Single-crystal X-ray diffraction and resonant X-ray magnetic scattering at helium-3 temperatures in high magnetic fields at beamline P09 at PETRA III

The resonant scattering and diffraction beamline P09 at PETRA III at DESY is equipped with a 14 T vertical field split-pair magnet. A helium-3 refrigerator is available that can be fitted inside the magnet's variable-temperature insert. Here the results of a series of experiments aimed at determining the beam conditions permitting operations with the He-3 insert are presented. By measuring the tetragonal-to-orthorhombic phase transition occurring at 2.1 K in the Jahn–Teller compound TmVO₄, it is found that the photon flux at P09 must be attenuated down to 1.5 × 10⁹ photons s⁻¹ for the sample to remain at temperatures below 800 mK. Despite such a reduction of the incident flux and the subsequent use of a Cu(111) analyzer, the resonant X-ray magnetic scattering signal at the Tm L_III absorption edge associated with the spin-density wave in TmNi₂B₂C below 1.5 K is intense enough to permit a complete study in magnetic field and at sub-Kelvin temperatures to be carried out.

1. Introduction

Single-crystal X-ray diffraction and resonant X-ray scattering are extremely valuable techniques for studying crystallographic phase transitions and other ordering phenomena such as orbital, multipolar or magnetic ordering as well as charge-density waves. These ordering phenomena usually show peculiar phase diagrams as a function of temperature, magnetic field or pressure and require that X-ray scattering experiments are made possible on samples subject to these external thermodynamic parameters. In the hard X-ray range, recent instrumental trends at third-generation synchrotron sources have been to push the feasibility of X-ray diffraction experiments in very high magnetic fields either using commercial superconducting (SC) magnets delivering permanent fields up to 17 T (Katsumata, 2005; Holmes et al., 2012; Strempfer et al., 2013) or developing pulsed field technology magnets producing peak fields of 30 T and beyond for a few milliseconds (Islam et al., 2012; Duc et al., 2014; Narumi et al., 2012). For what concerns the establishment of extremely low temperatures, a wide variety of cryostats are nowadays commercially available that permit temperatures down to 4.2 K to be achieved during X-ray experiments. Temperatures down to 1.5 K can be reached using special cryostats (Brückel et al., 1996) or SC magnets equipped with a ⁴He variable-temperature insert (VTI). However, while the technology to achieve sub-Kelvin temperatures is well established at neutron scattering facilities, i.e. the use of dilution refrigerators and ³He cold fingers, X-ray studies combined with such sample environments are very seldom at synchrotron or laboratory X-ray sources (Nakajima et al., 1995; Christiansen et al., 1996; Katsumata, 2005). One of the reasons for this is the low penetration depth of X-rays which imposes windows to be installed made out of low-absorbing but unfriendly materials like beryllium. Another reason is that local heating of the sample already occurs at temperatures below 100 K due to the high heat load of synchrotron X-ray beams. This imposes the photon flux to be reduced by one or two orders of magnitude depending on the incoming X-ray flux and depending on whether exchange gas is present or not in order to remain at base temperature. In the sub-Kelvin range, higher attenuation factors are expected.

In this article we describe principles of operation of the ³He insert at beamline P09 at the PETRA III storage ring and show that resonant X-ray magnetic scattering experiments are possible down to 360 mK in the hard X-ray range at a third-generation synchrotron source.

2. Helium-3 insert

Beamline P09 at the PETRA III storage ring is dedicated to resonant scattering and diffraction experiments in the hard X-ray range between 2.7 and 24 keV (Strempfer et al., 2013). It is equipped with a 14 T vertical field split-pair magnet that can be mounted on the heavy-load non-magnetic Huber six-circle diffractometer present in the second experimental hutch. The magnet is equipped with a ⁴He VTI operating at temperatures between 1.5 and 300 K and with successive beryllium and mylar windows allowing full optical access in the horizontal scattering plane. The closed-cycle ³He fridge is a top-loading 2 m-long insert custom-designed by Cryogenic Ltd to fit inside the 2.5 cm-diameter bore of the VTI. The sample holder is attached at the bottom of the ³He pot and cooled down by conduction. The sample holder and ³He pot are enclosed in an inner vacuum chamber (IVC) while the mini and main sorption pumps lie in the main VTI space. The IVC is equipped with a 1 mm-thick beryllium window over 360° opening and 28 mm height which makes it possible to mount several samples at once on the same holder. The IVC is loaded with a few mbar of ⁴He exchange gas at the start of operation which helps to cool the whole probe to base temperature using the cooling power of the VTI. Varying the temperature of the charcoal mini sorption pump (mini sorb) allows it to control the amount of ⁴He exchange gas into the IVC: at base temperature the ⁴He is completely removed so as to isolate thermally the ³He pot from the IVC. The charcoal main sorption pump (main sorb) controls the absorption and desorption of ³He gas from the main dump hence the amount of ³He condensing into the ³He pot. Varying the temperature of the main sorb allows it to vary the pumping pressure on the liquid ³He and therefore also the temperature of the ³He pot. The ³He pot, main sorb and mini sorb are equipped with separate sensors and heaters. The ³He pot temperature sensor is a CX1030 calibrated in magnetic field and temperature. Fig. 1 shows a photograph of the helium-3 probe with IVC removed, indicating the positions of the samples mounted on an oxygen-free Cu holder, the ³He pot, the main and the mini sorb. The ³He probe, VTI and magnet are controlled through a Labview program provided by Cryogenic Ltd. A dedicated Tango server establishes the communication between the PETRA III command line interface and this control software allowing for a complete remote control of all experimental variables.

Figure 1 Bottom of the ^3He probe with the IVC removed where the samples, ^3He pot, main and mini sorption pumps can be seen.

For the present set of experiments, the VTI was maintained at a temperature of 1.9 K and the ⁴He flow maintained at 12 mbar. About 10 mbar ⁴He exchange gas was loaded into the IVC. Once the ³He had condensed into the ³He pot and the base temperature was achieved, the temperature was controlled by varying the temperature of the main sorb while keeping the mini sorb cold. Continuous operation for about 8–12 h was possible. Then, a recondensation was required taking about 40 min. The temperature of the sample was monitored first by a commercial Cernox thermometer CX1030 from Lakeshore positioned close to the sample, and then by a CX1050SD sensor calibrated in a Quantum Design PPMS equipped with a ³He low-temperature stage at the HLD. The two CX1030 sensors (sample and ³He pot) systematically returned the same temperature pointing to the absence of any thermal gradient over the sample holder. However, only the CX1050SD thermometer returned the transition temperatures reported for the samples in the literature. After calibrating the CX1030 sensors versus the CX1050 sensor, the following correction, T = 0.06999 + 0.89643T_CX1030, was systematically applied to the temperature returned by the CX1030 sensors in zero field. In Fig. 2 we show a typical dependence of the temperature of the ³He pot versus the temperature of the main sorb during operation. The base temperature is 360 mK and the maximum temperature achieved here is 3 K. The temperature follows a sigmoidal trend in between these two extrema. The temperature dependence shown on Fig. 2 was found very reproducible for subsequent beam times. For measurements in the presence of a magnetic field, the magnetoresistance of the CX1030 thermometers has been fully characterized by Cryogenic Ltd. Henceforth, the zero field temperature can be retrieved. Since magnetic fields of 14 T induce temperature errors ΔT/T of the Cernox sensors of less than 1% at temperatures between 4 K and 20 K, the temperature of the main sorb does not need to be corrected for magnetic fields in this temperature range. On the other hand, since ΔT/T at 14 T is higher than 10% at temperatures below 1.4 K, the sample temperature must be corrected. This is done automatically within the temperature control software.

Figure 2 Temperature of the helium-3 pot as a function of the main sorb temperature.

3. Validating measurements

3.1. Jahn–Teller effect in TmVO₄

The rare-earth vanadate TmVO₄ undergoes a cooperative Jahn–Teller distortion at temperatures below T_c = 2.1 K, where the degeneracy of the electronic ground state of the Tm³⁺ ion is lifted by a crystallographic distortion (Cooke et al., 1970; Becker et al., 1972). Details of the tetragonal-to-orthorhombic second-order phase transition have been studied by X-ray diffraction on a single-crystal down to 1.7 K by Segmüller et al. (1974) at a laboratory X-ray source immersing the sample in a ⁴He bath and more recently by Suzuki et al. (2002) on a powder sample down to 250 mK at a laboratory X-ray source using a ³He–⁴He dilution refrigerator. The principal axes of distortion are [110] and [1,−1,0]. The (h,h,0) diffraction lines in the high-temperature state split into two lines (2h,0,0) and (0,2h,0) below T_c. The splitting varies continuously between 0.80 K and 2.15 K, making TmVO₄ an appropriate reference compound to investigate beam heating effects at those low temperatures.

The TmVO₄ single-crystal used in the present experiment was grown to a length of 12 cm and diameter of 8 mm by the optical floating zone technique at the DSME at Ames Laboratory. It was subsequently cut at DESY so as to obtain two parallel flat faces normal to the (1,1,0) direction. The sample surface was then highly polished in order to remove any surface damage. The sample, 1.5 mm thick and with  3.2 mm × 2.5 mm cross section, was glued with GE-varnish and mounted so that the (1,1,0) and (1,−1,0) reciprocal lattice directions were aligned in the horizontal scattering plane. The X-ray diffraction measurements were carried out at an incident energy of 8.664 keV and with a beam size of 200 µm × 70 µm (H × V) at the sample position. Higher harmonics were suppressed by mirrors and additional detuning of the monochromator. The detector horizontal slits were closed down so as to resolve the peak splitting. With an incident X-ray photon flux at P09 of about 1.5 × 10¹³ photons s⁻¹ at this energy (Strempfer et al., 2013) and considering the intensity reduction by the detuning and the attenuation of the incident beam by the magnet and IVC Be windows, the heat load on the sample is 13.1 mW. Calibrated attenuator foils upstream of the magnet were used to decrease the photon flux and the heat load on the sample. Independently of how much flux impinges on the sample, 99.99% (99%) of it is absorbed within a 67 µm (34 µm) depth due to the high absorption coefficient of the sample at this photon energy.

In Fig. 3 we show θ–2θ scans through the (220) Bragg reflection of the TmVO₄ tetragonal phase for decreasing incident beam attenuation factor of the incident beam between 10⁶ and 10². The ³He pot, main sorb and mini sorb were maintained at the base temperatures of 360 mK, 4 K and 3 K, respectively. The intensities are corrected by the monitor and attenuation factor values. The increasing local heating of the sample causes the (4,0,0) and (0,4,0) reflections of the orthorhombic phase to merge into the single (2,2,0) reflection of the tetragonal phase at attenuation factors lower than 300, indicating that the sample temperature exceeds T_c at these photon fluxes. For attenuation factors larger than 5000, there is no noticeable change in the peak positions, indicating that the sample temperature is lower than 0.8 K. Note that the temperature of the sensor shows only a small deviation of a few tens of mK from a base temperature of 360 mK over the whole attenuation range. The peak splitting between the (4,0,0) and (0,4,0) reflections was investigated as a function of temperature with an attenuation factor of 10⁵ (equivalent heat load of 0.12 µW) in order to rule out any beam heating effects. In Fig. 4 we show the associated static strain e₂/2 = (a − b)/a₀ as a function of temperature together with the results reported by other groups (Segmüller et al., 1974; Suzuki et al., 2002). As can be seen, all X-ray data compare well. Fig. 4 can be used to estimate the temperature of the sample in Fig. 3. In Fig. 5 we plot the static strain of the sample as a function of the logarithm of the attenuation factor and the associated true temperature. From a power fit through the data points, we find that an attenuation factor of the incident beam of 3500 is required to maintain the sample at temperatures below 1 K which corresponds to a beam heat load of 3.7 µW. A power-law extrapolation to lower temperatures suggests that an attenuation factor of 10⁴ (3 × 10⁴) is required to maintain the sample at temperatures below 700 mK (500 mK).

Figure 3 θ–2θ scans about the (220) Bragg reflection of the TmVO4 tetragonal phase for decreasing attenuation factor of the incident beam between 10^6 and 10^2 and the helium-3 pot maintained at a base temperature of 360 mK. Intensities are corrected by the monitor and the attenuation factor. The attenuation factor at which the scans were recorded and the temperature returned by the thermometer are indicated in the legend.

Figure 4 Static strain (a − b)/a0 as a function of temperature obtained in the present work compared with that reported by Segmüller et al. (1974) and Suzuki et al. (2002).

Figure 5 Static strain (left axis) and temperature of the sample (right axis) as a function of the logarithm of the attenuation factor.

3.2. Spin-density wave in TmNi₂B₂C

The rare-earth nickel borocarbides RENi₂B₂C (RE = Tm, Ho, Er, Dy) with body-centered tetragonal I4/mmm crystal structure (Lynn et al., 1997) have attracted much attention since their discovery in 1994 (Gupta et al., 1994; Canfield et al., 1998) because of the coexistence of superconductivity and antiferromagnetism at low temperatures. In TmNi₂B₂C, superconductivity sets in at 11 K and coexists with an antiferromagnetic order below T_N = 1.5 K (Movshovich et al., 1994). The magnetic structure has been investigated in detail by neutron diffraction down to 0.1 K in magnetic fields up to 6 T (Nørgaard Toft et al., 2004) and by resonant X-ray magnetic scattering (RXMS) down to 0.9 K (Mannix et al., 2004) at the XMaS UK CRG bending-magnet beamline at ESRF using a Displex cryostat equipped with a Joule Thompson (JT) stage flowing ³He through the JT stage. The Tm moments are aligned along the tetragonal c-axis forming an incommensurate spin-density wave (SDW) with ordering wavevector (0.094,0.094,0) and with an ordered magnetic moment on the Tm site of 3.78 µB at 360 mK (Lynn et al., 1997). For a magnetic field applied along the tetragonal a-axis, the ordering wavevector changes to (0.483,0,0) at 1 T and to (0.496,0,0) at 4 T.

Measurements at P09 were carried out using the same experimental set-up as described for TmVO₄ in the previous section. The TmNi₂B₂C sample is one previously used in a high-energy X-ray diffraction study at beamline BW5 at the DORIS storage ring at DESY (Andersen et al., 2006). The single-crystals are grown out of excess Ni₂B solution in a novel two-step process (Canfield & Fisher, 2001). The single-crystals grow as plates with the c-axis being perpendicular to the surface. The sample was oriented with the (1,1,0) and the (0,0,1) crystallographic axes in the horizontal scattering plane. This allows probing of the first-order magnetic satellites along the (1+τ, 1+τ, 10) reciprocal lattice direction and investigation of the magnetic field effects for the field applied along the (1,−1,0) crystallographic axis. In this geometry the incident X-rays are π polarized, i.e. the polarization vector lies in the scattering plane. The polarization of the scattered signal was analyzed using a Cu(1,1,1) crystal with 10% reflectivity. Linear polarization scans (Scagnoli et al., 2009) were carried out using two 400 µm diamond quarter-wave plates in series in 90° geometry (Strempfer et al., 2013; Francoual et al., 2013). All measurements were carried out with an equivalent attenuation factor of 10⁴.

In Fig. 6 we show a reciprocal-space scan along the (1+τ, 1+τ, 10) direction at 360 mK in the πσ′ channel at the resonant energy of 8.654 keV close to the Tm L_III edge. A non-zero signal is fully resolved peaking at τ = 0.0956 r.l.u. with an intensity at maximum of 25 counts counting 10 s per point. In Figs. 7(a) and Fig. 7(b) we show the results of a full polarization analysis of the magnetic superlattice peak and of the neighboring (1,1,10) charge Bragg reflection carried out at base temperature. The Stokes parameters of the scattered beam, and , are plotted as a function of the angle of linear polarization of the incident beam η. The Stokes parameters for the Bragg peak are fitted considering the expressions given for and in the case of Thomson charge scattering by Detlefs et al. (2012). In the horizontal scattering geometry, the Stokes parameters of the incident beam write as P₁ = −P_LINcos(2η) and P₂ = −P_LINsin(2η), with P_LIN = (P₁ ²+P₂ ²)^1/2. The Stokes parameters for the magnetic peak are computed considering the Jones matrix for resonant electric dipole exchange scattering given by Hill & McMorrow (1996). In the (u₁,u₂,u₃) coordinates system used in X-ray magnetic scattering experiments (Blume & Gibbs, 1988; Hill & McMorrow, 1996; Detlefs et al., 2012), the components of the magnetic moment can be written as S₁ = Ssin(α)cos(ψ), S₂ = Ssin(α)sin(ψ) and S₃ = Scos(α). For a magnetic moment aligned along the c tetragonal axis and for a scattering vector of (1.0956,1.0956,10), the polar angle α is 180 + 25.20° and the azimuthal angle ψ is 0°. The degree of incident linear polarization, P_LIN, is made to vary with η as a sine wave with 90° periodicity and 45° phase in accordance with known characteristics for the double phase-retarder set-up at P09 (Francoual et al., 2013). In Figs. 7(a) and Fig. 7(b) we plot with solid lines the computed values for and for the two different scattering processes for P_LIN = 0.91525 + 0.08225sin(2η + π/2). As can be seen, an excellent fit of the data is obtained which allows it to confirm the magnetic structure previously reported for TmNi₂B₂C (Nørgaard Toft et al., 2004).

Figure 6 q-scan along the (1+τ, 1+τ, 10) reciprocal lattice direction at base temperature in TmNi2B2C measured in the πσ′ channel with an attenuation factor of 10^4 at the resonant energy of 8.654 keV.

Figure 7 Stokes parameters and as a function of the angle of incident linear polarization η at base temperature (a) for the (1,1,10) Bragg peak and (b) for the (1+τ, 1+τ, 10) magnetic peak. The solid and dashed lines are the computed values for and , respectively, considering a charge scattering process at the charge Bragg peak and a resonant scattering dipole process at the magnetic Bragg peak assuming that the magnetic moment is aligned along c. The only fitting parameter is the degree of incident polarization, which is set to vary as a sine wave as a function of η (see text).

The (1+τ, 1+τ, 10) peak (Fig. 6) was investigated as a function of temperature at several different magnetic fields and as a function of magnetic field at several different temperatures by performing systematic q-scans along the scattering vector. The resulting integrated intensity I(1+τ, 1+τ, 10) is shown as a function of temperature for μ₀H = 0 T in Fig. 8(a) and as a function of magnetic field for T = 360 mK in Fig. 8(b). The reconstructed (μ₀H,T) phase diagram is shown Fig. 9. The temperature dependence of the integrated intensity is in good agreement with previous reports (Lynn et al., 1997; Mannix et al., 2004) in the full temperature range as shown in Fig. 8(a). The spin-density wave is suppressed at 1.6 K at zero field and at 1.4 T at zero temperature as extrapolated from Fig. 9. Here, the critical field for the field along (1,−1,0) is slightly lower than the one reported for the field applied along the (1,0,0) direction (Nørgaard Toft et al., 2004) which suggests a basal plane anisotropy. The error bars in Figs. 8(a) and 8(b) amount to 7% and 9% counting 20 s and 10 s per point, respectively. A 3% standard deviation would require counting 150 s per point for this sample and particular magnetic peak.

Figure 8 (a) Integrated intensity I(1+τ, 1+τ, 10) as a function of temperature in zero field compared with the neutron results from Lynn et al. (1997) and the RXMS results from Mannix et al. (2004). (b) Integrated intensity I(1+τ, 1+τ, 10) as a function of magnetic field at 360 mK.

Figure 9 (μ0H,T) phase diagram of the spin-density wave in TmNi2B2C for the magnetic field applied along (−1,1,0) (filled circles) as determined in the present work and for the magnetic field applied along (1,0,0) (empty circles) as reported by Nørgaard Toft et al. (2004). The solid lines are guides to the eye.

The reasonable signal-to-noise ratio obtained for the RXMS intensity in Fig. 6 together with the good quality of the data and the fits in Fig. 7(b) show that the helium-3 fridge described in §2 can be used to carry out RXMS studies in the sub-Kelvin temperature range despite the necessary reduction of the incident X-ray flux. Furthermore, the results shown in Figs. 8(b) and Fig. 9 demonstrate its usability in high magnetic fields.

4. Discussion

Heating by the synchrotron X-ray beam has been the object of detailed calculations and detailed experimental determinations mostly for biological samples immersed in a cooling gas stream (Mhaisekar et al., 2005; Snell et al., 2007, and references therein). In these models, the sample is cooled down by convection and its size is usually small compared with the X-ray attenuation length so that not all of the beam power is absorbed into the sample. Two temperatures are computed: the temperature of the outside surface of the crystal and the temperature inside. While the external temperature is altered by parameters such as the crystal size and the gas velocity, the internal temperature is influenced by the sample thermal conductivity and the beam size. Both temperatures are shown to depend linearly on the incident flux (Mhaisekar et al., 2005). For samples absorbing the full beam, more heat is generated internally and the temperature profile across the sample is less uniform (Kuzay et al., 2001). The temperature rise ΔT on a 1 mm glass bead with 0.97 W m⁻¹ K⁻¹ thermal conductivity in a 290 K N₂ gas stream absorbing 3.37 mW beam power is 20 K (Snell et al., 2007). ΔT increases to 80 K when the beam heat load is increased to 13.1 mW. In the set-up presented here, the heat balance problematic is different because the sample sits in a vacuum, it is cooled by conduction and only one of its sides is in contact with the cold surface. The outer surfaces lose energy by radiative transfer mechanisms, convection being absent due to the absence of exchange gas. Nevertheless, temperature increases in the same order of magnitude are expected from previous experiments using Displex closed-cycle cryostats. In the filled skutterudite PrRu₄P₁₂, the superlattice peaks associated with the transition at 60 K are invisible at a 5.5 K nominal temperature of the cryostat unless attenuating the incident beam by at least a factor of 20 at beamline P09 at 5962 eV (unpublished results). With the helium-3 fridge, we find a TmVO₄ sample temperature of 1.9 K and 1.1 K for an incident beam of 26 µW (attenuation factor 500) and 5 µW (attenuation factor 2650), respectively (Fig. 5). The dependence of the temperature rise on the incident flux is smaller than linear, which might be due to the combined effect of an increased sample thermal conductivity with decreased temperature and an increased cooling power at beam heat loads coming close to 25 µW. All thermometers show very little variation in their read-outs from the base temperature of 360 mK for beam heat loads up to 5 µW demonstrating that the beam heating is localized in that very narrow region around where the beam hits the sample. The temperature difference ΔT of 0.74 K between the sample temperature and the ³He pot temperature for 5 µW deposited energy is important, suggesting that the heat transfer is not very effective. The rate of heat transfer to the cold finger depends on the thermal conductivity κ of the sample, its absorption properties, its thickness and the quality of the thermal contact between the sample and the sample holder. TmVO₄ is a poor thermal conductor with κ = 0.1 W m⁻¹ K⁻¹ at 1 K and 0.02 W m⁻¹ K⁻¹ at 360 mK (Daudin et al., 1980). Furthermore, the sample used in our study is quite thick (∼1.5 mm). We expect a lower ΔT for thinner samples, for samples with higher thermal conductivity and for less absorbing materials. For instance, TmNi₂B₂C has 0.35 W m⁻¹ K⁻¹ thermal conductivity at 1 K (Hennings et al., 2002). Its attenuation length, μ_att, is 1.64 times shorter than that of TmVO₄ at the Tm L_III absorption edge. According to Fourier's law of heat conduction, the dependence of ΔT on κ is linear. Assuming the dependence of ΔT on μ_att to be linear (Mhaisekar et al., 2005), the temperature of a TmNi₂B₂C sample of a similar shape as the TmVO₄ sample used hereby would be 0.7 K rather than 1.1 K for a 5 µW beam heat load. The precise dependence of ΔT on ξ, μ_att and on the conduction shape factor are to be determined either numerically or empirically.

The nominal cooling power of the helium-3 refrigerator is 25 µW at 360 mK, 50 µW at 400 mK and 125 µW at 500 mK. The probe and sample thermometers behave in line with this, as shown in Fig. 3, i.e. they show a temperature of 400 mK for 50 µW incoming heat load. This means that the beam heat load needs to be attenuated at least down to 25 µW in order to avoid any rapid warm-up of the probe and ensure a long enough time window between two successive condensation cycles. We find that a beam heat load of 1.3 µW (attenuation factor 10⁴) guarantees continuous operation between 8 and 12 h, whereas at attenuation factors of the order of 10³ this time window is reduced to about 3 h. Therefore, independently of which sample is mounted onto the probe, an attenuation of the beam of four orders of magnitude is also required at P09 to ensure long operation of the helium-3 refrigerator.

Our study raises the long-standing issue about how to determine the true temperature of a sample in a low-temperature X-ray experiment when the sample is enclosed in a cryostat. As we have shown, thermometers are ineffective at measuring the temperature rise in that narrow region where the beam hits the sample. The best method is to use the X-ray probe to measure a quantity that has a known relationship to the temperature. For instance, Christiansen et al. (1996) have been able to return the true temperature of the sample by fitting the temperature dependence of the X-ray magnetic circular dichroism signal in plastocyanin as a function of magnetic field to the one expected for an isotropic S = 1/2 spin system. Wang et al. (2012) have used instead the imbalance between Stokes and anti-Stokes intensities in nuclear resonant vibrational X-ray spectroscopy. In this work, we have measured the peak splitting coming along with a crystallographic distortion to determine the true sample temperature in TmVO₄. During X-ray diffraction experiments, the real temperature experienced by the sample is usually checked by measuring the temperature dependence of a characteristic peak that appears below a known critical temperature T_c and comparing the temperature returned by the thermometer for that transition temperature with T_c. Unless the temperature rise ΔT is temperature independent as shown by Snell et al. (2007) for a glass bead between 220 and 290 K, this method allows it to determine ΔT and which attenuation factor of the incident beam to use only at T_c. Henceforth, it might be good practice to carry out a series of measurements attenuating by one more order of magnitude than has been determined experimentally and check for discrepancies.

A previous experimental study combining a synchrotron X-ray beam and a dilution refrigerator has concluded that ultralow-temperature experiments are better combined with a laboratory X-ray source (Suzuki et al., 2002). This is overlooking all the advantages of using a synchrotron X-ray beam, for instance the energy tunability and the polarization control of the X-ray beam, which permit investigation of the magnetic properties of materials (Lovesey & Collins, 1996). Furthermore, the higher flux at synchrotron sources enables insertion of additional optical devices into the X-ray beam before the sample such as a double-phase retarder for full polarization control of the incident X-ray beam, an X-ray beam diagnostic unit for polarization monitoring or compound refractive lenses for the purpose of focusing, as is done at beamline P09. We have proven hereby that a complete resonant X-ray magnetic X-ray scattering study is possible at sub-Kelvin temperatures at beamline P09. Both the resonant and non-resonant magnetic scattering intensities at first-harmonic satellites depend quadratically on the magnetic moment (Hill & McMorrow, 1996; Blume & Gibbs, 1988) and on the number of magnetic atoms in the unit cell. In TmNi₂B₂C, the ordered magnetic moment on the Tm site is 3.78 µB at 360 mK (Lynn et al., 1997) and there are two magnetic atoms per unit cell. The resonant enhancement of the magnetic scattering intensities at the L_II and L_III edges of the 4f lanthanide elements varies between 1 and 2500 for electric dipole E1 transitions and is the strongest for gadolinium and holmium (Hamrick, 1990). For thulium, the resonant enhancement is modest, of the order of 50 (Bohr et al., 1990). Even stronger resonant enhancements of the order of 3000 are expected for E1 transitions at the L_II and L_III edges of the 5d transition elements, at X-ray energies between 9.2 and 14.2 keV, as is exemplified in K₂ReCl₆ (McMorrow et al., 2003). At the M_IV and M_V edges of the 5f actinide elements, at X-ray energies between 3.3 and 3.9 keV, enhancements of the order of 10⁸ are even achieved (Hamrick, 1990). In TmNi₂B₂C, the count rate is 2.5 counts s⁻¹ at 8.654 keV at 360 mK using a Cu(111) analyzer with 10% efficiency and an attenuation factor of 10⁴. At a lower energy, the output count rate may be significantly altered due to absorption by the magnet and IVC windows. 3.7 times less intensity is transmitted at 5.5 keV than at the Tm L_III edge and 1000 times less at 3.5 keV. Also, though the beam power decreases , the photoelectric absorption increases slightly faster , ), bringing larger attenuation lengths, and the incident flux might further need to be reduced by a factor of ten in some extreme cases depending on the material. Yet these numbers are not prohibitive in light of the aforementioned resonant enhancements. Henceforth, TmNi₂B₂C is not a unique scientific case for which resonant X-ray magnetic scattering at helium-3 temperatures might be carried out successfully.

5. Conclusions

Beamline P09 at PETRA III at DESY is equipped with a helium-3 fridge that can be inserted into the variable-temperature insert of the 14 T magnet for the purpose of single-crystal X-ray diffraction and resonant X-ray magnetic scattering experiments. The helium-3 fridge insert has been commissioned and is now fully operational, permitting user-friendly experiments at temperatures between 0.4 K and 3 K in magnetic fields up to 14 T in a wide range of scattering angles. Beam heating effects have been fully characterized: an attenuation factor of the incident beam of 10⁴ is needed to ensure that the sample is at temperatures below 800 mK and that the helium-3 fridge remains cold for more than 10 h.