwave cycles. 3.6.5 Cell Noise The optimum noise cancellation discussed in the preceding section was not obtainable at every frequency with the equipment described. This was due to AM noise generated within the cell, which made the microwave noise before and after the cell of different character. Complete noise cancellation under this condition was not possible. For instance, the noise would increase typically 3X from optimum within 3 MHz; at any point with greater than optimum noise, conditions could not be improved by manipulation of either attenuator alone. The cell generated AM noise from FM noise. If there was FM noise, some of this was converted to AM noise if the cell had a transmission characteristic which was not flat or exhibited a non-linear phase shift [92]. The frequency region 62 of interest was 1 kHz each side of the microwave frequency at any particular time. Since the frequency was swept 3 MHz, it was desirable to have flat attenuation and a linear phase shift over this entire region. To a considerable extent the cell was "tuned" to minimize FM to AM noise conversion. This was done by inserting an X-band slide-screw tuner immediately before the tunable X-band sample detector mount and carefully tuning the three variables for minimum noise across the entire 3 MHz sweep range. With good tuning the balanced noise could be kept to within a factor of two of the detector noise over a 3 MHz sweep range. 3.7 Bolometer Detectors Bolometers were chosen as detectors in this experiment because of their long standing reputation as nearly pure Johnson (thermal) noise sources operating at an elevated temperature [5,24,91,92,93,97]. However, several theoretical papers [11,42,60] indicate that in addition to Johnson noise, noise due to statistical temperature fluctuation is generated. An attempt has been made by Li and Chen [51] to measure this, but a comparison with the work of Feher [19] will indicate more work needs to be done on this subject. Bolometer detectors are not normally used in microwave spectroscopy because of their low conversion gain, G, at the microwatt power levels used. The power levels are usually restricted because of saturation effects [97]. But bolometers could be used in this experiment because the power levels required for a good conversion gain were permissible due to 63 the small dipole matrix element of methane and the higher than usual cell pressure maintained. It will be shown that the same degree of power saturation given by one microwatt to the J = 0 -> 1 transition of OCS is given by 340 milliwatts to the most easily saturated component of the seven microwave methane transitions observed here. The bolometers purchased for use in this experiment were Narda type N610B/38B7 bolometers in a 1N23 style cartridge. Bolometers are microwave terminating resistors the resistance of which varies in proportion to the heat energy supplied by the microwaves. In order that the resistance change rapidly with variations in microwave power, the bolometers were constructed with a very small thermal capacity: the resistive element was a fine platinum wire typically 3/L> to l1 /2 microns (30 to 60 microinches) in diameter. Wire of such a small diameter is very difficult to handle, let alone draw [28]; therefore the platinum wire was encased in a silver jacket and drawn to the required diameter, mounted in the bolometer case and then enough of the silver was etched away with dilute nitric acid to give the proper resistance. The length of exposed platinum to give a room temperature resistance of 1 3 116 ohms varied from /2 mm for a /^ micron diameter wire to 2 mm for a IV2 micron diameter platinum wire. Although the resistance of a bolometer changes in accordance with the applied modulation, no signal voltage is generated. To recover a signal a d-c current was passed through the bolometer. In this experiment an approximately constant current was obtained by using a well shielded 12 volt 64 zinc-carbon battery and a 2000 ohm wire wound load resistance chosen for very low 1/f noise. The bolometers employed were designed to operate at a total resistance of 200 ohms. This gave the best impedence match for least reflections and optimum power transfer within the design frequency range of the tunable waveguide detector mounts used. At 200 ohms, the platinum wire is heated to 516°K or 1.7 times room temperature assuming a room temperature (300°K) resistance of 116 ohms. This figure is obtained from the approximate relation AT = AR(l+aT)/aR (3.1) where the 0°C temperature coefficient of resistivity, a = .0037 for platinum [44], R = 116 ohms and T = 27°C. 3.7.1 Bolometer Noise If the bolometer noise were purely thermal or Johnson noise, the noise voltage would be given by the expression E .ca = /4kTRAv (3.2) noise ' where k is Boltzmann's constant, T is the absolute temperature in °K, R is the resistance in ohms and Av is the bandwidth in Hertz. The noise voltage was expected to increase by a factor of 1.7 upon increasing the bolometer resistance from 116 ohms at 300°K to 200 ohms at 516°K, because both the bolometer resistance and temperature were increased by an approximate factor of 1.7. In practice this was roughly the increase in noise voltage found upon the application of 6 ma bias current in the absence of microwaves. The noise found was consistent with that given by a resistance of 200 ohms operated at approximately twice room temperature. 65 Feher [19] found using a d-c bias current of 8 ma, that the noise temperature of 20 bolometers obtained from four different manufacturers varied from 4 to 40 (one apparently- defective unit had a noise temperature of 1000). Some of these noise temperatures were significantly higher than the expected value of 2. Feher removed the air from one bolometer with a noise temperature of 10 and observed the noise temperature drop to the expected value of 2. Feher suggested that the unexpectedly high noise temperatures found for several bolometers might be due to air currents within the bolometer case. Li and Chen [51] have published a paper entitled "Noise in Metal Bolometers" in which they attempted to measure the statistical temperature fluctuation noise which has been shown theoretically [11,42,60] to be generated by a very small body in addition to Johnson noise. At a frequency of 1 kHz, they measured very high noise temperatures of 250 and 3500 at d-c bias currents of 6 and 8.7 ma, respectively. The bolometers used were three Narda N821B bolometers which are electrically equivalent to the N610B/38B7 bolometers used in this experiment. Li and Chen [51] comment that two of the three bolometers burned out before complete measurements were obtained. The work of Feher [19] suggests that the one bolometer on which they did complete all measurements was simply defective. Li and Chen [51] have stated "the heat loss of the (N821B bolometer) wire is mainly due to radiation". On the contrary, the heat loss was experimentally determined in our laboratory 66 • to be approximately 99% due to free air convection. A small hole was bored through the bolometer case and the bolometer was placed in a vacuum chamber. Upon evacuation only .166 mw of d-c power were required to heat the platinum wire to 200 ohms, but upon returning the bolometer to atmospheric pressure, it took 15.3 mw of power to bring the bolometer up to 200 ohms. Thus the combination of radiation and conduction down the leads accounts for only 1% of the heat loss. This is in agreement with the low emissivity of platinum at 492°K (219°C), which is only 3.9% [56]. Ironically, it had been shown previously by Langmuir [50] in 1912 and King [44] in 1914 that radiative losses account for only approximately 1% of the heat loss of a long platinum wire heated to approximately twice room temperature in still air. 3.7.2 Power Saturation Bolometers could be used in this experiment because the power levels required for a good detector conversion gain could be tolerated by methane without significant power saturation. The reduction in the peak absorption coefficient due to power saturation is given by [97]: Vax _ 1 (3.3) Y 2 o,max fl6iT SabvIt2TTT) 1 + 3ch where is the peak absorption coefficient in a radiation field of intensity I (in units of quanta per second per unit cross sectional area), y is the peak absorption o, max r r coefficient as the radiation intensity approaches zero, S&b is 67 the line strength [17,26,97], v is the frequency in Hertz, c is the speed of light, h is Planck's constant, x is the mean time between collisions, and t is the inverse rate of approach to equilibrium by collision (proportional to T). Good values for t and T are not readily available to calculate absolute degrees of power saturation, but they are proportional to the inverse of the line width, Av-1, which allows a relative calculation. The product of intensity and frequency, Iv, is proportional to (power/area). The reduction in the peak absorption coefficient due to power saturation can be written: Y max _ . 1 (3 4) Yo,max { S ^•power 1 + K [area:pressure2•(Av/P)2J where K is a constant dependent on the gas and (Av/P) is the pressure broadening parameter in MHz/torr. For a given molecule, pressure, cell and power level, the transition with the greatest value for S ^ will be the most easily saturated. Of the seven methane transitions observed here, the J= 18 E2 J = 18 E3 transition was the most easily saturated as seen by reference to the reduced line strengths tabulated in table IV of Dorney and Watson [17]: Sab/(0zy)2 = 6-73 xl°5: Using the value of 0xy given by Ozier [71]: 0xy = 2.41 X10"5 D, Sab = (6-73 X105)(2.41 X10-5)2 = S^IXIO-1* The line strength S ^ is given in terms of the reduced line strength S'^ by the relation [17]: Sab - Sa,bSab (3-5) 68 where g , *= 2 is the degeneracy common to the E levels. In cl, D this calculation only one. transition arising.from the doubly degenerate E level is of interest, so g_ , is taken as 1 cl t D and Sab = Sab = 3.91.X 10"•* in this case. The line strength used by Dorney and Watson [17]: 2 Sab = ^J(a|vf|b>| (3.6) faS is summed over all components a of a, and 3 of b and over the three space fixed directions, f; thus S&b corresponds to isotropic radiation. But the J = 18 transition under consideration has 2J' + 1 = 37 Stark components. The m = 18 components saturate most easily because the intensity of the individual Stark components varies as m2 [17,97]. A "worst case" calculation will be performed using an m = 18 Stark component. The line strength Sab (eq 3.6) is summed over all 37 Stark components for J = 18. J 'ab uab 2 m=-J SQK - S^^Tm (3.7) 1 1 f ( /s)J(J + D(2J + DS^ where S&b is the line strength for the m = 1 Stark component. The line strength for the m = 18 Stark component of the J = 18 E2 -»• J = 18 E3 transition of methane is: Qm=J _ T 2c m=l bab " J bab 1 = 3J[(J+l)(2J+l)]- Sab (3.8) = 3. 01X10"5 The line strength for isotropic radiation of the 69 J = 0 •*• 1 transition of OCS is [26]: SabS = ^ where the dipole moment, u, has been taken as .71 D [26]. For the same degree of power saturation, the expression: Sab-power (3.9) [area*pressure2•(Av/P)2J must be the same for the two gases, assuming AvOGS/t:OCS = AvCH,/t:CH,- (3'10) Using a pressure broadening parameter (Av/P) = 6 MHz/torr for OCS [26] and 2.7 MHz/torr for methane [76], the relation: pressureSab"'2•(Av/P)power 2 pressureSabS *P2•(Av/P)ower 2 5 3. 01X10- -power = .504(.001) 602-(2.7)2 62(6)2 gives a power of 340 mw required to give the same degree of power saturation in the m = 18 Stark component of the J = 18 E2 -»- J = 18 E3 transition of methane at 60 microns pressure as one microwatt produces in the J = 0 1 transition of OCS at 6 microns pressure. The observed J=18 E2 J=18 E3 methane absorption line was the superposition of (2J + 1) - m values. The lower values of m were more difficult to saturate and thus the 340 mw figure is conservative: at the power levels employed (^60 mw for the BWO's and 'vlOO mw for the klystrons) no significant amount of saturation for any of the observed microwave methane absorptions was expected. 70 3.7.3 Conversion Gain The previous calculation has shown that the microwave energy density in the cell could be maintained at a relatively high level without significant power saturation. This microwave power level permitted a good bolometer conversion gain, G, which has been given by Long [53] and Feher [19] as: 4P I2 (dR/dP^ 2 G = bolomet::er signal power output _ rf v ' ' bolometer signal power input R(l - I2dR/dP)2 P P G = 4(dR/dP)2 — (3.11) *1 _ J*c dR R dP, The conversion gain of the N610B/38B7 bolometers used can be shown to have a maximum in the vicinity of 10 mw microwave power. These bolometers have a typical room temperature resistance, Rc, of 116 ohms and typically require 15.3 mw of microwave plus d-c power to bring them up to a resistance of 200 ohms. The resistance of microwave bolometers in general follows the empirical relation [28]: 9 R = Rc + KP- (3.12) from which dR/dP « .9(R - R )/P = 5 ohms/mw (3.13) at 200 ohms. The conversion gain will be calculated as a function of microwave power, Pr£. while maintaining the resistance at 200 ohms. Substituting the values dR/dP = 5 ohms/mw and R = 200 ohms into the above equation, 4P JPJ 6- rf dc (3.14) 2 (40 - Pdc) 71 where Pr£ and P^c are expressed in milliwatts and P^c=15.3-Pr£ Using equation (3.14) the conversion gain, G, of the N610B/38B7 bolometers was calculated as a function of the microwave power for a total power of 15.3 mw. The results are tabulated in table VI which shows a maximum conversion gain in the vicinity of 10 mw microwave power. Table VI N610B/38B7 Bolometer Conversion Gain Prf (mw) G Pdc (mw) .001 .0001 15.3 .01 .001 15.3 .1 .0099 15.2 1.0 .087 14.3 10.0 .18 5.3 15.0 .014 .3 The bolometer signal is proportional to the product s G Pr£ which is tabulated in table VII. This gives a maximum signal strength in the vicinity of 9 mw microwave power and 6.3 mw d-c power at a bolometer resistance of 200 ohms. Experimental conditions were arranged such that 6 to 7 mw of d-c power were dissipated by the bolometers depending on the state of charge of the bias battery and 8 to 9 mw of microwave power were supplied to bring the bolometer resistance to 200 ohms. This gave near optimum conditions for a total bolometer power of 15.3 mw. 72 Table VII The Product G-P Prf (mw) G P ' rf PQC (mw) 1 .09 14.3 2 .30 13.3 3 .57 12.3 4 .89 11.3 5 1.17 10.3 6 1.42 9.3 7 1.62 8.3 8 1.74 7.3 9 1.80 6.3 10 1.76 5.3 11 1.63 4.3 12 1.41 3.3 13 1.09 2.3 14 .68 1.3 15 .17 .3 Table VIII shows the manner in which the relative bolometer signal output power increases as the microwave power is increased from 1 microwatt to 10 milliwatts by steps of ten. In this example, the microwave power increases by IO1*, but the bolometer signal output power increases by 1.8X107. On the other hand, the thermal noise remains constant because the total power is fixed at 15.3 mw. This underscores the importance of maintaining a high microwave 73 power level at the detector for optimum bolometer performance and demonstrates why bolometers cannot be used in microwave spectroscopy if power levels are restricted to microwatts because of power saturation. Table VIII Relative Bolometer Signal Output Power a Given by the Product (G-Prf) Microwave r p Power b'*rf 1 yw .0000001 10 yw .00001 100 uw .001 1 mw .09 10 mw 1.8 aFixed total power = 15.3 These calculations can be extended to show that the bolometer signal output can be increased, without an increase of microwave power, by increasing the d-c bias current. This has been done in table IX. The conversion gain increases 2.8 fold with a 20% increase in bolometer resistance. Since the platinum wire temperature is proportional to the resistance, the thermal noise voltage varies as the resistance using equation (3.2). With constant microwave power, the bolometer signal power increases as the conversion gain. The bolometer signal voltage increases as /R-P '. 7 since P . = E2 . /R. ° signal sig sig The expected increase in signal to noise with increasing bias 74 current is tabulated in table IX: a 53% increase in the signal to noise ratio is expected by simply increasing the bias current 46%,. Table IX Predicted Bolometer Behavior with Increased Bias Current R (ohms) . 200 220 240 P (mw) 15.3 20.3 25.0 total Prf (mw) 9.0 9.0 9.0 I (ma) 5.6 7.2 8.2 P (mw) 6.3 11.3 16.0 dc G .20 .38 .56 Relative thermal noise voltage 1.0 1.1 1.2 Relative signal power .... 1.0 1.9 2.8 Relative signal voltage 1.0 1.44 1.84 Relative signal/thermal noise 1.0 1.31 1.53 With the methane lines now firmly established and an adequate supply of bolometers the above method of improving the signal to noise ratio could be tested. This process cannot be carried too far because of the possibility of bolometer burn out or increased noise due to damaged solder joints [19,51]. 3.7.4 Bolometer Time Constant The bolometer time constant was important in that it gave an upper limit to the Stark modulation frequency. In addition it could have been important if it varied significantly with the microwave power level because the optimum phase adjustment 75 of the lock-in amplifier would have varied: if, for instance, the phase were adjusted with OCS at a bolometer resistance of 200 ohms, that same phase might not have been optimum while searching for a methane line with limited power. Cohn [13] state the "theoretical computation (of the time constant of a bolometer) is exceedingly difficult". Using the theory of the time constant of a hot-wire anemometer of Burgers [10] and Dryden & Kuethe [18] and the data of King [44], the time constant of a bolometer is calculated in appendix A. This calculation shows that whereas the bolometer time constant is 360 ysec when all the power supplied to the bolometer is d-c power, the time constant falls to 240 ysec under normal conditions of operation in this experiment, and rises slightly to 250 ysec in the case of limited microwave power which gives a bolometer resistance of only 180 ohms.' The phase retardation, <{>, due to the bolometer time constant, T (in seconds), is given by [61]: The bolometer time constant, T, is responsible for a reduction in detector signal voltage given by [61]: A = l/v/l+(27rfx)i! = .56 (3.16) The bolometer time constant of 240 ysec combined with a modulation frequency of 1 kHz gives a signal which is reduced in amplitude to 56% of what it would be if either the time constant or the modulation frequency were zero. The signal to noise ratio could be improved significantly by employing a bolometer with a smaller time constant or operating at a lower modulation frequency (provided 1/f noise of the microwave oscillator could be removed). 3.7.5 Bolometer Matching Impedence When matching the bolometer to the preamplifier of the phase sensitive detector, the dynamic impedence of the bolometer should be used for the best match. This can be found using the equivalent circuits which have been developed for bolometer detectors by Jones [43], Sorger [90], and Cohn [13]. The equivalent circuit of Cohn [13] is given in figure 3.6 where Ri and C are of dynamic origin and R is the d-c operating resistance. V is the open circuit signal voltage as the modulation frequency approaches zero and is the load impedence. AAA/V C 3 Figure 3.6 Bolometer Equivalent Circuit Given by Cohn [13] 77 Cohn [13] has given an expression for R^: 2R(R-R ) Ri = — (3.17) [Rc+(R/9>] which yields the value Ri = 244 ohms using a "cold" resistance, Rc, of 116 ohms and a d-c operating resistance, R, of 200 ohms. The sum of the d-c and dynamic resistances: R + Ri = 444 ohms is in good agreement with the d-c dynamic impedence [43], Z = 430 ohms, determined at an operating point of 200 ohms. The d-c dynamic impedence, Z, has been defined by Jones [43]: r, _ dE AE .099 / oo t, /o i ON HT = AT = TTJWI = 430 OHMS (3'18) The value of Z was determined using data obtained in our laboratory given in table X. Table X Bolometer Operating Values near 200 ohms R (ohms) E (volts) AE I (amps)' AI 197 1.711 .00868 .099 .00023 203 1.810 .00891 The resistance seen at the terminals of the bolometer at the modulation frequency, R^, is given by [13]: R = R + Ri = 200+^241 = 275 ohms (3.19) 111 l+(2TTfT)2 where f = 1 kHz, x = 240 ysec, and R and Ri are as previously defined. The value R^ = 275 ohms was used to match the bolometers to the phase sensitive detector.. 78 CHAPTER 4 EXPERIMENTAL CONDITIONS AND RESULTS 4.1 Introduction This difficult, challenging experiment can be broken into three parts. The first part was based on the quartic distortion constant, DT = 132,841 ±9 Hz determined by Ozier [70] and the (at that time) untried calculations of Dorney and Watson [17]. The difficulty of this experiment arose from the extreme weakness of the lines, and resulted in necessarily long search times. The initial search was conducted thirty megahertz each side of 10,570.2 MHz. Although this phase of the experiment did not give a methane absorption, it did produce a very sensitive, stable instrument. The second part of the experiment was based on the additional sixth degree theory of Kirschner and Watson [45] and the quartic distortion constant D^ and the sextic distortion constants H^ and determined by Curl [15]. With this data, the first four lines were observed within a period of a few months resulting in the publication by Holt, Gerry and Ozier [34]. The third part brought the experiment to a successful conclusion. Here the instrument was extended in frequency coverage and the noise cancellation scheme was employed. The theory was extended to eighth degree by Ozier [72] and three more lines were measured enabling an excellent fit of the 79 experimental lines to the theory [35]. 4.2 Initial Search Prior to this experiment, Ozier [70] had determined the J=2 E1 J=2 Fi splitting of methane [35]. From the splitting of 7.97046 ±.00054 MHz [35] and the f(J,K,t) a value of DT = 132,841 ±9 Hz was determined. At that time, D^,f (J, K , t) was considered to be the only tensor splitting term so the initial search was centered at 10,570.2 MHz. The values of the higher degree terms affecting the initial search frequency were unknown, but considering that at J=12, the rotational energy E = B0J(J+1) of methane is approximately 2.5X107 MHz, they were not considered to be insignificant. The search was arbitrarily restricted to thirty megahertz each side of 10,570.2 MHz because of the great amount of time required to search a given frequency region. To search 3 MHz required at least one day's time; if some feature in the 3 MHz wide sweep region looked interesting, it took on the average two more days of sweep time to reveal its true character. To search thoroughly the above region required the order of one month's time. During this period, many of the improvements described earlier were made to the instrument which warranted several more searches of this region. The J=12 line was later found to be at 10,321.91 MHz, approximately 250 MHz from the original search site. As seen in table XV, it was not a poor value of Drr, which held back initial success: use of only the well determined value of DT = 132,943.41 Hz gave a transition prediction of 10,578.34 MHz which is even further from the experimental frequency but 80 within the original search range. Clearly ignorance of the values of the sixth degree constants, and Hg,p, was a severe handicap. Moreover, even if both DT and had been known to current accuracy, initial success would not have been achieved. These distortion constants give a predicted frequency of 10,367.5 MHz, over 45 MHz away from the experimental frequency and outside the original 30 MHz sweep range. This review of the first part of the experiment is given to point out the experimental difficulty and the need to know where to look if success is to be achieved within a reasonable length of time. At least one year of persistent work would have been required to find the first line starting from 10,570.2 MHz using the 3 MHz/day search rate considered necessary. 4.3 Second Experimental Phase The success of the experiments of Curl et al [15,16] gave an improved value of the quartic distortion constant: Drj, = 132,933 ±10 Hz and the two sextic tensor distortion constants: H4T = -16.65 ±0.2 Hz and H&T = 10.2 ±1 Hz of methane. Using Ozier's molecular beam results [70] these distortion constants were corrected slightly (the corrections were inside Curl's error limits). With these distortion constants the frequency uncertainty of the microwave methane transitions between 7 and 15 GHz were reduced to less than 10 MHz each (see table XI). The J=12 E1 -»• J=12 E2 ground state Q-branch rotational transition of methane was found 81 within a week. The three remaining transitions listed in table XI were found within a few months. The four lines observed are reproduced in figures 4.1, 4.3, 4.4, and 4.5. The line strengths and experimental conditions are given in section 4.5. An energy level diagram based on [17] for the J=12 level of methane is given in figure 4.2, where the observed transition is indicated by the arrow, the irreducible representations, K, of the rotational wavefunctions are given in the point group 0, the parity is indicated by ±1, the levels within an irreducible representation are labelled in order of increasing energy by t, and the "zero" reference energy is given by the scalar terms in the Hamiltonian (approximately 25,000 GHz). All the transitions observed here were E type electric dipole transitions, which were the only ones to exhibit a first-order Stark shift. Table XI gives the observed transition frequencies with estimated experimental errors of the X- and P-band lines. Also listed in table XI are the predicted frequencies with standard deviations and differences based on Curl's distortion constants (as corrected): DT = 132,942 ±10 Hz, H4T = -16,8 ±.19-Hz, and H6T = 10.2 ±.9 Hz and the equation: v = [DT + H4TJ(J+l)]Af(J,K,t) + H6TAg(J,K,t), (4.1) where the frequency expression, Af and Ag were obtained from Kirschner and Watson [45]. Also listed in table XI are the predicted frequencies with standard deviations [52] and differences given by Tarrago et al [95]. Tarrago, Dang-Nhu and Poussigue [95] redetermined D^, H/rp and Hcrp based on the then total of seven radio frequency 12 Table XI Experimental Frequencies of X- and P-band AJ=0 Transitions of CH4 in the Ground Vibronic State together with Predictions based on the Distortion Constants D„, H,T, and HfiT of Curl [15] and Tarrago [95] t ti Experimental Predicted Freq Predicted Freq J KL+KL Diffa Diffa Frequency (MHz) Curl [15] Tarrago [95] 12 E^E2 10,321. 91 ±.1 10,322,. 0 ±4. 9 -.09 10,322. 20 ±1. 30 -.29 2 13 E^E 11,261. 37 ±.1 11,256., 0 ±6. 7 5.37 11,261. 03 ±1. 52 .34 2 3 14 E +E 7,861. 67 ±.1 7,855.. 8 ±8. 3 5.87 7,862. 01 ±1. 14 -.34 2 15 E^E 14,151. 81 ±.1 14,145. 3 ±8. 3 -6.51 14,151. 73 ±2. 19 .08 Difference = Experimental mean - Predicted mean oo N3 J I i : i : L 10321.0 10322.0 10323.0 FREQUENCY (MHz) Figure 4.1 512 Sample minus 512 Baseline 51.2 second Scans to give 1 2 l2 the J=12 E -*• J=12 E Transition of Ground State CEtt 84 JL +10 M X >- cr LU LU o- < o ° in k^> A Figure 4.2 The J=12 Rotational Energy Levels for 12CHi». Wg gives the zero reference energy. Figure 4.1 illustrates the transition. 11261 11262 11263 FREQUENCY (MHz) - Figure 4.3 512 Sample minus lk 1024 Baseline 51.2 second Scans to give the J=13 E1 •*• J=13 E2 Transition of Ground State 12CHi, CO Figure 4.4 1024 Sample minus 1024 Baseline 51.2 second Scans to give the J=14 E2 -> J=14 E3 Transition of Ground State 12CHi, 14151 14152 14153 FREQUENCY(MHz) *- Figure 4.5 512 Sample minus 512 Baseline 51.2 second Scans to give the J=15 E1 * J=15 E2 Transition of Ground State l2CHi, 00 88 and microwave lines. They obtained the values: DT = 132,858 ±13 Hz, H4T = -16.151 ±.062 Hz, and Hgrr, = 10.468 ±.027 Hz under the handicap of the time constant distortion of Ozier's line [34,35,71] and the lack of an eighth degree theory (although the sixth degree theory used did take into account the off diagonal elements of ft6). The differences between the experimental frequencies and the predictions of Tarrago et al are significantly less than those between the experimental frequencies and the predictions based on Curl's corrected constants as would be expected because four additional experimental frequencies have been used. However, the differences between the experimental frequencies and the predictions of Tarrago et al [95] are still up to three times greater than the experimental uncertainty. This is due to undetermined higher order distortion constants which have been assimilated into Drj, H^ and HgT. The size of the undetermined eighth degree terms together with the ability of effective lower order constants to predict frequencies in spite of these higher terms will be used in section 4.8 in evaluating the goodness of the six distortion constants evaluated there. 4.4 Final Experimental Phase The experiment was originally conceived for X-band (8-*--*-l2.4 GHz) and the spectrometer was constructed such that optimum performance was obtained while searching for.the J=12 and J=13 transitions. The J=15 transition, although outside of X-band at 14.15 GHz was found without difficulty using the original spectrometer without modification except 89 for the substitution of a P-band backward wave oscillator. The J=14 transition at 7.86 GHz apparently was beyond the range of the instrument with the BWO plug-ins available in our laboratory. The range of the instrument was lowered below 8 GHz in the manner described in chapter 3 to give the line at 7862 MHz. Phase lock stability was poor at 7862 MHz requiring the storage of each sweep in a buffer memory so that one mishap would not spoil hours (days) of work. This line was the most demanding because approximately 1000 scans were put into the signal averager one at a time. The lines at 18 and 19 GHz were definitely beyond the range of the original instrument; no d-c voltage could be applied to the microwave source to increase the frequency to the expected line frequencies. Klystrons had to be employed, and they proved to be much noisier than the BWO's used at lower frequencies, necessitating the development of a noise cancellation scheme. Phase stability had to be imparted to the klystrons. A synchronizer designed for use at lower frequencies on stand-by status in the department was modified to perform the task. The author wishes to thank Professors J.B. Farmer and C.A. McDowell for the use of their synchronizer. During modification of the spectrometer, Ozier [72] completed calculations of the tensor distortion energy of tetrahedral molecules carried to eighth degree in J and fit D,p, H^, Hgrj, and the octic constants L^, L^^, and Lg^- to the seven transitions then available. These were the molecular beam J=2 splitting [35] , the two 3=1 lines of Curl [15] and the four recently found microwave lines [34]. All 90 constants except Lgrp fit well, enabling the J=14 line at 19,288 MHz to be predicted to within a standard deviation (68% probability [52]) of ±1.5 MHz. A line was found within this 1.5 MHz range. No other transition was found in searches 20 MHz each side of the prediction. Fitting the above constants to eight lines allowed the 18 GHz J=16 and J=18 methane lines to be predicted to within a standard deviation, a, of ±.5 MHz. The J=16 line was found within ±.2 MHz. Finally 9 lines were used to predict the tenth, J=18, line to within a = ±.2 MHz. The final fit given in table XIV leaves little doubt as to the validity of the assignment and the theory. The three transitions observed in this phase of the experiment are reproduced in figures 4.6, 4.8 and 4.9. An energy level diagram for the J=14 level of methane in the ground state is given in figure 4.7. 4.5 Experimental Conditions Experimental conditions which varied from line to line are listed in table XII. All the microwave transitions of methane observed here were at a sample pressure of approximately 60 mTorr and a temperature of approximately 300°K. The microwave power level going into the cell was the full power level of the backward wave oscillator or klystron employed, typically from 60 to 120 milliwatts. The phase of the synchronous phase detector was set using an OCS sample, the gain was set at 1 microvolt and the output filter consisted of two simple R-C filters with a time constant of 1 second each in cascade. Figure 4.6 384 Sample minus 384 Baseline 51.2 second Scans to give 1 2 12 the J=14 E J=14 E Transition of Ground State CKk 92 A, A2 E F, £ 12 Figure 4.7 The J=14 Rotational Energy Levels for CH.». W£ gives the "zero" energy. Fig. 4.4 illustrates the upper, and fig. 4.6 the lower transition. 18562 ' 18563 FREQUENCY (MHz) — Figure 4.8 3328 Sample minus 3328 Baseline 51.2 second Scans to give VO U> the J=16 E2 -»• J=16 E3 Transition of Ground State 12CH^ Figure 4.9 3072 Sample minus 3A 4096 Baseline 51.2 second Scans to give the J=18 E2 •*• J=18 E3 Transition of Ground State 12Cm vO -P> Table XII Experimental Conditions 11 Y X10 Total Sweep AC(J,K,t) Stark Field Av for 'max Stark J Freq (MHz) (cm"1) Time (hr) [17] (kV/cm) m=l (kHz) 12 Ex+E2 10,321.91 5.09 77 4.025 1.06 104 13 11,261.37 3.54 73 12.857 .43 133 14 E2+E3 7,861.67 2.32 29 17.496 .34 145 14 E^E2 19,288.63 5.88 28 5.544 .85 115 15 E^E2 14,151.81 5.53 66 6.457 .81 127 16 E2-*-E3 18,562.40 2.06 47 17.494 .28 117 18 E2+E3 18,528.94 2.06 44 10.144 .47 116 VO 96 4.5.1 Line Strengths The line strengths in table XII are calculated in the manner of Dorney and Itfatson [17] with the same parameters, except for the line broadening parameter which has been given a value of 2.7 MHz/torr based on R branch transitions of CH3D [76] and the value of Drr,, which has been revised from .12 to .133 MHz. The lines observed in this experiment are the weakest ever observed with a conventional Stark modulated microwave spectrometer. 4.5.2 Sweep Times Each frequency sweep cycle consisted of two phases: the first in which the frequency was increased at a constant rate for 51.2 seconds, followed by a second phase in which the frequency was decreased at the same constant rate for another 51.2 seconds. The advantages of this "sweepback" mode were that time constant distortion did not effect the line center frequency and the synchronizer did not fall out of lock because of the application of a discontinuous frequency analog driving voltage. The total time spent sweeping a line varied because of line intensity, tests or experimental conditions. The lines at 10, 11, and 14 GHz received more sweep time, not out of necessity, but because it was here that tests were performed to test power saturation, adequate Stark modulation, and sample purity. Of the lines at 18 and 19 GHz, the 19 GHz line required less sweep time because of its greater line strength. The lines at 18 GHz required 6 days sweep time each taking into account the required baseline sweep time! The line at 7861.67 MHz, although of only 29 hours sweep duration, had to be done one sweep at a time. Although virtually the same strength as the 18 GHz lines, as good a signal to noise ratio was achieved in less time because the instrument was in its prime operating range (discounting the phase lock problem). 4.5.3 Stark Fields Stark modulation was supplied as a zero-based, 1 kHz square wave throughout the experiment. The Stark shift coefficients, C(J,K,t), for the E levels of methane have been calculated by Dorney and Watson [17]. Since the microwave electric vector was parallel to the Stark field, Am = 0 transitions were observed [97] which gave a Stark shift: x Avstark = AAC(J,K,t)mG ye (4.2) 5 -1 1 where AvStark is in kHz, A = 5.035 X10 kHz•D (kV/cm)- is the Stark effect constant [97], AC(J,K,t) is the absolute difference between the Stark shift coefficients for the levels concerned as tabulated by Dorney and Watson [17], m is the magnetic quantum number, 9xy = 2.41X10"5 D is the dipole z moment factor [17,71,100] and e is the electric field strength in kV/cm. Thus the components with biggest m show the largest shift. Since the Stark field had a direct bearing on the severity of the baseline slope, it was desirable to keep the Stark field to a minimum. Since the Stark component intensity varies as m2 for AJ = Am = 0 transitions [97], it was possible to leave a few of the Stark components under the line and yet 98 achieve significant modulation. Using the values of AC(J,K,t) from Dorney and Watson [17] listed in table XII, the frequency shifts of the m = 1 components are calculated from the relation: AvStark = 24.27[AC(J,K,t)]e (4.3) The assumption will be made that if a Stark component was shifted more than 300 kHz (IV2 times a typical line half-width at half-height) it modulated the line, but a shift of less than 300 kHz resulted in no modulation. From table XII the Stark components shifted least were those of J=12. In this case, the relative total Stark intensity was given by the sum of the squares of the integers up to and including 12, which is 650. Since two Stark components were moved less than 300 kHz, the relative Stark modulation which was ineffective is l2 + 22 =5. The percent Stark modulation was thus: 100(650-5)/650 = 99+% for J=12. The Stark modulation was more than 997o effective for the six other microwave transitions observed. 4.6 Line Identification A great deal of effort was expended to prove that the transitions were due to methane and not to a small amount of impurity with a relatively large permanent dipole moment. Impurities could be outgassing from various parts of the cell (perhaps remnants of samples used in prior experiments) or they could be leaking into the cell from the room. These concerns were heightened because the cell did show a definite, slow pressure rise during long scan times. These possibilities 99 were eliminated by evacuating the cell, isolating it and allowing the pressure to increase spontaneously over a long period of time to 60 mTorr at which time after a suitable search period, observing that no spectrum appeared. The impurities could have been purchased from the Matheson Co. with the sample. The impurity level in the Ultra-High Purity methane purchased was given as 3 parts per 10,000 maximum. The sample was frequently solidified with liquid nitrogen so that accumulated air (along with methane vapor) could be pumped off, but this would not remove the great preponderance of impurities which have a negligible vapor pressure at the temperature of liquid nitrogen. Instead of removing the impurities from the solid methane, they were kept there and the usable methane vapor pressure at liquid nitrogen temperature were used to give an absorption spectrum. This test eliminated the possibility of virtually every effective impurity except polar permanent gases such as carbon monoxide. The possibility that the lines were due to impurities were further reduced by the following considerations. The line strengths agreed with the calculated line strengths to within a factor of two, based on a comparison with known OCS line strengths [46], and taking into account the power saturation of OCS. The line widths agreed to within a factor of 2 of those expected from a pressure of 60 mTorr and a CH3D broadening parameter of 2.7 MHz/Torr [76]; the line widths were approximately eight times less than those of OCS under identical conditions indicating a very small transition 100 moment. The Stark fields required for complete modulation of the lines followed those tabulated by Dorney and Watson [17]. More proof that the lines were due to the centrifugal distortion dipole moment of methane is given by the excellent fit of all the microwave lines to the theory carried to eighth degree as shown in the next section. Subsequent to this experiment, Ozier, Lees and Gerry [77] have had success obtaining the 13CHi| spectrum slightly shifted in frequency from the above, giving additional proof that the transitions observed are due to methane and not an impurity. 4.7 Determination of Distortion Constants In addition to the seven transition frequencies determined here, the two J=7 double resonance transitions determined by Curl [15] and the J=2 ortho-para splitting determined by Ozier [35,70] were used in a weighted linear least squares fit to determine the six distortion constants listed in table XIII: DT, H^, H^T, L^T, L^T and LgT. The sextic distortion constants, and'H^, are smaller than the quartic distortion constant, D^., by an approximate factor of 10"*. The octic distortion constants, L^, LgT, and Lgrp, in turn are approximately 10K times smaller than the sextic distortion constants. This ratio between distortion constants which is close to that of Born-Oppenheimer [6] indicates that the distortion constants derived have a physical significance and are not merely fitting parameters [99]. The experimental frequencies together with experimental errors and differences between experimental frequencies and Table XIII Centrifugal Distortion Constants of .12CHi, in the Ground State Constant Value (Hz) 3 °LS DT 132 943. 41 .71 3.7 H -16. 983 9 6 4T .007 .023 H6T 11. 034 2 .008 6 .019 L4T 002 027 .000 024 .000 085 L —. .000 .000 12 6T 002 677 035 L —. 003 00 .000 18 .000 27 8T Standard deviation in Hz obtained by fitting frequencies to eighth degree theory by weighted linear least squares. Estimated standard deviation of absolute error in Hz. 102 frequencies predicted on the basis of the above distortion constants are listed in table XIV. The frequencies used in the linear least squares fit were weighted according to the square of the reciprocal of the experimental uncertainty to arrive at the standard deviations, cr^g, listed in table XIII. A comparison of experimental uncertainties will reveal that the J=2 splitting was given approximately 300,000 times the weight of an X-band microwave transition in spite of the fact that the J=2 experimental error expressed as a percentage is approximately 7 times that of an X-band microwave transition. This heavy weight on the J-2 splitting is justified by consideration that the energy of the J=2 levels is affected only by three of the six distortion constants: for J=2, ET = [DT + 6HArp The spherical tensor operators in the Hamiltonian of rank 6, 8, and higher do not affect the energy of the J=2 rotational level for the following reason. The J=2 rotational wavefunctions are spherical tensors of rank 2. The energy is a scalar quantity (spherical tensor of rank 0). A spherical tensor operator of rank I in the Hamiltonian can affect the energy of the molecule only if a vector of length I can combine with two vectors of length J corresponding to the rotational wavefunctions to give a triangle equality [84]. This is because the energy is given by E = <^a|P^|b^ which requires : rank 0 - rank J + rank I + rank J : where the addition is vector addition. If I > 2J, the triangle cannot be closed and the energy corresponding to the tensor operator ft. is zero for the J value under consideration. For J=2, Table XIV The AJ=0 Transitions of Ground State 12 J Exp. Freq. (MHz) Diff? 2 7.97046 ±.00054a .00003 7 F2+F2 423.02b ±.02 .008 2 1 7 FJ+F2 1,246.55b ±.02 -.003 1 2 14 E2+E3 7,861.67 ±.1 -.003 12 E*+E2 10,321.91 ±.1 -.003 13 EX-*E2 11,261.37 ±.1 .000 15 E^E2 14,151.81 ±.1 .003 18 E2+E3 18,528.94 ±.2 -.004 16 E2+E3 18,562.40 ±.2 .003 14 E^E2 19,288,63 ±.2 .000 a Ozier [70,35] b Curl [15] c Difference (MHz) = Predicted Frequency -Experimental Frequency based on the distortion constants given in table XIII 104 only fti» affects the energy. Regardless of whether the J=2 splitting is left out of consideration, the six distortion constants so obtained fall within the range of the least square deviations, aLg, listed in table XIII. Moreover, a fit to 8 lines omitting the high J (J=16 and 18) lines, or a fit to only the 7 microwave lines determined here, or a fit to all 10 lines weighted equally gave, in all cases, distortion constants within the least square deviations listed in table XIII. The fact that the constants did not change appreciably when one, two, or three transitions were left out of consideration shows that no one line (J=2 in particular) was distorting the fit. The fit was performed in the manner described by Ozier [72]. During the first iteration, all distortion constants except D^, were set equal to zero. The Hamiltonian then assumed the form: WT = DTJU (4.4) which was diagonalized to give the "zeroth-order" wavefunctions. The operators ftij, fi6 and ft8 then gave the diagonal matrix elements: ^u)0o = f, (^&)0 . - g and - = h. For each transition frequency a linear equation was written. For instance, for J=12, 2 2 v = [DT + H4TJ(J+1) + L4TJ (J+l) ]Af (4.5) + [H6T + L6TJ(J+l)]Ag + LgTAh 6 3 s 3*6322 X10 = 79570 DT + 12413 X10 H4T + 19364 X 10 L4T 3 3 - 4676 X10 H6T - 72946 X10* L6T - 20853 X10 LgT where the distortion constants are in Hertz. 105 Using from 7 to 10 linear equations of this type in 6 unknowns (only 3 unknowns for J=2), the values of the unknown distortion constants were determined by weighted linear least squares fit. The "zeroth-order" distortion constants thus determined were used as the basis for the second iteration. The Hamiltonian was set up using all six distortion constants and diagonalized to give a new set of wavefunctions. Matrix elements were then calculated in terms of the new wavefunctions, new linear equations were set up and another improved set of distortion constants determined by weighted linear least squares. The iteration was continued until the individual constants converged. For all the subsets of lines considered, the third iteration gave constants which agreed with those of the second iteration to within 10% of the standard deviations of the second iteration. 4.8 Influence of Dectic and Higher Order Distortion Constants It is interesting to consider the frequency contributions of the terms associated with the various distortion constants as a function of J. These are given in table XV. Table XV illustrates the expected: as J increases, the importance of higher degree terms increases [D is fourth, H is sixth and L is eighth degree]. Within a given degree, the terms associated with the lower rank tensor (given by the subscript 4, 6, or 8) are generally more important. But the trends are not always smooth in this irregular spectrum. Note the abnormally large values for the J=16 terms associated with . Table XV Frequency Contributions (in MHz) of the Terms Associated with the Distortion Constants Listed in Table XIII to the Transition Frequency t t' Exp Freq J DT H H L L L K +K 4T 6T 4T 6T 8T 7.970495 2 E1 7.976605 -.000611 -0- .000044 -0- -0- 2 2 423.02 7 F2 ->F 425.76 -3.05 .30 .02 -.004 -.002 1246.55 7 1255.32 -8. 98 .15 .06 -.002 .002 10321.91 12 E^E2 10578.34 -210.82 -51.55 3.92 '. 1.95 .06 11261.37 13 E*+E2 11452.13 -266.27 73.18 5.78 -3.23 -.22 7861.67 14 E2-*E3 7977.08 -214.01 98.60 5.36 -5.02 -.34 19288.63 14 E*+E2 19886.97 -533.53 -82.79 13.37 4.22 .39 14151.81 15 E^E2 14663.55 -449.59 -78.97 12.88 4.60 -.65 18562.40 16 E2->E3 18884.62 -656.22 335.68 21.30 -22.15 -.83 18528.94 18 E2+E3 19411.73 -848.13 -72.63 34.62 6.03 -2.68 107 The fourth and sixth degree distortion constants determined here can be compared with those determined by Tarrago et al [95] (indicated by a prime ' for clarity here). The distortion constants determined by Tarrago et al using the 3=2 transition of Ozier [35,70], the two 3=1 transitions of Curl [15] and the J=12,13,14,15 transitions reported here [34] are: Dj, = 132858 ±13 Hz, H4T = -16.151 ±.062 Hz and H£T = 10.468 ±.027 Hz. In this work, DT = 132943.41 ±.71 Hz (removed 6.54 a' from D^), H4T = -16.9839 ±.0076 Hz (removed 13.43 a' from H4T) and H6T = 11.0342 ±.0086 Hz (removed 20.96 a' from H£T). The reason the distortion constants determined here are so far from those determined by Tarrago et al [95] is because they incorporated the eighth and higher degree constants into D^,, H4T and H^ whereas here the eighth degree constants have been separated out (but tenth and higher degree terms are still incorporated). Table XVI gives predicted frequencies with standard deviations based on the constants given here and those given by Tarrago et al [95], The differences between the predicted and experimental frequencies are also listed, as are the total eighth degree term contributions to the frequencies (obtained from table XV). The effective distortion constants determined by Tarrago et al predict the frequencies reasonably well up through the maximum J value used in the fit (J=15) in spite of some large eighth degree contributions. Above J=15, the ability of Tarrago's constants to predict frequencies disintegrates rapidly. The six distortion constants determined here are a great Table XVI Predicted Transition Frequencies based on the Distortion Constants of Tarrago e [95] and the Distortion Constants determined here. , Experimental Prediction (MHz) Diffa Total Prediction (MHz) Diffa J Kt->Kt 8th , Freq. (MHz) based on this work degree based on Tarrago [95] 12 E^E2 10321. 91 1 10321. 91 + .014 -.003 5. 93 10322. 20 ±1. 30 .29 13 EJ+E2 11261. 37 1 11261. 37 + .014 .000 9. 23 11261. 03 ±1. 52 — .34 14 E2->E3 7861. 67 + _ 1 7861. 67 + .02 -.003 10. 72 7862. 01 ±1. 14 .34 14 E^E2 19288. 63 2 19288. 63 + .035 .000 17. 98 19288. 54 ±2. 76 — .09 15 E^E2 14151. 81 1 14151. 81 + .03 .003 18. 13 14151. 73 ±2. 19 — .08 2 3 16 E ->E 18562. 40 + # 2 18562. 40 + .06 .003 44. 28 18565. 42 ±3. 14 3 .02 18 E2+E3 18528. 94 2 18528. 94 + .22 -.004 43. 33 18520. 37 ±3. 64 -8 .57 Difference (MHz) = Predicted frequency - Experimental frequency Total eighth degree contribution equal to the sum of the absolute values of the terms associated with L^, L^T and LgT given in table XV. 109 improvement over the three determined by Tarrago et al [95]. The predicted frequency is equal to the experimental frequency to all the significant figures used in this experiment. The standard deviations are two orders of magnitude better for all the J values used by Tarrago et al. But just as the comparison of this work with that of Tarrago et al has shown (section 4.3), the absolute values of the six distortion constants tabulated in table XIII cannot be expected to be within the least square standard deviations, o^g, listed in table XIII because of the inclusion of tenth and higher order terms. Using trends shown in table XV, the magnitudes of the four dectic terms can be estimated as a function of J. For instance, the ratios DT/H4T and E^/L^rj, can be used to give an approximate ratio for L^/M^y. which can be used to approximate The contribution of terms in M^ and Mgy. are estimated to be of the order of 0.1 MHz for J=12 and 1 MHz for J=18; while the contributions of Mg^ and are estimated to be negligible. Based on these order of magnitude estimates, the absolute errors are given in table XIII. Although in some cases is several times a^g, the least well determined constant, Lgy., is not significantly poorer in terms of (known to within 9%) than in terms of a^g (known to within 6%). Based on this, no attempt to determine any of the dectic constants seemed warranted. Even though the contributions of the dectic terms up to J=17 are estimated to be up to 1 MHz, the effective constants determined here should be able to predict frequencies for 110 transitions up to J=17 to within 0.1 MHz. As J goes above the highest value used in this fit (J=18), the ability of the six constants determined here to predict accurately is expected to get poorer rapidly. Table XVII lists term values in MHz for all the rotational levels of methane up to J=21 with respect to the energy given by the scalar terms in the Hamiltonian. Here takes a maximum value of 2.4 MHz. Use of this table should give transitions to within 0.1 MHz for J < 16, to within 0.4 MHz for J < 18, to within 1.3 MHz for J < 20, and to within 3.4 MHz for J < 22. Ill Table XVII Tensor Distortion Energy E^ for 12CH^ in the Ground Statea t J Err. (MHZ) aE (MHz) J K* Err. (MHz) 0f K E 0 Ax -0- Fi1 -855.411 0. 003 2 1 Fi -0- 8 F2 1,323.223 0. 003 E 2 1,173.921 0. 003 2 F2 3.18820 0.00002 Fi2 678.946 0. 002 E -4.78230 0.00002 1 F2 112.200 0. 002 E 1 -1,393.578 0. 004 3 A2 47.78042 0.00023 Fi1 -1,451.344 0. 004 F2 7.96885 0.00004 -1,549.761 0. 004 Fi -23.89566 0.00011 Ai 9 A2 2,202.189 0. 005 4 F2 103.41425 0.00045 2 F2 1,966.574 0. 004 E -15.85721 0.00008 Fi3 1,665.052 0. 003 Fa -55.69044 0.00024 Ai 944.442 0. 004 Ai - 111.45700 0.00049 Fi2 174.479 0. 002 5 Fx2 218.4807 0.0009 E -14.596 0. 003 1 E 166.7862 0.0007 F2 -2,371.901 0. 005 1 F2 -111.0389 0.0005 Fi -2,473.351 0. 005 Fa1 -218.6326 0.0009 3 10 F2 3,157.929 0. 006 6 Ai 499.9253 0.0017 E 2 2,753.177 0. 005 Fi 380.8742 0.0013 Fi2 2,389.525 0. 004 2 F2 246.0790 0.0009 Ai 1,284.364 0. 007 1 A2 -260.9141 0.0011 Fi 126.668 0. 002 2 F2» -404.9655 0.0014 F2 -299.909 0. 004 E -452.4870 0.0016 A2 -3,729.734 0. 007 1 F2 -3,820.052 0. 007 2 7 Fi 814.164 0.002 1 E -3,861.732 0. 007 E 558.015 0.002 2 3 F2 391.136 0.001 11 Fi 4,535.927 0. 007 2 A2 80.229 0.001 E 4,246.546 0. 006 1 3 F2 -748.642 0.002 F2 3,583.832 0. 007 112 (MHz)+ a. •(MHz)- 2 3 Fi 1, 761. 548 0. 009 F2 5, 996. 395 0. 012 E 1 353. 813 0. 003 E 2 3 396. 758 0. 013 2 2 F2 634. 969 0. 004 Fi 2 631. 683 0. Oil A2 -1, 072. 371 0. 007 Ai 1 417. 734 0. 013 1 1 F2 -5, 706. 826 0. 008 Fi -4, 478. 110 0. 014 1 2 Fi -5, 777. 211 0. 008 F2 -4, 861. 682 0. 010 A2 -15 840. 180 0. 031 2 Ai 6, 553. 297 0. 009 1 F2 -15 874. 791 0. 032 Fi 3 6, 190. 678 0. 008 E 1 -15 891. 872 0. 032 3 F2 5, 722. 754 0. 008 2 A2 4, 916. 127 0. 013 A2 15 566. 36 0. 02 2 «» F2 2, 490. 198 0. 009 F2 15 110. 80 0. 02 E 2 2. 043. 089 0. 010 Fi A2 4, 634. 261 0. 013 1 Fi -21 ,167. 20 0. 07 2 F2 3, 188. 315 0. 007 3 Fi 2, 227. 735 0. 009 F2 19 830. 81 0. 04 Ai -2, 300. 808 0. 013 E 3 19 271. 74 0. 05 2 Fi -2, 821. 418 0. 008 Fi it 18 683. 31 0. 05 1 E -3, 033. 251 0. 007 Ai 2 11 671. 59 0. 03 1 3 F2 -11, 617. 690 0. 014 Fi 9 936. 68 0. 04 1 3 Fi -11, 662. 556 0. 015 F2 8, 309. 86 0. 04 A2 3 ,100. 74 0. 07 F2 11, 678. 606 0. 014 2 F2 1 ,316. 30 0. 05 3 E 258. 425 0. 015 2 11, E 709. 34 0. 04 3 Fi 10, 539. 840 0. 021 2 Fi -10,389. 76 0. 02 113 (MHz)->-o-T (MHz)- 1 l F2 -10,671. 70 0. 02 F2 -44, 813. 78 0. 53 E 1 -27,620. 56 0. 15 E l -44 818. 90 0. 53 Fx1 -27,630. 19 0. 15 5 1 32 Ai -27,649. 34 0. 15 Fi 39 289. 0. 30 E 3 38 768. 65 0. 31 5 25,212. 81 0. 5 17 Fi 08 F2 38, 224. 91 0. 33 E 3 2 24,730. 41 0. 09 A2 24 388. 52 0. 27 4 F2* 24,064. 01 0. 10 F2 22 369. 99 0. 28 Fi* 14,508. 30 0. 08 Fi it 20 088. 76 0. 28 E 2 11,576. 04 0. 09 Ai 14, 797. 03 0. 30 Fz3 10,083. 40 0. 09 Fi 3 10 126. 98 0. 30 2 A2 7,268. 66 0. 09 E 8 858. 77 0. 31 2 3 F2 592. 50 0. 09 F2 -4 347. 94 0. 22 Fi 3 -569. 15 0. 08 Fi 2 -5 374. 93 0. 23 Ai -14,569. 26 0. 02 E 1 -26 666. 72 0. 15 2 2 Fi -14,799. 92 0. 02 F2 -26 735. 82 0. 15 E 1 1 -14,910. 06 0. 03 A2 -26, 871. 25 0. 15 1 28 0. 1 F. -35,454. 29 F2 -55 856. 23 0. 92 Fi1 -35,468. 39 0. 29 Fi 1 -55, 863. 60 0. 92 18 A 2 5 i 31,896. 50 0. 16 20 F2 48, 207. 28 0. 53 Fi* 31,417. 67 0. 17 E * 47, 747. 63 0. 54 5 5 F2 30,866. 96 0. 18 Fi 47, 195. 85 0. 56 A 2 30,206. 78 0. 19 2 F2* 29 474. 97 0. 46 3 F2* 18,366. 30 0. 15 E 26 378. 01 0. 46 3 E 17,080. 93 0. 16 Fx* 24 115. 55 0. 47 Fi3 12,678. 37 0. 18 Ai* 18, 097. 89 0. 56 3 3 F2 8,543. 50 0. 18 Fi 11 180. 89 0. 48 2 E -1,448. 01 0. 15 3 F2 8 690. 79 0. 50 2 0. Fi -2,048. 51 14 A2 -7 780. 74 0. 31 1 -3,067. 85 0. 14 2 Ai F2 -8 676. 10 0. 32 Fi1 -20,060. 82 0. 06 E 2 -9 076. 79 0. 33 2 2 F2 -20,236. 33 0. 06 Fi -34 ,666. 31 0. 31 1 -44,803. 52 0. 53 1 32 A2 F2 -34 771. 79 0. FC J K Err, + (MHz) + aE E 1 -68, 775.40 1.52 Fi1 -68, 778.03 1.52 Ai1 -68, 783.30 1.52 2 21 A2 58, 795.40 0.87 5 F2 58, 363.19 0.89 Fi6 57, 892.50 0.91 Ai2 57, 373.15 0.93 Fi5 36, 119.35 0.74 E 3 34, 361.96 0.75 F2* 29, 966.21 0.73 Fi* 21, 217.52 0.89 E 2 10, 399.54 0.72 3 F2 8, 883.89 0.76 1 A2 6, 442.04 0.82 2 F2 -13, 222.60 0.43 Fi3 -13, 894.80 0.46 Ai1 -44, 186.52 0.60 Fi2 -44, 266.66 0.60 E 1 -44, 306.36 0.61 1 F2 -83, 749.82 2.43 Fi1 -83, 753.56 2.43 a Og is the standard deviation calculated by least squares analysis. 115 CHAPTER 5 CONCLUSIONS This experiment has produced the first microwave spectrum of a spherical top molecule using a general purpose instrument. The method does not rely on fortuitous coincidences and thus can be used to investigate the entire class of spherical tops provided the microwave transitions of interest are of sufficient intensity and the search region is reasonably well defined. The need to know where to look has been amply demonstrated by this experiment. The three octic distortion constants L^, Lgy. and LgT of 12CHit have been determined for the first time in this experiment. The least well determined distortion constant, Lgy, is known with an accuracy of approximately 1 part in 10 which is the same as Dy. was four years ago [17]. The six distortion constants determined here (table XIII) have permitted the accurate prediction of all the Q-branch transitions of methane to J=21 (table XVII). This accurate prediction could possibly result in the detection of methane in interstellar space using the techniques of radio astronomy. Interest has even been expressed in using the microwave absorptions of methane to identify this gas in oil well bores [23]. The six tensor distortion constants determined here (table XIII) will also be useful in the determination of the scalar distortion constants of methane when the transitions which depend on both the tensor and scalar distortion constants 116 are measured with sufficient precision (using laser techniques possibly). The transitions observed here should aid in the setting-up of new microwave spectrometers designed to detect absorptions due to very weak transition moments because of the difficulty of power saturating the methane lines. It is hoped this experiment, which successfully employed a simple microwave source noise cancellation scheme using thermal detectors combined with a modern system of data collection has served to advance the art of the detection of weak signals buried in noise. Although the microwave transitions reported here are the weakest ever observed with a conventional Stark modulated microwave spectrometer, it is not felt the ultimate limit has been reached. Longer path lengths, quieter detectors, and equipment of improved design should yield transitions even weaker than those observed here. The theory of the ultimate sensitivity of a microwave spectrometer has been summarized by Gordy [26], Townes and Schawlow [97] and Strandberg [91]. It is interesting to compare the sensitivity achieved with the spectrometer used in this experiment with an "ultimate" sensitivity calculated by Strandberg [91]. Strandberg's "ultimate" sensitivity is 10"11 cm-1 based on the following assumptions: a) a bandwidth of .01 Hz equivalent to a time constant of 314 seconds (RC = TT/B, ref.[91]), b) a cell path length of 60 m, c) a signal to noise ratio of 1, and d) a cell attenuation coefficient of 10"3 cm-1. In order to compare sensitivities, the amount of time required to scan a typical methane line 117 with Strandberg's hypothetical spectrometer must be determined. From figure 15 of reference [91] it appears the fastest tolerable sweep rate is that in which half the line width at half-height is swept in one time constant. Since a typical methane line half-width at half-height was 1 / I* MHz, the fastest sweep time would be 12 time constants or approximately 1 hour for maximum tolerable distortion of a methane line in a 3 MHz scan. If Strandberg's cell length were reduced approximately five fold from 60 meters to the 13 meters used in this experiment, approximately 52 = 25 hours of sweep time or 1 day would have been required to achieve the same "ultimate" sensitivity of 10"11 cm-1. Furthermore if a signal to noise ratio of 3 instead of 1 were desired, the sweep time would have to be increased to 32 = 9 days for a sensitivity of 10"11 cm-1. If Strandberg's cell attenuation coefficient (IO-3 cm-1) were increased to correspond with the best encountered in this experiment (1.4 X10-3 cnrt1), the "ultimate" sensitivity would have been degraded to 1.4 X10"11 cm-1 (equation X.8, reference [91]). Two days of continuous sample sweep time were required to observe the weakest methane lines (2 X10"11 cm-1) at a signal to noise ratio of 3. Four days of sweep time would have been required to reach the above calculated sensitivity of 1.4 X10"11 cm-1. These calculations show that Strandberg's "ultimate" spectrometer modified with a shorter, more lossy cell would require 9 days to produce the same signal to noise ratio which the spectrometer used in this experiment could 118 produce in 4. In other words, the experimental spectrometer used here was in a sense "better than ultimate". This can be explained in terms of another assumption of Strandberg: that the performance is limited by the source noise of a very low noise klystron, whose noise power is 160 db/Hz below the carrier (an exceptionally good klystron). The spectrometer used here employed noise-cancellation which reduced the effect of klystron noise significantly. The effect of klystron noise may have been reduced to the point that it was no longer the limiting factor. 119 APPENDIX A BOLOMETER TIME CONSTANT The bolometer time constant can be calculated using the hot-wire anemometer time constant theory of Burgers [10] and Dryden & Kuethe [18] together with the data of King [44]. The operation of a hot-wire anemometer is simple: air flowing over a hot wire removes heat energy cooling the wire, as the wire is cooled, the resistance falls which can be used to measure the air velocity. The effectiveness of forced convection cooling of a hot wire can be seen by blowing on the electric heating element of a domestic toaster. Hot-wire anemometers and bolometers are quite similar in that they both consist of fragile platinum wires of comparable dimension [57] heated approximately the same degree above ambient with air convection the dominant heat loss mechanism. The bolometer heat loss is due mainly to free-convection, whereas the hot-wire anemometer heat loss is due mainly to forced- convection. Hot-wire anemometer theory can be applied to bolometers by setting the forced air velocity to zero which leaves only free-convection terms. The heat loss of a bolometer due to free-convection has been shown in section 3.7.1 to be approximately 99% of the total heat loss. The rate of change of heat energy of a hot-wire anemometer has been given by Dryden & Kuethe [18]: 2 dH/dt = i R- (K + Cv^KT- T0) (A.l) where C and K are constants, V is the air velocity, T is the 120 temperature of the wire, and To is the temperature of the air in °K. In order to apply this equation to a bolometer, the air velocity must be set equal to zero and an additional term must be added to account for the heat supplied to the wire by the microwave power. When this is done, the foregoing equation becomes: 2 dH/dt = i R + Prf - K(T - T0) (A.2) In the steady state, dH/dt = 0; this yields: 2 i R + Prf = K(T - To) P = K(T - To) (A.3) where K has been given by King [44] for a wire of diminishingly small diameter: K = 2.5 X10-lt[l + .00114(T — To)] (A.4) in units of watts per °K per unit length. Before proceeding to the time constant calculation, these equations can be used to derive the "empirical bolometer resistance law " , 9 R = R0 + LP- (A. 5) given by Griesheimer [28], where L is a constant which varies from bolometer to bolometer. The constant K is evaluated at two points 100° and 200°C above ambient. The power dissipation per unit length of wire is then calculated from equation (A.3). These values are tabulated in table XVIII. Let the resistance of a bolometer be given as an exponential function of P: R = Ro + LPX (A,6) where the exponential x is to be determined. The value of R 121 Table XVIII Bolometer Heat Dissipation watts T - To (°C) K p [ watts 'C-unit lengthJ [unit length 100 2.785 X 10-* 2.785 X 10-2 200 3.07 X 10"* 6.14 X 10"2 can be obtained from the relation giving the temperature dependence of resistance: R = R0[1 + a(T - T0)] (A.7) where a is the temperature coefficient of resistivity (°C)_1. Combining the two preceding equations gives: X R0a(T -T0) = LP X Roa = LP /(T -T0) (A. 8) The left hand side of the above equation is a constant for the two points 100° and 200°C above ambient. Using values from table XVIII, the following equation is obtained: LP loo _ LP2 o o ToU 2W (A. 9) -2\X = 1/. vin-2\X 2(2. 785 X10"2)A = tc(6.14X10-2) (2.2046)x = 2 which gives the value: x = .877 Thus the resistance of a bolometer is given by: R = Ro + LP-87 7 (A. 10) in excellent agreement with equation (A.5). The above derivation is based on expressions and constants for the free convective heat loss of platinum wires given in the excellent paper by King [44]. 122 The bolometer time constant in seconds, x, can be calculated from an equation given by Dryden & Kuethe [18] modified to take into account additional incident microwave power: x = ms P = K' (T — To) = K'(R- R0)/aR0 (A.12) 2 where K' = KA. Using (A.12) and the relation P f = P - i R, the time constant becomes: msP/K' = msP/K' 2 2 P + i (R0 - R) (P/K')K' + i (-aR0)P/K' T = — (A:13) 2 K' - i aR0 Before the bolometer time constant can be calculated, the mass of the bolometer wire must be determined. The length may be determined from the heat loss because for very small wire diameters, the heat loss is independent of the diameter [44]. Once the length is known, the diameter can be calculated from the volume resistivity of platinum and the total resistance. With the length and diameter, it's a simple matter to calculate the wire mass. At 200 ohms the platinum bolometer wire is heated 216°C above an ambient of 27°C assuming a cold (27°C) resistance of 116 ohms; this is obtained from the approximate relation 123 AT = AR(1 + aT)/aR (A. 14) where the temperature coefficient of resistivity of platinum, a CC)"1, is given the value .0037 [44], R = 116 ohms and T = 27°C. It was experimentally determined in laboratory that 15.3 mw of power were required to increase the resistance of the bolometer from 116 to 200 ohms. From the relation P = K' (T — To) .0153 watts = K'(216°C) K* = 7.1 X10"5 watts/°C (A.15) for the Narda N610B/38B7 bolometers used The free convective heat loss per unit length, K, of platinum wire 216°C above ambient can be calculated from the work of King [44]: equation (A.4): K = 2.5X10-*[1 +.00114(216)] = 3.1 X10"* watts (A.16) cm- °C The length of the platinum wire in the bolometer is thus: length = — = 7-1 X1Q"5 = .229 cm = 2.29 mm (A. 17) K 3.1X10-* from equations (A.15) and (A.16). The resistance of the bolometer is given by: R = pa/A = .1(2290)/A (A.18) where p is the volume resistivity: p = .1 ohm-micron; I is the length in microns; and A is the area in microns2. Solving for A: A = 1.975 * 2 microns2 = 2X10"8cm2 (A.19) The resulting wire size: 2.3 mm X 1.6 microns diameter is reasonable. The mass of the wire is given by: m = l-A-density (A.20) 124 where the density is 21.4 gm/cm3. m = .229 cm(2 X10"8cm2)21. 4 gm/cm3 = 9.8X10-8gm (A.21) The time constant of the bolometer is now calculated for the case in which it is brought up to 200 ohms solely with d-c power, that is, i = 8.75 ma: T = ——— = 360 microseconds (A. 22) 2 K' - i aR0 where m = 9.8X10"8 gm; s = specific heat of platinum = .139 5 joules/gm-°C; K' = 7.1X10" watts/°C; R0 = 116 ohms; and a = .0037 CC)"1 from King [44]. The time constant obtained of 360 microseconds is in excellent agreement with the catalog value of 350 microseconds for a Narda N610B/38B7 bolometer. Cohn [13] finds the catalog value of the time constant of the electrically equivalent Sperry 821 to be reliable. If the bolometer is brought up to 200 ohms with 6.3 mw d-c bias power and 9 mw microwave power, the time constant ms T = = 240 microseconds (A.23) 2 K* - i aR0 where i = 5.6 ma. This is the time constant used in the text. It is useful to know the time constant when the bias current is maintained at 5.6 ma, but microwave power is limited such that the bolometer can be brought up only to, say, 180 ohms. At 180 ohms, the temperature rise is less: AR(1 (T - To) = AT = + aTo) = 64[1 +.0037(27)] aR0 .0037(116) AT = 164°C (A.24) A smaller temperature rise gives a smaller value for King's [44] constant K (A.4): 125 K = 2.5X10"*[1 +.00114(164)] = 2.96X10"* U-25) which in turn gives a smaller value of K' = K£ K' = .229(2.96 X10"*) = 6.8 X10"5 watts/°C (A.26) The time constant is: ms = 250 microseconds (A.26) 2 K' - i aR0 where m, s, a, and R0 are as previously given K' = 6.8 XIO"5 watts/°C i = 5.6 ma In the case of limited microwave power (bolometer resistance down to 180 ohms from 200 ohms), the time constant rises slightly from 240 to 250 microseconds. 126 APPENDIX B THE SYMMETRIC TOP ROTATIONAL WAVEFUNCTION PARITIES Kronig [49] found the wave equation for a diatomic molecule to be invariant under the transformation: x -»• x y -y z z tj) -+• IT + tj) x -x This transformation is given by equation (15) reference [49]. The symbols mip and a to correspond with current usage. The x,y,z are the molecule fixed Cartesian coordinates. The Euler angles tj>, 6, and x used here have been defined by Kronig [49] as follows: 6 is the angle between the positive space fixed Z-axis and the positive molecule fixed z-axis, cj> is the angle between the positive space fixed X-axis and the positive line of nodes, and x istn e angle between the positive line of nodes and the positive molecule fixed x-axis. The rotations are carried out according to the right hand rule in the following manner: first the molecular coordinate system is rotated by tj) about the space fixed Z-axis, then by 6 about the x-axis, and finally by x about the z-axis. The Euler angles used here are illustrated in Symon [94] where \\> replaces x used here. Wang [98] found the wave equation for an asymmetric top to be invariant under the same transformation (B.l). The 127 transformation consists of a reflection in the molecular x,z-plane (the reflection is parallel to the y-axis) followed by a two-fold rotation about the "molecule fixed" y-axis. This transformation is equivalent to a molecule fixed inversion. The transformation gives a space fixed inversion also. This is demonstrated by applying the above transformation to Kronig's equations relating the space fixed Cartesian coordinates to the Euler angles [48] and the molecule fixed Cartesian coordinates (equation (1) reference [48]): X = x(cosxcoscj) — cos9sinxsin Y = x(cosxsintj) + cos6sinxcos(f>) + y (— sinxsin<|> + cos6cosxcosc|>) + z(— sin6cosc|)) Z = x(sin6sinx) + y(sin8cosx) (B.2) + z(cosB) When subjected to Kronig's transformation (B.l), the above equations yield: X + -X Y + -Y (B.3) Z + -Z Thus Kronig's transformation gives both a space fixed and a molecule fixed inversion of all particles through the origin. The "Kronig's symmetry" found by Wang is parity. It will be shown that the Wang functions satisfy parity. This will be done using unnormalized symmetric top wavefunctions 128 [N =/(2J+l)/8TTz ]. The symmetric top wavefunctions are: D 6 u (B.4) m±K<* X> - e-^^dim±K.Ce" ) Under the inversion operation the Euler angles become: e + TT-0, <{>-> Using Wang functions defined in equation (2.12): iJmK1) = (B.6a) = [^(^.^-e.-x) ± D^_K(7r+c}),7r-e,-x)] (B.6b) (B.6c) ± e-i[tn(7r+*)-K(-x)]dJ (7T_e) m-K m i(m K = (-l) fe- *- ^d^K(Tr-e) ± e-^^^dl^CTt-e)] (B.6d) m-lT J i(m J J 1 D , + D (B.6f) PmK± m-K = (-1)' m-K - mK where e~lmiT = (-l)m has been used to obtain (B.6d) and J m J d^Ce) = (-l) " d v(TT-6) has been used [8] to obtain (B.6e). mix m—ix Equation (B.6f) can be re-written to show Wang's parity clearly: i|JmKi|JmK' +)> == (-1)"(-l)J|JmK|JmK' +>) (B.7) i | JmK") = - (-1) J | JwK~y where Wang's functions are: .A */ JntfO = (2)"/2[|JmK^£ > + | Jm-K) (B.8) 129 Bibliography [1] Aliev, M.R., Zh. Eksp. Teor. Fiz. Pis'ma. 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