bei48482_FM

Figure 7.2The two possible ori-

entations of the spin angular-

momentum vector are “spin up”

(ms^12 ) and “spin down”

(ms^12 ).

Sz

h

h

1

+ 2

S =^3 h

2

ms = –^12

ms = +^12

The space quantization of electron spin is described by the spin magnetic quantum

number ms. We recall that the orbital angular-momentum vector can have the 2l 1

orientations in a magnetic field from lto l. Similarly the spin angular-momentum

vector can have the 2s 1 2 orientations specified by ms^12 (“spin up”) and

ms^12 (“spin down”), as in Fig. 7.2. The component Szof the spin angular momentum

of an electron along a magnetic field in the zdirection is determined by the spin mag-

netic quantum number, so that

Szms  (7.2)

We recall from Sec. 6.10 that gyromagnetic ratio is the ratio between magnetic

moment and angular momentum. The gyromagnetic ratio for electron orbital motion

is e 2 m. The gyromagnetic ratio characteristic of electron spin is almost exactly twice

that characteristic of electron orbital motion. Taking this ratio as equal to 2, the spin

magnetic moment sof an electron is related to its spin angular momentum Sby

s S (7.3)

The possible components of salong any axis, say the zaxis, are therefore limited to

sz       B (7.4)

where Bis the Bohr magneton (9.274  10 ^24 JT 5.788  10 ^5 eV/ T).

The introduction of electron spin into the theory of the atom means that a total of

four quantum numbers, n,l,ml, and ms, is needed to describe each possible state of

an atomic electron. These are listed in Table 7.1.

7.2 EXCLUSION PRINCIPLE

A different set of quantum numbers for each electron in an atom

In a normal hydrogen atom, the electron is in its quantum state of lowest energy. What

about more complex atoms? Are all 92 electrons of a uranium atom in the same quantum

state, jammed into a single probability cloud? Many lines of evidence make this idea

unlikely.

e



2 m

zcomponent of

spin magnetic

moment

e



m

Spin magnetic

moment

1



2

zcomponent of

spin angular

momentum

Many-Electron Atoms 231

Table 7.1 Quantum Numbers of an Atomic Electron

Name Symbol Possible Values Quantity Determined

Principal n 1, 2, 3,... Electron energy

Orbital l 0, 1, 2,... , n 1 Orbital angular-momentum magnitude

Magnetic ml l,... , 0,... , l Orbital angular-momentum direction

Spin magnetic ms ^12 , ^12  Electron spin direction

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