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X-ray Detected Magnetic Resonance (XDMR) is a novel spectroscopy in which X-ray Magnetic Circular Dichroism (XMCD) is used to probe the resonant precession of

Orbital magnetism refers to a subtle interplay among several effects: Coulomb and spin-orbit interactions, hybridization and crystal fields. No reliable simulation of these orbital effects can be envisaged without an independent determination of the

In 2005, X-ray Detected Magnetic Resonance (XDMR) emerged as a novel spectroscopy in which XMCD was used to

Let us start with the important remark made by Kittel [_{z}_{z}^{spin}_{z}^{orbit}_{z}^{lattice}^{spin}^{orbit}^{lattice}_{z}^{orbit}_{z}^{lattice}^{lattice}

We are left here with an oversimplified picture in which essentially the spin part of the angular momentum is precessing. Van Vleck [_{1}_{j}_{j}

in which _{B}_{jk}_{jk}_{jk}^{2} for a pseudo-dipolar (superexchange) interaction and (^{4} for the pseudo-quadrupolar coupling [

Spin Hamiltonians have been extensively used for the analysis of electron magnetic resonance spectra [

For crystal fields weak compared to spin-orbit coupling,

If the crystal field is large compared to spin-orbit coupling, it splits the L-multiplet. If the lowest state is a singlet, the (quenched) orbital moment has no first order contribution. Recall that an orbitally threefold degenerate T-state mathematically is isomorphic to a P-state (

As noted by Blume _{z}

in which the Hamiltonian
_{αα}_{B}_{xx}_{yy}_{zz}

Orbital effects add much complexity in those systems where the ground state is orbitally degenerated while magnetism is dominated by exchange interactions. Indeed the standard Heisenberg–Dirac form of exchange is not anymore appropriate and additional terms should be considered [

It is the aim of the present paper to discuss the information content of XDMR spectra, emphasis being laid on the forced precession of orbital magnetization components. In Section 2, we briefly review the conceptual bases of XDMR and recast them in the framework of the linear and nonlinear theories of spin waves developed for FMR. What makes the difference between XDMR and FMR is clearly the element, edge and, in favorable cases,

In XDMR, not only the frequency of the microwave _{2}_{,}_{3}-edges being similarly associated with the photoionization of the spin-orbit split 2

Edge-selectivity of XDMR measurements is a further advantage linked to the conservation of angular momentum in the photoionization process of deep atomic core levels, e.g., the 1_{2}_{,}_{3} absorption edges that are split by spin-orbit, the Fano effect implies that part of the photon angular momentum is converted into spin _{3} edge; _{2} edge). No such conversion is possible at a K-edge (or L_{1}-edge) due to the absence of spin-orbit coupling in the core states and the photon angular momentum is entirely converted into _{3} and L_{2} edges can only be caused by a spin imbalance in the empty states because the orbital momentum transferred to the photoelectron has implicitly the same sign at both edges. This is all the physical content of what is known as the magneto-optical sum rules for XMCD [

In XDMR, it is more appropriate to use a

in which [Δ_{K}_{p}_{RX}_{0} _{F}_{RX}_{0} and _{F}_{z}_{p}_{z}_{p}

One could easily extend this result to molecular complexes where the final states are described with molecular orbitals, e.g., when the absorbing atom is in a tetrahedral ligand field. In all cases, however, K-edge XDMR experiments will refer to the forced precession of some energy-dependent _{z}_{2}_{,}_{3}-edges can similarly be related to differential operators involving spin and orbital magnetization DOS [

in which _{b}_{z}_{z}_{z}_{z}_{z}_{S}_{S}^{2} ^{2} is the electric quadrupole operator which is a traceless spherical tensor of rank 2. In cubic crystals, _{z}_{z}_{z}

As opposed to electron magnetic resonance experiments which can be pumped with poorly energetic photons (^{−1}), XMCD measurements require highly energetic photons with wave numbers in excess of 10^{7} cm^{−1}. Typically, the transition-time towards X-ray excited states is considerably shorter than the time-scale of spin-orbit interactions and of collective magnetic excitations involving exchange and dipole-dipole interactions. XMCD thus delivers a

This has some interesting implications illustrated with

We have also reproduced in _{z}_{g}_{g}_{2}_{,}_{3} edges of iron since 2

Let us associate a fictitious, ^{(}^{ℓ,s}^{)} to 〈_{z}_{z}^{(}^{ℓ,s}^{)} can indeed be described either by

in which the magnetic free energy _{Z}_{A}_{D}_{G}_{B}_{s}

in which _{r}_{1}. It will become clear later on that the transverse relaxation time T_{2} is affected by additional relaxation processes involving a redistribution of the absorbed energy among all degrees of freedom of the whole spin system.

As illustrated with

- In the _{RX}^{⊥} of the incident, circularly polarized X-rays is set perpendicular to both the static bias field _{0} and the oscillating pump field _{p}_{⊥} which oscillates at microwave frequency.

- In the _{RX}^{||}_{0}. If one assumes that _{s}_{eq}_{z}_{z}_{z}_{0}) and _{⊥}. Moreover, any information regarding the phase or helicity of the precession gets lost.

For a ferromagnetic thin film with uniaxial anisotropy and perpendicular magnetization, the opening angle of precession _{0} is a constant of motion that characterizes the precession dynamics: it can be determined by combining together XDMR and

whereas in the TRD geometry:

The Landau–Lifshitz–Gilbert equations lead to the following resonance equations for _{0}:

in which _{cp}

where _{u}_{A}_{1} refer to the uniaxial demagnetizing field and the cylindrical component of the magnetocrystalline anisotropy respectively. On the other hand, _{0} = −_{G}_{p}_{G}^{−6} for high quality YIG films. Similarly, one would show that:

In the low microwave power limit, _{0} → 1, the resonance condition _{0} = 0 will converge towards the usual Lorentzian lineshape whereas _{0} → _{0} → 0. However, for pumping fields in excess of 100 mG or higher, _{0} with multivalued (unstable) solutions that characterizes the so-called

Clearly, in TRD geometry, the oscillating signal Δ_{XDMR}_{RX}^{⊥}_{p}_{XDMR}_{RX}^{||}_{p}^{2}. At this stage, one should keep in mind that Δ_{XDMR}_{XMCD}_{0} is inverted, independently of the oscillating pump field. Indeed, the normalized ratios (_{0}) should be inverted in order to keep the product

In magnetically ordered systems, the precession is driven by non-uniform, time-dependent fields ^{(}^{ℓ}^{)} directly experience dipole-dipole interactions.

In this regime,

in which _{H}_{0} _{z}_{s}_{A}_{1}) and _{M}_{B}_{s}_{z}_{||}_{⊥}_{k}_{k}_{0}. Note that _{g}_{g}

We like to draw attention onto a further point. In addition to circularly polarized precession modes, elliptically polarized modes can be excited as well for which _{g}_{k}^{*}_{−}_{k}

It is preferable to reformulate the Landau–Lifshitz–Gilbert equations as a system of canonical Hamiltonian equations for spin wave amplitudes (_{k}^{*}_{k}_{k}

in which the relaxation frequency (_{rk}_{k}_{rk}_{k}_{k}_{k}^{*}:

The first term in the right-hand side of _{k}_{k}^{2} _{k}^{2}]^{1}^{/}^{2} in which:

where _{ex}_{k}_{k}_{eq}_{k}

The next terms refer to the perturbation induced by the microwave pump field. The first one (
_{⊥}

which causes the resonant excitation of the uniform mode (

It causes the parametric excitation of a pair of spin waves satisfying the conservation laws: _{p}_{1}+_{2} and _{1} = −_{2} =

All information about the interaction of spin-waves between themselves is contained in the perturbation term

In the quasi-particle language of second quantification,
_{1} = _{2} + _{3} for a 3-magnon splitting process, whereas (_{1} + _{2}) = (_{3} + _{4}) in a 4-magnon scattering process. Even though
_{1} · _{2})^{2}, it immediately appears, however, that the exchange cross section is zero if either _{1} or _{2} is zero: in other terms, exchange _{i}_{i}

There is no doubt that the local spin magnetization components ^{(}^{s}^{)} probed by XDMR are directly affected by collective excitations of exchange spin waves. Even though exchange interactions should have no direct effect on orbital magnetization components ^{(}^{ℓ}^{)}, one should not forget that ^{(}^{ℓ}^{)} and ^{(}^{s}^{)} are dynamically coupled by spin-orbit interactions at a time-scale much shorter than the time-scale of precession. Let us insist, however, that it cannot be taken yet as firmly established that ^{(}^{ℓ}^{)} couple to exchange spin waves.

Recognizing that dipolar interactions open a band of degenerate magnons is the clue to understand the relaxation mechanisms in FMR [_{2} in the Bloch–Bloembergen model.

Statistical thermodynamics led to a pair of important equations which proved to be very helpful in discussing the relevance of various relaxation processes:

where _{k}_{z}

Let us make a distinction between two different processes: (i) spin-orbit and dipole-dipole interactions can be strong enough to let the local orbital magnetization components ^{(}^{ℓ}^{)} couple to spin waves; (ii) energy can be redistributed within additional degrees of freedoms involving orbital ordering (O_{ord}

The starting consideration is that both spin exchange and charge motion strongly depend on the orbital state in chemical bonding. One may easily admit that the interaction energy can hardly be optimized simultaneously for all bonds: this leads to peculiar frustrations, oscillations and quantum resonances among orbital bonds. Let us consider first the superexchange term of the Hamiltonian of a magnetic system which would be orbitally degenerate:

For simplicity, we deliberately neglected Hund’s coupling in _{ij}^{γ}_{ij}^{γ}_{g}_{2} or _{2}_{g}_{g}

whereas: _{ij}^{γ}_{h}^{2}_{h}_{g}^{γ}_{ij}

where ^{x}^{z}^{z}_{x}_{2}_{–y}_{2}, whereas ^{z}_{3}_{z}_{2}_{–r}_{2}.

Regarding triply degenerate states (e.g., _{2}_{g}_{i}_{z}_{i}^{2} _{x}_{i}^{2} _{y}_{i}^{2}] and [(_{x}_{y}_{i}_{y}_{x}_{i}_{x}_{y}_{z}_{xz}_{yz}_{xy}

Collective excitations of the orbital ordering degrees of freedom were first envisaged by Cyrot and Lyon-Caen [

in which _{i}_{i}

Regarding magnetic systems with orbitally degenerate ground states, Khaliullin went to the following conclusions: (i) fluctuations are enhanced in both spin and orbit subspaces; (ii) given the strong magnetic anisotropy of bonds,

Already in the late fifties, Suhl realized that a parametric amplification of spin-waves could cause a saturation of the precession cone angle (_{0}) when the microwave pump field exceeded some critical threshold field _{th}

There are two nonlinear processes by which the uniform precession mode can couple to degenerate spin waves. In the three-magnon process of _{2} = _{3} = _{p}_{2} = −_{3} = _{3} = _{4} = 0) are destroyed whereas two degenerate magnons are created such that: _{1} = _{2} = _{3} = _{4} = _{p}_{2} = −_{1} = _{0}, whereas the pair of degenerate magnons created in a 4-magnon scattering process propagates along the direction of _{0}, _{k}_{0} = _{p}

Instability arises when the rate of creation of a pair of degenerate spin-waves (+

in which _{0}_{,}_{0}_{,k,–k}_{0} and _{k}

Thus, one may predict the number of degenerate magnon pairs to grow exponentially when:

whereas:

Here _{r}_{0} and _{rk}_{cp}_{th–}_{2} beyond which one should enter into the spin wave instability regime, provided that some useful expression of _{0}_{,}_{0}_{,k,–k}

in which _{k}

It immediately appears that _{0}_{,}_{0}_{,k,–k}^{dd}_{k}_{0}. In practice, there are tricks to suppress the subsidiary absorption, but there is no way to remove the second-order instability [

As pointed out by Schlömann [

in which Δ_{k}_{rk}_{zth}_{rk}_{k}_{0}. As far as Δ_{k}_{k}_{zth}_{k}

Let us emphasize that the two magnons that are simultaneously excited have the same energy (1_{p}_{⊥}_{⊥}_{z}_{XMCD}

One has to be aware that a specific beamline design and a fairly different instrumentation are required to carry out XMCD measurements with

Other considerations also come into play. In magnetic materials that suffer from large conductive losses (e.g., intermetallics or metallic multilayers with large magnetoresistance), the penetration of the microwave pump field is restricted to the skin depth which hardly exceeds 1

As a preamble, let us mention that, regarding XMCD experiments carried out with hard X-rays, we do not measure directly the absorption cross-section but rather the total X-ray fluorescence yield caused by the photoionization of the deep core level. Let us stress, however, that there is no X-ray fluorescence detector that can measure a small dichroic signal oscillating at microwave frequencies as expected in the transverse detection geometry (TRD). At the ESRF, high quality XDMR spectra were recorded in the TRD geometry using a novel heterodyne detection [_{f}

On Fourier-transforming _{f}

One may easily check that the half-width at half maximum of the gaussian envelope: Δ_{1}_{/}_{2} _{p}_{bpsk}_{N}_{N}

The ESRF heterodyne (or superheterodyne) detection scheme falls in the group of time-average measurement methods. Arena, Bailey _{p}

With excellent insulator properties and no detectable magnetic disorder, yttrium iron garnet (YIG) has played a historical role in the promotion and understanding of FMR. It has a cubic structure (space group:Ia3̄d; group N°230), each unit cell consisting of eight formula units Y_{3}[Fe_{2}](Fe_{3})O_{12} [_{c}_{B}

In Section 3, we shall reproduce XDMR spectra collected essentially on two iron garnet films that were grown by liquid phase epitaxy on oriented gadolinium gallium garnet (GGG) substrates: _{3}Fe_{5}O_{12} (YIG # 520); _{1.3}La_{0.47}Lu_{1.3}]Fe_{4.84}O_{12} (Y-La-LuIG). Note that in film ^{1}_{0}) rare earth cations (La^{3+}, Lu^{3+}) substitute for Y^{3+}. In Section 3.4, we shall also briefly discuss XDMR spectra recorded with a thin, polished platelet of gadolinium iron garnet (GdIG) single crystal.

We like first to show that XDMR spectra of unprecedented quality can now be recorded in the transverse detection geometry (TRD) on YIG and related thin films, thanks to our superheterodyne detection scheme that relies on the powerful concepts of either bpsk or qpsk (bi- or quadrature phase shift keying) used in satellite telecommunications. A vector detection scheme also allows us to recover the phase information that is preserved in a TRD geometry. This is illustrated with _{1} = 7113.91 eV) and the film was rotated by _{Y}^{0} in order to minimize the demagnetizing field anisotropy. We checked that the XDMR peak intensity varied linearly with the square root of the pumping power up to

Arrows in ^{(}^{ℓ}^{)} couple to non uniform magnetostatic spin waves through dipole-dipole interactions. Note that forward/backward MSW seem to have different phases. At this stage, one should realize that there is no chance to excite and detect standing waves resonances associated with magnetostatic modes unless there is a

In

Long before we started the XDMR experiments reported below, a _{z}^{3+}, Lu^{3+}, Y^{3+} [_{2}_{,}_{3}-edges would be less beamtime-consuming than similar experiments at the low-energy Y L_{2}_{,}_{3}-edges. Preliminary test experiments carried out on film ^{(}^{ℓ}^{)} measured at the Fe K-edge, _{0}^{(}^{ℓ}^{)} [^{(}^{s}^{)} measured at the La L_{2}_{,}_{3}-edges, _{0}^{(}^{s}^{)} [_{0}^{(}^{ℓ}^{)} [_{0}^{(}^{ℓ}^{)} [_{0}^{(}^{s}^{)} [_{0}^{(}^{s}^{)} [

We compare in _{RX}^{||}_{0}). The FMR and XDMR spectra both exhibit strong foldover distortions resulting into broad lineshapes (Δ_{0} _{0}^{(}^{ℓ}^{)} [^{(}^{ℓ}^{)}

From ^{(}^{ℓ}^{)}_{2}. Regarding _{z}^{(}^{ℓ}^{)} unaffected since two uniform magnons are replaced by two degenerate magnons, so that the total number of magnons is left unchanged. This, however, does not hold true if the life-times of uniform and degenerate magnons are different. Schlömann [

in which the amplitude of the uniform mode _{0}^{2} directly depends on the incident microwave power. Clearly, the higher the microwave power, the longer should be the lifetime of the pair of degenerate magnons. _{z}^{(}^{ℓ}^{)}, _{z}^{(}^{ℓ}^{)} exactly as observed in _{z}_{z}

In the Kasuya–LeCraw process which is regarded as the dominant spin-lattice relaxation process in YIG [_{z}_{z}

Mechanism (5B.b) deserves more attention in a context where valence or orbital fluctuations may cause a dynamical replacement of Fe^{3+} cations in either tetrahedral or octahedral sites with short living Jahn–Teller cations (Fe^{4+}, Fe^{2+}) that had unquenched orbital moments. Such valence fluctuations could be initiated or enhanced by the presence of small amounts of impurities (e.g., Pb^{2+}) introduced during the growth of the film by liquid phase epitaxy. In other terms, the ^{6}_{5}_{/}_{2} configuration for iron may not be time-invariant. Recall that the effects induced by Fe^{4+}, Fe^{2+} or other Jahn–Teller ions in doped YIG films have fed long debates that are not closed. In the case of film ^{3+} cations and of the small size of the Lu^{3+} cations which allow lutetium to partly replace iron in the octahedral sites. The non-uniform distribution of the rare earth cations induces fluctuations in the lattice parameter,

The replacement of all Y^{3+} ions with Gd^{3+} (^{8}_{7}_{/}_{2}) in the dodecahedral (24c) sites results in a more severe perturbation than that caused by the ^{3+} or Lu^{3+}. Even though YIG and GdIG have identical crystal structures and nearly the same Curie temperatures (_{C}_{B}_{B}_{cp}

We have reproduced in _{0}, the contribution of the uniaxial anisotropy was heavily reduced. In _{B}_{cp}_{cp}_{1}

For sure, the strong anomaly revealed by ^{(}^{ℓ}^{)} at the

As far as the proposed interpretation holds true, the clue to the present problem is in the scattering amplitude
_{B}_{cp}^{57}Fe spin-echo NMR spectra recorded well below T_{B}_{dc}_{ad}

For iron garnets, the shape anisotropy is very often regarded as the main cause for elliptical precession in FMR: this is typically the case of YIG thin films when tangentially magnetized.

Under the conditions of elliptical precession, the longitudinal component of the magnetization _{z}

Clearly, the detection of such a weak frequency-doubled XDMR signal was a critical test regarding the sensitivity of the superheterodyne detection scheme. It is noteworthy that some phase information can be recovered using a vector detection of the frequency doubled signal. It can be shown that _{z}^{(2)} as well as _{z}^{(0)} are time-reversal odd observables that can perfectly be detected by XMCD as long as the oscillating term preserves its time-even parity. This will obviously be the case if the precession helicity is inverted in order to keep 2

The 2 _{p}

With a pumping power as high as 1.25 W, the foldover lineshape of the FMR PSD spectrum was not unexpected. Rotating the film at the magic angle (_{H}_{p}

We have reproduced in _{p}_{0} _{RX}_{eq}_{0} but along the direction of some _{eff}_{eq}_{eq}_{H}_{0} = 54.74_{p}_{eff}_{p}_{⊥p}) that simultaneously excites the uniform precession mode at frequency _{p}

We were primarily interested in the excitation of a pair of spin waves (_{p}_{eff}_{σ}_{π}^{(}^{ℓ}^{)} measured at the Fe K-edge _{z}_{z}

So-far, we discussed XDMR spectra recorded only on ferrimagnetic films or crystals. In collaboration with the research center for the development of the far infrared region (University of Fukui, Japan), we are presently testing at the ESRF the feasibility of high-field XDMR experiments at sub-THz pumping frequencies [

At the ESRF, quite a few proposals were concerned with static XMCD studies on paramagnetic organometallic complexes. In practice, such experiments require low temperatures (_{0} _{7}](Bu_{4}N)_{3}). Recall that

Interestingly, the XMCD spectra of complexes _{z}_{z}^{4}_{1} slightly above the Kramer’s ground state ^{4}_{2}. The existence of a weak orbital moment is also supported by the rather large anisotropy of the EPR spectra: Δ^{2} = _{||}^{2}− _{||}^{2}^{51}_{7}_{/}_{2} nuclei. Actually, the very weak anisotropy of the EPR spectra (Δ^{2}

It immediately appears from _{2}_{,}_{3} edges are considerably more intense than in the previous case. Moreover, the XMCD spectra displayed in _{2}_{,}_{3}-edges: according to the magneto-optical sum rules, this is a typical signature of a large orbital moment 〈_{z}_{B}^{2} _{z}_{z}^{−4} _{B}

Unfortunately, the XDMR spectra of the paramagnetic complexes _{p}

What stimulates us to invest time and efforts in this challenging project is the hope that high-field X-ray detected EPR experiments should allow us to probe the precession dynamics of orbital magnetization components in paramagnetic species with a significant zero-field splitting (zfs), e.g., high spin complexes with integer spin that are EPR-silent at microwave pumping frequencies. Many examples can be found in a long list of Mn(III) complexes [^{(}^{ℓ}^{)} in the X-ray excited final states. In other terms, spin-orbit is responsible for both the large zfs of Re(IV) complexes and the large contribution of 〈_{z}_{z}

The intriguing case of the Van Vleck paramagnetism still deserves a few comments here: it can be best observed when the angular momentum

Whereas Curie’s paramagnetism reflects the alignment of ^{3+} with its ground state ^{7}_{0} separated from the first excited state by ^{−1} and for which a slow longitudinal relaxation process was proposed by Van Vleck [^{169}Tm. One of the best characterized crystals is thulium ethylsulphate, _{2}H_{5}SO_{4})_{3}.9H_{2}0 for which high-field EPR spectra have been reported at 1.2 K [

Since no static XMCD signal can be measured on AFM materials with antiparallel ground states, it is tempting to conclude that there is no hope to detect any XDMR signal as well. The situation may not be as desperate if one looks at what happens under the conditions of antiferromagnetic resonance (AFMR). For simplicity, let us consider a crystal with two AFM ordered sublattices and uniaxial magnetic anisotropy directed along the _{0} _{p}

in which _{E}_{E}_{1} _{E}_{2} _{A}_{A}_{1} _{A}_{2} _{N}_{||}_{⊥}

Anyhow, two distinct precession modes (^{+}; ^{−}) are excited which have the opposite helicity. Moreover, due to the existence of the anisotropy field (_{A}_{A}_{1}_{,}_{2} ), the precession cone angles _{1} and _{2} are no longer identical for the two magnetization components _{1} and _{2}. In the transverse detection geometry (TRD) illustrated with _{⊥}^{±}_{1}^{±}_{2}^{±}^{+} or ^{−} with the following lineshape [

in which _{cp}^{(}^{±}^{)} is the relevant circularly polarized component of the pump field, _{G}_{C}_{C}_{2} is most often expected to be in the sub-THz, if not in the far-infrared range because the exchange fields (_{E}_{1}, _{E}_{2}) are much stronger than the external bias field. Recall that there is, as yet, no X-ray detector that can measure a signal oscillating at sub-THz frequencies, our superheterodyne detection scheme being restricted to _{p}_{0} can be strong enough to shift the resonance frequency ^{−} down to the microwave range. Notice that, at the ESRF, the static bias field applied at the sample location could now be increased up to 17 T.

When the external field _{0} approaches the critical field _{C}_{1} = [2_{E}_{A}_{A}^{2} ]^{1}^{/}^{2} _{C}_{2}, the magnetic system undergoes a first order transition (spin-flop) resulting in the new situation sketched in _{1} +_{2} will be pumped:

in which: _{E||}_{E}_{A}_{E⊥}_{E}_{A}_{C}_{1} = [_{A}_{E||}^{1}^{/}^{2} and _{C}_{2} = [_{A}_{E}_{⊥}]^{1}^{/}^{2}. For completeness, it should be mentioned that there exists also a

Given that the precession cone angles _{1} and _{2} are different, one may question whether a (very weak) XDMR signal could be measured as well in the longitudinal detection geometry (LOD), _{C}_{2} is very large, ^{−} down into the microwave X-band. Unfortunately, one expects the XDMR signal measured in LOD geometry to be much smaller than in FMR because Δ_{z}^{±}_{0}]^{2} _{C}_{E}^{2}, in which _{C}_{E}

For XDMR experiments, it may be preferable to select AFM crystals in which the exchange field _{E}_{N}_{2} (_{N}^{−}) excited by a microwave pump field (F_{p}_{0} _{2} substrates [_{E}_{2} may not be the ideal candidate for Mn K-edge XDMR experiments due to the vanishing orbital moment 〈_{z}^{2+} ions in a high spin (^{6}_{z}_{z}^{6}_{1}_{g}

The case of KCuF_{3} (_{N}_{3} in the paramagnetic phase revealed the existence of a strong contribution of the Dzyaloshinskii–Moriya antisymmetric exchange interaction:

with _{jk}_{jk}

in which _{j}_{(}_{k}_{)} are the energy separations between the ground state orbital levels _{j}_{(}_{k}_{)} and the relevant excited states _{j}_{(}_{k}_{)}, _{j}_{(}_{k}_{)} being the effective superexchange constants at sites _{0} ⊥ _{3} should behave just like a

Under such conditions, one may reasonably expect a (weak) XDMR signal to be detectable at low temperature in TRD geometry.

Among transition metal oxides for which the low frequency mode ^{−} could be excited at microwave frequencies, single crystals of ilmenites (
_{3}, with
_{z}_{2}O_{3} (T_{N}_{2}O_{3} is rather low (_{C}_{1} _{⊥}^{(}^{±}^{)}will be large enough to yield a XDMR signal easily detectable at microwave pumping frequencies.

Still very little is known regarding AFMR in crystals involving 5_{2}ReBr_{6} (_{N}_{2} are easily available.

XDMR is a novel spectroscopy which is still in an early stage of development often dominated by severe instrumentation problems. In this review, we tried to convince the reader that XDMR could develop as a unique tool to study dynamical aspects of orbital magnetism including collective excitations. For the first time, direct experimental evidence was produced of the forced, elliptical precession of orbital magnetization components ^{(}^{ℓ}^{)}. There is no doubt left that locally, ^{(}^{ℓ}^{)}can couple to magnetostatic spin waves through dipole-dipole interactions but there is no experimental result which would definitively prove that ^{(}^{ℓ}^{)}could similarly couple to exchange spin-waves. On the other hand, we pointed out strong anomalies in the XDMR spectra which support our view that orbitons may contribute to new relaxation mechanisms. Several important questions were left aside, e.g., the question to know whether, locally, the spin and orbital magnetization components M^{(}^{s}^{)} and M^{(}^{ℓ}^{)}do precess in or out of phase and possibly around different effective axes. Also the whole problem of the magnetoelastic waves resulting from the modulation of the spin-orbit interactions by the lattice phonons will be discussed elsewhere. Nevertheless, in Section 4, we tried to anticipate over the emergence of XDMR as a parallel method to study orbital magnetism in paramagnetic as well as in antiferromagnetic phases.

The authors greatly acknowledge the valuable collaboration of Toshitaka Idehara and Isamu Ogawa (University of Fukui, Japan) in installing and operating a powerful gyroton source temporarily used at the ESRF for test XDMR experiments in the sub-THz frequency range.

^{237}Np

_{81}Fe

_{19}

_{7}]

^{3−}and [Re(CN)

_{8}]

^{3−}

_{7}]

^{3−}

_{3}

_{8}: A comprehensive test for the anderson-weiss model

_{2}

_{2}

_{3}

_{3}caused by the antisymmetric exchange interaction: Antiferromagnetic resonance measurements

_{3}crystal

_{2}O

_{3}

(_{6} at the Fe (24d) and Fe (16a) crystal sites in YIG: opposite signs were expected due to the antiferromagnetic coupling of the Fe cations in (24d) and (16a) sites; (

XDMR in transverse pumping mode with either longitudinal or transverse detection geometries for near perpendicular magnetization. The precession cone angle _{0} never exceeds a few degrees and was exaggerated for clarity. As compared to _{⊥}, _{z}

(

(

(

Complex vector XDMR spectra recorded in transverse detection geometry (TRD) with a thin single crystal platelet of GdIG cut parallel to the (110) plane: (_{cp}_{cp}

(_{max}

(_{2}_{,}_{3}-edges for a powdered pellet of _{7}](Bu_{4}N)_{3}. Note that the XMCD signal measured at the Re L-edges is much stronger than at the Co K-edge.

Precession modes in an antiferromagnet with an easy axis of anisotropy: (_{1} and _{2} are unequal in configurations (