Electron paramagnetic resonance spectra of trivalent chromium in MgO
by David Hugh Dickey
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Physics
Montana State University
© Copyright by David Hugh Dickey (1969)
Abstract:
The electron paramagnetic resonance spectra of dilute trivalent chromium at sites with axial symmetry
in single crystals of magnesium oxide has been studied in detail. In particular, the angular dependences
of resonance line intensities and field positions for Cr53 have been investigated.
Trivalent chromium substitutes for the divalent magnesium ion in Mg0, and a significant fraction of the
chromium ions are in association with a next-nearest-neighbor cation vacancy. The presence of the
vacancy causes a splitting of the ground state spin quartet into two doublets, and the resonance
spectrum can be analyzed in terms of the usual axial field spin-Hamiltonian. The chromium with which
one of the crystals was doped was enriched with Cr53 beforehand.
A dominant feature in the resonance spectra is the presence of so-called forbidden hyperfine lines.
These result from transitions for which the selection rules ΔMs = ±1, ΔmI = 0 are violated, and are
characterized by ΔmI = ±1, ±2, and ±3. Such lines have not previously been reported for chromium
spectra.
Calculations of line intensity have been made using perturbation theory and using a method involving
the magnetic field induced at the nucleus by the electron spin. The perturbation series do not converge
very rapidly, so the calculations have been carried to fourth order for both wave functions and energies.
Perturbation expressions for wave function admixtures are given, but even for fourth order the
agreement with wave functions obtained from a computer diagonalization of the Hamiltonian matrix is
poor. The method of induced fields, however, does show remarkable agreement with results from
diagonalization. I
I
ELECTRON PARAMAGNETIC RESONANCE SPECTRA OF
T R I VALENT CHROMIUM IN MgO
by
DAVID HUGH DICKEY
A t h e s i s s u b m i t t e d to the. Gr aduat e F a c u l t y i n p a r t i a l
f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r t he degr ee
of
DOCTOR OF PHILOSOPHY
in
I
Physi cs
. , '■
Approved:
/
't f f n iM f .
Head, Ma j o r Department
airman,
Examining Committee
Gra'du&fe Dean
MONTANA STATE UNIVERSITY;,
Bozeman, . Montana
June 1969
•
.
ACKNOWLEDGEMENTS
The a u t h o r
g r a t e f u l I y acknowledges
s u p p o r t o f the
National
Adm inistration
and f o r
Science
Foundation.
P r o f e s s o r 0.
Lyle
e a rlier
He i s
support,
the
National
extremely g r a te fu l
instrumental
in
the
The a u t h o r a l s o w i s h e s
Hammer f o r
financial
and Space
E . Drumhel I e r , whose a d v i c e
e n c o u r a g e me n t were
o f the w o rk .
Aeronautics
the
her e x c e l l e n t
typing
to
to
and
accomplishment
t h a n k Mr s.
o f the m a n u s c r ip t.
iv
TABLE OF CONTENTS
Chapter
Page
LI ST OF TABLES ....................................................
LI ST OF FIGURES
ABSTRACT
I
II
III
.
IV
V
VI
. . . . . . . .
v.
.......................
.................................... ....................... ....
INTRODUCTION
.
.
.
vi
.
. ' ..............................................
I
THE HAMILTONIAN
.......................................................... '
OBSERVED SPECTRA
..........................................................
CALCULATED SPECTRA - -
LINE . POSITIONS
v iii
. . . .
. 10
21
43
Di a g o n a l i z a t i o n o f t h e H a m i l t o n i a n M a t r i x
.
The P e r t u r b a t i o n Method
..........................................
Concl u s i o n s ....................................................
43
46
56
CALCULATED SPECTRA - -
57
LINE INTENSITIES
.
.
D i r e c t D i a g o n a l i z a t i o n o f 16 x16 M a t r i x
.
I n d u c e d F i e l d Method
................................................
The P e r t u r b a t i o n Method
......................................
Comp ar i s o n w i t h E x p e r i m e n t
..................................
60
63
70
83
S U M M A R Y ..................................................................................
88
APPENDIX
91
.......................................'.
.
... .......................
Appendix A P e r t u r b a t i o n C a l c u l a t i o n s
.
. .
A p p e n d i x B Comput er P r o g r a m .................................
Appendix C S pectrom eter D e s c r i p t i o n
.
. .
92
105
HO
LITERATURE CITED
117
.........................................................
V
LI ST OF TABLES
Table
I
11
III
IV
V
VI
VII
V III
IX
Pa ge
Hamiltonian
Crg2 Line
Parameters
Positions
M=+1^-1- C r 52 L i n e
C r 55 R e l a t i v e
Forbidden
R elative
. . . .
28
f o r v = 9 9 9 4 .0 MHz
. . .
45
Positions
.................................. ■
P ositions,
Doublet Separations
Intensity
Computed L i n e
Admixture
Line
.........................
Positions
Parts
. . . .
List
53
. . . . . .
o f H y p e r f i ne L i n e s
C oefficients
Spectrometer
0- 45°
and I n t e n s i t i e s
55
. . .
.
52
60
.
62
.................................................
80
........................................
114
vi
' LI ST OF FIGURES
Figure
1
Page
L a t t i c e Defects A s s o c ia te d w i t h the T r i v a l e n t
Chromi um I o n ........................ " ........................................... : .
3
2
Ener gy L e v e l s
4
3
The C r t-O H a m i l t o n i a n M a t r i x f o r M a g n e t i c
F ie la j Quantization
' .........................................................
T5
The C r 52 H a m i l t o n i a n M a t r i x f o r C r y s t a l
A x i S 3Q u a n t i z a t i o n
.................................. . ...................
I6
4
o f the
Cr^g Ground S t a t e
5■
Cr ^g Ener gy L e v e l s
6
Comp l et e S p e c t r u m f o r
7
Crgg S p e c t r a
8
Central
Field
9
Angular
Dependence o f
10
for
for
8 = 90°
0=0 °
.
.
.
.
........................ ....
.
.
. ...................................
22
6= 0 ° and 8 = 9 0 0
Spectrum
.
19
24
. ' ...................................... . .
Fine S t r u c t u r e
Groups
25
.
.
27
Dependence o f L i n e Wi d t h on Rate a t Whi ch
L i n e P o s i t i o n Changes w i t h O r i e n t a t i o n .
.
.
.
30
11
Angular
.
.
.
31
12
Mg =^ - J j - T r a n s i t i o n s
8 = 4 5 ° ...........................................
33
13
Mg= T j - * - C r g g
8 = 4 5 ° ...........................................
34
14
Mg=^^-^ T r a n s it io n s
for
0=40°
......................................
36
I5
I
I
M g = ^ -^ Transitions
for
8=36°
......................................
37
16
Mg=?;-)--^- T r a n s i t i o n s
for
0=32°
......................................
38
17
Mg=2^ - ^ T r a n s i t i o n s
for
8=28°
......................................
39
18
Chromi um S p e c t r u m t h r o u g h 3675 and 3705 G
Wi ndows
................................. . . . . . . . . . . .
19
D.ep.endence. o f C e n t r a l
Lines
for
for
Chromi um S p e c t r u m t h r o u g h
Field
Lines.
3735 G Window
.
.
.
40
.
41
vi i
List
of
F i gur es
Figure
20
Page
Schematic
Ener gy Le v e l
Di agr am f o r
Mg = +Jr and Mg = -^- L e v e l s
21
Ener gy L e v e l
Admixtures
the
...........................................
47
Di agr am Showi ng Wave F u n c t i o n
........................................................................
58
22
P l a n Vi ew o f x - z
Pl a n e
...................................... .
23
Line I n t e n s i t i e s
F i e l d Method
C a l c u l a t e d Us i ng I n d u c e d
. . ..........................................................
24
A n g u l a r Dependence o f L i n e I n t e n s i t i e s
2.5
Measur ed L i n e
m ^ - _ 2"j “ 2^ ~
26
Measur ed L i n e
m=-|-
27
28
Intensity
for
Doublets
Intensity
Allowed Line
. . .
69
82
the
........................................ ■ •
for
65
85
the
.......................................................
86
B l o c k Di agr am o f S u p e r h e t e r o d y n e
Spectrometer
...................................... ' .........................
Ill
AFC C i r c u i t
11G
Di agr am
.
.
........................ .......................
v iii
ABSTRACT
The e l e c t r o n p a r a m a g n e t i c r e s o n a n c e s p e c t r a o f d i l u t e
t r i v a l e n t chr omi um a t s i t e s w i t h a x i a l symmet r y i n s i n g l e
c r y s t a l s o f magnesi um o x i d e has been s t u d i e d i n d e t a i l !
In
p a r t i c u l a r , t h e a n g u l a r dependences o f r e s o n a n c e l i n e i n t e n ­
s i t i e s and f i e l d p o s i t i o n s f o r Crgg have been i n v e s t i g a t e d .
T r i v a l e n t chr omi um s u b s t i t u t e s f o r t h e d i v a l e n t magnesi um i o n
i n MgO, and a s i g n i f i c a n t f r a c t i o n o f t h e c hr omi um i o n s a r e i n
a s s o c i a t i o n w i t h a n e x t - n e a r e s t - n e i g h b o r c a t i o n vacancy.
The
p r e s e n c e o f t h e v a c a n c y causes a s p l i t t i n g o f t h e gr ound
s t a t e s p i n q u a r t e t i n t o two d o u b l e t s , a n d t h e r e s o n a n c e
s p e c t r u m can be a n a l y z e d i n t e r ms o f t h e u s u a l a x i a l f i e l d
s p i n - H a m i l t o n i an.
The chr omi um w i t h w h i c h one o f t h e c r y s t a l s
was doped was e n r i c h e d w i t h Crgg b e f o r e h a n d .
A dominant f e a t u r e in the resonance s p e c t r a i s the
presence o f s o - c a l l e d f o r b i d d e n h y p e r f i n e l i n e s .
These
r e s u l t from t r a n s i t i o n s f o r which the s e l e c t i o n r u l e s
AMg = ± 1, Am % = 0 a r e v i o l a t e d , and ar e c h a r a c t e r i z e d by
Amj = ± 1 , ± 2 , and ±3.
Such l i n e s have n o t p r e v i o u s l y been
. r e p o r t e d f o r c hr omi um s p e c t r a .
C a l c u l a t i o n s o f l i n e i n t e n s i t y have been made u s i n g
p e r t u r b a t i o n t h e o r y and u s i n g a met hod i n v o l v i n g t h e m a g n e t i c
f i e l d i n d u c e d a t t h e n u c l e u s by t h e e l e c t r o n s p i n .
The
p e r t u r b a t i o n s e r i e s do n o t c o n v e r g e v e r y r a p i d l y , so t h e
c a l c u l a t i o n s have been c a r r i e d t o f o u r t h o r d e r f o r bo t h
wave f u n c t i o n s and e n e r g i e s .
P e r t u r b a t i o n expressions' f o r
wave f u n c t i o n a d m i x t u r e s ar e g i v e n , b u t even f o r f o u r t h o r d e r
t h e a g r e e me n t w i t h wave f u n c t i o n s o b t a i n e d f r o m a c o mp u t e r
d i a g o n a l i z a t i o n o f the H a m ilt o n ia n m a t r i x i s po or.
The
met hod o f i n d u c e d f i e l d s , h o w e v e r , does show r e m a r k a b l e
a g r e e me n t w i t h r e s u l t s f r o m d i a g o n a l i z a t i o n .
I .
The t r i v a l e n t
electrons,
The c u b i c
degeneracy o f
as a g r o u n d s t a t e
thousands
s till
than
o f 4 . 2 0 A.
vacancies w i l l
in.the
The number o f a n i o n
vacancies
cation
vacancies
enhanced i n
charge
in
the c r y s t a l
A small
fraction
energy.
spin
sing le t
The s i n g l e t
d e g e n e r a c y whi c h can
or a c r y s t a l
identical
When i n
to
Stark f i e l d
that
of NaCl,
t h e r mo d y n a mi c
is
The p r e s e n c e o f t r i v a l e n t
concentration
is
suppressed
o f vacancies.
and t h e number o f
such a way t h a t
the t o t a l
conserved.
o f the
substitutional
in
association with
in
a next-nearest-neighbor
cation
is
distorted
and t h e
to
l y i n g - some
la ttic e .
the e q u i l i b r i u m
is
of
numbers o f b o t h magnesi um and oxygen
exist
c hr omi um a l t e r s
fie ld
cubic.
a la ttic e
small
a crysta llin e
as.,
the
trip le ts
fie ld
structure
equilibrium ,
for
an o r b i t a l
in
MgO has a c r y s t a l
spacing
incorporated
leaving
has a f o u r - f o l d
be removed by a m a g n e t i c
is
substitutes
F-state,
and two o r b i t a l
o f s y mmet r y l o w e r
with
the
unpaired d
removes t h e s e v e n - f o l d
o f wave numbers h i g h e r
ground s t a t e
only
it
fie ld
three
When i t
and becomes exposed t o
symmetry.
orbital
with
F ground s t a t e .
i n magnesi um o x i d e ,
magnesi um i o n
cubic
chr omi um i o n ,
4
has a
an i m p u r i t y
INTRODUCTION
a magnesi um v a c a n c y .
tetragonal
site,
spin
If
c hr omi um i o n s
are
a vacancy occu rs
the l o c a l
quartet
symmet ry
ground s t a t e
m
m
Zm
m
is
s p lit
Figure
I
i n t o , two d o u b l e t s
illu s tra te s
and c hr omi um i o n
o t h e r types
the
for
separated
re la tive
this
by a b o u t 0 . 2
locations
o f the
t y p e o f d e f e c t as We l l
which are ob served.
cm™ .
v ac anc y
as f o r
The p a r a m a g n e t i c
two
resonance
*
spectra
for
completely
w ill
t h e even i s o t o p e s
studied
review
concentrate
the only
t h e even i s o t o p e
m a i n l y on t h e
has a n u c l e a r
spin
so t h a t
pairs
so t h a t
of
the
four
there
into
in
but w i l l
is
Cr^g i s
9% a b u n d a n t , and
Cr^g i n t r o d u c e s
each o f t h e
a r e 16 s t a t e s
fie ld
transitions
states.
2 as a f u n c t i o n
o f c h r o mi u m ,
inte ractio n
presence o f a magnetic
lifte d
research,
T h i s wor k
of 3/2.
degeneracy o f
states,
^
s p e c t r u m o f Cr^g i n MgO.
odd i s o t o p e
The h y p e r f i n e
tional
by W e r t z ^ ™3 ^ and o t h e r s .
some o f
stable
o f c hr omi um have been
altogether.
the degeneracy
is
electronic
In the
completely
may be o b s e r v e d bet ween c e r t a i n
The 16 e n e r g y l e v e l s
of magnetic
four
an a d d i - ■
fie ld
for
are
shown i n
Fig.
the f i e l d
parallel
t o - *•
T h e r e ar e t h r e e even i s o t o p e s :
C r g g , C r g g , and C r g ^ ,
none o f w h i c h have a n u c l e a r s p i n .
They a l l have t h e same EPR
s p e c t r u m , s u p e r i m p o s e d , and i t w i l l be r e f e r r e d t o as t h e
Crgg s p e c t r u m .
• The t r a n s i t i o n s a r e o b s e r v e d w i t h an e l e c t r o n p a r a ­
magnetic resonance s p e c tr o m e te r .
The s p e c t r o m e t e r o p e r a t e s
a t a f i x e d f r e q u e n c y , ( o r p h o t o n e n e r g y ) and s p e c t r a ar e
t r a c e d o u t by s w e e p i n g t h e m a g n e t i c f i e l d .
The ' s p e c t r o m e t e r
i s d e s c r i b e d i n A p p e n d i x C.
o
X
O
X
O
X
O
X
O
X
O
X
x
O
X
O
X
O
X
O
X
O
X
O
O
X
O
-3-
O
X
O
X
O
X
X
O
X
X
O
X
O
X
O
O
X
O
C r 3+' )
/
/
/ ^
O
O
X
O
/
O
O
X
X
O
O
X
X
O
O
X
' -----N
/ r- 3 + \
Fe
I
i
I
i
I
O
I
i
I
I
i
I
I
I
O
I
I
I
I
I
C r3+J
O
0 //
C r 3+x,
X
O
X
O
X
O
X
O
X
O
O
X
O
X
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O
o
X
O
X
O
X
O
X
X
O
X
/
/
\
X
/
I
X
\
O
X
\
'C r 3 + I
/
O
x - Mg^+ , o - O ^
D e f e c t s ; from top
to bottom
1.
tetragonal
(firs t
2.
tetragonal
(second
Fig.
I
Lattice
kind)
kind)
Defects
Chromi um I on
2.
r h o mb i c
4.
cubic
Associated
with
the T r i v a l e n t
mT=
T
h v = 0 .33 cm
2D = 0 . 1 6cm
Jl
I 000
2000
3000
MAGNETIC FIELD
Fig.
2
Ener gy L e v e l s
of
the C r ^
4000
gauss
Ground S t a t e
5000
6000
-5the
axis
tions
joinin g
which are al l o w e d
Am^ = 0 a r e
Fig.
If
c hr omi um i o n and v a c a n c y .
2 are
by t h e
indicated.
the
the t e t r a g o n a l
fie ld
axis,
is
s ig n ifica n t
This w i l l
leve ls,
and numerous e x t r a
o c c u r s may change r a p i d l y
devoted
to
intensity
at
to
so-called
and p o s i t i o n
of
for
the
some a n g l e
re la tive
of
forbidden
Chapters
to
m i x i n g may
bet ween any p a i r
as t h e d i r e c t i o n
of
levels
| S ,Mg , I ,mj > = | M, m>.
lines
a t which a p a r t i c u l a r
the c r y s t a l .
an e x p l a n a t i o n
AMg = ±1,
wave f u n c t i o n
permit t r a n s itio n s
The ma g n e t i c , f i e l d
changed r e l a t i v e
eigenstates
directed
occur.
appear.
rules
The wave f u n c t i o n s
pu r e m a g n e t i c
the magnetic
selection
The 12 t r a n s i ­
of
w ill
transition
o f the
fie ld
is
IV and V ar e
t h e a n g u l a r dependence o f t he
the
various
allowed
and f o r b i d d e n
transi tions.
The H a m i l t o n i a n
for
t h e C r 53 gr ound s t a t e
in
a tetragonal
*
fie ld
w ritten
K
terms
of spin
variables
is
= geft-S + D [ S ; , - 1 s 2 ] + A ? - t - gn enM
The f i r s t
fie ld
in
term i s
interaction.,
and t h e l a s t
the
Zeeman e n e r g y ,
the t h i r d
term i s
term i s
the n u c l e a r
t h e second
'
’
is
the h y p e r f i n e
Zeeman e n e r g y .
(1.1)
the c r y s t a l
interaction^
For a
The sp i n - H a m i l t o n i a n ^ ^ i s a common f o r m u l a t i o n i n
/y\
p a r a m a g n e t i c r e s o n a n c e w o r k . . For i t s d e r i v a t i o n , see T a k e ,
C h a p t e r 3, o r H e c h t 5(S) C h a p t e r 5, f o r i n s t a n c e .
-6magnetic
fie ld
o f a b o u t 4000 g a u s s , t h e
re la tive
magnitudes
of
2000:1000:20:1,
o f the e i g e n f u n c t i o n s
of
calculate
line
the
eigenvalues
various
w ill
Three d i f f e r e n t
The f i r s t
allow
inte n sitie s,
a direct
16 x 16 H a m i l t o n i a n m a t r i x .
are a d j u s t e d
priate
by v a r y i n g
a tra nsition
the l i n e
fie ld
is
inte nsity
obtained
amount o f c o m p u t e r t i m e ,
so i t
finding
is
the l i n e
by t h e
constitutes
zation
of
u n til
the ap p ro ­
two f o r w h i c h
needed f o r
by-product of
a sig n ifica n t
has been used a t o n l y a few
inte nsities
in
by Bi r , ^
^
the
firs t
electronic
two t e r ms
matrix
Cr^g w i t h
mate e l e c t r o n i c
also
gives
of
Hamiltonian.
in
Eq. ( 1 . 1 ) ,
for
This
The m a t r i x
and
Cr ^ g •
accuracy.
used
the 4 x 4
Di a g o n a l i -
one t h e e l e c t r o n i c
reasonable
wave f u n c t i o n s ,
and i s
t h e Cr^g s p e c t r u m .
the complete H a m ilto n ia n
of this
functions
o f the f u l l
o f the H a m ilto n ia n
upon a c o m p u t e r d i a g o n a l i z a t i o n
representing
defined
the problem.
as a c h e c k on t h e o t h e r me t hod s .
met hod r e l i e s
matrix
the
positions.
bet ween t h e
method r e q u i r e s
one t o
solving
ar e a n a t u r a l
The second met hod was d e v e l o p e d
for
fie ld
Knowl edge
k no wl edge o f
The wave f u n c t i o n s
calculations
This
for
The e i g e n v a l u e s
wanted.
the d i a g o n a l i z a t i o n .
orie nta tions
is
while
d i agonalization
the magnetic
energy d i f f e r e n c e
permits
o f the l i n e
ar e c o n s i d e r e d
is
have a p p r o x i m a t e ,
respectively.
Hamiltonian
determ ination
met hods
method
this
t er ms
wave- -
Knowi ng a p p r o x i ­
one can c a l c u l a t e
the magnetic
r
- I -
fie ld
the
induced a t
four
possible
order of
axis
the nucleus
by t h e e l e c t r o n
electronic
states.
at
the nucleus
external
fie ld ,
for
is
not in
though,
so i t
general
w ill
nuclear spin
w ill
states
and may be r e l a t e d
to
be l i n e a r
the o ld
operator.
The t r a n s i t i o n
of
spin
nuclear
perturbation
term i s
series
is
met hod f o r
fie ld
does n o t c o n v e r g e
obtained,
the c r y s t a l
order Hamiltonian
fourth
a spatial
rotation
The new
rotation
bet ween a p a r t i c u l a r
be w r i t t e n
in
pair
t er ms o f
finding
is
the e i g e n f u n c t i o n s
and e i g e n ­
t h e use o f p e r t u r b a t i o n
interactions
as t h e
Since
and h y p e r f i n e
the
all
treated
crystal
Zeeman t e r m , t h e
very r a p i d l y .
and c a r r y i n g
the
are
theory.
as a
fie ld
perturbation
Adequate convergence
the d ia gon al
inte ractio n
perturbation
in
parts
of
the z e r o -
series
to
order.
Experimental
is
p roba bility
h o w e v e r , by i n c l u d i n g
potential
the
ones by t h e a p p r o p r i a t e
and h y p e r f i n e
as l a r g e
to
c o m b i n a t i o n s "of t h e o l d ,
on t h e Zeeman e n e r g y .
nearly
The i n d u c e d
elements.
o f the H am ilto nian
The c r y s t a l
ar e on t h e
a tra n sitio n .
can t h e r e f o r e
operator m atrix
The t h i r d
values
states
each o f
a quantization
parallel
suffer
system undergoes
in
These f i e l d s
the' n u c l e a r s p i n .
when t h e e l e c t r o n i c
rotation
spin
TOOK gauss , so t h e y a b s o l u t e l y d e f i n e
and r e p r e s e n t a t i o n
fie ld
-
not p o s s ib le
measur ement o f
line
a t many o r i e n t a t i o n s .
positions
and i n t e n s i f i e s
Su p e r i mp o s e d s p e c t r a
-8from extraneous
large
portion
paramagnetic
of the t e t r a g o n a l
severe l i n e
broadening
but c e r t a i n
orie n ta tio n s.
occurs
e v e r t h e y are p o s s i b l e ,
Chapters
met hods
IV and V w i t h
described
described
certain
in
Chapter
Forbidden
been r e p o r t e d
observed
(I
= 5/2)
in
in
resonance
in
though,
values
chr omi um s p e c t r u m :a t a l l .
of
using
^^
lines
transitions
although
( I = 7/2)
a variety
forbidden
s p e c t r a was f i r s t
function
of
neighboring
of a p a rtic u la r
are a dominant
have n e v e r b e f o r e
t h e y have been
and manganese ^ ^
hyperfine
lines
interaction
in
Rubi ns
expressions
give
forbidden
the
hyperfine
transitions
theory c a lc u la tio n
of
which
paramagnetic
by B l e a n e y and I ngr am ^ ^ ^
states
into
The a d m i x t u r e
m atrix
ground s t a t e
for
in
by B l e a n e y and Rubi ns
the presence o f o f f - d i a g o n a l
at
of crysta ls.
observed
state.
three
recorded
e x p l a n a t i o n was based upon a c a l c u l a t i o n
admixture
the
a r e shown and
spectra
wher e f o r b i d d e n
in
have been made w h e r e -
calculated
hyperfine
mask a
and t h e y a r e compar ed i n
including
1951 , and was e x p l a i n e d
Their
the
Obs er v ed s p e c t r a
III,
vanadium^"*
The e x i s t e n c e
the c r y s t a l
c hr omi um s p e c t r u m , and '
in
C r 53 s p e c t r a ,
spectra
in
Measur ement s
above.
orientations
feature.
ions
elements
Hamiltonian.
the r e l a t i v e
(111
in
1961.
o f the
t h e wave
results
o f the
from
hyperfine
B l e a n e y and
inte nsities
of the
ar e based upon a p e r t u r b a t i o n
the a d m i x t u r e s .
Their
expression
for
■9Am = I
and Am = 2 l i n e
f orb.
(
a ll.
inte nsities
3 Ds' i n 2 0
)
4g3H
,vs(s+r)
ar e
n2
C-I ( I + I )
- m( m- 1 ) ]
(1.2)
and
I
I
I / 3 D s i n 2 6 / n ,S(S + 1 ) n2
2'
4g3H ] L 1 3M(M-1 ) J
forb,
a ll.
S D A s i n 2 O n , ' S( S + 1 )
±
i
2
(4g3H)2^
x
for
[1(1+1)
T
l
I M,m> = 17r,m>-*f-7j-5m- l >
respectively.
large
may n o t c o n v e r g e .
expressions
It
dependence o f
transitions,
tra n sitio n s.
in
this
assume t h a t
tra nsition s,
D << gBH, b u t i n
the p e r t u r b a t i o n
be shown i n
wor k
the e l e c t r o n
tetragonal
regard
reported
(1.3)
I
i
| ^ - , mt l >-> |-- ^, m+l >
enough t h a t
w ill
- m(m+l) ]
series
C h a p t e r V how t h e above
a r e m o d i f i e d when D - gSH/ 4.
The p u r p o s e o f
o f C r 53 i n
and
These f o r m u l a s
many s y s t e m s ' D i s
with
- m ( m - l ) ] [1(1+1)
to
sites
simultaneous
is
to
analyze
paramagnetic
in
MgO.
electron
o r what have been c a l l e d
Such t r a n s i t i o n s
the a n g u la r
resonance
spectrum
The s p e c t r u m i s
and n u c l e a r
"forbidden
spin
hyperfine"
have n o t p r e v i o u s l y
c hr omi um spectra.-,, n o r has any p r e v i o u s
been made t o f i t
wave f u n c t i o n s .
sim ilar
spectra w ith
fourth
analyzed
order
been
attempt
perturbation
II.
THE HAMILTONIAN
When t h e c hr omi um i o n
i n MgO i s
associated w ith
nearest-neighbor cation
v a c a n c y , t h e Coulomb f i e l d
vacancy s p l i t s
quartet
The o r b i t a l
there
is
the spin
angul a r momentum o f t h e
no d i r e c t
admixture
ground s t a t e
of
higher
interaction
lying
to
states
into
gr ound s t a t e
o f the
two d o u b l e t s .
is
z e r o , so
cause a s p l i t t i n g ,
into
a next-
but
t h e g r oun d s t a t e makes
*
the s p l i t t i n g
possible.
inte ractio n,
the s t a t e s
±3/2 d o u b l e t
lying
Consistent with
with
below the
these d o u b le ts
has a t o t a l
nuclear spin.
The h y p e r f i n e
d e g e n e r a c y , and a m a g n e t i c
energy l e v e l s
in
values
Hamiltonian:
of
X =
the
a. m a g n e t i c
M5 r e ma i n d e g e n e r a t e , t h e
±1/2 d o u b l e t .
I n C r ^ g , each o f
degeneracy o f e i g h t
inte ractio n
fie ld
fie ld
lifts
parallel
to
in
Fig.
because o f t h e
removes some o f t h i s
what r e m a i n s .
a r e g i v e n by t h e
+ D[S^i- Is2] + A?-T - QnPnH-T
For t h e case i l l u s t r a t e d
wher e H i s
eq ual
an e l e c t r o s t a t i c
2 in
the z 1 a x i s ,
eigen­
.
the p r eced ing
this
The
Hamiltonian
(2.1)
chapter,
reduces
to
The a s s u m p t i o n t h a t t h e S t a r k e f f e c t a l o n e causes t h e
g r oun d s t a t e s p l i t t i n g i s a v e r y p o o r o n e .
D i s t o r t i o n of the
l o c a l c r y s t a l symmet r y r e q u i r e s t h a t t h e p r o b l e m be a p p r o a c h e d
using the t h e o r y o f c r y s t a l l i n e f i e l d s .
Sharma , t
) for
i n s t a n c e , d i s c u s s e s a number o f mechani sms i n v o l v i n g l o w sy mmet r y c r y s t a l l i n e f i e l d s and r e s u l t a n t a d m i x t u r e s o f e x c i t e d
s t a t e s i n t o the ground s t a t e .
-I I -
K
I s 2]
= gf3HSz + D [ $ z -
+
+ ASz I z - gn 3n H l z
(2 .2)
+ SyIy]
Rewriting
the
last
term w i t h
the d e f i n i t i o n s
S- = S% - iSy
S+ = Sx + TSy
(2.3)
, etc
we have
A t 5X 1X+ 5Y 1y ) '
We see t h a t
it
is
very
values
of
the
this
in
than
ignored.
Fig.
re la tive
2:
the n u c le a r
The e x p r e s s i o n
are d i s p l a c e d
is
axis.
is
to the
firs t
terms,
fie ld
lies
but
so t h e e i g e n ­
+ AMm - g n 3n Hm
t e r m as a p e r t u r b a t i o n
(2.5)
leads
Zeeman energy, and i s
of
Eq.
(2.5)
is
therefore
describes
in
to a .
t h e gr aph o f
H, t h e
intercepts
2
second t e r m d e p e n d i n g on M , and t h e
by t h e
structure
non-diagonal,
are a p p r o x i m a t e l y
a dominant term l i n e a r
The s i t u a t i o n
magnetic
(2.2)
2
o f o r d e r A / g (BH = .01 A, w h i c h
Eq. ( 2 . 5 )
there
hyperfine
Eq.
Hamiltonian
the non-diagonal
correction
smaller
term o f
g|3HM + D[M2 - j S ( S + l ) ]
'M ,m
Treating
last
small
(2.4)
5-tS+I. + S „I + )
is
is
at
described
by t h e t h i r d
term.
c o n s i d e r a b l y more c o m p l i c a t e d when t h e .
some a n g l e
One has t h e c h o i c e
re la tive
of takin g
either
to
the
crystal
the c r y s t a l
axis
-12(z '-a x is )
or
the magnetic
quantization
affect
the
ultim ate
the Z 1 axis
zation
of
however,
axis..
fie ld
The r e p r e s e n t a t i o n
re sults,
is
When u s i n g
most c o n v e n i e n t t o
as t h e q u a n t i z a t i o n
the a x ia l
symmetry,
the z 1 a x is
with
z axis.
If
tion
the magnetic
axis,
the
by a r o t a t i o n
rotation,
lies
t he m a g n e t i c
field
c a s e , owi ng t o
be c o n c e r n e d w i t h
direction
an a n g l e
diagonal i theory,
In e i t h e r
the x -z
second t e r m o f
through
should not
perturbation
so a c o o r d i n a t e
in
fie ld
take
axis.
one need o n l y
z and z 1 a x i s ,
i n which
the
one c hooses
as a
b u t as a m a t t e r o f c o n v e n i e n c e
direction
bet ween t h e
(z-axis)
may be used when ma k i n g a c o m p u t e r i z e d
the H a m i l t o n i a n .
it
dire ction
plane,
is
Eq. . ( 2 . 1 )
system i s
taken
as a q u a n t i z a ­
must be t r a n s f o r m e d
6 about the y - a x i s .
For t h i s
one has
so t h e c r y s t a l
fie ld
(2.6)
t e r m becomes
D [ S ^ s i n 26 + ( S S 7 + S7S j s i n e c o s e
X
XZ
ZX
(2.3),
chosen
ma k i n g an angl e. 0
Sz., = Sx SinG + Sz cos0
After
the angle
some m a n i p u l a t i o n ,
one o b t a i n s
and u s i n g
+ S7Cos2B - i-S2] . ( 2 . 7 )
i
Z
O
the s u b s t i t u t i o n s
o f Eq.
JL
-
§[(Sz
13
-
- I s 2 ) ( 3 c o s 26 - l ) + l ( s j
+ S ^ s i n 2Q
+ ^ ( S ^ S + + S;S_ + S+S^ + S _ S z ) s i n 2 8 ]
The h y p e r f i n e
iso tro p ic,
Eq.
and n u c l e a r
so t hey, w i l l
(2.2).
Zeeman t e r ms
(2.8).
ar e assumed t o be
keep t h e same f o r m as t h e y have i n
The e l e c t r o n i c
Zeeman e n e r g y has a s l i g h t
anisotropy,
t h o u g h , and i t
is
g-factor
t h e f o r m o f an a x i a l l y
in
accounted f o r
■
by w r i t i n g
symmetric
t he
t e n s o r , so
t h a t (25)
g(e)
= [ V c o s 2S t
^
g2s i n 2e] 1 / 2
-
where g p = g_,, z , , and g s = 9 % ' x '
m odifications,
for
(2.9)
magnetic
= gy ' y ' ‘
fie ld
With
quantization,
t h e above
the complete
Hami I t o n i an i s
K = g ( e ) 3 H S z + D[ S2 - 1 s 2] P 2 ( c o s 6) + 5 . [ ( S 2 + S2 ) s i n 2e
+ ( S z S+ + SZS_ + S+ Sz + S _ Sz ) s i n 2 6]
+ ASz Lz + § ( S + I _ + S _ I + )
where P2 ( c o s e )
is
Hamiltonian
used i n
Chapters
It
the
is
the
- QnBnHIz
Le gen dr e p o l y n o m i a l .
the p e r t u r b a t i o n
'
( 2 'T0)
This
form o f the
calculations
of
I V and V.
is
convenient to
relations
re-scale
the parameters
D and A by
-14D -»■ gp BD . ,
A -> gp 3A
so t h e f a c t o r
gp 3 w i l l
divided
This
out.
energies
in
be common t o
transform ation
t h o s e t e r ms
along
and may be
has t h e a d v a n t a g e t h a t
t h e H a m i l t o n i a n may be measur ed i n
Fo r q u a n t i z a t i o n
is
(2.11 )
the c r y s t a l
axis,
units
the
all
of gauss.
Hamiltonian
somewhat s i m p l i f i e d :
S + s
K = Sn-PHS2 ,.cose + g s 6H(—!—2— 1 J s i n e + D[ S2 , - j-S2 ]
+ ASz , I z , + | ( S + I_ + S _ I + ) - Sn BnH I z ,
The H a m i l t o n i a n m a t r i x
o r f r o m Eq.
(2.10),
can be c o n s t r u c t e d
dependi ng
The c o m p u t e r d i a g o n a l i z a t i o n
m atrix
has been w r i t t e n
the e ig e n v e c to rs
calculated
c ompar ed.
After
Eq.
the m a t r i x
ap p e a r s
dimensional
kets
the
vector
dimension
vectors
are the
for
crystal
the
eigenvalues
c h r o mi u m.
axis
kets
|M>.
fie ld
rescaling
vectors.
o f the m a t r i x
Eq.
as shown i n
In
(2.10)
Fig.
the m a t r ix
is
so
t h e o r y may
according
3.
to
The 16- •
defined
has the-
t h e absence o f n u c l e a r s p i n ,
reduced to 4 x 4 ,
This
smaller m a trix ,
quantization,
one
quantization,
is
and i t s
for
to o b ta in
t h e even i s o t o p e s
a r e made i n
ba.sis
shown i n F i g .
may be d i a g o n a l i z e d
and wave f u n c t i o n s
Such c a l c u l a t i o n s
equation
the f u l l
from p e r t u r b a t i o n
space i n w h i c h
I M, m> as b a s i s
pr ogr am f o r
f o r magnetic
be d i r e c t l y
(2.11),
from t h i s
upon t h e r e p r e s e n t a t i o n
uses.
that
(2.12)
Chapters
of
I V and V.
4
O
Dl
O
O
O
O
O
O
DZ
O
O
O
O
O ■ *
'0
O
§A
Dl
O
O
O
O
DZ
O
O
O
O
O
O
O
*
O
O
/3A
Dl
O
O
DZ
O
O
O
O
O
O
O
O
O
*
O
O
IaDl
DZ
O
o ■
O
O
O
O
Dl
§A
O
O
*
O
O
O
O
O
.0
O
O
O
O . • DZ
O
Dl
/3A
O
O
*
O
O
O
O
O
/BA
O
O
DZ
O
O
O
Dl
| a
O
O
*
O
O
O
ZA
O
DZ
O
O
O.
O
O
Dl
O
O
O
*
O
/3A
O
O
DZ
O
O
O
O
O
O
DZ
O
O
O
O
*
O■ O
O
-Dl
O
O
DZ
O
O
O
O
/3A
O
*
O
O
O
- Dl
VyI A
O
DZ
O
O
O
O
ZA
O
O
.0
*
O
O
O
-Dl
■
.
‘
O
O
O
O
/Ta
O
O
O
O
O
O
O
O
O
DZ -Dl
O
O
O
O
'O
O
O
O
O
O
O
O
* The g e n e r a l
DT=^DsinZQ
Fig.
O • O
DZ
3
.
O
DZ
O
| a - Dl
DZ
O
O
O
vT a
O
O
O
O
O
O
o
O
DZ
.
diagonal
element
. DZ = Z ^ D s i n^ Q,
. The Cr ^g H a m i l t o n i a n
is
■
O
O
'
.
.
IaO '
O
i A
O
*
O
O
O
-Dl
O
O
*
O
O
’
O
O
O
O
*
O
O
-DT
O
O
O
*
O
*
-Dl
O
O
O
l A
T HM+AMm+ ( M ^ - ^ D P g - r ^ H m .
r = { c o s 2Q+ ( g ^ / g ^ ) s i n 20 } 1 7 2 ,
Matrix
a
f o r Magnetic
Field
Fn= ^ 1 >
Quantization.
O O
Z' Z
3
I
Z’ Z
3
I
Z ’ "Z
3
3
Zi_Z
I
3
Z’ Z
I
I
Z' Z
I
I
Z ’ "Z
I
3
Z ’" Z
I
3
™Z ’ "Z
I
I
"Z ’ _ Z
I
I
-g ,
2
I
3
ro |
o ■
O
g
- g ,
PO| CO tv i| OO r \ 3| Ca) f \ ] | W
*
' 2
' 4
3 I
" Z ’ ~Z
3 I
"Z* Z
3 3
~Z ’ Z
M, m
Jj-HcosB +D
vI y H s i n e
0
0
Fig.
4
The C r ^
0
vI r H s i n G
-TjrH
COS 6 -D
y Hs i n 6
0
yHsine
0
+
!
0
4
—g-Hc o s 6 -D
vI y H s i n8
I
"2
vI y Hs i n e
--^H cos6 +D
3
"2
Hamiltonian M atrix
for
Crystal
Axis Q u a n t iz a t io n .
- 9 L-
M=
-.17The s m a l l e r m a t r i x
functions
for
is
also
Cr^g i n
the
use t h e . i nduced f i e l d
to
the v a rio u s
t e r ms
dipole
a r e AM = ± 1,
Am = 0.
and f i n a l
states
may be w r i t t e n
in
the magnetic
fie ld
Eq.
Hq , w h i l e
if
necessary to
a spe ct rum are r e l a t e d
the H a m i l t o n i a n .
The o b s e r v e d l i n e s
f o r wh i c h
the s e l e c t i o n
bet ween
t h e mi c r o w a v e p h o t o n
and
fie ld
is
a t which
parallel
a line
(2.11)
to
occurs
rules
in itia l
e n e r g y and
as
hv
(2.13)
the c r y s t a l
axis.
The
is
(2.14)
Hq - D( 2M-1 ) - Am
Referring
in
four
to
groups
central
lines
gr o u p s
The p a r a m e t e r s
Tower p a r t
3A.
is
Fig.
2,
According
a r e a t an a v e r a g e f i e l d
c omponent s
gr o u p
of
of four each.
ar e d i s p l a c e d
bet ween h y p e r f i n e
the M =
the
three
so t h e span o f a gr oup i s
negative
is
V
the o t h e r
separation
gauss,
the
in
H + D( 2M- I ) + Am
12 l i n e s
(2.14),
lines
(2.5)
wh i c h
Bir.'
The e n e r g y d i f f e r e n c e
from Eqs.
where Hq = h v / g ^ B .
are
a p p r o x i m a t e wave
calculations
spectra, i t
of
must equal
M, m;M-1 ,m
there
provide
intensity
transitions
- E
M-I ,m
"M,m
magnetic
actual
how t h e p o s i t i o n s
are magnetic
if
line
met hod o f
Before d is c u s s in g
indicate
used t o
a distance
in
any g r o u p
Note t h a t
to occur at
±2D..
is
to
of
The
A
D must be
lowest f i e l d .
gp , D and A may be d e t e r m i n e d
by
-18measuring
the average
o f any one o f t hem.
is
gs , and i t
positions
of
Fig.
at
fie ld
for
The o n l y
4, w r i t t e n
three
remaining
can be d e t e r m i n e d
8 = 90°.
the
gr o u p s
important
0 = 90°,
parameter
by measur ement o f
The s e c u l a r e q u a t i o n
for
and t he span
for
can be f a c t o r e d
C r^
line
the m a trix
into
t h e two
equations:
A2 + yHA - I
y 2H2 - D2 - yHD = 0
(2.16a)
(2.16b)
are
the energy l e v e l s
versus
H in
Fig.
two c o n s e c u t i v e
5.
of C r ^
If
D is
solutions
for
8 = 90°,
and a r e p l o t t e d
known, the d i f f e r e n c e
can be s e t equal
to the
between
phot o n
e n e r g y and t h e
resulting
equation
solved n u m e ric a lly
Me as ur ement o f
the f i e l d
a t which
the
w ill
H and t h e n c e y .
establish
Note f r o m F i g .
f i e l d , , but f o r
5 that
0 = 0,
it
I
the M =
occurred
,
In the f o l l o w i n g
spectrum is
discussed
used t o o b t a i n
3
on t h e
The r e s o n a n c e s p e c t r u m must t h e r e f o r e
dependence.
tra nsition
line
is
f o r yH.
observed
occurs
l ow f i e l d
have a s t r o n g
at hig h,
side.
angular
!
chapter the
experimental Iy
and t h e o b s e r v e d l i n e
numerical
values
for
observed
positions
the parameters
in
ar e
the
-6 LI OOO
Fig.
5
Cr^g Ener gy L e v e l s
2000
3000
MAGNETIC FIELD
gauss
for
8 = 90°
4000
-20Hami l torn" an.
.E xplicit
calculations
orientations
are c a r r i e d
out
in
of
Chapter
line
I V.
positions
at a ll
III.
The c r y s t a l
t a k e n . to
lie
fie ld
along
to the octa he dra l
lying
along
each o f
OBSERVED. SPECTRA
associated with
the z 1 axis
in
the preceding
sy mmet r y . of MgO, t h e r e
t h e x 1 and y 1 d i r e c t i o n s .
t h e e q u i v a l e n t axes a r e j u s t
associated with
a magnesi um v ac a nc y was
the
z 1 axis,
I ons
associated with
as numerous as the. ones
and t h e i r
resonance
on t h e s p e c t r u m we w i s h t o
the spectra
assocated w it h
d e n o t e d A,
ions
B and C, r e s p e c t i v e l y ,
change e v e r y 90° as t h e m a g n e t i c
X 1- Z 1 p l a n e .
its
axis
is
The c r y s t a l
perpendicular
upon wh i c h
from Semi-Elements,
Inc.
doped w i t h
Cr53.
enriched
approximately
70%,
inte nsity
of
spectrum c o n s i s t s
this
If
t h e A and B s p e c t r a
fie ld
is
rotated
in
in te r­
t he
to
recording
o f the
fie ld .
wor k was d o n e , o b t a i n e d
The i s o t o p e was e n r i c h e d
each l i n e
o f gr o u p s
component s
the magnetic
since
o f S a x o n b u r g , P e n n s y l v a n i a , was
is
the e q u i v a l e n t
hyperfine
spectrometer
observe.
z 1 , x 1 and y 1 axes ar e
but c o n s id e rin g
as many C r 53 l i n e s ,
the
spectra
The C s p e c t r u m has no a n g u l a r d e p e n d e n c e ,
a l wa y s
Owing
a r e e q u i v a l e n t axes
appear s uperi mposed
of
chapter.
of
flanking
spectrum f o r
that
s till
C r^
four
only
g a i n was i n c r e a s e d
are
four
times
about o n e - h a l f
line.
The r e s u l t i n g
equally
spaced C r 33.
each C r ^
9 =0
there
to
is
line.
shown i n
by a f a c t o r
of
A chart
Fig.
6.
The
25 a t f i e l d s
Cr ( c u b i c )
Cr
(rhombic)
-
22
-
Cr(A)
I 500
Cr(BfC)
2000
2500
Cr(BfC)
3000
MAGNETIC FIELD
Fig.
6
Comp l et e S p e c t r u m f o r
8=0
4000
qauss
-23b e l o w and above t h e c e n t r a l
tion
region.
The- M5 =
•I
in
the. c e n t r a l
region
without
added g a i n ,
and a r e n o t d i s t i n g u i s h a b l e
ar e a l m o s t t o o weak t o
The p o s i t i o n s
of
are
on t h e f i g u r e ,
indicated
five
lines
the
spectra
in
Figs.
is tic
the o u t e r
fiv e -lin e
absorption,
meters
and i n
gr o u p s
and i s
which
these
Fig.
is
fie ld
to
t h e A,
than
lines
6,
of l a t t i c e
the l i n e s
the
spectra" in
chr omi um l i n e s ,
there
7 is
displaced
iron.
Fig.
I .
spectro­
ar e numerous
study.
Some o f
i n cubic
others
sites
belong to
Other weaker l i n e s
associated with
in
derivative
some o f wh i c h ar e
we w i s h t o
vacancy, while
indicated
Fig.
the c h a r a c t e r ­
X- band m a c h i n e ,
ar e due t o Cr ^g and C r ^
chr omi um i o n s
defect
of
The s p e c t r o m e t e r
modulated,
7. and 8,
d i v a l e n t manganese and t f i v a l e n t
to
the
expanded s c a l e s
B and C s p e c t r a ,
Figs.
w h i c h have no a s s o c i a t e d
a r e owi ng
6.
Recordings
as t h e f i r s t
modulation.
In a d d i t i o n
strong
resolved.
such t h a t
a common d i s p l a y made f o r
A p p e n d i x C.
in
Fig.
ar e a p p a r e n t .
in
visib le
is
7 pa rticu la rly,
described
much more i n t e n s e
scale
in
be seen
B and C s p e c t r a
ar e shown w i t h
a superheterodyne, f i e l d
other, l i n e s
t h e A,
have been r e c o r d e d
using magnetic
used i s
of
but the
a t each l o c a t i o n
These s p e c t r a
of
lines
o f each g r o u p a r e h a r d l y
7 and 8 ,
I
-jr t r a n s i ­
the o t h e r types
An o b v i o u s
feature
t h e p r e s e n c e o f a second gr oup o f
toward
m id-field.
These s p e c t r a
of
-24-
- H -------------------1---------------------------------1----------------------------- S------------------ 1---------------- L 1800
1850
2750
4450
4500
5350
MAGNETIC FIELD
Fig.
7
Cr^3 Spectra
for
6=0°
-
gauss
and 6=90°
-
4
—
5400
Cr(cubic)
Cr (B)
\
— -— J
Cr (A)
— f3300
3400
3500
3600
MAGNETIC FIELD
Fig.
8
Central
Field
Spectrum
3700
gauss
-26a r e d e n o t e d A 1, B 1, and C ' , and Wer t z
has a t t r i b u t e d
them
(
to
tetragonal
defects
The H a m i l t o n i a n
smaller
for
Fig.
data.
and c u b i c
Eq.
that
D is
a t 1834 and 5383 gauss
in
(2.14).
the ir
Their
I
3
position
average p o s i t i o n
from
o f C r 52 and C r 5 3 , n e a r
by c l o s e - l y i n g
and were n o t used t o
manganese
establish
any o f
The v a l u e o f A was e s t a b l i s h e d
the M =
C r 33 g r oup f o r
The v a l u e o f
of this
o f a proton
g^ and D were c a l c u l a t e d
oblitera ted
liste d
the a id
These ar e t h e M -
tra nsitions
span o f
meas ur ement s a r e
t h e same e x c e p t
lines
for
parameters.
the. a v e r a g e p o s i t i o n
previous
to
chr omi um l i n e s
n e a r 3400 g a u s s .
I.
difference
values
ar e n e a r l y
measuring the
Fig.
22 g a us s .
probe.
The M =
Hamiltonian
shown i n
measur ed w i t h
and t h e
Experimental
3600 g a u s s ,
the
lines,
is
the C r ^
be 4 D, a c c o r d i n g
Hg.
these
of
resonance f i e l d
3 1
and M =
is
the A 1 ion
7 were c a r e f u l l y
magnetic
should
t h e second k i n d
i n m a g n i t u d e by a b o u t
The p o s i t i o n s
in
of
gg was o b t a i n e d
same g r o u p .
in Table
0 = 90°,
by m e a s u r i n g
The r e s u l t s
I , along w i t h
by
the
o f the-
results
of
authors.
The p o s i t i o n
measur ed f o r
was r o t a t e d
o f each o f t h e
each f o u r
re la tive
to
various
chr omi um l i n e s
degrees o f a n gl e a s . t h e ma g ne t ic
the c r y s t a l .
me as ur ement s
a r e shown i n
expected f o r
axial
Fig.
symmetry,
9,
there
for
is
The r e s u l t s
was
fie ld
o f these
t h e C r 32 l i n e s .
As
180° p e r i o d i c i t y
in
5500 *
o
5000 ' •
Experimental
C a l c u l ated
4000--
MAGNETIC FIELD
-
gauss
4500- *
3500 ‘
2500 *
2000
* "
Fig.
9
A n g u l a r Dependence o f
Fi n e S t r u c t u r e
Groups
-28TABLE I
HAMILTONIAN PARAMETERS
9s
I .97854±.00005
I . 98200+.0010
I .981711.00005.
- 879.76i.04
+1
r— '
r-.
CO
CO
.1
D
I . 9 7 8 8 2 ± . 00010
A
I 7.84+.10
the angula r
dependence.
t h a n 80°
it
was n o t p o s s i b l e
most o f
the
extreme
line-broadening
positions
C r 53 l i n e s
with
17 . 5 1 0 . 4 gauss
g r e a t e r than
to
with
angle.
The C r 52 l i n e s
could
however,
since
t h e y a r e more
numerous
forbidden
up i n t o
approach
in
the h y p e r f i n e
be o b s e r v e d
lines
of
line
Line widths
angles
not s p l i t
position
bec aus e o f t he
ty p ic a lly
broadening,
a b o u t 10° and
observe the
any c e r t a i n t y ,
gauss
- 886
M = ± g r o u p s
th e ir
I .9861.001
which o cc u r s whe reve r t h e
change r a p i d l y
near 45°.
■ I
17.73i.02
For a n g l e s
O
!+
O
O
9P
Low^)
77 °K
9 5 2 6 . 5 8 ± .01
9994.0 ±0.5 .
V
less
M a r s h a l l , e t a l ^ 2^
77 ° K
T h i s wor k
300°K
LO
'03
Parameter
t he
spacing
in
at
spite
intense
and do
as th'e C r 33 l i n e s ,
do.
Auzins
and Wer t z
^ ascribe
g ra in m is o r i e n t a t i o h , or s l i g h t
direction
o f the te tra g o n a l
the
local
axis.'
line
broadening
deviations
A convincing
in
of
to subt he
a r g u me n t f o r
-29this
idea
is
line
width
presented
is
changes w i t h
rapidly
tion
is
angle.
as 50 gauss
o f the
sh ift
plotted
the
the
per degree ,
axis
suppose t h a t
of
a t which
this
The.slope
spread
in
that
extra
the d i r e c t i o n
of
the c r y s t a l
ion
several
la ttic e
the
strength
its
direction-.
point
a change i n
fie ld
orientation,
t h o u g h , so t h e
a n g l e s ..
Fig.
c hr omi um A,
indicates
10
axis
of a rc .
w h i c h ar e
defects
could
amount .
but could
A t r i valent
much t o
s lig h tly
does n o t move v e r y
fact
reaches
The l i n e s
alter
alter
rapidly
a turning
are t h e r e f o r e
distinguished.
Th er e
s t r o n g manganese s p e c t r u m ,
c hr omi um s p e c t r u m c a n n o t be o b s e r v e d a t a l l
11 i s
a plot
o f t h e a n g u l a r dependence o f t h e
B , and C s p e c t r a
the
Fig.
seems r e a s o n a b l e t o
by t h i s
and i n
from the
in
13 mi n .
has s u b g r a i n s
b r o a d e n e d , and can be e a s i l y
interference
the d i r e c ­
away may n o t c o n t r i b u t e
fie ld ,
position
the t e t r a g o n a l
n e a r 6 = 40° a t a b o u t 3900 g a u s s .
serious
in
o f the l i n e
la ttic e
gr o u p o f l i n e s
not s e r io u s l y
is
s paces
o f the c r y s t a l
The M =
with
crystal
C r^
changes as
such a s p r e a d o f a b o u t
nearby o f f - a x i s
the
its
I
as 0.2 d e g r e e woul d
o u t o f a l i g n m e n t by even. 0.1 d e g r e e , i t
think
line
so a d e v i a t i o n
o f as s m a l l
angular
3
wher e t h e M =
rate
The p o s i t i o n
and i n d i c a t e s
than
10,
the
by 10 g a u s s .
a measur e o f
Rather
Fig.
versus
tetragonal
line
dire ction ,
in
regions
and t h e manganese s p e c t r u m ,
wher e u n o b s t r u c t e d
v i ews o f
the A
and
gauss
- 3 O-
LINE WIDTH
-
sI ope = 0.22 degr ees
dH/de
Fig.
10
gauss/degree
Dependence o f L i n e W i d t h
Line
Position
on Rate a t w h i c h
Changes w i t h
O rientation.
3500
3400
3600
MAGNETIC FIELD
Fig.
11
Angular
Dependence o f C e n t r a l
3700
gauss
Field
Lines
3800
s pec trum are p o s s i b l e .
In the
region
from
0 = 28° t o 45° t h e r e i s c o n s i d e r a b l e
mixing
of states,
and one s h o u l d
o f the
t y p e M,m =
1 3
1 1 > ±2">""2' ’ i Z" ’
and even d o m i n a t e
line
at
spectrum,
3879 g a u s s , f o r
two f o r b i d d e n
This
the
hyperfine
coincide.
about the c e n t e r ,
at
is
a sim ilar
w h i c h was doped w i t h
far
less
and shows f o u r
A and A '
A1 lin e
accounts
3848 g a u s s ,
for
fie ld
13.
Fig.
This
Returning
for
and t h a t
are accounted
in
Cr^g.
Crgg l i n e s ,
lines.
+
^
12.
The
,
of the
g - , and +4-,
where t h e A and B
Fig.
spectra.
12 s h o u l d - be s y m m e t r i c
To d e t e r m i n e whi c h
r e c o r d i n g was made on a
natural
chr omi um and t h e r e f o r e
spectrum is
to
Fig.
12,
inte nsity
of the
a t 3831
two w e a k e r Crgg l i n e s
The weak l i n e
in
in
one sees t h a t
t h e weak s h o u l d e r s
by t h e
recorded
Fig.
Fig.
at
12,
3876 i s
t h e Crgg
line
visib le
t h e ones ,at 3826 and 3888
superposition
-g- a l l o w e d l i n e and two Am = 2
forbidden
that
lines.
It
w ill
t h e m = ±^- a l l o w e d
and t h e one a t
be shown i n
lines
are
a t l ow
not accounted f o r .
lines,
the m =
at
and 3842 gauss
gauss a r e m = ±g- a l l o w e d
of
13,
t h e most p r o m i n e n t o f wh i c h ar e t h e
t h e abnor mal
Of t h e o t h e r C r 53 l i n e s
f a c t appear,
n o t because o f t he p r e s e n c e o f l i n e s
lines,are
extraneous,
Fig.
transitions
a superposition
9 = 45°,
sim ilar
contains
is
The s p e c t r u m o f
but
do i n
as shown i n
transitions
f r o m t h e A 1 and o t h e r
crystal
Such l i n e s
instance,
s p e c t r u m was r e c o r d e d
spectra
expect to observe
3869 gauss i s
the f o l l o w i n g
v e r y weak a t
this
chapters
a
45°
-33-
4---------------i---------------1-------------- !------------- ,V--------------1--------------- 1---------------f 3820
3830
3840
3850
\f
MAGNETIC FIELD
Fig.
12
Mg=
Transitions
3870
-
for
gauss
G= 4 5 0
3880
3890
45°
-34-
4------------------------- 1------------------------- !------------------------ !------------------------- 1-------------------------1------------------------- !------------------------- \—
3820
3830
3840
3850
3860
MAGNETIC FIELD
Fig.
13
Mg= \
^
C r 52 L i n e s
for
3870
-
gauss
6= 45°
3880
3890
3900
11
II
-35o r i e n t a t i o n ; even wea k er t h a n
t h e Am = 2 f o r b i d d e n
lines
which
o c c u r a t a p p r o x i m a t e ! y t h e same f i e l d .
Figures
14 - 17 were r e c o r d e d
and show t h e same g e n e r a l
spectra
is
do n o t c o i n c i d e
s till
lower
visible
fie ld .
inte nsities
o f the
features
in
Fig.
An a t t e m p t
the c a lc u la te d
are
in
From F i g .
these
14,
is
displaced
made i n
for
of the
gauss.
Figures
as i t
intense
it
Chapter V to
these
The v a r i o u s
chr omi um l i n e s
which
each c a s e .
One can see i n
Fig.
than
it
is
is
lines
These l i n e s
admixture
Rubi ns ^ ^
t he
in
figures
three
Cr^g t r a n s i t i o n s
to
which
m i g h t be p o s s i b l e
as i t
to
moves
bel ow 3800
o f the A spectrum
3675 , 3705 , and .3735 g a u s s .
are v i s i b l e
18 t h a t
are
labeled
the h i g h - f i e l d
c o n s i d e r a b l y wea k er a t a n g l e s
in
Am = I
n e a r 10°
near 45°.
The two s t r o n g ,
hyperfine
fit
lines
18 and 19 show p o r t i o n s
doublet
The A and B :
a b o u t 33 gauss t o w a r d
manganese s p e c t r u m a t f i e l d s
moves p a s t t h r e e wi ndows a t
forbidden
and 2 8 ° ,
but the B spectrum
o b s e r v e t h e A s p e c t r u m a t some o r i e n t a t i o n s
the
32°,
t h e two l i n e s .
11 one sees t h a t
through
36°,
as F i g i ' 1 2 . '
figures,
two h i g h - f i e l d
inte nsities
responsible
a t 0 = 40°,
narrow l i n e s
belonging
were f i r s t
1 964.
the
correctly
of neighboring
in
to
in
Fig.
19 a r e f o r b i d d e n
d i v a l e n t manganese s p e c t r u m .
explained,
hyperfine
states,
on t h e
basis
of
by D r u m h e l l e r and
f
40°
I /
3820
— I—
3330
-------------- 1--------------------------------------------------- 1—
3840
—
---------------------
3850
MAGNETIC FIELD
Fig.
14
Mg=
43870
4-
3860
-
Transitions
gauss
for
6= 40'
43880
+
4------------------- 1
-------------------H 3820
3830
3840
3850
3870
3860
v gauss
MAGNETIC FIELD
Fig.
15
1
2
■*
I
"2
Transitions
for
6
36°
— I--------------------f—
3880
3890
— F3900
-38-
3820
3850
3830
3860
MAGNETIC FIELD
Fig.
16
Mg= ^ ^
Transitions
-
3870
gauss
for
6= 3 2 0
3880
3890
3900
28°
-39-
4----3820
— I----------------- t--------------------1------------------- 1------------------- 1------------------- 1-------------------f—
3830
3840
3850
3860
3870
3880
3890
MAGNETIC FIELD
Fig.
17
Mg= \
Transitions
for
gauss
6= 28°
—
\—
3900
-40-
-41-
■ -42I n an a t t e m p t
to observe
forbidden
■ 3 I -
t h e M = ± ^ ± 2" g r o u p s , t h e a r e a s
gauss were c a r e f u l l y
0 = 0 to
15°
15°.
the l i n e
observation.
forbidden
broadening
There i s
in
ra tio
is
about
inte nsities
inte nsities
very poor,
vanish.at
has become t o o
some i n d i c a t i o n
these gr oup s,
and n o t h i n g
in
n e a r 1850 gauss and 5350
scanned a t o r i e n t a t i o n s
The l i n e
lines
hyperfine' lines
in
t h e r ange ■
0 = 0,
severe to
and a t
permit
the ir
o f the presence o f
but the s i g n a l - t o - n o i s e
d e fin ite
or accurate l i n e
can be d e t e r m i n e d
positions.
IV.
CALCULATED SPECTRA - -
The m a g n e t i c
occurs
is
sim ilar
levels
fie ld
determined
to
Eq.
w ill
either
The r e s u l t s
below,
using
of c a lcu la tio n s
compar ed w i t h
The H a m i l t o n i a n
of
so any o f a number o f
its
fine
inte ractio n
numerical
used,
of
is
but the f i r s t ,
in
correct
problem.
does n o t
required.
eigenvectors
of
Fig.
3.
be
positions.
Matrix
symmetric m a t r i x ,
factor
and t h e r e f o r e
p r o g r a ms
uses t h e method o f
but the
the H am iltonian
and Cr ^g w i l l
Two b a s i c
calculations
scheme,
and
the presence o f the h y p er-
for
were p e r f o r m e d s a t i s f a c t o r i l y
diagonalization
two
c o m p u t e r pr ogr ams may be used
f a v o r o f t h e one l i s t e d
t h e C r 52 (4 x 4)
vectors
a real,
Owing t o
which
of
Hamiltonian
3 is
bet ween t h e
Hamiltonian
both C r ^
the
line
Both o f t h e s e methods
observed l i n e
standard
the m a t r i x
solution
been d i s c a r d e d
of
Fig.
diagonalization.
the
for
experim entally
Diagonalization
for
theory.
resonance
the H a m i l t o n i a n ,
by d i a g o n a l i z a t i o n
or from p e r t u r b a t i o n
be d i s c u s s e d
of
.
o f e n e r g y by a r e l a t i o n
The e n e r g y d i f f e r e n c e
by t h e e i g e n v a l u e s
may be d e t e r m i n e d
matrix
at which a paramagnetic
from c o n s e r v a t i o n
(2.13).
are g i v e n
LINE POSITIONS,
Householder,
eigenvalues
to
to
give
t he C r 55 (16 x 16)
The met hod o f von Neumann, used i n
the
has
Al I
and e i g e n ­
Householder's
pr o g r a m f a i l e d
when a p p l i e d
have been
in Appendix B.
using
a
pr ogr am
-44liste d
i n A p p e n d i x B , has been v e r y s a t i s f a c t o r y
producing
Cr^g wave f u n c t i o n s .
The p a r a m e t e r s
f r o m meas ur ement s
arbitrary
angles
the value o f
tion
at
the H am ilto nian
0=0°
magnetic
fie ld
is
fie ld
the H a m ilt o n ia n ,
gauss.
eigenvalues
having
subtracted
difference
is
upon w h e t h e r
than the
the eigenvalue
energy.
conserved.
Co n v e r g e n c e i s
is
iteration s
into
the u n i t s
bet ween t h e a p p r o p r i a t e
subtracted
to
is
is
less
repeated u n t i l
rapid
H.
to d e fine
of
eigen­
or greate r
put
into
the
energy is
because t h e e n e r g y
Usually
a. l i n e
the three
a c c u r a t e l y measur ed f o r
of
f r o m H, d e p e n d i n g
fine
structure
three
position
Crgg l i n e s
9 = 0° and 9 0 ° ,
levels
or f o u r
to w i t h i n
gauss.
The p o s i t i o n s
the
p h o t o n e n e r g y and t h i s
bet ween a d j a c e n t
eq ual
suffice
and each o r i e n t a ­
energies
difference
process
gauss)
approximately
by t h e ph ot o n
The new v a l u e o f H i s
and t h e
(in
until
The method o f a d j u s t i n g
added t o o r
Hamiltonian
difference
d iffe r
for
by a d j u s t i n g
Hamiltonian
each l i n e
put a l l
from the
either
photon
the
positions
assume some v a l u e o f H and d i a g o n a l i z e
The e n e r g y d i f f e r e n c e
is
in
investigate.
to
The l i n e
be d e t e r m i n e d
T h i s must be done f o r
to
have been d e t e r m i n e d
and 9 0 ° .
can t h e r e f o r e
pair of
one w i s h e s
values
in
the magnetic
the a p p ro p ria te
energy.
for
have been
and t h e M =
O'. I
-45line
was l o c a t e d
listed
at
in
for
Table
6 = 30°.
11 a l o n g w i t h
These l i n e
positions
comput ed p o s i t i o n s .
1834 gauss and 5383 gauss were used t o d e f i n e
Hamiltonian,
with
so t h e
experiment,
other
lines
is
comput ed p o s i t i o n s
for
ar e
The l i n e s
the
t h e s e must agr ee
b u t t h e a c c u r a c y o f t h e a g r e e me n t f o r
a strong
argument f o r
the c o r r e c t n e s s
t he
o f the
C r 52 H a m i l t o n i a n .
TABLE I I
C r 52 LINE POSITIONS FOR v
Li ne
3> 1
'2 "2
3
I
"2 "2
4
* 4
,V
* 4
I
4
'
= 9 9 9 4 . 0 MHz-
Obser ved
Computed
0°
1834.25+0.10
1834. 31
90°
4475.75+0.10
4475.57
0°
5382.72+0.10
5382.70
90°
2750.31±0.10
2750.29
0°
3608.52+0.20
3608.50
30°
3837.38+0.20
. 3837.47
90°
3430.62+0.20
3430.64
Computed C r 52- l i n e
positions
for, a l l
angles
gauss'
ar e shown
the
preceding
chapter.
They a g r e e w i t h
me n t a l I y measur ed
positions
to w i t h i n
the e x p e r i m e n ta l
in
Fig.
9,' i n
experierror.
-46Line
positions
diagonalization
liste d
line
in
inte nsities.
Fig.
in
the M =
lines
only
Table V I I
in
for
t h e Cr^g s p e c t r u m were comput ed by
for
in
0 = 30° and 4 5 ° .
the next
A schematic
chapter,
These p o s i t i o n s
along w i t h
energy l e v e l
group.
c omput ed l i n e
diagram is
It
is
seen t h a t
perturbation
position's
spaced d o u b l e t s
be a r t h i s
theory, ^
^ and t h i s
The d o u b l e t
separations
width,
out.
and t h e
The s e p a r a t i o n
w ill
are t y p i c a l l y
effect
on t h e l i n e
The P e r t u r b a t i o n
The r e p r e s e n t a t i o n
of
is
be done i n
that
one w i t h
the magnetic
fie ld .
The p e r t u r b a t i o n
series
from the
exact
pair
of
used f o r
its
The H a m i l t o n i a n
<m | H , | ^ n>
n> +
m^n
(oT
En - Em
+ < n I H' U
>
t ha n t h e
amplitude.
Method
quantization
integral
the next
separation
the p e r t u r b a t i o n
used h e r e
from
less
so t h e y c a n n o t be r e s o l v e d , b u t t h e
does have an o b s e r v a b l e
tions
shown
t h e Am = ±1 , ±2, ±3
bet ween t h e two l i n e s ' o f a d o u b l e t can be p r e d i c t e d
line
comput ed
20, s ho wi n g t h e a p p r o x i m a t e r e l a t i o n o f t h e 16 l i n e s
o c c u r as a s e t o f c l o s e l y
section.
are
is
axis
is
in
the d i r e c t i o n
given
developed
calcula­
by Eq.
(2.10)
by i t e r a t i o n
equations:
(4.1 )
(4;2)
-47-
m= -
Icxi CO io u
m= -
ENERG
CO |c u
MAGNETIC FIELD
F i g . 20
Schematic
MS=+ 2~ and
Ener gy L e v e l
=
Levels
Di agr am f o r
t he
-48-
K
wher e
are
iterated
by r e p l a c i n g
next lo w e r.o rd e r
This
and H0 1n> = En ^ ° ^ | n > .
= H 0 + H.'
approximation,
can be s t a t e d
(s)
n> +
c ( S ) ■_ p ( o )
E n'"'
n
Carrying
the
and f o u r t h
E
on t h e
starting
righ t
I m>
m,&
the form o f a r e c u r s i o n
-
+ <n|H '|*n
for
to
energy,
(4.3)
(s-1).
third
(4.4)
order fo r
t h e wave f u n c t i o n
one o b t a i n s
nnHnUHt o
E E
nm nil
H H
nn mn
E 2
n m
( HnnHm t +Hmn Hn ^ Hiln
E ^E
nm ^nil
HmilHJlkHkn
m> F F F
nm n& nk
m,&,k
(4)
+ ■Z
m, A
+
.
(4.5)
^nm^m&^&n
E (°)+H
+Z - ^ rri--nI--+ E
E E
n
nn m, Enm
m,& nm nil •
Hn n Hmn
HnnHnniHm A n
'nm
E E ^
nm nil
HntnHm* HJ k Hkn '
E E E
m,&,k
nm nil nk
|n>,
re la tio n :
<m|H'^n ( s " 1 ) >
Z I m>
( s - 1) _ p ( o )
m^n
E
n
m
' Hn n Hmn
"nm
by t h e i r
with
.H - Hn
H
mil &n
mn .
v I .
n> + Z I m> — + Z I m>
E E
^nm nil
m
nm
m, il
(3)
+
in
ite ra tio n
order
and
T he ;; e e q u a t i o n s
HnnHmn'
'nm
Hnm(Hn n Hna.+HronHn i t )Ht n
Enm Ent.
(4.6)
-I
-49wher e a l l
sums e x c l u d e
The d e n o m i n a t o r s
energies
n , Efim= En
, and Hfiin = <mJ
have been w r i t t e n
by e x p a n d i n g
in
them a c c o r d i n g
t er ms
o f zero
Hamiltonian
to
(4.7)
can be c o n s i d e r a b l y
in
the
follow ing
sim plified
+ j
wher e a l l
diagonal
terms
and o f f - d i a g o n a l
e n e r g y and h y p e r f i n e
tion.
The number o f t e r ms
substantially
reduced, since
a r e more c o m p l i c a t e d
was used as t h e z e r o - o r d e r
elements
interaction
in
the
-
HIz
S i n 2GCsJ + S2 ]
ar e l umped i n t o
fie ld
energies
one w r i t e s
s i n 2 6 [ S z S+ + SZS_ + S+Sz + S Sz ]
Hamiltonian,
though,
if
way:
■ H0 = HSz + ASz I z + DP2 ( C o s G) [ S z - j S2 ]
H1 = | [ S + I_ + S _ I +] + I
1| n> .
order
yqr^r = I - x + X^ - ...
These s e r i e s
H
Eq.
(4.5)
Hpn = 0.
than
if
(4.9)
the zero
o f both
the
order
crystal
make up t h e p e r t u r b a ­
and
(4.6)
is
The z e r o - o r d e r e n e r g i e s
t h e Zeeman t e r m aI one
Hamiltonian.
The z e r o - o r d e r
are
EM m0 ^ = HM +' AMm + DP2 (CosG) [M 2- y S ( S + l ) ]
■( 4 . 1 0 )
-50wher e t h e
Eq.
parameters
(2.11)
With
A and D have been r e - s c a l e d
so t h a t a l l
Hnn = 0 ,
E (4 )
energies
the
expression
_ E ( o ) + £ Hmn +
n
n
-
„
E
m ^nm
HmnH«n
,
------------p
m
+ ■ ■
EnmEnjl
The v a r i o u s m a t r i x
give
a series
E.n i s
m, I
EnmEnJl
F
m,£,k
sim plified
(4 . I I )-
F-------------------
are w r i t t e n
using
I/H .
Em
f o r a r b i t r a r y S and I
o ’
-i
S = |- and M = ±^-, and i g n o r i n g
O p
A /H , t h e e n e r g y l e v e l s ar e
to
nm n£ nk
H1
- V,
found
to
o f gauss.
HnmHm lHt k H'kn
—F
for
E 1 =
± 2-,m
for
units
HnmHml,H&n
,
in
in
„
Z
elements,
The e n e r g y d e n o m i n a t o r s ,
expanded t o
are
according
Eq.
out
given
t e r ms
1 " - DP
™ 2 ±}Am - ' anHfiH
yP
Fig.
3.
( 4 . 1 0 ) - , may be
The r e s u l t i n g
is
in
expression
i n A p p e n d i x A, ■ For
smaller
than
4 3
D /H
or
±
" ^8TH- [ s i n 4e - 2s i n 220]
+ ^ - [ ± ( I 2- m 2 ) - 7 m ] + ^ l Z 2 [ 4 s i n 2 20 + s i n 40 ] ± ^ 7m [ 3 s i n 22 0 - s i n 40]
8H
SHz
,
[ 13
2HT
J
2
.
! 2- ^4) ]
W
SD4P2
—[ s i n 40- 8s. i n 220] ± — — 3 [ 8s i n 420 + l 6s i n 220
I 28H
BH'
s i n 40- s i n ^ 0]
x
(4.12)
I l l h
-S i­
de W i j n and van Ba! de.ren
all
these
ve rifie s
t e r ms
the ir
except
results
the denominators
levels
equation
for
to
has been f o u n d
a t which
V m V =
that
a particular
the d i f f e r e n c e
t hey , use Hq i n .
line
o c c u r s may
bet ween t h e two e n e r g y
p h o t o n e n e r g y , and s o l v i n g
For t h e t r a n s i t i o n
in
the c a l c u l a t i o n o f
4 O
D /H .
E q u a t i o n (4. 12. )
the e x c e p t io n
fie ld
the
H.
with
o f H..
by s e t t i n g
e q ual
t h e two o f o r d e r
instead
The m a g n e t i c
be f o u n d
^ have r e v i e w e d
1 1
the
resulting
, m 1, t h e s o l u t i o n „y
A p p e n d i x A 5 and i s
Ho
+ 9g pAe
+ 45— (2sin ^ 2 6 -s in ^ 6)
4Ho '
V
-
m- " ' >
1^ - 7 ( m - m 1 ) ]
,„„2
,
■ S fl2DP;,
„
„
- — 5— (m+m ' )'s i n ^20 + ------ 5— (m -m' -m-m 1)
+
)_( 5+m2+m ' 2 ) -
(m3+ m ' 3 ) - 10(m2- m ' 2 ) ]
4
+ —
2 _ [ 9 s i n 22 8 - s i n 4 6]
4H3
wher e H
0
= E , .
+ 2"5m
- E .
—2"5m
The C r g 2 l i n e
expression
is
position
by i g n o r i n g
all
- ^ - ? [ 2 4 s i n 4 2 6 + 3 s i n 8 0]
64H3 .
the
photon energy.
has been c a l c u l a t e d
t e r ms
( 4 . 1 3)
containing
A or
from t h i s
gn .
The
I i
52
-
calculated
in Table
position
is
-
compared w i t h
observed
line
positions
III.
TABLE I I I
M
I
i
+ 2'> - 2'
=
C r 52 LINE POSITIONS
Il
OT
O
O
o
LO
CD
!
- Il
CD
1
1
o
O
CO
CD
0 = 0°
calc.
3608
3842
3855
3736
obs.
3608
3838
3859
3735
The d i s c r e p a n c i e s
in
contributions
indicates
that
Cr^g l i n e
positions
compared
with
of
in
the
in
perturbation
Eq. (4 .1 3 )
positions
the f o u r
Table
IV w i t h
neighboring
it
converging,but
is
evident
to
slowly.
Eq.
the C r ^
When t h e
(4.12),
which
above d e s c r i b e
line.
lines
at
Relative
6 = 45° ar e
comput ed by d i a g o n a l i z a t i o n ,
found
lines.
the s e p a ra tio n
the
than the
converging.
positions
repulsion
that
3431
in
is
hyperfin.e
and w i t h
are mi xed ,
3437 gauss
w h i c h were i g n o r e d
positions
t h a n A owi ng t o t h e mu t u a l
Again
t e r ms
o f A gauss bet ween a d j a c e n t
states
'
somewhat s m a l l e r
series
re la tive
allowed
observed p o s i t i o n s ,
a separation
111 a r e
from the f o u r t h - o r d e r
The t e r m s
the
Table
0 = 90°
assumi ng o n l y
Since the
should
be g r e a t e r
o f the e n e r g y . l e v e l s .
perturbation
hyperfine
series
inte ractio n
is
operators
-53TABLE IV
C r 53 RELATIVE LINE POSITIONS', 6 = 45°
m = j
m = ^-
m -
3
m = -3
-2
29 . 0 9 G
p e r t u r b a t i on
-29.23
- 9.87
■ 9.57.
diagonalization
-31.15
-10.41
10.40
31 .22
A-separation
*
observed
-26.76
- 8.92
8.92
26 . 76
-32.0
-10.3
*
Corrected f o r
C r 53 and f o r Cr 53 .
are
involved
from very
it
high
is
a d i f f e r e n c e o f 0 ,. I gauss bet ween D f o r
(See Mar s hal I , e t a I , r e f . 2 6 ) .
possible
orders
doublets,
doublet
energy d i f f e r e n c e
a p p e a r as c l o s e l y
in
+^,m ^-^,m ',
can be f o u n d
to :
contributions
in
the
as AM ga us s .
previously
are l a b e l e d
separation
have s i g n i f i c a n t
h y p e r . f i ne l i n e s
as i n d i c a t e d
of a doublet
to
because t h e
d e n o m i n a t o r can be as s m a l l
The f o r b i d d e n
30.5
10. 7
Fig.
22.
spaced,
The two l i n e s
and +^-.,m‘ ->-^-,mi and t h e
f r o m Eq. ( 4 . 1 2 )
according
'
6H = Hm ,m I
,m
(4.14)
-54Equation
(4.14)
has been g e n e r a l i z e d
S i n A p p e n d i x A,
2gn^n
A 2DP9 .
+ ------ .(AS
4I-r
o
equation
tra nsition s.
6H
as
2- 3 ) (m2 - m 12) - ^
( I 2S 2- 5 ) (m2 -m
(4.15)
sh:
o
is
valid
only f o r
For S = ^
. 2g ne n
V
v a I -we o f .
+ ^H 0 ( 4 S 2- 1 ) ( m-m1)
Ho ( " l - ” ' )
P
This
and a p p e a r s
t o an a r b i t r a r y
( S 2=-^-)
the M =
i
it
T
I
electronic
reduces.to
Hq ( m-m 1 ) + ^- jj— (m-m 1 )
no
SA2 DP
—(m2 - m 12) - 5
which
is
and i s
also
vanishes
lated
con sistent with
consistent
as i t
f r o m Eq.
spectrum,
from the
*
should.
(4.16)
(m2 - m ' 2 )
the
in
the
equation
the
given
sense t h a t
Doublet
for
(4.16)
if
separations
six
doublets
and ar e compar ed i n T a b l e
V for
by W a l d n e r , ^ 2®^
m = m' ,
6H
have been c a l c u ­
of
the
chromi um
6 = 30° w i t h r e s u l t s
computer d i a g o n a l i z a t i o n . *
o
The s h o r t h a n d n o t a t i o n S has been used f o r
v a l u e S ( S + 1) i n s e v e r a l p l a c e s i n t h i s w o r k .
the e i g e n ­
L
-55TABLE V
FORBIDDEN DOUBLET SEPARATIONS
3 I
2 52
45
-0.10
° ) .
-
-
.30
The c o m p u t e r r e s u l t s
LO
(
I
di a g .
.33
.26
indicate
.63
-
-
.56
-
-0.18
-0.19
that
.51
the d o u b l e t
.90 G
.90
CM
CO
O
-
-1.61
-
CO
CO
(45°)
- I .64
.68
I
pe rt.
-
LO
- I .68
-
.30
I
(30°)
.21
-
CM
CO
diag.
.38
I
-
3 3
2 ,-g
'I
3
2 ’ "2
CM
CO
CO
(30°)
CO
O
O
pe rt.
.3 I
2 5"2
I
3
“ 2 *"2
1 I
2 ’ ~2
O
O
m,m 1
sp littin g s
a r e much more s t r o n g l y
a n g l e - d e p e n d e n t t h a n one w o u l d e x p e c t
from the
Eq.
third
term o f
(4.16).
m- dependenc e i s
w ea k er t h a n t h a t
in
Experimental
Eq.
(4.16).
the d o u b l e t
Figs.
s p littin g s
15 and 16 ,
doublet
resolved
doublet
is
evidence,
one sees t h a t
g r e a t enough t h a t
not re s o lv e d ,
and i s
Hamiltonian
last
however,
the
the
two l i n e s
in
though.
an i n d i c a t i o n
Fig.
This
results
that
s p littin g
are
the
may be more c o m p l i c a t e d
is
of
are
16);
indicates
the
that
In
t h e m,m'
=
actually
The - ^ , - ^ r
the o n ly
instance
at variance w ith
hyperfine
that
two t er ms
do have a s t r o n g m- d e p e n d e n c e .
where t h e d i a g o n a l i z a t i o n
data,
indicate
g i v e n by t h e
( a t 3826 and 3829 gauss
is
They a l s o
observed
term in the
t h a n has been assumed.
6 = 45° , t h e r e i s no e v i d e n c e t h a t e i t h e r o f t h e o b s e r v e d
At
I
-56Am = I d o u b l e t s
a r e s e p a r a t e d more t h a n T a b l e V i n d i c a t e s .
Conclusions
The c o m p a r i s o n
tion
theory with
zation
slowly.
Wi t h
results,
the
is
in
this
fraction
v e rific a tio n
the
satisfactory
line
from a diagonal!"-
that
although
converging
noted above,
line
o r d e r t e r ms
the
only very
the computer ■
the experimental
positions
calculations
calculated.
but are a l l
the p e r t u r b a t i o n
the z e r o - o r d e r
reached r e g a r d i n g
is
from p e r t u r b a ­
o f t h e s p i n - H a m i I torn" a h .
perturbation
case i n w h i c h
of
exception
of re la tive
as t h e h i g h e s t
are n o t e n t i r e l y
it
agree very c l o s e l y w i t h
types
discrepancies
indicates
converging,
single
and p r o v i d e
For t h e
calculated
those observed or c a l c u l a t e d
series
d i agonalizations
for
positions
o f the H a m ilto n ia n m a t r i x
perturbation
large
of lin e
energy.
inte nsities
the
a r e . a b o u t as
The r e s u l t s
one s h o u l d expe' ct
energy
' Sim ilar
in
considered,
is
a large
conclusions
the ne xt c h a p t e r .
ar e
LI
V.
CALCULATED SPECTRA - -
Corresponding
to
3 ar e e i g e n f u n c t i o n s
tions
of
the magnetic
M,m
The m a t r i x
can be w r i t t e n
eigenstates
of c o e ffic ie n ts
coefficients
function
I M.,m>.
^
m in
this
is
For o r d i n a r y m a g n e t i c
s ig n ifica n t
particular,
because o f
leve ls,
parts
o f the
states
The m a g n e t i c
paramagnetic
be n e g l i g i b l y
fie ld s,
of
selection
rules
one may w i s h
and each wave f u n c t i o n
s h o wi n g s p e c i f i c a l l y
is
is
In
hyper-fine
to
the a d m ixtu re
a set
investigate
labeled with
w ritten
are a p p r o p r i a t e
These s e l e c t i o n
levels
Each l e v e l
t h e wave f u n c t i o n
of neighboring
F i g u r e 21 i l l u s t r a t e s
tra nsition s.
t h e base s t a t e
| M, m+ l > .
resonance t r a n s i t i o n .
which
The wave
o t h e r base s t a t e s .
AMg. '= ±1 , Anij = 0.
w ithin
the
m may have admi xed s i g n i f i c a n t
| M , m - 1> and
dipole
though,
fie ld s,
and a l l
small.
be e s s e n t i a l l y
the p r o x i m i t y
^
la r g e magnetic
the H am iltonian
admixtures
t h e wave f u n c t i o n
the m a t r i x which
Fo r v e r y
case w i l l
may c o n t a i n
combina­
(5.1)
domi nate
e x c e p t C^ m w i l l
Fig.
| M, m>:
C^,
d i a g o n a l i zes t h e H a m i l t o n i a n .
t h e Zeeman e n e r g y w i l l
as l i n e a r
of
1,m 1>
'M' ,m'
M' ,m
LINE INTENSITIES
the an gle -d ep end ent H a m ilt o n ia n
which
h
in
its
to. a
rules
are
o f en er g y
the
allowed
wave f u n c t i o n ,
t h e f or m o f
of neighboring
Eq.
(5.1)
hyperfine
+ 6 0 |M,m+2>
v M,m
I M, m - I >
, m- I
\
ENERGY
I
Cl
CO
I
,m
^ M - I , m- I
Fig.
21
I M - 1 ,m>
+ a 2 I M- I , m - 1 >
|M -l,m-l>
+a
Ener gy L e v e l
Di agr am Showi ng Wave F u n c t i o n A d m i x t u r e s .
,m+2>
u
JLLL
-59states.'
are
Transitions
indicated
which
by t h e
arrows.
indicated
by v e r t i c a l
tions
indicated
are
The t r a n s i t i o n
by F e r m i ' s
golden
ar e a l l o w e d by t h e
Strong
arrows,
selection
tra nsitions
are those
and s o - c a l I ed. f o r b i dden t r a n s i ­
by t h e d i a g o n a l
arrows.
p roba bility
the v a r io u s
for
Tines
is
inte ractio n
given
rule:
= c o n s t x I < ^ f I H 1 I ^ 1H 2
The p e r t u r b a t i o n
rules
which
induces
(5.2)
the t r a n s i t i o n
o f the mi crowave m a g n e t i c
fie ld
is
t h e Zeeman
with
the e l e c t r o n
spin:
H f cosa)t
H1 = g3Hr f *S = g g c o s w t Hr f Sy = gp --------------- ( S+ - S_)
wher e Hr f
directed
is
the
along
the y - a x i s .
“ M.m-M'.m'
Equation
rules
(5.4)
cited
the f o u r
clearly
in
The t r a n s i t i o n
illu stra te s
since
lines
of
Fig.
Table VI.
etc.
of
is:
\
(5' 4).
the
selection
S+ and S_ can o n l y
by ±1.
Line
21 can be w r i t t e n
a . ,-3^,
and i s
probability
the o r i g i n
the o p e r a t o r s
f o r which M d i f f e r s
admixture c o e f f i c i e n t s
are I i sted
o f t h e mi c r o w a v e f i e l d ,
= consf x
above,
connect sta te s
for
amplitude
(5,3)
in
intensities
t e r ms o f t h e
The r e l a t i v e
inte nsifies
w r.\
:
-60- •
TABLE VI
RELATIVE INTENSITY OF HYPERFINE LINES
Transition
Intensity
M5IiH-M-I 5m
(l +Ooa2+Bc)B2+YoY2+*o*2+
M5ithM- 1 5m-l
(G3+Oo+G()6g+Yoag+
M,m-l ->M-l 5m
(B-J+Otg + BgS-J+Yga -]"1" • • • ) ^
M, m. - l +M- l 5m- l
(l +a^ot g + B-j Bg+Yj Yg + S-] Sg+ • • • )
Sim ilar
o
expressions
for
Am = 2 ,
3 forbidden
lines
may be
■readily obtained.
Predicting
the
re la tive
line
k n o wl e d g e o f t h e wave f u n c t i o n s ;
the admixture
available
to
tetragonal
coefficients
find
sites
inte nsities
spe cifically
a .,p .,'e tc .
these c o e f f i c i e n t s
i n MgO.
They are.;
a
a knowl edge o f
Thr ee met hods ar e
for
I)
requires
t h e Crgg i o n
direct
in
diagonal i z a t i o n
o f t h e c o m p l e t e 16 x 16 H a m i l t o n i a n m a t r i x ;
2)
i n d u c e d f i e l d s , d e v e l o p e d by. .B i r ; ^ ^ and. 3)
perturbation
theory..
These met hods a r e each d i s c u s s e d
follow ing
I )
D irect
a method u s i n g
in d e t a i l
in the
pages .
D iagonalization
The m a t r i x
of
Fig.
of
16 x 16 M a t r i x .
3 may be e a s i l y
diagonalized with
the
Jl
-61aid
of a d ig ita l
computer.
A standard
been used w h i c h c a l c u l a t e s
eigenvalues
vectors.
Specific
calculated
using
in
the
re la tive
calculated
the H a m ilto n ia n .
line
are
0' = 4 5 ° .
3 8 3 9 . 6 gauss
a n g l e s o f 30°
111 f o r
fie ld
the value
for
probabilities
Chapter
The m a g n e t i c
o c c u r s was used f o r
values
in
for
o f the
tra nsition
and 4 5 ° ,
the- + 2
"
the m a t r i x .
»
These
and 3 8 5 8 . 8 gauss f o r
The c o m p u t e r pr o g r a m used i s
The r e s u l t s
were
the parameters
a t which
of H in
0 = 30°
pr ogr am has
and n o r m a l i z e d e i g e n ­
transition
from the e i g e n v e c t o r s
values
libra ry
liste d
probability
in
A p p e n d i x B.
calculations
ar e
shown i n T a b l e V I I .
The e i g e n f u n c t i o n s
calculated
from which
are n o r m a l i z e d ,
group o f f o u r
Tines
in
the e i g e n f u n c t i o n s
each o f
the allowed
Table V I I
o f the fo r b id d e n
the c r y s t a l
lines
in
inva riant,
The sums i n
about 0.935
for
consist
inte nsity
Table V II
in te n sity
The number o f
total
be c o n s t a n t
o f each
sum t o u n i t y .
w o u l d have u n i t
so t h e
were
This
t a b l e f o r 0 = 0.
Each
i
o f a pu r e base s t a t e , and
would v a n i s h .
a group s ho u l d
changed.
rather
is
lines
should
a sim ilar
would
lines
inte nsities
so t h e t o t a l
can be seen by c o n s i d e r i n g
of
the
inte nsity
w h i l e ' a Tl
ions
of all
as t h e o r i e n t a t i o n
are not u n i t y ,
0 = 30° and a b o u t 0 . 9 2 5
however,
in
the
is
but
for. 0 = 45 ° .
.
The r e a s o n
from the
for
the dis c re pa nc y i s
I
I
group is l o s t
that
some o f t h e
O
the
groups
intensity
to
as w e l l
as
2"
-62TABLE V I I
M, m
.415
3827.4
. 363
H.
a 11 owed
CD
I
11
3806.5
0 = 30°
Remark
M1, m1
±2 I 2 - I2. J ±
2
O •—<
LO
T ransition
3=
COMPUTED LINE POSITIONS AND INTENSITIES
±
A
2J 2 - 2a 2
Am=I
3815.9
. 389
3837.7
.400
±
J- -L
2» 2 -2,-2
Am= 2
3825.3
.122
3848:1,
. 146
A A-^J. A
2, 2 -2,-2
Am=S
3834.8
.013
3858.5
,018
A A-> A A
2, 2 - 2, 2
Am= I
3817.5
. 390
3837.8
.400
3826.9
.059
3848. 1
. 026
. 368
3858.5
. 355
.120
3868.9
. 145
.121
3848.3
.145
. 368
3858.6
. 356
3847.3
. 060
3869.0
. 027
3856.7.
. 387
3879.3
. 397
3839. 7
.012 '
3858.8
. 017
A . U A A
2 , 2 ~2 , 2
Ii
E
<1
3836.3
A-* A A
2 -2 ,“ 2
E>
3
I!
ro
3845.7
JL J^-> -L —
2 5“ 2 " 2 a 2
CM
H
E
<]
A U A A
2, 2 “ 2 , - 2
a 11 owed
3828.6
A A-> A A
2 , _2 ~2 , 2
Am= I
38 3 7 : 9
A
2,
A
U
2 , - 2
A
A
“ 2 , - 2
a 11 owed
Ii
E
<
A
A »
2 , - 2
A->
A
2 , - 2
-
A
2 ,
A
2
CO
Il
E
<3
A
A »
A
A
2 , ~ 2
" 2 , - 2
.
A
A
2
Am=2
3849.0
.
11 9
3869.1
. 144
Am=I
3858.3
. 387
3879.4
/396
3867.7
.416
3889.8
. 366
“2
,
A
A - *
A
A
2 , - 2 '
- 2 , - 2
A
A *
A
A
2 , - 2
- 2 , - 2
a 11 owed
.
-63to
forbidden
lines
characterized
T
The m - tg- a l l o w e d
pa rticula rly
less
than
."allowed"
for
the ir
lines
6 = 45°,
b y 'AMs = ± 2 , ± 3 .
a r e seen t o be v e r y w e a k ,
and even t h e
a d j a c e n t Am = I
forbidden
0=0.
2)
Induced
Field
a r e b o t h much g r e a t e r
The e l e c t r o n
wave f u n c t i o n s
by t h e s e f i r s t
the nuclear
nuclear
than
The l a b e l s
state.
much s m a l l e r
states
rather
t h e n , except
the
and a r e n e a r l y
hyperfine
fie ld
energy.
determined mainly
On t h e o t h e r ha nd,
than
crystal
the h y p e r f i n e
depend e s s e n t i a l l y
than
and t h e
are t h e r e f o r e
two i n t e r a c t i o n s ,
spin
spin
electrons
lines.
Method
The Zeeman e n e r g y o f t h e e l e c t r o n s
energy i s
have i n t e n s i t y
and " f o r b i d d e n " a r e n e a r l y m e a n i n g l e s s
for
potential
lines
independent o f
the
n u c l e a r Zeeman
e n e r g y , so t he
on t h e
b e i n g d o m i n a t e d by t h e
state
o f the
external
magnetic
fie ld .
The h y p e r f i n e
equivalent
fie ld
to
than
the e x t e r n a l
quantization
direction
the a x i a l
sy mmet r y o f t h e
-functions
depend o n l y
and t h e
e n e r g y can be t r e a t e d
a Zeeman e n e r g y o f
i n d u c e d by t h e e l e c t r o n
much l a r g e r
valid
inte ractio n
external
the
the magnetic
The i n d u c e d
fie ld ,
and so i t
for
the
nuclear
c rysta llin e
If
in
spin.
on. t h e a n g l e
field..
nucleus
this
fie ld ,
as bei ng-
fie ld
defines
spin.
is
a very-
Owing t o
the.electron
bet ween t h e c r y s t a l
angle
is
taken
to
lie
wave
axis
in
the
-64x-z
plane,
direction)
the x-z
(with
then
the e xte rna l
the
induced magnetic
plane. . A plan
22 , w i t h
the v a r io u s
magnitude o f
electronic
the
view o f
directions
induced f i e l d
system i s
fie ld
field
the x -z
depends
w ill
plane
indicated.
axis
is
as t h e z -
also
Tie
in
shown i n
Fig.
The d i r e c t i o n
upon w h i c h
state
and
the
in.
The c omponent s o f
by c o n s i d e r i n g
magnetic
the
induced
fie ld
can be c a l c u l a t e d
the e q u iv a le n c e
t
^n^n ^ i n d
= At-T
(5.5)
W riting
o u t t h e d o t p r o d u c t and t a k i n g
o f both
sides
the e x p e c t a t i o n
value
one f i n d s :
- gn 3n H]. < 11 > = A < S . x I . >
,
i
= x,z
(5.6)
or
wher e t h e e x p e c t a t i o n
functions.
value
so t h e
induced
the q u a n t i z a t i o n
taken along
the
fie ld
direction
induced f i e l d ,
C r^
wave
and gn = - . 3 1 5 7 0 n . m. ,
has t h e v a l u e
t h e 4000 gauss e x t e r n a l
If
found w i t h
- 205320 g a u s s .
o f one o r t h e o t h e r
the o rder o f u n i t y ,
that
is
T a k i n g A = .001 648 cm
the c o e f f i c i e n t A/g^g^
expectation
v a l u e o f S.
component s
field s
for
of S is
are l a r g e
may be s a f e l y
The.
enough
neglected.
the nuclear- spin
the e i g e n f u n c t i o n s
on
is
are the
-65-
x
External
Crystal
Field
Axis
Axis
Induced F i e l d f o r
E le c tro n State M
H^ ^ ^ I n d u c e d F i e l d f o r
E l e c t r o n S t a t e M1
Fi g
22
Pl a n Vi ew o f x
z
Plane.
JJ.
-66
pure magnetic
states
defined
s y s t e m un d e r g o e s
tion
suffers
to
express
of
the o ld o n e s .
—n u c l e a r
in
the
w ritten
a rotation,
hyperfine
Zeeman f i e l d ,
effect
a tra nsition ,
t h e new n u c l e a r s p i n
By t r e a t i n g
uncoupled.
-
by t h e quant um number m^.
the e l e c t r o n i c
direction
I l
and i t
states
The t o t a l
quantiza­
becomes n e c e s s a r y ,
as l i n e a r
interaction
the e l e c t r o n i c
this
When
in
t e r ms
and n u c l e a r
wave f u n c t i o n
combinations
of a
syst ems
can t h e r e f o r e
(5.7)
wher e t h e M i n d e x on 8^ r e i n s t a t e s
nuclear states
r e ma i n
to
the
" " D n T y on" t h e e l e c t r o n i c
of a tra n sitio n
m. and m1 w i l l
' product of
taking
causing
system,
of
j M,m->M1 , m1
but the e l e c t r o n i c
Since-, t h e
a resonance t r a n s i t i o n
acts.
probability
p l a c e bet ween s t a t e s
nuclear spin
spin
<8
to
(M)
with
t h e square o f t h e
functions
the e l e c t r o n i c
may be c a l c u l a t e d
the
s p i n "system f the r e l a t i v e
be p r o p o r t i o n a l
the n u c le a r
is
electronic
the c o u p lin g
inde pe nde nt o f the n u c l e a r system.
microwave p e r t u r b a t i o n
wher e
be
as a p r o d u c t :
^Mm
states
are.
(
m
)
'■ and 8^ ,
scalar
( M ')
(M 1 ).
tra n s itio n .p ro b a b ility ,
f r o m C r 52 wave f u n c t i o n s .
:
(5.8)
and
Il
-67It
(5.8)
is
now n e c e s s a r y t o f i n d
by u s i n g
The f u n c t i o n s
the
8^ ,
( M
induced magnetic
the
in
the d i r e c t i o n
(W )
wher e Rpm. 1
( a M M, ) i s
for
They have t h e
R (I)
^Vm'
sh ifts
the m a t r i x
quantization
form.
operators
axis
axis
are d e f i n e d
in
(5.9)
rotation
through
an a n g l e
by M e s s i a h ,
some d e t a i l
by B i r . ^
^
(31 )
1
(a
I =
' a M5M'
( I - m )!
f
d ( I-m ')
( 1 + p ) ( m+m' ) / 2
wher e p = c o s a ^ M, .
element o f the
the q u a n t i z a t i o n
and a r e d i s c u s s e d
Jp O- Hi 1)
( I - m ) ! (I+nV ).!
( I + m) ! ( I - m 1')!"
I
(I + y )
(1-u)
The R - f u n c t i o n s
p r o d u c t we r e q u i r e
1/2
I+m
m-1
(5.10)
are n o r m a l i z e d :
m1 = I 5I - I 5. . . , - I
The s c a l a r
combinations
I 5 I - I 5. . . - !
These r o t a t i o n
instance,
have t h e i r
above.
of
Z R
pm
CtjvJ JvJ1.
described
' may be e x p r e s s e d as l i n e a r
V
o p e r a t o r which
field s
1)
(M)
e i g e n v e c t o r s ' 0p' ’ which
of
t h e s c a l a r p r o d u c t o f Eq.
(5.11 )
is
x ,2
I Rm , m ' ^a M5M ' )
(5.12)
11 I1
68-
-
wher e t h e
find
the
index
3
l(=|-)
value o f
directions
of
the
has been d r o p p e d .
^ l.
This
is
induced f i e l d
tion.
Since both d i r e c t i o n s
ea sily
found:
H (M)
aM3M1 = t a n
u s i n g . Eq.
(5.6)
with
the aid
products
to determine
degrees
o f angle
basically
section,
of th is
up t o
Cr ^g l i n e
II,
clo se ly with
for
section
various
is
positions
wher e i t
tra nsition
of
Eq.
(5.8),
m, i s
fie ld s,
out
positions,
and t h e s c a l a r
found f o r
each two
The c o m p u t e r pr ogr am i s
p r o g r a m used i n
only 4 x 4 in
the
preceding
dimension.
The
have a l r e a d y been shown i n
The t r a n s i t i o n
.do n o t
is
Fig.
t h e y agree very
and f o r b i d d e n
(which
bu t are o n l y the
line
were a l l
positions.
proba bilities
(5.13).
have been c a r r i e d
can be seen t h a t
allowed
proba bility
plane,
o f H.
I
I
M = + 2'y~2 gr oup a r e g r a p h e d as a f u n c t i o n
These t r a n s i t i o n
to
the t r a n s i ­
.
C r^
induced
8 = 90°.
experimental
the
values
eigenfunctions
except the m a t r i x
9 and T a b l e
the x-z
computer.
the d i ag o n a li z a t io n
calculated
itie s
the
component s o f
of nuclear
r e ma i n s
.H f(M ' )
of a d ig ita l
eigenfunctions,
in
and a f t e r
1 j ^ T M T “ t a n 1 JTTMrJ "
The c a l c u l a t i o n s
only
t h e a n g l e bet ween t h e
before
lie
It
lines
o f angle
include
the C r ^
squared s c a l a r
probabil­
o f t he
in
Fig.
23.
the
electronic
line
in te n sity)
products.
The
-69-
I - O - Vx
RELATIVE INTENSITY
0. 8
"
0.4 t
3 -3
0 . 2 -•
30
45
ORIENTATION
8
Fig.
23
Line
Intensities
Induced
Field
degr ees
Calculated
Method.
using
-70- ■
reason
for
p lo tting
me n t a l
line
the data
inte nsities
t h e C r 52 i n t e n s i t i e s .
in
this
a r e most e a s i l y
n o t be i n c l u d e d
in
that
the e x p e r i ­
measur ed r e l a t i v e
For c o m p a r i s o n w i t h
a n g u l a r dependence o f t h e e l e c t r o n i c
should t h e r e f o r e
form i s
experiment,
tra nsition
to
the
probability
the c a lc u la te d
C r ro l i n e
bo
intensities.
The i n t e n s i t i e s
agree ve ry
of
closely with
the f u l l
m atrix,
0 . 9 3 5 and 0 . 9 2 5 t o
The n o r m a l i z a t i o n
of the
shown i n
if
3)
they are
condition
be d i s c u s s e d
factors
the
r e d u c e d C r 52 l i n e
Eq.
(5.11)
calculated
after
is
intensity..
a restatement
regarding
Table V II
in
was used i n
Chapter
the c r y s t a l
fie ld
order Hamiltonian,
diagonal
terms
.In f a c t ,
with
function
series
IV.
follow ing
series
and h y p e r f i n e
so t h a t
Hnn = 0,
is
Lumpi ng a l l
sim plified
is:
the
inte nsities
to
experimental
section.
Method
The same p e r t u r b a t i o n
series
from d i a g o n a l i z a t i o n
section.
The P e r t u r b a t i o n
order
of
0 = 30° and 45°
r e d u c e d by t h e
argument g i v e n
Co mp ar i s o n o f t h e
ones w i l l
23 f o r
those o b t a i n e d
account f o r
norm alization
the preceding
Fig.
the
perturbation
t h e wave f u n c t i o n
higher
involved
is
this
section
o f the d i a g o n a l
interaction
the l a b o r
one o r d e r
used i n
in
into
parts
as
of
the zero-
Hamiltonian
series
considerably.
carrying
not too g r e a t .
has no
t h e wave
The f o u r t h - .
-71-
E | m>
m
Hnin'H&n
E
nm nA
m> Hm t Ht n Hkn
E I m>
F F F
m,&,k
nm n£ nk
The o p e r a t o r s
of
available
these o p e r a to r s
neighboring
in
alone
hyperfine
the p e r t u r b a t i o n
is
Hamiltonian
not capable o f c on nectin g
states.
That
is
(5.15)
<M ,m±l I H 1 I M, m> = 0
On t h e o t h e r
h a n d , two o p e r a t o r s
such as SZS+ f o l l o w e d
fine
is
states.
t a k e n as
(the
It
|M,m>,
fifth
a p p a r e n t t hen. , : t h a t
of
terms,
the
and n i n t h
t er ms
in
succession,
if
|n> i n
the f i r s t
line
can admi x t h e base s t a t e s
hyperfine
states,
t e r ms
in
except the f o u r t h
admi x n e i g h b o r i n g
second n e i g h b o r i n g
taken
by S _ I + can c o n n e c t n e i g h b o r i n g
second-order term)
The r e m a i n i n g
also
is
ar e
in
states.
(5.14).
Eq. ( 5 . 1 4 )
on ly the l a s t
|M,m±.l>.
and t h e e i g h t h ,
can
The a d m i x t u r e o f
| M, m± 2 > ,- can o n l y o c c u r
Eq.
hyper-
Third
in the
neighboring
states
I M W
-72c a n n o t be admi xed i n
order
term
admixture
consider
(a q u i n t u p l e
coefficients
five
of the
one needs o n l y
the
The a d m i x t u r e
using
only
resulting
(5.14)
sum) .
but could
a and g o f
t e r ms
fifth
in
in
Eq'.
Fig.
using
Eq.
formula w i l l
6 and y ,
a and 3 have been c a l c u l a t e d
by B l e a n e y and Rubi ns V ^ ^
(1.2).
To i l l u s t r a t e
be d e r i v e d
compar ed w i t h
To o b t a i n
terms.
below.
f o u r t h - o r d e r wave f u n c t i o n s
and t h e r e s u l t s
sections.
of
t he
22 t h e n , one must
(5.14).
and n i n t h
coefficients
the fo r m u la
be by t h e main f i f t h
In o r d e r to c a l c u l a t e
s e c o n d - o r d e r wave f u n c t i o n ' s
approach, th is
tions
Eq.
those
w ill
t he
The c a l c u l a ­
then
be o u t l i n e d ,
o f t h e two p r e c e d i n g
The a c t u a l
fourth-order
calculations
appear i n '
The l o w e s t o r d e r
perturbation
t e r m w h i c h can admi x
A p p e n d i x A.
neighboring
hyperfine
I n> = I M,m> and
states
is
the
|m> = | M, m - 1 > t h i s
second-order term.
ter m appears
as
IM m 1> T <M; tn- l I n ; U x f t l H 1 |M,m>
wher e t h e
E^( o )' 1S a r e g i v e n
I M+l,m-1> or
elements,
I M-I , m>.
using
Eq.
by Eq.
W riting
(4.9)
and
out
( 5. 1: 6)
(4.10)
and £ can o n l y be
the energies
(4.10),
Taking
and m a t r i x
one o b t a i n s
'I
-73si n28
| S ZS_+S_S
|M+1 , m- ] x'M+1 5m-l | S+ I _ | Mm>
HM+AMm+DP2 (M2 - S 2/ 3 ) - H ( M + l ) - A ( M + l ) ( m - l ) -DP2 [M2+2M+1-S2/ 3 )
I S I _ I M-I 5m><M-l sm.|S S_ + S_S |M,m>
_______________ - __________________ ^
" 4 _______________
HM+AMm+DP2 (M2 - S 2/ 3) - H ( M- l ) - A ( M - l ).m-DP2 (M2 -2M+1 - S 2/ 3 )
+
(5.17)
wher e t h e d e n o m i n a t o r
of
Em m - Em ni_1 and P2 i s
appearing
in
the l a s t
the eigenvalue
S (S + l ) .
h a n d l e d by u s i n g
the f i r s t
t e r m on t h e
the Legendre
polynom ial.'
two d e n o m i n a t o r s
The m a t r i x
a shorthand
for
righ t
is
is
2
The S ■
u n d e r s t o o d t o be
elements
the m a trix
a r e mos t e a s i l y
elements
of
S+ , S_ , e t c . :
S+ |M,m> = / S ( S + 1 )- M( M+l ) | M + l ;,m> = S+ | M+l , m>
S+ 1M+l ,m> = S+ + I M+2 ,m>
The s u b s c r i p t
script
becomes
denotes
+or
- w ill
the m a t r i x
(5.18)
, etc.
d e n o t e an o p e r a t o r ,
element.
while
The e x p r e s s i o n
t he s u p e r ­
for
a
JL
-74-
= AD s i n 29
8 AM '
......... ('2M+1 ) S-hS ^ T '
- H - A ( m - M j - D P 2 ( 2M+1)
D ' s i n2 9 [ ('2M-1') (' S~)2 -(2M+1 ) (S +
SM
H
+'
(2M-1 ) S~S~l ~
H+Am+DP2 ( 2 M - 1 )
)2I I "
D s'i'n20-I SM2 - S 2
4H
M.
The d e n o m i n a t o r s
assumption
greater
in
the
that
an e r r o r
have been a p p r o x i m a t e d w i t h
H >> D5A.
than A 5 bu t
late r
(5.19)
is
o n l y a f ew t i m e s
calculations
that
o f a b o u t t h e same s i z e
higher order
in
states
| M - I ,m+1> and
approximating
the
co e fficie n t
calculated
are
same m a n n e r .
It
w ill
approximation
be seen
introduces
as t h e c o n t r i b u t i o n
The wave f u n c t i o n
f SM2 - S 2]
for
from the
t h e s t a t e , | M 5m + l > may be
The p o s s i b l e
is
intermediate
The e x p r e s s i o n
very s i m i l a r
T+
the s t a t e
second-order
g for
|M+l 5m>.
the d e n o m in a to rs ,
D s i n 29
in
this
D.
i n d e e d much
terms.
The a d m i x t u r e
w ritten
In our c a s e , H i s
t he
for
8,
after
t o a:
(5. 2-0)
l a b e l e d M,m can t h e r e f o r e
approximation
be
as
D s i n 2 6 ^SM2 - S 2
) ( I " ]. M5m- 1 >
4H
I
M '■
I + I M5m + l >)
(5.21 ) .
H L ft
-75and f o r
the s t a t e
M- I , m-1
l a b e l e d M - I , m- I
I ,m-1 > '+
1
(I
I M - 1 ,m-2> .-
The t r a n s i t i o n , m a t r i x
The s q u a r e o f
is
I
+
this
I
M-I
| M- 1, m>) .
D s i n28
4H
3 M( M- 1 )
last
3M2- S 2
expression,
after
B l e a n e y and R u b i n s .
used t o d e r i v e
it
are:
states
is
of other
ignored.
I M+l ,m-1 >' can o c c u r
bute to
the
coefficients
States
2)
than
dividing
in
tra nsition
by
neighboring
hyperfine
like
second o r d e r and c o n t r i ­
matrix
element,
w o u l d be p r o p o r t i o n a l
| M,m-2> c a n n o t admi x i n
to
the
but t h e i r
t o AD/H
2
.
second-order.
| M, m± l > a d m i x t u r e
ignored.
The e n e r g y d e n o m i n a t o r s
are ap pr o xi m a te d .
i
Corrections
for
this
order, expressions.
w ill
appear i n
the
S
.
23
(S")2
The. a p p r o x i m a t i o n s
Admixture of s ta te s
Higher order c o r r e c t i o n s
are
3)
like
■ 3 ( M - 1 ) 2- S 2'
M-I
(5
of
The a d m i x t u r e
is
I S
the fo r m u la
1)
(5 2 2 )
e l e m e n t bet ween t h e s e s t a t e s
^ M , i J S+ " S- I ^ M - I ,m-1
3.Ds'i'n'20
4H
f SiH-U2-S2
Ds i'n'2 6
4H
higher
)
11 11
-76The a p p r o x i m a t i o n s
D is
su ffic ie n tly
smaller
sites
i n MgO, t h i s
tions
must be r e f i n e d
The n e c e s s a r y
t e r ms
in
is
t h a n H.
in
order to
are
t h e wave f u n c t i o n
re la tive
sizes
normalized,
and s i n c e
the r e l a t i v e
normalize
line
a and
preceding
section w i l l
The a d m i x t u r e
sig n ific a n t
appears
in
denominator,
t e r ms
safely
seventh
inte nsities.
by h i g h e r o r d e r
series.
and
(5.22)
sim ilar
be a l t e r e d .
an A
neglected.'
and n i n t h
2
16 a d m i x t u r e s
to
all
s h o u l d be
the a d m i x t u r e s ,
that
discussed
in
an
the
be u s e d .
second o r d e r
hyperfine
states
because Enm i s
te rm s , there w i l l
Because A i s
or A
ar e n o t
To r i g o r o u s l y
3
in
the ad m ix tu r e
co e fficient.
a l s o be an AM i n
so much s m a l l e r
Eq.
be c e r t a i n
(5.14)
the
2
2
t o A D, AD ,
t h a n D, t h o s e
t h e n u m e r a t o r s may u s u a l l y
There appear t o
t er ms , o f
is
o n l y AM gauss
and t h e n u m e r a t o r s may be p r o p o r t i o n a l
o r AD^ .
having
approxima­
the problem o f n o r m a l i z a ­
the denominator of
In the h i g h e r o r d e r
A ^ D , A^D^,
w ill
of neighboring
even i n
three
3 approach u n i t y near Q = 45 °,
Rather than c a l c u l a t e
norm alization
tetragonal
described
Eqs. (5 .2 1 )
inte nsities
inte nsity
and i t
is
t h e wave f u n c t i o n s , a l l
calculated.
that
t o be h a n d l e d because o f t h e
This
of
providing
p r e d i c t the l i n e
perturbation
D and H.
The wave f u n c t i o n s
and a l l
largely
p r o b l e m needs
of
valid
For C r 53 i n
not the case,
refinements
One f u r t h e r
tion.
above a r e q u i t e
elements
be
in
wh i c h can have an A
the
in
-77the d e n o m in a to r,
large
t h o u g h , and m i g h t
as AD^/ A^H^ ~ 4.
term w i l l
ninth
have A
in
term t h e r e
w ell.
It
If
the de n o m in a to r,
however,
c o n v e r g e because t h e s e
cancellation
according
-
+
the
t e r ms
k->&, and a d d i n g
I m>
+
E^E
E
nm n j n&
the e n t i r e
contributions
that
large
to c o n t r i b u t i o n s
and i f
cancel
seventh
the
f r o m t h e s e as
series
each o t h e r .
the seventh
the n i n t h
as
| k> = |m> i n
perturbation
can be seen by r e l a b e l i n g
t o A-»j,
S
- mj
m = |M,m±l>,
can be l a r g e
happens
lead
w ill
The
t er m
term:
E m> F E E E
nm n& nm nj
m£j
Hi M H« k Hk j Hj n
E
m> F F F F
nmn&^nk^nj
m&kj
(5.24)
kfm
The l a s t
if
sum can have no e l e m e n t s
I m> = I M , m ± l >.
interested
in
coefficient
-
E
^
E
j
Now E
n£
of
a particula r
admixture
I m> f r o m t h e f i r s t
EnmEn j En£ '
Hm j Hj n
EnmEn j
3
3
t h a n AD /AM -
.01,
D r o p p i n g t h e sum on m, s i n c e we are
Hm j Hj n H£n
Em£
greater
E
+
^
( n ame l y
two t er ms
|M,m±l>),
above
t he
is
Hm j Hj nH&m
Jjl EnmEn j En&
2n
Z
"n&
AM i f
jm>> and
.H
Jim
(5.25)
'nil
| n> a r e n e i g h b o r i n g
hyperfine
-78s t a t e s , so t h e
denominator
in
the . l a s t
t e r m above can be
expanded:
E
a
m
h
L
i
H .H .
mj j n
1
E . *
J E2
nm nj ^
Since n e i t h e r m nor n i s
the b r a c k e t cancel
admixture
Detailed
appear in
that
given
by Eq.
contribution
admixture
function
of
the
D
^
remaining
The a p p e n d i x
from c o n s i d e r i n g
two t er ms
and t h e c o n t r i b u t i o n
can be at. most
calculations
A p p e n d i x A.
derive
summed o v e r , t h e f i r s t
each o t h e r ,
co e fficie n t
(5.26)
E2
m£
also
to
/ -
.01.
t e r ms
in
gives
the
The r e s u l t s
from the
second o r d e r
of second-neighboring
t he
Eq.
(5.11)
corrections
e x a c t energy denominators
(4.10) .
in
as
have been c ombi ned w i t h
term.
state's,
After
the
adding the
t h e c o m p l e t e wave
is
M,m = |M,m> + a Mvj M , m - l >
+
Y
„
+ 3MjIJ M’ ni+1>
+ 6M>m| M, m* 2 >-
wher e
± 1 / 2 ,m
3D^ n29- I ~ [+2 + £ ( 4 - 9
H
± ^ ( - 8 + % i n 226
'
s i n 20)
I3
8 s i n e )]
(
5 27
.
)
79-
« ± 3 / 2 . ™ = 2T
f i
I _ f ±2 + f f ( - 4 + 7 s 1 " 2e>
\
± | j - ( S - ^ - s I n 2 Z e - ^ s I n 4 S)]
a
M,m
_ 3 Ds i n2 0
Y± 1 / 2 , m "
4H
' ± 3 / 2 ,m
_ D s i n26 T- T= r D s i n20
16H
1 1 L H
I + I ++
M,m
T- T= r 3 D s i n 2 6
1 1 L 2H
■
rr
+
“
(5.28)
t h e wave f u n c t i o n
the m a tr ix
I + , I ~, etc.
i3M m-1
15 9 i v e n
have been e v a l u a t e d
values
0 = 45°.
the
contained
Fo r i n s t a n c e ,
The c o e f f i c i e n t s
the t a b l e
state
H = 3840 gauss
the
are l i s t e d
and d i ag ona l i z a t i o n
each case i s
results
in
coe fficient
a,
3, y and 6
The same
the computer
.
0 = 30° and 3859 gauss
i n T a b l e V I I I 3 and may be
comput ed v a l u e s .
have been r e s c a l e d
| M3m> i n
for
only
the
f o r M = ±^- and 0 = 30° and 4 5 ° .
The r e s u l t s
compared t h e r e w i t h
in
mI ~ / 1+ -
is
o f H have been used her e as were used i n
diagonalizations:
for
AtanB1
H j
Y Msm
The m- dependenc e o f
elements
% A t a n Q1
4H ■J
so t h a t
unity,
The comput ed v a l u e s
the
c o e ffic ie n t of
so p e r t u r b a t i o n
may be d i r e c t l y
compar ed. . .
results
Some
TABLE V I I I
ADMIXTURE COEFFICIENTS
COE F .
0 = 30°
0 = 45°
.
pert.
di ag .
pert.
diag.
ai
3
~Z> T
+ .632
+ .499
+.693,
+ . 435
YI
2’
+ . I 56
+ .145
+ .205
+ .111
+ . 732
+. 6 60
+ .800
+ .557
-.632
-.595
-.693
- .49 9
+ .156
+
. 173
+ .205
+ .126
+ . 632
+ .595
+ .693
+ . 496
•- . 732
- . 660
- . 800
-.555
+ .156
+ .169
+ .205
+ .123
-.632
- .496
- .693
- .432
+ .156
+ .141
+ .205
+ .107
- .423
- . 405
- .453
-.394
3
2 '
i
%
Bi
i
23 2
4 . i
ai
i
23 2
'
Bi
3
2 3" 2
s
i,-i
I
Ot
3
" T 3
Y l
T
3
2 3 T
a i
i
.204
. 164
+
. 095
-
.510
+ .423
+
.455
+
.453
. 155
+
.103
+
. 204
' +.201
-.453
-.653
+
.155
-
. 490
'
+
'
-.522
+
-.722
-~2 3 T
B_ I
2 3
Y - i2
a i
3
I
2
.659
2
-Y-i
-.451
-,423
i
~ T 3~ T
6
+
i
+
.
+
. 490
+.155
:
. 720
+
.510
+
.522
+
+
.103
+
.204
+ .201
5 - T 3- T
B i
3
" T 3" T
+
.423
+
.401
+
. 453 .
+
.53 0
6
+
. 155
+
. 093
+
. 204
+
.162
1
3
" T 3" T
-81 coefficients
since
they
have been o m i t t e d
vanish.
from the Ta ble,
(8
is
the admixture
3
= • j so t h e r e
Agr e e me n t bet ween t h e
the signs
two met hods
and r e l a t i v e
R elative
admixture
line
coefficients
and Am=!
is
magnitudes
inte nsities
such as 8
is
of
the s t a t e
for
allowed
for
t h e Am=2 and 3 t r a n s i t i o n s .
ra th e r poor,
although
ar e c o r r e c t .
to
the
formulas
t r a n s i t i o n s , and w i t h
f r o m t he
in. T a b l e VI
sim ilar
The c a l c u l a t i o n s
formulas
have been
made f o r
each f o u r degrees;, o f a n g l e
up t o 44 d e g r e e s ,
only
the
of Table V I I .
for
starting
the ir
firs t
four
f r o m m = -^. )
sum ( w h i c h
transitions
They have been n o r m a l i z e d
should
normalized)
and d i v i d i n g
n e a r 0 = 0,
but
normalized
with
be u n i t y
to about
re la tiv e .in te n s itie s
t h o s e comput ed u s i n g
those c a l c u l a t e d
the
from the
fourth-order
results
off-diagonal
matrix
this
One may c o n c l u d e
angle.
of
by f i n d i n g ,
The sum i s
6 = 44°.
in
Fig.
were
unity
The
24,
along
B l e a n e y and Ru b i n s .
for
me t h o d .
in. p o o r e s t
elements
gr aphed
results
from the induced f i e l d
are
(Those ■
i n d u c e d f i e l d - met hod and w i t h
perturbation
results
sum.
1.8 f o r
24 t h a t
perturbation
but
t h e wave f u n c t i o n s
are
formula
One can see f r o m F i g .
the
if
each by t h e
increases
T '5 \
)
no s t a t e
have been c a l c u l a t e d
according
1/2,3/2
in
the
that
angles
up t o a b o u t 1 0 ° ,
agr ee q u i t e
well
Near 9 = 4 5 °
agreement,
Hamiltonian
with
t he
because
the
are l a r g e s t
the p e r t u r b a t i o n
the
method
at
-82-
O
Am=O
fourth
order
Am=I
fourth
order
Am= 2
fourth
order
Am= 3
fourth
order
induced
fie ld
second o r d e r
- '
0
- ■
perturbation
Intensity
0. 6
method
4
Relative
.
0. 2 -
ORIENTATION
Fig.
24
Angular
Dependence o f
8
d e gr e es
Line
Intensities.
I
-83could
improve mark ed ly
or
the
if
i f . D were s m a l l e r by a f a c t o r
e x p e r i m e n t were done a t
a proportionately
s in (20°),
higher
frequency.
The s e c o n d - o r d e r
at
small
made i n
4)
angles,
the
and t h i s
denominators
Co mp ar i s o n w i t h
Because t h e
and t h e
ties.
either
tions
would
width
of the m =
occur
in
overlapping
separation
also
the p e r t u r b a t i o n
I
of
3
3 1
, ~2^~2
a single
separation
a n g u l a r dependence,
the
We make t h e s i m p l e
separation
amplitude
o f about 0.9
is
by 20%.
gauss
for
inte nsi­
a p p e a r t o be more c o m p l i ­
or d i a g o n a liz a t io n
calcu la­
the l i n e
i n c r e a s e much
so we may assume t h a t
this
does n o t have a marked a n g u l a r
60°.
the e q u i v a l e n t d o u b le t
of
individual
d o u b l e t does n o t
lin e ,
doublets,
ar e a n g l e - d e p e n d e n t ,
E x p e r i m e n t a l I y , we see t h a t
t h e r a n g e 10° t o
the completeness
line
lines
separation
doublet
dependence i n
doublet
(5.19)..
Experiment
indicate.
beyond t h e w i d t h
strong
Eq.
do n o t ag r e e even
to the approxim ation
has been made t o measur e t h e i r
cated than
spectrum,
in
and d o u b l e t
The d o u b l e t
particula r
results
can be t r a c e d
forbidden
line-w id th
no a t t e m p t
perturbation
On t h e o t h e r end o f t h e
separation
and t h i s
fact
is
casts
seen t o have a
some d o u b t on
Hamiltonian.
approximation
constant
This
and i s
requires
two l i n e s
that
the
high-fieId
such as t o " r e d u c e
a doublet
the
separation
each w i t h , a w i d t h
o f 2 . 0 gauss
-
and i m p l i e s
a doublet
line
observed d o u b l e t w id t h
consistent with
the
t h e measur ed l i n e
lines,
The c u r v e
calculated
the f i g u r e
experimental
for
that
corrected
at approximately
The m =
so i t s
tions.
inte nsity
by a f a c t o r
o f 1.7 to
difference
re la tive
from the
to
the C r ^
of angle
induced f i e l d
comparison.
with
line
in
Fig.
met hod i s
The a p p r o x i m a t e
25.
shown,
shape o f t h e
the c a l c u l a t e d
c u r v e , and
p r o b a b l y be e x p l a i n e d by t h e
amplitude
presence o f the m =
has n o t been
forbidden
doublet
t h e same f i e l d .
allowed
is
line
does n o t o v e r l a p
much e a s i e r
has been s c a l e d
up by t h e
plotted
26.
Fig.
bet ween Cr ^g and Cr^g
as a f u n c t i o n
The a m p l i t u d e o f t h i s
in
increasing
up a g a i n
curve agrees w e l l
the
After
the do u b le t
t h e measur ed C r 52 l i n e
for
therefore
account f o r
t h e downward d i s p l a c e m e n t c o u l d
fact
The
by 20% t o
amplitude
has been p l o t t e d
gauss.*
a b o u t 2 . 5 gauss and i s
scaling
amplitude
in
o f about 2.5
above a p p r o x i m a t i o n s .
the p o p u la t io n
the doublet
-
width
amplitude
s e p a r a t i o n , and t h e n
account f o r
is
84
t o compare w i t h
line
factor
The e r r o r
other
in
re la tive
of 1.7,
to
lines,
the c a l c u l a ­
the Cr^2 l i n e
and t h e
results
the experimental
measur ement s
*Two u n r e s o l v e d l i n e s do n o t have a t o t a l w i d t h equal t o
t h e i r r e l a t i v e displa cement plus a s i n g le l i n e w i d t h .
Bl oember gen and Royce v2 9 ) have measur ed t h i s e f f e c t i n r u b y ,
u s i n g an a p p l i e d e l e c t r i c , f i e l d t o p r odu c e t h e d i s p l a c e m e n t .
-85-
1
.
0
*
-
RELATIVE INTENSITY
O. 8 - •
0.6--
0.4-'
ORIENTATION
Fig.
25
Measur ed L i n e
[Ti=
-
I
2
-
3
3
gj ~2
de gr e es
Intensity
1
^~2
for
Doublet.
the
RELATIVE INTENSITY
-86-
0.8
- •
0. 6
- -
0.4 - •
0. 2 -
ORIENTATION
Fig.
26
Measur ed L i n e
Allowed
degr ees
Intensity
Line
for
the
is
quite
large,
calculated
of
the
and a g a i n much- o f t h e d i s c r e p a n c y w i t h
c u r v e may be a t t r i b u t e d
t o t he v a r i a b l e
amount o f e x p e r i m e n t a l
that
are
d e p e n d e n t on o r i e n t a t i o n .
indeed s t r o n g l y
the al low e d
the ir
other
line
to the
bet ween c a l c u l a t i o n s
inte nsities
inte nsities
than 0 = 0
in te n sity
line
v e r i f i c a t i o n , t en ds
t o s u p p o r t the. i d e a
angles
intensity
Crg2 l i n e .
The l i m i t e d
that
the
i s. t h a t
the
are s h a r p l y
C r^
reduced at
t h e y must g i v e
based on t h e
up much o f
induced f i e l d
of
the v a l i d i t y
o f t h e f o r m e r me t h o d .
perturbation
calculations
n e a r 0 = 45°
in
system.
ag r eement
method w i t h
the H a m ilto n ia n
On t h e o t h e r
may n o t be r e l i e d
spectrum
The r eas on
f o r b i d d e n . l i n e s . . The. c l o s e
t h o s e based on a d i a g o n a l i z a t i o n
this
in
pr ov es
hand,
upon f o r
angl es
VI.
The e l e c t r o n
C r 53 i n
an a x i a l
paramagnetic
crystal
The us u a l
perturbation
to
order
fourth
SUMMARY
fie ld
resonance spectrum o f t r i v a l e n t
has been a n a l y z e d
approach
has been m o d i f i e d
in
an e f f o r t
to
dependence o f l i n e
positions
and i n t e n s i t y .
s p littin g
in
and f o r
does n o t c o n v e r g e f a s t
met hods
describe
have been u s e d ,
the
adequately
t h e case c o n s i d e r e d
Zeeman s p l i t t i n g ,
spectrum.
this
in
is
the a n g u la r
h a l f o f the
the p e r t u r b a t i o n
A ltern atively,
and have been f o u n d t o
The f i r s t ,
and e x t e n d e d
The gr ound s t a t e
approximately
reason
enough.
explain
d e ta il.
direct
series
two o t h e r
adequately ■
d i a g o n a l i z a t i 0n o f
the H am ilto nian m a t r i x ,
has been used as t h e norm a g a i n s t
which
a r e me a s u r e d .
to
t h e o t h e r met hods
test
the v a l i d i t y
results
to
method,
called
of
experimental
the
appearance o f s o - c a l l e d
very e x is te n c e
anisotropy
the ir
is
in
fie ld
forbidden
is
has Been f o u n d t o
inte n sitie s.
hyperfine
the major reason f o r
spectrum,
simultaneously
is
the
lines.
the s t r o n g
Their
inte nsity
and an e x p l a n a t i o n
an e x p l a n a t i o n
the complete spectrum.
its
The second a l t e r n a t e
me t h o d ,
of lin e
been used
by c o m p a r i n g
o f the observed s p e c tr a
o f the allowed
inte nsities
inte nsities
me a s u r e me n t s .
description
A dominant f e a t u r e
has a l s o
the s p i n - H a m i l t o n i a n
induced
p r o d u c e an a c c u r a t e
It
of
The f o r b i d d e n
of
-89tra n s itio n s , characterized
by Am1 = ±1 , ±2,
the c r y s t a l
fie ld
and m a g n e t i c
Competition
bet ween t h e s e two q u a n t i z a t i o n
c o mp l e x admi xed wave f u n c t i o n s
diagonal
t e r ms
operator.
e x p lic it
expressions
hyperfine
Detailed
point
for
to a po s s ib le
the h y p e r f i n e
with
strong
enough t o
on mj
to
Sim ilar
explain
e t al
Two p o s s i b l e
the
results
spectrum at
that
of
i n t e r a c t i o n , , and i f
the p a r t i c u l a r
is
not
t h o u g h , and a
have t h e p r o p e r dependence
effects
which are
have been o b s e r v e d
in
observed.
spectra
o f the
i n MgO .
additional
experiments
The f i r s t
K-band f r e q u e n c i e s . .
perturbation
the
t h e Cr^g
anisotropy
the d is c re p a n c ie s
would not
work.
in
which
c o u l d be an i n t e r a c t i o n
for
this
the
have measur ed a s l i g h t
account
i s o e l e . c t r o n i c v anadi um i o n
po sitio n s'o f
incompleteness
The h y p e r f i n e
discrepancies
and second
some d i s c r e p a n c i e s
fie ld .
quadrupole in te r a c tio n
in
Hamiltonian
of f i r s t
re la tive
has a q u a d r u p o l e. moment t h e r e
the c r y s t a l
results
t h e wave f u n c t i o n .
inaccuracy or
anisotropy
nucleus
into
has r e v e a l e d
M arshall,
in
interaction
o f the
Hamiltonian.
axes
met hod has been used t o d e r i v e
states
resonance l i n e s
axes a r e n o t c o - l i n e a r .
the a d m ix tu r e
examination
a r i s e , when
because o f t h e p r e s e n c e o f o f f -
the h y p e r f i n e
The p e r t u r b a t i o n
neighboring
various
in
fie ld
etc.,
is
a r e s u g g e s t e d by
an a n a l y s i s ' o f t h e
(24 GHz)
It
is
expected
t h e o r y wou l d be c o n s i d e r a b l y more e f f i c i e n t
-90in
this
case.
The second i s
o f an a p p l i e d
external
could
ve rify
hope t o
an i n v e s t i g a t i o n
ele ctric
fie ld
the te tra g o n a l
as the. o r i g i n
of
the A 1 s p e c t r u m .
lines
in
the C r ^
visib le
more c o m p l i c a t e d
predict
some o f
The a n a l y s i s
result
la ttic e
these
would
from d e fe c ts
on t h e s p e c t r u m .
defect of
One
t h e second k i n d
T h e r e ar e o t h e r ,
weaker
s p e c t r u m whi ch, a r e p r e s u m a b l y due t o
defects,
spectra
be v a l i d
and i t
may be p o s s i b l e t o
on t h e b a s i s
only f o r
more d i s t a n t
chromium-vacancy p a i r .
o f the e f f e c t
than
o f the
Stark e f f e c t
t h o s e s p e c t r a wh i c h
two u n i t
cells
from the .
APPENDIX
APPENDIX A
Line
Perturbation
function
and
series
This
evaluation
of
Hamiltonian
for
t h e e n e r g y and wave-
state
have been g i v e n
in
appendix w i l l
give a d e ta ile d
outline
these e x p r e s s i o n s ,
o f Chapter
The p e r t u r b a t i o n
to o b tain
Hamiltonian,
Calculations
expressions
of a p a rtic u la r
(4.5).
authors
Position
the-axial
(4.6)
of
the
fie ld
II.
approach
the
has been used by a number o f
energy l e v e l s
but l i t t l e
o f the a x i a l
a g r e e me n t e x i s t s
de W i j n and van B a l d e r e n ^ ^
f o u n d t h a t most o f
using
Eqs.
in
have r e v i e w e d
the disagreements
fie ld
th e ir
results,
the s i t u a t i o n ,
result
from omission
and
of
i
some i m p o r t a n t t e r m s .
which
includes
all
They g i v e
t e r ms
and t h i s
expression w i l l
t e r ms
4
3
D / ( gBH) w i l l
to
in
them f r o m s i x t h
if
and t h e
series
be e v a l u a t e d
expressions
energy.
energy l e v e l s
perturbation
degenerate
levels.
DXA^~X/ (g 3 h )
be v e r i f i e d
the e l e c t r o n i c
the unperturbed
unperturbed
of order
for
below.
the energy
( x =- 0 t o 3 ) ,
In a d d i t i o n ,
ignoring
the
contributions
order.
The p e r t u r b a t i o n
evaluated
an e x p r e s s i o n
c a n n o t be r i g o r o u s l y
Zeeman e n e r g y a l o n e
The r e a s o n o f c o u r s e
is
t a k e n as
that
t he
retain
the nuclear
spin
expressions
are d e r i v e d
as s u mi n g non--
The p r o b l e m i s
are s i m p l i f i e d
is
circumvented
a t t h e same t i m e
if
all
degeneracy,
and t he
diagonal
parts
11
LI _?j.
-93of
the c r y s t a l
fie ld ,
Zeeman e n e r g i e s
S tarting
ar e
with
s ummat i ons u s i n g
in
Eq.
hyperfine
included
Eq.
in
(4.11),
inte ractio n,
and n u c l e a r
the
unperturbed
then,
we m u s t . d o
as t h e p e r t u r b i n g
Hamiltonian
energy.
the various
the expression
(4.9):
K '
= J-(S+ I _ + S_I + ) + I
+ j
The f i r s t
Si n20
( S 2S+ + SZS_ + S+ Sz + S_SZ )
sum in. Eq.
m = I M+l , m - 1 > ,
(4.11)
I M-I ,m + 1> ,
I M - I ,m> , I f n =
S i n 2O (S 2 + S2 )
I M, m>.
energy d i f f e r e n c e s
in
must be e v a l u a t e d
|M+2, m>,
Keepi ng
| M- 2 , m> ,
( A . I).
for
| M+l , m> and
i n mi nd t h e s i g n
the denominators,
this
o f t he
sum i s
z V i = / V>
( S " ) 2 ( I + )2
^ Enm
4 LH - r nH+A( m- M+l ) +DP 2 ( 2 M - l ) ~
(s+)2( i - ) 2
i
H - r n H + A ( m - M- l ) + D P 2 ( 2 M + l ) j
+ D2S i n ^ r
( S - ) 2 ( S =)2
( S + ) 2( S + + ) 2 n
32
LH+Am+DP2 ( 2 M- 2 ) ' " H+Am+.DP2 ( 2M+2J -1
•
, D2Sin2 26 r
( 2M- 1 ) 2 ( S" ) 2
(2M+1 ) 2 ( S +)2
16
LH+Am+DP2 (2M-1 ) ” . H+Am+DP2 ( 2M+1 ) J 5
wher e Tfi = 9n 3n/ 9 p ^ »
as d e f i n e d
expanded
half
of
by Eq.
to give
the f i r s t
and t h e e i g e n v a l u e s
(5.18).
a series
S+ , S ~ , e t c . ,
The d e n o m i n a t o r s
in
I/H .
t e r m becomes
are
can now be
Fo r e x a m p l e ,
the f i r s t
•
'
. 0)
-94'a
Ur^
^ [ T - p - ( m - M + l ) - ^ p - ( 2 M - l ) + n-(m-M + ] ) + - r r ^ ( 2 M - ] )
T p (S
wher e F H has been i g n o r e d . ■■ We r e t a i n
order
D
^ / or A ^/H ^.
be d i s c u s s e d
before w r i t i n g
The t h i r d
tors,
order
because t h e
returning
possible
to
the
this
term cannot
in itia l
state
those
sums o f
Eq.
involve
the
must f o r m a c l o s e d
o n ly at the end.
Summing o v e r t h e s e s i x
D3
(4.11)
hyperfine
l o o p s a r e M -> M±1 -> M±2 -* M, M ->- M±1
nm m& £n
E E•
nm nil
t er ms
up t o
w ill
one o u t any f u r t h e r .
three operators
M -> M±2 -> M±1 ■+ M.
Z
m,&
The r e m a i n i n g
only
opera­
loop,
The o n l y
M+l
loops,
M., and
we o b t a i n
2
?
s i n 20s i n ^0
2 ( 2 M+ 1 ) (S + ) 2 ( 2 M + 3 ) ( S ++)2
[ {H+Am+DP2 ( 2 m + l ) }{2H+2Am+2DP 2 ( 2M+2) }
2 ( 2 M - 1 ) ( S " ) 2 ( 2 M - 3 ) ( S =)2
{H+Am+DP2 ( 2 M - 1 ) H 2H+2Am+2DP2 ( 2 M- 2 ) }
2 ( 2 M + l ) ( S+ ) 2 ( 2 M - 1 ) ( S -)2
{H+Am+DP2 ( 2M+1) HH+Am+DP 2 (2M-1 ) }
]
( A. 3)
wher e t h e n u c l e a r Zeeman e n e r g y has been i g n o r e d .
ing
t h e s e d e n o m i n a t o r s , we may i g n o r e
they r e s u l t
in
the terms
In e x p a n d ­
i n Am, s i n c e
3 3
a t e r m o f o r d e r AD /H .
In the l a s t
t e r m o f Eq.
(4.11),
2
2
t i on o f o r d e r AD /H
from those
one may g e t a c o n t r i b u -
combinations
for
wh i c h
-95l
= |M,m±l>,
since
an A f r o m t h e
w ill
numerator.
sum and t h e o t h e r
be o n l y +AM gauss and w i l l
The r e m a i n i n g
fourth
order
parts
term w i l l
order
to ev a luate .
contributions
differences
in
the e l e c t r o n i c
T a k i n g Z = | M, m±l >
^nm^m&^&k^kn
v
^
.
m,£,k
U
C
C
in
_v
r
Z
a-nd o f o r d e r D^/ H^
in
all
the f o u r t h
by t h e a p p r o p r i a t e
Zeeman e n e r g y .
the t r i p l e
L c
l
nm nii'nk
= Z
[ Z
Z bnS.
m
I
.
sum, we f i n d
v
^nm^m&
l
c
l
nm
k
n&m
v ^nk^kA
r
J
nk
Ijm -£]2
t Om
I ,AD s i n
m {~-8
■ '
The d e n o m i n a t o r s
may be a p p r o x i m a t e d
the t r i p l e
make c o n t r i b u t i o n s
o f o r d e r A ^ / H^ and A^D^/ H^ w h i c h we i g n o r e ,
w h i c h we w i s h
of
cancel
2 8 \ 2 r ( S + ) 2 l + (2M+l)
L
~
, !,A D sin 2 8 , 2 r ( S " ) 2 I ~ ( 2 M - l )
+ AM^
8
; L
H
, ( S- ) 21+ ( 2 M - 1 ) n 2
+
R
j
, (S+ ) 2 I " ( 2M+1) n2
-H
J
AD2 s i n 2 28 [ ( D 2- ( I + ) 2 IC ( S+ j 2(2M + l ) - ( S ")2 (2M-1 )]2
64 H2M
( A . 4)
since
for
Z = | M,m+1> , m can o n l y
Thi s, c o n t r i b u t i o n
la st
line
so f a r
derived
of
Eq.
w ill
(A. 2 ) .
combi ne w i t h
All
of
the
have, been done p r e v i o u s l y ,
be l ow have n o t .
be
|M+T,m> o r
| M-I ,m+1>.
a s i mid a r t e r m f r om t h e
calculations, outlined
b u t t h e t e r ms
in
4 3
D /H
-96The f i r s t
fourth
down as a p r o d u c t
of
o r d e r sum i n
the
Eq.
(4.11)
may be w r i t t e n
second o r d e r e n e r g y t i m e s
a sim ilar
expressi on:
v
H2
H2
H2
mn nJln
H2
"An
Z "mn
A ^nA
"nm
D
16
+ s 1 n 22 6 [ < 2 M l ! ( O i
- i p i l i i l i V ]
s i n 4 e [ ( s - ) 2 ( s = >2 + ( S t ) 2 I f i ) 2 ]
4H^
4H^
+ s i n 2 2 e [ ( 2 M --U 2 <-S-' ) 2 + ( Z m l 2 ( S t ) 2 ]
wher e a l l
t e r ms
contributions
fourth
order
w ill
loops
A have been i g n o r e d .
combi ne w i t h
sim ilar
These
ones f r o m . t h e
other
sum.
F in ally,
trip le
involving
( A. 5)
we must e v a l u a t e
sum w h i c h do n o t
to c o n s id e r,
involve
such as M
Q
Two a r e m u l t i p l i e d , s i n
s i n ^ 6s i n^20 .,
a b o v e , we f i n d
After
those
A..
remaining
parts
Ther e ar e t w e n t y
o f the
d iffe ren t
M±1 -> M±3 -> M±2 -> M f o r
A
0 , two by s i n 28, and s i x t e e n by
adding
the
contributions
calculated
e x a mp l e .
IL
- 97 ■
e M5IT
1
= HM + AMm + DP2 (M2- 1 s 2 ) - Tp Hm
+ ^
[ ( s ~ ) 2 ( i + ) 2 - ( s + .)2 ( r ) 2 ] + - 2-32H
y -S4 -,[ ( S ~ ) 2 (S ) 2 -(s+) 2 ( s + + )2 ]
+ ^
29. [ ( 2M-1 ) 2 ( S- ) 2 -
— 9[ ( m - M + l ) ( S ' ) 2 ( I +)2 -
A2DP
[ ( 2 M- 1 ) ( S " ) 2 ( I + )2 -
(2M+1 ) 2 (S+ ) 2 ]
.
( m - M - 1 ) ( S + ) 2( I " ) 2 ]
(2M + 1 ) (S + ) 2C
D 2]
4H
-
[ ( s - ) 2 ( s = ) 2 _ ( S+ ) 2 ( S+ + ) 2]m
2>2WC
+ AD2 S i n2 28 [ ( S+ ) 4 /M _ 2 ( S+ ) 2 ]m
+ D ^ s i n 2 sin22_8[(2M+T)(2M+3)(S+)2(S++)2
64H^
+ (2M-1 ) ( 2 M - 3 ) ( S “ ) 2 (S= ) 2- 2( 2M + 1 ) ( 2 M - 1 ) (S+ ) 2 ( S " ) 2 ]
3
. 4
D P9S i n 0
9 _ 9
+ 9 4.+ 9
------ ^ - 9 ------ [ ( M - 1 ) ( S - ) 2 ( S " ) 2 - ( M + 1 ) ( S + ) 2 ( S+ + ) 2 ]
I 6H^
D3P9S i n2 29
,
9
------------------- [ ( 2 M-1 ) 3 ( S" ) 2 1. 6, '
v + 0
( 2 M+ 1 ) 3 ( S D
U ./
-98-
4096H
+ 2 ( S + ) 4 ( S + + ) 4- 2 ( S " ) 4 ( S = ) 4 ]
4 . 4
+ P. s i n 29[ ( 2 M- 1 ) 2 ( S ~ ) 2 ( 2 M - 3 ) 2 ( S= ) 2-( 2M + 1 ) 2 ( S+ ) 2(2M+3)2(S++)2
512Hd
- 2 ( 2 M - 1 ) 4 ( S ~)4 + 2 ( 2M+1) 4 ( S+ ) 4 ]
+ • d 4 s 1 H4QsJ J i 2-2e.[ ($ ~ ) 2 ( s = ) 2 (.2M - 5) 2 ( S3 ~ ) 2
3072Hd
-
(S+ ) 2 (S+ + ) 2 ( 2 M + 5 ) 2 ( S 3+)2
+ ' { 2 ( 2 M - 1 ) + ( 2 M - 5 ) } 2 ( S " ) 2 ( S= ) 2 ( S 3")2
- {2(.2M+1 ) + ( 2 M + 5 ) } 2 (S+ ) 2 (S+ + ) 2 (S 3+ )2
+ 3 ( 2 M - 1 ) ( 1 0 M + n ) ( S " ) 2 (S+ ) 2 CS+^)2
- 3 ( 2M+1) ( I OM-1 1 ) ( S+ ) 2 ( S~ ) ( S= ) 2
+ I 2 ( 2 M + 1 ) 2 ( S ~ ) 2 (S +)4 - 1 2 ( 2 M- I ) 2 (S+ ) 2 ( S ~)4
+ 3 ( 2 M - 3 ) 2 ( S - ) 2 ( S = ) 4 - 3 ( 2 M + 3 ) 2 (S+ ) 2 (S + + ) 4
+ 9 ( 2 M + 1 ) 2 (S+ ) 4 (S ++)2 - 9 ( 2 M - 1 ) 2 ( S " ) 4 ( S = ) 2]
-99
D4P ^ s i r i 4 O
3--[(M-l)2(S-)2(s=)2_(M+T)2(S+)2(S++)2]
8H
4
p
?
D^ p ^ s i n ^ Z O
+ ------2-[ ( 2 M - 1 ) 4 ( S - ) 2- ( 2 M + l ) 4 ( S+ ) 2 ]
16Hd
D4 P9S i n 2O s i n 2ZO
9
---------- — *--------------- [ ( 4 M+ 3 ) (2M+1 ) (2M + 3) (S ) 2 (S
64H^
9
)2
+ (4M-3 ) ( 2M-1 ) ( 2 M- 3 ) ( S " ) 2 (S ) 2-8M(2M+l ).(2M-T) (S + ) 2 ( S " ) 2 ]
(A. 6 )
This
expression
the r e s u l t
is
To f i n d
has been e v a l u a t e d
given
in
in
for
wh i c h a t r a n s i t i o n
t h e two e n e r g y l e v e l s
to
t h e m i c r o w a v e p h o t o n e n e r g y , and t h e
is
solved
expression
for
H.
is
resulting
I
I
For t h e t r a n s i t i o n . M,m =
can be f o u n d f r o m Eq.
=H
2>m
3
S = ^" anc*' M •=
I
and
Eq. ( 4 . 1 2 ) .
the magnetic f i e l d
o c c u r , the d i f f e r e n c e
for
r,m
+ ^ p - ( S i n 4 O - Z s i n 2ZO)
(4.12),
s e t equal
expression
1, t h i s
and i s : '
+ §-(m+m 1) - T H ( m - m ' )
n
+ ^ - ( 2 I 2-m2 -m 12-7m+7m1)
tan? .
?
4
SA2 DP,
+ — 5—(m+m 1) ( 3 s i n 2 20 - s i n40 ) +------o— ( -m2+m ' 2+m+m '.)
8 H2
ZH^
- — n [ - ^ m 2+ ^ m 12+ m ( I 2-m 2+p)+m 1( I 2~m 12+|-) I
4H^
■
^
^
can
-100-
( s i n ^ e - 8s i n 22e) +
q ( 8s i n 428
64HJ
+ 1 6 s i n 22 0 s i n ^ 0 - s i n ^ 0 ) '
T a k i n g h v / g p B = Hq , an a p p r o x i m a t e
H
-
H 0 D
- ^
j -
( m + m 1 )
(A. 7)
solution
for
H is
(A. 8 .)
-
o
Us i ng
this
approximation
in
then expanding the small
■ n u m e r a t o r s , one o b t a i n s
Considering
on n u c l e a r
spin,
only
Crgg l i n e s
Differences
in
group.
t e r ms on t h e
Eq.
t hose terms
in
of
re la tive
Crgg l i n e s
For t h e f o r b i d d e n
Eq. ( A . 7 ) ,
into
the
Eq. ( A . 6 ) w h i c h depend
an e x p r e s s i o n
to
these displa cements
pairs
right
of
(4.13).
one can f i n d
ment o f t h e
particula r
the denominators
,nvrM-l ,m 1 " ^M, m' - >M- l , m
the d i s p l a c e ­
the e q u i v a l e n t
give
w ithin
hyperfine
for
Crgg T i n e .
the s e p a r a t i o n
t h e same f i n e
doublets
bet ween
structure .
the s e p a ra tio n
is
^
2
A ( 2M- T) (.m' - m ) + 2 r n H(m-m' ) - ^ H- [ ( S " ) 2 ( 2 m, -2m)
+ ( S + ) 2 (m2- m - m 12+m 1) + ( S = ) 2 ( m ' 2+m' -m2m) ]
+ ----- =2^-[ ( 2M-1 ) ( S- ) 2 ( 2m 1- 2m)
4H
/
-101+ (2M-H ) ( S + ) 2 (m2- m - m , 2+m' ) + ( 2 M - 3 ) ( S
) 2 (m 12+m' -Hi 2-In) ]
3
+ — o.[2m( M-m) ( S ) 2-2m 1(M-m 1) ( S ) 2+(S
4 Fr
+
) 2 (m 12+m 1-Hi 2-Hi)
(M-m)(S+ ) 2 ( r ) 2-(M-m)(S= ) 2 (I + ) 2
+ ( M- m 1) ( S = ) 2 ( I , + ) 2- ( M - m 1) (S + ).2 ( I ' " ) 2
+ (S+ ) 2 ( I ' ) 2- ( S + ) 2 ( I ' " ) 2+ 2 ( S = ) 2 ( I + ) 2- 2 ( S = ) 2 ( I , + ) 2] . ( A . 9 )
I f M=
2", t h e n ( S )
= S +^-, ( S )
t h e above e x p r e s s i o n
using
third
in
term above,
Admixture
Eq.
( A . 8) f o r
one o b t a i n s
C oefficient
The c o m p l e t e wave f u n c t i o n
is
given
wish
in
in
Eq.
|M,m±2> i n
(5.14)
and one o f t h o s e
remaining
is
S
a nd
for
that
by Eq.
coefficients
denominator
for
l a b e l e d M,m
(5.14).
hyperfine
by a p a r t o f a n o t h e r .
considered
are:
We
t h e base s t a t e s
Onl y c e r t a i n
can admi x n e i g h b o r i n g
t e r ms w h i c h m u s t - b e
the
Eq. ( 4 . 1 5 ) .
the s t a t e
equation.
cancelled
H in
After
Calculations
order approximation
t o comput e t h e a d m i x t u r e
IM,m±1> and
t e r ms
fourth
=
can be c o n s i d e r a b l y c o n d e n s e d .
the approximation
o f the
= (S )
o f the
states,
The
-102-
(A. I 0 )
k/m
The f i r s t
important
sum above has been e v a l u a t e d
corrections
energy denominators
second sum a b o v e ,
Following
in
the
C h a p t e r V,
admixture
derive
in
this
C h a p t e r V, b u t
f r o m c o n s i d e r i n g more e x a c t
sum.
S im ilarly,
in
e v a l u a t i n g ■t h e
one mus t c o n s i d e r d e n o m i n a t o r c o r r e c t i o n s .
procedure
the
in
used i n
higher order
co e fficie n t
for
the
t h e second o r d e r
calculation
sums have been e v a l u a t e d .
state
|M,m-T> i s
[ ( 2 M - 1 ) ( S " ) 2- ( 2 M + 1 ) ( S+ ) 2J I '
D2 P2 s i n 2 9
[ ( 2 M- I ) 2 ( S ~ ) 2- ( 2 M + 1 ) 2 (S+ ) 2] I ~
SH2M
+ D2S i n 2Q s i n28 [ ( M - I ) ( S ' ) 2 ( S = ) 2 - 2M( S+ ) 2 ( S - ) 2
I S H2M
+ ( M + l ) (S+ ) 2 ( S+ + ) 2] I '
D3P2S i n20
[ ( 2 ' M- l )3 ( S" ) 2 - ( 2M+ 1 I 3 (S+ ) 2 ] ! "
The
'
H
-103D3p,sin28sin28
_ _ _
----------^ ---------------- [ ( M - 1 ) ( 4 M + 3 ) ( S " ) Z ( S =)2
I S Hl3M
- 8M2 ( S " ) 2 (S + ) 2+.(M+ l ) ( 4M- 3) (S + ) 2 ( S+ + ) 2] I "
3
4
- P.. s j .n
1n 2 9 [ 2 ( S M - I ) ( S + ) 2 (S~) 2 ( S= ) 2+( 6M+7) (S + ) 2(S++) 2(S3+)2
S l Z H l3M
- 2(6M+l)(S-)2 (S+)2 (S++)2-(6M-7)(S-)2 (S=)2 (s3-)2
+ 4 ( 2 M + 1 ) ( S " ) 4 ( S +)2 - 4 ( 2 M - 1 ) ( S + ) 4 ( S ")2
+ ( 2M+3) ( S+ ) 2 (S ++)4 -
( 2 M - 3 ) ( S " ) 2 (S =)4
+ 2 ( 2 M - 1 ) ( S " ) 4 ( S =)2 - ’ 2 ( 2 M + 1 ) ( S + ) 4 (S+ + ) 2] I "
3
3
- P----14 ---' 9- [ 2( 2M+l ) (2M+3) (M+l ) (S + ) 2 (S ++)2
• I ^ S H l3M
- 2 ( 2 M - 1 ) ( 2 M - 3 ) ( M - l ) ( S " ) 2 ( S= ) 2- 2 ( 2M+1) (2M-1 ) ( S " ) 2 (S + )2
+ (2 M- I ) 3 ( S -)4 The a d m i x t u r e
in
sign,
is
coefficient
and I + r e p l a c e s
sign a ris e s
it
( 2M+1 ) 3 ( S+)4 ] I "
AM f o r
in
for
the
I"
the energy
a and -AM f o r
the
order
sum:
in
| M, m + l > i s
each t e r m .
in
opposite
The change i n
each d e n o m i n a t o r :
8.
hyperfine
admi x ed by t h e second and f o u r t h
|M, m-2> o n l y
state
which occurs
The s e c o n d - n e i g h b o r i n g
the s t a t e
(A. IT.)
three
terms
loops
states
in
can o n l y be
Eq.- ( A . 1 0 ) .
ar e p o s s i b l e
in
For
the t h i r d
M,m+M+1 ,m-l-*-M+2 ,m-2->M ,m-2 ; M,m->M+ 1 , m-l ->M-l ,m-l->M,m-2;
-104,m-^M-l , m - l ^ - M , m - 2 .
order
with
sum t h e r e
2 2
A M in
than the
general
the ir
third
remaining
are s i x
loops,
two l o o p s
same a d m i x t u r e
four
denominators
order
expression
For t h e
o f wh i c h r e s u l t
for
the
in
the f o u r t h
i n t er ms
and wh i c h a r e t h e r e f o r e
contribution.
vanishes
in
The c o n t r i b u t i o n
any case f o r
| M, m- 2> a d m i x t u r e
S =
3
larger
from the
The
co e fficient
is
Y = ft^-s-1^ 2e[ 3M2 - S ( S+ 1) ] r r
I 6H^M
+
AD2 s i n 2 200. _ 0
255H2M
[ ( S " ) Z ( S - ) 2(2M-l)(2M-3)
C
-
(S+ ) 2 (S+ + ) 2 ( 2M+1)
(2
M
+
3
)]l"r
+ D ^ i l g i | i [ ( S- ) 4 ( 2 M- l ) 2+ ( S + ) 4 (2.M+l ) 2
I 28H ^
- 2 ( S " ) 2 (S+ ) 2 ( 2 M - 1 ) ( 2 M + 1 ) ] I " I =
For t h e a d m i x t u r e
t wo t e r m s
in
coe fficient
above change s i g n
each t e r m i s
replaced
t h e Am = 2 i n t e n s i t y
of
the
but the
by I + I + + .
formula
given
state
last
( A. 1 2 )
| M, m+2>,
does n o t .
These r e s u l t s
the f i r s t
The I
I
a g r ee w i t h
by B l e a n e y and R u b i n s .
(Il)
APPENDIX B
PROGRAM T0 COMPUTE [PR LINE POSITIONS AND INTENSITIES
PeR THE M= I /S TO M= . 1 / 2 TRANSITIONS OP CR53 IN MGO
DIMENSION A (136 >> V ( 16>1 6 ) , R( 2 5 6 ) , TME(16)
EQUIVALENCE ( V/ R)
" ■
READ IN CRYSTAL AND EXPERIMENTAL PARAMETERS
DAT A A l , GLD/ GN/17 »8, I'. 001602, " » 8 8 7 . 1 , + 8 ' 6 8 8 9 E " ' 0 5 /
DATA OOf HH' P H OT / 4 5 , , 3 8 5 0 , , 3 6 0 8 * 5 /
0=08*3.1415927/180*
P 2 = . 5 * ( 3 " * COSIO) * * 2 » 1 , )
N= 16
ND = I
ESTABLISH LOOP TO COMPUTE LINE POSITIONS DESIRED
DO 99 L =5,8.
DO 99 M=9,12
22=1000
10 H=HH*SQRT(COS( 9 ) * * 2 * G » * 2 * S I N ( 6 ) * * 2 )
'
define
matrix
elements
of
the
h a m i l t qn i a n
DO 20 1=1, 1 36
20 A ( I ) = O . ,
A(
l)=l,5*H +2*25*A l+D *P2+1.5*0N *H H
A ( 3 ) = l ' 5 * H + 0 " 7 5 * A l + D * P 2 - k ,5%GN*HH
A ( 6 ) = I • S^ H. Q»7 5 * A l + D*P2w • 95*.GN*HH
A ( I O) = I »5 * H ,5’2 «25' *AI +D* P2 r I ■>5*GN*HH
A ( 11) = , 5 * SORT(3»> * D * S I N ( 2 , * 8 )
A( 12)= I-*5» AI
A( 15)a,5*H+,75*Al*D*P2*l,5*GN*HH
A ( 17) =A I 11)
' •
A ( 18) = SORT ( 3 . )*A1
A ( 21 ) = 9 5 * H * ?2 5 * A l eiD^PS* «5*GN*HH A ( 24) =A( U )
A ( 2 5 ) =A( 12)
A ( 28 ) = «5 * H " (• 2 5 * A l ' , D*P2s' »5*GN*HH
AC 3 2 ) =AC I l ) A . ( ' 3 6 ) = , 5 * H * , 7 5 * A 1 " D * P 2 « 1 , S * G N %H H
AC 4 0 ) = , 5 * SQr T ( 3 . ) * d * S I N ( 0 ) * * 2
A I 45 ) =A C 1 5 ) . H ®3, O^GNiK-HH
AC 4 8 ) =AC 40)
AC 5 3 ) = A ( I S)
AC 55) = A ( 2 1 ) . H " l f O* GN* HH "
AC 5 7 ) =AC 40)
AC 6 2 ) = 2 " * A 1
'
AC 6 6 ) =AC 2 8 ) . H ^ l o 0*GN*HH
AC 6 7 ) =A C 40)
.
AC. 7 2 ) =AC 18)
.AC 78) =AC 3 6 ) . H *3» 0*GN*HH
AC 8 6 ) s A ( 40)
.,
V
•/
I
- I 06A ( 8 7 ) s * A ( 11)
A( 91)ba(
A ( 9 8 ) =A( 40)
A ( 1 0 0 ) =A( 12)
A ( 10 1 ) BnA ( 11)
A ( 1 0 5 ) =A(
3),3,*H
A ( I l l ) = A t 40)
A ( 115)= A( 18)
A ( 1 16 ) = A ( 11)
A ( 1 2 0 ) BA ( 6 ) » 3 , * H
A ( 1 2 5 ) =A( 40)
A ( 1 3 1 ) =A( 12)
A ( 1 3 2 ) = - A ( 11)
A ( 136 ) - A ( '10 ) v3 * *H
'
30
32
40
41
45
50
51
69
"»3 • 0*6N*HH
*1,0*0N*HH
+ 1 . 0*eN*HH
+3»0#GN*HH
DECIDE WHETHER UINE POSITION BNUY BR UlNE- POSITION
and l i n e i n t e n s i t i e s wi u u Be computed
I F ( (U + M) ' G E , 20) (SB TB 50
MV = I
DIAGONAUIZE' THE MATRIX
CAUU SE I GEN(Aj R ^ M V )
UU=( U* U+L) / 2
MM=( M#M+M) / 2
TEST ENERGY CONSERVATION '
Z l s A ( U U ) - A ( M M ) . PHOT
I F ( A B S ( Z l ) , GE. ABS(Zg)) GO TB 99
;
I F ( A B S ( Z l ) , U T , + , I ) G0 TB 40
REDEFINE THE MAGNETIC FIELD AND RE.DI'AGBNa u I ZE
HH =HH" Z I
/
ZS=Zl
/
ND=ND+!
GB TB 10
ADD THE EIGENVALUES
(TRACE SHOULD BE ZERG)
SUM=O.
DO 41 I = I j N
11 = ( I * I + I >/2
SUM =SUMfA( I I ) .
PRINT TRANSITION, FI ELD, T r ACEj AND' .NUMBER OF
diaggnauizations ■
WRITE( 10 8j 9 1 ) JUj MjHHj SUM, ND
I F( MV h I ) 5 l ' 9 9 , 5 1 .
MV=O
GB TB 30
' .
•
WRITE( 1 0 8 , 9 2 )
,
..................
PRINT EIGENVALUES AND EIGENVECTORS FOR LAST LINE ,
DB 70 J = I j N U J = ( J * J f J ) XH
WR I TE( I OSj 9 3 ) j J j A ( J J ) j ( V ( I j J ) j I 9 I j 8) • •
-10770 WR H E ( 1 Q8, 9 4 M V (
1=9, 16)
COMPUTE TRANSITION MATRIX ELEMENTS FSR ALL 16 LINES
OS 80 K=5,R
OS 79 J = l , 4 ■•
TME(J)=O*
OS 78 1=1, 4
78 T M E ( J ) = T ME ( J ) + V ( I , K ) * V ( 1 + 4 , 1 3 » J ) *SQRT( 3 * ) * » 5 + V <I + 4 ; K )
I
*V(13»I,13"J)+V(13,I,K)*V(17^I,13MJ)*SQRT(3,)*,5
79 CONTINUE
80 W R I T E d O S , 9 5 ) , K, ( T M E ( J ) / J = 1^4)
91 F0RMAT(1HO/1 2 , 2X, 1 2 / I
H=• , F l O « 3 , I
SUMsl ,
I
F6 «3, I
N O = ' / 14)
92 FSRMATdHo, i
EIGENVALUES I , 30X, , EI GENVECTORS! / )
93 FORMAT (-!HO, 13, F l 2 «3, 4X, 8 F 11 «7 ) "
94 FORMATdH , 1 9 X , 8 F l l . 7 )
95 FORMAT( ! H O / 1 2 , 8 X , 4 F 18»8)
99 CONTINUE
100 CALL EXIT
END'
10
15
20
25
30
35
40
SUBROUTINE SE I GEN( A , R , N , MV)
GROUP: ' m a t r i c e s
IF MV=I , SUBROUTINE WILL’ COMPUTE ONLY EIGENVALUES
' IF MV=O, EIGENVECTORS ARE Also COMPUTED'
DIMENSION A ( I ) / R ( I )
I F( MVml ) 1 0 , 2 5 , 1 0
IQ = -N
, ■
DO 20 J =1 , N
I Q = I Q+N
DO 20 11? I , N
IJ=IQ+!
R(IJ)SO,O
I F ( I - J ) 20,15,20
R(IJ)=I,O
CONTINUE
ANORM=O,O
DO 35 1 = 1 , N
DO 35 J = I , N
I F ( I - J ) 30,35,30
I A= I + ( J * J - J ) / 2
, ANORM=AN8RM + A ( I A ) * A ( I A >
CONTINUE
IF(ANORM) 1 6 5 , 1 6 5 , 4 0
ANORM=I,414+SQRT(ANORM)
ANRMX=ANBRM*!.0E«6/ FL8AT( N)
IND = O
THReANORM
-10845
50
55
60
THR=THR/FL8AT(N)
L=I'
M=L+ I
MQ=(M+N-M)ZS
LQ=( L+L ^ L ) / 2
LM=L+MQ
62 I F ( ABs(A(LM))=THR) 1 3 0 , 6 5 , 6 5
65 IND=I
LL=L+LQ
MM=M+MQ
X = 0 , 5 * ( A( L L ) =A ( MM) )
68 Y = - A ( L M ) / SOFT(A ( L M) +A ( L M ) +X+ X)
IF(X) 70,75,75
70 Y = -Y75 SINX =YZ. S Q R T < 2 . 0 * ( l , 0 + ( SQRT( 1 8O= Y * Y ) ) ) )
Sl NXS=SI NX+SINX
78 CQSX= SQRT(IeO=SINXS)
C0SX2=CQSX*C9SX
SINCS =Sl NX+CQSX
I LQ =N+ (L = I )
IMQ=N+ ( M - I )
DQ 125 1 * 1 , N
IQ=(I+I-I)ZS
IF (I-L) 80,U 5,80
80 I F ( I - M ) 8 5 , 1 1 5 , 9 0
85 I M = I +MQ
QG TQ 95
i
90 IM=M+ IQ
95 I F ( I - L ) ' 1 0 0 , 1 0 5 , 1 0 5
100 I L = I +LQ
GQ TS H O
I o 5 I L = L + IQ
I l O X s A ( I L ) +CQSXeA(I M)+SINX
A( I M ) = A ( I L > * S I N X +A ( I M) +C0SX
A(IL)=X
H S IF(MVrl) 120,125,120 '
120 I L R = I L Q + I
IMR=IMQ+ I
X = R ( I L R ) +CQSXmR(IMR)+ SINX
R ( I M R ) = R ( i L R ) * S l N X +R ( t MR)+CQSX
R(ILR)=X
125 CONTINUE
X = 2 . O+A ( L M ) +SINCS
Y = A ( L L ) +CQSX2+A(MM)+STNX2»X
.- X = A ( L L ) +SlNXg+A(MM)+CQSXS+X
A ( L M ) H A ( L U » A ( M M ) ) +Sl NCS+A(LM)-+ (C9SX2eSl NX2)
■ A( L L ) =Y
A(MM)=X
- 109
130 I F ( M- N ) 1 3 5 / 1 4 0 / 1 3 5
135 M=M+ I
G9 TB 60
HO I F a - ( N - I ) ) 145/150/145
H S L =L + I
GB TB 55
150 I F ( I N D - I ) 1 6 0 , 1 5 5 / 1 6 0
155 IND = O
Ge TB 50
160 IF(THR-ANRMX) 1 6 5 / 1 6 5 / 4 5
165 IQ = -N
DB 185 I = I z N
IQ=IQ + N.
LL- I + ( I * H I ) / 2 ’
J Q=N* ( 1 - 2 )
DB 185 J - I / N
JQ =JQ+N
MM=J+<J*J-J)/2
IF tA ( L L ) H ( M M ) ) 170/185,185
170 X=A(LL)
A(LL)=A(MM)
A(MM)=X '
IF (M V -I) 175/185/175
175 DB 180 K«I / N
I L R M Q +K '
IMR =JQ+K.
X =R ( I L R ) , ■t
R( I L R H R t j MR)
i s o R(I MR) =X
185 CONTINUE
RETURN
END
APPENDIX C
THE SPECTROMETER
The s p e c t r o m e t e r may be b r i e f l y
heterodyne
fie ld
s y s t em w i t h
modulation.
The k l y s t r o n s
operate
operates
frequency
90 KHz, and t h e f i e l d
Figure
basic
a t a n o mi n a l
27 shows a b l o c k
parts
of
as a s u p e r ­
balanced m ixer d e t e c t i o n
IF a m p l i f i e r
is
described
diagram o f
and m a g n e t i c
a t about- 10 GHz, t h e
30 MHz, t h e s t a b i l i z a t i o n
modulation
is
a t 400 Hz.
the e n t i r e
system.
The
the s p e c tr o m e te r a r e :
1.
Signal
and l o c a l
2.
Microwave b r i d g e ,
oscilla to rs,
with
operating
fe rrite
at X-band.
circula tor
and s l i d e -
screw t u n e r .
3.
R e s o nan t c a v i t y ,
containing
4.
Ma g n e t ,
5.
System t o m o d u l a t e m a g n e t i c
adjustable
peak-to-peak
from 0 to
Mi c r o w a v e m i x e r ,
7.
Automatic
8.
400 Hz phase s e n s i t i v e
9.
Chart re c o rd e r.
description
elsewhere.
fie ld
pream plifier,
frequency c o n tro l
and s e v e r a l
o f the
It
up t o
50 gauss
and IF a m p l i f i e r .
system.
designed w i t h
hand-made c i r c u i t s .
instrument
in
(AFC)
detector.
s p e c t r o m e t e r was o r i g i n a l l y
tee b r id g e
10 k i l q g a u s s .
a t 400 Hz.
6.
This
t h e s a mp l e .
its
has been r e b u i l t
A complete
original
with
a magi c
f o r m ap pea r s
a c i r c u l a t o r bridge
-
I
- CIRCULATOR
SST - SLIDE SCREW TUNER
C - TEq11
CAVITY
M - MIXER-PREAMPLIFIER
A - ATTENUATOR
AO - AUDIO OSCILLATOR
F i g . 27
Block
111
T
-
-
ISOLATOR
SK - SIGNAL KLYSTRON
LK - LOCAL KLYSTRON
I F - 30 MHz AMPLIFIER
AFC - FREQUENCY STABILIZER
L I A - LOCK-IN AMPLIFIER
Di agr am o f S u p e r h e t e r o d y n e
Spectrometer.
-112and c o m m e r c i a l
replaced with
lock-in
one o f b e t t e r
o f the ins tru m e n t
o f two,
it
am plifier.
is
much l e s s
sensitive
The s p e c t r o m e t e r ' s
is
fed
microwave
balanced b r i d g e .
cavity
in
arm.
operation
power,
Balance
the b r id g e
The c a v i t y ,
fie ld
absorption
this
in
results
The m a g n e t i c
entering
is
to v i b r a t i o n
and t he
can be g e n e r a l l y
fre q u e n c y modulated
cavity
wh i c h
obtained
to produce
containing
is
described
a t 90 KHz,
a part
of a
by a d j u s t i n g
a slide-
s t a n d i n g waves i n t h e
the sample,
o f microwave
a re la tiv e ly
fie ld
large
is modulated
so t h a t
frequency.
s e n s itiv ity
power i n
change i n
the re sonant c o n d i t i o n
the m ix e r
is
therefore
d e p e n d i n g on t h e
the
the
( a b o u t one gauss
The m i c r o w a v e power r e t u r n e d
the am plitude
is
in
o f a p p r o p r i a t e m a g n i t u d e such as t o
paramagnetic
a t 400 Hz,
the
The b a l a n c e can be u p s e t by power a b s o r p t i o n
the c a v i t y .
magnetic
Although
i s much g r e a t e r .
to a r e f l e c t i o n - t y p e
screw t u n e r
design.
has been
has n o t been i mp r o v e d by more t h a n a f a c t o r
convenience o f o p e ra tio n
as f o l l o w s :
The AFC c i r c u i t
is
a dc
cause
sample,
bridge
shifted
in
varied
frequency
begins
to
signal
d rift
balance.
at th a t
from the c a v i t y
and
a m p l i t u d e m o d u l a t e d a t 400 Hz,
strength
o f the
paramagnetic
t h e 90 KHz f r e q u e n c y m o d u l a t i o n may be
the re tu rn e d
and
peak-to-peak)
absorption.
The phase o f
in
if
from the
the
signal
klystron
cavity frequency.
M
-113The d i r e c t i o n
klystron
o f the
frequency
phase s h i f t
is
depends upon w h e t h e r t he
above o r b e l o w t h e c a v i t y ' s
resonant
frequency.
The r e f l e c t e d
a mi crowave
klystron
than
signal
from the
oscillates
the s i g n a l
am plified
of
power, w i t h
at
to
the
IF a m p l i c i c a t i o n ,
" c o n t r o l " signal
Hz.
locked
to
to
the s ign al
the c a v i t y
using
klystron.
frequency.
resonance
the magnetic
time axis
fie ld
The o n l y
produced
is
is
the
fie ld
a t 400
am plified.
with
voltage
the
a
wh i c h
is
klystron
containing
phase-sensitive
s o u r c e as a r e f e r e n c e ,
displayed
on a s t r i p - c h a r t
is
on a c h a r t
slowly
recorder.
increased
so
can be r e a d as a
axis.
The c omponent s
IX.
t h e AFC
keeps
pre­
stage
signal
The 400 Hz s i g n a l
the f i e l d - m o d u l a t i o n
is
last
and f u r t h e r
inform ation
I n an e x p e r i m e n t ,
magnetic
This
is
producing
resonance
a correction
dc o u t p u t
the
detected,
filte re d
and t h e r e s u l t i n g
that
The l o c a l
At the
phase compar ed i n
to generate
the paramagnetic
detected,
is
mi xed w i t h
n o m i n a l l y 30 MHz h i g h e r
IF a m p l i f i e r .
t h e 30 MHz i s
is
is
The 30 MHz b e a t s i g n a l
signals
signal
90 KHz r e f e r e n c e
applied
oscilla to r.
a t 90 KHz, and t h e
Each o f t h e s e
The c o n t r o l
local
modulations,
a frequency
k l y s t r o n ..
and f e d
its
of
the s p e c t r o m e t e r are l i s t e d
important c i r c u i t
t h e AFC.
w h i c h was n o t
This c i r c u i t
was a d a p t e d
in Table
commercially
f r o m one
-114TABLE IX
SPECTROMETER PARTS LIST
Component
Manufacturer
Model
2
2
I
I
I
I
I
I
I
3
I
I
I
I
I
I
I
Varian
H e wl e t t - Packar d
H e wl e t t - P a c k a r d
M i c r o w a v e Components
M i c r o w a v e Components
LEL, I n c .
LEL, I n c .
H e wl e t t - P a c k a r d
Hewlett-Packard
hand-made
hand-made
Kepco
Kepco
Varian
hand-made
Princeton
X-I 3
X-382A
X-532B
X-331
X-360B
X - BH-.2
IFG0B-50
200 AB
130 BR
-
-
-
-
I
I
I
2
I
I
I -
-
-
-
-
Klystrons
Attenuators
Frequency Meter
D e t e c t o r mount
Topwal I C o u p l e r
Mix e r-p re a m p li f i e r
IF A m p l i f i e r
Audio O s c i l l a t o r
Oscilloscope
350 v dc power s u p p l i e s
300 v dc power s u p p l y
7 . 5 v dc power s u p p l y
6 v dc power s u p p l y
Magnet s y s t e m
AFC c i r c u i t
Lock-in a m p lifie r
K l y s t r o n power c o n t r o l
c irc u it
M a r g i n a l o s c i l l a t o r (NMR)
S tr ip - c h a r t recorder
Frequency c o u n t e r
I s o l ators
Circ u la to r
S l i d e screw t u n e r
C y l i n d r i c a l TEq i ^ c a v i t y
hand-made
hand-made
Varian
Hewlett-Packard
A irtron
Mi cr omega
Waveline
hand-made
-
-
-
—
ABC7 . 5- 2
PRM6-25
V- 3400
— —
121
G-14A
5245L
890350
XL-21.0
683
-
-
described
by B e r r y and B e n t o n , a n d
voltage
source
c irc u it
diagram is
superior
to
for
the
signal
shown i n
klystron
Fig.
28.
t h e one used p r e v i o u s l y
introduce
a base l i n e
supplying
a correction
sh ift
on t h e
voltage.
includes
t h e dc
re flector.
The.
T h i s AFC c i r c u i t
in
that
it
is
does n o t
c h a r t r e c o r d e r when i t
i
+ 15v
--*
OUT
116
Fig.
28
AFC C i r c u i t
Di agr am
Q l ,2,3
Q4
2N4124
2N4126
• LITERATURE CITED
1.
J . We r t z and P. Auz i n s ,
2.
P. A u z i n s
and
J.
W e r t z , J . Chem. P h y s . 43_, 1 229
(1 96 5 ) .
3.
P. A u z i n s
and
J.
Wertz, B u l l .
Am.
707
4.
W. Low,
T05, 801
(1957).
5.
J . van We i r en ge n
and J . R e n s e n , P a r a m a g n e t i c Re s o n a n c e ,
Vol . I , W. Low, Ed.., Academi c P r e s s , New Y o r k , 1 963.
6.
B . B l e a n e y and K.
7.
G. P a k e , P a r a m a g n e t i c R e s o n a n c e , C h a p t e r 3 ,
New Y o r k , 1962.
8.
H. H e c h t , M a g n e t i c
W i l e y and Sons ,
9.
G. Bi r , So v . P h y s . - S o l i d
10.
G.
11.
B . B l e a n e y and R. R u b i n s ,
77 , 1 03 (1961 ) .
12.
D. D i c k e y ,
13.
D. D i c k e y and J . Drumhel I e r , Phys.
14.
B. B l e a n e y and D. I n g r a m ,
A205 , 336 (1 951 ) .
15.
E.
Fr i edman
16.
V.
F o l e n , Phys. Rev.
17.
D.
Lyons and R.
18.
J.
Drumhel I e r
Phys.
Rev.
P h y s . Rev.
Stevens,
Rep.
T06 , 484
P h y s . S o c . 9_,
Pr o g .
Benjamin,
Resonance S p e c t r o s c o p y , C h a p t e r 5,
New Y o r k , 1967.
M. S . T h e s i s ,
W. Low,
State
5^, I 628 (1 9 6 4 ) .
P r o c . Phys.
Mont ana S t a t e
State
Soc.
Phys.
125 , 1 581
and R. R u b i n s ,
5,
2637
( L ond on)
U niversity
Rev.
1 61 , 279
P r o c . Roy . S o c .
K e d z i e , Phys.
19.. M. OdehnaT, Czech.
(1 964)
P h y s . J6_, 108 ( 1953 )
B i r and L . S o c h a v a , S o v . P h y s . - S o l i d
(1 9 6 4 ) .
and
(1 9 5 7 ) .
(1966).
(1 967 ) .
(London)
R e v . 1 20 , 408
(1 9 6 0 ) .
(1 9 6 2 ) .
Rev . T 4 5 , 148
Phys.
(1 9 6 6 ) .
Rev. I 3 3 , Al 099
J ., Phys . BI 3 , 566 (1 9 6 3 ) .
(1964).
" •
K u n i i , S. T o b i t a
2T, 479 (1 966) .
118
—
—
—
"
20.
S.
21.
D. T a n i t no t o and J . Kemp, J . P h y s . Chem. Sol . :27_, 887
22.
B. C a v e n e t t , P r o c . P h y s . S o c .
23.
J . D r u m h e l l e r 5 He! v . Phys . A c t a '37^ 689 ■ (1 9 6 4 ) .
24.
R.
25.
W. L o w5 P a r a m a g n e t i c Resonance i n S o l i d s , C h a p t e r 9,
Ac ademi c P r e s s , New Y o r k , 1960.
26.
S. M a r s h a l l , J . Hodges and R. S e r w a y 5 P h y s . R e v . 136,
Al 024 (1 9 6 4 ) .
27.
H. de W i j n and R.
1381 ( 1 9 6 7 ) .
28.
F . W a l d n e r 5 He! v . P h y s . A c t a 5 ZS_, 756
unpublished corrigendum.
29.
N. Bl oember gen and E. R o y c e 5 P a r a m a g n e t i c R e s o n a n c e ,
V o l . I I 5 W. Low, E d . , Academi c P r e s s , New Y o r k , 1 963.
30.
A.
31.
I . G e l ' f a n d , R. M i n l o s and Z . S h a p i r o 5 R e p r e s e n t a t i ons
o f t h e R o t a t i o n and L o r e n t z G r o u p s , S e c . 7,
M a c m i l l a n , New Y o r k , 1 963.
32.
J . Berry
• ■
and E. H i r a h a r a 5 J . P h y s . S o c . ( J a p a n )
(London)
84^ I
Sharma and T . D a s 5 P h y s . R e v . 149 5 257
(1 965)
(1 9 6 4 ) .
(1 966 ) .
van B a l d e r e n 5 J . Chem. P h y s . 4 6 ,
(1 962 ) ;
also
M e s s i a h , Quantum Me c h a n i c s , A p p e n d i x C55 W i l e y and
S o n s 5 New Y o r k 5 1965.
and A.
—
B e n t o n 5 Rev.
Sci .
Inst.
36_, 958 (1 9 6 5 ) .
'
MONTANA STATE UNIVERSITY LIBRARIES
762
001 0721 6
%
T
I
D378
D559
Dickey,
cop.2
Electron paramagnetic
resonance spectra of
^ r ^valent chromium in
David Hugh
NAMK
AND
D. 3 . S i e u e r s
W j I A
l W
ADDRKf
C'
/Kl
IxiorWu
I
-j t T . __