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A thermodynamic explanation of the Invar effect

Nature Physics volume 19, pages 1642–1648 (2023)Cite this article

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Abstract

The anomalously low thermal expansion of Fe–Ni Invar has long been associated with magnetism, but to date, the microscopic underpinnings of the Invar behaviour have eluded both theory and experiment. Here we present nuclear resonant X-ray scattering measurements of the phonon and magnetic entropies under pressure. By applying a thermodynamic Maxwell relation to these data, we obtain the separate phonon and magnetic contributions to thermal expansion. We find that the Invar behaviour stems from a competition between phonons and spins. In particular, the phonon contribution to thermal expansion cancels the magnetic contribution over the 0–3 GPa pressure range of Invar behaviour. At pressures above 3 GPa, the cancellation is lost, but our analysis reproduces the positive thermal expansion measured separately by synchrotron X-ray diffractometry. Ab initio calculations informed by experimental data show that spin–phonon interactions improve the accuracy of this cancellation over the range of Invar behaviour. Spin–phonon interactions also explain how different phonon modes have different energy shifts with pressure.

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Fig. 1: Phonon DOS of Invar at different pressures.
Fig. 2: Phonon entropy of Invar as a function of pressure.
Fig. 3: Magnetization of Invar as a function of pressure.
Fig. 4: Entropy contributions and thermal expansion.
Fig. 5: Calculated phonon DOS below the Curie transition.
Fig. 6: Mean energy of longitudinal mode and longitudinal IFCs.

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Data availability

All relevant data from experiments and calculations are available via Zenodo at https://doi.org/10.5281/zenodo.7987363.

Code availability

The computer code used to generate the numerical results is based on the Vienna ab initio simulation package and TDEP, both of which are publicly available. The formulations and algorithms necessary to reproduce the theory results of this study are described in the Methods section and ref. 27.

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Acknowledgements

This work was supported by the National Science Foundation under grant no. 1904714 (S.H.L., P.G., C.M.B-C., C.N.S. and B.F.). Work at Boston College was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, under award no. DE-SC0021071 (M.H. and D.B.). O.H. acknowledges support from the Swedish Research Council (VR) program 2020-04630. M.H. and D.B. acknowledge the Boston College Linux clusters for their computational resources and support. Calculations were also performed in part using MATLAB. This research used resources of the APS, a US DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357 (S.H.L., P.G., C.M.B-C., G.S. and B.F.). HPCAT operations are supported by DOE-NNSA’s Office of Experimental Sciences. Use of the COMPRES-GSECARS gas-loading system was supported by COMPRES under NSF Cooperative Agreement EAR-1606856 and by GSECARS through NSF grant EAR-1634415 and DOE grant DE-FG02-94ER14466. This research also used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory (S.H.L., C.M.B-C. and B.F.). We thank D. Rhiger for informing us of his dissertation research, C. Li for assistance with the pressure cells and J. A. Kornfield for use of the DSC. We also acknowledge the help from the beamline scientists at the APS (E. E. Alp, J. Zhao, B. Lavina and M. Hu) and D. Abernathy at Oak Ridge National Laboratory.

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Author notes

  1. These authors contributed equally: S. H. Lohaus, M. Heine.

Authors and Affiliations

  1. Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA, USA

    S. H. Lohaus, P. Guzman, C. M. Bernal-Choban, C. N. Saunders & B. Fultz

  2. Department of Physics, Boston College, Chestnut Hill, MA, USA

    M. Heine & D. Broido

  3. Advanced Photon Source, Argonne National Laboratory, Argonne, IL, USA

    G. Shen

  4. Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, Rehovoth, Israel

    O. Hellman

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Contributions

S.H.L. designed and performed the experiments, analysed the data and compiled the paper. M.H. performed and analysed the calculations. P.G. and C.M.B-C. helped execute the beamline experiments at the APS. C.N.S. analysed the neutron scattering data. The remote XRD experiments were locally handled by G.S. O.H. developed and supervised the computational analyses. D.B. conceptualized and supervised the computational efforts. B.F. conceptualized and supervised the study. S.H.L., M.H., D.B. and B.F. wrote the paper, with contributions from all authors.

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Correspondence to S. H. Lohaus or B. Fultz.

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Lohaus, S.H., Heine, M., Guzman, P. et al. A thermodynamic explanation of the Invar effect. Nat. Phys. 19, 1642–1648 (2023). https://doi.org/10.1038/s41567-023-02142-z

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