.. _zoo_mcphase: ******* McPhase ******* ========= ====================================================================== ========= ====================================================================== Status Verified Links `Web `_ Languages C++, Fortran, Perl, Java ========= ====================================================================== Spin Hamiltonian ================ The spin Hamiltonian is in general: .. math:: \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 :math:`\hat{\mathcal{H}}(n)` denotes the Hamiltonian of a subsystem :math:`n` (e. g. an ion, or cluster of ions). The second term describes a bilinear interaction between different subsystems through the operators :math:`\hat{\mathcal{I}}_{\alpha}^n`, with :math:`\alpha = 1, 2, \dots, m`. The operators :math:`\hat{\mathcal{H}}(n)` and :math:`\hat{\mathcal{I}}_{\alpha}^n` act in the subspace :math:`n` of the Hilbert space, i. e. :math:`[\hat{\mathcal{I}}_{\alpha}^n, \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0`, :math:`[\hat{\mathcal{H}}(n), \hat{\mathcal{I}}_{\alpha}^{n^{\prime}}] = 0` and :math:`[\hat{\mathcal{H}}(n), \hat{\mathcal{H}}(n^{\prime})] = 0` for :math:`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 .. math:: \hat{\mathcal{H}}_{Z-J}(n) = - \sum_{\alpha=1,2,3} g_n \mu_B \hat{J}^{n}_{\alpha} H_{\alpha} where :math:`\hat{\mathbf{J}}^n` is a total angular momentum operator, is included. Or the term of the form .. math:: \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 :math:`\hat{\mathbf{S}}^n` and :math:`\hat{\mathbf{L}}^n` are the (inverse) spin and orbital angular momentum operators of the ion :math:`n`, respectively. Note that by "external magnetic field :math:`\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 :math:`\overline{n}_{\text{demag}}` times the magnetisation :math:`\mathbf{M}` of the sample, i. e. :math:`\mathbf{H} = \mathbf{H}_{\text{applied}} - \overline{n}_{\text{demag}} \mathbf{M}`, e. g. for a spherical sample :math:`\overline{n}_{\text{demag}} = 1/3\,\overline{I}`, where :math:`\overline{I}` is the identity matrix. The unit of :math:`\mathbf{H}` is usually chosen to be Tesla, which actually refers to the quantity :math:`\mu_0 \mathbf{H}` (in SI units). Magnetic exchange ----------------- Isotropic magnetic exchange interaction (e. g. Heisenberg exchange) is written as .. math:: \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 :math:`\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x`, :math:`\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y`, :math:`\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 .. math:: \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 :math:`\hat{\mathcal{I}}_{1} \leftrightarrow \hat{S}_x`, :math:`\hat{\mathcal{I}}_{2} \leftrightarrow \hat{S}_y`, :math:`\hat{\mathcal{I}}_{3} \leftrightarrow \hat{S}_z`, :math:`\hat{\mathcal{I}}_{4} \leftrightarrow \hat{L}_x`, :math:`\hat{\mathcal{I}}_{5} \leftrightarrow \hat{L}_y`, :math:`\hat{\mathcal{I}}_{6} \leftrightarrow \hat{L}_z`. Dzyaloshinskii-Moriya interaction --------------------------------- DMI can be written in the form of the two-ion interaction as .. math:: \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 :math:`\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x`, :math:`\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y`, :math:`\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z` and :math:`\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 .. math:: \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 :math:`\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x`, :math:`\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y`, :math:`\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z` and .. math:: \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 .. math:: \hat{\mathcal{H}}_{CF1-S}(n) = \sum_{lm} B_l^m \hat{O}_{lm}(\mathbf{J}^n) or with Wybourne operators :math:`\hat{T}_{lm}^n` and Wybourne parameters :math:`L_l^m` as .. math:: \hat{\mathcal{H}}_{CF1-W}(n) = \sum_{lm} L_l^m(n) \hat{T}_{lm}^{n} where :math:`\hat{T}_{lm}^n` is an operator equivalents of real valued spherical harmonic functions .. math:: \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})] for the ion :math:`n`, where index :math:`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 .. math:: \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 .. math:: \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 :math:`\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{O}_{lm}(\mathbf{J}^n)` or :math:`\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{T}_{kq}^n` and index :math:`\alpha` runs over :math:`m` pairs of :math:`(l,m)` or :math:`(k,q)` respectively. Interaction parameters are either :math:`\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{ll^{\prime}}^{mm^{\prime}}(nn^{\prime})` or :math:`\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow \mathcal{K}_{kk^{\prime}}^{qq^{\prime}}(nn^{\prime})`, where index :math:`\beta` runs over :math:`m` pairs of :math:`(l^{\prime},m^{\prime})` or :math:`(k^{\prime},q^{\prime})` respectively. Exchange striction ------------------ .. math:: \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}} Phonons ------- A three dimensional Einstein oscillator (for atom :math:`n`) in a solid can be described by the following Hamiltonian .. math:: \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 :math:`\hat{\mathbf{u}}` is the dimensionless displacement vector :math:`\hat{\mathbf{u}} = \hat{\mathbf{P}}_n / a_0 = \Delta \hat{\mathbf{r}}_n / a_0`, with the Bohr radius :math:`a_0 = 0.5219` Angstrom), :math:`m_n` the mass of the atom :math:`n`, :math:`\hat{\mathbf{w}}_n = d\hat{\mathbf{u}}_n/dt = \mathbf{p}_n / a_0` the conjugate momentum to :math:`\hat{\mathbf{u}}_n` and :math:`\overline{K}(nn)` the matrix describing the restoring force. External force :math:`\mathbf{F}(n)` can correspond to the electric field :math:`\mathbf{E}_{\text{el}}`, i. e. :math:`\mathbf{F}(n) = q_n \mathbf{E}_{\text{el}} a_0`, where Bohr radius :math:`a_0` is included in order to yield :math:`\mathbf{F}_{\text{el}}(n)` in units of meV. Coupling such oscillators leads to the Hamiltonian .. math:: \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 :math:`\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{\mathbf{u}}^n_{\alpha}` and interaction parameters are :math:`\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{\alpha\beta}(nn^{\prime})`. Single-ion electrostatic and spin-orbit interaction --------------------------------------------------- .. math:: \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 :math:`\nu_n,Z_n` and :math:`\mathbf{R}_n` denote the number of electrons, the charge of the nucleus and the position of the ion number :math:`n`, respectively, for each electron being :math:`p` the momentum, :math:`m_e` the mass, :math:`e` the charge and :math:`\mathbf{r}` the location. Spin orbit coupling is written in terms of the orbital momentum :math:`\mathbf{l}` and spin :math:`\mathbf{s}` of the individual electrons. Magneto-elastic interaction --------------------------- .. math:: \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) and .. math:: \hat{\mathcal{H}}_{ME2} = - \sum_{\substack{n