TUNABLE LASER SPECTROSCOPY OF THE NaNe A fi-X
2
+ S)'TE7
by
RIAD AHMAD
BS.,
BITAR
CAIRO UNIVERSITY
(1969)
M.S., AMERICAN UNIVERSITY OF BEIRUT
(1973)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1977
Signature of Author .e.r. e. .
Ph.sics,.u.u.
Deparment ofPhvsics,
12, 1977
August 12,
1977
Certified by.
David Pritchard, Thesis Supervisor
Accepted by .
.
.
.
Chairman, Physics Graduate Committee
ARCHIVES
sE P'
1~77
2
TUNABLE LASER SPECTROSCOPY OF THE NaNe A 2-
X E+
SYSTEM
by
Riad N. A/G. Ahmad Bitar
Abstract
We have produced the van der Waals molecule NaNe in
a supersonic expansion, and have studied the A2H
transition using a single mode cw dye laser.
--
X2
Vibronic
bands of both 23Na20Ne and 23Na22Ne have been completely
assigned using analysis based on the long-range R
behavior of the excited state A2H potential. This analysis has allowed a definitive determination of the well
depth and location of theA 23/2 state (D
at reA = 5.1
8.0
.9 cm~
=
.1 a ) and the ground X 2E state
at reX = 10.0± .1 a 0 ).
140 ±3 cm 1
(DeX =
The recent pseudo-
potential calculations of Malvern and Peach agree much
better with these results than earlier pseudopotential
calculations by Pascale and by Baylis.
The interpreta-
tion of several recent line broadening and collisions
experiments is discussed in light of our findings.
3
DEDICATED TO MY FAMILY WHO SUFFERED FROM
DEPORTATION FROM THEIR HOMELAND PALESTINE.
4
TABLE OF CONTENTS
Page
ABSTRACT
2
LIST OF FIGURES AND TABLES
8
CHAPTER I
INTRODUCTION
--
Chapter II
11
THEORY OF THE EXPERIMENT
--
19
II.
1. COUPLING OF ROTATION AND ELECTRON
MOTION
23
II.
2. HAMILTONIAN
24
II.
3. PERTURBATION
25
CHAPTER III -III.
III.
EXPERIMENTAL SETUP AND PROCEDURE
1. TUNABLE DYE LASER
SCAN UNIT
27
32
35
A.
EXTERNAL
B.
MODIFICATIONS TO A SPECTR
PHYSICS CONTROLLER MODEL 481
39
C.
PROCEDURE OF LOCKING
41
2. FREQUENCY MEASUREMENT SYSTEM
A.
FABRY PEROTS
(ETALONS ASSEMBLY)
46
49
B.
MARKER GENERATOR
54
C.
TEMPERATURE AND PRESSURE CONTROLLER
57
D.
ALIGNMENT AND SIGNAL PROCESSING
OF FREQUENCY MEASUREMENT SYSTEM
61
E.
CALIBRATION OF FREQUENCY MEASUREMENT SYSTEM
63
5
CHAPTER III.
3. SUPERSONIC JET MACHINE
67
VACUUM SYSTEM
76
A.
CHAPTER IV
--
IV. 1.
IV. 2.
B.
MOLECULAR BEAM ASSEMBLY
80
C.
INPUT/OUTPUT LASER OPTICS
93
D.
FLUORESCENCE COLLECTION
100
E.
BAFFLING
107
F.
DETECTION ELECTRONICS
111
G.
ALIGNMENT, RUNNING AND DATA
TAKING PROCEDURE
113
SYSTEM
122
RESULTS AND ANALYSIS
137
J',
J" ASSIGNMENT AND FINDINGS
A.
ROTATIONAL CONSTANTS
B.
A-DOUBLING PARAMETER
C.
ROTATIONAL TEMPERATURE
NaNe
D.
MULTIPLET SPLITTING "A" AND
VIBRATIONAL SPACING "AG
v
"
E.
SUBBAND PERTURBATION
161
F.
INTENSITY ANOMALIES
161
LONG RANGE AND ISOTOPE SHIFT
162
(B.,
146
DV
154
"T R"
OF
155
159
ANALYSIS (Comparison with Dunham
Expansion Formulae)
A.
VIBRATIONAL QUANTUM NUMBER
ASSIGNMENT
166
B.
DISSOCIATION ENERGY OF THE XSTATE (D
)
174
C.
THE X-STATE WELLDEPTH D
176
ex
6
Page
D.
THE A-STATE WELLDEPTH D
eA
AND BAND ORIGINS
177
E.
EQUILIBRIUM INTERNUCLEAR
DISTANCE Re
179
F.
LONG RANGE ATTRACTIVE COEFFICIENT C
181
THE X-STATE POTENTIAL
183
6
G.
CHAPTER V --
185
DISCUSSION
A.
DISCUSSION OF THIS EXPERIMENTAL
FINDINGS
185
B.
COMPARISON WITH OTHER WORKS
187
C.
SUGGESTIONS FOR FUTURE
EXPERIMENTS
192
196
APPENDIX A
A.
1.
HUND'S CASE
(a)
196
A.
2.
HUND'S CASE
(b)
196
199
APPENDIX B
B.
1.
NON-ROTATING HAMILTONIAN
199
B.
2.
BASIS SET OF NON-ROTATING
HAMILTONIAN
201
B.
3.
EIGENVALUES OF NON-ROTATING
HAMILTONIAN
202
B.
4.
HYPERFINE STRUCTURE
204
B.
5.
ROTATING HAMILTONIAN
205
B.
6.
BASIS SET OF ROTATING HAMILTONIAN
208
B.
7. EIGENVALUES OF ROTATING HAMILTONIAN
214
APPENDIX C
C.
209
1.
LAMBDA DOUBLING
214
7
Page
C.
2.
CENTRIFUGAL DISTORTION
216
C.
3.
VIBRATIONAL CROSSING OF 2H
218
WITH 2H1/2
REFERENCES
221
ACKNOWLEDGEMENTS
224
8
LIST OF FIGURES AND TABLES
FIGUR ES
Page
1
--
NaNe Potentials
12
2
--
Exploded View of NaNe Potentials
14
3
--
Electron Distribution For the X, A, and B-States
21
4
--
Vector Diagram of Hund's Case
5
--
Vector and Energy Diagram, Including the Hyperfine
6
--
Experimental Setup
29
7
--
External Scan Unit Circuit
37
8
--
Modification Circuit to 481 Spectra Physics Controller 40
9
--
Block Diagram of Laser Locking
42
10
--
Power Meter Circuit
43
11
--
Two Successive Frequency Scans
48
12
--
Fabry Perot Etalon
50
13
--
Marker Generator Circuit
55
14
--
Temperature Controller Circuit
60
15
--
Block Diagram of the Frequency Measurement System
62
16
--
Absorption Scan of Hot and Cold Beam
73
17
--
Vacuum System
77
18
--
Power Supply For Pumps
79
19
--
Flanges Design Show
20
--
Oven Assembly
81
21
--
Cross Section in Oven and Oven Holder Gas
82
22
--
View of Oven, Oven Holder, Source Flange, and
Oven Positioner
83
23
--
Vacuum Electric Feed Through
87
24
--
Power Supply of Oven Heaters
88
25
--
Oven Positioner
90
26
--
Auxiliary Laser Optics
94
27
--
Laser Beam Alignment Optics
97
28
--
Block
29
--
Assembly of Collection Optics
103
30
--
PMT Flange and Drawer
105
(a) and (b)
Flexibility For alignment
Diagram of Detection Electronics
L97
206
80
102
9
Page
31
--
Laser Baffling System
108
32
--
Collection Optics Baffling System
110
33
--
Diode Pump Circuit
112
34
--
123
Absorption Scan of Hot Na
2
35 -- Absorption Scan of Cold Na
2
36-- Absorption Scan of A213/22(5)+X2E()
37
--
125
of NaNe
Absorption Scan of A 21/2(4)+X E (0) of NaNe and
A
E +
E +X
u
g
22
+X2Z (0)
--
127
of Na1/
37'-- Time Chart Recorder Absorption Scan of A H 1 /2 (4)
38
126
Plot of Line Width Vs.
J'
128
130
39-- Energy Level Diagram of A 2H+XE
Transition
J(J+l)
139
40
--
A and B-States Rotational Energies Vs.
41
--
X-State Rotational Energies Vs. N(N+1)
151
42
--
Plot of Energy Deficit Vs. J+1/2, A Doubling
156
43
--
Plot to Find Rotational Temperature of NaNe
158
44
--
Possible Transitions From X to A-Potential
165
45
--
Long Range Fit of G(v')
46
--
Isotope Shifts Between Na
v'--v"=0 Transition
47
--
Potentials of the X and A-State of NaNe
191
Depth and Location For the X and A-State Potentials
188
and Bvvs. V'.
Ne and Na20Ne for a
150
170
171
TABLES
1
--
of NaNe
2--
Absorption Spectrum of A2H+X E of NaNe
131
3
--
Progressions of a Band
142
4
--
First and Second Combination Differences for v"=0
147
5--
Rotation Analysis of A
148
6--
Spectroscopic Constants
152
7
Multiplet "A" and Vibrational Spacing AGv+ /
1 2
160
--
2H/2(4)+X2E(Q)
8--
Long Range Fit
169
9--
Prediction of Unobserved Bands
179
Scattering Experiments Involving Ne
193
10
--
10
AT DAY'S END
Is anybody happier because you passed his way?
Does anyone remember that you spoke to him today?
The day is almost over, and its tolling time is through;
Is there anyone to utter now a kindly word of you?
Can you say tonight, in parting with the day that's slipping
fast,
That you helped a single brother of the many that you passed?
Is a single heart rejoicing over what you did or said;
Does the man whose hopes were fading, now with courage look
ahead?
Did you waste the day, or lose it? Was it well or sorely
spent?
Did you leave a trail of kindness, or a scar of discontent?
As you close your eyes in slumber, do you think that God
will say,
"You have earned one more tomorrow by the work you did today"?
11
I.
INTRODUCTION
This thesis work is a study of a weakly-bound diatomic
molecule-NaNe.
It is well known that molecules (we
specialized our discussion on diatomic molecules) may be
bound by both ionic and covalent forces.
have long been discussed by the chemist
These forces, which
(PAU25), involve
either the transfer of an electron from one atom to the
other (ionic bond) or the sharing of an electron between the
atoms
(covalent bond).
These forces cause a bonding energy
which is strong enough to keep the molecules bonded under
laboratory conditions standard temperature and pressure
(STP)
to allow their studies by a number of chemical and physical
techniques.
These binding energies roughly vary from 10
to -1 ev for different diatomic molecules.
(N2 )
If neither the
covalant nor ionic bonding mechanisms exists in a particular
diatomic system, that system is frequently called "repulsive"
even though the interatomic potential curve has a shallow
minimum.
These shallow minima (binding energy) typically
vary from 1 to 100 mev for different molecules and sufficiently
strong to bind atoms together, and can therefore produce a
class of molecules distinct from those bonded by ionic and
covalent forces.
The attraction causing this minimum is called the
dispersion forces, and molecules bound by it alone are
called van der Waal's molecules.
Figure 1 shows the ground,
first, and second excited state potentials of NaNe.
One
12
A2 T
2
'-I
a,
LU
Na Ne
2.5
Na (3P) +Ne (3S)
2.0+
1.5 -t
x 2 E+
1.0 4-
.5t
Na(3'S ,)+Ne(3S
0.0 r
2
4
6
8
10
12
14
R(a.u.)
Figure 1 --
NaNe Potentials
16
18
20
22
)
13
can barely
see these minima on such normal molecular scale.
To see these minima, the energy scale is expanded by a
factor of 103 as can be seen in Figure 2.
For a more detailed
discussion about these dispersion forces, readers are
advised to consult with Hirschfelder, et al, London, and
Margenan
(HIR54, LON30,
MAR31).
One more thing
characterizing van der Waal's molecules is the fact that
the interatomic separation (5 to 15 Bohr) is quite larger
than the separation of molecules which are ionic or
covalent bound (2 to 5 Bohr).
The binding energy
(1 to 100 mev) of Van der Waal's
molecules is so small that they
dissociate under normal
laboratory conditions Troom = 300K = 25 mev.
For this
reason, they are just beginning to be studied experimentally
(SMA77,
KLE74,
nozzle supersonic jet
EW175,
EW176)
beam techniques
as new
(CAM70)
permit the
relatively high density of their production and the
development of new spectroscopic techniques permit their
detection.
Such supersonic jet and a high resolution,
doppler-free technique are discussed in Chapter III.
Once we introduce ourselves to Van der Waal's molecules,
one should ask:
molecules?
What is
so interesting about these
To answer this question, we do not claim that
all features of these molecules will be listed, but one
can see that these molecules retain their atomic identity
14
A2 1/
2 , 3/2
'-4
U
k
B2 +
4)
w
NaNe
Na (3P3/
16973
2 ) +Ne
(3S0)
)+Ne(3S )
Na(3P
16943
16913
16883
16853
16823
X2 +
30.
Na (3S1/
2 ) +Ne (3S0)
0.0
2
4
6
8
10
12
14
16
18
20
R(a.u.)
Figure 2 --
Exploded View of NaNe Potentials
22
15
(internuclear separation 5-15 Bohr) and the
to high degree
molecule can be treated as two atoms perturbing each other
slightly --
a view point of interest to atomic-physicists.
These molecules provide a convenient setting in which to study
both dispersion and overlap forces.
They are also excel-
lent systems to study the dependence of molecular parameters on internuclear separation because these molecules
sample a much wider range of internuclear separation than
do conventional more tightly bound molecules.
Alkali-rare gas diatomics like NaNe have the additional
attraction that they are one-electron molecules;
this makes
it simpler to study their interatomic potentials using ab
initio
(SAX77, MAL77)
and pseudopotential
(BAY69, PAS74, BOT73)
methods.
Interatomic potentials are needed for the understanding of various physical processes that occur when two
atomic particles colide,
such as excitation transfer,
quenching of excited states, and pressure broadening of
spectral lines.
As an example, there are experimental studies
of line broadening of alkali atoms in rare gas, culminating
in the elegant far-wing fluorescence studies of York, et al
(YOR75) in which a new method has been developed for
determining interatomic potentials.
There are also some
differential, and total scattering experiments
BUC58, D-UR68)
deduced.
(CAR75,
from which interatomic potentials have been
A further stimulus to the study of alkali-rare
16
gas molecules
(Van der Waal's molecules) is the large
body of previous work on optical pumping of alkali atoms
in rare gas buffers.
As a consequency, if the molecular potentials are found
through a definitive method, one could easily verify the
validity of ab inito and pseudopotential methods
MAL77, BAY69, PAS74),
(SAX77,
and also check the theories used to
find potentials in the scattering and the far-wing fluorescence experiments.
We feel that the interaction potentials of such molecules are best studied by spectroscopic measurements of the
vibrational and rotational parameters of the bound molecular states since a wide array of proven spectroscopic
techniques have been developed for the determination of
these interatomic potentials with great accuracy.
Such
technique, which is described in Chapter IV of this thesis
work, employs high resolution laser spectroscopy of the
ultracold molecules which may be formed in a supersonic jet.
The interatomic potentials of NaNe molecules, especially
the first excited state
(A2H) have been the subject of several
investigations and much controversey.
performed by Baylis
Theoretical work
(BAY69), Pascale and Vandeplanque
(PAS74), Bottcher, et al
(BOT73) and Malvern and Peach
(MAL77), are in disagreement with the differential scatter-
17
ing experiment performed by Carter, et al
(CAR75), on
the excited state of sodium (P3 / 2 or P1/2 state) but seem
consistent with line broadening experiments performed
by York, et al
(YOR75) and McCartan and Farr
(MCC75).
More detailed discussion about the discrepancies of the different theoretical and experimental work can be found in the
discussion section, Chapter
V.
Another obvious motivation
for NaNe besides the motivations for the study of van der
Waal's forces in general, is the number of experiments
involving scattering of Na in the 3P state from Ne
APT76, PH177).
(PIT76,
One practical motivation is the existence
of a nicely scanable single mode dye laser which operates
best near the 35 -
3P transition of Na --
suggesting that
we study a weakly bound sodium-containing molecule.
Chapter II contains a brief theoretical discussion
of the fine structure of diatomic molecules concentrating
on phenomena observed in the state under consideration
(X2 + and A
).
No attemptis made to list all the dif-
ferent phenomena which one expects to see in diatomic
systems.
Chapter III contains the different pieces of the
experimental setup
sonic jet).
(laser, frequency standard, and super-
18
Chapter IV contains the results and describes the
analysis of the spectrum.
Chapter V contains the findings of this experiment
and compares them with the findings of other relevant
work.
Some suggestions for future experiments are also
outlined.
19
II.
THEORY OF THE EXPERIMENT
In this chapter we describe the Hamiltonian of the
system and outline in detail the different features which
2
2
are observed in the NaNe spectrum (A H+X E).
Historically,
diatomic systems have been treated quantum mechanically
a long time ago
(early in the Twenties) by many authors,
(KRO30,DUN32, MUL31, HER 50),
and still, they are the
target of many recent investigations
(HOU70, ZAR73) most
of them not inspired by lasers.
In this chapter, we shall concentrate on the phenomena
which appeared in the observed NaNe spectrum.
To under-
stand such spectrum, the problem is divided into three
parts.
The first is to understand an unperturbed Hamil-
tonian which is approximately the sum of three noninteracting parts; electronic, vibronic and rotational
interaction.
Once this Hamiltonian and its eigenvalues
are understood, one could start examining small interactions,
caused by perturbations of interacting eigenstates of the
system.
These are necessary to understand the deviation
of the eigenvalues from the ideal situation (unperturbed
system).
The third part is to discuss the coupling of
angular momenta
in both
the x- and A-state.
From the above it seems understandable
(to every
careful reader) that the aim of this Chapter is to write
20
down the quantum mechanical formalism of the different
features of the observed spectrum without getting much
involved in the theory of the diatomic molecures.
The best way to introduce the theory is to ask few
questions and to lay the background about the subject
through the answers.
want to study?
To start, let us ask:
We want to study the fine structure of the
A 2-X E+ transition and possibly the B E--X2+
next question is: what are these states (A2
B2 +)?
what do we
transition. The
2 + and
To answer this question, let us look to the atomic
configuration which represents these states.
The atomic
configuration with the electron distribution of each atomic
state which forms the above molecular states are shown
in Figure
to (HER50).
3.
For more detailed studies one should refer
The letter X always refers to the ground
state; A and B refer to the first and second excited state.
Accordingly, C, D, etc. refers to the third, fourth, etc.
excited states allowed for the X state.
The spin multi-
plicity (2S+1=2 for NaNe) is added to the term symbol
(E,
R) as a left superscript. E
and H are the electronic
term label of the state and they represent the projection
"A" of the electronic orbital angular momentum "L" along
the internuclear axis.
E stands for A=0 and H stands for
A=l.
The coupling of both vectors A and
to the internuclear
axis produces a resultant angular momentum Q along the internuclear axis.
Thus the quantum number Q is given by
21
+
X2
a)
= Na(32 S1/2) + Ne (3'S0)
L=0
A= 0
S=1/2
E=?
±1/2
AR
A 2H
b)
Na(32 P) + Ne(3'S0)
L=l
A=l
S=1/2
E=1/2, -1/2
0=3/2, 1/2
A
c)
B E
B
a(3 P) + Ne (3'S0)
AR
2S+1 -
Figure 3
--
L=l
A= 0
S=1/2
E=? ± 1/2
B
2
Schematic representation of the electron
A 2 H, and B 2E
distribution for the X 2E,
states of an alkali-noble-gas diatomic system.
22
Q= IA+EI,--
IA-EI
If A= 1 and E = 1/2 there are two values,1/2 and 3/2,
Q.
for
As a result of the interaction of S with the magnetic
field produced by
A, these
different values of
A+E
corre-
spond to somewhat different energies of the resulting
molecular states A2H3/2 and A2 1/2
with A$ 0 splits into
multiplet (2S+l) components.
Finally, the subscript "+"
ator.
In a
Thus an electronic term
on the Z state is
a symmetry
oper-
diatomic molecule (or a linear polyatomic
molecule) any plane containing the internuclear axis is a
plane of symmetry.
Therefore the electronic eigenfunction
of a non-degenerate state
(Estate) either remains unchanged
or changes sign when reflected at any plane passing through
both nucleio
In the first case, the state is called a E+
state, and in the second case it is called a E~
For degenerate states
its
IAI
(1,A ---
state.
are doubly degenerate since
can be +A and -A while a Z state has A=0) linear
combinations can easily be found such that the total eigenfunctions remain unchanged or go over to their negatives
by the
reflection, i.e.
+ = XeiA$ + Re-iA$
A$
= Xe
where X and R
and
- Xe -iA$
are functions of all coordinates except $.
23
Accordingly, the y
H~
and A+
A-
and y
may therefore distinguish H
energies.
II.
1
Z+ and Ei states
basis functions similar to
but with the difference that
H + and R~
,
have exactly equal
This so far explains the states observed.
COUPLING OF ROTATION AND ELECTRONIC MOTION
The type of coupling of the different kinds of
angular momenta in a diatomic system is a prerequisite to
understand before we start talking about the best basis
set which can describe
the Z
or H-state and calculating
rotational energy level expressions for certain limiting
situations known as Hund's coupling cases.
Hund's coupling cases, two of them
Of the five
(case (a) and case
(b))
are going to be discussed due to the fact that the X and Astate of the NaNe molecules can be described fairly well
by them.
We begin by summarizing the types of angular momenta
considered and the notation to be used.
angular momenta in the molecule
-
tronic orbital angular mementum L.
nuclear rotation R
J
-
The different
electron spin S,
elec-
Angular momentum of
from a resultant
(total) designated
i.e.
+
+ +++
J = L+S+R
(2-1)
The total angular momentum apart from spin is called N and
is given by
24
N = R+L
The quantum number associated with L, S, R, J, and N
are L, S, R, J, and N respectively.
Their projection on the
internuclear axis is designated by A,Z,
respectively.
--
, Q, and A
The nuclear spin angular momentum is not
considered now and will be treated separately.
It is
to be noticed that the projection of the rotational angular
momentum, R, along the internuclear axis is zero.
Hence,
no quantum number is introduced for this projection.
Hund's case
(a) and (b) are discussed in Appen-
dix A.
II. 2.
HAMILTONIAN
The Hamiltonian of a diatomic molecule can be considered
to consist of two main parts (HOU70, KRO30) excluding perturbation:
H =
rotating part and non-rotating part, i.e.
ev+Hr
(2-3)
where "H ev" is the electronic-vibrational
part of the
Hamiltonian alone while "Hr " is the rotational part of the
Hamiltonian, which involves the rotational variables and
the total angular momentum "J".
The dynamics of any
diatomic molecule may be thought of as a sum of translational
in space
(nct considered in this calculation), rotation
in space, vibrational of the nuclei, and electronic motion.
25
In addition, there may be electron spin and nuclear spin
interactions.
The vibrational and electronic motion as
well as the electronic spin interaction are considered to
be H ev,
while the rotational motion is considered in Hr.
Dividing the Hamiltonian into two parts implies that one
can choose a basis set in which the quantum numbers of the
nonrotating molecule are good.
lev; r>
Such basis functions
be written as simple products of the form
lev, r>
=
(2-4)
iev>Jr>
where the functions lev> are eigenfunctions of Hev and
the functions
Ir> are appropriate rotational wave functions
for the states mentioned above
(LAN58).
The rest of this Chapter is to understand and specify
how one can get the eigenvalues from both equations
and (2-4).
II.
3.
(2-3)
The treatment can be found in Appendix B.
PERTURBATIONS
Perturbations of different origins have been observed
in the NaNe of A
+X2 Z+
system.
These perturbations are:
interaction between B 2 Z+ state and A2 1/2 substate causing
A-doubling; centrifugal distortion due to vibration-rotation
interaction,
and 2R1/2 and 213/2 substates crossing belong-
ing to different vibrational levels.
26
Normally, deviation of the observed data from the fit
by the unperturbed Hamiltonian can be remedied by increasIn
ing the dimension of the effective Hamiltonian matrix.
our observed and analyzed spectrum, as an example, we have
found that measuring the energy spacing between two successive
levels of the 2 R/2 state from two different sets of lines
does not yield the same energy value, which implies that the
(double) degeneracy of this state has been split by interaction with the nearby B 2
state.
See Section IV. 1.
B
In Appendix C, the three types of perturbations,
which are mentioned above, will be discussed without much
emphasis on the mathematical derivation.
Sufficient
references will be suggested for readers who are after
more specific details.
27
III.
EXPERIMENTAL SET UP AND PROCEDURE
This chapter is to introduce the reader to the
different parts of the experimental set up which can
obtain a high resolution resonance fluorescence spectra,
and to describe the procedure of taking data.
We shall
state the objectives first and then describe the system
which is used to achieve these objectives.
The objective is to detect and measure the molecular
spectrum of NaNe which is accessible by the laser.
The
basic idea of the experiment is to perform a sub-Doppler
absorption measurement using a tunable laser source and
detecting the absorption by the subsequent fluorescence.
The apparatus consists of four components:
a.)
A source of NaNe molecules;
b.)
A tunable source of light
(laser) signal
frequency, capable of sweeping mode
frequencies and narrow enough to resolve the
rotational absorption
c.)
spectrum of NaNe;
A detector for the fluorescence which also
displays the data;
d.)
A frequency measurement system for the laser
light which enables one to measure the frequency
of the absorption lines.
28
An experimental setup which can exhibit the above idea
will be
able to scan through the absorption spectrum
and measure the frequency of each line.
such
an
experimental set up.
Figure 11611 shows
In Figure 6 it is shown
that the laser light, molecular beam and direction of
detection of fluorescence are perpendicular to each
other.
The necessity for both the laser light and the
molecular beam to be perpendicular to each other is to
eliminate the first order Doppler broadening of the
transition.
The direction of fluorescence detection
is not necessary to be exactly perpendicular to either
the laser beam or the molecular beam,
but was most
easily designed in this configuration.
To implement the method described above, let us
start with the demand of making NaNe molecules.
It is important first to realize that weak van der Waal's
forces
(like the one between Na and Ne) can support bound
molecular states of the atom pair.
The binding energy
of NaNe molecule, as obtained from this experiment, is
about
.1 mev
(-8 cm)
and can support 2 or 3 vibrational
levels and up to 12 or so.rotational levels.
The van
der Waal's forces between sodium and neon atoms are
inadequate to form molecules which are stable under
normal laboratory conditions
(standard toom temperature
29
INTENSITY AND FREQUENCY
CONTROL SYSTEM
'ifIF
I4
AR-ION
I
LASER I
DYE LASER
FREQUENCY REFERENCE
SYSTEM
L
CHART
RECORDER
OPTICAL BAFFLES
PHOTOMULTIPLIER
FLEMOLECULAR BEAM
(OUT OF PAGE)
FIGURE 6 --
Experimental Setup
30
and normal pressure),
make these molecules.
so another technique is needed to
Such a technique obviously
should allow the molecules to survive at least until
they reach the interaction region once they are formed.
Supersonic jets with free expansion
(CAM70, SMA76) is
the type of technique which we employed to fulfill the
There is one big advantage to
above requirement.
performing the experiment in a beam:
narrow.
the linewidth is
There is no collision broadening since there are
no collisions; furthermore, the Doppler width is
greatly reduced since the molecules are all moving in
nearly the same direction.
Our supersonic jet
machine,
which is to be described in Part Three of this Chapter,
produces a stable cold beam of NaNe molecules.
The next demand is to have a source of excitation
light adequate to probe energy transitions between
ground and first excited state.
have a line-width
Such a source should
less than 300 MHz
be capable of sweeping frequency
(0.01 cm~ 1) and
smoothly. There are
two reasons why a tunable laser is ideal for this
application.
moves rapidly
The first is that the molecular beam
(-105 cm/sec) requiring the laser's high
intensity to excite the molecule before they leave the
interaction region.
The second is that the incident
31
light must be extremely well collimated and baffled to
avoid laser scattering into the fluorescence detector.
We used a Spectra Physics 580 c.w. dye laser.
This
laser is a single mode laser with 50 MHz line-width,
but needed considerable modification to sweep smoothly
over appreciable
(-
1 0 A) spectral range.
These modifi-
cations are discussed in detail in Part One of this
Chapter.
The next demand is to monitor the absorption.
The
way we chose to accomplish this was to build a
tower of collection optics in order to collect the
fluorescence.
Whenever
an absorption occurs it will be
followed immediately by a flourescence.
The fluorescence
is collected and converted to an electric signal which is
recorded.
This florescence collection optics as well
as the electronics are discussed in Part Three of this
Chapter.
The last demand is to measure the laser frequency.
The laser frequency is measured by the frequency measurement system which is discussed in Part Two.
This system
monitors the laser frequency at which each absorption
line occurs.
A description of an experimental run will be discussed
32
in Section III. 3. G.
Before going into the details of the pieces of
the experimental set up, a briefing of how the experiment
will be operated is helpful at this point.
The tunable
dye laser which is capable of going single mode and
sweeping the laser frequency smoothly incidents on a
molecular beam perpendicularly to reduce doppler broadening of the absorption lines.
A small proportion of this
light goes to the frequency measurement system.
quency measurement
The fre-
system produces markers to serve
as a scale measure for the laser light frequency.
While
the laser is in the process of sweeping, it excites the
molecules to different energy levels.
The photon detector
detects the fluorescence which occurs immediately after
the absorption of laser light.
In this way we measure
in fact the absorption frequency by the frequency
measurement system since it measures the light of
absorption and the fluorescence detector just tells us
that an absorption just occurred.
III.
1.
TUNABLE DYE LASER
The source of light in this experiment is a
Spectra Physics
580 dye laser operated with Rhodamine 6G
dye dissolved in methanol and pumped by a Spectra Physics
166 argon ion laser.
This single mode laser has a line-
33
width of about 40 MHZ and tunable between 5600 to 61001A. It
is operated on single mode by tuning three optical
elements
(prism, intercavity etalon, and cavity length).
These optical elements provide a rough, course, and fine
(1500, 5, and .015 cm~
) selection of frequency.
the laser continuously about 5 cm
-l
We can sweep
(150GHZ) by coordinat-
ing the tuning of the three optical elements.
Locking the
three optical elements for frequency sweeping is done by
applying a saw-tooth voltage
piezo and a proportion
prism piezo.
to the cavity and etalon
of the etalon voltage to the
These voltages will displace the cavity length
"kc" and etalon,
length "kE" by "A9c"
and rotate the prism by "AOp".
and "AE
These displacements
are linearly proportional,
i.e.
similar proportionality
defined between AZE and AO ).
E
and AOp are done
These displacements
synchronosly
is
Atc'
AtE
(FSR)
E(FSR)
E
A.
c
(A
in a way to keep the central frequency at
the maximum transmission of each of them. [To sweep about
one etalon mode spacing, the scan generator of Spectra
Physics model 481 applies a multiple integer
of 1/2
wave length sawtooth voltage on the cavity while only
one ramp voltage is applied on the etalon.]
proportion
A small
of the etalon ramp voltage rotates the prism
to keep the central etalon frequency on the peak transmission of the prism.
For more detail
on frequency
sweeping,
34
the reader is advised to consult with Spectra Physics
model 580 single frequency tunable dye laser book.
Undesirable mode hopping occur-s .at the reset
process of each cavity ramp
(i.e. when the cavity fly-
back to its original length) causing jumps in frequency
during the scan
(5 cm~
) instead of smooth sweeping.
Such problem is caused by the differentiated ramp
voltage of the etalon which is applied on the cavity
integrator while the etalon integrator is left with no
correction signal (an error signal) to keep it locked
during the cavity flyback.
To stop undesirable
mode hopping, we used a phase detector in a phase lock
loop to generate an error signal which required an
addition of an external scan unit to the 481 electronics.
Also we modified the 481 controller to accept signals
from both the phase detector and the external scan unit.
The modification as well as the external scan unit and
the operation of the phase detector will be described in
the following Sections.
Locking the three optical elements
(prismcavity,
and intercavity etalon) of the modified 580 dye laser
works well over several hours.
Sweeping the frequency
does not cause cavity mode hopping during the scan.
The
procedure of locking will be discussed in Section C of
this part.
35
III.
1. A.
EXTERNAL SCAN UNIT
This electronic unit is added to give a better
control of locking the cavity to the inter-cavity etalon.
It essentially takes a DC-error signal, amplifies it and
corrects the etalon length to have both the cavity and
the etalon on the maximum transmission of the light mode
which oscillates inside the cavity.
To understand the need for such a unit, let us describe
the sweeping process and its problem.
For sweeping the
frequency, about 160 ramps are applied on the electric
piezo of the cavity and just one ramp on the electric
piezo of the intercavity etalon in each scan.
Each ramp
on the cavity is suppose to change its length by Akc at
the same time the length of the etalon is displaced by
ALE.
By the end of 160 ramps on the cavity the etalon
has only one ramp and it is displaced roughly by
160 AYc
The problem occurs at the end of each cavity ramp.
Instead
of allowing the next light mode to oscillate another mode oscillates or the frequency switches back and forth between two
modes due to the sudden change from k +6k
c
some besterises in the piezo electric
c
to k
c due to
inducers of
the cavity and the addition of the differential of the
etalon ramp to the cavity integrator without compensating
for that by applying a correction signal
(error signal)
36
to the etalon integrator.
Such frequency jump is due to
the lack of proper locking of the cavity to the etalon.
This cavity mode hopping (jumping) is prevented
by applying an amplified DC error signal to the
intercavity etalon with a certain proportionality.
The
error signal is generated by a phase detector from the
dithered dye laser light which is detected by a power
meter and processed by the phase detector.
(The function
of both the phase detector and the power meter will be
described in Section C.)
The DC error signal corrects
for a change in k c by a proper change in kE in a
way to keep both on the maximum transmission of the
oscillating mode.
This kind of locking was satisfactory
and kept the sweeping smooth over couple of hours.
The circuit diagram of the external scan unit is
shown in Figure
7.
This circuit is homemade.
consists of three main parts:
"scan rate control",
"sweep amplitude control" and a "jumper".
of each is as follows:
The function
the "scan rate control" selects
different rates of the scan.
0.05
It
The rate ranges from
up to 20 GHZ/sec and it is marked accordingly on
the front panel.
Control" number
It is fine
(6) (see Figure
adjusted by a "Fine
7).
"Scan Rate Control"
is connected to a two position switch marked with stop
Scan Rate Control
Sweep Amplitude Control
To Etalon Scan Gen rator
To Cavity Scan Generator
t.,o
N6
tiSV
lot(
3.0
DC-Error Signal
FIGURE 7 --
External Scan Unit
Jumper-
38
start on the front panel.
This switch allows to stop
the sweeping by grounding the cavity scan generator.
The 'sweep amplitude control' is incorporated with the
"scan amplitude" of 481 controller to select frequency
scan amplitude.
up to 10 cm ~.
The selection range is from 0.1 cm~1
The frequency scan amplitude is finely
tuned with a fine control number
(2) on Figure
7.
The scan rate control and sweep amplitude control are
coupled together through a 10MO resistor.
The DC
error signal is amplified and fed through the "sweep
amplitude control" to the etalon scan generator.
The
DC error signal which goes to the etalon scan generator
can be varied with the fine control
number
(9)
(2).
Switch
is a two-way switch providing locking and
unlocking positions.
Locking occurs when the DC-error
signal is connected to the etalon scan generator.
Finally the "jumper" provides a pulse of about 10 volts
onto
the etalon by discharging the .02pf capacitor
which is charged and triggered externally.
This jumper
moves the intercavity etalon to the next cavity mode
when the jumper is triggered.
The process is to change
the etalon peizo voltage by the same amount needed to
have the next cavity mode centers on the peak of the
etalon transmission.
39
III.
1.
B.
MODIFICATIONS TO THE MODEL 481 CONTROLLER
Modifications include the following changes in
481 controller:
panel
(a)
The variable switch on the front
(of the 481 controller) is replaced by double pole
double through [DPDT] switch to provide internal and
external sweep to the unit.
(b) A hole was drilled above
the etalon heater on the rear panel to accommodate a
five wire plug recepticle to supply the "External Scan Unit"
with +15, +10,
0, -8 and -15v from the 481 controller.
(c) The output of the Blanking is converted to etalon
scan generator BNC.
The signal from the "external scan
unit" BNC of Figure 8 is wired to the etalon scan generator BNC of the 481 control.
(d) the input of the
blanking is converted to cavity scan generator BNC.
signal from the external scan unit BNC
(Figure
The
7) is wired
to the cavity scan generator BNC of the 481 controller.
(e)
The fine control of the vertical gain unit is
replaced by BNC connected internally to the piezo drive.
of the intercavity etalon and connected externally to the
phase detector which provides a square wave with adjustable frequency and amplitude.
are shown in the
Part of these changes
circuit diagram of Figure
8.
The
dotted lines are the original lines in the 481 controller.
The additions are shown with solid lines.
Cavity Scan
Rate
Etalon Scan
Rate
Green
Blue
Blue
Black-
Blue
--
-----
C206
AID
White
Black
FIGURE 8 --
Modification Circuit to 481 Spectra Physics Controller
41
This modification couples both scan generators,
the cavity and the etalon, to those of the "external
scan unit". see Figure
throw" switch
9.
The "double pole double
(DPDT) of Figure 8 allows the user to do
the coupling or use the 481 controller for internal sweep
by isolating the "external scan unit".
III.
1. C.
PROCEDURE OF LOCKING
Procedure of locking the cavity to the
intercavity etalon
diagram
figure
9.
is shown in the block
Mainly, we superimpose
a square
wave generated with the phase detector onto the
ramp voltage of the inter-cavity etalon peizo.
The
amplitude of the square wave is adjustable and usually
chosen to cause the etalon central frequency to oscillate
back and forth with about ±20% the mode spacing
of the cavity length (about .1 GHZ).
(FSR)
This results in
amplitude modulation of the laser power since laser
power is maximum when the etalon is exactly on-resonance
with a phase depending on which side of the transmission
peak of the etalon the cavity mode is on.
The modulated
dye laser light is detected with a photodiode.
detected light is amplified with a power meter.
The
The
circuit diagram for this-power meter is shown in Figure 10
Dye
Pum Laer580
Pum Lse .Laser
Phase
Detector
\Power
-Meter
481 Scan
Generator
External
.Scan
lnit
Scope
FIGURE 9 --
Block Diagram of Laser Locking
tie/k
Fvok L-0
t
To Phase
4
Detector
-
To Argon Ion Laser
3o X4
F
L
2-00 L
4v
1001T
FIGURE 10 --
Power Meter Circuit
.o9-
44
A
portion
of the amplified signal is sent to the phase
detector and another portion is sent to the argon ion laser.
The signal which goes to the argon ion laser stablizes
the dye laser light intensity with a feedback loop.
[Details of the stabilization of dye laser intensity is
described by Apt
(APT76)]. The portion which goes
to the phase detector converts the amplitude
modulation frequency signal into a DC error signal which is
wired back to the external scan unit then channeled to
the etalon scan generator.
The sign of the DC error
signal is now a guide to center the cavity mode on
This technique of
the peak transmission of the etalon.
locking was successful to several hours of running.
However, a sudden disturbance can cause the etalon to
jump to another cavity mode and might disturb
the
locking.
Locking procedure is as follows:
has the following adjusting knobs.
the phase detector
The amplitude gives
a control on the square wave amplitude which is to be
summed on the ramp of the etalon piezo drive.
The
frequency controls the square wave frequency.
The Phase
controls the phase of the DC error signal relative to
the square wave signal.
DC gain controls the DC level
of the DC error signal.
The
carrier null is to suppress
the modulation frequency from the output error signal.
45
The first step of locking is to have the proper
settings of the phase detector.
the square wave
This is done as follows:
is sent to the scope and its amplitude and
frequency are set at about .06 volt and 200 Hz respectively.
The output DC error signal is displayed on the scope and
adjusted with the carrier null to have the smallest oscillation and its phase relative to the square wave is adjusted by the 'phase control" to be approximately zero.
For locking, connect as shown in Figure 9.
The
scope displays the DC error signal of the phase detector.
If the DC error signal oscillates, the frequency should
be changed until a DC signal with the least noise is
displayed.
This avoids frequencies, e.q. 180HZ, where
the ion laser is noisy.
Put the locking switch of the
external unit on locking position and watch the DC error
signal.
If the whole DC level oscillates, the amplitude
of the square wave should be increased until it stops.
It is wise though to look for the dye laser mode
structure while increasing the square wave amplitude
(you
should never increase the amplitude to a value which
will allow the laser to oscillate between more than two
cavity modes.
Locking implies a constant DC level on
the scope.
is helpful sometimes to give a negative
It
DC value for the DC error signal to stop the DC error
46
signal from oscillations.
The next step is to sweep the frequency of the laser and
to watch
if it sweeps the whole scan smoothly.
This
is done by pushing the button of the "single sweep" of
the 481 controller down and to put the "start-stop"
switch of the external scan unit on start.
If it fails
to lock at the last 1/4 part of the scan, it may mean
that the signal to noise ratio of the power meter is comparable or the intensity of the laser output has
decreased near the end of the scan.
This can be cured
simply by increasing the output Argon Ion laser intensity
or by adjusting the dye laser.
If this is not possible,
then the dye laser intensity at which it is stablized
should be decreased to a value at which the whole scan
will be done without getting out of laser stabilization.
Once this is done, the whole locking procedure could be
followed as described above.
III.
2.
"FREQUENCY MEASUREMENT SYSTEM"
It is indicated at the beginning of this Chapter
that the laser light frequency has to be measured in
order to find the absorption lines of A 2[X2 E transitions.
Obviously, a system which can monitor a tunable laser is
very
important for.many applications.
It
is
also mentioned
47
that the laser light is a single frequency of 20 MHZ line
width and scanable.
Therefore,the objective is to
construct a system to monitor the
ately.
Accuracy of part per 10
laser frequency accur(e.g. 0.002 cm~
) makes
the analysis of the data easier.
The system which was built consists of three parts:
two fabry perot
(etalons), marker generators, and
temperature controllers. This system takes a portion of
the laser beam and converts it into two kinds of markers
pulses which are summed onto the laser scan.
The two
kinds of markers are of opposite polarity, the first is
to find the laser frequency to the accuracy mentioned and
the other helps to overlap one scan with the one before
or after.
Thus, each scan will have the molecular spec-
trum with both kinds of markers super-imposed on it.
A scan sample is shown in Figure 11 which illustrates
how
it measures the frequency and traces the end of one
scan with the beginning of the one after.
The first
parts
(fabry
three sections describe,
the three main
perots, marker generator, and temperature
controller) of the frequency measurement system.
fabry
perots
indicators.
provide light
The
pulses to serve as frequency
The marker generator converts the light
pulses into sharp electric pulses
(Markers).
The tempera-
lst Scan
0Frequency
FXGURE, 11
--
-*.increases.
from left to right
2dSa
Two successive frequency scans. The frequency of the line scan
indicated by v helps to overlap and identify the markers in
the two scans.
49
ture controller controls the pressure and temperature of
both etalons to define a constant physical dimension for
the etalon cavity.
Two additional sections are included; the first is a
detailed description of the whole frequency measurement
system, the second describes two ideas to calibrate
the system for absolute frequency measurements and the
limitations of this system for frequency measurements.
III.
2. A.
FABRY PEROTS
(ETALONS ASSEMBLY)
In the frequency measurement system interference
fringes are produced as the laser frequency scans across
the regularly spaced
(in the wave numbers) transmission
perots etalon.
peaks of a fabry
These fringes occur at con-
stant spacing and used for frequency measurements.
Most optics
textbooks, as well as Fabry and Perot have discussed
fully this interferometer
(FAB99).
Most of what will
follow is a description of the installed Fabry perots and
the alignment procedure.
Figure 12
shows a cross section through the differ-
ent elements of the etalon used.
plates
It consists of two glass
(mirrors) with plane surfaces.
The inner surfaces
have dielectric coatings with 98% reflectivity centered
near 5900* A.
The
glass plates are made slightly prismatic
Glass Plate
(Mirror)
Projection Ring
1J
/
/
7
K
N
\
Cape
A
reflecti ve coating
0O
,rXi
/
//
/7/7'
//
7/
FIGURE 12 --
Fabry Perot Etalon
-~
/
Etalon Cavity
51
in order to prevent the reflections of the outer uncoated
surfaces from producing more interference effects.
The
inner surfaces of the glass plates are separated with
a hollow cylinder made of invar
(etalon cavity) to main-
tain a fixed distance "D" between the mirrors.
The invar
material has very small thermal expansion in comparison
to other materials.
It is necessary that the cavity
suffers very little change with temperature fluctuation
from day-to-day because the cavity length "D" defines the
reproducability of the frequency light pulses.
The
cavity has a tapped hole designed to connect the cavity
to a
pressure scanning system.
this experiment.
This hole was sealed for
The top part of the cavity has a projecting
ring used to secure complete parallelism between the two
mirrors.
The parallelism is achieved by fine sanding of
the projection ring.
The cavity also has three tapped
holes in each side equally spaced to fasten the caps
to the cavity body.
Each mirror is kept in place with
a cap mirror holder bolted to the cavity body.
The
joints between the cavity and the cap and between the
mirror and the cap are sealed with 0-rings.
Sealing
the cavity maintains constant density inside it and
stabilizes the optical cavity length against atmospheric
pressure changes.
Alignment is the most crucial element for good per-
52
formance of this spectrometer.
To crudely test the
parallelismof the mirror surfaces, the following experiment
is performed.
Figure
11.
The etalon is assembled as shown in
As He-Ne laser beam is shined normally
onto the etalon mirror, the light transmitted through the
etalon is projected on a screen.
If the transmitted
light is of multible reflection behavior
(e.g. a series of
spots) it indicates that the reflecting mirror surfaces
are not parallel.
In this case, the projection ring
should be sanded smoothly with a fine sand paper to collapse the battern of multiple reflections.
This step
should be continued until the image is just one transmitted
spot.
This implies that the light is just reflecting
on itself.
Now intercepting the He-Ne beam with a lens will
diverge the light and will produce an interference rings
pattern on the screen.
(The same can be achieved by
diffusing the light with a lens paper.)
The interference
rings are not supposed to change size as the etalon is
moved
across the light in case of complete parallelism between
the entire surface area of the mirrors.
better parallelism
If the rings move,
can be obtained by shining the dye
laser beam onto the etalon normally.
Sweeping the dye laser
will produce an incomplete ring pattern, their center is
outside the illuminated mirror area and the pattern moves
53
in a certain direction while the laser is sweeping.
A
fine sanding along the opposite direction of the rings
movement. will focus the transmitted light onto a spot.
This
step is very delicate and sanding should be done
very smoothly and carefully.
The transmitted spot light
can be improved by sanding different directions of the
projection ring to produce a spot circular in
which appears very fast when the sweep is
shape
on advance.
The important measurable quantity for the perfor-
ance of the interferometer is a quantity called the
"finesse".
range
The finesse is defined to be the free spectral
(FSR) over the width at half maximum
the transmitted light pulses.
(FWHM) of
The FSR is the frequency
spacing between two successive transmission peaks.
bigger the finesse,
etalons.
D-sg
ED-short
The
the better the alignment of the
The finesse achieved for the long/short etalon
-.
3.33747;:-
8.31) is about 25/45
long/short etalon is found to be
.
The FSR of
(15.8/l.9t .1)GHZ from
the calibration.
The limitation of the accuracy of the frequency measurements is to be discussed in the calibration section.
54
III. B.
MARKER GENERATOR
The marker generator is an electronic package
which converts the light pulses into electric pulses
which
serves as frequency markers.
five stages.
It consists of
These stages are mainly detection,
cation, flitering, differentiating,
amplifi-
generating the
step type signal,and firing stages.
The circuit diagram and the signal at each stage are
shown in Figure 13.
an SGD -
The detector
circuits are based on
040 A photodiodes.
The next state is an amplification one.
controlled with a five position switch.
The gain is
The offset is
controlled by biasing the non-inverting input to a proper
voltage which is taken from ± 15 volt power supply.
volt, terminals.
secure
each is
The ±15
grounded through 12 KQ resistor
to
the op amp from any sudden increase in
voltage.
The next stage is a filtering one.
Its purpose is to
eliminate high frequency noise which is undesirable at the
firing stage because it might produce multiple firing.
It mainly filters out the high noise frequency of the
main signal.
The values of R. and C. in the feedback
loops are determined solely by knowing the width of the
input signal.
The width of the input signal "D" is just
55
I
GC
cpsp
Deflection
Amplification
33
Filtering
X1
Lni
Filtering
CN
XK
Differentiating
+
StepwiseSignal
Firing
FIGURE 13 --Marker Generator Circuit
+
56
D =
FSR
1
F
T
F
-
(where "FSF"/F is the free spectral range/
finesse of the etalon and T is the sweep rate of the
laser frequency.)
Typical values for-
long and short etalons are
respectively.
( 5 8.
"T' is .2GHZ/sec.
FSR
and
X
-) for both the
(4)
1.9)s
By knowing the value
of "D" for both the short and long etalong signals
the time constants of the feedback loop is chosen to filter
the high frequency noise i.e. R and C are 1 p.f and 150kQ
for the short etalon and
etalon.
.lpf and 200 kQ
for the long
It is to be noticed that the filtering stage
will move the position of the peak of the signal towards
the right.
This is obvious because of the charging
process of the capacitor "C'.
The next stage is a differentiating one.
an RC circuit.
output signal.
"C"
It is just
also depends on the width of the
C' is chosen to give a time constant
about the same as the width of the output signal. "C'"
is o.5/.05 pf for the short/long etalon signals.
The next stage is a stepwise signal.
It converts
the differentiated signal to a stepwise kind of signal
which switches from -15v to +15v when the differentiating
signal switches form +ve to -ve.
The diode prevents the output from making multiple
swinging between -15 and +
15v due to noise at the input.
57
The offset is to position the place of flipping
the output
signal from -15 to +15 volt.
The last stage is a firing stage. The 0.lpf capaciter
discharges through the diode by the time the stepwise
signal switches from -15 folt to +15 volt.
This firing
produces markers for frequency measurement system.
OP amp number 5 is
The
added to put a threshold on the
amplitude of the signal which is to be fired.
This is
done by taking the filtered signal and putting an inverted
portion of this amplitude at the diode "D 2 ".
This helps
to put a limit on the small amplitude signals which might
occur due to noise in the circuit or small fluctuations
in the dye laser intensity.
Both of the output markers of the short and long
etalon are coupled together through lMQ resistor and just
one BNC output appears on the front panel of the marker
generator.
The marker generator designed in this way
functions properly.
It has knobs in the front panel to
prevent multiple firing and to adjust the marker amplitude
and the amplitude threshold.
III.
2.
C.
TEMPERATURE AND PRESSURE CONTROLLER UNIT
The drift in the position of the markers
relative to a known frequency caused by fluctuations in
58
room temperature and pressure was the reason to build this
unit.
Temperature and pressure fluctuations from one day
to another or even during the same run of the experiment
produced a change in the etalon's optical length "D"
which in turn will move the marker position in frequency.
To obtain reproducibility in the marker position,
the cavity of the etalon is sealed with o-rings
(see
Figure 12) and its temperature was kept constant with an
oven, whose temperature is controlled with the temperature controller.
Keeping the oven
temperature constant
ensured a constant pressure and temperature for the
etalon cavity.
Before going into the details of this unit, let us
It is mentioned earlier in this part
justify its need.
that the accuracy of the frequency measurements is in
the order of
Let us calculate the drift in the
.lGHZ.
marker position which is caused by a 3*C change in room
temperature.
The drift "Av" in the frequency v is
AT
Av
-=
YT
where Y = 5x10-6 degree
expansion of invar and
temperature.
.45 GHZ
is the coefficient of linear
AT
-T
is the fractional change in
Using v to be 1015HZ gives a drift
Av equal to
which is about five times bigger than the accuracy
59
needed.
Before the etalons were temperature and pressure
contrQlized,a drift in the order of few markers was
observed in both increasing and decreasing direction of
the frequency in one day measurements.
After the
unit was installed, minimal drift was observed in the
marker position which is less than .001 cm~1
(.lGHZ).
The circuit diagram of the temperature controller unit is
shown in Figure 14.
It is similar to the one designed by
Spectra Physics for the 580 dye laser,
additions.
Q103
except for
two
(the first addition) is connected to
the output of C2106 and biased through 800 0 and a lamp
to -15 v.
This addition gave better indication of the
etalon temperature.
Now the unit has
one lamp to
indicate if the oven is cold and the other if the over
is hot.
The second addition was to replace a +15 volt
power supply to U102 and Q102 by 30 volt power supply.
This replacement was necessary to supply more power
to the over in order to be able to change the oven
temperature over 100 K.
The oven is powered by the greatly amplified output
of wheatstone bridge with a
etalon via one leg.
thermistor attached to the
The oven is a tantalum wire
wound along the etalons.
The wire is covered with an
asbestos sheet to isolate the oven from the outside and
-15V
+15V
+30V
w
SAWAMtOAW
#V~f
4IU44PA"
to
cal~
02WAr
a
*~t
AM
AAO
1
gri-t
747C
-4
471LC
CA
4/v
4
&
XIA?
r/e
0
I 1~1
Af
FIGURE 14 --
Temperature Controller Circuit
J
140
LZJ
,.I1
P-
H1
1L
P-0-
VP&MA
61
to minimize heat loss by conduction.
The bridge is
biased with a zener reference voltage supplied by U102.
The
bridge output is amplified by AR102 and fed to the
power amplifier U102 which is boosted by Q102.
The
output of the error amplifier AR102 is also fed to the
error indicator light comparators which consists of AR101
and diodes GR106-107.
The error light goes on when
the thermistor bridge is unbalanced.
III.
2.
D.
ALIGNMENT AND SIGNAL PROCESSING OF THE FREQUENCY
MEASUREMENT SYSTEM
This section is a description of how the whole
frequency measurement system works.
diagram of this system.
light
is
Figure 15 is a block
A small portion of the dye laser
sent to a beamsplitter which reflects
about 30%
of the light onto the short etalon and transmits the rest
to a mirror which reflects the rest onto the long etalon.
Both the mirror and the beam splitter
on a brass metal plate.
are each mounted
This plate does not provide
any fine adjustment to direct the laser beam onto the
etalons.
The beam is directed instead with "m ".
Both
etalons are mounted each on an aluminum. plate with two fine
threaded screws.
The screws can fine,
adjust the
surface of the etalon mirror to reflect the beam on itself.
This is done by looking
at
the reflection of the
laser light onto a piece of paper.
The transmitted light
Photo
Diode
Marker GeneratoJ
Mixer
0
-)
Recorder
C%4
0
,Temperature
Controlle:
/
/
m
Figure 15 --
Block Diagram of the Frequency Measurement System
63
signal is collected with a 15 mm focal length lens and
focused onto the detector of the SGD-040A photodiode
which is part of the marker generator.
mounted on an aluminum plate.
The lenses are
The photodiodes are fixed
to a plexiglass mount which is in turn fixed to an
aluminium plate.
The plexiglass mount can rotate and
translate as well to provide two degrees of freedom to
maximize the signal.
All the plates which support S,
M 2 , long/short etalons,
lenses and photodiodes are bolted
on a vertical aluminium plate which in turn is bolted to
the laser table.
The electric detected signal is sent to the marker
generator which processes it and produces sharp markers.
The markers are sent to a mixer which takes the molecular
signal and the markers and displays both on the x-y
recorder.
A sweep sample is shown in Figure 11
displays both kinds of markers
markers) with Na 2 spectrum.
which
(long and short etalon
The short etalon markers
appear with the polarity (up) while the long etalon
markers appear with -ve polarity
III.
2. E.
(down).
CALIBRATION OF THE FREQUENCY MEASUREMENT SYSTEM
The system described in the previous sections
produces frequency markers during the laser sweep.
Yet, the
64
frequency of these markers and the accuracy of the measurements are not known.
To measure any marker frequency
absolutely with the system described in the previous
sections we need to have at least one frequency
frequency domain of the dyelaser,
long etalon.
"v" in the
and the "FSR" of the
The "FSR" cannot be known without at least
two known frequencies "Vi"
and "v 2".
The accuracy of
frequency measurements of the markers though depends on
three main factors:
the reproducibility of the marker
position, the dispersion effect of the dielectric coating
of the etalon mirrors, and the accuracy of the lines
which are going to be used for the calibration.
There are two kinds of systems which can be used
for the calibration; the first is an atomic system; the
second is a molecular system.
In both cases the
calibration is achieved by using the absorption of the atomic
lines
or the molecular lines.
The disadvantage of using
molecular spectrum for calibration is that there is
difficulty in assigning the line transition in order to
know its frequency and the limited accuracy in the
measurements of those lines.
The advantage thPugh
that the spectra can fill the whole frequency domain
which makes the marker frequency measurements more
accurate.
is
65
The system which is used for calibration is the atomic
D-lines of sodium (32 P 3 / 2 and 32 P 1 / 2 ).
The absorption of
these lines are recorded simultaneously with the molecular
spectrum of NaNe.
The atomic D-lines are split
hyperfine interaction of 32 S1/ 2 .
by the
This splitting is
1.7716 GHZ and the line frequency of 2 1/2 and 2 3/2 are
16973.379 and 16956.183 ± .001 cm
respectively.
The
center of gravity of the absorption of these two lines are
used for the calibration.
That is, the number of FSR's
is counted between P 1 / 2 and P3/ 2 to calculate the FSR of both
the long and short etalon.
is found to be
The FSR of the long/short etalon
(1.9/15.8) GHZ.
This kind of calibration
of the frequency measurement system is adequate for frequency measurements in the neighborhood of these two frequency lines.
The NaNe spectrum is observed between the two
sodium D-lines and about 25 cm
1
to the red of P 1 /2.
The first limitation on the accuracy of the frequency
measurements is the accuracy of the lines used for
calibration.
The accuracy of either P1/2 and P3/2
not better than .001 cm 1.
is
Therefore, the absolute
frequency measurements are not better than .001 cm1
but the relative frequency measurement is indeed about
5 x 10
cm
(error of FSR of etalon).
just the accuracy (.0014 cm~
This number is
) divided by the number of
66
FSR between the two lines
(P1 / 2 ' P3/2).
The error in ab-
solute frequency measurements which one might expect to
get by using this calibration is
Avn = t(Av 0 + nAFSR) = 0.001 + 0.002 = 0.003cm
(For measurements about 25 cm
1
1
away from the line (P1 / 2 )
n FSR is 0.002 cm~1 which is as far as our spectrum
is observed.)
The second limitation on the accuracy is the repro-
ducibility of the marker position within the run.
The re-
preducibility of the markers in position is found better
than .002cm~1 in two runs with three days periods between
the two runs.
So the effect of the reproducibility on
the accuracy was not a limiting factor for one run
measurements at all.
The third limitation on the accuracy was the dispersion effect of the dielectric coating of the etalon
mirrors.
The dielectric coating produces different
phases at different laser frequencies.
Accordingly,
this will make the FSR of the etalon a function of the
frequency
(v) instead of being constant.
This effect is
very small over a small frequency range since the dispersion
of the dielectric coating is a smooth and very slowly
changing function with frequency.
67
The three kinds of limitation on the frequency
measurements are found not to produce more than .003 cm
accuracy.
Accordingly, we can claim that the frequency
measurement system which is described in the previous
sections allows us to measure the NaNe spectrum to .003cm
1
accuracy.
III.
3.
SUPERSONIC JET MACHINE
Supersonic jet machines provide an intense beam
and a free expansion.
These properties are essential pre-
requisites to have a beam of NaNe molecules able to
survive
until at least the time of measurements.
Before
explaining what good the above two properties will do
for a beam of NaNe, an estimate for the type of signal
we expect to get will help in verifying the need for
an intense beam with a free expansion.
Signal Estimate:
The signal "N "
c
(the number of
counts per second which is seen by the photo-detector)is
the multiplication of the number of molecules "M" available
in a certain molecular level at the interaction region by
the geometry of the collection optics "G", by the fraction
of molecules which get excited by the laser beam, - i.e.
N
c
=
MG.
(3-1)
68
The function M depends mainly on the flux of NaNe
as well as on the number of eigenvalues "F"
INaNe
in the ground state of the molecule and its rctational
temperature.
Assuming the beam is not really
enough, i.e. KT ot
D
(D
cold
is the well depth of
X2 E+) gives
M = VNaNe
F
Q
(3-2)
is the fractional solid angle of the molecular beam
at the interaction region.
The flux FNaNe can be
estimated from measurements done on KAr by Mattison et
al
(MAT74).
They found that rKAr/ FK ~ 10~4 for a molecular
This number is smaller for
beam of K and KAr.
NaNe
rNa
DeKAr
because NaNe has shallower potential than KAr (
rDeNaNe
Assuming (2)10
for :NaUe, equation (3-2) becomes
=5)
PNa
M = rNa
(33)
(2) 105 F
The flux of Na "rNa"
through the nozzle hole of area "A"
is
1
rNa
(3-4)
2 n*v*A
where n and v are the density and thermal velocity of the
atoms, equation (3-4) is correct if the beam undergoes a
forward jetting condition.
equation
(3-3) yields
Substituting equation
(3-4) in
69
M = 10- 5 n*v*A* 1/4F
(3-5)
Typical values for the constants in equation
10 7
n = 10
cm
3
5
cm
v = 10 cm/sec,
and F = 20.
-6
A = 7x10
(3-5) are
2
cm,
Q.0l,
To calculate n and v we assumed a temperature
Substituting these constants in
of 400*C and Q = 0.01.
equation (3-5) gives
M
- 10 7 #/sec
(3-6)
The geometry factor G depends on the area and the
position of the collection lens relative to the interaction region and the attenuation of the fluorescence
light due to reflection at a number of optical surfaces
through the collection direction of the fluorescence.
Assume this attenuation is 1/10.
"G" also depends on
the quantum efficiency of the photo tube "E",
G = 10HR 2
i.e.
(4HR 2
(3-7)
HR2
41R2 2
is the portion of the fluorescence light
collected by the collection lens.
Assuming 6 = 0.1
and a collection lens of 1-f-number and R
= 2" located
at R2 = 2.5" away from the interaction retion.
Therefore
G = 1.6 x 10-3
Finally, the fraction of the molecules "a"
get
which
excited depends on the rate of absorption "R"
70
multiplied by the time "T" the molecules spend in the
interaction region.
The rate of absorption "R" depends
between the ground and first excited state
on the overlap
eigenfunctions as well as on the laser intensity "I".
With 500 mw/cm 2,
you can certainly saturate the most
intense lines and get 1 photon/molecule, i.e.
rnl
(3-9)
Substituting equations
in equation
(3-9),
(3-8) and
(3-6)
(3-1) gives
1.6* 10- 31.6*104
N Cl*10
(3-10)
counts/sec...
This estimation of the signal "N c" which one hopes
to measure has been done by assuming that a- every molecule
which is formed is stable until it interacts with the
laser light b- the molecular beam undergoes a forward
jetting condition i.e. the flux through the hole nozzle
is given by
-
nvA.
There is a considerable overlap between
the ground and first excited state eigenfunctions.
Of course, the signal can be enhanced by allowing
more flux of Na or Ne through the hole nozzle.
has a limiting effectiveness. The limitation is
This
how low
the pressure in the vacuum system can be kept in order
to prevent the molecules from the invasion of the background gas.
Putting more flux through the nozzle implies
71
a build up in the background pressure which in turn needs
a pumping system of high throughput.
In fact high flux
through the nozzle by applying high pressure behind the
nozzle hole will create a kind of barrel shock (CAM70,
SMA76) with a mach disk
nozzle hole.
The
(shock) at distance X from the
shape of the barrel shock and the
position of mach disk depends on the nozzle diameter "D"
the pressure of the oven "P"
X = .67
5
P
P
and in the vacuum system
1/2
Figure below shows a schematic of
JiET
the barrel shock.
BOUNDARY
FEFLECTFD
SHOCK
NACH DISK (SHOCK )
BARREL
SHOCK
72
This equation is derived by equating the force on the
mach disk by the change of momentum of the jet.
constant .67
of the jet.
The
depends on the solid angle of conversion
It is possible to reach a mode of operation
where the expanding gas in the region upstream of the
shock structure is unaffected by the background and is
equivalent to the same region of a free jet expanding
into an infinite vacuum
[see equation (3-11)].
The
free expansion of the gas from the nozzle produces an
extensive cooling in the translational degrees of
freedom by the effect of binary collision
cooling
(AND67).
This
proceeds until the expansion has produced such
a degree of cooling that a significant number of collisions
no longer occurs.
The cooling of these molecules will
enhance the signal because the low rotational energy
levels are the ones which are mostly populated and
dissociation is harer by collision.
Coming back to the
first paragraph,we have showed the necessity for an
intense beam because of the small signal and the free
expansion to prevent the molecules of disappearing at
the region of interaction because of background collision
with the beam.
To see these effects, Figure 16 shows two
scans of the same domain in frequency for Na 2 with free
expansion and without it.
A 'vv
Ii''~i~'r
4,'
COLD
ii
m2
I
11,A
~f1~
ii'
HOT
FIGURE 16 --
Absorption Scan of Hot and Cold Beams
*1'
'4
p.,I
44I
I
4
j
j~
j
'~
74
Other limiting factors which might demolish the signal
and precautions should be taken against them are accidental
scattering of incident photons
detector.
(laser) into the fluorescence
Black body radiation and stray light scatters
into the fluorescence detector, and signal from both the
D-lines of sodicum (P 1 /
2 1'
cence collection optics.
system
It seems that an optical isolation
(baffles) like the one of Smalley
Pruett and Zare
(10
16
(SMA76) and
(PRU76) is to be installed in the Super-
sonic jet machine.
photons
P 3 / 2 ) collected by the fluores-
This system should baffle the incident
/sec) by a factor of at least 10
-12
.
Details
of the baffling and optical isolation system is discussed
in Section III. 3. E.
Baffling the Lorentz tail of both
D-lines is impossible, and our only hope is that the part
of the spectrum very near the D-lines will not be severely
needed for the data analysis.
Pumps of a moderate pumping speed but high throughput
are superior in handling the gas flow from a high pressure
nozzle.
To decide on the pumping system, let us write
down the typical values for the vacuum chamber pressure
"P"
(.1 torr),
the mach disk "X" (.5cm),
of the nozzle hole D(30y).
the diameter
Equation (3-11) can be used to
find what nozzle pressure is needed to get a supersonic
jet with the above specification, i.e.
75
P
= 1/(.67)
2
X2
(2E)
[P]
- 7x10
4
torr - 100 atom.
To obtain a supersonic molecular beam which undergoes a
free expansion to a distance X=.5cm with a background gas
pressure P-0.1 torr, a 100 atm. gas pressure "P" is
needed behind the nozzle.
To find the pumping through
put "T", which is the background gas pressure "P"
multiplied by the pumping speed "S", we need to find the
flow of gas through the nozzle
, i.e. r= 3x109 E #/sec
(T is measured in k torr/sec).
r is given by equation
(3-4) i.e.
r = 1 A*nxv =
4
iD2
* P
K
5KT) 1/2.
For neon
gas pressurized to 100 atm behind a nozzle heated to
about 400,4 r is approximately 4x1020 #/sec.
the throughput Tis aboutl3.3 torr k/sec.
the throughput
simply suggests
the pumping system
with
should
a pumping speed equal to
background pressure.
Accordingly
This value for
that the throughput of
be better than 13.3 torr Z/sec
133 £/sec at 0.1 torr
For this system, a
4" stroke
"Ring Jet" booster diffusion pump backed with two Welch
#1398 Duo-Seal pumps connected in parallel are installed
to achieve the above requirements.
Under the above estimations and assumptions,
the supersonic jet machine is designed and constructed.
The following sections will give a full description of
this machine with detailed outline of the alignment of
76
the different pieces of this machine.
A separate section
is devoted to describe a typical run.
III.
3. A.
VACUUM SYSTEM
The vacuum system is a rectangular aluminium
box
(see Figure 17) welded together out of 1" thick
aluminium plate.
The inner dimension of the box is
about 25.5" x 22.9" x 15.0".
we have:
baffle
a-
In the base plate of the box
two 4' holes, one of which mates to a water
(to prevent the oil vapor
from getting into
the inner box), which in turn mates to a
booster
oil
diffusion pump with a '14
pumping speed.
used
4" stokes
'/
sec.
The other hole will eventually be
for another pump and it is now sealed with a
flange which has a DV-6 gauge tube for measuring the
pressure inside the box.
b-An
6" hole is available for
signal collection in future experiments
The diffusion pump is backed with two 1397B DUOSEAL rotary pumps connected in parallel and each can provide
a pumping speed of about 7 Z/sec.
The pressure of the fore-
line is monitored with a DV-24 gauge tube. The diffusion
pump oil
is
heater.
The potential drop across the heater was
Apiezion 201 heated with a 220 volt (ac)
electric
increased to about 245 volt which improved the pumping
speed and throughput by a factor of 1.5.
The circuit shown
Fluorescence Collection
Direction
Oven assembly
2
/
LIMEI-
N
N
Water Baffle
Di ffusion
Pump
To 1397B DUO-Seal Pump
To 1397B DUO-Seal Pump
FIGURE 17 --
Vacuum System
78
in Figure 18 is used to provide two power lines of 220
volts each and another 245 volt
line.
One of the 220
volts runs the rotary pump and the 245 volts supplies
power to the diffusion pump heater.
In the top plate, we have a 6" hole which is sealed
with the hardware of the fluorescence collection optics.
The side plates are made identical.
Each side has two 4"
holes, one of which is used for the laser beam.
The
other is closed with a plexi-glass flange and allows a
view to the interaction region.
This
-
used for another laser beam in the future.
hole will be
The front
is closed with a rectangular flange and the back flange
is welded onto the box and contains a 57" hole.
8'
This
hole is sealed with the source flange which supports
the rare gas and oven assembly, as well as the oven
alignment assembly.
All flanges for this chamber are
made of aluminium square plates.
The sealing mechanisms and
movement of flanges for alignment are made possible with
the design shown in Figure 19.
From Figure 19 it is easy
to see that the center of the flange can move ±6
x-direction and ±A inthe y-direction.
in the
The tube can move
inside the flange along the third orthogonal axis.
Con-
straint on this movement depends on the type of function
the tube has.
This constraint will be mentioned whenever
. Neutrol
0"
115V
_____Ei_
Ln
2115V
>1
14
£
I
120V
120V
65K
1
208V
To Rotary
Pump
0~~
N
W
65K
1
W
208V
Not Used
1.15
3
3
41
21
4
To Fore Line
Gauge
7
To Chamber gauge
Controller
Neutrol
To Diffusion Pump
Variable from.
"' 208 to 25V
FIGURE 18 -- Power Supply for Pumps
80
z
X
Y
K-
6
K'
Symmetry
Plane
I
-4K
26
6 is the treedom in the X direction
A is the treedom in the Y direction
FIGURE 19 --
Flanges Design Show Flexibility For Alignment
80
it is found necessary in the content.
The background pressure achievable in this system
without gas load is typical 10- 3torr.
A typical value
for the pressure with load (100 atmosphere gas pressure
is about .
behind a nozzle of 30 vim diameter hole)
0.1 torr in the
foreline.
vacuum chamber and about 750p in the
The reading of the pressure gauges are always
corrected to read the true pressure of the gas
(Ne)
which is under running.
III.
3. B.
MOLECULAR BEAM ASSEMBLY
This section includes two parts.
The first
part will be a description of the different pieces which
are installed in the supersonic jet beam assembly for a
nicely operating molecular oven.
These pieces are
catagorized as an oven assembly (see Figure 20),
holder and inert gas assembly
(see Figure 21),
an oven positioner system (see Figure 22).
oven
and
Most of these
pieces are fixed into the source flange.
The second part will be a description of the procedure for loading the oven.
Most of the molecular beam assembly is shown in
Figure 22.
Figure
20.
The oven assembly is shown in detail in
The oven
and the nozzle heater jacket are
made of 17-4-PH stainless steel rods.
After matching the
81
Nozzle Heater
Copper Washer
J
NozzleT
Stainless st el
mesh
Thermocoupi e
Nozzle
Heater Jacket
Stainless
steel filter
Sodium Boat
Drilled screw
Thermocoup e
Body Heater
Body Heater
Jacket
01
Radiation
Shield
FIGURE 20 --
Oven Assembly
Heat
Senk
Molecular Oven
Oven Holder
Tantalum-Ceramic
Heaters
Drilled Gas
Inlet
Nozzle Heate r
Block
Weld
CN
00
/
High Pressure
Gas Inlet Tubing
Au-Ni Braze
Sodium Boat
3Op Nozzle Hole in
Mo lybdenum Disk
Sintered Stainless
Steel Filter
FIGURE 21 --
Cross Section in Oven and Oven Holder Gas
High Pressure
Gas Coupling
Vacuum Chamber Wall
lo -I
Vacuum Elec trical Feed-Through
/
//
X-Y Positioni ng
Device
Source Flan,
,1,
30V Nozzle
Oven Holder
5.'
Cajon Fitting
\x
E
Sodium Oven
/
/
/
Nozzle Heater Jacket'
/
'I'
FIGURE 22 --
View of Oven, Oven Holder, Source Flange, and Oven Positioner
84
The
two pieces, they are heat-treated to harden them.
nozzle is a 2.0 mm diameter molybdenum disk with an
30p hole in the middle.
It is a commercial Ted Pella,
Inc., Costa Mesa, Calif. product.
The nozzle is sealed
to the oven by a thin copper washer which fits inside
the front opening of the nozzle section.
Molybdenum matel is found to be relatively resistant to
erosion by sodium.
Stainless steel nozzles were found
to have an irregular hole after a few runs with sodium.
The nozzle heater jacket is screwed onto the oven
threads.
Molybdenum disulphide powder is spread on the
threads to serve as a lubricant.
The screwed nozzle
heater jacket seals the front oven to the nozzle disk.
The sodium boat has a space for a 5pm pore filter at the
front.
The back has a threaded hole for a screw which
has a .020" hole and 8mm long.
High purity sodium liquid is poured inside the boat.
The filter is then fitted inside the boat.
The boat
assembly goes inside the oven as shown in Figure
21
The hole in the screw keeps Na vapor from diffusing to the
cool parts of the gas handling system.
A cleaned
stainless steel mesh is also inserted behind the boat
to slow the sodium vapor of diffusing backward.
85
The nozzle heater and the back heater are closely
wound coils of 0.01" Tantalum wire.
The coils are wound
on a ceramic rods and cemented inside a ceramic tube
with a Saureisen #1 cement, a high temperature ceramic.
This is to isolate them from the metal jacket.
The coils
do not short to each other because of the oxide layer which
Five/four heater coils are connected
forms on the surface.
together in series by crimping a copper-nickel tube
around the junction to form the back/nozzle heater.
The
back heater jacket is made of a sheet of copper which
surrounds the middle oven body.
three steel screws.
It is held in place with
Five copper tubes equally spaced are
welded around the copper cylinder for the heaters to set
in.
Two chrome-Alumel
thermocouples are attached to the
oven; one is kept between two steel washers which is
bolted to the middle hole of the body heater jacket;
the other is cemented to the front of the nozzle jacket
see Figure 20.
The thermocouple, leads are inserted in
ceramic nuts to isolate them, and their ends are
connected to a connector by crimping the wire inside the
connecter.
couple
Both the nozzle/back heaters and thermo-
connecters are plugged to a vacuum electrical
feed-through connector which is itself sealed to the
86
vacuum chamber with a 'cajon' fitting.
Figure 23
shows
a cross section of the vacuum electrical feed-through
connector.
The source flange supports the molecular
oven assemble, the vacuum electrical feed-through
connectors, and the xy-positioning device.
Most of these
are shown in Figure 22.
The heater leads are wired to a power supply whose
circuit is shown in Figure 24.
The thermocouples leads
are wired to an Omega-type thermocouple meter shown
also in Figure
24.
Radiation shields are made of 0.002
and shaped as shown in Figure 20.
stainless foil,
The shields cover both
the nozzle heater jacket and the oven body heater jacket
as well as all heaters and thermocouple leads.
shields
These
cut down the black body radiation from the oven,
and also reduced the power required to heat it.
heat sink which is shown also in Figure 20 is
The
placed around
the back side of the oven and its leads are bolted onto
the source flange.
This heat sink is made of copper and
the leads are thick sheets of fine copper wires.
The
heat sink keeps the Teflon "o" ring temperature down.
Figure
22
show part of the oven holder which
connects the oven assembly to a stainless steel tubing
used for the flow of neon as well as a translation of
movement for aligning the nozzle.
The oven holder is a
"O.D.,
8 mil Wall,
SS Tube
RTV Filled Cavity
I
0 0
o
0
N-
Epoxy
Male Amphenol Pin
Plexiglass,
\ Drilled for Pins
Female Amphenol Pin
Plexiglass
Epoxy
Wire - Soldered to Pins
Scale: 2 times actual size
FIGURE 23 --
Bacuum Electric Feed Through
Double
15A,
115U
5A Fuse
Vaziac
Isolation
0-5 amp (AC)
4
Neon Lamp
Heater
1 out
out 1
out 2
Heater
2 out
Cxo
out 3
47
I
00
Heater
3 out
OMEGA
Thermocouple
meter
F
-%
",
115U
Figure 24 --
0-150 V(AC)
Power Supply of Oven Heaters
89
The
machined stainless steel piece.
treaded end of the
oven is screwed into the oven holder where a Teflon
0-ring provides high pressure seal. Molybdenum power is
used to lubricate the oven and the oven holder
The stainless steel tubing is
The inner tube is an
ill
tube is an f
two coaxial tubes.
" O.D and 0.016" I.D.
O.D. and
3"1
"
I.D.
treads.
The outer-
The inlet is brazed
onto the oven holder with Au-Ni braze as shown in Figure
22.
The outlet is welded on a high pressure gas
coupling.
The outer tube is used to strengthen the
capability of holding the oven assemble without suffering
much bending; also to move the oven around for alignment
purposes.
The outlet is connected to an
"
O.D. and
.016" stainless steel tube which runs to a preheated ziolite
column which in turn is connected to a regulator.
The
ziolite column is used to trap vapor water from neon gas.
The regulator has a 3000 PSI output and 5000 PSI input
gauges.
The regulator is connected to a neon cylinder.
The neon gas is a first type run.
The cylinder contains
about 75% of National neon (20 Ne and
22Ne) and 25% of
helium.
The oven front can be moved along three orthogonal
axis as shown in Figure
supports the oven
25.
The Cajon connector which
and provides a vacuum seal is drilled
to increase its inside diameter.
This allowed movement
90
Oven Holder and
Gas Handling Tube
x
X
Source Flange
N
XY-Translator
y
Z-translation
FIGURE
25 --
Oven Positioner
Cajon Fitting
91
of the oven front about 1/2" in each direction
without hitting the cajon.
(X and Y)
X and y direction oven
movement is accomplished with an xy-translator (see
Figure 22)
which is bolted on a "U" type aluminum
piece secured to the source flange.
The displacement
of the oven front is constrained to an 1/25 radians
angle.
This was barely sufficient to fine tune the
oven for maximizing the signal.
The oven can be moved along the third orthogonal
axis
(z-axis) with a crude z-translator made from two
threaded rods.
This movement in the z-direction is
to investigate the molecular density and temperature
as a function of downstream distance.
To run stably without clogging the oven,
Na
must be very clean and carefully loaded.
To accomplish
this, we follow the following procedure.
The oven
assembly
(oven body, sodium boat, boat screw, a small
stainless steel funnel, and radiation shields) are cleaned
thoroughly with a detergent in an ultrasonic cleaner.
The
hole is dryed with a heat gun.
the nozzle
(see Figure 20)
ing of the oven front.
are
The copper washer and
fixed inside the small open-
Both the oven and nozzle heater
jacket treads are lubricated with a moly powder
(be
thorough or the threads will seize upon disassembly).
Then
92
the nozzle heater jacket is screwed onto the oven front
to a degree where you feel that the molybedenum nozzle
is pushing on the copper washer.
After the nozzle is
tightened to the oven body, a diffraction test for the
nozzle hole is necessary to be sure that the nozzle
hole is open.
A Ne-We laser light incident on the nozzle
hole projects a diffraction pattern on a screen in front
of the nozzle.
The pattern should exhibit
nice
diffraction circles if the nozzle hole is completely
clean.
The nozzle hole diameter "D" can be checked with
the diffraction formula sine
1. 22
D
=
Once the nozzle hole is tested, the accessories used in
oven loading (electric heater, funnel, hammer, screw driver,
ampule of sodium, sodium boat, stainless steel filter,
stainless steel mesh) are set inside a chamber filled with
dry nitrogen which does not react with sodium (Na).
The ampule of pure sodium is melted with the electric
heater.
With a tissue
paper holding the two ends of
the sodium ampule, a torque should be applied at the
neck of the ampule to break it.
As fast as possible,
liquid sodium should be poured inside the sodium boat
with the help of the funnel.
The stainless steel filter
is hammered inside the sodium boat to close it.
The whole
boat is inserted inside the oven and pushed down with the
help of the screw driver and
soft hammering.
93
After that a small piece of stainless steel mesh is
inserted inside the oven and the oven back is closed
with a rubber stopper.
The loaded oven is then taken from the nitrogen
chamber and screwed to the oven holder.
The back
heater as well as the heated sink are fixed onto the
oven body.
The cone part of the radiation shields
are blackened with lamp black generated from an open
to flame of Trichlorethylene. This reduced the scattered
laser light from that part.
The neon gas is then
leaked to the oven and the gas flow through the nozzle
hole should be checked before attempting to run the
experiment.
III.
3.
C.
INPUT/OUTPUT LASER OPTICS
This section will include three parts -input/out laser optics, auxiliary optics
(channels
proportions of the laser beam into several devices),
and the alignment procedure of the input/output light.
Small portions of the laser light are sent to
different devices
(the frequency standard, power meter,
spectrum analyzer and sodium cell) as shown in Figure
"S
"
is a 12.7mm thick glass plate of
error.
wave front
It splits the laser beam into three portions.
26.
To E 2
Tol Frequency Measurement
System
To E
M
A4S
Adapter
A
S
3
S1
S
2
To supersonic
Jet Machine
Tunable Dye Laser
M1
Spectr
Analyze
01%
N-cel
Na cell
Scope
x .4-
FIGURE 26 --
Auxiliiiry Laser Optics
Power
Meter
95
The first portion is a reflection of the front surface of "S"
which is focused onto a photodiode "P", with a lens "L".
The signal of
I"P
is sent to a power meter which monitors
the intensity of the laser beam [the laser beam intensity
is stabilized with a feedback loop applied onto the
Argon laser.
The procedure and the process of stablization
Apt (APT76).]
is described by
The second
portion is a reflection of the second surface
of "S"
This light is sent to the frequency measurement system
by a bounce on a mirror m.
Mirror "im"
moves the
laser beam laterally into two orthogonal directions by
two fine screws.
This light is split into two with a
beam splitter S4, one goes to the short etalon "E",
and the other bounces m 2 and goes to the long etalon
The frequency standard is described in Part 2 of
1E2"'
this Chapter.
"S
The third portion is a transmission through
sent to another beam splitter "S2"'
"
a small portion to a sodium vapor cell.
"S
reflects a
The Na-cell is
used for alignment purposes as well as for counting modes
to the red or to the green of the sodium P1/2 or
P3/2
lines.
Fluorescence in the cell, which is heated to about
551C, is clearly visible when the laser is at the correct
frequency.
"S 31" is another beam splitter which reflects a small
portion of the laser into a Spectra Physics 470, 8GHZ spectrum
96
analyzer.
This spectrum analyzer is swept by the horizontal
scope drive and displays the mode structure of the laser.
This is necessary to be sure that the laser is operating
at a single frequency and is also convenient for observing
sweeps or mode hops.
The optical system which directs the laser beam into
and out of the supersonic jet machine
is shown in Figure 27.
(interaction region)
Earlier in the introduction a
geometry factor has been introduced in equation
(3-1) to
describe in a way the shape of the laser beam at the
interaction region.
to converge with an
The laser beam is shaped intentionally
radians angle for three reasons;
the first is to reduce doppler broadening of the absorption lines; the second is to increase the absorption
region
to enhance
the signal,
the third is to increase
the intensity of the laser light at the region of interaction.
The system which will be described here has the
above three concepts in mind.
The laserafter passing through the beam splitter
S3
shown in Figure 26,
"A"
passes through an adapter
"A".
Adaptor
determines the size of the laser beam and purifies it
from the unwanted light caused by double reflection through
the laser intercavity etalon and blocks the majority of the
uncollimated fluorescent emission from the dye cell, and
Chamber
Output
Baffle Arm
Input Baffle Arm
oven
M2
F2
wall
F
M
1
FIGURE 27 --
Laser Beam Alignment Optics
1
A(ADapter)
98
specially defines the output beam.
The laser light is
then diverged with a 170mm focal length lens "F
located about 5cm away from the adapter
mounted on an xy-translator.
"A".
Lens "F
" is
This mount can move the lens
smoothly which in turns helps to focus the laser light on
the lens axis.
The expanded laser light makes two bounces
at "m " and "m 2 "
("
1
" and
"m2 " are Spectra Physics
front
surface reflectors for 450 angle of incident) before it
is directed to the supersonic jet machine.
m2
Both m 1 and
are mounted on a mirror mount which has two fine screws
to direct the beam onto two orthogonal axes.
Density
filters "D" are used sometimes to attenuate the laser beam
The attenuation is necessary to decrease the intensity
of strong fluorescence and keep the photocathode in thermal
equilibrium.
cathode,
If a strong signal is seen by the photo-
the dark current becomes several times larger
than its original value until the photocathode returns to
thermal equilibrium.
The reflected expanded laser light
hits a 495mm focal length lens "F 2"
F2 is mounted
inside an input baffle arm against an 0-ring fitted in a
baffle.
It serves as a vacuum seal.
The seal is made
possible by the difference in pressure between the vacuum
chamber pressure and the atmospheric pressure.
The input
baffle arm consists of two tubes of different diameters.
The outer tube which has a bigger diameter slides on the
99
other and can be tilted with four screws.
This tilt helps
to let "F 2 " focus properly on the molecular beam.
F2
focuses the laser light at 1/4" away from the intersection with the molecular beam and provides a 1/15 rad.
converging angle which constrains the interaction region
The laser light exits
to be within this angle.
through
a 3/8 thick plexiglass exit window which is glued onto
the output baffle arm at about 56" angle to provide a
to the vacuum system.
seal
The reflected light at the
exit window gets absorbed with the 70cm output baffle arm.
The exited laser light is absorbed by black flock paper
which is mounted on the wall in front of the exit window "w".
Simple lens equations are used to find the distance "D"
between the lenses "F"
and "F 2 ""(D=92.0)";
the beam
size at the adaptor A(2.5mm) and the distance between lens
F2 and the focusing point
(25 1/4").
The alignment procedure of the input/output laser beam
into the supersonic jet machine is done along the following
steps.
1.
Both the input/output baffle arm axes are aligned
to pass through the center of the hole of the vacuum chamber.
2.
Both input/output flanges which hold the input/out
arm baffles are tightened firmly.
100
3.
Both lenses "F " and "F2
(Figure 27 )
are
removed and the laser beam is directed to go through the
axis of the input/output arms.
and "im
mirrors "im"
This is done by both
which change the inclin-
"
2
ation of the laser beam
relative to the input
baffle arm axis.
4.
If the laser beam does not pass through the
axis of the output baffle arm, the flange which holds this
arm should be displaced until the light passes in the middle
of the exit window "w" as well as the entrance.
5.
holder.
Lens
"F2"
is to be mounted in its input baffle arm
'This arm should be tilted slightly until the focused
beam emerges in the middle of the tube of this arm.
6.
Lens
"F 2 " is then mounted and adjusted in space
by the xy-fine adjustment in order that the diverged
laser light hits the lens F2 in the middle.
The size of
the laser beam can be defined with the adaptor "A
III.
3. D.
"
.
FLUORESCENCE COLLECTION OPTICS
The system which collects fluorescence is designed to
have two properties.
The first is enough flexibility
(degrees of freedom) to focus the interaction region properly on the photocathode of the detector.
This objective
was achieved by designing movable hardware pieces which
101
hold the optical components.
will follow.
Description of these pieces
The second is a way to prevent the scat-
tered light from reaching the detector.
This is achieved
by baffling the scattered light at the focus of the collection lens which must therefore have reasonable optical
quality.
A schematic diagram of the optical system used for
detecting the fluorescence and processing the signal is
shown in Figure
The collection lens "F1
28.
",
50.8
mm focal length f/l.0 fresnel lens, is mounted on the
baffle "A" which is in turn fixed to a movable tube shown
in detail
in Figure
proper focusing
29.
Tube "D"
of lens F 1 -
by 3 screws equally spaced.
protect lens F
slides on tube "E"
for
It is fixed in position
Conical
baffle "A" helps
from sodium contamination on its surface.
When the molecular beam jets well, little sodium is
deposited on the lens edge as well as on the tongue of the
baffle A.
Tube "E" extends outside the vacuum chamber
where it can slide inside a rectangular flange G 1 which
seals the vacuum system with two o-rings.
A flange G 2 is
welded onto tube "E" and has an 1/2" hole which is covered
with a 66-2450 Rolyn heat reflecting filter (cut off below
0
6500 A) sealed to it with RTV silicon rubber.
Tube "E" is
supported by three long screws which fix both flanges G 1
and G 2 together.
These screws also function as fine
Y
PMT
l
.J
- T (Filler)
F2
B(Baffle)
x
D (Filter)
F1
PowerCutr
supply
Photon
isc
----- DiC
Electrometer
iod
Pum
XY
ecorde
Refrigera or
Power
SupplyInvertor
Figure 28 --
Td
Amp and
Block Diagram of Detection Electronics
e
103
FA-t
FIGURE 29
-- Assembly of Collection Optics
104
adjustment for focusing of lens F when the experiment is
running.
Rough focusing is done by observing an object
placed at the interaction region through the lens F1 and
sliding tube "D" in and out tube E, as well as tilting
tube E with the three long screws "S" until a sharp goodquality image is observed at the baffle "B".
On the top of flange G
baffle "B" is attached to
the movable arms of an xy translator
(mounted on flange
G 2 ) so that it can be moved to coincide with the image of
the interaction region.
The magnification of the collec-
tion lens "f," is around 8 so that the image of the interaction region is 2 x 5 mm.
The fluorescence light which passes through the
baffle is a 89 mm focal length f/l.6 lens and focused on
a 31034 RCA photomultiplier tube.
Focusing is done with
the help of a tube "I" and both flanges "G "1and "G51*
4
Tube I is welded onto a flange G 4 .
Flange G 4 has 3 1/4"
where a drawer can slide through.
hole and a slot
(The flange drawer and the tube assembly is shown in
Figure
30.)
The drawer is rectangular in shape and has
an opening to install filters or pin holes.
By sliding
the drawer on the flange G 4 , the photomultiplier cathode
can view through the pin hole or the filter and can be
blocked as well.
This flexibility is necessary to test
/Mi
7
LC)
0
H
FIGURE 30 --
PMT Flange and Drawer
106
for blackbody radiation, to attenuate large signals and
to peak up the signal.
The flange G 4 also has two 1"
clearance holes for two 1/4" screws.
These screws pass
through the holes to two-treaded holes in the flange on
the PMT cooler.
The PMT can be moved in the area
between the hole and the screw.
This movement is used
to maximize the signal by proper focusing of the fluorescence on the photocathode.
The 31034 RCA photomultiplier tube is cooled with a
TE-104
(Products for Research, Inc.)
and powered with a 1118
refrigerated chamber
JF DC power supply.
The dark
count rate is around 35 counts/sec.
Alignment:
The alignment procedure of the collection optics is
as follows.
A sharp edged object is positioned to block
the lower half part of the laser beam at the interaction region.
Flange G
is then positioned so that the image of the sharp
edge passes through the 1/2" hole in flange G 2 .
This
image is focused on the position of baffle "B" by sliding
tube D up and down.
Baffle B can be moved horizontally
by the xy translator for fine centering of the image.
The photocathode of the PMT is moved up and down to set
on the image focused by F 2 .
"I" over tube H
(Figure 29).
This is done by sliding tube
Usually a small signal is
allowed to leak to the photocathode by attenuating the
107
laser beam.
Final adjustment must be done when the chamber is
under vacuum because evacuating the chamber moves the
optics slightly.
This will be described in Section III.
3.G.
III. 3. E.
"BAFFLE SYSTEM"
The signal of NaNe estimated in Eq. 3-10 was found
in the order of 105 counts/sec.
This suggests that the
fluorescence detector should not view any scattered or
stray light whatever its source.
used two baffling systems:
To achieve this, we
one to cut down the laser
scattered light; the other to baffle the collection lens.
The laser baffling system is shown in Figure
It consists of an input and output arms.
31.
Tube "B" which
holds the lens Fslides over ring "C" and can be tilted
as well for alignment purposes.
Both tubes "A" and "B"
contain six brass disks painted black, mounted coaxially
and located at fixed distances along the tubes "A" and
"B".
Black flock paper of about 97% absorbtivity is
used to position each disk along the tube and to absorb
scattered light.
The output arm is exactly similar to
the input arm except that a Brewster angle (56*) plexiglass exit window replaces the lens F 2 .
The window is
glued to tube B with an epoxy to provide a vacuum seal.
N
NOW
ov-en-
.. 6J
0,
0
r-
I't
-
-- .2 aI
A
FIGURE 31 --
Laser Baffling System
J
1
~1
109
The lens "F 2", tube 4 and flange E provide a vacuum seal
to the vacuum system.
The design of this baffle system is done on the
assumption that scattered light from the lens should not
hit the oven which is located 1/4" off the optic axis.
Light which is scattered outside the converging cone of
the laser light gets baffled with baffles 2-5.
is to pass through.
The rest
Baffle 1 is only to intercept light
scattered from these baffles.
Multiply scattered light
of baffle 2 is not baffled when the oven is less than 1/4"
away from the laser light.
The main laser beam passes
cleanly through the reaction zone and the aperture of
baffle 1' of the opposite arm.
Baffle 2' of the opposite
arm intercepts this light and reflects it away from the
interaction zone.
The collection optics baffling system is shown in
Figure
32.
This baffling system is designed to
reduce the scattered background light (whatever its
source) through the collection optics.
two baffles "A" and "B",
It consists of
Baffle A is shaped
and a hat C.
to be a cone with its top cut and shaped to have tongue.
Its bottom is formed to have a seat for the collection
lens "L" and it extends after that to form a base which
has two holes.
The holes are used to
fasten baffle "A" to tube D
(Figure
29) with two long
110
(Baffle)
B
(Filter)
F
(Baffle)
A -
oven
(lens)
zn.Q
o
(Interaction region)
(Hat)C
FIGURE 32 --
Collection Optics Baffling System
111
screws which are used also for holding the hat "C".
Its
inside is covered with
black flock paper and its out-
side is painted black.
Baffle B is located at the focus-
ing of the collection lens L.
It is made of
cardboard with a 2 x 3 mm aperature.
black
The hat C is also
shaped to be a cone and made of black flock paper.
It
is located in a position that the lens L just can view
the inside of the hat C.
After all these efforts the scattered light level
could be kept typically to about 40-50 counts/sec when
the oven is more than 1/4" away from the interaction
region.
If the oven is moved closer to the interaction
region, the number of counts of the background increases
rapidly, implying that baffling should be installed
closer to the focus.
III.
3. F.
DETECTION ELECTRONICS
The PMT anode signal is processed as shown in the
block diagram of Figure
photon discriminator
28 .
This signal is
sent to a
(511 Mech-Tronics) whose output is
sent to an 5321 B Hewlett Packard counter and to a diode
pump.
The diode pump produces a DC voltage proportional
to the pulse rates.
Figure
33.
Its circuit diagram is shown in
The output of the diode pump is channeled
to an electrometer which is our analog signal monitor and
112
0.0037pf
Out
In
61S
FIGURE
33 --
1 OkA
Diode Pump Circuit
0. 05pf
113
to an amplifier and inverter.
The output signal of the Amp and inverter is sent to
a mixer for the frequency markers and the fluorescence
signal.
The mixer is just a potentiometer.
The output
of the mixer is sent to an xy recorder and time chart
recorder.
The xy recorder is swept by a small portion of
the intercavity etalon PZT voltage of the dye laser.
The
xy and time chart recorders display the same molecular
spectrum.
The time chart recorder has a control
which gives the advantage of expanding the spectrum in
order to locate the center of the lines more accurately.
It gives more resolution to distinguish lines which are
nearly overlapping each other,
and it
has a linear distance
vs. frequency relationship.
III.
3.
G.
ALIGNMENT, RUNNING AND DATA TAKING PROCEDURE
This section describes the procedure of a typical
run.
This includes preliminary alignment of the system,
checking the jetting
condition of the
oven,
fine align-
ment and data taking procedure.
The preliminary requirements for alignments are described at the end of the previous sections, e.g. the
input/output laser alignment can be found in the laser
input/output section; the alignment for fluorescence signal can be found in the fluorescence collection optics.
114
After defining the interaction region, the molecular beam
should be aligned to pass through it.
During this align-
ment process a slow flow of neon should be kept on to
prevent any flow of air back through the nozzle to the
sodium.
If this is not done, an ox-idation as well as
hydration of sodium might clog the nozzle as well as the
filter pores.
Having accomplished the preliminary alignment, the
vacuum chamber is closed and pumped out.
Evacuating the
chamber will displace the input baffle arm, the tubes
which hold the fluorescence collection optics, and the
source flange which holds the molecular oven assemble.
This will upset the preliminary alignment and further
alignment is unavoidable.
A fine alignment is done on an atomic signal
or P
of sodium lines) and on a molecular signal
dimer or sodium neon lines) as well.
(P12
(sodium
Before getting to
this step, it seems unrealistic to fine align the experiment without being sure that the oven is jetting properly.
Proper jetting implies a one-directional flow of the fluid
(gas mixture of both sodium and neon) through the nozzle
hole.
If this criterian
is fulfilled, we should expect
that the flux of the fluid F to be given by
F
=
ynvA
(4-12)
115
where y is a constant and A is the area of the nozzle
hole.
n and v are the density and mean thermal velocity
of the fluid.
The constant y is bigger or equal to 1/4
for a nozzle hole which is open and has negligible length.
The flux (throughput) is measured by different ways, e.g.
it is the product of the chamber
(or the fore-line) pres-
sure times the pumping speed, i.e.
P
= S P
c c
= S P
where Pc
=
4 x 1021 P
=
c
(4-13)
4.2 x 1020 Pf
f are the chamber/fore-line pressure in torr,
and Sc Sf = 133/14 are their corresponding pumping speeds
in Z/s.
The numbers in Eqs. 4-13 are just the pumping
speed of particles per torr.
Flux can be measured with the change in pressure
with time "dP/dt", i.e.
='
=
5 x 1018 d414
the "5 x 10 18" depends on the volume of the vacuum
chamber.
The throughput can be measured with the sodium Dlines signal or the Lorentz tail of the D-lines
P3/2 '
i.e.
(P1 / 2 or
116
F = 2.8
x
10 7
P R
P
Ne
P
PT1/2Na
(4-15)
and
F = 2.96 x 1012
(RAv)2 PNe
1/2
Na
4-16)
where
PNe is the neon pressure in PSIG.
PNa is the sodium pressure in torr.
P
is the laser power in mw.
R
is the distance between nozzle hole and
interaction region.
Av
is the off-tuning of the D-line central
frequency.
6
is the baffle size at the xy translator.
(See Figure
T
is the oven temperature.
Ne
is the number of counts read by the photocathode.
Eqs. 4-13 and 4-14 can be used to find the valve of Y
before accomplishing the fine alignment, while Eqs.
4-15 and 4-16 can be used after the fine alignment is
accomplished.
Having the vacuum chamber at several microns of
pressure, a test should be performed to be sure that the
117
nozzle is not partially clogged.
This can be done by com-
paring the flux measured by either Eq. 4-13 or 4-14 with
the expected flux from Eq. 4-12.
If y turns out to be
less than 1/4, two things might occur:
either the nozzle
hole is clogged or the neon gas cannot get to the nozzle
hole because the sodium might clog the filter pores
through the loading.
Ify turns out to be bigger than 1,
it implies the neon gas is leaking through the position
of the vacuum seal at the nozzle.
In this case more
tightening of the nozzle heater jacket to the oven and
the oven itself to the oven holder will be necessary.
But if y is less than 1/4, it is more advisable to heat
up the oven and check y again as before.
From experi-
ence,
y was
it was found that most of the time
1/4 before heating the oven, and got
less than
close to 1/3 after
If heating the oven does not bring y above 1/8 or
that.
so, it is not worth the effort of running the experiment.
Instead an effort should be made to clean the oven and
the nozzle.
It is found that cleaning the oven with diluted
sulfuric acid and brushing the inside of the oven with a
stainless steel mesh helped
to remove, any
chemical deposited inside it and eventually helped minimize
(amost
to zero)
the frequency of clogging the nozzle.
118
We should mention at this point that the old stainless steel nozzle was inefficient in getting a supersonic
jet.
The reactive sodium was probably the factor of erod-
ing the nozzlehole-and
than 1/8.
lowering the constant y to less
Molybdenum nozzles,
which are drilled
to
decrease the nozzle hole length, were definitely producing supersonic jet with y
-
1/3.
Having a molecular beam jetting properly is the
first step for a fine alignment of the experiment.
The
next step is to remove the upper part of the fluorescence
optics tower (from PMT to tube "I";
see Figure
29 ) and
then to set the laser frequency on either the sodium P1/2
or P3/2 line.
This can be done by watching the sodium
fluorescence in the sodium cell
(see Figure
26).
The
next step is to look at the sodium fluorescence in the
beam and to move the nozzle up and down until the two
beams
(laser and molecular) intersect each other.
One
can see the interaction region through a mirror fixed
inside the machine which reflects the sodium fluoresence
through a plexiglass flange.
The fluorescence can be
seen clearly through the fluorescence collection lens.
If the fluorescence light is not seen illuminating the
middle of the fluoresence collection lens, the flange,
which
holds the collecting lens assembly, should be dis-
placed by soft hammering on it until the light is clearly
119
illuminating the middle of the collection lens.
Aligning
the fluoresence through this direction requires imaging
the fluoresence light at the position of the baffle on
the xy translator (see Figure
29).
This can be done by
focusing the eye at the baffle position and moving the
head back and forth.
If the image did not move, it
implies that it focuses at the baffle.
If it moves, the
collection lens should be moved up or down with the three
fine alignment screws
tionary.
(Figure
29) until it appears sta-
The baffle should then be placed on the xy
translator with its diagonal parallel to the laser beam.
The baffle is to be translated until the fluorescence
image passes through the baffle hole.
All this should be
done while making sure that the laser drift
corrected
is
always
by tuning the laser back to a maximum
illumination of fluorescence
in
the sodium cell.
The next step is to install the PMT in its position
with its photocathode closed.
of transmission ~ 10~
(see Figure
27'
A neutral density filter
should be placed in the laser beam
before exposing the photocathode to the
fluorescence light; otherwise it will saturate the photocathode electron emission which implies an increase in
the anode dark current.
should not exceed 10~
Maximum anode current of PMT
amp.
is read with the electrometer
The PMT anode signal which
(Figure
28)
is
maximized
120
by tuning the baffle with the xy-translator and moving the
PMT with the two screws attached to its flange
Figure
kept
29).
(see
Again the laser frequency should be
at the maximum fluorescence of the sodium cell.
If the laser frequency drifts, the fluorescence region
will move accordingly.
At this point jetting condition
of the oven can be tested by using Eq. 4-15 and 4-16.
Still no claim can be said that the system is completely aligned.
The last possible fine alignment should
be done on a molecular line like those of Na 2 or NaNe.
The laser frequency is then moved two etalon modes
(10 cm~
) to the red of P /2
The density filters 10-
are then removed and a sodium neon line is
chosen
for fine alignment.
This is done by fine tun-
ing the oven xy-translator and sweeping the laser across
the NaNe line.
This will direct different parts of the
molecular beam to the optical interaction region.
change in fluorescence signal can be viewed
The
on the
electrometer.
The anode signal is processed as shown in Figure
28.
The noise is minimized by choosing a proper time constant
at the signal amplifier and inverter.
Both the xy and
time chart recorders are used to monitor the signal.
this point, it is important to stress
At
the idea of
changing both the back oven and nozzle heater temperatures
121
to give a better signal-to-noise ratio.
What is meant
by noise here is the one due to the blackbody radiation
and Lorentz wings of P1/2 and P3/2 sodium D-lines.
This
is important because the molecular sodium neon lines lie
in the vicinity of the sodium D-lines.
If the signal is
not maximized relative to the Lorentz tail, it will be
difficult
to find the molecular lines in that domain.
The data is obtained with the following parameters
of the supersonic jet:
oven/nozzle temperature 350/400*C,
pressure of neon gas - 100 atmospheres, chamber pressure
.1 torr, distance between the nozzle exit and the interaction region 0.5 cm.
The laser power was about 10 mw.
122
IV.
RESULTS AND ANALYSIS
Looking to an absorption spectrum of a diatomic
molecule is quite
messy
for the first time especially
to a beginner in the field.
1
an A 1E
1
u
X E
g2
Figure 34 is an example of
transition of Na 2 .
This part of the spectrum
can be more complicated if the rotational temperature of
the molecule is allowed to warm up.
Figure35
shows the
same frequency region with higher rotational temperature.
The analysis of a molecular spectrum will usually reveal
lots
of characteristics
about the different kind of
physical phenomenon which take. place in the region of
observation in the molecular states..
The following sections are a full description of
the data obtained and of the method utilized to resolve
the spectrum into molecular parameters which can very
well describe its different features.
Before going into
the specifics of the analysis, a general description of
the data will be outlined in the following paragraphs.
The molecular excitation spectrum which is detected
in the frequency domain from 16975.59 to 16929.75cm~1 is
a mixture of both A H1+X E
of NaNe and-A
Na 2 in addition to the D-lines of sodium
ug
(2 P
1 +
-
1/ 2
'
2P
3 / 2 ).
This frequency domain is about 25 and 20 cm'1 to the red
and the blue of 32 P 1 / 2 of Na.
The NaNe lines are
discriminated from the rest by observing the line intensity
(N
PL
u
LaT
T r~11 V~n I
13
A
I
L JA-IA
J
''rai
8
9
10
14
A~L-IL~irr I. hA.
'r~1 1~
12
I
I
t0
I
4
9
8
7
10 GHz
FIGURE 34 --
Absorption Scan of Hot Na
2
I I ilL
3
P
2
6
5
R
4031
2
13
14
9
10
I
I2
I-
I
I
I
4
x
0
X0
Io
5
6
8
6
7
8
9
10
11
12
4f[
II
F
2
4031
5
T2
II
R
P
.li
I
__j
I
-200
1
1
-180
-160
I
-140
I
I
.
I
-100
-120
FREQUENCY (GHz)
FIGURE 35 --
Absorption Scan of Cold Na
2
-80
-60
-40
-zo
I
V0 3.8
125
dependence on the neon pressure in the oven and the hyperfine
structure exhibited with the NaNe spectrum.
The
increase in neon pressure suppressed the Na 2 lines and enhanced the signal of NaNe lines.
The NaNe lines were
found to consist of two components split by the hyperfine
structure of the 32
1.8
/2
sodium ground state, which is fv
GHz, and with almost the same 5:3 intensity ratio.
To decide whether a weak line is NaNe or Na2' because one
of the hyperfine components is overlapped with another
line, it is judged by comparing them with the spectrum of
Na2 which does not have NaNe lines.
The entire observed spectrum of NaNe consists of about
760 lines with about 4x10 5 counts/sec for the strongest
line observed.
Figure 36 represents A H3/
transition of Na 20Ne and Na
does not have any Na 2 lines.
Al
1/2
(v'=4)+X2+
transition of Na 2 .
Ne.
(v=5)+X2 + (v"=)
This sample of the spectrum
Figures 37 and 37' represent
transition of Na 20Ne and A' E ++X' E +
v"=0)
u
The intensity ratio of
2
I(NaNe)
I(Na 2 )
q
i
roughly 1/8 as shown in Figure 37.
Near the atomic D-linesof sodium
(1.5cm~)
it
was hard
to get a good NaNe spectrum because of the strong Lorentz
tail of the D-lines in comparison with the NaNe signals.
In that region of the spectrum, neutral density filters were
used to decrease the incident laser light.
Accordingly
bad relative and absolute intensity measurements resulted.
3.5
p2 2.4.5 55 6.5
i
I
7.5
I
I
8.5
I
11.5
1
10.5
9.5
I
I
1.5 2.5 3.5 4.5 5.5
6.5
7.5
8.5
I
I
I
I
II I
3.5
4.5
5.5
.5 1.5 2.5
R
1
R
1
.5
1 1.5
I
1
3.5
I
2.5
I
4.5
I
9.5
I
I
6.5
No
10.5
20
Nq
I
7.5
8.5
1
1
5.5
6.5
9.5
1
7.5
8.5
i
i
3
'O
0
x
21-
U
4n
0
t.0
N
z
0
i
X
0I-
R21
R2
I
1.5
.5
1.5
I1
1I I1
1.5 2.5 3.5
01
P2
I
.5
i1
1
3.5 4.5 5.5
1
-40
FIGURE 36 --
2.5
1I
4.5
I
1
6.5
7.5
I
8.5
-
1
No
1
6.5
1
5.5
1
9.5
i
l
I
0
60
20
40
-20
FREQUENCY FROM BAND ORIGIN (GHz)
80
100
i
Absorpiton Scan of A 2I
i
i
(5)+X 2 Z(0)
of NaNe
22
Ne
2.5
p 1.5/3.5
'2
5.5
4.5
1.5 2.5
p I
3.5
4.5
I
I
I
Q.
6.5
.8.5
1
7.5
1
5
1.5
1.5
6.5
5.5
3.5
2.5
1.5
.5
9.5
5.5
4.5
3.5*
2.5
9.5
8.5
7.5
6.5
5.5
5.5
4.5
10.5
9.5
8.5
7.5
6.5
80
x 64C
z
0
0
2-
a.
0
I
7
8
S _
-10
,
10
II
I
VO
20
FIGURE 37 --
|
40
I
9
3
4
5
6
3
I
I
8
6
7
80
60
FREQUENCY (GHz)
Absorption scan of A
ID
r
2
t1l/2 (4) X Z(Q)
100
5
PI
1
4031
2
120
of NaNe and A*Eu
i
140
-XE
of Na2
I
00
I.-
CN
-
-I 1
-I~
-
I
H-
I
-
r~
I
-~
~
IL
Ii~
11I
-- A
'*1
-~*~-:1
I
I
' 4p
%tk.
FIGURE 37' --
Time Chart Recorder Absorption Scan of A H1/2 (4)
X2E (0)
129
Electronic predissociation is observed in the
2 3(v'=5)
- 2 + (v'=0) transition.
Overlapping of a certain
electronic state 213/2 (that is, of its vibrational or
rotational levels)by the dissociation continuum belonging
to another electronic state 2R1/2
observed lines in that band.
caused broadening to the
The line width is found to
increase with increasing J' as shown in Figure
38.
The spectrum of both Na 20Ne and Na 22Ne species is
transferred onto a roll of chart recorder paper. On the chart
ity
the intensity, and the reliabil-
the position,
recorder roll,
of the stronger hyperfine component of NaNe lines
are written under each line.
The reliability number 1,
2, or 3 is assigned to each line depending on whether the
line is clear, partly obscured or
doubtful.
This is
subsequently used in our evaluation of the standard
deviation calculation of the lines.
of the line frequency,
intensity,
Table 2
is a list
andreliability.
Having the data properly labeled, the next need is
to assign J',
J", v' and v" as well as to determine
whether the transition is A H3--X E+,A H
3/2
B 2E+X E+
for each line.
This
1/2 +X E
or
information for each line
will be prerequisite before a serious attempt can be made
to find the molecular parameters of the molecule.
parameters are necessary to find the potentials
X2 E+f A2H,
and B2E+
for NaNe.
The
130
1.81
0
1.5-
1.2"
0
G
0.9-+
~z
H
0
0
0.6-
0. 3.
I
I
6
8
J
FIGURE 38 --
Plot of Line Width vs. J'
a
10A
12
131
TABLE 2-- Absorption Spectrum of A
-
2
+X
of NaNe
-------.------
)
---- --- --- --- --
Line
582. 8 0
583. 1.7
585. 00
587,
588.
5 89.
591..
57
48
91.
61.
591.. 83
593. 1.3
.594. 71.
597. 52
599. 27
6 0 0.. 1.7
604 39
t6.0 7. 73
608. 1.7.
61.3. 81.
61.8. 23
618. 68
61.9. 90
620. 78
525. 55
6.3 0. 89
6I1.
1 67
6.2.
5.
67<7.
139.
64 6.
1.1.
78
74
91:
647, 8 3
656. 37
664. 74
665. 91
I6F66C.2f 1;'1.
667. 76
669. 22
W
()'l
1.59 2 9.
16929.
16929.
16929.
16929.
1.6929.
.233
45
2
361.
:
477
641.
697
7P9
896
910
992
1.
94
45
81
2
2
13
16930.
169~I:0.
1 693 :.
1692.
271.
1.54
1
37 9
81
1:6
I
I
4;9
715
1.69- 0. 91.8
16930. 945
16931. 383
16931. 582
1693:1. 611
1.6931. 689
1.6931. 744
1.69.2. 147
16932. 185
169?32. 434
16932. 463
1.6932. 695
16932. 79? .
1.6932. 945
1F.9~
]: 31I3 fr
459
1:134. 00
16934. 529
1-6914.
1. -.9 4.
1.6 934.
1.6934.
604
#6--23
7?1.
814
67.
27
673. 00
1 697:4. 881.
16935. 053
675. 28
677. 56
679. .1.
1.69.5. 1.97
16935. 24?
1.6935. 4.39
16935. 81.2
16935. 902
1.6936. 553
16937 ?71.
1.69 9. (6
1.6938. 04':
1. .697:8. 4 4
1.6938. 591
1.6938. 746
1.6938. 91.0
4,
1.6939
F
16940. 812
16940. 582
1 694(. 61.
i685.
00
686. 41.
696. 67
708. 02
909. 0 0
720.
727.
728.
731.
733.
746,
751..
759.
760.
20
1.7
83
30
87
C7
25
00
75
1.
2
:1
1
2
1
1.6929.
1 6929.
1 929.
16930. 092
I'F9*3
-- --- --
79
1.26
147
1.63
92
157
183
:145
:
60 'I
130.40"
1
16931* I7
1.
186
77
2
77
2
1
2
2
1 72
49
1.49
146
)
1
1032.633
49. 1
57
171
S5
2
1.
1.
7:6
05
1.57
120
141
50
&
1.
1
1
6P.~"f
5
1.8
2
-f
3:
1.
48
3
84
88
1.1.5
1.12
159
96
*9:
6
47
79
F
2
48
46
4 3
1
1
1
1
3
3
1.
6
47
57
59
1
3
1.
)
132
767, r 0
7 72. 2 3
775. 44
77.91
778. 8 P
7-81. 88
78i2..537
782. 9
86
786. 1.~3
784.
786.-9
787. 46
788.21
55
1.6941.. 48
16941. 54:
16941. 699
1.6941. 756
16941: 951.
1641. 992
A'4
11.
P4
159
91':
1.6942. ?21
16942. 2:8
159!
:1.1
55
141
11.9
77
188
824
97
796. 77
1.6942.
1.6942.
16942.
1.6942.
1.6942.
1.6942.
16942.
1.6942.
1.6942.
1.6942.
1.6942.
846
895
85
1 89
797. 46
1 6942.
99
3
800. 28
16942. 888
1694. 1.17
1 6942:. 282
788.48
789. 97 92. 0
792. 79
792. 85
795.6
PtF
795.IS4
796. 00
798.4
-81 . 1-*.:7
8 03. 9 A
804.8 3
806. 22
80 6. 4
8A .
809.
8 Ci9.
809.
81A.
81.2.
796.
:2,
22
5A
92
93
81.
77
p.72
8.1.4. 52
87A8. 65
28F
81.5.
88:1.. 06
816. 19
816.
73
84 -C1
817. 15
88. 9.
822. 05
8 2.-. 40
823.5 8
825. 69
828. 30
84.2.2
829. 45
830. 74
84.
Q4
:5
354
369
461.
594
643
646
787
627
186
1.2
51.
62
1.694:.969
16944. 828
16944. 892
16944.
16944.
1.694.4.
16944.
16944.
.16944.
1.6944.
1.25
1.6
1.86
348
352
292
496
16944. 582
1.6944. 594
1.6944. 789
16944. 89 3
16944. 918
16944. 965
1.6945. 847
16945. 068
16945. 192.
16945. 221
16945. 342
16945. 75
16945. 42
1 6945. 547
16945. 684
1 945. 662
16945. 775
.5:
S
1.
1
2
.
66
168
1 74
684
785
729
16942:. 855
1.6943. 912
1.
1
66
494
582
16943.
16943.
16942.
16943.
1.6943.
16943.
8i5. 3 2
875. 92
4
11.9
16943. 486
1.694:. 348
16943.374
16945.294
841.
89
71
174
18:t
71.
71,
814. 0.:
P838. I61
8 :9. 2
1.75
182
1 87
1.81
78
141.
44
119
129
146
1
1
2.
I
I
i
1.
S:
1.
1.
1.41
119
69
1.77
177
29
1.26
224
I
:1
1~
I
I
1.97
1.75
2.
23:5
I
I
?87
1.88
124
?
1 419-461
69 I. 30
~1.
I .
16942.821
16942. 1.41
4.
1~777 7o
1.
I
133
84:. 63:
1.6945. 863
844. 9
845.32
P.4 6. 75
847. 1.5
850. 26
851. 90
:16945. 945
855. 42
857. 20
858.24
858. 50
860. 51.
8164. 67
16945. 971,
16946. 062
1.6946. 088,
1.6946. 285
16946.
:9
:16946.
1.6 946.
:1 6946.
16946.
1.6946.
477
1 6946.
9.4.
61.1.
725
789
907. 29
9C9. 55
16950.041
911.
91.8.
92.
923.
9.30.
1695P. 1.82
1.695c.687
16958. 721
1-695 17. 910
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137
In the following sections, the method of analyzing the
data and extracting the molecular parameters will be
presented in the order in which they are used to make
assignments of both the J and v quantum numbers.
The first
part (IV-1) tells how one can follow branches and assign the
j-quantum numbers from each line.
It also explains how
the bands are grouped to belong to A2113/
A2H
<-X Z
1/2
transition.
The second part
and
2Z+
(IV-2) explains the
long range analysis compared with isotope shift to assign
v' for A H-bands.
It also contains a fit to the vibra-
tional eigenvalues using a Dunham vibration formula.
IV. 1. J' , J" --
ASSIGNMENT AND FINDINGS
The assignment of the J-quantum numbers is done
unambiguously by tracing lines which belong to the same
branch and constructing first and second combination
differences for the ground state.
Before explaining
procedure of the J-assignment, it is necessary to discuss
the selection rules of the allowed transitions and the
number of branches which are expected to be observed.
As explained in the theory Chapter II-1, the
ground state
(X2 Z+) is a Hund's case
2
(b)
and the first
excited state (A H7 is a good Hund's case (a).
each level of X2E+
interaction
Since
is split by Spin-rotation (S-R)
(p-doubling) and each A 2H is split by spin-
orbital interaction into two sub-states A2 11/2 and
138
A 2H3/2
(each level in these sub-states is doubly degenerate),
the allowed transitions are those of AJ=0,±l or AN=O,±l,±2
which allows 12 branches to be possible.
Figure 39 shows
the twelve allowed transitions from X 2E to A 2H- state.
The
NaNe spectrum is found (as will be seen below) to display
eight branches instead of twelve branches because the spin
rotation (S-R) interaction of the X2E is beyond the measurement resolution.
So
one should expect to find
sub-bands
which are split by spin-orbit interaction (~l2cm1 ) and each
sub-band will display four branches (progressions).
These
branches are the one described by solid lines in Figure 39.
To show how one can recognize rotational progressions,
let us first find the frequency of any line in a band.
If
one neglects the A-doubling in the A 2R state and spin doubling
in X2E+
state, that is, using for the A2H state
F'(J) = B
1
J(J+1),
F
-eff
2
J(J+l)
(4-1)
-eff(41
and for the X2E+
F1
2 (J) = B
state equation 2-25, i.e.
(J) = B"(J-
)
1
1
1
(J+ 21
B"N(N+1)
(4-2)
3
F 2 1" (J) = BO(J+f) (J+-f)
where B
eff
=
B' (1± L)[
AA~ [see Section II-2.G. and (HERSO)]
One obtains for the frequency of the branches
139
2
N'
(5)
(F 2 )
'
*1*
4%-
____T
(4)
211/2
(3) 2%
(Z)
I-
=
172
(Fl)
N,
7,
(4)
4 /
P77J+
+
aY (3)
II
II
L
OPI
i>
I
II
I;
I
I
(2)
'.1
+
+
II
I
I
II
I
I
L
II
N"
I
'7,,
I
E
0OI
I II ii
II
i
F 1
F2
4
2
8
F,
21/2
1
0.
+
-
F,
IIi
2
2.
1
Y2.4
F1
FF2
F
-'I 1
I
7,&.
2
2
+ Band. In actual cases the
(n)Energy Level Diagram for the First Lines of a
39 -spin-doublet splitting in the upper state is often much larger than shown, while, on the other hand,
the spin-doublet splitting in the lower state and the A-doublet splitting in the upper state are in
general much smaller. If the 2 H state belongs strictly to case (a), the dotted and broken-line transitions are of the same intensity as the full-line transitions; however, in going over to case (b), they
change over into the satellite branches given there.
FIGURE
140
',(1) +
P1 (J)
Q1(J) = VO(
F1'(J
+ F1'(J)
F1"(J)
-
R1 (J) = vo(l) + Fl'(J + 1)
= ( + 2Bet.(
+ F1'(J
PO(
-
(1
-
IB"
-
= vo(l)
-
Beff.('J +
1)
-
-
(Beff.()
-
(Beri.(D' - B")j2
1
+ 1B" + Beft.( J + (Beg.(C-
B" )j2
Fi"(J)
+ IB" + 3Bent.(C'J + (Beiti'I - B"
1
P 1 2 (J)
B"
1) - Fi"(J) = o(1) +
-
)j2
F2 "'(J)
+ 2B")J + (Bet.(1) - B" )j2
Q 12 (J) = 'o(1) + Fl'(J) - F2"(J)
-
P 1 (J + 1)
R 12 (J) = O(1) + F1 '(J + 1) - F2 "(J)
-
Q 1(J + 1)
W.-3)
P 2 (J) = vo(2)
+ F2 '(J - 1)
Q2 (J) = vo() + F2'(J)
F2"(J)
-
R2 (J) = vo(2) + F2'(J +
o(2) + 2Beitf
2
+ F2 '(J
1) -
1) -
-
Vo(2) + F2'(J +
=(
+
2
2Bef.( )
Fi"(J) -
Fi"(J) z R 2 (J
1) -
-
B"
)J
2
F 2 "'(J)
2
"B" + (3Bef.(C
) -
Q21 (J) - V(2) + F2 '(J)
=
(Be.2)
(2B" - Be, 1 U )J +
B" -
(-
R21 (J)
F2 "(J)
2
B" - (Bei.(2 + 2B")J + (Beff. (2) - B" )J
(-
P 21 (J) = Vo(2)
-
-
2
2B")J + (Bef.(C)
Q2 (J -
B" )j2
1)
1)
Fi"(J)
2
+ kf.B" + 3Ben.( J + (Beu.(
2
) -
B" )j2
Here J is the quantum number in the ground state
V0
and v
2)
For small
(X2 Z)
are the origins of the two sub-bands.
(S-R) interaction in x 2+, the branches Q12'
R 1 2 ' P 2 1 , and Q21 coincide with Pl, Q , Q , and R2
1
2
respectively.
NaNe.
while
In fact this is what has been found for
The resolution of this experiment is 0.003cm~ 1
(S-R) interaction is definitely less than 10 MHz.
The splitting is found
(MAT74).
(.24) MHz for KAr by Matison
Mulliken (MUL31) has given more accurate
formulae than those of equation
the A-type doubling of the 2
of 2 Z state.
(4-3) taking account of
state and the spin splitting
141
Think of three lines which belong to the same branch and
whose J" are i-1, i, and i+l.
Call this branch X (X might
be any one of the above twelve branches).
the second difference A2 X
(X i+-Xi)-(X.-X _ ),
If one calculates
which is defined to be
one should find that this value is a
constant and equal approximately to
2
(B
-B").
(1)
Beff
is
approximately equal to B( 2 ) for bands with the same vibra-
eff
tional quantum numbers and high multiplet spacing relative
to B'.
This property can be used successfully to sort out
lines which belong to a certain-branch in a band.
In
practice A 2 Xi changes very slowly with i because the distortion
coefficient D 's are not included in the rotational energy levels
equations
(4-1) and
(4-2).
This property has been employed
to sort out lines which belong to different branches for
the bands which are easiest to analyze, namely the bands
2H1/2(v'=3,4) and 2H 3 / 2 (v'=5).
Table 3
which shows four recongized progressions.
is a band sample
Those parts of
the spectrum which have lines belong to different bands or
are crowded in a small region of frequency space were left
to be analyzed using the combination differences methods.
These tests will be a powerful technique to assign lines and
have branches for the more complex parts of the spectrum.
First, we must learn how to sort out lines into
branches, to assign the j-quantum numbers, and what to
name each progression of lines
(branch).
If we have an
xi
i
I
I
z
t
I
____________
z.
I
4
1
40.45
2
47.15
6.70
3
55.42
8.27
4
65.39
4
4
*
I
A4
f
I
31.06
.32
1.70
33.04
1.98
1.66
5.33
1.76
36.80
3.76
1.78
51.90
7.00
1.63
42.24
5.44
1.68
1.59
60.51
8.61
1.61
49.50
7.26
1.82
11. 45
1.62
70.83
10.32
1.71
58.24
8.74
1.48
92. 79
13. 01
1.56
82.75
11.92
1.60
68.70
10.46
1.72
107. 29
14. 50
1.49
96.30
13.55
1.63
81.30
12.60
2.14
123.50
16.21
1.71
111.76
15.40
1.91
1.91
1.79
39.53
3.61
6. 63
1.60
44.90
58. 50
8. 24
1.62
1. 64
68. 33
9. 83
104.28 14.53
1. 53
79. 78
8
120.28 16.00
1. 47
9
137.80 17.52
1. 52
3. 23
1. 57
43. 63
5. 02
9.97
1. 70
50. 25
5
76.75 11.36
1. 39
6
89.75 13.00
7
3 --
~,
1f9.
-xI~I
30.74
35.92
38. 61
TABLE
A)(j
34.01
35. 38
1
3 3
F 2.
H
N)
143
idea about the relative size of
(B' -B")
then a
guess might turn out to be helpful to name the progression.
If B'-B">O, then one can safely say that R 1 (J)>Q(J)>P(J)
>P 12(J).
This gives us some kind of an intuition of naming
each branch if
these progressions are traced.
To proceed
in the assignment, the combination differences are very
decisive way to assign the J-quantum numbers and to
name these progressions
(branches).
A good place to start
with in order to understand how these combination differences work is to consult Herzberg, page 175
(HER50).
Essentially what I did is to measure differences in the
ground state eigenvalues like A F" (N), A 2F" (n).
These
values are defined as
A F" (N)
=
F" (N+l)
-
F" (N)
and
A 2F" (N)
= F" (N+1)
-
F" (N-l)
(4-4)
It is easy to find out from Figure 39 that A 1 F"(N) and
A 2 F"(N) for the 2 1/2 sub-band are
A 1F
(N)
= R 1 (N)-Q1 (N+l) = Q 1 (N)-P1 (N+l) =
(4-5)
P 1(N)-P 12(N+l)
A 2 F"(N)
= R 1 (N-l)-P1 (N+l) = Q 1 (N-l)-P 1 2 (N+1)
and for the 2I3/2
sub-band are
A 1 F " (N)
2 N+l)
R 2 1 (N)-R
(4-6)
= R2 (N)-Q2 (N+l)
Q 2 (N)-P
2
(N+l)
(4-5')
144
A 2F"(N)
Equations
=
R 2 1 (N-l)-Q
(4-5),
(4-6),
2
(N+l)
(4-6')
R 2 (N-l)-P 2 (N+l)
(4-5') and (4-6') are quite independent
of the formulae (4-1) and (4-2)
(the rotational
levels eigenvalues) and would also hold when irregularities
(perturbations) occur.
Equations
(4-5) and (4-6) are
used to assign the lines and name the branches
of 2111/2
sub-band, by -permuting the progressions and forming the
combination differences. Equations
(4-5) will yield three
equal values for A 1 F"(N) except for N=O where just two
values can be obtained for A 1 F"(0) by using R 1 , Q, and P1
branches as can be seen in Figure 39.
Equations
(4-5')
yields the same result except for N=O it gives one value
for A 1 F"(<Q)
and for N=l it gives two values for A 2F"(1).
This difference between equations
(4-5) and (4-51) for
N=O and N = 1 can be employed to distinguish between
2T3/2 and 2 11/2 sub-bands.
It is clear that equations
(4-5) and (4-5') yield 6 equal values of A F"(N) for any N
except N=O and N=l.
These values
(A1 F"(N))
can be
obtained from the absorption lines of just one band of the
A H+X E transition.
When forming these combination differences
(A1 F"(N)
for the 21/2 sub-band it is found that these values do
not agree very well with each other and an energy defect
is calculated which is due to an interaction between the
21/2
substate
and the B 2 E-state.
This interaction is
145
called A-doubling.
The P and R branches always have an
upper state with different symmetry from the one giving rise
to the lines of the Q-branch.
bination relations
It follows that the com-
(4-5) no longer hold exactly and the
so-called combination defect "&"
R(N)-Q(N+l)
occurs, i.e.
= Q(N)-P(N+l)+s
(4-7)
As shown in Figure 39 for N=l, the combination defect "e"
is equal to the sum
J
of the A-splittings of the terms with
= 1-1/2 and 2-1/2.
.
Table
These defects
From equation
that "E" is the sum
successive levels.
(4-3)
(4-6) and
A 2 F"(N),
is
clear
The splitting of one level is very
Equation C.2.27 can be used to
find the A doubling parameter
ences
and Figure 39 it
of the A-type splittings of two
nearly one-half of this.
Equation
E are listed in
"q+ 1P"
(4-6') second combination differ-
can be used as a better check for the
assignment since they are insensitive to the combination
defects "E".
Both R
This can be seen clearly in Figure 39.
and P 1 branches always come from the same level in
the A-state.
The same thing is true for Q
If one band is clearly established
numbers and branches assigned),
and P 1 2 branches.
(has
its J-quantum
combination differences
can be used to assign new bands since all of the assigned
transitions come from v"=O.
This is true because the
beam was cold enough to enhance the v"=O population levels.
146
The way to use the known values of the combination differences to assign new bands is to measure frequency spacing
between lines near the suspected band origin, and to
match these measurements with what has been found from
the assigned bands.
This way was successful to resolve
the bands which overlap each other like the
2 1Hl/
2
(v'=6)
2 3/2(v!=3,4).
Table 4
shows the result of the method described above.
It lists essentially the first combination difference
A 1 F" (N)
with the combination defect
"e" for three sub-bands
and the second combination difference "2F"(N) for two subbands.
We can construct similar tables for the combination
differences of the excited state A 21/2'
A 2 F' (J))
3/ 2
(A 1 F' (J) and
exactly the same way as those of Table
4.
A
sample of the observed bands with the quantum number assignment are listed in Table.
Figure
IV.l. A.
5.
4
This band is shown in
37.
ROTATIONAL CONSTANTS B.7, Dv
The direct advantage of forming the combination differences A 2 F"(N) and A 2 F'(J) is that the upper and lower
rotational terms, equations
from each other.
(4-1) and
Since equation
(4-4)
depend only on the lower state and
only on the upper state.
(4-2) are separated
A 1 F"(N) & A 2 F"(N)
A 1 F'(J) & A 2 F' (J) depend
By adding corresponding A 1 F(J)
or A 2 F(J) values, we can obtain the position of the rota-
A22
3/2
+
2 +
XI
Al2
t
N
5-0 band
4-0 band
A 2F"
A F"
R 21-Q2
R2-P 2
R -P
A 2F"
1
N
H
2
0.481
3
+
A2 3/2
X2E+
4
I
3-0 band
0
0.290-
X2E+z
H
A 2F"
1
+
4-0 band
1.
5-0 band
A F"
A F"
A F"
A F"
A F"
A F"
R1 -Q 1
QP PQ-P
1
R -Q
Q -P
R 21-R
R
0.108
0.082
0.012
0.117
0.087 0.030
0.100
2
0.287
0.294
0.212
0.178
0.034
0.223
0.170 0.053
0.193
0.191
0.002
0.475
0.482
0.478
0.309
0.267
0.041
0.327
0.260 0.067
0.286
0.288
-0.002
0.667
0.667
0.666
0.670
0.416
0.357
0.050
0.437
0.340 0.097
0.381
0.378
0.003
4
0.852
0.853
0.854
0.852
0.513
0.435
0.078
0.533
0.418 0.115
0.488
0.468
0.01
5
1.033,
1.026
1.028
1.029
0.603
0.511
0.092
0.632
0.496 0.136
0.559
0.549
0.009
6
1.198
1.182
1.198
1.192
0.705
0.599
0.106
0.728
0.567 0.161
0.643
0.644 -0.001
7
1.365,
1.367
1.363
1.363
0.668
0.113
0.823
0.636 0.187
0.722
0.722
-0.001
8
1.512
1.514
1.518
1.525
0.781
0.866
0.739
0.127
0.906
0.696 0.210
0.798
0.791
0.008
9
1.656
1.647
1.725
1.648
1.645
0.930
0.773
0.157
0.859
0.857
0.002
10
1.748
TABLE
4
148
Reliability
P 12
16945.047
.068
.193
.254
.342
.375
.432
.547
.604
.662
.775
.863
.945
1
1
1
1
2
2
2
3
4
2
5
46.088
.285
.389
.611
.789
.807
.934
1
1
1
1
1
1
1
47.242
.430
.453
.588
.963
1
1
1
1
1
48.154
.344
.787
.979
1
1
1
1
49.201
.707
.898
1
1
1
50.182
.721
.910
1
1
1
51.830
1
>1
TABLE
5 --
1
1
2
1
1
1
1-
_ __2_
R12
R1
1
0
2
1
3
0
6
4
1
3
5
2
8
4
6
3
5
9
7
4
6
8
5
7
9
8
6
10
7
9
8
Rotation analysis of the A 2IT/2 (v'24)+X 2+ (v'=0)
X is the wavelength in cm~
,
value of the rotational quant
the ground state.
P
2
to R
is the
2number N" for
149
tional levels in the X-state and A,B state as well.
This
is shown in Figures 40 and4l where the rotational eigenvalues F (J) and F (N)
are plotted
versus J (J+l) and N (N+l),
(remember that J is a good quantum number in the A-state
a good quantum number in the X-state) .
while N is
two isotope bands of Na
marked by the letter
Ne
i.
Notice that
are analyzed in Figure 40 and
The smoothness
of the lines
in the figures show in fact, the goodness of the analysis.
They also show that F(J) or F(N) cannot be represented
only by a linear function of J(J+l) or N(N+l) because
the molecule cannot be treated as only a rigid rotator
and instead
a rotation vibration interaction term
should be added.
It has been shown earlier in Section
II-.3.B. that this kind of interaction will lower the
eigenvalues by a term which goes approximately as either
D
[J(J+1)] 2 or Dv [N (N+1)]
(see equation C.2.34.
these terms are combined with equations
If
(4-1) and
(4-2),
one should get
F (J) =B
F(N)
=
J(J+1)-D [J(J+1)] 2 and
eff
B N(N+l)-D
v2
The eigenvalues of each band (Figure 40 and
submitted to
(4-8)
[N(9+1)]
41)
are
a linear least square fit of a polynomial as
the one given by equations
(4-8).
The linear least square
fit yielded the molecular spectroscopic constants shown in
Table 6 which reproduces the data to within ±.003cm~1 except
150
(v)+FfJ)
(cm- )
2
B Z (v)
2
3/2
(6)
75-2
3 S1/
3 p 1/2
2n
2
16970
(5)
3/2
2 3/2 (4)
651t
2 1/2(6)
2
)
60
2
-"
r-,
3 p1/
-
..2- / 2 ( 4
3 S1/ 22
2
3/2
55,
+
2
16950
452
n2
.
(3)
2
-1
(3)
(3)
40-
35
30
16925.0(
1
'
20
40
I
I.
610
80
1
100
J(J + 1)
FIGURE
40 --
A and B States Rotational Energies
vs. J(J+1)
120
5 t
4
3
2
1.
20
10
30
50
40
N(N + 1)
FIGURE 41
--
X-State Rotational Energies
vs. N(N(N+1)
60
70
80
90
Multiplet
Splitting
Vibra-
tional,
Quantun
Number
A2 111
A 2 I3 /2
2
Band
Origin
Band
Origin
(v)
(cm-1)
(cm-1)
from
from
1/2
of Na
(cm-1 )
1/2
D'
X
D
(eff
10 4(cm-
Bef f
(CM-1
23
1/2
Of Na
D' ffX10 5
B' eff
+2 p
B
D"x105
cm
cm
3/2
(cm
(cm-1 )
)
0
0.0489071 3.00
2
N~
UL)
3
-26.30
0.128480(6)
4.509(6)
-31.64
0.130683(3)
5.684(3)
11.86
4
-10.92
0.103676(6)
4.759(6)
-16-.31
0.104798(3)
4.605(3)
11.81
5
-1.68
-6.06
0.092299(3)
6.325(3)
6
7
+2.49
0.31
0.058257(3)
6.652(3)
H
6.826(6)
0.057335(6)
15.02
8
B
e
ae
.216324(3)
cm
.0246449
cm
5.0X10- 5
-1
cm
5.099(a0 ) a (2.697)
re
A
TABLE --
&
6
.4
A
I
1
.4 ______
153
near perturbations.
These constants should be regarded as
effective constants because the interaction with the
excited B E
state, and the observed perturbation between
21/2 and 213/2 substates belonging to different vibrational
levels are not indluded in equation (4-8).
To see these
perturbations, one should look to the column of the energy
and to the crossing of 21(/24)
defect "E" in Table 4
with 2 3/2(3)
and 21/2 (6)
with 2113/2 (4) in Figure 40-
The band crossing perturbation is discussed in Herzberg,
page 282,
(HER50) and in
Appendix C.3.
At this stage of the analysis, one could find three
useful spectroscopic constants for the A-state: namely Be
a,
if one uses an approximation formulae for By,
and
T e,
B
=B-a
B
i.e.
v
B
e
ee
(V+
2 +Y e (v+)22
(4-9)
is the constant which should be used to find the
equilibrium separation r e, i.e B
B e=
is given by
4 cRe2
(By gives the
mean
(4-10)
value
of -7
during the vibration).
A rigorous calculation done by Hyllerass
the above concepts.
The linear least
(HY135), justifies
square fit
the 213/2 substate to equation (4-9) yielded Bel
which is shown in Table
constants B
6.
of the B
ae and ye
These constants reproduce the
to better than 3xl0
cm
.
(The vibrational
of
154
numbering of these bands will be discussed in (4-2)).
B
If
is substituted in equation (4-10) it yields 5.099 a0
(2.597 *A) for the position of the potential minimum.
In Part IV-2, we show how one can find B e(v'=0) from long
range analysis.
Not enough information is available for the X-state
to find Be accurately,
but still
be too much different from B
e.
our B
(v"=0)
should not
In Section IV. 2.
G. we show a
method which can minimize the uncertainty in re .
Four characteristics helped in assigning bands to
either 2H1/2 or 2H3/2 substates.
The characteristics are
A-doubling, the value of Bv, combination differences,
the value of
the multiplet A.
Strong A-doubling is
found for 2l1/2 bands and zero is found for 2T3/2 bands.
rotational
constant B
is found almost the same for the
two substates which have the same v'.
The first and second
combination differences for the X-state have different
behavior for J"=l/2 and 1-1/2 as explained in IV-l.
Finally, bands with the same v' have a multiplet splitting
equal
approcimately 2/3 the energy between 32P1/2 and
323/2 of Na.
IV. 1.
B.
The
A-DOUBLING PARAMETERS
The effect of A-doubling was observable on the
2 1/2 substate especially for the bands v'=3 and v'=4.
155
This effect can be seen quite clearly in Table 4 -for the
2H1/2 substate the energy defect "E"
varies linearly with J
[see equation (4-7)]
[see equation C.2.291while the 2 3/2 sub-
state shows unobservable effect at least to the measurement
accuracy
(.003 cm 1) limit.
If one uses equation C.2.31
as a good approximation for Hund's case (a).
A linear
least square fit of a straight line can be used to find
the best value for the parameter
and 0.0115945 cm
q+P/
2
0.007547
The fityielded
.
for v=3 and 4 respectively. Figure 42 shows
a linear plot for the energy defect
(e) versus J+I.
As mentioned in Appendix C.1, equation C.2.30 can be
used to roughly find the potential of the B2E state
the measured values of q, A, B ,
separation
one can find the
v(H,E) of the 2111/2 from the
separation v(11,Z)
(SMA77). From
can be used with the
B 2 E-state.
This
H/1/2 potential to
find two points on the B E-potential.
IV. 1. C.
ROTATIONAL TEMPERATURE
"T R"
of NaNe
The rotational temperature "TR" of the molecular
beam of the NaNe species can be found from the line intensities and the J',
J"-band assignment.
The line intensities
of the A2+2 Z transition in diatomic molecules was worked
out by Earls
(EAR35).
Earls found the following formulas
for the line strenth "I".
0.24
0.20
0.16
4-4
>1
m
LC)
C.).
0.12
0.08
0.04
1
2
3
5
4
J + 1/2
FIGURE 42
--
Plot of Energy Deficit
vs. J+
6
7
8
9
10
157
for J>-2.
sn....s
->*nIntensityR, ) (21+1)2 (2J+1) U(4f+4J+1-2X)
P2
RR
PPP1 2
2
(2J+1) F (2J+1) U(4
QR12
Qp21
P
32(J+1)
2 1J
R
+4J-7+2X)
32(J+)
(4_11)
(2J+1)[(4P2+4JT-1)
Q2
R
RQ
4- U(SJP+12J2-2J+ I -2X)]
2
2+)(j+J1
PQ1
Q,
F,U(8J3+12J
T
2
-2J-7+2X)]
32J(J+1)
P: (2J+1)2±(2J+1) U(4J 2+4J-7+2x)
R:
cP21J
QR1
RRR2
PPP12
Pj
R1
32J
(2J+1)2:F(2J+1)U[4J+4J+1 -2X]
32J
3
where
U=[4--4x+(2J+1)2J-i.
and X =A/Bu
for J=
A'PPp2=
j>OPQ12
X<O
x~GQ
=
p,
Q1=T
P2 =QPII i
=RQ21=1
P-.RR2R = Rj
RQ21 =Q,~
R,
=QR%2-i
Q2 =PQ12-f.
_B 2 N (N+1)
combining these formulas with the Boltzman factor e
T
it is possible to plot the per state population
(intensity divided by strength) as a function of -N(N+l).
A typical graph is shown in Figure 43.
the data for the 2113/2(5')
band.
This plot represents
The slope was found to
give a rotational temperature "TR" about 1.6 ±0.1K.
points 0
are those for R21 branch, the points X are those
of Q2 and P 2 1 branches.
2
H1/1
The
2 (3)-band
Similar plot is done for the
and an 2.3K value is obtained for TR.
This
4
3
2-0
N (N+l)
10
FIGURE 43 --
20
30
40
50
60
70
Plot to Find Rotational Temperature of NaNe
80
9
159
might reflect the fact that the data were not taken under
identical operating conditions since attemps were taken
to improve the signal by making fine alignment while the
experiment was running.
IV. 1. D.
MULTIPLET SPLITTING "A"
~AG"v~/
IA iv+1/2
AND VIBRATIONAL SPACING
The multiplet splitting "A" is obtained from
Figure 40.
If one goes back to equation (2-10) where the
electronic energy of a multiplet term E
is given by AEA,
it is easy to find that the multiplet splitting
spacing) of 2H1/2 from 2 3/2 is just A.
(energy
This multiple "A"
measures the spin-orbit interaction in the A H state at dif-
ferent v'.
These values are listed in Table
7.
The large
uncertainty in'the 2H1/2 (v'=5) band origin reflects the presence
of the intense 32S1/2 3P1/2 atomic transition of Na.
The vibrational spacing AGv+1/2 is also obtained from
Figure 40.
If equation B.2.11 is examined, one could easily
get
AG +1/2=(W e-
x +wy)-(2w x -3w
v~l 2ee
e
e2
y ) (v+1/2)
(4-12)
2
+W y (v+1/2)
AGv+1/2 measures the amount of the deviation from a simple
harmomic oscillator potential which predicts a constant
value for AGv+ 1 /2 .
in v.
To a first approximation, AGv+ 1 / 2
The values of AGv+1/2 are listed in Table
7.
is linear
160
v
A
3
11.86
4
5
A2 l/2
A 2 H 3/2
G (4) -G (3)
15.38
15.33
11.81
G(5)-G(4)
09.24
10.25
12.82+
G(6)-G(5)
±0.13
6.37
0.13
6
AGV+1/
15.02
TABLE 7
2
4.17-.13
161
IV. 1.
E.
SUBBAND PERTURBATION
As discussed in II.3.C., perturbation might
take place due to crossing of bands belonging to different
substates.
This perturbation shifts the rotational
eigenvalues and sometimes produces intensity anomalies.
Figure
40 shows places where crossing of 2113/2 bands with
2 1/2-bands occur.
J=10 1
e.g. 2 1/2(4) with 213/2(3) near
and 2,(6)
2
1/2'
Equation C.2.37
the perturbed levels.
where
w
with 2 3/(4)
13/2
near J = 51
er
2
can be used to find the position of
The shift is given by [1
4|w
+6
-
is the vibrational overlap of the matrix element
and 6 is the separation of the unperturbed levels.
To
calculate this shift, one needs to find the correct
potentials then to find the matrix elements w
,
one
should solve the Schrodinger equation to find the vibrational
wave functions.
Of course, this is to be done on the
computer.
IV. 1.
F.
INTENSITY ANOMALIES
Intensity anomalies are deviations of band
intensity or some lines in a band from the values predicted
by the line intensity formulae
[see equation
(4-11)].
In principle these can be predicted by introducing perturbations or a proper new scheme of angular momentum coupling.
We have observed intensity anomalies in the two bands near
162
the dissociation of v'=6 for both 2 1/2 and 2 3/2'
These bands were to have only two branches.
The branches
are assigned as R 2 1 'Q2 and R2' 2, by using the combination
differences of the ground state.
The absence of two branches may be explained on the
basis of intensity anomalies induced by the nearby B 2E
state or by using a more atomic basis for those levels,
e.g. Hung's case
case
(e) which permits a passage from Hund's
(a) to a more atom-like coupling scheme for the highest
vibrational levels of the excited state in which
Na becomes a good quantum number.
j
of the
Any of these suggestions
to explain the intensity anomalies needs a deep and a
lengthy investigation which is not in our hands at this
point.
IV. 2.
LONG RANGE AND ISOTOPE SHIFT ANALYSIS
[Comparison with Dunham Expansion Formulae for G(v)]
So far we assigned the J-quantum numbers for the
absorption lines, but the vibrational quantum numbers
are not assigned yet.
It is essential to know the vibra-
tional numbering of these bands to be able to construct
the first excited state potential
(A2H).
The method, used
to assign the vibrational numbering, is to use asymptotic
formulae for the vibrational energy near dissociation
(long range formulae) (LER70) and isotope shift formulae
(STW75).
The assignment is affirmed by using Dunham's
163
equation B.:2.11
f or the vibrational energy.
As it
will
turn out, the long range formulae combined with isotope
shift calculation is a very reasonable method;
not'just
to find vibrational numbering, but to make possible the
following findings,
One could estimate the well depth
DeA of the A-state, the dissociation energy DoA of the
X-state as well as its well depth D eX
the location of
the well depth reA, and the constant "C6"1 of the first
leading term of the dispersion potential (VC
6 R -6).
To proceed, one should describe what is meant by
long range analysis method and question the legitimacy
of its use.
This method is concerned with the influence
of the long range part of the interatomic potential on the
distribution of levels near the dissociation limit of
diatomic molecules.
The method described in this section
is based on expressions relating level energies to the
detailed nature of the long-range potential.
The assumption of R-6 potential is quite satisfactory
since the second order perturbation theory gives rise to
the dispersion terms R-6,
R 8, and R_10 which always
contribute to the potentials, but still the leading term
is
R 6 near the dissociation.
The legitimacy of using this
technique lies in the fact that all observed bands are influenced mainly by the long range forces because of two facts:
the first, the X-state interatomic
separation
(re)
is
164
much bigger than the A-state interatomic separation which
does not allow transitions
fact.
to low v'.
Figure 44
shows this
The second, the NaNe molecules are not bound by
either ionic or covalent forces and overlap between electronic clouds of the two atoms is highly improbable.
This
implies that the leading attractive force is C6R-6 from
second perturbation theory calculation
(HIR54).
To account for these long range forces which dominate
near dissociation, I will summarize the needed equations
even though vast literature can be found about the subject
(LER70, LER70, HIR54, STW73, STW72).
If two neutral atoms
with at least one in an S-state, are sufficiently far apart
that their electron cloud overlap
is negligible, the inter
action potential of the long-range region can be accurately
approximated by V(R) D-Cn/Rn where D is the dissociation
limit, and n is some weighted average
(in general non-integer
and approximately equal to 6) of the powers of the locally
important terms.
The asymptotic allowed rotationless vibra-
tional eigenvalues G(v,J=O) corresponding to integer values
of v and the rotational constant "B " are derived by LeRoy
and Bernstein
(LER70). They found
G(v) = D-[(vD-v)Hn
B
= Qn(vDv)
4
/n-
2
n/n- 2
(4-12)
2
(4-13)
165
A2
Na Ne
Lu
All
16973
Na(3P)+Ne(3S30)
v
16943
4
v =3
'16913
16883
V3l =
16853
16823
X 2 E+
30
Na (3S 1/2)+Ne (3S0)
0.0
2
i
i
i
i
i
i
i
4
6
8
10
12
14
16
j
18
i
i
20
22
R(a.u.)
FIGURE 44 --
Possible Transitions from X to A Potential
166
where
where
H
H1
n
r1(1+
Q n = ( 4Cy )
F(
I (1+-) (1)
n
2
r(
)
n
n-2
=
n)
2n
11
2)1 )(
(4-14)
1/n
n
1/(C
(4-15)
2/n
n 21
and vD is an integration constant.
If the competing (leading)term of the long range is
assumed to be VR-6 , the eigenvalues "G(v)"
[equation (4-12)]
and the rotational constant B [equation (4-13)] can be
represented by
G(v) = D-[(v D-v) H 6 ] 3
B
Bv
where H6
Q=
(4-16)
Q66[[ v
-1(4-17)
VDv
=
1 /6
(196 94336) P 1/2 C6
(4-18)
2
p-3/2 C-1/
546.658
6
6
(4-19)
Units of energy, length, and mass are taken to be
cm
,
A
and a.m.u. respectively.
Dunham expansion formulae B.2.11
will be used when
applicable to compare long range analysis findings with
those of equation
IV. 2. A.
B.2.1l..
VTBRATIONAL QUANTUM NUMBER ASSIGNMENT
We have the basic formulae IV. 16 to start the long-
range analysis and Eq. B.2.l1
to make the comparison.
The
first step is to locate the band origins of the A H-state from
167
the observed bands of the A2IT/2
stated in Part 11.3, the B
with A 2 H1
and A2 H
As
-substates.
E state interacts strongly
substate, especially near the dissocation
part of the potential, while its interaction with the
2113/2 substate is unobservable within the experiment
resolution
A2 H
3/2
(0.003 cm 1).
Thus one believes that the
substate is pushed up in energy from the un-
perturbed A T level by the spin-orbit interaction
which is equal to 5.73 cm
does indeed
Thus the A"H3/2
.
represent the A 2 H bands to an additive constant
cm~
).
(AZA)
(5.75
To make this assumption clearer, the band origins
of the A H potential can be deduced from the observed
multiplet splitting near the bottom and at the dissociation.
At the dissociation, the molecule asymptotically
approaches the sodium D-lines
ground state
(31Se0) of Ne.
(32 P 1 / 2
3/2) of Na and the
The D-lines are split by
fine structure into 2P1/2 and 2 3/2 components.
The
splitting is 17.196 cm 1. and the ratio of splitting is
2:1, i.e. the 2P1/2 is pushed by 11.464 cm~ 1
to the red,
while the 2P 3 / 2 is pushed by 5.732 cm1 to the blue of
the center of gravity of the 3 2P-line.
Near the bottom,
the multiplet splitting is about 11.86 for v'= 3 and 4
(see Table
7).
Approximately half of this
2
2
is the energy between the A it and A H
.
splitting
(5.8cm
As a conclusion
)
168
the
A H
3 2 -bands
are - 5.7 cm~
off the blue with
respect to the A 2TH-bands as a result of an extrapolation
between the two asymptote
This concludes that both potentials
the bottom).
(A2 H and A 2 H 3 / 2 ) are
estimation of
done
Dox,
very similar.
DeX,
by using the A2I3/2
To proceed, Eq.
[D -
2 0 G(v)]
(the dissociation and near
1/2
DeA*
Consequently the
r eA
and C 6
bands.
(4-16) is rewritten again as
(20vD)
20H6 and different D's are tried
to obtain a straight line fit between [D and
( 20vD
5.1955 cm-
-
v).
are best
Table 8
G(v)] 1/3
shows the fit of "D" equals
with the expected errors in each value.
Figure 45 shows the straight line fit.
The fit yielded
the following average values
(20vD-
v0 )
= 6.1239 t
.0004
D 0H(4-20)
20
=
.533
1
± .000070 (cm A)1/3
TABLE
8
1
(cm')
from
V
- 2P
1
E Y. (cm1)
3
Y.-Y
± AS
+ AY.
SE
1
(cm') 3
1
1
xl10 6(cm- )3
x104(cm-1 )3
6.13184
8
.32
0.0004
VD -V
+
A(VD
-
-31.64
3.327394
V +1
-16.31
2.784655
.54274
5.12839
24
.64
0.0006
V +2
- 6.06
2.241303
.54335
4.12298
40
1.19
0.0009
V + 3
0.31
1.697432
.54387
3.12102
80
1.19
0.0007
V
m
2
3
[5.2-G(v)]1
1
0
0
0
V)
o
[D-G(V)]
3.0
0.15
* Isotope
N
E
C)
0.1
2.0
E
to
N
m
(9
0
N
H
0
1.0
0.0
0.05
'I
I
'I
2
3
4
5
6
'
'
7
8
V
FIGURE 45 --
Long Range Fit Of G(v')
and B
'
vs. v'
9
0.0
171
One way is to
Our aim is to find the value of v 0 .
scale the calculated values of 20HD and 20H6 to 22VD and
22H6 and to use Eq. 4-16 to predict the band origins
of
To find the frequency of the isotope bands
the isotopes.
2 2 G(v')
"i,one needs to know the band origins
t
and the
difference between the ground state band origins
20G
(0) -
G (0) as shown in Figure
46.
The vibra-
tional quantum numbers v scales
like
Na
NOW
Ik
(LER70).
1
20+
f
11
20 2
2 2v 1
+~)
)T
22 2
A ,
G (U;
i.e.
1
'I
VD =
(20v+12
12
-
1
(4-21)
11
and 22H 6scales
[see Eq. (4-18)]
1
22H
20
2
6= 122 Ip
H6
(4-22)
Assuming a set of values 2,3,4
4x
10,
(C
for vo and substituting the values of
20.v
20H
6
(Eq. 4-20) in Eqs. 4-21
Figure 46
and 4-22 will predict corresponding values for 22v
one value for 22H 6 .
and
The 22.D and 22H6 values are then
substituted in Eq. 4-16 to obtain the isotope band origin
)
172
G(5).
These values are shown in Table below.
TABLE
22.
22 GW
VD
0
+2)
2
8.34151
-6.9716
3
9.36899
-7.2041
4
10.39200
-7.4010
The assignment of
.5297708
v =3 is consistent with the observations.
0
It gives
22.
VD
=
9.3690
(4-24)
and
2 2 G(5)
The
= -
(4-25)
7.2041 ± .0076
G(5) differs from the isotope band origin which is
observed at
V
= -
by .0634 ± .008 cm-1 .
7.1407
(4-26)
± .003
This difference arises from the
isotope shift in the GX(0) level as can be seen clearly
in Figure
46.
To confirm the above assignment of "3" for the vibrational quantum number
v 0(see Table 8),
Eq.2-ll is linear least squares fitted.
the following constants:
Dunham's
The fit yielded
173
W
e
=
weXe =
w y
44.2873 cm'
-4.57934 cm~
=
(4-27)
.157344 cm~-
These constants reproduce the observed band origins to
the experiment's accuracy.
gins
=
20 G(v)
W (1-p) (v+
where .p =
G(v)]
-
)
-
2 2 G(v)
for v= 5 is given by 20G(v) - 22G(V)
WeXe (1-02
/20 /22p.
The difference in band ori-
v + 1)2 + W Y
(1-p3) (v + 1) 3
Therefore
= - 7.206 ± .004 cm~-
(4-28)
Again this value differs from the isotope band origin
(Eq. 4-26) by .0653 ± .005 cm~
which accounts for the
isotope shift in the G (0) level.
This method has
affirmed the assignment of 3 for the first band observed.
(The first band observed is assigned by v 0
in Table 8.)
After the vibrational numbering is established, a
return to the long-range Eqs. 4-16 and 4-17 enables one
to find quite a few important molecular parameters by
suitable extrapolation.
the following sections.
These findings are summarized in
174
(Dx)
DISSOCIATION ENERGY OF THE X-STATE
IV.2.B.
The parameter D in Eq. 4-16 is the sum of
) and the dissociation energy of the X-
v(32 S 1 / 2 +32P
3/ 2
state D O.
(Band origins of NaNe are measured from
P3/
v(32 1/232
2
).)
From Table 8,
fact the dissociation energy Dox ~
the value of D is in
5.2 cm
.
We gave an estimate for DOX above, the question is:
how good is it?
To answer this question,
we inves-
tigated the error by making similar extrapolations
calculated eigenvalues of pseudopotentials
from
Lennard-Jones 8-6 and Durham type of poten-
(BAY69;
PAS74),
tials.
We found that pseudopotentials deviates with
+ 0.8 cm
while (R 6,R~
-0.8 cm 1.
) and Durham deviates with about
This kind of calculation makes us feel that
the dissociation cannot be known better than
D
=
5.2 ± 0.8 cm~
(4-29)
with this kind of extrapolation.
The next thing to be checked is the value of DoX
5.2 cm 1.
The check is to start with Eq. 4-12.
The
first and second derivative of the potential gives the
following equation
(LER70).
E'(v) (E"(v))~1 =
-
[(n-2)/(n+2)] (vD-v)
(4-30)
175
The necessary values for n and vD may be obtained
from a least square fit to Eq. 4-30.
[E(v) is very well
obtained from the linear least square fit of Eq. B.2.11 at
(see Eq. 4-27).] The
least between the observed bands
fit of equation 4-30 yielded n = 6.143 and vD = 9.046.
It is to be noticed that "n" can be a noninteger number
(LER70) since it represents the effective part of the
attractive force.
Having fixed the n and vD values thus obtained,
Eq. 4-12 becomes linear in a new independent variable w,
i.e.
G(v)
=
where w = {[n-2)/2n] (vD _
k
n
~
22-
D -
(4-31)
wkn
2n/(n-2) and
r (1
P+
n
1/n)
FF(1/2+1/n) Cn 1 /n
n
A least square fit to Eq. 4-31 yielded the values of
D = 5.13 cm~
.
This value is in good agreement with the
value obtained by a straight line fit Eq. 4-29.
Eq. 2-11
(Dunham's equation) combined with least
square fit findings Eq. 4-27 was also used to find D.
The prediction of Eq.
Dx =
which
is
B.2.11 is
5.11 cm 1
again in agreement with Eq. 4-29.
(4-31)
176
THE X-STATE WELL DEPTH
IV.2.C.
The well depth of the X-state DeX can be estimated
from the difference of the deduced band origin 22G(v)
and the observed frequency of the isotope bands
v.
The
difference is nothing but the difference in energy of
V'" = 0 of Na 20Ne and Na
Figure
Ne as can be seen clearly in
46, i.e.
V
2 2 G(5)
Eq. 4-28,
- 22G(5)
=
2
2
e
i"(1-
(4-32)
p)
has been deduced in two ways, Eq. 4-25 and
(from long-range fit and Dunham's fit).
Sub-
stituting the values of 22G(5) in Eq. 4-32 yields the
following two estimates for we":
From longe-range fit
w e" = 5.15± 0.66 cm
From Dunham's fit
we" = 5.31± 0.41 cm
-l
-l
e
One also can obtain w e" from
fit to a model potential.
Jones
(8-6)
are tried;
(4-33)
rotational eigenvalues
Two potentials, Morse and Lennard
both fitted the observed
rotational energies to the experimental accuracy.
parameters were adjusted to obtain the fit.
Three
It is con-
ceivable to fit the data with such model potentials,
since the maximum N" observed is 11, while N" = 15 is
probably still bound, and just two constants D" and B" can
v
v
represent the data within error. From the Morse fit we
177
obtained
"y
W "Xe + I
w "
=
G" (0)
=
2.87 ± .03
(4-34)
cm~
and
1
we"
0.82122x
7
6
j2
6.5 cm
1
(4-35)
Combining the values of Eq. 4-33 and Eq. 4-34 with the
estimated value of the dissociation DeX
(Eq. 4-28) yields
the following estimates of the well depth of the X-state.
The fit
to different formulaes is
Eqs. Used
D
(cm 1)
shown in Table below:
Type of Fit
4-34, 4-31
7.98
±0.9
4-33, 4-12
7.78
±1.04
Morse potential plus
Dunham's formulae
Long-range formulae
4-33, B.2.11
7.77
+ 0.9
Dunham's formulae
These fits cannot give the well depth directly
because the observed transitions occur from levels above
1
1
the well depth minimum by at least - T w"
IV.2.D.
THE A-STATE WELL DEPTH
1
-
i
w1"Xe
(DeA) AND BAND ORIGINS
Essentially we have two independent equations
and
(2-11)
(4-16) to estimate both the well depth "DeA" and the
band origins of the A-state.
4-16 to v'
-
yields 143 cm
An extrapolation of Eq.
for the well depth.
178
This value should be thought of as an upper bound since
the vibrational bands near the bottom of the potential
get affected by the repulsive part of the potential
(LER70)
which is not accounted for in the derivation of
Equation
(4. 6) .
The maximum value of G
D
(v) in Equation B.2.11 is
e since no discrete vibrational levels lie above the
asymptote
(HER50).
Also "D "
can be found by sub-
stituting the predicted values of G(v)
Equation
DeA
(4-37),
(Table
9) in
i.e.
1
2 wA
We -
1
Te
4 W Xe
+
1
89
wwee
y
+
=
G (v+l) -G (v)
v=O
(4-37)
The band origins of the levels v' = 0, 1, 2, 7, 8,
and 9 can be predicted by equations B.2.11 and
Table 9
(4-16).
shows such prediction and lists the observed
bands.
IV. 2.
E.
EQUILIBRIUM INTERNUCLEAR DISTANCE r
The equilibrium distance of X-state "r eX
found from the rotational eigenvalues fit to
Jones
(6-8) and Morse model potentials.
obtained from the fit is
uncertainty in r eX
is
the Lennard
The value
reX = 1O.O±O.la . The big
reflects the fact that different
1
G(v) (cm
)
From 2sl/ 2
Fit to
Eq. (4-9)
Fit to
Eq. (4-17)
Fit to
Ea. (2-11)
Fit to
Ea. (4-16)
0
-116.28
-116.65
0.204015
.203210
1
-80.64
-80.81
0.179470
.179030
2
-52.74
-52.80
0.155027
.154850
V
-4v
3/2
Of Na
B
V
3
-31.64
-31.64
-31.64
0.130683
0.130683
.130670
4
-16.31
-16.31
-16.38
0.104789
0.104798
.106490
5
-6.06
-6.06
-6.05
0.082299
0.082299
.0823110
0.31
0.31
0.058257
0.058257
.0581306
7
3.65
3.66
0.034318
.033950
8
4.92
4.97
0.0104776
.0097695
9
5.05
5.20
unbound
1
-137.30
0.31
6
2
TABLE
9 --
Fit to 2 3/2
substate.
-143.
.216324
unbound
.215300
H-
180
model potentials produces different values of
r eX.
The equilibrium distance of the A-state r eA is
obtained by two ways.
of equation
(4-9).
The first is the least square fit
The fit yielded the spectroscopic
constatnts shown in Table
6.
The parameter Be
corresponds to an reA ru 5.099 ± 0.1 a .
The second
is an extrapolation of equation (4-17) to v' = 1/2.
The linearity of equation (4-17) is shown in Figure
The least square fit of equation
(4-17) yielded
20 6 = 0.02418 cm 1 and vD = 8.40403
(4-38)
Using equation (4-38) and extrapolate to v'
reA
eA
5.11 ± .la,.
0
Both fits [equations
predicted values for B
yields
(4-9) and (4-17)
for v' = 0, 1, 2, 7, 8.
values are listed in Table 9
of B '
=
These
with the observed values
181
IV. 2. F.
LONG RANGE ATTRACTIVE COEFFICIENT
The long range attractive coefficient"C
6
is related to both the coefficient 20H6 and 20Q6 by
equations
(4-18) and
(4-19).
These coefficients are
obtained from the slope of the fit which are given by
equations
(4-20) and
(4-38).
Using equation
(4-18) and the
value in equation (4-20) give
C 6 = 415.7 ± 0.3 Hartree
(a0 ) 6
(4-39)
Using equation (4-19) and the value in equation
(4-38)
give
C 6 = 86.902 ± 22.33 Hartree (a0 ) 6
(4-40)
Obviously there is a factor of 4.78 discrepancyin the two
values.
This discrepancy is explained by LeRoy
(LER70)
as errors introduced by the approximations fundamental to
the derivationof equations
(4-12) and
(4-13)
should be
relatively more serious for the latter. Another reason
lies behind the fact that v D
determined by both
equations are in disagreement [see equation (4-20) and
which affects directly the coefficients Q6 and H 6 .
(4-38)]
Also, a
small deviation from R-6 potential produces large changes
in the constant coefficient of Q6 while that of H6 remains
with a very small
B
rather than G(v)
change.
The effect is dominant in
which is reflected in the big
error in equation (4-40).
182
To make a reasonable estimate for C 6 , one could
combine equations
which is independent of vD'
[D -
(4-17) to yield an equation
(4-16) and
G(v)]
=1/3
= H
i.e.
B
6v
=
(3.648233)
10-2 P C 1 / 3 B
6
v
(4-41)
This equation has the benefit of providing one extra point
at
which in
the dissociation
A least square fit of equation
C6
=
65 ± 17
Hartree a
6
fixes
fact
v D to its true value
(4-41) yielded
(4-42)
0
One could estimate the maximum value of C 6 by considering V=D -
R6
and the probability is all concentrated
at the outer turning point of the potential.
For the
vibration eigenvalues G(v) one has
C6
= D-
G(v)
(4-43)
R6
V
where R
is the outer turning point (see Figure below).
The rotational constant B
corresponding to Rv
h2
B'
h
1
(4-44)
V
ID
R
183
This value of B
B
and
for the band G(v).
(4-44)
is obviously less than the observed
If one combines equation (3-43)
", one could estim-
and assumes the observed "B
ate an upper limit for C 6 ' i.e.
1.6621x10-2 C1/ 3 p B =
6
v
[D-G(v)]1/3
which gives
6
C 6 < 687.7 Hartree a6
6
IV.
2. G.
0
THE X-STATE POTENTIAL
The best and reliable way to find a diatomic
potential is the Rydberg-Klein-Rees
RKR- method.
(RYD31, KLE32, REE47)
Since only one vibrational state v'=O was
observed in the X-state, it is impossible to use the
above method to construct this potential.
Instead we
have adopted a parametized form for the potential and
varied the parameters to fit the observed eigenvalues.
The resulting potential is not necessarily a unique
potential for the X-state although it reproduces the
observed rotational eigenstates.
The method to find such potential is as follows:
we use the computer to generate a potential
with its r eX10.1
(see IV. 2. E.)
a
(e.g. Morse)
and its well depth DeX~7.9cm~1
and equation
(4-36).
the rotational eigenvalues of v'=O.
We then ask
for
If the first few
generated eigenvalues do not fit the observed eigenvalues,
184
we adjust rg
until they do.
parameter "reX
This step fixes the first
of the potential.
The next step we
adjusted the curvature of the potential until all the
generated rotational eigenvalues agree with the observed
Such potential
eigenvalues.
V= -[14.22
_ 1066.18
R6
Units are cm~1 and a
R8
0
is given by equation
(4-45).
(4-45)
185
V.
DISCUSSION
In the following sections the work of this experiment
will be discussed and compared with other work done on the
same molecules.
In addition, possible new work with our
apparatus will be also outlined.
A.
DISCUSSION OF THIS EXPERIMENTAL FINDING
We will report on observations,
define absolutely),
system
(which we could not
and on further possible work on the same
(NaNe) which will make the findings of this work
more definitive.
We have observed a band about 2.82 cm~
of the P3/2 lines with a B v=.019 cm -.
last band shown in Figure 40
to the blue
This band is the
(labeled by B E).
It does not
belong to 213/2 state because the long-range formula 4-16
and Dunham's equation 4.11 do not predict its position.
It also shows a very small rotational constant which is in
disagreement with both equations 4-9 and 4-17.
This
evidence together with the fact that only second combination
differences of the X 2
(v=0) are found for this band makes
one conclude that this band should be B E(v')-X E+ (v"=0)
transition.
The well depth DeB of this state must therefore
exceed 2.4 ± .8 cm- 1.
This number is inferred from the
dissociation energy of the X-state
band origin of this transition
(5.2 cm~
(2.8 cm1 ).
) and from the
186
More investigations and work should be done along
the following lines:
We could not analyze the lines of the transition
1.
2
1/2(4,6),
21 3 / 2 (31 4)-X 2Z(O) in the region of the sub-
band perturbation because we found just a few lines.
These
perturbed lines occur in the vicinity of the strong D-linos
of sodium.
Also, the beam was cold enough to prevent high
The perturbed lines still need
population of high J".
further investigation.
We could not also observe v"=l even though we
2.
believe it is bound.
Again, the cold beam did not allow
enough signal to be seen.
The 2 H/2(v'=5) band origin lies in the vicinity
3.
(1 cm~
) of 32 1/2
+
32 1/2 of Na.
Most of its rotational
structure gets buried in the Na transition line.
Although
lines to the blue and the red of this D-line could be seen
clearly.
0.3 cm~
to the blue of the D-line, the spectrum
consists of three bands, two of them
2 H3/2(v'=4)
and
2H/2 (v'=6) which are perturbed around J=5-1/2, and the
third is the progression of 2111/2 (5).
This complexity
prevented us from analyzing the 211/2(5) at high J and
proceeding from there to find the band origin.
4.
The bands A 2H(v'=0,1,2) were not seen because of
the small Frank-Condon factor.
Improvement in the experi-
187
ment by increasing the beam intensity and better baffling
will allow an enhancement in the signal to noise ratio
which might permit observation of v'=2.
5.
Further investigations need to be done to explain
22
the intensity anomalies for the 2 1/
2
(v'=6) and 2
3 /2
(v=6).
These bands were found to have two branches, the 21/2 had
P
and R 1 branches, the 213/2 had R21 and R2 branches.
The absence of the other two branches in each subband may
be interpreted as a transition from Hund's case
(a) to a
more atom-like coupling, where j of the atom becomes a good
quantum number.
These ideas need more effort and consider
ably more work to check their validity.
B.
COMPARISON WITH OTHER WORKS
Molecular spectroscopy is the definitive method for
determining interatomic potentials; there is no question that
its techniques work well for our NaNe spectrum.
This permits
a clearcut comparison of our results with the four pseudopotential calculations for NaNe, BAY69, PAS74, BOT73, MAL77.
Forthermore, our results bring into sharp question some
aspect of the experiments or of their interpretation about
the excited state potential of NaNe.
involved diverse phenomena:
excited Na from Ne
from Ne
These experiments have
differential scattering of
(CAR85), far-wing emission of excited Na
(Y)R75(, and shift and widths from absorption of light
close to the Na D-lines
(MCC76).
188
Table 1 shows a compendium of theoretical and
experimental values for the well positions and depths
and A 2 H states.
for the NaNe X 2
TABLE 1
Depth and Location of Minima for
A 23/2 and X2 + Potential Curves of NaNe
X 2 E+
Method
(cm
D
A2H3/2
r e(o)
1)
Ref. BAY69
1.8
12.9
Ref. PAS74
1.8
13.13(*12)
Ref. BOT73
Ref. MAL77
This work
Scattering
<40
17.8
8.0 ± .9
11 + -4
D e(cm 1)
13.7
&.2
r (ao)
8.5
9.0
>9.45
10
10.0
9.1
89.5
.1
6.0
140 ± 3
5.1
120± 15
8.0 ±.3
.1
189
Our results show that the recent pseudopotential calculations of Malvern and Peach
(MAL77) are much better
able to predict the excited state interaction potential
for NaNe than earlier pseudopotentials of Baylis
Pascale
(PAS74) and Bottcher
(BAY69),
(BOT73).
The X-state parameters found from earlier scattering
data
(CAR75) are in reasonable accord with this more
definitive work:
D eX agrees within error, and although
reX deduced from the scattering data is too low, we feel
this disagreement reflects the departure of the true
potential from the Morse van der Waal's shape assumed
earlier.
The excited state well depth determined from
the scattering agrees with the result reported here but
We feel that this discrepancy
reA definitely does not.
most probably arises from failure of the elastic approximation used in the scattering analysis:
Although Saxon,
Olson and Liu (SAX77) have shown that this approximation
2
works well for the A I state of NaAr
Reid
(with DeA ~
-1
500 cm
)
(RE175) has shown that it fails miserably for NaHe
(using DeA -
38 cm
1
).
Apparently NaHe is an interme-
diate case; close coupling calculations using the potential parameters proposed here would be valuable, and
might also resolve discrepancies in the velocity dependence of Na-Ne fine structure changing collisions
PH177).
(APT76,
Figure 47 shows the potentials of the different works
which are mentioned above.
A
IC
ID 84
AIT
I 7,050
17,000
)6,950
'5
H
C"
5
E
10-r
5
p
-u
16,900
CD
m.
NaNe
I
PI
H
I
5
R (W.u.)
-15
.1
15
-I 19
2C
--- "
10
\7/
L
FIGURE 47 --
L
XA
20
Potentials Of X and A-State of NaNe
A.
Present Work
B.
Ref . CAR75
C.
Ref. MAL77
D.
Ref.
PAS74
191
The present experiment calls into question the ability of spectral line shape data to yield useful information on interatomic potentials for weakly bound systems.
The conclusion of Lwin, McCartan and Lewis
(LW176), based
on line shifts and broadening, that the NaNe potentials
are more repulsive than those of Baylis
Pascale and Vandeplanque
(PAS74) is incorrect.
conclusion of McCartan and Farr
well depth is 0.5 cm
(BAY69) or
The
(MCC76) that the B E
is inconsistent with our observa-
tion of a band bound by 2.4 ±.8 cm-1
in the B E potential.
The failure of York, Scheps and Gallagher
(YOR75) to
observe a pressure dependence of the far red wing fluorescence of Na-Ne is puzzling in view of the success of
their analysis for NaAr.
It seems most probable to us
that they did not take emission profiles at sufficiently
low perturber pressure for NaNe to see anything except
the high pressure
(thermalized) emission.
This suggests
that the failure of Gallagher and co-workers to observe
pressure dependence in LiHe, LiNe
may stem from the same problem.
(SCH75) and NaHe
(YOR75)
This implies that the
systems NaHe, LiNe and LiHe may also have D eA's which are
a sizeable fraction of kT in the experiments by Gallagher
and co-workers
(YOR75, SEH75).
The failure of the pre-
ceding line shape experiments to indicate that the excited
state of NaNe is many times more attractive than predicted
192
by pre-1977 calculations does not invalidate those experiments --- rather it raises the challenge of finding
what went wrong in their interpretation plus the hope of
finding out more about line broadening and/or parts of
the NaNe potential curves which are not determined in
this experiment
of the X-state).
(e.g. the B-state and repulsive regions
It also remains to be seen if more
sophisticated theories of line broadening can explain the
line shapes observed by the preceding workers using
potentials consistent with the findings of this work.
The interatomic potentials of NaNe, particularly the
A H states, are relevant to experiments involving scat(PIT67, CAR75,
tering of Na in the 3P state from Ne
APT76, PH177).
These experiments are listed in Table 10.
They are relevant also to the determination of the perturbed line shape of the Na resonance radiation
lines) in Ne buffer gas
C.
(the D
(YOR75, MCC76, HOP75).
SUGGESTIONS FOR FUTURE EXPERIMENTS
Extension of the present studies to other rare gases is
obviously fruitful, given the existence of theoretical
predictions
(BAY69, PAS74, BOT73, MAL77).
Extensions to
a lighter system (helium) is of interest since most theoretical work has predicted that the X 2I
state is
bound.
This system requires more pressure of helium since these
molecules are less bound than NaNe (probably by a factor
193
Collisions Experiments (Na* + Ne)
Author
Experiment
Kind
Na*( 2P112 ) + Ne t Na*(2P3/2)
+Ne
Pitre & Krause
Carter et al.
Na*(2 P3/2) + Ne
-+
Na*(2 P3/ 2 ) + Ne
+
Total (cell)-1967
Na* + Ne
Differential
(Beam) - 1975
Na*( 2P1/2 )
Vel. Dep. on Total
(cell) - 1976
Apt & Pritchard
+ Ne
Phillips & Kleppner
TABLE 10
--
Na*(2 P3 /2 ) + Ne
+ Ne
Na*(2 P1 /)
Vel. Dep. on Total
(Beam) -1976
Scattering Experiments Involving Ne
194
of 10).
This will require more pumping spped which means
an additional work on the pumping system.
Extension to heavier NaX systems should imply a more
bound B2E state.
This should make it possible to determine
both excited state potentials
(A2H, B 2 Z).
Such a determin-
ation will make it possible to check the existing data on line
broadening (YOR75, SCH75) as well as many elastic and
inelastic scattering processes.
Such experiments will
entail modifications to the pumping system to permit recir-
culation of gas
(Kr and Xe are expensive).
Studies of
alkali rare gas molecules containing alkalis beside Na
are also of interest since they can be analyzed by the
same methods we applied to Na.
These are experimentally
more difficult because of difficulty of obtaining lasers
at the proper wavelength for excitation.
Extension to another type of van der Waals molecules
like dimmers of alkalis in the triplet states is another
interesting experiments.
More highly excited states of NaX molecules may be
studies by locking a second cw laser to one of the lines observed in the present study and scanning the present laser
to excite molecular levels which decay through intermediate
levels coming from Na(4P)+Ne.
These intermediate levels
could be detected with low background by isolating their
UV fluorescence.
195
An rf resonance experiment can be performed to study
the hyperfine and spin-rotation interactions in NaX
molecules.
The idea is to use the laser signals to
select and analyze a molecular state, which can then be
studied by rf resonance.
The state which is depleted by
the laser can be repopulated by the rf frequency and reexcited again.
Such experiments may need a skimmer so
that the high intensity beam can be introduced into a
long high-vacuum resonance region without destroying the
molecules.
196
APPENDIX A
A.
HUND'S CASE
1.
(a)
When the spin-orbit interaction is very large in
comparison with the interaction of the nuclear rotation
with the electronic motion, i.e. when the multiplet A-is
large compared to the off-diagonal element of B .
the rotational constant),
then the angular momentum
and R form the
couples according to Hund's case (a).
resultant J.
(B is
Figure 4 shows a vector diagram for this
case of coupling.
The vector J is constant in
magnitude and direction in the body fixed coordinate system.
and R
precess about this vector.
L and S precess
about the internuclear axis but at different frequencies.
According to the discussion at the beginning of this
for A 21H-state.
Chapter, Q is either 1/2 or 3/2
Since
4.
is the component of J along the internuclear axis, it
follows that
5
is half integral, i.e. J = N+R.
J cannot be smaller than its projection 0.
J
=
Q1, Q+i, 0+21
Naturally,
Therefore
0
Bear in mind that each J level is doubly degenerate because
of the two parity function H + and H.
A.
2.
HUND'S CASE
(b)
When the coupling of the spin to the internuclear
axis is smaller than the nuclear rotation interaction with
197
R
Hund's Case
(a)
A
+
,w WT-**"
LF
S
Hund's Case
Rd
(b)
J
Sd
FIGURE 4 --
Vector Diagram of Hund's Case
(a) and
(b)
198
the spin S, then the angular momentum couples according to
Hund's case (b), i.e. when the multiplet "A" is small
compared to BJ.
Also, when A = 0 and S /
0, the spin vector
S is not coupled to the internuclear axis at all.
means that 0 is not defined at all.
This
1.
In this case:
The wavefunctions for the nonrotating molecule are
usually characterized by three good quantum numbers:
A
(the projection of L along the internuclear axis),
S
(the spin quantum number).
2.
The angular momenta
R and S form the resultant J in case of A = 0.
shown in the vector diagram Figure 4.
and
form N.
and
This is
(In case
A/0,
Then the angular momentum N and S form the
total angular momenta J.)
The possible values of J for a given N are, according
to the principles of vector addition, given by J =
IN-SI
(N+S)
for S = 1/2, each N level is doubly degenerate
and "J" is half integral.
Normally, the coupling of S to
N causes a small splitting of the levels with different
J and equal N which increases with increasing N.
199
APPENDIX B
B.
1.
NONROTATING HAMILTONIAN
The non-rotating Hamiltonian is represented by
"H
ev
"
in the equation (2-3).
This Hamiltonian is the sum
of two parts, the electronic and the vibrational part. The
electronic part is concerned mainly with the electron dynamics.
Electronics energies and wave functions for the non-
rotating molecule can be determined from ab initio (SAX77)
or psuedopotential
PAS74).
(semi-empirically) treatments
(BAY69,
Both calculations require sophisticated computer
programs and many hours of computer time.
(Interested readers
should consult with BAY69, PAS74 and SAX77 references for
more information.)
The vibrational part is concerned
with the vibrational motion of the nuclei along the internuclear axis.
Vibrational energies and wavefunctions for
the non-rotating molecule can be determined more easily
than electronic energies and wave functions by solving a
one-dimensional Schr5dinger equation.
Thus, conventional
vibrational eigenvalue formulas will be taken into account
explicitly.
The simplest starting point of treating "H ev" is to
assume that it is separable to electronic "H ",
vibration
"H .
and spin part H .
imation
(BOR27)]. Classically, this means that the equations
vib ",
s
[This is Bohn-Oppenheimer's approx-
for each of the coordinates, and the energy is the sum
of
all, i.e.
H
ev
= H
e
+H .
vib
+ H
s
(b.2-5)
200
"H " describes the electrons coordinator.
e
This part of the
Hamiltonian is not going to be discussed any further.
But
readers who are interested can be referred to HER50,
BAY69, PAS 74,
SAX77.
Hvib is the vibrational part of the Hamiltonian.
The
potential energy curve V(R) arising from the variation
of total electron energy and eigenfunction with internuclear
separation R, acts as the potential for the Schrodinger
equation governing the radial motion of the nuclei, i.e.
H~
t2
2
d
a-
Hvib = -f -R
H
+ V(R)
(B.2-6)
is the electronic spin interaction.
This interaction
is represented here by the simplified form valid only for
diagonal matrics elements A'i-i
where ' and S
are operators
representing the total electronic orbital and total
electronic spin angular momenta respectively.
H
is the sum of Hso + H
orbit interaction, H 5 5
HSR is
+ HSR where H
In fact,
is the spin-
is the spin-spin interaction, and
the spin-rotation interaction.
H
is identically
zero and H SR will be disregarded because it is beyond
the measurement resolution.
operator
The spin-orbit interaction
9 is used to compute spin splittings within
a given spin multiplet.
To summarize, the non-rotating Hamiltonian (B.2-5) can
be rewritten as
201
H
ev
=H
B. 2.
e
+
[
d2 + V(R)] + AL-S
(B.2-7)
2yp dR2
BASIC SET OF NONROTATING HAMILTONIAN
The basis function lev>[Eq. (2-4)] for the
non-rotating molecule are not considered in detail in calculations of rotational energy levels.
Consequently, these
basis functions are often represented by symbols containing the quantum numbers used to describe the basis set,
e.g. JQ>,
IASE>, or ILASE>.
More details on the subject can
be found in HOU70, ZAR73, and HER50.
The vibrator eigenfunctions are usually found after
the potential energy curve is constructed.
The best
way to find the potenial energy curves for A 2 H and X2E
states is to use the observed data from the experiment
to generate the potential by the RKR procedure,
KLE32, REE47).
(RYD31,
Then use the potentials to find the
vibrational eigenfunctions by solving Schrodinger's
equation numerically by the computer.
To summarize, 1.
case (b).
E
+
the X2 E-state is strictly Hund's
The basis functions are IASE>=
101E>
1 2. The A2 T-state is split by S-L coupling into
A 2H/2 and A 23/2 substate.
case (a).
It is a very good Hund's
The basis set functions are IASE> -
where A = ±1 and E =
1
AE>
202
B.
3. EIGENVALUES OF NONROTATING HAMILTONIAN
Matrix elements of nonrotating molecule Hamiltonian
operator, see equation (B.2-7), in the basis set
(see B. 2.)
IAS>
are taken to have the following form.
<ASEI
H
ev JASE>
E
(B.2-9)
.
+ E
S(B29
el + E vib
The exactness of this equation occurs when the functions
involved in the matrix element are exact eigenfunction of
H
ev
E is the electronic part of energy and will be left
el
as it is due to lots of complication to find this term.
E
is the electronic energy of a multiplet term.
by
(HER50, HOU70),
is given, to a first approximation
It
ES = AAE
(B.2-10)
Precise energies of the multiplet components represented by
the wave functions
IAS7>
sion ±AAE because
IASE> are only approximate eigenfunctions.
For X E-state, E
A2 Ztt
deviate somewhat from the expres-
is zero because A = 0 while the
2
by 1
A E-state is split into two A Rl/2 and A 13/2 by
from the position of the A 21-state.
A
This multiplet
splitting "A" is well verified by observations on NaNe
system except for the part of the potential near the
dissociation.
For that part, the multiplet splitting A
3
approaches the atomic sodium fine structure splitting -A.
203
E vib is the vibrational energy of the molecule.
In
any real molecule, rotation and vibration are occurring
simultaneously, and the spectrum will reflect this fact
in both the vibration eigenvalues, which are not equally
spaced, and the rotational eigenvalues, which are not
rigid rotator.
A harmonic oscillator is characterized by a parabolic
potential curve while real molecules are an harmonic.
Dunham
(DUN32) assumed a vibrating rotator model for a
diatomic molecule.
He expressed the energy levels of
the vibrating rotator as a power series and showed coefficient relation to the molecular parameters for nonrotating molecules as
E Gv)=W
Evib =G(v) =w
1
1 2
- wexe(v+ )2+weye(v+
(v+-)
13 +.2
(.-1
To sum up, the energy eigenvalues of the nonrotating
molecule in the
E
ev
(a) X2E
= E
el
state is
+ G(v)
(B.2-12)
(b) A 2 13/2 state is
E
ev
= E
el
+ G(v) + 1/2 A
(B.2-13)
(c) A 211/2 state is
E
1/2 A
ev = E el + G(v) -
To calculate the term value v(v", v')
of a transition
from X to A state, one can use expressions in equation B.2-11
i.e.
204
A2fl
(a)
v(v";
(b)
2+X2 +
= Te + G(v')
v')
-
G(v")
-
1/2A
and
A R 3/ 2
v(v",
v')
(B.2-14)
= Te + G(v')-G(v")
+ 1/2A
where G(v) is given by equation B.2-11
The first term Te in equations
(B.2-14) is the electronic
energy separation between the X and the A state (the
separation between the minima of the two potential curves.)
v' and v" are the vibrational numbers of the A and X
state respectively.
B.
4.
HYPERFINE STRUCTURE
It is observed experimentally that the weak interaction of NaX molecules
("X" is an inert gas atom stands
for Ne and Ar so far) preserved the atomic property of
the hyperfine interaction in the ground state 32S1/2 0f
sodium into two components of F equal to 2 and 1.
F is the quantum number of the coupled electron and nuclear
spin.
The splitting is 1.772 GHz.
of NaX molecules exhibits a
Each absorption line
1.8 GHz splitting with the
same intensity ratio as those of F=2 and F=l in Na.
Intuitively, one could visualize that sodium atoms with
both F=2 and F=l combine with the inert gas atoms to form
molecules with P rather than
coupled to the rotation
205
to form molecular J.
This is explained in the vector and
energy diagram shown in. Figure 5.
In such scheme, molecular F, J, and R are good
4-
quantum numbers.
4
.4
Accordingly, S and I form F, and R
and F form the resultant J, i.e. J = R+F and F = S+I.
The hyperfine interaction yI-S splits the X2z+
state into two substates described by one potential, one
with F=l and the other with F=2.
is so small
The 6F-R interaction
[about 6 = .024 MHZ in KAr (MAT74)] which
leaves each R level with a 2F+l degeneracy.
The discus-
sion will proceed on the assumption of neglecting the
hyperfine splitting in calculations in the rest of
this thesis reminding that the hyperfine is strongly
observed in A 2H
E
transition of NaNe.
The first
excited state hyperfine structure was not observed.
Such
interaction would be less than the measurement resolution.
B.
5. ROTATING HAMILTONIAN
"H r" for any diatomic molecule can be written
as a sum of two products.
Each product consists of
a rotational constant "B" for the molecule and the square
of the component of rotation vector along x or y axis
(Z-axis is the internuclear axis of the molecule).
thus written as
Hr is
206
F
R
6
y
R
-
-
1.8GHZ
ll
FIGURE 5 --
Vector and Energy Diagram, Including the Hyperfine
207
or
SR2
(B. 2-15)
H
r
B(R
=
2
x
2,
+ R )
y
(There is no rotational angular momentum about its internuclear axis).
If R is expressed in terms of J, L, and S for the
purposes of calculation, one gets
H
r
=
B(J -J
B(L+S
where J+=Jx iJy
xy
z
)+B(L -L
z
)+B(S -S
z
) +
+LS+)-B(J+L_+J_L+)
-
L+=L +iLy, and S
_x
S ±iS
=
B(J+S-+JS_)
x
(B.2-16)
y
Equation (B.2-16) shows that both L and S affect the rotational energy levels only through four cross terms JxL
JyLy i
S ,
S
since the selection rules for non-
vanishing matrix elements of Lx , L
x y
and AZ = +1
Hamiltonian
respectively.
and S , S
x
are AA =
y
If both L and S in the
(B.2-16) can be ignored, then one might expect
the rotational Hamiltonian to be given by approximately
2 2
z
B(J -J )
Mulliken
for the A-state.
Such expressions are found by
(MUL30) when the separation be A and X-state
is large and "A" is larger compared with BJ.
Our ultimate goal is to diagonalize the matrix of
the complete Hamiltonian given by equation 2-3.
complete Hamiltonian as described by equation
The
(2-3) is
208
the summation of both the rotating and nonrotating
Hamiltonian which can be represented as
H = H +H
e
V
(B.2-17)
+AL-S+B(r)R
Notice that spin-rotation interaction is rejected
This interaction is measured by
because it is small.
Mattison et al on KAr and found to be 0.24(l)MHz.
B.
6. BASIS SET OF ROTATING HAMILTONIAN
The wave functions associated with the rotational
part can be characterized by one parameter 0',
and two good
quantum numbers J and M; where J specifies the total angular
momentum (see equation
(2-1))
in the molecule, M specifies
the projection of the total angular momentum along some
laboratory fixed z-axis
-J);
(M takes on the values J, J-l
and 0' helps to characterized the rotational wave
functions of the diatomic molecule.
is the projection
of J along the internuclear axis which is accidentally
equal, for a linear molecule, to the projection of L and
S on the same axis.
From what is mentioned above,
Q
is not a quantum number for the rotational wave functions,
since it does not correspond to an eigenvalue of some
operator acting on the rotational wave functions.
(Notice
that (L +S ) operator, when acting on a nonrotating moleZ
cule basis
Z
set
function,
can be replaced by 0) .
209
A basis set for the complete problem consists of
products of the basis functions for the nonrotational
problem
and basis functions for the rota-
|v>|nA6E>
tional problem I|JM>.
(Notice that this is not the only
choice of basis set which one could choose).
Such
functions are represented by
|v>nASE>
+
t
electronic
vibration
B.
1JM>
f
rot ation
7. EIGENVALUES OF A
(B.2-18)
ROTATING MOLECULE
Here we will not get involved in the derivation
of matrix elements of the rotating part of the Hamiltonian
equation B.2-16; an interested reader can consult MUL31,
HOU70, HIL28, and ZAR73.
Instead, we will summarize the
important steps taken to calculate matrix elements for
211
and 2 Z states.
Matrix elements of the rotational Hamiltonian
(equa-
tion (B.2-16) can be obtained from general considerations
of the matrix elements of an angular momentum operator.
As an example let us consider the spin angular momentum S
<SES 2
<SE S
2
=
S(S+l)
SE> =
< E±lSI
E>
(B. 2-19)
=
i[(S
E) (S E+1)]1/2
210
The non-vanishing matrix elements of the components
of the orbital angular momentum L
can be obtained from
equations (B.2-19) by placing S by L and E by A everywhere.
For J, one should replace S by J and E by Q everywhere
in the third
except that S+ must be replaced by J
equation of equations
<JQ±l JI
(B.2-19), i.e.
Q> = t[(J~+)(J±Q +1)]1/2
The strange behavior of J is discussed by Van Vleck
(VAN51).
The next thing one should find is the basis set for
and 2Z
both 2H
states;
for 21 state one gets from
equation (B.2-18)
2 3/2
=
211 1/2>=IV
2H
>
IV>
,,I 2
IV>
2
for 2
JM>
1-11
4/2
2 -3/2
j> 13
2
JM >
3
_ JM>
_1
f
2
=
one gets from equation
2 1/2> =
2 E1/>
H -
v>10
Iv
(B- 2-20)
> -. 1 JM>
2
(2-18)
JM>
1 -1-
2
2
(B. 2-21)
J
2
Rotational Eigenvalues:
To calculate matrix elements by using the complete
Hamiltonian equation (2-5) and
immediately that
(2-l
one should realize
211
(1)
the matrix element
<vIASEI H e+Hv+B(r) (L -L z
(B.2 22)
TAS>
e+G(v)
This is by definition, T e is the electronic energy
and G(v) is the vibrational energy given by equation (B.2.11).
(2)
the matrix element of B(r)
<vi
B(r)Iv>
= B
Computing the matrix elements one will obtain the following matrix element for the 2H
state
e
1
T +G(v)+ A+B[J(J+l)-7/4],
1 (J3]1/211
-B [(J- -) (J+r)]
-[-.)+.
-B[(J-1) (J+
T +G (v)
, -B[(J-1)
0
0
A+B [J (J+l)+1]
0, 0, T+G(v)+B[J(J+1)-7/4
0, O, -B[(J- 1)(J+3)]1/2,
)]l1/2
)o2,
(J+3)]1/2
T+G(v)+B[J(J+1)+!]
Solving the secular equation produces energy levels
which are doubly degenerate pairs.
The energies obtained
from the secular equation are
E=
T +G(v)+B[(J+I) -1]-
2
For Hunds case (a)
[A(A-4B)+4B
where A>>BJ,
approximate the radical in equation
[A(A-4B)+4B
(J+T)]l/ 2 = A{l-
it
is convenient to
(2-23)
+
(J+1)231/2
i.e.,
B(
(J+)2
2
keeping only terms through order B/A, we obtain the two
energy levels for the 2H3/2 and 2H1/2 substates
E
= T +G(v)+PA+B[J(J+l)
E2 = T +G(v)-
-7/4]
1
=(B.2-24)
2
A+B[J(J+L)+1/4]
(B.2-23)
212
These expressions for the energies of the
2 1H-state
agree
with expressions B[J(J+l)-Q 3 ] for the rotational eigenvalues given by Herzberg
(Her50) apart from an extra
The B[J(J+1)-Q 2] eigenvalues can easily be obtained
1/2B.
if one ignores all terms of equation B.2-16
except the
first one.
B 2 are retained,
one would
If terms of the order (j)
find that the coefficients of J(J+l) in equation (B.2-24)
must be replaced by B(l+R) and B(1-E) instead
shows that B for
2B2
A
of B.
This
H1/2 differs from that of 2 3/2 by
12~/
factor.
2EZRotational Eigenvalues:
Hund
(HUN27),
VanVleck
(VAN29),and Mulliken
have shown that in the case of 2E
(MUL30)
states the rotational
Hamiltonian eigenvalues are given by
1
= BvN(N=l)+yN
F
(B.2-25)
1
F2 = BvN(N+1)-ly(N+l)
1
where F 1 refers to the components with J = N+and F2
refers to those with J=N--.
The splitting constant y
is the effective spin-rotation interaction which is very
small compared to By.
Expressing equation
(B.2-25) in terms
of J and neglecting the spin-rotation term, one gets
F
F
=B
1 =
2
B
(J+ )(J+3)
(J+) (J- )
J+1 )(-1
y(+v *
fJ--T
(
(B.2-251)
213
The complete energy term for the 2Z state is therefore
E
= T +G(v) + F
(B.2-26)
E2 = T +G(v) +
F2
where G(v) is given by equation (B.2-11) and F
are given by equation (B.2-25').
and F2
214
APPENDIX C
(A) DOUBLING
LAMBDA
C. 1.
Electronic states with
erate, one for each sign of A
IAI/0
are doubly degen-
if one does not consider
interaction between the nuclear rotation with the orbital
angular momentum L.
In general, the degeneracy is lifted by
the mentioned interaction with the B E state giving rise
to a splitting which increases with increasing J.
This
splitting is called A-doubling and can be interpreted as a
second order interaction of the H state with all, especially E states.
Van Vleck
(VAN29) and Kovacs
(2 1 /2'
multiplet states
(a) have a
(KOV58) have shown that
213/2) belonging to Hundcs case
A-type splitting of different variation with
J for the different multiplet components.
For 21H states,
the A-type doubling of the 211/2 component varies linearly
with J; whereas for the 213/2 component it
varies approximately
with the third power of J and for small J is very small
compared to that of 2H1/2
Therefore, to a
approximation Van Vleck found
21/2
substate is
F=
e Ff
Where F
"e"
also
the A-type splitting for the
e
and F
f
=
(q+)
(C.2-27)
(q+?)j
are the rotational energies of the sublevels
and "f",(q+P/2) is
the A-doubling parameter.
Van Vleck
obtained comparatively simple formulae for the
constant q and p when the H state considered lies close to
215
one particular E state but far away from other E states
and when both E
and H state have the same well-defined
integral orbital angular momentum L.
In this case,
assuming L is due to one electron only and is thus equal
to ' of this electron, Van Vleck found
2B 2
~
q
~
P
(P+1)/V(HE)
(C.2-28)
2AB Z(9+l)/v(HrE)
Where v(H,E) is the separation of the H from the E
assuming that B
is the same in the two states.
should be made here about equation (C.2-28),
nearby E state which causes A-doubling
is
i.e.
state
A remark
if
the
weakly bound
(or repulsive),
its potential might be roughly deduced by
using equation
(C.2-28)
1
(q+ -P)
2
and B
with the measured values
v
More recent work on A-doubling has been reported by
Hougen
(HOU70) and Zare (ZAR73).
An approximate formulae
for A-type splitting is given by considering second order
perturbation of the rotating Hamiltonian and the nearby
E-state.
It is found that the splitting
F -F
where
= 2(J+f)
[2
(HE)+aS(HE)]
(C.2-29)
(HE) and ao(HZ) are defined as
S(HE)
and
~q/2
P/2
o (ll)
P
Bv(21
v(H E)
AB £(Z+l)
v
V
(l,E)
(C.2-30)
216
The approximate equality at the right
of equation (C.2-30)
and (C.2-28)X& called the "pure precession" relation and is
valid whenever a H and a E state differ by single molecular
orbital and for this orbital k is a good quantum number.
It should be noticed that for Hund's case
(a) molecules,
where A>>B,vthe leading term of equation
(C.2-30) is
and equation (C.2-29)
ac(HE)
can be approximated as
Fe-F
~
C.2.
CENTRIFUGAL DISTORTION
(C.2-21)
P(J+- )
As mentioned earlier in this part, centrifugal
distortion results from the fact that the rotational
constant "B" is not a constant at all, but rather a func-
tion of "r" and as a result can have matrix elements offdiagonal in v.
Eq.(C.2-18)
The basis set that we have been using
contains terms such as
IASZ>lv>IQJM>.
For
a consistent theory, it must be possible to obtain the
centrifugal distortion by finding matrix elements 'of the
rotational Hamiltonian to use to correct the energy terms
by second order of perturbation theory.
(This is appro-
priate since differences between vibrational energy levels
are much larger than those between rotational energy levels.)
Hougen
(HOU70)
andZare (ZAR73)
have shown how this
approach
accounts for the centrifugal distortion and more recently
Zare
(ZAR73) has calculated this term for the 2if/2' 3/2
217
and 2E states.
At this stage, it seems a summary of Zare's
calculations is unavoidable.
What one would like to do is to examine the rotational
[Eq.(B.2-16)] to obtain the precise centrifugal
Hamiltonian
distortion correction to the rotational eigenvalues of
each state (2I1/2' 2H3/2,
and
2
In general, one
+)E.
should find the centrifugal correction E
<ZSAv <MJPIHyI;E'S'A'>!v'>IM'J'ov><'I<'t<' Hrl>>>
G.(v) - G (v' )
< '
<1<
_
E2
_
=
to be given by
v'v1
~
-D
<v B (r) lv'><v'|B (r ) v>.
G(v)-G(v')
|>1><I<l
[<I<l
< I
IZ'S'A'>jM'J'Q'><'I<'IH
E <ZSAI<MJQIH
I> I> (C. 2-32)
r
r
I>
>
|>1>]
where by definition D is given by
-D
-D=
v'><v' IB(r) v>
~ 3<vlB(r)G (v)
E
-G (v' )(C
,
(C.2-33)
2-3
Using equation (2-32) and (2-33) Zare has found that centrifugal correction terms for
(1)
2 1/2 state is
-D{J+)+
(2)
[)+(J+~-)12 -l]}
2 3/2 state is
-D{[(J+ 1)1 2~22]2+ [(
(3)
2+
j 122_1]
(C.2-34)
state is
1 2
=-D{
(J+~-)
T;(-l)
J+S
1
'(J+ 2 )}
2
218
It is to be
.noted
that equation
(C.2-34) is indeed
the correct form and differs from the distortion coefficients
normally adopted by molecular spectroscopists which goes like
2
[J(J+l)]2.
Traditionally all rotational energies are
expanded in series of J(J+1).
C.3.
VIBRATIONAL CROSSING OF 2H312 WITH 2H/2
This kind of perturbation occurs due to crossing
of vibrational levels governed by the AJ=O selection rules.
The perturbation appears as a displacement and an intensity
anomaly for a number of successive J values.
The displace-
ment and intensity anomalies usually have a resonance-like
behavior when the deviation increases and decreases
rapidly with increasing J values.
Such perturbations
have been discussed fully in many books
KRO30),
(KOV37, HER50,
and just a summary of the perturbations which we have
seen is going to be discussed.
The substates perturbation which we observed was generated
.substate with
from vibrational bands crossing of the 2H1/2
Some of the rotational levels were
the 213/2 substate.
found
displaced.
By perturbation theory one can find
off-diagonal matrix elements "w.."
13
the two
which connect
perturbing states, i.e.
1.
=3 <A.S.
.I<v
1
<M J.Q
IHIj
H
JjMj
.>I
E-
-Aj>
where H is the Hamiltonian of the perturbed state.
(C.2-35)
219
The conditions for non-vanishing w..
(the selection
rules for perturbations) have been derived by Kronig
can be accounted for from equation(C.
2
. 3 5 ).
(KR028) and
These selection
rules are:
(1)
J.-J
(2)
A.-A.
0
=
0,
+i
(3) Both states must be of +ve or -ve parity, i.e.
+
4
+,
and +--/+-
-++-,
[in absence of electric
field]
(4) For identical nuclei, both states must have the
same symmetry in the nuclei,
i.e.
s+/-+a.
If the Hamiltonian in equation (2-3) breaks down into
two parts, rotating and nonrotating, one should find a
strong perturbation occurs only if the vibration eigenfunctions have suitable matrix elements of 1/r 2.
Eventually
this part will be evaluated through a computer program
which generates vibrational eigenfunctions from the
RKR potential.
The rotational part will just give the
normal eigenvalues
(see equations
(C.2.24) and
(C.2-25).
To find the eigenvalues of perturbed levels, one
should diagonalize the two by two matrix, and find the
position of the new levels.
If we do that, we should get
a matrix
E -E
1
W..
Ei
ji
2-
= 0
(C.2-36)
220
and new eigenvalues
E =
where E
(E+E 2 )t
v4wij 12+62
(C.2.37)
and E2 are the unperturbed eigenvalues, 6=E -E
2
is the separation of the unperturbed levels.
equation
E-E 2
According to
(C.2.37), the shifts of the two levels E-E
are equal in magnitude
2
opposite sign.
[/41w.,l
1J
2+62
-6]
For a given
and
but of
The larger the matrix element w
greater are these shifts.
1
the
o.., they are largest
fJ
for 5=0 and decreases with increasing 6.
221
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ACKNOWLEDGMENTS
During the period of my graduate work, a number of
people deserve words of thanks.
My advisor, Professor
David E. Pritchard has provided me with lots of his
experiences.
He developed in me the attitude of working and
deciding on things alone to achieve a planned goal.
He
supplied the necessary equipment for this experiment.
He
also suggested and worked very closely with me the long
range isotope shift
analysis part.
His many helpful
suggestions concerning this work were definitely fruitful.
W. P. Lapatovich helped in the design of the supersonic
jet machine and in running the computer programs
which constructed the ground state Lennard-Jones
Potential.
(6.8)
Professor Robert W. Field and I. Renhorn have
been very helpful in discussing the findings of the analysis.
My discussion with them was very fruitful and beneficial.
AlbertChang worked closely with me on the frequency
measurement system, especially the temperature and
pressure controller unit.
long range analysis fit.
R. McGrath has helped on the
I am grateful to the friendship
of the many past and present group members.
I am also
very grateful to the M.I.T. Community in general.
I
also express my gratitude to the members of the Research
Laboratory of Electronics for the services and advice
that they have provided.
Special thanks goes to my wife
225
who has shared the pain and joy of these years of building,
data taking, analyzing, and writing, as well as taking
care of our baby.
Her love and encouragement have made
the pains milder and the joys sweeter.