... Rotter¹

Independent international Scientific Consultant, email: mcphase@icloud.com

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... Le²

Dept. Physics and Astronomy, Seoul National University, Seoul 151-742, Korea

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... Keller³

Universität Regensburg

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... Hoffmann⁴

University of Oxford, Physics Department, Clarendon Laboratory, Parks Road, Oxford, UK

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... Hoffmann⁵

Forschungszentrum Jülich, D-52425 Jülich, Germany

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... Rotter⁶

Wien, Austria

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...jensen91-1. ⁷

The Hamiltonian of the standard model of rare earth magnetism is described in section 5.

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...#tex2html_wrap_inline11207#⁸

Note that these conditions are essential and put a limit to the applicability of the theory, for example in the case of charge transfer excitations from one subsystem to the next.

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... coupling⁹

For further information on the notation and symmetry restrictions to the parameters in the Hamiltonian refer to [1].

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... energy.¹⁰

The procedure is described in detail in [20].

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... density¹¹

see section 19

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... Hamiltonian¹²

...within the $\vert JLSm_J \rangle$ ground state multiplet of the 4f electron wave function. The first term in equ. 25 describes the crystal field, the second the effect of a magnetic field (Zeeman term). The strength of the crystal field is given by the crystal field parameters $B_l^m$.

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... model¹³

point charges on the neighbouring atoms, for details on these calculations see [27]

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... structure¹⁴

see section 19)

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...cfze). ¹⁵

In addition to $\langle \mathbf J_i \rangle$ the module also returns the partition sum $z$ and the magnetic energy $u=\sum_{i=1}^{2J+1} p_{i} \epsilon_{i}$.

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.... ¹⁶

Note that if you use the module cfield, the choice is more unconventional:$\vec a\vert\vert y$, $\vec b\vert\vert z$ and $\vec c\vert\vert x$ Tools for rotating crystal field parameters are described in appendix I.

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... data¹⁷

section 17

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...  ¹⁸

To trigger the Monte Carlo calculation set NOFMCSTEPS to a nonzero value. Then after MAXNOFMFLOOPS has been reached the program switches into Monte Carlo walk mode with randomly choosing energy eigenstates and computing total energy of a supercell (NOFMCSTEPS times per spin in the supercell). The initial quantum state is chosen such as to match most closely the meanfield state/ initial configuration. The single ion program is called with $T=0$ and delivers a random energy eigenstate and the energy of the Hamiltonian is updated according to the Metropolis algorithm [2]. The final state is saved in mcphas.sps and averaged energy, magnetic moment are calculated and stored in mcphas.fum.

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... field¹⁹

for electric fields $\hat H(s)$ will contain also the electric field, compare section 11

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... (MAXNOFMFLOOPS).²⁰

Note two special cases:If MAXNOFMFLOOPS=1, then the mean fields are calculated from the initial values of the spin configuration and free energy is evaluated immediately without checking convergence. If MAXNOFMFLOOPS=2, then from calculated mean fields the spins are calculated and free energy is evaluated, without checking convergence. For MAXNOFMFLOOPS$>$2 free energy is only evaluated if the iteration reached the convergence limit specified by MAXSTAMF.

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... edited).²¹

Alternatively, one may make use of the program spins.

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...Eel). ²²

$R^{\alpha}R^{\delta}\omega_{\sigma} \epsilon_{\sigma\alpha\beta}\delta_{\beta\... ...eta}\delta_{\alpha\delta})= (\mathbf R . \mathbf \omega )^2-R^2\omega^2 \neq 0$, if $\mathbf R$ is not parallel to the rotation axis. Thus for some rotation the elastic energy will depend on rotation angle if transversal springs are introduced.

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... using ²³

S. Bluegel, Juelich, private communication

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... first²⁴

Note that we do not calculate all the terms in the braces in the first line of equation 4. In particular using the central field approximation the first summation produces Hydrogen-like energy levels, called configurations, which are split into terms by the second summation. What we refer to as the Coulomb interaction is only this second summation, and we shall consider the lowest energy configuration only. This configuration corresponds to the outer most electrons in ion, and is labelled $nl^\nu$, for example, $4f^1$ for Ce$^{3+}$.

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... properties²⁵

For the case of $LS$-coupling this corresponds to ignoring the Coulomb and SO interactions, which are in this limit both much larger than the CF. For $jj$-coupling, the Coulomb interaction is treated as small and neglected, but the spin-orbit interaction is considered.

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...#tex2html_wrap_inline13473#²⁶

Also denoted $m_J$.

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...racah49-1352²⁷

Equation 11

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... formulae²⁸

Except for the case of the rotation group in 3 dimensions, SO(3) (whose representations are labelled by the angular momentum quantum numbers, $L$ and $J$), where they are simply the $3j$ symbols

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...allison74²⁹

http://cpc.cs.qub.ac.uk/cpc/cgi-bin/showversions.pl/?catid=acry

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...elliot57-509³⁰

Eqn 16 for SO, and 25-27 for CF, and reproduced in more modern notation in appendix C of [52]

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...racah49-1352³¹

Eqns 66 for the conversion of the Racah parameters to Slater integrals, 63 for the $\hat e_0$ operator, 69 for $\hat e_1$, 73-74 for $\hat e_2$, and 78, 28 and 80-87 for $\hat e_3$

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...judd88-1³²

Eqn 7-58

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... intervals³³

Depending on the Java setup on your computer, a conflict may occur between the McPhase Java programs and other Java programs, such as the Matlab GUI. If you find that no graphics windows open, and have other Java programs running in the background, try to close all other Java programs.

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...#tex2html_wrap_inline14095#³⁴

in princpiple it is possible to go beyond this approximation, this would involve an inversion of the complete phonon dynamics, i.e. all $K_{\alpha\beta}(ij)$ ... this has not been programmed

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...wybourne65³⁵

Note that table 6-1 and equ 6-7 are not correct in this reference.

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...#tex2html_wrap_inline15001#³⁶

Note that these conditions are essential and put a limit to the applicability of the theory, for example in the case of charge transfer excitations from one subsystem to the next [73].

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... susceptibility³⁷

the $-$ on top of $\chi$ indicates matrix notation for $\chi_{\alpha\beta}^{ss'}$

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... investigated³⁸

supplementary material - screen shot movie comparing the speed of traditional Green's function method and DMD

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... cell³⁹

In case of magnetic order. In general this will be the unit cell of the Bravais lattice in section M, which is a superlattice of the crystal lattice.

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... elements⁴⁰

using the appropriate expression for the matrix elements of of the scattering operator as given in [32, equ. (11.86) or (11.87b)].

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