UI
m-4- ;
D
R00
V
36- 1
NUCLEAR QUADRUPOLE ....YPERFIN
NYP!RFINI::STRUCTURE
STRUCTURE
IN SLIGHTLY ASYMMETRIC ROTOR MOLECULES
GEOFFREY KNIGHT, JR.
B. T. FELD
La6/A
CO'
I
TECHNICAL REPORT NO. 123
JUNE 10, 1949
RESEARCH LABORATORY OF ELECTRONICS
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
Y~~.bl
The research reported in this document was made possible
through support extended the Massachusetts Institute of Technology, Research Laboratory of Electronics, jointly by the Army
Signal Corps, the Navy Department (Office of Naval Research)
and the Air Force (Air Materiel Command), under Signal Corps
Contract No. W36-039-sc-32037, Project No. 102B; Department
of the Army Project No. 3-99-10-022.
Preface
The following report contains excerpts from the doctoral thesis of
Geoffrey Knight, Jr., "Interaction of Nuclear Electric Quadrupole Moments
with Molecular Rotation in Slightly Asymmetric Rotor Molecules." It is concerned with the theory for the interpretation of the hyperfine structure
patterns of rotational transitions in slightly asymmetric rotor molecules.
The approach has been to approximate the molecular wave functions in terms
of symmetric rotor wave functions, and to derive explicit expressions for
the quadrupole interaction in terms of the quantum numbers appropriate to
the molecular states and of a single asymmetry parameter which measures the
departure of the molecule from a symmetric top.
The material is being prepared for publication in The Physical Review.
This report is issued in order to make the results available to workers in
the field as rapidly as possible.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Research Laboratory of Electronics
June 10, 1949
Technical Report No. 123
NUCLEAR QUADRUPOLE HYPERFINE STRUCTURE
IN SLIGHTLY ASYMMETRIC ROTOR MOLECULES
Geoffrey Knight, Jr.
B. T. Feld
NUCLEAR QUADRUPOLE HYPERFINE STRUCTURE
IN SLIGHTLY ASYMMETRIC ROTOR MOLECULES
Introduction
We wish to determine the energy of a nuclear electric quadrupole in the
field produced by the electrons and the other nuclei of a molecule. This
problem has been solved for the atomic case by Casimir (1) in an investigation of the source of deviations from the interval rule in the hyperfine
structure of atomic spectra. Nordsieck (2) has extended the solution to
the linear molecules HD and D 2 and Jauch (3) and others have solved the problem for symmetric top type molecules. Recently Bragg (4), making use of
previous investigations of the wave mechanical and matrix mechanical descriptions of a rotating asymmetric top (15), has obtained two solutions for the
general case of asymmetric top type molecules; one formal, and the other involving interpolation in published tables of line strengths.
The objective of this paper is the evaluation of Bragg's formal solution
for slightly asymmetric molecules so that the pattern of the interaction is
made clear, and so that numerical results can be obtained more readily. The
formulae obtained should be useful in the identification of the fine structure of rotational transitions observed in microwave spectroscopy (5), of
the subsidiary minima observed in molecular beam measurements (6), and ultimately of the hyperfine structure of electronic transitions in molecular
spectroscopy. Also they should simplify the evaluation of the products of
certain molecular constants with the nuclear quadrupole moment, once identification has been made.
I. Formulation of the Problem
Classically the energy of a charge distribution p(r') in a field Eext
produced by other charge far away from the given distribution is (7):
Q
1 ~
W = Ze ,ext+P.
r=O + P ' Eext
E=O - E
V"extO
+
..
S
where Ze is the total charge, P = r'p(r')dr'
is the dipole moment, (which
is zero for a nucleus), and Q =
(3r'r' - r't)
p(r')dT' is the quadrupole
moment dyadic of the charge distribution. We will be concerned here only
with the last term.
Classically the quadrupole moment dyadic is the charge-weighted average
of the simple dyadic (3r'r' - r'2). The corresponding quantum mechanical
average for a nuclear quadrupole is:
<Im i
IQ
J
imI
, 2
I,mi(r ' e,)r'
4I,
A
rUr
-
)
(r1,8e1,)r12 sine' dr'de'de'
-1-
J
where T',mI is
the wave function describing the state of the nucleus and
Ur, is the unit vector in the r'
must have the form
4I,m
direction.
(mi)
(I,,)
g(r)YI
completely specify the state; the
if the nuclear spin I and its component m
integral then becomes:
< Im I IQ Im,>
Jg*(r )rs g(rt)dr '
8
A
A
(mI)
YI
($,$
(m I
)(3Ur ,Ur,
-
)
1)Y I
(8',$')sin,'
d'd4'
A
and V'.
A
We would like to replace Ur, in this expression by some operator with
This can be done if we note that
known eigenvalues belonging to I,m.
since U,
depends only on 8'
3/2(II + II) - I21 has the same angular dependence as (3Ur,Or, -1) so that
its substitution as the operator in the matrix element in place of 3r'r' r'21 will change only the integral over r'.
<ImIlQIImI>
=
(II +
const <I
The symmetrized form II + II (tilda
I) - I21 Im>
.
transposed) must be used to correspond
i.
to Qij = QJi since IiIj A IjI i , i
To evaluate the constant we may consider the expectation value of the
I:
ZZ component of Q for the state m I
eQ
I
zzI II>
- I2II)
=
const<IIj3I
=
const I(2I - 1)
where
eQ =
4I. I (3Z'2 _ r'2 )
I,
dT'
is called "the" quadrupole moment.
The other factor appearing in the quadrupole term of the energy is VE.
If it is noted that classically VE is the charge weighted average of the
simple dyadic -/r
that
5
[3r r - r2 1], we find by the argument just applied to Q
=op
J'(2J'_
(J'J'
-2-
+ JJt)
-
12 1]
where
q-
(3cos(zr)
dT
and J' is the total angular momentum of the molecule exclusive of I.
The operator for the quadrupole term of the energy is (-1/6) times the
dyadic dot product of these two operators:
1
Wop = -
=1
.
Qop : VEop
e2 C1Q
(2I - 1)J,(2J'
K1
(
-)
1i
2
J
3
+
$')
I
2j,2l
)
and the energy of a nuclear quadrupole in the electric field in the interior
of a molecule, when the nucleus has spin I and the rest of the molecule as
a whole has angular momentum J', is the expectation value of this W
in the
~' -1.
op
state F = I + J', assuming this quadrupole interaction is small compared with
the fine structure so that F remains a good quantum number. This expectation
value is
-.
-_~.
e2Q
w
C(C + 1)-
)J'(J' + 1)]
I(i2+
where
C
F(F + 1) - I(I + 1) - J'(J' + 1)
I, J', and C are numbers that specify nuclear and molecular states. The
constant Q is the quadrupole moment of the nucleus, since it depends only
on the nuclear spin, I, it can have only one value for a given nucleus in
its ground state. Therefore it is a number also; a number that we hope to
be able to determine by measuring W. Thus of the various quantities that
appear in this formula only eq requires further investigation.
II.
Evaluation of eq by Wave Mechanics
By definition eq is the charge-weighted average of the function
(3 cos 2 (Z,r) - l)/r for the rotational state J',J' · ZU = J'; i.e.
eq J
Jj
mole
cos r3
er)
- l)
dT
mole
where r is the radius vector from the quadrupolar nucleus to the volume
element d which contains an amount of charge t'*dT. In so far as the molecule approximates a rigid body, the charge density 9 * can be split into
two factors, one depending on the position of d in a molecule fixed system
with coordinates r, ", 0" and the other depending on the orientation of
-3-
the molecule in space. This orientation is most simply described in terms
of Euler's angles (8):
Here A, B, and C are the principal axes of the
z
-..
.
m
molecule. tOu is the intersection of the AB
B
and XY planes so it is perpendicular to both
c
C and Z.) Then
(*v))J
/(
= ()p(r,6",O")
(44)JJ(OOP)
where J is the rotational angular momentum of
the molecule.
A
2Z3
N
Likewise the function 3 cos (Z,r) - /r 3
can be expressed in terms of the polar coordinates of r and Z in the molecule fixed coordinate system by the addition
theorem of spherical harmonics (9).
The combination of these two expansions gives:
+
j'*4
[*
{
[
J
+
* *
9PFqfp
sin29" sin2O sin(O +
[ 3 sin2e"sin29 cos2(+
3
.4
0"
d dr
1)
upJJdTpdTrot
(a)
The averages over the molecular charge are characteristic constants of
each molecule in a given electronic and vibrational state; we will not consider them further.
The evaluation of the rotational integrals for slightly asymmetric molecules is the actual contribution of this paper.
C
N
A
Polar angles of r and z in the molecule-fixed coordinate
system. Note that the polar coordinates of z in the ABC
where
and
are the Euler's
system are
and n/2 angles of ABC in the space-fixed system.
-4-
The wave functions ~JJ that we require are solutions of the following
wave equation which describes the rotation of a rigid asymmetric body (10)
(11i):
(A + b cos2e)
++
¢ os2)
(Ao
(As- b
2b
1
a (siea '3)
2
22cose a2i
2-1
- cose
s
+
+ (1 + b cos2*) a
2
(1 + 2cot2e) a
2b cos2* cote ~ + b sin2
- 2b sin2* cote csce Up + N' = 0
where
Ao= o +
b-'-
;
X= L
_-
C=
F ~-a
+·
This equation cannot be solved in closed form. However, the part of it
that remains when b, the parameter of asymmetry, equals zero, can be solved
by separation of variables. The result is the wave function for the symmetric top (12)(13)
O
(e
)
ei(Ew)
=R
F(-J + /2-
.
+
1,J +
K
cosj
/2;
ose
- MI; 1
+
1I
MI
co
where F(a,b;c;x) is the hypergeometric function,
- IK + MI + IK - MI + 2,
and NJKM is the normalization constant. K and M are separation constants
that must be integer if f is to be single-valued. The third quantum number,
J, is a positive integer such that IKI
J > IMI.
LzJKJ
JKM = M"JKM
h
and
LCjK =c
_At1 JM
=
K+JKM
so Mh is the component of angular momentum along the Z
ponent along the symmetry axis. Therefore, since J is
each of these two components, J(J + 1) + 2 must be the
angular momentum.
In our problem we need to deal only with the case
tion then simplifies to:
-
(2J(+ 1)n2
(J +
)!(J - K)2
[1 + cose] J -
-5-
axis; ihis the comthe maximum value of
square of the total
M
K [1 - cos]
J; the wave func-
J--
ei(K + J)
If this solution is substituted into the asymmetric top Hamiltonian
from which the wave equation was derived, we find that (14):
a JKJ
=
-
1) + K ]JJ1
[oJ(J+
1
[( + K + 1)(J + K + 2)(J - K - 1)(J
+ l)(J - K+
(J -
)
+
2)(J + K - 1)(J + K)]
K-2J
in units of h 2 CI2.
The first term on the right is the energy of a symmetric top type body.
If we write Ha = Hs + H', we see that the only non-vanishing matrix elements
of H' for symmetric top states are:
<J K + 2J IH'I JKJ>
-bF(J,K + 1)
<J K - 2 IH'IJKJ>
-bF(J,K - 1)
and
where
F 2 (J,n)
[J(
+ 1) - n(n+
1)] [J(J+ 1)
-
1)]
n(n-
Since the symmetric top wave functions form a complete set (13), we can
expand the asymmetric top wave functions in terms of them. We have just seen
that the Hamiltonian is diagonal in J and M, so the expansion has the form:
NT~=K'
z
SKJKJ
K
Then for any function P(O,*,4j),
'
SKXSKI,
Y';JPJTJdT
9 '%NK
P
,;;d
.
(b)
K,Kt
We will apply this equation to each of the terms of eq in turn.
4J*
1.
The
(3
sine d
d
d
dependent part is
2
The
os2 6 - 1) ,J
-iK,iKId
2r KE
dependent part is
C2 (J,K)
The substitution X-
(1 + cose)J+K(
- cose)J'K(3 cos2
1/2(1 - cosG) reduces this to:
-6-
-
1) sin
d9
C2(,)j'
-
C2(J,K)2+2+
[2(l -
2
2 x]
)]J+t[
(1 - x)J+K(x)J-K
2
x
[1
6x(1
-
x)]
-
2dx
(1- - x)J+K+l (x)JK+l dx.
6
According to integral No. 482 in Pierce's Tables this is:
2(2J + 1)!
(J + K)'(J(J + K).(J - K)'
I2J+
2J
J
2.
K)
[-6 (J + K+
1)(J - K+ 1)
)2J + 3
+
sin29 sin(* + *')
sin
3
d
d d'
The 9 dependent part is
e'ik
sin(
')eiK'd
+
= 0
for K' A K + 1. We will find that if 5, T 0, S.+l,T - 0
Therefore, all terms of the second group in eq vanish, either because
of the vanishing of the integral or because of the vanishing of SK, TSK+1,
.
sin2
JOI
3.
The
Cos2(
1
+ *s)
dO
d
T
dependent part is:
f1 '
e-iX. [e2(
+')
+ e-2i(+*')]elKd2'
2
2l('-K+2)
1
)
The
,K'J sin
K+2
5K-2
dependent part of
f
IGJ
TJJ sin2 e
9
JK+2J dT
is
5
C(J,K)C(J,K + 2)(1 + cosO)J+K+l(l - cos8)j 'K1
= C(J,K)C(J,K + 2)
i22(1
)J+K
x +I xJ-KI
In the K + 2 case this is:
-7-
x(1
sin3
dO
x)2dx
T
22J
8
C(
XJ-K dX
K+2)J++2
(2J
+ )!
(J - K - 1)(J
tJ + K~jkW-KT4(V +
-K)
2)
J+ K+ 2)!(J (2+
2J+)1
2F(JK + 1)
4
40(2J + 2)(2J+ 3)
In the K - 2 case it is:
8 ·
2 2J
(1 - x)J+K
C(J,K)C(J,K - 2)
(2J
4- (T3 K kI
1))
1) +
(
+
xJK+2 dx
++
:-K + 2)
HT+ K)(J
K)J
K+ 2)!1
K)(J-
+ 3)
2F(JK - 1)
4
-J
+ 2(2J +
t=
)
Combining these results with Eq. (a), page 4, and with Eq. (b), page 6, and
noting that
S
BK, TSK-2,
K
Ffi,-l)F(JK)
(J
+ 1
2J
=J
2
+
3
K
we finally find that
eq = (+
1 )( 2 J+3)
+ 1)
j
T
3K
J(J+l)
*cs
l
dr
-23K,
K+ 2 F(J,K + 1) .
* 3 sin2e3 cs2d
Bragg's
formula
(4).
This is identical with
Only the SK,'s, which depend on the degree of molecular asymmetry,
remain to be evaluated. They cannot be obtained in a closed form in the general case, but if the molecule is nearly symmetrical we can find an approximate solution by means of perturbation theory.
We wish to find the eigenfunctions of the Hamiltonian H = H + H' where
a =E
E = AoJ(J + 1)
H' is small, of the order b. We know that H°
+ K where n represents J, K, J and a- K/IKI is the quantum number of degeneracy. Let H - E and expand 'Pin terms of the symmetric top wave functions:
-
7
an,a'4n,a
n,a
(These an,,, will become the SKT1s when they are properly normalized.)
Then
-8-
I
-
(H-H°W = H'
gives
n,a
na
nm,a
n
j n,
aT
(E - En) ani
am <n IH'IX
*(1)
)
m,
If b, and therefore H', is small, we expect the energy values and the
eigenfunctions to differ very little from those of the symmetric top. Therefore as a first approximation we will assume that am, << a,= if E Ep.
Then Eq. (1) becomes
~
n,a _a
\7-P<no
a ,a
IH'I PB>
E - E°
p
n
Substitution of this approximation back into Eq. (1) gives as the second
approximation:
(Ep -
) a
r<nalHc'Ip>
ap
-
L3'
<ncaIH'I my><m lH'IPa>)
+
m
Ep-
y'sp
j
(2)
For the case n = p this is
a,
aIH'ImY
<PClI p>+
<mIH'IP>
'YAAp
L
(E
-
Ep)6
-
0
m
(3)
The solution of Eq. (3) will give the ratio ap, 1 /a, 2 as well as the
energy Ep. Eq. (2) will then give a,a n p in terms of one undetermined
constant, say ap 1' the value of which will be fixed by normalization.
In applying these results to our problem we must consider these equations for each value of p in turn, making use of the fact that
<JKJIH'I
JK + 2J> = -bF(J,K + 1)
are the only non-vanishing matrix elements of H'.
For K = 0 Eq. (3) gives
E = E0 +
E
<O 0
IH'I2>0 2 + <IHI
- E22
For K = 1 the secular determinant is:
-9-
E
2>2
- E2O-
(E1
E0)
<1 IH'I 3)>2
- <1 IH'I - 1>
o
3
BO
1
=0
- <lI'l- 1>
(E1
-
E)
-
<-1I H'I-
.
3)2
D1 3
Now
<1 IH'I 3> = -bF(J,2) - -<l IH'I - 3>
SO
E1+
=
E
+
_.
- 1> + <O1 IH 1
~~E1
- E3
<llH
The first reduces Eq. (3) to a1l
For K
.
a_; the second gves a
-all.
2 the elements Tii of the secular determinant are:
<21H'I >2
T 1 1 = E 2 - E - <2IH'Io>
E2 - E o
= _-2 H'I ><0 IH'I - 2)
T12
E 2 - E°o
T2 2
= E2 -
E2
2
<-21H'I0)
(
<-2E E2 - Eo
T 21
=
=
4
T21
<-2 IH'I-
2
)4>
.
Nov
<2 H'I10) = -bF(J, 1) - <-21H'0)
and
<2 H' 4>
=
-bF(J,3) = <-21H'I- 4)
80
E2+
-
E2+ o2 IHI
E 2 - E4
+
is the solution of the secular equation. The first reduces Eq. (3) to
a22 = a 2 ; the second gives a 2 = -a2.
For K = 3 the elements Tii of the secular determinant are:
-10-
11
<3 HII 1>2
E 03
E01
3
3
<3 IH'I 5>2
E - E
3
5
T1 2 = 0 = T21
T 2 2 - E3 - E - <-31 I.I- i>21
E33) E°1
-31 E-
5 )2
E03~ E 5
so
E3+ =
_
<31'HII 5>2
O31H'
>2
+
0
E 0 - E5
L
3~ 5
The substitution of this result into Eq. (3) does not give any relation
between a33 and a33 , but we note that if terms of the third order approximation, namely
<31'1 ><l IH'I - ><- IHII- 3>
had been included the result a3 - + a33 would have been obtained and E3+
would no longer have been degenerate.
For K > 3 the solution of II is:
+ <KIHIKE
- 2>2 + <KIHI K + 2>2
4 EK 2
E4KEK+2
EX
a
a +
K
Thus
0
o-
2
(J,
2
E1+ = E1 + bF(J,O) - b 2 F
(J. 2)
8
F 2 (12
' -) +
-2
= E 20
2 - b
0
.
E3,
·
·
Eo
3 +b
·
·
2
·
2
F2(J,.2)
- b2
8
·
·
-11-
·
·
u42 a··
b2
b F2(J,X + 1)
l)
b2 F2(JK
E + b
EK± E+
-oE+
The + subscript corresponds to the symmetric-antisymmetric combinations
Substitution of these values of
of degenerate symmetric top wave functions.
E into Eq. (2) and subsequent normalization give the S'8s which can then
The resulting values of eq depend only
be substituted into Bragg's formula.
We obtained the
on rotational quantum numbers and molecular constants.
following values of the SKT's and formulae for eq:
b F(Jl)
2o
K = 0
2o
300
) lb
-b b42 ?
=
-2o
a2 V
=
eq = -
..
az
(J,
1
)(2J + 3)
(J +
2V 2
b2F2(J,2)1
a2 V
aSy'
lax'
531
K - 1
b F(J,2)
31
J2
SllS
11
S-3
az
=
b2
+
F2(J,2)
1 2'
___
1
3 - J(J + 1) +
(J + 1)(2J + 3)
2 F(J,O)F 2(J,2)
-
F(J,2) [1 8 + bF(J.0)1
[a2
_F(J,O) +bF2(,,2
+
[1 + bF(JO)j
b 2 F2j,2)
12_
IF
±
3-31
2V
8
1
I-
-11 ="
eq =
F(J,)
[b F2 (J,1)]
-
..
1)2
+j~i1
f~-j(j2
* [-(J
+
b
a2
-a2
a2
y
32or O
or 0
I
-12-
2
}
8
K = 2
KS22
=2
=_
s2,2
1
b2
T2
/-2
-
I
1
2 - J(J + 1) + 3 b2F2(J3) aV
-b2(J, 1
42
ax
'7
So,2
b 2 :2,.3)
1
288
:
=0
-
1
b 2 F 2 (J3)
=-2
+ 288
b
F2
1
ii-
(J
12
- J(J +
(J "1)(2 +
3
ay
12
-
$-2,2
S_4,2
-
12
F(, 3)
=
,
s2,2
a2 v
2v
'I
Bt,2
F2V,
.. b
12
F2 (J,3) -
a2V
=
-az
+
288
(J + 1)(2J + 3
K= 2
. . eq
2 (T')
b F(J,3)
S-4,2
a2v
a7
ib
+
'-b
7 2-=F(J, )
-2,2
. eq = a-Z
+2Jj ¢J'
U
2 L 288
F
S_4
o,2
2
b
-IZ
-13-
)+
b2F2J,3)]a 2
K >2
SK+2K
= r
b F(J.K + 1)
4
-T2
-2
a vb
*
+
)
4K
- )
F 2(jK_
2 L 32( + 1)
32(K - 1)
F(J
eq = 72 - (J +
K
+
)
(..
)(32J +)
(K+
+
1)
_ 1-2)F~~~2(2~iJ
- 1)
a2V;
-b
2(K
K -)
-12)
1
b--~2IF2(jK + 1
aJK
[b
, 'bL
a2v
Z
K(+ 1)
+ )
These values of eq should be in error only by terms involving b to the
third and higher powers.
The following table and graphs give the coefficients of
2V
2V
Kax2
in these formulae for eq. (It is important to note the term that is independent of b and which, therefore, does not appear in the expression for
eq when K 1.) Calculations are simplified by use of the identity
+ 1)ner
- nCross
I (15).
For
byrefJ,n)ce
I of
Table
toKing,
or by reference to Table I of King,
arner and Cross I (15).
-14-
00
co
o
te
ai
A~
,j
~4
H
O
0o
i
'*
C
CO
cu
I0
N
vo
\D t
in
0
i
O
o
oMJ
'0
in
f
fn
am
H
I
s
CU
o
0
*
OH
I
I
I
U
U
a
C
oO
CO
o
in
0
LA
n
*
HO
H0
O
00
aO
0
0000
0
'
M
0
LH
-!
O\
D
0
0
0
0
0-I
_
0
OO
C*
o
cuo
cQ __
I W
_ _ __
0
0
0
ic
0
o
O
N
I
I
0
-I
I
c
o
o
c;
IO
0*
O-!
O
CU
cm
CU O 0I~
-15-
0
At
In
H
0
t
c\
CUH 0
0
!
\
o
o-
ts
K\
W
c\
CU XD
CO
*
** UL
Ln
'
O
LA
0
cu
H
*t
in
I.\
'0
~0
tt-
CO
co
-I
-3
-4
-61
2
Fig. 1
3
4
Magnitude of coefficient of -b
5
2
-
6
/
ay'
-16-
7
8
vs. K for different values of J.
;
'
K=3
*1
-2
-3
-4
-5
-6
J-
Fig.
2
I
1
-
I
lI
I
I
2
3
4
5
Magnitude of coefficient of -b
-,V)
2ax
-17-
ay
2
vs.
-
I
I
6
7
K=2
8
J for different values of K.
III.
Application of These Formulae to Experimental
Measurements of Rotational Fine Structures
It was shown in Part I that the energy of a single nuclear quadrupole
moment in the electric field in the interior of a rotating molecule is:
C(C+
2e
uQ
1)-(
)I(I
+
2I(2I1 - 1J2J -
q(J,)Q
+
)
1)
where
c
)-I(I + 1)-J(J + 1)
F(J +
so that in general each term for given J is split into a number of levels.
Therefore, in any of the allowed transitions there will be fine structure.
The magnitude of the separations of these fine-structure components will
be simply:
Shu
[WQ(FlJ
1
Tl )
-
WQ(F PJ
2
2
'T 2
-
[WQ(F3-JlTl) -
Q(.4'J2 'V2 )]
3/2.
1
J - 2 transition for I
As an example let us consider the J
The rotational energies of the initial and final states are shown schematically (for a more complete diagram see Herzberg, "Infrared and Raman Spectra").
JK IK I
f/
K= ±2
-~220
2
221
0
K=:I
212
-I
202
-2
K:=O
K:±+_l
J= I
110
IllI
0
101
-I
K=:O
Here the first vertical group of levels refers to symmetric top energies;
the second group to asymmetric top energies. In the next column the large
numeral gives the J value and the small numerals give the K values of the
limiting prolate and oblate symmetric tops. The numbers in the column
labeled T give the order of the energy levels.
v = K_1 - K1 .
Now each of these levels is split further by the quadrupole interaction.
In our example these separations from the unperturbed rotational levels are:
-18-
I
J
F
3/2
1
5/2
1 e2 q(1,T)Q
3/2
-e2q(1,'r)Q
2
WQ
1/2
5 e2 q(J1,)Q
7/2
I e 2 q(2,,)Q
5/2
- ;
e 2 q(2, )Q
3/2
0
1/2
7 e 2 q(2,r)Q
and the frequency fine structure is:
1 q(l,rl)
-
q(l,'rl) +
q(2, T2 ) - -At i
5
q(2,,
2
)
+ 5 q(2,r2 )
-q(l,rT)
-q(1,r1 )
-
-Au I
-
'
= -AI- 3
J q(l,T1 )
J
q(1,
1
) -
q(2,r 2 ) = -AVI
1
'7 + 'ff
where A- h/e 2 Q.
If we now subtract one of these lines, say the
+ ,
from each of the
others in turn, we obtain:
* q(2,'r 2 )
= -A
J.
5
- ~ J~~-u
7 = -AD1
-
q(l,tl)
+
q(2,r 2 ) = -A -I.
-19-
X
q(1,rl1 ) +
q(2, 2 )
q(l,r1 ) +
q(2,T2 ) = -A(
q(l,
1
)
-
.@
-A
-A
q(2,T 2 ) = -A(ol
-A=
l
-;
7
3
-A
5
These correspond to the numbers obtained in the measurement of fine
structure; that is, the separation of each line from a given line.
We have here the only possible pattern for J = 1 J = 2 fine structure,
essentially in terms of the single parameter q(2, 2)/q(1, l), as far as
ratios of frequency differences are concerned. Similar patterns can be
written for all specific cases of J1 l J 2.
In order to identify an experimentally determined set of fine lines for
a slightly asymmetric molecule once the J1 and J2 values are known, we may
need only the zero-order approximations to eq, i.e. those without b if b is
sufficiently small. In that case the ratio q(J,0
2
2 )/q(Jls
1
1) reduces to a
set of numbers, one for each pair of values of 1 and 2 and the ratios of
frequency differences are completely determined. There is one exception;
if either T1 or 2 corresponds to K - 1 the ratio of the q's is not in general a pure number if
2
aX'
ax1 -
as yais comparable to a2
azI
so there is nothing to be gained by substitution for the eq's.
Once identification has been made the last set of equations can be
solved for eq(Jl,'rl)/A and eq(J 2,T 2 )/A and our formulae can then be used to
obtain values of
a 2v
a2v
eQ
,
eQ --y
axe1ay?
, and
a 2v
eQ -
aZ2
Very recently Mr. J. H. Goldstein (16) at Harvard has measured the separations of the components of two of the rotational lines of vinyl chloride,
C2N3C35, which he has identified as llo0
211 and 111 + 212 transitions,
and very kindly permitted us to use his data.
-20-
c 2N3c1
Transition
Component
110-' 211
5/2 - 7/2
111 -'212
35
Separatioi from
Principal Component
(5/2- 7/2) in Mc/sec
0
5/2 - 5/2
- 6.35*
11
3/2
5/2
-14.31
"2
3/2 - 3/2
- 9.79
03
1/2 - 3/2
4.51
04
1/2 - 1/2
10. 89
05
0
5/2 - 7/2
5/2 - 5/2
- 7.79
3/2 - 5/2
-14.37*
3/2 - 3/2
- 8.82
3.10
l3
04
10.85
05
1/2 -3/2
1/2 --1/2
°2
*The frequency of one line in each group has been altered
slightly; 6.65 -*6.35 and 14.17 -14.37.
When these frequencies are substituted into the last set of equations,
we obtain
q(1,O)/A = -5.28 + 0.03 ergs
q(l,l)/A = -6.36 + 0.03
q(2.0)/A = +7.32 + 0.05
q(2,-1)/A = +8.88 + 0.04
where the root mean square deviations of the various determinations of the
q's are given.
Our formulae are
eq(l,O)
eq(l, 1)
eq(2,-1)
eq(2,o0)
1 a2v
= + Ta z 2
r+
(a2v
a 2w
ay 2 _
axl
ay
1 ra2v
al
a2v
a 2v
2v
1 aa
+ 2v
7 L3z'2
\ax'
1
a 2v
(a2v
7
LaZ 1
-21-
aX1
ax'
as~~i
-
/-~z2
a2v
aS
)1
.
Substituting the experimental results gives
eQ
= -58.2 + 0.3 Me/sec
-82v-
azl
eQa v
=
-56.7 + 0.5 Mc/sec
=
-57.5 + Mc/sec (average)
a2v
aY'
a2
\ax'
-
5.4 + 0.6 Me/see
0.
5.5 + 1.0 M/sec
= -
Applying Laplace's equations, V2 v = 0, we find that
2
eQ
= +26.0 Mc/sec
a 2v
eQ --ay'
= +31.5 Mc/sec
e ax,
These average values agree perfectly with Goldstein's evaluation using
one of Bragg's formulae and interpolation in the CHK tables. However, the
3-percent difference between the two values of eQ a2 v/az 2 may be significant, possibly indicating a second order effect.
Using the value Q = -0.079 x 10-2 4 cm2 we find (17)(18):
+4.8 x 106 ergs/cm 2
-a
e
2
a2v
e ---
= -2.2 x 1066
ax'
e
a2
-2.6 x 106
Appendix
Relation between the Parameters of Asymmetry K and b
and c
Label the moments of inertia so that A
1/C.
K
Then
2-
2(b
(a
_ + c)
a-e
c-+
a-c- (c
;
If B
C,
b2= 2a
-b a -b
2 - (a + b)
o-b
(b+ e)
-
-a
'
2
-
a
-
b -1/B,
Let a -1/A,
a)
-
K+l
b-c
b
If A~ B,
B < C.
c)+
b-o
-c
b - c
a- c
- cb -s
) - (b -c)
K-
1
K
I
K-+ 1
Note that Jbl g 1/7 over half the range of K and Ibl < 1/3 for any degree
of asymmetry.
-22-
References
(1) H. B. G. Casimer, Archives du Musee Teyler 8, 201 (1936)
(2)
A. Nordsieck,
Phys. Rev. 58, 310 (1940)
(3) J. M. Jauch, Phys. Rev. 72, 715 (1947)
(4) J. K. Bragg, Phys. Rev.
(5)
J. Bardeen and C.
4, 533 (1948)
. Tawnes,
Phys. Rev.
,
97 (1948)
(6) B. T. Feld and W. E. Lamb, Phys. Rev. 67, 15 (1948)
(7)
J. Schwinger, Unpublished lecture notes
(8)
e.g. J. C. Slater and N. Frank, "Introduction to Theoretical Physics,"
McGraw-Hill Co., New York (1933)
(9)
e.g. E. U. Condon and G. H. Shortley, "Theory of Atomic Spectra," Camb.
Univ. Press (1935)
(10)
See also H. A.
, 60 (1927).
E. E. Witmer, Proc. Nat. Acad. Sci.
Kramers and . P. Ittmann, Z. f. Phys. 53, 553 (1929); 58, 217 (1929);
60, 663 (1930). 0. Klein, Z. f. Phys. 58, 730 (1929)
(11)
F. Reiche and H. Rademacher, Z. f. Phys. E,
(12)
R. de L. Kronig and I. I. Rabi, Phys. Rev. 29, 262 (1927)
(13)
See also L. Pauling and E. B. Wilson, "Introduction to Quantum
Mechanics" 275, McGraw-Hill Co., New York (1935), and H. Margenau and
G. M. Murphy, "The Mathematics of Physics and Chemistry" 352, Van
Nostrand, New York (1943)
(14)
S. C. Wang, Phys. Rev.
(15)
G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys. 11. 30 (1943)
4441 (1926); 41, 453 (1927)
4, 243 (1929)
(16) J. H. Goldstein and J. K. Bragg, Phys. Rev. 75, 1453 (1949)
(17)
L. Davis, Jr., B. T. Feld, C. W. Zabel, and J. R. Zacharias, Phys. Rev.
56, 677 (1939)
(18)
L. Davis, Jr. and C. W. Zabel, Bull. Am. Phys. Soc.
29, 1948)
*
*
-23-
23, No. 3 D10 (April
__