W ¼ À l
! Á H
! ¼ À l
! Á H cosð l
! HÞ
ð 4:2Þ
(b) Both the orbital and the spin angular momentum undergo quantization
(Figs. 4.1 and 4.2), as well as their components in a given space direction, being
their values
LðL þ 1Þ
1=2 h=2p SðS þ 1Þ
1=2 h=2p
M L 2L+1 values from –L to +L
M S 2S+1 values from –S to +S
(c) The moment of the magnetic dipole is proportional both to the orbital and the
spin angular momentum.
The property associated to the rotation of the electron negative charge, and the
related relations are
l z ¼ cM L h=2p
l z ¼ cM s h=2p and if c ¼ À ge=2mc b ¼ eh=4pmc !
l z ¼ ÀgbM S
From the a and c considerations, the energy of one electron with M S = – ½ is
W ¼ gbHM S ¼ Æ1=2gbH D ¼ gbH
Based on points a–c, the spin properties of the system under investigation
determine its behavior; thus, the Hamiltonian equation for a spin only case results in
S z w i ¼ M s w i S z w i M s Æ 1=2
ð
Þ¼Æ 1=2M s w i M s Æ 1=2
ð
Þ
Configuration of
minimum energy
Maximum energy
H
z
W = -μH
μ
μ z
μ
μ
W = -μH cos θ = -μ z H
θ = 180º
W = +μH
θ
N
N
N
a = 0
S
S
S
H
H
(a)
(b)
(c)
Fig. 4.1 Energy interaction of one electron in a magnetic field ([2], pp. 8, 11, 13)
4.1 Interaction Between Electrons and Magnetic Field
67