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X-ray diffraction by magnetic charges (monopoles)

Magnetic charges, or magnetic monopoles, may form in the electronic structure of magnetic materials where ions are deprived of symmetry with respect to spatial inversion. Predicted in 2009, the strange magnetic, pseudoscalars have recently been found different from zero in simulations of electronic structures of some magnetically ordered, orthorhombic, lithium orthophosphates (LiMPO4). We prove that magnetic charges in lithium orthophosphates diffract x-rays tuned in energy to an atomic resonance, and to guide future experiments we calculate appropriate unit-cell structure factors for monoclinic LiCoPO4 and orthorhombic LiNiPO4.

Lithium orthophosphates of divalent manganese (lithiophilite), iron (triphylite), cobalt, and nickel are members of the olivine family of structures (some minerals in the family of structures possess the colour olive-green), with magnetic motifs indexed on the chemical unit cell (Γ-point) [11]. Cell dimensions of lithium orthophosphates permit Bragg diffraction at L-edges, which are 2p → 3d dipole-allowed in the soft x-ray region with a wavelength λ ≈ 14 Å. X-ray diffraction is strongly enhanced by L 2,3 edges of 3d transition metal ions and may display a weak E1-M1 event and, in consequence, magnetic charge [7,12,13].
In this communication, we report electronic structure factors for transition metal ions in lithium orthophosphates that describe diffraction, of both neutrons and x-rays, and dichroic signals. Results make use of established chemical and magnetic symmetries. Our findings with regard to magnetic charge accord with ab initio simulations of electronic structure in orthorhombic crystals [14]. It is shown here that, x-ray diffraction by LiNiPO 4 can confirm magnetic charge predicted in simulations of this compound. A second example of this kind is LiCoPO 4 . For this compound magnetic charge is allowed in the monoclinic structure proposed by Vaknin et al [15], while it is forbidden by symmetry in the undistorted, orthorhombic structure used in [14].
Orthorhombic, magnetic lithium orthophosphates are discussed in Sections 2, 3 and 4. In the next section their point symmetries and multipoles, followed, in Section 3, by electronic structure factors for bulk properties, neutron diffraction and x-ray diffraction. Resonant x-ray Bragg diffraction is the subject of Section 4. The same topics are covered in Section 5 for the monoclinic compound LiCoPO 4 . Conclusions are gathered in Section 6.

Orthorhombic space groups and point-symmetry
The chemical structure is orthorhombic Pnma-type with metal ions using sites (4c) that are not centres of inversion symmetry. Crystal class D 2h (mmm) is centrosymmetric.
There are three magnetic motifs of interest for lithium compounds LiMPO 4 with M = Mn, Fe, Co and Ni, and we label them by the corresponding parity-odd, one-dimensional irreducible representations (irrep) that are Γ − 1 , Γ − 2 , and Γ − 4 . Here we list the motifs, space-groups, crystal class, and point symmetry of sites (4c) used by metal ions. (Magnetic space-groups are specified in terms of the Miller and Love and Belov-Neronova-Smirnova notations. [16,17]) Magnetic crystal classes that appear here occur in the 58 classes that allow a linear magneto-electric effect.
Electronic degrees of freedom of the magnetic ions are encapsulated in irreducible, spherical multipoles. Our notation for a generic spherical multipole where the positive integer K is the rank and Q the projection, which satisfies − K ≤ Q ≤ K [8,18]. Angular brackets 〈 ... 〉 denote the time-average of the enclosed quantummechanical operator, i.e., a multipole is a property of the electronic groundstate. Parity-even multipoles are denoted by 〈T K Q 〉 and they have a time signature σ θ = (− 1) K . In x-ray scattering, they arise in a description of Thomson scattering and resonant Bragg diffraction enhanced by a parity-even event, e.g., E1-E1 and E2-E2 events. They arise also in a description of neutron diffraction [8]. In this case, K is odd, and for nd-ions K has has a maximum value 5 (triakontadipole) and for nf-ions K has a maximum value 7 (octaeicosahecatontapole). For parity-odd events in x-ray diffraction our notation for multipoles is 〈U K Q 〉 for time-even, polar and 〈G K Q 〉 for time-odd, magneto-electric [8,18]. The time-odd pseudo-scalar 〈G 0 0 〉 is magnetic charge, 〈G 1 〉 is an anapole (toroidal dipole), and the time-even pseudo-scalar 〈U 0 0 〉 is chirality (helicity). The five standard dichroic signals (linear, circular, natural circular, magneto-chiral and non-reciprocal linear) can be written in terms of multipoles, and expressions for them are found in reference [3].
Magnetic charge can be represented by the scalar product of spin, S, and position, R, operators with 〈G 0 0 〉 = 〈S•R〉 [3,9]. It is allowed in an E1-M1 event, where it contributes to diffraction in which polarization is rotated, namely, σ→π and π→σ ( Figure 1) [1]. The physical content of multipoles in resonant Bragg diffraction is brought out in the sum-rules they obey. For diffraction enhanced by L-edges, parity-even dipoles (K = 1) satisfy, where 〈L〉 3d is orbital angular momentum in the 3d-valence state, and quadrupoles (K = 2) satisfy, where the tensor product {L ⊗ L} 2 has for its diagonal component (Q = 0) a value {L ⊗ L} 2 0 = [3L 2 z − L(L + 1)]/√6, which demonstrates affinity to a standard, parity-even quadrupole operator. For magneto-electric dipoles one finds, which demonstrates that 〈G 1 〉 is related directly to standard spin and orbital anapoles.

consequences of point-group symmetry
There are two site symmetries to consider, namely, m y and m y '. Site The identity m y 〈O K Q 〉 = 〈O K Q 〉 leads to the condition, where σ π = ± 1 is the parity-signature of 〈O K Q 〉. The magnetic dipole, σ π = + 1, 〈T 1 〉 points along the b-axis, as expected. Parity-odd (σ π = − 1) multipoles with K odd are purely real. In particular, the c-axis component of the anapole, 〈G 1 0 〉, is allowed and magnetic charge is not allowed, 〈G 0 0 〉 = 0, for point-group symmetry m y .
Environments at sites 2, 3, and 4 are related to site 1 by rotations by 180 o about crystal axes, and they are 2 z , 2 y , and 2 x , respectively. In ensuing calculations, we will use 2 z 〈O K Q 〉 = (− 1) Q 〈O K Q 〉, and 2 x 〈O K Q 〉 = (− 1) Q 2 y 〈O K Q 〉.

charge diffraction
Pure charge diffraction by the magnetic ions is created by parity-even and time-even multipoles, with σ θ = (− 1) K = + 1 for K even. Corresponding multipoles are purely real for both sites symmetries and, which is purely real, for the chemical structure is centrosymmetric. Standard extinction rules for Miller indices satisfy Ψ K 0 (charge) ≠ 0, whereas Templeton & Templeton scattering, created by angular anisotropy in electronic structure, is labelled by Miller indices and Q ≠ 0. Thus, to avoid charge scattering from orthorhombic structures consider reflections (h, k, 0) with h odd that give Ψ K 0 (charge) = 0.

magnetic diffraction
All three structure factors are proportional to 〈O K Q 〉 which is subject to either (2.2) or (2.3). Structure factors for Γ − 1 and Γ − 2 are distinguished solely by point-group symmetry, i.e., properties of 〈O K Q 〉, and for this reason we only  (3.6) and the expressions applies to orthorhombic crystals containing Mn, Fe or Co ions. Magnetic dipoles, and all other parity-even and time-odd multipoles, possess σ θ σ π = − 1 and make no contribution to bulk properties. The same is true of magnetic charge for Γ − 2 , because 〈G 0 0 〉 = 0 by virtue of the point-group symmetry, m y . But there is a net magnetic charge for Γ − 1 , together with all other magneto-electric multipoles with Q even and not forbidden by 〈G K 0 〉 = 0 for K odd in point-group symmetry m y '. Turning to Γ − 4 , for Ni ions. Here, also, magnetic moments fully compensate, together with all other parity-even magnetic multipoles. There is no net magnetic charge, and a net anapole moment resides along the b-axis [21,22].

Orthorhombic unit-cell structure factors
Unit-cell structure factors for x-ray diffraction, F, are very easily derived using A K,Q = A K,−Q = (Ψ K Q + Ψ K −Q )/2 with A K,0 = Ψ K Q , and B K,Q = − B K,−Q = (Ψ K Q − Ψ K −Q )/2 and B K,0 = 0. Unit-cell structure factors for E1-E1 and E1-M1 events are found in references [1,23]. A K,Q and B K,Q , are calculated relative to states of polarization and a Bragg wavevector (h, k, l) shown in Figure 1; specifically, the setting of the crystal at the origin of a rotation of the crystal by an angle ψ around the Bragg wavevector (azimuthal-angle scan).
Γ − 2 : Site symmetry m y does not allow magnetic charge. In Section 5 we examine consequences of a lower symmetry structure for LiCoPO 4 proposed by Vaknin et al [15] and find that magnetic charge is allowed and contributes to diffraction at space-group forbidden reflections. The Bragg wavevector (h, k, 0) is aligned with − x in Figure 1 by rotation by an angle α about the c-axis (z-axis). We find, For the particular case of (h, 0, 0) α = π. For this reflection, there is no diffraction in channels with unrotated polarization, F π'π = F σ'σ = 0, while for the channel with rotated polarization F π'σ = F σ'π and, This is a singular benefit of the fast-collision approximation, which amounts to neglect of angular anisotropy in the wavefunction for the intermediate state that accepts the photon [8]. Loss of the benefit results in a vastly more complicated expression for the scattering length, the starting point for an electronic structure factor, that might not be justified [24,25]. the Bohr radius is a o and ∆ is the energy of the resonance.

LiCoPO 4
Vaknin et al [15] show that in LiCoPO 4 the Co magnetic dipole departs from the b-axis and the appropriate magnetic space-group is monoclinic P2 1 '11 (unique axis a), crystal class 2 x ', and not Pnma1'. Here, we demonstrate that magnetic charge is allowed and contributes to diffraction for this monoclinic structure. The chemical and magnetic structures (Γ 2 ) of LiCoPO 4 are depicted in The site symmetry 1 (identity) allows magnetic charge, of course. Two environments inside (2a) are related by the 2 x ' symmetry element. We go on to find, Bulk properties are determined by, X-ray charge scattering is absent for h odd and l = 0, the same condition as in Pnma, and electronic structure factors reduce to, for space-group reflections (h, k, 0) with h odd, with a similar expression for Ψ K Q (2). Applying (5.4) to parity-even (t), polar (u) and magneto-electric (g) multipoles, Ψ K Q (t) = − Ψ K −Q (t) and Ψ K 0 (t) = 0, (5.5) Ψ K Q (u) = − (− 1) K Ψ K −Q (u) with Ψ K Q (u) = 0 for K even, It follows from (5.5) that the magnetic dipole parallel to the c*-axis does not contribute to diffraction and charge scattering is absent, as expected.
Contributions from polar dipoles are parallel to the b-axis and c*-axis, and the anapole contribution is parallel to the a-axis.

Conclusions
Magnetic properties of LiMPO 4 with M = Mn, Fe, Co and Ni have been discussed, with emphasis on properties visible in neutron and x-ray scattering.
Particular attention is given to magnetic charges, or magnetic monopoles, that we predict in calculations of resonant x-ray Bragg diffraction amplitudes for LiCoPO 4 (monoclinic) and LiNiPO 4 (orthorhombic). Recent, independent ab initio simulations of the electronic structure of LiNiPO 4 support the existence of magnetic monopoles in this compound [14]. All being well, resonant x-ray Bragg diffraction experiments will soon be made on both the monoclinic and the orthorhombic compounds. However, magnetic dipole moments associated with Co 1 and Co 2 nearly compensate, as shown in the cartoon, due to an antiferro-magnetic exchange between the two sub-lattices.