# 4 The single-ion spin Hamiltonian

## 4.1 Multi-electron crystal field levels

Multi-electron states that diagonalize the crystal-field Hamiltonian are indicated with capital letters \(A\), \(E\) and \(T\) in Fig.4.1 and represent the counterpart of Russell-Saunders terms (see next section) for the \(O_h\) symmetry group. The ground states of a transition metal ion with several electrons in the 3d shell and embedded in an octahedral ligand field is determined by progressively filling the single-electron levels. One proceeds similarly to the case of the free ion in which this procedure follows the prescriptions of the Hund’s rules. As imposed by the \(O_h\) symmetry, single-electron levels are the doublet \(e_g\) and the triplet \(t_{2g}\) and are indicated with small letters. The resulting multi-electron configurations are indicated with capital letters, following the denomination of the relative irreducible representations of the \(O_h\) group. Multi-electron configurations – from (3d)\(^1\) to (3d)\(^{10}\) – and the corresponding names are given in Fig.4.1. It is evident that for the d\(^1\), d\(^2\), d\(^3\), d\(^8\), d\(^9\) configurations^{12}the filling of single-electron levels basically proceeds according to the first two Hund’s rules, with the additional prescription that \(t_{2g}\) levels be filled first. For electronic configurations between d\(^4\) and d\(^7\), the electrons may either occupy different orbitals to maximize the total spin (1st Hund’s rule) or populate the lowest-lying \(t_{2g}\) levels as much as possible, thus forming spin singlets (Pauli principle). These two choices produce either high- or low-spin configurations, respectively. This is possible because for transition-metal ions the intra-atomic exchange interaction and the crystal-field strength are of the same order of magnitude, as illustrated in Scheme left. Therefore, the high-spin configuration fulfills the 1st Hund’s rule at the expense of crystal-field energy (weak crystal field regime), while the low-spin configuration violate the 1st Hund’s rule, i.e. frustrates the intra-atomic exchange, to lower the crystal-field energy (strong crystal field regime).

We remind that the perfect octahedral symmetry \(O_h\) corresponds to having \(Dt=Ds=0\) in the matrix defined in Table. Consistently with Eq.(3.12), the splitting between the levels \(e_g\) and \(t_{2g}\) is in this case uniquely determined by the strength of the octahedral crystal field \(Dq\). In practice, \(Dq\) is assumed as a fitting parameter that can be determined from spectroscopic measurements. For instance, it is found experimentally that for different ligands the \(Dq\) values can be ordered as follows:

I\(^{-}\) \(<\) Br\(^{-}\) \(<\) Cl\(^{-}\) \(<\) S\(^{2-}\) \(<\) N\(^-_3\) \(<\) F\(^- <\) OH\(^-\) \(<\) OAc\(^-\) \(<\)O\(^{2-}\) \(<\) H\(_2\)O \(<\) NH\(_3\) \(<\) bipy \(<\) CN\(^-\) \(<\) CO.

This trend is known as *spectrochemical series* and provides a useful reference frame for the rationalization of the magnetic behavior of different compounds based on transition metals.

### Jahn-Teller distortion

Up to the third electron, the progressive filling of multi-electron configurations produces the row on the top of Fig.4.1, in which only levels of the \(t_{2g}\) triplet are occupied. With the forth electron one starts filling the \(e_g\) levels. As long as the octahedral symmetry is kept rigid, both the \(d_{z^2}\) and \(d_{x^2-y^2}\) levels have the same energy. This degeneracy is not removed by spin-orbit coupling at the first order in perturbation theory. In fact, the expectation values of the angular-momentum components sum up to zero for the three electrons in the \(t_{2g}\) multiplet. The same quantities evaluated on the \(e_g\) doublet vanish as well. Instead, a slight elongation of the octahedron along the \(z\) axis would push the ligands away from the \(d_{z^2}\) orbital – which points right along the \(z\) axis – thus reducing the energy of this level. This is actually what happens spontaneously in the electronic configuration d\(^4\). This distortion is realized by increasing the distance of two chemical bondings at the expense of creating strain in the lattice. The optimal elongation is thus a compromise between an increase in the lattice strain and a decrease of the energy of the level \(d_{z^2}\), i.e. the Coulomb repulsion between the electron occupying it and the ligands lying along the \(z\) axis. On the contrary, the energy of the \(d_{x^2-y^2}\) level increases due to the elongation of the ligand octahedron. But this has no energetic consequence because in the d\(^4\) configuration the \(d_{x^2-y^2}\) orbital remains unoccupied. This mechanism is called *Jahn-Teller distortion* and is not effective in non-degenerate cases (labeled with \(A\) in Fig.3.3, because there is no energy to decrease by deforming the octahedron of ligands. It is, instead, particularly important for \(E\) configurations, which are said to be unstable with respect to Jahn-Teller distortion. In this case, the actual symmetry seen by the ion is lowered to \(D_{4h}\) and the resulting scheme of levels is consistent with a non-degenerate ground state: the Jahn-Teller distortion thus removes the two-fold degeneracy of \(E\) multi-electron configurations (in \(O_{h}\) symmetry). We note from the diagram of levels in Fig.3.3 that the distortion splits also the \(t_{2g}\) levels. However, the “center of gravity” of their energy remains the same. Therefore, the splitting of the \(t_{2g}\) levels does not affect the global energy, in the specific case of d\(^4\) ions. The Jahn-Teller distortion is prominent in Mn\(^{3+}\); this magnetic ion is ubiquitous in molecular magnets and multiferroics, like some the rare-earth manganites whose structure is sketched in Fig.3.2 left.

Similar electrostatic arguments as those explained above justify the opposite splitting of \(e_g\) and \(t_{2g}\) levels by compression of the octahedral ligand structure, illustrated in Fig.3.3.

### Co\(^{2+}\) ion in octahedral crystal field

The arguments provided before for one electron in the \(t_{2g}\) multiplet can be extended to all the three-fold degenerate configurations in Fig.4.1 labeled with \(T\). To them an effective orbital momentum \(L'=1\) can formally be assigned. The prime superscript in \(L'\) is important to remind that one of such multiplets does not behave as a Russell-Saunders term \(P\) in every respect. As mentioned, within a \(T\) multiplet the matrix elements of spin-orbit coupling are not zero. Thus, spin-orbit interaction actively works to lower the orbital degeneracy, though it may not manage to quench it completely. As a result, \(T\) multiplets usually have a very complex magnetic behavior, characterized by a large magnetic anisotropy. A typical example is Co\(^{2+}\), which would have a \(^4T_{1g}\) ground state in perfect octahedral symmetry. The total twelve-fold degeneracy (\(L'=1\) and \(S=3/2\)) is partially removed by the spin-orbit coupling, leaving two quartets and two Kramers doublets. One Kramers doublet has the lowest energy and it is separated from the other states by an energy gap of the order of 10 meV. This is the reason why Co\(^{2+}\) in octahedral environment of ligands is usually modeled as an effective spin 1/2 with a very anisotropic \(g\)-tensor. A spin chain compound containing Co\(^{2+}\) ions with this coordination is sketched in Fig.3.2 right: The combination of anisotropic \(g\)-tensor and helical structure is at the origin of the magneto-chiral dichroism observed on this compound.

## 4.2 The single-ion spin Hamiltonian

The hierarchy of terms in the Hamiltonians(3.1) and(2.12) highlights that, already at the atomic level, magnetism results from a delicate interplay between competing interactions. When scientists need to compare theoretical models with experimental measurements – usually performed on specimens consisting of many magnetic atoms – this delicate interplay is summarized in a Hamiltonian that depends only on spin coordinates. The energy terms appearing in this spin Hamiltonian are *effective*, in the same meaning as the intra-atomic exchange interaction defined in the Heisenberg Hamiltonian(2.18). In general, the spin variables themselves – entering spin Hamiltonians – can be *effective*, in the sense that they may also incorporate some orbital contribution. In the forthcoming chapters the spin-Hamiltonian approach will be largely employed to distinguish the behavior of different magnetic systems at finite temperature and describe their dynamic response.

Let us start by reproducing a textbook calculation that clarifies the origin of magnetic anisotropy and the tensorial nature of the Landé factor.

We consider a transition-metal ion and recall, for convenience, the ordered contributions to its Hamiltonian:
\[\begin{equation}
\tag{4.1}
\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_{\rm ee} + \mathcal{H}_{\rm cf} + \mathcal{H}_{\rm so} + \mathcal{H}_{\rm Z}\,.
\end{equation}\]
We express the orbital part of the wave function on the symmetry-adapted basis
\(|\Gamma,\gamma\rangle\), on which the leading free-ion contribution \(\mathcal{H}_0\) and the crystal-field Hamiltonian \(\mathcal{H}_{\rm cf}\) are simultaneously diagonal.

Concretely, the notation \(|\Gamma,\gamma\rangle\) represents multi-electron configurations like those sketched in Fig.4.1, including excited states. We would like to treat the spin-orbit and the Zeeman interactions,
\[\begin{equation}
\tag{4.2}
\begin{split}
&\mathcal{H}_{\rm so}= \lambda\,\hat{\mathbf S}\cdot \hat{\mathbf L} \\
&\mathcal{H}_{{\rm Z}} = \mu_B \left(\hat{\mathbf L} + g_s\, \hat{\mathbf S}\right) \cdot \vec B \,,
\end{split}
\end{equation}\]
as perturbations and project out the dependence on the orbital coordinates. The final goal is to obtain an effective Hamiltoninan that retains only the dependence on spin coordinates. For simplicity, we focus ourselves on an orbitally non-degenerate ground state defined by the multiplet \(|\Gamma,\gamma,S,M_s\rangle=|\Gamma,\gamma\rangle\otimes|S,M_s\rangle\). The spin part of the Zeeman interaction acts only on spin coordinates and is not affected by the integration over orbital coordinates; therefore, we shall write it as it appears in the second line (\(\mathcal{H}_{{\rm Z}}\)) of Eq.(4.2). The remaining parts of the Hamiltonians(4.2) depend on \(\hat{\mathbf L}\) and do not give any correction to the energy of the ground-state to the first order of perturbation theory in the absence of orbital degeneracy (as assumed). To the second order of perturbation theory, instead, one has
\[\begin{equation}
\tag{4.3}
\mathcal{H}_{\rm eff}= g_s\mu_B \,\hat{\mathbf S}\cdot\vec B -
\sum_{\Gamma',\gamma'}\frac{|\langle\Gamma',\gamma'| \mu_B \,\hat{\mathbf L}\cdot\vec B +\lambda\,\hat{\mathbf S}\cdot \hat{\mathbf L} |\Gamma,\gamma\rangle|^2}{E_{\Gamma',\gamma'}-E_{\Gamma,\gamma}}\,,
\end{equation}\]
where the sum runs – in principle – over all the excited states \(|\Gamma',\gamma'\rangle\) with energy \(E_{\Gamma',\gamma'}>E_{\Gamma,\gamma}\). The square in Eq.(4.3) can be expanded to yield
\[\begin{equation}
\tag{4.4}
\begin{split}
\mathcal{H}_{\rm eff}&= g_s\mu_B \,\hat{\mathbf S}\cdot\vec B -
2\mu_B\,\lambda \sum_{\alpha,\nu} \Lambda_{\alpha\nu} B^\alpha \hat{S}^\nu \\
&-\lambda^2 \sum_{\alpha,\nu} \Lambda_{\alpha\nu} \hat{S}^\alpha \hat{S}^\nu
-\mu_B^2 \sum_{\alpha,\nu} \Lambda_{\alpha\nu} B^\alpha B^\nu
\end{split}
\end{equation}\]
with
\[\begin{equation}
\tag{4.5}
\Lambda_{\alpha\nu} = \sum_{\Gamma',\gamma'}\frac{\langle\Gamma,\gamma|\hat{L}^\alpha |\Gamma',\gamma'\rangle\langle\Gamma',\gamma'|\hat{L}^\nu |\Gamma,\gamma\rangle}{E_{\Gamma',\gamma'}-E_{\Gamma,\gamma}}\,.
\end{equation}\]
In the remainder of the course the term in Hamiltonian(4.4) which has a quadratic dependence on the \(\vec B\) components will be neglected. This energy term is responsible for a temperature-independent
contribution to the susceptibility called Van Vleck susceptiblity.
The other terms of the spin Hamiltoninan \(\mathcal{H}_{\rm eff}\) are usually written as
\[\begin{equation}
\tag{4.6}
\mathcal{H}_{\rm eff}= \mu_B\,\sum_{\alpha,\nu} g_{\alpha\nu} B^\alpha \hat{S}^\nu
- \sum_{\alpha,\nu} D_{\alpha\nu} \hat{S}^\alpha \hat{S}^\nu
\end{equation}\]
where
\[\begin{equation}
\tag{4.7}
g_{\alpha\nu} = g_s\delta_{\alpha\nu} - 2 \lambda \Lambda_{\alpha\nu}\qquad \text{and}\qquad
D_{\alpha\nu} = \lambda^2 \Lambda_{\alpha\nu}
\end{equation}\]
are the \(g\)-tensor (or Landé tensor) and the magnetic-anisotropy tensor, respectively.
These tensors are practically 3-by-3 symmetric matrices (\(\delta_{\alpha\nu}\) is the 3-by-3 unit matrix). The \(g\)-tensor accounts for the fact that the magnetic response of an atom embedded in a crystal may depend on the direction along which the \(\vec B\) field is applied (with respect to the crystal frame).
The anisotropy tensor accounts for the fact that the magnetic moment (effective spin) may prefer to lie along some crystallographic directions. When magnetic atoms are arranged in complicated crystal structures, it may happen that the \(g_{\alpha\nu}\) and \(D_{\alpha\nu}\) tensors are not diagonal simultaneously on the same reference frame for all the equivalent magnetic atoms. In that case, the reciprocal canting of the anisotropy axes may be taken into account by rotating the \(g_{\alpha\nu}\) and \(D_{\alpha\nu}\) tensors. For instance, this is crucial to reproduce the magnetism of the helical Co(hfac)\(_{2}\)(NITPhOMe) spin chain, whose structure is sketched in Fig.3.2. To discuss some general features of collective models of magnetism we will limit ourselves to considering collinear uniaxial anisotropies that contribute to the spin Hamiltonian with a term like
\[\begin{equation}
\tag{4.8}
\mathcal{H}_{\rm an}= - D (\hat{S}^z )^2\,.
\end{equation}\]

In this expression the magnetic anisotropy constant \(D\) can be thought of as the only non-zero term of a diagonal \(D_{\alpha,\nu}\) tensor. By adding a constant to the Hamiltonian(4.8), the magnetic-anisotropy energy can be expressed as a traceless tensor: This is the most common convention in molecular magnetism and in the chemistry literature^{13}.

Before proceeding with the concrete example of Mn\(^{3+}\), we would like to remark that it is the spin-orbit interaction \(\mathcal{H}_{\rm so}\) that “communicates” the information about the crystal frame to the spin coordinate. For instance, the true Zeeman interaction acts isotropically along every component of the *true* spin operator \(\hat{\mathbf S}\), consistently with the fact that \(g_s\) is a scalar. In this formalism some residual orbital contribution to the Zeeman energy manifests itself in the anisotropy of the \(g_{\alpha\nu}\) tensor. This is one of the reason why the spin variables in Hamiltonian(4.6) are called *effective*.

### Anisotropy of the Mn\(^{3+}\) ion in oxides

A straightforward application of the formalism developed above is represented by the Mn\(^{3+}\) ion in an elongated octahedral environment of ligands.The electronic ground-state configuration \(|\Gamma,\gamma\rangle\) is shown in Fig.4.2a and originates from the Russell-Saunders term \(^5D\). For simplicity, we consider only excited states of the type \(|\Gamma',\gamma'\rangle\) that originate from the same ground-state term, namely with maximal spin \(S=2\) as prescribed by the 1st Hund’s rule. Given that^{14}
\[\begin{equation}
\begin{split}
& |\langle d_{yz}|\hat{L}^x |d_{x^2-y^2}\rangle|^2 = 1 \\
& |\langle d_{xz}|\hat{L}^y |d_{x^2-y^2}\rangle|^2 = 1 \\
& |\langle d_{xy}|\hat{L}^z |d_{x^2-y^2}\rangle|^2 = 4
\end{split}
\end{equation}\]
are the only non-zero matrix elements within the subspace \(S=2\), the excited states to be considered are those shown in Fig.4.2b-d. The transitions that produce those configurations from the ground state \(|\Gamma,\gamma\rangle\) are indicated with arrows to identify the relevant matrix elements. Then the application of Eq.(4.5) produces the following components of the \(\Lambda_{\alpha\nu}\) tensor
\[\begin{equation}
\tag{4.9}
\begin{split}
& \Lambda_{x x}= \frac{1}{\Delta E(d_{yz}\rightarrow d_{x^2-y^2})} \\
& \Lambda_{y y}= \frac{1}{\Delta E(d_{xz}\rightarrow d_{x^2-y^2})} \\
& \Lambda_{z z}= \frac{4}{\Delta E(d_{xy}\rightarrow d_{x^2-y^2})} \,.
\end{split}
\end{equation}\]
The values \(\Delta E(d_{xy}\rightarrow d_{x^2-y^2})=18000\) cm\(^{-1}\) and \(\Delta E(d_{xz}\rightarrow d_{x^2-y^2})=\Delta E(d_{yz}\rightarrow d_{x^2-y^2})=21000\) cm\(^{-1}\) are obtained from spectroscopic measurements, while the spin-orbit coupling constant for Mn\(^{3+}\) is \(\lambda=90\) cm\(^{-1}\). Putting these numbers into Eqs.(4.7) and(4.9) the values \(g_{xx}=g_{yy}=1.99\) and \(g_{zz}=1.96\) are obtained, in fair agreement with the experimental ones. For what concerns the anisotropy, the scalar constant in Eq.(4.8) is given by the difference \(D=D_{zz}-D_{xx}\). In this case, Eqs.(4.7) and(4.9) yield \(D=1.4\) cm\(^{-1}=2\) K that is significantly smaller than the value typically observed in experiments. This result can be improved taking into account excited states for which \(S\ne 2\), i.e., originating from other Russell-Saunders terms than the \(^5D\) Hund’s ground state. In particular, we focus on the excited state corresponding to the electronic transition \(d_{xy}\rightarrow d_{z^2}\) in which two electrons – paired in a spin singlet – occupy the orbital \(d_{z^2}\). This configuration originates from an eigenstate of \(\mathcal{H}_0\) which is not the Hund’s ground state but rather the \(^3H\) term (\(L=5\) and \(S=1\)). Therefore, the difference of Coulomb interaction associated with the two terms \(^5D\) and \(^3H\) shall also contribute to the denominator in Eq.(4.5). One finally obtains a scalar anisotropy constant \(D=4.9\) cm\(^{-1}=7\) K, remarkably larger than the previous estimate and in better agreement with experiments.

## 4.3 Transition-metal vs rare-earth ions

Towards the end of the previous chapter we mentioned that the application of the three Hund’s rules yields a satisfactory estimate of the magnetic moments of rare-earth ions. In this regard, rare earths in solids behave similarly to free ions but pretty differently from transition-metal ions in solids. This happens because in rare-earth ions the unpaired electrons occupy the 4f shells which lie inside the filled (5s)\(^2\) and (5p)\(^6\) shells (the pair of (6s)\(^2\) electron is usually lost to form a stable ion). Consequently, the electrons that build up the atomic magnetic moment are weakly affected by crystal fields. The contributions to the electronic Hamiltonian are ordered by descending strength as follows \[\begin{equation} \tag{2.12} \mathcal{H} = \mathcal{H}_0 + \mathcal{H}_{\rm ee}^1 + \mathcal{H}_{\rm so} + \mathcal{H}_{\rm cf} + \mathcal{H}_{\rm Z}\,. \end{equation}\] Note that with respect to the hierarchy defined in Eq. (3.1) for transition metals (3d elements), the relative importance of spin-orbit and crystal-field interaction is inverted. For rare earths in solids spin-orbit coupling can be treated on the same footing as for the corresponding free ions. In practical terms, this means that the third Hund’s rule is applied before considering crystal-field effects.

Figure4.3 summarizes the differences between 3d and 4f ions in this respect and the orders of magnitude of the relevant interactions in the atomic Hamiltonian.

We conclude this brief excursus on rare-earth ions mentioning that crystal-field effects on the lowest lying multiplet are usually taken into account expressing the corresponding term in the Hamiltonian (2.12) by means of Stevens operator equivalents of \(\hat{\mathbf J}\):

\[\begin{equation}
\tag{4.10}
\mathcal{H}_{\rm cf}= \sum_{k,q} B_k^q \hat{\mathbf O}_k^q\,.
\end{equation}\]
Tables of the \(\hat{\mathbf O}_k^q\) operators can be found in the literature. Below we provide few of them to convey a concrete idea of their formal dependence on the components of \(\hat{\mathbf J}\):

\[\begin{equation}
\label{Stevens}
\begin{split}
&\hat{\mathbf O}_2^0 = 3 (\hat{J}^z)^2 - \hat{\mathbf J}^2 \\
&\hat{\mathbf O}_2^2 =\frac{1}{2} \left[(\hat{J}^{+})^2 +(\hat{J}^{-})^2\right] \\
&\hat{\mathbf O}_4^4 =\frac{1}{2} \left[(\hat{J}^{+})^4 +(\hat{J}^{-})^4\right] \\
&\dots
\end{split}
\end{equation}\]

From Eq.(4.10) the matrix elements \(\langle J',M'_J|\mathcal{H}_{\rm cf}|J,M_J\rangle\) can be determined, similarly to what done for transition metals with Eqs.(3.10) and the following matrix.
Applying to rare earths (\(l=3\)) the same arguments developed after Eq.(3.4) for the case \(l=2\), one concludes that only \(k\le 6\) will enter the sum in Eq.(4.10) in the present case.
Which coefficients \(B_k^q\) are actually allowed to be different from zero will be defined by the symmetry of the specific environment seen by the rear-earth ion.

We omit the principal quantum number 3 and parentheses from the electronic configurations (3d)\(^{n_e}\) to simplify the notation.↩︎

In the chemistry literature the opposite sign is used in Eq.(4.8), i.e., an easy axis anisotropy would correspond to a negative \(D\).↩︎

Matrix elements are obtained using the fact that \(\hat{L}^x = (\hat{L}^+ + \hat{L}^-)/2\) and \(\hat{L}^y = (\hat{L}^+ - \hat{L}^-)/(2i)\) and that \(\hat{L}^{\pm}=\hat{l}_1^{\pm}+\hat{l}_2^{\pm}+ \dots +\hat{l}_{n_{\rm e}}^{\pm}\). Concisely, this yields \(\hat{L}^+ |l_i,m_i\rangle = \sqrt{(l_i-m_i)(l_i+m_i+1)}|l_i,m_i+1\rangle\) and $^- |l_i,m_i= |l_i,m_i-1$.↩︎