CRYSTALLINE STARK SPLITTING
AND MICROWAVE RESONANCE ABSORPTION
IN PARAMAGNETIC SALTS
C. KITTEL AND J. M. LUTTINGER
TECHNICAL REPORT NO. 49
September 10, 1947
RESEARCH LABORATORY OF ELECTRONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
I_I __(\_l·_II__III__X_- --- -II1I-II--·---
Rserch Iiboraory of nlectrmics
chnical BRport No. 49
OUSCbLLM
8epteber 10, 1947
ARK SDPLITING AID XIICBlOWTVV
IN IUfIMIC SA
m
ABSPlI0
by
0.
Ittel and J. n. lattinger
A theoretical discussion is given of the applicatio
of microwave absorption measremuents to the determination of
enlrg levels in single crystals of pacraei
salts. Selectio rules are given for magetic dipole transitions between sublevels in the presence of rystalle electric fields of cubic
tetragonal, trigmnal and rhombie sy trT. The Zeeman effect in
the presene of crystalline electric fields is disoused in sow
detail for three speal olses: a) J
/2, cubic field;
b) J
7/2, cubic fielad
a) J
3/2, triaonal field. These
eamples ma correspond to certain matically dilute salts containin
r
Fe ,
't and Or + ion, respectively, provided that
exang interactions o ot plar an important role Numerical
tvalues are tabulateds for the rlative frequencies and lie
trengths asoiated with magnetic dipole transitions in the
oases disused,.
he character
of the spectra may han
radlally between the limit where the Zeeman splitting is
ml
in
ceaparison with the Stark splitting, and the opposite lrit where
the Stark splitting ma be neglected.
..___
I
--____r_---·--rr--rn-·rr·rr-·-
-r-l-rrrrrura--;rrwunw-rsi
I'·UICI-L
.*yllll^---··----
__1_
_
_1_11
_1_
I)
-
I.
II, iTROD1,CI-IJTi
'fle present kno-rleuge
parai. agnetic salts is
of measuremnents of
of
taute
eived fromn t-e i'lter-rftatJ on
largely
speci.ic he.ets,
s,
agnietic susce-l.ili'iitie
a'.sor-ption snectra.
anld otical
quan-rt';iln
' of t.e
Ihe difficl-tic
ivting
il
a consistent account of the vrious ?heno-n:'le '1.ve been
3
19)39 Strasbouirg confece
sized by Van vlec'- at te
:Tnd by
,
Penney and I rnch5
The recent exoeri ietal
discovery b,
Zavoiskr
6
of
Dar&-lmagetic resonlence absor-t-ion provides a new and direct
metod for th
in
a;l-terials.
pDara.lagnetic
field.
(1)
s
: nt~ situated
.lced in an r-f circu.it elei e,
The Theory of ilectric and ilag.--netic SusoTU Univers . t,, Pre ss, 1' 2
tre s
L T e i;
'Ver
n1 a1d
,
. ianet
rss,
(Ca;- bridge Un-1 Iversity Prsne..
3:
.a.P.r
d Strasbourg, Le,r.nt s'e
Uiversite
.
lec{e oo--pc=ration I'i
ntiona!
era t
netis-e (Institt
Tue
i,
Pris,
(4)
C .,Gortt
(5)
1)
ti.G..Penney
(6)
uantum transi-t.ions
J.IH.Van Vieck,
ceoDt ibilit ie
(2) T.B .C . Cas iir,
(3)
c'serving
ic
freql-ue-ncy or icrowave electronagn-et
a ra.do
The salt
o
Th e -leth,od ccons;ists
easure:ents the
by mreans of' electrsicel
ilsduced b
energy levels
ivestigation of closely spaced
(
1 940)
Plran-o-etic
ibon]_slevier,
iel:,
Amsterda-n,
SOC.
nrid G.J. T yrinch, Froc. Po,.
,
Cleo:k.
V.,r
see also JTi.ian VecL*,
(9
Z, l/o2)
10
P
Freed, Rev. i Io......>;.
, 2
9
U..R.
s
J. hys.
E. Zavoisyy,
so R L :orow
se e a-'
57
C(19-6);
1
1, 170,
Phrrs. Rev.
________ __ _,_
,_____..
A170, 112 (1939);
S.
,7 (1937);
5, 47 ( 19
a ,d
5);
D Ha l.idav,
70, 433 (1946).
..I_._.. .IP-^-----------
-.1.--
1-·-----111·11111li
As th -e stre-.th
between the pole pieces of an electro.;magnet.
owe:r, absorptio:.! in the salt
of the static field is varied the
found to -pass through a well-defined rlaxii.mm.
is
reorted by Zavoisky thee posi-
In ti-he rsurceents
tion ofl the resonance satisfics close'v the Larm:,or equation
for electron spins:
2/i
where
HI
= frequency;
T'
2. 80 mc/Oe
-
field; e Letic
lc
d.c.
ectron:c
charge in e.s.u.; i1 = electronic i.ass; c = veloci;ty of liht.
The aLicability of zq. (1) to
wSi 1ch te
free waEs pointed out by
is
sin
the eapi'erii:ent is
condition,
lctron states il
so lHds
renke 7.
in
Under this
to thie nuclear imagnetic
a.aloous
resona;ice ec-xer.lent
i
On te
se,,. ;ictt-re o-
-?re
are affected i
ics
energy levels of the paramagl-letic
inhlo.ogeneous crstall iire
iway by te
o.or ,anit
whi-;ch are caulsed largely b
sflittiig
This
f:,
the
tlhe
effect situs
energy levels of the free ioiis.
an irm-
ctric f ields,
.io lents
the dole
Thie crystalline Strrlr
hydration-.
:r.r:la.gC::etic solids the
.ater of
do-goerrate
of a s ingle
level into several co;0.:ionrents r-cl;es possible the occurrence
an r-f field of .cagnetic
coI:ponenits,
second ef-Lect
the
of the s...Di-ilttig is
eeilan efect il
_
-1
J.
C---·Y~·-·~·-·
Frenkel,
L
·-
P·Il
a stac.tic
to c'
.ne
i-lagnetic f'eld.
the clractr
A
of
-:agnetic
ield:
under certain
a stctic
t er:ect to fiZnd severa3 lines in the absorp-
conditions we r.-
(7)
di)ole traisitions between the sit
tlie absence o
i
eve
in
J.
-
Phys. U .S..R
L_/I__
__^·1_II·
9,
_1_··_ II_·I
2US (L9i5
)_ll____l_·rL_____.·II_
tion s ectrum,
in
con-ltrast to te
In Sect:ion II
sin-lc line found by Za.voL-
-er
of this
we discuss sono a. Lects
a.bsence of the
of 'iicrovave absortic-on spectra in te
effect (Ho=O).
I', Seciion
line Stark er:,ecvs a re co
i-
the
icrowave
"ioa1_s
refer-_ to sizLu..le crJsta.s,
tifields along-u crcystal
,
ra-;.
.o.L
,
-ive
iCl r-
staiC
., ..
for etor~owdered seci_,,ens of th;llese salts
s
.
Sitla-
of olIr clcu'la-
. l
wi.th
of hih
xe.
.C..J.
a-nid wou)tld p-roba'-,
rticular
.si-.ered.'r ;h.ree
of
.-
iltOSt
co.l)ined Zecl:.an an-d crystal-
III te
t- o is
7eeman
..
and r-f
'.'r,~,'-1
s itIa
t,
ion
)1
extrez_:
-_...i.rt
co.<ica-ed,
'si
- tt-,
1..
l
crys ba Is.
It
oss ib e thati; E,
is
betweern ions.
is perla-ls one of
c r, sona 1,)c e .pc;-i
deter;inod by oar' i:nc
t ii
fc cases of crvstai:ne
are prese-ltec. only as
calcuktLaio
c:11le
n
_
.e
I
ar
.. t; n
coU-lig,
ilof.straio ,. of thle
coru:.'on situat io-.
ap:'p-,11cab le
l,tejl
-
'
JVarious
_
·
_I
'lt
'Tct.tS
e s
...
5~-~ 2.
of cCage
ffcts
in a
ind of behavior
f
t
ll' O
Co
-)u-1d.i
.cr whrlich is
·IIIII^_
·· Y*l·Y·-Y--LI_-LIIIC--
our
strong ox1st
are
to a.p-ear
the Physical
leview.
_,
.
-·II^-Ql----·
ci-
in this --:aer
-i.
th--.o
i
ex- -g
cup.lig.
ex.chnnge interactions;
.22
..
considered ')7r J. HI. Vcl Valck,
in
i-)i'
he
slitti.c :siored
trctoCd in t1ro absoe-lcc o_:
to be e
ly Sstrorg
i-.;,ss
.
i n-,ortant
.toi'e
wt
-ase
revalence of s.cl
Thle
t-he
i i;
ntlber of salts
l
rill be
effect of the St-rk -ltti;
s
change coupli:
n
-··1_811
-
II.
4
-
-IICROU'
AVE ABSORPTIOi CAUSED BY
CRioTALLI. STARK SPLITTING
The usual
tcavrk. sDlittnLs in
crystals are of tie
order of sor.e hundred or thousand ci-;
in
stances, how.ever, the splittings are o
the order of one c
Tnis
is
the case tith somie o
cooling expori:elnts.
the sl
certain circlum
.
s sUu-diied inti
Gorter 4 gives the fol.low-
For exa:p;].e,
ing overall splittings of the ground state:
Gd2(SO
Gd2
(CO4)3
10 cm -1
0.0
1
.081.2O
1) TH
,0
CrK(SO, ) . 12 HOO
O.
c
Fo.iH
0.13
c1
4
(04)2
* 12 H20
Splitt!i.gs in th e ne'!ihborhood (.o
principle accessiblie
methods.
1
are in
to investir-ction 'by -means of :,iicro,!?ave
The selection rulehs in
nagnetic dole
one c 'l
transitions
enral
Electric
oermit a nul:ilcr of
diole
trai.sit ions are
forbidden by the parity- rule, siice t he parity of initial and
final states
re id:"ntici
the same dgeneratoe
i:hei
level o
tles
stat;s originate from
the free ion.
Electric quadru-
nole transitions are less DrobalAc tIan ;arinet.c di-Dole transitions
y a factor of the order of (atoi.i.c radius/wrave-
length) 2 x (Debye unit/Bonr i, ncton) 2 ,
of 10 - 12 for
= 1 cm'.
c
ilagnitudo of the
The
rhich is of the order
iffect
agnSitude and dtectab-ilitv of the resonance
absorption effect :lay be estima.loued by reasoning similar to
_
II
_·LII1__IPI___ILII__
LIL--
..LY-----
---
XIII_·Y*·LIIIIYyl-l------··-···l·
- 5Torrey, Purcell
that given b
:for the case of nu-
Pould
and
TVe shall consider thice
clear resonance.
-owerr absorIbed by a
syste-m wihich has ;two eignstates 1 and 2 wit-
X0
ferinlg by the cnergy
elgenvalues dif-
The trnsibion probab
between
lit-iy
states i and 2 for a silngle sste'-i per unit ti:.le
..
r x-polar-
izod radiation is
tx2 is
where
U-.12 2
(2r-/' 2 )
p
(2)
the :lmatrix clemeiint of the
-anletic
lo-.ilet OJC-r-
the total oerg;v
ra-
i1n the i.c.d..t
tor p=
g BJx and I is
diation
ield per unit volume per unit frequoinccy interval.
In th
ground state is
the
uilmber of srstems in
eqilibiriu-l the
ri.lcl
uimber of syste-mls in tlie ex-
greater than th-ie
cited state; the excess is
Ij -1 '22)
(-T
for
lle tota
is
{o0'/kT << 1; h.:'re
kT(
IT
)
Yl(I
e
ncr
o
The
' s',,steis.
-,given by
porer a'Osorption is
2"'T (yjo) I
h f t'i
Su:?-oseO tiChat the badw.;wcTridti
()
p
i,n
olectrom-lagnettic
cdnt
to m.-;Le-
(e
widtl
of 'lhe
resc.na.lce lim-
tic nom.icnt interactions)
is Am
(>> Af); then only the fraction
wave is
Af a-n;
2tr Af/zlo
witi-ii
t
te
toof +thetoal
baididith . o
-un--ber
-lthe
of sstemi;-ls
r-f
be tl.e total energy density in
raiation
ave their resonances
feld.
Lo t U = If
the icident radiation fiCl.d;
then(8)
H.C.Torrey,
.;ii.Purcell and R.V.Found, Phys. Rev.
9,
68 0
(1946)
·rY---------·----^-
·-- ·
IIYIIC-·P--C-·--rrrrs-
·Il·rr--rrY1_.·u------·-^r^---rr_
-^------u-·--··-··-r-rr--r^-suuurx
--
--··lll··l-·m·-·
--
-----
-62 T2
l2 12 n
2
p _
5n
U
1=12
kT Ao
is
the integrated power absorption per unit volume,
is
the number of para-iagnetic io:ls ner unit voluame.
where n
Nlow the QI of the systclhl is given by
o
2
if
q.
From
200, w-rhicl
is
casil
-Awo
300=
or T
one hs,
(6)
= 1 c- ,
ald
, -:s;cc
te
detectable
rhich '.'iay be of theC
cavity Q's
n2
2/a3
so that
interactions,
-Q
considered as caused by magnetic dipole
the line width is
oe
C:pt
Oi
tis
lre o.t
orcder of 1000 or
wavelen-gth.
Selection Rules
'Te
out .1ailb- the allowrd transitions are
have C.oited
iasj;na tic dipole tra':-sitions.
scelection ru]3.es
groum,
o'r -tioret-ical
itllods.
ti nos ccrrespond'..n.cl: to the enicrgy levels i
electric fields transforisentations
accordi.ng -to
The ch.r.ci;ters of te
,h
in the paers of Ct
The1
caigenfulnc-
the cr-rstalline
the irreduciblo1nrere-
of the symmetry ,g:roup of thi
Pi
Tisza
10
field.
crst11
irreducible rcrcsentatlons -re
'ol
the
One c n proceed to calculate
and Jahn.
(given
I0
The character
of thCe magnetic ;mont operator under a rot;ati-on t-:.rogh an
angle cp is given by y;(y)
= 1 + 2 cos (,
siicethe
Arn. der hysiil: [53 3, 133 (29)
(9) i.A.Beoth,
2, + (1933)
ysik
(10) L, Tisza, Zoits. f.
, 117 ('93)
(11) H. Jahn, Proc. Roy. Soc. A
-
agnetic
- 7-
mor-ent trarnsfor-.Ims as an axial vector.
It
is
a
unda;eontal resil't of grou-, t'heor
tat
transitions are allowed betwreen levels m and n only if
x
rn
here ra
a contaLi s '6ie identical representation rl;
x
Pr
The allowed tran-
is the represe-tatioun of an axial vector.
sitions found frol t-he character calculations are gi.ven bclow;
the represcntations are lab.elled according to te
Bcthei
.
notation of
Tn t 1he cases of the one and two-valued rhn-.bi'ic arnd
character tbles
are £ivCi
the
iven in detail br Btle,
trigonal groums, which wo-re inot
in Ap-ni- A.
indicatcs tat lt
Th e nolaticn.
th-e enorr-v
-
vel hose eigenfunctions belong to Pc has allowcd rianetic dipole transitions to and from the en-crgoy llevels whoso cigenfunctions belong to P
Cubic Gr"oup
4
1,3,4
and P b
-
a .al,)i
--
2,3,,;
TrOa-,A( = rP + '5
Tetragono¢_ Group ("axia
Transitions for J:
Transitions
or JJ
Transitions for JX Jy
1_
I----II
3(z
-P
-,
1-+
Transitions
Transitions for
for J:
Jz: F
C _ __I_
=
r
,22;
)
,33
+ r,
r
r,
¥)
;Pi'5
s;'+ P66,
2,3
r7;
6; r 7-
P3;;
P;6;
4-+
Pf- P 33 ; 3- '3; P4- ~ 5
xyl~~~~~~~~5
IIY·-LII-·_1---CI--L·II(IIILY·_L_
7
6;
"7
r7 r77"-
F50
r5-
+
T- IP·-
r5'
F
I
_I_
-8-
Rhombic Group (raxial - r2 +
Transitions for
Transitions for r3: r
,:P 1 4-
Transitions for
P(x
34-
4; P5-
r3; r 2--
F 4; r5
P4;
£3;
f -L F2;
2:
+ r
3
)
+( (Z)
'5
I5
15 -
P.
r
The selection rules for the tctragonal case have
been given
reviously by Bthe1 2 .
rules nay be il-
The application of tlhorse scction
lustrated by the case of the Gd ++ ion, for whic
J =-7/2.
According to Bethe 9 the ground state of thec ion slits
cubic field into a two-fold degenerate
a two-fold degenerate
evel
in a
levcl belonging to
6
;
belonging to r7; and a forr-old
degenerate level. belonging to r.
transitions between P 6 and Fr,
The selection rules Deriit
and between
7 and rP,
but not
between P 6 and rg.
If the overall splitting of the ground state in
Gd2 (S04 )3
. 8H20 is 1.0 cm
1,
as given by Gorter4 , the allowed
transitions will correspond to rwavelengths of 1.6 cmi and. 2.7
cm.
This follows froml the fact that,
if
only the cubic term
of the fourth degree in t he crystalline potential is
ered1 3 , the splitting is
consid-
in thec ratio of 5 to 3, with the four-
fold level between the two two-fold levels.
Intensity Factors
Eq.
(5)
is
a quite general expression for the pover
absorption when the bandwidth Af
of the incident eleoctromagne-
(12) H. Bethe, Zeits, f. Physik 60, 21 (1930)
Purell,
J. C er. Pahys. 5,
(13) I-I. il. Hebb and. I.
"~ -"
.-"~I^IUI"IPsl"L-IP--·-···IIU·II·
·-·-·-----I
I---"p
-'
338 (1937)
~I '----"I--'
-9
small in coi.marison with the effective bandwidth
tic wave is
.00In the absence of
0
o of the resonance, and for kT >>
detailed knowiedge regarding the deelndence of Awo on the cryand applied magnetic fields, we shall su-,-pose for the
stalline
a constant for any secific
A0 is
sake of discussion tat
Under this ass-,lmtion t he de-endence of te power absorp-
salt..
:a-gntic field intensity is determined by
tion on frequncy and
co 0 2 1
or
tho fact
2
2
in
(5).
E.
7refer to discuss the di;:ension-
For convenience we
less quantity:
This
ratr;
o
lar lmoi:olriltUl
actor.
sitsit
be called the inte
ill
(7)
In)/12
)2 I/ (J
(
0i
Ter c J is
lust be defined
-is an energy whic!h
closcly rel.at;ed to the pure-
is
for each s ccific '--roLle. - it
thec angu.-
1-l cr-ystalline slitt-ing.
operators J
=
se CD7oa-tIly.
Jx
Jx
orc convceni.nt;
is
In gencral it
= J.
J
iJy
y anc
+
..
iJ,,
-
to Tor' with the
J . or J y
thlan with
.
;e ha s
°xx
~
2
('
(
-t
J-
J_
;
now from the reality of Jx and Jy,
(I J -in)--
(9)
(n J-m)
so that
(mirQxn)
=
((ra.>Jin) + (n+J |r.i)
.trix
This rclation is used to calculate t;lJx fro
tose
)
'
(10)
elenlts
of
of J.
Tables of nunmerical valucs of the int'liS ty factor
are given in the following sections along pith the
__
______L-·-·.sl---·
I
^·-LYI"-IUW.1"··-L-*Y^I·
--
.
cnergy
- 10 -
differcnces associated with the various transitions.
III.
It
CCiBI:LD CSTA.LL T
is
ANID ZEiIA1.T
SPLITi],4G
cncral 'fas.sion
ossible to discuss in a
not
the belvior of energy
lcvels in
of co:lbi.ned
-brsencc
thc
crystalline electric fields anld external static
fields.
-.cnetic
is necessary to find in detail for each
Rather, it
quantun
individual case the solution to te
mchanlicl
er-
turbat ion proble-m.
Level sJli -ttinus corrcsponding to
quencies are believed to arise only in
uations
(a)
rc-
scle-lct s!icial sit-
includiiLg:
When te
ground sta--e of the frce ion is
fiel.d causes slit-ting only i
the crystalline
uwith sin-orbit cou-lin
(b)
icroave
T
mlove 'the degeleracy
highlr
hishr sylnmtc'cry.
symm,1etry.
ilay have a sall
conj.unction
t
s
small.
crystalline field of loJ srym te-try :.a -y re-
ll
A s
Te rsultnt
an S state,
eft
argser crystallinc fcl!d of
by a
Ft'or
For ex~lo,;,
trional
aaxa;
D)redoinantly
, cl;u
_
.i vcs rise to a
..
whi
con
c field
....
furt-cer splitting.
Hebb and Purcell13
magnetic coolin
tin
is
e xperiments discuss exa:'.?hles we;re t:he slit~manitude o o
of the order of
split;ting is
of
in their theortical stud
found,
for oxamcel,
o;nc c
-,
Tyoc (a)
in salts of trival-l't
ado-
++
linium, whorie tle
(
5/2 ) .
(14)
14
J.
;,round level is
Type ( b ) sitt
in
is
3/7;
found
H. Van Vlccik and "T. G. Penney,
--------·-C-·-·-i--w·--rr-rP---;I^·
-·-c-·--··r-r---i·*P*I-··IWII-L-II
and tih
and in Fe
++
or exale
in
potassium
Phil. HIag.
17, 9'61 (1934)
-
chromic alum (KCr(SO)
2
1
-
. 12 H 2 0),
whore there is thourrht to
be a small tri-oonal field superposed on the predoviinant cubic
The possible transitions and corrusnonding
field.
intensity
factors for the cases just cited arc discussed in detail below.
J = 7/2 (Gd+ + + )
Case I.
in th-e absence o
is
external -!magnctic
split by a cubic electric field into two two-fold levcls and
one four-fold level.
According to Bethe (cf reference 9,
155) the two-fold lvecls belong to r6
level to Pr.
in
fields this state
tIe
p.
and I'7, the four-fold
assulre that the splitting1 5
tabes place as
, which is consistent with therc orik of Hobb and
Figire
Purcell1 3
The zero order
rwcave functions belonging to these
reprosentations are listed below.
(15)
__
However, the situat-iion in the Gd + + salts ay not be so
he re indebted to Profssor P . Bclinfa-tec for
siL.ule.
eli of tie energy
corresondencc regardilng a detailed trat
Van Dyk and hi'r'sel.
level schceme in -)re2)aration b','
I
I
I
_I_____lll___l_·LI_.La-LIIULII-
I..^ylllll^^--^·.-L1-·ll
____ I___
_I
- 12
*a
(1)
_1
2
o5/2
)
)
)
) r7
)
- o.3/ 2]
l1
"a (1)
(i) =_r1
K-5 / 2 - 03/23
2[32
1
37/ 2
fib
- (5/3 52
1
(2)
= (7/12)
Vb
(3)
C c;7/2 - (5/3
-7/2
T1
~. [-t.5
YVb
;-.
+ 3 21
2
+ 3
C ) /"
)
1
1
- 7/
2
+ (7/3 5 )
'/2]
)
)
1
(2)
( 5/12 )
ena denotes
q7/2
+ (735
1/2]
a function with J = 7/2, iIj
the secular equcation in the ?rescncc
-^.··C--IIl
=I.
,JWlobtain for
of an (001) directed mag-
netic field,
__ 1_1 ------YI---C
ICIIIII·_Y-I
(11)
I-------·
- 13 -
(1)
%I (1)
* (3
_C_
I (1)
)
a
(
a
V3
)
0
=0
-E
*d?(1 )
T 35 a
0Q
-E
U1
0
1
0
-
(12)
V3a
3 a
-
- E
3
0
a (1)
~a
_3a
"
_.
L
i: -_-~I
3^ a -
- '
Here a = gBH(g=2 since L = 0), and there is a sii ilar block
involving (-a)
for (a) and the reiining functions.
sult has been
sscntially
conne ction.
other
ii = E/
,x= a/
iven by Hebb ad
Introducing diCcsionss
one finds for th-l
Purcell in anquantities
cigcnvalues -
____ _·-LI-lll-_l--_I
1(1·III1
·--CII^·IIY
·-
-1
I1I ^--·-LIYI-L-P-_--PI^·-·Yllll^llllli··-
This re-
CI
-
Z
i-13
97/2
7/2
x
/22
r1
3
/
2
+ 1
-2x2
+2
+
1
~X+
2
2
3
3
1-3/2 -2
'
T
"-
12 x
2x
5
-+
12
(5
3
13
l/3
rr,-7/277
2
+
12 x 2
+
,
T
2 ~/
2
2
221
,
+
2
1
+
14O2
tT
3 x)
2
x +
2
9
+ 12x2
2
x-x+ --1
1
-5/2
13
,
lj22 - - -
140
3 x);9
(3
tT.2
22
14
2x)2
x
X
)13
+ 12 x
2
+ 140
2
2
-9
lts being labeled according to the levels they r.rescnt
the
These are plotted in Figure 2.
in very large fields.
The
numerical values of the frouoncies anrd intensity factors
m-.trix
"
corrsspondin- to nol-v.anisin
found'
in Table I,
while
Figurc
leonts are to be
3 sives plots of inten-si-ty fac:lagntic
nd frequencies of typical lines vs, aied
tors
field.
The sum of the squares o
of J
x.
is
all the matrix
elements
exactly 84 for J = 7/2, indDcpcndent of teo vaclu
This result followls froml
t
stability (cf. refcrence 1, p.
of
priaciple of sctroscopic
139), and provides a very useful
erxprcesThe authors have available copies of the aaiytic
sions for tic wave functions and imtrix: clelicnts as fmrnctions of x for distribution to anlon,
---I
-IIIL--ll_·_Il_^·llllll^·--Y··-·)··111----·-·
111 -rm. -·-Il-·.X·--
o
;cay rcculi.r
them.
---
-
- 15-
chieck on the nunmricai calculations.
For strong
rix CeCijenCts
,a;:ntic
aplroac
From Table I we
fiCilds the
igcnvalcus and *mat-
i,..values charactristic
gOst values of the intositJr
-a8ctors bClOn
ized transitions
-±
usual selectio '
7/2
halch_I
t
r- -i-)o]ar-
tso
1, in agreeCent with the
n oxa;-l le of such a transo .c
.
5/2, for whiich the values of the freq.uency
sity factor are slown in Figure 3.
weak for all
: an exanple is
independenlt of x; an
Figure 3.
3/2i--*
oints o
(Figure 2) corresord to
lines
Case II
J = 5j2 (Fe
+++
All o
of the
intl;sity factor,
te
The cse of J = 5/2 is
the
1/2 sho1m
lectric fild
and also by Kronizg and Bouwikarp
in
ienvaluo diagram
)
vr-
si.ila,
to t'-hat of J
The roots of the secular equation for an (001)
field and a c..bic
z-
-ero frequency.
of
l ++
and inten-
-7/2.
xanlle is the line 7/2--
The crossing
is
A number of lines are
pola rized lines have conistant values
7/2 .
free ions.
that -for liare x the lar-
see, for e:a:1nl.,
or
o
havc
been
ive
b
ianetic
Db.r
15
.
iJe shall not discuss this state in any detail, but
morcly give rsults.
(15)
(i)
The energy levels
are
P. Dbye, Ann. dor Physik C[5 32,85 (93J)
R. do L. Kronig and C. J. Bouwar -p, PhysF;c
___lll__l__l^_lg__lIII/YII-·U·-XWY-LI-
·.I___.I_IUIIIIII1--^-
....
,
·.C^IIIX-LI I-1I_----·1II*I_1-
290 (1939)
-1 IP_-·II-
I
I
16
-
-
E1/2 = 1 + ½ x
-
£_1/2
E5
/
x1
1
2+ + (3/4)2
+x 2
_
2
_~~~~~~~~~~~
(14.)
+a /2
2--
6
c3/2
3/2
-
-2
+
-x
(3/)2 2
+(3/1)2
2* [x
+
2
x2 _- +
(3/4)2
2
where as before x = gi BH/c,= E/o (38 beilg the overall
splitting in the crystalline electric field alone).
case under coisideraion,
just as in
In the
the 7/2 case, L = 0 so
In Figure 4 one finds these energies -- ottod as
that g = 2.
a function of extcrnal field.
Table II
gives the numerical
values of the frequencies and intensity factors.
ri ci-ple of sec-tIroscopic stability apliod to
The
the J = 5/2 case tolls us that th
the natri;
elemeints of J+ is
Case III (KCr (SO 4 ) 2
s
35, independcent of x.
12 H 2 0)
According to Hebb and PFurcell the Cr +
salt is
stal lin
in
a 4
F3 / 2
field.
If
ion in tis
state, wich is not split by the cubic cry-
llnt a
one assumes that thlCre is also nrese
s;lal.l field of trigoc.!al sy,rlnlitry the axis of which
coinc-ides
s of the c)ubic lattice
with onile of the body &iaeoni.
to be
xpected fron I.-ra. studies of 'the
3/2 state does split and give
crowave absorntion.
_1-m-I
·IIIC··-----·--..l·(Yen·
all
of the suares of
i:ow'r
in
rlse to
sle t)
t hen--,
the J
or
ossibilitis
.h.s alum the Cr++
· · __I-IIYI·CII*-·---1111-
(th.is is
mi-
ions lie on
i "~^~~I'~-"I-~x'-I~
I-·lilllllU--·l·
--^-(·1111_--·1-8·L
-
-17
Tf one rsolves t, :a-cc-e
z1e cubic lattices.
vicc into
on iiontrenio'at:ing sirm-
i.e.,
a face centercd cu:tbic 3lattice,
lat-
the four sii!ple cu ic lattices thei cc finds tat
the trigonal
xis cf' th. electric ficld is in ceach c aso along
he1
dig.onal of tc unit cbeo.
a different body
tlcso four Si: 1 lZe Cibji
SYvstee il.CC:,-wldC.;
c
for' th
Thki''
..
t'rixol
1
.
notcnt ic] only the soco 'ci
rstalline
orcer dor1mer
~s we!r'
j .1
V,
whore te
xi-s 5is he.co t a
z
tont ial iias
iorc
as a result o
t'she (001)
- '
our dr
...s
nr'
c ry.
gr
,
h ;:ir,
ho'>
rcticon arnacn
trf''oj-eal'
lc
ra
t
f iitccoiponni,_
arrays,
OX
placed
f" -l
c
lc .o
;
is,
. ro.tios
o.-;.o
i.
. d shou.d coe
1 t;-T
f:"icI
-
di-
ni'
ct.t,
..sTidoI
,I e .;n,,o>.
V,:o
onlly
t;he 0i.
;.c 'it1- triC-'onacl ac:i
in
directio n..
the (.11)
-to solve te-ia -;robl
ow .ve
uie
3/2 -erbturbe d
r
;'=
trig
F3
to be, alo n
_ ___IIIIII__^__/IILIslP·II)··--_llll-
of
.n ..t o,
with J
. iitrcction
H
For a
s;,y:'
syi:l- try
3.Ce:1s,
,:)"i-.tic
ri
nd thl-e
axis.
Jlst
'
-T
Tlls
..
ax.De')
it h
er \S.
o3f te states.. as fu ct. Gios of t)
th ;ield
(15)
l ( . dhi"s-1o
i ' . . d'irc.
1... ti.
cauic
the st--rcngth o
- 2z'"
h
h.n:
s-Tm:l.CIr7
each Coi r:onrtt si:e-.r
onel
2
-
cian tri.goaal (i
this apa-ront ail
in
2
= D (
+
- HE-*
+
X(i
'Takiwi
state g
2/5.
t
dir ctior,
(l."
.
J)
(16)
the d:iroction of
ua.-ti%.zation
and choosing t C x and y axes
.(l--·ILXY-··_II21YL-l
III-CPII^
-^-_I1_I1·1----
1
I
I
---·-
I
------
-1--11
i
-
such that the comupontent of H
to th'.- z axis lis
r-lcricr
equally between thci.l we have
=
H . J
Hmg
(Jx + J
C (1 (l-i) J
iaag
where
+ J )
+
+
or
(1+i)
2
(17)
Jz )
J_i
( -,
z
~
1
a.
The level schemne in the absence of Hmag is
levels are lcboeled br thci
Here
-,hi
5.
in Fiur
wirakz fielc
Ct as Cor:'
,
ing now TH tio we
nruag
Irntroduc
quantum numbers.
ssown
secu-lar
equation:
3X/2 _1_1L/2
:_
31?
1
-+
3/2
I- 2
A."
i4
IL i
X-s
1
l+i 3 2
x
x
1/2
-
3
_
I-
--
Exp-anding
I
--
---
= 0,
_~.r
/
-
1+i
0
L-'L.C -x:-.
,
--
Y
:7
--
3
-
I-L
1
-
X
#-
2
llI
IO (I-i)
_ I'_
_
_
_
1
2-
x
-
3
x
-
3
(i9)
l-
..
gives
l _ 1 (
2
+ 15x I2 )5,,-
thle roots of whiclich
__III_-YIYI-··---Q---·ll..
0
x
1
X
-
_ E/5
where
(i-i)
I
(i+i)
-
- 3/2
0
2 "
0
- 3/2
/2
C
1_
0
1/2
-
I
12
___
.-CII--ICII--·l-CLII-Y
C2
T6
(1
+
("lo~-
-
2
) = 00_ (20)
(20)
ro
1-1II_ - l_·I-_I-IY*-U·----·--YI.IC---
11II
111-11
- 19
(-2 2
-
c(-L
(1
whre
sity rules it
It
E(i1)
should be r
1fild alon1
nu-bers.
ing
1mu\-ric
he
1then
Por an r-l
,roill.
p:.
for example we wll _. not h ave onlw
C.
s a,e-,
inti li
f
-f
the
field.
irected ai o-g ti':o (001)
ficL.i is
r-f
+ Jy + JZ
of J
lents
the ;.m.rix cc
to
The wave fu.rctlions and tkcn correspond-
i vJ
tile r',t
Jz
t'.
' J+ relative
re ;'ust. ko-.O the directi-Lon of
lines
for exa pl,
_or
are Ji Z
(
of J+,
re to be frced in Apc-Lx
lec:t-ts
difforont
axis
1u c tions
trlx "i l-;nts
thcls
i ntr-
order to cilculatc thC relative institiCs
In
If,
vr v c
usJ d i;' t-
axi.s
clcier-jns -prscet.
JZ
t
oleibu;ud thai;
a-ro-riate
-n ~
To find thle slcctin
also the
ad
nbi-:c (00)
1/2
en-rgies with their
of ql%.1;"tizabati
the sstc.,
-2
+
(21)
)
)
)
)I
x ) 1 /2
+
necessary to find th
is
belonging to
- 6x /
+h'n15x 2 + 6x ,1
Ae '.a:velaojc.d tc
strong field quat-ul
x2 )!/2
x -/1+
(1 + 15x
- -2)
WflC1' wehvcbl'
1
x2 +
(1 +
=
c(3)
=
T.
,l
., 1 C. .l c t Ln L
z
iv"
clcu-
A sil
shJow.rs
I(MiTr11.,
2
1(
+. I
+
Rc
JT+
1t )12 +
(i:J+ II ) (Z[,
t
(E
jM' I+lj 1)2 + 1(M/j1M1)~ .
JIj+(()
) ((-:J
[
)
+
I Ii,-).1IJZ
(1t
+
(:
J
1
_) ( ,Jz
Id,))
(22)
+Im
(I!t
_
.l;-')(Z'jJzl
____·_II LLLII·IOTII-I-I..I.11-^·1111-
' )';"
----··
LII)I·.(IIYI-1111--
|
__I
) ((IlT Jz II1' ))
-I·I.I--P·I-UtUY
Y-·
-
I-
- 20 -
actors for tls
All of the intensity
This is
s ia ll,
fu;nction of x is
scon
Jfrom'J
the usual sili.-f
spctru-. of
d
Co
.
CL-S.0
which is
expected to b
s
ph
. again t.1. th
Tic ions
oaraa~lnt.
lilgr between
considord,
ay well
that at .
of t is
Jct
were m ade on m.
not
salts which
thte
r
'
i;m.:ortancc
i
of
llriday,
lof
CII--·-·-C
----
t-
of
it.
ill:r
sing sici.r2;
One sou
..also
ith the static
crt-stls
iigh syi.v-lctry.
H. Van Vleck, C. J.
Tisza for sv..vra'
a,-c.
nd
discussioc~ns-,
assistance.
John Si-moil Gugglhin;
the1l
o' whlch
We wish to thank Professors J.
sCOmp~Utaj eonrll
ing
:lost
::-ost likely to bc realized
I- 7
s:
ald r-fn m-ag.letic ficlds a.lolg
cr--s'.'
a.ws of
AClKi[
O(iLEDGi . iTTS
and L.
te;zure of kwh-icl
I
One of us (C .... )
eo;.cl-l
tiS
.--I·---*-·-)·111·_1·11----1-
it
h- - mea sure-
Th
salts
wc have
Ind eed,
sort occ urs in
cgnetIic:_
llr-colocenratcl
situation which we hav,.- treated is
in
rspctrum.
uln-row ard
ad
t
of Cavoi.sY
Ilents
the ieffc-ts
g.e'.:
si
anud n.a.-.y resitllt in a .Ll,-r
e cou.-
C.tir:uily negl.cted.
h-s .,en
This coupling iin :'cnry salts -lay su
ie-ld.
L exh.e
LCf
ii.lunC
i
a ratirhor
_.ct - c
com.lic.tod function of the cxt;rnal- static
T1e wish to :
based on
the :icrowave absrp'tion
tn-a
crystals is
ara;nagnetic
as a
6.
tle abovc treat-'Icnt,
odel,
field
le.
A plot of the oi_.
given in Fi-;ur
IV
is
-the r.f.
?prtly the consequ.ence of talin
parallel to the static field.
It
are very
cas;
Fo-end.ti'c:.:
t
is
Gortcr
Kock for
indcidbt cd to the
n for a Fcellowshio dur-
rork was carried out,
-·---
-1*·14_--^-------_1_1·I---I
11
1___.__
A-oiend i
A
Character Tablc - ------Trigonal Double------ Group
(after Jahn - )
-r
----------
rl
E
R
2C!I
2CI7
3C2
3C?
1
1
1
1
1
1
1
1
1
-1
2
-i
r3
2
2
-1
-L
0
0
r4
2
-2
-2
2
0
0
I'5
2
-2
1
0
0O
NUote
-
Pr
is
-1
O
O
used to denote two conjugte-co;miplcx one-diiien-
sional reeT'c sent-at ions.
.Cl.for the trigonali group is
The splitting sc
j =O
.1l
2
'5
2 + P3
1
3
r 4+
2
2
r5
'1 + 2113
2
2
P44 + 255
3
+
r1
1 + 2'2
2 +2 233
_ ;~~~~~uxw
~
as follows -
~
c~· -·--·--1.·l---x,*lr~r,~·*u · ln.--·------~---
~~~~~~~
~~____.~~~
~~~~~___~~~~~~___~~~
---------
Appendix A - 2
Character
Table rC-·-·IY-I--I-----·
- Rhombicc Double .Grou
--
(aXtrr Bb
BctcB )
__
1
T
EJ
R
2C1
1
1
i
1
1
-1
1
1
-1
r
3
r4
r5
5
2
1
.1
-2
0
2C
2C3
1
1
-1
-!
-1
-1
.1
-1
0
.....
w
I.
II~I
~
-·
-_1^_·
1·~-
Center of Gravity SumI Rule for Crystalline Fields
Consider an unperturbed syste>.it wlhic-h has a definite
J.
angular momentum-.
It
undergoes a perturbation due to an ex-
ternal non-singular (electric) field satisfying Laplece.s equation.
gravity of te
slow thiat center o
e shall
2J+41 comlno-
nents of the originally degenerate energy level is not shifted,
to the
irst order.
erturbed system;
Let the cigenfuncti.ons of the
J+l).
(i=l;.,
be 4ri
levels
'Then the center of gravity of te
will be given by
g
i
S V
VW 2J:-' = -Z
where V is the
.etl..io1
repre
(I-- -J,...,
use the eigenfunctions
is
unri a-nd
i._vari'alt
Te could therefore
der choice of re;,resencat.;tion.
system.
HIcwever t his is simply
tial.
-erturring poten
the trace of V in te
dt
+J) o
just as well
te unperturbed
That is,
." 2Jv+~
dr
V t:i
"
2J-1 J' d V (Z T
't1T form, tne
is rotationally invariant since te
Z i"ii
M4
irreducible representabasis vectors of a (2J+1)-dieisionai
iNow
tion of the rotation group, and in this
the sum indicated is
tion tht
reprcsentation sace
just a scalar product.
V sat isfies La-lace's euat ion (which is
invariant) we have that V
variant solutions of Laplace's ecuttion
;.-*NlrnB^.Ws*IY·;·Y
-.--·I-·
l·-.·L-r·III*-sU--LYIII^
rotationally
ulst tracnsform as one couplonent of
some representation of the rotation group.
shifts all
ioWr by the assunlp-
Since te
ar. a constant
onlr in(ich
levels equally and just aounts to a ciialee of the
_--1BC.I*-
IM1IP
IIIIIWW
-i-iY1I-(.
Appendix B - 2
oncrgy zero
oint) and
/r (whlich is
s igulr)
.e cC;: l sa
the represc-lnttion to which V belongs does not conCi.i:'
dent ity representct
ion.
tiieorotical argu.ont'
Thercfor
according to
the itegral
quired result has been
inl i vanis'CS
stablished.
theore- h. s bly
bioon g ivan prvio.s
A
the
that
the
i-
. ust-l rOUp
.:nd
te
re-
ercaker for-, of this
hisy KyIlch
Cf for exa.-DI.e, H.
yri,
J T..
tr
a'ad
Kibal , r.a.tu.
Chemisti2y (Joiln ilae,
1ieo York,
-rI. 1 4.!.)
l .
G. J.
ynch, Trans. Farladay Soc. 33, 1402 (1237).
1
_ _
I_
Ip _I_ 11 ·I1I1I___1III
l--------Y·lll
PII-_I^-III-.·L-__X1_---·--_11
_
_
_
Appendix C
This appendix contains the wave functioi-sand matrix
element for K Cr Alum in an
wave fctions
(001) static -cagnti
The
icld.
arc given by
*
b
(I ) "3/2
w
+
where
17
s
f
1
- 1/2
CI1
+
-3/2
--
[·
1at,1, ++ I~l~rT
Ibt 12 +- I cl." I
+
2
I IIl
eI)
,
(1 + i)
I
x
1
1
2 /2)
- i(3
-
-('
+ 7 x
( -
(l+i)
-
I)
aII-
2x
1
(1 i) (33b2 /2)
-- e(II))
hCmar
elee2
The
atrixcle3nts o
2,
2
x
J
2 (a,
2
and J
..
are now
b
*
7 cl
t CII
~~~ ~-_
3
- r' bI.l '
a,14;
I
T
I I1
_.-12
`~-~~
(
t
I i
,
I 1"'i, )
L(1+1a
1
2
+
1b
.,
2 +
I
Iai,
14I I
c 111I 2 )I (+
1
3 2 a1
-1:) I(1+|9.1,|2
L
_
I
__
...
,2
C·---·l-·llll-·l-;P·l-r--·--·l
+
+ 2al '
----c '
I .1 T I
,i
bI, + 32
II'
hA1
+
2,
+
2I
1
+,
(1,- I Ii +
+
tcI. 2)(T
PY-9LY·IIIX---
+
I,
'--c- ^L-··-·-·L--··---LII-I
-·L-
11
t2+b, 12+1 C 12)
-------.^C--L-_·--·i-_··--L-·I-
2
r6
-t
36
.
56
if
r7
Figure 1. Energy levels of 8S state after splitting by electrostatic potential of
form V = D(x4 +
4
+ z4); the over-
all separation is 8&
·-
- ---·
--
·--- ·---·---
----- ·------·--------·-------------·-
19
20
16
12
8
4
o
4
I-
Z
0
I)-
0 -4
Z
-8
-12
-16
-r,'
0
14
12
10
8
6
4
2
MAGNETIC FIELD STRENGTH IN UNITS OF 6 /gpB
Figure 2.
___I_·I
ICI__^^__IIIIQ__I_
X --
Eigenvalues of 8S state as a function of the strength of
the static magnetic field in the (001) direction; a constant cubic electrostatic field is also present. The
subscripts on the tts indicate the value of the magnetic
quantum number M characterizing the statefor x )) 1.
--I·LV^
*I*·I^···II··I·IIIPI-··--_-·slP··I
An
e
2
100
QUO
90
1
-C
80
La6
0
70
Z
z
z
60 °
W 1
Z
U
J
I.-
z
0
50
O
z
a0co
40 Z
a.
C)
Z
U
30
LLJ
D
Cr
Li~
20
10
0
2
4
6
85
1
12
MAGNETIC FIELD STRENGTH IN UNITS OF 6/gP9B
Figure 3.
_
0
14 X
Frequency and intensity factor for ome typical
absorption lines resulting from an S, state; the
intensity factor is proportional to the power
absorption and is defined by Equation (7).
11_^1__1_1______1__·.--I·*U·IY·II---
I__IIIX---·
I
I
-
20
16
-12
8
co
L0 4
z
Do
0 -4
wL-4
Z
-8
-12
-16
-20
0
14
12
10
8
6
4
2
6/9PB
OF
UNITS
IN
MAGNETIC FIELD STRENGTH
Figure 4.
3EigenvaluBs of S state as a function of the
static magnetic field in the (001) direction.
_ I_
I
_
_
__III__IIYI___-
X--
__
3
2
3
2
5
Figure 5.
Energy levels for
1
1
2'
2
Cr(S0 4 )2 . 12 H
presence of axial electrostatic fiedi.
_
-- ---
F___
III·L-·--^-LIIII··^111-
in
- -·---
I
E
u
16
12
8
i0
0
L)
-4
z
z
-
0
Il)
O
w
-4
z
i
-8
-12
-16
-'>n
O
2
4
6
8
10
12
14
X-O
MAGNETIC FIELD STRENGTH IN UNITS OF CpB/6V3
Firure 6.
I
_
III
12 R2 0 as a function of the strength
igenvalues for K Or(S0 4 ) 2
of the static magnetic field in the (001) direction.
_
I__
11___
1_II
-C-L----WI·-O····^IP
-
I
-
_-
-
TABLE I
Frequencies and Intensity Factors for Transitions in J - 7/2 Case: Cubic Field
Table is labelled in terms of dimensionless quantities x - E b and x - gH/
The intensity factor is defined by Eq. (7) and is given in this case by
I(mJzn)12
()
Transitions
2
2
,whereo is as shown in Figure 1.
- ~ I(mIJ+In)12 ( )2 and Gz =
. The static magnetic field is in the Z direction; this is ctso t-e (00%) dredon.
Quantity
1
x
I
....
.5
.7
1.0
1.5
.23
.07
.71
.59
1.22
1.35
1.76
2.08
2.60
2.93
4.05
3.74
3.33
8.10
4.13
6.57
5.09
5.04
6.19
3.78
7.98 11.19 14.53
2.47 1.34
.82
*
.
.3
.I
2.0
5.53
4.15
7.5
10.0
8.51
4.51
14.51
4.73
22.00
4.82
29.50
4.86
44.50
4.88
21.36
.39
35.22
.14
52.65
.09
70.11
.04
105.07
.02
5.22
7.65
107.71 232.68
10.11
407.68
15 .07
907.61
3.0
5.0
I
I
TI
2
0
1
1
da
2.73
9.27
2.33
9.97
2.09
10.49
1.99
11.14
1.98
12.74
2.19
17.28
2.53
24.04
3.36
43.85
0=
.37
.09
1.09
.73
1.78
1.63
2.44
2.46
3.40
3.35
4.95
4.14
6.47
4.49
9.49
4.76
15.49
4.89
23.00
4.93
30.50
4.93
45.50
4.92
5.00
5.84
48.86 63.20
6.53
77.32
7.72
105.98
9.85
170.79
12.40
270.04
14.93
390.78
19.96
697.32
.41
.04
1.53
.36
4.16
1.04
9.88
1.77
17.24
2.15
24.68
2.33
39.62
2.50
11.41
2.07
13.53
1.28
18.16
.55
27.88
.18
40.24
.07
52.68
.04
77.62
.02
5.00
5.16
76.14 89.59
5.47
105.55
6.28
144.42
8.15
247.28
10.60
419.62
1-
2
5
1
2
2
1
e
s
2
2
2
2
2
14
2
2
0
0
.-1
2
2_
2
2
-2
.66
.26
.66
.25
.21
.02
.52
.14
7.99
.10
8.08
.87
8.34
1.95
8.74
2.72
5.03
58.41
5.04
62.23
5.02
66.32
5.00
69.50
.10
.02
.29
.19
.44
.47
.55
.77
.61
2.02
.49
.67
.18
.09
.65
1.28
2.56
19.61
5.03
75.75
7.51
169.42
4.91
17.94
4.81
15.90
4.80
13.62
4.89
11.32
5.20
8.25
6.05
4.77
7.18
2.92
9.79
1.36
15.47
.49
22.81
.22
30.23
.12
5.11
5.41
5.80
6.29
19.43 20.13 19.88 18.72
7.20
15.83
9.05
10.68
11.18
7.08
15.79
3.53
25.47
1.34
37.81
.6D
50.23 75.16
.34
.15
2.61 3.49
18.41 34.87
4.18
51.40
5.35
85.33
7.44
166.07
9.97
298.44
12.49
467.63
17.49
918.01
.82
2.47
.14
.08
2.10
16.46
4.57
78.19
7.06
186.47
12.04
543.19
12.18
3.55
14.99
3.62
20.81
3.43
28.22
3.29
35.66 50.61
3.18
3.07
8.18
.27
12.99
. 17
22.81
.08
35.22
.04
47.66
.02
72.61
.01
3.18
18.21
2.14
8.16
.10
.02
2.43
10.34
4.94
42.84
9.96
173.77
.42
.08
9.58
2.92
2.56
21.19
2.31
18.08
2.09
15.21
1.79
11.36
1.30
6.15
8.05
.09
8.25
.68
8.56
1.44
8.94
2.10
9.60
2.80
10.84
3.35
.25
.02
.85
.12
1.56
.23
2.34
.31
3.60
.35
4.96
4.84
4.69
53.93 49.01 44.13
4.51
39.58
4.21
33.49
3.70
25.04
g
2.96
3.04
3.32
3.75
4.58
35.0 35.0
35.0
35.0
35.0
6.24
35.0
8.06
35.0
11.87
35.0
19.72
35.0
29.65
35.
39.61
35.
59.57
35.
AZ
3.09
35.0
3.41
35.00
3.87
35.00
4.43
35.00
5.39
35.00
7.14
35.00
9.00
35.0
12.85
35.0
20.71
35.0
30.64
35.
40.61
35.
60.57
35.
4.81
75.0
4.52
75.0
4.36
75.0
4.34
75.0
4.58
75.0
5.57
75.0
7.00
75.0
10.44
75.0
18.03
75.0
27.84
75.0
37.75
75.0
57.66
75.0
5.21
75.0
5.70
75.0
6.24
75.0
6.84
75.0
7.81
75.0
9.54 11.36
75.0
75.0
15.13
75.0
22.91
75.0
32.79
75.0
42.72
75.0
62.65
75.0
3Z
_
45.16
.05
2.84
24.52
a
_
12.51
469.24
1.95
9.63
5.84
.32
_.
_
13.07
18.04
639.47 1220.2
1.44
5.02
0
2
3.98 4.40
35.13 40.73
.89
1.82
0
2
3.17 3.56
28.02 31.75
.30
.20
Z
-2
2
_____
15.0
2 ++
~2
_
r
_________________---
------
_
_
_
_
TABLE II
Frequepcies and Intensity Factors for Transitions in J - 5/2 Case: Cubic Field
is the overall splitting
Table is labelled in terms of dimensionless quantities c - E/S and x - gpf/8, where
1
4in the electric field alone. The intensity factor is defined by Eq. (7) and is given in this case by x
and 0z - (m[Jzn)2
'A
Quantity II -I
As
2
1
AS
2
2
1
AE
2
2
2
1
2
2
2
2
2
2
2
AC
AC
.24
.02
.75 1.29
1.85
.32
2
2
2
As
e
2.0
3.0
5.0
r
~
1.00
1.67
.71
.30
.62
2.57
.94
.17
.77
2.70
4.15
5.62
8.59
.64
.76
7.5
i
10.0
-[
][r
(tOOl direcle
15.0
-
7.53
13.15
5.04
50.82
113.40
12.52
313.44
.92
14.56
1.09
22.04
1.18
29.53
1.23
44.52
1.29
3.70 4.25 4.62
26.54 35.52 42.24
5.59
62.08
7.56
10,03
201.32
12.53
113.92
313.76
17.52
613.55
.46
3.29
20.29
2.97
14.00
2.99
11.93
3.12
9.72
3.39
4.00
5.29
7.61
5.33
3.56
6.70
2.83
9.62
2.23
15.57
1.85
23.04
1.69
30.53
1.61
45.52
1.54
3.11
5.28
3.44
5.45
3.91
4.89
4.54
3.82
5.70
2.34
7.94 10.33
15.21
.27
25.13
.10
37.58
50.06
.02
75.04
.01
1.06
.89
1.67
2.60
6.99
61.02
9.49
12.00
4.97
112.68
179.91
17.00
361.10
2.84
3.71
2.91
2.71
.46
.17
.17
.02
z
3.45
1Ar
3.15
18.08
2.91
4.63
2
.21
1.37
E7-x
VO-
3.04
16.00
AC
As
ez
-,
1.01
.09
I,
--
.88
.78
.37
e~
i7
.61
.25
As
OX
x
-
.23
.02
.10
tke
t'The static magnetic field is in the Z direction;this ilso
(mlJ+ln)12 ()2
Transition
.67
.42
3.28
20.00
3.90
20.00
20.00
2.75
20.00
2.37
20.00
22.66
1.10
.62
2.70
8.60
3.36
13.93
3.92
4.97
19.14
30.79
3.14
1.82
3.70
.98
4.94
.43
6.33
.23
9.21
.10
15.13
.04
22.58
.02
.76
.70
.08
.01
1.03
1.34
3.01
11.34
14.18
20.00
10.25
20.00
2.16
.62
5.30
20.00
2.24
2.37
20.00
20.00
4.58
.58
.36
.16
6.40
8.31
6.40
8.31
20.00
20.00
10.25
10.25
20.00
3.00
20.00
4.58
20.00
6.40
20.00
.04
30.06
.01
45.04
37.89
8.00
80.07
13.00
211.38
22.11
20.00
32.08
20.00
42.06
20.0
62.04
20.
18.14
20.00
28.09
20.00
38.07
20.0
5.51
.00
58.0
20.
-.
--
--
--------
------ ·------
--~~1~·--sll"~ll~~~~-I--
--·------
I