Appendix L: Atomic Magnetic Dipole Moment

Appendix L: Atomic Magnetic Dipole Moment Consider an electron of mass me moving in a circular orbit of radius r around a nucleus (see Fig. L.1). If o0 is the angular velocity and T is the time period of revolution of the electron, then the orbital current produced by the electron is given by IL ¼ 

e eo0 ¼ 2p T

(L.1)

The area of the circular loop is given by A ¼ pr2

(L.2)

From elementary electricity, the magnetic moment produced by the current loop is given by mL ¼

IL A c

(L.3)

where c is the velocity of light. Substituting the values of IL and A from Eqs. (L.1), (L.2) into Eq. (L.3), one gets mL ¼ 

e p 2me c f

(L.4)

with pf ¼ me o0 r2

FIG. L.1

(L.5)

An electron of mass me is moving with velocity v in a circular orbit of radius r around the nucleus in the xy-plane.

623

624

Appendix L

Here pf is the angular momentum of the electron (multiplication of moment of inertia me r2 and the angular velocity o0 ). From Bohr’s quantization rule for orbits, one can write the orbital angular momentum as pf ¼ ħL

(L.6)

where L is the orbital angular momentum (in units of ħ) and has integral values as 1, 2, 3,…. From Eqs. (L.4) , (L.6) one can write ! mL

¼ mB L

(L.7)

eħ 2 me c

(L.8)

Here mB is called the Bohr magnetron and is defined as mB ¼

The negative sign in Eq. (L.7) indicates that the orbital magnetic moment is in a direction opposite to the orbital angular momentum. The above expression is valid only for the orbital motion.