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Magnetism of Ni3Al and Fe3Al under extreme pressure and shape deformation: an ab initio study
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) e205–e206
Magnetism of Ni3Al and Fe3Al under extreme pressure and shape deformation: an ab initio study & a,*, D. Leguta,b, M. Fria! ka, J. Fialab M. Sob b
a & zkova 22, Brno CZ-616 62, Czech Republic Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zi& & Department of Chemistry of Materials,Faculty of Chemistry, Brno University of Technology, Purkynova 118, CZ-616 62 Brno, Czech Republic
Abstract Total energies and magnetic moments in Ni3Al and Fe3Al are calculated along the tetragonal and trigonal transformation paths at various volumes by means of the FLAPW method. Whereas magnetism does not play an important role in stabilization of the L12 structure in Ni3Al, the magnetic effects in Fe3Al are vital. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Bb; 75.50.Cc; 62.20.Fe; 81.30.Hd Keywords: Ab initio calculation; Magnetic phase transformation; Volume and shape deformation
Intermetallic compounds Ni3Al and Fe3Al constitute a basis of many technologically important materials. They are ferromagnetic at ambient temperatures, however, very little is known about magnetism in the heavily deformed regions of extended defects in these compounds (e.g. grain boundaries, antiphase and interphase boundaries, dislocation cores, crack tips, etc.). The purpose of this study is to explore magnetic behavior of defect-free Ni3Al and Fe3Al at high-strain tetragonal and trigonal deformation. Such configurations may be found in the regions of extended defects. We consider two simple transformation paths connecting cubic structures. For elemental metals, they correspond to the BCC–FCC transformation path via tetragonal deformation corresponding to extension along the [0 0 1] axis (the usual Bain’s path) and to the trigonal deformation path that corresponds to uniaxial deformation along the [1 1 1] axis; this latter path connects the BCC, simple cubic and FCC structures [1]. Analogous deformation paths may be devised for intermetallic compounds with B2, L12 or D03 structures [1,2]. In the L12 (Cu3Au) structure, the atoms are at the *Corresponding author. Tel.: +420-532 290 455; fax: +420541 212 302. & E-mail address: [email protected] (M. Sob).
FCC positions with the (0 0 2) planes occupied alternatively by Cu atoms and by Cu and Au atoms in the same ratio. We may consider this structure as tetragonal or trigonal with the ratio c/a=1. Now, performing a tetragonal deformation, the cubic symmetry of the L12 structure is lost and becomes tetragonal, even for c=a ¼O2=2; when atoms are at the BCC positions, but because we have two kinds of atoms, the structure does not retain the cubic symmetry. When we perform the trigonal deformation, the structure becomes trigonal except for the case of c/a=0.5, when we encounter a simple-cubic-based structure that, indeed, has a cubic symmetry. For c/a=0.25, the atoms adopt the BCC positions, but the symmetry of the structure remains trigonal. Similar paths may be constructed for the D03 structure. All the structures appearing along those paths will be characterized in more details in a subsequent publication [2]. The total energy and magnetic moment of Ni3Al and Fe3Al compounds were calculated using the FLAPW code WIEN97 described in detail in Ref. [3]. For exchange-correlation energy, the generalized gradient approximation (GGA) was employed. The details of the calculations may be found in Ref. [2]. Fig. 1 displays the total energy of Ni3Al as a function of the volume and c/a for the trigonal deformation. We
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.598
ARTICLE IN PRESS & et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e205–e206 M. Sob 0. 00 9
0.07
0. 36
0.1 2 0.18 0.3 0.24
0.5
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1
0.0
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8 27 0.0
7
0.88 0.7
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45
54
0.03 0.045 6 0.054 0.063 0.072
0.009
0
0.018
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V / Vexp
0
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12 0.
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2
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NM
0.009
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8
8 01 0.
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NM
0.96
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FM
27 0.0
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5
0.063 0.054 0.045 0.036 0.027
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NM 1
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0.07 0.12
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1.06
24 0.
1.05
0.12
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0 0. .036 0 0. 45 05 0.0 4 63 0.0 72 0.081 0.09 0.
e206
1.1
Fig. 1. Total energy (per formula unit) of Ni3Al as a function of volume and c/a ratio, characterizing the trigonal deformation, calculated within the GGA. The energy is measured relative to the energy of the equilibrium FM L12 state (the minimum at c/a=1). Only states with the minimum energy are shown. The contour interval is 20 mRy. Thick line shows the NM/FM phase boundary. The ground-state minimum at c/a=1 and the saddle point at c/a=0.5 (outside the figure area) are dictated by symmetry [1].
show only those states the energies of which are the lowest for a given configuration. A nearly vertical border divides the area of Fig. 1 into the ferromagnetic (FM) and nonmagnetic (NM) regions. All energy profiles corresponding to a constant volume exhibit the symmetrydictated maximum at c/a = 0.5 and the symmetrydictated minimum at c/a=1. In the contour plot (Fig. 1), there is a saddle point for c/a=0.5 and V/VexpB1.2 (outside the area of the figure). The minimum at c/a E0.27, V/VexpE1.01 is not dictated by the symmetry. Fig. 2 shows the total energy of Ni3Al as a function of the volume and c/a for the tetragonal deformation. Here NM regions extend to both sides of the FM ground state. However, there are no energy extrema and saddle points in those NM regions. It is interesting that the transition from the FM to NM state during both the trigonal and tetragonal deformation is essentially continuous, without any discontinuities in magnetic moment. Xu et al. [4] have shown that the energy gain in Ni3Al associated with magnetism is about an order of magnitude smaller than that due to the structural differences. Our calculations show that the NM/Ni3Al in the L12 structure is stable with respect to tetragonal and trigonal deformations (the shear moduli C0 and C44 are nearly the same for the NM and FM states). Therefore, magnetism does not appear to play an important role in the control of phase stability. This is in sharp contrast with iron, where the onset of ferromagnetism stabilizes the BCC structure and NM BCC states are not stable with respect to tetragonal deformation [5]. Trigonally deformed Ni3Al in the D03 structure also exhibits NM/FM phase boundaries in the (c/a, V/Vexp) plane. However, Fe3Al keeps its ferromagnetic ordering both in trigonally deformed D03 structure and in
0.8
0.9
1
1.1
1.2
1.3
c/a
c/a
Fig. 2. Total energy (per formula unit) of Ni3Al as a function of volume and c/a ratio, characterizing the tetragonal deformation, calculated within the GGA. The energy is measured relative to the energy of the equilibrium FM L12 state (the minimum at c/a=1). Only states with the minimum energy are shown. The contour interval is 3 mRy. Thick lines show the NM/FM phase boundaries. The only symmetry-dictated extremum is at c/a= 1.
trigonally and tetragonally deformed L12 structure in a wide range of volumes and deformations [2]. It may be concluded that whereas in Fe3Al the magnetic effects are vital for stability of the ground state and ferromagnetism is preserved for large volume and shape deformations, it transpires that in Ni3Al magnetism is not very important in phase stability considerations and, indeed, it is lost in large regions of the (c/a, V/Vexp) plane. This research was supported by the Grant Agency of the Academy of Sciences of the Czech Republic (Project No. IAA1041302), by the Grant Agency of the Czech Republic (Project No. 202/03/1351), and by the Research Project Z2041904 of the Academy of Sciences of the Czech Republic. A part of this study has been performed in the framework of the COST Project No. OC 523.90. The use of the computer facilities at the MetaCenter of the Masaryk University, Brno, and at the Boston University Scientific Computing and Visualization Center is acknowledged.
References & [1] M. Sob, L.G. Wang, V. Vitek, Comp. Mater. Sci. 8 (1997) 100. & [2] D. Legut, M. Sob, M.Fri!ak, J. Fiala, in preparation. [3] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Technical University of Vienna 1997 -Improved and updated Unix version of the original copyrighted WIEN-code, P. Blaha, K. Schwarz, P. Sorantin, S. B. Trickey, Comput. Phys. Commun. 59 (1990) 399. [4] J. Xu, B.I. Min, A.J. Freeman, T. Oguchi, Phys. Rev. B 41 (1990) 5010. & [5] M. Fri!ak, M. Sob, V. Vitek, Phys. Rev. B 63 (2001) 052405.
Journal of Magnetism and Magnetic Materials 272–276 (2004) e205–e206
Magnetism of Ni3Al and Fe3Al under extreme pressure and shape deformation: an ab initio study & a,*, D. Leguta,b, M. Fria! ka, J. Fialab M. Sob b
a & zkova 22, Brno CZ-616 62, Czech Republic Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zi& & Department of Chemistry of Materials,Faculty of Chemistry, Brno University of Technology, Purkynova 118, CZ-616 62 Brno, Czech Republic
Abstract Total energies and magnetic moments in Ni3Al and Fe3Al are calculated along the tetragonal and trigonal transformation paths at various volumes by means of the FLAPW method. Whereas magnetism does not play an important role in stabilization of the L12 structure in Ni3Al, the magnetic effects in Fe3Al are vital. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Bb; 75.50.Cc; 62.20.Fe; 81.30.Hd Keywords: Ab initio calculation; Magnetic phase transformation; Volume and shape deformation
Intermetallic compounds Ni3Al and Fe3Al constitute a basis of many technologically important materials. They are ferromagnetic at ambient temperatures, however, very little is known about magnetism in the heavily deformed regions of extended defects in these compounds (e.g. grain boundaries, antiphase and interphase boundaries, dislocation cores, crack tips, etc.). The purpose of this study is to explore magnetic behavior of defect-free Ni3Al and Fe3Al at high-strain tetragonal and trigonal deformation. Such configurations may be found in the regions of extended defects. We consider two simple transformation paths connecting cubic structures. For elemental metals, they correspond to the BCC–FCC transformation path via tetragonal deformation corresponding to extension along the [0 0 1] axis (the usual Bain’s path) and to the trigonal deformation path that corresponds to uniaxial deformation along the [1 1 1] axis; this latter path connects the BCC, simple cubic and FCC structures [1]. Analogous deformation paths may be devised for intermetallic compounds with B2, L12 or D03 structures [1,2]. In the L12 (Cu3Au) structure, the atoms are at the *Corresponding author. Tel.: +420-532 290 455; fax: +420541 212 302. & E-mail address: [email protected] (M. Sob).
FCC positions with the (0 0 2) planes occupied alternatively by Cu atoms and by Cu and Au atoms in the same ratio. We may consider this structure as tetragonal or trigonal with the ratio c/a=1. Now, performing a tetragonal deformation, the cubic symmetry of the L12 structure is lost and becomes tetragonal, even for c=a ¼O2=2; when atoms are at the BCC positions, but because we have two kinds of atoms, the structure does not retain the cubic symmetry. When we perform the trigonal deformation, the structure becomes trigonal except for the case of c/a=0.5, when we encounter a simple-cubic-based structure that, indeed, has a cubic symmetry. For c/a=0.25, the atoms adopt the BCC positions, but the symmetry of the structure remains trigonal. Similar paths may be constructed for the D03 structure. All the structures appearing along those paths will be characterized in more details in a subsequent publication [2]. The total energy and magnetic moment of Ni3Al and Fe3Al compounds were calculated using the FLAPW code WIEN97 described in detail in Ref. [3]. For exchange-correlation energy, the generalized gradient approximation (GGA) was employed. The details of the calculations may be found in Ref. [2]. Fig. 1 displays the total energy of Ni3Al as a function of the volume and c/a for the trigonal deformation. We
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.598
ARTICLE IN PRESS & et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e205–e206 M. Sob 0. 00 9
0.07
0. 36
0.1 2 0.18 0.3 0.24
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54
0.03 0.045 6 0.054 0.063 0.072
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27 0.0
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0 0. .036 0 0. 45 05 0.0 4 63 0.0 72 0.081 0.09 0.
e206
1.1
Fig. 1. Total energy (per formula unit) of Ni3Al as a function of volume and c/a ratio, characterizing the trigonal deformation, calculated within the GGA. The energy is measured relative to the energy of the equilibrium FM L12 state (the minimum at c/a=1). Only states with the minimum energy are shown. The contour interval is 20 mRy. Thick line shows the NM/FM phase boundary. The ground-state minimum at c/a=1 and the saddle point at c/a=0.5 (outside the figure area) are dictated by symmetry [1].
show only those states the energies of which are the lowest for a given configuration. A nearly vertical border divides the area of Fig. 1 into the ferromagnetic (FM) and nonmagnetic (NM) regions. All energy profiles corresponding to a constant volume exhibit the symmetrydictated maximum at c/a = 0.5 and the symmetrydictated minimum at c/a=1. In the contour plot (Fig. 1), there is a saddle point for c/a=0.5 and V/VexpB1.2 (outside the area of the figure). The minimum at c/a E0.27, V/VexpE1.01 is not dictated by the symmetry. Fig. 2 shows the total energy of Ni3Al as a function of the volume and c/a for the tetragonal deformation. Here NM regions extend to both sides of the FM ground state. However, there are no energy extrema and saddle points in those NM regions. It is interesting that the transition from the FM to NM state during both the trigonal and tetragonal deformation is essentially continuous, without any discontinuities in magnetic moment. Xu et al. [4] have shown that the energy gain in Ni3Al associated with magnetism is about an order of magnitude smaller than that due to the structural differences. Our calculations show that the NM/Ni3Al in the L12 structure is stable with respect to tetragonal and trigonal deformations (the shear moduli C0 and C44 are nearly the same for the NM and FM states). Therefore, magnetism does not appear to play an important role in the control of phase stability. This is in sharp contrast with iron, where the onset of ferromagnetism stabilizes the BCC structure and NM BCC states are not stable with respect to tetragonal deformation [5]. Trigonally deformed Ni3Al in the D03 structure also exhibits NM/FM phase boundaries in the (c/a, V/Vexp) plane. However, Fe3Al keeps its ferromagnetic ordering both in trigonally deformed D03 structure and in
0.8
0.9
1
1.1
1.2
1.3
c/a
c/a
Fig. 2. Total energy (per formula unit) of Ni3Al as a function of volume and c/a ratio, characterizing the tetragonal deformation, calculated within the GGA. The energy is measured relative to the energy of the equilibrium FM L12 state (the minimum at c/a=1). Only states with the minimum energy are shown. The contour interval is 3 mRy. Thick lines show the NM/FM phase boundaries. The only symmetry-dictated extremum is at c/a= 1.
trigonally and tetragonally deformed L12 structure in a wide range of volumes and deformations [2]. It may be concluded that whereas in Fe3Al the magnetic effects are vital for stability of the ground state and ferromagnetism is preserved for large volume and shape deformations, it transpires that in Ni3Al magnetism is not very important in phase stability considerations and, indeed, it is lost in large regions of the (c/a, V/Vexp) plane. This research was supported by the Grant Agency of the Academy of Sciences of the Czech Republic (Project No. IAA1041302), by the Grant Agency of the Czech Republic (Project No. 202/03/1351), and by the Research Project Z2041904 of the Academy of Sciences of the Czech Republic. A part of this study has been performed in the framework of the COST Project No. OC 523.90. The use of the computer facilities at the MetaCenter of the Masaryk University, Brno, and at the Boston University Scientific Computing and Visualization Center is acknowledged.
References & [1] M. Sob, L.G. Wang, V. Vitek, Comp. Mater. Sci. 8 (1997) 100. & [2] D. Legut, M. Sob, M.Fri!ak, J. Fiala, in preparation. [3] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Technical University of Vienna 1997 -Improved and updated Unix version of the original copyrighted WIEN-code, P. Blaha, K. Schwarz, P. Sorantin, S. B. Trickey, Comput. Phys. Commun. 59 (1990) 399. [4] J. Xu, B.I. Min, A.J. Freeman, T. Oguchi, Phys. Rev. B 41 (1990) 5010. & [5] M. Fri!ak, M. Sob, V. Vitek, Phys. Rev. B 63 (2001) 052405.