548
10 Quantum Statistics
The decrease in the number of electrons in m s =
1
2 states and the increase in the
number of electrons in m s = −
1
2 states are both given by the number of electrons
making the m s = +
1
2 → m s = −
1
2 transition. Moreover, as the area under an
vs. plot between two energy values gives the number of translational states
in that energy interval, we may therefore estimate the net magnetic moment for
temperature T = 0 K by twice the product of the magnitude, μ 0 , of an individual
electron magnetic moment and the number of spin transitions, given by the area
lying between the curve in Panel (b) of Fig. 10.6 and the energies F and
F + μ 0 B: this area may be approximated by the rectangular area
1
2 μ 0 BB(( F ) =
μ 0 BB 0
√ F . Thus, the net magnetic moment M ce is given by
M ce 2μ
2
0 BB 0 (V )
√
F .
(10.4.56a)
In order to facilitate a comparison between the magnetic moment of a normal
paramagnetic material with that of a typical (non-ferromagnetic) metal, it will be
convenient to replace 2 0 (V ) by 3N ce (( F )/[2
3
2
F ] via Eq. (10.4.42) to give the net
magnetic moment due to the conduction electrons as
M ce μ
2
0 B
3N ce (( F )
2 F
.
(10.4.56b)
Upon comparing this expression for M ce with the Curie law expression,
M ≡ χ
(m) (T , ρ)B =
Nμ 2
m B
3k B T
,
of Eqs. (8.4.14a) and (8.4.8) for the magnetic moment of a normal paramagnetic
material, we see that these two expressions are indeed rather similar, with the
principal difference between them being replacement of the thermal energy k B T
by the Fermi energy F . It will thus already be clear from this 0 K result for M ce
that the magnetic moment induced in a metal by a magnetic field is, by virtue of
the difference between a typical thermal energy and F , many orders of magnitude
smaller than the magnetic moment induced in a bulk substance whose constituents
contain unpaired electrons. Physically, the difference can be associated with the
fact that the spins of all unpaired electrons in a normal paramagnetic substance (by
virtue of their association with individual atoms at distinct spatial locations) are
able to align along the direction of the applied magnetic field, while only the spins
of those conduction electrons associated with energy levels that lie no more than
μ 0 B below the Fermi level of a metal may ultimately be both unpaired and aligned
along the direction of the applied magnetic field.
For nonzero temperatures, the number of continuum translational energy states
for m s = −
1
2 electrons is given by
N(m s = −
1
2 ) = 0 (V )
∞
−μ 0 B
( + μ 0 B)
1
2 f (() d ,
(10.4.57a)
10 Quantum Statistics
The decrease in the number of electrons in m s =
1
2 states and the increase in the
number of electrons in m s = −
1
2 states are both given by the number of electrons
making the m s = +
1
2 → m s = −
1
2 transition. Moreover, as the area under an
vs. plot between two energy values gives the number of translational states
in that energy interval, we may therefore estimate the net magnetic moment for
temperature T = 0 K by twice the product of the magnitude, μ 0 , of an individual
electron magnetic moment and the number of spin transitions, given by the area
lying between the curve in Panel (b) of Fig. 10.6 and the energies F and
F + μ 0 B: this area may be approximated by the rectangular area
1
2 μ 0 BB(( F ) =
μ 0 BB 0
√ F . Thus, the net magnetic moment M ce is given by
M ce 2μ
2
0 BB 0 (V )
√
F .
(10.4.56a)
In order to facilitate a comparison between the magnetic moment of a normal
paramagnetic material with that of a typical (non-ferromagnetic) metal, it will be
convenient to replace 2 0 (V ) by 3N ce (( F )/[2
3
2
F ] via Eq. (10.4.42) to give the net
magnetic moment due to the conduction electrons as
M ce μ
2
0 B
3N ce (( F )
2 F
.
(10.4.56b)
Upon comparing this expression for M ce with the Curie law expression,
M ≡ χ
(m) (T , ρ)B =
Nμ 2
m B
3k B T
,
of Eqs. (8.4.14a) and (8.4.8) for the magnetic moment of a normal paramagnetic
material, we see that these two expressions are indeed rather similar, with the
principal difference between them being replacement of the thermal energy k B T
by the Fermi energy F . It will thus already be clear from this 0 K result for M ce
that the magnetic moment induced in a metal by a magnetic field is, by virtue of
the difference between a typical thermal energy and F , many orders of magnitude
smaller than the magnetic moment induced in a bulk substance whose constituents
contain unpaired electrons. Physically, the difference can be associated with the
fact that the spins of all unpaired electrons in a normal paramagnetic substance (by
virtue of their association with individual atoms at distinct spatial locations) are
able to align along the direction of the applied magnetic field, while only the spins
of those conduction electrons associated with energy levels that lie no more than
μ 0 B below the Fermi level of a metal may ultimately be both unpaired and aligned
along the direction of the applied magnetic field.
For nonzero temperatures, the number of continuum translational energy states
for m s = −
1
2 electrons is given by
N(m s = −
1
2 ) = 0 (V )
∞
−μ 0 B
( + μ 0 B)
1
2 f (() d ,
(10.4.57a)