Parity-odd multipoles, magnetic charges, and chirality in hematite α-Fe_ {2} O_ {3}

Abstract

Collinear and canted magnetic motifs in hematite were investigated by J. Kokubun et al. [Phys. Rev. B 78 115112 (2008) using x-ray Bragg diffraction magnified at the iron K-edge, and analyses of observations led to various potentially interesting conclusions. We demonstrate that the reported analyses for both nonresonant and resonant magnetic diffraction at low energies near the absorption K-edge are not appropriate. In its place, we apply a radically different formulation, thoroughly tried and tested, that incorporates all magnetic contributions to resonant x-ray diffraction allowed by the established chemical and magnetic structures. Essential to a correct formulation of diffraction by a magnetic crystal with resonant ions at sites that are not centers of inversion symmetry are parity-odd atomic multipoles, time-even (polar) and time-odd (magneto-electric), that arise from enhancement by the electric-dipole (E1)–electric-quadrupole (E2) event. Analyses of azimuthal-angle scans on two space-group forbidden reflections, hexagonal (0,0,3)h and (0,0,9)h, collected by Kokubun et al. [Phys. Rev. B 78, 115112 (2008)] above and below the Morin temperature (TM=250 K), allow us to obtain good estimates of contributing polar and magnetoelectric multipoles, including the iron anapole. We show, beyond reasonable doubt, that available data are inconsistent with parity-even events only (E1-E1 and E2-E2). For future experiments, we show that chiral states of hematite couple to circular polarization and differentiate E1-E2 and E2-E2 events, while the collinear motif supports magnetic charges.

Full text

Parity-odd multipoles, magnetic charges and chirality in haematite (α–Fe2O3)
S. W. Lovesey
ISIS Facility & Diamond Light Source Ltd, Oxfordshire OX11 0QX, United Kingdom
A. Rodríguez-Fernández and J. A. Blanco
Departamento de Física, Universidad de Oviedo, E–33007 Oviedo, Spain
(Dated: November 3, 2010)
Collinear and canted magnetic motifs in haematite were investigated by Kokubun et al .[Phys.
Rev. B 78, 115112(2008)]. using x-ray Bragg diffraction magnified at the iron K-edge, and analyses
of observations led to various potentially interesting conclusions. We demonstrate that the reported
analyses for both non-resonant and resonant magnetic diffraction at low energies near the absorption
K–edge are not appropriate. In its place, we apply a radically different formulation, thoroughly tried
and tested, that incorporates all magnetic contributions to resonant x-ray diffraction allowed by the
established chemical and magnetic structures. Essential to a correct formulation of diffraction by a
magnetic crystal with resonant ions at sites that are not centres of inversion symmetry are parity-odd
atomic multipoles, time-even (polar) and time-odd (magneto-electric), that arise from enhancement
by the electric-dipole (E1) - electric-quadrupole (E2) event. Analyses of azimuthal-angle scans on
two space-group forbidden reflections, hexagonal (0,0,3)hand (0,0,9)h, collected by Kokubun et al .
[Phys. Rev. B 78, 115112(2008)] above and below the Morin temperature (TM= 250 K), allow us
to obtain good estimates of contributing polar and magneto-electric multipoles, including the iron
anapole. We show, beyond reasonable doubt, that available data are inconsistent with parity-even
events only (E1-E1 and E2-E2). For future experiments, we show that chiral states of haematite
couple to circular polarization and differentiate E1-E2 and E2-E2 events, while the collinear motif
supports magnetic charges.
PACS numbers: 78.70.Ck, 78.20.Ek, 75.50.Ee, 75.47.Lx,
arXiv:1010.1686v3 [cond-mat.mtrl-sci] 2 Nov 2010

2
I. INTRODUCTION
Enigmas about ichor-like haematite (α–Fe2O3) and famed lodestone, both real and concocted, have been worried
and written about from the time of Greek texts in 315 B.C., to William Gilbert of Colchester, the father of magnetism,
in the 16th. C., to Dzyaloshinsky in 1958 who gave a phenomenological theory of weak ferromagnetism. Haematite is
the iron sesquioxide that crystallizes into the corundum structure (centro-symmetric space-group ]167,R¯
3c) in which
ferric (F e3+,3d5) ions occupy sites 4(c) on the trigonal c-axis that are not centres of inversion symmetry. For an
extensive review of the history and properties of haematite see, for example, Morrish1and Catti et al .2
At room temperature, the motif of magnetic moments is canted antiferromagnetism with moments in a (basal) plane
normal to the c-axis. Weak ferromagnetism parallel to a diad axis of rotation symmetry, normal to a mirror plane
of symmetry that contains the c-axis, is created by a Dzyaloshinsky-Moriya antisymetric interaction D·(S1×S2)
between spins S1and S2and the vector Dis parallel to the c-axis, Dzyaloshinsky,3Moriya.4The Morin temperature
250 K, at which moments rotate out of the the basal plane to the c-axis, may be determined from the temperature
dependence of magnetic Bragg peaks observed by neutron diffraction. Rotation of the moments takes place in a range
of 10 Kin pure crystals but the interval can be much larger, ≈150 K, in mixed materials.5Ultimately, moments
align with the c-axis and create a fully compensating, collinear antiferromagnet with an iron magnetic moment = 4.9
µBat 77 K. We follow Dzyaloshinsky3and label collinear (low-temperature phase) and canted (room-temperature
phase) antiferromagnetism as phases I and II, respectively, see figure 1. In phase I haematite is not magneto-electric
unlike eskolaite (Cr2O3), which also possesses the corundum structure and collinear antiferromagnetism.
FIG. 1. Magnetic and chemical structure of haematite, space group R¯
3c. The red and the yellow dots represent oxygen and
iron sites, respectively. The left line denotes the magnetic motif along the c-axis below the Morin temperature (Phase I). The
right line denotes the motif above the Morin temperature, where iron moments are contained in the a-bplane (Phase II).
Finkelstein et al .6and Kokubun et al .7studied haematite by x-ray Bragg diffraction, with Bragg intensities enhanced
by tuning the energy of the primary x-rays to the iron K-absorption edge. In these experiments, attention is given to
Bragg reflections that are forbidden by extinction rules for the space-group. Often called Templeton and Templeton
reflections,8the reflections in question are relatively weak and arise from angular anisotropy of valence states that
accept the photo-ejected electron. Following rotation of the crystal about a Bragg wave-vector aligned with the c-axis,
Finkelstein et al.6observed a near six-fold periodicity of the intensity that is traced to a triad axis of rotation symmetry
that passes through sites occupied by resonant, ferric ions. In general by measuring intensities, collected at space-
forbidden reflections, we can obtain information of high-order multipoles existing in the materials such as magnetic

3
charge (or magnetic monopole),9electric dipole,10 anapole,11,12 quadropole,13 octupole14,15 and hexadecapole.16,17
Therefore, these weak reflections are extremely sensitive to charge, orbital and spin electron degrees of freedom and
haematite is no exception.18
We apply an atomic theory of resonant Bragg diffraction formulated for the corundum structure,19 to data gathered
by Kokubun et al.7at forbidden reflections (0,0, l )hwith l= 3(2n+ 1) and infer from available data relative values of
atomic multipoles of the resonant ion. A successful story emerges with scattering represented by a mixture of parity-
even and parity-odd (even or odd with respect to the inversion of space) multipoles at sites in the structure occupied
by resonant iron ions, which are not centres of inversion symmetry. Parity-odd multipoles arise in a resonant event
using the electric dipole (E1) and electric quadrupole (E2) - corresponding multipoles are labelled polar (time-even)
or magneto-electric (time-odd) - while parity-even multipoles arise from E1-E1 and E2-E2 events. A chiral state
of haematite is demonstrated by a predicted coupling of resonant intensity to circular polarization (helicity) in the
primary beam, and the effect also differentiates between E1-E2 and E2-E2 events. The two parity-odd multipoles
of rank zero correspond to chirality and magnetic charge20,21 and both pseudo-scalar monopoles are present in the
electric dipole-magnetic dipole (E1-M1) amplitude for resonant scattering by haematite in phase I.
Our communication is arranged as follows. Section 2 contains essential information and definitions. Unit-cell
structure factors for Bragg diffraction enhanced by E1-E1, E1-E2 and E2-E2 listed in an Appendix are exploited
in Sections 3 and 4, which report the successful analysis of Bragg diffraction data gathered on haematite at room
temperature and 150 K, well below the Morin transition. Thereafter, in Section 5, there are simulations of resonant
intensity induced by circular polarization in the primary x-ray beam which signals existence of a chiral state. Section
6 addresses magnetic charge found in the E1-M1 structure factor, and not visible in a dichroic signal. A discussion of
findings in Section 7 concludes the communication.
II. BASICS
There are four contributions to the amplitude of photons scattered by electrons calculated in the first level of
approximation in the small quantity (E/mc2), where E is the energy of the primary photon, namely, Thomson
scattering, spin scattering and two contributions with virtual intermediate states, one of which may become large
when E coincides with an atomic resonance. Of particular interest with magnetic samples is a celebrated reduction
of the amplitude, derived by de Bergevin and Brunel,22 which occurs at large E. In this limit, all three contributions
excluding Thomson scattering add to give so-called magnetic, non-resonant scattering made up simply of spin and
orbital magnetic moments. De Bergevin and Brunel’s result is not valid at low energies, and certainly not below an
atomic resonance, as is at once obvious from steps in its derivation.23
In an analysis of x-ray Bragg diffraction data for haematite collected at space-group forbidden reflections we use
the spin and resonant contributions to the scattering amplitude. The spin contribution Gs=i(E/mc2)(e×e0)·Fs(k)
with k=q−q0where eand q(e0and q0) are, respectively, the polarization vector and wave-vector of the primary
(secondary) photon, and the Bragg angle θthat appears in structure factors for resonant scattering is defined by
q·q0=q2cos(2θ). Fs(k)is the unit-cell structure factor for spin magnetic moments. The measured energy profiles of
reflections (0,0,3)hand (0,0,9)hshow a single resonance in the pre-edge region, devoid of secondary structure, which
is modelled by a single oscillator centred at an energy ∆=7.105 keV with a width Γ, to an excellent approximation.7
In this instance, the resonant contribution to scattering is represented by d(E)Fµ0νwhere d(E)=∆/[E−∆ + iΓ] and
Fµ0νis a unit-cell structure factor for states of polarization µ0(secondary) and ν(primary). We follow the standard
convention for orthogonal polarization labels σand π;σnormal to the plane of scattering and, consequently, πin
the plane. Unit-cell structure factors listed in an Appendix are derived following steps for the corundum structure
found in Lovesey et al.19 The generic form of our Bragg scattering amplitude for haematite at a space-group forbidden
reflection (no Thomson scattering) is,
Gµ0ν(E) = Gs
µ0ν+ρ d(E)Fµ0ν,(1)
where ρis a collection of factors, which include radial integrals for particular resonance events, that are provided
in an Appendix.
Atomic multipoles hTK
Qiin parity-even structure factors, for E1-E1 and E2-E2 events, have the property that even
rank K are time-even (charge) and odd rank K are time-odd (magnetic). For enhancement at the K-absorption edge,
all parity-even atomic multipoles relate to orbital degrees of freedom in the valence shell - spin degrees of freedom are
absent Lovesey.24 Thus, for enhancement at the K-absorption edge, multipoles hTK
Qiwith odd K are zero if the ferric,
3d5(electron configuration 6S) of the iron ion is fully preserved in haematite. The measured iron magnetic moment
4.9µBat 77 Kindicates that the orbital magnetic moment is small and likely no more than ≈2% of the measured
moment.2,5

4
III. PHASE I
We report first our analyses of data gathered by Kokubun et al .7on haematite at 150 K. With 100% incident
σ-polarization and no analysis of polarization in the secondary beam, the measured intensity of a Bragg reflection is
proportional to,
I=|Gσ0σ(E)|2+|Gπ0σ(E)|2,(2)
For a collinear antiferromagnet, in expression (1) for Gµ0ν(E) one has Gs
σ0σ= 0 and in the channel with rotated
polarization,
Gs
π0σ= 4 sin(θ)sin(ϕl) (E/mc2)fs(k)hSzi,(3)
where ϕ=−37.91 deg., the Bragg angle θ= 10.96 deg. (34.77 deg.)for a Miller index l= 3 (9),hSzi ≤ 5/2is
the spin moment and fs(k)is the spin form factor with fs(0) = 1. Note that |Gs
π0σ|2∝sin2(θ)above is not the
expression in equation (20) in Ref. (7), which is derived by use of an abridged scattering amplitude that is not valid
in the experiment.22
At resonance, the spin contribution Gs
π0σis suppressed compared to the resonant contribution by a factor Γ/∆≈
10−4and it may safely be neglected.
Confrontations between our theoretical expressions for the azimuthal-angle dependence of Bragg intensity with
corresponding experimental data reported in Ref. (7) reveal a 30 deg. mismatch of origins in the azimuthal angle.
Our origin ψ= 0 has the a-axis normal to the plane of scattering19 whereas Kokubun et al .7specify an origin such
that the a-axis is parallel to q+q0giving a nominal mismatch in the origin of ψ, between theory and experiment,
of 90 deg. The actual mismatch, 30 deg., revealed by our analysis of data is likely to arise in the experiments by
mistakenly using for reference a basal plane Bragg reflection off-set by 60 deg. In this and the following section we
reproduce data as a function of ψoff-set by 30 deg. compared to data reported in Figures 5and 10 in Ref. (7).
In light of the established negligible orbital magnetism in haematite, parity-even, time-odd atomic multipoles (K=
1 & 3) are set equal to zero. Looking in the Appendix one finds Fµ0ν(E1−E1) = 0. Additionally, Fσ0σ(E2−E2) = 0
and Fπ0σ(E2−E2) produces Templeton - Templeton scattering proportional to [hT4
+3i0cos(3ψ)], where ψis the
azimuthal angle. Inspection of data for phase I reproduced in figure 2 shows that an E2-E2 event on its own is not an
adequate representation. The missing modulation is produced by the E1-E2 event that introduces a polar quadrupole
hU2
0iin phase with the parity-even hexadecapole.25 Figure 2 displays satisfactory fits of {| Fσ0σ|2+|Fπ0σ|2}, using
equal measures of E1-E2 and E2-E2 events, to data from azimuthal-angle scans performed at reflections (0,0, l)h
with l= 3 and 9. The influence of the polar quadrupole is very notable for l= 9 because for this Miller index the
hexadecapole is suppressed, with the ratio at l= 9 to l= 3 of tan(ϕl)equal to 0.15. Relative values of multipoles
inferred from fits to the low temperature data are gathered in table I. Values of hT4
+3i0and hU2
0iin phase I are found
to be of one sign and in the ratio 20 : 1, with near equal magnitudes of the polar quadrupole and magneto-electric
octupole, hG3
+3i0. If |ρ(E2−E2)/ρ(E1−E2) |≈ 1.0, as suggested by our estimate, magneto-electric multipoles are
≈5% of the dominant parity-even hexadecapole, hT4
+3i0.
Without polarization analysis, it does not seem possible from azimuthal-angle scans to distinguish between E1-E2
and E2-E2 events. However, as shown in Section 5, the two events can be distinguished with circularly polarized
x-rays.
The failure of pure parity-even structure factors E1-E1 plus E2-E2 to explain the data is most pronounced for l= 9.
To illustrate the extent of the failure, figure 3 displays a fit to intensity at l= 9 with an amplitude made of equal
amounts of E1-E1 and E2-E2 unit-cell structure factors, and the quality of the fit is clearly inferior to the one shown
in figure 2.
IV. PHASE II
In this phase, above the Morin transition, iron magnetic moments lie in a plane normal to the c-axis. We choose
orthonormal principal-axes (x,y,z) with the x- and z-axes parallel to the crystal a- and c-axes, respectively. The
crystal a-axis is parallel to a diad axis of rotation symmetry, normal to the mirror plane that contains the trigonal
c-axis.
The spin contribution Gs
σ0σ= 0, while the corresponding π0σ-scattering amplitude can be different from zero and,
notably, it depends on azimuthal angle. We find,

5
20
15
10
5
0
Intensity (arb. units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,3)h E0=7.105 keV T=150 K
1.2
1.0
0.8
0.6
0.4
0.2
Intensity (arb. units)
200150100500-50-100-150
Azimutal Angle (degree)
α−Fe2O3 (0,0,9)h E0=7.105 keV T=150 K
FIG. 2. Azimuthal-angle dependence of intensity of Bragg reflections (0,0, l)hwith l= 3 and l= 9 for phase I (150K).
Continuous curves are fits to structure factors for E1-E2 and E2-E2 events with magnetic (time-odd) parity-even multipoles
set to zero. Inferred relative atomic multipoles are listed in Table I. Experimental data is taken from Kokubun et al .7
TABLE I. Relative values of atomic multipoles for collinear antiferromagnetism in phase I (at ≈100 K below the Morin
transition) and canted antiferromagnetism in phase II (room temperature). The magnitude of the dominant hexadecapole,
hT4
+3i0, is set to +10.00. The estimate hU2
0i= +0.50 inferred by fits to data for phase I is also used in analysis of data for
phase II. Values for other multipoles are inferred by fitting to data equal measures of E1-E2 and E2-E2 structure factors listed
in an Appendix, with time-odd figures (magnetic) multipoles in E2-E2 set to zero. Fits are displayed in figures 2 and 4.
With our definition, real h...i’ and imaginary h...i” parts of a multipole are defined through hGK
Qi=hGK
Qi0+ihGK
Qi00 with
hGK
Qi∗= (−1)QhGK
−Qi, and identical relations for the other two multipoles, hTK
Qiand hUK
Qi. All multipoles with projection Q
= 0 are purely real. Using radial integrals from an atomic code factors in equation (1) are in the ratio ρ(E2−E2)/ρ(E1−E2) ≈
−0.98, which is no more than a guide to the actual value in haematite. This ratio is not eliminated in listed values of multipoles.
Multipole Phase I Phase II
hG1
+1i0−0.50(2)
hG2
0i0.11(2) −
hG2
+1i00 − −0.38(3)
hG3
+1i0−1.07(6)
hG3
+3i00.41(2) 2.45(5)
Gs
π0σ= 4 cos(ψ)cos(θ)sin(ϕl) (E/mc2)fs(k)hSyi,(4)
and |Gs
π0σ|2∝cos2(θ)from eq. (4) is not the same as the corresponding result, equation (19) in Ref. (7) for
reasons spelled out in Section 3.
Away from a resonance, the result (4) predicts a two-fold periodicity of intensity as a function of azimuthal angle,
which is in accord with observations in Ref. (7). Spin moment in the mirror plane hSyiis close to 5/2while
spontaneous magnetization, directed along a diad axis, is ≈0.02% of the nominal value. From (3) and (4) we see that
the ratio of |Gs
π0σ|2for phases I and II depends on tan2(θ)which takes the value 0.04(0.48) for l= 3(l= 9). For
l= 3, Kokubun et al .7report intensity between 150 K (phase I) and 300 K (phase II). Starting from ≈210 K a large

6
1.2
1.0
0.8
0.6
0.4
0.2
Intensity (arb. units)
200150100500-50-100-150
α−Fe2O3 (009)h E0=7.105 keV T=150 K
2.0
1.5
1.0
0.5
Intensity (arb. units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,9)h E0=7.105keV RT
FIG. 3. Azimuthal-angle dependence of intensity of the Bragg reflection (0,0,9)hfor phases I (150K)and II (room temperature).
Continuous curves are fits to parity-even structure factors E1-E1 and E2-E2 including all magnetic multipoles. Experimental
data taken from Kokubun et al.7appears also in figure 2 and 4.
increase of intensity is observed over an interval of ≈40 K. Rotation of magnetic moments from the c-axis to basal
plane, between phases I and II, takes place in a range of 10 K in pure crystals but the interval can be larger in mixed
materials as commented above.
Slightly away from the resonance, interference between the non-resonant, spin contribution (4) and d(E)Fπ0σmay
enhance intensity in a Bragg peak if (E−∆)[Gs
π0σ/(Fπ0σ)0]i0. We find [Gs
π0σ/(Fπ0σ)0]is of one sign for l= 3 and l= 9
provided that fs(k), the spin form factor, is of one sign. At face value this finding is not at one with Kokubun et al .7
who discuss a sighting of slight enhancement of the intensity on the low-energy side of the resonance for l= 9 that is
apparently absent, or completely negligible, for l= 3.
Figure 4 shows fits of E1-E2 and E2-E2 structure factors to data gathered at l= 3 and l= 9 in phase II (room
temperature). As before, in our analysis of data gathered on phase I, parity-even multipoles with odd K are set to
zero. Time-even contributions to structure factors, determined by chemical structure, are taken to be the same in
phases I and II. Consistency with this assumption, about chemical structure, implies for phases I and II the same
values of hT4
+3i0and hU2
0i. Inferred relative values of time-odd atomic multipoles for phase II are listed in Table I,
with values of hT4
+3i0and hU2
0iin the ratio 20 : 1. Relative to the magnitude of hU2
0i, none of the magneto-electric
multipoles are negligible in phase II. Figure 3 contains a fit of pure parity-even structure factors, E1-E1 and E2-E2,
to data for the reflection l = 9, and the quality of the fit is clearly inferior to that reported in figure 4 with E1-E2
and E2-E2 structure factors.
V. CHIRAL STATE
A chiral, or handed, state of a material is permitted to couple to a probe with a like property, in our case circular
polarization (helicity) in the primary beam of x-rays. In our notation, the pseudo-scalar for helicity, P2, is one of
three purely real, time-even Stokes parameters. Intensity induced by helicity in the primary beam is,17
Ic=P2Im {G∗
σ0πGσ0σ+G∗
π0πGπ0σ},(5)

7
20
15
10
5
0
Intensity (arb. units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,3)h E0=7.105 keV RT
2.0
1.5
1.0
0.5
Intensity (arb. units)
200150100500-50-100-150
Azimutal Angle (degree)
α−Fe2O3 (0,0,9)h E0=7.105 keV RT
FIG. 4. Azimuthal-angle dependence of intensity of Bragg reflections (0,0, l)hwith l= 3 and l= 9 for phase II (room
temperature). Continuous curves are fits to structure factors for E1-E2 and E2-E2 events with magnetic (time-odd) parity-even
multipoles set to zero. Inferred relative atomic multipoles are listed in Table I. Experimental data is taken from Kokubun
et al .7
where the amplitudes Gµ0νare given by eq. (1) and * denotes complex conjugation. Icis zero for Thomson
scattering since it is proportional to (e·e0)and diagonal with respect to states of polarization.
Let us consider the fully compensating collinear antiferromagnet (phase I). For both E1-E1 and E1-M1 events there
are no contributions diagonal with respect to states of polarization and Icis zero. Using structure factors listed in
the Appendix for the E1-E2 and E2-E2 events we find,
Ic(E1−E2) = −P2(8√2
5)ρ2(E1−E2) |d(E)|2sin(3ψ)cos3(θ) (1 + sin2(θ))cos2(ϕl)hG3
+3i0hU2
0i,(6)
and,
Ic(E2−E2) = −P24ρ2(E2−E2) |d(E)|2sin(6ψ)sin(θ)cos6(θ)sin2(ϕl)hT3
+3i00 hT4
+3i0,(7)
The predicted intensities are significantly different - notably in dependence on the azimuthal angle - and offer a
method by which to distinguish contributions from the two events. Intensities (6) and (7) depend on long-range
magnetic order, with Ic(E2−E2) = 0 if the ferric ion is pure 6S. The polar quadrupole in (6) is a manifestation
of local chirality19,25 whereas the pseudoscalar hU0
0i, discussed in the next section, is a conventional measure of the
chirality of a material. While for phase II, we find that Ic is given by,
Ic(E1−E2) = P2(8√2
5)ρ2(E1−E2) |d(E)|2cos2(ϕl)cos2(θ)hU2
0i{ 1
√3sin(ψ) [ −3
√5(cos(3θ) + cos(θ)) hG1
+1i0+
+(cos(3θ)−cos(θ)) hG2
+1i00 −1
√5(cos3(θ)+2cos(θ)) hG3
+1i0]−sin(3ψ)cos(θ) (1 + sin2(θ))hG3
+3i0},
(8)

8
100x10-3
80
60
40
20
0
-20
-40
-60
Intensity (arb.units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,3)h E0=7.105 keV T=150 K
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
Intensity (arb.units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,9)h E0=7.105 keV T=150 K
FIG. 5. Simulation of the azimuthal-angle dependence from eq. (6) for a circular polarized light of Bragg reflections (0,0, l)h
with l= 3 and l= 9 for Phase I. Continuous curves are simulations made with the values of the multipoles from the E1-E2
event gathered in Table I. For the E2-E2 event Icis zero because our magnetic (time-odd) parity-even multipoles are zero for
a ferric ion. Zero Icdoes not mean zero intensity for Ic is only the circular polarization contribution to intensity.17
Ic(E2−E2) = −P2(1
√2)ρ2(E2−E2) |d(E)|2sin2(ϕl)hT4
+3i0{4sin(ψ)cos4(θ) [ −1
√5sin(θ) (8cos2(θ)−5)hT1
1i00 +
+r3
5sin(θ)cos3(θ)hT3
+1i00 ]−4√2sin(θ)cos6(θ)sin(6ψ)hT3
+3i00 },
(9)
VI. MAGNETIC CHARGE AND CHIRALITY
The pseudo-scalar monopoles hG0
0iand hU0
0ihave particularly simple and interesting physical interpretations.
Both monopoles are allowed in haematite structure factors for the E1-M1 event, as we see by inspection of relevant
expressions in the Appendix. A conventional measure of the chirality of electrons in a molecule or extended media
is hS·pi/| hpi |, where Sand pare operators for spin and linear momentum, and, not unsurprisingly, hU0
0iis
proportional to hS·pi/|p|. It is well-known that, hU0
0icontributes to natural circular dichroism.26 On the other
hand, hG0
0i, a magnetic charge, does not contribute to dichroic signals but it can contribute in scattering. Such is
the case for gallium ferrate,27 and phase I of haematite. Magnetic charge, and the magneto-electric quadrupole, are
present in the amplitude for back-scattering with q=−q0.
VII. DISCUSSION
We report successful analyses of resonant Bragg diffraction data gathered by Kokubun et al.7on haematite in
the collinear (phase I) and canted (phase II) antiferromagnetic phases, with no analysis of diffraction according to

9
1.0
0.5
0.0
-0.5
-1.0
Intensity (arb.units)
α−Fe2O3 (0,0,3)h E0=7.105keV RT
6
4
2
0
-2
-4
Intensity (arb.units)
200150100500-50-100-150 Azimutal Angle (degree)
α−Fe2O3 (0,0,9)h E0=7.105keV RT
FIG. 6. Simulation of the azimuthal-angle dependence from eq. (8) for a circular polarized light of Bragg reflections (0,0, l)h
with l= 3 and l= 9 for Phase II (room temperature). Continuous curves are simulations made with the values of the multipoles
from the E1-E2 event gathered in Table I. For the E2-E2 event the Ic is equal to zero because our magnetic (time-odd) parity-
even are zero. Zero Icdoes not mean zero intensity since Icis only the circular polarization contribution.17
polarization of the x-rays. We infer good estimates of iron atomic multipoles, and find large amounts of parity-odd
multipoles. Of particular importance to a successful analysis is a polar quadrupole, a measure of local chirality,25
and, in phase II, magneto-electric multipoles that include the anapole. Slight departures between our theory and
experiment could be due to a less than ideal crystal, as witnessed in the extended interval of temperature for rotation
of magnetic moments between phases I and II.7
Future experiments might employ polarization analysis that will allow closer scrutiny of unit-cell structure factors
for haematite we list in an Appendix, which are derived from the established chemical and magnetic structures of
haematite. We predict for phase I that scattering enhanced by the E1-M1 event contains monopoles that represent
chirality and magnetic charge.
Our analyses of data are based on an atomic theory of x-ray Bragg diffraction19 with unit-cell structure factors that
are fundamentally different from corresponding structure factors employed by Kokubun et al.7One difference arises
in the treatment of non-resonant magnetic scattering. We use the exact expression, due solely to spin moments, while
Kokubun et al .7mistakenly - because it is not valid in the investigated interval of energy - use an abridged amplitude
by de Bergevin and Brunel22 that is a sum of the exact expression and the high-energy limit of two contributions to
scattering that involve intermediate states (one of the two is capable of showing a resonance).Treating the resonance
as a single oscillator, in accord with the reported energy profile, our structure factors for resonant diffraction are
completely determined with no arbitrary phase factors, unlike the analysis in Ref. (7). This difference in the analyses
is a likely explanation of our evidence that published data for azimuthal-angle scans are miss-set by 30 deg. Our
treatment of magnetic (time-odd) contributions to scattering is another major difference in the analyses. Whereas
Kokubun et al .7allow only the dipole in the E1-E1 event we consider all permitted time-odd contributions in both
parity-even and parity-odd events. Time-odd multipoles from parity-even events, hTK
Qiwith odd K, are related to
orbital magnetism when the intermediate state in resonance is an s-state, as is the case in the experiments in question
with absorption at the iron K-edge. The available evidence is that orbital magnetism of the ferric ion in haematite is
negligible, as expected for an s-state ion, and the same can be said of the parity-even, time-odd multipoles, including
the dipole which at resonance is the only source of magnetic scattering considered in Ref. (7). From our analysis,
we conclude that magnetic scattering at resonance is provided by magneto-electric multipoles in an E1-E2 event. We

10
demonstrate beyond reasonable doubt that, allowing magnetic hTK
Qidifferent from zero the available data are not
consistent with diffraction enhanced by purely parity-even events, E1-E1 and E2-E2.
In summary, we have derived information on the relative magnitude of multipoles for the antiferromagnetic phases
of haematite (above and below the Morin Temperature) These estimates are obtained from analyses of experimental
azimuthal dependence gathered in resonant x-ray Bragg diffraction at space-group forbidden reflections (0,0,3)hand
(0,0,9)h. A chiral electron state is proposed from a predicted coupling of resonant intensity to circular polarization
in the primary beam. This effect allows differentiating between contributions of the E1-E2 and E2-E2 events. In
addition, pseudo-scalar monopoles (chirality and magnetic charge) are present in the E1-M1 amplitude for resonant
scattering by haematite below the Morin temperature.
VIII. ACKNOWLEDGMENTS
Professor Gerrit van der Laan provided values of atomic radial integrals for a ferric ion. We have benefited from
discussions with Dr. A. Bombardi and Professor S. P. Collins, and correspondence with Dr. F. de Bergevin. One
of us (SWL) is grateful to Professor E. Balcar for ongoing noetic support. Financial support has been received from
Spanish FEDER-MiCiNN Grant No. Mat2008-06542-C04-03. One of us A.R.F is grateful to Gobierno del Principado
de Asturias for the financial support from Plan de Ciencia, Tecnología e innovación (PTCI) de Asturias.
Appendix A: Unit-cell structure factors
Some factors in eq.(1) contain a dimensionless quantity ℵ=m∆a2
o/¯h2= 260.93 where aois the Bohr radius and
∆ = 7.105keV . Radial integrals for the E1 and E2 processes at the K-absorption edge are denoted by {R}sp and
{R2}sd. Estimates from an atomic code are {R}1s4p/ao=−0.0035 and {R2}1s3d/a2
o= 0.00095, and it is interesting
that the magnitudes are smaller than hydrogenic values with Z= 26 by a factor of about three. More appropriate
values of the radial integrals will be influenced by ligand ions. The M1 process between stationary states of an isolated
non-relativistic ion is forbidden because the radial overlap of initial and final states in the process is zero, on account
of orthogonality. For an M1 process in a compound the radial integral, denoted here by {1}γγ , is an overlap of
two orbitals with common orbital angular momentum, Γ, which may be centred on different ions. The magnitude of
{1}γγ is essentially a measure of configuration interactions and bonding, or covalancy, of a cation and ligands. Factors
appearing in eq.(1) are,
ρ(E1−E1) = [{R}sp/ao]2ℵ,(A1)
ρ(E1−M1) = q{R}sp{1}γ γ ,(A2)
ρ(E1−E2) = [q{R2}sdRsp /a2
o]ℵ,(A3)
ρ(E2−E2) = [q{R2}sd/ao]2ℵ.(A4)
Haematite structure factors Fµ0νfor forbidden reflections (0,0, l)hwith l= 3(2n+ 1) and enhancements by E1-E1,
E1-M1, E1-E2 and E2-E2 events are listed below. In these expressions, the angle ϕ=−πu, where u= 2z−1/2=0.2104
for α−F e2O3, the angle θis the Bragg angle, and hTK
Qi,hGK
Qiand hUK
Qiare the mean values of the atomic tensors
involved.
1. Collinear antiferromagnet, phase I
(E1-E1)
Fσ0σ(E1−E1) = 0 (A5)

11
Fπ0σ(E1−E1) = −2√2sin(ϕl)sin(θ)hT1
0i(A6)
Fπ0π(E1−E1) = 0 (A7)
(E1-M1)
Fσ0σ(E1−M1) = 0 (A8)
Fπ0σ(E1−M1) = 2√2
√3cos(ϕl){2√2[−sin2(θ)hG0
0i+icos2(θ)hU0
0i] + (2 + cos2(θ))hG2
0i+icos2(θ)hU2
0i} (A9)
Fπ0π(E1−M1) = 0 (A10)
(E1-E2)
Fσ0σ(E1−E2) = −4√2
√5sin(3ψ)cos(ϕl)cos(θ)hG3
+3i0(A11)
Fπ0σ(E1−E2) = 2
√5cos(ϕl){−[3cos2(θ)−2]hG2
0i+icos2(θ)hU2
0i+√2sin(2θ)cos(3ψ)hG3
+3i0}(A12)
Fπ0π(E1−E2) = −4√2
√5sin(3ψ)cos(ϕl)cos(θ)sin2(θ)hG3
+3i0(A13)
(E2-E2)
Fσ0σ(E2−E2) = −√2sin(3ψ)sin(ϕl)hT3
+3i00 (A14)
Fπ0σ(E2−E2) = r2
5sin(ϕl){sin(3θ)hT1
0i − sin(θ)[3cos2(θ)−2]hT3
0i −
−√5
4cos(3ψ)[[3cos(3θ) + cos(θ)]hT3
+3i00 −i[cos(3θ)+3cos(θ)]hT4
+3i0]}(A15)
Fπ0π(E2−E2) = −1
√2sin(3ψ)sin(ϕl)sin(4θ)hT3
+3i00 (A16)
2. Canted antiferromagnet, phase II
Time-even contributions to structure factors, determined by chemical structure, are the same in phases I and II.
Thus the structure factor with polar multipoles, Fµ0ν(u), for phase II is identical to the foregoing expression for phase
I. With the convenience of the reader in mind, structure factors for parity-even multipoles, Fµ0ν(t), are given in full
although contributions only with K = 1 and 3 differ from foregoing expressions.
(E1-E1)
Fσ0σ(E1−E1) = 0 (A17)
Fπ0σ(E1−E1) = 4cos(ψ)sin(ϕl)cos(θ)hT1
+1i00 (A18)
Fπ0π(E1−E1) = 4sin(ψ)sin(ϕl)sin(2θ)hT1
+1i00 (A19)

12
(E1-M1)
Fσ0σ(E1−M1) = 8sin(ψ)cos(ϕl)cos(θ[−hG1
+1i0+hG2
+1i00 ](A20)
Fπ0σ(E1−M1) = 4cos(ψ)cos(ϕl)sin(2θ)[hG1
+1i0](A21)
Fπ0π(E1−M1) = −8sin(ψ)cos(ϕl)cos(θ)[hG1
+1i0+hG2
+1i00 ](A22)
(E1-E2)
Fσ0σ(E1−E2) = 4√2
√5cos(ϕl)cos(θ){1
√3sin(ψ)[ −3
√5hG1
+1i0− hG2
+1i00 +1
√5hG3
+1i0]−sin(3ψ)hG3
+3i0}(A23)
Fπ0σ(E1−E2) = 2r2
5cos(ϕl)sin(2θ){cos(ψ)
√3[3
√5hG1
+1i0−2hG2
+1i00 −1
√5hG3
+1i0]−cos(3ψ)hG3
+3i0}(A24)
Fπ0π(E1−E2) = −4√6
5cos(ϕl){r5
3cos(θ)sin2(θ)sin(3ψ)hG3
+3i0+
+sin(ψ)[cos(3θ)[hG1
+1i0−√5
3hG2
+1i00 ] + 1
3[cos3(θ)+3cos(θ)]hG3
+1i0]}(A25)
(E2-E2)
Fσ0σ(E2−E2) = sin(2θ)sin(ϕl){sin(ψ)[ −2
√5hT1
+1i00 −r6
5hT3
+1i00 ] + √2sin(3ψ)hT3
+3i00 }(A26)
Fπ0σ(E2−E2) = −sin(ϕl){cos(ψ)×[2
√5cos(3θ)hT1
+1i00 +r6
5cos(θ)(1 + sin2(θ))hT3
+1i00 ] +
+cos(3ψ)[√2cos(θ)(3cos2(θ)−2)hT3
+3i00 ]}(A27)
Fπ0π(E2−E2) = 1
√2sin(ϕl)sin(4θ){sin(ψ)[−4√2
√5hT1
+1i00 +r3
5hT3
+1i00 ]−sin(3ψ)hT3
+3i00 }(A28)
1A. H. Morrish, Canted Antiferromagnetism: Hematite (World Scientific Publishing, Singapore, 1994).
2M. Catti, G. Valerio, and R. Dovesi, Phys. Rev. B 51, 7441 (1995).
3I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).
4T. Moriya, Phys. Rev. 120, 91 (1960).
5E. Krén, P. Szabó, and G. Konczos, Phys. Lett. 19, 103 (1965).
6K. D. Finkelstein, Q. Shen, and S. Shastri, Phys. Rev. Lett. 69, 1612 (1992).
7J. Kokubun, A. Watanabe, M. Uehara, Y. Ninomiya, H. Sawai, N. Momozawa, K. Ishida, and V. E. Dmitrienko, Phys. Rev.
B78, 115112 (2008).
8D. Templeton and L. Templeton, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 38, 62 (1982).
9S. W. Lovesey and V. Scagnoli, J. Phys.: Condens. Matter 21, 474214 (2009).
10 J. Fernández-Rodríguez, J. A. Blanco, P. J. Brown, K. Katsumata, A. Kikkawa, F. Iga, and S. Michimura, Phys. Rev. B 72,
052407 (2005).
11 S. W. Lovesey, J. Fernández-Rodríguez, J. A. Blanco, D. S. Sivia, K. S. Knight, and L. Paolasini, Phys. Rev. B 75, 014409
(2007).
12 J. Fernández-Rodríguez, V. Scagnoli, C. Mazzoli, F. Fabrizi, S. W. Lovesey, J. A. Blanco, D. S. Sivia, K. S. Knight,
F. de Bergevin, and L. Paolasini, Phys. Rev. B 81, 085107 (2010).
13 S. B. Wilkins, R. Caciuffo, C. Detlefs, J. Rebizant, E. Colineau, F. Wastin, and G. H. Lander, Phys. Rev. B 73, 060406
(2006).
14 J. A. Paixão, C. Detlefs, M. J. Longfield, R. Caciuffo, P. Santini, N. Bernhoeft, J. Rebizant, and G. H. Lander, Phys. Rev.
Lett. 89, 187202 (2002).
15 S. W. Lovesey and K. S. Knight, J. Phys.: Condens. Matter 12, 2367 (2000).

13
16 Y. Tanaka, T. Inami, S.W. Lovesey, K.S. Knight, F. Yakhou, D. Mannix, J. Kokubun, M. Kanazawa, K. Ishida, S. Nanao,
T. Nakamura, H. Yamauchi, H. Onodera, K. Ohoyama, and Y. Yamaguchi, Phys. Rev. B 69, 024417 (2004).
17 J. Fernández-Rodríguez, S. W. Lovesey, and J. A. Blanco, Phys. Rev. B 77, 094441 (2008).
18 S. W. Lovesey and K. S. Knight, J. Phys.: Condens. Matter 12, L367 (2000).
19 S. W. Lovesey, E. Balcar v K. S. Knight, and J. Fernández-Rodríguez, Phys. Rep. 411, 233 (2005).
20 S. W. Lovesey and E. Balcar, J. Phys. Soc. Jpn. 79, 074707 (2010).
21 S. W. Lovesey and E. Balcar, J. Phys. Soc. Jpn. 79, 104702 (2010).
22 F. De Bergevin and M. Brunel, Acta Crystallogr. A 37, 324 (1981).
23 S. W. Lovesey, Rep. Prog. Phys. 56, 257 (1993).
24 S. W. Lovesey, J. Phys.: Condens. Matter 10, 2505 (1998).
25 V. E. Dmitrienko and E. N. Ovchinnikova, Acta Crystallogr. A 57, 642 (2001).
26 S. P. Collins, S. W. Lovesey, and E. Balcar, J. Phys.: Condens. Matter 19, 213201 (2005).
27 U. Staub, Y. Bodenthin, C. Piamonteze, M. García-Fernández, V. Scagnoli, M. Garganourakis, S. Koohpayeh, D. Fort, and
S. W. Lovesey, Phys. Rev. B 80, 140410 (2009).