The Zeeman Effect

The Zeeman effect is the name given to the interaction between an electron and an external magnetic field. For a radical tumbling freely in solution, it can be written as 

For a pair of radicals, the Zeeman effect serves to energetically separate the triplet RP into three sublevels, written as T+i, Tq, and T i. The energy of the T + i state increases, while that of the T 1 state is reduced by an equal amount. The S and Tq states possess no magnetic moment in the direction of the applied field and thus are unaffected. This is illustrated in Fig. 8.2b. 

The effect of the application of an external field on an electron is well described in many EPR texts, for example, Ref. 15. A vector picture is often used that, while approximate, describes the interaction sufficiently for most situations. The electron magnetic moment experiences a torque that causes it to process around the direction of the applied magnetic field at the Larmor frequency. 

The orientation of electron spins in this manner influences the electron-electron dipolar interaction described above. For strong magnetic fields, the diffusive 

When atoms are subjected to external magnetic fields, their spectral lines become split into several components whose separations depend on the field strength B. In atoms with zero net electronic spin (S = 0), the splittings turn out to be identical for all spectral lines. For weak, static magnetic fields B, the interaction energy for a spinless atom of angular momentum J = L with the field is 

Since ranges from -f L to — L, a level with orbital angular momentum L becomes split into (2L -f 1) components separated in energy by the amount ehB /lm. A spectral line that appears at energy AE in zero magnetic field then appears at energy 

In atoms with nonvanishing spin as well as orbital angular momentum, the situation is far more interesting. The Hamiltonian for such atoms in external magnetic fields assumes the form 

Using the approximate value 2.0 for the electron spin g factor g results in the common expression [1] 

Finally, the Zeeman correction to the energy in a state with angular momentum quantum numbers L, S, J, Mj becomes in the weak-field limit 

Splitting of the spectral lines into three components according to the normal Zeeman effect occurs only for atoms with singlet lines (terms with A = 0). Singlet lines are the main resonance lines of the alkaline earth metals (Be, Mg, Ca, Sr, Ba) and the Zinc group metals (Zn, Cd, Hg). 

For the normal Zeeman effect the energy levels are expressed as  

In strong magnetic fields the magnetic energy of an atom will be higher than the splitting of the lines. Then the L-S interaction will become negligible and the anomalous Zeeman effect resembles the Zeeman triplet (the Paschen-Back effect). 

In 1896, Zeeman observed that application of an external magnetic field caused a splitting of atomic spectral lines. We shall consider this Zeeman effect for the hydrogen atom. We begin by reviewing magnetism. 

Magnetic fields arise from moving electric charges. A charge Q with velocity v gives rise to a magnetic field B at point P in space, such that 

Consider the magnetic (dipole) moment associated with a charge Q moving in a circle of radius r with speed v. The current is the charge flow per unit time. The circumference of the circle is lirr, and the time for one revolution is lirr/v. Hence / = Qv/lirr. The magnitude of m is 

The magnitude of L is given by (5.95), and the magnitude of the orbital magnetic moment of an electron with orbital-angular-momentum quantum number / is 

Now consider applying an external magnetic field to the hydrogen atom. The energy of interaction between a magnetic dipole m and an external magnetic field B is [Halliday and Resnick, Eq. (33-12)] 

FIGURE 6.14 Probability densities for some hydrogen-atom states. [For accurate stereo plots, see D. T. Cromer, 1 Chem. Educ., 45,626 (1968).] 

Two electric charges + Q and — Q separated by a small distance b constitute an electric dipole. The electric dipole moment is defined as a vector from — Q to +Q with magnitude Qb. For a small planar loop of electric current, it turns out that the magnetic field generated by the moving charges of the current is given by the same mathematical 

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Not only can electronic wavefiinctions tell us about the average values of all the physical properties for any particular state (i.e. above), but they also allow us to tell us how a specific perturbation (e.g. an electric field in the Stark effect, a magnetic field in the Zeeman effect and light s electromagnetic fields in spectroscopy) can alter the specific state of interest. For example, the perturbation arising from the electric field of a photon interacting with the electrons in a molecule is given within die so-called electric dipole approximation [12] by ... 

Magnetic circular dicliroism (MCD) is independent of, and thus complementary to, the natural CD associated with chirality of nuclear stmcture or solvation. Closely related to the Zeeman effect, MCD is most often associated with orbital and spin degeneracies in cliromophores. Chemical applications are thus typically found in systems where a chromophore of high symmetry is present metal complexes, poriihyrins and other aromatics, and haem proteins are... 

In the previous section it has been shown that the measured sample absorbance may be higher than the true absorbance signal of the analyte to be determined. This elevated absorbance value can occur by molecular absorption or by light scattering. There are three techniques that can be used for background correction the deuterium arc the Zeeman effect and the Smith-Hieftje system. 

In practice, the emission line is split into three peaks by the magnetic field. The polariser is then used to isolate the central line which measures the absorption Ax, which also includes absorption of radiation by the analyte. The polariser is then rotated and the absorption of the background Aa is measured. The analyte absorption is given by An — Aa. A detailed discussion of the application of the Zeeman effect in atomic absorption is given in Ref. 51. 

The number of energy levels found to date, with the aid of the Zeeman effect and the isotope shift data, is 605 even and 586 odd levels for Pu I and 252 even and 746 odd for Pu II. The quantum number J has been determined for all these levels, the Lande g-factor for most of them, and the isotope shift for almost all of the Pu I levels and for half of those of Pu II. Over 31000 lines have been observed of which 52% have been classified as transitions between pairs of the above levels. These represent 23 distinct electron configurations. 

Energy level splitting in a magnetic field is called the Zeeman effect, and the Hamiltonian of eqn (1.1) is sometimes referred to as the electron Zeeman Hamiltonian. Technically, the energy of a... 

Hedeishi and McLaughlin [457] have reported the application of the Zeeman effect for the determination of mercury. Atomic absorption and atomic fluorescence techniques using closed system reduction-aeration have been applied widely to determine mercury concentrations in natural samples [458-472]. 

The need for improved background correction performance has generated considerable interest in applying the Zeeman effect, where the atomic spectral line is split into several polarised components by the application of a magnetic field. With a Zeeman effect instrument background correction is performed at, or very close to, the analyte wavelength without the need for auxiliary light sources. An additional benefit is that double-beam operation is achieved with a very simple optical system. 

A further approach to correction for broad band interference utilizes the Zeeman effect. Under the effect of a strong magnetic field atomic orbitals can be split into sets with energies higher or lower than the original. The orbitals responsible for the broad band absorption remain largely unaffected. 

Figure 12.2 Magnetic field dependence of the energy levels of ortho- and para-H2. Parahydrogen (p-H2) is a singlet that is unaffected by the magnetic field, whereas orthohydrogen (o-H2) is a triplet. Its energy levels split, showing the Zeeman effect.

The magnetic hyperfine splitting, the Zeeman effect, arises from the interaction between the nuclear magnetic dipole moment and the magnetic field H at the nucleus. This interaction gives rise to six transitions the separation between the peaks in the spectrum is proportional to the magnetic field at the nucleus. 

The observation of natural ORD or CD requires lack of symmetry in the molecule, but any molecule may exhibit magnetic circular dichroism (MCD). It constitutes a molecular analogy for the Zeeman effect in atomic spectra. Measurements in this area may well reveal substituent interactions which are masked in normal UV spectra. Extensive definitive papers of great interest which well illustrate this have appeared on MCD of pyridine derivatives, measured in cyclohexane, acetonitrile, and alcohol or aqueous acidic solutions for protonated... 

Other types of background correction have also been developed. The Zeeman effect background correction system started gaining popularity in the early 1980s. An atomic spectral line when generated in the presence of a strong magnetic field can be split into a number of components... 

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