1. science
2. physics

INTRODUCTION
TO
ATOMIC SPECTRA
BY
HARVEY ELLIOTT WHITE, PH.D.
Assi stan t Professor of Ph ysics, at the
Univ ersity of Cali f orni a
McGRAW-HILL BOOK COMPANY, INc.
NEW
YORK A ND
1934
LONDON
CH APTER I X
HYDRO GEN FINE STRUCTURE AND THE DIRAC ELECTRON
Even t he hydrogen spectru m, t he simplest of all sys tems, is obse rve d .
to have a fine structure . At an early date M ichelson studied t he Bal mer
lines with an in terferometer and found t hat both H" and H(:l were close
doublet s with sepa rati ons of only 0.14 and 0.08 A, or 0.32 and 0.33 em:",
respecti vely . M an y subseq uent in vesti gat ions by others have confirmed
these resu lt s (see Fig. 9.1).
H2
a
FIG. 9 .1.-P hotographs of t he H" lin e of both of t he h ydrogen isotopes H I a nd H2.
HI
~
(A fter
L ewi s an d S peddi ng.)
The mo st informing observations t hat have been made on hyd rogenlike atoms ar e t hose of P aschen I on t he singly ioniz ed helium line M6 86.
This line (see Fi g. 2.8) corresponds to t he first mem ber of t he so-called
Paschen series of hydrogen. Hi storically Paschen 's observations were
mad e and published at a most opportune time, for in t he next issue of
t he A nnalen der Physik Somm erfeld ind epend ently predi cted ju st such a
fine str ucture by an extension of t he Bohr atom to include elliptic or bits
and t he special t heory of rela ti vit y." While t he quantum mec hanics
gives a mo re perf ect acco unt of t he observed fine st ru cture, t he development of t he orbital model is interesting in t hat it lead s to t he same energy
levels .
9.1. Sommerfeld Relativity Correction.- T he extensio n of Bohr's
atomic m odel by Sommerfeld to include ellip ti c orbits adds no new
energy levels to t he hydrogen atom (see Sec. 3.3). F or a give n total
quantum number n, all ellipt ic orbits s, p, d, . .. ha ve just t he sa me
1 P AS C H E N ,
F ., Ann. d. P hys., 60, 90 1,1916.
A., Ann. d. P hys., 61, 1, 1916.
132
2 S OM MERFELD,
SEC. 9.1]
133
HYDROGEN FINE STR UCTURE
energy as the Bohr circular orbi t with the same n.
numbers is
W
RZ2
- -- - - n 2
-T,
he
This energy in wave
(9.1)
where R is the Rydberg constant
271"2
me4
R =,
Ch
(9.2)
(1+ : )
3
h is Planck's constant, c the velocity of light, m and e the mass and charge
of the electron, and M the mass of the nucleus with charge Z e.
Bohr pointed out in his earliest papers t hat the relativistic change in
mass of the orbital electron should be taken into account in computing
the energy levels. Introducing elliptic orbits, Sommerfeld applied the
special theory of relativity to the
electron mass. Due t o t he different
velocity of the electron in orbits of
the same n but differing azimuthal
quantum number, the ma ss of the
electron and hence the resultant
energy levels are all different. If the
rest ma ss of the electron is m o, its I
mass when moving with velocity v is
given by t he spe cial theory of relat ivit y as
V
C22)!.
m~ = m o 1 -
(
(9.3)
FIG. 9.2 .- S ch ema ti c r epres entation of
t he preces sion of an electron or bit due t o
t he rel ativity change in m as s of the electron wi th vel ocity. (A f ter S ommerfe ld .)
As a result of this change in mass,
which is greatest at perihelion and
greatest for the most ellip tic orbits, there is an advance of the perihelion,
or a precession of the electron orbit, similar t o that of a penetrating orbit
in the alkali metals (see Fig. 7.4), or to that of the planet Mercury moving
about the sun. This pre cession is shown schematically in Fig. 9.2.
While the derivation of Sommerfeld's equation for t he change in energy
due to this precession is out of pla ce here, we shall find use for it in making
comparisons with the quantum-mechanical results.! According to the
Sommerfeld theory the te rm values of hydrogen-like atoms are given by
T = _ W = J1.C{l
hc
h
+
2
a
Z
2
}-l + J1.C,h
(9.4)
(n- k + V k 2 - a 2Z 2 ) 2
1 For a deri vation of Sommerfeld 's relativistic fine-structure formula see " Atomic
Structure and Spect ral Lin es," p . 467, 1~23; also A. E. Ruark and H . C. Ur ey , "Atoms,
Mol ecules and Quanta," p . 132.
134
INTROD UCTIO N TO ATOMIC S P EC T RA
where
a
[C HA P .
IX
is the fine-structure constant
a
2
4 •
4
= ~~~ = 5.30 X 10- 5
(9.5)
and
i}I rn
J.t=
M+m'
For convenien ce of evaluation , Eq. (9.4) ha s been expanded by Sommerfeld into a conve rging series,
= RZ2+ Ra2Z 4[~ - ~J
T= _ W
he
n2
n4
k
4
+ Ra4Z6l!(~)
3 + ~(~)2
- ~(~)
+ ~J
n
4 k
4 k
2 k
8
6
+ R:6;T~(iY + ~(iY + ~on3
-¥(iY
+ 15(!!:)
8 ~k
+ ....
_ 35]
64
(9.6)
The first term of t his expa nsion is t he same as that derived by Bohr
for circular or bits, neglecting relati vity, and gives t he major par t of, the
energy . With n = 1, 2, 3, .. . , and with Z = 1 for hydrogen, Z = 2
for ionized helium, and Z = 3 for doubly ionized lithium, t his t erm gives
t he following values :
T AB LE 9. 1 . -TE R M VA LUES, NEGLECT ING FIN E-STR UCT UR E C ORRECTION S
H yd rogen
Hydrogen
Ionized
Doubl y
helium
ionized lithium
(isotope mass I) (isotope mass 2)
R = 109677.76 R = 109707.56 R = 109722.4 R = 109728.9
For n
=
1
2
3
4
5
6
7
109677.76
27419.4
12186 .4
6854. 8
4387 .1
3046 .6
2238 .3
109707.56
27426.9
12189.7
6856 .7
4388 .3
3047.4
2238 .9
438889 .6
109722.4
48765.5
27430.6
17555.5
12191.3
8956. 9
987560 .1
246890 .0
109728 .9
61722 .5
39502 .4
27432 .2
20154 .4
To each of these valu es correct ions from Eq. (9.6) mus t be added.
For sm all values of Z t he first te rm involving Z4and a2 is the only one
of importance and t he t hird and succeeding terms may be neglected.
In x-ray spect ra , however, Z becomes large for the heavier elements
and terms in a 4 and a 6 mu st be taken into account (see Chap. XVI) .
135
H YDR OGE N FINE STRUC TURE
SEC. 9.2]
The corrections to be ad ded to eac h of the above give n terms are t he refore give n by
(9.7)
where k is Sommerfeld's azimuthal quantum number 1, 2, 3, .. . for
s, p, d, . . .. For all allowed values of nand k the correction is positive
and is to be ad ded to t he terms in Table 9.1. For either of the two
hydrogen isot op es,
(9.8)
These corrections are shown graphica lly in F ig. 9.3 . The straight
lin es at t he top of each of the four diagrams represent the first four terms
n -I
ll.Tr +ll.T -ll. T
1.45 crrr i
k-l
35
D.38cm- 1
4 8 O.16cm- 1
25 \.18cm- t
15 7.26 cm-'
FIG . 9.3 .-Fine st r uc t ure of the hydrogen ene r gy lev els. f:;.Tr and f:;.T I ,s r epres ent
the relativity and the sp in-or bit cor rec t ions r esp ectively, The d a sh ed lin es r epres en t
Sommerfeld' s r elativity co rrections .
of hydrogen given by 'fable 9.1. The shifted levels for each value of
nand k are shown by t he dotted line s wit h the term value increasing
down ward . T he left-hand side of each diagram has to do with the
spinning-electron picture of the atom and will be taken up in the following
section . For ion ized helium and doubly ionized lithium t he intervals
give n in Fig. 9.3 must be multiplied by 16 and 81, respect ively .
9.2. Fine Structure and the Spinning Electron.- Wit h the int rodu ction of the spinning electron and the quantum me chanics another account
of t he hydrogen fine structure has been given. H eisenberg and Jordan!
1 HEISENBERG,
W .. and P .
JORDAN ,
Z eits. f . Phys., 37, 263, 1926,
136
I NTRODUCTIO N T O A TOMIC SPEC TRA
[CHAP.
IX
h ave shown from a quan tum-mechanical t reatme nt t hat Sommerfeld's
relativity correction for hydrogen-like at om s sho uld be
tlT r
Ra
2Z 4
(
= ~ l
+1! -
3)
(9.9)
4n '
where l = 0, 1, 2, . .. for 8 , p, d, . . . electrons . A gene ral com pa rison of all clas sical with quantum-mechanical result s for t he same ph enomenon shows t hat t he class ica l values k, k 2 , and k 3 may usually, but no t
always, be replaced by l
!, l (l 1), and l (l !)(l 1), resp ectively,
to obt ain the qu antum-mech anical result s. Sommerfeld's relativ ity
equation is a good example of t his ; k in E q . (9.7) re place d by l
! gives
Eq. (9.9).
To t he qu antum-mechanical relativity cor rec tion [E q. (9.9)] a seco nd
t erm due to the spin-orbit in t er action must be adde d . This in t er action
energy h as alrea dy been calculate d in Sec. 8.6 and show n to be given by
E q . (8. 17) :1
Ra 2Z 4
}*2 _ l*2 - 8*2
tlT I •8 = - n 3l(l + !)(l
1)
2
(9. 10)
+
+
+
+
+
+
Applying t he first correction tlT r to t he hydrogen terms of Table 9.1,
eac h level n is split in t o n com pone nts as show n at t he left of eac h of t he
four di agrams in Fi g. 9.3. Applying now t he spin-or bit in t eracti on
tlT I • ea ch of these terms, wit h t he exception of 8 terms, is split into t wo
parts just as in t he alkali met als. In eac h case t he level with } = l
-!
has been shift ed up, and t he one with } = l - ! has been shifted down ,
t o t he nearest Sommerf eld level. In ot he r words, levels with t he sa me
j values come together at t he older rela ti vity levels k, where k = j
!.
The remarkable fact that Sommerf eld 's formula derived fr om t he rela tivit y
t heory alone should give t he sa me resul t as t he newer t heo ry , whe re both
relativity and spin are t aken in to account, is a good exa m ple of how
t wo in correct ass umpt ions can lead to t he cor rect res ult. While t he
number of numeri cally differ en t ene rgy levels is t he sa me on both t heo ries,
there is an exp eriment al method for sho wing t hat t he first t heory is not
correct and t hat t he lat t er ve ry probably is correct. This will becom e
apparent in Sec. 9.4.
Sin ce the newer theory lead s to Sommerfeld's equation, t he sum of
Eqs. (9.9) and (9.10) should reduce to Eq. (9.7) . Since j t a kes on values
l
-! or l - ! only, E q . (9.10) is split in t o t wo equations :
8,
+
+
+
tlT I •8
=
tlT I •• =
n 32 (l
+ !) (l +
Ra 2Z
+n 2l (l + !)
3
1)
for
for
j = l
j = l -
!.
+ -!,
(9.11)
(9. 12)
1 Equation (9.10) was first derived on t he quantum mechanics by W. Heisenberg
and P . Jord an , loco cit.
SEC.
137
HYDROGEN FINE S T RUCTU RE
9.3)
The sign in front of the right-hand side of each of these equations has been
changed to conform with Eq. (9.9), where a positive sign means an
in crease in the term value, or a decrease in the energy. Adding Eq.
(9.9) to each of Eqs. (9.11) and (9.12),
j = l
for
j
for
=
If k in Sommerfeld's equati on is repl aced by land l
these equations will result. If again k is replaced by j
3~ D~1
0.046 -,
0 036-,
~(\B~l~
P
3 p," P%
I, '.'
35' S' 2l'
,
)/21
I
II
I
I
_!.
(9.13)
2
+ 1, respectively,
+ 1, which is just
r
II
O.364cm;'
0.364
."-6563
>"486\
FIG. 9.4 . -Sch em a ti c dia grams of th e lin es H" and
the same as replacing l
single equation
l
+~,
H~.
in the Balmer series of hydrogen.
+ 1 and l of Eq. (9.13) by j + ! , we obtain the
sr
Ra 3 ( _I_ _ ~) .
2Z 4
=
j
n
+!
4n
(9.14)
9.3. Obse rved Hydrogen Fine Structure.- Schematic diagrams of
the theoretical fine structure of the first two lines of the Balmer series
of hydrogen are shown in Fig. 9.4. Applying selection and intensity
rules, both H" and H II should be composed of two strong components and
three weaker ones. Nei ther one of these patterns has ever been resolved
into more than t wo components. The best results to date are those of
Lewis, Spedding, Shane, and Grace;' obtained from H2, the behavior of
1 LEWIS , G .
F. H .. C. D.
N. , and F . H.
and N . S .
SHANE ,
Phys. R ev., 43, 964, 1933; also
Phys. Rev., 44, 58,1933.
SPEDDING ,
GRACE ,
SPEDDING,
138
I N T ROD UCTIO N T O A T OilHC S P EC T RA
[C HAP. IX
t he t wo kn own hydrogen isot op es. Usin g Fabry a nd Perot et alons,
ph ot ographs simila r to t he one shown in the center of Fig. 9.5 have been
obtaine d. For t his phot ogr aph t he first order of a 30-ft. grating mounting (of t he Lit trow type ) was used as t he auxiliary disper sion instrument.
M icrophotometer cur ves of both H! and H2 are reprodu ced abov e and
below ea ch pa ttern. It is to be not ed t hat t he compone nts of H ~ ar e
considerably sharper than H~, and t hat a t hird com pone nt is beginning to
show up. The broadening is du e to t he D oppler effect and should be
greater for t he ligh t er isot ope.
FIG . 9 .5.-Fine st ruc tu re of H& a nd H i', fro m the Balmer se r ies of the t wo h ydrogen
iso topes. M icrop h ot o rnet er cu r ves above an d below wer e m ade from t he in t er fer en ce
pa tterns in the cen te r . ( A f ter S p edd inq , Sha n e, and Grac e.)
Theor eti cal intensities for t he fine structure of hydrogen were first
calculated by Sommerfeld a nd Unsold I in 1926. E xpe rime n tally it is
found t hat t he relative in t ensiti es of t he two ma in com po ne nts, t he only
ones resolved , de pend lar gely upon t he condit ions of excitation . In
some inst ances t he supposedly wea ker of t he t wo lin es will be t he stro nger,
as it is in Fig. 9.5.
In going to higher members of t he Balmer series t he separatio n of t he
t wo strong compone nts of eac h member a pproaches t heo retically and
expe rime ntally t he sepa ration of t he com mon lower state , 0.364 em- I.
This in t erval occurs in eac h doublet between the faint er of t he t wo strong
lines and the next t o t he weak est sat ellit e.
9.4. Fine Structure of the Ionized Helium Line J-,4686.-A better
detailed agreem en t betw een observati on and t heory has been found in the
hydrogen -like spectrum of ioniz ed helium. A microphot ometer curve
of t he lin e M686 is give n at the bot t om of Fig. 9.6. This line corresponds
with the first line of t he P aschen series in hydrogen (see Fi g. 2.8). Wi th
I
S OI\lIllE RF' E LD,
see also
A., a nd A. UNSO LD, Z eits. f. Phys. , 36, 259, 1926; 38, 237 ,
Eo , Ann. d. Phsjs. 80, 437, 1926.
SCH ROD INGE R,
19~6;
I N1 'RODUCTIO N TO ATOMIC S PEC T RA
140
j = I -
[C HAP.
IX
!
tfl =
- i (l + u ) ' M ep r_l' M rF_1- 1.
tf2 = i(l - u - 1) . MoPr~ll. M rF_ 1- 1.
tf3 =
M ePr . M rG- 1- 1.
tf4 =
M OP/+l . M rG- 1- 1.
(9. 18)
In t his form eac h wave function in pol ar coordinates is written as t he
product of two functi on s, on e of whi ch gives tf as a functi on of t he angles
<I> and 8 alone and t he othe r as a function of r alone . },tIe and .M r are t he
angular and radial normalizing fact ors, respecti vely. The sphe rica l
harmonics P/ (cp,8 ) ar e defined by
d )l-U(cos 8 - 1) 1
+ u ) ! sin- 8( d cos
8
2 l!
. eiu<P.
2
P 't = ( - I )u(l
u
1•
(9.19)
The r adial functi on s F and G are functi on s of t he electron-nuclear
di stance alone and are giv en by
and
whe re
0"1
=
0"2
=
(N
+ l + 1) .
+ 1, 2kr),
+ 1, 2kr),
r P-l • e:" . IF 1 ( - n r, 2p
- n r ' r P - 1 . e-kr . IF 1 ( -n r + 1, 2p
(9.21)
(9.22)
a nd
nr
=
n -
N =
k
=
Il - 11=
(9.23)
(9.24)
(9.25)
radial qua ntum number,
+ 1) 2 - a ]l,
[n; + (l + 1) 2 + 2p n r]l ,
p = [(l
2
IN'
al
al = 0.528 Angstroms,
a2 =
5.30 X 10- 5,
(9.26)
wh ere al is t he radius of t he Bohr first circ ular orbit and a is t he finestru ct ure constant. The fun ctions IF 1 ar e of t he form of series
+
+
+
_
_ a_
a( a
1) 2
a (a
1) (a
2) . 3
• • •
) -1+
(
C
IF1a,b,c
b 'l!c+ b (b+l )'21 +b(b+l ) (b+2 ) '3!c +
(9.27)
which t erminate for nega t ive in t eger values of a.
9.6. The Angular Distribution of the Probability Density Pe.- W it h
t wo set s of four wa ve functi ons, t he soluti ons and hence t he probability
den sity # * must be di vided into t wo part s, one for j = l + ! and one
for j = l - !. Corres ponding to give n n, l, and i. t here are 2j + 1
magn eti c states for whi ch t he magnetic quantum number m takes values
from m = + j to m = -j. H ar tree! has shown that f or given n, l, and
j the magnetic states with equal but oppo sit e m have the same probabilitydensity di stribution. He has also shown t hat the angular di stribution
1
HARTREE ,
D . R., Proc. Camb. Phi l. S oc., 26, 225, 1929.
SEC.
141
HYDROGEN FINE S T RUCTU RE
9.6]
fo r two electrons with the same n, i, and m, and l = j ± ! is the sam e.
Af3 an example of t hese two t heorems, the angular charge dis tri butions,
as t hey are ca lled by H art ree, for t he m agn et ic states m = ± ~ of a 2P~
t erm are no t only t he sam e but are also id enti cal with t he two magnetic
states m = ±t of t he 2D~ te rm. It sho uld be point ed out, however, t hat
t he radial charge distributi ons of t he 2p and 2D te rms are differen t.
Wi th t his sim plification of t he problem t he charged distributions need
be det ermined for j = l + ! and positive m on ly .
11\
:--_<:::-_ _----:-...:::".._ _1rn-
For j = l
wri t t en
wy;* =
+!
t he prob ability density
P o ' P ; = M~[pr+ lPr,~\
r f*
'%
on t he Dirac t heory is
+ pr-tlPr-tl*J . M~[F1 + Gil,
(9.28)
The first par t of the solu tio n is a function of t he ang les cp and () alone .
(9.29)
where t he angula r normalizing fa ct or in t he form give n by R oess! is
M~ =
[41T(l
+ u + 1) !(l
- u) !]-l.
(9.30)
From Eqs. (9.19) and (9.29) it may b e seen t hat t he ang le cp occ urs
in ea ch polynomial in t he form of an exp onenti al e,m'!' a nd is a lways
1
RoE SS,
L. C ., Ph ys. Rev., 37, 532, 1931.
142
[CH.\ P. IX
I NTROD UCTIO N TO A TOM IC S P EC T RA
t o be multiplied by it s complex conjugate e-im<P . This gives a constant
for all m values. Exactly the same result is obtained on the Sehrodinger
theory, which indicates t hat the probability density is symmetrical
about t he cp (magnet ic) axis for all states. Leaving out t he fa ctor
1/471" in t he normalizing factor M~, the probability den sity P B for a
number of states is given in Table 9.2 and is shown gra phically in Fig .
9.7. 1
Hartree ha s shown t hat for electrons with the same n, l, and j , the
probability density summed over the states m = +t to m = +j pre sents
spherical symmetry. In this case spherical symmetry is shown by
m= +i
k
(9.3 1)
P B = constant ,
m= + l
given in the last column of Table 9.2 and graphically by the shaded
TADLE 9.2. -THE PROBABILITY DENSITY AS A F UNCTION OF THE ANGLE 0
Energy states
i
m
-
PB
m=+ i
~
PB
m=+ l
--2S. or 2Pj
!
2P i or 2D i
3
2
!
s
~
!
1
3
! sin 2 0
!(3 cos- 0
2
2D 1 o r W I
2
1
2
2F! or "G!
7
2
1
+
1
H sin" 0
i
-l~
!
2
Jg'>- sin' 0
i sin 2 0(15 cos" 0 1)
i(5 cos' 0 - 2 cos! 0
1)
2
~
+ 1)
+
sin' 0(35 cos ! 0
1)
0(21 cos ' 0 - 6 cos - 0
1'«(175 cos" 0 - 165 cos' 0
cos 2 0
9)
H sin "
3
+
+
+ 1)
+ 45
4
areas under t he straight lines in Fig. 9.7. Since the probability density
for the negative m states is the same as that for the corresponding
positive m states, the sum of t he probability densities for all negative m
states is also a constant. This means that four electrons in 2P i states
or two electrons in 2P. states will form a spherically symmetrical charge
distribution. 2 Another and similar consequence of t he Dirac theory
is that a single electron in a 2P. state is two electrons in a 2P i sta t e
present spherical symmetry. Not only are all S states, formed from a
1
2
WHITE, H . E ., Phys. R ev., 38, 513, 1931.
These correspond to the jj-coupling states
(!!! I). and
(!!)o in Fig. 14 .21.
SEC.
9.6]
HYDROGEN nNE STRUCTURE
143
single s electron or from a complet ed subshell of any type of electrons,
spherically symmetrical, as on the Schrodinger theory, but, also, all
FIG. 9.8.-T he angula r factor P8 of t he prob a bili t y de nsity"''''' plotted in angular
coordinates. Above an d below t he quantum-mechanica l electro n di stributi on s the co r responding classical elec tron orb its are sho wn or iented in each case according t o t he model
l ", 8*, j *, and m.
one-elect ron systems with but one valence electron and t hat in a 2Pt
state. The norm al stat es of B, AI, Ga, In, and TI are examples of
t his.
144
I NTROD UCTIO N TO ATOMIC S P EC T RA
[C HAP. IX
If angular coordinates are used in plotting Po a number of in teresting
correlations with t h e classical orbit s may be made. Su ch a ngula r
distributions are shown in Fig. 9.8 for s, p, d, and f electro ns . Above
an d below ea ch density figure the corresponding classical orbits a re shown
orient ed in ea ch case according to model a (see Fig. 4.8). On t his model
t he a ng ular mom entum vect or s l * = Vl (l
1) and s* = v s (s
1)
a re com bine d to form t he result an t j * = vj(j
1), j* in t urn being so
oriented t hat it s projecti on m on t he 'P axis takes half-int egr al values
m = ±t, ±t, ±-t, ... ±j. It m ay be seen fr om t he figures t hat in
t he precession of l* about j* and t he sim ultaneo us pr ecession of j * a bo ut
t he 'P (m agnet ic) axis t he electron orbit fills out a figure in space not
greatly unlike t hat of t he quantum mechanics. The den sity curves
are symmetrica l abo ut 'P. The orbit normal l * is show n for t he fou r
p ositions it takes when in t he sa me meridian plane (the plane of t he
pap er ). In order t o illu strate an orbit rather than its st raight-line
proj ec tion, t he orbit plane is tipped slight ly out of the normal t o l *.
Whil e t he classica l mod els b, c, and d may also be com pared with t he
prob ability curves of Fig. 9.8, mod el a seems to be in t he best gene ra l
agreement. It sho uld be poin t ed out t hat t he electro n is not -confined
to t he shade d areas in eac h prob abilit y curve, bu t t hat t he magnit ude
of a lin e joining t he center to any poin t on t he curve is a measure of t he
pr ob a bili ty of t he elect ron bein g found in t he dire cti on of t hat line.
From t he physical stan dpoint it is in tere sting t hat Po becom es zero
for (J = 0 and 7r only. The electron can pass t he refore fr om any region
to another without going t hrou gh a node of zero probability. This is
also t r ue for the radial fact or as will b e sho wn.
9.7. The Radial Distribution of the Probability Density Pr.- For
j = l
t t he probability-density fa ctor P ; is [see Eq . (9.28)1,
+
+
+
+
P r = M ;(F1
The radial nor malizing fa ct ors
+ Gn.
M; have been
give n by R oess as
+ + n )(2k )2p+l r (2p + n, + 1)
+ l )J2[ (N + l + 1) 2 + nr(nr + 2p ) ]'
2 _
(N
p
M r - n r !2N [r (2p
(9.32)
r
(9.33)
whi ch from t ables of t he gamma function ar e readily evaluated . The
radial fun ction Fr, as compared with Gr, is extremely small and for
hydrogen is of t he order of m agnitude of t he square of t he fine-struct-ure
consta nt. If a is set equal t o zero t hrougho ut t he radial equati on s
t he ga mma functi ons be come sim ple fact orials, Fi va nishes, and P ;
redu ces to t he radial factor of t he Schrodinger t heo ry (R n.D2 (see Chap.
IV) .
Since t he radial densiti es on t he Schrodinger and Dirac t h eories
are so ne arly id enti cal, it is difficul t to show t heir differences graphically
SEC.
9.71
145
HYDROGEN FINE STRUCTURE
However, by splitting up M;(F; + GD into two parts M;F; and M;G;,
curves may be given for each on different scales. In Fig. 9.9, for example,
curves for the 4p, 2P j state are drawn. The factors M;Gi and M;Fi
are shown by the dotted lines in the top and middle figures, respectively.
Multiplying by 47l'r 2 the density-distribution curves (shaded areas) are
obtained. The main reason for representing the density distribution
in this way is to show that the zero points of each of the two upper
0.02
QOI
o
30
- r _ _ 400,
08,10';__
02,10 "
04
30
40a;
11
08
"'' '*,
FIG. 9.9 .-The radial fa ctor P T of the probability density
for the st a te 4p,2P j.
The spin correction (middle figure) added to the Schrodinger di stribution (u p pe r figure)
gives the Dirac distribution (lower figur e).
curves, other than those points at r = 0 and r = 00, occur at different
values of r . While the resultant curve 47l'r 2M;(Fl + Gf) in the lower
figure is almost identical with the top figure for 47l'r 2M;G;, it does not
come to zero at the two points near r = 6 and r = 14. The difference
at these two points only has been exaggerated in the bottom figure . The
slight shift in the probability density, due chiefly to the addition of the
F function, produces the fine-structure shift of the hydrogen energy
levels. This will be given in Sec. 9.9. The classi cal 4p orbit shown in
the last figure is drawn according to model a. As in the case of the
Schrodinger theory, the density distribution differs greatly from zero
only within the electron-nuclear distance of the corresponding classical
orbits.
SEC. 9.9]
14"(
HYDROGEN PI NE STRUCTURE
t he probabilit y densit y ifiifi* on Schrodinge r's t heory . Introdu cing t he
electron spin into t he classica l model, t he orbit in a weak field no longer
pr ecesses about t he magneti c axis but about t he spin-or bit resultant j*,
and j* in t urn precesses about cp (or H ). Correspond ing to t his change in
space quantization is t he cha nge in t he (J fa ctor of t he probabilit y density
# * brough t abo ut by int rodu cing t he spin t hro ugh D irac's theory.
R adially, however, lit tle cha nge has occurred in either t he orbital or
t he quantum-mec ha nical model ifiifi*. T his might have been expected,
since t he main part of t he energy in hydrogen is given by t he radial
fact or in ifi an d t he introdu cti on of t he spin has added only the finestructure corr ect ions to t his energy.
9.9. The Sommerfeld Formula from Dirac's Theory.- Gordon 1 has
shown t hat Somm erfeld's fine-structure formula [Eq. (9.4) ] may be
derived fr om Dirac's t heory of t he hydrogen atom. Gord on obtains
for t he t erm values of E q. (9.4),
(9.34)
where j' is a new qua ntum nu mber given for the va rious terms by t he
va lues give n in T able 9.3.
TABLE 9.3.-Q UANTUM N UMBERS FOR THE HYDROGEN ATOM ON D IRAC'S THEORY
l
j
2S 1
2P I
2P i
2Di
2D,
2F,
2Fl
0
0
1
2
2
3
3
.!. ~
!
.~
a
2'
1!
s
!
!
+ 1
-2
+2
-3
+3
-4
2
r
- 1
Since j' occurs as t he squa re only, t he minus sign can be neglected.
Due to t his fact j' 2 can be re placed by (j + t)2, and Eq. (9.34) becomes
T
= _J.LC{1 +
h
2
(n - j -
t
2
a Z
+ vj +
t )2-
}-l + J.Lc.
a 2Z 2 ) 2
h
(9 35)
.
E xpanding, following Sommerfeld's treatment, " and dropping all but
t he first t wo terms,
(9.36)
The first te rm gives t he Balmer te rms and t he second t he corrections
given by E q. (9. 14) . -t- ~w!
1
2
GORDON, W., Z eits. f . Phus., 48, 11, 1928.
See SOMMERFELD, A., "Atomic Structure and Line Spec tra," p. 477, 1923.
148
INTROD UCTION TO ATOMIC S PEC T RA
[C HAP.
IX
Problems
1. Ca lculate the fine-structure pattern for th e electron transit ion n = 3 to n = 2
for ionized helium. Det ermine the wave-length at which this line is to be found , and
compare the fine-structure intervals in Angstroms with those of H a in hydrogen.
2. Con struct radial-densit y-distribution curves for a 4d,2D l and 4d ,2D i st ate s
(see Fig. 9.9).