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INTRODUCTION
TO
ATOMIC SPECTRA
BY
HARVEY ELLIOTT WHITE, PH.D.
A ssi stant P rofessor of Phueics , at th e
University of Califo rnia
McGRAW-HILL BOOK COMPANY, INc.
NEW YORK AND
1934
LONDON
CHAPTER X
ZEEMAN EFFECT AND THE PASCHEN -BACK EFFECT
10.1. Early Discoveries and De velopments.- I n t he year 1896
Zeeman 1 discovered t hat when a sodium flam e is placed between t he
poles of a powerful elect romagnet t he t wo lines of t he first principal
doublet are considerably broadened. Lorentz pointed out t hat t his
phenomenon is in harmony with t he electron t heo ry of mat t er and
radiati on proposed by him self. H e predicted from t heoretica l considerations t hat t he light from t hese lines should be polariz ed by t he
magn eti c field , circularly polarized if viewed in a dir ection parallel to
t he lines of for ce, and plane polariz ed if viewed at ri ght angles to t he field.
These predictions were later verified by Zeeman by mean s of Nicol
pri sms as analyzers.
It has been shown fr om t he simple classical t heory of Loren tz t hat,
if a ligh t source be placed in a magnetic field , t he moti ons of t he elect rons
should be modified in such a way as to cha nge t heir periods of motion.
In t he simple case of an electron movin g in a circular orbit, t he plan e
of which is normal to t he field dir ecti on H , t he electron will be speeded
up or slowed down by an amount which depends up on t he magnetic
field strength H , t he ma ss and charge on t he electron, and t he velocity
of ligh t. 1r. class ical treatment of t his pr oblem shows t hat if 1'0 represents
t he orbit al frequen cy of t he electron wit hout field , t he frequency in
t he pr esence of a field will be given by 1'0 ± ~ v, where [see Eqs. (3.50)
and (3.59)]
~l'
eH
= --.
411"mc
(10.1)
If t he field is normal t o and up from t his page of t he book, t hen
electrons moving in a counterclockwise dir ection in t he plane of t he
paper are speeded up by an amo unt ~v and t hose movin g in a clockwise
direction are slowed down by the same amount . It will now be shown
how t hese modified mo ti ons have been employed in giving a classical
explanation of t he normal Zeeman effect .
In the following explan ation we are concerned with an ass embly
of electrons mo ving in orbit s oriented at random in space. We start
by selecting one of these orbits and resolve t he motion into t hree components along three mutually perpendicular axes (see Fig. 10.la). The
1 ZEEMAN ,
P. , Phil. Mag. , 6, 43, 226, 1897.
149
150
[CH AP. X
I NTROD UCTION TO A T OM IC S PEC T RA
m oti on of the electro n is her e pictured as consisting of three sim pleharmonic mo tion s, one along the x axis, one along the y axi s, and on e
along the z axi s. When this resolution is repeated for all of the electrons,
the average am plitude of all the motions along each axi s will be t he
same . If n ow, in t he absence of a field , t he electrons are em itting
light an d we obse rve t he radia tion in t he x directi on, onl y t he ligh t
from t he y and z motions will be observed . Sin ce t hese motion s a re proj ections fr om all orien tations , t his ligh t will be unp olarized. Thus, in t h e
a bsenc e of a field, the light observed in any direction is unpolarized.
y
y
y
• (cJ
(0)
(d)
Y--H-- - X
z
y
It--* , *I --X
Z
)C---+-- X
=Ef1=. j · +·;i=s
(e)
~
(A)
e
-l'-
5
liJ
5
(5)
FIG . ID.I.- S ch cm a ti c d iagrams for t he classical explanation of t he no r mal Zeeman e ffect .
If now we return to Fig. 10.la, which represen t s t he t hree h armoni c
compo nents of a single orbital electron, a nd apply a magne tic field
in t he directi on of t he z axis, t he x a nd y motions will be modified a nd
t he z m otion will rem ain unchanged. M oving t ransvers ely to t he field,
t he x a nd y mo tions will take the form of ro settes as shown in Fig. 10.lb. 1
This is t he same t y pe of motion as one en counter s in t he Faraday effect .
These ap pare ntly com plex motions ca n be described to be t ter advantage in terms of circ ular motions somew hat as follo ws. The sim pleharm oni c motion giv en by t he y com pone nt of Fig. 10.la, for example, is
eq uivalent to t he resultant of t wo equal but opposit e circ ula r mo tions
y+ an d y- as show n in Fig. 1O.lc. Simil arly t he x com ponen t ca n be
represented by t wo opposite circ ular motions x+ and x-. When the field
is applied, t he x+ and u" rot ations will be speeded up by a n a mount
t111 [see Eq. (10.1)] and t he x- a nd y- ro t a ti ons will be slowed down by
t he sa me amo unt. (T he faster x+ mo ti on combine d with t he slower xm otion results in F ig. IO.lb.) The x+ an d y+ mo tions and t he x- a nd y1 The electrons are he re movi ng at right angles to t he field , and t hey t he refo re
expe rience a force at righ t ang les to their moti on t he directi on of wh ich dep ends upon
the directi on of t he field and t he direction of t he motion.
SEC. 10.1]
ZE E M A N EFFECT AND T HE PASCHEN-BACK EFFECT
15 1
motions are now com bined, as shown in Fig. IO.ld, t o form plus and
m inus re sultant s. Thus t he m otion of a single electro n in a magnetic
field is repr esent ed by a lin ear motion along t he field direction wit h
unchanged frequency Po and two circular motions at right angles to
t his, one with the freq uency Po + ~P and the other with frequency
Po ~P.
On summing up such motions for all of the electrons, the
resu lt will be the same as if one -third of t he electrons are moving with
unchanged freq uency along t he z axis, one-third moving with a counterclockwise circ ular motion norm al to z of frequ ency Po + ~P, an d t he other
t hird moving with clockwise circ ula r motion normal to z of frequency
Po ~ P (see Fig. IO.le) .
We are now interested in t he nature of t he light t hat should classically
be radiated from these motions. Whe n viewed in t he direction of t he
field , only t he circula r motions are observe d and t hese as right- and
left-handed circularly polarized light (F ig. IO.IE). Since light is a
transverse wave motion, t he z motions will not emit ligh t in t he field
direct ion . When viewe d perp endicular to t he field, t he z motions are
observed as plane-polarized light with the electric vector vibrating
parallel to the field, and the circular motions, seen edge on, are observed
as plane-p olarized light with the electric vector at right angles to t he
field. A spectrum lin e vie wed normal to H sho uld therefore reveal
t hree plane-polarized components (see F ig. IO.I A), a center u nshi ft ed
line and two othe r lines eq ua lly displ aced one on either side. This is
called a normal tri plet. T he abbreviation p stands for a vibration
parallel to the field and s (senkrecht) stands for a vibration normal to
t he field, T he experimen t al agreement with the dire ction of rotation
of the circ ularly polarized components is proo f that the radiation is due
to moving negative electric cha rges .
In Zeeman's early inv estigations he was not able to split any lines int o
doubl et s or triplets, but he did find t hat t hey were widened and t hat
t heir outside edges were polarized as predi cted . H e was later able to
photograph the two outer components of lines in a number of the elements, Zn, Cu, Cd, and Sn, by cutting out the p components with a
Nicol pr ism. P rest on ! using greater dispersion and reso lving power
was able to show not only t hat certain lines were split u p into triplets
when viewed perpendicular to t he field, bu t t hat ot hers were split int o
as many as four and eve n six components (see Fig. 10. 2a). He also
point ed out that the pattern of all lines (usually called Z eeman patterns)
belonging to the same series of spe ctrum lines was t he same and was
.characteristic of that series. This is now know n as Pr eston' s law. With
Prest on 's law firml y established, t he Zeem an effect has been , and still
is, a powerful too l in spectrum analysis .
1
PRE ST ON ,
T. , P hil . Maa.. 46, 325, 1898.
152
I NTROD UCTION TO A T OM IC S PEC T RA
[C HAP.
X
From Lorentz 's classical trea tment of t he Zeeman effect (F ig. 10.1)
t he shift AI' of t he s components from t he un shifted p comp onent (see
E q . 10.1) is given by
He
AI' = - = 4.67 X 10- 5 • H cm- 1 = L cm- 1 ,
471'"mc 2
(10. 1a)
where AI' is in wav e numbers, H is t he field in gauss, c is t he velocity
of light , and e is t he cha rge on t he electron in electrostatic uni ts. Zee-
..
I
i
II
I
..
. -,
8 Wove-Lengths
..,p
~
l, I' 'r~I
I',
II'
,
~
:::0
oo::t". . _
- ---_._-_
<D
;:0
~
IIId
~ IIl!
I
I
II
"
I p
!
Zeemon Effect
In
.
:0
-e-- - --..."..
II
I
1\,I
'I P I
,
,
-Ii
~
;
p
,
the Chromium Spectr um
FIG. l O.2a. - Anom alous Zeem a n effect as observed in t he neutral spectrum of ch ro mi u m.
(A f ter Babcock.)
Zinc Singl et
Sodium Principal Doublet
No field
Weak f ield
l....1...J
Nor ma l T r ip let
L...L..J
Anomalo us
L....L.....J
Patterns
Zinc Sharp Trl ple+
No f ield
WeClk field
L...l-J
l..-L-J
l-J-l
Anoma lous PaHerns
FIG. 1O.2b.- N or mal and anomalous Zeem an effect.
field.
Viewed pe rpen dic ula r to t he m agn et ic
man pat terns showing just t hr ee lines with exactly t hese separations .
are called normal triplets. All other line groups, as, for example, t he
complex pa t terns obser ved in t he chromium spect ru m (see Fig. 10.2a),
SEC.
ZEEMA N EFFECT AND THE P A SCHEN-BA CK EFFECT
10.21
153
a re said to exhibit t he anomalous Z eeman effect. One of t he mo st important of t he early in vestiga tions of t he ano malous Zeem an effect was
ca rried ou t by Paschen and Runge." Each member of t he principal
seriesof sodium, copper, and silv er wa s observed t o have 10 components
as shown in F ig. 10.2b. The sh arp-series t riplets in mercury are still
more complicated, the strongest lin e in each triplet having nine components, the middle line six, and t he weakest line only t hree. This
last line does not form a normal triplet since it h as twice the nor mal
separation.
P:--1.----~---+----r-
I
I
I
I
5
P:-r-..-I~
5
I
}~~~m
3 p- 3 S
I
I
I
,
-0
I
I
I
I
I
I
I
0
I
P:--T-y----r---'---'---.L-.j-,.-..--)
5
I
I
I
,
- 0
+CI
I
I
'I
P
I
P:-----r---L.--.--5
I
5
I
o
I
I
I
I
I
I'
+0
I
I
I
5,---- -.'- -"-- "'T1-'" -
Normal Tri plet
Nor mal Triplet
FIG. l O.3 .- Sch em a t ic r epresen t a t ion of anomalous Zeem a n patterns vi ew ed pe rpe n dic u la r to t he field and showing polari zation (p-components above and s-components
be lo w) . r ela t ive in t ensi ti es (h eig hts of lin es) . a nd in terv al s (dot t ed lin es) .
In 1907 Runge made an import ant contribution t o a t heo re tical
explanation of the anomalous Zeeman effect by announcing that all
known pat terns-could be expressed as rational multiples of the n ormaltriplet sepa rations . If, for exa mple, L represen t s t he shift of t he s
componen t s from t he unshif t ed line, as given by Loren tz's formula,
t hen t he princip al- or sharp-series doublet s (see Fig. 10.3) m ay be
expressed as
± i L, ±tL
±lL, ± ! L, ± %L
28i - 2P i,
281 _ 2P~ ,
and the principal- or shar p-series triplet s a re express ed as
81
81
38
1
3
3
-
3P 2 ,
3P 1,
3P O,
0, ±!L ± l L ± ! L ± -! L
± !L ± tL
0, ± t L.
± ~L
t his is now known as Runge's law.
10.2 . The Vector Model of a One-elec tron Sys tem in a Weak M ag n etic Field.-Soon afte r t he discovery of t he a nomalous Zeeman effect
1
P ASCH E N ,
3. 441. 1902.
F ., an d C.
R UN GE ,
Astrophy s. J our., 16, 235, 333, 1902; Phys. Z eits.,
154
INTROD UCTION TO ATOMIC S PEC T RA
[CHAP. X
came the development of the Lande vector model of the atom and the
calculat ion of t he famous Lande g factor . The accuracy with whi ch
t his model, with it s empirical rules, accounted for all observed Zeeman
pat terns, and predi cted others which were later verified, is one of t he
No Field
Weak Field
~-----m=t
_.2
Z
FIG . l OA .- Split ting u p of an ene rg y lev el in a weak m a gn eti c field.
drawn for t he case wh er e j = ~ .
This figure is
marve ls of scientific history. With t he adv ent of t he spinning electro n,
and later t he qu an tum mechani cs, Lande's vector model gave way to a
more satisfactory and, at t he same t ime , a simpler semiclassica l model.
It is with t his simplified model and its adequate account of t he Zeeman
effect t hat we are concerne d in t his
cha pt er.
Experimentally it is obse rve d that
in a weak field each spectrum line
is split up into a number of com ponents forming a symmetrical Zeeman
pattern, and t hat, in general, t he wid th
of any give n pattern is no t greatly
different from that of a normal t riplet .
Theoretically t his effect is at t ributed
to a splitting up of t he energy levels
s*
into
a number of pr edetermined
FIG. 1O.5. -C lassica l p recession of a
single valen ce ele ct ron around t he field equ ally space d levels (see F ig. 10.4).
d irection H .
T ransiti ons betw een t wo sets of t hese
levels, subject t o certain selection and intensity rules, give rise to t he
observed spect ra l frequencies .
Before at tempting a calculation of t hese Zeeman level s it is well
t hat we formulate some picture of t he atom in t erms of t he semiclass ica l
model. In Fig. 8.8, we have seen t hat t he orbit al an gular-momentum
vector l* and the spin angular-momentum vecto r s* precess with uniform
speed around their resultant j *. Wh en t he atom is placed in a weak
magneti c field, t he magneti c moment }J.j associated with t he total mec ha nical moment Pt = j *hj 21r causes t he atom to pr ecess like a to p ar ound
SEC. 10.3)
ZEEMAN EFFECT AND 1'HE PASCHEN-BACK EFFECT
155
the field direction H (see Fig. 10.5). The qua ntum conditions imposed
upon this motion (see Sec. 9.6) are t hat the projection of the angula r
momentum j *h/27r on the field direction H will take only t hose values
given by mh/27r, where m ±!, ±1, ±t . . . ,±j. In other words t he
projection of j* on H takes half-int egral va lues fr om + j to - j only .
T he discret e orientations of t he atom in space, and t he small change
in energy due to t he precession, give rise to t he various discrete Zeeman
levels. While the number of t hese levels is determ ined by the mechanical
moment j *h/27r, t he ma gnitude of
j*
t he sepa rations is determined by t he
,
field strengt h H and the magneti c
/ '>.
moment u. In field-fr ee space an energy
/1
",
level is defined by t he three quantum
~:'- 7r-<.- - - - - - , 1*
numbers n, I, and i- In a weak magne"7- -t ic field an additional orfourth quantum
* ~A~\ _~
number m is necessary to define the
S
- . \ ._ _
state.
'"
\
10.3. The Magnetic Moment of a
"
Bound Electron.- To determine the
/ 1 ~,
magnitude of t he separations between
/ 1,'1\"
///! 1\ " ' ,
Zeeman levels, it is essential t hat we
// 1 ' I \ , "
/
1' 1 \ \
"
first det ermine t he t otal magnetic
/ 11 I' 1 \ '' " "
/
moment of t he atom. In t he simplest
/ ~\ I
I
-J \
,
"
/
{:-- -: ..... 1- -\ - - \
',..us
case to be considered here t he atom I- - - ---':::-T=-=F~-'\ ~- - - -~
core and nucleus will be ass umed to --- - - - --;""-_ J
L _ _ -- --I
,I ', " II- \\
/
have zero magnetic and mechanical
I
moments so that any moments attriJ
I' ,
\
/
/
I '
\
I
buted to the atom mu st be assigned
J
II ', ' \~\ I /
I
to a single valence electron. Later it
I
1- - - -<- >:L1 - - - - , :,u.
will be shown where t hese assumptions
--- - ::uT >--- Is
are justified.
FIG. lO.6.- Vector m od el sho win g
According t o t he classical t heory, t h e m a gn e ti c an d m echani cal m oments
the ratio between t he magneti c and of a sin gle valen ce ele ct ro n .
mec hanical moment s of an electron in an orbit [see Eq. (3.52)] is given by
,
).1.1
e
PI = 2mc'
(10 .2)
J ust a s observations of the fine structure of spectrum lines show
that the mechanical moment of t he spinning electron is given by s*h/27r ,
where s* = V s(s + 1) and s = ! , so the anomalo us Zeeman effect
shows that t he ratio betw een t he magnetic and mechanical moments
for the spinning electron is just t wice that for the orbital-motion, i. e.,
).I..
p.
= 2_ e_ .
2mc
(10.3)
156
I NTROD UCTION TO A T OM IC SPEC T RA
[C HAP.
X
This result has also been der ived t heo retically on t he qu antum me chanics
(see C hap. IX).
A schematic vector di agram of t he magn eti c and mech anical moments
is shown in Fig. 10.6. H ere it is seen t hat t he resul t ant magneti c
mo me nt 1J. 1•• is not in lin e with t he resul t ant mechanical mo me nt j* h/ 27r.
Since t he resul t an t mechanica l mom ent is in varian t, 1*, s*, 1J. 1, IJ.. , and
1J. 1•• precess around j*. As a resul t of t his precession, only t he com ponent
of 1J. 1•• parallel to j* contributes to t he magn eti c moment of t he atom.
This may be seen by resolvin g 1J. 1•• in to t wo compone nts, one parallel to
j * and t he other perp endicular. The perpendicular com pone nt, owing
to t he continual change in dir ecti on, will average out t o zero . The
parallel com pone nt lJ.i may be evaluat ed as follows:
By Eqs. (10.2) and (10.3) IJ. I and IJ.. ar e give n as
IJ. I
h
e ergs
- -and
27r 2me gauss
= 1*- ' -
h
e erg s
- - '
27r 2me gauss'
= 2· s*_ · -
IJ..
(10.4)
and t he ir com pone nts along j* are given as
Component
.
Component
h
2~ cos (l*j *)
7r me
h
e
= 2· s*2 . -2 cos (s*j* ).
7r me
IJ.l
JI. .
= 1*2 .
(10.5 )
Adding t hese, we obtain
IJ."
1
= [1* cos (l*j *)
+ 2s* cos
(s*j*)]~
. _ c_ .
27r 2me
(10.6)
Since t he last t wo fact ors in t his equation are equivalent t o one Bohr
magneton [see Eq. (3.57)], t he qua ntity det ermined by t he bracket
gives t he total magneti c mo me nt of t he atom i~ Bohr magn et on s. . This
bracket term is readily evaluated by setting it equal to j* t imes a constant
g,
j * . g = 1* cos (l*j *)
+ 2s* cos
(s*j*) .
(10.7)
Making use of the vector mod el and t he cosine law t hat
S*2 = 1*2
+ j*2 -
we obt ain
21*j * cos (l*j*) ,
1* cos (l*j* )
j* 2 + 1*2 _ S*2
2j *
.
s* cos (s*j*)
j *2 _ 1*2
2j *
Simila rly,
+ S*2
Substi tuting t hese t wo cosines in E q. (10.7), :we get
j *2 S*2 - 1*2
g = 1+
2j*2
;
+
•
(10.8 )
(10.9)
(10.10)
(10.11)
SEC.
lOA]'
ZEEMAN EFFECT AND THE PASCHEN-BACK EFFECT
in terms of the quantum numbers land j and the spin
=
g
1
+ j(j + 1) + 8 (8 + 1) 2j(j + 1)
l(l
157
8,
+ 1).
(10.12)
The importance of t his g factor cannot be overemphasized, for it
gives directly t he re lative separations of t he Zeem an levels for t he
differe nt terms.! We shall now see how this comes about.
10.4. Magnetic Interaction Energy.- By Eqs. (10.6) and (10.7) the
ratio between the total magnetic and mechanical moments of the atom,
J.L j and Ph is just
J.L j
Pi =
e
g . 2mcf
(10.13)
where Pt = j*h /271".2 The precession of j * around H is the result of a
torque acting on both l* and 8 *. Due to the electron's anomalous sp in
magnetic moment, 8* tends to precess twice as fast around H as does l*.
If t he field is not too strong, the coupling between l* and 8* is sufficiently
strong to maintain a constant j*, so that this resultant precesses with a
compromise angular velocity, by Larmor's theorem [Eq. (3.58) ], given
by g times the orbital precession angular velocity
w
L
=
e
Hg- ·
2mc
(10.14)
The t otal energy of the precession is given by the precessional angular
ve locity WL times the component of the res ultant me chanical moment
j* h/271" on t he ax is of rotation H: 3
LlW
=
:Lj*~
cos
271"
(j*H)
=
u . g_e
_j*~ cos
2mc 271"
(j*H ).
(10.15)
1 The values of g giv en by Eq. (10.12) are exactly the same as those given by
Lande's model.
2 In any expe riment lik e the St ern-G erla ch experimen t (Z eits. f . Phy s., 8, 110,
1922), performed for t he purpose of det ermining the magneti c a nd mechanical mom ent
of the a tom, the moments J1. j a nd Pi a re ori ented at some a ngle with the field just as
in the Zeeman effect (see Fig. 10.7). Wh at on e m easures in t his exp eriment is the
component J1. of the resultant magneti c moment along H. By theory we say the
component of J1. j will be J1. = J1.j cos (j *H ), and th e component of j*h/27r along H will
be mh/27r, where m takes values differing from each other by unity from m = +j
to m = - [ ,
3 The magnetic energy can be considered as the energy of a permanent m agn et of
moment J1., at an angle 6, in the field H , or as th e added kin eti c ene rgy of the elect ron's
orbital mo tion. If , in the ca se of a cir cular orbit normal to th e field , E represents
the kinetic energy before the field is applied , and E' = H (w W L ) 2 the kinet ic energy
after, then t he change in energy is ju st t;E = E' - E = Vwl
IwwL. Since the
added field does not change t he size of th e orbit, I remains cons t ant . With w > > WL,
the first t erm is negligibly small and the ene rgy change is given by the product of the
me chanical moment, I w = j*h/27r, and W L .
+
+
158
I NTROD UCTION TO A T OM IC S PEC T RA
[CHAP. X
In terms of the magnetic quantum number m, j *h/211" . cos (j *H) is just
equal to mh/211", so t hat!
e
h
eh
6.W = H· g'm' - = m ' g ' H - - ·
2mc
211"
411"mc
(10.16)
Dividing by he, the in teraction energy in wave numbers becomes
6.W
- -6.T
he -
=
He
411"mc-
m . g . - - ., em- i.
(10.17)
With g = 1 t his equation redu ces to Loren tz 's class ical formula [Eq.
(10.1)].
Sin ce t he field H is t he sa me for all levels of a given atom, it is convenient to express the Zeeman splitting in te rms of what may be called
the Lorentz unit, L = He/47l"mc2 , and writ e simply
- 6.T
= m .
g . L cm-l.
(10.18)
It should be emphasized t hat 6.T is t he cha nge in energy for each m
level from t he original level, and t hat the shift is proporti onal to t he field
st rengt h H. Wi th m = ±t, ± ~, ±t, . . . , ±j , t he g factor is seen
t o be of primar y import an ce in t he splitting of eac h level. Valu es of
t he g fa ctor for doublets are given in Table 10.1 and, along with t hem,
values of t he corresponding splitting fact ors mg.
T AB L E ID.l.- T H E LANDi: g FACTORS AND THE SPLlTTiN G FACTORS mg FOR D O U B L E T
TERMS
l
T erm
D
2S!
2P !
mg
g
~
±I
1
I
i
i
H
2P ~
2D ,
2D i
t
± i, H
2
t
3
2F i
2Fi
4
2G i
2G,
l
~
i
JjjQ.
±i, H
±! , ±!, ±\Il.
\
_ "7, ±~, ±J.f
+
± }, ±¥, ±1,}l,
±~,
±J§~,
±i, ±V,
±~"
-.,
±.2j, ± -2l
±~Jf,
+ ""
±~Il.
Con sid er, for example, t he splitting of a 2p , level in a weak m agne ti c
field. With j = i , t here are four magneti c levels m = t , t, - t , and i ,
shifted from the field-free level by mq = t, i , -t, and -t- A gra ph ical
1
with
Here tn, t he magnetic quantum number in the numerator, must not be confused
tn, the mass (If t he electron in the denom inator.
SEC. 10 .5]
ZE E M A N EFFECT AND THE PA SCHEN-B A CK EFFECT
159
represen tati on of t his splitting is shown in Fig. 10.7, where t he vect or
j * is shown in t he four allowed positions. Multiplying
by g and project ing t he pr oduct on t he H axis, the displacem ents mg shown at the
left are obtained.
R eferring now to t he spip-orbit in t er acti on energy [Eqs. (8.17) and
(8.20)], wh ich gives rise to t he doublet fine structure, and to t he eq uation
r
H
---
mg = ~ -----4
3"
4
3"
6
-3- - - - ---_..- "' ,-
FIG . 1O.7.-Sehematie orientation diagram of an atom in a 2PI state showing t he resul t a n t
Zeem an levels in a weak magnetic field .
just derived fr om t he Zeeman splitting, t he complete te rm value T of
. any magnetic level may be wr itten
T = To -
r -
mg . L .
(10.19)
T o is t he te rm value of t he hy potheti cal center of gravit y of t he field-free
doublet , r is t he fine-structure shift , and mg . L is t he m agneti c shift .
10.5. Selection Rules.- As an exam ple of t he calculati on of Zeeman
pat t erns, conside r t he simple case of a princip al-seri es doublet lik e the
sodium ye llow D lin es XA5890 and 5896. The g fact ors for t he ini ti al
2P i and 2P I states and for t he final 28 i state (see T abl e 10.1) are j, J, and
2, respecti vely . The splitting of eac h of t hese levels is shown schematically in F ig. 10.8, starting with t he field-free levels at t he left . The
dot t ed lin es in eac h case represent t he centers of gravity of the associated
levels.
The t heo retical selection rules for t ra nsit ions betwe en levels, in
agreement with observations, may be stated as follows: I n any transition.
the magnetic quan tum number m charujes by + 1, 0, or -1 , i .e.,
L'J.m
= 0, ± l.
(10.20)
For the stronger of t he t wo field-free line s t here are six allowed t ra nsitions and tw o forbidden t ransit ions . F or the other line there ar e four
160
I N T R ODUCT IO N TO ATOMIC SPECTRA
[CHAP .
X
allowed transit ions . T he obse rved and ca lculated patterns a re shown
at the bot tom of t he figure.
Polarizati on rul es der ived fr om the classical t heo ry as well as t he
qu ant um mechanics are found to hold experime ntally . These rul es
may be stated in t he followin g way :
{t.m
Sm.
Viewed II t o field {t.m
V'
d
t fi ld
iew e .1 0 e
=
=
=
Sm =
± 1; plan e polariz ed
0; plan e polariz ed
.1 to H ; s com pone nts
II to H ; p com pone nts
ecom pone nts
± 1;
circ~larly polarized;
0; forbidden :
p com pone nts
10.6. Intensity Rules.- T he in t ensity rules for field -fr ee ene rgy
levels, first deri ved fro m experime ntal obse rvations by Burger , Dorgelo,
6
I
2
P31. ·
,2
1/
r . ah
--2-- --8--
(::-
\'
To- - - - ,:.- - - - - - - \ \
'.
PY2
---- - --
- - - -
--
- -
6
\
2
2
8
I
r · -"
.'----~:-( -
-
- -
/
- -
-
- - -
- - -
-
- 4-
'4
4
- 4
mg
1'2
%
%
-);2 - 2;3
12 - 93
liz
>3
:.. V
2 - 1'3
1
-- - - --- -
m
~2
liz
- -
P
s
- 1'2
' \
Calc
(
Obs
F IG. 1O.8 .- Zeeman effec t of a principal-series doublet,
and Ornst ein (see Sec. 8.3) are readily sh own t o follow directly from
the int ensit y rules for t he same levels in a weak magnetic field . In
short t hese rul es may be stated as follows :
Th e sum of all the transitions starti ng f rom any initial Z eem an level
is equal to the sum of all transitions leaving an y other level having the same
n and l values. Th e sum of all transitions arriving at an y Z eeman level
is equal to the sum of all tran sitions arriving at any other level having the
same nand l values.
F or any given field-free spectru m line t hese rules are bet t er expressed
in t erms of formulas whi ch have been der ived from t he class ica l! as well
1 For a classica l d eriva t ion of t he in t ensit y ru les, bas ed on Boh r 's correspondence
pr in eiple, see J. H . Van Vleck , " Qua ntum Principl es and Lin e Spectra ," N at. Research
Council, Bull., Vol. 10, 1926.
SEC.
10.61
ZEEMA N EFFECT AND THE PA S CHEN-BA CK EFFECT
as t he qu an tum-mechanical theories.
ated as follow s:
Transition j
-t
j
. .on J. - t J. +
T ransiti
m -t m
{ m -t m
±
1, I
-t
These formulas may be abbrevi-
= AU ± m
,I = 4Am 2 •
l{m
m+- 1,, II
m -tm
161
=
B (j
= 4B (j
+
l ) (j
±m+
+m+
+ m).
l )(j ± m
l )(j - m
(10. 2 1)
+ 2).
+ 1).
(10.22)
A and B are constants t hat need not be determined for relati ve intensiti es
within eac h Zeem an pa t t ern.
These formulas take in to acco unt t he
Zeeman Patterns
For Doublets
F IG. l O.g.-Zeem a n patterns for a ll of the com m on ly observed doublet transitions.
dots represen t no rmal t rip let separations.
r
,
T he
fa ct tha t, when the radiation is. obser ved perpendicular t o the field
direction, only half of t he ligh t making up t he s com ponents of a pat t ern
is obser ved. Obse rvations parallel to t he field give t he othe r half of t he
s com pone nts' in t en sity. The a bove given rul es and formulas applied
to the principal doublet in Fig. 10.8 giv e t he relative in t en siti es shown
at the top of t he arrows. The sa me values a re indicat ed by t he heigh ts
of t he lin es sho wn at t he bot t om of t he figure. The sum of all transitions
st a rt ing from any lev el in t he figure is 12, a nd t he sum a rrivi ng at eit he r
of the lower lev els is 36. (I n obtaining these values t he s com pone nts
mu st be multiplied by 2.) E xperimen t ally , t he p 0 1' 8 com pone nts may
be ph ot ogr aphed separately by inser ting a Nicol pri sm in t he pa th of
t he ligh t coming fr om t he source bet ween t he poles of an electromagnet.
...
162
[CHAP. X '
I N TROD UCTION TO A TO MIC SPECTRA
Zeem an patterns for a number of do ublet transitions a re given in
Fig. 10.9. These calculated pa t terns are in excellent ag ree ment wit h
all experiment al observations . It is par t icularly in t eresting to see how
closely t he differ en t lines grou p t hemselves arou nd t he normal- triplet
separations, whi ch are indicated in t he figure by dot s. Normal Zeem an
t riplets , as will be seen lat er , arise in special cases and in parti cul ar for all
lines belongin g to a singlet series, where t he g fact or for bot h t he initi al
a nd final states i's unit y.
A method fr equen tly employed for a rapid calculation of Zeem an
pa tterns will be give n briefly as follows. The separation fa ct or s m g,
for both t he ini ti al and final states, are first written down in two rows
with equal values of m directly below or a bove eac h othe r . For a
2D i-2p ! t ra nsition t hey are :
m =
mg initial state
t
¥
mg fina l state
t
!
lK
* --t -- t
-~
-
~
J1'.Q
~D<t><t><l
/
-~
~
- j
- g
In t his array, t he ve rtical arrows indicat e t he p compo nents, Sm = 0,
and t he diagonal arrows t he s com ponents, :::"m = ± 1. The differe nces
expresse d wit h a least commo n de nominator are as follows :
Vertica l Differen ces
p Components
+ 1\' +1\' -1)"
- l~rr
D iago nal D ifferences
s Components
±H, ±-H, ± -H, ± -H
In short t hese may be abbrev iated,
A
_
1->V -
(± 1), (±3), ± 15, ± 17, ± 19, ± 21
- 1
L ern , (
15
the four p com ponents being set in pa ren theses, followed by the eig ht
s com ponents (see Fig. 10.9).
A sim ple qu alitative rul e for t he in t ensities has been give n by Ki ess
a nd Meggers as follows: If t he j values of t he two combining terms
are equal, t he ve rtica l differen ces at t he end of t he scheme, a nd t he
diagonal differences at t he center, give t he strongest p a nd s com pone nts,
respectively. If the j v alues are not equal, as in t he case shown a bove,
t he vertical differences in t he middle of t he sche me a nd t he di agon al
differen ces at t he ends give t he strongest p and s com pone nts, resp ectively.
10.7. The Paschen-Back Effect.- I n deriving t he interact ion ene rgy
between an atom containing one single valence electro n and a n external
magnetic field, it was ass umed t hat t he field was weak as com pared
with the internal fields du e t o t he spin and orbital motion of t he electron. When t he extern al field becom es greater t ha n t hes e int ernal
SEC. 10. 71
ZEEMA N EPPECT A ND THE P A SCHEN -BA CK EFPECT
fields t he in t ernal motions are greatly per turbed a nd t he at om gives
rise t o the so-called Paschen-Back effect.
Just as t he doublet fine-structure sepa rations are a meas ure of t he
cla ssical frequen cy with whi ch l* and s* pr ecess around t heir resultant
j* (see Sec. 10.4), so t he Zeem an sepa rations of t he sa me ene rgy states
in a weak magn eti c field are a measure of t he fr equ en cy with whi ch j*
precesses around H, I n calculating
the Zeem an separations in Sec. 10.4, it
was tacitly ass umed t hat t he pr ecession
of l" and s* a round j* was mu ch faster
t ha n t hat of j* around H, This was
ne cessary in order t hat t he componen ts
of l* and s* normal to j* aver ag e to
m
zer o and do n ot apprecia bly per turb
t he other precession. If now t he field
H is in crea sed until t he t )"o precessions
are of t he sa me order of magni tude,
FIG. lO.lO. -Vector model fo r the
then t he Zeem an levels of t he doublet Paschen-Ba
ck effec t where the field is
will begin t o ove rla p, t he re will be no so st ron g that 1* a n d s* precess independaveraging t o zer o, a nd E qs . (10.17) ently around t h e field direction H .
and (10.19) will not hold. Under these conditions t he coupling betw een
l* a nd s* will be parti ally broken dow n, the classica l motions of l* a nd
s* will becom e com plicated , a nd j * will no lon ger be fixed in magni tude.
As t he field H is still fur ther in creased , l* and s* will soon becom e
H
H
m
_~H~~'_. ,
5*
S'
,
1
I
,
\
I
-T
r
:
:
--'
s=!
I-I
1-2
FIG . lO.l1 .- Sp a ce-quan tiza ti on di agr ams fo r p a n d d ele ctrons in a strong m a gn et ic field ,
P asch en -B a ck effec t .
quantized sepa rately and precess more or less indep enden tly around H
(see Fig. 10.10 ). This is t he P aschen-Ba ck effect .'
The quantum conditions in a st rong Paschen-B ack field a re : (1)
The projection of l* on H takes in t egral va lues from m, = +l to m, = - l.
(2) T he projecti on of s* on H takes one of t he two values m, = +1, or
1 P AS CH E N ,
F ., and E .
B AC K ,
Ann. d . P hys., 39, 897, 1912; 40, 960, 1913.
164
I N TROD UCTION TO A T OM IC SPECT RA
[CH AP.
X
- t o For a p electron with l = 1, s = t, t here are six possible states
ml = 1, 0, and -1, when m. = t or - t o Space-q ua ntization diagrams
of these cases are given in F ig. 10.11. Sin ce for every elect ro n t here
1 va lues of mi , t here are 2 (2l
1) comare tw o va lues of m . and 2l
binations of t he qua ntum nu mb ers corres ponding to 2 (2l + 1) different
+
+
states of t he atom. AJ3 might be expected, t his is exactly t he num ber of
wea k-field levels.
The total ener gy of t he atom in a field stro ng enough to give t he
P aschen-Back effect is m ad e up of t he t hree par t s: (1) the energy due
to the precession of l* around H ; (2) the energy due to the precession of s*
around H ; (3) the interaction energy between l* and s*. By Lar mor's
theorem [Eq . (3 .58)], t he pr ecessional angular velocit ies are give n by H
t imes t he ra tio betw een t he magneti c and mechani cal moment s:
Wl*
= H _ e_ and
2m c
w.*
= 2H _ e_ .
2m c
(10.23)
Sin ce the ratio between t he magneti c and mechanical moment for t he
spin of the electron is t wice the or bit al ra ti o, s* should, on t he classica l
picture, precess t wice as fast as l*. Multiplying each of t hese ang ular
velocit ies by t he pr ojecti on of t he angula r momentum on H [see E q .
(10.15)], one gets t he first two te rms of t he energy :
e
h
e
h
H - l*-2 cos (l*H ) = H2- m l-2'
2mc 11'
mc
11'
e
h
e
h
= 2H
s*2 cos (s*H) = 2H- m . .
2mc 11'
2mc 211'
~ W I . 1l =
(10.24)
~Ws ,
(10 .25)
H
The sum of t hese t wo energ ies acco unts for the main energy shift
fr om t he unperturbed energy level and is
~ W ll
=
(m l
eh
+ 2m .)H;c·
'±1I'm c
(10 .26)
Di viding by he, t he te rm shift in wav e nu mbers becomes
- ~Tll =
(m l
He
+ 2m' )41I'mc• cm-l,
(10.27)
or in Lorentz uni ts of H e/411'mc2 ,
(10.28)
To t his magn eti c energy t he small correction te rm du e t o t he interaction be tween l* and s* must be added. Although t hese two vect ors
precess independently aro und H , each moti on still produ ces a magneti c
field at t he electron which perturbs t he motion of t he other. This
in teracti on ener!&, t hough small as com pa red with t hat due to t he
external field, is of t he same order of magnit ude as t he fine-structure
SEC. 10.8)
ZEEM A N E FFEC T AND THE PA SCHEN-B A CK EFFECT
165
do ublet separations in field-free space, whic h by Eqs. (8.14), (8. 18), and
(8.19) are given by t he r fact or,
r = - t..T l •• = al*8* cos (l*8*) ,
(10.29)
where
(10.30)
In field-free space, t he angle between l* and 8 * is constant and t he
cosine term cos (l*8*) is eas ily evaluated . In t he presen t case, however,
t he angle is continually changing, so t hat an average value of t he cosine ·
m ust be calculated. From a well-know n t heorem in t rigono metr y it
may be shown t hat wit h 8* and l* pr ecessing independen tly with fixed
a ngles aro und a t hird directi on H,
cos (l*8*) = cos (l*H) . cos (8*H) .
(10.3 1)
M aking t his substit ution in (10.29),
I'
= -t..T l •• = al* cos (l*H)8* cos (8*H) .
These are ju st t he projections of l * and
8*
(10.32)
on H , so t hat
(10.33)
Adding t his term to Eq. (10.28), t he total energy shift becomes
- t..T cm-l = (ml
+ 2m. )L + amrni •.
(10.34)
We may now write dow n a general relation for t he te rm value of an y
strong fieldlevel,
(10.35)
where To is t he term value of t he hypotheti cal center of gravit y of t he
fine-st ructure doublet .
10.8. Paschen-Back Effect of a Principal-series Doublet.-As an
example of t he P aschen-Back effect, conside r first t he calculation of t erms
and te rm separations involve d in a principal-seri es doublet 2Sl-2P I . ~.
The fine-st ructure sepa rations du e to t he interaction of l* and 8* in
field-free space are given in Col. 2, T able 10.2 (see F ig. 8 .9). In t he
next t hree columns t he wea k-field energies are calculated (see Fig. 10.8).
In t he last five columns t he stro ng-field energies are calcula ted , using
Eq. (10.34) .
The values tab ulated are shown schematically in Fig. 10.12. At the
left t he undisturbed fine-structure levels and t he observed transit ions
are show n. The wea k-field Zeem an levels are next show n with the
observed Zeeman pat t erns below. In t he strong field t he P aschen-Baok
levels are show n with, and without, the small l*8* coupling cor rection
166
[C HAP. X
I NTROD UCTION TO A T OM IC S PEC T RA
The allowed t ransit ions and t he calculated pat tern are shown
below.
In deri ving the above equat ions for the Zeeman and Paschen-Back
effect s, t he atomic system was assumed t o be in one of t wo ideal situations .
In t he first case t he field was ass umed so weak t hat
t he resultant
of l* and s*, was in vari an t as regards magni tude and inclinati on t o t he
field axis. In t he second case t he field was assumed so strong t hat
l * and s* pr ecess independently around H . The question of intermediate
arn /rn, .
r,
TABLE 10.2.-WEAK- AND STilONG-FIELD ENEilGIES Fa ll A PilIN CIPAL-SEIlIES DOUBLET
T er m
r
m
+a/2
mg
g
- -
'p !
'-
-
+i
+!
+ i
- .,
+!
+!
+!
- .,
3
- 1
+1
- !
+!
0
_ .1
3
- 1
+1
- 1
+2
+1
+a/2
0
+!
0
0
-a/ 2
-a/ 2
- .,
- .,
1
-!
- 1
0
_ Ii
1
- ~
- 2
+ a/ 2
0
+!
+!
+1
0
0
- !
-
- 1
0
-
2
-
'3
1
2
+!
+!
~
- j
- - - - --
H
+1
0
!
+ !.
-a
m=
ml+ 2m . am,m 8
ml + m.
m.
-- - -
- 32
- - - - - - -
2S !
ml
+~
1
'p !
Strong field
(Pas chen- Bac k effect )
Weak field
(Zee ma n effect )
No field
- - ---
1
2
0
-!
1
2
fields, t herefore, arises, and one asks, how does eac h weak-field level
go over to a corresponding strong-field level? Darwin's treatment
of this problem, whi ch will not be given here, answers t his qu estion in a
very sim ple m anner. I According to t he classical law of t he conse rv ation
of angul ar momen tum, t he sum of t he pr ojecti ons on H of t he various
angula r-mome ntum vectors must rem ain t he sa me for all field strengt hs .
Sin ce in weak field t his sum is given by rn and in stro ng field by rn / + rn"
we may wri t e, as part of t he correlation rule, m = rn/
m .; This
alone is not sufficient to corre lat e all weak- and strong-field levels, sin ce
in most ins tances t here will be more t han one level with t he sa me m.
value. The more specific rule, in keeping with t he quan tum mechanics,
may be stat ed as follows: L evels with the same m never cross.
An ingenious method for ob taining t he same correlation has been
given by Breit." An array of weak- and strong-field quantum numbers
+
DARWIN, C. G., Pr oc. R oy. So c., A, 116, 1, 1927.
• BREIT, G., Phys. R ev., 28, 334. 1926.
1
S E C.
10.8)
ZEEMA N EFFECT AND 'i 'HE P A SCHEN-BA CK EFFECT
167
is written down as follow s (see Fig. 10.13). Values of m , a re written
down in t heir regul ar order in a horiz on t al row a nd v alues of m, in a
vert ical colum n. The array is next filled in with all possibl e sums of
Strong
Fi eld
- --
ml + ms
_. + 1 +
-
'Qh
=
m
Yz -+0/2
No Fie l d
rQI2
2
2 _
-
f- .
- --:;r'"
2
2
0-1'2- -Yz
---
%
-1- V2--tz
-
,
2
--
_
~-
.rr. =me
.
__.-
- -
- - -- -. I -
=o=crr
,P
SIS
SI S
FIG . 1O.12.-Energy levels fo r a principal-se ries do ublet starting wi th no field at t he
left a nd ending with a strong field (Paschen-Back effe ct) at t h e r igh t . Allo we d transitions
are shown be low. ~
mz an d m. . These sums are t he weak-field qu antum num be rs, di vided
in t o two part s by t he dot t ed lin es. Each weak-field level m is to be
corre lated wi th t he strong-field level give n by t he value of mz directly
above , and t he value of m, directl y to t he ri gh t of t he m value . The
2P I' m = 4 state, for exam ple, goes t o t he state m.; = 1, an d m. = t ·
P
mj
m
=
I
=t
:~
0
d
-1
t -t t
t--=t'l -~
-t
2P! : ZP3 m s
212
ml= Z
.2.
m=2
0
1.
z
1
Z
-I
-2
_1 - ~
Z
2
I
Z
~~ f-r-~~ --·:T: -%-t
ZD
i2D
r3 :' 5'2
ms
FIG. 1O.13.- C orrela ti on of weak- and strong-fie ld quantum numbers and energy lev els.
(After B reit .)
It is obv ious t hat t he re are t wo way s of drawin g t he L-sh ap ed do tted
lin e. Of t he two ways only t he one sho wn will give t he correct cor re lat ion for doublet s fr om a single electron.
168
I NTROD UCTION TO ATOMIC S PEC T RA
[C HAP.
X
10.9. Selection Rules for the Paschen-Back Effect.- Selection rules
for t he P aschen-B ack effect , derived fr om t he classical t heory may be
stated as follows : In any t ra nsition
tsm , = 0,
± 1 and S m , = O.
(10.36)
When t hese rules are applied to a given dou blet they are found to lead,
in a very strong field, to a pat t ern closely resembling a normal Zeeman
t riplet. The radi at ed frequ encies of t he pri ncipal-series do ublet con
side red above are assembled aga in in F ig. 10.14. T he res triction S m, = 0
impli es t he necessary conditio n t hat t he polarizati on of any give n
line be ret ained t hro ughout all field strengths. The p and s components
'2. 5 1 2- '2 P3h.
25Y2- '2. PI!c.
I"/z
a
No Field
We" k Field
--v~
-- -----
»> -:
/'
/'
Strong
Fi eld
S
Ld..
p
-'-L
---"'--'
N_ormal Tr i pletFIG . lO.14.-Principal-seri es d ouble t in various magneti c field stre n gths .
effec t .
P a soh en-Ba ck
in weak field become p and s com ponents, respectively, in t he strong
field or else disappear as for bidd en lines. For t he s com ponents, t he
fine-structure separation is just t wo-thirds t hat of t he field -fr ee doublet .
It sho uld be m ad e clea r t hat as t he field stre ngth in creases and t he
q uantum number j disappear s, and, as j and it s projecti on m are repl aced
by m l and m., t he selection ru les for j can no longer be expected to
hold. This is in agreement with observations made by P aschen and
Back! who wer e t he first t o observe experiment ally t he so-called normal
triplet of a principal-series doub let in a very strong field. Since t he
strong-field levels mu st be widely sepa rated as com pa red with t he fine
struct ure, only t he very nar row fine-structure doublets m ay be expecte d
to be carried over t o t he P aschen-Back normal t riplet with t he ordinarily
obtained field strengths . Using t he princip al-seri es doublet of lithium,
with on ly a separation of 0.34 wave numbers, P aschen and Back required
t he very strong field of 43000 gau ss to obser ve it as a m agn eti c t riplet.
10.10. The Zeeman Effect, and Paschen-Back Effect, in Hydrogen.Although t he hydrogen atom is t he simplest of all atom ic systems, t he
1 P AS CHEN ,
F ., and E .
BACK ,
Phusica, 1, 261, 1921. v
SEC. 10.11 ]
ZEE M A N EFFECT AND T HE PASCH EN-BACK EFFECT
169
Zeeman effect in hydrogen is not very simple. In Sec. 9.3 we have
seen how each of the Balmer terms contains a fine structure which
is made up of dou blet s 28, 2p, 2D, etc. When placed in a weak magnetic
field each of t hese do ublet levels, as in sod ium, should undergo a n anomalous splitting (see F ig. 10.8). As a resul t of t his splitting eac h finestructure com ponent of a lin e lik e H a , }'6563, should reveal an anomalous
but symmetrical Zeeman pattern (see Fig. 10.9). With H, made up
of seven different transitions, t here would be, in t his
case, seven patterns of t he type show n in F ig. 10.9,
all lying wit hin a n int erval of abo ut half a wave
number (see Fig. 9.4). It is now easy to see why t he
a
Zeeman effect of hydrogen has not been observed.
I n a strong magnetic field, t he magnetic levels
of each do ub let begin to overlap eac h ot her, until
in a field strength of seve ra l t housand gauss the
Paschen-Back effect sets in. Un der these conditio ns each doublet level, combining wit h another
doublet level, gives rise to approximately a normal
triplet like the one shown in Fig. 10.12. T he H,
b
line, for example, will be made u p of t hree normal
triplets 28_ 2P , . 2p_ 28 , and 2p_2D, superimposed
almost on top of each other. In a field of 32000
gauss, P aschen and Back! observed ju st t hat; a
FIG.1O .15 .-Paschenwell-resolved triplet with practically the ' class ica l Back effect of hydrogen
separation . In such a field the pattern is some H a, A6563. (a) Enlarged
from photograph in the
six t imes..as wide as t he field-free line . A phot o- published paper of Pasgraph ic reproduction of this triplet is given in chen and Back. (b) Same
as a with photographic
Fig. 10.15. Normal triplets have also been observed paper moved parallel to
by Paschen and Back for H 13 and H y •
lin es during enlargemen t .
10.11. A Quantum-mechanical Model of the Atom in a Strong Magnetic Field.-In this chapter the Zeeman effect, as well as t he P aschenBack effect, has been treated chiefly from t he standpoint of the sem iclass ica l vector model. Quantum-mechanical treatments of the sa me
problems have been made by different investigators and found to lead
to exa ctly the same formula . Chronologica lly, t he more acc urate
quantum mec hanics led t he way to a simpler formulat ion of Lande's
vector model. We have seen in Chaps. I V and IX how, in field-free
space, this model is surprisingly sim ila r to t he quantum-mechanica l
model of probability-density distributions for t he electron. As would be
expected from energy relations, t he weak- and strong-field distribu tions
for the electron on any model should be lit tl e different from eac h other
1 PASCHEN, F., and E. BACK, Ann. d. Phys., 39, 897, 1912. T h e photograph in
F ig . 10.15 is a copy of the photograph given in Pla t e VIII of Ann. d. Phys., Vo l. 39,
1912.
I
170
INTRODUCTION TO ATOMIC SPECTRA
[CHAP. X
radi ally . Angul arly, as show n by t he vector model, t hey should cha nge
considerably .
Space-q ua ntized to a va nishi ng magnetic field t he field-free states,
oriented as shown in F ig. 9.8, may be taken to re present t he weak-field
states, t he Zeeman effect , of t he atom . Assuming separate angular
distributi on curves for t he spin of t he electro n, simila r to t he orbital
curves in Fi g. 4.3, pict ures ana logous to t he in dependent pr ecession of
1* and s* around the field direction H may be formulated for stro ng
fields, i.e., t he P aschen-B ack effect .
In passing, it should be mentioned t hat a give n state of t he atom
in a weak field is specified by t he qua ntum numbers n, l, i. and m , and in
a stro ng field by n , l, ml , and m .. For intermediate fields eit he r set
may be used, although, to use an expression introduce d by M ullike n,
i, tn i, and m. are not good quantum numbers.
Problems
1. Compute the Zeeman pattern (sepa ra tions and intensities) for the doublet
t ra ns it ion "Gr- 'H ,,"
2. Find the total wid th in wave numbers of t he Zeeman patt ern of P rob . 1 in a
weak field of 500 0 ga uss.
3. Compute t he weak- and strong-field en ergies for a d iffuse-ser ies doublet, and
tabulate th em as in T abl e 10.2. Plot t he in it ia l a nd fina l st a t es, a s shown in F ig. 10.1 2,
a nd indicate the allowed t ransitions by a rrows.
4. Plo t , as in Fig . 10.14, the field-free lin es, t he weak-fi eld lines, and the st rongfield lin es of t he above exa mple. [No 'rnt-e-Ce rt nin compone nts of th e forbidd en
transition 'Pi- ' D j appear in st rong field s and should be indicat ed (see Fig. 13.1 4)].
6. What field st rength would be requ ired to ca rr y the first m ember of the princip al
se ries of sodium over to the P a sch en- Ba ck effect whe re t he separa tion of t he resultan t
normal triplet (see F ig . 10. 14) is fou r times the fine-stru ct ure separation of t he field free doublet?