8.5 Paramagnetism in Spin Systems
455
Table 8.1 Properties of the free lanthanide tripositive ions
Ln 3+ ion
La 3+
Ce 3+
Pr 3+
Nd 3+
Pm 3+
Sm 3+
Eu 3+
Gd 3+
Ground level
1 S 0
2 F 5
2
3 H 4
4 I 9
2
5 I 4
6 H 5
2
7 F 0
8 S 7
2
¯
h 2 A so /cm −1
0
640
375
300
–
236
227
0
μ eff (expt)
0
2.46
3.61
3.63
–
1.66
3.61
7.97
g J
1
6
7
4
5
8
11
3
5
2
7
3
2
2
μ J
eff (Curie)
0
2.535
3.578
3.618
2.683
0.845
0
7.937
Ln 3+ ion
Tb 3+
Dy 3+
Ho 3+
Er 3+
Tm 3+
Yb 3+
Lu 3+
Ground level
7 F 6
6 H 15
2
5 I 6
4 I 15
2
3 H 6
2 F 7
2
1 S 0
¯
h 2 A so /cm −1
−270
−364
−520
−823
−1375
−2950
0
μ eff (expt)
9.81
10.6
10.5
9.46
7.51
4.47
0
g J
3
2
4
3
5
4
6
5
7
6
8
7
1
μ J
eff (Curie)
9.721
10.646
10.61
9.581
7.561
4.436
0
into the atomic interior effectively shields the 4f electrons from the environment
created by surrounding ligands in the solid state. This class of compounds is
referred to as the rare earth compounds. 8 Because the 4f electrons are well shielded
from surrounding ligands in solid lanthanide salts, the ligand fields in these solids
can be treated as only minor perturbations to the electronic Hamiltonians for
electrons in the valence shells of lanthanide ions. This shielding of the 4f electrons
allows a meaningful comparison to be made between results obtained for free (i.e.,
spherically symmetric) lanthanide atoms or ions and experiment.
Let us now consider specifically the Lanthanide series of tripositive ions. The
Curie magnetization for this series of ions can be calculated directly once the atomic
term symbols are known. The relevant information has been entered into Table 8.1,
together with the experimental values for this series of ions.
The Curie magnetization for a collection of paramagnetic ions (such as Gd 3+ ),
all found in unique levels (for Gd 3+ , the 8 S 7
2
level), is given by
M J (T , B) N
J (J + 1)g 2
J μ 2
B B
3k B T
= χ
(m)
J (T )B,
(8.5.59)
in which case μ J
eff is obtained from Eq. (8.5.58) as
μ eff (L, S, J ) = g J (L, S)
J (J + 1).
(8.5.60)
8 See, for example, the recent article ‘Rare earths in a nutshell’ [J. Zhou and G. A. Fiete, Physics
Today 73, 66 (2020)] on the importance of the rare earths for many technological products.
455
Table 8.1 Properties of the free lanthanide tripositive ions
Ln 3+ ion
La 3+
Ce 3+
Pr 3+
Nd 3+
Pm 3+
Sm 3+
Eu 3+
Gd 3+
Ground level
1 S 0
2 F 5
2
3 H 4
4 I 9
2
5 I 4
6 H 5
2
7 F 0
8 S 7
2
¯
h 2 A so /cm −1
0
640
375
300
–
236
227
0
μ eff (expt)
0
2.46
3.61
3.63
–
1.66
3.61
7.97
g J
1
6
7
4
5
8
11
3
5
2
7
3
2
2
μ J
eff (Curie)
0
2.535
3.578
3.618
2.683
0.845
0
7.937
Ln 3+ ion
Tb 3+
Dy 3+
Ho 3+
Er 3+
Tm 3+
Yb 3+
Lu 3+
Ground level
7 F 6
6 H 15
2
5 I 6
4 I 15
2
3 H 6
2 F 7
2
1 S 0
¯
h 2 A so /cm −1
−270
−364
−520
−823
−1375
−2950
0
μ eff (expt)
9.81
10.6
10.5
9.46
7.51
4.47
0
g J
3
2
4
3
5
4
6
5
7
6
8
7
1
μ J
eff (Curie)
9.721
10.646
10.61
9.581
7.561
4.436
0
into the atomic interior effectively shields the 4f electrons from the environment
created by surrounding ligands in the solid state. This class of compounds is
referred to as the rare earth compounds. 8 Because the 4f electrons are well shielded
from surrounding ligands in solid lanthanide salts, the ligand fields in these solids
can be treated as only minor perturbations to the electronic Hamiltonians for
electrons in the valence shells of lanthanide ions. This shielding of the 4f electrons
allows a meaningful comparison to be made between results obtained for free (i.e.,
spherically symmetric) lanthanide atoms or ions and experiment.
Let us now consider specifically the Lanthanide series of tripositive ions. The
Curie magnetization for this series of ions can be calculated directly once the atomic
term symbols are known. The relevant information has been entered into Table 8.1,
together with the experimental values for this series of ions.
The Curie magnetization for a collection of paramagnetic ions (such as Gd 3+ ),
all found in unique levels (for Gd 3+ , the 8 S 7
2
level), is given by
M J (T , B) N
J (J + 1)g 2
J μ 2
B B
3k B T
= χ
(m)
J (T )B,
(8.5.59)
in which case μ J
eff is obtained from Eq. (8.5.58) as
μ eff (L, S, J ) = g J (L, S)
J (J + 1).
(8.5.60)
8 See, for example, the recent article ‘Rare earths in a nutshell’ [J. Zhou and G. A. Fiete, Physics
Today 73, 66 (2020)] on the importance of the rare earths for many technological products.