non resonant Gd resonance Eu resonance
5. Non-resonant magnetic x-ray diffraction
Resonant magnetic x-ray scattering has the convenience of easily detectable signals due to the resonance enhancement and provides element specific information. Therefore many magnetic x-ray diffraction studies nowadays deal with resonance exchange scattering. However, for transition metal ions, only K-edges lie in the range of hard x-ray wavelengths, where atomic resolution is achievable. Due to the dipolar and quadrupolar selection rules and the small overlap between core 1s-states and the magnetic sensitive 3d or 4p energy bands, resonance enhancements are negligible at transition metal K-edges and one is left with neutron- or non resonant magnetic x-ray scattering. These techniques have the advantage that they measure directly the order parameter, in contrast to resonant scattering, where transition matrix ele-ments are involved, which are not known a priori. Moreover, the form of the cross section (6) suggests that spin S and orbital L angular momentum can be determined separately by means of polarisation analysis. Take the example of a collinear antiferromagnet for which charge – and magnetic reflections are well separated in reciprocal space and for which we can align the moments along the û2 – axis, i.e. S(Q) = (0,S(Q),0) and L(Q) = (0,L(Q),0). At the antiferro-magnetic reflections charge scattering vanishes and, according to (6), antiferro-magnetic scattering only occurs in the σ→σ’ and the π→π’ polarisation channels. In the former, scattering is sensitive to S(Q) only, in the latter, it is sensitive to a linear combination of S(Q) and L(Q), thus allowing a unique determination of the ratio between both contributions. While such a model independent separation of S and L cannot be achieved with neutron scattering due to the fact that the neutron cross section is proportional to L+2S, it provides very important information e.g. to verify band structure calculations. Such a separation of S and L by means of polarisation analysis of non-resonant magnetic x-ray diffraction has for example been done in Ho [5], NiO [20] or Cr [21].
When discussing non-resonant magnetic x-ray diffraction, one has to start with the beautiful pioneering work by Gibbs et al. [4] and Bohr et al. [34] on Ho metal. Together with the fol-lowing work [35], this is one of the most comprehensive studies of magnetic x-ray scattering so far undertaken. The basic magnetic structures in the heavy rare-earth metals are known for
quite a long time from the pioneering experiments by Koehler et al. The results are illustrated in figure 18.
Fig. 18: On the left, the hexagonal close packed hcp crystal structure realised in the heavy rare-earth metals is illustrated. On the right, the magnetic structures of the heavy rare-earth metals are sketched. From left to right: basal-plane ferromagnet, helix structure, cone struc-ture and the c-axis modulated CAM strucstruc-ture.
The moments in any particular hexagonal base-layer are parallel and only the relative align-ment of different planes, when propagating along the hexagonal c-direction are illustrated. In Holmium, a helix structure is realised below the Néel temperature of 132 K down to 20 K, where a cone structure develops. In the high temperature region, the modulation period in-creases continuously with decreasing temperature, which indicates that the helix is incom-mensurate with the crystal lattice, i. e. the ratio between magnetic period and lattice period cannot be expressed as a simple rational number. At 20 K, a lock-in transition occurs and the magnetic modulation becomes commensurate with the chemical lattice with a period of 6 unit cells, i. e. 12 atomic layers, corresponding to τ = 1 . The beauty of the x-ray experiment was 6 to show that much of what was originally thought to be an incommensurate phase in fact contains a series of subtle lock-in transitions. These details could be revealed because of the high wavevector resolution of 10-3 Å-1 for the synchrotron x-ray experiments. Subsequently, these observations have led to a new model of magnetism in Holmium and other rare-earth
sume that the helix axis is aligned parallel to the scattering vector Q as was the case in the experiments by Gibbs et al. Then, due to the paramagnetic modulations, satellite peaks de-velop on either side of the charge Bragg peaks. Figure 19 shows the original measurement by Gibbs et al. in the vicinity of the 0004 Bragg peak in Ho.
Fig. 19: Scattering in the vicinity of the 0004 Bragg peak in Holmium at 17 K. The open circles show the scattering without polarisation analysis, while the filled circles show the scattering observed with a polarisation analyser aligned to observe only σ-π' scattering.
Without polarisation analysis two peaks are visible: one sharper satellite reflection and one, which is significantly broadened. Both reflections occur at rational positions indicating a commensurate structure. According to (6) and (7), magnetic scattering can be identified with the help of a polarisation analyser, aligned to observe the off-diagonal σ-π' scattering. This experiment is shown by the filled circles. Obviously, only the sharp peak is of magnetic ori-gin, while the broader feature has to be assigned to charge scattering. In the Appendix, we also derive the charge scattering, due to magnetoelastic interactions for a spiral structure. The result is that for a pure spiral structure, no induced strainwave should occur. To resolve the puzzle, we can look at the temperature dependence of the magnetic modulation vector in hol-mium, deduced from x-ray and neutron data (see figure 20). With the high resolution x-ray
data, several lock-in transitions as well as a hysteresis behaviour can be observed at low tem-perature. Figure 21 shows several scans in the low temperature range.
Fig. 20: The temperature-dependent magnetic modulation wavevector of holmium from x-ray and neutron data according to [4].
Fig. 21: Magnetic and lattice modulation diffraction peaks in holmium at various temperatures (taken from [34]). The diagram on the right illustrates the model for magnetic ordering that can
describe the observations.
A consistent model that can describe these observations assumes the existence of discommen-surations. The spin slip structure depicted on the right of figure 21 results from a competition between exchange energy and the sixfold basal plane anisotropy. According to this model, the basal plane anisotropy determines the orientation of the magnetic moment of two adjacent layers, which adopt a parallel magnetisation. The magnetisation of the next double layer is rotated by 60° and lies again along an easy in-plane axis. This pattern repeats until at one point exchange energy wins and a spin slip is introduced with only one layer and not a double layer pointing along the easy axis. The change in magnetoelastic coupling at spin slip sides causes a small distortion of the chemical lattice, with a period equal to the spin slip period λS. As fig. 21 shows, the magnetic period λM and the spin slip period λS are different, giving rise to the two peaks observed. While the magnetic period is long range ordered, the broad charge peak indicates that the spin slip structure is short ranged with a finite correlation length. With this spin slip model, Bohr et al. [34] could calculate the allowed wavevectors for magnetic and spin slip scattering and there is a remarkable agreement with the experimental data. We do not want to discuss this example in more detail, but conclude that the high resolution of magnetic x-ray scattering compared to magnetic neutron scattering immediately gave rise to a much deaper understanding of magnetic structures and interactions in rare-earth metals.
The second example we want to discuss deals with the separation of spin and orbital angular momentum in the monoxides of 3d transition metals [36] and we have selected in particular the example of NiO [20]. Why are we interested in these very simple compounds? After all, MnO was the first material for which the antiferromagnetic structure has been determined with neutron scattering, a main corner-stone for the nobel prize of Cliff Shull. The reason is that electron correlation effects become particularly obvious in the transition metal oxides.
Prominent examples of systems, which cannot be described in a simple Fermi liquid picture are the high TC cuprates or the colossal magnetoresistance manganites, which show some very interesting and unexpected behaviour, due to electron correlations. MnO, FeO, CuO, CoO and NiO can thus be seen as test samples for band theory models. In the most simple electronic band model, the oxygen p-states are fully occupied, while the metal s-states are empty and the d-states are partially occupied. With such a picture, one would expect a partially filled band at the Fermi level, i. e. the oxides would be expected to be conductors. In reality, they show insulating behaviour, which cannot be reproduced by simple density
functional theory. Moreover it is common believe that the strong crystal field on the 3d orbitals leads to a suppression of the orbital momentum, the so-called quenching. For these reasons, it is of interest to perform a separation of spin- and orbital moment magnetisation density in NiO. We have already mentioned that this is possible, when discussing the amplitude (6) of non-resonant magnetic scattering. How this is done in detail for NiO will be calculated in our exercise class. Here, we just want to present the results obtained in [20].
NiO has the NaCl fcc-structure with a = 4.177 Å at room-temperature. The ground state con-figuration of the Ni2+ ion has the 3d8 configuration. Below TN = 523 K, NiO orders in the type II antiferromagnetic structure, where ferromagnetic planes are stacked antiferromagnetically along the [111] axis with their magnetic moments, aligned in the [111] planes, along one of the
[
112]
directions. The magnetic structure of MnO is depicted in figure 22.Ni
Fig. 22: Magnetic unit cell of NiO.
The separation of spin and orbital contributions can be done by means of polarisation analy-sis. Here, we assume that the radiation incident on the sample is fully polarised in the hori-zontal plane. In the actual experiment, performed at the magnetic scattering beamline ID20 at ESRF, the degree of polarisation in the horizontal plane was found to be 99.5 %. As men-tioned in the section on experimental considerations, pyrolytic graphite is an ideal material to be used as a polarisation analyser. In the actual experiment, PG006 was used and the incident beam energy was tuned to 7.84 keV to fulfill the 90° reflection condition at the analyser crys-tal. A crystal was prepared with a [111] face. Specular magnetic reflections (1/2 1/2 1/2), (3/2 3/2 3/2) and (5/2 5/2 5/2) could be measured as a function of the angle of rotation ψ around the scattering vector. The scattered magnetic intensity depends on this angle ψ since during rotation we change the components of S and L, which enter the scattering cross section.
Be-fore we proceed with the discussion of the results, we have to give some explanation about the magnetic domain structure that can develop in such a crystal as it is cooled from the cubic paramagnetic phase into the orthorhombic antiferromagnetically ordered low temperature phase. Since in the cubic phase 4 symmetric equivalent [111] directions exist ([111], [ 1 11], [1 1 1], [11 1 ]) the magnetic propagation vector can align along one of these directions.
Therefore during cooling down, the crystal will be spontaneously develop 4 so-called K-domains corresponding to the symmetry equivalent [111] directions, which can be realised in different parts of the crystal. In addition the magnetic moments are aligned in the [111] planes along one of the [11 2 ] directions. Since in the cubic state, the [111] axis is a three-fold symmetry axis, there exist three equivalent possible spin directions perpendicular to the [111]
axis, the so-called S-domains. Depending on crystal faults (surfaces, small angle grain boundaries, impurity atoms etc.), but also on random processes, an arrangement of these K- and S-domains develops in the sample in the low temperature phase. During the experiment it turned out that the sample just developed a single K-domain with propagation vector parallel to the surface normal, at least in the near surface region of roughly 40 micrometer probed in the experiment. However, when measuring the scattered intensities from a given K-domain, all contributions from the associated S-domains add incoherently. The S-domain distribution within a K-domain can be studied by rotating the sample about the surface normal. Such an experiment is shown in figure 23.
Ψ (deg.)
Fig. 23: Normalised integrated intensities of the 3/2 3/2 3/2 reflection as a function of Renninger angle for the two polarisation states σ→π and σ→σ and the total scattering, respectively.
Ψ
From such experiments, the S
2L ratio can be determined. It is plotted in figure 24 as a function of sinθ/λ.
Fig. 24: Measured variation of L(Q) / 2 S(Q) as a function of sinθ/λ = Q/4π. The continuous line is the dependence estimated from theory, adjusted to fit through the data with a contraction of the wave function by 17 %.
By normalising to the charge reflections, the spin and orbital formfactors can be determined.
They are depicted in figure 25.
Fig. 25: Spin and orbital moment form factors of NiO. Again the continous lines are the calculated variations of S(Q) and L(Q) with an expansion of the Q-scale by 17 %.
Let us summarise the most important results of this investigation: First of all, there is a rather large contribution of orbital moment in NiO. In fact, the extrapolation to Q = 0 gives an ef-fective ratio 0.34
S
L = . This large contribution clearly contradicts the common believe that in such transition metal compounds, the orbital angular moment is largely quenched due to the crystal field. Furthermore, the results also indicate a contraction of the atomic wave function for Ni in NiO, if the experimental values are compared with the predictions of theory. The
atomic wave functions are contracted by 17 % as compared to the free ion. We can conclude that the magnetisation density in a simple system like NiO is not yet fully understood and the results of these studies have to be taken into account, when models for the electronic and magnetic structure and properties of transition metal compounds are being made.
So far, we have discussed the magnetic scattering of x-rays in the "conventional energy range" between 4 and 20 keV. We now want to discuss an alternative method, namely the non-resonant magnetic scattering of very hard x-rays with energies above 80 keV [22-25].
From (9) it follows that with high energy x-ray diffraction one can determine the spin density distribution independent of the polarisation of the incident beam and without analysis of the final polarisation after scattering. While in neutron diffraction only the total magnetic moment, proportional to the sum L+2S is accessible, x-ray diffraction in the conventional energy range requires polarisation analysis to separate the spin momentum S(Q) from the orbital angular momentum density L(Q). The additional principal feature of high energy magnetic x-ray diffraction is the drastic increase in penetration depth. For 3d transition metals, the absorption length 1 µ increases from some µm at 8 keV to several mm at 80 keV.
This leads to a volume enhancement of the signal which is, however, partly compensated by the λ2 term for the reflectivity. Moreover, true bulk properties become accessible, a feature especially important for studies of magnetic disorder phenomena. Magnetic x-ray scattering can be studied in transmission geometry. Corrections for absorption, extinction, beam foot print etc. are simple and therefore, by normalising the intensity of the magnetic reflections to the intensity of the charge reflections, absolute values for the spin moment can be determined [25]. Neutron and photon experiments of bulk properties from the same crystal become possible, where one advantage of the x-ray study is the high intrinsic resolution of about 10-3 Å-1 longitudinal and 2·10-4Å-1 transversal.
Fig. 26: Schematic sketch of the three crystal diffracto-meter for high energy x-ray scattering BW5 at HASYLAB/Ham-burg (http://www-hasylab.desy.de).
Figure 26 depicts a typical experimental set-up. The diffractometer of the beamline BW5 at HASYLAB receives a white x-ray beam from a 2T high field wiggler. Inclined water cooled Cu plates limit the beam dimensions to about 4 x 4 mm2. To reduce the heat load on the opti-cal elements, a water cooled 1 mm thick Cu window absorbs all radiation of energy smaller than 60 keV. For the experiments on MnF2 described below, we employed annealed Si 311 crystals with a mosaic width of 10´´ in Laue (transmission) geometry as monochromator and analyser crystals. The analyser can be used to increase the momentum space resolution and to reduce the background, but should not be employed for the collection of integral intensities.
Iron collimators with a quadratic cross section of 50 x 50 mm2 and a free bore of 10 x 10 mm2 are positioned between monochromator and sample, sample and analyser and analyser and detector to reduce the background. With an energy sensitive Ge solid state detector, the ine-lastic background from Compton scattering and fluorescence can be efficiently suppressed.
The sample is mounted inside a cryostat with Al windows and large tails to avoid Al back-ground scattering to enter the detector.
The potential of magnetic high energy x-ray diffraction can be demonstrated with experiments on the antiferromagnetic model system MnF2 [24]. Its structure is shown in figure 27. With an energy of 80 keV, a high peak count rate of 12.000 counts/sec, a good peak-to-background ratio of 230:1 and an excellent Q space resolution can be obtained for the magnetic 300 Bragg reflection. Figure 28 shows a measurement of the temperature dependence of the sublattice magnetisation. In the critical region close to the Néel temperature TN, the reduced sublattice magnetisation m =M(T) M(T =0) follows very accurately a power law behaviour
m(τ)= D⋅τβ (23)
as a function of the reduced temperature τ =
(
TN −T)
TN. The value of the critical exponent of β=0.333(3) corresponds well to the predictions of the Ising model.Fig. 27: The crystalline and magnetic unit cell of MnF2.
τ
T[K]
Intensity [a.u.]
m
0.1 1
0.0001 0.001 0.01 0.1
β = 0.333(3) T = 67.713(2) KN
0 100 200 300 400 500
0 20 40 60 80
Fig. 28: Critical behaviour of the sublattice magnetisation of MnF2 in a double logarithmic plot in reduced variables. The insert shows the temperature dependence of the intensity of the magnetic 3 0 0 Bragg reflection of MnF2 measured with 80 keV photons.
MnF2 is a classical model antiferromagnet with localised spin moments and therefore is an ideal test material for any new technique in magnetism. The 3d metal Cr on the other hand shows very intriguing magnetic properties. Chromium is an itinerant antiferromagnet exhibiting an incommensurate spin density wave (SDW) below TN = 311 K [26]. Above the spin flip transition at TSF = 123 K the spin density wave is transversally polarised, whereas below TSF the polarisation becomes parallel to the modulation wave vector, which leads to a longitudinally polarised SDW. The SDW gives rise to magnetic satellite peaks at positions
corresponding to the magnetic propagation vector qm. These satellites could be readily meas-ured, despite the small root-mean-squared moment of 0.43 µB, see figure 29. The temperature dependence of the intensity and propagation vector for the 1-δ 0 0 satellite is plotted in figure 30. One can clearly see the spin flip transition at 123 K in a drastic drop of the satellite intensity as the spin moment rotates from an orientation perpendicular to the scattering plane to an orientation within the scattering plane. Therefore at the spin flip transition, the spin component S2 - and according to (9) the Bragg intensity - vanishes.
−120 −80 −40 0 40 80 120
10 20 30 40 50 60
70 T=124.5 K
ωs [arcsec]
Intensity [counts/s]
T=121.6 K T=112.0 K
Fig. 29: Diffraction geometry (left) and 1-δ 0 0 magnetic satellite (right) of Cr. The measurements were done with photons of energy 100 keV. While above the spin flip transition in the transverse SDW, moments perpendicular to the diffraction plane exist, these moments come to lie within the diffraction plane as the SDW becomes longitudinal polarised below TSF.
The main aim of our study of chromium was, however, to determine the relative contribution of spin and orbital angular momentum to the SDW. As discussed above, the spin momentum is directly accessible with high energy x-ray diffraction, while neutrons measure the combi-nation L+2S. Therefore by combining results of measurements of both techniques, spin S and orbital L momentum densities can be determined separately. To this end, we measured 5
The main aim of our study of chromium was, however, to determine the relative contribution of spin and orbital angular momentum to the SDW. As discussed above, the spin momentum is directly accessible with high energy x-ray diffraction, while neutrons measure the combi-nation L+2S. Therefore by combining results of measurements of both techniques, spin S and orbital L momentum densities can be determined separately. To this end, we measured 5