## Introduction

Strong coupling between magnetism and lattice vibrations can lead to many emerging phenomena, as have been shown in unconventional superconductors, heavy-fermions, multiferroics, and other new functional materials1,2,3,4,5. Due to strong spin–orbit coupling, correlated 5f-electron spin systems represent a perfect platform to scrutinize the phenomena related to spin–phonon interactions, especially when interacting with other degrees of freedom such as multipolar ordering or Jahn–Teller interactions6,7. An excellent example is uranium dioxide (UO2). This antiferromagnetic Mott–Hubbard insulator8 is the main nuclear fuel and the most studied actinide material to date9. Its correlated ground state is characterized by a competition among non-collinear magnetic dipoles, electric quadrupoles, and dynamic Jahn–Teller distortions10,11,12. Recently, it has been shown that, due to the magnetic symmetry of the non-collinear 3-k antiferromagnetic order (shown in the inset of Fig. 1a) and strong magneto-elastic coupling, UO2 undergoes a trigonal distortion under magnetic field and becomes a piezomagnet with exceptionally large coercive characteristics13. In piezomagnetic crystals, a magnetic moment can be induced by the application of physical stress14,15. This phenomenon has captured attention in recent years as a mechanism that could be used, in combination with multiferroics and piezoelectrics (especially at the nanoscale), to achieve control of magnetism by electric fields16. Piezomagnetism is also utilized in geology where the so-called volcano-magnetic effect is used for monitoring volcanic activities17,18,19. Despite intensive work, the microscopic nature and crystallographic evidence of piezomagnetism are still elusive20,21.

Here we show a direct micro-structural probe of piezomagnetism using single-crystal X-ray diffraction. By using a high-resolution back-reflection geometry setup (see Supplementary Fig. 4 for details), we were able to resolve small changes in the [888] Bragg peak of UO2 when the magnetic field is applied along the [111] direction, which corresponds to the longitudinal lattice response along the parallel cube diagonal of its crystal lattice. In the paramagnetic state (above 30.5 K), the [888] Bragg peak moves toward larger 2Θ values (see “Methods”) with the applied magnetic field, exhibiting negative magnetostriction. When the UO2 crystal is cooled below the magnetic transition temperature (below 30.5 K), the application of a magnetic field causes positive magnetostriction in agreement with the measurements of the macroscopic variation of the sample length using a fiber Bragg grating (FBG) technique13. When the magnetic field direction is reversed, the sample initially compresses until a critical field is reached and then rapidly expands. The overall behavior resembles the magneto-elastic “butterfly” previously seen with the FBG magnetostriction measurements13 and represents piezomagnetic switching between two magnetic structures connected by the time-reversal symmetry. X-rays offer additional insight not available via bulk FBG methods—specifically, we observe a splitting of the [888] peak under the magnetic field. This arises from time-reversed (TR) domains of 3-k magnetic structures that have different responses to the applied magnetic fields. To the best of our knowledge, this study represents the first crystallographic observation of piezomagnetism and the switching effect in general, and in a 5f-electron spin system in particular.

## Results and disscusion

### The thermal expansion and magnetostriction in the paramagnetic state of UO2

The presence of a sudden volume collapse in the unit cell of UO2 at low temperatures has been known for some time22,23. The small discontinuity corresponds to the rapid and simultaneous magnetic, electrical, and structural transition at TN = 30.5 K. Comparing the relative change in the d-spacing corresponding to the [888] peak with temperature, which was taken using back-reflection  geometry X-ray diffraction, to previously taken dilatometry on single crystal UO2 in the <111> direction13 in Fig. 1a, one can see that the diffraction reproduces the expected behavior. The conversion from the angle of the Bragg peak to micro-strain (or p.p.m.) is made using: ΔL/L = [sin θ0/sin θ′]−1, where θ′ is the Bragg angle of the [888] peak at a given temperature or applied magnetic field; θ0 is its zero-field value. The precision of both techniques is comparable, showing approximately a −30 p.p.m. strain change in the length along the <111> direction at 30.5 K. This shows that the volume collapse at TN can be observed as the material goes from a paramagnetic state to the ordered 3-k antiferromagnetic state. When subjected to a pulsed magnetic field in the paramagnetic state (T = 40 K), the diffraction shows a small contraction of the unit cell along the <111> direction of about 50 p.p.m. at the maximum field of 21.2 T, as seen in Fig. 1b, which also matches the FBG magnetostriction measurements13. The structure then reversibly changes back after returning to zero field.

### The magnetostriction in the antiferromagnetic state of UO2

When repeating the measurements in the magnetically ordered state, a different and unexpected response occurs. Represented schematically in Fig. 2a (not to scale), FBG magnetostriction measurements in a pulsed magnetic field below TN revealed that the bulk crystal expanded in the <111> direction, the reverse of the non-magnetic state. While measuring diffraction in the back-reflection geometry, the sample was cooled to 15 K, then subjected to a pulsed field in the <111> direction. Under the application of the magnetic field, the [888] peak splits and re-converges as the applied field rises and falls, reversibly returning to its original state after the pulse (see Supplementary Fig. 7 for more details). This surprising result contrasts with the single peak seen in the non-magnetic state. Moreover, the split peaks shift in opposite directions, i.e. higher and lower scattering angles (2Θ), indicating simultaneous contraction and expansion of the same crystal lattice. This can only occur if X-ray diffraction originates in two types of physically distinct, three-dimensionally ordered regions of the sample with a robust local symmetry producing fully coherent Bragg peaks. Furthermore, one of the domains (red peak) is much more sensitive to the applied magnetic field, i.e. reaching 700 p.p.m. strain in 20 T, than all the other peaks measured. Remarkably, the two regions respond with opposite sign just as one would expect from two magnetic domains related by time reversal in a magnetic material. This is a very unusual observation, hard to explain in the context of the fcc crystal structure of UO2 yet a natural consequence in a piezomagnetic system. We, hence, believe that the splitting arises from TR domains of the 3-k magnetic structure that have opposite responses to the applied magnetic fields. The integrated intensities of the peaks along 2θ shown in Fig. 2b reveal that there is a larger volume fraction of the sample that corresponds to expansion (blue peak), and a smaller component that corresponds to contraction (red peak). The peak positions of these two separate components plotted in Fig. 2c show that, though the contracting component has a lower intensity, the absolute value of its corresponding strain is larger. The peak positions in the rising and falling field overlap symmetrically as well, showing no hysteresis. Repeatedly applying the pulsed field in the same field direction produces the same result. As can be seen from the figure, the magnetostriction obtained here, governed by the blue peak, agrees well with the FBG pulsed field magnetostriction measurements (black line)13.

When the sample is subsequently exposed to a pulsed magnetic field applied in the direction opposite that of the previously applied field, startlingly different behavior is seen. Represented schematically in Fig. 3a, the FBG magnetostriction measurements show that in this reversed field direction in the magnetic state, the response along the <111> direction is a linear contraction, displaying broken time-reversal symmetry, a characteristic of the piezomagnetic effect13. After reaching a certain temperature-dependent critical applied field strength24, the sample rapidly returns to the previous expansion response seen in the single direction applied field used in the first part of the experiment, then follows that path back to the same initial state at zero field. Looking at the integrated intensities of the diffracted peaks in Fig. 3b, the behavior of the peaks is very different from the single field direction. As seen there, the overall response is compressive at approximately −10 T (see point b in Fig. 3). But then, at increasingly negative field values, the data return abruptly to behavior similar to that seen in Fig. 2b, albeit with different relative intensities. The key point is that upon returning from −20 T to zero, the overall compressive strain is not recovered near −10 T (point f in Fig. 3). Immediately switching the field direction again shows this same new behavior, resulting in a butterfly-like loop13. The peak is clearly seen to split again (first as a broadening of the peak and then as a separate peak), this time in a positive field following immediately after negative pulses at 25 K (a positive magnetic field part is shown in the Supplementary Fig. 9). The FBG experiments have shown that critical field of about 11 T is expected at this temperature, as marked by the solid line in Fig. 3b (see Supplementary Fig. 11 for more details and ref. 24). This behavior provides direct evidence that the “butterfly” hysteresis observed in the FBG experiment can therefore be related to piezomagnetic domain evolution revealed by X-rays. Figure 3c shows the relative positions of the peaks (blue squares and red circles). The lines are visual guides. The higher angle and intensity “blue peak” (point b in Fig. 3) undergoes a switching from contraction to expansion at the critical field of ~−11 T. This demonstrates that the sign of the strain in the dominating peak is rapidly reversed in this region so that the blue peak matches the butterfly pattern observed in the magnetostriction taken at the same temperature. If so, then only magnetic domains corresponding to the blue peaks have sufficiently low critical fields to exhibit the rapid reversal of strain behavior expected from the piezomagnetic effect. It has been proposed that, in UO2 crystals subject to pulsed magnetic fields of reversed polarity, the magnetic subsystem switches between two states connected by time reversal13. In Fig. 3c, this switching phenomenon between the two states is directly observed. It has to be noted that while the coloring of the peaks is not unique, the key point is that upon an initial field reversal only a compressive strain is observed while on the subsequent return to zero field, two peaks are seen. This behavior is the same whether the sample was first trained in positive or negative fields. The two types of behavior apparently cross on initial field reversal, and therefore, that the most sensible labeling of the peaks is as indicated in the text above (see Supplementary Fig. 12 for more details).

In general, the piezomagnetic response, related to UO2’s fcc, $$Pa\bar 3$$ structure, can be explained using a model Hamiltonian that includes a strong magnetic anisotropy, elastic, Zeeman, Heisenberg exchange, and magnetoelastic contributions to the total energy. This simple model, where the degrees of freedom are the orientation of the magnetic moments of the four U atoms in the $$Pa\bar 3$$ unit cell and the shear components of the strain tensor, successfully reproduces the intriguing experimental observations (see Supplementary materials in ref. 13 for more details regarding the model used). Minimizing the total energy with respect to the elastic shear components allows obtaining their dependence on the applied magnetic field:

$$\varepsilon _{xy} = \frac{E}{{c_{44}a^3}}M_{{\mathrm{st}}}H_z$$
(1)

with similar expressions for the other components. Here $$E = 0.280{\,} {\mathrm{meV}} {\mathrm{T}}^{ - 1}$$ is the strength of the magnetoelastic interaction13, $$c_{44} = 60 {\,}{\mathrm{GPa}}$$ and a = 5.47 Å are the experimental shear elastic and lattice constants respectively, and $$M_{{\mathrm{st}}}$$ is the staggered magnetization that is different from zero below the Néel temperature. The change of sign of the staggered magnetization, which can be positive or negative depending on which AFM structure connected by time-reversal symmetry is stabilized, allows understanding the observed “butterfly” behavior of the magnetostriction (Fig. 3c).

The phenomenology of peaks splitting under applied magnetic field in UO2 is contrary to the more common case of magnetic detwinning. More typically, the application of magnetic fields to materials containing symmetry-related twin domains will result in a free energy difference between the domains, favouring one over the other and driving the material into a single-domain state. Conversely in the case of UO2, at least in the low-field regime, it is clear that the applied magnetic field drives opposing magnetostrictive responses in different TR domains without converting one domain entirely into the other. As seen in Fig. 1, only a single peak, shifted by magnetostriction, appears above the Neel temperature. Also, a single peak is observed above TN under high pressure (see Supplementary Fig. 2 for more details) and below TN in the absence of magnetic fields. Because the two-peak structure seen in Figs. 2 and 3 is only observed in the magnetic-field-polarized antiferromagnetic state, it is likely that the two magnetic domains consist of distinct TR versions of the 3-k structure. The piezomagnetism of UO2 (ref. 13) occurs as the 3-k structure of the ordered phase switches between TR versions under the action of applied magnetic fields. The red circles represent the negative magnetostriction of the −TR state in positive fields. That this state has not fully converted to the +TR state by a field of +20 T indicates that either this TR state is pinned, or that the switching field in this domain is larger than 20 T. The robustness of minority domains to field-conversion may be closely related to the exceptionally hard piezomagnetic response of UO2. Interestingly, the presence of two 180° antiferromagnetic domains has been observed before in another piezomagnet, MnF2, via polarized neutron tomography25,26. Furthermore, the domain configuration was determined to be sensitive to the strain condition of the specimen. It was suggested that for given local stress, the domain configuration (type A and/or B) depends on the direction of the applied magnetic field, such as the domain type being reversed when the magnetic field points out in the opposite direction25. A single domain state is possible if the stress is uniform over the entire crystal. This is a condition difficult to achieve, however, since an increase in the number of domains to attain small-scale variations of stress is prevented by the large wall anisotropy energy. In our experimental configurations, the strain effects in UO2 might play an even bigger role. A potential domain pinning and strain effect imposed by the stycast epoxy (see Fig. 3 in the Supplementary Information) could increase the piezomagnetic switching field so that the preparation of a single TR state becomes increasingly difficult. It has been shown that in other antiferromagnets, such as NiO crystal27, the domain structure has been very sensitive to mechanical stress and magnetic fields, where the twin wall dynamics appear to be limited by a spin-rotation energy loss. We speculate that the sensitivity of the switching field to mechanical stress might be related to the proximity of the mixed antiferromagnetic/antiferroquadrupolar state of UO2 to the non-magnetic, purely quadrupolar phase28. A virtual transition into the quadrupolar state allows for the conversion from one TR state to the other without the need to overcome an anisotropy barrier. In this scenario, internal constraints modify the local distortions that give rise to electric-field gradients and hence reduce the presence of the purely quadrupolar phase. Experiments are underway to explore the sensitivity of the switching field to uniaxial strain in this material. We would like to point out that the 888 peak splitting effect (the presence of two magnetic domains) observed in UO2 is real and does not originate in any way from the superposition of many fields–pulse combinations.

### Conclusion

We report the low-temperature single crystal X-ray diffraction studies of UO2 in pulsed magnetic fields. The high-resolution single crystal diffraction allows us to study details of subtle unit-cell distortions below and above the structural and magnetic phase transition. We unveil direct microstructural observations of the piezomagnetic and switching effects being a direct consequence of the non-collinear 3-k magnetic order that breaks time-reversal symmetry in a non-trivial way. These results will help in better understanding the strong coupling between magnetic and structural degrees of freedom in this important nuclear material. We also observe the presence of magnetic domains with distinct magnetic-field evolution in the magnetically ordered state of UO2, both when the field is applied repeatedly in a single direction and when the field direction is alternated. We argue that the origin of this behavior is related to the presence of the distinct TR versions of the 3-k structure. This behavior, especially the role of quadrupoles on the piezomagnetic properties, should be investigated in the future to further explore the details of the field-induced broken symmetries.

## Methods

It is worth noting that the quality of the data obtained in the experiment was only possible by bringing together a combination of unique factors. Through the use of an exceptional quality single-crystal sample (evidenced in very narrow diffraction peaks, see Supplementary Fig. 1), high-intensity synchrotron radiation, high, pulsed magnetic fields, and a fast prototype compound area detector, we were able to achieve high precision in our measurements during the short time frame of a pulsed fields (using one of the two choices of ~7 and 10 ms in total duration [start-to-finish] currently available). The single crystal of UO2 that was used in these studies is from the same batch as used in previous measurements (refs. 5 and 13). The crystal was aligned and cut into a plate approximately 500 × 500 × 200 µm, with the [111] crystal face normal to the plane. This allows us to study the longitudal magnetostriction of the UO2 crystal along the desired <111> crystolagraphic direction. The experiment was carried out in a dual cryostat single solenoid pulsed-magnet system described in ref. 29, in beamline 6 ID-C of the Advanced Photon Source at Argonne National Laboratory.

The detector used was a compound-type Mixed-Mode Pixel Array Detector (MM-PAD) with a silicon sensor30,31. This allowed for high dynamic range (up to 2 × 107 x-rays/pixel/frame at 15.85 keV), low background single photon counting, and a fast frame rate so that multiple images at different rising and falling field strengths during pulses could be obtained. The angular 2Θ range of the beam, where 2Θ is the angle between the incident and scattered rays, was calibrated on the area detector using the [888] peak of a silicon reference crystal, mounted at the center of sample theta rotation of the cryostat. The collection window for each frame was about 140 µs at a frame rate of 1 kHz, and the detector array was ca. 4 cm × 6 cm. For temperature scans in zero field, the sample was stabilized at various temperatures and then rotated through the Bragg condition at several hundred discrete sample theta positions as frames were taken at each position, then the frame intensities were added to obtain the full [888] peak profile. The center of mass in sample θ and Bragg peak position 2θ of the profile were found and converted to the change in d-spacing to find the change in unit cell length along the <111> direction. A similar process was used for measurements in the applied field, with fewer discrete sample θ rotation positions, limited by the long duty cycle of the magnet for each individual pulse, as elaborated on in the supplementary material (Supplementary Figs. 48).

The capacitor bank used in these experiments provided a maximum field of about 21 T and a pulse width of ~7 ms, as shown in Supplementary Fig. 5, with the ability to internally reverse the direction of the supplied current to the magnet coil. The reversed field part of the experiment was carried out in a similar manner, but after a first pulse in the positive direction (which was then discarded), successive pulses were taken in alternating directions along the <111> direction of the sample crystal. The resulting reversed-field direction frames were then analyzed the same way as the single direction pulses (Fig. 3b and c).