Download Doublet Fine Structure and the Spinning Electron

INTRODUCTION
TO
ATOMIC SPECTRA
BY
HARVEY ELLIOTT WHITE , PH.D ,
Assistant P rof essor of P hy sics, at the
Univ ersity of Cali fornia
McGRAW-HILL BOOK COMPANY, INc.
N EW Y ORK AN D LONDO N
1934
CHAPTER VIII
DOUBLET FINE STRUCTURE AND THE SPINNING ELECTRON
From t he very earli est obser va ti ons of spectra l series it has been
known t hat each member of certain genera l t y pes of series shows fine
st ruc t ur e while t hose of others do not. Each member of some of t he
series in the alkali metals, for example, is a close doublet (see F ig. 1.9),
whereas in the alkaline ear ths it is a close t riplet (see Fig. 1.10). So
far as energy level s are concern ed, term analy ses of so-called doublet
spectra show t hat P, D, F, G, . . . leve ls are probably double, whe reas 8
levels are always single (see Figs. 5.1, 5.2, 6.4, and 6.5) . It is with this
fine st ru ct ure t hat we ar e concerned in this chapter, for in its exp lanation
we are led to a new conce pt, the spinning electron.
8.1. Observed Do ublet Fine Structure in the Alkali Metals and the
Boron Group of Elements.- Before taking up t he qu estion of the origin
of fine st ruct ure, it is important that we become more or less familiar
with t he doublet spectra of t he alka lies and another grou p of elements
not yet considered . These are t he elements in t he t hird group of the
periodic t able (Table 5.1), boron, aluminum, gallium, indium, and
t hallium . Lik e t he alka li metals eac h of t hese atoms gives rise t o four
chief series of spectrum lines: sha rp, principal, diffuse, and fundament al.
From t hese observed series, energy level diagr am s are finally const ruct ed
as shown in Fig. 8.1. The chief difference bet ween t he alka li metals and
what we shall hereafter call t he boron grou p of elements is best described
in te rms of t he ene rgy levels rather t ha n in t erm s of t he spectrum lines.
In t he alka li met als a 28 term lies deepest, followed by a 2p term
as t he first excited state (see Fig. 5.2); whereas, in t he boron group a 2p
term lies deepest, followed by 28 . This observed experimental fa ct
is the basis, in the Bohr-Stoner scheme of t he building up of the elements,
for the addition of p electrons in these elements (see T able 5.1). The
complete electron configurations for the boron group are written as
follows :
B
AI
Ga
In
TI
5,
13,
31,
49,
81,
l s2
l s2
l s2
l s2
ls 2
2s 2
2s 2
2s 2
2s 2
2s 2
2p
2p6
2p6
2p6
2p6
3s 2 3P
3 s 2 3p6 3d 10 4s 2 4p
3 s 2 3p6 3d lO 4s 2 4p6 4d 10
5s 2 5P
2
10
2
10
3 s 3p6 3d 4s 4p6 4d 4j14 5s 2 5p 6 5d 10 6s 2 6p
All other subshells being complet e, it is t he single unbalanced p electron
which, whe n the atom is excited, takes on t he various possible energy
114
115
DOUBLE T FINE STRUCTU RE
SEC. 8.11
st ates shown in Fig . 8.1. In boron , for example, t he first excited 28
state finds t he elect ron in a 3s orbit , t he I s and 2s subshell being alrea dy
filled. There being no d electrons in boron and no virtua l d orbits
with a total qua ntum number n lower t han 3, t he excitati on of t he va lence
electron to t he first 2D state places t he electron int o a 3d orbit .
Neglectin g for t he moment t he doub let nature of t he different levels,
it should be noted t hat t he 2F terms in all five elements, just as in t he
alka li metals, are nearl y hydrogen-lik e, indicating non penetrating
0
r E
o
0
0
0
0
45
N
tJ)
OJ
::>
<,
35
~
~
ll-
0
'0
~
'<:
"I-
_
_
!:\J
~
I'<)
'",.,.,<::>":t-
3p 32p
4p
o
65
:::::
55
-
!;:?
~
~
~
0
0
"!"
E
L
~
"~
~
~2p
2p
5p 5
6p
0
=
0
0
'"
2p
2'2 P
FIG. S.l.-E ne r gy lev el diagrams of the bo ro n gro up of elements.
orbits. It is to be noted too t hat for a given total qua ntum number n
t he order of binding, which is also t he order of in creasin g penetrati on, is
2F, 2D, 2p , and 28.
The large st fine-structure sepa ra tions in each elemen t of t he boron
group are to be found in t he normal states 2P . The very narrow 2P t_2 P!
interval in boron widens wit h each succeeding element un til in t ha llium
it has become almost as large as t he gross-st ruct ur e int erval 6 2P-7 28 .
The first sha rp series doublet s given in Table 8.1 indicate the enormous
spreading out of t he first 2p te rms in going to higher atomic numbers. One
line of t he thallium doubl et is in t he visible green region of t he spect rum
and t he ot her member is in t he ultra-violet over 1500 Aaway.
Ex actly t his same spreading out of t he fine st ructur e is observed in
t he alkali met als in going from lithium to caesium. This may be seen
directl y from t he principal series as t hey are plot ted in Fig. 8.2. The
sodium doubl ets are relati vely mu ch narrower t ha n shown. The fine-
116
INTROD UCTION TO A T OM IC S PEC T RA
[C H A P .
VIn
TABLE 8.l.- D o UBL ET SEPARATIONS FOR THE FIRST MEMBERS OF THE SHARP SERIES
B
Al
On
In
TI
2P r2S;
2P;_2S;
~A
~v
A
A
A
em'<
2497 .82
3961.68
4172.22
4511. 44
5350.65
2496.87
3944 .16
4033.18
4101. 87
3775.87
0 .95
17.52
139.04
409.57
1574.78
5 .20
112.07
826.10
2212.63
7792 .42
structure in t er vals for t he first member of the principal series in each of
t he alkalies and ionized a lkaline earths are obs erve d as follows:
Li I
KI
Rb I
Cs I
57.9
237 .7
554.0 em "?
MgII
Ca II
223 .0
Sr II
800.0
Ba II
91.5
0.338
Nal
17.2
Be II
6.61
1691.0 cm "
!
In lithium t he 2 2 S-2 2P in terval (see Fig. 5.1) is over 4000 t im es t he
fine-structure in terval 0.338 cmr", whereas in caesium t he corres pond ing
inter val is only 20 times larger.
Principal Series
JIIIIJ-=::==:::=;::=======
I
: ; : :=======
I
in[]
ilDIJ
========~
j
LK
C
mIJIIJ
I
Rb
_ _---1-1mIT]1---_IJ'!
,
40,000
Na
'=:;;:==~:;;:=====~
==== =~~
!
Li
,
!
!
!
,
,
,
!
,
,
,
!
_ [[
,
3 0,000
c mr !
20,000
-4- Rodlated Freq,uenc y -
F IG. 8 .2.- Illus t r a t in g fine structure in the prin cip al ser ies of the a lka li m etals.
an d p otassium doublets are narrow er than sho wn .)
!
Cs
J
10,000
(Sod iu m
A general survey of the energy lev el di agrams (F igs. 5.1, 5.2, 6.4, and
6.5 ) will enable certain general conclusions conce rning fine-structure
intervals to be drawn: First , corresponding doublet separat ions increase
with atomic number. Second, doublet separations in the ionized
alkaline earths are larger than the corresponding doublets in the alkali
metals. Third, within each element doublet separations decrease in
going to higher members of a series. Fourth, within each element
DOUB LET FINE STR UCTURE
SEC. 8.2 )
117
P doublets ar e wider t han D do ublets of t he same n, and D doublets
are wider t han F doublets of t he same n . The last two stat eme nts
are illu strat ed schematically in Fig . 8.3 by giving t he four lowest m embers
of t he t hree chief term series. It should be mentioned in passing that
excep t ions to these rules are well known. These exce ptions, however,
are fully accounted for and will be treated as special cases in Chap. X IX.
2p
2D
2F -
2D
-
-
-
_
2p =======
2p-
F i G.
-
-
-
2D=
=
=
=
S.3 .-Schem rt ic r epresen t ation of rel a tive t erm sep a rations in the differ ent series
8.2. Se reetion Rules for Doublets.- I n the doublet spect ra of atomic
systems containing but one valence electron the small let t ers s, p, d,
I, . . . . for the different electron orbits are replaced by the corr esponding
ca pitals S, P, D, F , .. . , for the terms. The small superscript 2 in
fr ont of eac h term indicat es t hat t he level in question, including S levels,
has doublet proper ti es and belongs to a doublet system. Although all
S levels are single, t heir doublet nature will lat er be seen to reveal itself
when the atom is placed in a magnetic field. In order to distinguish
bet ween two fine-structure levels having the same nand l values, t he
cumbrous but theoretically important half-integral subscripts are used.
T his subscript to eac h term, first called t he i nner quantum number by
Som merf eld , is of im portance in atomic structure, for it gives t he total
angular momentum of all t he extranuclear electrons (see Sec . 8.4).
The inner quantum numb er is frequently referred to as the electron
quantum number j or the term quantum number J .
Observation shows that, for t he transition of an electron from one
energy state to another, definite selection ru les are in operat ion. This
is illu st rat ed schematically in Fig. 8.4 by six different sets of combin at ions. From t hese diagrams, which are based upon experimental
observati ons, selection rules for dou blets may be summarized as follows :
In any electron transit ion 1
l changes by
+ 1, or
- 1 only,
and
j changes by 0,
+ 1, or
-1 only.
1 Violations of eit he r of these select ion ru les are attributed to the presence of an
ex ternal electric or magnetic field (see Chaps. X and XX) or to quadripole rad ia t ion.
118
I NTRODUCTIO N T O A T OM IC SP EC T RA
[C HAP. VIn
The t ot al quantum number n has no restrictions and may change by any
integral amount. The relative in tensities of the radiated spect rum
lines ar e illustrated by t he heights of t he lines dire ctly below each t ransition arrow at t he bot t om of t he figure, Combinations bet ween 2p
and 28 always give rise to a fine-structure doublet, whereas all other
combinations give rise to a doubl et and one satellit e. In some doublet
spectra, 2G and 2H terms are kn own. In designating any spectrum
line like M 890 of sodium (see Fi g. 5.1), t he lower state is writ t en first
followed by t he higher state t hus, 3 28 1-3 2P I . The reason for t his order
goes back t o t he very earliest work in atomic spectra (see Cha p. I). Spect rum lines in absorption are writ t en in t he same way, t he lowest level first.
~ ap
sf,
Y2
2p
2S
""'_...L-'o
3/2
2p
ao
~
2p
toP
20.'l'z
20
ll.0
4
3
2p
toP
2f
2~
20
.-.l...Lf~
F 2D5h
ao
~h~ '~ ~l 3/~
toP
~
toP ll.O-v ~t.o to P
t.DKF
Mt.D
FIG . SA .- Illustrating se lec t io n and intensity rules for double t com bin a t ion s.
8.3. Intensity Rules for Fine-structure Doublets.-Gener al observat ions of line intensiti es in doubl et spectra show t hat certain in tensity
rules may be formul ated. These in tensity rules are best stated in te rms
of the quantum numbers of t he elect ron in the initi al and final energy
states involved . The st rongest lines in any doubl et ar ise from transiti ons
in which j and l cha nge in t he same way. Wh en t here is more t han one
such line in t he same doublet, t he line involvin g t he largest j va lues is
strongest . F or example, in t he first principal- series doublet of Fi g. 8.4. t he
line 28 1- 2P j is st ronger t ha n 281-2Pl since in t he form er l changes by
-1 (l = 1 to l = 0) andj cha nges by -1 (j = t to j = t) . As a second
example, consider a member of t he diffuse series in which there ar e t wo
strong lines and one satellit e. Fo r t he two strong lines zP I-zD t and
zP 1-zD I, j and l both cha nge by -1. The st ronger of t he t wo lines
zPj_z D ! involves t he larger j va lues. F or t he fain t satellit e zP I- zD I,
III = -1 and t;.j = O.
Quan ti t ati ve rul es for t he relative int ensiti es of spectrum lines
were discovered by Burger, Dorgelo, and Ornstein.' Wh ile these rules
1 B URGER, H . C., and H . B . D OR GELO, Z eits. f . Ph ys., 23, 258, 1924;
L. S., and H . C. B U RGER, Zeits. f. Phys., 24,41 , 1924 ; 22, 170, 1924.
ORN STEIN,
DOUBLET nNE S T RUCT URE
SHc.8.3)
119
apply to all spectra in general, they will be st ated here for doublets
only. (a) The sum of t he intensitie s of t hose lines of a doublet whi ch
come from a common initi al level is pr oportional t o t he quantum weight
T ABLE 8.2 .- INTENSITY MEASUREMENTS IN THE PRINCIPAL SERIES
M ETALS
(Af ter S ambursky)
OF
THE ALKALI
El emen t
Combination
Wa ve-leng t hs
Na
3 2S - 3 2P
3 2S-4 2P
3 2S-5 2P
5890 : 5896
3302 : 3303
2852 : 2853
4 2 S-4 2P
4 2S -5 2P
42S-6 2P
4 2S -7 2P
7665:
4044 :
3446 :
3217 :
7699
4047
3447
3218
2
2 .2
2 .3
2 .5
:
:
:
:
5 2S -5 2P
5 2S -6 2P
5'S -7'P
5'S - 8'P
5 2S -9 2P
5 2S -10 2P
7800 :
4201 :
3587 :
3348 ;
32 28:
3157 :
7947
4215
3591
3351
3229
3158
2
2 .7
3 .5
4 .3
5
3
: 1
: 1
: 1
: 1
: 1
:1
6 2S-6 2P
6'S- 7 2P
6 2S - 8 2P
6 2S-9 2P
6 2S -10 2P
6' S - IJ2P
6"S -12 2P
62 S -13 2P
8521 :
4555 :
3876 :
3611 :
3476:
3398 :
3347 :
3313 :
8943
4593
3888
3617
3480
3400
3348
3314
2 :
5:
10 :
15 .5 :
25 .0 :
15 .8 :
5 .7 :
4 .5 :
K
Rb
...
~
Cs
Ratio
2: 1
2 : 1
2 : 1
1
1
1
1
1
1
1
1
1
1
1
1
of that level. (b) The sum of the intensities of those lines of a doublet
whi ch end on a common level is proportional to the quantum weight
of that level. The quantum weight of a level is given by 2j + 1. This,
it will be seen in Chap. X, is the number of Zeeman levels into which a
level j is split when the atom is pla ced in a magnetic field.
In applying these intensity rule s, consider again the sim ple case
of a principal-series doublet. H ere there are two lines starting from
the upper levels 2P i and 2P I and ending on t he common lower level
2Si . The quantum weight s of t he 2p levels are 2(-V + 1 and 2(t) + I ,
giving as t he intensity ratio 2:1. The same ratio results when the 2S
level is above and the 2p level below .
120
I NTRODUCTIO N T O A T OM IC S PEC T RA
[C HAP. VI n
Quanti tati ve measurements of line int ensities in some of t he alkali
spectra are give n in T abl e 8.2. 1
The par ti cular investi gations of Sambursky on t he principal series
of N a, K, Rb, and Cs show t hat, whil e t he first membe rs have , in agreem ent with obse rvations ma de by ot hers , t he t heo retica l ratio 2 :1, higher
m emb ers do not. This is expecially t rue in caesium wh ere t he observa t ions have been extende d to t he eighth member. In caes ium t he in tensity
ratio starts with 2 and rises to a maximum of 25 in t he fifth member, t hen
drops quit e abruptly to 4! in t he eighth member. > Co nside r nex t the
diffu se-series doublets whi ch involve t hree spectrum lines. The following
combination scheme is found to be par ti cularl y useful in represen ting
all of t he transit ions between ini ti al and final states. A diffuse-serie s
doublet is written
2P i 2P!
4
2
2D B6 X
0
Z
2D~4
Y
The numbers dir ectl y below and to t he right of t he t erm sym bols ar e
t he qua ntum weights 2j + 1. Let X, Y, and Z represen t t he unknown
intensities of t he t hree allowed t ra nsitions and zero t he forbidden transit ion . From t he summation rules (a) and (b) t he following relati ons
are set up: The sum of t he lines starting from 2D j is to t he sum starting
from 2Di as 6 to 4, i .e., Y
ending on
2P~
~ Z = ~;
and, similarly, t he sum of t he line s
is to t he sum ending on 2P! as 4 is to 2, i .e., X
i
Y
=~.
T he smallest whole numbers which satisfy t hese equations ar e X = 9,
Y = 1, and Z = 5. If the 2D te rms are very close together so t hat t he
observed lines do not resolve t he satellit e fr om t he main line, as is usually
t he case, t he t wo lines observe d will have t he int ensity ratio 9+ 1 : 5
or 2 : 1, t he same as t he principal-serie s or sharp-series doublets. Intensity
m easurements of the diffu se series of the alk ali met als by Dorgelo" confirm
t his.
A favorable spect rum in which the satellite of a diffuse-series doublet
can be easily resolved, with ordinary inst rument s, is that of caesium.
The first three memb ers of t his series ar e in the infra-red and ar e not
readily a ccessible to phot ogr aphy. The fourth member of the series,
composed of the three lines >.>. 6213, 6011, and 6218 has been observed
SAMBURSKY, S. , Z eits. f. Ph ys., 49, 731 , 1928 .
The ano ma lous in ten sities observed in caes ium have been given a sa t isfac to ry
explanation by E . Fe rm i, Zeits. f . Phys., 69, 680, 1930 .
3 D OR G ELO, H. B., Z eits. f . Phys., 22, 170, 1924 .
1
2
121
DOUBLET FI NE S T RUCTUR E
SEC. 8.41
TABLE 8.3.- INTENSITY M EASUREMENTS IN THE DI FFUSE SERIES OF THE ALKALI
M ETALS
(A f ter Doryelo)
El ement
Combin a t ion
Wave-lengt hs
Rat io
Na
3 2P-4 2D
3 2 P-5 2 D
5688 : 5682
4982 : 4978
100: 50
100 : 50
K
4 2 P-5 2 D
4 2P- 6 2D
5832 : 5812
5359 : 5342
100 : 51
100: 50
Rb
5 2 P-6 2 D
5 2P-7 2D
5 2P- 8 2D
6298: 6206
5724 : 5648
5431 : 5362
100 : 51
100 : 52
100: 52
to have t he int ensit y rati os 9:5.05:1.17. Theoreti cal intensiti es for t he
combination 2D-2F ar e given by the followin g formulations:
2Dl 2Dj
16
4
X
8
2F j
y + Z - {\
2F , 6 Y Z
X + Y
6
8r-O-
- Z-
=4
The smallest whole numbers satisfying t he equations in t he center ar e
X = 20, and Z = 14. The results given in T abl e 8.2 show t hat one
cannot always expect t he intensit y rul es to hold. The t heoretical
in tensiti es are ext reme ly useful, however, in making iden tifications in
spectra not yet analyzed.
8.4. Tne Spinning Electron and the Vector Mode1.-With t he
co-development of complex spectrum ana lysis and t he Lande vector
model, it becam e necessary t o ascribe to eac h atom an angular momentum
in additi on to the orbi t al angular momentum of the va lence elect rons.
At first t his new an gular momentum was ascribed to t he atom core
and assigned various values suitable for t he pr oper explanation of t he
various types of spectral lines: singlets, doublets, t riplets, qu ar tets,
quintets, et c. Due to t he insight of t wo Dut ch ph ysicists, Uhlenbeck
and Goud smit ., ' and ind epend entl y Bichowsky and Urey,? t his new
angular momentum was assigned to t he valence elect rons . In order
t o account for doubl et fine st ructure in t he alka li metals, it is sufficient
to ascribe t o t he single va lence electron a spin s of only one-ha lf a quantum
unit of angular mom entum,
s;7r = ~ . 2:'
This half-in t egr al spin is
not to be taken as a qua ntum nu mb er t hat takes different values like n
1 UHLENBECK, G. E ., and S. GOUDSMIT, Nolu runssenschaften, 13, 953, 1925 ; Nature,
117,264, 1926.
2 BI CHOWSKY, F . R. , an d H . C. U HEY, Proc. Nat . Acad. s«, 12, SO, 1926.
122
INTRODUCTION TO ATOMIC SPECTRA
[CHAP.
VIII
and l but as an inherent and fixed property of the electron. The total
angular momentum contributed to any atom by a single valence electron
is therefore made up of two parts: one due to the motion of the center
t*
s*
F IG. 8.S.-Spin a nd orbital m oti on of t he electron on t he cla ssical theory.
of mass of t he electron around the nu cleus in an orbit, and t he other
due t o the spin mo tion of t he electron a bout an axis t hroug h it s center of
mass (see Fig. 8.5 ). Disregarding nuclear spin t he at om core, as we shall ~
see later, contributes nothing to the total angular momentum of the atom.
By analogy with t he quantum-mech anical developments in Chap. IV,
we re turn now to the orbital mod els a, b, c, and d (F ig. 4.8) to find a
«uit a ble method for combining these two angular momenta. For this
Vector
model Q
Vector
model b
s
s
j
1
1
j
j .,1 +
4
j ., t -
i
J" 1
+t
j - l -!
FIG. 8.6.-Vector mod els a a nd b for the co mposition of t he elect ro n sp in a nd or bit.
purpose models a and b are both frequently used Of these models, a
is generally preferred since it gives in many cases, but not always, t he
more accurate quantum-mechanical results. Models c and d have been
rejected because of the many fortuitous rules ne cessarily introduced
to fit the experimental data, and they are of historical interest only.
SEC.
8. 51
123
DOUBL E T PINE S T R UCTU R E
Vector diagrams for the composition of orbit and spin, on models a and b,
are given in F ig. 8.6 for t he two possible states of the d electron .
On m od el b t he spin angula r momentum s . hj21r is a dded vectoria lly to
t he or bital ang ula r m omen tum l· h/ 21r to for m t he resu lt ant j- h j 21r, whe re
j = l ± s, On mod el a t he spin angul ar m om en tum s* . h j21r is a dde d
vectoria lly to t he orbit al ang ula r mcmentum l* . hj21r to form t he resul tant
j * . h/21r, where s* = vs (s + 1) , l* = VlU + 1), j* = v j(j + 1) , and
j = l
s, It sho uld be not ed t hat two is t he maximum num ber of j
values, differing from each other by uni t y , t hat are possibl e on eithe r
mo de l and t hat for s electrons there is but one possibility." For s, p, d,
j , . .. orbits l = 0, 1, 2, 3, .. ' . The q uantum conditions are that
j shall take a ll possib le half- int egral values only, i .e., j = t ,...~, t ,h .. . .
Fo r a d electron l = 2, e = t, l* = V6, and s* = tv'3. The on ly
possible orientations for land s, or for I" a nd s*, arc such t hat j = ~
and~ , and j* = tv'35 and tvf5. Fo r a p electrorrZ = 1, e = t,
l* = V~ and 8* = tV3. T he on ly po ssible or ientations for l an d 8, or
for l* and 8*, are su ch t hat j = -~ and t , andj* =~VI5 and!V3": For
an 8 electron l = 0, 8 = t , I" = 0, and 8* = t V 3. The on ly possib le
value for j is t , and j * = ! V 3.
8.5. The Normal Order of Fine-structure Doublets.- In t he do ublet
energy levels of atomic sys te ms conta ining bu t one valence electron
it is generally, but not alw ays, ob served that the fine-structure level
j = l - t lies deeper than the corresponding level j = l + t. Fo r
example, in the case of p an d d elec trons, 2P j lies deeper than 2P l a nd
2D l lies deep er t han 2D! . T his r esul t is to be expected on t he classical
t heory of a sp inning electron and on t he quantum me chanics. Classically
we may think of t he electron as having an orbital angular momentum
and a spin angular momentum. Due to the cha rge on the electron
eac h of these t wo motions produces magnetic field s.
Due to t he orbital mo t ion of t he electron , of cha rge e, in the radial
electric field of t he nucleus E, t here will be a magnetic field H at t he
electron normal to the plane of the orbit." That t his field is in the direct ion of t he orbital mec hanical moment l may readily be seen by im agini ng
t he electron at rest and the pos it ively charged nucleus moving in an
orbit aro und it. I n t his field t he mo re stable state of a given doublet
will t he n be t he one in wh ich ,t he spin ning electron, t hought of as a
small magnet of moment J.l.s, lines u p in t he direction of H . I n F ig .
8.7 t he electron sp in moment J.l.s is seen to be parallel to H in t he state
j = l - t, and antiparallel to H in t he state j = l
t. Of the two
+
+
1 T he sma ll letter 8 used for electron spin mu st not be confuse d with the sma ll
letter for 8 elec t rons.
2 This field is not apparent to an observer at rest with the nucl eus but would he
experienced by an observer on the electron .
124
I NTROD UCTIO N TO A T OM IC S PEC 7'RA
[C HAP. VIn
possible orientations t he one j = l - -! is classically t hen t he more
st able and t herefore lies deeper on an energy level diagram. On t he
vect or mod el a (see Fi g. 8.6) t he same conclusion is reached and we can
say t hat of t he t wo states t he one for which t he spin moment J.!.. is more
nearl y par allel to H lies deeper.
Although most of t he doubl et spectra of one-elect ron systems ar e
in agreement with t his, t here are a few exceptions to t he rule. In
caesium, for example, t he 2p and 2D te rms are norm al, and t he first 2F
term is inverted. By inverted is mean t t hat t he te rm j = l + -! lies
dee pest on . an energy level diagram. In rubidium t he 2p terms are
norm al, and t he 2D and 2F te rms are inv erted. In sodium and potassium
A j =1 + S
B j= 1- s
FIG. 8 .7. -I llustrating t he m echan ica l a n d m a gn et ic moments of t h e sp inning elect ron
for the two fine-structure states j = I
t and j = I - t. T h e vectors are d rawn a ccordin g to t h e classica l model (b) .
+
t he 2p te rms are norm al, and t he 2D, and very pr obably t he 2F, te rms
are inverted. F or a possible explanation of t he inversions see Sec. 19.6.
., It C2,n be shown quantum mechanically t hat, neglecting disturbing
influences, doubl et levels arising from a single valence electro n will be
norm al. Wh ere resolved, all of t he observed doubl ets of t he boron
group of elements, t he ionized alka line earths, and t he more highly
ioniz ed atoms of t he same type are in agreement with t his.
8.6. Electron Spin -orbit Interaction.- T he problem which next
presen ts itself is that of calculating t he magnitude of t he doublet sepa rations. Experiment ally we have seen t hat doubl et- term sepa rations, in
general, vary from element to element, from series t o series, and from
member t o memb er. Any expression for t hese separations will therefore involve the atomic number Z, t he quantum number I, and the
quantum number n.
A calculation of the interacti on energy due to t he addition of an
elect ron spin to t he atom model ha s been made on t he qu antum mechanics
by Pauli;' D arwin," Dirac," Gordon, ' and ot hers . By use of t he vector
W. , Zeits. f . Phys., 43, 601, 1927.
C. G., P roc. Roy. Soc., A, 116, 227, 1927; A, 118, 654, 1928.
s DIRAC, P. A. M. , P roc. Roy. Soc ., A, 117, 610, 1927 ; A, 118, 351, 1928.
4 GORDON, W., Zeits. f. Phys., 48, 11, 1929.
1
PW
2
D ARWIN,
Ll ,
124
I NTROD UCTIO N TO A T OM IC S PEC7'RA
[C HAP. VIII
possible orientations t he one j = l - t is classically t hen t he more
stable and t herefore lies deeper on an energy level diagr am . On the
vect or model a (see Fi g. 8.6) t he same conclusion is reached and we can
say t hat of t he t wo states t he one for which t he spin moment }J.. is more
nearly parallel t o H lies deeper.
Although most of t he doubl et spect ra of one-elect ron systems are
in agreement with t his, t here are a few exceptions t o t he rule. In
caesium, for example, t he 2p and 2D te rms are normal, and t he first 2F
te rm is inv erted. By inverted is mean t t hat t he te rm j = l + t lies
deepest on . an energy level diagram . In rubidium t he 2p terms are
norm al, and t he 2D and 2F te rms are inverted. In sodium and potassium
B j-l s
A j =l+s
r
FIG. 8 .7.- I llustrating t he m echan ica l a nd m a gn et ic moments of t he sp inning elect ron
for t he two fine-structure states i = I
! and i = I - t. T h e vectors a re drawn a ccording to t h e classical m od el (b) .
+
t he 2p te rms are normal, and t he 2D, and very pr obabl y t he 2F, terms
ar e inverted. F or a possible explan ation of t he inversions see Sec. 19.6.
., It can be shown quantum mechani cally t hat, neglecting disturbing
influences, doubl et levels arising from a single valence electro n will be
norm al. Wh ere resolved, all of t he observed doublets of t he boron
group of elemen ts, t he ionized alkaline earths, and t he more highly
ioniz ed atoms of t he same t ype are in agreement with t his.
8.6. Electron Spin-orb it Interaction.- T he problem which next
presents itself is that of calculati ng t he magnitude of t he doublet separations. Experiment ally we have seen t hat doublet-term sepa rations, in
general, vary from element t o element, from series to series, and from
member t o member. Any expression for t hese separations will therefore involve the atomic number Z, the quant um number I, and the
quantum number n.
A calculation of the inter action energy due to t he addition of an
electron spin t o t he atom mod el has been made on t he quantum mechanics
by Pauli;' Darwin," Dirac," Gordon, ' and ot hers . By use of t he vector
P o\.UL 1, W. , Z eits. f. Ph ys., 43, 601, 1927.
D ARWIN, C. G. , Proc. Roy. Soc., A, 116, 227, 1927; A, 118, 654, 1928.
s DIRAC, P. A. M ., Proc. Roy. Soc., A, 117, 610, 1927 ; A, 118, 351, 1928.
4 GORDON, W., Z eits. f. Ph ys., 48, 11, 1929.
1
2
SE C.
8.6]
125
DOUBLET FI NE S T RUCTU RE
model a semiclassical calculation of t he in teraction energy may al so be
made whi ch leads t o t he sa me result. Because of it s sim plicity t his
treatmen t will be given here. Dirac's qu an tum-mechanical t reatme nt
of t he electro n will be give n in t he next chapter.
On t he classical mod el of hydrogen-like atoms t he single electro n
mo ves in a central for ce field with an orbital 'angular mo me ntum
h
l *- = mr X v
211"
(8. 1)
+
wh ere l * = Vl (l
1), m is t he mass of t he electro n, v it s ve locity, a nd
r the radius vector . According to class ica l electromagnetic t heo ry , a
charge Z e on t he nucleus gives rise to an electric field E at t he electro n
given by
E =
Ze
T'3 r.
(8.2)
M ovin g in t his field t he electron expe riences a magneti c field give n by
H = E X v.
c
(8.3)
Ze
H = - 3 r X v.
cr
(8.4)
From t hese t wo equations,
Applying Bohr's quantum ass umption [Eq. (2.20)]
27rmr X v = l* h,
(8.5)
t he field becomes
(8.6)
In t his field t he spinning electron, lik e a small magn eti c top, undergoes a Larm or precession around t he field directi on. From Larmor 's
t heorem [Eq. (3.58) ] t he ang ular ve locity of t his precession sho uld be
giv en by the product of t he field stre ngth H and t he rati o between t he
magneti c and mechanical mom en t of t he spinning electron ;'
WL
= H . 2- e = l * -h . -Z e . -13 . 2-e ·
2mc
211" mc r
2mc
(8.7)
Th i s i s just twice the ordinary Larmor precession, giv en in E q. (3.58)
1 The ratio between t he mag netic and mechanical moment of a spinn ing electron
is just twice the cor respondi ng rat io for the elect ron's orbital motion. T his is in
ag reement wit h result s obtained on the quantum mechanics and accounts for the
anomalous Zeeman effect to be treated in Chap. X .
126
INTROD UCTION TO ATOMIC SPECTRA
[CHAP.
VIII
A relativistic treatment of this problem by Thomas' has revealed, in
addition to the Larmor precession W L, a relativity pr ecession WT, one-half
as great and in the opposit e direction. The resultant precession of the
spinning elect ron is therefor e one-half W L, i.e., ju st equal to t he ordinary
Larmor precession:
(8.8)
Now, the in teraction energy> is just t he product of the precessional
angular velocity Wi and the proj ection of t he spin angular momentum
on l*:
D. WI .s
With the value of
Wi
=
Wi •
h
s* 21r cos (l*s*) .
·(8.9)
from E q. (8.8),
Z e2
h2 1
2 2 ' 4 2 • 3 ' l*s* cos (l*s*) .
m e
1r r
D. WI,s = -2
(8.10)
In this equation for t he in t eracti on energy t he last two factors are
still to be evaluat ed . In general t he electron-nuclear dis tance r is a
functi on of Z, n , and l and changes con tinually in any given stat e.
Be cause t he in t eracti on energy is small com pared with the t ot al energy
of the electron's moti on the average energy D. WI,. may be calculated by
means of per turbation t heo ry. In doin g t his, only t he average value
(~) need
be calculated .
From per turbation t heory and t he quantum
me chanics (see Sec. 4.9),
(~)
zs
(8.11)
where al is t he radius of the first Bohr circ ular orbit,
h2
al = 41r 2me 2 '
(8.12)
For t he las t fa ctor of Eq. (8. 10) we t urn to the ve ctor model of t he
atom. In calculating the precessional frequen cy of s* around... t he field
produced by t he orbital mo tion t he vector l* was ass ume d fixed in space.
L. H ., Nat ure, 117, 514, 1926.
The interaction ene rgy here is just t he kineti c ene rgy of the elect ron 's precession
around t he field H . If W = V w2 represent s t he kinetic ene rgy of t he spin of the
electron in th e ab sence of t he field H , and W' = V (w + w')" the kin eti c energ y in th e
pr esen ce of t he field , t hen the change in ene rgy is jus t Jl W = W' - W = V W / 2 +
Iww' . Sin ce I rem ain s constan t and w > > w', t he first t erm is negligibly small and
the ene rgy is given by th e product of w' and t he mec hanica l mom ent I w = s*hJ21r = p••
1 THOMAS,
2
..
SE C.
12i
DOUBLET FINE S T R UCT URE
8.6]
One mig ht equally well have calculated the precession of the orbit in t he
field of the spinning electron. It is easily shown that this frequency is
just equal to the ordinary Larmor pre cession and to t he precession of the
electron around l*. In field-free space both orbit and spin are free to
move so t hat l* and s* will precess aro und their
.*
mechanical res ultant j* . By the law of conservaJ
tion of angular momentum thi s res ultant j * and
/
hence the angle between l* and s* must remain
/
\
invariant. The ve ctor model t herefore takes the
/
\
form shown in Fig. 8.8. With the angle fixed the
/
\
/ _ - -t-- \""
cosine does not need to be averaged and l*s* cos
\
~--I--\
\
/
(l*s*) is calculated by the use of the cosine law
j * = l*2 + S*2 + 2l*s* cos (l*s*)
\ /
(8.13)
\
/ \
from which
l*s* cos (l*s*)
j*2 _ l*2 _ S*2
2
.
(8.14)
Substituting Eqs. (8.11) and (8.14) in Eq. (8.10),
-
Z e2
h2
za
~ WI ,s = 2m 2e2 . 4 1r 2 • a~nal (l + 1) (l + 1)
j*2 _ l*2 _ S*2
2
(8.15)
FIG. S.S. -C lass ica l pre cession of electron sp in 8 *
and or bit I " around their
mech anical resultan t i" .
Upon substituting the Rydberg constant R = 21r 2me4 / eh 3 and the
sq uare of the fine-structure constant a 2 = 41r 2e4 / e2h 2, the energy becomes
Ra 2ehZ 4
j* 2 _ l*2 _ S*2
(8.16)
~ W l, 8 = n 3l(l + t)(l + 1) .
2
and, dividing by he, t he term shift in wave numbers becomes
Ra 2Z 4
j*2 - l*2 - S*2 = -r.
~TI . 8 =
3
n l (l + 1 )(l + 1)
2
(8.17)
T his spin-orbit interaction energy, often referred to as the I' factor,
is written in short as
(j *2 _ l*2 _ S*2)
(8.18)
I' = a '
= a . l*s* cos (l*s*),
2
where
Ra 2Z 4
a = n 3l (l
+ 1 )(l + 1) cm-l.
(8.19)
M easured fr om t he series limit down, t he term va lue of any finestructure level will be given by
T = To -
r,
(8.20)
128
INTROD UCTION TO ATOMIC SP EC T RA
[CHAP. VIII
where T o is a hypothetical term va lue for the center of gravity of t he
doub let in question and I' gives the shift of each fine-st ru ct ur e level from
To. r va lues for 2p , 2D, and 2F terms are shown in Fig. 8.9. For an s
orb it l = 0, j = s, and t:.T = O. This is in agreement with observation
that all S states are single. The two states of a doub let are seen by t he
figure to be given by the difference bet ween t heir r va lues. It is to be
noted that Z occurs in the numerator of Eq. (8.19) , and nand l occur in
t he denominator. This is in agreement with experimental observations
1~
2P%
I
,..-
I
I
To ~------­
I
\
2'D
/
5Jz
CI
--(- - - - - - -
\
\
\
30/
,
/2
I
\
\
,
t hat doub let -term separations (1) increase with increasing atomic number,
e.g., in going from Na I to K I , or from Na I to Mg II; (2) decrease with
increasing n, i. e., in going to higher members of a given series; (3) decrease
with increasing l, i. e., in going to different series p, d, I, etc. As t he interaction energy gets smaller and smaller in a given series, I' will approach
zero and the levels will gradually come together at their center of gravity
T o.
8.7. Spin-orbit Interaction for Nonpenetrating Orbits.- In ap plying
t he spin-orbit-interaction energy formula [Eq. (8.17)], as just der ived, to
t he observed data, it is necessary t hat we first simplify the exp ress ion
and substitute the known phy sical constants. For any give n doub let,
land s have t he same val ues whereas j = l + ! for the upper level and
l - ! for the lower level. The successive sub stitution of t hese values
for j in the last factor of Eq. (8.17) gives, by subtraction,
Ra 2Z 4
t:.v = n3l(l + ! )(l
1) = n 3lRa(l + 1) cm".
2Z 4
(
+ 1) l + 2
Inserting the value of the Rydberg const ant R = 109737 cm ?
fine-structure constant a 2 = 5.3 X 10- 5,
Z4
t:.v = 5.82 n 3l (l
+ 1) em- I.
(8.21)
and the
(8.22)
For hydrogen-like systems the effect ive nuclear charge is given
simply by the atomic number Z.
SEC.
8.8]
129
DOUBLET FI NE S T RUCTU R E
In t he previous cha pters we have seen how for other atomic systems
the deviations of the term values from t hose of hydrogen-like atoms are
attribut ed to a polariz ation of t he atomic core or generally to a quantum
defe ct [see Eq. (5.4)],
T =
RZ2
RZ2
(8.23)
(n - fJ. ) 2 = -n-;ff'
where Z = 1 for neut ral atoms, 2 for singly ionized atoms, etc. Inst ead
of at tributing the in creased binding of t he electron t o a defect in t he
qu antum number n, one may argue t hat it should be attribut ed to a
screening of t he va lence electron from t he nu cleus by t he interve ning
core of elect rons, and t hat Z should be repla ced by Zeff where Zeff = Z - CT,
Z is the atomic number, and CT is a screening constant :
(8.24)
By exactly t he sa me reasonin g one may write t he fine-structure
doublet formul a [Eq . (8.22)] as
AlJ =
Rcx 2 (Z - S)4
(Z nal(l +I) = 5. 82 n 31(l
S)4
+ 1)'
(8.25)
Mo st of t he doubl et s to whi ch t his formula applies are kno wn only for
singly and multiply ioniz ed at oms. Although its general application
will be left to Cha p. XVII on I soelectroni c Sequences, it should be
remarked here t hat t his for mula gives doubl et intervals in remarkably
good agree ment with experime ntal observations.
8.8. .S pin - orbit Interaction for Pene tr ating Orbits.- I n t he pr ecedin g
cha pter on penet rating and nonpenetrating orbits we have seen how
pen etrating or bits may be considere d as ma de up of two par ts, an inside
segment of an ellipse and an outside segment. In at t emp ting to apply
Eq. (8.25) to t he doublets of pen etrating orbits, much bett er agreement
with t he observed va lues is obtained, especially for t he heavier eleme nts,
by aga in considering separately t he inner and t he outer par t of t he orbit
(see Fi g. 7.6). Wh at ever atomic model is formul at ed, t he electron in a
deeply penetrating orbit is by far t he greater par t of t he t ime in an outer
region where t he field is nearl y hydrogen-like. If t he elect ron remain ed
in an oute r orbi t like the outer segment, t he doublet formula [Eq. (8. 21)]
would be
(8.26)
whereas, if it remained in an inner orb it like t he inner segment, t he
formul a would be
RCX 2Z 4
(8.27)
AlJ i = n~l(l + ' I )
130
INTRO D UCTION TO ATOM I C S P ECTRA
[C HAP. VI II
To bring t hese t wo formulas t oget her for t he actual orbi t , t he motion
in ea ch of these segments is weighted according to the time spent in each.
Now the time t required to traverse t he whole path is so nearly equal to
t he t ime to requi red to traverse a complete outer ellipse t hat we may
write, t o a first approximation,
n 3h 3
t = to = 411'" 2me
0
4Z2
(8.28:·
OJ .
This equation for t he period of an elect ron in a K epler ellipse was left
as an exercise at t he end of Cha p. III. The time required to traverse a
completed inn er ellipse is
(8.29)
Now t he resultant frequ ency separat ion ~ V for t he actua l orbit will
t imes t he fractiona l time tof t spent in t he outer segment, plus
~V i times t he fractional time t;it spent in t he inn er segment :
be
~ Vo
~V
=
tv
6.v ot
+ ~V '·-t.t ·
(8.30)
Making the ap prox imation that tv = t and subs tit uting the values of
t, and t i from Eqs. (8.26), (8.27), (8.28) , and (8.29), we get
~vo, ~V i,
A
_
uV -
Ra2Z~
+ 1) (Z'.
n~l (l
+ ZOJ .' ) em
- 1
.
(8.3 1)
For heavy elements t he effective charge Z ie at deepest penet ra ti on is so
much greater t han t he effective charge Zoe outside, t hat t he formula may
be simp lified to
_ Ra 2Z;Z;
- 1
~V - n~l(l + 1) em .
(8.32)
This equation was derived from t he qu an tum t heory and used by Lande!
before t he ad vent of t he spinning electron an d t he newer qua ntum
mechani cs. In calculating Zi for a number of atoms Lande showed t hat
t he pen et ra ti on in many cases is almost comp lete, Z i being almost equa l
to t he atomic nu mb er Z (see T able 17.4A ). It is to be noted t hat no is
t he effect ive qu an tum nu mb er.
Inser ting screening constants for each of t he Z' s, in Eq. (8.32),
_ Ra 2(Z - 80) 2(Z - 8i)2
-1
(8.33)
~V n~l (l + 1)
em .
The application of t his formula to observed doubl ets, in genera l, will be
set aside, to be taken up aga in in treating isoelectronic sequences of atoms
in Cha p. XVII.
1
LANDE, A.. Zeits , f.
Ph y ~. ,
26, 46, 1924.
SEC.
8.8]
DOUBLET PI NE S T RUCTU RE
131
P roblems
1. Compute doublet-term sepa ra t ions for t he non penetrating 2p states of lith ium
and singly ioniz ed beryllium. Assum e complete screening by t he core electrons.
Compare t he calcula t ed values with the ob serv ed va lues given in Sec. 8.l.
2. Determine th eoreti cal intensit y ratios for t he doublet t ransiti ons 2F ,_2G j •
3. Construct vector-model diagrams for 'F!, 2F l , 'G! and 'G~ states ba sed on mod el
a, Fig. 8.6.
4. Determine t he electron spin-orbit pr ecession fr equ ency w/ 27r for a 41 state in
potassium. Assume com plete screening by t he 18 core electrons.
6. Comput e a t heo retica l doublet separation for the 6 2P state in caes ium. Assume
complete pen et ra tion wh en t he electron is insid e t he core, i .e., Si = 0, an d pe rfect
screening wh en it is ou t side, i .e., So = Z - 1. T he effective quantum nu mber no
can be determined from t he obse rve d te rm values (use t he center of gravity of th e
doublet ). All ot he r facto rs rem aining the sa me, wh at value of Si will give the obse rve d
doublet separa t ion ?