46
2 One Magnetic Center
This expression shows that the magnetic susceptibility is inversely proportional to
the temperature and is known as the Curie law:
χ =
C
T
(2.37)
where C is a constant that only depends on the spin quantum number of the ground
state.
2.7 (a) Show that the denominator in Eq. 2.33 is equal to 2S + 1 when E (0) = 0
and verify the expression for the summation over M S of M 2
S for singlet, triplet
and quintet states. (b) Many researchers in the field of molecular magnetism use
the cgsemu (centimeter-gram-second electromagnetic units) system instead of
the standard units defined by the international systems of units SI. In this
alternative unit system, the value of N A μ 2
B /3k is equal to 0.12505 (nearly
1/8). Calculate C for the S-values that can be found for the TM ions and the
lanthanides.
Virtually always deviations to the Curie law are observed at low enough temperatures, because the magnetic centers in any real system are never truly isolated but
interact with their environment. Moreover, spin-orbit coupling can also introduce
extra interactions not covered by the Curie law. The interactions with other magnetic
centers will be addressed in more detail in Chap. 3, but we describe here a mean-field
approach to include their effect. In this rather crude approximation, each magnetic
center experiences an internal field due to the average interaction with the other
centers in addition to the uniform external field. This internal field depends on the
average magnetization (M) of the material and is known as the Weiss field
H = H ext + H int = H ext + λM
(2.38)
Combining this expression with the Eqs. 2.27 and 2.37 and assuming that H is aligned
along z, one obtains
χ =
M
H
=
M
H ext + λM
=
C
T
(2.39)
Then, with H ext = M
(T − Cλ)/C
, the measured susceptibility χ ext can be written
as
χ ext =
M
H ext
=
C
T − λC
(2.40)
known as the Curie-Weiss law. λC is the Weiss constant and often written as .
Positive values of are indicative of ferromagnetic interactions and a material with
dominating antiferromagnetic interactions will show a negative .
2 One Magnetic Center
This expression shows that the magnetic susceptibility is inversely proportional to
the temperature and is known as the Curie law:
χ =
C
T
(2.37)
where C is a constant that only depends on the spin quantum number of the ground
state.
2.7 (a) Show that the denominator in Eq. 2.33 is equal to 2S + 1 when E (0) = 0
and verify the expression for the summation over M S of M 2
S for singlet, triplet
and quintet states. (b) Many researchers in the field of molecular magnetism use
the cgsemu (centimeter-gram-second electromagnetic units) system instead of
the standard units defined by the international systems of units SI. In this
alternative unit system, the value of N A μ 2
B /3k is equal to 0.12505 (nearly
1/8). Calculate C for the S-values that can be found for the TM ions and the
lanthanides.
Virtually always deviations to the Curie law are observed at low enough temperatures, because the magnetic centers in any real system are never truly isolated but
interact with their environment. Moreover, spin-orbit coupling can also introduce
extra interactions not covered by the Curie law. The interactions with other magnetic
centers will be addressed in more detail in Chap. 3, but we describe here a mean-field
approach to include their effect. In this rather crude approximation, each magnetic
center experiences an internal field due to the average interaction with the other
centers in addition to the uniform external field. This internal field depends on the
average magnetization (M) of the material and is known as the Weiss field
H = H ext + H int = H ext + λM
(2.38)
Combining this expression with the Eqs. 2.27 and 2.37 and assuming that H is aligned
along z, one obtains
χ =
M
H
=
M
H ext + λM
=
C
T
(2.39)
Then, with H ext = M
(T − Cλ)/C
, the measured susceptibility χ ext can be written
as
χ ext =
M
H ext
=
C
T − λC
(2.40)
known as the Curie-Weiss law. λC is the Weiss constant and often written as .
Positive values of are indicative of ferromagnetic interactions and a material with
dominating antiferromagnetic interactions will show a negative .