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Phonon softening and martensitic transformation in α-Fe
ELSEVIER
Physica B 234-236 (1997) 897-899
Phonon softening and martensitic transformation in a-Fe J. N e u h a u s a'*, W . P e t r y a, A. K r i m m e l b aTU Miinchen Physik Department El3, D-85747 Garching, Germany bHahn-Meitner-Institut, D-14109 Berlin, Germany
Abstract
The phonon dispersion of the low-temperature BCC-phase of pure iron (a-Fe) was investigated as a function of temperature. A strong softening of the entire Tl [~0] and T1[~2~] branch is observed on approaching the martensitic a-y-phase transition temperature. The eigenvectors of these phonons are in the direction of displacements needed for the transformation to the FCC- (7-) phase. This indicates low potential barriers for displacements towards the closed-packed structure and can be interpreted as a dynamical precursor for the martensitic phase transition.
Keywords: Martensitic transitions; Phonon dispersion; Lattice dynamics
Iron exhibits two structural (martensitic) phase transitions in its solid phase. On lowering the temperature the high-temperature BCC f-phase transforms to the closed-packed FCC 7-phase. With further decreasing temperature, however, Fe transforms back to the open BCC structure (a-phase) at 1184 K. This unusual behaviour is generally interpreted in terms of magnetic contributions to the free energy [1,2] where contributions of the vibrational entropy were supposed to be negligible. To test this hypothesis we measured the phonon dispersion of a-Fe at a temperature of 773, 1043 and 1173 K in the main symmetry directions and in the off-symmetry direction [~2~]. As our paper deals only with the structural phase transition we will call the entire low-temperature BCC-phase a-phase. High purity single crystals of the s-phase 5.5 mm in diameter and of variable length were grown by the recrystallization technique at the Max-PlanckInstitut fiir Metallforschung, Stuttgart. A standard resistance furnace has been used to heat the crystals under vacuum and absolute temperatures are precise * Corresponding author. 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 1 1 85-4
within 4-8 K. For all measurements pyrolytic graphite monochromator and analyser were used in constant final wave vector mode. The measurements were performed at spectrometer E7 at the HMI, Berlin. The obtained phonon frequencies are depicted in Fig. 1. The phonon dispersion has been parametrized by Bom-von Kfirmfin force constants which have been fitted up to the fifth nearest-neighbour shell to the measured phonon frequencies. The fit is shown in Fig. 1 as solid lines. By means of this parametrization the phonon density of states (DOS) has been calculated [3], which then served to calculate the vibrational entropy Svib. Similar calculations were done on the basis of available phonon data for a-Fe at 293 K [4] and 933 K [5]. Within the a-phase decreasing frequencies of the entire phonon dispersion are observed when passing from RT to T~. Most pronounced, however, is the decrease of the transverse branches T l [ ~ 2 ~ ] and/'1 [~0]. This softening has a nonlinear temperature dependence around the ferromagnetic transition as observed earlier by Satija et al. [6] and Vall~ra (Fe95 Sis ) [7]. This nonlinearity is observed mainly for the low-frequency modes as depicted in Fig. 2. The
898
J. Neuhaus et al./ Physica B 234-236 (1997,) 897~99 F
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b
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i
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i
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600
800
1000
1200
TEMPERATURE
,
8-
=o:o
/ K
Fig. 2. Temperature dependence of selected phonon modes in ~-Fe. Solid lines are a guide to the eye.
6 4 2 0
'
'
'
'
0.5<_
0
~.._> 1.0 <....~0.5
0
REDUCED WAVE VECTOR
Fig. 1. Phonon dispersion of ~-Fe for different temperatures. The solid lines represent a BvK fit including interactions up to the fi~h nearest-neighbour.
perature independent phonons in the y-phase [9] we find AS~i~b=0.14(2) which compares to a smaller AS~r=O.O91kB/atom [10]. This agree well with earlier findings in group 3 and 4 metals, where the high-temperature phase is mainly stabilized by the increase in vibrational entropy [11-13]. We would like to thank E. Giinther and U. El3mann from the Max-Planck-Institut ftir Metallforschung, Stuttgart for providing the ~-Fe single crystals. References
corresponding shear constants c' and c_ are involved in the transformation mechanism (Bain path) from BCC to FCC. Total energy calculations along the B a i n path [8] s h o w e d a linear d e p e n d e n c e o f c ~ on the e n e r g y difference b e t w e e n the B C C and F C C structure. The p r o n o u n c e d softening can therefore be interpreted as a d y n a m i c a l precursor for the structural ~ v-transition. The precise k n o w l e d g e o f the p h o n o n dispersion allows us to calculate the excess vibrational entropy at the transition temperature Tar. R e l y i n g on t e m -
[1] H. Hasegawa and D.G. Pettifor, Phys. Rev. Lett. 50 (1982) 130. [2] G. Grimvall and I. Ebbsj6, Phys. Scr. 12 (1975) 168. [3] G. Gilat and L.J. Raubenheimer, Phys. Rev. B. 144 (1966) 390. [4] C. van Dijk and J. Bergsma, Neutron Inelastic Scattering, Vol. 1 (IAEA, Vienna, 1968) p. 233. [5] A.M. Vall~ra, PhD Thesis, Cambridge University (1977). [6] S.K. Satija, R.P. Com~s and G. Shirane, Phys. Rev. B 32 (1985) 3309. [7] A.M. Vail&a, J. de Phys. C6, Suppl. 12 (1981) 398. [8] V.L. Sliwko, P. Mohn, K. Schwar-z and P. Blaha, J. Phys.: Condens. Matter 8 (1996) 799.
J. Neuhaus et al./ Physica B 234-236 (1997) 897-899
[9] J. Zarestky and C. Stassis, Phys. Rev. B 35 (1987) 4500. [10] C.Y. Ho (ed.) Properties of Selected Ferrous Alloying Elements (Hemisphere Publ. Co., New York, 1989). [11] W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H.R. Schober and G. Vogl, Phys. Rev. B 43 (1991) 10933.
899
[12] A. Heiming, W. Petry, J. Trampenau, M. Alba, C. Herzig, H.R. Schober and G. Vogl, Phys. Rev. B 43 (1991) 10948. [13] F. Giithoff, W. Perry, C. Stassis, A. Heiming, B. Hennion, C. Herzig and J. Trampenau, Phys. Rev. B 47 (1993) 2563.
Physica B 234-236 (1997) 897-899
Phonon softening and martensitic transformation in a-Fe J. N e u h a u s a'*, W . P e t r y a, A. K r i m m e l b aTU Miinchen Physik Department El3, D-85747 Garching, Germany bHahn-Meitner-Institut, D-14109 Berlin, Germany
Abstract
The phonon dispersion of the low-temperature BCC-phase of pure iron (a-Fe) was investigated as a function of temperature. A strong softening of the entire Tl [~0] and T1[~2~] branch is observed on approaching the martensitic a-y-phase transition temperature. The eigenvectors of these phonons are in the direction of displacements needed for the transformation to the FCC- (7-) phase. This indicates low potential barriers for displacements towards the closed-packed structure and can be interpreted as a dynamical precursor for the martensitic phase transition.
Keywords: Martensitic transitions; Phonon dispersion; Lattice dynamics
Iron exhibits two structural (martensitic) phase transitions in its solid phase. On lowering the temperature the high-temperature BCC f-phase transforms to the closed-packed FCC 7-phase. With further decreasing temperature, however, Fe transforms back to the open BCC structure (a-phase) at 1184 K. This unusual behaviour is generally interpreted in terms of magnetic contributions to the free energy [1,2] where contributions of the vibrational entropy were supposed to be negligible. To test this hypothesis we measured the phonon dispersion of a-Fe at a temperature of 773, 1043 and 1173 K in the main symmetry directions and in the off-symmetry direction [~2~]. As our paper deals only with the structural phase transition we will call the entire low-temperature BCC-phase a-phase. High purity single crystals of the s-phase 5.5 mm in diameter and of variable length were grown by the recrystallization technique at the Max-PlanckInstitut fiir Metallforschung, Stuttgart. A standard resistance furnace has been used to heat the crystals under vacuum and absolute temperatures are precise * Corresponding author. 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 1 1 85-4
within 4-8 K. For all measurements pyrolytic graphite monochromator and analyser were used in constant final wave vector mode. The measurements were performed at spectrometer E7 at the HMI, Berlin. The obtained phonon frequencies are depicted in Fig. 1. The phonon dispersion has been parametrized by Bom-von Kfirmfin force constants which have been fitted up to the fifth nearest-neighbour shell to the measured phonon frequencies. The fit is shown in Fig. 1 as solid lines. By means of this parametrization the phonon density of states (DOS) has been calculated [3], which then served to calculate the vibrational entropy Svib. Similar calculations were done on the basis of available phonon data for a-Fe at 293 K [4] and 933 K [5]. Within the a-phase decreasing frequencies of the entire phonon dispersion are observed when passing from RT to T~. Most pronounced, however, is the decrease of the transverse branches T l [ ~ 2 ~ ] and/'1 [~0]. This softening has a nonlinear temperature dependence around the ferromagnetic transition as observed earlier by Satija et al. [6] and Vall~ra (Fe95 Sis ) [7]. This nonlinearity is observed mainly for the low-frequency modes as depicted in Fig. 2. The
898
J. Neuhaus et al./ Physica B 234-236 (1997,) 897~99 F
F
N
H
10
F
P I
[~,}~1o
4!,
T~Ip Tply
_
= ~3K
0.5L[~0]
6
N "1" I->(5 Z I.U
2 0
I
I
0.5T2[,~,0]
I
:
l
I
I
I
I
I
I
I
0.5T~[~,0]
7
6
I
I
o~-Fe T=10431 o2Pji~
BCC-Fe
-
O LU n" LL
5
>- 6 0 Z W
0 W n* LL
2 0
b
i
i
i
i
400
600
800
1000
1200
TEMPERATURE
,
8-
=o:o
/ K
Fig. 2. Temperature dependence of selected phonon modes in ~-Fe. Solid lines are a guide to the eye.
6 4 2 0
'
'
'
'
0.5<_
0
~.._> 1.0 <....~0.5
0
REDUCED WAVE VECTOR
Fig. 1. Phonon dispersion of ~-Fe for different temperatures. The solid lines represent a BvK fit including interactions up to the fi~h nearest-neighbour.
perature independent phonons in the y-phase [9] we find AS~i~b=0.14(2) which compares to a smaller AS~r=O.O91kB/atom [10]. This agree well with earlier findings in group 3 and 4 metals, where the high-temperature phase is mainly stabilized by the increase in vibrational entropy [11-13]. We would like to thank E. Giinther and U. El3mann from the Max-Planck-Institut ftir Metallforschung, Stuttgart for providing the ~-Fe single crystals. References
corresponding shear constants c' and c_ are involved in the transformation mechanism (Bain path) from BCC to FCC. Total energy calculations along the B a i n path [8] s h o w e d a linear d e p e n d e n c e o f c ~ on the e n e r g y difference b e t w e e n the B C C and F C C structure. The p r o n o u n c e d softening can therefore be interpreted as a d y n a m i c a l precursor for the structural ~ v-transition. The precise k n o w l e d g e o f the p h o n o n dispersion allows us to calculate the excess vibrational entropy at the transition temperature Tar. R e l y i n g on t e m -
[1] H. Hasegawa and D.G. Pettifor, Phys. Rev. Lett. 50 (1982) 130. [2] G. Grimvall and I. Ebbsj6, Phys. Scr. 12 (1975) 168. [3] G. Gilat and L.J. Raubenheimer, Phys. Rev. B. 144 (1966) 390. [4] C. van Dijk and J. Bergsma, Neutron Inelastic Scattering, Vol. 1 (IAEA, Vienna, 1968) p. 233. [5] A.M. Vall~ra, PhD Thesis, Cambridge University (1977). [6] S.K. Satija, R.P. Com~s and G. Shirane, Phys. Rev. B 32 (1985) 3309. [7] A.M. Vail&a, J. de Phys. C6, Suppl. 12 (1981) 398. [8] V.L. Sliwko, P. Mohn, K. Schwar-z and P. Blaha, J. Phys.: Condens. Matter 8 (1996) 799.
J. Neuhaus et al./ Physica B 234-236 (1997) 897-899
[9] J. Zarestky and C. Stassis, Phys. Rev. B 35 (1987) 4500. [10] C.Y. Ho (ed.) Properties of Selected Ferrous Alloying Elements (Hemisphere Publ. Co., New York, 1989). [11] W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H.R. Schober and G. Vogl, Phys. Rev. B 43 (1991) 10933.
899
[12] A. Heiming, W. Petry, J. Trampenau, M. Alba, C. Herzig, H.R. Schober and G. Vogl, Phys. Rev. B 43 (1991) 10948. [13] F. Giithoff, W. Perry, C. Stassis, A. Heiming, B. Hennion, C. Herzig and J. Trampenau, Phys. Rev. B 47 (1993) 2563.