Magnetism in the Localised Electron Model

1 7.1 Magnetism of a Free Atom or Ion

1.1 1.1 A SINGLE ELECTRON

1.1.1 1.1.1 Orbital magnetic moment

In chapter 2 the magnetic moment m associated with a current density j occupying a volume V is given as:

(7.1)

Now consider an electron within an atom. Let v be its velocity, and r its position at a given time, thus:

(7.2)

where − e is the charge of the electron (e = 1.6 × 10⁻¹⁹ C). The distribution δ (r) has dimensions of inverse volume due to its integral over space being unity.

Putting this expression into equation (7.1), one obtains the orbital magnetic moment (i.e. that corresponding to the movement of the electron in its orbit):

(7.3)

where £_o = r × m_e v is the orbital angular momentum of the electron, and m_e is its mass. This general result shows that the orbital magnetic moment of a charged particle is proportional to its angular momentum.

It is straightforward to arrive at equation (7.3) using the simple minded representation given in figure 7.1of an electron travelling with...