Department of Physics, Middlebury College

1992-93

Modern Physics Laboratory



XIV. The Zeeman Effect in Atomic Mercury: Measurement of Atomic g-factors



Discussion

The neutral mercury (Hg) atom in its ground state has 80 electrons in the configuration 1s22s22p63s23p63d104s24p64d104f145s25p65d106s2 in which the n = 1, 2, 3, 4, and 5 electrons form an inert core for two 6s valence electrons. The optical emission spectrum of Hg results from transitions of the two valence electrons between various excited two-electron configurations. The Hg spectrum therefore has many features in common with the two-electron helium system. Complete discussions of the two-electron system, the Hg spectrum, and the Zeeman effect may be found in Refs. 1-7. A brief summary will be given here.

In a helium-like system, the total angular momentum J of the atom is determined solely by the total angular momentum of the two valence electrons, since the orbital and intrinsic spin angular momenta of the electrons in the closed-shell, inert core are coupled to zero. In the Russell-Saunders or LS coupling scheme, the orbital angular momentum quantum numbers 1 and 2 of the two valence electrons are coupled to form a resultant angular momentum quantum number L, and similarly, the intrinsic spin angular momentum quantum numbers s1 and s2 are coupled to a resultant intrinsic spin angular momentum quantum number S. When the conditions for the LS coupling approximation are satisfied, the operators and commute with the Hamiltonian operator for the atomic system and the allowed energy levels may be labeled directly in terms of the angular momentum quantum numbers L and S, but not the individual quantum numbers 1, 2, s1, and s2. The total angular momentum operator also commutes with and therefore the total angular momentum quantum number J = L + S may also be used to label atomic energy levels. The two-electron wavefunction can therefore be written LSJMJ, where the magnetic quantum number MJ is the eigenvalue of the operator z.

The angular momentum addition theorem restricts the possible values of an angular momentum quantum number L resulting from the sum of two individual angular momenta 1 and 2, to the range

| 1 - 2 | ¾  L  ¾  | 1+ 2 | .

A similar restriction governs the sum of s1 and s2 to form S, and the sum of L and S to form J. These angular momentum restrictions may be used to predict the quantum numbers of the low-lying excited states of the neutral Hg system. If one considers only single electron excitations of the 6s2 ground state, the lowest excited state configurations should be 6s6p, 6s6d, 6s7s, 6s7p, and 6s7d. The Russell-Saunders states LSJMJ resulting from these configurations are given in Table I. In Table I, the traditional spectroscopic notation 2S+1LJ has been used to denote a state with quantum numbers L,S, and J. For a two-electron system, s1 = 1/2 and s2 = 1/2, so that the total intrinsic angular momentum quantum number S of the atom is limited to the values 0 and 1, corresponding to what are called singlet and triplet terms, respectively. Since the electric dipole selection rules (to be discussed below) only allow transitions among singlet states or among triplet states, it is useful to group the singlet and triplet states of the two-electron system separately.

The emission spectrum of neutral Hg has been studied extensively and many lines have been identified as transitions between the terms expected from Table I. An energy level diagram of Hg indicating many of the observed optical transitions is given in Fig. 1.3

The electric dipole selection rules allow transitions that involve only the following changes:3
(1) S = 0,
(2) L = ±1, 0,
(3) J = 0, ±1, but not J = 0 --> J = 0,
(4) MJ = 0, ±1, but not MJ = 0 --> MJ = 0 when J = 0.

Note that although only triplet-triplet and singlet-singlet transitions are permitted by the electric dipole selection rules, occasional "crossing" transitions such as the 2536.5 Å triplet-singlet line shown in Fig. 1 are observed. Such forbidden transitions can be used to study the extent to which the assumptions of the LS coupling approximation are violated8

The most important technique for determining the angular momentum structure of atomic energy levels is the observation of the Zeeman effect on spectral lines. In the Zeeman effect, an atom is placed in a strong, homogeneous magnetic field B = B z. The interaction of the total atomic magnetic moment µ with the external magnetic field B produces a change in the energy of the atom by an amount that classically is given by

E = - µ * B .        (1)

Classically, the magnetic moment of an atom is due to the motion of its component charges so that for a single electron with charge -e in arbitrary motion in an atom, it can be shown that the atomic magnetic moment µ is related to the electron orbital angular momentum by1

µ = - e/2me         (2)

TABLE I. Possible LS coupling terms for the ground state and low-lying excited state configurations of the neutral Hg atom.

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Configuration

Singlet (S = 0)

Triplet (S = 1)

  Terms

  Terms

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6s6s

61S0

  -

6s6p

61P1

63P2,1,0

6s6d

61D2

63D3,2,1

6s7s

71S0

73S1

6s7p

71P1

73P2,1,0

6s7d

71D2

73D3,2,1

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so that classically
E = + e/2me z B.

In quantum mechanics, the classical dynamical variable z must be treated as an operator z with eigenvalue m in the state m. The quantum mechanical result for the first order energy shift of the electron energy is then given by

E

 = 

+ e/2me m B

 = 

+ e/2me m B

 = 

µB m B

(3)

where µB = e/2me = 9.27 x 10-24 J/T is called the Bohr magneton.

To extend this argument to a real atom such as Hg, two essential modifications must be made to the reasoning that led to Eq. (3).

(1) The intrinsic magnetic moment of the electron has not been incorporated in the derivation of Eq. (3). Classically, one expects that the electron's intrinsic magnetic moment is related to its intrinsic spin angular momentum S by a relationship similar to Eq. (2), however, the internal dynamics of the electron cannot be described successfully by classical physics and the electron intrinsic spin is observed experimentally to have a magnetic moment nearly twice as large as suggested by Eq. (2), namely

µs = - gs e/2me s

where gs 2.

(2) In the multi-electron atom, the net atomic magnetic moment results from the sum of the intrinsic and orbital magnetic moment contributions of all electrons. Although an inert, closed-shell core contributes no net magnetic moment to an atom, there remains the difficulty of calculating the intrinsic and orbital magnetic moment contributions of the valence electrons. Even so, one would still expect that the net atomic magnetic moment would be related to the total angular momentum J by a relationship like Eq. (2), namely

µJ = - gJ e/2me J

where gJ is a dimensionless factor of order unity that would depend in some essential way on the quantum mechanical coupling of the angular momenta of the valence electrons.

A quantum mechanical calculation that incorporates these two modifications and revises Eq. (3) yields

EMJ = gJ µJ B MJ        (4)

where the Landé g-factor gJ is given by

gJ = 1 +

J (J + 1) + S (S + 1) - L (L + 1)

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2J (J + 1)

    (5)

Eqs. (4) and (5) provide an important tool for determining the angular momentum structure of atomic energy levels from experimental spectra. To see this, consider the Zeeman effect of the 4046.6 Å transition between the 73S1 and 63P0 terms of atomic Hg shown in Fig. 1. From Eq. (4) we see that the 73S1 term splits into three levels with MJ = +1, 0, -1 and the relative magnitude of the splitting is set by g(3S1) = 2, calculated from Eq. (5). The 63P0 state has just one level with MJ = 0 and g(3P0) = 0. The magnetic splittings of the 73S1 and 63P0 terms are shown in Fig. 2 where the three allowed electric dipole transitions are also indicated. Figs. 3(a) and 3(b) show experimental spectra for the 4046.6 Å, 73S1 --> 63P0 transitions with no magnetic field and with a magnetic field B = 29.0 kG, respectively. The observed splitting of the zero-field line into three lines is called the normal  Zeeman effect because of Zeeman's early classical prediction that spectral lines should be split into just three components when an external magnetic field is applied.

When the angular momenta of the two levels involved in a radiative transition are not as simple as in the example given above, the observed spectra can be considerably more complex than expected classically and are often called anomalous  Zeeman spectra. Figs. 4(a) and 4(b) show experimental spectra for the 4358.4 Å, 73S1 --> 63P1 and 5460.7 Å, 73S1 --> 63P2 transitions of atomic Hg in a magnetic field B = 29.0 kG. These anomalous spectra can be understood by constructing appropriate energy level diagrams similar to Fig. 2, using Eqs. (4) and (5) and by applying the electric dipole selection rules.

Electric dipole transitions are also known to have the following polarization properties:

 
(1) MJ = ± 1 circularly polarized ( case),
(2) MJ = 0 linearly polarized along z direction ( case).

By placing a linear polarizer between the Hg source and spectrometer one can see how the various Zeeman transitions are affected. The polarizations of the emitted radiation ( or ) are often indicated in the manner shown in the lower part of Fig. 2. Fig. 5(b) shows the Zeeman splitting of the 5460.7 Å, 73S1 --> 63P2 transition for a magnetic field strength B = 29.0 kG; Figs. 5(b) and 5(c) show spectra obtained with a linear polarizer selecting the and transitions, respectively.

Procedure

A photograph of the Zeeman effect apparatus used in this experiment is given in Fig. 6.

(1) Turn on the cooling water to the Varian V-4005 electromagnet and verify that there is sufficient water flow. Be sure to remove the Hg Pen-Ray lamp from between the electromagnet pole pieces before turning on the magnet power supply. This is to assure that the Hg lamp is not crushed by the pole pieces as they are drawn together when the magnet current is turned on.

(2) Turn on the electromagnet power supply and advance the magnet current to 40 A. With the current at 40 A, determine if the Hg lamp can be placed between the pole tips. If the lamp cannot be placed between the pole pieces you will have to remove the Hg lamp, turn the magnet current down to zero and rotate the pole piece adjustment so that the pole pieces are farther apart. With the Hg lamp removed, turn the magnet back up to 40 A. See if the Hg lamp can now be put between the pole pieces.

(3) Measure the magnetic field at the center of the region between the pole tips using the Hall effect gaussmeter. Be sure that the gaussmeter has been calibrated using the calibration magnet. With 1 in tapered pole pieces separated by 0.25 in you should measure a magnetic field of nearly 27 kG on the gaussmeter.

(4) Adjust the lenses used to transport light from the Hg lamp to the SPEX 1704 spectrometer. You should be able to obtain adequate light intensity with entrance and exit slit settings of less than 10 µm. Adjust the lock-in amplifier output for proper operation with the Metrabyte board of the IBM PC/XT computer. Boot up the computer and prepare the McSPEX program for data acquisition.

(5) Obtain spectra for the following three Hg lines:
(a) 4046.6 Å 3S1 --> 3P0,
(b) 4358.4 Å 3S1 --> 3P1,
(c) 5460.7 Å 3S1 --> 3P2.

In Fig. 1 you can see that these transitions are from the same initial state to three members of a fine structure multiplet. These three transitions are each to be observed under the following four conditions:

 
(a) magnet off / no polarizer,
(b) magnet current at 40 A / no polarizer,
(c) magnet current at 40 A / polarizer aligned for transmission of light that has linear polarization parallel to B,
(d) magnet current at 40 A / polarizer aligned for transmission of light that has linear polarization perpendicular to B.

When finished, you should have a total of twelve spectra stored on disk.

(6) When step (5) is completed, immediately remove the Hg lamp and measure the magnetic field between the pole tips with the magnet current set at 40 A. Record your magnetic field reading Fig. 6. Photograph of the optical configuration used in the Zeeman effect measurements. in your laboratory notebook.

Analysis

(1) Prepare Zeeman effect energy level diagrams like Fig. 2 for the three transitions 3S1 --> 3P0,1,2 observed in the lab. Indicate electric dipole allowed transitions with their emitted polarizations as is in Fig. 2. Do the spectra of Figs. 4(a) and 4(b) agree with what you expect from your energy level diagrams? In particular, why does there seem to be a peak missing in the middle of Figs. 4(a)?

(2) The Hall effect gaussmeter is inaccurate above 20 kG so that your gaussmeter readings are only rough indicators of field strength; however, your experimental spectra can be used to measure B if the gJ-factor of one of the states is known by an independent technique. Luckily for you, the gJ-factor of the 3S1 term has been measured to be g(3S1) = 2.0050 ± 0.0047 in an independent experiment.9 This is so close to the Landé value expected from Eq. (5) that you will take g(3S1) 2 in the analysis of this experiment.

Taking g(3S1) 2, the splitting of your 3S1 --> 3P0 Zeeman spectrum can be used to measure B using Eq. (4). From your experimental data, determine B and an associated uncertainty. To do this, convert all transition wavelengths (in Å) to transition wavenumbers 1/ (in cm-1). This will simplify your calculations because Zeeman energy splittings are proportional to differences in transition wavenumbers, not differences in transition wavelengths. Be sure to make a corresponding conversion of your uncertainties.

How does this value for B compare to the Hall effect gaussmeter readings you recorded? Be sure to use all  experimental splittings in your determination of B and its uncertainty.

(3) With the value of B determined in step (2) and g(3S1) 2, use your experimental 3S1 --> 3P1 and 3S1 --> 3P2 Zeeman spectra to determine values for g(3P1) and g(3P2), and their respective uncertainties. To do this, convert all transition wavelengths (in Å) to transition wavenumbers 1/ (in cm-1). Be sure to make a corresponding conversion of your uncertainties. Use all experimental splittings in your determination of g(3P1) and g(3P2) and their respective uncertainties. How do your values for g(3P1) and g(3P2) compare to the Landé g-factor predictions of Eq. (5)? Consult Ref. 10 for an efficient way to tabulate your results.

(4) For each of the three transitions 3S1 --> 3P0,1,2, discuss briefly the spectra taken with the linear polarizer. Show that your experimental polarization spectra correspond to those expected for the and transitions drawn in the energy level diagrams of step (1).

References
1. R.A. Serway, C.J. Moses, and C.A. Moyer, Modern Physics  (Saunders College Publishing, Philadelphia, 1989), pp. 216-230.
2. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (John Wiley & Sons, New York, 1985), pp. 364-370.
3. A.C. Melissinos, Experiments in Modern Physics  (Academic Press, New York, 1966), pp. 43-52, 283-294, and 320-339.
4. H.G. Kuhn, Atomic Spectra  (Academic Press, New York, 1962).
5. G. Herzberg, Atomic Spectra and Atomic Structure  (Dover, New York, 1944).
6. H.E. White, Introduction to Atomic Spectra  (McGraw-Hill, New York, 1934).
7. G.K. Woodgate, Elementary Atomic Structure  (Clarendon Press, Oxford, 1980).
8. T.D. Donnelly, Intermediate Coupling and the Paschen-Back Effect in Atomic Mercury, B.A. thesis, Middlebury College, 1990.
9. Th.A.M. Van Kleef and M. Fred, Physica 29, 389 (1963).
10. M.J. Kaufman, Zeeman and Paschen-Back Measurements in Atomic Mercury, B.A. thesis, Middlebury College, 1987.

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