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6.10 ZEEMAN EFFECT

How atoms interact with a magnetic field

In an external magnetic field B,a magnetic dipole has an amount of potential energy

Umthat depends upon both the magnitude of its magnetic moment and the orien-

tation of this moment with respect to the field (Fig. 6.15).

The torque on a magnetic dipole in a magnetic field of flux density Bis

Bsin 

where is the angle between and B. The torque is a maximum when the dipole is

perpendicular to the field, and zero when it is parallel or antiparallel to it. To calcu-

late the potential energy Umwe must first establish a reference configuration in which

Umis zero by definition. (Since only changesin potential energy are ever experimen-

tally observed, the choice of a reference configuration is arbitrary.) It is convenient to

set Um0 when  2  90 , that is, when is perpendicular to B. The poten-

tial energy at any other orientation of is equal to the external work that must be

done to rotate the dipole from  0 2 to the angle that corresponds to that

orientation. Hence

Um



 2

 dB



 2

sind

B cos  (6.38)

When points in the same direction as B,then UmB, its minimum value. This

follows from the fact that a magnetic dipole tends to align itself with an external mag-

netic field.

The magnetic moment of the orbital electron in a hydrogen atom depends on its

angular momentum L. Hence both the magnitude of Land its orientation with respect

to the field determine the extent of the magnetic contribution to the total energy of

the atom when it is in a magnetic field. The magnetic moment of a current loop has

the magnitude

IA

where Iis the current and Athe area it encloses. An electron that makes frev/s in a

circular orbit of radius ris equivalent to a current of ef(since the electronic charge

is e), and its magnetic moment is therefore

efr^2

Because the linear speed of the electron is 2frits angular momentum is

Lmr 2 mfr^2

Comparing the formulas for magnetic moment and angular momentum Lshows

that

L (6.39)

e



2 m

Electron magnetic

moment

Quantum Theory of the Hydrogen Atom 223

θ B

Figure 6.15A magnetic dipole of

moment at the angle relative

to a magnetic field B.

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