EXPERIMENTS INMODERN PHYSICSSecond Edition

Adrian C. MelissinosUNIVERSITY OF ROCHESTER

Jim NapolitanoRENSSELAER POLYTECHNIC INSTITUTE

@ACADEMIC PRESSAn imprint of Elsevier Science

Amsterdam Boston London New York Oxford Paris San DiegoSan Francisco Singapore Sydney Tokyo

218 6 High -Resolut ion Spec troscopy

even with modest finesse F = 100, the resolution (see Eq. (4.62» is~ v = 30 MH z. Thus for)." = 500 nm, namely v = 6 x 1014 Hz

v 7-= 2 x1O .~ v

In the following two sections the Zeeman effect and the theory of hyper­fine structure are discussed in some deta il. We also discuss the isotope shiftand present data on the shift between the spectral lines of hydrogen anddeuterium. We then describe a measurement of the Zeeman splitting of the546.I-nm green line of Hg, using a Fabry-Perot etalon. The final sectionis devoted to a measurement of the hyperfine structure of rubidium usingDoppler-free saturation spectroscopy.

The bibliography on atomic spectroscopy is vast and because of the"reach" of laser experiments it is kept up-to -date . A list of suggestedreferences is given at the end of the chapter.

6.2. THE ZEEMAN EFFECT

6.2.1. The Normal Zeeman Effect

As already discussed in Section lA, the solution of the Schrodinger equa­tiorr' yields "stationary states" labeled by three integer indices, n, l, andm, where l < nand m = -t, -t + 1, ... , l - 1, l . For the screenedCoulomb potential, the energy of these states depends on nand l but noton m; we therefore say that the (2t + 1) states with the same nand l indexare "degenerate" in the m quantum number. Classically we can attributethis degeneracy to the fact that the plane of the "orbit" of the electron maybe oriented in any direction without affecting the energy of the state, sincethe potential is spherically symmetric.

If a magnetic field B is switched on in the region of the atom, we shouldexpect that the electrons (and the nucleus") will interact with it. We needonly consider the electrons outside closed shells, and assume there is onesuch electron; indeed the interaction of the magnetic field with this electron

3"Quantum Mechanics" A. Das and A. Melissinos, Gordon and Breach (1.986),New York. Or any other text on quantum mechanics .

4For our present discussion this interaction of the nucleus with the external field is sosmall that we will neglect it.

6.2 The Zeeman Effect 219

J1.

",

/",

FIGURE 6.1 Magnetic moment due to a current circulating in a closed loop.

yields for each state an additional energy !1E, given by

!1E = mIlBR. (6.5)

Thus, the total energy of a state depends now on n, I, and m, and thedegeneracy has been removed.

To see how this additional energy arises we consider the classicalanalogy . See Fig. 6.1. The orbiting electron is equivalent to a current

density''

J(x) = -ev8(x - r) ,

where r is the equation of the orbit and x gives the position of the electron;the negative sign arises from the negative charge of the electron. Such acurrent density gives rise to a magnetic-dipole moment

1f 1JL = - x x J(x) d 3x = -- e(r x v).2 2

5Por a circular orbit, the electron is equivalent to a current I = !1Q /!1T = ef T =eta/ 2rr , where to is the angular frequency co = v/ a; a is the radius of the orbit. However , aplane closed loop of current gives rise to a magnetic moment JL = I A, where A is the areaenclosed by the loop; in our case A = rra 2, hence

ev 2 evaJL = 2rra n a 2

The angular momentum for the circular orbit is L = ms im , hence

eJL=-L

2me

as in Eq. (6.1).

(6.6)

(6.7)

220 6 High-Resolution Spectroscopy

However, the angular momentum of the orbit is given by

L = r x p = me(r x v),

so that

e eli/L = -- L = -- lUL,

2me 2m e

where we expressed the angular momentum of the electron in terms of itsquantized value L = l (h /2rr )UL and UL is a unit vector along the directionof L. The energy of a magnetic dipole in a homogeneous field is

eE = -/L' B = - L· B

Zm, '

but the angle between L and the external field B cannot take all possiblevalues." We know that it is quantized, so that the projection of L on the Z

axis (which we can take to coincide with the direction of B since no otherpreferred direction exists) can only take the values m = -l, -l + 1, .. . ,l - I, l. Thus the energy of a particular state n, l , m in the presence of amagnetic field will be given by7

where''

En, l ,m = -En ,l + mBJLB , (6.8)

eliJLB = --.

2me

In Fig . 6.2 is shown the energy-level diagram for the five states with givennand l = 2, before and after the appl~cation of a magnetic field B. We notethat all the levels are equidistantly spaced, the energy difference betweenthem being

Let us next consider the transition between a state with ru , l;, m ; and onewith n I> lI> m f· As an example we choose lt = 2 and lf = I, so that the

6This was first clearly shown in the Stem-Gerlach experiment. W. Gerlach and O. Stern,Z. Physik 9, 349 (192).

7The energy in the field is positive because the electron charge is taken as negative.8me in this expression is the mass of the electron , not to be confused with the magnetic

quantum number m.

6.2 The Zeeman Effect 221

No field With field

FIGURE 6.2 Splitting of an energy level under the influence of an external magnetic field.The level is assumed to have I = 2 and therefore is split into five equidi stant sublevel s.

(a) (b) (c) mj+2 fa+1

£j=20

fa

tfa

- 1 La- 2A

I mf

<:::: + 1 Tb£'=10

i b

FIGURE 6.3 Splitting of a spectral line under the influence of an external magnetic field.(a) The initial level (I = 2) and the final level (I = I ) with no magnetic field are shown.A transition between these levels gives rise to the spectral lines. (b) The two levels afterthe magnetic field has been applied. (c) The nine allowed transitions between the eightsublevels of the initial and final states.

energy-level diagram is as shown in Fig. 6.3: without a magnetic field inFig. 6.3a, and when the magnetic field is present in Fig. 6.3b.

However, for an electric-dipole transition to take place between twolevels, certain selection rules must be fulfilled: in particular,

D.I = ±l. (6.9)

Thus, when the field is turned on, we cannot expect transitions between them sublevels with the same I, since they do not satisfy Eq. (6.9). Further,the transitions between the sublevels with Ii = 2 to the sublevels with

222 6 High-Resolution Spectroscopy

If = 1 that do satisfy Eq. (6.9) are now governed by the additionalselection rule9

tim = 0, ±l , (6.10)

and thus only the transitions shown in Fig. 6.3c are allowed.Let the energy splitting in the initial level be a, and in the final level

be b, and let A be the energy difference between the two levels when nomagnetic field is applied. Then the energy released in a transition i -+ fis given by

E; - Ef = Aj f + m ia - m f b. (6.11)

These energ y differences for the nine possible transitions shown in Fig. 6.3care given in matrix form in Table 6.1; x indicates that the transition isforbidden and will not take place .

At this point the reader must be concerned about the use of a and b;according to our previous argument (Eq. (6.8)), as long as all levels are sub­ject to the same magnetic field B, their splitting must also be the same, and

a = b = fJ-BB.

Thu s, we see from Eq. (6.11) (or Table 6.1) that only three energydifferences are possible

E; - E f = A + a(m f - m j ) = A + a Sm,

where tim is limited by the selection rule, Eq. (6.10), to the three values+1, 0, -1 . Consequently, in the presence of a magnetic field B, the single

TABLE 6.1 Allowed Transitions from Ii = 2 to If = I and the Corresponding Energies

m ofm of initial state

final state + 2 +1 0 - I -2

+1 A+2a -b A+a- b A - b x x

0 x A+a A A - a x

- I x x A+b A-a+b A -2a+b

9The selection rules of atomic spectroscopy are a consequ ence of the addition of angularmomenta. In this specific case the selection rules indicate that we consider only electric­dipo le radiation.

(6.12)

224 6 Hig h- Resol utio n Spe ctrosco py

intrinsic magnetic moment of the electron (associated with its spin) andwill be discussed in the following sections.

6.2.2. The Influence of the Magnetic Momentof the Electron

In Section 1.6 it was discussed how the intrinsic angular momentum (spin)of the electrons 8 couples with the orbital angular momentum of the elec­trons L to give a resultant J ; this coupling gave rise to the "fine structure"of the spectra .I I The projections of J on the z axis are given by mJ, andwe could expect (on the basis of our previous discussion) that the totalmagnetic moment of the electron will be given by

JLBJL = Ii: J.

Consequently, the energy-level splitting in a magnetic field B would be inanalogy to Eq. (6.8) :

(6.13)

(6.14)

These conclusions , however, are not correct because the intrinsic mag­netic moment of the electron is related to the intrinsic angular momentumof the electron (the spin) through

e enJLs = 2 -- 8 = 2 -- sus

Zm, Zm;

and not according'{ to Eq. (6.6). Consequently, the total magnetic momentof the electron is given by the operator

JL = (JLB/n)[L + 28]. (6.15)

II We will use the following notation: L, S, J represent angular momentum vectors thathave magnitude /i~, /iJs(s + I), /iJ j(j + I). The symbols I, j, etc. (s is alwayss = 1), are the quantum numbers that label a one-electron state and appear in the abovesquare root expressions. The symbols L, S, J , etc., are quantum numbers that label a statewith more than one electron and are then used instead of I , s , j .

12The result of Eq. (6.14) is obtained in a natural way from the solution of theDirac equation ; it also emerges from the classical relativistic calculation of the "Thomasprecession."

6.2 The Zeeman Effect 225

We can think of JL as a vector oriented along J but of magnitude

(6.16)

(6.17)

The numerical factor g is called the Lande g factor and a correct quantum­mechanical calculation gives'

j (j + 1) + s (s + 1) - l (l + 1)g=l+ 2j(j+l) .

The interesting consequence of Eqs. (6.16) and (6.17) is that now thesplitting of a level due to an external field B is

(6.18)

and in contrast to Eq. (6.8) is not the same for all levels; it depends onthe values of j and l of the level (s = ! always when one electron isconsidered). The sublevel s are still equidistantly spaced but by an amount

Consider then again the transitions between sublevels belonging to twostates with different l (in order to satisfy Eq. (6.9)). However, since we aretaking into account the electron spin, l is not a good quantum number, andinstead the j values of the initial and final levels must be specified. If we

13This result can also be obtained from the vector model for the atomic electron. InFig. 6.5 the three vectors J, L, and 8 are shown, and L and 8 couple into the resultant J, sothat

J = L+8.

By taking the squares of the vectors, we obtain the following values for the cosines

j2 + 12 _ s2cos (L, J) = 2/j

From Eq. (6.15) we see that

J1, /J1,B = 1cos (L, J) + 2s cos (8, J).

Thus

J1, j2 + 12 - s2 2j2 + 2s2 _ 2/2 j2 + s2 _ 12g = - = + = ! + =----------,.-----

J1,Bj 2j2 2j2 2j2

Finally we must replace j2 , s2, and P by their quantum-mechanical expectation valuesj (j + I), etc., and we obtain Eq. (6. 17).

226 6 High- Res oluti on Spect rosco py

L

FIGURE 6.5 Addition of the orbital angular momentum L and of the spin angularmomentum S into the total angular momentum J, according to the "vector model ."

choose for this example Ii = 1 and If = 0, we have the choice of ji = ~

or j i = 1, whereas h = 1.Transitions may occur only if they satisfy, inaddition to Eq, (6.9), also the selection rules for j

I:1j = 0, ±1 not j = 0 -+ j = O. (6.9a)

Furthermore the selection rules for m j must also be satisfied; they are thesame as given by Eq . (6.10)

Sm , = 0, ±1. (6.10a)

InFig. 6.6 the energy-level diagram is given without and with a magneticfield for the doublet initial state with I = 1, and the singlet final state, I = O.Six possible transitions between the initial states with j = ~ to the final

state with j = 1are shown (as well as the four possible transitions from

j = 1to j = 1)'By using Eg. (6.17 ) we obtain the following g factors

1= 1 j = 3 s = I g = 42 2 3

1= 1 j I s= I g= 2- 2 2 3

1=0 j I I g = 2.= 2 s = 2

The sublevels in Fig. 6.6 have been spaced accordingly.In Table 6.2 are listed the six transitions from j = ~ to j = 1in anal­

ogy with Table 6.1. However, since now a i= b, the spectral line is splitinto a six-component (symmetric) pattern. Thi s structure of the spectralline is indicated in the lower part of Fig. 6.6; following adop ted conven­tion, the components with polarization parallel to the field are indicatedabove the base line , and with polarization normal to the field, below. 14 Asbefore the parallel components have I:1m = 0, the normal ones I:1m ± 1.

14It is also conventional to label the parailel components with tt , and the normal onesby a (from the German "Senkrecht") .

6.2 The Zeeman Effect 227

TABLE 6.2 Allowed Transitions from ji = ~ to j f = i and the CorrespondingEnergies

mj of

final state

I+­

2

I

2

m j of initial state

3 I I 3+- +- - - --

2 2 2 2

3a b a b a bA+--- A+--- A- --- x

2 2 2 2 2 2a b a b 3a b

x A+-+- A--+- A--+-2 2 . 2 2 2 2

g =2

g= ~

g = §

,

----'----"":::::

/"

------- -----------<"

ilm= O

ilm= ±1

FIGURE 6.6 Energy levels of a single valence electron atom showing a P state and an Sstate. Due to the fine structure, the P state is split into a doublet with j = ~ and j = i.Further, under the influence of an external magnetic field each of the three levels is splitinto sublevels as shown in the figure where account has been taken of the magnetic momentof the electron. The magnetic quantum number m j for each sublevel is also shown as is theg factor for each level. The arrows indicate the allowed transitions between the initial andfinal states, and the structure of the line is shown in the lower part of the figure.

The horizontal spacing between the components is proportional to thedifferences in the energy of the transition, and the vertical height is pro­portional to the intensity of the components; the relative intensity can bepredicted exactly since it involves only the comparison of matrix elementsbetween the angular parts of the wave function .

(6.19)

228 6 High-Resolution Spectroscopy

As the magnetic field is raised, the separation of the components contin­ues to increase linearly with the field until the separation between Zeemancomponents becomes on the order of the fine-structure separation (spacingC in Fig. 6.6). At this point the Zeeman components from the j = ~ ---+ !and j = ! ---+ ! transition begin to overlap; clearly the perturbation causedby the external magnetic field is on the order of the L . S energy and affectsthe coupling of Land S into J; J ceases to be a "good quantum number."

For very strong fields, Land S become completely uncoupled, so thatthe orbital and intrinsic magnetic moments of the electron interact with thefield independently, giving rise to an energy shift

!:J.E = _/-LB L. B-2 /-LB S· B - aL· SIi Ii

= -/-LBB(m/ + 2m s ) - am/m s ·

In this region one speaks of the Pashen-Back effect. The reader can findmore details in the references, in particular in the classic text by Condonand Shortley.

So far we have discussed the case where the atom has only a singlevalence electron. In Section 1.6 we considered also atoms with two valenceelectrons and saw that for Hg the total angular momentum J = L+S, whereL results from the coupling of II and 12 and S from the coupling of SI andS2. In this case the g factor is still given by Eq. (6.17), but by using L, S,and J, the quantum numbers for the coupled angular momenta.

An interesting case arises in the 579.07-nm yellow line of Hg, which isdue to the transition from the 6 I D2 state to the 6 I PI state. (See Fig. 1.24for the energy level diagram of Hg.) As the reader should verify, by usingEq. (6.17), the g factors of the initial and final state are both equal to I.Thus we have exactly the situation shown in Fig. 6.3, and the line splitsinto three components (normal Zeeman effect) .

6.3. HYPERFINE STRUCTURE

Spectral lines, when examined under high resolution, do show structureeven in the absence of an external magnetic field. As already mentionedthis hyperfine structure arises from the interaction of the atomic electronswith the nucleus. The largest effect arises from the magnetic-dipole momentof the nucleus, but the effect of higher order moments are also observed.A related effect is the isotope shift, which shifts the spectral lines betweenisotopes, i.e., atoms of the same element but with nuclei of different mass.