bei48482_FM

250 Chapter Seven

table—hydrogen, lithium, sodium, and so on—are of this kind. They have single

electrons outside closed inner shells (except for hydrogen, which has no inner elec-

trons) and the exclusion principle ensures that the total angular momentum and mag-

netic moment of a closed shell are zero. Also in this category are the ions He, Be,

Mg, B^2 , Al^2 , and so on.

In these atoms and ions, the outer electron’s total angular momentum Jis the vector

sum of Land S:

JLS (7.16)

Like all angular momenta, Jis quantized in both magnitude and direction. The mag-

nitude of Jis given by

Jj(j 1 ) jlsl ^12  (7.17)

If l0, jhas the single value j^12 . The component Jzof Jin the zdirection is given by

Jzmj mjj,j1,... , j1, j (7.18)

Because of the simultaneous quantization of J, L,and Sthey can have only cer-

tain specific relative orientations. This is a general conclusion; in the case of a one-

electron atom, there are only two relative orientations possible. One relative orien-

tation corresponds to jls, so that J L, and the other to jls, so that

J L.Figure 7.15 shows the two ways in which Land Scan combine to form J

when l1. Evidently the orbital and spin angular-momentum vectors can never

be exactly parallel or antiparallel to each other or to the total angular-momentum

vector.

Total atomic

angular momentum

J

S

L

J

S

L

j = l + s =^3 _

2

j = l – s =^1 _

2

Figure 7.15The two ways in which Land Scan be added to form Jwhen l1, s^12 .

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