< The magnetic interactions | Index | Static dipoles >
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Neighbour sites Electron wave functions in a solid overlap and hybridize to give rise to bands. Let us neglect this and force a simple one electron picture onto the solid. If we start considering magnetic neighbours to the site of the probe nucleus, the spatial average of the dipolar field over the electron wave function will not introduce drastic changes: we may approximate the result with a point-like electron spin at the neighbour site. After treating on-site electrons we shall come back to nearest and next nearest neighbours, considering the effect of the overlap.
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The length of r is of the order of the lattice spacing
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On-site unpaired electrons Consider all of them as one total angular momentum J1, corresponding to a magnetic moment m1=gγeħJ1. The classical dipolar field Bd(r) diverges for r→0, but classically two bodies cannot be exactly in the same point. However quantum electrons have a finite probability of being at r=0, if they are in 1s states around the nucleus. Although the field diverges, the contribution of the 1s wave to the interaction energy, known as the Fermi contact term (-Ec), is finite. The hyperfine field values is this energy term, divided by the probe's magnetic moment, Bhf=Ec/|m|. The field vector is along the source moment and it can be calculated as
{${\mathbf B}_{hf}= \frac 2 3 \mu_0 |\psi_{1s}(0)|^2 \gamma_e \hbar {\mathbf s}_1$}
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The length of r is of the order of the atom radius
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Unpaired electrons are normally found in transition metals, 3, 4, and 5d states, and in rare earths or actinides 4 and 5f states. These hydrogen-like wave functions would have strictly zero probability density at r=0, and the 1s state is doubly occupied, therefore the two corresponding contact terms cancel. However, in the presence of unpaired spins in the valence wave functions, the atomic many-body Coulomb interaction polarizes the core states, resulting in a large unbalance of the 1s spin-up and spin-down occupation. The simple Fermi formula retains its validity replacing ψ1s(0) with
{$|\Psi(0)|^2=\rho_\uparrow(0)-\rho_\downarrow(0)$}
and γeħs1 with m1=γeħS.
Hence the contact term is
{$ E_c = \gamma\hbar{\mathbf B}_d \cdot{\mathbf I} = A {\mathbf S} $}
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When the unpaired electron wave function Ψ(r) is not spherically symmetric
an additional on-site term must be considered, the so-called pseudo-dipolar term. Due to the tensorial nature of Bd, that depends on the relative orientation of r and m1, the average of Bd(r) over Ψ(r)
{$ \delta{\mathbf B}_d = \int |\Psi({\mathbf r})|^2 {\mathbf B}_d({\mathbf r}) d{\mathbf r} $}
also depends on the orientation of the source magnetic moment m1. Therefore one can write
{$ \gamma \hbar \delta{\mathbf B}_d \cdot {\mathbf I}= \delta {\mathbf A}\cdot {\mathbf S} $}
and the pseudodipolar hyperfine coupling δA is a traceless tensor.
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Neighbour sites, revisited. Transfer of spin polarization happens non only from valence to core states of the same ion, but also across bonds to nearest and next nearest neighbour ions. In this way a Fermi contact term arises e.g. on the La nucleus in a LaMO magnetic perovskite, where M is a magnetic 3d ion. In pseudo-cubic LaMnO3 139La experiences eight contributions from neighbour Mn, of the type
{$ ^{139}{\cal H} = -\sum_{i=1}^8 A {\mathbf S}_i $}
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In the same lattice, by the same route, the Mn-O-Mn bond provides 55Mn with additional (transferred hyperfine) couplings to neighbouring Mn spins
{$ ^{55}{\cal H} = -(A+ \delta{\mathbf A})\cdot{\mathbf S} -\sum_{i=1}^6 A_i {\mathbf S}_i $}
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