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Figure 3
Curves of f 0 (E) (bottom) and f 00 (E) (top) for Fe3+ of Ni ferrite related
to the Kramers–Kronig transform. The vertical lines indicate photon
energies Eon and Eoff which were used for X-ray diffraction experiments.

symmetry. The bond distances and angle of Fe(A)—O,
Fe(B)—O and /[Fe(A)—O—Fe(B)] are 1.8906 (3) A

2.0594 (5) A and 123.63 (2) , respectively. The thermal parameters are: 11 = 0.001408 (4) and 12 = 0 for the A site; 11 =
0.001893 (3) and 12 = 0.000184 (4) for the B site; 11 =
0.001861 (7) and 12 = 0.00012 (1) for the oxygen site, where
the constraints are 11 = 22 = 33 and 12 = 13 = 23.
Having a high photon energy, sufficient to promote an
electric transition in an atom, the resonant scattering becomes
dominant close to the absorption edge and the real term of the
anomalous scattering factor f 0 (E) in (3) has a deep minimum
at the Fe K edge. As shown in Fig. 3, the pre-edge structure
appears in the f 0 (E) curve after the Kramers–Kronig dispersion transform. Although a spectroscopic study of absorption
does not detail the site-separated information between the A
and B sites, the diffraction technique is of advantage in
distinguishing the scattering from the two sites. The f 0 term in
the atomic scattering factor f was determined site-independently for independent j described in (2). Then, the multiplicity parameter of f 0 was refined with a scale factor in the
least-squares calculation to minimize the sum of squared
residuals in the function wi(|Fobs| |Fcalc|)i2. The best-fitting
curves for the A and B sites were obtained from the intensity
data measured at Eon and Eoff (Fig. 5). The variation of residual factors has a minimum against parameter f 0 , indicating
good convergence. The f 0 values obtained for the Fe ions are
summarized in Table 1, and are 7.063 and 6.971 at Eon and
6.682 and 6.709 at Eoff for the A and B sites, respectively.
J. Synchrotron Rad. (2012). 19, 759–767

Figure 4
Crystal structure of magnetite in the origin at centre ð3 mÞ. (a) Overview
of the crystal structure and (b) schematic drawing of the linkage and
bonding around the A and B sites.

These values are almost identical between the two sites but the
A site has more negative values. The differences f 0 on f 0 off
are reasonably negative, 0.381 and 0.262 for Fe in the A
and B sites, respectively. Two types of anomalous scattering
factors were obtained by the following approaches: (i) structure-factor analysis of magnetite and (ii) Kramers–Kronig
transform of NiFe2O4 . The difference in f 0 between the two
approaches are 0.811 and 0.812 at Eon and Eoff , respectively. It should be noted that the f 0 values of magnetite are
smaller than those of NiFe2O4, owing to the presence of one
extra electron of Fe2+ in magnetite.

5. Electron-density distributions on pre-edge resonance
The partitioning of f 0 in reciprocal space is helpful to visualize
part of the electron densities resonating at the pre-edge, which
are Fourier synthesized from X-ray diffraction data of
magnetite. The difference between observed and calculated
structure factors appears on the difference-Fourier map to
calculate the Fourier coefficients [|Fobs(hkl)| |Fcalc(hkl)|].
Since the calculated model generally does not yield the whole
scattering power of all the atoms in the structure solution, the
difference-Fourier method is widely used to complete the
structural model by supplying the difference in electron
density between obs(r) and calc(r). When the X-ray resonant
scattering effect is applied to squeeze part of the electrons
Okube, Yasue and Sasaki

Fe K pre-edge peak of magnetite