Effects of uniaxial pressure on the spin ice Ho2Ti2O7

The spin ice materials Ho2Ti2O7 and Dy2Ti2O7 are experimental and theoretical exemplars of highly frustrated magnetic materials. However, the effects of an applied uniaxial pressure are not well studied, and here we report magnetization measurements of Ho2Ti2O7 under uniaxial pressure applied in the [001], [111] and [110] crystalline directions. The basic features are captured by an extension of the dipolar spin ice model. We find a good match between our model and measurements with pressures applied along two of the three directions, and extend the framework to discuss the influence of crystal misalignment for the third direction. The parameters determined from the magnetization measurements reproduce neutron scattering measurements we perform under uniaxial pressure applied along the [110] crystalline direction. In the detailed analysis we include the recently verified susceptibility dependence of the demagnetizing factor. Our work demonstrates the application of a moderate applied pressure to modify the magnetic interaction parameters. The knowledge can be used to predict critical pressures needed to induce new phases and transitions in frustrated materials, and in the case of Ho2Ti2O7 we expect a transition to a ferromagnetic ground state for uniaxial pressures above 3.3 GPa.


I. INTRODUCTION
Higly frustrated magnets display a rich variety of exotic groundstates and excitations 1-4 . A primary reason for this diversity is that the physical properties of the system are frequently determined by a delicate balance of weaker interactions of similar strength. Following Anderson's original classification 5 there are two major classes of frustrated systems; one in which the lattice plays a dominant role in frustrating the system, and another where frustration arises due to competing interactions. Recently it was realized that even in the first case, competing interactions may unexpectedly refrustrate the system for the spin ice materials Ho 2 Ti 2 O 7 (HTO) and Dy 2 Ti 2 O 7 (DTO) 6 . Therefore, it is of interest to find physical realizations of a wide variety of Hamiltonians, since even small alterations of the interactions may yield novel physical properties and phases.
In order to explore the vast parameter space of frustration there is a constant drive to synthesize new and promising frustrated materials 1,7,8 , each representing a unique set of interaction coefficients determined by, for example, crystal fields, ionic magnetic moments, inter-ion distances and atomic overlaps. A different approach to probing frustration is to alter the interactions of a given material. One way to do so is the application of external pressure, which alters the position and thereby also the atomic overlaps of the ions of the material. While there are many observations of novel states and phenomena induced by pressure [9][10][11] , progress is hampered by a number of experimental challenges that are exacerbated when studying phenomena at cryogenic temperatures. On the theoretical side there are open questions, since it is not easy to predict the effects of external pressure on, for example, exchange interaction parameters.
In the present study, we apply uniaxial external pressure to increase the parameter space we can explore using the parent material HTO. The physical properties of the multiaxial Ising materials HTO and DTO show a strong directional dependence on, for example an applied magnetic field 12,13 . Therefore we choose to use uniaxial pressure rather than isotropic hydrostatic pressure in this study. Our starting point are relatively straight forward magnetization measurements to determine the evolution of the exchange parameters under pressure and then we validate our results by comparing the derived model to neutron scattering experiments. In this study, we demonstrate the feasibility and limitations of describing the measurements of HTO under pressure using an effective model with a single pressure dependent parameter. While our application of relatively low pressure did not induce any changes in the order of the material, it shows the feasibility of exploratory experimental studies using magnetization measurements. Such measurements can be used to determine the evolution the model parameters and enable theoretical predictions of pressures necessary to alter the delicate balance of interactions in arXiv:2006.02756v1 [cond-mat.str-el] 4 Jun 2020 frustrated materials enough to cross phase boundaries and induce new phases.
Over the last 15 years HTO and DTO have become model compounds for studying classically frustrated systems featuring residual ground state entropy 14 , topological magnetic monopole excitations 2 , and slow lowtemperature dynamics 15,16 . The theoretical description in terms of the dipolar spin-ice model (DSM) [17][18][19] captures the experimental features at a quantitative level and DTO and HTO are now some of the best characterized frustrated materials. This makes them an ideal starting point for a systematic exploration of the evolution of frustrated systems under pressure. In an earlier study, measurements of the pressure induced changes in the magnetization of DTO were reported 20 . These results were first modeled theoretically using a nearest-neighbor exchange interaction model 21 . Recently, the effects of the dipolar interactions were included in the theory, and the measurements were modeled using a single parameter in the DSM 22 . Since DTO has a large neutron absorption cross section, neutron scattering measurements were performed on HTO, rather than DTO, but the observed changes in scattering intensity were modeled within the same framework.
One shortcoming of the previous study 22 was that magnetization measurements of HTO under pressure were missing, and it was not possible to connect the parameters determined from the evolution of the magnetization, measured for DTO, with the changes in the scattering intensity, determined for HTO 22 . In this study, we remedy the situation and report magnetization measurements on HTO, which we can use to independently determine the pressure dependent parameter in the theory for HTO. Our main result is that the independent analysis of both the magnetization and the neutron scattering measurements lead to the very similar values for the pressure dependent parameter. We therefore verify the feasibility of our approach to use more straight forward magnetization measurements to determine the evolution of model parameters before performing the more demanding neutron scattering experiments under pressure.

A. Magnetization measurements
High-quality samples are essential in high-pressure studies, and we synthesized HTO single crystals using the floating zone technique at the image furnace at Lund University. The crystals were grown from the bottom up with a growth rate of 3 mm/hr. Visual inspection of the as-grown crystals revealed no color change along the growth direction as seen by others. 23 We ascertained the high quality of the single crystals using x-ray and neutron diffraction 24 and will publish the details of the growth and resultant crystalline state elsewhere. To ensure an optimum shape for the (a) [ directions. The quality after cutting was asserted using x-ray diffraction. The crystalline direction of the major axis was ascertained after cutting the crystals. For each crystal, we applied uniaxial pressure and magnetic field along the cylinder major axis. We measured the magnetic moment at p 001 = p 111 = 0, 1.0 ± 0.2 and 1.5 ± 0.3 GPa for the [001] and [111] crystalline direction and at p 110 = 0, 0.5 ± 0.1, 1.0 ± 0.2 and 1.5 ± 0.3 GPa for the [111] crystalline direction. We also measured the DC susceptibility at pressures p 001 = p 111 = 0, 0.5 ± 0.1, 1.0 ± 0.2 and 1.3 ± 0.2 GPa using a probing field of 0.01 T along the cylinder major axis.
Using an epoxy resin (Stycast 1266, Ablestick Japan Co., Ltd.) we placed the crystals into a piston-cylinder type of pressure cell (CR-PSC-KY05-1, Kyowa-Seisakusho Co., Ltd.), which can be inserted into a superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS-XL). The combination of Stycast and the above uniaxial pressure enables the Poisson effect to be excluded and thus the shrinkage ratio in the direction perpendicular to the load can be ignored 25 . The quoted value of the pressure is at liquid helium temperature, after considering the thermal shrinkage. The pressure value at liquid helium temperature was estimated from the shift of the superconducting transition temperature of lead under applied pressure 26 . Due to the symmetric design of the upper and lower part of the sample chamber the magnetic background was negligible for the large magnetic signal of HTO. The measurements at different pressure were performed on the same respective crystal for each direction and a maximum field of H ext = 7 T was used.

B. Neutron scattering
Magnetic neutron diffraction experiments were performed at the Institut Laue Langevin (ILL) using the polarized diffuse scattering instrument, D7 27 , with nominal incident wavelength λ = 4.86 ± 0.1Å. We recorded neutron diffraction profiles for HTO in the (h, −h, l) scattering plane under uniaxial pressure along the [110] crystalline direction and with incident neutron polarization along the [110] crystalline direction. For this experimental study, we cut our synthesized HTO crystals into cylinders of diameter 3.0±0.05 mm and height 2.0±0.05 mm, still with major axis along [110] but with larger diameter and smaller height than the crystals used in the magnetization measurement. Pressure was applied using a CuBe anvil-type pressure cell with a CuBe window of minimal thickness of 2.5 mm for the entire scattering plane. Further details of the pressure cell are provided in appendix C and a detailed overview of the pressure cell will be published elsewhere. The force was calibrated prior to the experiment using an in situ transducer for deviations experienced at cryogenic temperatures. The pressure was deduced from the force, allowing for some uncertainty via friction of the piston. We measured both the neutron spin-flip and non-spin-flip scattering as a function of the sample rotation about the major axis of the cylinder ([110] crystalline direction). The data were cor- rected for detector and polarization analyzer efficiencies using standard samples of vanadium and amorphous silica, respectively 27 . Background measurements were performed at 300 K with an Al sample, closely matched in dimensions to the HTO sample, to determine scattering from the pressure cell. The background was entirely symmetric in the sample holder rotation angle and the rotation-averaged background at ambient pressure was subtracted from the measured signal.

C. Experimental results
Figures 1 and 2 show the measured magnetic moment and susceptibility at different pressures for the indicated directions. We see a change in the magnetic moment on the order of a few percent under the application of pressure. For the [001] measurement, Fig. 1(a), the magnetic moment is monotonically reduced for all fields when pressure is applied. For the [111] measurement, Fig. 1(b), the magnetic moment is reduced both in the high and the low field limits, but for an intermediate field around 1 − 2 T the magnetic moment increases as pressure is applied. For the [110] measurement, shown in Fig. 1(c), there is an overall decrease when pressure is applied, but the decrease is not monotonic at high fields, as it is for the other directions. There is also a large jump between the p 110 = 0 measurement and the p 110 > 0 measurements in the high and low field regions. We will discuss the implications of these features further in section IV A. Figure 2 shows the susceptibility, χ ext , measured with field and pressure in the [001] and [111] crystalline directions. There is a noticeable change in the susceptibility when pressure is applied to the sample, but compared to the magnetic moments in Fig. 1 the effects are much less clear. For example, in our measurements using a small applied field of 0.01 T, the recorded change in the susceptibility is not generally monotonic with respect to the applied pressure.  Figure 3 shows the spin-flip neutron scattering structure factor, S(Q), at ambient and at an applied pressure of 0.35 GPa at T = 1.5 K 28 . The noted pressure was applied at 300 K and the subsequent pressure change as a result of thermal contraction could not be determined in situ. However, prior temperature dependent calibration measurements of the pressure cell provides an indication that the noted pressure is approximately correct. A uniaxial pressure cell with in situ pressure determination for diffuse neutron scattering and polarization analysis is under development. In Fig. 3(a,b), the measurements are taken with two different samples, with equivalent sample dimensions, as the first sample broke when pressure was applied. The samples were cut from the same single crystal. We note a change in the intensity of the diagonal satellite peaks (h ≈ ±1.5, l ≈ ±1.5) when pressure is applied. To better demonstrate these subtle changes, we have integrated the shaded regions over Q, see Fig 3, and fitted a Gaussian curve to the integrated data. We find that the intensity of the diagonal satellite peaks increases by about 4% when pressure is applied, and discuss this further in section IV B.

A. Model and simulation method
A theoretical model for the evolution of the interactions in classical spin ice under uniaxial pressure was proposed in an earlier investigation 22 . We adapt this model to the current measurements of HTO and refer to it as the dipolar spin-ice pressure model (DSPM). The DSPM is an extension of the standard dipolar spin-ice model (s-DSM) 29 . The Hamiltonian for the DSPM for classical unit Ising spins S i on the pyrochlore lattice is defined as (1) The strength of the dipolar interactions is taken to be D = 1.41 K 30 , a is the nearest neighbor distance, r i is the position of spin S i and r ij = r i − r j . The angled brackets in the first sum denotes summation over nearest neighbors. The antiferromagnetic nearest neighbor interaction J(i, j) is caused by oxygen-mediated superexchange. The value of J at ambient pressure is set to J p=0 = 1.56 K 31 . In the DSPM it is assumed that J ⊥ = J p=0 = 1.56 K if the neighbors i and j are in a lattice plane perpendicular to the direction of applied pressure and J(i, j) = J if they are not, (J 1 ,J 2 for pressure in the [110] direction as there is less symmetry) see Fig. 4. Compression of the crystal is modeled with zero Poisson ratio, which is also the case in the current experimental setup for the measurements of the magnetic moment 25 . As the lattice is compressed, the local Ising axis of the spins is modeled to keep pointing towards the centers of the compressed tetrahedra. The dipolar interaction becomes stronger along the direction of compression, growing as |r ij | −3 with decreasing distance. We define the lattice compression κ as the relative length contraction along the direction of pressure. At high fields, the saturated magnetic moment is dependent only on κ, due to the change in angles of the Ising moments. The high field saturated value of the magnetic moment therefore sets the lattice compression, and we use the low field data to determine exchange interactions.
Monte Carlo (MC) simulations using the Metropol-isHastings algorithm and single spin flip updates are used to investigate a number of different system sizes with periodic boundary conditions. Since the lowest temperature used in the experiment was 1.5 K, monopole excitations are still prevalent and loop flips 32 are not needed. We use the 16-particle standard cubic unit cell. All supercells are cubic of size L 3 unit cells, L ∈ [1, ..., 8]. Ewald summation is used to effectively account for the longrange conditionally convergent dipolar contributions 33 .

B. Theoretical misalignment effects
The DSPM describes the changes in the magnetic moment observed in measurements 20 of DTO under uniaxial pressure 22 quite well. Particularly when pressure and field is applied along the [001] or [111] crystalline directions the model works very well. However, when pressure and field is applied in the [110] crystalline direction for DTO, anomalous behavior is observed which cannot be explained by the DSPM 20,22 , since the pressure-induced change in the magnetic moment does not saturate at high fields 20,22 .
In this study, we attempt to explain this previously noted discrepancy for the measurement in the [110] crystalline direction and speculate that it can be an experimental artefact due to misalignment of the crystal. Next, we outline a brief discussion of what effects might be expected from crystal misalignment.
When measuring the magnetic moment of a single crystal under uniaxial pressure, there are three quantities that need to be aligned: the crystalline direction, the direction of the uniaxial pressure and the direction of the magnetic field.
In order to investigate the influence of misalignment, we consider the possibility that the crystalline direction matches the direction of pressure perfectly, and that the magnetic field is misaligned with respect to the direction of pressure. This assumption minimizes the number of new parameters introduced to the theoretical model, and we believe that it still captures the essential effects of misalignment. In the experiment, on the other hand, we may expect the misalignment between the crystal direction and the applied pressure to be more significant. Since the crystal is surrounded by more compressible Stycast it is also possible that this misalignment may change during a series of compressions. See appendix A for fur-ther discussion. . We assume that the misalignment angle δϕ can vary with application of pressure. For a field in this direction there are two degenerate ground states for perfect alignment (double headed arrows in (a)), and only one ground state when the field is misaligned (b). We denote the misalignment at ambient and applied pressure by δϕ 0 and δϕ p respectively.
Misalignment has particularly strong effects when the measurement is conducted with pressure along the [110] direction 34 . This is because half of the spins in the material have a local Ising axis perpendicular to the [110] direction, so that at high fields, already a small misalignment will strongly affect the dynamics of these. We therefore adjust the applied field so that it is misaligned by an angle δϕ with respect to the [110] direction and remains perpendicular to [001], as this is the simplest way for the field to couple to the previously unaffected spins. Fig. 5 illustrates the scenario. The magnetic moment is measured along the direction of the field, and when the field is misaligned we start to observe the dynamics of the previously perpendicular spins. This is not the case when we measure along the [ Fig. 7(c).

C. Demagnetizing corrections
In order to account for the macroscopic boundary effects, we perform a demagnetizing transformation. The samples used were cylindrical with an aspect ratio of γ ≡ height diameter = 1.50 ± 0.05 at ambient pressure. How-ever, as the the sample is compressed, the aspect ratio decreases. By fitting κ to the change in saturated magnetic moment we determine γ. An important aspect, often overlooked, is that the demagnetization factor N is a function of both γ and the internal volume susceptibility χ V int , N = N (γ, χ V int ) as has been recently calculated and verified 35,36 . In high-susceptibility materials (χ int > 1 ), such as HTO, the susceptibility dependence of N is very significant and we include it in the analysis. We perform a demagnetization transformation according to where the intensive quantities M V and χ V are the magnetization and volume susceptibility respectively. It is important to differentiate these from the external extensive quantities M = M V V (γ(p)) and χ ext = χ V ext V (γ(p)), measured in the experiment. The Monte Carlo calculations, performed using periodic Ewald boundary conditions, yield internal quantities.
Since the susceptibility of classical spin ice is isotropic the susceptibility dependence of N calculated in Ref. 35 can be used for HTO, as demonstrated in Ref. 36. Using the tabulated values of N (γ, χ V int ) we perform an interpolation using cubic splines 35 . As an example, we show the temperature dependence of N for an HTO cylinder of aspect ratio γ = 1.5 in Fig. 6, where we use the experimentally measured intrinsic susceptibility for HTO 37 at zero field to determine the temperature dependence.

D. Calculation of susceptibility
For the susceptibility measurements, we define the relative change in the susceptibility with respect to the applied pressure as ∆χ ext /χ ext ≡ (χ p ext − χ 0 ext )χ 0 ext . We measured the relative change ∆χ ext /χ ext as a function of temperature. This quantity is highly sensitive to changes in J and κ, and in order to gain some physical insight, we consider the high-temperature limit, T → ∞. In this non-interacting limit the relative change depends only on κ. For the crystalline directions along which measurements were performed we find that where the superscripts denote the crystalline direction of pressure and field.
From these formulae, we see that in both directions, the susceptibility at high temperature is reduced under application of pressure. This is a purely geometrical effect stemming from the fact that the Ising moments tilt away from the axis of pressure when the lattice is compressed.

E. Calculation of magnetic structure factor
We calculate theoretical predictions for the spin-flip magnetic structure factor according to where the scattering wavevector is denoted by Q, the normalized polarization vector of the incident neutron beam is given by P ⊥ Q. The component of the spin perpendicular to the wave vector is defined as N is the number of particles in the supercell and f (|Q|) is the magnetic form factor for Ho 3+ . The angled brackets · · · denote the thermal average which is calculated using the MC method.

A. Magnetization and susceptibility
In Fig. 7 Fig. 7(a,b), κ has been set so that the change in calculated saturation magnetic moment agrees with that of the experiment. With this constraint, there is only one free parameter, J , which we determine from the best fit for each case. We find that for the [  In the [111] direction the DSPM predicts a spurious shoulder at 1 T, and the experimentally observed minimum centered at H ext = 3.5 T is not accounted for in the model. Figure 7(c) shows the change in magnetic moment for the [110] measurement. We note that the change has hardly saturated as a function of field. Seemingly, we need a field of about 7 T to saturate the change in the [110] direction, higher than the 5.5 T needed for the [111] direction. This feature is not possible to reproduce in the DSPM in which the field needed for [111] saturation is higher than that needed for [110] saturation, since the saturated [111] state requires that we break the two-in two-out ground state of spin ice 38 .
As discussed in section III B, we speculate that the high-field unsaturated behavior can be due to misalignment of the crystal, since measurements are particularly sensitive in the [110] direction. In Fig. 7(c) we illustrate the influence of misalignment. We first fit a curve with κ = 1%, as an estimate of the compression based on the other directions at 1 GPa (0.8% and 1.5% for [001] and [111] respectively). In the DSPM, the change in magnetic moment will saturate at about 1 T (same order of magnitude as in the [001] case) regardless of the parameters used. For simplicity we assume that J 1 = J 2 = J which gives a rough value of J = 1.2 K when we fit this single parameter to match the sub 1 T experimental results. This DSPM fit is shown as the dashed line in Fig. 7(c). For all fields above 1 T, the DSPM gives a constant saturated change in magnetic moment dependent only on κ. Clearly this is not what we see in the experiment, where the change keeps varying up to fields of about 7 T.
If we introduce crystal misalignment according to section III B, we can partially reproduce the high-field trend observed in the measurement. Within this framework, we have two additional parameters δϕ 0 , δϕ p as discussed in section III B for the misalignment of H with respect to the [110] direction, at zero(0) and applied pressure(p) respectively. To get a decrease in magnetic moment when pressure is applied, we find it necessary that the field couples stronger to the perpendicular spins in the reference measurement. Hence, to keep the model as simple as possible, we set δϕ p = 0 • . We then adjust δϕ 0 in order to fit the experimental curve above 1 T. We find that δϕ 0 = 0.75 • gives the best fit. We do not get the exact same features as those observed, but we do demonstrate that the decrease in magnetization above 1 T could be due to misalignment. For further discussion of more elaborate misalignment models under pressure we refer to appendix A.
Finally, we use our obtained values of J and κ to predict the change in susceptibility as a function of temperature. In Fig. 8 we show the measured relative change in external susceptibility plotted against temperature for an applied pressure of 1 GPa. The theoretical predic-  Fig. 7(a,b), respectively. In contrast to the measurements of the magnetic moment, the susceptibility measurements are more challenging due to the low field used and a higher sensitivity to uncertainties in the demagnetizing factors. Due to the fluctuations in the [001] measurement, Fig. 8(a), we include a trend line by fitting rational polynomials to the susceptibility. The qualitative curve shape is the same for both theory and experiment, but they differ by an overall shift along the vertical axis. At the lowest temperature T = 1.83 K we expect that the relative change in susceptibility should coincide with the H ext → 0 limit in the relative change in the magnetic moment Fig. 7(a), which for 1 GPa is −2%. There is therefore a discrepancy between the two experimental measurements. From theory we also expect that the susceptibility should be reduced at high temperature when pressure is applied, see Eq. (3). Since the susceptibility measurements in Fig. 2 were nonmonotonic in pressure we suspect a systematic error in the experimental data and note that if we shift the susceptibility measurement performed under pressure by −2% the data matches the theoretical prediction as well as the magnetization measurement performed at T = 1.83 K.
For the [111] direction there is a more significant mismatch between theory and experiment. In particular, theory would predict a maximum in ∆χ/χ near 5 K, which is not present in the experiment. We would also expect from theory that the relative change should saturate to a lower value. Indeed, with the assumption of Ising spins it directly follows from Eq. (3) that ∆χ/χ must reach a negative value at high temperature. The discrepancy is most likely due to insufficient accuracy in the present susceptibility measurements.  Fig. 9, simulated at T = 1.5 K and p = 1.5 GPa. In the right half we show the change in the experimental 39 scattering at p = 2.2 GPa and T = 1.5 K measured in previous work 22 . In the outer region we see eight patches of increased intensity which have been marked and coincide with the theory prediction. In the inner region there are two regions of increased intensity not seen in the the experiment, but we note, as mentioned in previous work, that the experimentally measured signal was poorly sampled at wave vectors shorter than 1.5Å -1 due to windows in the scattering plane of the previous CuBe pressure cell 22 used in that neutron scattering experiment. A key result of this study is therefore that the scattering profile calculated from the parameters obtained from the fit to the magnetic moment, Fig. 7(a), matches the best profile that can be obtained by freely adjusting J to the experimental data, as was done in the previous study 22 . Furthermore, we use the DSPM to calculate the spin-flip structure factor in the (h, −h, l) plane with pressure along [110], shown in Fig. 3. We take the parameters fitted to the magnetic moment, Fig. 7(c), and calculate S(Q) from Eq. (4). In order to better show the changes in scattering intensity, we integrate the signal around one of the satellite peaks as shown by the shaded region in Fig. 3. In Fig. 10 we see Gaussian fits to the experimental 28 data (solid lines) and note a clear increase of about 4% at the top of the diffuse peak. The theoretical estimate from the fit to the magnetic moment (J = 1.2 K,κ = 1%) is shown together with the theoretical prediction at zero pressure (dashed lines). We note that the theory captures the increased peak intensity and conclude that the model describes the observed phenomena both for magnetization and neutron scattering measurements in several crystalline directions.

C. Evolution of the model parameters
Using the relatively straight forward magnetization measurements we can obtain the model parameter dependence on pressure. In Fig. 11 we depict the pressure dependence of J and κ for the [001] direction. We include the current data points at pressures of 1.0 GPa and p = 1.5 GPa, as well as the result of the previous neutron study at an applied pressure of 2.2 GPa 22 . The three data points show a near-linear dependence on pressure, and all three points lie on a curve in the (J , κ)-space for which the system is on the border between two different types of dipolar chain 29 ground states 22

V. CONCLUSIONS
We have performed measurements of the field-induced magnetic moment, magnetic susceptibility and neutron structure factor of HTO under applied uniaxial pressure. Through extensive MC calculations we demonstrate that a dipolar spin ice model, with a pressure-tuned nearest neighbor interaction is able to capture the most essential features of the measurements of HTO. The framework is extended to include effects of misalignment, and we have found that misalignment can, to some extent, describe the anomalous effects observed in the magnetic moment for the [110] direction in both HTO and DTO.
The T -dependent pressure induced changes in the susceptibility turns out to be a more sensitive quantity to both measure and model than the field-induced magnetic moment. The low field used experimentally results in a weak signal which is easily overshadowed. Theoretically, the susceptibility of spin ice has proven to be a sensitive function of the intrinsic competing interactions 40 , and the measurements are sensitive to sample shape 36 , with the optimal sample size probably spherical. To perform susceptibility measurements on a spherical sample under pressure is highly challenging, and we therefore have to contend with our present results. Still, the qualitative curve shape for the susceptibility in the [001] direction is captured rather well by our model, but there are significant discrepancies in the [111] measurement.
Our main result is that the model parameters derived from the measurement of the magnetization also captures the most salient features of the pressure induced change in the neutron scattering structure factor. That the same model describes both bulk properties and spin-spin correlation functions lends credibility to the theory. Therefore we hope that using relatively straight forward magnetization measurements to determine the pressure dependence of interaction parameters can prove useful also when it comes to other classes of frustrated materials. Increased theoretical predictive power supporting demanding neutron experiments under high pressure would benefit many investigators in the field. With new, and more intense, neutron sources under construction we expect this to be a research topic of increasing importance in the near future.
In the neutron scattering experiment for the larger 3 mm diameter crystals, the nuclear Bragg peaks observed on D7 are consistent with the alignment. However, the instrumental parameters of D7: incident beam size, divergence and detection resolution, do not provide a very accurate determination of the absolute alignment. The nature of the neutron scattering experiment also prevents the use of stycast to mitigate against the Poisson expansion. To include such expansion of the sample in the simulation would introduces additional unknown model parameters, which from these few measurements is hard to determine. We therefore chose not to include this effect, and we find that this approximation still gives valid prediction for the neutron scattering intensity. However, we mention that the physics that can be reached by including this effect can also be reached by allowing J ⊥ to vary, and it will not introduce any new effects, apart from having more free parameters allowing for an easier fit to data. The neutron scattering uniaxial pressure cell has been developed to enable neutron scattering experiments in the cold energy spectra range, 2 λ 20Å, for diffuse magnetic scattering profiles, inclusive of weak inelastic scattering features, with polarisation analysis. The uniaxial pressure cell should be able to provide pressures up to 2 GPa at cryogenic temperatures. As such, the requirements include a scattering window that covers a wide angular range with a very clean background profile and a non-magnetic cell that enables polarisation analysis. These requirements place stringent restrictions on the materials for the manufacturing of the cell and lead us to focus on the optimization of an anvil type cell. An engineering overview of the cell employed during the D7 experiment is shown in Fig. 14. The main body is manufactured out of CuBe with non-magnetic stainless steel anvils. Force is applied at room temperature and recalibrated using a calibration profile developed within our team. Pressure is deduced from the known contact area. The calibration profiles are not perfect and may lead to some uncertainty in the exact pressure applied at the lowest temperatures. Complete details of the cell will be published elsewhere and an uniaxial pressure cell with in situ pressure determination is under development ensuring we maintain the aforementioned parameters.
FIG. 14: Engineering drawings and overview of the uniaxial pressure cell.