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Spin Hamiltonian

The spin Hamiltonian is in general:

\[\hat{\mathcal{H}} = \sum_{n=1}^{N} \hat{\mathcal{H}}(n) - \dfrac{1}{2} \sum_{n,n^{\prime},\alpha,\beta} \mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]

The first term \(\hat{\mathcal{H}}(n)\) denotes the Hamiltonian of a subsystem \(n\) (e. g. an ion, or cluster of ions). The second term describes a bilinear interaction between different subsystems through the operators \(\hat{\mathcal{I}}_{\alpha}^n\), with \(\alpha = 1, 2, \dots, m\). The operators \(\hat{\mathcal{H}}(n)\) and \(\hat{\mathcal{I}}_{\alpha}^n\) act in the subspace \(n\) of the Hilbert space, i. e. \([\hat{\mathcal{I}}_{\alpha}^n, \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0\), \([\hat{\mathcal{H}}(n), \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0\) and \([\hat{\mathcal{H}}(n), \hat{\mathcal{H}}(n^{\prime})] = 0\) for \(n \ne n^{\prime}\).

Next we give specific examples which fill the above expression with life.

Zeeman energy

When an external magnetic field is applied the term of the form

\[\hat{\mathcal{H}}_{Z-J}(n) = - \sum_{\alpha=1,2,3} g_n \mu_B \hat{J}^{n}_{\alpha} H_{\alpha}\]

where \(\hat{\mathbf{J}}^n\) is a total angular momentum operator, is included.

Or the term of the form

\[\hat{\mathcal{H}}_{Z-LS}(n) = - \sum_{\alpha=1,2,3} \mu_B \Bigl( 2 \hat{S}^n_{\alpha} + \hat{L}^n_{\alpha} \Bigr) H_{\alpha}\]

where \(\hat{\mathbf{S}}^n\) and \(\hat{\mathbf{L}}^n\) are the (inverse) spin and orbital angular momentum operators of the ion \(n\), respectively.

Note that by "external magnetic field \(\mathbf{H}\)" we refer to the magnetic field in the sample, which is the applied field (e. g. generated by a coil) diminished by the demagnetizing tensor \(\overline{n}_{\text{demag}}\) times the magnetisation \(\mathbf{M}\) of the sample, i. e. \(\mathbf{H} = \mathbf{H}_{\text{applied}} - \overline{n}_{\text{demag}} \mathbf{M}\), e. g. for a spherical sample \(\overline{n}_{\text{demag}} = 1/3\,\overline{I}\), where \(\overline{I}\) is the identity matrix. The unit of \(\mathbf{H}\) is usually chosen to be Tesla, which actually refers to the quantity \(\mu_0 \mathbf{H}\) (in SI units).

Magnetic exchange

Isotropic magnetic exchange interaction (e. g. Heisenberg exchange) is written as

\[\hat{\mathcal{H}}_{X-J} = - \dfrac{1}{2} \sum_{nn^{\prime}, \alpha\beta=1,2,3} \mathcal{J}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]

where the operators are understood as \(\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x\), \(\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y\), \(\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z\).

Intermediate coupling

For some rare earth ions and for transition metals or actinides it is necessary to include more single-ion ion states with different L, S into the calculation. Two-ion intermediate interaction can be written as

\[\hat{\mathcal{H}}_{IC} = - \dfrac{1}{2} \sum_{nn^{\prime},\alpha\beta=1,\dots,6} \mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]

where the operators are understood as \(\hat{\mathcal{I}}_{1} \leftrightarrow \hat{S}_x\), \(\hat{\mathcal{I}}_{2} \leftrightarrow \hat{S}_y\), \(\hat{\mathcal{I}}_{3} \leftrightarrow \hat{S}_z\), \(\hat{\mathcal{I}}_{4} \leftrightarrow \hat{L}_x\), \(\hat{\mathcal{I}}_{5} \leftrightarrow \hat{L}_y\), \(\hat{\mathcal{I}}_{6} \leftrightarrow \hat{L}_z\).

Dzyaloshinskii-Moriya interaction

DMI can be written in the form of the two-ion interaction as

\[\hat{\mathcal{H}}_{DMI} = - \dfrac{1}{2} \sum_{nn^{\prime},\alpha\beta=1,2,3} \mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]

where the operators are understood as \(\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x\), \(\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y\), \(\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z\) and \(\mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n})\) is a skew-symmetric matrix.

Classical dipole dipole interaction

Classical magnetic dipole-dipole interaction can be written in the form of the two-ion interaction with the matrix parameter

\[\hat{\mathcal{H}}_{CDD} = - \dfrac{1}{2} \sum_{n\ne n^{\prime},\alpha\beta=1,2,3} \mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]

where the operators are understood as \(\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x\), \(\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y\), \(\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z\) and

\[\mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) = \dfrac{\mu_0\mu_B^2 g_n g_{n^{\prime}}}{2\pi} \Biggl( 3\dfrac{(R_{n^{\prime}}^{\alpha} - R_{n}^{\alpha})(R_{n^{\prime}}^{\beta} - R_{n}^{\beta})}{|\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}|^5} - \dfrac{\delta_{\alpha\beta}}{|\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}|^3} \Biggr)\]

Crystal field

Single-ion crystal field is written with Stevens operators as

\[\hat{\mathcal{H}}_{CF1-S}(n) = \sum_{lm} B_l^m \hat{O}_{lm}(\mathbf{J}^n)\]

or with Wybourne operators \(\hat{T}_{lm}^n\) and Wybourne parameters \(L_l^m\) as

\[\hat{\mathcal{H}}_{CF1-W}(n) = \sum_{lm} L_l^m(n) \hat{T}_{lm}^{n}\]

where \(\hat{T}_{lm}^n\) is an operator equivalents of real valued spherical harmonic functions

\[\begin{split}\hat{T}_{l0} &= \sqrt{4\pi/(2l+1)} \sum_i Y_{l0}(\Omega_{i_n}),\\ \hat{T}_{l,\pm|m|} &= \sqrt{4\pi/(2l+1)} \sum_i \sqrt{\pm 1} [Y_{l,-|m|}(\Omega_{i_n}) \pm (-1)^m Y_{l,|m|}(\Omega_{i_n})]\end{split}\]

for the ion \(n\), where index \(i_n\) runs over all electrons in the open shell of an ion (d, f). Two-ion crystal field can be written with the Stevens operators as

\[\hat{\mathcal{H}}_{CF2-S} = -\dfrac{1}{2} \sum_{nn^{\prime}} \sum_{ll^{\prime}} \sum_{mm^{\prime}} K_{ll^{\prime}}^{mm^{\prime}}(nn^{\prime}) \hat{O}_{lm}(\mathbf{J}^n) \hat{O}_{l^{\prime}m^{\prime}}(\mathbf{J}^{n^{\prime}})\]

or with the Wybourne operators as

\[\hat{\mathcal{H}}_{CF2-W} = - \dfrac{1}{2} \sum_{nn^{\prime}} \Biggl[ \sum_{kk^{\prime}} \sum_{qq^{\prime}} \mathcal{K}_{kk^{\prime}}^{qq^{\prime}}(nn^{\prime}) \hat{T}_{kq}^n \hat{T}_{k^{\prime}q^{\prime}}^{n^{\prime}} \Biggr]\]

where operators are defined as \(\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{O}_{lm}(\mathbf{J}^n)\) or \(\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{T}_{kq}^n\) and index \(\alpha\) runs over \(m\) pairs of \((l,m)\) or \((k,q)\) respectively. Interaction parameters are either \(\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{ll^{\prime}}^{mm^{\prime}}(nn^{\prime})\) or \(\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow \mathcal{K}_{kk^{\prime}}^{qq^{\prime}}(nn^{\prime})\), where index \(\beta\) runs over \(m\) pairs of \((l^{\prime},m^{\prime})\) or \((k^{\prime},q^{\prime})\) respectively.

Exchange striction

\[\begin{split}\hat{\mathcal{H}}_{XS} = - \dfrac{1}{2} \sum_{\substack{nn^{\prime},\alpha\beta\alpha^{\prime}\gamma=1,2,3\\\beta^{\prime}=1-6}} \Biggl( \dfrac{\partial \mathcal{J}_{\alpha\beta}}{\partial \epsilon_{\beta^{\prime}}} + \dfrac{\partial \mathcal{J}_{\alpha\beta}}{\partial R_{nn^{\prime}}^{\alpha^{\prime}}} \dfrac{\partial \epsilon_{\alpha^{\prime}\gamma} R_{nn^{\prime}}^{\gamma}}{\partial \epsilon_{\beta^{\prime}}} \Biggr) \epsilon_{\beta^{\prime}} \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}\end{split}\]

Phonons

A three dimensional Einstein oscillator (for atom \(n\)) in a solid can be described by the following Hamiltonian

\[\hat{\mathcal{H}}_{P1}(n) = \dfrac{a_0^2\hat{\mathbf{w}}_n^2}{2m_n} - \dfrac{1}{2} \sum_{\alpha\beta=1,2,3} \hat{u}_{\alpha}^n K(nn) \hat{u}_{\beta}^n - \sum_{\alpha=1,2,3} F_{\alpha}(n) \hat{u}_{\alpha}^n\]

Here \(\hat{\mathbf{u}}\) is the dimensionless displacement vector \(\hat{\mathbf{u}} = \hat{\mathbf{P}}_n / a_0 = \Delta \hat{\mathbf{r}}_n / a_0\), with the Bohr radius \(a_0 = 0.5219\) Angstrom), \(m_n\) the mass of the atom \(n\), \(\hat{\mathbf{w}}_n = d\hat{\mathbf{u}}_n/dt = \mathbf{p}_n / a_0\) the conjugate momentum to \(\hat{\mathbf{u}}_n\) and \(\overline{K}(nn)\) the matrix describing the restoring force.

External force \(\mathbf{F}(n)\) can correspond to the electric field \(\mathbf{E}_{\text{el}}\), i. e. \(\mathbf{F}(n) = q_n \mathbf{E}_{\text{el}} a_0\), where Bohr radius \(a_0\) is included in order to yield \(\mathbf{F}_{\text{el}}(n)\) in units of meV.

Coupling such oscillators leads to the Hamiltonian

\[\hat{\mathcal{H}}_{P2} = - \dfrac{1}{2} \sum_{n\ne n^{\prime},\alpha\beta=1,2,3} \hat{u}_{\alpha}^n K_{\alpha\beta}(nn^{\prime}) \hat{u}_{\beta}^{n^{\prime}}\]

where operators are defined as \(\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{\mathbf{u}}^n_{\alpha}\) and interaction parameters are \(\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{\alpha\beta}(nn^{\prime})\).

Single-ion electrostatic and spin-orbit interaction

\[\hat{\mathcal{H}}_{E-SO}(n) = \sum_{i_n=1}^{\nu_n} \Biggl[ \dfrac{(p_{i_n})^2}{2m_e} - \dfrac{Z_n e^2}{4\pi \epsilon_0 |\mathbf{r}_{i_n} - \mathbf{R}_n|} + \zeta_n \mathbf{l}^{i_n} \cdot \mathbf{s}^{i_n} \Biggr] + \sum_{i_n > j_n=1}^{\nu_n} \dfrac{e^2}{4\pi \epsilon_0 |\mathbf{r}_{i_n} - \mathbf{r}_{j_n}|}\]

Here \(\nu_n,Z_n\) and \(\mathbf{R}_n\) denote the number of electrons, the charge of the nucleus and the position of the ion number \(n\), respectively, for each electron being \(p\) the momentum, \(m_e\) the mass, \(e\) the charge and \(\mathbf{r}\) the location. Spin orbit coupling is written in terms of the orbital momentum \(\mathbf{l}\) and spin \(\mathbf{s}\) of the individual electrons.

Magneto-elastic interaction

\[\begin{split}\hat{\mathcal{H}}_{ME1}(n) = - \sum_{\substack{\alpha=1,\dots,6,\\\gamma=1,2,3}} G^{\alpha\gamma}_{\text{mix}}(n) \epsilon_{\alpha} \hat{u}_n^{\gamma} - \sum_{\alpha=1,\dots,6, lm} G^{\alpha\gamma}_{\text{cfph}}(n) \epsilon_{\alpha} O_{lm}(\hat{\mathbf{J}}^n)\end{split}\]

and

\[\begin{split}\hat{\mathcal{H}}_{ME2} = - \sum_{\substack{n<n^{\prime},lm,\\\alpha=1,2,3}} \Gamma^{\alpha\gamma}(nn^{\prime}) \hat{u}_n^{\alpha} O_{lm}(\hat{\mathbf{J}}^{n^{\prime}})\end{split}\]

Elastic energy and external stress

Classical elastic energy and external stress can be included as

\[\hat{H} = \dfrac{1}{2} \sum_{\alpha\gamma=1-6} c^{\alpha\gamma} \epsilon_{\alpha} \epsilon_{\gamma} - \sum_{\alpha=1,\dots,6} \sigma_{\alpha} \epsilon_{\alpha}\]

Convention

Spin normalized	no
Multiple counting	yes