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MHexaFerrite

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The physics of M hexa-ferrite

The aim of this page is to illustrate the relation between crystal structure, magnetic structure (the orientation and magnitude of magnetic moment at each Fe site) and the hyperfine field detected by NMR on ⁵⁷Fe nuclei.

The ⁵⁷Fe nucleus

NMR is extremely difficult in standard Fe compounds (see the table below), since the signal is provided only by one every fifty nuclei and the frequency is very low: 10T yield a Larmor frequency of 10 MHz (remember, the FID is proportional to {$\omega^3$}).

Crystal structure

The crystal structure of BaFe₁₂O₁₉ was determined by x ray and neutron diffraction. It is summarized in this 1977 paper. Try and read the BaFe₁₂O₁₉ part. Details will become clearer after the XRD experiments. You can visualize the structure by installing the jmol viewer, downloading the appropriate cif file, and open it from the File drop-down-menu of jmol. Right-click on the display and select Symmetry/Reload [1 1 1] to view a full cell. Rotate with the mouse to grasp the 3D structure. Locate the different iron ions

Magnetic state of iron ions

To understand magnetism in BaFe₁₂O₁₉ calculate the valence of Fe, assuming ionic configurations, with Ba²⁺ and O^2-. You get Fe³⁺ for all 12 Fe in the formula. Check the Hund rule filling of 3d orbitals in Fe³⁺: it is a 3d⁵ configuration and by Hund's rule all ions should have maximum spin. Assuming this and the quenching of orbital angular moment (explanation forthcoming) their magnetic moment should be proportional to the total spin S=5/2.

One has further to consider that the wave function associated with the 3d⁵ consideration has spherical symmetry, therefore the hyperfine coupling that originates the strong hyperfine field (50 T) seen by the ⁵⁷Fe nucleus must be isotropic. It is the same whichever the local orientation of the spin.

We must note that assuming 12 equivalent Fe³⁺ ions we would predict equal maximum magnetic moments for all sites, (according to quantum mechanics {$|\mu_S|=g\mu_B\sqrt{S(S+1})$} should amount to {$5.92 \mu_B$}). Beware: the neutron data already indicate some deviation since they refine (Table 10) five slightly different values of the moment (five inequivalent sites), and smaller than expected {$4.1\le g\sqrt{S(S+1)}\le 4.9$}. NMR also indicates the need for still more ingredients: if we had only five inequivalent 3d⁵ ions, with the simple ingredients described above we would see five sharp NMR resonances. Instead we see five peaks, but also broad features.

Collinear spin structure

The 1977 paper states that the absence of a [001] peak in neutron diffraction data, plus the assumption of ferrimagnetism implies a collinear structure, i.e. all spin either parallel of antiparallel to the easy direction (001). We now see what NMR expectations this predicts.

NMR spectra from collinear spins: external field dependence

With an isotropic hyperfine interaction {$ -\gamma\hbar A {\mathbf I}\cdot{\mathbf S}$} the hyperfine field {${\mathbf B}_{hf}=A{\mathbf S}$} is antiparallel to the Fe total spin {$\mathbf S$} (keep in mind that the sign of {$A$} is negative for on-site isotropic hyperfine coupling, since the spin polarized valence function determines an opposite polarization of the s wave-function, with non-vanishing density at the nucleus).

Therefore, when the field penetrates the sample we expect a vector composition {${\mathbf B}=\mu_0{\mathbf H}+ {\mathbf B}_{hf}$}.

For BaFe₁₂O₁₉ this implies that for {${\mathbf H}\parallel(001)$} the NMR resonance shifts with the field, whereas for {${\mathbf H}\perp(001)$}, since {$B_{hf}\ll \mu_0 H$}, the frequency shift is too small to be seen.

Non collinear spin structures

This means that each Fe spin can be directed in a different direction such that the sum of all magnetic moments in a cell is only along (001) (to agree with magnetization data which indicate (001) as the easy axis).

NMR spectra from non collinear structures

Each NMR resonance frequency should shift maximally when the field is applied parallel to the local spin direction. So a similar thing should happen but with different crystal orientations.

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