research papers
Linear polarization scans for resonant Xray diffraction with a doublephaseplate configuration
^{a}European Synchrotron Radiation Facility, BP 220, 38043 Grenoble Cedex 9, France
^{*}Correspondence email: scagnoli@esrf.eu
An invacuum doublephaseplate diffractometer for performing polarization scans combined with resonant Xray diffraction experiments is presented. The use of two phase plates enables the correction of some of the aberration effects owing to the divergence of the beam and its energy spread. A higher rate of rotated polarization is thus obtained in comparison with a system with only a single retarder. Consequently, thinner phase plates can be used to obtain the required rotated polarization rate. These results are particularly interesting for applications at low energy (e.g. 4 keV) where the absorption owing to the phase plate(s) plays a key role in the feasibility of these experiments. Measurements by means of polarization scans at the uranium M_{4} edge on UO_{2} enable the contributions of the magnetic and quadrupole ordering in the material to be disentangled.
Keywords: Xray diffraction; magnetic scattering; phase plate; polarization.
1. Introduction
Resonant Xray diffraction has become in the last decade a powerful tool for the study of charge, magnetic and orbital et al., 1988; Murakami et al., 1998). By tuning the energy of the incident photons to a given it provides sitespecific information and also, in general, an enhancement of the observed signal of several orders of magnitude. The most striking examples are the resonances at the uranium M_{4} and M_{5} edges (Isaacs et al., 1989). As a result, even usually weak signals such as those from Xray magnetic scattering can be measured and give precious and complementary information to that obtained by neutron scattering (Scagnoli et al., 2006; Fabrizi et al., 2009).
in a wide variety of physical systems (GibbsIt is not always straightforward to establish the source of the resonant diffracted intensities, however. Several contributions might be present at the same time, such as the anisotropy of the tensor of susceptibility (Templeton & Templeton, 1982). In addition, different resonant events, e.g. dipole–dipole (E1–E1), quadrupole–quadrupole (E2–E2) and dipole–quadrupole (E1–E2), might be present (Paolasini et al., 1999). In order to extract valuable information and to disentangle different contributions that might be present at the same time, polarization analysis of the scattered beam and scans of the scattering wavevector, the socalled azimuthal scan, can be used. Hill & McMorrow (1996) have given a full account of the polarization dependence of the Xray magnetic scattering length.
While azimuthal scans often provide unique information for disentangling different contributions, they are severely limited by the setup geometry and sample environment. In a conventional setup the diffraction geometry is limited to only two of the four polarization channels. Rotating the polarization of the incident Xrays offers the appealing possibility to access the other two polarization channels.
Recently, a new method of extracting information by means of resonant Xray scattering has been presented (Mazzoli et al., 2007). It consists of studying the polarization dependence of the diffracted intensities as a function of the direction of the (linear) polarization of the Xrays impinging on the sample. Such a scan is referred to as an incident polarization scan or, in short, a `polscan'. A subtle interference between purely dipole (E1–E1) and purely quadrupole (E2–E2) transitions, leading to a phase shift between the respective scattering amplitudes, was observed. This effect may be exploited to disentangle two closelying resonances that appear as a single peak in a conventional energy scan, in this way allowing the different multipole order parameters involved to be singled out and identified. The same method was used in TbMn_{2}O_{5} to refine the spin orientation of the terbium ion (Johnson et al., 2008).
Resonant Xray diffraction is mostly employed to study 3d transition ions and 4f systems. Thus the energy range generally used is between 3 and 10 keV. In this energy range the most effective way to rotate the Xray polarization is to use as a phase plate a perfect crystal close to a Bragg reflection (Golovchenko et al., 1986; Hirano et al., 1991). theory predicts that the plate shifts the relative phases of two orthogonal polarization components. The phase shift Δφ depends on the thickness of the plate and on the offset from the Bragg position. The main drawbacks are the imperfect collimation and chromaticity which lead to some depolarization. This can be reduced by using a thicker plate which in turn may strongly absorb the beam. Even using diamond, the lightest perfect crystal, at very low energies the plate thickness represents a major constraint to the feasibility of experiments. Absorption might dramatically reduce the signal below the detection threshold. In an effort to minimize the plate absorption, two diamonds, instead of a single one, can be used. The two retarders compensate one another in some of the depolarizing effects. With the same equivalent thickness a better rate of rotated polarization is thus obtained (Okitsu et al., 2001). Therefore, the same rotated polarization rate can be achieved using two thinner diamonds.
To summarize, for resonant Xray diffraction experiments with incident polarization scans it is important to achieve the best performance out of the phase plates by minimizing their absorption and enhancing the rotated polarization rate. These two requirements are both met by the combined use of two phase plates.
The layout of the paper is as follows. §2 introduces the and describes how to perform a polarization scan by means of diamond phase plates. §3 illustrates the advantages of using a double phase plate compared with a setup with a single phase plate. In §4 we describe the invacuum diffractometer designed to align the two phase plates and to perform polarization scans. §5 gathers the results obtained at different energies and quantifies the different performances of a single and a doublephaseplate setup. §6 describes a practical application: polarization scans, performed with a pair of 0.05 mmthick diamonds, are employed at the uranium M_{4} edge in UO_{2} to disentangle the magnetic and quadrupolar ordering which take place below the antiferromagnetic ordering temperature. Finally, §7 contains our conclusions.
2. and polarization scans
The polarization state of an Xray beam is most easily described in the Poincaré–Stokes notation (Blume & Gibbs, 1988; Born & Wolf, 1999). The P_{1}, P_{2} and P_{3} describe the rate of linear polarization in the horizontal plane, in a plane rotated by 45° around the beam with respect to the horizontal, and circular polarization, respectively. For a plane wave with electric field amplitudes E_{h} and E_{v} in the horizontal and vertical planes, respectively, the are given by (Fano, 1957; Blume & Gibbs, 1988)
We define the density matrix
where I = E_{h}^{2} + E_{v}^{2} is the intensity of the beam, P = (P_{1}, P_{2}, P_{3}) is the Stokes polarization vector, and σ represents the Pauli matrices
Note that this is not the standard convention for the Pauli matrices.
The natural horizontal polarization of synchrotron radiation from a planar undulator source is described by
i.e. (perfect) linear polarization in the horizontal plane. Let M describe the diffraction process of an Xray optical element, such as a phase plate, a sample or a polarization analyzer. The density matrix of the scattered beam is then given by
The transfer matrices for Thomson, nonresonant and resonant magnetic scattering have been discussed in detail (Blume & Gibbs, 1988; Hill & McMorrow, 1996).
We recall that the relative phase shift, Δφ, in the transmitted beam between the Xrays with polarization perpendicular (σ) and parallel (π) to the of the phase plate is given by Giles et al. (1994),
where F_{h} and θ_{B} are the and the of the excited reflection, r_{e} is the classical electron radius, V is the volume, λ is the Xray wavelength, and t is the thickness of the crystal. Δθ represents the misalignment relative to the Bragg condition. It is therefore desirable to use thin crystals (to minimize absorption), and not to work too close to the Bragg condition, i.e. at Δθ much larger than the divergence of the beam and the mosaicity of the crystal. At a given wavelength, one thus has to select a crystal with a large per unit volume, F_{h}/V, small mosaicity, and low Xray absorption. Highquality diamond crystals perfectly fulfil these requirements. In particular, the [111] reflection gives the best results as it has the largest value for the structure factor.
For an Xray phase plate with vertical
the matrix is given bywhere Δφ is given by (9). Upon rotating the through χ around the beam, the matrix transforms as
where
The resulting polarization parameters are
We note that for all Δφ and χ the beam is perfectly polarized [ = 1]. Furthermore, for Δφ = 0, ±π, ±2π,…, the beam is linearly polarized (P_{3} = 0). In this setting the phase plate may be used to rotate the plane of polarization around the beam. The direction of the photon polarization is rotated by an arbitrary angle ζ = 2χ (Fig. 1). For the production of circularly polarized Xrays (Δφ = ±π/2), following (15) the has to be rotated around the beam to form an angle χ = 45° with the plane of the synchrotron and Δθ_{circ} ≃ 2Δθ, where Δθ represents the misalignment relative to the Bragg condition for rotated linear light.
3. Double phase plates compared with a single one
In this section we show that a doublephaseplate device can be operated in four modes, one of them equivalent to a single plate and each of the other three compensating for two of three defects, which are the divergences in two perpendicular directions and the achromaticity (energy dispersion).
The doublephaseplate diffractometer can be operated in different modes: let the angle between the beam and the lattice planes of a plate be
The sign of the angular offset Δθ from the θ_{B} is the sign of the phase lag of the π polarization relative to the σ polarization. We observe that rotating χ, i.e. the azimuth of the plate about the beam, by 180° is equivalent to a sign change in θ [but not in Δθ defined in (16)]; and rotating χ by 90° exchanges the σ and π polarization directions. Consider two phase plates, subscripted 1 and 2, of equal thickness, tuned at offsets Δθ_{1} = Δθ_{2} = Δθ. They produce the same phase lag, and the signs of Δθ_{1} and Δθ_{2} should be such that those lags add. In the first configuration χ_{1} = χ_{2}, θ_{1} = θ_{2}, then in the three others χ_{2} is rotated by 90° in sequence while leaving χ_{1}, θ_{1}, θ_{2} unchanged. Call these configurations (I), (II), (III) and (IV) (Fig. 2). Configuration (I) is equivalent to one single phase plate whose thickness is the sum of both.
In configuration (III) (χ_{1} = χ_{2} + 180°), the σ and π directions are again common to both plates, and, with Δθ being the same, the phase lags add as in (I), but, when referred to a common axis, θ_{1} and θ_{2} are now opposite. A ray inclined by δθ in the common diffraction plane sees the lattice planes at the angles
The offset errors ±δθ are opposite and so are the phase lag errors, which cancel to first order. A ray inclined in the perpendicular direction has no θ offset to first order. Therefore all angular deviations are ineffective to first order. If a ray with the correct direction has an energy with a = θ_{B} + δθ, we may substitute for θ_{B} in (16),
This defect produces phase lag errors from both plates, which add. It is not compensated for.
In configurations (II) and (IV), in order to add phase lags, and because the σ and π directions are exchanged between the two plates, the signs of angular offsets should be opposite,
Now a ray at a different energy undergoes two opposite phase lag errors which compensate to first order. In (II) a ray deviating by δα in the horizontal plane in Fig. 2 travels at θ angles
so that the errors in offsets are opposite and compensate each other. A ray deviating by δβ in the perpendicular direction travels with angular errors whose relative signs are changed from the above and do not compensate. In (IV) the role of both directions is exchanged: compensation is therefore a deviation as δβ but not for one as δα. The results are summarized in Table 1, where a (+) sign indicates a compensation and a (−) sign indicates no compensation.

The discussion began with θ_{2} = θ_{1} in configuration (I). If instead the initial setting is θ_{2} = −θ_{1}, the properties of (I) and (III) are simply exchanged, as those of (II) and (IV). A further effect should be accounted for, the correlation between chromaticity and angular deviation in the vertical plane, produced by the monochromator. This correlation is modified, and its sign possibly inverted, by any focusing element inserted between the monochromator and the phase plates. This is not discussed here.
The compensation of defects was demonstrated as effective (Okitsu et al., 2001). A preliminary experiment was also performed on ID20 (Giles et al., 1997), showing at 10.44 keV an improvement of the polarization, in the extreme case, from = −0.6 to = −0.8.
4. Doublephaseplate invacuum diffractometer
To maximize the amount of rotated and transmitted light, we have developed an invacuum doublephaseplate diffractometer, following the results of Okitsu et al. (2001). It is currently installed at the magnetic scattering beamline ID20 (Paolasini et al., 2007) of the European Synchrotron Radiation Facility, Grenoble, France.
Fig. 3 shows a schematic view of the diffractometer. It is composed of six circles (θ_{1}, θ_{2}, χ_{1}, χ_{2}, α and μ) plus two motorized translations (x and z) in order to align the two retarders with suitable accuracy.
The fine positioning of the two diamond phase plates (to ensure the correct phase shifts, Δθ_{1} and Δθ_{2}) is obtained by use of two Newport Microcontrole URS100 rotation stages with resolution 0.2 mdeg (0.72 arcsec). The rotation stages are equipped with Renishaw encoders having a resolution of 0.1 mdeg (0.36 arcsec). The rotation of the direction of the polarization is obtained by two concentric Huber rotation stages (model 410 for the first phase plate and 408 for the second). Therefore, the two phase plates can be rotated about the beam direction independently by sweeping angles χ_{1} and χ_{2}. This degree of freedom plays an important role as different doublephaseplate configurations are possible in order to correct, for each configuration, some but not all of the aberrations present in the Xray beam (see §3). The α, μ, x and z are needed in order to have the χ_{1}, χ_{2} axes of rotation exactly parallel to the beam direction. There are two translations (x and z) which form a lefthanded orthogonal reference system with the beam direction (y). Finally two rotations, combined with the translations, enable the rotation axis defined by the two Huber rotating stages to be placed parallel to the beam direction.
The diamond phase plates have polished (110) faces. They are aligned in the sample holder in a (110)– zone, so that the reflection can easily be used in the symmetric Laue geometry.
Diamonds of different thickness are currently available: 2 × 0.05 mm, 2 × 0.1 mm, 2 × 0.2 mm, 0.3 mm, 0.4 mm, 0.5 mm, 0.72 mm, 1.2 mm. The phase plates (except 0.72 mm and 1.2 mm) were made from locally dislocationfree highly pure (nitrogen content less than 0.1 p.p.m.) typeIIa highpressure hightemperature material, manufactured by Element Six Technologies, Johannesburg, South Africa.
Fig. 4 shows whitebeam topography of the new highquality diamond compared with an old phase plate.
4.1. Thickness of the phase plate
When selecting the material, thickness and Bragg reflection of the phase plate for a given working energy, one has to compromise between absorption and the quality of polarization. The absorption depends on the photon energy and material only and is thus straightforward to calculate. The quality of polarization depends on more parameters: the use as quarter or halfwave plate (i.e. phase shift Δφ = π/2 or π), the divergence and bandwidth of the incident beam, and the crystal quality, see (9). One way of quantifying this problem is to specify the desired working point Δθ, and to calculate the corresponding thickness,
and compare it with the attenuation length. Modifying the working point Δθ or the required phase shift Δφ multiplies the entire curve by a corresponding factor, but does not change the energy dependence.
As an example, we compare diamond, silicon and germanium quarterwave plates (Δφ = π/2), choosing a working point at Δθ = 0.02° (see Fig. 5). Except for the lowest photon energies E < 3.5 keV, C* (diamond) offers the best performance. At 4.35 keV, a phase plate of thickness 80 µm gives an Xray path length in the crystal of ∼120 µm, compared with an attenuation length of ∼100 µm. Note that the optimum thickness for a silicon or germanium phase plate below 3.5 keV is well below 10 µm. Beryllium has a high potential as phase plate material (Giles et al., 1995). To date, however, single crystals of sufficient quality are not widely available. For a given material, e.g. diamond, the choice of the Bragg reflection remains. We find that for low energies the (111) reflection gives the best performance, whereas for higher energies the (220) reflection performs slightly better, as the smaller is compensated by the larger Bragg angle.
5. Results
In the present work we present systematic checks with various energies and plate thicknesses. Three different Xray incident energies were chosen: 3.720, 5.570 and 7.200 keV, which lie close to the uranium M_{4}, vanadium K and iron K edge, respectively. In order to measure the rotated polarization rate for each energy the appropriate polarization analyzer crystal was selected [Au(1 1 1), graphite (0 0 4), MgO (2 2 2), respectively]. Incoming Xray energies match Bragg's law for the polarization crystal dspacings in order to have the equal to 45° to fully suppress the polarization component in the polarizer scattering plane.
The polarization state of the Xrays after the phase plate(s) is evaluate by means of
where I_{0} and I ′ represent the intensity of the Xrays before and after the phase plate(s), η represents the rotation of the analyzer crystal about the incoming beam direction (Fig. 1). and are two of the three introduced previously (primes here indicate that they are evaluated after the retarders). Several rocking curves of the polarizer crystal are collected as a function of η. The integrated intensities are then used to estimate I_{0}, and .
The estimated values of the rotated linear polarization (socalled halfwave mode) for the different photon energies and phase plate configurations are given in Tables 2–5. Configuration (I), equivalent to a single phase plate, gives the poorest rotated polarization rate for all the different incident energies. Configurations (II) and (IV) provide the best results with an improvement in the transmitted polarization rate of roughly 10%. The main error corrected by the presence of the two phase plates is the energy spread of the Xrays coming from the monochromator (see Table 2).




This is illustrated in Fig. 6. Plate 1 is fixed as outphasing by π/2 and transforming the linear polarization into circular. The θ angle of plate 2 is scanned. When it passes through positions outphasing by ±π/2 the linear polarization is either rotated by π/2 (at the minimum of ) or restored to its initial direction (at the maximum). The data are intensities measured in the unrotated channel, see (22) with η = 0, suitably normalized and shifted. Cases (I) and (IV) are shown, together with fitted curves. The fitting function is obtained from , which depends on Δθ through (9) and (13), by the following procedure. Equation (9) is extended to represent the total outphase from plates 1 and 2, so that is now a function of Δθ_{1} and Δθ_{2} (see details in Appendix B). That function is convoluted with two resolution functions in (Δθ_{1} + Δθ_{2}), (Δθ_{1} − Δθ_{2}). Two angular spreads are then used, just one being compensated. Since this calculation goes beyond the first order considered in §3, some small depolarization arises even from the compensated spread. In configuration (I) the compensated spread is forced to 0, while the uncompensated one, owing to all effects, δE, δα, δβ (see Table 1), is 0.0025 (2)° root mean square (r.m.s.). In configuration (IV), the uncompensated δα is 0.0015 (2)° r.m.s. and the compensated (δE^{2} + δβ^{2})^{1/2} is 0.0020 (4)° r.m.s. Fig. 7 shows the measured in configuration (IV), with Δθ_{1}, Δθ_{2} adjusted to the total halfwave shift and rotating χ_{1} and χ_{2} together (ζ = 2χ_{1}, see end of §2). Fig. 8 compares the singleplate configuration with the best result obtained with the twophaseplate configuration. In the examined energy range the doublediamond setup provides a 10% improvement compared with a singlediamond system.
In order to underline the importance of this result, which at first glance might not justify the effort of realising a more complicated setup, it is interesting to calculate the thickness required by a singlephaseplate system to produce a 10% higher polarization rate. It is shown in Appendix A that the depolarization produced by a phase plate is inversely proportional to the square of its effective thickness. Using the value for in Table 2, taken at 3.720 keV, the required thickness t_{ I}^{*} to have the same polarization rate as configuration (IV) with thickness t_{IV} would be
t_{ I}^{*} is obtained considering that given the polarization rate corresponding to the thickness t_{I} = t_{IV} we are looking for the thickness (t_{ I}^{*}) that will produce the wanted polarization rate . The absorption factor for that t_{ I}^{*} would be roughly 500, about the square of the absorption factor 17 occurring for t_{IV}. Therefore, the use of the double phase plates represents a significant advantage.
6. Resonant Xray diffraction on UO_{2}
In order to test the doublephaseplate setup we performed a resonant Xray diffraction experiment on a sample of UO_{2} at the U M_{4} edge (3.728 keV). Below the Néel temperature, T_{N} = 31 K, it assumes a complicated magnetic structure of the 3k variety (Willis & Taylor, 1965; Frazer et al., 1965; Burlet et al., 1986). Important theories have been developed since the late 1960s (Allen, 1968a,b; Sasaki & Obata, 1970; Siemann & Cooper, 1979; Solt & Erdös, 1980) up until more recent times (Kudin et al., 2002; Laskowski et al., 2004; Magnani et al., 2005). All these theories emphasize the importance of the interplay between the Jahn–Teller and quadrupolar interactions in UO_{2}. The evidence for the ordering of the quadrupoles was provided recently by resonant Xray diffraction at the uranium M_{4} edge (Wilkins et al., 2006).
Following Wilkins et al. (2006) we write the resonant diffraction amplitude for an electric dipole (E1) event as
where the terms f_{n} are given by the following equations,
F_{1q} is the resonant energy factor (Hill & McMorrow, 1996), is the direction of the and is a tensor of rank two. and represent the direction of the incident and diffracted photon polarization, respectively.
The term f_{0} represents Thomson (charge) scattering and equals zero for spacegroup forbidden reflections. The term f_{1} probes a tensor of rank one, with odd timereversal symmetry arising from a net spin polarization. The term f_{2} represents a traceless symmetric tensor of rank two that can arise from an asymmetry intrinsic to the (Templeton scattering or anisotropic tensor susceptibility) or it can be due to antiferro order of electric quadrupole moments. It possesses a timeeven symmetry. It can be shown (Wilkins et al., 2006) that the reflections arising from quadrupolar ordering coincide with those due to magnetic dipole ordering. The experimental challenge for Wilkins et al. (2006) was therefore to separate the two contributions: magnetic dipole and electric quadrupole. They achieved this result by means of diffracted polarization analysis. Indeed, for σpolarized incident Xrays (the polarization direction is perpendicular to the diffraction plane) all the scattering from the magnetic structure is in the rotated channel σ–π′, while the signal from the electric quadrupole might be present in both rotated σ–π′ and unrotated σ–σ′ channels.
In order to test the doublephaseplate setup under `real' conditions we performed first the same experiment as carried out by Wilkins et al. (2006) by measuring the azimuthal dependence (scan about the scattering wavevector) of the (112) forbidden reflection in both σ–π′ and σ–σ′ channels. Then we took advantage of the possibility to change the direction of the incident photon polarization. The goal of these measurements was not to provide better evidence for the quadrupole ordering but to check the sensitivity of the method, given the remarkable difference in the strength of the two signals.
Results from the azimuthal scan follow the expected dependence and are displayed in Fig. 9. The contributions of two transverse domains are taken into account as an incoherent sum of intensities. We show the ratio σ–σ′ over σ–π′ here as first it gives a straightforward estimate of the magnitude of the two resonant processes. This will play an important role in the analysis of the polarization scans. Secondly, some systematic errors such as the change in the absorption of the sample and a partial illumination of the sample, which might occur as a function of the azimuthal angle, are eliminated by the ratio. These errors are normally significant for offspecular reflections.
The use of phaseplate polarimetry and polarization scans enables the experimenter to eliminate such errors. The measurements are performed with the sample at rest; the sample does not move and only the direction of the incident polarization together with the position of the analyzer crystal is changed. Moreover, polarization scans are very sensitive to the simultaneous presence of two sources of scattering that may (Mazzoli et al., 2007) or may not, as we will see with UO_{2}, interfere. In this section, unprimed values for P_{1} and P_{2} refer to the describing the polarization state of the beam after the phase plates, and primed values refer to the polarization state of the beam after being diffracted by the sample (see Fig. 1).
We have selected three different positions in azimuth, ψ = −95°, ψ = −11° and ψ = 32°. At each position a polarization scan was performed. ψ = −95° was chosen as no quadrupolar signal is expected, ψ = −11° is where the quadrupolar scattering is maximum and ψ = 32° is where the ratio between the quadrupole and the magnetic signal is expected to be maximum (compatible with our experimental geometry limitations). Results are presented in Figs. 10, 11 and 12 for the three azimuthal angles, respectively. For each direction of the polarization (represented by the angle ζ) the analyzer stage is rotated discretely (angle η) and the polarization analyzer is rocked. The set of integrated intensities obtained by the latter scans are subsequently fitted using (22). The fit enables the extraction of the and The experimental values of the are then compared with their expected values. The latter can be readily calculated once the magnetic and quadrupolar structure factors are known. An instructive example on how to perform such a calculation can be found by FernándezRodríguez et al. (2008). For ψ = −95°, no quadrupole contribution is expected and therefore all the observed intensity comes from magnetic scattering. Comparing predictions with experimental results is straightforward. Indeed the agreement is rather good as can be seen in Fig. 10.
Now we turn our attention to the other two polarization scans which are illustrated in Figs. 11 and 12. At first we tried to fit the data only with the magnetic contribution (continuous line). The agreement with experimental results is already remarkable. Only minor differences are visible. However, these tiny differences are exactly the signature of the presence of the quadrupolar ordering. Including the quadrupolar contribution to the calculated scattering amplitude leads to an improvement in the description of the experimental data. The model calculation including both contributions (without interference) is represented in the figures by a dashed line. One possible explanation for the lack of interference is that this term may be cancelled between two domains. The calculations are performed by utilizing a single parameter, which represents the ratio between the quadrupole and magnetic contribution in the scattering amplitude. In our case this parameter has been determined by the azimuthal angle scan illustrated in Fig. 9. The small difference between the two models reflects the weakness of the quadrupole scattering compared with the magnetic one. From Fig. 9 one can readily estimate that the contribution of quadrupole scattering barely exceeds 4% of the magnetic signal. Nevertheless, such a small signal gives a sizeable contribution in the polarization scans.
To conclude, with the use of polarization scans we have disentangled the magnetic and quadrupole contributions of the diffracted intensity. This result proves important as it shows the sensitivity of this technique, even in the case where one signal is only a few percent of the other.
7. Conclusions
The use of a doublephaseplate setup produces a 10% higher rotated polarization rate compared with a system with a single phase plate of equal total thickness over the energy range examined (3.7–7.2 keV). These results are particularly interesting for resonant Xray scattering applications, especially at the uranium M edges, where, to obtain the same rate of rotated polarization, the transmitted beam is attenuated 50 times less thanks to the presence of the two retarders.
In order to test the setup in the most interesting energy range, we have performed a resonant Xray scattering experiment on UO_{2} at the uranium M_{4} edge. By taking advantage of the possibility to rotate the direction of the incoming photon polarization we show that it is possible to disentangle the quadrupole and magnetic diffraction contributions. Therefore, polarization scans might be regarded as a complementary and/or alternative means to investigate samples by use of Xray diffraction. Whilst this approach shows all its benefits when interference occurs between two scattering events, this is not the case for UO_{2}.
APPENDIX A
Phase plate thickness and depolarization effects
If we recall the expression for given in (13), with φ = Δφ to ease notation, we have
to have linear light rotated by 90°, where χ will be ±π/4 and φ* = −π. If we expand in the vicinity of φ* we have
where δφ represents a distribution about the actual setting φ*. Then,
Now if we rewrite (9) as φ = kt/Δθ with k constant and we differentiate on both sides, we obtain
Looking back at (30) we obtain
which shows that the degradation in the produced rotated light is inversely proportional to the square of the effective thickness of the retarder.
APPENDIX B
Convolution of angular spreads
The aim is to calculate the Stokes parameter for a beam made of angularly spread rays, after going through two phase plates. Two approximations are made: first, the δE and the vertical deviation δβ is ignored, though such a correlation is produced by the monochromator. Equation (9) is rewritten for two plates as
close to the is neglected; second, the correlation between the energy deviationK_{1} is different from K_{2} if the plate thicknesses are different. Through (13) and (33) is a function of (Δθ_{1}, Δθ_{2}) and, since the total intensity is invariant for small Δθ, the value of is obtained by summation of its values for individual rays. We represent the beam as a Gaussian distribution in (Δθ_{1} + Δθ_{2}) and (Δθ_{1} − Δθ_{2}),
where s and d are r.m.s. spreads in the two directions. The sum of Δθ is not compensated whereas the difference is. They represent deviations along horizontal δα, vertical δβ and energy δE, combined as indicated in Table 1; energy is converted to angles through Bragg's law. The integration of over the resolution function (34) is performed numerically.
Acknowledgements
We wish to thank R. Caciuffo for fruitful discussions and for providing the UO_{2} sample. We also thank J. Hartwig who provided the phase plates and carried out their topographic characterization.
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