- Email: [email protected]
Elastic instabilities, phase transitions and soft modes in cubic media
Physica B 316–317 (2002) 186–189
Elastic instabilities, phase transitions and soft modes in cubic media ! c T. Paszkiewicza,*, M. Pruchnikb, P. Zielinski a
! ul. Rejtana 16 A, PL-35-310 Rzeszow, ! Poland Institute of Physics, University of Rzeszow, b Institute of Theoretical Physics, University of Wroc!aw, Wroc!aw, Poland c ! Poland Institute of Nuclear Physics, Krakow,
Abstract The low symmetry phases and the related soft modes are found for symmetry breaking equitranslational phase transitions of cubic media. The existence of the only possible longitudinal soft modes is predicted in two corners of the stability triangle. The phase velocities of soft modes are expressed by eigenvalues of the tensor of elastic constants. r 2002 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ks; 64.60.i; 69.70.Kb Keywords: Ferroelastic phase transitions; Soft modes
1. Introduction The vanishing of the bulk modulus B; the shear modulus Z and the Poisson ratio sP mark instabilities of elastic media. Three of them are related to eigenvalues of the tensor of secondorder elastic constants C2 : Vanishing of each of the eigenvalues marks mechanical instability. The corresponding eigentensors of C2 are strain tensors (called also small deformation tensors), providing information about the lattice distortion leading to new, less symmetric phases. The allowed low symmetry phases related to cubic media are monoclinic, orthorhombic, rhombohedral, tetragonal and triclinic. With exception of isomorphous discontinuous phase transformation, the above phase changes are accompanied by soft modes. *Corresponding author. Fax: +48-17-852-67-92. E-mail address: [email protected] (T. Paszkiewicz).
The elastic properties of cubic media are characterized by three parameters s1 > 0; s2 and s3 [1]. The parameters s2 and s3 can only vary inside the stability triangle bounded by s3 ¼ 53s2 16 (ab-side), s3 ¼ 23s2 23 ðac-side) and s2 ¼ 1 ðbc-side) lines. The ðs2 ; s3 Þ coordinates of corners of the stability triangle are: a ¼ ð0:5; 1Þ; b ¼ ð1; 1:5Þ; c ¼ ð1; 0Þ: Tensors of elastic constants C2 are repre* 2 ; while small sented by a 6 6 matrix C deformation tensors e by six-component vectors ðexx ; eyy ; ezz ; eyz ; exz ; exy Þ:
2. Changes of symmetry on borders of the stability triangle We shall study symmetry changes taking place when one comes across one of the borders of the
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 4 5 4 - 4
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189
stability triangle. An elastic instability is related to vanishing of at least one of eigenvalues of C2 : Lowering of symmetry is defined by eigentensors, i.e. eigendeformations, corresponding to these vanishing eigenvalues. These eigenvalues are: l1 ¼ C11 þ 2C12 ¼ 3B; l2 ¼ l3 ¼ C11 C12 ¼ 2Cs ; l4 ¼ l5 ¼ l6 ¼ 2C44 ¼ 2ZCs : Vectors defining the crystalline elementary cell a1 ; a2 ; a3 determine a matrix A* with elements * ab ¼ aðbÞ ða; b ¼ 1; 2; 3Þ being the projections ðAÞ a of the vectors aa onto the reference axes. A spatially homogeneous deformation changes both vec* namely aa -aa þ daa tors aa and the matrix A; * * * ða ¼ 1; 2; 3Þ; A-A þ dA: The symmetric deformation tensor e* with elements eab is related to A* 1 and dA* ð*eÞab ¼ eab
3 1X ¼ ½ðA* 1 Þag dA* gb 2 g¼1
þ dA* ag ðA* 1 Þgb :
ð1Þ
For simple cubic media with the lattice constant a the matrix A* is proportional to the unit matrix I;* therefore it is a simple task to find daa ða ¼ 1; 2; 3Þ corresponding to eigenvectors zi ði ¼ 1; y; 6Þ and to their linear combinations. They are listed in Table 1. The eigenvector z1 represents an isotropic expansion or contraction without any change of symmetry of the cubic medium. Thus, the phase transition is isomorphous [2]. Deformations corresponding to the remaining eigenvectors zi ði ¼ 2; y; 6Þ; occurring when the system approaches the ac and bc sides of the stability triangle, break the initial cubic symmetry of the medium. The resulting new phases belong to different crystallographic systems given in Table 1 in the column ‘‘n.ph.’’. The point groups marked in Table 1 are the maximal subgroups of the group m3m compatible with the appropriate deformations. In contrast to the isomorphous phase transition, the symmetry breaking phase transitions are accompanied by soft modes [3]. At the corner a the eigenvalues l1 and l2 ¼ l3 vanish, whereas at b l1 ¼ lp ¼ 0 ðp ¼ 4; 5; 6Þ: Since l1 and l2 ; l3 as well as l4 ; l5 and l6 belong to different irreducible representations these degeneracies are accidental. At the corner c l2 ¼ l3 ¼
187
lp ¼ 0 ðp ¼ 4; 5; 6Þ: These degeneracies are also accidental.
3. Isomorphous phase transitions and soft modes Assume that the mode j with wave vectors ðjÞ 0pkpkm ; the frequency oðk; jÞ and the polarization vector eðk; jÞ belonging to a narrow neighborhood of the G point, propagating in nðjÞ s direction ðjÞ soften: kðjÞ ¼ kn : The corresponding critical s s temperature is Tc : The part of uj;k ðrÞ related to this soft mode defines the deformation tensor (Eq. (1)). For T ¼ Tc and in the limit jkðjÞ s j-0 the matrix e* representing the strain tensor is ðjÞ ðjÞ ðjÞ eab ¼ 12 ½eðnðjÞ s ; jÞ#ns þ ns #eðns ; jÞab :
With pffiffiffi the fully symmetrical strain ð1= 3Þ½111000 Eq. (2) takes the form 0 1 0 1 1 0 0 nx e x nx e y nx e z B C B C @ 0 1 0 A ¼ @ ny e x ny e y ny e z A : 0 0 1 nz e x nz e y nz e z
ð2Þ z1 ¼
ð3Þ
Eq. (3) has no solution. Therefore, generally, there is no soft mode related to the isomorphous phase transition. This is in agreement with our numerical resultsFthe corresponding slowness surfaces do not change their linear dimensions when approaching the ab side of the stability triangle. Thus, only cubic materials with the elasticity parameters lying in the corners a and b of the stability triangle undergo the structural isomorphous phase transitions accompanied by soft modes. We can write ðjÞ eab ¼ ejj na nb þ 12 ½e> ðnðjÞ s ; jÞ#ns
þ nsðjÞ #e> ðnðjÞ s ; jÞab :
ð4Þ
Eq. (4) defines the strain engendered by a single phonon mode j with the propagation vector ns and with the polarization vector e: We see that only longitudinal modes have nonvanishing trace of the strain defined by Eq. (4), whereas every pure transverse phonon is related with a traceless strain. This follows from the fact that for the transverse phonons ejj the parallel component of the polarization vector vanishes, i.e. ejj ¼ en ¼ 0: On the
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189
188
Table 1 Distortions of the unit cell of cubic crystals related to various eigenvalues of the tensor of second-order elastic constants ev
dax
day
daz
n.ph.
p.g.
d.a.
z1 z2 z3 z4 z5 z6 a ðz P46þ z5 Þ i¼4 zi
eax# eax# eax# 0 ea#z ea#y ea#z eað#y þ z# Þ
ea#y ea#y ea#y ea#z 0 eax# ea#z eaðx# þ z# Þ
ea#z 0 2ea#z ea#y eax# 0 eaðx# þ y# Þ eaðx# þ y# Þ
Cub Ort Tetr Ortb Ortb Ortb Mon Rhom
m3m mmm 4=mmm mmm mmm mmm 2=m 3% m
F F z x y z ð1 1% 0Þ ð1 1 1Þ
ev is the eigenvector, n.ph. the new phase, p.g. the point group, d.a. the distinguished axis, Ortb the base centered orthorhombic structure with the centered face perpendicular to the distinguished axis. The general combination of zi ði ¼ 1; y6Þ produces a triclinic structure 1% ðCi Þ: a Choosing ðzi þ zj Þ; iaj ði; j ¼ 4; 5; 6Þ with various allowed i and j one gets different equivalent distinguished axes.
other hand e> ðnðjÞ s ; jÞ ¼ 0 for the pure longitudinal mode. 3.1. Relation of soft modes to eigenvalues of the tensor of elastic constants Using Eq. (2) we shall find the modes related to * 2 : Substituting the the eigendeformations of C expression (2) into eigenequation C2 zi ¼ li zi and multiplying both sides of the obtained equation by n; we get an equation, solutions of which read: ðlÞ 2 ðlÞ ðlÞ ðlÞ eðnðlÞ s ; j ¼ lÞ ¼ ns with c ðns ; j ¼ lÞns ¼ ðl1 =rÞns ; ðsÞ or eðnðsÞ ; j ¼ t Þ ¼ e ðn ; j ¼ t Þ with s > s s s ðsÞ ðsÞ c2 ðnðsÞ s ; jÞe> ðns ; j ¼ ts Þ ¼ ðli =2rÞe> ðns ; j ¼ ts Þ; where s ¼ 1; 2 and i ¼ 2; y; 6: Thus, the phase ðlÞ velocities related to the eigenvalues i are cðn ffi pffiffiffiffiffiffiffiffiffi plffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ;j ¼ lÞ ¼ l1 =r and cðnðts s Þ ; j ¼ ts Þ ¼ li =ð2rÞ: Above the parameter r is the medium mass density. Consequently, we see that soft modes can only be either pure longitudinal or pure transverse. However, any longitudinal mode involves nonvanishing trace of the strain tensor, i.e. a change of volume. Therefore, a longitudinal mode can only be soft whenever the bulk modulus also softens. This happens in the exceptional case of corners of stability triangle.
(cf. Section 2) soft longitudinal and soft transverse modes may coexist. For the longitudinal mode the trace of eigendeformation z1 does not vanish. Hence, this deformation changes the volume, but since l1 vanishes, such a change costs no energy. These properties explain experimentally observed fact that cubic-to-tetragonal phase transitions (for them l2 ¼ 0Þ; generally, does not involve longitudinal soft modes [4]. At the corner c of the stability triangle the pure transverse soft modes belonging to different irreducible representations (compatible with soft modes in isotropic media) may coexist. Deformations arising in phase transitions corresponding to the corners b and c of the stability triangle consist simultaneously of both shears and volume changes.
Acknowledgements Useful discussions with Prof. A.G. Every (University of Johannesburg) are acknowledged. This work was supported by Poland National Research Committee under grants Nos. 2 P03B 038 18 (T.P. and M.P.) and 2 P03B 072 18 (P.Z.)
3.2. Soft modes at the corners of the stability triangle References Note that due to the accidental degeneracies at the corners a and b of the stability triangle
[1] A.G. Every, Phys. Rev. B 22 (1980) 1746.
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189 [2] J.C. Tol!edano, P. Tol!edano, Landau Theory of Phase Transitions, World Scientific, Singapore, 1987; R. Folk, H. Iro, F. Schwabl, Z. Phys. B 25 (1976) 69; R. Folk, H. Iro, F. Schwabl, Phys. Rev. B 20 (1979) 1229.
189
[3] K.H. Michel, J. Naudts, J. Chem. Phys. 67 (1977) 547; K.H. Michel, J. Naudts, J. Chem. Phys. 68 (1978) 216. [4] R.D. Lowde, R.T. Hartley, G.A. Saunders, M. Sato, R. Scherm, C. Underhill, Proc. Roy. Soc. A 374 (1981) 87.
Elastic instabilities, phase transitions and soft modes in cubic media ! c T. Paszkiewicza,*, M. Pruchnikb, P. Zielinski a
! ul. Rejtana 16 A, PL-35-310 Rzeszow, ! Poland Institute of Physics, University of Rzeszow, b Institute of Theoretical Physics, University of Wroc!aw, Wroc!aw, Poland c ! Poland Institute of Nuclear Physics, Krakow,
Abstract The low symmetry phases and the related soft modes are found for symmetry breaking equitranslational phase transitions of cubic media. The existence of the only possible longitudinal soft modes is predicted in two corners of the stability triangle. The phase velocities of soft modes are expressed by eigenvalues of the tensor of elastic constants. r 2002 Elsevier Science B.V. All rights reserved. PACS: 61.50.Ks; 64.60.i; 69.70.Kb Keywords: Ferroelastic phase transitions; Soft modes
1. Introduction The vanishing of the bulk modulus B; the shear modulus Z and the Poisson ratio sP mark instabilities of elastic media. Three of them are related to eigenvalues of the tensor of secondorder elastic constants C2 : Vanishing of each of the eigenvalues marks mechanical instability. The corresponding eigentensors of C2 are strain tensors (called also small deformation tensors), providing information about the lattice distortion leading to new, less symmetric phases. The allowed low symmetry phases related to cubic media are monoclinic, orthorhombic, rhombohedral, tetragonal and triclinic. With exception of isomorphous discontinuous phase transformation, the above phase changes are accompanied by soft modes. *Corresponding author. Fax: +48-17-852-67-92. E-mail address: [email protected] (T. Paszkiewicz).
The elastic properties of cubic media are characterized by three parameters s1 > 0; s2 and s3 [1]. The parameters s2 and s3 can only vary inside the stability triangle bounded by s3 ¼ 53s2 16 (ab-side), s3 ¼ 23s2 23 ðac-side) and s2 ¼ 1 ðbc-side) lines. The ðs2 ; s3 Þ coordinates of corners of the stability triangle are: a ¼ ð0:5; 1Þ; b ¼ ð1; 1:5Þ; c ¼ ð1; 0Þ: Tensors of elastic constants C2 are repre* 2 ; while small sented by a 6 6 matrix C deformation tensors e by six-component vectors ðexx ; eyy ; ezz ; eyz ; exz ; exy Þ:
2. Changes of symmetry on borders of the stability triangle We shall study symmetry changes taking place when one comes across one of the borders of the
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 4 5 4 - 4
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189
stability triangle. An elastic instability is related to vanishing of at least one of eigenvalues of C2 : Lowering of symmetry is defined by eigentensors, i.e. eigendeformations, corresponding to these vanishing eigenvalues. These eigenvalues are: l1 ¼ C11 þ 2C12 ¼ 3B; l2 ¼ l3 ¼ C11 C12 ¼ 2Cs ; l4 ¼ l5 ¼ l6 ¼ 2C44 ¼ 2ZCs : Vectors defining the crystalline elementary cell a1 ; a2 ; a3 determine a matrix A* with elements * ab ¼ aðbÞ ða; b ¼ 1; 2; 3Þ being the projections ðAÞ a of the vectors aa onto the reference axes. A spatially homogeneous deformation changes both vec* namely aa -aa þ daa tors aa and the matrix A; * * * ða ¼ 1; 2; 3Þ; A-A þ dA: The symmetric deformation tensor e* with elements eab is related to A* 1 and dA* ð*eÞab ¼ eab
3 1X ¼ ½ðA* 1 Þag dA* gb 2 g¼1
þ dA* ag ðA* 1 Þgb :
ð1Þ
For simple cubic media with the lattice constant a the matrix A* is proportional to the unit matrix I;* therefore it is a simple task to find daa ða ¼ 1; 2; 3Þ corresponding to eigenvectors zi ði ¼ 1; y; 6Þ and to their linear combinations. They are listed in Table 1. The eigenvector z1 represents an isotropic expansion or contraction without any change of symmetry of the cubic medium. Thus, the phase transition is isomorphous [2]. Deformations corresponding to the remaining eigenvectors zi ði ¼ 2; y; 6Þ; occurring when the system approaches the ac and bc sides of the stability triangle, break the initial cubic symmetry of the medium. The resulting new phases belong to different crystallographic systems given in Table 1 in the column ‘‘n.ph.’’. The point groups marked in Table 1 are the maximal subgroups of the group m3m compatible with the appropriate deformations. In contrast to the isomorphous phase transition, the symmetry breaking phase transitions are accompanied by soft modes [3]. At the corner a the eigenvalues l1 and l2 ¼ l3 vanish, whereas at b l1 ¼ lp ¼ 0 ðp ¼ 4; 5; 6Þ: Since l1 and l2 ; l3 as well as l4 ; l5 and l6 belong to different irreducible representations these degeneracies are accidental. At the corner c l2 ¼ l3 ¼
187
lp ¼ 0 ðp ¼ 4; 5; 6Þ: These degeneracies are also accidental.
3. Isomorphous phase transitions and soft modes Assume that the mode j with wave vectors ðjÞ 0pkpkm ; the frequency oðk; jÞ and the polarization vector eðk; jÞ belonging to a narrow neighborhood of the G point, propagating in nðjÞ s direction ðjÞ soften: kðjÞ ¼ kn : The corresponding critical s s temperature is Tc : The part of uj;k ðrÞ related to this soft mode defines the deformation tensor (Eq. (1)). For T ¼ Tc and in the limit jkðjÞ s j-0 the matrix e* representing the strain tensor is ðjÞ ðjÞ ðjÞ eab ¼ 12 ½eðnðjÞ s ; jÞ#ns þ ns #eðns ; jÞab :
With pffiffiffi the fully symmetrical strain ð1= 3Þ½111000 Eq. (2) takes the form 0 1 0 1 1 0 0 nx e x nx e y nx e z B C B C @ 0 1 0 A ¼ @ ny e x ny e y ny e z A : 0 0 1 nz e x nz e y nz e z
ð2Þ z1 ¼
ð3Þ
Eq. (3) has no solution. Therefore, generally, there is no soft mode related to the isomorphous phase transition. This is in agreement with our numerical resultsFthe corresponding slowness surfaces do not change their linear dimensions when approaching the ab side of the stability triangle. Thus, only cubic materials with the elasticity parameters lying in the corners a and b of the stability triangle undergo the structural isomorphous phase transitions accompanied by soft modes. We can write ðjÞ eab ¼ ejj na nb þ 12 ½e> ðnðjÞ s ; jÞ#ns
þ nsðjÞ #e> ðnðjÞ s ; jÞab :
ð4Þ
Eq. (4) defines the strain engendered by a single phonon mode j with the propagation vector ns and with the polarization vector e: We see that only longitudinal modes have nonvanishing trace of the strain defined by Eq. (4), whereas every pure transverse phonon is related with a traceless strain. This follows from the fact that for the transverse phonons ejj the parallel component of the polarization vector vanishes, i.e. ejj ¼ en ¼ 0: On the
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189
188
Table 1 Distortions of the unit cell of cubic crystals related to various eigenvalues of the tensor of second-order elastic constants ev
dax
day
daz
n.ph.
p.g.
d.a.
z1 z2 z3 z4 z5 z6 a ðz P46þ z5 Þ i¼4 zi
eax# eax# eax# 0 ea#z ea#y ea#z eað#y þ z# Þ
ea#y ea#y ea#y ea#z 0 eax# ea#z eaðx# þ z# Þ
ea#z 0 2ea#z ea#y eax# 0 eaðx# þ y# Þ eaðx# þ y# Þ
Cub Ort Tetr Ortb Ortb Ortb Mon Rhom
m3m mmm 4=mmm mmm mmm mmm 2=m 3% m
F F z x y z ð1 1% 0Þ ð1 1 1Þ
ev is the eigenvector, n.ph. the new phase, p.g. the point group, d.a. the distinguished axis, Ortb the base centered orthorhombic structure with the centered face perpendicular to the distinguished axis. The general combination of zi ði ¼ 1; y6Þ produces a triclinic structure 1% ðCi Þ: a Choosing ðzi þ zj Þ; iaj ði; j ¼ 4; 5; 6Þ with various allowed i and j one gets different equivalent distinguished axes.
other hand e> ðnðjÞ s ; jÞ ¼ 0 for the pure longitudinal mode. 3.1. Relation of soft modes to eigenvalues of the tensor of elastic constants Using Eq. (2) we shall find the modes related to * 2 : Substituting the the eigendeformations of C expression (2) into eigenequation C2 zi ¼ li zi and multiplying both sides of the obtained equation by n; we get an equation, solutions of which read: ðlÞ 2 ðlÞ ðlÞ ðlÞ eðnðlÞ s ; j ¼ lÞ ¼ ns with c ðns ; j ¼ lÞns ¼ ðl1 =rÞns ; ðsÞ or eðnðsÞ ; j ¼ t Þ ¼ e ðn ; j ¼ t Þ with s > s s s ðsÞ ðsÞ c2 ðnðsÞ s ; jÞe> ðns ; j ¼ ts Þ ¼ ðli =2rÞe> ðns ; j ¼ ts Þ; where s ¼ 1; 2 and i ¼ 2; y; 6: Thus, the phase ðlÞ velocities related to the eigenvalues i are cðn ffi pffiffiffiffiffiffiffiffiffi plffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ;j ¼ lÞ ¼ l1 =r and cðnðts s Þ ; j ¼ ts Þ ¼ li =ð2rÞ: Above the parameter r is the medium mass density. Consequently, we see that soft modes can only be either pure longitudinal or pure transverse. However, any longitudinal mode involves nonvanishing trace of the strain tensor, i.e. a change of volume. Therefore, a longitudinal mode can only be soft whenever the bulk modulus also softens. This happens in the exceptional case of corners of stability triangle.
(cf. Section 2) soft longitudinal and soft transverse modes may coexist. For the longitudinal mode the trace of eigendeformation z1 does not vanish. Hence, this deformation changes the volume, but since l1 vanishes, such a change costs no energy. These properties explain experimentally observed fact that cubic-to-tetragonal phase transitions (for them l2 ¼ 0Þ; generally, does not involve longitudinal soft modes [4]. At the corner c of the stability triangle the pure transverse soft modes belonging to different irreducible representations (compatible with soft modes in isotropic media) may coexist. Deformations arising in phase transitions corresponding to the corners b and c of the stability triangle consist simultaneously of both shears and volume changes.
Acknowledgements Useful discussions with Prof. A.G. Every (University of Johannesburg) are acknowledged. This work was supported by Poland National Research Committee under grants Nos. 2 P03B 038 18 (T.P. and M.P.) and 2 P03B 072 18 (P.Z.)
3.2. Soft modes at the corners of the stability triangle References Note that due to the accidental degeneracies at the corners a and b of the stability triangle
[1] A.G. Every, Phys. Rev. B 22 (1980) 1746.
T. Paszkiewicz et al. / Physica B 316–317 (2002) 186–189 [2] J.C. Tol!edano, P. Tol!edano, Landau Theory of Phase Transitions, World Scientific, Singapore, 1987; R. Folk, H. Iro, F. Schwabl, Z. Phys. B 25 (1976) 69; R. Folk, H. Iro, F. Schwabl, Phys. Rev. B 20 (1979) 1229.
189
[3] K.H. Michel, J. Naudts, J. Chem. Phys. 67 (1977) 547; K.H. Michel, J. Naudts, J. Chem. Phys. 68 (1978) 216. [4] R.D. Lowde, R.T. Hartley, G.A. Saunders, M. Sato, R. Scherm, C. Underhill, Proc. Roy. Soc. A 374 (1981) 87.