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The symmetry of martensites
THE SYMMETRY
OF MARTENSITES
JOHS W. C/U-IN Department of Materials Science and Engineering. Massachusetts Cambridge. MA 02139. U.S.A. (Receiced
Institute of Technology.
16 November 1976)
symmetry of martensite is usually less than that of the related equilibrium phase-being a subgroup of both the parent crystal and the transformation strains. The attainment of the fuller symmetry is prevented by local or long-range ordering in the parent phase. Shuffles are necessar?. but may not be sufficient, for increased symmetry (supergroup formation) in cases where there IS more than one atom per unit cell. Abstract-The
R&urn&La symt-trie de la martensite est gkniralement inftrieure & celle de la phase d’tquilibrc correspondante; elle constitue un sous-groupe du cristal initial et un sous-groupe des diformations produites par la transformation. L’ordre local ou B longue distance de la phase initiale empiche d’obtenir la symitrie compl&e. Des accommodements sont nkcessaires. mais pas toujours suffisants. pour augments: la symt-trie (formation de super-groupe) dans les cas oti il ya plus d’un atome par maille. Zusammenfassung-Die Symmetrie von Martensit ist iiblicherweise geringer als die der zugehijrigen Gleichgewichtsphase, die eine Untergruppe sowohl des Mutterkristalls als such der Umwandlungsverzerrungen ist. Das Erreichen der hijheren Symmetrie wird verhindert durch lokale oder weitreichende Ordnung in der Mutterphase. Verschiebungen sind notwendig, magen aber nicht ausreichend sein fir eine hijhere Symmetrie (fjbergruppenbildung) in FLllen, in denen mehr als ein Atom pro Einheitszelle vorliegt.
INTRODUCTION Martensitic reactions are phase changes in which a new phase is formed by a large strain which occurs so rapidly that there is no time for diffusive motion of the individual atoms. The resulting martensitic phase is always one which is closely related to a stable or metastable equilibrium phase in that system. Martensite differs from that phase in that it often has less symmetry [I]. In this paper we examine how the strain destroys symmetry elements of the parent phase and could create new symmetry elements in the martensitic phase. We shall find that formation of the new symmetry elements frequently cannot occur and that the symmetry of the martensite phase is a subgroup of both the symmetry of the parent phase and the strain. Because of the strain mechanism and the absence of diffusion, there is an atom-by-atom correspondence between positions in the. parent phase and the new one. This correspondence may be described by a unit cell transformation, a homogeneous (lattice) strain and a periodic inhomogeneous strain called shuffles. Additionally, in alloys there is also a small irregular displacement of individual atoms that results from adjustment to the different sizes of their neighboring atoms. The unit cell transformation leads to a matrix which relates the unit cell in the parent phase to what will be the unit cell in the martensite. Such matrices are used in crystallographic studies where different
choices of unit cells are often made and some are tabulated [2). For ferrous martensites, there is a change from a face-centered lattice to a body-centered one and the matrix and its inverse are:
The determinant gives the ratio of the number of atoms in the two unit cells. The matrix is used to find corresponding indices of planes and reduced positions in the reciprocal lattice, while the transpose of the reciprocal matrix is used to find corresponding reduced position in the unit cell and zone axes. For ferrous martensites, the matrix is called the Bain correspondence [3]. The homogeneous strain, S. sometimes called the lattice strain or the Bain strain. relates the new lengths (rather than displacements) to the old. The inhomogeneous part (shuffles) is often described by static displacement waves. In addition, for accommodation with the adjacent untransformed phase, there is twinning or plastic deformation by slip. Rotation also occurs. but this has no effect on symmetry. In what follows, we will assume that the accommodation is by twinning and that the twin also forms martensitically from the parent phase by the same mechanisms. The stable or metastable equilibrium phase closely related to the martensite is usually well known. In ferrous martensites it is b.c.c. The martensite is usually 721
722
CAHN:
THE SYMMETRY
only a slightly distorted version of this phase and diffusive atom motion, as in tempering, often produces this phase. Martensite is here considered to be an unstable version of this phase with a non equilibrium arrangement of atoms resulting from an absence of a mechanism of equilibration. SUBGROUP FORIMATION FOR HOMOGENEOUS STRAIN We first consider how the symmetry of a crystal is affected by an arbitrary homogeneous strain. Except for certain special values of the strain, rhe symmetry elements of the resultant must be a subset of both the symmetries of the original crystal and of the swain. Translation symmetry is not lost in a homo-
OF MARTENSITES
metry elements appear and the structure is b.c.c. For thermodynamic reasons. b.c.c. is the equilibrium phase and in pure iron the strain would carry the structure exactly to this point of supergroup formation. If alloys. however, there are reasons why the supergroup formation cannot occur. For instance. the j-fold axis carries [OOI] into [lOO] or [OIO]. However, [OOl] corresponded to [OOl) in the original f.c.c., while [loo] and [OIO] were (l/2) [I IO] and (1;2) [IiO]. Carbon atoms could sit in the octahedral hole between two atoms separated by [OOI], while none could sit in (l/2) [ 1lo]. which is a close-packed nearneighbor separation. In a substitutional alloy, if local order is such that first- and second-pair correlations in the original f.c.c. are different, then the 3-fold axis also cannot appear, Even if c = CI,the resulting structure will always be tetragonal. Even if some physical properties accidentally conformed to a higher symmetry, the majority would not. With a homogeneous strain, supergroup formation does not occur in the case of pure compounds where some atoms do not reside at lattice points. For example, a homogeneously strained ammonium ion loses its tetragonal symmetry. In this case, shuffles can restore the higher symmetry. A NaCl-type structure which is Fm3m undergoing an ideal strain that would carry the anions into a b.c.c. structure would always be tetragonal because while one species could occupy the sites at (000) [and (l/2 l/2 l/2)] in a b.c.c. lattice, the other would occupy an ordered one-third of the six (l/2 0 0)-type sites. A random occupation among the sites, as is thought to occur in AgI, would create a b.c.c. structure. Supergroup formation always involves creation of symmetries that were not present in the original phase. It can occur in pure metals and in ideally random alloys. When it does not occur, the symmetry of the martensitic cr;ystal remains that of the subgroup.
geneous strain. If the three principal strains are unequal, the point group symmetry of the strain is Ds,, (mmm); if two of the principal strains are equal, it is D,, (infinite rotation axis with a perpendicular mirror plane). The point symmetry of the structure as well as that of any general or special position becomes one which is a subgroup of both the original symmetry and the strain symmetry. For each symmetry plane or axis, the surviving symmetry is that which is a subgroup of both the original crystal structure and the strain. The surviving symmetry elements are thus relatively few in number. For a strain of type Drh, the symmetry axis of the crystal parallel to the unique strain axis (infinite rotation axis) survives. Mirror planes parallel and perpendicular to this axis survive. Even-fold rotation axes perpendicular to this axis are reduced to 2-fold axes. All other elements are destroyed. For a strain of type DZh,mirrors and evenfold axes aligned to the principal strain axes survive as mirrors and 2-fold axes. All others are destroyed. The Bain strain that would carry f.c.c. to b.c.c. is a homogeneous strain belonging to D,,. The infinite rotation axis coincides with the Cfold cubic axis of the f.c.c. phase, and remains a 4-fold axis. The other RECIPROCAL SPACE DESCRIPTION two 4-fold axes coincide with the mirror plane of the Intensity in reciprocal space is related to the Fourstrain and survive as 2/m. The 3-fold axes are lost ier transform of scattering intensity in the real cryssince they do not coincide with any symmetry directal [4]. As shown in the Appendix, a homogeneous tion of the strain. Unless new symmetry elements appear, the resultant structure is tetragonal D4,, strain therefore distorts reciprocal space by the transpose of the reciprocal of the strain without changing (4jmmm). the distribution in intensity. Because the reciprocal Supergroup formation for homogeneous strain lattice always has the same point group symmetry as the crystal, we can re-examine the symmetry of The equilibrium phases corresponding to martensites are usually highly symmetrical phases with symthe martensitic crystal in reciprocal space. The transpose of the inverse of the strain has the metry elements in addition to those which were pressame symmetry as the strain. The reciprocal space ent in the original phase. If we apply an arbitrary of primitive (P), rhombohedral (R), and end-centered strain belonging to D,, to pure f.c.c. iron with the rotation axis along the cubic axis, we obtain a bodycrystals is, respectively, P, R, and end-centered. The centered tetragonal structure. For certain values of reciprocal space of face-centered (F) crystals is bodycentered (I) and vice versa [YJ. Thus, in ferrous marthe strain, c and a are equal and four 3-fold axes tensites, we start with f.c.c. which has a b.c.c. reciprocal appear along [I 111 directions which were [lOI] lattice. If the martensite were b.c.c., it would have an 2-fold axes in the f.c.c. and were lost during the strain. f.c.c. reciprocal space. Reciprocal space here clearly unFor this very definite value of the strain, new sym-
CAHN: THE SYMMETRY dergoes a distortion which is the inverse of the strain in real space. If there is local order or clustering, it results in variable diffuse scattering in reciprocal space. This diffuse scattering originally has the symmetry of the point group of the crystal. After the martensitic transformation, the symmetry of the diffuse scattering belongs to the subgroup and reciprocal space can never conform to the supergroup. Perfectly ordered alloys and compounds have no diffuse scattering. However, the structure factors and thus the intensity of the Bragg peaks vary with hkl. After a Bain strain the positions of the peaks may be those of the more symmetrical structure, but the varying intensities of the peaks prevent supergroup formation. Non-stoichiometric compounds and partially ordered alloys have both diffuse scattering and variable intensities at Bragg peaks to prevent supergroup formation. SHUFF-LES Shuffles affect translational symmetry as well as point group symmetry and can form sub- and supergroups for both kinds of symmetry. The atomic movements due to shuffles can be described in terms of a small number of displacement waves in either the original or the homogeneously strained structure. Each wave is given by a wave vector & and a displacement vector G. For example, shuffles imposed on the homogeneously strained structure lead to positions ?::I: .,I r, - f; = cE‘scosjs*?;. (2) S
All these p, vectors must be commensurate with the reciprocal lattice vectors; otherwise there would be no periodicity in the martensite. If any of the & differ from a reciprocal lattice vector, a new reciprocal lattice is formed having additional points to contain the 8;. This increases the number of atoms in the unit cell of the martensite and implies a loss of translation symmetry. The shuffles that occur when h.c.p. with two atoms per primitive cell is formed from either b.c.c. - or f.c.c. with one atom per unit cell is due to a j& which is l/2 of a [ 11l] reciprocal lattice vector. The symmetry of strains due to a shuffle caused by a single wave is determined by the angle between Bs and Zs, being D,, if they are parallel, DZh (mmm) if perpendicular (with the three mirror planes perpendi’cular to &. ES and in the plane containing both), Cl,, (2/m) if neither parallel nor perpendicular (with the mirror plane containing both). Shuffles cause subgroup formation in the same way as do homogeneous strains, but they are also important in supergroup formation. Supergroup formation results because shuffles by rearranging atoms within a unit cell alter the unit cell structure factors and, therefore, intensities at various reflections. If the homogeneous strain has placed the reciprocal lattice points in a lattice with
OF MARTENSITES
72.7
higher symmetry. but with intensities that do not correspond to the higher symmetry, shuffles will enable the crystal to achieve the higher symmetry. Whether or not these actually occur will depend on the existence of a mechanism for the required shuffles. Some large amplitude shuffles can probably occur only with thermal activation. In certain transformations (e.g. h.c.p. to f.c.c.), the shuffles must cause extinctions at certain reciprocal lattice points to form the structure with new point group symmetries or to reduce the number of atoms per unit cell. Shuffles have no effect on diffuse scattering due to substitutional local order. In cases of such local order, supergroup formation remains impossible. An example of martensitic transformation with shuffles is the b.c.c. to B in which Z is along [111] and I(Iis 2!3 of a [ Ill] reciprocal lattice vector. The shuffles as well as any homogeneous strain belong to D,, aligned along the 3-fold axis. The resulting structure is RJm with three atoms per unit cell regardless of local order. For b.c.c. to h.c.p., both the homogeneous strain and the shuffles (g = l/2 of [ITO] with i along [l 10)) belong to D2,,. Without supergroup formation, the resulting martensite will be orthorhombic. For f.c.c. to h.c.p.. the homogeneous strain and shuffles (p = 1.2 of [ill] with Z along [112]) also belong to DLh. but oriented along [ 1111. Without supergroup formation, the resultant structure is monoclinic. REVERSIBLE MARTENSITES Reversion of martensite to form the parent phase again can often occur martensitically, that is, with an exact atom by atom correspondence. If the martensite has achieved the high symmetry of the equilibrium phase, there are usually several crystallographitally equivalent strains for the reversion. If the martensite has the lower symmetry of the subgroup, only one of these strains reproduces the original parent phase. In this case. the strain is the exact inverse of the original strains. All other transformation strains produce a low-symmetry version of the parent phase presumably of higher free energy. This energy difference should favor the exact inverse, as is observed in ordered martensites. DISCUSSION The symmetry changes due to martensitic transformations in either direct space or reciprocal space make it clear that there are many reasons why martensites have less symmetry than the same phase would have if equilibrated. The symmetries of martensites are usually subgroups of both the symmetries of lattice strain and the original phase. The lattice strain and shuffles are usually known. In some cases were there is ambiguity it has been suggested that the experimentally-found martensite symmetry might resolve it [63.
CAHN:
724
THE SYMMETRY OF MARTENSITES
The inability to form a symmetry supergroup in alloys with local order described by pair correlations was the basis of a paper by Cahn and Rosenberg [6]. Another description of the cause of tetragonality due to clusters was given by Speich and Winchell [q. Since pairs of atoms are a subset of all clusters, it would seem that the Speich and Winchell description would cover all cases described by Cahn and Rosenberg. That this is not so can be demonstrated by a single counter example. In the Speich and Winchell treatment, the point symmetry of the center of any cluster is compared in the austenite and martensite to see if it “can support tetragonality”. Any three adjacent atoms is a linear cluster centered on m3m in both structures2regardless of whether they line up along [lOO], [llO] or any other crystal direction. According to Speich and Winchell these cannot support tetragonality. But we have shown that any local pair-wise order between the members will produce tetragonality. Structures of pure elements with one atom per unit cell can give rise to martensites with high symmetries, but shuffles are required for obtaining high symmetries if the parent structure has more than one atom per primitive unit cell. Solid solutions whether interstitial or substitutional with local order will never give high-symmetry virgin martensite. Diffusive motion is required. Effect of long-range
order
Long-range ordering in the parent phase is itself a case of subgroup formation both for translation and point groups. To understand the subsequent changes in symmetry because of the martensitic transformation, we must begin with the symmetry of the ordered phase. Shuffles are essential for supergroup formation, but if ordering is imperfect supergroup formation is prevented. Acknowledgements-The interest and patient criticism by Bernhardt J. Wuensch, Jr., about the crystallographic aspects were gratefully received and were essential for the development of many of the arguments. I am also grateful to Morris Cohen, Sam Allen, and Walter Owen for their critical reviews. The work was 73076%AOI-DMR.
supported
by
NSF
Grant
No.
Burke), Vol. 3, p. 1433. Plenum (1973).
2. International Tables of X-Ray Crystallography. Vol. I, Table 2.5.1, p. 21. Kynoch Press, Birmingham (1965). 3. A. Kelly and G. W. Groves. Crystallography and Crystal Defects, Chapter 11. Addison Wesley. Reading (1970). B. E. Warren, X-Ray Diffraction, Chapters l&12. Addison Wesley, Reading (1969). International Tables of X-Ray Crprallography, Vol. 1. Table 2.42, p. 13. Kynoch Press, Birmingham (1965). J. W. Cahn and W. Rosenberg, Scripta Met. 5, 101 (1971). P. G. Winchell and G. R. Speich, Acru Mer. 18, 53 (1970).
APPEhDIX The E&t
of a Homogeneous Strain on Scattering Intensity in Reciprocal Space
Given the scattering factor, j”, and the position. i,, of every atom in the original phase, the scattering intensity I as a function of position Z in reciprocal space is given by C41:
(Al) With a homogeneous deformation S, the position of the nth atom becomes 7; and (r; L rQi = &jr. - rm)j.
REFERENCES Met& AIM& Science
H. M.
642)
The inverse of S is S-’ and has the following properties
sijs,:qf = S,‘Sj, = &. Let us define a I? space deformed by a strain (S-i)’ is the transpose of the inverse of S I;’ = (S-i)+ I;
(A3) which (A4)
k;(rA - rk)i = S; ‘kjSit(r, - r,,Jlr = 6jtkj(r, - rmjr = k,(r, - r,)j.
(0
li’qr:, - ?k.)= L(i, - ?“).
(A6)
It follows that
The scattering intensity at i;’ from the strained crystal I&‘) is identical to that of the unstrained crystal at x r,(p)
=
1
c f, f~e’iY;:. - 2”)
1 m
= i(k).
A word about the strain (S-l)‘:
1. P. G. Winchell and M. Cohen, Trans. Am. Sot. 55. 347 (1962); M. Cohen, Trans. metall. Sot. 224, 638‘ (1962); J. C. Williams, in Titanium and Technology (edited by R. I. Jaffee and
Press, New York
If S contains a rotation, (S-i)+ contains the same rotation. The length of each principal axis of (S-‘)’ is the reciprocal of that of S. Deviations from this rule occur due to inhomogeneous deformation. Shuffles are considered in the text. Diffusion is assumed not to occur. Size effects cannot be prevented. Thus, the theorem holds only for the Bragg peaks and that part of the diffuse scattering not due to size effects.
OF MARTENSITES
JOHS W. C/U-IN Department of Materials Science and Engineering. Massachusetts Cambridge. MA 02139. U.S.A. (Receiced
Institute of Technology.
16 November 1976)
symmetry of martensite is usually less than that of the related equilibrium phase-being a subgroup of both the parent crystal and the transformation strains. The attainment of the fuller symmetry is prevented by local or long-range ordering in the parent phase. Shuffles are necessar?. but may not be sufficient, for increased symmetry (supergroup formation) in cases where there IS more than one atom per unit cell. Abstract-The
R&urn&La symt-trie de la martensite est gkniralement inftrieure & celle de la phase d’tquilibrc correspondante; elle constitue un sous-groupe du cristal initial et un sous-groupe des diformations produites par la transformation. L’ordre local ou B longue distance de la phase initiale empiche d’obtenir la symitrie compl&e. Des accommodements sont nkcessaires. mais pas toujours suffisants. pour augments: la symt-trie (formation de super-groupe) dans les cas oti il ya plus d’un atome par maille. Zusammenfassung-Die Symmetrie von Martensit ist iiblicherweise geringer als die der zugehijrigen Gleichgewichtsphase, die eine Untergruppe sowohl des Mutterkristalls als such der Umwandlungsverzerrungen ist. Das Erreichen der hijheren Symmetrie wird verhindert durch lokale oder weitreichende Ordnung in der Mutterphase. Verschiebungen sind notwendig, magen aber nicht ausreichend sein fir eine hijhere Symmetrie (fjbergruppenbildung) in FLllen, in denen mehr als ein Atom pro Einheitszelle vorliegt.
INTRODUCTION Martensitic reactions are phase changes in which a new phase is formed by a large strain which occurs so rapidly that there is no time for diffusive motion of the individual atoms. The resulting martensitic phase is always one which is closely related to a stable or metastable equilibrium phase in that system. Martensite differs from that phase in that it often has less symmetry [I]. In this paper we examine how the strain destroys symmetry elements of the parent phase and could create new symmetry elements in the martensitic phase. We shall find that formation of the new symmetry elements frequently cannot occur and that the symmetry of the martensite phase is a subgroup of both the symmetry of the parent phase and the strain. Because of the strain mechanism and the absence of diffusion, there is an atom-by-atom correspondence between positions in the. parent phase and the new one. This correspondence may be described by a unit cell transformation, a homogeneous (lattice) strain and a periodic inhomogeneous strain called shuffles. Additionally, in alloys there is also a small irregular displacement of individual atoms that results from adjustment to the different sizes of their neighboring atoms. The unit cell transformation leads to a matrix which relates the unit cell in the parent phase to what will be the unit cell in the martensite. Such matrices are used in crystallographic studies where different
choices of unit cells are often made and some are tabulated [2). For ferrous martensites, there is a change from a face-centered lattice to a body-centered one and the matrix and its inverse are:
The determinant gives the ratio of the number of atoms in the two unit cells. The matrix is used to find corresponding indices of planes and reduced positions in the reciprocal lattice, while the transpose of the reciprocal matrix is used to find corresponding reduced position in the unit cell and zone axes. For ferrous martensites, the matrix is called the Bain correspondence [3]. The homogeneous strain, S. sometimes called the lattice strain or the Bain strain. relates the new lengths (rather than displacements) to the old. The inhomogeneous part (shuffles) is often described by static displacement waves. In addition, for accommodation with the adjacent untransformed phase, there is twinning or plastic deformation by slip. Rotation also occurs. but this has no effect on symmetry. In what follows, we will assume that the accommodation is by twinning and that the twin also forms martensitically from the parent phase by the same mechanisms. The stable or metastable equilibrium phase closely related to the martensite is usually well known. In ferrous martensites it is b.c.c. The martensite is usually 721
722
CAHN:
THE SYMMETRY
only a slightly distorted version of this phase and diffusive atom motion, as in tempering, often produces this phase. Martensite is here considered to be an unstable version of this phase with a non equilibrium arrangement of atoms resulting from an absence of a mechanism of equilibration. SUBGROUP FORIMATION FOR HOMOGENEOUS STRAIN We first consider how the symmetry of a crystal is affected by an arbitrary homogeneous strain. Except for certain special values of the strain, rhe symmetry elements of the resultant must be a subset of both the symmetries of the original crystal and of the swain. Translation symmetry is not lost in a homo-
OF MARTENSITES
metry elements appear and the structure is b.c.c. For thermodynamic reasons. b.c.c. is the equilibrium phase and in pure iron the strain would carry the structure exactly to this point of supergroup formation. If alloys. however, there are reasons why the supergroup formation cannot occur. For instance. the j-fold axis carries [OOI] into [lOO] or [OIO]. However, [OOl] corresponded to [OOl) in the original f.c.c., while [loo] and [OIO] were (l/2) [I IO] and (1;2) [IiO]. Carbon atoms could sit in the octahedral hole between two atoms separated by [OOI], while none could sit in (l/2) [ 1lo]. which is a close-packed nearneighbor separation. In a substitutional alloy, if local order is such that first- and second-pair correlations in the original f.c.c. are different, then the 3-fold axis also cannot appear, Even if c = CI,the resulting structure will always be tetragonal. Even if some physical properties accidentally conformed to a higher symmetry, the majority would not. With a homogeneous strain, supergroup formation does not occur in the case of pure compounds where some atoms do not reside at lattice points. For example, a homogeneously strained ammonium ion loses its tetragonal symmetry. In this case, shuffles can restore the higher symmetry. A NaCl-type structure which is Fm3m undergoing an ideal strain that would carry the anions into a b.c.c. structure would always be tetragonal because while one species could occupy the sites at (000) [and (l/2 l/2 l/2)] in a b.c.c. lattice, the other would occupy an ordered one-third of the six (l/2 0 0)-type sites. A random occupation among the sites, as is thought to occur in AgI, would create a b.c.c. structure. Supergroup formation always involves creation of symmetries that were not present in the original phase. It can occur in pure metals and in ideally random alloys. When it does not occur, the symmetry of the martensitic cr;ystal remains that of the subgroup.
geneous strain. If the three principal strains are unequal, the point group symmetry of the strain is Ds,, (mmm); if two of the principal strains are equal, it is D,, (infinite rotation axis with a perpendicular mirror plane). The point symmetry of the structure as well as that of any general or special position becomes one which is a subgroup of both the original symmetry and the strain symmetry. For each symmetry plane or axis, the surviving symmetry is that which is a subgroup of both the original crystal structure and the strain. The surviving symmetry elements are thus relatively few in number. For a strain of type Drh, the symmetry axis of the crystal parallel to the unique strain axis (infinite rotation axis) survives. Mirror planes parallel and perpendicular to this axis survive. Even-fold rotation axes perpendicular to this axis are reduced to 2-fold axes. All other elements are destroyed. For a strain of type DZh,mirrors and evenfold axes aligned to the principal strain axes survive as mirrors and 2-fold axes. All others are destroyed. The Bain strain that would carry f.c.c. to b.c.c. is a homogeneous strain belonging to D,,. The infinite rotation axis coincides with the Cfold cubic axis of the f.c.c. phase, and remains a 4-fold axis. The other RECIPROCAL SPACE DESCRIPTION two 4-fold axes coincide with the mirror plane of the Intensity in reciprocal space is related to the Fourstrain and survive as 2/m. The 3-fold axes are lost ier transform of scattering intensity in the real cryssince they do not coincide with any symmetry directal [4]. As shown in the Appendix, a homogeneous tion of the strain. Unless new symmetry elements appear, the resultant structure is tetragonal D4,, strain therefore distorts reciprocal space by the transpose of the reciprocal of the strain without changing (4jmmm). the distribution in intensity. Because the reciprocal Supergroup formation for homogeneous strain lattice always has the same point group symmetry as the crystal, we can re-examine the symmetry of The equilibrium phases corresponding to martensites are usually highly symmetrical phases with symthe martensitic crystal in reciprocal space. The transpose of the inverse of the strain has the metry elements in addition to those which were pressame symmetry as the strain. The reciprocal space ent in the original phase. If we apply an arbitrary of primitive (P), rhombohedral (R), and end-centered strain belonging to D,, to pure f.c.c. iron with the rotation axis along the cubic axis, we obtain a bodycrystals is, respectively, P, R, and end-centered. The centered tetragonal structure. For certain values of reciprocal space of face-centered (F) crystals is bodycentered (I) and vice versa [YJ. Thus, in ferrous marthe strain, c and a are equal and four 3-fold axes tensites, we start with f.c.c. which has a b.c.c. reciprocal appear along [I 111 directions which were [lOI] lattice. If the martensite were b.c.c., it would have an 2-fold axes in the f.c.c. and were lost during the strain. f.c.c. reciprocal space. Reciprocal space here clearly unFor this very definite value of the strain, new sym-
CAHN: THE SYMMETRY dergoes a distortion which is the inverse of the strain in real space. If there is local order or clustering, it results in variable diffuse scattering in reciprocal space. This diffuse scattering originally has the symmetry of the point group of the crystal. After the martensitic transformation, the symmetry of the diffuse scattering belongs to the subgroup and reciprocal space can never conform to the supergroup. Perfectly ordered alloys and compounds have no diffuse scattering. However, the structure factors and thus the intensity of the Bragg peaks vary with hkl. After a Bain strain the positions of the peaks may be those of the more symmetrical structure, but the varying intensities of the peaks prevent supergroup formation. Non-stoichiometric compounds and partially ordered alloys have both diffuse scattering and variable intensities at Bragg peaks to prevent supergroup formation. SHUFF-LES Shuffles affect translational symmetry as well as point group symmetry and can form sub- and supergroups for both kinds of symmetry. The atomic movements due to shuffles can be described in terms of a small number of displacement waves in either the original or the homogeneously strained structure. Each wave is given by a wave vector & and a displacement vector G. For example, shuffles imposed on the homogeneously strained structure lead to positions ?::I: .,I r, - f; = cE‘scosjs*?;. (2) S
All these p, vectors must be commensurate with the reciprocal lattice vectors; otherwise there would be no periodicity in the martensite. If any of the & differ from a reciprocal lattice vector, a new reciprocal lattice is formed having additional points to contain the 8;. This increases the number of atoms in the unit cell of the martensite and implies a loss of translation symmetry. The shuffles that occur when h.c.p. with two atoms per primitive cell is formed from either b.c.c. - or f.c.c. with one atom per unit cell is due to a j& which is l/2 of a [ 11l] reciprocal lattice vector. The symmetry of strains due to a shuffle caused by a single wave is determined by the angle between Bs and Zs, being D,, if they are parallel, DZh (mmm) if perpendicular (with the three mirror planes perpendi’cular to &. ES and in the plane containing both), Cl,, (2/m) if neither parallel nor perpendicular (with the mirror plane containing both). Shuffles cause subgroup formation in the same way as do homogeneous strains, but they are also important in supergroup formation. Supergroup formation results because shuffles by rearranging atoms within a unit cell alter the unit cell structure factors and, therefore, intensities at various reflections. If the homogeneous strain has placed the reciprocal lattice points in a lattice with
OF MARTENSITES
72.7
higher symmetry. but with intensities that do not correspond to the higher symmetry, shuffles will enable the crystal to achieve the higher symmetry. Whether or not these actually occur will depend on the existence of a mechanism for the required shuffles. Some large amplitude shuffles can probably occur only with thermal activation. In certain transformations (e.g. h.c.p. to f.c.c.), the shuffles must cause extinctions at certain reciprocal lattice points to form the structure with new point group symmetries or to reduce the number of atoms per unit cell. Shuffles have no effect on diffuse scattering due to substitutional local order. In cases of such local order, supergroup formation remains impossible. An example of martensitic transformation with shuffles is the b.c.c. to B in which Z is along [111] and I(Iis 2!3 of a [ Ill] reciprocal lattice vector. The shuffles as well as any homogeneous strain belong to D,, aligned along the 3-fold axis. The resulting structure is RJm with three atoms per unit cell regardless of local order. For b.c.c. to h.c.p., both the homogeneous strain and the shuffles (g = l/2 of [ITO] with i along [l 10)) belong to D2,,. Without supergroup formation, the resulting martensite will be orthorhombic. For f.c.c. to h.c.p.. the homogeneous strain and shuffles (p = 1.2 of [ill] with Z along [112]) also belong to DLh. but oriented along [ 1111. Without supergroup formation, the resultant structure is monoclinic. REVERSIBLE MARTENSITES Reversion of martensite to form the parent phase again can often occur martensitically, that is, with an exact atom by atom correspondence. If the martensite has achieved the high symmetry of the equilibrium phase, there are usually several crystallographitally equivalent strains for the reversion. If the martensite has the lower symmetry of the subgroup, only one of these strains reproduces the original parent phase. In this case. the strain is the exact inverse of the original strains. All other transformation strains produce a low-symmetry version of the parent phase presumably of higher free energy. This energy difference should favor the exact inverse, as is observed in ordered martensites. DISCUSSION The symmetry changes due to martensitic transformations in either direct space or reciprocal space make it clear that there are many reasons why martensites have less symmetry than the same phase would have if equilibrated. The symmetries of martensites are usually subgroups of both the symmetries of lattice strain and the original phase. The lattice strain and shuffles are usually known. In some cases were there is ambiguity it has been suggested that the experimentally-found martensite symmetry might resolve it [63.
CAHN:
724
THE SYMMETRY OF MARTENSITES
The inability to form a symmetry supergroup in alloys with local order described by pair correlations was the basis of a paper by Cahn and Rosenberg [6]. Another description of the cause of tetragonality due to clusters was given by Speich and Winchell [q. Since pairs of atoms are a subset of all clusters, it would seem that the Speich and Winchell description would cover all cases described by Cahn and Rosenberg. That this is not so can be demonstrated by a single counter example. In the Speich and Winchell treatment, the point symmetry of the center of any cluster is compared in the austenite and martensite to see if it “can support tetragonality”. Any three adjacent atoms is a linear cluster centered on m3m in both structures2regardless of whether they line up along [lOO], [llO] or any other crystal direction. According to Speich and Winchell these cannot support tetragonality. But we have shown that any local pair-wise order between the members will produce tetragonality. Structures of pure elements with one atom per unit cell can give rise to martensites with high symmetries, but shuffles are required for obtaining high symmetries if the parent structure has more than one atom per primitive unit cell. Solid solutions whether interstitial or substitutional with local order will never give high-symmetry virgin martensite. Diffusive motion is required. Effect of long-range
order
Long-range ordering in the parent phase is itself a case of subgroup formation both for translation and point groups. To understand the subsequent changes in symmetry because of the martensitic transformation, we must begin with the symmetry of the ordered phase. Shuffles are essential for supergroup formation, but if ordering is imperfect supergroup formation is prevented. Acknowledgements-The interest and patient criticism by Bernhardt J. Wuensch, Jr., about the crystallographic aspects were gratefully received and were essential for the development of many of the arguments. I am also grateful to Morris Cohen, Sam Allen, and Walter Owen for their critical reviews. The work was 73076%AOI-DMR.
supported
by
NSF
Grant
No.
Burke), Vol. 3, p. 1433. Plenum (1973).
2. International Tables of X-Ray Crystallography. Vol. I, Table 2.5.1, p. 21. Kynoch Press, Birmingham (1965). 3. A. Kelly and G. W. Groves. Crystallography and Crystal Defects, Chapter 11. Addison Wesley. Reading (1970). B. E. Warren, X-Ray Diffraction, Chapters l&12. Addison Wesley, Reading (1969). International Tables of X-Ray Crprallography, Vol. 1. Table 2.42, p. 13. Kynoch Press, Birmingham (1965). J. W. Cahn and W. Rosenberg, Scripta Met. 5, 101 (1971). P. G. Winchell and G. R. Speich, Acru Mer. 18, 53 (1970).
APPEhDIX The E&t
of a Homogeneous Strain on Scattering Intensity in Reciprocal Space
Given the scattering factor, j”, and the position. i,, of every atom in the original phase, the scattering intensity I as a function of position Z in reciprocal space is given by C41:
(Al) With a homogeneous deformation S, the position of the nth atom becomes 7; and (r; L rQi = &jr. - rm)j.
REFERENCES Met& AIM& Science
H. M.
642)
The inverse of S is S-’ and has the following properties
sijs,:qf = S,‘Sj, = &. Let us define a I? space deformed by a strain (S-i)’ is the transpose of the inverse of S I;’ = (S-i)+ I;
(A3) which (A4)
k;(rA - rk)i = S; ‘kjSit(r, - r,,Jlr = 6jtkj(r, - rmjr = k,(r, - r,)j.
(0
li’qr:, - ?k.)= L(i, - ?“).
(A6)
It follows that
The scattering intensity at i;’ from the strained crystal I&‘) is identical to that of the unstrained crystal at x r,(p)
=
1
c f, f~e’iY;:. - 2”)
1 m
= i(k).
A word about the strain (S-l)‘:
1. P. G. Winchell and M. Cohen, Trans. Am. Sot. 55. 347 (1962); M. Cohen, Trans. metall. Sot. 224, 638‘ (1962); J. C. Williams, in Titanium and Technology (edited by R. I. Jaffee and
Press, New York
If S contains a rotation, (S-i)+ contains the same rotation. The length of each principal axis of (S-‘)’ is the reciprocal of that of S. Deviations from this rule occur due to inhomogeneous deformation. Shuffles are considered in the text. Diffusion is assumed not to occur. Size effects cannot be prevented. Thus, the theorem holds only for the Bragg peaks and that part of the diffuse scattering not due to size effects.