Chapter 2
One Magnetic Center
Abstract This chapter discusses some of the magnetic phenomena that can be
observed in systems with a single paramagnetic center. After shortly reviewing the
basics of the magnetic moments of a free atom, we analyze the effect of spin-orbit
coupling and an external magnetic field on the M S levels of the ground state of larger
systems. In a step-by-step procedure, we will first derive the model Hamiltonian
to describe the magnetic anisotropy without external field, the so-called zero-field
splitting. Secondly, the role of the external field is explored and a relation is established with the magnetic susceptibility, a macroscopic quantity. The chapter is closed
by discussing the model Hamiltonian that combines the zero-field splitting and the
anisotropy of the g-tensor to complete the description of the splitting of the M S levels
in systems with one, anisotropic, magnetic center.
2.1 Atomic Magnetic Moments
The two main sources for the magnetic moment of a free atom or molecule are the
electronic spin moment and the angular moment. The motion of electrons relative
to the nucleus in atoms with filled shells and in closed shell molecules leads to zero
spin moment and zero angular moment. Therefore, such atoms and molecules can
only have an induced magnetic moment when placed in an external magnetic field.
The simplest system with an intrinsic non-zero magnetic moment is an isolated
one-electron atom or ion, treating both particles as point charges and neglecting the
possible nuclear spin. The motion of the electron around the charged nucleus induces
a microscopic current that produces a microscopic magnetic field. This leads to a
so-called orbital magnetic moment which is proportional to the angular moment of
the electron. Taking z as our quantization axis, its z-component equals
m z =−μ B m l
(2.1)
where m l is the magnetic quantum number of the electron: m l =−l, −l+1 ...l−1, l.
The quantity μ B = eh/4π m e (1/2 in atomic units) is the elementary unit of magnetic
moment, called the Bohr magneton. Its value is 9.27 × 10 −24 JT −1 .
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_2
33
One Magnetic Center
Abstract This chapter discusses some of the magnetic phenomena that can be
observed in systems with a single paramagnetic center. After shortly reviewing the
basics of the magnetic moments of a free atom, we analyze the effect of spin-orbit
coupling and an external magnetic field on the M S levels of the ground state of larger
systems. In a step-by-step procedure, we will first derive the model Hamiltonian
to describe the magnetic anisotropy without external field, the so-called zero-field
splitting. Secondly, the role of the external field is explored and a relation is established with the magnetic susceptibility, a macroscopic quantity. The chapter is closed
by discussing the model Hamiltonian that combines the zero-field splitting and the
anisotropy of the g-tensor to complete the description of the splitting of the M S levels
in systems with one, anisotropic, magnetic center.
2.1 Atomic Magnetic Moments
The two main sources for the magnetic moment of a free atom or molecule are the
electronic spin moment and the angular moment. The motion of electrons relative
to the nucleus in atoms with filled shells and in closed shell molecules leads to zero
spin moment and zero angular moment. Therefore, such atoms and molecules can
only have an induced magnetic moment when placed in an external magnetic field.
The simplest system with an intrinsic non-zero magnetic moment is an isolated
one-electron atom or ion, treating both particles as point charges and neglecting the
possible nuclear spin. The motion of the electron around the charged nucleus induces
a microscopic current that produces a microscopic magnetic field. This leads to a
so-called orbital magnetic moment which is proportional to the angular moment of
the electron. Taking z as our quantization axis, its z-component equals
m z =−μ B m l
(2.1)
where m l is the magnetic quantum number of the electron: m l =−l, −l+1 ...l−1, l.
The quantity μ B = eh/4π m e (1/2 in atomic units) is the elementary unit of magnetic
moment, called the Bohr magneton. Its value is 9.27 × 10 −24 JT −1 .
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_2
33