Force constant vs. bond length relations [pt.1] : The use of frequency modulation to generate a
reference beam for atomic absorption analysis [pt.2]
by Andrew Mackie Held
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree. of
DOCTOR OF PHILOSOPHY in Chemistry
Montana State University
© Copyright by Andrew Mackie Held (1972)
Abstract:
In the first part quadratic, cubic, and quartic force constants have been calculated from the known
spectroscopic constants of both ground and excited states of those diatomic molecules composed solely
of elements of the first row of the periodic table. The calculated force constants have been compared
with bond lengths, and correla tions between the force constants and bond lengths have been noted.
Force constants were also calculated using theoretical data, and the theoretical constants have been
compared with the experimental con stants. The force constant behavior is interpreted in terms of
free-electron theory and a constant repulsion curve.
In the second part the feasibility of applying various physical effects in the frequency modulation of a
light beam or source has been determined, as related to the generation of a reference beam for atomic
absorption analysis. The Zeeman effect was found to be the best suited for such a purpose, and an
apparatus for determining mer cury, which utilizes this effect, was constructed and tested. Because of
its background correction capability, the instrument can easily and reliably detect nanogram quantities
of mercury without the use of chem ical separation techniques. I.
II.
FORCE CONSTANT vs. BOND LENGTH RELATIONS
THE USE OF FREQUENCY MODULATION TO GENERATE A
REFERENCE BEAM FOR ATOMIC ABSORPTION ANALYSIS
by
ANDREW MACKIE HELD
A th e sis subm itted to the Graduate Faculty in p a rtia l
f u lf illm e n t o f the requirements fo r the degree.
of
DOCTOR OF PHILOSOPHY
in
Chemistry
Approved:
Head, Major Department
Chaii^pan, Examining Committee
MONTANA STATE UNIVERSITY
Bozeman, Montana
March, 1972
iii
ACKNOWLEDGMENT
The author would lik e to express h is g ra titu d e to h is parents
and to Ray W o o d riff f o r t h e ir continued support.
He wishes to acknowledge also the fin a n c ia l assistance p ro v i­
ded by the National Defense Education Act and the Endowment Re­
search Foundation o f Montana S tate U n iv e rs ity .
iv
TABLE OF CONTENTS
P a rt I .
Force Constant vs. Bond Length Relations
L is t o f F ig u r e s .................... ................... ... ........................................... ...
A b s t r a c t .................................................................... ...
In tro d u c tio n
................................
................................ v i i i
. . . . . . . . .
............................ I
Chapter I . H is to ric a l R e v ie w .............................................. ....
Chapter I I . T heoretical
. vi
3
Review and General T h eoretical Results . 8
Chapter I I I .
Procedure
. . . . . . . . . . . . . . . . . . . . .
Chapter IV .
Data A nalysis
............................ . . . . . . . . . . .
Chapter V. Conclusion . . . . . . .
Appendix A
...........................................
Appendix B
............................
.................... . . . . . . . . .
............................
27
28
70
72
. . . . . . . . . . .
73
Appendix C
..........................
74
Appendix D
...........................................
75
Appendix E
................................................ . . . . . . . . . . . . . .
76
Appendix F
.................... ........................... ............................................... ...
Appendix G
...........................................
Appendix H
Appendix I
............................................................
■ . . . . / ..........................
References C i t e d .................... . . ............................ ...
. 79
86
93
94
95
V
P art I I . The Use o f Frequency Modulation to
Generate a Reference Beam f o r Atomic Absorption Analysis
In tro d u c tio n .................................................... . . . . . . . . . . . . .
107
Chapter I . T h e o r y ........................ ................... ... ................................... ...
. HO
Chapter I I . Experiments Using the Zeeman E ffe c t
.1 2 1
Chapter I I I .
Conclusion
References Cited .
. ........................
. . . . . . . . . . . . . . . . . . . . .
135
137
Vl
LIST OF FIGURES
Page
1.
Hydrogen-hydrogen bond, long range graph
................
. . . . .
29
2.
Hydrogen-hydrogen bond, s h o rt range graph . . . . . . . . . .
30
3.
Hydrogen-hydrogen bond, th e o re tic a l . ....................
. . . . . .
37
4.
Force constant d e via tio n fo r hydrogen . ................
. . . . . .
38
5.
P o te n tia l energy curves fo r hydrogen
. . . . . . . . . . . .
39
6.
Second d e riv a tiv e s o f the p o te n tia l c u r v e s ...........................
7.
Helium -helium b o n d .................... ... ........................................... ...
42
8.
Force constant d e v ia tio n fo r helium molecules ....................
. . 44
9.
Helium-hydrogen bond
. 40
. . . ................................................ ....
10. Lithium-hydrogen and lith iu m - lith iu m bonds
11. Hydrides o f b e ry lliu m , boron, and carbon
46
. . . . . . . . .
48
. . . . . . . . . .
49
12. Boron-boron . . . . . . . . . . . . . . . . .
................
. . .
50
13. Carbon-carbon . . . . . . . . . .
. . . . . . . .
53
. . . . . . . . .
56
15. N itro g e n -n itro g e n ........................................................ .... . .. . . . .
58
16. Carbon-nitrogen and boron -nitroge n bonds
. . . . . . . . . .
59
. . . . . . . . .
61
. . . . . . . . . . . . . . .
62
14. Carbon-boron
. . . . . . .
....................
............................
17. Nitrogen-hydrogen and oxygen-hydrogen bond
18. Oxygen-oxygen bond
19. Nitrogen-oxygen bond
20. Carbon-oxygen bond
....................
. ........................................ ...
. . . . . . . . . . . .
63
................
. . . .
65
21. Boron-oxygen and beryllium -oxygen bonds . . ................
. . . .
66
v ii
Page
22. Bonds co n ta in in g flu o rin e .................................................... . . . .
23. Diagram o f experimental apparatus . . . . .
24. Diagram o f observed sign al
........................
68
. . 130
........................................................ ...
. 132
25. S e n s itiv ity vs. magnetic f ie l d s t r e n g t h ...................................
. 133
26. O ptical d e nsity vs. q u a n tity o f mercury ........................................ 134
v iii
ABSTRACT
In the f i r s t p a rt q u a d ra tic , c u b ic , and q u a rtic force constants
have been c a lc u la te d from the known spectroscopic constants o f both
ground and e x c ite d s ta te s o f those diatom ic molecules composed s o le ly
o f elements o f the f i r s t row o f the p e rio d ic ta b le . The calcu la ted
force constants have been compared w ith bond le n g th s, and c o rre la ­
tio n s between the fo rc e constants and bond lengths have been noted.
Force constants were also ca lc u la te d using th e o re tic a l data, and the
th e o re tic a l constants have been compared w ith the experimental con­
s ta n ts . The force constant behavior is in te rp re te d in terms o f fre e e le c t ron theory and a constant re pulsion curve.
In the second p a rt the f e a s i b i l i t y o f applying various physical
e ffe c ts in the frequency modulation o f a lig h t beam o r source has
been determ ined, as re la te d to the generation o f a reference beam fo r
atomic absorption a n a ly s is . The Zeeman e ffe c t was found to be the
best s u ite d fo r such a purpose, and an apparatus fo r determ ining mer­
cu ry, which u t iliz e s th is e f f e c t , was constructed and te s te d . Because
o f i t s background c o rre c tio n c a p a b ility , the instrum ent can e a s ily and
r e lia b ly detect nanogram q u a n titie s o f mercury w ith o u t the use o f chem­
ic a l separation techniques.
PART I
FORCE CONSTANT vs. BOND LENGTH RELATIONS
INTRODUCTION
Many chemists and p h y s ic is ts have attempted to fin d a general
e m p iric a l r e la tio n between the fo rce constant, or s tiffn e s s , o f a
molecule and the in te rn u c le a r d is ta n c e , s in c e , as H e rz b e rg ^ s ta te s ,
. no s a tis fa c to r y th e o re tic a l re la tio n between the force con­
s ta n t and the bond length has been found".
Recently e f f o r t has been
centered on the determ ination o f re la tio n s h ip s fo r p a r tic u la r atom
p a ir s , and re la tio n s have been found which f i t the experimental data
q u ite c lo s e ly .
To date only the atom p a irs carbon-oxygen, n itro g e n oxy­
gen, and n itro g e n -n itro g e n have been f i t t e d to such a r e la tio n .
w ork, th e n , is an attem pt to enlarge the number o f
This
known in d iv id u a l
re la tio n s and to observe any general patterns a ris in g from them, w ith
the u ltim a te goal in mind o f o b ta in in g a general e m pirical force con­
s ta n t vs. bond length r e la tio n .
These e m p irica l re la tio n s and those
f o r h ig h e r-o rd e r force constants w i l l be o f use in determ ining the
general behavior o f m olecular p o te n tia l energy curves and h o p e fu lly
w i l l guide the way toward the development o f a s a tis fa c to ry th e o re ti­
cal understanding o f the re la tio n between the force constant and the
bond le n g th .
One o fte n hears o f the importance o f chemical in t u it io n in study­
in g the chemical bond.
Y et, we seem to e a s ily fo rg e t th a t the flow ers
o f chemical in t u it io n need the f e r t i l e s o il o f a large body o f fa c ts .
The co m p ila tio n and study o f data on diatom ic molecules has been
-2 -
considered to be too fa r a fie ld f o r the chem ist, beneath the d ig n ity
o f the p h y s ic is t, and one o f those th in g s fo r which the sp e ctro sco p ist
never has enough tim e.
The vast body o f knowledge which we possess
concerning the diatom ic molecules has no t been c o lle c te d and used as i t
should in the study o f the nature o f the chemical bond, nor has i t been
used in the study o f sim ple polyatom ic molecules.
never land between the diatom ic and polyatom ic
ic a l in t u it io n
It
is in th is never-
molecules, where chem­
f o r lack o f e a s ily accessible data
cannot
be a p p lie d ,
th a t I have tr ie d to reach.
The cataloguing o f th is vast body o f data is the f i r s t step in the
development o f a " s a tis fa c to r y th e o ry ."
dust as the data c o lle c te d by
Tycho Brahe served f o r tw e n ty -fiv e years as the basis f o r K epler's
s tu d ie s , the c o lle c te d data can be analyzed u n til a ll the m olecular pro­
p e rtie s are c o rre la te d w ith one another.
We may then hope th a t the ex­
pla n a tio n s f o r these c o rre la tio n s w i l l provide the basis f o r
some fund­
amental changes, ju s t as Newton's, p re d ic tiv e laws o f motion superseded
the explanations o f Kepler.
CHAPTER
H is to ric a l
I
Review
The in tro d u c tio n o f quantum mechanics in the 1920's made possible
the in te rp r e ta tio n o f m olecular spectra in terms o f fundamental molec­
u la r constants.
A. K r a t z e r ^ 1 in 1920, was the f i r s t person to ob­
serve a re la tio n between v ib ra tio n a l frequency and bond length o f a
m olecule.
In 1925 R; M e c k e ^ noticed th a t the v ib ra tio n a l frequency
and the bond length fo llo w e d approxim ately the re la tio n
(bond len gth) (v ib ra tio n a l frequency) = Constant
fo r d iffe r e n t e le c tro n ic sta te s o f the same molecule.
(1 -1 )
In 1929
P. M o rs e ^ found th a t a s in g le r e la tio n ,
q
(bond le n g th ) (v ib ra tio n a l frequency) = Constant,
(1 -2 )
could be used fo r the e le c tro n ic sta te s o f d iffe r e n t diatom ic mole­
cu le s, g iv in g a probable e r ro r o f less than fo u r percent.
Thus, from ,
a s in g le e m pirical r e la tio n , i f the bond length were known fo r one
s ta te o f a diatom ic m olecule, the v ib ra tio n a l frequency o f th a t s ta te
could be determined w ith in a few pe rce n t, or vice versa.
Morse also
noted th a t the best f i t was obtained f o r molecules which were symmetric
and th e re fo re p o stulated the need fo r the in tro d u c tio n o f an "unsymmetry
fa c to r" o f the form ( ^ n ^ / D i i i + r r ^ ] ) ^ where m-j and
are the masses o f
“4 -
the atoms, to c o rre c t fo r the d e v ia tio n shown by very unsymmetric mole­
cules such as the diatom ic hydrides.
He observed th a t the d e via tio n o f
a s ta te from the re la tio n corresponded to a poor f i t o f the data fo r
th a t s ta te to a p a r tic u la r p o te n tia l energy curve which is now commonly
c a lle d the Morse p o te n tia l fu n c tio n .
The Morse re la tio n was la te r im­
proved by C l a r k ^ to
Re3V
v 2 = C,
(1 -3)
where n is the number o f e le ctron s outside the closed atomic s h e lls , C
is a constant th a t is the same f o r molecules w ith the same closed atomic
s h e lls , and Rg is the e q u ilib riu m bond len gth.
fu r th e r m o d i f i e d ^ t o
The re la tio n was then
make i t hold fo r various is o to p ic molecules
by in tro d u c in g the fo rce constant, Kg , in place o f the v ib ra tio n a l f r e ­
quency, wg.
The re la tio n then becomes
Rg6V
= C.
(1-4)
Another re la tio n having a s lig h t ly d iffe r e n t form was proposed by
B a d g e r^0,
Ke(Re -
Ci1 j ) 3
= C.
(1 -5)
Here d . . is a constant th a t is d iffe r e n t f o r each type o f molecule, and
C is the same f o r a ll molecules.
This r e la tio n , known as "Badger's r u le , "
-5 -
was found to be a p p lic a b le to polyatom ic molecules as w e ll as to d ia ­
tomic molecules and has found wide use.
G lockler and Evans^
la te r
improved the accuracy o f Badger's ru le by g iv in g C s lig h t ly d iffe r e n t
values f o r d iffe r e n t types o f molecules.
S ta rtin g in the 1930's, many attempts were made to fin d accurate
but general and sim ple fu n ctio n s r e la tin g p o te n tia l energy to in te ra ­
tomic distance and the fo rce constant.
H u g g in s ^ 1^
, by m odifying
the Morse p o te n tia l fu n c tio n , obtained a fo rce constant re la tio n which
was la t e r fu r th e r s im p lifie d to the form
In(K e) = ARe + B
by Wu and C h a o ^ ^ .
(1 -6)
At about the same time L in n e tt^ ® ’ ^
suggested a
new p o te n tia l fu n c tio n which gave a fo rc e constant re la tio n c lo s e ly re ­
sembling the p u re ly e m p irica l re la tio n which had been proposed by Morse.
The v a lid it y o f the various fo rce constant and p o te n tia l fun ction s which
were p o s tu la te d , however, could not be te ste d sim ply because o f a lack
o f d e ta ile d in fo rm a tio n on the fo rce constants and p o te n tia l fun ction s
o f most molecules.
Subsequent attempts to develop v a lid fu n ction s were
also hampered by a la ck o f data to provide adequate te s tin g .
Attempts have also been made to re la te force constants to m olecular
constants o th e r than the in te ra to m ic dista nce.
R elations re la tin g the
fo rce constant and bond length w ith the heat o f d is s o c ia tio n were
-6 -
f l 8 19)
proposedv ’ 7 b u t co n firm a tio n o f the re la tio n s again could not be
made because o f la rg e u n c e rta in tie s in the values o f the d is s o c ia tio n
energies o f many im po rta nt molecules.
A lso, the d is s o c ia tio n energy
depends upon the long range force fie ld s to a much g re a te r e xte n t than
do the fo rce constant and bond le n g th .
The la s t major e f f o r t to de­
vise a general fo rc e constant re la tio n was made by W alter G o r d y ,
who took in to account bond o rd e r, N, and e le c tro n e g a tiv ity , x , as w e ll
as bond le n g th , r , and obtained a form ula,
Ke = aN(xaxb/ r 2) 3/4 + b,
(1 -7 )
w ith an average d e v ia tio n o f two percent f o r seventy-one cases.
In the la s t ten years spectroscopic constants have been determined
f o r a la rg e number o f sta te s o f some o f the sim p le r molecules.
This has
encouraged several authors to reform ulate new force constant re la tio n s
by concentrating on s in g le molecules f o r which a la rg e amount o f data is
present.
In 1961 Ladd, e t. a l. ^ 2^
c o lle c te d the a v a ila b le data on the
carbon-oxygen bond and f i t the fo rce constants and bond distances to the
re la tio n
AR
(1- 8)
They la t e r extended th is work to a study o f the nitrogen-oxygen bond (2 2 )
and obtained an accurate re la tio n o f the same form fo r i t .
In 1966
-7 -
f 23)
J. C. D eciusv 1 published a s im ila r study o f the n itro g e n -n itro g e n
bond and found a re la tio n o f the same form which was v a lid over several
orders o f magnitude in the fo rc e constant.
No one, however, has published a comprehensive and up-to-date col
le c tio n o f the known spectroscopic constants, nor has anyone published
comprehensive study o f the fo rce constant re la tio n s f o r o th e r types o f
chemical bonds, though some attempts have been made to observe c o rre la ­
tio n s between the spectroscopic and m olecular constants o f ground-state
m olecules.
Herschbach and L a u r ie ^ ^ in 1961 stu d ie d the dependence o f
enharmonic p o te n tia l constants on bond length fo r some ground-state mol
ecules.
A ls o , in 1964 J o h n s o n ^ ^ suggested the extension o f force
constant re la tio n s to the noble-gas o r Lennard-Jones type atom p a irs .
In 1968, d u rin g the course o f th is work, Calder and R eu den berg^) ex­
amined th e sp e ctro sco p ic constants and .fo rce constants o f ground-state
diatom ic molecules and determined some re la tio n s between the constants.
CHAPTER
II
T h eoretical Review and General Theoretical Results
The fo rce constant o f a m olecular bond is defined in terms o f the
p o te n tia l energy fu n c tio n o f th a t molecule.
The problem o f understand­
in g m olecular fo rce constant r e la tio n s , th e re fo re , is e s s e n tia lly one
o f understanding the p o te n tia l fu n ctio n s o f molecules.
The p o te n tia l
fu n c tio n is most re a d ily defined w ith in the context o f the Born-Oppenheimer a p p ro xim a tio n ,1 which e s s e n tia lly sta te s th a t the atomic nuclei
are i n f i n i t e l y heavier o r slower than the e le c tro n s .
Under the Born-
Oppenheimer approximation nuclei move s u f f ic ie n t ly s lo w ly so th a t they
can be considered to be a t re s t fo r a given e le c tro n ic s ta te .
Then
f o r any in te rn u c le a r d ista n c e , R, there w i l l correspond an actual en­
ergy E ( R ) .
The energy E(Rj) fo r a given in te rn u c le a r se p a ra tio n , R^,
is c a lle d the "p o te n tia l energy," though the term is inaccurate in
th a t E ( R j ) includes the k in e tic energy o f the e le c tro n ic motion.
Thus,
the p o te n tia l fu n c tio n o r p o te n tia l energy curve, which is defined as
E ( R ) , includes both k in e tic and p o te n tia l energies.
E ( R ) , however,
acts as a p o te n tia l energy fo r the nuclear m otion, so th a t there is
1The Born-Oppenheimer approximation is sometimes c a lle d the a d ia b a tic
approxim ation; however, in modern u s a g e ^ ^ i t is considered to be a
special case o f the a d ia b a tic approxim ation.
-9 -
sorae ju s t i f i c a t i o n fo r the use o f the term "p o te n tia l e n e rg y ."1
For a diatom ic molecule the p o te n tia l fu n c tio n , E(R), c lo s e ly ap­
proximates a parabola near the minimum.
The nuclei o f the molecule be­
have, th e re fo re , very much lik e a harmonic o s c illa t o r near the e q u ilib ­
rium p o s itio n .
The fo rce con stan t, being sim ply the curvature o r second
d e riv a tiv e o f the curve a t the minimum, allows the determ ination o f the
shape o f the p o te n tia l curve near the e q u ilib riu m p o s itio n .
A study o f
the r e la tio n o f the harmonic o s c illa t o r force constant to oth er molecu­
la r constants then gives in s ig h t in to the behavior o f p o te n tia l energy
curves near the e q u ilib riu m p o s itio n .
The harmonic o s c illa t o r approximation i s , though, o f lim ite d value
in any a n a lysis o f the shape o f the p o te n tia l curve; a much more f le x ­
ib le and accurate approximation may be used by making what is commonly
c a lle d the "fo rc e constant" the f ir s t - o r d e r term in a power serie s ex­
pansion about the e q u ilib riu m in te rn u c le a r d ista n ce , Rfi.
This ordina ry
force constant is then c a lle d the q u a d ra tic o r harmonic force constant,
and the h ig h e r order c o e ffic ie n ts in the expansion are c a lle d the c u b ic ,
q u a r tic , e tc ., force constants and represent the c o n trib u tio n o f curves
o f the form
E =
(R - R g)^,
E =
(R - R g ) \ . . .
(2-1)
1The issue is fu r th e r confused by the fa c t th a t through the use o f the
Hellman-Feyman theorem one may c a lc u la te the "p o te n tia l energy" using
only e le c tr o s ta tic re pulsions and a ttra c tio n s .
“ 10-
to the shape o f the p o te n tia l curve near the minimum.
An n th -o rd e r
force co n stan t, the n, is also the n th -o rd e r d e riv a tiv e o f the poten­
t i a l curve a t the minimum.
The shape o f a p o te n tia l curve near it s
minimum may then be e a s ily analyzed, since the repre senta tion as a
power se rie s allow s the decomposition o f the curve form in to c o n tr i­
butions from various elementary curves.
To s im p lify the problem o f comparing the r e la tiv e shapes o f the
p o te n tia l curves o f d iffe r e n t m olecules, the power expansion may be
c a rrie d ou t in terms o f reduced coordinates.
Such an expansion is
c a lle d a D u n h a n /^ fu n c tio n and w i l l be used here.
The nomenclature
and formulas needed to make the expansion are given in Appendix A.
We must remember, though, th a t the p o te n tia l curve and force con­
s ta n t are o n ly mental c o n s tru c ts , and in fo rm a tio n 'c o n c e rn in g molecules
must u ltim a te ly be obtained from spectroscopic data.
Formulas g iv in g
the values o f the Dunham c o e ffic ie n ts o r force constants in terms o f
spectroscopic constants have been d e r i v e d ^ ® a n d are reproduced in
Appendix B.
In th is paper a ll forces are expressed in mi 11i dynes and
bond lengths in Angstroms.
The various spectroscopic constants needed are in tu rn determined
from the actual spectra.
Analysis o f the d e ta ile d s tru c tu re o f the
spectra o f diatom ic molecules leads to the determ ination o f the e le c ­
tr o n ic energy le v e ls , in c lu d in g the energy le v e ls o f the v ib r a tio n a lIy
and r o ta tio n a lly e x c ite d s ta te s .
Using quantum mechanics, the
-1 1 -
v ib ra tio n a l and ro ta tio n a l energy le v e ls fo r each e le c tro n ic s ta te may
represented as a power se rie s in terms o f the v ib ra tio n a l and ro ta ­
tio n a l quantum numbers, y and J:
E (v ,J )/h c = we (v + 1/2) ~ Wqxe ( v + 1 /2 )2 + • • • +
( 2- 2 )
+ ByJ(J + I ) - CvJ2( J + I ) 2 + . . .
Wfi and WfiXe are the e q u ilib riu m v ib ra tio n a l spectroscopic constants.
By and Cy are ro ta tio n a l spectroscopic constants.
w i l l be used subsequently in the place o f E/hc.
The "term value" G
Once the energy le v e ls
are known from the spe ctra the e q u ilib riu m constants
mined.
can
be
d e te r­
The ro ta tio n a l e q u ilib riu m co n sta n ts, Bfi and a f i , are determined
from the power se rie s expression
By = Bfi - afi(v + 1/2) + .* •
(2 -3)
The equation
Bfi (cm- 1 ) = 16.857/iiRfi2(A)
(2 -4)
allows the determ ination o f the e q u ilib riu m in te rn u c le a r dista n ce , once
the reduced mass is known.
Quite o fte n (n e a rly always in the case o f
polyatom ic molecules) only the fundamental v ib ra tio n a l frequency, v (l-O )
o r A G (l/2 ), is known.
I t is often used as an approximation fo r wfi.
A
-1 2 -
b e tte r approximation is
A G (l/2) = we - 2Wexe
(2-5)
wexe may sometimes be estim ated from the Morse re la tio n
(2-6)
where Dg is the e q u ilib riu m d is s o c ia tio n energy, o r from the well-known
ru le th a t WgXeZwg is only s lig h t ly s m a lle r than OigZBg .
S im ila r ly , often
only Ry _ q , o r Rg , is known, so th a t Rg cannot be determined (again
th is is n e a rly always the case w ith polyatom ic m olecules).
One is o fte n
forced to use AG(1Z2) and Rg in determ ining the force constant and bond
le n g th .
No mention has been made in the lit e r a t u r e o f the magnitude o f
e r ro r in cu rre d by the use o f AG(1Z2) and Rg , and t h e ir s u b s titu tio n fo r
wg and r g has im p lie d th a t the e r r o r is ra th e r s m a ll.
Expressions fo r
the magnitude o f the d e v ia tio n o f AG(1Z2) from wg and o f Rg from Rg are
derived in Appendix C.
a lly less than 0.02.
For a normal molecule Wg Xg Zwg and Otg ZBg are usu­
Using the expression derived in the appendix we
then fin d th a t the use o f AG(1Z2) gives a force constant which is low by
e ig h t percent o r le s s , whereas Rg is only o n e -h a lf percent o r less l a r ­
ger than Rg .
Whether the use o f AG(1Z2) and Rg in combination w ith each
o th e r would re s u lt in a large e r r o r would depend upon the slope o f the
force constant vs. bond length curve.
■13-
I t is po ssible to study the r e la tiv e shapes o f m olecular p o te n tia l
curves th e n , p ro v id in g th a t a s u f f ic ie n t number o f spectroscopic con­
sta n ts are a v a ila b le fo r those molecules.
In recent years large amounts
o f new spectroscopic data have become a v a ila b le , making i t possible th a t
an in-d ep th analysis m ight reveal some basic new re la tio n s governing the
shapes and behavior o f p o te n tia l curves.
I t w i l l be advantageous to t r y a t the very s t a r t to determine what
a d d itio n a l fa c to rs can infuence the re la tio n between fo rce constant and
bond le n g th .
I t is a p p ro p ria te , then, to ask at th is p o in t what molec­
u la r p ro p e rtie s m ight be expected to have an in flu e n c e upon the re la ­
tio n s .
Any p ro p e rtie s which would e ff e c t the shape o f the p o te n tia l
curve would also be expected to have an in flu e n c e upon the force con­
s ta n t and the bond len gth.
The d is s o c ia tio n energy o f d iatomic molecules has been re la te d to
the fo rce constant and bond length by several aut hor s^ 6
I t is
w e ll known th a t a high d is s o c ia tio n energy is associated w ith a large
fo rce constant and a small in te rn u c le a r distance f o r a normal chemical
bond.
The d is s o c ia tio n energy also has long been recognized as a funda­
mental parameter in the determ ination o f the shape o f the p o te n tia l en­
ergy fu n c tio n .
able
Thus, the d is s o c ia tio n energy may be considered a prob­
c o n trib u tin g fa c to r to force constant r e la tio n s , and i t s in flu e n c e
should be examined c a r e fu lly .
However, r e lia b le values f o r d is s o c ia tio n
energies are not g e n e ra lly a v a ila b le .
A lso, i t is almost c e rta in th a t
-1 4 -
the in flu e n c e exerted by the d is s o c ia tio n energy w i l l be extremely v a r i­
able.
Whenever accidental "le v e l crossing" occurs the d is s o c ia tio n en^
ergy o f the s ta te w i l l be anomalous, and the degree o f in flu e n c e o f the
d is s o c ia tio n energy w i l l depend on whether o r not i t is possible to take
the le v e l crossing in to account.
The d is s o c ia tio n energy is also d e te r­
mined by long range forces, so th a t sometimes the d is s o c ia tio n energy
w i l l be anomalous even i f le v e l crossing does not occur.
Thus a careful
analysis o f the in flu e n c e o f the d is s o c ia tio n energy would be somewhat
premature.
The degree o f io n ic character o f a bond is also an im portant fa c to r
in determ ining the shape o f a p o te n tia l curve.
Io n ic bonds are known to
have p o te n tia l curves o f a d iffe r e n t shape o r form than covalent
(30 31 32)
bonds' * ’
.
About the only experimental measure o f io n ic character
is the d ip o le moment; however, io n ic it y is a ra th e r v a ria b le fu n c tio n o f
in te ra to m ic d ista n ce , so th a t the change in dip o le moment w ith in te r a ­
tomic dista n ce , i . e . the dip ole d e riv a tiv e , would be im portant as w e ll.
I t may be noted th a t f o r ground s ta te molecules io n ic it y c o rre la te s
ra th e r w e ll w ith the e le c tro n e g a tiv ity d iffe re n c e between the c o n s titu ­
ent atoms.
Excited sta te s w i l l vary considerably in io n ic character
even f o r homonuclear diatom ic molecules, making the d ip o le moment and
e le c tro n e g a tiv ity concepts d i f f i c u l t to use.
Since data concerning d i­
pole moments and dip o le d e riv a tiv e s are ra th e r sparce, a system atic
study o f the in flu e n c e o f io n ic cha racte r or io n ic it y cannot be made.
-1 5 -
The number o f e le ctron s and protons in a m olecular system is , o f
course, o f extreme importance fo r any m olecular pro p e rty.
Likewise, the
e m p iric a l in fo rm a tio n on force constants and bond lengths c o lle c te d here
w i l l show c o n c lu s iv e ly th a t the number o f protons present influences the
r e la tio n between force constant and bond length s u f f ic ie n t ly to make
c la s s ific a t io n o f the re la tio n s in terms o f atomic number a b solute ly es­
s e n tia l.
The in flu e n c e o f the atomic number upon the atomic core ra d ii
o r ra d ia l e le c tro n d e n s itie s is apparently the cause o f the strong de­
pendence upon the atomic numbers.
The number o f e le ctro n s present,
though, w i l l be shown to have no apparent e ffe c t upon the re la tio n s .
Since the fo rce constant and bond length may be defined in terms
o f the v ib ra tio n a l and ro ta tio n a l spectroscopic co n sta n ts, any re la tio n s
between the spectroscopic constants w i l l be re fle c te d in the force con­
s ta n t vs. bond length r e la tio n s , and vice versa.
o f fo rc e constant
Thus the determ ination
re la tio n s provides a means o f studying the re la tio n s
between spe ctroscopic constants as w e ll.
The to t a l e le c tro n ic energy a t the e q u ilib riu m distance might be
expected to have some in flu e n c e upon the force constant re la tio n s , since
i t plays an im po rta nt ro le in th e o re tic a l problems.
I t w i l l be shown
th a t t o t a l energy c o rre la te s ra th e r po o rly w ith e q u ilib riu m distance and
fo rce co n sta n t; however, i t w i l l be o f in te r e s t to check f o r a c o rre la ­
tio n o f t o t a l energy w ith the d e via tio n o f force constants from "ex­
pected" values.
-1 6 -
F in a lly 5 th e re are many th e o re tic a l p ro p e rtie s such as "overlap
p o pula tion" and quantum number which one could attem pt to re la te to the
force constant and bond le n g th .
Such an attem pt, the author fe e ls ,
would not be in keeping w ith the e s s e n tia lly phenomenological approach
taken here, i f the attem pt were made on a large scale .
The re la tio n s
obtained w i l l be c e r ta in ly useful in e va lu a tin g various th e o re tic a l
approaches and approxim ations, even though th is th e sis is to be consid­
ered an attempt to generate representation-independent concepts.
There­
fo re , the c o m p a tib ility o f only some o f the sim ple r m olecular models
w ith the fo rce con stan t re la tio n s w i l l be evaluated.
An e f f o r t w i l l be made to provide a reasonable th e o re tic a l context
fo r the in te r p r e ta tio n o f force constant re la tio n s .
Since fo rc e con­
sta n ts are d e riv a tiv e s o f the to ta l energy, i t is necessary only to re­
la te the d e riv a tiv e s to the values o f some fundamental th e o re tic a l con­
s tru c ts in order to provide such a th e o re tic a l context.
The to ta l po­
te n tia l and k in e tic energies are two such fundamental con stru cts.
Ap­
proximate e le c tro n ic c o n trib u tio n s may be obtained from one-electron en­
ergies determined from simple quantum-mechanical c a lc u la tio n s .1 The en­
ergies may then be re la te d to the energy d e riv a tiv e s through the v i r i a l
theorem.
The v i r i a l theorem, though, does not by i t s e l f allow
the
unique determ inatio n o f the k in e tic and p o te n tia l energies from the
1The n u c le a r-n u c le a r p o te n tia l is sim ply ZiZg/R where Z1 and Z2 are the
atomic numbers. The nu cle a r k in e tic energy is n e g lig ib le under the BornOppenheimer approxim ation.
-1 7 -
energy d e riv a tiv e s .
By making some reasonable assumptions, however, the
v i r i a l theorem may be used to determine the behavior o f the force con­
s ta n ts , o r energy d e riv a tiv e s .
The v i r i a l theorem determines the r e la tiv e magnitudes o f the k in e t­
ic
and p o te n tia l energies o f any m olecular system.
The general forms
o f the theorem given below ^34^ are v a lid fo r any diatom ic molecule w ith ­
in the Born-Oppenheimer a p pro xim atio n.1
RE' + 2E = V,
RE' + E = -T ,
(2-7)
(R f' + 2T) + (RV' + V) = O
T is the to ta l k in e tic energy, V is the to ta l p o te n tia l energy, and der­
iv a tiv e s are in d ic a te d by primes.
the in te rn u c le a r dista n ce , R.
The theorem is v a lid f o r any value o f
D iffe r e n tia tio n o f the v i r i a l theorem a l­
lows expressions f o r the p o te n tia l co n stan t,
to be d e rive d :
thu s.
(2-8)
e
e
and
V =V
nuclear
+ V
e le c tro n ic
(2-9)
combine to give
ReKe = - W
i
'
V
Re2 + ( v,e le c tr o n ic ' R
e
Again the nuclear k in e tic energy is neglected.
( 2- 10)
- 18 / 33 )
Borkman and P a rrv ' have re c e n tly used th is theorem to in te r p r e t
the behavior o f p o te n tia l constants and p o te n tia l curves.
and th a t o f KnolI
T h e ir work
w i l l serve la rg e ly as the basis f o r the present
in te rp r e ta tio n through the v i r i a l theorem.
Borkman and Parr based
th e ir analysis upon the fa c t th a t the Born-Oppenheimer e le c tro n ic energy
s a tis fie d the d if f e r e n t ia l equation
R2E" + 4RE' + 2E = (-1 /R )(R2T )'
,
which may be derived from the v i r i a l theorem.
(2-11)
The author has shown in
Appendix D, however, th a t the same method may be used to generate the
more general d if f e r e n t ia l equation
R2E" + (n + 2) RE' + nE = -(R nT )V R n" 1
where n is g re a te r than one.
,
(2-12)
I f the r ig h t hand side o f equation (2-12)
is se t id e n tic a lly equal to zero, then a ll o f the s o lu tio n s o f the equa­
tio n w i l l be re c ip ro c a l fu n ctio n s o f the form
V = A/R + Vfi/Rn + Vq
(2-13)
T = TnZRn + T
O
(2-14)
and
9
-1 9 -
where
VfiZTfi - -(n - 2 )/(n - I )
(2-15)
I f the r ig h t hand side o f equation (2.-12) is a constant, then the s o lu tio n s w i l l be o f the form
.
V = A/R + Vn/Rn + Vq
(2-16)
and
T = BZR2 + TfiZRn + T0
.
(2-17)
Thus the d if f e r e n t ia l equation o f Borkman and Parr forms a p a r tic u la r ly
simple case in which the homogenous s o lu tio n s are o f the form
V = -AZR + Const.
(2-18)
O
T = BZR + Const.
(2-19)
and
The general s o lu tio n to equation (2-12) is
E(R) = AZR + BZRn - (lZR)/T(R)dR
.
( 2- 20 )
Since the general s o lu tio n o f the v i r i a l equation is
E(R) = AZR - (lZR)/T(R)dR
9
(2- 21)
-2 0 -
i t is seen th a t the method used by Borkman and Parr introduces e xtra n ­
eous s o lu tio n s .
The method employed by Knoll avoids th is d i f f i c u l t y
by using the v i r i a l theorem d ir e c t ly ; th e re fo re , i t is to be p re fe rre d .
The fo rc e constant re la tio n s derived by Borkman and P arr then arose
e s s e n tia lly from t h e ir assumption th a t
2
( 2- 2 2 )
ra th e r than from t h e ir treatm ent o f the v i r i a l theorem.
an assumption equation (2 -1 3 ).
I introduce as
This assumption, because o f i t s more
general n a tu re , can be expected to f i t the data much b e tte r than the as­
sumption o f Borkman and Parr.
I t should be noted here th a t assuming a fu n c tio n a l form fo r the
k in e t ic energy, p o te n tia l energy, o r to ta l energy a u to m a tic a lly allows
the d e te rm in a tio n , v ia the v i r i a l theorem, o f the form f o r the oth er
two types o f energy.
Thus i t r e a lly makes no d iffe re n c e whether one
works w ith T(R), V(R), o r E(R) in determ ining force constant re la tio n s
o r in determ ining energy re la tio n s from the force constant re la tio n s .
From the assumption o f Borkman and P arr the re la tio n s
(2-23)
and
Me = 36Ke/Re
2
(2-24)
-2 1 -
and
KeMeZLe2 - I
(2-25)
are obtained by d iffe r e n tia tio n o f the v i r i a l theorem and s u b s titu tio n .
From the a u th o r's assumption the re la tio n s
Le = -(n + 4) Ke/Re
,
Me = (n2 + .Zn + 18) Ke/Re2
(2-26)
,
(2-27)
and
KeMeZLe2 = I + (2 - n)Z(n + 4 )2
(2-28)
are obtained, and again the re la tio n s o f Borkman and Parr form a special
case where n is equal to two.
x
There is another useful theorem which re la te s the energy to any
parameter being used in the determ ination o f the energy.
I t is c a lle d
the Hellman-Feynman theorem and u s u a lly is w ritte n in the form
I r = / p*n I V t •
(2-29)
The form ula is very useful because we can s u b s titu te the in te rn u c le a r
distance o r the fo rce constant in as a parameter.
By using the in t e r ­
nuclear distance as the parameter one obtains the " e le c tr o s ta tic
theorem," which e s s e n tia lly states th a t the force on the nucleus can be
-2 2 -
determined by simple e le c tro s ta tic s as the sum o f the n u c le a r-n u cle a r
re p u lsio n and an e le c tro n -c lo u d a ttr a c tio n .
also be de rived from th is th e o re n /35\
The v i r i a l theorem may
Thus, by using the v i r i a l
theorem we are a c tu a lly also using the Hellman-Feynman theorem.
Before pursuing complicated o r a b s tra c t models i t is appropriate
to ask i f th e re are any simple c la s s ic a l models which one may use to
account f o r the force constant re la tio n s .
Indeed, there are several
very sim ple c la s s ic a l models from which to choose to account fo r the
basic exponential re la tio n between fo rce constant and bond length.
The s im p le s t model o f a ll is th a t o f two b a lls and a s p rin g .
I f one
va rie s o n ly the length o f the s p rin g , then the re la tio n
Ke = AY/Rg
(2-30)
h o ld s, since Young's Modulus, Y, and the c ro s s -s e c tio n a l area, A, are
constant and represents the force o f r e s titu tio n per u n it length o f
s p rin g .
I f we r e s t r i c t the sp rin g so th a t i t has the same amount o f
.
m a teria l no m atter what i t s e q u ilib riu m length i s , then the re la tio n
becomes
Ke = VY/Re 2
,
where the volume, V, is constant.
(2 -3 1 )
S t i l l , the c la s s ic a l b a ll and spring
model is u n s a tis fa c to ry , as there is no way to fu r th e r modify the model
w ith o u t making i t seem somewhat c o n triv e d .
There does e x is t an extrem ely
simple
quantum - mechanical
-2 3 -
model which, w ith the use o f a few parameters, a c tu a lly reproduces both
q u a lita tiv e ly and q u a n tita tiv e ly the basic re la tio n between the force
constant and bond length in molecules.
model.
I t is c a lle d the fre e -e le c tro n
Some attempts have been made to exp lain the basic force con­
s ta n t vs. bond length behavior using simple fre e -e le c tro n models^
5
’
'.
The.model, however, has not been c a r e fu lly and ex­
te n s iv e ly tre a te d in any o f the cases.
There are d iffe r e n t types o f fre e -e le c tro n models, the le a s t so­
p h is tic a te d o f which is the p a rtic le -in -a - b o x model.
I f we consider
the valence or bonding e le ctron s in the molecule as being constrained
to move only along the axis in a region whose boundaries are at the
n u c le i, then we have a f a i r l y s a tis fa c to ry one dimensional p a r tic le in -a -b o x model.
I t gives the re la tio n
Ke - (h2/4me)/R e4
.
'
(2-32)
I f we allow motion in a ll three dimensions and fu r th e r assume th a t the
boundaries in the la te r a l dimension remain constant, then we fin d th a t
the fo rce constant again varies as the inverse fo u rth power o f the in te rn u c le a r distance.
tio n (2 -3 1 ).
In fa c t , the re la tio n obtained is e x a c tly equa­
I f the la te r a l boundaries are allowed to expand and con­
tr a c t along w ith the a x ia l boundary one s t i l l obtains an inverse fo u rth
power r e la tio n !
I t is im portant to remember th a t th is model only as­
sumes th a t the bonding o r valence e le ctro n s behave as fre e electrons
-2 4 -
con fin e d to the region between the n u c le i.
in v o lv e d .
No oth er assumptions are
The model may be m odified by changing the number o f e le c ­
trons p a r tic ip a tin g or by changing the rules fo r determ ining the bound­
a rie s .
I t is very easy to reproduce the em pirical fo rce constant to
bond length re la tio n s by making such m o d ific a tio n s .
The inverse fo u rth
order dependence o f the fo rce constant applies fo r any shape o f box one
chooses, as is shown in Appendix E.
One m ight hope also th a t there is some connection between the form
o f the p o te n tia l energy curves o f in d iv id u a l states and the form which
the fo rc e constant vs. bond length re la tio n takes f o r the m u ltitu d e o f
s ta te s .
There are many s a tis fa c to ry mathematical forms f o r describing
a p o te n tia l energy c u r v e ^ ^ J ? , 29,38-43)^
^ost o f these forms can be
made c o n s is te n t w ith the e m pirical force constant re la tio n s because a
separate se t o f parameters is used f o r each s ta te .
I f one could re ­
duce the number o f e m p irica l parameters in the p o te n tia l energy curves
to the p o in t where a s p e c ific force constant bond length w i l l be pre­
d ic te d , then one m ight gain some in s ig h t in to the fa c to rs which could
re la te the fo rce constant behavior to the form o f the p o te n tia l energy
curve.
Such models w i l l be discussed la te r .
Another problem re q u irin g some th e o re tic a l consideration is whether
the fo rc e constants o f polyatom ic molecules and c ry s ta ls are comparable
in the same sense as those o f diatom ic molecules.
For polyatom ic mole­
cules the p o te n tia l energy is w ritte n out in terms o f a “ General Valence
-2 5 -
Force F ie ld " in which there is a force constant fo r each in te rn a l coor­
dinate and an in te ra c tio n force constant f o r each combination o f two
in te rn a l coordin ates.
The fo rce constant then becomes a symmetric ma­
t r i x o f fo rc e co n stan ts, in which the diagonal constants are the force
constants f o r the in te rn a l coordin ates, and the o ff-d ia g o n a l elements
correspond to co rre ctio n s fo r the n o n -a d d itiv ity in combinations o f the
in te rn a l co o rd in a te s.
Since the in te rn a l coordinates are sim ply changes
in the bond lengths and bond angles one would hope th a t the force con­
s ta n t corresponding to a p a r tic u la r bond would be analogous to the d ia ­
tom ic fo rc e constant fo r th a t bond.
From an in t u it iv e p o in t o f view one
m ight wonder i f th is is a c tu a lly tru e , since there is no p h y s ic a lly rea­
liz a b le polyatom ic v ib ra tio n which can sim ulate com pletely the diatom ic
v ib r a tio n .
However, the force constants in the GVFF approximation c o r­
respond t o v ib ra tio n s th a t are not p h y s ic a lly re a liz a b le e ith e r , so th a t
the diagonal s tre tc h in g force constants do correspond to diatom ic v ib ra ­
tio n s .
The o ff-d ia g o n a l elements represent coupling between the idea­
liz e d in d iv id u a l o s c illa to r s .
There is one catch, though.
When a d ia ­
tom ic m olecule v ib ra te s , the non-binding ele ctron s are fre e to re la x into
t h e ir optimum p o s itio n s fo r each value o f bond le n g th .
fo llo w s i t s
re action coordinate during the v ib ra tio n .
The molecule then
By in tro d u c in g the
coupling constants in the GVFF method, one freezes the ele ctron s in the
o th e r bonds, so th a t the id e a liz e d v ib ra tio n o f a s in g le bond involves no
re la x a tio n o f the e le ctro n s outside the bond in question.
Instead th is
-2 6 -
re la x a tio n o f the ele ctro n s and n u c le i
in te ra c tio n constants.
is
handled by the o ff-d ia g o n a l
Thus, the id e a liz e d diatom ic molecule o f the
GVFF method does not fo llo w the actual reaction coordinate.
Force con­
sta n ts computed by th is method, th e n , would be expected to be higher
than those o f the corresponding actual diatom ic m olecules, e s p e c ia lly
i f many o th e r bonds are adjacent to the bond in question !
This d i f ­
ference apparently has not been considered by any previous workers.
I t is po ssib le to remove th is c o n s tra in t from the polyatom ic force con­
sta n ts i f the in te ra c tio n fo rce constants are known.
One then has the
problem o f determ ining i f a motion along the reaction coordinate is a
b e tte r approximation than the constrained motion.
I f another bond is
being ap preciably strengthened w h ile the bond in question is s tre tc h in g ,
then the unconstrained fo rce constant w i l l be appreciably lower than
th a t f o r the diatom ic molecule!
L a ttic e v ib ra tio n s in c ry s ta ls do not admit a sim ple in te rp re ta ­
tio n , p r in c ip a lly because o f the importance o f long-range forces in
c ry s ta ls .
CHAPTER
III
Procedure
No comprehensive lis t in g s are a v a ila b le o f the spectroscopic con­
sta n ts o f the known diatom ic m olecules;
th e re fo re , a lit e r a t u r e search
was conducted to o b ta in a l l o f the spectroscopic data needed to d e te r­
mine the Dunham constants ao , a-], and a2 fo r as many o f the e le c tro n ic
sta te s as possible o f a ll o f the known diatom ic molecules and polyatom ic
m olecules.
Data on polyatomic molecules are, however, in almost every
case in s u f f ic ie n t fo r the c a lc u la tio n o f these constants.
The search
encompasses a ll o f the lit e r a t u r e lis t e d by Chemical A bstracts through
December, 1970.
The spectroscopic constants needed were Wg5 WgXg jOtgs
Bg , and r g and are lis te d in Appendix F fo r most o f the diatom ic mole­
cules f o r which the constants have been determined.
The Dunham and
M aclaurin expansion c o e ffic ie n ts , o r force constants, were ca lcu la te d
f o r these molecules using the SDX Sigma 7 computer on campus and are
lis te d in Appendix 6.
For those molecules having a s u f f ic ie n t ly larg e
number o f known e le c tro n ic states a least-squares analysis o f the
q u a d ra tic force constant vs. bond length re la tio n was c a rrie d out. De­
v ia tio n s o f p a r tic u la r force constant values from the “ expected" le a s tsquares values were then ca lcu la te d and compared w ith the higher-order,
force constants and w ith any m olecular p ro p e rtie s which m ight be ex­
pected to in flu e n c e them.
The re s u lts o f the least-squares c a lc u la tio n s
are summarized in Appendix H.
CHAPTER IV
Data Analysis
Hydrogen-hydrogen Bond
The hydrogen molecule is the sim p le s t m olecular system known and
w i l l be tre a te d in some d e t a il, e s p e c ia lly from the view point o f th e o ry ,
since i t is the only molecule fo r which accurate wave fu n ction s may be
obtained w ith o u t the use o f a large amount o f computer tim e.
A lso, cur­
re n t progress toward b e tte r quantum mechanical c a lc u la tio n s is leading
to in c re a s in g ly complex wave fu n c tio n s , making i t more and more d i f f i ­
c u lt to re la te them to q u a lita tiv e ideas about chemical bonding.
Only
f o r hydrogen does the development o f conceptual o r i n t u it iv e in te rp re ­
ta tio n s seem l i k e l y to have a d ir e c t o r apparent re la tio n to the wave
fu n c tio n s .
The only molecules in existence which have hydrogen-hydrogen bonds
are the hydrogen molecule and the hydrogen molecule io n s.
The re la tio n
between fo rce constant and bond length f o r the hydrogen-hydrogen bond,
as obtained from a le a s t squares a n a ly s is , i f is given in Appendix H and
shown g ra p h ic a lly in Figure I and Figure
2.
These graphs e x h ib it
some s c a tte rin g , as do a ll force constant vs. bond length p lo ts .
w ould, o f course, wish to know the cause o f th is s c a tte rin g .
One
A lso,
were the cause known, appropriate c o rre c tio n s might be made to give a
more n e a rly exact r e la tio n .
I propose now to c o rre la te the deviations
o f the in d iv id u a l sta te s from the "expected" force constants to other
-2 9 -
F ig u r e I .
H y d ro g e n -h y d ro g e n b o n d , lo n g ra n g e g rap h
-3 0 -
F ig u r e 2 .
H y d ro g e n -h y d ro g e n bond, s h o r t ra n g e g rap h
-3 1 -
p ro p e rtie s o f the molecule.
Because an o rdina ry least-squares analysis
weights the end po ints h e a vily i f they are somewhat is o la te d , the end
p o in ts o fte n w i l l no t be included in the analysis o f the d e via tio n s.
It
is noted th a t the t r i p l e t states have a p o s itiv e average d e v ia tio n ,
whereas the d e via tio n o f the corresponding s in g le t sta te s is negative,
in d ic a tin g th a t t r i p l e t sta te s tend to have higher fo rce constants than
s in g le t sta te s f o r a given e q u ilib riu m in te rn u c le a r distance.
No cor­
r e la tio n o f the de via tio n s w ith any o th e r pro p e rtie s is evident.
The sim p le st molecule w ith in the hydrogen group and the sim plest
molecule o f a ll is the hydrogen molecule io n ,
•
From Figure 2 we
n o tice th a t the force constant and bond length o f the hydrogen molecule
ion are very close to those o f many o f the e xcite d sta te s o f the hydro­
gen m olecule.
Thus, the re la tio n between force constant and bond length
would appear to be s t r i c t l y a fu n c tio n o f the number o f protons present
in the system ra th e r than o f the number o f ele ctron s present.
With th is
in mind, l e t us re-examine some models.
The sim p le st model which w i l l d u p lic a te the fo rce constant behavior
is the fre e -e le c tro n model mentioned p re v io u s ly .
Assuming th a t one elec-
in the molecule behaves as a p a rtic le -in -a - b o x w ith length R equal to
the in te rn u c le a r d ista n ce , the a d d itio n o f an e le c tro n to the molecule
im m ediately poses the problem o f where to put the e x tra e le c tro n .
If
the e x tra e le ctro n is added to the box both the k in e tic energy and it s
d e riv a tiv e must double, and the force constant must th e re fo re also
-3 2 -
double.
Since th is does not happen, we must conclude th a t there is not
room f o r more than one e le ctro n in the box.
The e x tra e le c tro n must
then occupy a region whose boundaries do not change s iz e w ith in te rn u c le a r distance.
This model presents an in te re s tin g physical p ic tu re .
I f an e le c ­
tro n is out o f the b in d in g o r fre e -e le c tro n region the s iz e o f the re ­
gion i t occupies can be expected to be determined la rg e ly by the nuclear
a ttr a c tio n p o te n tia l o f a s in g le atom, and th e re fo re is f a i r l y constant.
E le c tro n -e le c tro n repulsion presumably keeps the e le ctro n s from occupy­
in g the same region sim ultaneously.
This model,when assuming v a ria tio n
in the longitudinal d ire c tio n o n ly , p re d ic ts the fo rce constant re la tio n
K6 = 12.05Re" 4
(4 -1)
fo r the hydrogen-hydrogen bond, which gives very high fo rce constants
compared to the e xp e rim e n ta lly determined r e la tio n ,
Ke = 1-94 Re" 3' 9
.
(4-2)
Another possible explanation is th a t the force constant is d e te r­
mined by the n u cle a r-n u cle a r repulsion p o te n tia l.
Since the a d d itio n o f
an e le c tro n brings about no change in the n u c le a r-n u cle a r repulsion po­
t e n t ia l, the force constant re la tio n would not be a ffe c te d .
This expla­
nation re q u ire s , however, th a t the e le c tro n p o te n tia l have a n e g lig ib le
curvature a t the e q u ilib riu m d ista n ce , an u n r e a lis tic assumption.
A lso,
-3 4 -
the fo rc e constant must vary as the inverse th ir d power o f distance,
though the bare model is unacceptable, the explanation may be m odified
by assuming th a t e le c tro n -e le c tro n re pulsion terms, in conjunction w ith
the n u c le a r-n u c le a r re p u ls io n , give ris e to a common re p u lsive curve o f
the form AR n.
There is some ju s t if ic a t io n fo r b e lie v in g th a t th is to ­
ta l re p u ls io n curve remains somewhat constant from s ta te to s ta te , since
i t is determined in large p a rt by the non-bonding e l e c t r o n s \
In
th is case the force constant obeys the re la tio n
Ke = n(n + 1 )A /Rn + 2
(4-3)
i f the cu rva tu re o f the to ta l a ttra c tio n p o te n tia l is small re la tiv e to
the cu rva tu re o f the re p u lsiv e p o te n tia l.
The assumption o f n e g lig ib le
c u rv a tu re , though u n r e a lis tic as b e fo re , has some p r o b a b ility , since the
e le c tro n ic a ttra c tio n p o te n tia l in a molecule always has a .p o in t o f in ­
fle c tio n where the curvature is zero.
I t is also w e ll known th a t a re­
p u ls iv e p o te n tia ls o f the form AR~n n e a rly always give an e x c e lle n t f i t
to the a ctu a l p o te n tia l energy c u r v e s ^ ’ 38s3
seems to serve ra th e r w e ll f o r most molecules.
9
Thi s explanation
I t requires th a t the ad­
d itio n o f an e le c tro n not change the curvature o f the e le c tro n -e le c tro n
re pulsion te rm s; th a t i s , the e le c tro n -e le c tro n re pulsion terms r e s u lt­
in g from the a d d itio n o f an e le c tro n must have a lin e a r dependence upon
the in te r n u c le a r distance.
One im m ediately notices th a t according to
th is model a l l the e le c tro n repulsion terms fo r the hydrogenrhydrogen
-35must be lin e a r , since fo r Hg+ the to ta l re pulsion is the same as the
n u cle a r-n u cle a r re p u ls io n .
A lso, the c o n trib u tio n o f the nuclear-
nu clear repulsion to the force constant in H2+ is 3.87 mdynes/A,
whereas the actual force is only 1.56 mdynes/A, making i t necessary to
includ e the curvature o f the a ttr a c tiv e p o te n tia l.
In the case o f a ttr a c tiv e p o te n tia l having a s ig n if ic a n t in flu e n ce
upon the force co n stan t, the form o f the force constant re la tio n is no
longer a simple inverse power r e la tio n , and f i t t i n g o f th e o re tic a l re ­
la tio n s to e m p iric a l re la tio n s becomes s lig h t ly com plicated.
I f the
p o te n tia l energy curve is expressed as a do u b le -re cip ro ca l fu n c tio n ,
one may cast the force constant re la tio n in to a simple re c ip ro c a l form.
A d o u b le -re cip ro ca l fu n c tio n w i l l be tested on the ground s ta te o f ca r­
bon monoxide.
For the hydrogen-hydrogen bond one may ra th e r e a s ily c a lc u la te the
exact p o te n tia l energies i f the e le c tro n -e le c tro n coulomb repulsion and
s p in - o r b it in te ra c tio n are no t includ ed .
By comparing the actual
r e la ­
tio n and p o te n tia l energy curves w ith those a ris in g from the treatm ent
n e g le ctin g e le c tro n -e le c tro n re pulsion the in flu e n c e o f the e le c tro n e le c tro n re pulsion m ight be made c le a re r.
Exact p o te n tia l energies are
a v a ila b le fo r the various states o f the hydrogen molecule i o n ^ ^ .
These exact energies may be used as m olecular o r b ita l e n e rg ie s , and the
p o te n tia l energy curves o f the hydrogen molecule states may be calcu­
la te d by assuming the wave fu n c tio n o f the hydrogen molecule to be a
-3 6 -
simple product o f the exact one-electron wave fun ction s o f H0+.
L.
Once
the energies are ca lcu la te d they can be f i t t e d to an a n a ly tic a l expres­
sion from which the force constant may be c a lc u la te d .
The re s u lts are
compared g ra p h ic a lly w ith the e m p irica l data in Figure 3.
We observe
th a t the molecules no t having e le c tro n -e le c tro n repulsion e x h ib it much
la rg e r s c a tte rin g o f the force constants than do the actual molecules.
Consequently, the spin in te ra c tio n s and e le c tro n -e le c tro n coulomb terms
decrease the s c a tte rin g ra th e r than add to i t !
The very nature o f the
exact quantum mechanical s o lu tio n s , i t seems does no t allow an exact
re c ip ro c a l re la tio n between the force constant and the bond length.
I t is in te re s tin g in th is connection to examine to what e xte n t the s c a t­
te r in g o f the force constant values is re la te d to the shapes o f the po­
te n tia l energy curves.
That there is some s o rt o f connection is re a d ily
seen by observing the re la tio n o f the fo rce constant d e v ia tio n to the
constant a-j and by p lo ttin g the c o n trib u tin g one-electron p o te n tia l
curves and t h e ir second d e riv a tiv e s .
Figure 4 shows the c o rre la tio n
between the force constant d e v ia tio n and the constant a] f o r both the
e m pirical and ca lcu la te d data.
There is no obvious c o r re la tio n , though
the e m pirical and ca lcu la te d data seem to e x h ib it q u ite d iffe r e n t trends
In Figure 5 the ca lcu la te d p o te n tia l curves fo r
are drawn.
T h eir
shapes are seen to be q u a lita tiv e ly d iffe r e n t, e s p e c ia lly f o r those de­
riv e d from united atom sta te s having d iffe r e n t angular momenta.
Figure 6 the second d e riv a tiv e s o f the p o te n tia l curves o f Hg
p lo tte d .
In
are
From i t we see th a t the p r in c ip le c o n trib u tio n to the force
-37-
O
zP n
O
2Ptr
Jdcr O
F ig u re 3 .
H y d ro g e n -h y d ro g e n b o n d , t h e o r e t i c a l
-3 8 -
O
—
ce/c.
IS — H z
expL.
-2 .0 4 - if +
A
A
A
A
- 0. 4-
-02
A
0.4
O
d e v ia tio n
. of
A-
•Force,
Qd
O
-LZ
O
O
-ZO
4 -
o
A
'S 1
F ig u r e U .
F o rc e c o n s ta n t d e v ia t io n f o r h y d ro g e n
constant
-39-
Rydbergs
Bohrs
F ig u r e 5 .
P o t e n t i a l e n e rg y c u rv e s f o r h y d ro g e n
- 40“
-
1.0
- E
it
0 .0
-
--0 .5
1 .0
--1 .5
_ -
2 .0
0.0
i
F ig u r e 6
----- R
Bohrs
Second d e r i v a t i v e s o f th e p o t e n t i a l c u rv e s
-41 -
con stan t, a fte r the n u cle a r-n u cle a r re p u ls io n , is from the ground-state
o r b it a l.
The c o n trib u tio n from o th e r H2* - type o r b ita ls is seen to be
small in comparison and, stra n g e ly enough, o f about the same magnitude
as the change in the ground-state c o n trib u tio n .
No mention has been made o f the negative ion sta te s o f hydrogen.
Though experimental observations on these states are in c o n c lu s iv e ,
c a lc u la tio n s in d ic a te th a t the force constants and bond lengths are
o fte n n e a rly the same as those o f the corresponding sta te s o f hydrog e n ^ 7^.
A lso, the best ca lcu la te d values^48) fo r the e x c ite d states
o f the hydrogen molecule show about the same amount o f s c a tte r as do
the experimental data, w ith no apparent system atic d iffe re n c e s between
the ca lcu la te d and experim ental values. •
Helium-helium Bond
The re la tio n between force constant and bond length f o r the helium
helium system is shown g ra p h ic a lly in Figure 7.
The re la tio n is seen to
be q u ite d iffe r e n t from th a t o f the hydrogen-hydrogen system.
The o b v i­
ous explanation fo r th is change is th a t the hydrogen molecules, being
one and two e le c tro n systems, have no core, o r non-valence e le c tro n s .
The presence o f core e le ctro n s may be expected to have a la rg e in flu e n c e
upon the force constant re la tio n s because the in te rn u c le a r distance at
which re p u lsive forces become
strong
now
is
determined
by
the
repulsion o f the core e le ctro n s as w e ll as the n u c le a r-n u c le a r re p u l­
s io n .
The force constants f o r the helium -helium bond f i t the fre e -
e le c tro n model much b e tte r than do those f o r the hydrogen-hydrogen bond.
-42-
SINGLET
TRIPLET
F ig u r e 7 .
HeI iu m - h e Iiu m bond
-4 3 -
The helium -helium curve also shows some new r e g u la r itie s .
F ir s t ,
the t r i p l e t sta te s f o r each c o n fig u ra tio n have a lower force constant
to bond length r a tio than the corresponding s in g le t sta te s in a ll cases
except the 3d 6 s ta te s , in which the s in g le t and t r i p l e t have almost
id e n tic a l ra tio s .
Second, the d e v ia tio n o f s in g le fo rc e constant values
from the least-squares values c o rre la te s f a i r l y w ell w ith the Dunham
constant a ^, which measures the cubic c o n trib u tio n to the p o te n tia l en- '
ergy curve.
Figure 8 shows th is c o rre la tio n .
A high value fo r a-j is
associated w ith a r e la tiv e ly low value o f the force c o n sta n t, in d ic a t­
ing th a t high values o f the cubic fo rc e constant tend to depress the
q u a d ra tic fo rce constant.
I t is shown in Appendix I t h a t , i f one
assumes the v a lid it y o f equation (2-13) i
the k in e tic energy can be
expressed in the form
T = T0 + T1ZCR-3aT 4)
,
(4-4)
which shows a d ir e c t re la tio n between the cubic force constant and the
form o f the k in e tic -e n e rg y curve.
The He^
++
ion in the ground s ta te has not been observed, though
i t can be expected to be sta b le even, though its d is s o c ia tio n energy is
ne gative ^4^ .
I have ca lcu la te d th e .q u a d ra tic and cubic fo rce constants
from the th e o re tic a l data and fin d th a t the q u adratic constant is an­
omalously low and the cubic constant anomalously high.
Since the shape
o f the p o te n tia l energy curve is also somewhat anomalous, th is comes as
-44-
TRlPLET
SINGLET
-0 .1 5
F ig u r e 8 .
F o rc e c o n s ta n t d e v i a t io n f o r h e liu m m o le c u le s
-45no s u rp ris e .
Helium-hydrogen Bond
The ground s ta te o f the HeH m olecule, lik e the He2 ground s ta te , is
unstable.
Accurate experimental data are not a v a ila b le f o r th is system.
Exact c a lc u la t io n s ^ ^ show th a t the HeH+^ molecule has only one bound
s ta te , the 2pc s ta te .
The HeH+ molecule is known e xp erim e nta lly only
from mass spectroscopic studies. Therefore, the th e o re tic a l values o f
(51)
M ichelsv 1 are used. For the e x c ite d states o f the HeH molecule the
c a lc u la tio n s o f Michels and H a r r i s ^ ) have been used.
Because there
are only a few normal sta te s fo r the helium-hydrogen bond, none o f which
have been e m p iric a lly determined, a r e lia b le force constant vs. bond
length re la tio n cannot be e sta b lish e d .
However, i t is in te re s tin g to
see i f the re la tio n obtained from the c a lc u la tio n s is close to being a
mean o f the hydrogen-hydrogen and helium -helium re la tio n s .
Inspection
o f Figure 9 reveals th a t the three sta te s having s h o rt bond lengths are
very close to the hydrogen-hydrogen curve, w hile the s ta te w ith the long
bond length has an anomalously high fo rce constant, as seems to often be
the case w ith very long bonds.
L ith iu m -lith iu m Bond
Lith ium is the
element o f low est atomic w eight
c lo s e d -s h e ll core o f non-valence e le c tro n s .
which contains a
As would be expected, the
Pauli re pulsion between the core ele ctro n s prevents the form ation o f
strong s h o rt bonds.
The re s u lt is th a t the force constant vs. bond
-
O HeH
2Z
46
-
-4 7 -
length re la tio n is not e a s ily compared to the re la tio n s fo r o th e r bonds.
L ith iu m -h e liu m Bond
The lith iu m -h e liu m molecule has only very s lig h t ly bound states
having d is s o c ia tio n energies o f less than 0.07 eV.
and th e re fo re
w i l l n o t be considered.
Lithium -hydrogen Bond
The re la tio n between force constant and bond length f o r th is bond
is shown along w ith th a t f o r diatom ic lith iu m in Figure 10.
Again only
three s ta te s are known and these sta te s f a l l in a region apart from most ■
bonds.
B eryllium -hydrogen Bond
To date the only observed states o f diatom ic molecules containing
b e ry lliu m bonded to i t s e l f or to a lig h t e r atom are fo u r states o f BeH
and BeH .
They are shown in Figure 11.
Boron-boron Bond
S u rp ris in g ly , only two sta te s are known fo r the diatom ic boron mo­
le c u le .
We must th e re fo re use the re s u lts o f th e o re tic a l c a lc u la tio n s i f
we are to fin d out anything about the boron-boron force constant re la tio n .
F o rtu n a te ly , Bender and Davids on
have conducted a very thorough
th e o re tic a l in v e s tig a tio n o f the e le c tro n ic states o f Bg.
Using a curve­
f i t t i n g ro u tin e we can estim ate the e q u ilib riu m force constants and bond
lengths from th e ir re s u lts .
The values obtained are p lo tte d in Figure 12.
The two observed states are denoted by large c ir c le s .
Extremely larg e
“ 48-
F ig u r e 1 0 .
L ith iu m -h y d r o g e n and l i t h i u m - l i t h i u m bonds
-49-
Figure 11.
Hydrides of beryllium , boron, and carbon
-50-
-
12
O
- 10
Afe= M ,
TTo
TTo o
'z; o <p
1 .5
Figure 12.
1.6
Boron-boron bond
------- R0------ ^
1.8
1.9
o
-5 1 -
d e viation s from the least-squares values e x is t, making the le a s tsquare re la tio n dubious even though over fo r ty e le c tro n ic states were
used.
Again the cause o f the la rg e deviation s is o f p a r tic u la r in ­
te re s t.
In th is case the large d e viation s are shown c h ie fly by h ig h ly
enharmonic g s ta te s .
Boron-hydrogen Bond
Boron has not been known to bond to atoms lig h t e r than i t s e l f o th e r
than hydrogen.
Spectroscopic constants have been determined f o r three
sta te s o f boron h yd rid e , though the re s u lts are not known to be accurate.
The data in d ic a te the force constant re la tio n shown in Figure 11.
The boron-hydrogen system is the f i r s t th a t we have encountered
fo r which polyatom ic molecules are known. . He must now ask i f we can
expect the same c o rre la tio n s between fo rce constant and bond length f o r
polyatom ics.
P art o f the problem is th a t o f d e fin in g the force con­
s ta n t f o r polyatom ic molecules.
The constant which corresponds to the
q u a d ra tic fo rce constant o f diatom ic molecules is the diagonal bonds tre tc h force constant which appears in a norm al-coordinate a n alysis.
Since norm al-coordinate analyses have not been performed on many po ly­
atomic molecules, we are ra th e r lim ite d in the s e le c tio n o f polyatomics
we may use.
Thus, though the s p e c ie s .BHg and BgH^ are well-know n, there
is not s u f f ic ie n t in fo rm a tio n on them f o r the c a lc u la tio n o f meaningful
force constants.
The analysis o f force constants fo r polyatom ic molecules is generally
-5 2 -
too complex (a c tu a lly the amount o f data is too sm a ll) to allow the con­
s id e ra tio n o f anharmonic terms.
We have already seen th a t the enhar­
monic terms play an im portant p a rt in the re la tio n s .
We know th a t the
the consideration o f anharm onicity leads to force constants which show
considerably less s c a tte rin g than force constants determined w ith o u t
anharmonic c o rre c tio n s .
Carbon-carbon Bond
The graph o f force constant vs. bond length fo r the diatom ic carI
bon species is shown in Figure 13. Only one s ta te , the E Z s ta te , seems
to deviate s ig n if ic a n t ly from the p a tte rn , though the sta tu s o f the b3ir
s ta te is not known because o f i t s is o la tio n .
As expected, the E s ta te
is h ig h ly anharmonic, w ith a very high a-j value.
The b s ta te also has
an unusually high anharm onicity, making i t suspect as w e ll, though i t s
anharm onicity is o f a d iffe r e n c t character than th a t o f
the E s ta te .
The E s ta te also has the highest z e ro -p o in t energy and low est d is s o c ia ­
tio n energy o f any o f the s ta te s .
The E s ta te has not been assigned an
e le c tro n c o n fig u ra tio n , as there are several p o s s ib ilit ie s .
The sim p le st polyatom ic molecule w ith a carbon-carbon bond is acet­
ylen e.
A carbon-carbon s tre tc h in g force constant wich includes anhar­
monic c o rre ctio n s has been determined by Suzuki and O verend^3^.
I t is
s lig h t ly la rg e r than the force constant expected from the graph o f d ia ­
tomic force constants.
The high value may be the r e s u lt o f the neglect
o f re la x a tio n e ffe c ts th a t is inh ere nt in the treatm ent o f polyatom ic
-53-
O
Figure 13.
Carbon-carbon bond
-5 4 -
molecules.
A lso, the exponent value fo r the diatom ic re la tio n appears
to be low and m ight, th e re fo re , give low force constant values fo r very
small in te rn u c le a r distances.
Ethylene has also been stu d ie d c a r e fu lly by ta k in g anharmonicity
in to a c c o u n t T h e
carbon-carbon s tre tc h in g force constant is con­
s id e ra b ly hig her than the least-squares value.
This time the only ap­
parent explana tion is th a t the d iffe re n c e is due to the re la x a tio n e f­
fe c ts discussed a t the end
the theory se c tio n .
For molecules o f
which anharm onicity has not been considered, the agreement w ith the
diatom ic values should g e n e ra lly be b e tte r because the e r ro r w i l l com­
pensate f o r the d iffe re n c e due to the re la x a tio n e ffe c t.
fo rce constant determined fo r
w ith the dia tom ic re s u lts .
Thus, the
is in f a i r l y good agreement
The force constant fo r ethane^58^, which
includes a c o rre c tio n fo r anharm onicity, also appears to be high.
Q uite re c e n tly a new determ ination has been made o f the force
constant o f the carbon-carbon bond in g ra p h ite ^59^ , which afford s us
our f i r s t chance to compare our re s u lts w ith those fo r a s o lid .
When
compared w ith the e m p irica l curve, the force constant determined fo r
g ra p h ite is seen to be rid ic u lo u s ly low!
Close perusal o f the a r t ic le
re ve a ls, however, th a t the force constant was c a lc u la te d d ir e c tly from
the observed Raman v ib ra tio n a l frequency; thus, the fo rce constant cor­
responds d ir e c t ly to the normal mode and is th e re fo re not the force con­
s ta n t f o r a simple bond s tre tc h , but ra th e r a bond s tre tc h in combination
-55w ith adjacent bond compressions.
That i s , the force constant represents
an extreme form o f re la x a tio n o f e le ctro n s from one bond to another, re ­
s u ltin g in a fo rce constant considerably lower than th a t fo r a diatom ic
molecule.
The force constant f o r diam ond^^, which was determined w ith
a valence fo rce f i e l d ra th e r than d ir e c t ly , is in good agreement w ith the
diatom ic curve.
Carbon-boron Bond
Diatomic boron carbide was no t known to e x is t u n t il 1964, and there
s t i l l are no spectroscopic data a v a ila b le f o r th is species.
Kouba and
O h r n ^ ^ have re c e n tly attacked th is problem from a th e o re tic a l basis
and c a lc u la te d th e o re tic a l spectroscopic constants f o r a ll the possible
sta te s o f diatom ic boron carbide.
T h e ir data are used to obtain the
t o t a l l y th e o re tic a l graph o f force constant vs. bond length displayed
in Figure 14.
Several o f the states have ra th e r abnormal force constants.
The fo rc e constants o f the states having large in te rn u c le a r distances
c o rre la te ra th e r po orly w ith distance.
Carbon-hydrogen Bond
Simple compounds o f carbon w ith b e ry lliu m , lith iu m , and helium
not known.
are
In the case o f the carbon-hydrogen bond the bond distances
fo r the d ia tom ic sta te s are a ll la rg e r than those f o r the polyatom ic mol­
ecu les, as is shown in Figure 11.
Rather than make a comparision between
the dia tom ic and polyatom ic force constants fo r th is bond, we are perhaps
b e tte r o f f using them tog eth er to get a force constant re la tio n which is
-5 6 -
O
Figure Ilu
Carbon-boron bond
-5 7 -
v a lid over a wide range o f distance.
f o r the polyatom ics^
Again the force constant values
appear to be s lig h t ly high and are cor­
rected f o r anharmoni ci ty .
N itro g e n -n itro g e n Bond
Here we encounter our f i r s t bond f o r which an e m p irica l force
constant vs. bond length re la tio n has been re'ported^23^.
The reported
r e la tio n was determined by using force constants c a lc u la te d from the
frequency o f the fundamental 0 - I tr a n s itio n ra th e r than from the
e q u ilib riu m fo rce constants used here.
Figure 15 shows the re la tio n
and fo rc e constants as determined by the author.
The polyatom ic force
constants f a l l below the least-squares lin e , except fo r
How­
eve r, the fo rce constants o f a ll o f the polyatomics having th is bond
are d i f f i c u l t to analyze and have been determined by n e g le ctin g the
anharm onicity; th e re fo re , the fo rce constants can be expected to be low.
Remembering th a t the e x tra atoms on polyatom ic molecules would be ex­
pected to increase the fo rce constant o f a bond, i t comes as no s u rp rise
th a t N2H^+4", which has s ix e x tra atoms, has a r e la tiv e ly high force con­
s ta n t.
Carbon-nitrogen Bond
The force constant vs. bond length graph in Figure 16 shows two
ra th e r abnormal s ta te s .
One s ta te has such a high force constant th a t
one must s e rio u s ly doubt the v a lid it y o f the spectroscopic data from
which i t was determined.
A reana lysis o f the data would c e rta in ly be.
-5 8 -
Figure
Nitrogen-nitrogen
-59-
HCN-
Figure 16.
Carbon-nitrogen and boron-nitrogen bonds
-6 0 -
i n o r d e r , as often the spectrum o f one molecule is mistaken f o r that o f
another.
The other abnormal s ta te i s , as might be expected, abnormally
harmonic.
Schaeffer and Hei I
have calculated t h e o r e t i c a l spectro­
scopic constants f o r many o f the states o f CM.
The t h e o r e t i c a l force
constant r e l a t i o n shows an extremely large s c a tt e ri n g .
Boron-nitrogen Bond
The two known states o f diatomic b o r o n - n i tr id e have been included
i n Figure 16.
They e x h i b i t force constants which are lower than the
expected carbon-nitrogen values; t h u s , the usual p e ri o d ic trends f o r
the fo rc e constant r e la t i o n s i s i n t h i s case reversed.
Nitrogen-hydrogen Bond
The nitrogen-hydrogen r e l a t i o n i n Figure 17 covers only a very nar­
row range.
None o f the diatomic states f o r t h i s bond seem to e x h i b i t
any abnormal behavior.
Oxygen-oxygen Bond
Figura
18 shows the various states o f diatomic oxygen.
Again no
marked i r r e g u l a r i t i e s are present.
Nitrogen-oxygen Bond
This bond may vary considerably i n s t i f f n e s s and bond length, as is
shown in Figure 19.
Many simple polyatomic molecules contain the n i t r o ­
gen-oxygen bond, so t h a t there has been considerable i n t e r e s t in the
bond's c h a r a c t e r i s t i c s .
Ladd, e t. a l .
in 1966 surveyed the known
diatomic and polyatomic states and calculated a force constant vs. bond
-61-
i \NH
L S a (O H )
1 .2
Figure 17.
------ Re
Nitrogen-hydrogen and oxygen-hydrogen bonds
------Re
Figure 18.
Oxygen-oxygen bond
-63-
Figure 19.
Nitrogen-oxygen bond
-6 4 -
length r e l a t i o n .
The data ava ila ble has not .changed much since then.
Carbon-oxygen Bond
Ladd, e t . a l .
bond i n 1964.
( 21 )
studied the force constants f o r the carbon-oxygen
They were the f i r s t to recognize t h a t the force constant
o f a bond i s f a i r l y i n s e n s i t i v e to the h y b r i d i z a t i o n .
The s c a tt e r i n g
o f the force constants o f the various states p l o tt e d i n Figure 20 f o l ­
lows the usual trends.
The ground sta te o f carbon monoxide has been exte ns iv el y studied
and th er ef ore was chosen as a t e s t case f o r the determination o f a
double-reciprocal fun ction to represent the po te n ti a l curve and force
constant r e l a t i o n .
The p o te n ti a l curve is best represented by a func­
t i o n having exponents equal to 2.95 and 1.85 f o r the rep ulsive and
a t t r a c t i v e parts re sp e c ti v e ly .
I f the a t t r a c t i v e p o rti on doesn't con­
t r i b u t e to the curvature at the minimum, then the force constant r e l a ­
ti o n must vary as
remains constant.
This follo ws also i f the a t t r a c t i v e portion
—6
Since the force constant varies as R , the use o f
the common repulsive po te n ti a l i s not completely s a t i s f a c t o r y .
Boron-oxygen Bond
Three states are known f o r diatomic boron oxide, and they are repre­
sented in Figure 21.
•
Beryllium-oxygen Bond
Four states are known f o r diatomic be ry ll iu m oxide, and they are com­
pared wi th the boron-oxygen bond in Figure 21.
The usual trend toward
-65-
Figure 20.
Carbon-oxygen bond
-6 6 -
Figure 21.
Boron-oxygen and beryllium-oxygen bonds
-6 7 -
reduction o f the force constant as the atomic number increases is e v i ­
dent.
Hydrogen-oxygen Bond
E l e c t r o n i c states o f the OH molecule are represented i n Figure 17.
The least-squares f i t f o r the force constants is extremely i n t e r e s t i n g
in t h a t i t pre dict s a curve which i s inc o ns is te n t with the curves f o r
the o t h e r hydrides.
We must be c a r e f u l , th e n , not to always believe
the least-squares r e s u l t s , as they are constrained to fo l l o w an expon­
e n t i a l fu n c ti o n having a constant exponent.
The data f o r OH in d ic a te
c l e a r l y t h a t a d i f f e r e n t form o f function could give a much b e t t e r f i t
to the data.
We have seen before t h a t long bonds tend to have a high
for ce,constants r e l a t i v e to the least-squares curve; again, the use of
a d i f f e r e n t form o f fun ction would seem appropriate i f a r e l a t i o n v a l i d
over a l arg e range is to be found.
Thus, a r e l a t i o n o f the form
Ke = ARe" 1 + BRe" n
(4-5)
would be s u i t a b l e from an experimental p o in t o f view as well as from a
t h e o r e t i c a l p o i n t o f view.
Fluorine Bonds
The force constants and bond lengths f o r the f l u o r i n e - c o n t a i n i n g
diatomic molecules are p l o t t e d in Figure 22.
Again a reciprocal r e l a ­
t i o n between atomic number and force constant appears.
Here, though,
the r e l a t i o n breaks down f o r the f l u o r i n e - f l u o r i n e , n i t r o g e n - f l u o r i n e .
-6 8 -
O
ij£ ___________ ___ fe— ^__________ KU
Figure 22.
Bonds containing fluo rin e
-6 9 -
and e sp e c i a l l y oxygen-fluorine bonds, f o r which very few data are a v a i l ­
able.
Thus one remains uncertain as to whether any meaning should be
attached to the apparent v i o l a t i o n o f t h i s " r e c i p r o c i t y " ru le.
CHAPTER
V
Conclusion
This study, though f o r the author a long and sometimes arduous task,
merely scratches the surface o f a f i e l d o f study which has in recent
years been sadly neglected.
The force constant r e l a t i o n s f o r bond com­
binations in c l u d i n g elements not in the f i r s t row o f the pe riod ic chart
have not y e t been determined.
Many bond properties have not been r e l a ­
ted with the force constant behavior, though the existence o f such r e l a ­
ti on sh ip s i s very probable.
tions need to be t r i e d .
A lte rn ate fun cti on al forms f o r the r e l a ­
The f r e e - e l e c tr o n theory, f o r example, lends
i t s e l f p a r t i c u l a r l y well to the equation
Ke = A(Re - d ) - m
where m i s set at fo u r.
,
The parameter d could be used as a measure o f
the sum o f the i o n i c r a d i i o f the atoms,
d may be obtained from the
usual exponential fun ction by use o f the equation
d = (-mRe/n + RfJ
,
where n i s the exponent o f the normal exponential r e l a t i o n .
The compari­
son o f diatomic force constants wi th .those, o f polyatomic molecules could
be c a r ri e d f u r t h e r i n cases where the force f i e l d s o f the polyatomic mol­
ecules are well known.
Also, the manner in which force constants change
as ca lc u la ti o n s are changed is o f increasing i n t e r e s t to t h e o re ti c i a n s .
- 71-
A universal formula f o r the force constant vs. bond length r e l a t i o n has
not been found which works appreciably b e t t e r than those presently in
use.
The s c a t t e r o f force constants o f i n d iv i d u a l states from one an­
other was shown to be l a r g e l y systematic ra th er than random.
The
changes brought about by changing the number or d i s t r i b u t i o n o f protons
in a system were found to e x h i b i t d e f i n i t e but not absolute trends.
APPENDIX A
POWER SERIES EXPANSION OF THE POTENTIAL CURVE
MacTaurin expansion
(3)
( 2)
E(R)
Ee
T
(R
Re)2 + T
(4)
( R - Re) 3 + 5 r
(A-T)
( R - Re) 4 +
Ee v y = quadratic force constant, Kg
(3)
E x y = cubic force constant, L
e
e
= q u a r t i c force constant, M.
Dunham or reduced-coordinate expansion
R - R.
2
2
E(r) = aQr (I + a^r + a^r +
Ke
ao = 2 # Re
2
L
aI
3K
(A-2)
M
Re
a2 “ 12K
, 2
e
APPENDIX B
EXPANSION COEFFICIENTS IN TERMS OF SPECTROSCOPIC CONSTANTS
Dunham c o e f f i c i e n t s :
a
o
a
I
(B - I)
*e«e
-I
B a1 2
a2
(B-2)
2V e
(B-3)
4
Maclaurin c o e f f i c i e n t s :
to 2
we
(B-4)
L
e
3K " B6
dKe
n>E
a,8
i
no
2Be Re 2
(B-5)
Re Be 2
6m 2
e
Cl)
I CO
iSf"
I o=®
5 a /
2V e
4
3Be
I •
(B-6)
APPENDIX.C
ERRORS INCURRED IN THE USE OF AG(1/2) AND Rq
AGO/2 ) ~ we ' 2wexe
we
™
2wExB
(C -I )
we
i s well known^ ^ , and
Ro/Re = ^Be/Bo)1/2
(C-2)
follo ws from equation (2 -4 ). I f ae/Be «
I substitution of
B0 = Be - ( l / 2 ) a e
(C-3)
i n t o (C-2) y i e l d s
R0 ZRe * I + =-e /4 B e
(C-4)
.
Likewise, assuming wexe/ we to be sm all , s u b s t i t u t i o n of
K = Const, w2
(C-5)
i n t o (C -I ) gives
K(AG)/Ke = I - 4wexe/we
.
(C-6)
I f the r a t i o s wexe/we and OieZBg are f i x e d , then the use o f the approxi­
mate constants w i l l give a force constant r e l a t i o n having the same ex-,
ponentia! dependence as the e q u i l i b r i u m r e l a t i o n .
APPENDIX D
DERIVATION OF EQUATION (2-16)
According to the v i r i a l theorem
RE' + E = -T
.
(D-I )
D i f f e r e n t i a t i o n o f (D -I ) gives
RE" + 2E' = - T 1 .
(D-2)
M u l t i p l i c a t i o n o f (D -I ) times nRn “ 1 and o f (D-2) times Rn and
adding gives
-(RnT ) V R n " 1 = R2E" + (n + 2 ) RE' + nE
a f t e r c o l l e c t i n g l i k e terms.
(D-3)
APPENDIX E
FORCE CONSTANT RELATIONS FOR THE FREE-ELECTRON THEORY
Suppose th a t
T = AR'2 = ARe" 2
f o r R near to Re-
( E - I)
Then
(dT/dR)R = -2ARe' 3 .
(E-2)
6
S u b s t it u t i o n o f (E-2) i n t o the v i r i a l r e l a t i o n
ReKe = ' ( r ) Re
(E" 3)
gi ves
K = 2AR " 4
e
e
.
(E-4)
For a one-dimensional p a r t i c l e in a box
T = (h2/8m)R'2 / ,
(E-5)
2
where h /Sm is constant and R is the length o f the box.
I f the length
R i s taken as equal to R , then an inverse fourth order dependence o f
the force constant on Re is implied.
Likewise, assuming a three-dimen­
sional box having constant l a t e r a l dimensions give the r e l a t i o n
-7 7 -
T = (h^/8m)Re~^ + Constant,
which gives the exact same force constant r e l a t i o n as the one-dimen­
sional box, since the r e l a t i o n is obtained by d i f f e r e n t i a t i o n o f the
expression f o r k i n e t i c energy.
I f the l a t e r a l dimensions are i n pro­
po rtion to the length one obtains
T = (h2/8m )(l + [ReZ i y 2 + [ReZ y 2JRe" 2
where Rq and R^ are the l a t e r a l dimensions o f the box.
(E-7)
Again one obtains
an inverse fo u r t h - o r d e r dependence f o r the force constant.
I now introduce as a theorem the statement t h a t any box which main­
tains a constant shape w i l l give an inverse fou rth order dependence f o r
the force constant, no matter what i t s i n i t i a l shape i s .
I f we can prove
2
t h a t the energy i s proportional to 1/h , where h i s a one-dimensional
scale f a c t o r which measures the change i n size o f the box along each d i ­
mension, then the theorem w i l l f o l l o w from the argument used at the be­
ginning o f t h i s section.
An increase i n size o f any box can be compen­
sated f o r by changing the coordinate system by a scale f a c t o r ,
X1
= x/h
,
(E-8)
so tha t i n the new coordinate system the box appears not to have changed
si ze.
By transforming coordinates the po te n ti a l energy and boundary con­
d i t i o n s are e f f e c t i v e l y unchanged.
The only change i s i n the form o f the
-7 8 -
Laplacian operator, which becomes m u l t i p l i e d by the constant 1/h .
Re-
w r i t i n g the wave equat ion, we obtain
V2IMx1) = Ii2K E X x 1)
,
(E-9)
which i s i d e n t i c a l to the o r i g i n a l wave equation,
V2iK x ) = KEiMx) ,
o
except t h a t the energy E i s replaced by h E1.
E'
E
which was to be proven.
(E-IO)
Thus
(E -Il)
A P P EN D IX
STATE
BE
WE
WEXE
F
AE
BE
-R E F
M2
XlS
X2$
A3S
BlS
BlSt
BlS '
C3P
ClP+
c ip D lPDIP*
D3P
D4P +
D4PD5PD5P+
ElS
E3S
IS P ;ip J3DJlD N3P
,7 4 1 6
1,0600
,9 8 8 7
1,2925
1,1380
1,1110
1,0376
1,0220
1,0310
1,0420
1,0140
1,0496
1, 03 00 .
1,0580
1 ,0 4 3 0
1* 05 40
1,0120
1,1070
1*0 7 00
1,0690
1,0540
1,0540
1*05 70
4 4 o l,2 0
2297,00
2664,80
1375,40
2074,90
2197*50
2465,00
2420,40
2431*20
2361,50
2358,20
2371,60
2329,70
2330,20
2323,60
2316,30
1784,40
2195,80
2253,55
2259,15
2345,26
2341,15
2322,00
21,340
3,0620
1,4000
62,000
71,650
1,6710
20,420
,9 2 0 0
13,400
1,4900
2 o3500
68,100
61,400
1,4250
5 5 , 6Q0
1,7600
60*900
1,8300
1 ,9 6 0 0
69,100
68,500
2,0000
66,270
1,5450
63,000
,6 4 0 0
63,300
1,1100
1,4500
62,100
63,900
-,5 400
48,110.
»6760
65,800
1,5150
1,5060
67,050
78,410 . 1,5800
6 6 o560
1,6900
63,200
1,7200
62,900
1,3000
6 0 » 853o
29,8000
34,2160
20o 035o
25,8000
27,1000
31,0700
32,0000
31,5000
30,8000
32,5000
30,3640
31,6000
29,9000
30,8000
30,1000
16,3690
27,3000
29,2200
29,2600
30,0850
30,0800
29,9300
68
64
64
64# 6
64# 6
64# 6
64# 6
64/6
64# 6
64# 6
64# 6
64
64# 6
64# 6
64# 6
64# 6
67
I
65
65
65
65
I
7,7870
7,4030
7,0680
7 , 365q
7,1560
7,2300
7,7100
7,4470
7,0050
7,3410
69*7
69 *7
69-7
6 9 -7
6 9 -7
69-7
69 =7
6 9 -7
69 =7
6 9 -7
HE2
AlS
B lP
CIS
DlS
FlP
FlD
A3S
B3P
C3S
D3S
I »0400
1,0660
1,0910
1,0690
1,0840
1,0790
1*04 50
1*06 30
1,0960
1* 07 10
1861,30
1765,80
1653,40
1746,40
1670,60
1706,60
1809,90
1769,10
1583,80
1728,00
35,100
34,390
41,000
3 5 *5 4 0
40,030
35,060
38,900
35,020
52,700
36,130
,2 2 8 0
,2 1 6 0
,2 4 5 0
»2180
,23 50
,22 50
,2 4 4 0
,2 2 0 0
,31 00
«2240
-8 0 -
state
F3S
F3P
F3D
X2S
RE
I o09.10
I 0 O8 6 O
1,0790
1,0810
WE
1635,80
1661,50
1706,80
1698,50
WEXE.
44,410
44,790
35,100
35,250
REF
69 *7
69*7
69 n 7
69=7
AE
,2 4 6 0
,2 3 3 0
,22 90
,2 2 4 0
BE
7,0710
7,1360
7,2300
7,2110
,0 0 7 0
0 OOS4
,0 0 8 0
,6727
,4 9 7 4
,5572
.,2130
«*,0783
,9 8 6 0
7,5130
2*8190
3,3830
,3 0 0 0
,32 90
,29 30
,1 2 5 0
10,3080
10,4700
10,8000
7,1830
I
I
I
I
,0 1 4 0
,0 1 1 0
1,2120
1,1600
I
I
,4 1 2 0
,8 3 5 0
,4 8 5 0
,4 3 2 0
,3 9 0 0
1 2 *0 2 1 0
12,2950
12,3390
1 2 *4 1 0 0
12,7570
Liz
X lS
AlS
B lP
2,6720
3,1080
2,9360
351,43
255,46
269,69
2,592
1,574
2 0 744
I
I
I
UlH
XlS
AlS
BlP
1,5950
2,5960
.2,378 0
1405,65
234,41
215,50
23,200
28,950
42,400
I
I
74
BEH
X2?
A2P
XlS
AlS
1,3430
1,3330
1,3120
1,6090
2058,60
2087,70
2221,70
1476,10
35,500
39,800
39,800
14,800
B2
X3S
A3S
i ,5 8 9 0
1,6250
1051,30
937,40
9,400
2,600
.
BH
xis+
AlP
BlS+
CIS +
C ID
1,2324
1,2190
1,2160
I # 2130
1,1960
2366,90
2251,00
2399,90
2474,70
2610,00
49,390
56,670
69,520
54,420
46»620
75
75
75
75
75
-81 -
STATE
RE
NE
WEXE •
AE
BE'
REF
1*89 80
1*61 60
1*7 8 30
1* 83 30
1*79 30
1*632 0
1* 49 80
1* 75 30
1.1920
1*448 0
1 * 87 70
I #74&0
76
76
76
76
76
76
76
76
76
77
78
78
CS
XlS
BlP
ClP
DlS
ElS
X3P
A3S
A3P
B3P
F3S
CE
CE=
1*24 25
1*318 4
1* 25 52
1*23 78
1*2 5 17
1*31 19
1*3 6 93
1*26 60
1*53 50
1*39 30
1*223,3
1*26 82
18 54 *7 0
16 08 *3 0
1809* 10
18 29 *6 0
1671* 50
1641.30
1470*40
1788.20
1106* 60
1360* 50
1968.70
17 81 *0 0
13 *3 40
12 *0 80
15 *8 10
13 *9 70
4 0 *0 2 0
11*6 70
11 *1 90
16 *4 40
39.260
14.800
14 *4 30
1 1 *5 80
*0170
*0170
*0180
*0200
*0420
*0170
*0160
*0160
*0240
*0400
*0180
*0170
BC
ES*
ES I
EsE
ES3
EPl
EPE
EP3
EDI
EDE
4S1
4SE
4SE*
4S3
4S5
4P1
4P3
4P6
4F1
6P1
1*693 0
1*6870
1*43 80
2*4380
1*54 40
1*85 90
1*599 0
1*68 70
2*3370
1*66 50
1*63 60
1*672 0
2*2570
2*3920
1*460 0
1*818 0
2*3230
1* 71 30
1*703 0
*0192
8*490
949*00
1*0340.
9l7*00
13*2 40
*0l54
I*0 4 i0
15 04 *0 0
23*640
$0192
1* 43 20
680*00
5*370
*0003
»4990
»0172
11 94 *0 0
17*2 40
1 * 24 30
4*980
883*00
«0028
*8 580
6 4 *9 2 0
1339* 00
»0316
1 * 15 90
»0134
944*00
11*3 80
1* 04 10
390*00
15 *8 70
=»0270
*5420
991*00
10*3 90 . ■*0128
1*069 0
1032* 00
11*6 10
*0120
1*1 0 70
8 8 8 . 0 0 ' 6 2 *1 4 0
»0414 . 1*05 90
723.00
«0006
3.160
*5 810
1103* 00
6*340, =»0016
*5 180
1441* 00
30.780
* 02 4 6
1,3890
*0188
702*00
17*8 60
o.89 6 q
4.330
668*00
=«0023
. «5490
*0151
920*00
1.0100
13*9.70
9,790
973.00
«0121
1* 02 10
61 '
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
CH
XSP
AED
CES +
1*1 2 00
1*1020
1*114 0
2858*50
2930*70
2840*20
63*000
96 * 650
2 5 *9 6 0
*5300
«6970
,7 1 8 0
14.4570
14,9360
1 4 *6 03 0
79
79
79
-8 2 -
STATE
*S
*P
RE
I »1309
1*23 45
1,2320
WE
2739o70
18 65 *3 5
2075*50
WEXE
»000
15,850
76,300
AE
*4917
,9414
,6 2 0 0
BE
14«1770
11,8990
11,9370
REF
80
80
81
1,7600
*0170
«0170
*0180
*0170
»0080
*0200
1*7980
*0170
»0180
»0190
*0200
»0195
*0060
«0190
I o9 9 9 q
1,4800
1*56 40
1,6170
1,4980
1,1530
1,9200
I «4550
1,4740
1,6370
1,8250
1,9320
2*0830
«9000
*7 800
82
83
83
83
83
84
84
82
83
83
83
I
I
I
I
»0173
*0175
,0221
,00 64
,01 87
»0191
,01 88
,03 20
*0020
■ 1 ,8 9 9 6
1,7165
1,9700
1,4870
1,3834
1,6770
1,8964
1,9030
1,4030
I
I
I
85
86
87
88
88
88
,67 00
.,7320
»6050
16,7800
16,6800
16,7300
89
89
90
NS'
XlS
AlS
AlP
AlPa
WlD .
8 ' IS
CHS
A3$
B3S
B3P
C3P
X2S
B2S
SG+
3DG
1,0970
1,2750
1,2200
1,2200
I *2680
I «4450
1 ,1 2 0 0
1,2860
1,2780
1,2130
1 * 14 90
1*11 60
1*0750
1 ,6 1 0 0
1*7500
2359,61
1530* 25
1666,05
1694,21
1559,24
762,90
2193,10
1460,60
1516,88
1733,40
2047,20
2207,19
2419,84
1135,00
684,00
14,456
12,070
13,550
13,950
11,890
.3,950
22,070
13 ,8 5 1
12,180
14,120
2 8 *4 5 0
16 *1 36
23,190
6,300
20,900
CN-
X2S
A2P
B 25
E2S
F2D
+ BlP
+ AtS
+FlS
+ CIS
1*1718
1*23 27
1* 15 06
1*3 2 10
1*371 0
1,2470
1*172 7
1*17 10
1*36 30
2068*70
1814,43
2164,13
1681* 43
1239,50
1688,35
2033,05
2670*50
1265,00
13,144 .
12,883
20,250
3,600
12,750
15,120
16,140
46,900
11,000
NH
AID
A ID *
B lS +
1,0330
1,0370
1*03 50
3314,00
3 3 62 * 00
3347,00
63,000
16,000
70,700
-83-
STATE
BIS#
X3So
X3S«
RE
1*03 50
1*03 70
1* 04 10
WE
3396*00
3 2 03 * 20
3300*00
WEXE
13.000
7 8 *3 0 0
20,000
AE
* 7120
*6480
*7600
1 6 *7 30 0
16 *6 70 0
16,5600
*0160
*0110
*0i?0
*0180
»0205
*0200
*0190
»0160
*0220
1,4457
,8190
I ,4 2 6 4
1*4 0 04
* 826o
I , 692q
1*0 6 17
1 * 10 47
1,2873
*0204
,01 78
*0164
«0116
*0189
»0210
»0160
*0182
1,2520
1,7050
1*9 9 52
1 * 07 60
1 * 17 70
1*3320
1 * 99 20
.1,9863
95
I
I
I
I
96
I
I
,02 05
* 0261
*0196
»0175
»0187
«0195
*0420
*0168
*0181
*0200
1 * 61 04
1*9 6 12
1,9533
1,9313
1*3 4 53
1*6911
1 * 98 60
1*3 0 99
1,2848
1,8890
97
97
97
97
97
97
97
97
97
97
be
REP
89
91
89
92
X3S
B3S
AID
B lS
ISU
X2P
ASP '
A4P
B4S
1*20 74
1* 60 40
1*215 5
1* 22 67
1*59 70
1*116 0
1*4090
1* 38 10
1* 28 00
1580* 36
676*80
1 5 09 * 30
14 32 *6 9
650*50
19 03 *8 5
895*60
1035,69
1196.77
12,070
9.293
1 2 *9 00
13 *9 50
1 7 *0 40
16.180
13*4 00
1 0 *3 90
17*0 90
I
92
I
I .
93
1 ,9 4
I
I
I
N0
G2S
X2P
A2S
■B2P
B»2P
B 12D
D2S
E2S
1*34 26
1*15 08
1*063 7
1*44 80
1*38 50
1*3 0 20
1*06 50
1*0661
1085*54
1903*85
2371,30
1036.96
1 0 38 ,4 1
12 17 *4 0
2327*00
2373.60
11 *0 §0
1 3 *9 70
14 *4 80
7'* 603
7 * 6Q3
15 *6 10
2 3 *0 0 0
1 5 *8 00
CS
AlP
B lS
CIS
is
A3S
A3P
B3S
030
53S.
J3S
1*23 56
1*119 7
1* 12 19
1*1 2 83
1*3 5 19
1*20 58
1*11 30
1*3 7 00
1*3834
1*14 10
1515.40
2112*70
2175.92
2169,82
1230* 65
1743.55
2188.00
1152.58
1113.67
2196.00
17,200
15,220
14 *7 60
13 *2 94
11 *0 13
14 *4 70
3 0 *0 0 0
7* 28 1
9*596
15 *0 00
•
-84-
STATE
XSS
ASP '
BSS
SD
RE
Io llB S
I «2439
I »1688
1*34 60
WE
2214.24
1562,06
1734.18
1144,00
WEXE
15,16413,532
27,927
33,300
AE
,0 1 9 0
,0 1 9 4
,0 3 0 3
,0 2 4 0
BE
1.9772
1,5894
1,7999
1,3570
REP
97
97
97
97
,0 1 6 5
,0 1 9 6
,0 1 8 0
1,7800
1,4132
1,5030
I
I
I
,0 1 9 0
,0 1 6 3
• 0154
,01 60
1,6510
1 ,3 6 6 1
1,5760
1,3970
I
I
X98
,7 1 4 0
,8 0 7 0
,0 7 8 0
,8 3 4 0
,7 6 4 0
18,8710
17,3550
4,2470
13,6620
16,7920
oOUO
1,0800
I
,0 0 9 7
1,1040
101
BQ
XSS+
ASP
BSS+
1,2050
1,3520
4,3110
1885,44
1260.70
1280,30
11,770
11.160
10,070
BEB
xis
AlP
BlS
A3P#
1,3310
I ,4 6 3 0
1,3620
1,4480
1487,32
1144,24
1370,82
1213,00
11,830
8 , 4 15
7,745
.8,600,
BH
XSP
ASS
C3S+
*3P
*35 =
,9 7 0 6
1,0120
2,0450
I «1400
1»0290
3735.20
3180,60
1232,90
2145,20
3077,30
82,810
94,930
19,100
84,500
58,600 .
I
I
99
100
100
. ft
BlP
1,2820
1139,80
9,700
BF
2P<*
1,3210
1211,00
5,150
-85-
STATE
RE
WE
WEXE
AE
BE
REF
,0 1 5 0
,0 1 4 5
1*20 60
1,2377
102
102
*0184
00245
o0250
I , 417o
i»7250
1,3200
103
103,
105
,0 1 6 0
,0 1 8 0
,01 78
,0 1 9 4
,01 74
*0169
o 1580
,02 00
,01 76
1,5170
1*42 26
1* 65 90
1* 62 38
1 ,6 3 8 1
1*62 90
1* 41 30
1*63 85
1*651 8
106
106
106
106
106
106
106
106
106
,01 76
,0 1 7 5
*0140
1 * 48 90
1*4 2 00
1,5700
107
107
108
,7 9 8 0
«0173
20*9560
4*0263
109
HO
NF
X3S
BlS
i «3170
I »3000
1141,37
1197*49
8*990
8,640
CF
X2P
A2P
B2S*
!« 2 6 7 0
!« 1 4 8 0
!#3180
1308,10
1779,00
1191,00
11 *1 00
29*800
19*4 00
BF
XlS
AlP
3S1S
3P1S
4P1S
3P1P
A3P
B3S*
D3P
I »2630
!» 3 0 4 0
i«2076
I »2207
I «2153
1*2 1 90
1*3080
1,2150
1«2103
1402*10
1264*96
1692*90
1613.20
1666» 60
1673*00
1 3 23 * 90
1629*30
16 96 *8 1
11 *8 00
12*5 00
12 *4 00
14,500
18,700
14,000
9,200
22,260
11,000
BEF
X2S
A2P
C2S
1,3600
1*394 0
1,3250
1266*90
1171*20
1419*70
9,350
8,530
9,900
HF
X is
»9168
2,0916
vis
*ST8P* O
4138*32
1158*50
89,880
17,750
A P P EN D IX
STATE
AO
Al
A8
G
KE
LE
me
H2
XlS
X2S
A3S
BlS
BIS?
B lS
C3P
CiP*
C1P«
DIP *
D IP *
D3P
D4P*
D4P«
D5P°
D5P*
ElS
E3S
ISP=
IIPU3D°
UlD =
NSP
1*581
ofi79
I »031
»469
,829
e885
»971
,909
,9 3 2
,3 9 9
,850
*920
»853
,90S
»870
o8S5
»966
»877
»86 3
»866
»908
»905
»895
e I 96 1
»60
®I »63
= 4 o53
o|»77
a2» 17
°S»61
0 I »69
= 4 e75
*1*81
=1,74
= 3, »66
*1 ,2 5
= I u 48
a I »59
= A77
=1*75
=1,74
= I o66
= 1*69
=1*73
*1 ,7 4
=1*56
1*90
I o83
1,94
2*23
1*00
4,22
1*91
2*43
2*53
2*61
2*40
2*00
* 62
1*33
1*82
= *67
1* 87
2*19
1*92
1* 80
2*27
2*39
1*65
5,748
lfl565
2,109
. »561
1,280
1,434
1* 804
1, 7 4 1
1* 753
1,656
1,653
1*670
1,608
1*611
1*600
1*594
I * 886
1*431
1,508
1* 516
1 ,6 3 4
1,629
1*601
=37,36
= 7* 10
=1 0* 45
?1 *9 9
=5 * 9 8
= 8* 41
=8 , 3 8
= 8* 6 5
= 8» 9 l
=8 , 6 5
f 8,53
=7 * 9 3
=5, 8 5
= 6* 7 7
=7 * 3 3
=3 * 4 9
=9 , 7 9
= 6» 76
=7 , 0 3
=7 , 2 1
= 8» 05
=8 * 0 8
= 7* 1 0
237*90
3 o *5 4
50,24
8*99
11*90
S g , 84
38,37
48 *5 2
5 0 *0 2
47,84
46 *2 5
3 6 *3 7
11*28
23 *0 5
32 *1 9
=1 1*61
41 *3 3
30 *7 6
30 *4 2
28,72
4 0 *0 0
4 2 *0 8
2 8 *3 3
4 * 08 5
3 ,6 8 1
3,227
3,599
3,296
3,436
3,864
3,694
2,961
3,522
=25,53
=2 2* 38
=20,87
=21*91
=20,78
=21,25
=2 4* 83
= 2 2* 62
=2 1* 62
=21,67
I 29»70
106,30
9 9 *0 5
10 0* 82
9 2 *7 7
10 4* 53
123* 05
10 7* 84
114* 74
10 1* 42
HE2
AlS
B lP
CIS
DlS
FlP
Fl D
ASS
BSP
CSS
D3S
2*209
2,091
I »921
2,056
I »937
2,000
2»110
2*087
I «778
2,020
*2,17
*2,16 "
*2 « 35
=2,17
=2 , 2 8
*2,22
*2 » 24
= 2« 17
= 2» 67
*2,20
2.e 86
2*73
3*04
2*67
2* 7 6
2*95.
2,90
2,75
3* 8 8
2*75
-8 7 -
STATE
F3 S
F3P
F3D
X2S
AO
1* 87 9
1*921
2 *001.
1*987
Al
*2*34
” 2*27
=2 *2 5
»2 * 2 2
A2
2*67
2*24
3* 0 7
2*90
KE
3 * 15 7
3 * 25 8
3*437
3*400
LE
=2 0 * 3 3
*20*40
*21*47
* 20 «9 4
ME
84 *8 4
74 *2 4
10 8* 77
«255
,135
,1 5 0
9*55
* * 25
o* 33
*85
* 43
»53
1 * 02 7
»029
«024
°3 * 64
s *02
9 o12
11 *52
*37
e64'
2*264
2,327
2*637
1*164
« 9 *9 6
, = 1 0* 70
o i l * 64
0 3 *4 6
38 *3 9
42 * 26
40 *4 4
9*77
3,587
2*849
* 18 *08
*11*98
6 3 *7 5
64*57
3.047
2*754
3*135
3* 33 1
3.708
*15*76
*20*83
*17*59
. *17*77
*18*99
69*92
1 9 4* 09
68*91
78 *56
86*41
101*20
L I2
Xis
AlS
B lP
*912
»652
* 648
«1 «91
= 1* 93
” 2*16
I »97
2*54
2*54
UIH
XlS
AlS
BlP
1*306
*097
*068
=1 * 8 8
o* 62
= 4® 09
2*38
7*32
12 *6 0
BEH
XPS
A2P
XlS
AlS
2*042
2*067
2*269
1* 506
= 1* 97
=2 * 0 4
*1*93
*lo 6 0
2*55
2*69
2*20
1*81
B2
X3S
A3S
4*528
3*762
=2 * 6 7
*2*28
3*74
4* 9 9
BH
XlS +
AlP
BlS+
ClS+
CUD
2*314
2*046
2*318
2* 451
2*652
* 2 * 12
*3*07
*2*27
* 2 * 16
°2 #04
2*90
8*73
2*71
2*89
2*78
-88-
STATE
AO
Al
'
AE
KE
LE
ME
CS
Xi s
B lP
ClP
DlS
ElS
X3P
A3S
A3P
B3P
F3S CS
CS"
9,000
7,948
9,115
9,068
7,738
8 =197
7,167
9 , Q58
5,101
6,348
10=254
9 =OS I
eS =46
®2o 74
® 2,7l
=2,82
” 4,64
®2 =75
=2,75
” 2 =55
” 4,12
=5,33
” 2 , $8
” 2,66
2»87
4»43
3 =25
4,83
12*03
4,66
4,45
1* 89
” , 79
28,64
3,83
4,39
11,659
9,146
11*571
11,838
9,877
9,525
7,645
11,303
4,330
6,542
13,704
11,218
=69,22
* 5 7 , Ig
=74,87
"80*77
"109,83
®59 , 8 1
"46,02
"68,35
"34,83
"75*04
"89,94
"70*46
2 6 0* 18
280*00
286*36
447*35
909,77
309*37
217,96
15 9* 68
"1 7 = 4 0
1158*81
420*69
367,56
3,018
2*819
7,587
1 =548
4,778
2,611
6 =009
2=987
=510
3,291
3,501
28645
1 =754
4,077
6 a965
1 =653
1 =496
2 =836
3,175
«2 0* 54
«l5o90
=52,97
=2 * 1 7
=29,85
"6,57
« 7 0* 46
"1 5 = 6 5
3 =26
'=16,90
= 1 7 o 13
"30,68
"2*83
,49
"58,14
O1 0 e20
o*29
"16,24
"16*12
1 6 3* 77
48 *69
131*94
"17*38
88*41
«7*51
323=81
44 =80
12,79
52 =33
29,91
148=75
=7.37
"69*67
229,53
25 *1 6
"1 7 = 4 0
48,05
52 =45
"26,47
«32,34
□ 3 0 ,8 8
136* 53
17 0* 19
113=67
BC
23+
2S1
2S2
2S3
2P1
2P2
2P3
201
202
4S1
4S2
482+
483
4S5
4P1
4P3
4P6
4F1
6 P1
4.325
4*011
7 * 84 4
4 * 602
5,695
4,512
7.682
4*251
1,394
4*562
4*685
3*698
4 * 468
11 *6 63
7,423
2*731
4,036
4*161
4,604
®3,84
=3*17
»3,35
” 1* 1 4
” 3,22
” 1,56
=6 , 2 5
=2,95
4*97
=2 , 8 5
=2,67
= 6 9 46
° 1 ,SI
*10
° 4,06
” 3= 74
= 9 15
” 3,27
=2 * 8 8
12,96
4*10
3*00
«5«56
3 » 68
"*83
11.48
3 * 56
11 .4 1
3*67
1 =91
13 =10
«1 =78
"8 = 1 5
5,85
4=19
«5 =23
4*14
3,99.
.
CH
XS P
A2D
C2S*
2,807
2 =856
2 =743
« 2 *2 1
«2,53
=2=59
3 =19
3 =66
2 =66
4 =475
4,703
4.421
-89-
state
*s
tD
AO
2 o629
1,452
1,792
Al
92 *1 2
*3,07
■ »2*51
AS
5*60
5*27
3,58
KE
4,111
1,906
2,361
UE
« 2 3* 09
«14,20
«14,40
ME
216,12
■ 7 9 *0 4
6 6 *8 9
' NE
XlS
AlS
AlP
A1P*
WlD
B» IS
Cf i s
A3S
B3S
B3P
C3P
X2S
B2S
S3*
3DG
1 3 ,8 3 1 * 1 7 4 , 2 1
7,857
°2 o98
8 , SI 3
=2 , 9 3
8,815
=2 , 9 4
8,059
” 2,97
2,507
=-1.77
12,439
«2,98
7,281 «207,75
7,752
=2 , 9 8
9,114
«2,94
11,404
«2,95
1 2 ,5 2 1
” 2,97
13,959
«2 ,8 1
7,108
=2 , 4 0
2»979
=4* 5 6
37932,19
5 o66
4,95
5,08
5.72
1 ,6 1
3,46
53943,34
5,58
5,06
, 46
5,47
2,47
2,54
8,13
2 2 , 9 8 6 «-10951, I o 8 6 9 4 4 6 2 , 0 0
9 e666
403,79
=67,76
*85,32
11,842
472,95
11,845
” 85,74
485,24
10,025
« 7 0 ,4 1
428,35
2 2 *2 2
2,401
” 8,80
19,833
656* 41
” 158*4 7
.8,805
=4267,26 3446411,00
9,492
388*97
«66,36
12,389
511*05
«90,10
17,275
72*00
” 132,90
20,107
«160,59
1058* 94
2 4 *1 59
” 1 8 9* 62
618,57
” 24,54.
5,484
64,50
1,945
6 1 *9 6
«15,g i
dN
X2S
A2P
BPS
EPS
F2D
+BlP
+ AlS
+ F lS
+CIS
11,187
9,524
11,805
9,441
5,515
8,441
10,823
18,609
5,664
” 2,66
«2,80
«3 *0 5
«1 ,8 1
=3 , 0 2
*2,91
*2,77
*4,93
«1 ,2 1
4,22
4,77
4.81
2.49
5,25
4,58
3,93
13,99
«3,38
16*2 94
1*2,535
17,835
1 0 ,8 2 1
5*868
10,856
15 *7 40
27,142
6,097
=110,87
«85,30
” 1 4 2, 0 1
” 44 o51
«38,76
«76,03
” 111,59
” 343,01
«16,30
600*43
4 7 2 ,1 1
776,80
184,99
19 6* 49
383,86
539,22
3322*05
*133*29
6*092
6,258
6,208
” 40,94
« 4 4 o80
=39,69
287*15
21 0* 61
227,01
NH
AID
A ID *
BIS*
3,250
3,365
3,325
«2,31
«2,47
« 2 ,2 1
4,19
3,02'
3,26
-90-
STATE
AO
3 ,4 2 3
B IS *
X3S*
3 ,0 5 6
X3S* . 3 •2 6 6
Al
= 2 ,4 4
= 2 *2 4
= 2 ,5 2
A2
. 2 ,9 4
3 ,1 7
3« 13
KE
6*39 1
5 ,6 8 4
6 ,0 2 7
UE
= 4 5 ,2 o
= 3 6 ,9 2
= 4 3 ,8 4
ME
2 1 0 *3 2
2 0 0 *9 5
2 0 9 *1 2
5 ,8 1 1 1 ,7 6 9
2 *1 5 9
2 =59
6 *0 0 1 0 ,7 3 5
9 ,6 7 4
6 *0 9
1 ,9 9 5
8«90
6 ,5 6 1 7 ,0 8 2
3 ,7 7 9
7 ,0 4
5 ,0 5 6
7 ,0 4
7 *7 8
6 ,7 4 4
= 8 8 ,2 1
= 1 1 ,5 1
=8 2*18
=75*51
= 1 5*95
=147*71
• = 2 8 ,2 9
»3 5*84
=67* 66
5 6 2 *5 5
2 6 *0 5
5 2 2 *8 9
4 6 9 ,9 4
8 3 ,5 8
1 0 7 9 ,4 9
1 6 0 *7 8
2 2 3 ,9 6
38 4*51
62
X3S
B3S
AID
B lS
ISU
XSP
ASP
A4P
B4S
8,57.9
2 ,7 7 7
7 ,9 3 0
7 ,2 7 8
2 ,5 4 4
1 0 ,6 3 8
3 ,7 5 2
4»822
5 ,5 2 5
=3 ,0 2
°2 q85
= 3 *1 0
=3 ,1 9
° 4 ,2 6
= 3 ,2 2
=3 ,5 2
=3 ,2 6
= 3 ,6 5
NB
SgS
XgP
AgS
BgP
B igP
BfgD
D2S
E2S
4 ,6 7 4
1 0 ,5 5 7
1 3 ,9 9 5
4 ,9 6 2
4 ,5 4 9
5 ,5 2 5
1 3 ,4 9 9
1 4 ,0 8 5
= 3 ,3 5
= 2 ,9 4
= 2 ,6 3
= 2 ,7 3
= 3 ,3 6
» 3 ,4 0
= 2 ,5 6
= 2 ,8 2
8* %7
5 ,3 6
3 *8 0
4 *6 2
9 ,8 2
6 *6 5
«52
4 *6 7
5 *1 8 6
15 *9 42
2 4 ,7 3 8
4 ,7 3 4
4 ,7 4 3
6 ,5 1 9
2 3 ,8 0 2
2 4 ,7 8 5
= 3 8 ,8 7
= 1 2 2 ,3 1
= 1 8 3 ,3 7
= 2 6 ,7 9
= 3 4*53
= 5 1 ,0 9
= 1 7 1 ,9 0
= 1 9 7 ,0 2
28% ,92
7 7 4 ,7 6
9 9 5 *8 6
125*06
2 9 1 *2 4
3 0 6 ,8 6
13 0*60
1 2 2 2 ,5 1
A lP
B lS
CIS
IS
A3S
A3P
B3S
D3D
E3S
J3S
7 ,0 8 1
1 1 ,3 0 2
1 2 ,0 3 6
1 2 ,1 0 6
5 ,5 9 0
8 ,9 g 7
1 1 ,9 7 0
5 ,0 3 6
4 ,7 9 4
1 2 ,6 7 7
=3 *0 0
= 3 *3 9
= 2 e86
= 2 *7 0
= 3 ,1 2
= 2 ,9 8
= 4 ,8 8
= 2 ,8 8
= 3 ,0 4
= 3 ,0 5
4 *1 0
9 *1 9
, 5*21
4*51
6 ,7 2
5 ,4 1
1 9 ,7 4
6 *6 4
6 *5 4
6 *3 4
9 ,2 7 6
18 *0 2 9
1 9 ,1 2 6
1 9 ,0 1 8
6 ,1 1 7
1 2 ,2 7 9
1 9 ,3 2 6
5 ,3 6 6
5 ,0 0 9
1 9 ,4 7 5
» 6 7 *4 9
= 1 6 3 ,7 2
= 1 4 6 ,4 2
=136*41
= 4 2 ,3 8
= 9 1 ,0 8
= 2 5 4 ,3 7
=33*81
»3 2 *9 7
= 1 5 6 *2 5
2 9 9 *1 4
1 5 8 5 ,1 1
9 4 9 ,6 5
808»10
2 7 0 ,0 1
5 4 7 ,9 1
3 6 94*73
2 2 7 ,9 5
2 0 5 ,3 1
1138*91
-9 1 -
state
X2S
A2P
BgS
20
AO
1 2 *3 1 3
7 ,6 2 3
8 ,2 9 7
4 ,7 8 9
Al
« 2 ,7 9
« 3 *0 0
«3 *7 0
°3 * 48
AS
KE
4 *6 2 1 9 ,8 0 3
5 ,5 8
9 ,8 5 6
6 ,7 6 1 2 ,1 4 7
o l* 1 8
5 ,2 6 7
.
Ce
” 1 4 8 ,6 3
« 7 1 ,3 5
« 1 1 5 ,3 3
« 4 1 ,0 7
ME
8 8 2 ,0 2
426*91
7 2 0 ,9 8
« 4 1*28
» 8 9 ,6 6
»4 1 *5 2
« 3 8 ,9 4
4 8 3 *I 9
2 5 8 *9 7
2 0 4 *4 8
7 ,5 1 1
4 a447
6 ,3 8 4
4 ,9 8 9
■ « 4 6 ,1 8
« 2 4 ,3 1
» 3 3*98
» 2 7 ,4 7
2 3 0 *2 0
11 9 *0 6
16 6*14
1 3 4*86
7 ,7 9 4
5 ,6 5 3
,8 5 0
2 ,5 7 4
5 ,2 9 0
« 5 4 ,1 6
1 » 4 0 *5 6
» 2 *3 5
« 1 7 ,6 0
» 3 6 ,8 5
3 3 6 *7 8
2 4 3 *4 4
3 ,$ 6
102*45
2 8 8 *4 3
B0
xas*
ASP
BSS+
9 ,9 1 7
5 ,5 8 5
5 ,4 1 6
« 2 ,6 4
° 3 »Q6
« 2 ,7 0
4 * 28 1 3 ,6 6 0
6 ,4 6
6 ,1 1 1
4 ,6 5
6 ,3 0 2
BE9
X lS
A lP
B lS
A3P*
6 ,6 5 3
4 ,7 5 9
5 ,9 2 1
5 ,2 3 0
« 2 ,7 3
» 2 ,6 7
«2 e42
« 2 ,6 6
4o'52
4 o78
4 ,0 2
4 ,7 2
9H
XgP
A2S
C3S*
+3P
+33 «
3 ,6 7 1
2 ,8 9 5
1 ,7 7 7
1 ,6 7 3
2 ,8 0 0
«2 *2 5
« 2 ,4 2
®I «89
«2 *6 0
«2 *3 9
3 ,3 9
3 ,6 8
1 *4 6
4 ,3 1
4 ,8 1
P2
B lP
5 ,9 7 3
« 3 ,2 8
7 ,4 6
-
7 ,2 6 9
» 5 5 ,7 9
3 9 5 *9 8
7 ,5 6 0
= 4 4 ,7 5
2 7 9 *7 4
• 9F
2P*
6 ,5 9 6
« 2 ,6 1
5 ,3 8
-9 2 -
state
AO
Al
A2
KE
LE
ME
6 ,1 8 5
6 ,8 0 9
= 4 1 *7 3
® 4 5 .39
2 5 6 ,5 8
2 7 9 *4 2
7,471
6»0l
3,29 1 3 ,8 2 6
8 ,7 1 .6 ,1 4 4
.= 5 3 * 0 3
= 1 2 4 ,3 3
=53*81
3 3 5 *7 2
4 1 3 *5 8
369*71
= 5 0*30
= 4 3 *4 5
= 8 2 *5 6
= 7 8 ,1 8
= 7 8 ,8 4
. = 7 8 ,4 5
= 3 0 4 ,8 9
= 8 1 ,3 6
= 8 2 ,7 3
20 7*91
2 0 7 ,4 6
4 8 3 *1 7
441*61
4 2 9 *7 6
3 6 1 *8 0
2 1 296 *9 8
2 0 9 ,5 7
5 3 5 *4 2
NP
X3S
SIS
5 *3 6 4
5 *7 5 3
= 2 ,9 6
= 2 ,8 9
6 , QO
5 ,7 8
CF
X2P
ASP
B2S«
5 ,9 9 6
9«m
5 ,3 3 6
?3«00
o 3 ,4 4
= 3 *8 5
BP.
X lS
A lP
3S1S
3P1S
4P1S
3P1P
A3P
B3S*
D3P
6 *4 3 5 . « 2 ,6 2
5 ,5 8 5
= 2 ,8 8
8 ,5 7 8
= 2« 82
7 ,9 5 8
=2 ,9 8
8 ,4 2 0
=2 ,8 0
8 ,5 3 2
= 2 ,7 8
6 ,1 6 0
o l 8 ,4 6
8 ,0 4 5
= 3 ,0 2
8 ,6 5 5
=2 ,8 2
3 ,4 3
4*47
4»99
5 ,1 3
4»64
3 ,9 0
4 2 1 ,6 8
2 ,3 7
5 ,5 3
8 *0 6 8
6 ,5 6 9
1 1 ,7 6 5
1 0 ,6 8 2
1 1 ,4 0 2
1 1 ,4 8 4
7,201
1 0 ,9 0 0
1 1 ,8 1 8
•
BEF
XSS
A2P
C2S
5 ,3 5 3
4 ,7 9 7
6 .3 7 5
=2 ,6 8
= 2 ,6 9
= 2 ,3 4
4 ,7 7
5«07
2 ,6 6
5 ,7 8 8
4 ,9 3 7
7 ,2 6 2
= 3 4 ,1 7
= 2 8 ,6 2
= 3 8 ,5 4
1 7 8 ,9 6
15 4*50
1 3 2 ,2 1
9 ,6 5 6
,7 5 7
= 7 1 ,2 0
? l* 3 l
4 8 0 *7 6
=2*33
Hf
X lS
vis
oSTBPa
4 ,0 5 8
1 ,6 5 5
0
= 2 ,2 5
= 1 ,2 1
3 *4 9
= 1 ,1 2
APPENDIX H
force constant relations obtained by least - squares analysis
MOLECULE
EXPONENT
CONSTANT
H2
3.91
1.94
He2
5.27
5.05
HeH
3.53
2,50
L i2
4.36
18.0
LiH
7.97
38.9
BeH
3.83
B2
BH
2.91
17.5
7.72
14.3
C2
BC
5.07
36.6
5.75
60.0
CH
7.31
9.8
N2
7.10
46.8
CN
6.27
48.6
NH
17.9
7.16
11.1
°2
NO
5.95
32.8
5.99
35.1
CO
640
39.1
BO
7.45
53.2
BeO
5.09
31.6
OH
2.75
55.0
CF
5.94
31.2
BF
6.93 *
43.0
APPENDIX I
R elation o f Cubic Force Constant to K in e tic Energy
I f one assumes th a t the k in e tic energy o f a molecule obeys the
equation
T = T o + T1R '"
(I-I)
near the e q u ilib riu m d ista n ce , then by d iffe r e n tia tin g , we fin d th a t
T1V T 1
n + I
-R
(1-2)
From d iffe r e n tia tio n o f the v i r i a l theorem, re la tio n s between the force
constants and the k in e tic energy d e riv a tiv e s can be obtained^33^ , which
when combined give the re la tio n
<TV e
3 + 3a
(1-3)
By combining (1-3) w ith (1-2) one obtains
n = -Sa^ - 4
(1-4)
thus r e la tin g the form o f the k in e tic energy curve to the cubic force
constant.
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Herzbergg G.
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2.
K ra tz e r3 A.
1950.
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P rin ce to n ,
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The In fra re d Rotation Spectrum o f the Halogen Hydrides.
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Mecke3 R.
Zum Aufbau der Bandenspectra.
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4.
Morse3 P /M .
II.
5.
Diatomic Molecules According to the Wave Mechanics.
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The P e rio d ic ity o f Morse's Function.
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C lark, C. H. D. and K. R. Webb.
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VI.
In te r r e la tio n o f E q u ilib riu m Bond Constant and In te r -
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7.
Systematics o f Band-spectral Con­
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V II.
E m pirical Form o f R elations In v o lv in g the Group Number.
Trans.
The
Faraday Soc. 37, 299-302 (1941).
8.
A lle n , H. S. and A. K. Longair .
In te rn u c le a r Distance and V ibra­
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(1935).
9.
C la rk, C. H. D.
The Relation Between V ib ra tio n a l Frequency and
Nuclear Separation f o r Some Simple Non-hydride Diatomic Molecules.
P h il. Mag. 18, 459-470 (1934).
-9 6 -
10.
Badger, R. M.
A R elation Between In te rn u c le a r Distance and Bond
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11.
Badger, R. M.
J. Chem. Rhys. 2, 128rl31 (1934).
The R elation Between the In te rn u c le a r Distances
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G lo c k le r, G. and G. E. Evans.
Distance.
13.
J. Chem. Rhys. 3, 710-714 (1935).
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M olecular Constants and P o te n tia l Energy Curves
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14.
Huggins, M. L.
J. Chem. Rhys. 3, 473-479 (1935).
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15.
Force Constants and In te rn u c le a r
Wu9 C. K. and S. C. Chao.
L I.
J. Chem. Rhys. 4, 308-312 (1936).
The R elation Between the Force
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L in n e tt, J. W.
II.
Constant
A mod­
Rhys. Rev. 71, 118-121 (1947).
The R elation Between P o te n tia l Energy and In te ra ­
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36, 1123-1134 (1940)..
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L in n e tt, J. W.
The R elation Between P o te n tia l Energy and In te ra ­
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II.
Trans. Faraday Soc.
3 8 ,1 - 9 ( 1 9 4 2 ) .
18.
S utherland, G. B. B. M.
The Determination o f In te rn u c le a r Distances
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161-164 (1940).
J. Chem. Rhys, 8,
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19.
Fox, J. J. and A. E. M artin.
Relations Between In te rn u c le a r Dis­
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20.
Gordy. W.
J. Chem. Soc. 884-886 (1939).
A R elation Between Bond Force Constants, Bond O rders,
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Ladd, J. A ., W. J. O rville-Thom as, and B. C. Cox.
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III.
Molecular
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Ladd, J. A. and W. J. O rv ilie-Thomas.
Bond S tru c tu re .
V.
M olecular Parameters and
Nitrogen-oxygen Bonds.
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23.
Decius, J. C.
R elation Between Force Constant and Bond Length f o r
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24.
J. Chem. Phys. 45, 1069 (1966).
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Anharmonic P o te n tia l Constants
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Johnson, H. S.
C o n tin u ity o f Bond Force Constants Between Normal
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C alder, G. V. and K. Ruedenberg .
Q u a n tita tiv e C o rrelatio ns Between
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Kolos, W. A d ia b a tic Approximation and i t s Accuracy.
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Dunham, J. L.
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-1 0 2 -
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The Spectrum and S tru ctu re o f the
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The Spectrum o f the CN+ Mol­
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-1 06-
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PART
II
THE USE OF FREQUENCY MODULATION TO GENERATE A REFERENCE
" BEAM FOR ATOMIC ABSORPTION ANALYSIS
INTRODUCTION
The broad a p p lic a tio n o f atomic absorption spectroscopy to chem­
ic a l a n a lysis was. f i r s t re a liz e d by A. W a ls h ^ in 1955.
Since then
atomic absorp tion analysis has become the p rin c ip a l method used in
tra ce-e lem e nt a n a ly s is .
Research in atomic absorption analysis is
p re se n tly centered upon devisin g ways to increase the s e n s it iv it y and
accuracy o f the method.
In recent years the s e n s it iv it y o f the method
has been increased to the p o in t where q u a n titie s as small as 10
grams
may be d e te cte d , and s e n s it iv it ie s are approaching the th e o re tic a l
lim it s .
To increase the accuracy o f the method i t is necessary to de­
velop a way o f c o rre c tin g f o r the unwanted absorption and s c a tte rin g by
the sample.
Chemical treatm ent o f the sample to separate the m a trix
c o n s titu e n ts from the element to be analyzed is a g e n e ra lly u n s a tis fa c ­
to ry approach, though a c e rta in amount o f chemical treatm ent can be very
h e lp fu l.
The second p o s s ib ilit y is to make a c o rre c tio n f o r the un­
wanted absorption by the use o f a reference beam.
Double-beam o r r e fe r ­
ence-beam methods, however, have been d i f f i c u l t to apply e ffe c tiv e ly to
atomic absorp tion sp e ctro chemical a n a ly s is .
At present there are no
t r u ly s a tis fa c to r y methods o f compensating fo r the absorption and s c a t­
te r in g o f l i g h t by species o th e r than, the analyte atoms.
Two basic ap­
proaches may be taken to provide such a background c o rre c tio n .
A c la s s ic a l double-beam spectrom eter may be constructed using a
beam s p l i t t e r and beam condenser.
The successful use o f a double-beam
-1 0 8-
system requires th a t one be able to d u p lic a te p re c is e ly the chemical
composition o f the sample w hile the element being analyzed is com­
p le te ly removed.
This cannot u su a lly be done.
One must also d u p lic a te
the co n d itio n s under which the sample is introduced in to the absorption
c e ll or region.
Since atomic absorption is observed only in atomized
samples, d u p lic a tio n o f such con dition s can also be extrem ely d i f f i c u l t .
The magnitude o f the background absorption may also be estim ated
by m onitoring absorption a t a wavelength near the wavelength being used.
This method has the drawback th a t absorption o f l i g h t by the unwanted
species is fre q u e n tly frequency dependent, g iv in g an e r ro r in the back­
ground reading.
N evertheless, th is approach is fa r more p ra c tic a l than
the use o f a dual-beam spectrom eter.
The c u rre n tly a v a ila b le in s tr u ­
mentation u tiliz e s th is method to obtain background c o rre c tio n s .
In
order to m onitor absorption a t a second wavelength one must (a) add a
second l i g h t source, (b) merge the beam from the second source w ith th a t
from the p r in c ip le source, and (c) provide a means f o r separately de­
te c tin g the T ig h t from the second source.
These requirements may a ll be
met by sim ply frequency modulating the p r in c ip le l i g h t source and using
an A. C. o r phase s e n s itiv e d e te cto r.
However, p re s e n tly o th e r methods
are being used, f o r c o rre c tin g atomic absorption measurements.
( 2 3)
The p rin c ip a l method' 9 ' being used is to mount a deuterium lamp
near the source f o r the atomic lin e , merge the beam from the deuterium
lamp w ith th a t o f the prim ary lamp, and d e te ct the reference l ig h t by
-1 0 9-
the use o f a chopper arrangement s im ila r to th a t used in most separateddouble-beam systems.
The. frequency spread o f the l i g h t detected from
the deuterium lamp is determined by the sp e ctral s l i t w idth o f the mono­
chromator and includes the sample o r resonance frequency.
Since lig h t
from the deuterium lamp is p a r t ia lly attenuated by the analyte atoms,
i t is not a tru e reference beam.
Abbott and S k o g e rb o e ^ have used the continuum
emission o f e le c ­
tro d e le ss discharge lamps as a re fe re n ce , thereby e lim in a tin g the need
f o r a second lamp when the e le c tro d e le s s discharge lamp emits the lin e
desired f o r atomic absorption a n a ly s is .
Other m ethods^) have been suggested f o r making th is c o rre c tio n ,
but they are not commonly used.
CHAPTER
I
Theory
There are many po ssible methods fo r frequency modulating a beam o f
l i g h t o r a l i g h t source.
For each method one must devise a means o f as­
sessing i t s e ffe c tiv e n e s s r e la tiv e to o th e r methods.
For th is reason
I have chosen two elements, sodium and mercury, to use as examples (o r
te s t cases) fo r each method.
These two elements represent extreme ends
o f the spectrum f o r many c r i t i c a l elemental p ro p e rtie s .
wave length o f emission and atomic w eight.
For example,
They are also g e nerally the
e a s ie s t elements to use in the actual te s tin g o f po ssible methods be­
cause o f t h e ir high vapor pressures.
For both sodium and mercury a min­
imum wavelength s h i f t o f about 0.05 Angstroms is needed f o r c le a r
o lu tio n o f the reference and sample beams.
res­
The methods which may be
used w i l l be categorized by the physical e ffe c ts they take advantage o f.
Doppler E ffe c t
The Doppler e ffe c t is fa m ilia r to everyone as the increase o r de­
crease in the apparent frequency o f sound em itted by a source tr a v e l­
lin g toward o r away from the observer, re s p e c tiv e ly .
The e ffe c t may be
observed f o r both transverse and lo n g itu d in a l waves.
I f . f is the ob­
served frequency o f the waves and f
is the frequency o f the waves as
measured by an observer a t re s t w ith respect to the source, then
f = f 0( l + v /c )
,
(1 -1)
-
111
-
where c is the v e lo c ity o f propagation o f the waves and v is the re la ­
tiv e v e lo c ity o f the source toward the observer.
For li g h t waves one
may also w rite
A = A0U - v /c ) ,
(1-2)
where v /c is much sm a lle r than one and A is the w a v e le n g th ^ .
The Doppler e ffe c t is one o f the p r in c ip a l causes o f lin e broaden­
in g .
Since the average v e lo c ity o f the atoms depends on the tem perature,
the magnitude o f the Doppler broadening f o r an atomic lin e also depends
on
the tem perature.
By in cre asing the temperature o f a source one may
widen the lin e considerably.
I f one could ra p id ly change the tempera­
tu re o f an atomic l i g h t source, one could a lte rn a te ly em it a narrow and
then a broad lin e , thereby generating a sample-and-reference beam fo r
atomic absorption a n a ly s is .
Simply broadening a lin e , however, as pre­
v io u s ly in d ic a te d , does not generate a tru e reference beam.
I w ill
th e re fo re not consider methods which produce an e ffe c t in a random fash­
io n , c re a tin g a simple broadening o f the lin e .
Nearly a ll e ffe c ts may
also be produced in a s e le c tiv e fa s h io n , c re a tin g a change in the f r e ­
quency o f the l i g h t w ith o u t appreciably changing the w idth o f the lin e .
Pursuing th is lin e o f thought, we t r y to devise a l i g h t source in which
a ll the e m ittin g atoms are tr a v e llin g a t high v e lo c ity toward or away
from the d e te c to r.
A hollow-cathode lamp may be constructed which w i l l
allow the ions to accelerate through a voltage drop before acq uiring an
-112-
e le c tro n and e m ittin g lig h t .
Such a lamp is e a s ily constructed and used
f o r dem onstrating the Doppler e ffe c t in h y d ro g e n ^ .
However, to modu­
la te the l i g h t from the lamp one must e ith e r change the voltage drop o r
pressure in the c e ll ra th e r q u ic k ly .
Large changes in the in te n s ity o f
the l i g h t would be expected to accompany such changes, and emission from
unaccelerated atoms would always be presen t, so the frequency modulation
could not be achieved w ith o u t the a d d itio n o f cumbersome side e ffe c ts .
A frequency s h i f t o f 0.05 Angstrom would re q u ire th a t the ions experience
a voltage drop o f about 400 v o lts f o r mercury and about 10 v o lts fo r
sodium.
A ra p id change in path length could also be brought about by the
use o f ro ta tin g or v ib ra tin g m irro rs .
t ic a l a change o f the order o f 10
5
For such a method to be prac-
cm/sec. would be needed.
The use o f
a fro n t-s u rfa c e m irro r on a p ie z o e le c tric c ry s ta l w arrants in v e s tig a tio n
in th is re gard, as there would be no side e ffe c ts expected fo r such a
device.
Dynamic R e fra c tiv e Index Change
G eneralizing upon the discussion o f the Doppler e ffe c t in the pre­
vious s e c tio n , we consider the e ffe c tiv e o p tic a l path length ra th e r than
the physical distance o f the o p tic a l path.
C a llin g L the physical d is ­
tance and n the index o f re fra c tio n o f the medium through which the ra ­
d ia tio n passes, we define the o p tic a l path le n g th , x , by the equation
x = nL.
(1-3)
-1 1 3-
In the previous sectio n the case where n is held constant and L is var­
ied was discussed as a method fo r producing .a change in path length.
In th is sectio n the case where L is held constant and n is varied is
discussed.
Since the o p tic a l path length is changed, a frequency s h i f t
must occur.
The same equations apply fo r the usual Doppler e ff e c t,
except th a t x is used instead o f L.
The problem o f de visin g a method fo r frequency modulating a l i g h t
beam is now reduced to sim ply fin d in g a way o f ra p id ly changing the in ­
dex o f re fra c tio n o f p a rt o f the medium through which l i g h t passes.
This method has the advantage th a t i t involves the l i g h t beam i t s e l f
ra th e r than the source o f l i g h t and thereby minimizes chemical side e f­
fe c ts .
The well-known dependence o f the r e fra c tiv e index o f a medium up­
on the transverse e le c t r ic f i e l d , c a lle d the Kerr e f f e c t , allows one to
b rin g about a frequency modulation o f l i g h t by applying an e le c tr ic
f i e l d which is amplitude modulated.
When an e le c t r ic f i e l d is imposed
on an is o tr o p ic d ie le c t r ic , the d ie le c t r ic reacts as a u n ia x ia l c ry s ta l
w ith i t s o p tic axis p a ra lle l to the e le c t r ic f ie ld .
L ig h t p o larized
p a ra lle l to the f i e l d w i l l th e re fo re see an index o f r e fr a c tio n , Hp,
which is d iffe r e n t from the index o f r e fr a c tio n , n$ , seen by l ig h t
p o la riz e d pe rpendicular to the f ie ld .
The in d ic e s , according to the
Langevin-Born t h e o r y ^ , are determined by the equations
np = n0 = 2ABE2/3
(1-4)
-1 1 4-
and
ns = n0 - ABE2/3
,
(1-5)
where B is the K err constant fo r the d ie le c t r ic and E is the e le c tr ic
f i e ld .
Kupper and F u n f e r ^ , using a tra v e llin g -w a v e K err c e ll,
created a frequency s h i f t on the order o f 0.4 Angstrom.
Likew ise,
S t a u f f e r ^ ) , using a s im ila r Kerr c e ll arrangement, has observed a
broadening o f two mercury lin e s having a separation o f 0.045 Angstrom
to the p o in t where they could ha rd ly be resolved.
B urdette and
H ughes^^) re c e n tly measured the d iffe re n c e in frequency o f the per­
p e n d icu la r and p a ra lle l p o la riz e d components o f the l i g h t beam in a
Kerr c e ll.
Using the Dopple r - s h if t form ula
A = A0( I - n 'L /c )
(1-6)
along w ith the Langevin-Born th e o ry, the expression f o r the change in
frequency o f l ig h t p o la riz e d p a ra lle l to the f ie l d becomes
Av = 4BLEE73
.
(1-7)
Thus, the frequency s h i f t can be increased by superposing a la rg e r con­
s ta n t p o te n tia l as w ell as by in c re a sin g the amplitude and frequency o f
the a lte rn a tin g p o te n tia l.
Equation (1 -7 ) must be considered only an
approximate e q uatio n, however, since the Langevin-Born theory breaks
down fo r very high frequencies o f the a lte rn a tin g p o te n tia l.
To achieve
a frequency s h i f t o f more than 0.01 Angstrom a voltage drop o f at le a s t
10,000 v o lts per centim eter and a powerful high-frequency voltage source
-
115
-
o f about 2,000 megacycles per second would be needed.
A pulsed e le c t r ic
f i e l d could be generated, though, w ith o u t the use o f expensive equipment.
A transverse magnetic f i e l d may be used in an analogous manner to
induce b ire frin g e n c e in m a te ria ls .
This e ff e c t, c a lle d the Cotton-
Mouton e f f e c t , allows one to produce a frequency s h i f t in much the same
manner as w ith the Kerr e ffe c t.
weaker than the Kerr e ffe c t.
The Cotton-Mouton e ffe c t is much
High-frequency, a lte rn a tin g magnetic fie ld s
are also d i f f i c u l t to o b ta in .
Thus, CottonrMouton e ff e c t appears to be
less advantageous to use than the Kerr e ffe c t.
A lo n g itu d in a l magnetic f ie ld also produces a change in the re ­
fr a c tiv e in d ice s o f a m a te ria l.
The e ff e c t, c a lle d the Faraday e ff e c t,
re s u lts in equal bu t opposite changes in the indices o f re fra c tio n , n+
and n_, fo r r ig h t - c ir c u la r ly and l e f t - c i r c u l a r l y p o la riz e d l i g h t in the
medium.
A changing lo n g itu d in a l magnetic f i e l d s p lit s a monochromatic
beam o f l i g h t in to two components having d iffe r e n t freque ncies, an id e a l
s itu a tio n fo r the generation o f a reference beam.
norm ally considered in terms o f the
which i t produces.
ro ta tio n
The Faraday e ffe c t is
o f plane p o larized l i g h t
The magnitude o f the ro ta tio n is p ro p o rtio n a l to the
Verdet co n stan t, V, o f the m a te ria l.
The angle o f r o ta tio n , <j>, is given
by
( 1- 8)
where H is the magnetic f i e l d in te n s ity and L is the path le n g th ^ . The
-1 1 6-
ro ta tio n may also be determined from the Fresnel equation^12^ ,
<J) = (n_-n+)L/2A .
(1-9)
For any o f the methods discussed the r a tio (A - A0)/Ao must be g reate r
than 10
—5
fo r the method to be u s e fu l.
th a t d n /d t must be a t le a s t 3*10^.
This means, using equation (1-6),
A ccord ingly, i f an a lte rn a tin g mag­
n e tic f i e l d o f about one megacycle per second or a pulsed f ie ld having
a d u ra tio n o f about one microsecond is used, then the magnetic f ie ld at
i t s maximum must produce a change in the index o f re fra c tio n o f about
0.03 times the normal value w ith a 10 centim eter c e ll.
cannot be achieved w ith normal d ie le c tr ic s .
Such changes
T herefore, a useful f r e ­
quency s h i f t is not e a s ily created using the Faraday e f f e c t , though
in spe cial re s tr ic te d cases the Faraday e ffe c t could be used.
M echanically induced changes in the re fra c tiv e index are very
small and occur ra th e r s lo w ly , so th a t the frequency s h ifts so generated
are n e g lig ib le .
In te rru p tio n ,B ro a d e n in g
I t is w e ll known in ra d io science th a t any amplitude modulation o f
a wave can also be considered as a frequency modulation in which "side
bands" are generated.
I f a c a r r ie r wave o f frequency vQ is s in u s o id a lly
am plitude modulated a t a frequency v ^, then the c a r r ie r wave is a c tu a lly
decomposed in to two waves having frequencies o f vQ + Vffl and Vq - v .
The observed amplitude modulation is then the re s u lt o f the beating o f
-
the two s id e bands.
117
-
Therefore, i f one can produce a high-frequency am­
p litu d e modulation o f a l ig h t beam, then one has a u to m a tic a lly also pro­
duced the desired frequency m odulation.
le a s t 10
9
cyle s/se c.
For our purposes v
must be a t
(13)
E. Rupp' ' in 1928 performed an experiment to
v e r ify t h is phenomenon.
He passed l i g h t from a th a lliu m resonance lamp
through a Kerr s h u tte r operated a t IO9 cycles/sec. and found th a t the
l i g h t was n o t appreciably absorbed by th a lliu m vapor.
One cannot be.
su re , however, whether the frequency change he observed was produced by
the s h u tte r a ctio n o r by the change in index o f re fra c tio n in the Kerr
c e ll.
A t any ra te , there is no doubt th a t, i f one could devise a sh u t­
te r having a frequency o f IO^9 c y c le s /s e c ., then a n e a rly id e a l re fe rs
ence beam f o r atomic absorption would be re a liz e d .
S c a tte rin g Processes
S c a tte rin g processes provide a simple and extrem ely e ffe c tiv e
means o f changing the frequency o f l i g h t .
U n fo rtu n a te ly , the in te n s ity
o f s c a tte rin g is always low compared to the in te n s ity o f the in c id e n t
r a d ia tio n 9 making the use o f s c a tte rin g extremely d i f f i c u l t in pra ctice ..
There are many types o f in e la s tic s c a tte rin g in which larg e frequency
s h ifts are produced, c h ie f among them being the Raman e ffe c t.
Many
e la s tic s c a tte rin g processes also produce larg e frequency changes;
Thompson s c a tte rin g from high speed e le ctro n s can even r e s u lt in the
production o f X rays.
-1 1 8-
Stark E ffe c t
An e le c t r ic f i e l d w i l l s p l i t the energy le v e ls o f atoms and
consequently re s u lt in a change in the frequency o f emission o r
absorption o f the atoms.
The e ffe c t need not be considered in d e ta il
because (a) the s p l i t t i n g even fo r larg e e le c t r ic fie ld s is very small
and .(b) i t re s u lts in the d is s o c ia tio n o f e le ctron s and d iffu s e spectra
before the s p l i t t i n g becomes la rg e .
t r i c f i e l d o f 100,000 V/cm.
In mercury, fo r example, an e le c ­
produces a frequency s h i f t in the 2537
Angstrom lin e o f only 5 x 10"^ Angstrom^
Zeeman E ffe c t
The o rig in o f the Zeeman e ffe c t is the change due to a magnetic
f i e l d in the energy le v e ls o f the atoms.
I t was f i r s t observed as a
s p l i t t i n g in to three components, o f the emission lin e o f mercury in a
magnetic f i e ld .
La ter observations on o th e r s p e ctra l lin e s revealed
s p lit t in g " in to more than three components; such s p lit t in g s were c a lle d
the anomalous Zeeman e f f e c t , whereas the t r i p l e t s p l i t t i n g was designa­
ted the normal Zeeman e f f e c t , since i t is explainable in terms o f c la s ­
s ic a l th e o ry.
B a s ic a lly , the normal Zeeman e ffe c t is due to the o r b ita l
magnetic moment o f the e le ctro n s and the anomalous e ffe c t is due to the
spin magnetic moment o f the e le c tro n s .
When a sp e ctra l lin e s p lit s in a magnetic f i e l d the components in ­
to which i t s p lit s are p o la riz e d e ith e r p a ra lle l o r perpendicular to the
magnetic f i e l d i f viewed pe rpendicular to the f ie ld .
-1 1 9 -
The form er are c a lle d the it components and the l a t t e r the o components.
Drawings are commonly used to describe the nature o f the Zeeman s p l i t ­
tin g f o r a p a r tic u la r lin e ; it components are drawn above the horizon­
ta l and a components below, the length o f the v e r tic a l lin e s being pro­
p o rtio n a l to in te n s ity and frequency being p lo tte d along the abscissa.
Thus the drawing fo r a normal Zeeman s p l i t t i n g is
I
I
I
I f the s p e c tra l lin e is viewed along the magnetic f i e l d , only the a
components are v is ib le , and they become l e f t and r ig h t c ir c u la r ly p o la r­
ize d .
According to the c la s s ic a l th e o ry ^
the displacement o f Av, o f
each a component is given by
Al) - eB/47Tme
,
(1-10)
where e and me are the charge and mass o f the e le c tro n and B is the mag­
n e tic f lu x d e n sity.
The magnitude and cha racte r o f the s p lit t in g o f a sp e ctra l lin e
depends upon how the energy le v e ls are s p l i t fo r both o f the e le c tro n ic
sta te s in v o lv e d in the tr a n s itio n .
For the mercury 2537 Angstom lin e a
simple t r i p l e t is observed in which the s p lit t in g is 3/2 la rg e r than
the c la s s ic a lly p re d icte d value.
f o r the tr a n s itio n is 3/2 .
two
a
components
and
In
We th e re fo re say th a t the g value
the
case
two; tt components
of
the
are
sodium
found,
D-j
the
lin e ,
a
components being displaced 4/3 o f the expected c la s s ic a l va lu e , w hile
-1 2 0-
the TT components are displaced a sm a lle r amount.
enough to be useful are not easy to o b ta in .
Again s p littin g s large
The fie ld s required to d is ­
place the a components 0.05 Angstrom are about eleven kilogauss fo r
mercury and about two k i logauss fo r sodium.
There are many ways in which one m ight take advantage o f the
Zeeman e f f e c t to produce a reference system.
One m igh t, fo r example,
apply a lo n g itu d in a l magnetic f i e l d (though in some cases a transverse
f i e l d could be used) to the sample, thereby changing i t s
resonance f r e ­
quency ra th e r than th a t o f the l i g h t and using the Zeeman e ffe c t in
a b so rp tio n .
Some o f the possible experimental c o n fig u ra tio n s using the
Zeeman e f f e c t w i l l be described in the next chapter.
A p ra c tic a l system
f o r the a n a lysis o f sodium which u t iliz e s a magnetic f i e l d has already
been b u i l t by D. W o o d riff^
CHAPTER I I
Experiments Using the Zeeman E ffe c t
Mercury has a s u f f ic ie n t ly high vapor pressure (10
-
1
Torr) at room
temperature th a t a one centim eter c e ll w ith a drop o f mercury in i t w i l l
absorb a larg e p o rtio n o f the resonance l ig h t from a mercury lamp.
For
the sake o f convenience, mercury was used as a te s t element in a ll o f
the experiments.
For mercury a vapor lamp may be used in place o f the
usual hollow -cathode lamp.
I t has the advantage o f being small and re­
la t iv e ly inexpensive.
The f i r s t experiments were conducted using a m odified Beckman DB
spectrophotom eter.
The m odified instrum ent has been described by
Woodrif f , C ulver, and O l s o n ^ I t is h e lp fu l to know some o f the de­
t a i l s o f how the m odified DB works in order to understand how and to
what e xte n t i t can be used in ta kin g advantage o f the Zeeman e ffe c t.
The instrum ent in the double-beam mode measures the r e la tiv e in te n ­
s it ie s o f two beams which are generated by a v ib ra tin g m irro r system
a fte r the l i g h t has passed through a lo w -re s o lu tio n g ra tin g monochro­
mator.
The r a tio ( I yi - I s ) / I r is measured, where I f is the in te n s ity
o f the reference beam and I g is the in te n s ity o f the sample beam.
By
pla cin g two Glan-type p o la riz e rs in the sample compartment, the in ­
strument th e o r e tic a lly becomes capable o f measuring e ith e r the degree
o f p o la riz a tio n o f a s in g le beam o r the re la tiv e in te n s itie s o f two
col in e a r p e rp e n d ic u la rly p o la rize d beams.
Generally the magnetic f ie ld
-122is h o r iz o n ta l, so th a t a h o riz o n ta l p o la riz e r is used f o r the sample
beam and a v e r tic a l p o la riz e r fo r the reference beam.
Many more p ra c tic a l problems a ris e when p o la rize d l i g h t is used
than when p o la riz a tio n is neglected.
O ptical components, fo r example,
can ro ta te o r p a r t ia lly de polarize the p o la riz e d lig h t .
The DB, how­
e ve r, was te ste d and found to cause ro ta tio n o f less than one degree
and only about two percent d e p o la riz a tio n .
Another common p ra c tic a l problem in atomic absorption analysis
is the presence o f in te r fe r in g emission due to o th e r elements or oth er
lin e s o f same element.
For the DB no change in the s e n s it iv it y fo r
the sim ple atomic absorption o f mercury was observed as the s l i t w idth
was changed, in d ic a tin g th a t there are no appreciable in te r fe r in g lin e s
w ith in about 50 Angstroms o f the 2537 A resonance lin e .
A t h ir d problem, which n a tu ra lly becomes very im portant when a
magnetic f i e l d is p re se n t, is the s u s c e p tib ility o f the in te n s ity and
lin e w id th o f the resonance lin e to change as con dition s change.
The
apparent w id th o f the resonance Tine was found to depend la rg e ly upon
the tem perature o f the l i g h t source.
Since the absorbing atoms are a t
room temperature in the atomic absorption apparatus used by author, a
considerable increase in s e n s it iv it y can be achieved by m aintaining the
lamp a t a low tem perature.
mercury vapor lamp.
An a ir stream was th e re fo re used w ith the
Likew ise, a w ater cooled hollow-cathode lamp would
be expected to give a narrow resonance lin e and high s e n s it iv it y .
The
-1 2 3-
in te n s ity o f resonance emission g e n e ra lly increases as the temperature
o f the lamp incre ase s; however, f o r mercury an increase in the tempera­
tu re appears to decrease the in te n s ity i f the temperature is above a
c e rta in le v e l.
This e ffe c t may be the r e s u lt o f d i s t i l l a t i o n o f mer­
cury away from the e m ittin g p o rtio n o f the lamp.
The mercury Pen-ray
lamp, when operated on D.C. c u rre n t, shows a marked decrease in in te n ­
s i t y a f t e r opera ting f o r several m inutes; again, d i s t i l l a t i o n o f mer­
cury is th e suspected cause.
In te n s ity o f emission also generally
increases as the c u rre n t o r voltage is increased.
The actual in te n ­
s i t y o f a lamp, th e re fo re , is determined by the combined influences
o f c u rre n t and temperature.
A magnetic f i e l d also causes a large
change in em ission in te n s ity .
For example, pla cin g the mercury Pen-
ray lamp in an 8kG. magnetic f i e l d more than doubles the stea dy-state
in te n s ity o f the lamp.
I f th is magnetic f i e l d is then turned o f f
q u ic k ly , the in te n s ity w i l l momentarily double again before tu rn in g
to i t s o r ig in a l in te n s ity .
S im ila r ly , i f the f ie ld is above about 4
kilogauss and is in cre a sin g ra p id ly , the in te n s ity momentarily drops
before going to the ste a d y-sta te value.
Since the behavior o f emission
sources in magnetic fie ld s is extrem ely im portant fo r th is p ro je c t, I
s h a ll undertake to e xp la in these phenomena.
One p o ssib le cause fo r the change in ste a d y-sta te in te n s ity w ith
magnetic f i e l d is the o r b itin g
lin e s .
o f a ll the ions about the magnetic f i e l d
I f the ra d ii o f the o rb its are about as larg e o r la rg e r than the
-1 2 4-
mean fre e path len gth f o r c o llis io n s w ith the atoms whose ra d ia tio n is
being m o n ito re d , then
the in te n s ity o f the resonance ra d ia tio n w i l l in ­
crease as the magnetic f ie l d is in c re a s e d .1 However, i f the magnetic
f i e l d is increased to the p o in t where the o rb its become much sm aller
than the mean fre e path le n g th , then the c o llis io n ra te w i l l go down
and the in te n s ity w i l l s t a r t to decrease as the magnetic f ie ld is
increased.
For the mercury discharge lamp the in te n s ity stops in cre a s­
ing w ith the magnetic f i e l d a t about e ig h t to ten k ilo g a u s s .
I f the
pressure in the lamp is assumed to be about one m illim e te r o f mercury,
the n, by the o r b itin g e le c tro n approach, the in te n s ity reversal should
occur a t approxim ately ten kilo g a u s s , in agreement w ith the experimental
re s u lts .
A lso , the c u rre n t in the lamp decreases s lig h t ly as the mag­
n e tic f i e l d is incre ase d, so th a t an increase in the c o n d u c tiv ity o f
the plasma is not responsible f o r the increased in te n s ity .
For the hollow -cathode lamp there are a d d itio n a l fa c to rs which
should be taken in to account in any explanation o f the in te n s ity changes
in a magnetic f i e l d .
Much time and e f f o r t would be required to d e te r­
mine e xp erim e nta lIy the r e la tiv e magnitude o f c o n trib u tio n s from various
fa c to r s ; th e re fo re , I w i l l sim ply enumerate the p o s s ib ilit ie s .
F ir s t ,
*In any discharge lamp the in te n s ity o.f the resonance ra d ia tio n is gov­
erned la rg e ly by the frequency w ith which fre e e le ctro n s c o llid e w ith
gaseous n e u tra l atoms. A p a r tic le which is tr a v e llin g in a h e lic a l path
sweeps ou t a la rg e r area per u n it time than a p a r tic le moving in a
s tr a ig h t lin e and is less lik e ly to c o llid e w ith a w a ll.
-1 2 5-
in any commercial hollow -cathode lamp ferrom agnetic m a teria ls are used
to support the cathode and anode and are used in the socket pins.
This
means (a) th a t i t is d i f f i c u l t to put the lamp in a high magnetic f i e l d
w ith o u t in tro d u c in g considerable s tra in on the lamp, and (b) th a t the
magnetic f i e l d in the hollow-cathode w i l l not be homogeneous.
Second,
the m a te ria ls used to make the cathode are not u s u a lly disclosed and,
in f a c t , are o fte n considered a tra de s e c re t.
Whether the cathode
m a te ria l is diam agnetic, paramagnetic, o r ferrom agnetic is very im­
p o rta n t in p re d ic tin g the behavior o f the lamp in a magnetic f i e l d ,
since the magnitude and homogeneity o f the f i e l d is determined la rg e ly
by the magnetic p ro p e rtie s o f the cathode.
F in a lly , the magnetic pro­
p e rtie s o f the atoms o f the e m ittin g element and o f the c a r r ie r gas
are im p o rta n t, e s p e c ia lly when the f i e l d is not homogeneous.
A d e ta ile d account o f the behavior o f the atoms in a glow d is ­
charge in a magnetic f i e l d is beyond the scope o f th is paper and some­
what superfluous in view o f the lack o f extensive experim ental eveden ce on the behavior o f hollow-cathode lamps in a magnetic f ie ld .
Only one p a p e r^ ^
has been published in th is area, and i t does not
tr e a t the problem thoroughly.
cathode lamp an
For the commercial mercury hollow -
increase in the s te a d y -s ta te in te n s ity o f about 15
percent is observed when a 400 gauss f i e l d is applied .
This behavior
is c o n s is te n t w ith several mechanisms.
The anomalous time-dependent behavior o f the mercury vapor lamp
- 1
2
6
-
has been determined (a) not to be caused by the e ffe c t o f the changing
magnetic f i e l d on the e le c tro n ic s , (b) not to be caused by changes in
the c u rre n t, (c) to occur w ith both A .C. and D.C. c u rre n t, (d) to occur
w ith both resonance and non-resonance lin e s o f both argon and mercury,
and (e) to occur only when the magnetic f i e l d is over about 5 kilogauss
and to increase as the f i e l d is fu r th e r increased.
The o r b itin g be­
h a v io r o f e le ctro n s in a slo w ly varying magnetic f i e l d is e a s ily d e te r­
mined from the fa c t th a t the magnetic moment o f the e le c tro n remains
constant as the f i e l d is changed.
Thus, When the magnetic f ie ld is
increased the e le ctro n s gain energy, whereas when the f i e l d is decreased
the e le ctro n s lose energy to the magnetic f ie ld .
We observe, however,
th a t as the. f i e l d is increased, less energy is em itted from the atoms,
so the changes are t r u ly anomalous!
An increase o r decrease in the os­
c i l l a t o r strengths o f the atomic tra n s itio n s is extrem ely u n lik e ly ; so
the e ffe c t must be re la te d to the frequency w ith which the atoms c o llid e
w ith e le c tro n s .
Since the discharge seems to develop some in s t a b ilit ie s
near 5 k i logauss, which probably are re la te d to the magnitude o f the
Lorentz fo rc e ,
f = q M
,
one would suspect th a t th is force m ight also be im portant fo r the tim edependent phenomena observed.
We know th a t according to Maxwell's equa­
tio n s , a changing magnetic f i e l d must induce a cu rl in any perpendicular
-1 2 7-
e le c t r ic f i e ld .
Using these equations one fin d s th a t the Lorentz
fo rce pushes the io n s and ele ctro n s toward a region o f low e le c tr ic
f i e l d when the magnetic f i e l d is in c re a sin g and toward a region o f
high e le c t r ic f i e l d when the magnetic f i e l d is decreasing.
Thus the
discharge is probably more d i f f i c u l t to sustain w h ile the magnetic
f i e l d is in cre a sin g and e a s ie r to su sta in when the f i e l d is decreas­
in g , accounting f o r the observed anomolous changes in in te n s ity .
F in a lly , a s p e c ia l problem which arises whenever mercury is de­
termined by atomic absorp tion is the in te rfe re n c e o f ozone.
Although
i t is known th a t ozone has a broad and strong continuum absorption
band centered a t about 2550 Angstroms, the in te rfe re n c e o f ozone in
the atomic absorp tion analysis o f mercury has been mentioned only once
to the a u th o r's kn o w le d g e ^8^.
A pparently no one has considered th a t
the mercury lamp m ig h t have an e ff e c t upon the a i r in the o p tic a l path.
Most mercury lamps have qu artz windows which tra n s m it the 1850 Angstrom
lin e o f mercury as w e ll as the commonly known 2537 Angstrom lin e , thus
produce ozone photochem icalIy fo r the surrounding a i r v
'.
The ozone
concentration would be expected to b u ild up to a f a i r l y high level i f
the a i r in the o p tic a l path is confined o r stagnant.
In the apparatus
used by the author a 2 percent increase in apparent in te n s ity was found
by sweeping fresh a i r through a o n e -h a lf meter confined se tio n o f the
o p tic a l path.
,
A ls o , flu c tu a tio n s o f about one percent were observed when
a i r was swept from the lamp toward the o p tic a l path.
Therefore, in any
-1 2 8-
accurate analysis care should be taken to avoid th is in te rfe re n c e by ex­
clu d in g oxygen o r the 1850 Angstrom ra d ia tio n from the o p tic a l path.
Because o f the ra d ic a l changes in in te n s ity which take place when a
mercury lamp is placed in a magnetic f i e l d , a ll the experimental c o n fig ­
u ration s which re q u ire th a t the lamp be placed in a ra p id ly changing .
magnetic f i e l d were discontinued.
C onfigurations which re q u ire an A. C.
magnet were also e lim in a te d because o f the high cost o f co n stru ctin g
such a magnet w ith a peak-to-peak f i e l d in the kilogauss range.
The
only c o n fig u ra tio n s , then, which were s e rio u s ly considered a fte r pre­
lim in a ry study were those re q u irin g
a fix e d magnetic f i e l d around the
lamp o r the sample chamber o r both.
With a s in g le D. C. magnet two
basic c o n fig u ra tio n s are p o ssib le .
F ir s t , the magnet may be placed
around the lamp, causing the lamp to em it normal resonance li g h t p o la r­
ized p a ra lle l to the f i e l d as w e ll as p e rp e n d ic u la rly -p o la riz e d fr e ­
q u ency-shifte d li g h t o f equal in te n s ity .
Second, the magnet may be
placed about the absorption c e l l , causing l i g h t p o la riz e d p a ra lle l to
the f i e l d to be absorbed in a normal manner w h ile l i g h t p o la riz e d per­
pe n d icu la r to the f i e l d is absorbed on ly a t s h ifte d frequencies.
The f i r s t experiments were conducted using a hand magnet rated at
4300 gauss1 w ith a one-centim eter spectrophotometer cuve tte serving as
the sample chamber.
These f i r s t experiments were encouraging in th a t a
1The f ie l d in te n s ity o f the magnet was determined w ith a commercial .
H a ll- e ffe c t gaussmeter borrowed from the physics department.
-1 2 9-
d e fin ite increase in the percent transm ission o f the sample was observed
when the magnetic f i e l d was added.
However, the change was too small
(about twenty percent o f maximum) to make the method r e a lly useful f o r
analysis w ith o u t s e n s itiv e e le c tro n ic equipment.
F o rtu n a te ly , a
Varian se rie s v a ria b le D.C. electrom agnet was g ra c io u s ly made a v a ila b le
to us by Dr. John Hanton, so th a t the e ffe c ts o f a varying magnetic
f i e l d upon the equipment used in these experiments could be measured.
I t was then p o s s ib le , in tu rn , to p re d ic t w ith some confidence the type
o f c o n fig u ra tio n which would be the most useful to c o n s tru c t.
fig u ra tio n f i n a l ly chosen fo r te s tin g is shown in Figure 23.
The con­
The de­
te c to r assembly co n sists o f an IP-28 p h o to m u ltip lie r tube (as w ell as
a power supply and o s c illo s c o p e ) w ith a 2520 Angstrom in te rfe re n c e f i l ­
t e r se rvin g as the monochromator.
The use o f an in te rfe re n c e f i l t e r
norm ally is no t recommended because o f the danger o f o th e r atomic lin e s
in te r fe r in g w ith the desired lin e .
However, by using the Zeeman e ffe c t
apparatus such lin e s do no t in te r fe r e .
The key idea o f the system is
the use o f a r o ta tin g p o la riz e r to create an A.C. sig n a l a t the d e te c to r
which corresponds to I g - I^ .
I t is also useful to keep tra c k o f the
absolute value o f I y,, so th a t the r a tio ( I y. - I g jZ I y, may be measured.
The sample chamber is sim ply a glass tube w ith s i l i c a p la te s on each
end.
I t is about fou rte en inches long and has connecting tubes fo r
the entrance and e x it o f gas.
tio n
Compressed a ir is bubbled through a s o lu ­
con tainin g the sample and one m i l l i l i t e r o f twelve
percent
- 1 30-
R o ta t i n g P o la r i z e r
M agnet
Lamp — »- ( ^
)~
Detector
A b s o rp tio n C e ll
—
In te rfe r e n c e F i l t e r
Babble Chamber
F ig u r e 2 3 .
D iag ra m o f e x p e r im e n ta l a p p a ra tu s
-1 3 1 -
stannous c h lo rid e s o lu tio n made up to a to ta l volume o f about t h ir t y
m i l l i l i t e r s w ith d i s t i l l e d w ater.
The a ir then c a rrie s the mercury
vapor picked up from the s o lu tio n over to the sample chamber.
A mix­
in g tim e o f about two minutes is used to allow a ll o f the mercury to
be reduced to the elemental s ta te before le t t in g the a i r bubble through
it.
A flo w ra te o f about 0.4 cubic fe e t per hour and a magnetic f i e l d
o f 14,000 Gauss were used in most o f the experiments.
The signal from
the p h o to m u ltip lie r appears on an o s c illo s c o p e , and any absorption due
to mercury shows up as a t h i r t y cycle-per-second sine wave, whereas
the background absorption shows up only as a change in the D. C. sig n a l
le v e l.
The sig n a l observed using a 14,000 Gauss magnetic f i e l d and a
one-centim eter c e ll co n ta in in g sa tu rated mercury vapor is shown in
Fi gure 24.
The s e n s it iv it y o f the apparatus was measured as a fu n c tio n o f the
magnetic f i e l d s tre n g th ; a p lo t o f I / I
, which is a measure o f se n si­
t i v i t y , vs. the magnetic f i e l d stre n g th is shown in Figure 25.
Most o f
the ir r e g u la r it ie s in the curve are due to the h yp erfin e s p lit t in g o f
the mercury lin e ^ 2^ .
The absolute s e n s it iv it y o f the apparatus a t
14,000 Gauss is illu s t r a t e d by the c a lib r a tio n curve in Figure 26.
One
o f the fa c to rs preventing the method from being more accurate is the
r e la t iv e ly high noise le v e l o f the o u tp u t s ig n a l.
This noise o rig in a te s
la rg e ly from the f l notations o f lamp in te n s ity , power-supply v o lta g e , and
p h o to m u ltip lie r a m p lific a tio n , and could probably be reduced by a fa c to r
-132-
Lamp I n t e n s i t y
F ig u r e
2k
D ia g ra m o f o b s e rv e d s ig n a l
-1 3 3 -
k ilo g a u s s
F ig u r e 2 $ .
S e n s i t i v i t y v s . m a g n e tic f i e l d s t r e n g t h
- 134-
Box h e ig h t r e p r e s e n ts maximum
n o is e l e v e l .
F ig u r e 26. O p t ic a l d e n s it y v s . q u a n t it y o f m e rc u ry
-135o f ten o r more by th e in s t a lla t io n o f b e tte r e le c tro n ic equipment.
Background sig n a ls a re observed in almost every d e te rm in a tio n , even
though reagent-grade m e rcu ric c h lo rid e was used, in d ic a tin g th a t such
an instrum ent would be very u n re lia b le in the single-beam mode.
Absorbances as g re a t as 0.2 were observed f o r w ater vapor alone.
Other
workers have also n o tic e d larg e background absorptions in the determ in­
a tio n o f mercury by atom ic a b so rp tio n ^
I have added im p u ritie s
such as benzene and to lu e n e to the samples and have observed only in ­
creases in the background a b so rp tio n , thus s u b s ta n tia tin g the hypo­
th e sis th a t only mercury w i l l absorb so as to give an
A. C. s ig n a l.
CHAPTER I I I
Conclusion
The experim ents, both h yp o th e tica l and re a l, described herein in ­
troduce a new method fo r changing atomic absorption analysis from a
single-beam to a double-beam method.
The success o f the experiments
using the Zeeman e f f e c t , along w ith comparison to re s u lts not c o rre c t­
ing fo r background a b so rp tio n , demonstrates the
in g
a
advantage
of
us­
double-beam system, a t le a s t in the analysis o f mercury.
I have
shown th a t many problems must be overcome in order to b u ild a re lia b le
frequency-modulated apparatus, bu t th a t, i f the problems can be over­
come, as in the case o f mercury analysis o f the Zeeman e ff e c t , th a t an
a n a ly tic a l instrum ent o f high accuracy and r e l i a b i l i t y can be b u ilt .
The e lim in a tio n o f background absorption allows one to c o n fid e n tly mea­
sure (w ith o u t perform ing chemical separations) q u a n titie s o f mercury
which are orders o f magnitude sm a lle r than the q u a n titie s which can be
determined by the single-beam method.
The apparatus is extremely sim­
ple in th a t i t performs a dual-beam experiment w ith only a s in g le
source and a s in g le d e te cto r.
With only mercury 199 isotope in the
lamp, the magnetic f i e l d needed to e ff e c t a complete separation o f the
s p l i t lin e s from the absorption p r o f ile can be reduced to a b o u t^ H e n
kilo g a u ss, so th a t the apparatus can be made p o rta b le .
A sm a lle r magne­
t i c f i e l d could also be used w ith an evacuated absorption c e ll s im ila r
137
to th a t constructed by Goleb^23^ , since the absorption p r o f ile is much
narrow er in a low-pressure c e ll.
S im ila r ly , resonance lamps^24’ 25^
em it very narrow lin e s , thereby a llo w in g the complete separation o f
Zeem an-split lin e s w ith a lower magnetic f ie ld .
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762 1001 0772 9
cop. 2
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