Magnetism in the Localised Electron Model
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7
MAGNETISM IN THE LOCAL/SED ELECTRON MODEL
(This chapter can be skipped on first reading; it assumes that the reader is aware of the basics of quantum mechanics).
In this chapter the two contributions (spin and orbital) to the angular momentum (and thus to the magnetic moment) of an electron are introduced. The way in which each electron in a free atom or ion contributes to the total magnetic moment is described using quantum mechanics. While most elements have a magnetic moment when isolated, only a few of these retain a moment when they form part of an arrangement of atoms such as a molecule, liquid or solid. Such elements are those which have unfilled 3d, 4f or 5f shells of electrons, and thus appear in the iron, lanthanide (or rare earth), and actinide series of elements. Their properties are briefly described.
1. MAGNETISM OF A FREE ATOM OR ION 1.1. A SINGLE ELECTRON 1.1.1. Orbita[ magnetic moment In chapter 2 the magnetic moment m associated with a current density j occupying a volume V is given as: m=lJrxj(r}dV
2
(7.1)
V
Now consider an electron within an atom. Let v be its velocity, and r its position ata given time, thus:
j(r')
= - e V O(r'- r)
(7.2)
o
where- e is the charge of the electron (e = 1.6 x I0-19 C). The distribution (r) has dimensions of inverse volume due to its integral over space being unity.
252
MAGNETISM - FUNDAMENT ALS
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): m0
= - (e/2) rxv = - (e/2me) ! 0
(7.3)
where ! 0 = r x me v is the orbital angular momentum of the electron, and fie is its mass. This general result shows that the orbita[ 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 7.1 of an electron travelling with velocity v on a circular orbit of radius r.
Figure 7.1 - Schematic representation of the orbita/ moment of an electron
In order to develop this idea further, one needs to make use of quantum mechanics. The stationary states of an electron experiencing the potential of the: nucleus and the average potential from all of the other electrons are characterised by 4 quantum numbers n, .e, m, and cr. Remember that: • the principal quantum number n takes the values 1, 2, 3, 4, ... + for a given n, the orbita[ angular momentum quantum numbe~r .e can take the integer values such that O:::; e : :; n -1. For e =O, 1, 2, 3, ... the states are known as s, p, d, f, ... • for a given .e, the magnetic quantum number m can take the integer values such that-.f:S:m:S:.f, • the spin quantum number cr can take the values ± 112. The orbital angular momentum associated with these states is written as: (7.4) where li is Planck' s constant divided by 21t, li = h 121t = 1.054 x 1O··34 J. s, and .e is a dimensionless vector operator often called the orbital angular mome:ntum. The values of .f2 and .e z are characterised by two integer qu
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