Nanocalorimeter platform for in situ specific heat measurements and x-ray diffraction at low temperature

Modern x-ray sources such as third-generation synchrotrons and free electron lasers produce photon beams of extreme brightness.¹ The advent of these high luminosity technologies opens new research opportunities.² For instance, in the realm of solid-state physics, which is the background of the work described here, synchrotron-based x-ray diffraction experiments were instrumental in identifying the coexistence of charge and/or spin density wave order and superconductivity in various cuprate^3–7 and iron-based superconductors,^8,9 illuminating the role of nematicity in pnictides,¹⁰ and discovering field-induced intertwined orders in a Mott-Kitaev system.¹¹ A prerequisite for the study of phase transitions and the emergence of competing/coexisting order parameters is the precise knowledge of the temperature and thermodynamic state of the sample. In fact, high photon flux can easily induce significant and uncontrolled sample heating especially of small samples at low temperatures,^12,13 potentially leading to erroneous interpretations of the experimental results. Therefore, the concept of multi-modal measurements in which x-ray scattering is combined with the simultaneous measurement of a different quantity such as the specific heat has attracted considerable attention.

Here we demonstrate a nanocalorimeter platform for in situ specific heat measurements and x-ray diffraction. This device is based on the concept by Tagliati et al.¹⁴ and allows for the precise spatial and thermal control of the sample in the x-ray beam. Membrane-based nanocalorimeters¹⁵ have emerged as a powerful technique for the thermal characterization of phase transitions in small single crystals, thin films, or nanoparticles. Owing to their small mass, a particular strength of these nanocalorimeters is their ability to rapidly ramp the temperature, which has led to wide-range applications in scanning calorimetry, for instance, on polymer films.¹⁶ Several commercial devices are now available.^17,18

Simultaneous synchrotron x-ray and calorimetry measurements were first performed more than thirty years ago at the Stanford Synchrotron Radiation Laboratory by Russell and Koberstein.¹⁹ A commercially available differential scanning calorimeter unit was modified to be mounted vertically on the beamline, and small angle x-ray scattering experiments were carried out on a low-density polyethylene disk of 1 mm thickness. This pioneering work was followed by several research groups utilizing various designs of calorimeter cells and diffraction geometries^20–27 to study the phase transitions in a number of organic molecules, polymers, and liquid crystals. However, samples investigated in these reports are nearly always of polycrystalline nature, with a typical mass in the range of tens of milligrams, limited by the relatively large addenda of calorimeter cells. Rapid progress in microfabrication in the last two decades has led to the development of nanocalorimetry.^{14,15,28–32} Membrane-based nanocalorimeters using micromachined SiN_x membranes offer a calorimeter cell with both vanishingly small thermal mass and a naturally transparent platform for sensitive x-ray work. Fueling the renewed interest in combined synchrotron x-ray and calorimetry experiments with much reduced sample mass and improved sample uniformity, Vlassak and co-workers utilized a customized parallel nano-scanning calorimeter array to study phase transitions at elevated temperatures in metallic films of merely several micrograms in mass.^33,34 More recently, Ivanov and co-workers designed a sample-in-air setup around a commercially available membrane-based nanocalorimetry platform, to be used on the micro-focused beamline at the European Synchrotron Radiation Facility (ESRF).^35,36 They were able to demonstrate both DC operation and AC operation of the calorimeter with fast temperature ramps and characterize the localized beam heating at room temperature. Meanwhile, to the best of our knowledge, combined synchrotron x-ray and calorimetry experiments dedicated to micrometer-sized, single-crystalline samples at low temperatures and/or in magnetic field have never been attempted.

In this paper, we first introduce the nanocalorimeter device and setup optimized for the study of phase transitions in small single crystals at cryogenic temperatures and in applied magnetic fields. This design has several advantages. At 3rd generation synchrotron sources, linear dimensions of the x-ray beam are routinely of the order of 100 μm and can be substantially less (<5 μm) if additional focusing is employed. In particular, with absorption lengths for hard condensed-matter samples being of the order of several micrometers at typical x-ray energies of ∼10 keV, the volume that is illuminated by the x-ray beam is well matched to the typical sample size (∼100 μm) guaranteeing that features seen in x-ray scattering and calorimetry originate from the very same material. We next discuss temperature control and beam-heating related applications including photon-beam diagnostics and conditioning. The devices have exquisite temperature sensitivity, and beam-induced heating is easily quantified. Furthermore, such heating enables a straightforward way to center the sample in the incident photon beam. An ultimate in situ use of calorimetry concurrently with x-ray studies would rely on x-ray absorption cross section and subsequent transfer of absorbed photon energy from the electronic system to the phonons. This allows us to study x-ray absorption effects via calorimetry. In particular, we explore a novel variant of x-ray absorption spectroscopy by monitoring the sample temperature as the photon energy is scanned through an absorption edge. We demonstrate this mode of measurements at the Cu K-edge of an elemental Cu-foil as well as an YBa₂Cu₃O_7−δ (YBCO) single crystal. Finally, we demonstrate the precise temperature correlation between structural and thermal properties, by monitoring the evolution of the lattice constants and heat capacity in the non-equilibrium super-heated region at the first-order magneto-structural phase transition in the magnetocaloric material Fe₂P. This latter study outlines an approach for very low temperature scattering studies in high magnetic fields.

The calorimeter cell used in this study^14,37 consists of a Ge_1−xAu_x (GeAu) thin-film thermometer, an offset (dc) heater, an ac heater, several insulation layers, and a thermalization layer as shown in Fig. 1(a). The calorimeter chip contains two cells, a sample cell and a reference cell, enabling differential operation and enhanced resolution. The functional layers of the calorimeter chip are fabricated using a bilayer photolithography lift-off process on a 150-nm silicon nitride-coated silicon wafer. All leads are between 50 and 100 nm thick and consist of a titanium adhesion layer and a gold layer in varying thicknesses deposited by e-beam evaporation. The heaters are made only from titanium. Various SiO₂-insulation layers [seen as rounded rectangles in Fig. 1(a)] and the GeAu thermometer are sputtered. The GeAu target has a nominal composition of 17% Au. After annealing at 190 °C for 1 h, the GeAu films display a semiconducting-like temperature dependence of their resistances. These thermometers are well suited for cryogenic applications since they have high and almost temperature independent sensitivity dn(R)/dn(T) ≈ −1, whereas thermocouples, for instance, loose sensitivity at low temperatures since the Seebeck coefficient (of all materials) tends towards zero. After all front-side deposition is completed, 1-mm² membranes under each cell are back-etched in a KOH bath. Samples are mounted on the membrane either with the help of a micro-manipulator or by hand using a single-haired brush. The specific heat measurements are based on the ac steady state method.^38,39 An oscillatory heater power of frequency ω and amplitude P_ac induces an oscillatory temperature of amplitude T_ac with a phase lag ϕ. In a description of lumped elements, the heat balance equation can be solved in analogy to electric RC-circuits yielding

Here, C is the heat capacity of the cell (including contributions from the sample, thermometer, heater, and support), K is the thermal link of the cell to the thermal bath (the thermal relaxation time is given by τ = C/K), and ϕ is the phase between temperature and heater power oscillations. This analysis assumes a uniform temperature distribution inside the cell, that is, ωτ_i$≪$ 1, where τ_i is an internal thermal time constant. Optimal operating conditions are found at tan(ϕ) ≈ 1. At too high frequencies, the sample becomes thermally disconnected from the calorimeter platform, while at too low frequencies, T_ac approaches a constant value and becomes insensitive to the heat capacity.

The increase of the sample temperature above the bath temperature is given by

where the P_ac-contribution represents the average power due to the ac heating, P_offset is the power delivered by the offset heater, and P_beam is the absorbed x-ray beam power that has been converted to phonons.

We use a SynkTek MCL1-540 multi-channel lock-in system⁴⁰ to supply the bias signals for the heaters and thermometers and to monitor the amplitudes and phases of the ac-signals. Experiment control, data acquisition, and data analysis are achieved with a custom LabVIEW measurement program.

The nanocalorimeter has a thermal link in a vacuum of 5 μW/K at room temperature, decreasing to 10 nW/K at 1 K, enabling wide-range offset scanning using sub-mW powers. The sample and reference GeAu thermometer is designed as a thin-film square resistor with about 1 kΩ resistance at room temperature, increasing (with a slope dnR/dnT close to −1) to about 0.7 MΩ at 1 K. Thermometer calibration was obtained with a small, low-frequency (1 Hz) ac bias with self-heating below 1 mK against a calibrated Cernox thermometer. During normal operation, the thermometers are biased with a dc voltage of the order of 0.1 V, limiting the self-heating to a few percent of the absolute temperature. The thermometer bias is continuously but slowly adjusted to maintain good working conditions. For ac calorimetry, the ac heater power is automatically adjusted to maintain a temperature oscillation amplitude corresponding to approximately 0.2% of the absolute temperature. Near phase transitions, the ac power is kept constant to both further reduce the temperature oscillation amplitude and avoid possible adjustment effects. The frequency is adjusted to maintain a constant phase lag between ac heater power and thermometer response. The phase lag is chosen near the point tan(ϕ) = 1 to maintain both good absolute accuracy and high resolution. The temperature resolution is normally better than 1 in 10⁵ but is in many cases limited by x-ray beam power fluctuations.

The calorimeter chip is installed at the tip of a He-4 flow cryostat (Variable Temperature Insert, VTI) for measurements at low temperatures, see Fig. 1(a), in such a way that its face is parallel to the VTI axis. The VTI, shown in Fig. 1(c), is in turn inserted into a split-pair, vertical-field superconducting magnet with a maximum field of 4.5 T. The magnet is mounted on a two-circle diffractometer with a circular segment to tilt the magnet by ±3.4°, located in the 6-ID-C experimental station of the Advanced Photon Source (APS); see Fig. 1(b). In this geometry, the magnetic field is parallel to the calorimeter membrane.

A Si (111) double-bounce monochromator is used to select a photon energy with a 0.01% bandwidth. For the diffraction experiment, we use 11.22 keV x-rays, while the absorption measurements are performed at the copper K-edge energy (8.979 keV). The beam is focused in the vertical plane using a bent ultra-low expansion mirror, while a second smaller mirror further downstream is used to deflect the beam to make it horizontal and to further reduce higher harmonic contamination. The horizontal size of the incident beam is controlled by slits in front of the sample and adjusted—as needed—to the dimensions of the samples. A point detector (NaI-scintillator) is used for recording the scattering intensity of charged Bragg peaks. The incident beam intensity is measured with a helium gas filled ion chamber.

Having the sample in close contact with an offset heater and a thermometer provides numerous advantages for beamline experiments. It allows for a direct and unambiguous measurement of the sample temperature. This property should be contrasted with standard approaches for thermometry, where the thermometer is placed away from the sample and hence not sensitive to temperature gradients or heating produced by the x-ray beam. Our setup is particularly useful for initially positioning the sample in the x-ray beam. Whereas the traditional approach requires a tedious search for the sample, especially when cryostats or other constraints do not allow for a transmission profile, a fast x-y mesh scan of the sample temperature is sufficient to locate and center the sample in the beam; see Fig. 2(a). Indeed, as soon as the x-ray beam starts illuminating the sample, the temperature of the latter rises, goes through a maximum, and decreases again when the beam falls off the sample. One should note that the temperature rise due to x-ray absorption in the thin layers comprising the calorimeter cell is usually negligible as their illuminated mass is much lower than the sample mass. The positioning scans in Fig. 2(a), however, show that the heating effect of the beam on the sample may be significant. This is generally true at low temperatures,¹² but especially so when the sample platform is a weak thermal conductor such as a membrane-based calorimeter.²¹ Precise temperature information becomes particularly relevant when pinpointing the location of phase transitions. Fortunately, our local thermometry approach still allows measuring and controlling the true sample temperature (vs. the cryostat temperature in the traditional approach) while accounting for the heating produced by the beam. The relatively weak thermal link of the platform may actually be beneficial since any thermal gradient between sample and platform thermometer will be insignificant.

The offset heater enables efficient acquisition of x-ray diffraction (XRD) data over large temperature ranges. Since only the calorimeter cell is heated and not the entire cryostat as in the case of a conventional setup (using 30-50 W), displacement of the sample in/out of the beam due to thermal expansion of the cryostat and the need to frequently re-center do not arise. The diffraction data on Fe₂P, discussed below, were acquired using this approach. We estimate a factor of two time-saving for the zero-field data over a conventional setup which would have required warming and stabilizing the cryostat and checking the positioning and alignment of the sample at each temperature step. By heating only the sample, the temperature ramping also reaches a rate normally not accessible in diffraction experiments. For example, without harming the sample, we have reached cool-down sweeps in vacuum with rates above 100 K/s; see Fig. 2(b).

Figure 2(c) shows the beam-induced heating for different beam attenuator settings. The important design parameter describing the response of the calorimeter cell to an x-ray beam is the thermal link, K, between the cell and the thermal bath, i.e., the Si-frame of the chip. Following Eq. (3), a deposited x-ray beam power of P_beam will generate a temperature rise of ΔT = P_beam/K. With a thermal link K = 0.5 μW/K at 10 K, increasing to 0.8 μW/K at 15 K, the temperature rise of 5 K shown in Fig. 2(c) corresponds to an absorbed beam power of P_beam ≈ 3.2 μW. At room temperature, the same beam load would cause a sample temperature rise by 0.6 K. High K-values, realized, for example, through thicker Au-leads to the thermometer and heaters (see Fig. 1), are desirable when the lowest possible base temperature is important, whereas a low value is beneficial when measuring x-ray absorption via the temperature rise of the sample (see below). In an optimized design, these requirements need to be balanced against good calorimeter operation as described by Eqs. (1) and (2). Figure 2(c) also shows the one-to-one correlation between the rise in sample temperature and the incident x-ray flux as quantified by the intensity of the (4, 2, 0) Bragg peak of Fe₂P. Since the temperature change of the sample is directly proportional to the number of absorbed photons, it may serve as a detector of absorption edges when sweeping the energy of incident photons, as shown by experiment on Cu and YBCO below.

The high sensitivity of the sample temperature to absorbed x-rays opens a new approach to directly observe resonant effects of Bragg peaks and x-ray absorption spectroscopy. We demonstrate this feature by measuring the Cu K-edge in a 10-μm-thick elemental copper foil and in a YBCO single crystal. The YBCO crystal is naturally untwinned and has a 253 μm × 123 μm rectangular shape with 12 μm thickness. It displays a sharp superconducting transition at 90 K (see the lower inset of Fig. 3). Both samples are mounted such that the sample normal and incident beam suspend at an angle of ∼40°. We monitor the change of the temperature of the Cu-foil and of the YBCO crystal while the incident photon energy is scanned through the absorption edge. In our configuration, the temperature rise across the absorption edge for the Cu-foil and YBCO crystal is 0.8 K and 0.15 K, respectively, at a base temperature of 225 K. In order to discern changes in the x-ray absorption profile, the thicknesses of the Cu-foil and of the YBCO crystal were chosen to be close to their nominal absorption length (which changes from 30 μm to 6 μm for Cu and from 18 μm to 12 μm for YBCO across the Cu K-edge). For further analysis, we normalize the temperature readings to the incident photon flux as measured with an ion chamber in order to cancel the noise stemming from beam fluctuations and drifts. Following a common procedure, we subtract the low-energy baseline, normalize the absorption anomaly to unity, and present the data together with literature x-ray absorption data obtained in the conventional way in Fig. 3. For both samples, the absorption edges are clearly visible and match reported values.^41,42 The chemical shift of the absorption edge between pure copper and Cu in YBCO, stemming from the +2 oxidation state of the Cu-ion, agrees well with literature results.^43,44 Furthermore, the shoulder in the rising edge of elemental Cu representing the 1s to 4p transition (marked by arrows in Fig. 3) is seen in both data sets. We note that in comparison with elemental copper, the noise level in the YBCO data is larger. This is in part due to the lower concentration of Cu in the sample. In addition, whereas the temperature measurement itself contributes negligible noise, the major noise seems to originate from vibrations of the sample with respect to the beam center coming from vibrations due to the magnet that is cooled by a closed-cycle cryostat. This effect is more pronounced for the YBCO sample since its size is comparable to the vertical extent of the beam. By contrast, the Cu-foil is larger than the beam, and therefore absorption data are less sensitive to the relative beam position.

However, clear differences between data measured using our thermal method and conventional absorption spectroscopy appear above the threshold. In particular, x-ray absorption near edge structure (XANES) and white line features are strongly suppressed in the thermal data. We note that the thermal and the conventional methods probe slightly different quantities. In the conventional method, using an in-line transmission setup, the transmitted x-ray intensity is recorded as a function of incident photon energy. Since the contribution due to coherent and incoherent scattering is small in the energy range around 10 keV, the measured transmission is a good representation of the dominant photoelectric absorption. In the thermal method, the energy absorbed by a sample from the incident x-ray beam via photoelectric effects must be converted to phonons through scattering processes in order to be registered as a temperature increase. These processes include scattering of photoelectrons created by the primary beam as well as those excited by reabsorption of subsequent x-ray flourescence photons at lower energies. Therefore, fluorescence photons that escape would not contribute to heat generation (i.e., will not be visible via a temperature change of the sample). In our geometry, the amount of deposited energy decreases with distance away from the incident sample surface on which the incident beam impinges, reaching a minimum at the sample-membrane interface. As the incident energy is scanned through the copper K-edge, the absorption length decreases and more fluorescing photons are excited closer to the incident surface, which have a higher probability of escaping from the sample. It then seems plausible that these two effects counteract each other above the threshold, making the temperature change of the lattice weakly energy dependent. This is similar to “saturation effects” known in absorption measurements using fluorescence yield methods on grazing incidence.⁴⁵ For the same reason, the normalization applied to the data in Fig. 3 causes the 1s → 4p feature in the thermal Cu-data to appear at a higher relative level than in the conventional data.

We demonstrate multi-modal caloric and XRD measurements by recording the specific heat and lattice constants of an Fe₂P crystal across the first-order magneto-structural transition at the Curie temperature, T_c ≈ 218 K. This material crystallizes in the hexagonal P$6¯$2m structure in which the Fe-atoms occupy two inequivalent sites. Ferromagnetic order in which the Fe-moments are oriented along the hexagonal c-axis sets in at T_c. Strong magneto-elastic coupling drives the usually second order ferromagnetic transition to first order,^46–48 which makes this material interesting for magnetocaloric applications.^49–52 A bar-shaped crystal with a hexagonal cross section and an overall size of 60 × 60 × 200 μm³ was mounted onto the nanocalorimeter such that its crystallographic (2, 1, 0) direction was normal to the membrane and the c-axis (long direction of the bar) was perpendicular to the magnetic field.

The in-plane hexagonal lattice parameter, a, was obtained using longitudinal (“radial”) scans through the (4, 2, 0) Bragg peak as a function of temperature and magnetic field. Transverse (“rocking”) scans performed by scanning the angle of incidence through (4, 2, 0) Bragg angle (Θ_B) revealed peak widths (FWHM) of 0.01° (or less), underlining the high crystalline quality of the sample. Figure 4(a) shows the temperature dependence of a while increasing the temperature in zero-field and in applied fields along with selected literature values.⁴⁷ Data above 100 K were taken with the aid of the offset heater and at a relative stability of the sample temperature of better than 10 ppm. Remarkably, for this material, the in-plane lattice parameter decreases with increasing temperature contrary to what is expected on the basis of thermal expansion but consistent with pronounced magneto-elastic coupling in which the appearance of a finite magnetic moment induces an expansion of the lattice. At 218.5 K, a first-order discontinuity in a is clearly resolved on warming. This transition is accompanied by significant fluctuation effects in a, which are strongly enhanced in an applied magnetic field. This finding is consistent with magnetization measurements,⁵³ which yielded large field-induced magnetic moments above T_c. Figures 4(a) and 4(b) show the temperature dependence of the lattice parameter and of the specific heat on expanded temperature scales measured on cooling and warming using the offset heater. The hysteresis associated with the first-order transition is clearly visible. In order to determine the maximum extent of the hysteresis, we positioned the system on the $4,2,0$ reflection and ramped the temperature up/down using offset heating while monitoring the $4,2,0$ peak intensity. The temperatures at which the intensity dropped to zero are indicated by arrows in Fig. 4(b). Whereas the hysteretic behavior seen in the lattice parameter is fairly symmetric between cooling and warming, surprising asymmetry arises in the specific heat [Fig. 4(d)]. These data were taken at a frequency of 0.5 Hz and at an oscillating temperature of T_ac ≈ 50 mK. On increasing temperature, the specific heat traces out a sharp peak before the latent heat is released and the specific heat drops to the value in the paramagnetic state [near label “F” in Fig. 4(d)]. We note that as the ac-method is non-adiabatic, it is not well suited for obtaining a measure of the latent heat. The step in specific heat exactly coincides with the step in the lattice parameter. On cooling, the specific heat is almost featureless apart from a small anomaly near 217.5 K where it jumps to the low-temperature trace coinciding with the jump in a.

In order to gain further insight into this unusual behavior of the specific heat, we performed simultaneous transverse scans as a function of increasing temperature, shown in Fig. 4(c), at the temperatures labeled by stars and letters in Fig. 4(d). At low temperatures, the rocking curves feature a clear peak splitting. This splitting does not exist in the paramagnetic phase. However, it persists to low temperature deep in the FM phase. This splitting is rather small (peak-to-peak is of the order of 0.03°), and it is indicative of low-angle boundaries between transformative twin domains. In other words, while the detailed structural determination is beyond the scope of this report, the structural symmetry of the FM phase is different from that in the PM phase brought about by magneto-elastic coupling. Intriguingly, the data in Fig. 4(c) reveal that the spike in the specific heat coincides with the near-disappearance of this peak splitting (curves D and E), suggesting a superheated region at the FM-to-PM structural transition. The mechanisms underlying the evolution of this superheated region are unknown at present and await further scattering studies including reciprocal-space mapping. Upon cooling, the twinning only reappears after the lattice parameter has transitioned back to that of the low temperature phase. The results shown in Fig. 4 exemplify the unprecedented control that can be achieved over the thermal state of the sample using a nanocalorimeter platform, enabling a precise one-to-one correspondence between caloric and structural data and new insights into coupled phase transitions and non-equilibrium, i.e., superheated, states.

We demonstrate that the use of a nanocalorimeter platform enables the precise determination and control of the sample’s temperature and thermodynamic state concurrently with x-ray scattering measurements in a split-pair vertical-field magnet. This capability allows for quantification of beam heating effects, fast and precise positioning of the sample in the x-ray beam, and fast acquisition of x-ray scans over a wide temperature range without the need for repeated re-centering. We demonstrate a novel approach to x-ray absorption spectroscopy on an YBa₂Cu₃O_7−δ crystal and a copper foil, by monitoring the change in sample temperature as a function of incident photon energy. Finally, we illustrate the new insights that can be gained from in situ structural and thermal measurements by investigating the superheated state occurring at the first-order magneto-elastic phase transition of Fe₂P.

As we have demonstrated, an in situ capability to determine and control equilibrium and non-equilibrium states of x-ray probed volume of a sample under investigation would be indispensable in scattering studies of novel phenomena. Such phenomena often require very low temperature and magnetic fields.^54,55 One may expand incoherent x-ray scattering investigations of critical phenomena at a very small reduced temperature, $1−T/Tc$⁠, to those exploiting coherent-scattering methods at future light sources. The fast temperature ramping capability of the calorimeter itself could be used for quenching experiments and relaxation studies (e.g., ionic ordering essential for carrier-concentration control, redistribution of twin boundaries, and shape-memory effects) in operando.

The use of nanocalorimetric platforms presented here promises to usher in alternative approaches to photon-beam characterization and conditioning. For example, due to its extreme sensitivity to heat deposited by x-rays on a small mass, one can use a micro-particle patterned on the membrane to measure the beam profile. This can be accomplished by 2D rastering the membrane through an incident beam, complementing knife-edge and/or pin-hole (aperture) scans typically carried out to obtain such information. Unlike a knife-edge scan which provides a derivative of the profile and a pin-hole scan which suffers from diffraction and transmission, calorimetry images the beam directly. A more intriguing and simple application of our calorimetric platform might be a high-precision temperature control for changing the relative energy (∼meV) of a small perfect crystal (e.g., Si) relative to an analyzer.^56,57 In a back-scattering geometry, ΔE/E = αΔT, where α is the thermal expansion coefficient, providing a means of invoking relative energy change between a high-energy resolution monochromator and an analyzer. With a precision of 10 ppm, we may speculate that energy scans in steps of <100 μeV should be possible.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The calorimeter integration was supported through the APS visiting scientist program (A.R.). Use of the Center for Nanoscale Materials and Advanced Photon Source, Office of Science user facilities, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. K.W. acknowledges support from the Swiss National Science Foundation through an Early Postdoc Mobility fellowship. Z.D. acknowledges support from the Swedish Research Council (VR) under Grant No. 2015-00585, co-funded by Marie Skłodowska-Curie Actions (Project No. INCA 600398). D.C. acknowledges support from the Royal Swedish Academy of Sciences. A.R. acknowledges support from the Swedish Research Council (VR) under Grant No. 2016-04516. The authors thank Roland Willa for helpful discussions.

We pay tribute to the late Gopal K. Shenoy for many insightful discussions on the general issue of the temperature of the x-ray probed volume.