Abstract
We use the recentlyproposed compressible cell Isinglike model to estimate the ratio between thermal expansivity and specific heat (the Grüneisen parameter Γ_{s}) in supercooled water. Near the critical pressure and temperature, Γ_{s} becomes significantly sensitive to thermal fluctuations of the orderparameter, a characteristic behavior of pressureinduced critical points. Such enhancement of Γ_{s} indicates that two energy scales are governing the system, namely the coexistence of high and lowdensity liquids, which become indistinguishable at the critical point in the supercooled phase. The temperature dependence of the compressibility, sound velocity and pseudoGrüneisen parameter Γ_{w} are also reported. Our findings support the proposed liquidliquid critical point in supercooled water in the NoMan’s Land regime, and indicates possible applications of this model to other systems. In particular, an application of the model to the qualitative behavior of the Isinglike nematic phase in Febased superconductors is also presented.
Similar content being viewed by others
Introduction
Because it is biologically fundamental to the maintenance of all life, liquid water is one of the most important substances on the planet. Water exhibits a number of anomalous physical properties (see Fig. 1, refs^{1,2} and references cited therein), and over the last 25 years, much attention has been paid to the study of water on its socalled supercooled phase. The initial work on supercooled water in 1992 used molecular dynamics simulations^{3}. A subsequent research has explored the NoMan’s Land region in the phase diagram (see Fig. 1 and ref.^{2}). This topic has generated much debate (cf. refs^{2,4,5,6,7} and references therein).
One scenario describing supercooled water assumes the existence of two liquid phases at lowT, being each phase associated with either a high or lowdensity^{6}. Recently fs xray scattering was used on water droplets to determine the maximum isothermal compressibility, the correlation length, and the structures of water and heavy water. Experimental evidence of a secondorder critical endpoint in the Widom line was found^{7}, but no clearcut divergence in the quantities was observed. Here we study the liquidliquid critical point for supercooled water by analysing the behavior of the Grüneisen parameter (\({{\rm{\Gamma }}}_{s}\)), see Methods. Such approach has already been successfully applied to other systems^{8,9,10,11}. In the case of supercooled water, we find evidence supporting a liquidliquid critical point. We use a recentlyproposed compressible cell Isinglike model^{12,13,14} to obtain \({{\rm{\Gamma }}}_{s}\). Essentially, the model proposed in ref.^{12} assumes the coexistence of two possible volume values for each cell on the lattice, represented by \({v}_{}={v}_{0}\) and \({v}_{+}={v}_{0}+\delta v\). These volumes are responsible for the change between high and lowdensity liquids in the system. The two free volumes, i.e., the volume where a particle inside the cell can move, for each cell are \(0 < {\dot{v}}_{+} < {v}_{+}\) and \(0 < {\dot{v}}_{} < {v}_{}\) and their ratio is \(\lambda ={\dot{v}}_{+}/{\dot{v}}_{}\), see Methods.
It is worth mentioning that a structurally similar model was originally proposed in ref.^{15} and employed in the study of a large number of fluids, cf. ref.^{16}. Taking such studies into account, we emphasize that the originality of the present work lies not only on the choice of the model for the analysis of supercooled water, but also on its application in the analysis of \({{\rm{\Gamma }}}_{s}\) to a regime where experimental results are lacking as a consequence of the rapid crystallization of water under such conditions^{17}. Our analysis of \({{\rm{\Gamma }}}_{s}\) is complemented by the discussion of the pseudoGrüneisen parameter (\({{\rm{\Gamma }}}_{w}\))^{18}, see Methods.
Results and Discussion
The obtained expressions for the observables (see Methods), namely the isobaric thermal expansion α_{p}, the isobaric heat capacity c_{p} and the isothermal compressibility \({\kappa }_{T}\) together with \({{\rm{\Gamma }}}_{s}\) enable us to study the behavior of the system on the verge of the critical point. Because the equations for c_{p}, α_{p}, \({\kappa }_{T}\) and T depend on the pressure and volume of the system, they constitute a parametric system. This characteristic of the model prevents us from obtaining an analytical expression for v. Note that Eq. (3) (Methods) clearly indicates a transcendental equation for v. We thus analyze the behavior of the various observables by varying v, which causes variations in T. We fix the critical point parameters by employing the corresponding expressions, see Methods. The parameters were adjusted^{12} so that \({T}_{c}\simeq 180\,{\rm{K}}\), which is in the No Man’s Land region^{2}, but a bit lower than T_{c} reported in ref.^{7} and that shown in Fig. 1. Note that the free parameters of the model reported in ref.^{12} could be changed in order to explore other systems of interest. Here, we focus on the analysis of \({{\rm{\Gamma }}}_{s}\) and \({{\rm{\Gamma }}}_{w}\) (see Methods) to the supercooled phase of water. Also, for the sake of completeness, we stress that we have recalculated both thermal expansion and specific heat, already reported in ref.^{12}. Figure 2(a,b) show the p–v phase diagram for a range of temperatures and the T–v diagram. Note that when T = 0 K the resulting mapping \(p(T=0,v)\) is a straight line. This is obtained using Eq. (3). When the temperature is high, the pressure for \(v\approx {v}_{0}\) is higher than the case for low temperatures. For \(v\approx {v}_{0}+\delta v\), however, higher temperatures decrease the pressure for fixed values of v. Figure 2(b) shows that in a particular range of values of volume, for given pressure values, physical temperature values are inaccessible. Figure 2(a) shows that the point where the pressure is the same for every temperature value (blue vertical line) is the limiting value for the volume (v) for which physical values of the temperature are obtained. As discussed above, we cannot analytically obtain an expression \(v(T,p)\) because Eq. (3) is transcendental in v. Hence, we have a mapping of these physical quantities [see Eq. (4)], and we can find the corresponding v and T values for each pressure value (p). The same holds true for any other desired order of these three parameters. Figure 3(a–f) depict the behavior of the observables for the system considering 16 pressure values, varied in uniform steps from p = 1.17 kbar to 0.17 kbar. The panels a) and b) show the observables α_{p} and c_{p}, which were presented and discussed in ref.^{12} for a different range of pressure values. Here, we focus on an analysis of these observables near the critical point. Remarkably, the absolute values of α_{p} and c_{p} increase significantly for \(p={p}_{c}\) and \(T={T}_{c}\), a fingerprint of a phase transition and/or critical point. Figure 3(c) shows the behavior of \({{\rm{\Gamma }}}_{s}={\alpha }_{p}/{c}_{p}\), see Methods. Note the effect of pressure on \({{\rm{\Gamma }}}_{s}\) and its distinct behavior upon approaching the critical point, when comparing with c_{p} and α_{p}. In the immediate vicinity of the critical point, \({{\rm{\Gamma }}}_{s}\) is extremely sensitive to thermal fluctuations. Figure 3(d) shows the socalled pseudoGrüneisen parameter \({{\rm{\Gamma }}}_{w}={w}^{2}{{\rm{\Gamma }}}_{s}\)^{18} (see Methods). Note that for the data set corresponding to \(p={p}_{c}\), \({{\rm{\Gamma }}}_{w}\to 0\) for \(T={T}_{c}\). The vanishing of \({{\rm{\Gamma }}}_{w}\) can be understood in terms of the behavior of the normalized speed of sound w/w_{c} (where \({w}_{c}\approx 6.513\,{\rm{m}}\,{{\rm{s}}}^{1}\)), shown in Fig. 3(e). In the vicinity of the critical point, \({w}_{c}\to 0\), whereas its value for temperatures far from T_{c} increases to approximately 100w_{c}. Physically, this finding suggests that, near the critical point, the propagation of sound waves is significantly suppressed. Interestingly, an anomalous behavior of the sound velocity was also observed close to the Mott critical endpoint in strongly correlated electronic systems and associated with a diverging compressibility of the electronic degrees of freedom^{19,20}. Figure 3(f) shows that the compressibility also presents an enhanced behavior near the critical point. Our findings are in perfect agreement with those reported in ref.^{21} for the various observables.
The Maxwellrelation \({(\frac{\partial v}{\partial T})}_{p}=\,{(\frac{\partial S}{\partial p})}_{T}\) and the negative thermal expansivity shown in Fig. 3 indicate that the entropy (S) of the system is enhanced when approaching the liquidliquid critical point, i.e., by applying pressure, the high and lowdensity phases mix and the entropy increases. It is noteworthy to point out that we also find this in the finiteT critical endpoint reported for molecular conductors^{8,9} and the quantum critical points in heavyfermion compounds^{22,23}. The high and lowdensity phases produce two different energy scales. Because the degree of Hbonding depends on temperature and pressure, a scaling cannot be applied successfully^{24,25}. Reference^{6} indicates that water molecule interactions create an open Hbond structure that has a lower density than other configurations. We can capture the energy scales associated with the Hbond configurations that correspond to the low and highdensity phases using a compressible Isinglike model and two accessible system volumes. In particular, the capture of the energy scales associated with Hbonds is, in our analysis, represented by the vanishing of one of the possible volumes associated with the sites. Using the Landau theory^{26}, we find that, by decreasing the order parameter fluctuations, a divergence in both the correlation length^{7} and relaxation time^{27} are expected. Reference^{28} reports a connection between the entropydependent relaxation time and \({{\rm{\Gamma }}}_{s}\). We here suggest that this is also true for supercooled water.
In what follows, we use the compressible cell Isinglike model to study the Isingnematic phase recently detected in the lowdoping regime of Febased superconductors^{29}. An electronic nematic phase is essentially a melted stripe phase^{30}. Figure 4 shows that as the pressure is increased for \(v={v}_{0}+\delta v\), the temperature decreases. The limiting volume value for such a behavior is \(v={v}_{0}+0.17\delta v\) for \(\lambda =0.2\).
In the case of the proposed nematic phase in Febased superconductors, the pressure variation is caused by the chemical pressure introduced in the system by the doping effect on the crystal lattice. As the pressure (doping) is varied, the critical point signature vanishes (see Fig. 3). We obtain the same behavior shown in Fig. 4 (red curve) experimentally for the 122 doped Febased superconductors^{31}. In particular, the thermal expansion signatures are suppressed upon doping^{31}. Comparing the pressure versus temperature phase diagram reported in ref.^{31} for the 122 doped Febased superconductor with our results, we see that the regime that better illustrates the nematic phase is the one where \(v={v}_{0}+0.7\delta v\), since the critical point signature is shifted for lower values of T as p increases (see Fig. 4). Because there is a substantial number of free parameters that compose the current Isinglike model, we leave the fitting of the experimental results reported in ref.^{31} to future research. Here we used the compressible cell Isinglike model to simulate the doping effect in single crystals by assuming there are only two different volumes in the melted electronic nematic phase^{30}. When the system is doped, the electronic nematic phase associated with two coexisting volumes (see the figure in ref.^{30}) is suppressed, and the reported superconductivity appears, e.g., for Ba(Fe_{1−x}Co_{x})_{2}As_{2} single crystals^{31,32}. Yet, it is worth mentioning that for the limit case where \(v={v}_{0}+0.17\delta v\), we have no pressure variations for a wide range of temperature values. Since pressure and volume are conjugated variables, this behavior can be associated to the Invar effect, which has been widely investigated in ironnickel alloys, see e.g.^{33}.
Finally, we highlight our main findings. We have used an energyvolume coupled Isinglike model to calculate the Grüneisen parameter for the liquidliquid transition in supercooled water^{12}. We find that the behavior of the Grüneisen parameter is enhanced near pressure and temperature values that display anomalous behavior and thus supports the presence of a liquidliquid critical point governed by two distinct energy scales. Yet, such proposal is corroborated by the singular behavior of the isothermal compressibility, sound velocity and pseudoGrüneisen parameter in the vicinity of the liquidliquid critical point. Since the first submission of this manuscript, the compressible cell Isinglike model employed here has been used to describe the twocriticalpoint scenario^{34}. In addition to exploring the critical behavior of water and its other phases, our model can also be applied to other systems by adjusting its parameters. The application of the model to describe the nematic phase in the lowdoping regime of Febased superconductors revealed that the lowdoping regime is welldescribed by choosing values near the upper boundary values of the volume of each cell, namely, \(v\approx {v}_{0}+\delta v\). The latter corresponds to a lowerdensity configuration, in agreement with the theoretical description of the nematic phase for Febased superconductors^{32}. Our analysis of the Grüneisen parameter \({{\rm{\Gamma }}}_{s}\) and pseudoGrüneisen parameter \({{\rm{\Gamma }}}_{w}\) can be applied to investigate the critical behavior in any twostate system. One needs only to adjust properly the critical parameters according with the system of interest.
Methods
We recall some of the results obtained in the model proposed in ref.^{12}, which consist the basis of our analysis.
The system has N sites and coordination number c, where c is an adimensional parameter responsible for dictating the influence of the interaction among the sites when compared to its intrinsic energy \(cN{\varepsilon }_{0}/2\), where \({\varepsilon }_{0}\) is an arbitrary energy value. In each site, we suppose the existence of a cell. Each cell is characterized by its volume (and, consequently, its density). The interaction between sites is dictated by a constant energy coupling \(\delta \varepsilon \). The total energy of the system is \(E\{{n}_{i}\}\), where \({n}_{i}=1\) if the volume of the respective cell is \({v}_{0}+\delta v\) and \({n}_{i}=0\) if its volume is v_{0}. The expression for the energy reads
The volume of the system is the sum of the volume of each cell. Since the minimum volume that each cell occupies is v_{0}, all cells contribute to the total volume with a magnitude of Nv_{0}. Adding the contribution of the K sites having a volume \({v}_{0}+\delta v\), the expression for the total volume reads^{12}
Thus each particle is located in a site, and the volume has two possible values. We associate these two volumes with the low and highdensity phases and thus with two distinct energy scales. The association of different energy scales with the volume of each cell and, consequently, by their densities, is the key to understanding why \({{\rm{\Gamma }}}_{s}\) is enhanced near the liquidliquid critical point. The energy has two boundary values, corresponding to two limiting configurations of the system. When \(K=0\), \({E}_{max}=cN{\varepsilon }_{0}\mathrm{/2}\) results, whereas for \(K=N\), \({E}_{min}=[cN{\varepsilon }_{0}\delta \varepsilon (N\mathrm{1)}N\mathrm{]/2}\). The associated minimum and maximum values for the volumes are \({V}_{min}=N{v}_{0}\) for \(K=0\) and \({V}_{max}=N({v}_{0}+\delta v)\) for \(K=N\). Physically, the limiting cases represent the scenarios where all cells occupy the minimum (maximum) volume, corresponding to \(K=0\) (\(K=N\)). We obtain all the observables related to the system from Eqs (1) and (2), cf. ref.^{35}. We carry out an isothermalisobaric analysis and sum \({e}^{E/{k}_{B}T}\) and \({e}^{pV/{k}_{B}T}\) to the partition function, where k_{B} is the Boltzmann constant and p and T are the pressure and temperature of all possible microstates of the system, respectively.
The resulting partition function \(Z=Z(N,p,T)\) has the same mathematical structure as the Ising canonical partition function. Because we have not yet solved the threedimensional Ising model, we use an approximate meanfield solution^{12} to obtain the observables. The meanfield theory can be applied to a wide range of systems, including the Ising model and the van der Waals theory for liquidgas systems^{35}. Using it we replace the functional integral \(Z=N\,\int \,(Dm){e}^{E[m,H]}\) with the maximum value of the integrand, the socalled saddlepoint approximation. The parameter m is the orderparameter density, and Dm is the volume element. Because this approximation assumes that the only important configuration near the critical point is the one of uniform density, we expect that, because the density fluctuations in the order parameter are strong in this regime, this study of critical phenomena will exhibit artifacts. However ref.^{12} indicates that consistent results can be obtained in this framework. The equation of state for the system is^{12}
from which we deduce
We use Eq. (3) to determine the critical point coordinates \({p}_{c}=({v}_{c},{T}_{c})\), following ref.^{35}:
Thus,
and
We apply these conditions and the critical point parameters are:
Employing the basic thermodynamic relations^{35} and using \(f(v)=\,\mathrm{ln}(\lambda \frac{{v}_{0}+\delta vv}{v{v}_{0}})\)^{12} we obtain the isobaric thermal expansion α_{p}, the heat capacity c_{p} and the isothermal compressibility \({\kappa }_{T}\)
From Eqs 8 and 9 we see that the Tdependence of α_{p} and c_{p} are distinct. Thus, we expect a different singular behavior of these observables upon approaching the critical point, which can be explored by means of the Grüneisen parameter (see below). This is one of the main findings of this work, see Results.
We use Eqs (8) and (9) to determine the expression of the ratio \({{\rm{\Gamma }}}_{s}={\alpha }_{p}/{c}_{p}\) and the pseudoGrüneisen parameter \({{\rm{\Gamma }}}_{w}={w}^{2}{{\rm{\Gamma }}}_{s}\)^{16,18,21}, where w is the speed of sound, with
The calculations are straightforward and we obtain:
Both quantities, namely \({{\rm{\Gamma }}}_{s}\) and \({{\rm{\Gamma }}}_{w}\) were used in our analysis, see Fig. 3(c,d).
References
Debenedetti, P. G. & Stanley, H. E. Supercooled and glassy water. Phys. Today 56, 40–46 (2003).
Gallo, P. et al. Water: a tale of two liquids. Chem. Rev. 116, 7463–7500 (2016).
Poole, P. H., Sciortino, F., Essmann, U. & Stanley, H. E. Phase behaviour of metastable water. Nature 360, 324–328 (1992).
Kumar, P. et al. Relation between the Windom line and the breakdown of the StokesEinstein relation in supercooled water. Proc. Natl. Acad. Sci. USA 104, 9575–9579 (2007).
Franzese, G. & Stanley, H. E. The Widom line of supercooled water. J. Phys.: Condens. Matter 19, 2051261–20512616 (2007).
Gallo, P. & Stanley, H. E. Supercooled water reveals its secrets. Science 358, 1543–1544 (2017).
Kim, K. H. et al. Maxima in the thermodynamic response and correlation functions of deeply supercooled water. Science 358, 1589–1593 (2017).
Bartosch, L., de Souza, M. & Lang, M. Scaling theory of the Mott transition and breakdown of the Grüneisen scaling near a finitetemperature critical end point. Phys. Rev. Lett. 104, 2457011–2457014 (2010).
de Souza, M. & Bartosch, L. Probing the Mott physics in κ(BEDTTTF)_{2}X salts via thermal expansion. J. Phys.: Condens. Matter 27, 0532031–05320327 (2015).
de Souza, M., Menegasso, P., Paupitz, R., Seridonio, A. & Lagos, R. E. Grüneisen parameter for gases and superfluid helium. Europ. J. Phys. 37, 0551051–05510516 (2016).
Gomes, G. et al. The magnetic Grüneisen paramter for model sytems. arXiv:1903.04549v1.
Cerdeiriña, C. A. & Stanley, H. E. Isinglike models with energyvolume coupling. Phys. Rev. Lett. 120, 1206031–1206035 (2018).
Cerdeiriña, C. A., Orkoulas, G. & Fisher, M. E. Soluble model fluids with complete scaling and YangYang features. Phys. Rev. Lett. 116, 0406011–0406015 (2016).
Cerdeiriña, C. A. & Orkoulas, G. Compressible cell gas models for asymmetric fluid criticality. Phys. Rev. E 95, 0321051–03210515 (2017).
Holten, V., Anisimov, V. A. & Sengers, V. A. Towards a supercooled water guideline. International Association for the Properties of Water and Steam (Technical Report) (2012).
Mausbach, P., Köster, A., Rutkai, G., Thol, M. & Vrabec, J. Comparative study of the Grüneisen parameter for 28 pure fluids. J. Chem. Phys. 144, 2445051–24450511 (2016).
Moore, E. B. & Molinero, V. Structural transformation in supercooled water controls the crystallization rate of ice. Nature 479, 506–508 (2011).
Knopoff, L. & Shapiro, J. N. PseudoGrüneisen parameter for liquids. Phys. Rev. B 1, 3893–3895 (1970).
Fournier, D., Poirier, M., Castonguay, M. & Truong, K. D. Mott transition, compressibility divergence, and the P − T phase diagram of layered organic superconductors: an ultrasonic investigation. Phys. Rev. Lett. 90, 1270021–1270024 (2003).
de Souza, M. et al. Anomalous lattice response at the Mott transition in a quasi2D organic conductor. Phys. Rev. Lett. 99, 0370031–0370034 (2007).
Arp, V., Persichetti, J. M. & Chen, G. B. The Grüneisen parameter in fluids. J. Fluids Eng. 106, 193–200 (1984).
Zhu, L., Garst, M., Rosch, A. & Si, Q. Universally diverging Grüneisen parameter and the magnetocaloric effect close to quantum critical points. Phys. Rev. Lett. 91, 0664041–0664044 (2003).
Küchler, R. et al. Divergence of the Güneisen ration at quantum critical points in heavy fermion metals. Phys. Rev. Lett. 91, 0664051–0664054 (2003).
Roland, C. M., Bair, S. & Casalini, R. Thermodynamically scaling of the viscosity of van der Waals, Hbonded, and ionic liquids. J. Chem. Phys. 125, 1245081–1245088 (2006).
Cook, R. L., King, H. E. & Peiffer, D. G. Pressureinduced crossover from good to poor solvent behavior for polyethylene oxide in water. Phys. Rev. Lett. 69, 3072–3075 (1992).
Kadanoff, L. P. et al. Static phenomena near critical points: theory and experiment. Rev. Mod. Phys. 39, 395–431 (1967).
Rønne, C., Åstrand, P. O. & Keiding, S. R. THz spectroscopy of liquid H_{2}O and D_{2}O. Phys. Rev. Lett. 82, 2888–2891 (1999).
Casalini, R., Mohanty, U. & Roland, C. M. Thermodynamic interpretation of the scaling of the dynamics of supercooled liquids. J. Chem. Phys. 125, 0145051–0145059 (2006).
Fernandes, R. M., Chubukov, A. V. & Schmalian, J. What drives nematic order in ironbased superconductors? Nature Phys. 10, 97–104 (2014).
Fradkin, E. & Kivelson, S. A. Electron nematic phases proliferate. Science 327, 155–156 (2010).
Bud’ko, S. L., Ni, N., Nandi, S., Schmiedeshoff, G. M. & Canfield, P. C. Thermal expansion and anisotropic pressure derivatives of T _{c} in Ba(Fe_{1−x}Co_{x})_{2}As_{2} single crystals. Phys. Rev. B 79, 0545251–0545256 (2009).
Ni, N. et al. Effects of Co substitution on thermodynamic and transport properties and anistropic H _{c2} in Ba(Fe_{1−x}Co_{x})_{2}As_{2} single crystals. Phys. Rev. B 78, 2145151–2145159 (2008).
van Schilfgaarde, M., Abrikosov, I. A. & Johansson, B. Origin of the Invar effect in ironnickel alloys. Nature 400, 46–49 (1999).
Cerdeiriña, C. A., Troncoso, J., GonzálezSalgado, D. & Stanley, H. E. Water’s twocriticalpoint scenario in the Ising paradigm. J. Chem. Phys. 150, 2445091–2445098 (2019).
Huang, K. Statistical Mechanics (Wiley, 1987).
Acknowledgements
M. de S. acknowledges financial support from the São Paulo Research Foundation–Fapesp (Grants No. 2011/220504), National Council of Technological and Scientific Development–CNPq (Grants No. 302498/20176), T.U.V.S.O.T.E., the Austrian, Academy of Science ÖAW for the JESH fellowship and Serdar Sariciftci for the hospitality. The Boston University Center for Polymer Studies is supported by NSF Grants PHY1505000, CMMI1125290, and CHE1213217, and by DTRA Grant HDTRA11410017.
Author information
Authors and Affiliations
Contributions
G.O.G. carried out the calculations and generated the figures. G.O.G. and M.deS. wrote the paper with contributions from H.E.S. H.E.S. critically examined the manuscript. M.deS. conceived and supervised the project.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
O. Gomes, G., Stanley, H.E. & Souza, M.d. Enhanced Grüneisen Parameter in Supercooled Water. Sci Rep 9, 12006 (2019). https://doi.org/10.1038/s41598019483534
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598019483534
This article is cited by

Unveiling the Physics of the Mutual Interactions in Paramagnets
Scientific Reports (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.