Anharmonicity and Quantum Effect in Thermal Expansion of an Invar Alloy

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1 Anharmonicity and Quantum Effect in Thermal Expansion of an Invar Alloy 1 Institute for Molecular Science (IMS), Okazaki The Graduate University for Advanced Studies (Sokendai) Toshihiko YOKOYAMA 1,, Keitaro EGUCHI

2 Introduction: Thermal expansion of Invar alloy 1897 C. E. Guillaume 190 Nobel Prize Physics C. E. Guillaume, CR Acad. Sci. 15, 35 (1897). Weiss high-spin & low-spin model R. J. Weiss, Proc. R. Soc. London. A 8, 81 (1963). fcc FeNi alloy 36Invar Fe 64.6 Ni Invar Fe 58 Ni 4 Kovar Fe 54 Ni 9 Co 17 45Permalloy Fe 55 Ni 45 Inconel Fe 8 Ni 7 78Permalloy Fe Ni 78 Fe HS more stable, longer distance LS more unstable, shorter distance Higher T, larger LS concentration Compensate for normal thermal expansion Almost no thermal expansion from 0 to 400 K

3 Purpose of this work 1. Thermal expansion originates from thermal vibration Quantum effect at LT?. Transformation of HS LS only in Fe atoms Local thermal expansion especially around Ni? Path-Integral Effective Classical Potential (PIECP) method EXAFS & PIECP 3. Anharmonicity without thermal expansion Normal thermal expansion originates from anharmonic interatomic potentials

4 Measurement & analysis of EXAFS PF-PAC No. 010G551 Beam time used : 48 hours EXAFS function k 3 χ(k) and their Fourier transforms KEK-PF BL9C Si(111) Fe 64.6 Ni 35.4 foil (8µm) commercial transmission mode I 0 N 100% I Ar 50% + N 50% Calibration of T 1 st NN 1 st NN *calibrated Si diode *Cu foil EXAFS 3 rd NN 3 rd NN NS 4 χ k = e kr + φ k C 3k kr 3 0 Ck 3 ( ) sin [ ( ) ] C =< ( r R) > 3 C3 =< ( r R) > Mean square (cubic) relative displacements Abs. distance FEFF Relative distance Empirical

5 PIECP MC simulations Path Integral Effective Classical Potential (PIECP) A. Cuccoli, R. Giachetti, V. Tognetti, R. Vaia, and P. Verrucchi, J. Phys.: Condens. Matter 7, 7891 (1995). T. Fujikawa, T. Miyanaga, and T. Suzuki, J. Phys. Soc. Jpn. 66, 897 (1997). T. Yokoyama, Phys. Rev. B 57, 343 (1998). Convenient quantum mechanical simulation method in anharmonic systems Total energies using present EAM potentials Monte Carlo simulations FeNi 500 atoms (15 fcc lattices) Periodic conditions Fe:Ni=64.6:35.4 random distribution 11 samples Atomic potentials Embedded Atom Method (EAM) Empirical potentials for metals M. S. Daw and M. I. Baskes, Phys. Rev. B 9, 6443 (1984). S. M. Foiles, Phys. Rev. B 3, 3409 (1985). T=0 K, perfect fcc lattice System State R e (A ) E(R e ) (ev) Remark fcc Fe fcc Fe 65 Ni 35 HS LS more stable LS by 8 mev HS HS more stable LS by 5 mev fcc Ni

6 Results of thermal expansion by EXAFS analysis Relative interatomic distance for the 1 st & 3 rd NN shells Bond (interatomic) distance & lattice distance R bond < u > = Rlattice + R lattice MSRD for 1 st & 3 rd NN shells R bond 1 st NN R lattice Vibration perpendicular to the bond < u > from Θ D 3 rd NN Fe 1 st NN almost no thermal expansion Ni 1 st NN smaller than fcc Ni 3 rd NN associated with lattice

7 Computational results for thermal expansion Quantum Both bond & lattice distances agree well with experiments Classical At LT (<100 K) large expansion deviates from experiments at LT Conclusion 1: Quantum effect Very important at LT Without quantum effect, normal thermal expansion

8 Analysis of each component in 1 st NN shells Fe-Fe almost no thermal expansion Large deviation from HS-only results that may show hypothetical normal thermal expansion Ni-Fe, Ni-Ni smaller than normal due to change of Fe potential Expansion of Ni-Fe larger than Ni-Ni Fe surrounded by many Ni shows HS Ni-Ni potential softer Ni-Fe more likely to follow lattice EXAFS cannot distinguish 1 st NN Fe-Fe & Fe-Ni or Ni-Ni & Ni-Fe PIECP MC can easily distinguish To show hypothetical normal expansion, case of HS-only Fe was calculated (dashed lines) Conclusion : Local expansion Ni also shows very small expansion due to the change of Fe potentials.

9 Asymmetry or anharmonicity 1 st NN Both Fe & Ni show rather normal C 3 Conclusion 3: Asymmetry in no thermal expansion system Even in no thermal expansion system, RDF is asymmetric as usual Anharmonicity exists normally 3 rd NN Both Fe & Ni show no asymmetry No chemical bond in 3 rd NN shell gives no anharmoinicity RDF for non-interaction From central limit theorem, RDF should be gaussian T. Yokoyama, T. Ohta, and H. Sato, Phys. Rev. B 55, 1130 (1997).

10 Summary Combining EXAFS & PIECP, thermal expansion of Invar alloy Fe 64.6 Ni 35.4 is studied. 1. Thermal expansion originates from thermal vibration Quantum effect at low temperature? Quantum effect at LT is essential. Without quantum effect, normal thermal expansion. Transformation of HS LS only in Fe Local thermal expansion especially around Ni? Fe 1 st NN almost no expansion Ni 1 st NN smaller expansion than normal due to Fe potentials 3. Anharmoncity in the no thermal expansion system Although thermal expansion inherently originates from anharmonicity, there can be clearly found the asymmetry of the RDF (=3 rd -order anharmonicity) even in the system without thermal expansion. T. Yokoyama and K. Eguchi, Phys. Rev. Lett. 107 (011)