INTRODUCTION
TO
ATOMIC SPECTRA
BY
HARVEY ELLIOTT WHITE, PH.D ,
A ssistant Pr ofessor of P hy sics, at the
University of California
Mc GRAW-HILL BOOK COMP ANY, INc.
NE W
Y ORK AN D LO NDO N
1934
CHAPTER XV
THE ZEEMAN EFFECT AND MAGNETIC QUANTUM NUMBERS
IN COMPLEX SPECTRA
It is well kn own to spectroscopists t hat t he Zeeman effect plays
an important part in t he making of any analysis of a complex spe ctrum.
This is due larg ely to t he fa ct that t he Zeeman pat terns for all types
of t ransit ions can be pr edi cted with considerable accuracy. Let us
start t herefore with a calculation of Zeem an pat terns for va rious t ypes
of t ra nsit ions and t hen compare t he resul ts with t he observed patterns.
To do t his we must first obtain an expression for t he magnetic energy
of an atom in an y stationa ry state.
H
H
--- - ------
-JS;</'
f./ . . . )\ -,
'I,
,"1' -
- \ I
\
/'
S~
~
\~*- ~
I
Jj -C oup 1ine
LS-Coup1in9
F lO. 15.1.- Cl a ssical models fo r the mo tion of complex atoms in a weak m a gn e ti c field .
15.1. Magnetic Energy and the Lande g Factor.- I n th e previous
cha pte r we have seen t hat t he vector mod el for complex spectra is
essen tially t he same as t he model for t wo valence elect ro ns. The
extension in to complex spect ra was made by considering all but one
of t he electrons as a parent te rm or configuration t o whi ch t he remaining
electron is added (see Fi gs. 14.2 and 14.3).
In a weak magnetic field t he atom as a whole becomes quantized and
oriented t o t he field H, in such a way t hat t he pr oje ction of J *h/27r on His
equ al to Mh /27r (see Fi g. 15.1). Fo r te rms with whole-integral values
of J the magneti c qu an tum number j);! = 0, ± 1, ± 2, . .. ,±J. For
terms with half-integral values of J , A! = ±!, ±~, ±t, .. . , ±J.
Tbe magneti c quantum numbers for a 5D 3 term, for example, are 1~ = 3,
286
SEC. 15 .1]
ZEEM A N EFFECT , MAGNETi C Q UA NT UM N UM BE RS
287
2, 1, 0, - 1, - 2, -3, and for a 4P~ term they are M = -fr, t , !, -!, -t,
- i. Just as in t he case of one electron the magnetic energy is determined
by M, H, andthe Lande g factor [see Eq. (10.17)]:
He
-I1T = M . g . 471'"mc2 cm- 1 = M.: g . L cm: ' .
(15.1)
For a given field strength H t he separations of consecutive magnetic
levels for one te rm, relati ve to t hose of another, will be given by t he
g factor of each respecti ve level. Just as in one- and t wo-elect ron
systems the g factor is the ratio between the to tal magnetic moment
of the atom in Bohr magnetons and the total mechanical moment in
units of h/271'" .
If the coupling is LS (i.e., all spins coupled together to form S*
and all orbits coupled toge ther t o form L *, and S " and L* in turn coupled
together t o form J *), t he g factor is obtained in exactly the same manner
as it is for a sing le electron. S * is thus treated like the spin of a single
electron and L * like t he corresponding orbi tal momentum vector . From
Eq . (10.11) we write
J *2 + S*2 - L *2
(15.2)
g = 1+
2J*2
.
I
Should the coupling on the ot her hand be Jj (i .e., the spin and orbi t
of t he last bound electron coupled tog ether to form jf, and if in turn
coupled together with J ; of t he parent term of t he ion t o form J *),
the g fa ctor is obtained in exa ctly t he same manner as it is for two
electrons in .ij -coupling [see Eq. (13.14)]. J; of the parent term is
now tre-ated like i-, t he spin-orbi t resultant of a single electron to be
coupled with t he jt of t he other elect ron, so t hat
J *2 + i t 2 - J ; 2
J *2 + J ; 2 - j f 2
(15.3)
g = g1
2J*2
+ gr
2J* 2
'
where gt is t he g factor for the single added electron derived from Eq.
(10.11), and gP is t he g fa ctor for t he parent term derived from Eq . (13.8)
or (13.14). It should be poin ted out t hat t he parent term may arise
from an LS- or jj-coupling scheme even t hough t he added electron j i
is coupled to it finally in Jj-coupling. Tables of g factors for LS-coupling
are given in the Appendix. For Jj-coupling t here are so many possibilities that Eq. (15.3) must be used.
A graphical representation of t he ma gnetic energy corresponding
to the vector model, given by E q . (15.1), is shown in Fig. 15.2 for a 4D j
term. The vector J * is quantized with respect t o t he field direction H
so t hat it s proj ect ion M = t , %
, to, ! , - ! , - 'h - -fr, - to Multiplying
J* by the g fa ctor l.rQ. and projecting on Hwe obtain t he values of Mg .
These are proportional to t he magneti c energy. The result is eight
288
I NTRODUCTION TO ATOMIC SPECTRA
[C HAP. XV
equally spaced levels each separated from t he nex t by g . L. If t he Mg
values are multiplied by He/47rmc2, the figure will give the intervals in
em>'.
H
--
M9 " 70/ ,4 - - - 50hl
- - - - 101= 7/Z
30/14
5(1
'0/'4
- 1/2
3/1
4
D ~o/14-
- - - - - _ - -::'T;
I/1~~~::::::-::l
- 3/ z
- :lO/ '4 - - - - - -:'5/2 - 50/14
.= 7/ 2
- 70/14 - - - -
F lO. 15. 2.- S ch em a ti c r ep rese nta t io n of Zee m an sp litting for a ' D 7. term in
fi ~d .
II
weak m agneti c
-
16.2. The Calculation of Zeeman Patterns.- As an example of the
calculation of Zeeman pa t terns we sha ll first consider the t heoretical
pat terns of 783 - 7P 2.3 .4 , observed pat terns for whi ch appear in chromium
(see Fig. 14.1). Since 8 and P terms never have more than one and
t hree fine-st ru cture levels, respe ctively, this mu lt ip let cont ains but
three lines. Although t he lines look jus t lik e those of a principal-series
triplet, the Zeeman pa tterns are entirely different from those shown in
3
2
~~ .mg m.l
'-Ill
2
1
7/4
-,-2
0
m'4
'p. o
-3
-4
m-3
-7/4
44\2
~
-14/4
-2
-21/4
-3
6019
m·J
4
2
1
'S30
7.
1
;Ss 0
0
-2
-2
-4
-3
-6
A~ CQlculQl ed-A~
Ob><!rwd
1'1
1'3
0
- 7/J
"""2
-2
-14/3
~ 'mg
•2
m-3
2
I
0
'S"o
6'"'9
-l
-'I
-J
W3'mg
-I
-C9A2
-2B,14
,
m·2
'p. o
-2~.
0
-I
2
-I
EII/r2.me
I.
' I', 0
4
-I
-l
7.
0
-I
-2
-3
-2
-2
'3 '
-3
W~
I.
FIG. 15.3.-Zeeman effect, t h eo retical and ohserved patterns , for a TS , - 'P ." " m u ltiplet.
(Obseroed pattern s ar e f rom original spectrograms by H . D . Babcock .)
F ig. 13.4. The intervals between t he t hree lines indicate L8-coupling
wit h a rati o approximately 4:3. T he g factors for 783 , 7P 2, 7P 3, and 7P l
(see Appendix) are 2, i , H , and t , respectively. A graphical construe-
SEC.
15.3]
289
ZEEMAN EFFECT, MAGNETIC QUANTUM NUMBERS
tion, such as that shown in F ig. 15.2, gives t he intervals shown in Fi g.
15.3.
With t he selection r ule t hat in any transit ion t he magnetic qua ntum
numb er M changes by + 1, 0, or -1 only, t he Zeeman patterns shown
below in t he same figure are obtained . Another method is to write
down t he My va lues for both t he initi al and fina l states of a give n line
with equal va lues of M directly above and below each ot her, as follows :
2
J.,f
1
4
2
t
Vertical differences, f::"M
f::" JI
=
-1
- 2
-J.,f
- 3
°
-2
-4
- 6
0, give the p components at
= 0, ±tL, and ± l L em-I.
D iago na l differen ces, f::"M =
f::"JI
°
°
± 1, give the s components at
= ±1L, ± ~L , ± -~L , ± -IL, and ± ~ L em-I.
T hese fracti ons for t he whole pattern are written together as
A
_
u Jl -
+ (0), (1), (2), 4, 5, 6, 7, 8L - 1
_
3
cm .
Similarly t he patterns for t he two other lines are calculated as
7S
3
_
7P ,
3
f::"JI
= + (1), (2), (3),2 1,22, 23, 24, 25, 26L .
f::"v
= +
_
-
12
cm
- 1.
,
(0), (1), (2), (3), 4, 5, 6, 7, 8, 9, 10
- 1
L em .
4
The obse rved patterns of t hese three lines, reproduced at t he bot tom
of Fi g. 15.3, are from original spectrog rams taken by Babcock at the
Mt. Wilso n Observatory. T hese are some of t he earliest anomalous
Zeeman patterns to be ph otogr aphed in any laborat ory.
15.3. Intensity Rules and Zeeman Patterns for Quartets, Quintets,
and Sextets.- T he intensity rul es for the Zeeman effect in complex
spectra are t he same as t hose for one- and two -electron systems. These
rules, derived from t he classica l orbital model of the atom and t he sum
rules, are given in Sec. 13.3. Since t hese rules involve only the J 's
and M's of the initi al and final states t hey are valid for all coup ling
schemes . T he relative in tensiti es of t he different components of a
given pattern are usually represented by the heights of the lines in the
schematic diagrams. It is to be noted that the component J =
to
°
290
J = 0 of 783
INTRODUC1'ION TO ATOMI C S PEC T R A
-
7P 3
[C HAP. XV
(F ig. 15.3) is zero as give n by the in ten sity formulas,
i.e., the transition is for bid den.
A nu mber of pa t t ern s t ypical of quartet and sext et m ult iplets ill
gene ra l are give n in F ig. 15.4 for vana di um . P a t t erns typica l of quintet
mu lt iplets are give n in Fig. 15.5 for chromium . The calcula t ed pa tterns
are give n above or below each observed pat t ern. T he five lines }'>-.4352,
4344, 4339, 4337, and 4340 are t he five diago nal lin es of a 5D - 5F
m ultiplet. It is to be not ed t hat, as t he nu mber of compone nts decr eases
in going from one pa t t ern to t he next , t he sepa ra t ions increase. The line
>-.4340, (5D o - 5F 1) , alt ho ugh pa rtly masked by >-.4339, is of particul ar
in ter est in t hat it is one of t he few spectrum lines kno wn whi ch remains
single in all field strengths . The reason for t his is t hat t he M g values
for both t he ini tial and final states are zer o. Unresolved pa t t erns like
t hose of >-.4352 a nd >-.4344 a re called blend s.
The pat t ern >-.4613 in chromium has 2.5 t imes t he separation of a
normal t riplet. In general the Zeem an pat t erns for differ en t com binat ions ar e so differ en t fr om on e anothe r t hat t he ass ignme nt of a com plet ely
resolved pat t ern to a defini te m ult iplet is un ambi gu ous. The importance
of reso lved pat t erns in making t he analys is of an uncha r ted spectru m , as
well as chec king t he a nalys is of a spectrum already made, cannot be
ov ere m ph asized.
15.4. Paschen-Back Effect in Complex Spectra.- So far as experime ntal observat ions are concern ed t he P aschen-Back effect in com plex
spectra is practi call y un kn own. This is due in general t o t he fact t hat
eve n with t he strongest magnet ic fields attainable, t he mu lt ip let separa t ion s are many t imes t he Zeeman sepa rations of anyone level. There
is lit tl e do ubt, however, t ha t if, and when, st rong eno ug h field s are
obtained t he obse rve d splitting will be ve ry closely t hat predi ct ed by
t heory. There is some justification for t his as we shall see in t he study
of t he P aschen-Back effect of hyperfine st ru ct ure in Chap . XIX.
In a wea k magneti c field a com plex atom is quantized with respect
to H more or less as a rigid whole. When t he field becom es strong
eno ugh so t hat t he Zeeman levels of ea ch fine-structure level begin
to ove rlap t he pa t t erns of t he neigh boring levels of t he same multiplet,
t he strong-field P aschen-Back effect sets in a nd 8 * a nd L *, or J ~ and
it., becom e qu an tiz ed sepa rately with t he field directi on H . The quant um conditions under t hese circumstances are t he sa me as those for two
elect rons : the proj ecti ons M of 8* and L *, or J~ and it , on H t ake integral
or half in tegral v alues only, dep ending upon wh ether 8 a nd L , or J p and
ii, are whole or half in t egral res pective ly . When t he field becomes
still stronge r and t he magnetic levels of one multiplet begin to overlap
t hose fr om t he ne ighboring mult iplets of t he same configuration, t he
complete P aschen-Back effect sets in and eac h 8;* and l;* becomes qua nt ized ind ependen tl y with the field H. The qua ntum condit ions again
291
ZEEMAN EFFECT, MAGNETIC QUANTUM NUMBERS
SEC. 15.4]
ZEEMAN PATTERNS
OF
VANADIUM
Mill
4F!j</F!j<, ::\4595
Y
~!jf~* ~
4F",,_4GtI'.! ,\4421
' 0,3,5,7,91 m5, ~~4143.4!l47,4'1 51 ,. (glU93).8 M~3. 2052('7
t (7)
tr
~l'i6f12 .\4444
43
~f1-6p~
t(2)021 2(532 40
F IG. 15 .4.-Zeem an pa t terns, observed a nd ca lcu lated , for typical q u a r t e t a nd se xte t
multipl et lines in va na d iu m . (Observed patt erns are fro m origin al sp ectrograms by H . D .
Babcock .)
ZEEMAN PATTERNS OF CHROMIUM
~(Ql2l4\5,7. 9,I1,13
6
~.,
~ illDl.1. 2.3
'1lIDlj21 3.4,5.6,7
' mwllJ.la1Ul.27,1Il.3l.J6 tgIWjl! lQJUl.11l4,wt8
2
'4
;t; ~
~'I~I~'" ~IAI~"
III
20
10
~ -JJw- -J-ln- -t- ~ ~
5
0,-1>,
%52
t «WJ,1,3,5
2
1&12
50, ·5p,
'(21,~), 7,~II,13
6
§
50j 5p'
t (2),3,5
2
SOlF,
4340
0
0'
~
~
2
SO/P,
5S/p~
4496
' 10¥1i"1J,3,45,6,7
3
F IG. 15.5.-Zeeman patterns o bse rved a n d ca lcu lated, for typ ical q u intet multiplet lines
in ch ro m iu m . (Observed patt erns ar e [rom. original sp edroqrams by H . D . B abcock. )
292
INTRODUCTION TO ATOMI C S PEC T R A
[C HAP. XV
are just t he same as t hose for two-electron systems. Each spin and
eac h orbit has a com pone nt a long H. The sum of t hese com pone nts
1:m' i + ~mli gives t he magn eti c quantum number M of t he entire
atomic system in t hat state .
When conside ring t ransitions betw een levels in weak, strong , and
very strong fields, eac h multiplet in LS-coupling in a strong field will
go ove r to a pa t t ern resem bling a normal t riplet (see Fi g. 13.9). In
a very st ro ng field all t he lin es arising fr om t he configuration of electrons
will spread out in t o a pa t t ern again resembling a nor m al triplet (see
Fi g. 13.14). P a t t erns sim ila r to t ho se of Figs. 13.11 a nd 13 .15 will
resul t in t he case of J j -coupling. We shall now make use of t he weakand strong-field and very strong field q ua ntum numbers to ca lculate
the all owed spe ct ral te rms ar ising fr om give n electron configurations .
m s ,= 1;2
-V2
M
s= 1
__::p_
0
--,
Msp: 0 : -1
Msp= 1
0
-1
'h
Ms = ¥2
yz
-Yz
-Yz
Ms = ~
-- -
- - - - ....,
Sing. 'Trip. ms z'
Ms =
~
S3
ML p" 2
.Q.
m12 = 2
0
-I
-2
ML p = 2
0
-1
-2
0
mIL
?
Yz Vz
-~
-Yz
Daub.
mS
-Y2 i-% -Yz
,
Daub. ,Quar. m
I
0
Msp= 0
~
0 -I
-2
0
-I
1
1-3
,, F
-1
M = 3
2
ML = I
0 - I ~- 2
,
P I D
I
3
L
---- ....
M-L-= -----2 _____
1 0, - 1 I' - 2 0 .P
__
mt3
FIG. 15.6.- C ombin a tion of ve ry strong field qu a ntum numbers for t he elect r on co nfig uration dsp .
16.6. Derivation of Spectral Terms by Use of Magnetic Quantum
Numbers.- I n deriving spectral te rms fr om magn etic q uantum nu mbers,
one may st art with t he very stro ng field where t he coupling betw een
a ll spins and orbits is brok en down . Co ns ide r as a first example t hree
a rbitrary electrons, no t wo of whi ch are eq uivalent, suc h as 8, d, and
p . F or t hese electrons we sha ll have 81 = !,82 = ! ,83 = !, II = 0, l2 = 2,
and L« = 1, for whi ch t he magn eti c qu antum number s are m" = ±!,
m sa = ±!, m., = ±!, m Il = 0, m l, = 0, ± 1, ± 2, and ml, = 0, ± 1.
E xtending t he scheme of Breit (see Sec. 13 .10), we first com bine the
spins 81 and 82 as sho wn at t he left in Fi g. 15.6. The resul t ant values of
M sp a re cut a part by t he L-sh aped dot t ed lin e to for m paren t va lues for
t he t hird electron . Combining eac h run of l1f sp values with m •• we
obtain t hree runs of M s va lues. These correspond, as shown, to strongfield quantum numbers for doublet , quartet, and do ublet te rms. In
a sim ila r fas hion t he l's are com bined in t he lowe r arrays . me. with ml,
forms t he run M Lp = 2, 1, 0, - 1, -2, to form t he paren t values for t he
SEC. 15.6]
ZEEMAN EFFECT, MAGNETIC QUANTUM NUMBERS
293
third electron. Com bining t hese with mz t he strong-field runs of
values M L = 1 to - 1, 2 to - 2, and 3 to -3, corres ponding to P, D ,
and F terms, respectively, are obtained . In te rms of t he branchin g
rule t hese resultant values would be written down from the par ent
terms as follows:
sd,
sdp,
ID
I
3D
1\ A
2p
2D?F
2p 2D 2F
P arent te rms
4p 4D 4F
F ina l te rms
The first and fourth arrays give t he first row; and t he second , third,
and fifth arrays give t he last row. If now each final ru n of M s va lues
is combined with each run of M L va lues, all of t he weak-field quantum
numbers of t he above fina l terms are obtained. T his has been done
M = 9/ 7/ 512 312 V2 _1/2-3/2 312
-- - - 2- - -2- - - - - - - - - - -:\
5h 3/2 ~ -'h. - 3/2 :-% ~
tv1 = 5h % Ih -lh -3/2 1-% :-7
12 -V2
I
I
M = 3h '12 -Vz -312 I -5h :-7
h :-% 312
4
4
14
14
F%~ F%I F%: F%. Ms
M= ~
- -- -
- -
- - - - - -- - /)
I
- ----- -- -- - -~
FIG. 15.7.- C om bin a t ion of the strong-field quantum numbers of a 4F term showing the
formation of re sultant weak-field quantum numbers.
for t he 4F term in Fig. 15.7. The others will be left as exercises for t he
reader.
15.6. Equivalent Electrons and the Pauli Exclusion Principle.-In
the calculation of spectral terms ar ising from two or more equivalent
electrons we must take t he Pauli exclusion principle int o account and
start with t he ve ry strong field quantum nu mbers (see Sec. 13.11).
Consider as an example t hree equivalent p elect rons . We first write
down t he six possible states for a single p electron in a very strong field.
They are:
m =
1.
mi
1
8
>
2
(a)
!
o
1
"2
-1
(b) (c)
- ! -!
o
(e)
-1
(f)
(15.4)
Since t he exclusion pr inciple requires t hat no two electrons have
all qua ntum nu mbers n, l , m. or mz alike we collect all possible com-
I NTR ODUCTION TO A TO MIC S P£CT RA
294
[CH<\ P. XV
binations of t he above states t hree at a ti me, with no two alike.
are:
15.1
TABLE
abe
abd
abe
ab!
Wri ting M s for
For
For
For
For
~m.
Ms =
M» =
acd
ad!
ace
a4
ade
aef
!,
1
"2,
1
= - "2,
M s = - !,
M;
be!
bde
bd!
be!
bed
bee
~m l,
and M L for
They
cde
cd!
cef
de!
t he resul ts are tabulate d as follows:
°
ML =
M L = 0, 2,1,0,-1 ,-2, 1,0,-1
M L = 0, 2,1,0,-1,-2, 1,0,-1
°
ML =
(15.5)
These are just the strong-field values for 4S , 2D, and 2p terms (see Table
14.3).
In a similar calculation for four equivalent p electrons we write
down t he same six possible states for one p electron [E q. (15.4)] and
take all possibl e combinations of four states at a time with no t wo alik e.
They are :
T AB L E
abcd
abee
abe!
obde
abd!
abe!
15.2
aede
aed!
aee!
adef
bede
bed!
bee!
bdef
cdef
When m. and m l va lues are summed for each combination and tabulated
one finds :
For u , =
1,
For M s =
0,
For M.« = - 1,
ML
=
1,0,-1
M L = I , O , - I , 2,1 ,0,-1,-2,
M L = 1,0,-1
°
corresponding t o 1S, 3p , and 1D terms. These are exa ctl y t he te rms
arising from t wo similar p electrons (see T able 14.3) .
F or five equivalent p electrons t here are just six combinations, taking
five states at a t ime. They are
abede abedf abeef abdef aedef bedef
When summed and tabulated as before one finds:
For M s =
F or M s =
t,
- t,
M L = 1,0,-1
M L = 1,0,-1
These are just t he sa me values given in Eq. (15.4) for one electron and
give rise to a 2p term.
For six equivalent p electrons there is but one combinat ion of the
very st rong field quantum numbers, taking six states at a time, with no
SEC. 15.6]
ZEEMAN EFFECT, MAGNETIC QUANTUM NUMBERS
295
two alike, and that is abcdef. The sum gives M s = 0 and M L = 0, a
'8 0 term. Here we see that six p electrons with the same total quantum
number, by Pauli's exclusion principle, close a subshell and have zero
as a resultant angular momentum (see Table 14.3). The table of
p electrons is symmetrical about the center where the greatest multiplicities and the greatest number of magnetic levels arise.
In the calculation of terms arising from equivalent electrons in
jj-coupling we need only go to the strong-field quantum numbers.
The reason for this is that here each electron is specified by four quantum
numbers and it is not necessary to go to very strong fields to separate
them as in LS-coupling. As a single example of the calculations let
us consider three equivalent p electrons. The six possible states for
a p electron in a strong field are
j =
mi=
-~
~-
.~
~
t
-t
(a)
(b)
(c)
1
1
2
-l
(d)
1
2
(e)
1
2
1
- 2
(f)
Since the Pauli exclusion principle requires that no two electrons have
all the quantum numbers n, l, i. and m ., alike, we collect all possible
combinations of the above given states three at a time, with no two alike.
These will be just the combinations given in Table 15.1. Summing up
the values of m , for each of the 20 combinations the results may be
tabulated as follows :
Forj,= 4,
Forjl= ~,
For j,= 4,~
j 2=l, j a= t, . m i= t,t, -t,- t
j2=4, h=t, mi=!, -!,~, ~- ,! , -!,- 'i, - t, t ,!, - !,-4
.
1
J2= 2' j3=!, mj=-U ,-!,- l
These correspond to the five t erms (~ ~. '¥-) l, (t t !)!, (t t !)i, (l H)l,
and (t ! !) 1, whi ch go over in LS-coupling to 2Pl' 2P h 2D" 2D 1, and
48 1 (see Fig. 14.22).
We shall now turn to the more complicated cases of equivalent d
electrons. Two equivalent d electrons have already been treated in
Sec. 13.11 and shown to give rise to '8, 3p , ID, 3F, and 'G terms. For
three or more electrons we continue the same scheme by first writing
down t he 10 possible states for one d electron:
m, =!
m, = 2
(a)
1
2
1
(b)
!
o
!
-1
(c) (d)
For d 3 we take these three
combinations that have to be
sums will correspond exactly
d4, taking four at a time with
binations to be evaluated.
1
1
2
- 2
-2
2
(e)
(f)
1
-OJ
1
(g)
-! -! -!
o
(h)
-1
(i)
-2
(15.6)
(j)
at a time with no two alike and find 120
collected and segregated. The resultant
to the terms given in Table 14.'3. For
no two alike, there are 210 possible comSuch calculations become tedious and
296
I NTRODUCT ION TO ATOMIC SPECTRA
[CHAP. XV
cum bers ome, t o say t he least, bu t for tunately a shortha nd method
has been discovered by Gibbs, Wilber, and Whi te, 1 and by Russell."
We sha ll first apply t his scheme to t hree equivalent d electrons .
R eferring to Eq. (15.6) take only t hose combinations of numbers
three at a time where all t hree tn. values are plus an d t hose where all
t hree are minus. These in terms of t he letters are
abe
abd
abe
aed
aee bde i gh ihj gij
ade cde igi i ij hij
bed
igj ghi
[h i ghj
bee
Summing t he corres pon ding va lues of t he qua ntum numbers we
obtain for t he plus va lues
J11 s =
!,
ll'h = 3, 2, 1, 0, - 1, - 2, - 3,
1,0,-1
and for t he minus va lues
Ms
= - !,
kh
(15.7)
=
3, 2,1 ,0,-1,-2,-3,
1,0, -1.
These correspond to parts of 4p and 4F terms; t he rest of which
will be obtained from t he following process . Of t he 100 remaining
combinations to be ma de an d summed eac h one will have t wo of t he
tn. values plu s and t he ot her one minus, or vice versa. If in each of
t hese 100 combinations t he t wo m, values t hat are alike are considered
alone, t hey will contain just t he combinations t hat give t riplets for two
equi valent d elect rons : t hese are 3p and 3F . The t hird electron in all
combinations has an opposite sign and corresponds to t he values for a
single d electron. Combining the triplets t hat arise from d 2 wit h t he
doubl et t hat ar ises from d in all possible ways to form dou blets, we
obtain just t he 100 combinations desired. This is done qui ckly by t he
br an ching rul e as follows:
3p
/\
This final set of t erms has M s values of +! and - !. Striking out
one P and one F t erm to go with Eq . (15.7), we have left 2p, 2D, 2D, 2F,
2G, and 2H . The resultan t te rms for t hree equivalent d electrons are
d3 ,
2D,
2p, 2D, 2F, 2G, 2H,
-r , 4F.
Assume, as another example, t hat t he correct te rms have been derived
for d, d 2 , d 3 , and d4 and we wish to calculate t he te rms for do. R eferring
1 G IBBS,
R . C., D. T . WILB ER, and H. E. W
H. N., Ph ys. R ev., 29, 782, 1927.
2 R U SS E LL,
HITE ,
Phys. R ev., 29, 790, 1927.
SE C.
15.61
ZEEMAN EFFEC T, MAGNETIC QUANT UM NUM BERS
297
to Eq. (15.6) we take only t hose combinations five at a time where all
five rn. values are plus, or all five minus. These are abcde and fgh~j.
Summing these we have for t he plus values
M» =
t,
and for t he minus val ues
Ms
= - t,
(15.8)
These are just one-third the values needed to form a 68 term.
There will now be a set of combinations from Eq . (15.6) in whi ch
four spins will be plus and t he fifth one minus, or vice versa. That
part of the combinations in whi ch four of the spins add gives quintets.
From T able 14.1 we find for d 4 one quintet only, "D . The fifth electron
with oppo site sign corr esponds to 2D. Combining t hese in all possible
ways we get
Striking out the 8 t erm to go with Eq . (15.8), we have left parts
of 4p , 4D, 4F, and 4G te rms . It should be poin t ed out here t hat we now
have t hose part s of t he 68 te rm for which lVI s = t , t, - t , and - ~ , and
those parts of each quartet te rm for whi ch M s = t and - t o
Again there will be a set of combinations from E q. (15.6) in whi ch
three spins will be plu s and t he t wo others minus, or vice versa . The
combinati ons of d 3 hav e already been shown to give "P and 4F te rms,
and t he combinations of d 2 to give 3p and 3F terms . Combining t hese
in all possible way s to form doublets,
Striking out one 8 term to complet e the 68 term above, and one
each of P , D, F, and G, to complete t he 4p , 4D, 4F, and 4G te rms, respe ctively, we have t he remaining te rms as doublets. The resul tan t te rms
arising from five equivalent d elect rons are, t herefore,
do,
2D,
2p , 2D, 2F, 2G, 2H,
28 , 2D, 2F, 2G, 21, 4P ,4F, 4D, 4G,
68 .
Since a subshell of d elect rons lacking n electrons to complet e it
will give rise to exactly t he same terms as a configur ation of n equi valen t
298
INTRODUCTION TO ATOMIC SPECTRA
[CHAP. XV
electrons, written symbolically dl O- n = d n , the lower half of Table 14.3
is symmetrical with the upper half.
A continuation of this process for equivalent f electrons leads to
the terms given in the tables in the Appendix. The number of possible
combinations for each configuration of equivalent electrons is given in
parentheses at the beginning of each row . These numbers are computed
from the well-known combination-theory formula for p things taken q
at a time.
Number of combinations =
1(
q. P
p~
)1
q.
(15.9)
This number is equal to the number of Zeeman levels for the entire
configuration. For d 2, for example, the terms are lS, 3p, -D , 3F, and
lG, for which there are 1 + 9 + 5 + 21 + 9 = 45 Zeeman levels.
Without the above presented shorthand process of making several
thousand combinations, the calculations for equivalent f electrons would
become very laborious.
The largest number of Zeeman levels and the highest multiplicity to be
derived from equivalent electrons are from half a subshell. Furthermore,
the term of highest multiplicity is an S term and, wherever observed,
always lies deepest. On the quantum me chanics the configuration for
an S term is spherically symmetrical about t he nucleus (see Cha p. IV).
This is part of the explanation of the increased binding of d electrons
at the expense of 8 electrons at Cr and Mo in the first and second long
periods (see Fig. 14.18) . An 8 electron removed from a spherically
symmetrical shell 8 2 leaves a spherically symmetrical distribution.
Problems
1. Compute and plot the Zeeman patterns (int ervals and relative intensities)
for the transitions 7G 7 - 7F., 7G. - 71''0, 7G. - 7F., and 8PI - 8S 1.
2. Starting with the very strong field quantum numbers, derive t he terms ari sing
from the elect ron configurations 3d4p5p and 3d'4p (see Sees. 15.5 and 13.11).
3. Using the short hand method outlined in Sec. 15.6, calculate the spe ctral t erms
ari sing from (a) four equivalent d electrons, (b) two equivalent! electrons.