Rotating magnetocaloric effect in polycrystals—harnessing the demagnetizing effect

Climate change and the increasing demand for energy globally have motivated the search for a more sustainable heat-pumping technology. Magnetic refrigeration stands as one of the most promising alternative technologies for clean and efficient heat pumps of the future. The rotating magnetocaloric effect (RMCE) has previously been studied in materials with magnetocrystalline anisotropy due to its potential to improve devices by requiring only a single magnetic field region, but these materials are fragile and costly to obtain, making them inviable for applications. It has been shown that by exploiting the demagnetizing effect, an RMCE is, in fact, attainable in any polycrystalline magnetocaloric sample with an asymmetric shape, without requiring magnetocrystalline anisotropy. Using gadolinium as a case study, we provide a theoretical framework for computing the demagnetizing field-based RMCE and present thorough experimental verification for different magnetic field intensities and a wide temperature range. Direct measurements of the RMCE in gadolinium reveal that a significant adiabatic temperature difference (1.2 K) and refrigerant capacity (7.44 J kg−1) can be attained within low magnetic field amplitudes (0.4 T). Utilizing lower magnetic field intensities in a magnetocaloric heat pump can significantly diminish the need for permanent magnet materials, thus reducing the overall device cost, size, and weight, ultimately enhancing the feasibility of mass-producing such devices.


Introduction 1.Magnetocaloric refrigeration
The conventional magnetocaloric effect (MCE) consists in an adiabatic temperature change (∆T ad ) exhibited by a material when subjected to a change in magnetic field intensity.Magnetic refrigeration based on the MCE has been intensely studied over the past 25 yr due to its advantages over current heat pumping and refrigeration technologies, which are still dominated by vapor compression systems based on ozone layer-depleting and high global warming potential gas refrigerants [1][2][3][4][5].On the other hand, magnetocaloric refrigerators and heat pumps rely on solid-state materials which do not impact the environment through harmful gas emissions and have shown competitive performances, attracting significant attention and investment recently [6][7][8].Numerous magnetocaloric refrigeration prototypes have been reported, characterized, and reviewed over this period [4,9,10].Typically, the magnetic field sources in magnetocaloric devices are Halbach arrays of permanent magnets, namely Nd-Fe-B alloys, passively generating magnetic fields ranging from 0.8 T to as high as 2.4 T [10].However, the permanent magnet material is generally the most expensive component of a magnetic refrigeration prototype, so that decreasing the maximum field intensity required can drastically reduce size, mass, and cost of a device [11][12][13].Additionally, in conventional prototypes, since the magnetic field of permanent magnets is always 'on' , the magnetic field strength is modulated by moving the magnetocaloric material in and out of high and low field regions (or, conversely, moving the regions and keeping the magnetocaloric material fixed).Creating these zones through which the magnetocaloric material passes implies a large and complex arrangement of permanent magnets, and their corresponding movement driven by a motor also has a significantly detrimental impact on device efficiency, as power consumption operating the magnets can reach up to 78% of the total input power [14,15].
The rotating magnetocaloric effect (RMCE) consists in an adiabatic temperature change (∆T rot ad ) exhibited by a material when subjected to a change in the orientation of a magnetic field of constant intensity.The RMCE allows the use of simpler, more compact, and potentially more energy-efficient device architectures since they do not require both high and low magnetic regions to induce a ∆T ad [16][17][18].The RMCE's potential has been briefly explored since as early as 1971, mainly for cryogenic refrigeration, but re-emerged in 2010 with the report of the giant RMCE near room temperature [19], sparking a new wave of interest in the topic [20,21].The RMCE had virtually only been studied in single-crystalline materials with magnetocrystalline anisotropy, resulting in a positive ∆T rot ad when rotating the magnetic field from the hard to the easy crystal axis of magnetization [19,[22][23][24][25][26][27].The effect is not observable in isotropic polycrystalline materials since the random orientation of the crystalline grains results in an equal representation of the easy and hard magnetization axes in any given direction.However, the difficulty in producing large single-crystals has so far hindered its consideration for industrial devices and only a single report on the performance of a refrigeration device employing the RMCE in a single-crystal has been published [17].There have been some recent efforts in developing highly textured polycrystals exhibiting an RMCE, however, despite being easier to manufacture than single crystals, texturization still requires costly and time-consuming processing [28][29][30][31][32].

From the demagnetizing effect to a RMCE
Magnetocrystalline anisotropy is not the only physical basis available for inducing an RMCE.A distinct method, relying on the demagnetizing effect, is possible: changing the relative orientation of a sample with a non-spherical shape (e.g. a rectangular prism, thin plate, cylinder, etc) in regards to an external field (H ext ) of constant intensity.The sample will have different demagnetizing factors along the differently sized dimensions so changing its relative orientation to H ext (either by rotating the sample or the magnetic field) will change the internal field (H int ) intensity, since the demagnetizing field (H d ) intensity also changes.Since H int is the relevant field for determining magnetization and the MCE (as opposed to H ext ), such a change of H int will induce a ∆T rot ad .The demagnetizing field-based RMCE holds the unique advantage of occurring in any magnetocaloric sample with a rotationally asymmetric shape, including polycrystals without any magnetocrystalline anisotropy, providing a much more viable approach for applications.
Although the demagnetizing effect is often accounted for in magnetocaloric studies (e.g., to correct for H int ), to the authors' knowledge, the demagnetizing effect has only been shown to result in an RMCE in two previous works: first, in a 1984 patent by Barclay et al [33], measuring the RMCE in gadolinium with 0.3 T, and very recently in a 2023 research paper by Badosa et al [34], measuring the RMCE in gadolinium with 1.6 T.
In this work, we reveal the non-trivial magnetic field dependence of the demagnetizing field-based RMCE in gadolinium.This non-intuitive relationship suggests that a magnetocaloric device employing an RMCE should not aim for the highest possible magnetic field to maximize performance, in contrast with devices employing the conventional MCE, also benefiting the total cost of manufacture.Additionally, we provide a theoretical framework for computing the demagnetizing field-based RMCE given knowledge of the conventional MCE of a material, which can accelerate the assessment of this effect in additional material families using the already available results and experimental setups.We measured the RMCE directly under different magnetic field intensities up to 1 T, and indirectly through magnetization measurements up to 2 T. We compare the results with simulations of a spin 7/2 Ising model.The experimental details and results are preceded by a revision of the fundamentals of the demagnetizing effect, setting up the terminology and theory that enables understanding and modeling the demagnetizing field-based RMCE in this and future related studies.

Theory
A ferromagnetic or paramagnetic body magnetized by an applied H ext generates another external field (the stray field) parallel to H ext .The same magnetization generates an internal field opposite to the external one, the demagnetizing field, H d , which is related to the magnetization and H ext by the demagnetizing factor.The total internal field will then be given by difference between the opposing fields H ext and H d .
The general demagnetizing factor is a tensor that depends on the shape of the sample and on the position within it.However, it is possible to consider an effective demagnetizing factor as a position-independent constant, D, that quantifies the average H d along each orientation of a homogenously magnetized material.In this scenario, H d = DM (H int ), so that H int is given by where M is the field-dependent magnetization.In SI units, D is a unitless constant that assumes values between 0 and 1.
Let us consider a right rectangular prism with orthogonal sides, thickness a, width b, and length c.We will define the aspect ratio (AR) as the ratio between the length and the thickness of the prism (c/a).
The demagnetizing factors of right rectangular prisms have been thoroughly described previously [35].The demagnetizing factor along each side decreases with its size, so for rectangular prisms obeying a ⩽ c, then D a ⩾ D c , where D a and D c are the demagnetizing factors when the field is parallel to the sides a and c, respectively.
For positive functions of magnetization, decreasing D in equation ( 1) while H ext is kept constant leads to a larger H int .In the case of a rectangular prism, this means that changing the relative orientation of the magnetic field and the sample by 90 • , to become parallel to a larger side (e.g.rotate from H ext ∥ a to H ext ∥ c) leads to a decrease of D (from D a to D c ), and, consequently, to an increase of H int (from H int,a to H int,c ), since the H d field is reduced (from H d,a to H d.c ).Therefore, to increase H int , either H ext is increased (corresponding to the conventional MCE), or H d is reduced (corresponding to the demagnetizing field-based RMCE).
Figure 1 shows a schematic of the demagnetizing field-based RMCE in a rectangular prism with different dimensions.The rotation of the orientation of H ext to become parallel to a larger side reduces D, resulting in a positive internal field change (∆H int > 0) and, consequently, in a positive temperature difference if performed adiabatically (∆T rot ad ), or in a negative entropy change, if performed isothermally ∆S rot M ).The expression for ∆T ad is typically written only as a function of the final magnetic field intensity, H f , being assumed that the initial field intensity is zero.However, to describe the demagnetizing field-based RMCE, it is useful to write it as a function of the initial field, H i , which strongly influences it [36]: According to equation ( 2), the conventional MCE when applying H ext ∥ c and H ext ∥ a is ∆T ad (T, H int,c , 0) and ∆T ad (T, H int,a , 0), respectively.Since H int,c >H int,a , these quantities can be related to the RMCE by splitting the integral which gives ∆T ad (T, H int,c , 0) at H = H int,a : where the second term will correspond to the RMCE observed when rotating H ext ∥ a to H ext ∥ c.Note that the initial temperature of this term is shifted by ∆T ad (T, H int,a , 0), as represented by T ′ .Thus, the final expression for the RMCE in a material changing internal field from H int,a to H int,c is corresponding to the difference between the conventional MCE in each H ext orientation, with a shift in temperature.
Finally, this formulation also makes it clear that the demagnetizing field-based RMCE is only available so far as the conventional MCE is also available and differs between the two perpendicular orientations of field application.

Experimental and computational methods
Gadolinium is the benchmark magnetocaloric material due to its second-order magnetic phase occurring near room temperature, with Tc = 295 K, resulting in a significant MCE in this temperature range.Direct temperature measurements were performed via thermocouple, and indirect entropy change estimations via magnetization measurements in a SQUID magnetometer.All the experimental data were corroborated with simulations of an hexagonal close-packed (HCP) spin 7/2 Ising model.
Two rectangular prism-shaped polycrystalline gadolinium samples were used.Sample 1, with a = 2.1 mm, b = 7.0 mm, and c = 22.0 mm (AR = 10, D a = 0.695, D b = 0.234, and D c = 0.071, obtained through the measured dimensions as described in [35]) was used in the direct temperature measurements.The RMCE was induced by rotating the magnetic field orientation between parallel with the longest side (c) and the shortest side (a), to induce the greatest change of demagnetizing factor.A smaller sample, sample 2, was used in the magnetization measurements, with a = 1.2 mm, b = 1.0 mm, and c = 5.0 mm (AR = 5, D c = 0.096, D b = 0.490, D a = 0.414).Sample 2 was measured in two configurations: field parallel to the long side (c) and field parallel to one of the short sides (a) which had similar demagnetizing factor.Both samples were fabricated from commercial polycrystalline gadolinium (99.9% purity based on trace rare-earth analysis) 1 mm-thick, 25 × 25 mm 2 plates acquired from Sigma Aldrich (CAS number 7440-54-2).The plates were reshaped by cutting with a diamond-wire saw and polished with SiC abrasive paper for superficial oxide removal.X-ray diffraction measurements revealed no significant texture and negligible Gd 2 O 3 phase fraction.
The direct temperature measurements were conducted in a cryostat under vacuum (P < 10 −4 mBar) for appropriate temperature control.Temperature was measured with a 25 µm-diameter wire type-K thermocouple, whose sensing junction was glued in between two 1 mm-thick gadolinium plates (comprising the two halves of sample 1) with GE-varnish (see figure S1 of the supplementary material).The absolute temperature was calibrated through a reference junction in an ice bath.The magnetic field sources were an 8-segment Halbach array of permanent magnets, with a hollow cylinder shape, generating a 1 T field within, and an electromagnet, supplying controllable magnetic fields up to 0.8 T. The magnetic field applications and removals were performed manually consistently in less than one second, corresponding to field-application rates on the order of the ∼1 T s −1 .The field-rotation measurements were performed by manually rotating the field source in less than 3 s.These time intervals had a negligible effect on the adiabaticity of the measurements.Each ∆T ad (T i ) data point was extracted from the temperature versus time raw data by subtracting the temperature immediately before the field application/rotation (T i ) from the value measured around half a second after application/rotation (T f ), to avoid any influence of parasitic induction voltages in the thermocouple signal from the field change (see figures S2 and S3 of the supplementary material).The data under rotation within a 1 T field had a slightly increased uncertainty due to technical reasons related to the experimental setup, thus, to facilitate visualization, the raw ∆T ad (T i ) data is aggregated in 2 K intervals, with each point representing the average value and the maximum error bars in each direction.
The magnetization measurements were performed on a Quantum Design MPMS 3 SQUID magnetometer in fields up to 2 T. The isothermal magnetization versus field curves were obtained in two different orientations: field parallel to the larger side, c, and to the side a of sample 2. The measured absolute values of magnetization were corrected for geometric factors related to the sample installation according to a method previously described in [37].
The temperature and field-dependent thermodynamic properties of an HCP lattice of spin 7/2 Ising spins were estimated via Monte Carlo sampling of its energy-and magnetization-dependent Joint Density of States (JDOS) [38].The JDOS calculation for even relatively small lattices is challenging [39], but has some advantages over alternative approaches.Namely, the inputs of temperature, applied field, and the exchange interaction, J, used in the calculation of the desired thermodynamic properties are only included a posteriori to the JDOS estimate itself, avoiding the critical slowing down and clustering problems of more common importance sampling methods such as the Metropolis method.The full energy-magnetization phase space of the spin 7/2, 128 site, 12 nearest-neighbor HCP lattice here considered is quite large, with 897 × 18 817 (E × M) total states considered.To estimate the JDOS of this model lattice, the Flat Scan Sampling method [38] was employed, where each (E, M) macrostate was sampled 10 6 times.No energy or magnetization binning was employed.The magnetic exchange parameter J was chosen to lead to the experimentally observed T C value of Gd, resulting in J ∼ 5.3 meV.This corresponds to an energy difference between the AFM and FM arrangements of ∼63.5 meV atom −1 (considering 12 equivalent nearest-neighbor interactions), which is in good agreement with those estimated by full potential DFT calculations, ranging from ∼56 meV atom −1 [40] up to 69 meV atom −1 [41], with several reports in between [42,43].The total specific heat (with phonon and magnetic components), C p , was estimated considering the phonon contribution of a Debye model with T D = 169 K [44] and the magnetic specific heat of the HCP spin 7/2 Ising lattice.The simulated magnetization values, M, are functions of H int and temperature.The temperature and magnetic field dependence of both C p and M were considered to obtain ∆S M and ∆T ad and through equations ( 5) and (7), respectively.The demagnetizing factor was implemented by numerically solving equation ( 1) for the desired demagnetizing factor.For each H ext , equation ( 1) is solved to find the corresponding H int value, where M(H int ) is known from the simulations.In the case that DM (H int = 0) > H ext , there is no positive solution of H int .This condition can be related to the formation of magnetic domains in a material at low-field values in the ferromagnetic temperature region: though the magnetization of the material is microscopically significant, magnetic domains are formed to minimize the macroscopic field, so that H int is approximately null until H ext is strong enough to align the moments (once as given by equation ( 1) when H int = 0.The raw simulated data M(T, H int ) and C p (T, H int ) can be seen in figure S4 of the supplementary material.As expected from the previous discussion, the conventional MCE is maximum when the magnetic field is applied along the largest dimension of the sample (c), due to the lower demagnetizing factor in this configuration.This curve has a maximum of 3.55 K at 291.5 K which fits in the higher end of previously published values of the maximum MCE of Gd in a 1 T field, which, depending on experimental conditions, sample shape, sample purity, the sensor used, and how the sensor is assembled, spans from about 2.5 K [45,46], to around 3 K [47][48][49], up to 3.5 K [50], 3.6 K [51] and 3.8 K [52].The curve corresponding to the field applications along the shorter side exhibits lower values throughout the entire temperature range due to the larger demagnetizing factor, especially in the ferromagnetic region, where the larger magnetization yields a higher demagnetizing field.Remarkably, the measurements under field rotation also reveal very significant values, exhibiting a peak value of 1.27 K at 283 K, and displaying a flatter profile as compared to the curves of the conventional effect.Although the amplitude of the temperature difference on rotation is always lower than that obtained for the field applications parallel to the larger side, it surpasses the ∆T ad values obtained for H ext ∥ a at temperatures below 284 K.

Results and discussion
Together with the experimental data, figure 2 also shows the difference between the conventional MCE with H ext ∥ c and H ext ∥ a ('A-B', black dashed line), and the curve resulting from using equation (4) (black solid line) on the same data.Both curves show a noteworthy agreement with the experimental results obtained for the RMCE, as expected from equation (4).The two curves only differ by a horizontal shift of ∆T ad (T, H int,a , 0) (the MCE obtained for H ext ∥ a at each point), which is relatively small in the considered temperature window.This means that as a first approximation, the RMCE could be roughly considered from the direct subtraction of the conventional MCE along both orientations at each temperature and field value.Despite the noteworthy agreement, both the direct difference ('A-B') and the equation ( 4) curves in figure 2 show some discrepancy with respect to the experimental data obtained on field rotation around 290 K, however, this is likely just an experimental artefact as opposed to a phenomenon which our formulation does not account for.We include a detailed justification in section 4 of the supplementary material.
Figure 3 shows the experimental and simulated sets of magnetization isotherms as a function of H ext intensity obtained for both orientations of sample 2, revealing a very good agreement between the sets, and validating the use of the HCP spin 7/2 Ising model detailed in section 3. The demagnetizing effect is most noteworthy at low temperatures (when magnetization is largest) and low fields (when H ext and H d are comparable), which is reflected in the differing initial M (H ext ) slopes at low temperatures.The magnetization data can be used to obtain the field and temperature-dependent magnetic entropy change curves, ∆S M ( T, H f , H i ) .The magnetic entropy change is given by In contrast to what was shown in equation ( 4), the entropy change upon field rotation is given directly by without temperature shifts, since the temperature in equation ( 5) is constant.Additionally, by combining the C p obtained from the simulations, it is also possible to compute the simulated adiabatic temperature change curves from magnetization data through  The conventional MCE exhibits the expected rising profile with increasing magnetic field.The RMCE, on the other hand, approximates its maximum value at lower fields.The maximum measured ∆T rot ad under a 0.4 T field was 1.15 K, corresponding to 90% of the maximum measured value under a 1.0 T field (1.27 K).In the simulated data, this unforeseen field dependency was even more pronounced, with the maximum ∆T rot ad value for 0.4 T reaching 1.51 K, surpassing the maximum obtained value for 1 T, 1.39 K.This points the existence of a sweet-spot region in low fields where the RMCE can be most useful.It is particularly interesting from the point of view of applications to use lower fields since that represents a massive reduction of the required permanent magnet material.For example, reducing the desired maximum field of a cylindrical Halbach array with an inner diameter of 1.3 cm from 1 T to 0.5 T reduces the total volume of permanent magnet material by over 80%.Furthermore, only a rotary motion and a single magnetic field region are required, as opposed to the field modulation necessary for inducing the conventional MCE, enabling simpler device designs.
To highlight the magnetic field-dependent behavior, ∆T rot ad is also shown as a function of H ext intensity for different temperatures in figures 5(a)-(c).While the RMCE's intensity increases monotonously with magnetic field intensity above the phase transition temperature (at 300 K in figure 5(c)), it is more significant at lower temperatures, in which it reaches its maximum at low fields then plateaus or slightly decreases (at 280 and 290 K in figures 5(a) and (b), respectively), signaling a non-monotonous field dependency.This is also apparent in the simulated ∆S rot M curves, shown in the insets of figures 5(d) and (e), where the curve for 2 T is above the curves for 0.4 T and 1 T, but is surpassed by the curve for 1 T below 295 K and by the curve for 0.4 T in a small window around 288 K.
Figures 5(a)-(c) also compares our experimental and simulated results with the two previously published results on the RMCE of gadolinium [33,34].While [33] does not provide sample dimensions, [34] features measurements in a very thin gadolinium sample (a = 0.5 mm, b = 25 mm, c = 25 mm, AR = 50), with a sharper demagnetizing factor change induced by rotation (D a = 0.928 and D c = 0.035) as compared to ours.These sample dimensions should, in principle, lead to a more significant RMCE (about 40% more significant at 290 K and 1 T, according to our simulations), closer to the theoretical limit which would be displayed by an infinite plate, with D a = 1 and D c = 0. We suspect that this discrepancy may be due to experimental conditions (thermocouple dimensions, thermal coupling to the sample) or sample purity (as the authors mention due to the low magnetization measured), since the reported conventional MCE is also in the low end of what is expected for gadolinium, peaking at 2.54 K for a 1.1 T field application.
One important parameter for evaluating magnetocaloric materials is the refrigerant capacity (RC), which quantifies the total heat pumped in a thermodynamic cycle operating between two heat sink temperatures, T cold and T hot [53].It is defined as: which is readily generalized to the RMCE by substituing ∆S M for ∆S rot M .The RC is also useful as a means to quantify the MCE or RMCE's profile in temperature.The RC of the RMCE near room temperature, for T hot = 300 K, and two different cold-end temperatures, T cold = 280 K and T cold = 290 K is shown in figures 5(d) and (e).Similarly to ∆T rot ad , the RC obtained between 280 K and 300 K increases quickly with magnetic field and then plateaus, while that calculated between 290 K and 300 K increases more slowly with field, reflecting the attenuated dependence of the RMCE with field above the transition temperature.Crucially, these results show that a significant RC can be achieved even using low fields, with the experimentally obtained RC between 280 K and 300 K reaching 17.6 J kg −1 for 0.4 T and between 290 K and 300 K reaching 7.4 J kg −1 for the same field intensity.
Finally, to underline the relevance of the demagnetizing field-based RMCE for magnetocaloric heat pumping applications, we note that our experimentally measured ∆T rot ad values for a gadolinium sample are comparable to all those previously seen in the magnetocrystalline anisotropy-based RMCE materials near room temperature.If normalized to the magnetic field intensity, then our reported values for gadolinium surpass all previously obtained ∆T rot ad for single crystals/texturized materials, as they utilize larger magnetic field intensities.Furthermore, since the demagnetizing factor is highly dependent on sample shape, the demagnetizing field-based RMCE could be improved by increasing the sample's AR.For example, the maximum simulated value of ∆T rot ad for an infinite plate (with the minimum and maximum demagnetizing factors D c = 0 and D a = 1) is 2.09 K, 38% larger than that obtained for sample 1 (1.52 K).Thus, a comparable increase may be expected experimentally by increasing sample AR.Namely, reducing sample 1's thickness from 2 mm to 0.5 mm would increase AR from ∼10 to 40, corresponding to D c = 0.027 and D a = 0.896.In figure 6, we compare the maximum field-normalized ∆T rot ad observed experimentally in sample 1 and in simulations for an infinite plate with previously reported values of the magnetocrystalline anisotropy-based RMCE.

Conclusions
We have provided a theoretical framework through which a demagnetizing field-based RMCE in a polycrystalline sample of asymmetric shape can be understood.Direct temperature measurements of a gadolinium sample showed a maximum adiabatic temperature change on field rotation of 1.27 K within a 1 T field.The RMCE approaches maximum intensity within low field intensities, with the maximum experimentally measured adiabatic temperature change on field rotation reaching 1.15 K for 0.4 T, 90% of the maximum value observed for 1 T. Additionally, the field-normalized adiabatic temperature changes on rotation surpass previously reported values in single crystals or textured polycrystals near room temperature, with further improvements being possible by increasing the sample's AR.This motivates the use of low magnetic fields in applications, which could significantly reduce the amount of permanent magnet material used.
Relative to the MCE, the RMCE may contribute to simplifying the design and reducing the overall size, mass, and cost of magnetic refrigeration devices.Until now, only materials with magnetocrystalline anisotropy have been considered for this purpose, but the cost and the technical challenges associated with large single crystals manufacture have so far hindered advancements toward implementation.Since the demagnetizing field-based RMCE can be obtained in any magnetocaloric material family, it is a highly viable pathway for refrigeration devices, warranting further exploration in different materials, namely those presenting a giant magnetocaloric effect, and motivating additional efforts on designing device architectures and material shapes to exploit this effect.

Figure 1 .
Figure 1.Thermodynamics of the demagnetizing field-based rotating magnetocaloric effect and relationships between the relevant properties.(a) An entropy versus temperature diagram showing the adiabatic temperature difference resulting from the rotation of an external field (from Hext ∥ a to Hext ∥ c) with respect to an asymmetric sample, increasing its internal field (from H int,a to H int,c ).Vectors of the external, internal, and demagnetizing fields when (b) Hext is parallel to side a and when (c) Hext is parallel to side c.(d) Relationships between the different quantities, where Da and Dc are the effective demagnetizing factors for field application along a and c, respectively.

Figure 2
Figure2compares the directly measured conventional MCE for a 1 T field application along the largest side (H ext ∥ c), along the shortest (H ext ∥ a), and the RMCE for a field rotation from H ext ∥ a to H ext ∥ c.The difference between the conventional MCE obtained in each orientation is also shown.As expected from the previous discussion, the conventional MCE is maximum when the magnetic field is applied along the largest dimension of the sample (c), due to the lower demagnetizing factor in this configuration.This curve has a maximum of 3.55 K at 291.5 K which fits in the higher end of previously published values of the maximum MCE of Gd in a 1 T field, which, depending on experimental conditions, sample shape, sample purity, the sensor used, and how the sensor is assembled, spans from about 2.5 K[45,46], to around 3 K[47][48][49], up to 3.5 K [50], 3.6 K [51] and 3.8 K[52].The curve corresponding to the field applications along the shorter side exhibits lower values throughout the entire temperature range due to the larger demagnetizing factor, especially in the ferromagnetic region, where the larger magnetization yields a higher demagnetizing field.Remarkably, the measurements under field rotation also reveal very significant values, exhibiting a peak value of 1.27 K at 283 K, and displaying a flatter profile as compared to the curves of the conventional effect.Although the amplitude of the temperature difference on rotation is always lower than that obtained for the field applications parallel to the larger side, it surpasses the ∆T ad values obtained for H ext ∥ a at temperatures below 284 K.Together with the experimental data, figure 2 also shows the difference between the conventional MCE with H ext ∥ c and H ext ∥ a ('A-B', black dashed line), and the curve resulting from using equation (4) (black solid line) on the same data.Both curves show a noteworthy agreement with the experimental results obtained for the RMCE, as expected from equation (4).The two curves only differ by a horizontal shift of ∆T ad (T, H int,a , 0) (the MCE obtained for H ext ∥ a at each point), which is relatively small in the considered temperature window.This means that as a first approximation, the RMCE could be roughly considered from the direct subtraction of the conventional MCE along both orientations at each temperature and field value.Despite the noteworthy agreement, both the direct difference ('A-B') and the equation (4) curves in figure2show some discrepancy with respect to the experimental data obtained on field rotation around 290 K, however, this is likely just an experimental artefact as opposed to a phenomenon which our formulation does not account for.We include a detailed justification in section 4 of the supplementary material.Figure3shows the experimental and simulated sets of magnetization isotherms as a function of H ext intensity obtained for both orientations of sample 2, revealing a very good agreement between the sets, and validating the use of the HCP spin 7/2 Ising model detailed in section 3. The demagnetizing effect is most noteworthy at low temperatures (when magnetization is largest) and low fields (when H ext and H d are comparable), which is reflected in the differing initial M (H ext ) slopes at low temperatures.

Figure 2 .
Figure 2. Direct measurements of the magnetocaloric effect in a gadolinium sample with µ0Hext = 1 T applied along the larger side, c (in triangles), and along the shorter side, a (in squares).Direct measurements of the rotating magnetocaloric effect achieved by rotating the external field with constant intensity (µ0Hext = 1 T) from along side a to along side c (in circles) with vertical error bars.The black dashed line shows the difference between the magnetocaloric effect along both orientations, which coincides nicely with the curve obtained for the rotating magnetocaloric effect.The inset shows a schematic of sample 1's shape and size.

Figure 3 .
Figure 3. Magnetization versus external magnetic field measurements at different temperatures for a gadolinium sample with (a) Hext parallel to the larger side c, and (b) with Hext parallel to the short side a.The experimental data are the solid lines whereas the dashed lines represent the results obtained from the HCP spin 7/2 Ising model simulations.The inset in (b) is a schematic of sample 2's shape and size.

Figure 4 .
Figure 4. Direct and indirect measurements of the conventional and rotating magnetocaloric effect in gadolinium.The adiabatic temperature change observed when (a) applying an external magnetic field and when (c) rotating the external field.Analogously, the isothermal entropy change when (b) applying an external magnetic field and when (d) rotating the external field is shown.In dashed lines are the simulated values obtained from the HCP spin 7/2 Ising model, and in solid lines are the experimental results, both measured or calculated for sample 1.

Figures 4 (
Figures 4(a)-(d) shows the conventional MCE and RMCE, comparing the experimental and simulated values of ∆T ad , ∆S M , ∆T rot ad , and ∆S rot M obtained for different H ext intensities.There is a clear agreement between the experimental and simulated curves, despite some overestimation of ∆T ad and ∆S M in the simulated data around the phase transition temperature, 295 K.The conventional MCE exhibits the expected rising profile with increasing magnetic field.The RMCE, on the other hand, approximates its maximum value at lower fields.The maximum measured ∆T rot ad under a 0.4 T field was 1.15 K, corresponding to 90% of the maximum measured value under a 1.0 T field (1.27 K).In the simulated data, this unforeseen field dependency was even more pronounced, with the maximum ∆T rot ad value for 0.4 T reaching 1.51 K, surpassing the maximum obtained value for 1 T, 1.39 K.This points the existence of a sweet-spot region in low fields where the RMCE can be most useful.It is particularly interesting from the point of view of applications to use lower fields since that represents a massive reduction of the required permanent magnet material.For example, reducing the desired maximum field of a cylindrical Halbach array with an inner diameter of 1.3 cm from 1 T to 0.5 T reduces the total volume of permanent magnet material by over 80%.Furthermore, only a rotary motion and a single magnetic field region are required, as opposed to the field modulation necessary for inducing the conventional MCE, enabling simpler device designs.To highlight the magnetic field-dependent behavior, ∆T rot ad is also shown as a function of H ext intensity for different temperatures in figures 5(a)-(c).While the RMCE's intensity increases monotonously with magnetic field intensity above the phase transition temperature (at 300 K in figure5(c)), it is more significant at lower temperatures, in which it reaches its maximum at low fields then plateaus or slightly decreases (at 280 and 290 K in figures 5(a) and (b), respectively), signaling a non-monotonous field

Figure 5 .
Figure 5. Magnetic field dependence of the rotating magnetocaloric effect in gadolinium.The adiabatic temperature change obtained indirectly from the HCP spin 7/2 Ising model simulations (dashed lines) compared to the experimental results (square markers) for sample 1 and previous works [33, 34] at (a) 280 K, (b) 290 K, and (c) 300 K.The refrigerant capacity obtained experimentally and in simulations for sample 1 for a hot end temperature of 300 K and a cold end temperature of (d) 280 K and (e) 290 K.

Figure 6 .
Figure 6.Field-normalized values of the maximum RMCE observed near room temperature in single crystal/highly texturized samples from previous reports [19, 24, 44, 54-56] and polycrystalline gadolinium in this work.