Phase transitions with no group-subgroup relations between the phases

Transcription

1 Phase transitions with no group-subgroup relations between the phases Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy International School on the Use and Applications of the Bilbao Crystallographic Server Lekeitio, Spain, 2-27 June 29

2 Buerger's classification of structural phase transitions reconstructive: primary (first-coordination) chemical bonds are broken and reconstructed discontinuous enthalpy and volume changes first-order thermodynamic character (coexistence of phases at equilibrium, hysteresis and metastability) displacive: secondary (second-coordination) chemical bonds are broken and reconstructed, primary bonds are not small or vanishing enthalpy and volume changes second-order or weak first-order thermodynamic character order/disorder: the structural difference is related to different chemical occupation of the same crystallographic sites, leading to different sets of symmetry operators in the two phases vanishing enthalpy and volume changes second-order thermodynamic character M. Catti Lekeitio 29 2

3 Symmetry aspects of Buerger s phase transitions Displacive and second-order phase transitions: - the space group symmetries of the two phases show a group/subgroup relationship - the low-symmetry phase approaches the transition to higher symmetry continuously; - the order parameter η measures the 'distance' of the low-symmetry to the high-symmetry (η=) structure T-driven transition: usually the symmetry of the l.t. phase is a subgroup of that of the h.t. phase p-driven transition: it is hard to predict which one of the two phases (l.p. and h.p.) is more symmetric M. Catti Lekeitio 29 3

4 Reconstructive phase transitions: - the space group symmetries of the two phases are unrelated - the transition is quite abrupt (no order parameter) but: - any kinetic mechanism of the transformation must be based on an intermediate structure whose space group is subgroup of both space groups of the two end phases - the intermediate state transforms continuously from one to the other end phase, according to the change of the 'reaction coordinate', or kinetic order parameter Examples of simple reconstructive phase transitions: HCP to BCC, FCC to HCP and BCC to FCC in metals and alloys rocksalt (Fm3m) to CsCl-type (Pm3m) structure in binary AB systems: C.N. changes from 6 to 8 zincblende (F43m) to rocksalt (Fm3m) structure in binary AB systems: C.N. changes from 4 to 6 M. Catti Lekeitio 29 4

5 Mechanisms of reconstructive phase transitions and symmetry of the intermediate states G (S.G. of phase ) H (S.G. of intermediate state) G 2 (S.G. pf phase 2) H G, H G 2, G G 2 () Let T, T 2 and T be the translation groups of G, G 2 and H, respectively, and T T 2. Then: T T, T T 2 (2) In the simplest case T =T 2, so that G and G 2 have the same translation group (i.e., the primitive unit-cells of phases and 2 have the same volume, except for a minor difference due to the V jump of first-order transitions). The translation group of H may coincide with that of G and G 2 (T=T ), but it may also be a subgroup of it (T T, i.e., the volume of the primitive cell of the intermediate state is an integer multiple of that of the end phases, called the index i k of the superlattice). M. Catti Lekeitio 29 5

6 The index of the superlattice T of T is equal to the klassen-gleich index of the subgroup H of G. In the general case, we have then that: i k, = T /T = V/V, i k,2 = T 2 /T = V/V 2 ; hence: i k, /i k,2 = V 2 /V. V, V and V 2 are the volumes of the primitive unit-cells associated to subgroup H and groups G and G 2, respectively. As the volume per formula-unit should be the same in all cases, it turns out that: V/Z(H) = V /Z(G ) = V 2 /Z(G 2 ); it follows that: i k, = Z(H)/Z(G ), i k,2 = Z(H)/Z(G 2 ), i k, /i k,2 = V 2 /V = Z(G 2 )/Z(G ). (3) In other words, the ratio of the two k-indexes of the subgroup H is inversely proportional to the ratio of the corresponding numbers of f.u. in the primitive unit-cell volumes of G and G 2. M. Catti Lekeitio 29 6

7 If a conventional centred (non-primitive) unit-cell is used, then the relations V c = f c V, Z c = f c Z should be used, where f c is the number of lattice points contained in the conventional cell. The relation (3) gives the first general constraint on the determination of the common subgroups H. The second important constraint concerns the atomic displacements during the reconstructive phase transition: Atoms must remain in the same types of Wyckoff positions of H along the entire path G G 2. If that were not true, then the H symmetry would be broken to allow atoms to change their Wyckoff positions. As a consequence, the Wyckoff positions of corresponding atoms in G and G 2 must transform into the same Wyckoff position of the common subgroup H. M. Catti Lekeitio 29 7

8 Systematic search for the common subgroups H of the symmetry groups G and G 2 : ) Method of Stokes and Hatch (Phys. Rev. B (22)) The first step of a systematic search of the possible intermediate states involves the search for all common superlattices of phases and 2. Q Q 2 Q Q 2 - {a } {a}, {a 2 } {a}, {a } {a 2 } Q and Q 2 are the transformation matrices from the primitive unit-cells of phases and 2 to the primitive cell of the intermediate structure. Their components must be integer numbers. det(q ) and det(q 2 ) are the indexes i k, and i k,2 of the intermediate superlattice with respect to the lattices of phase and 2, respectively. Q Q - 2 is the transformation matrix relating the lattices of the two end phases, for the transition mechanism considered - Important for a comparison with the experimental relative crystallographic orientation of the end phases (if available)! M. Catti Lekeitio 29 8

9 Search for common superlattices: all possible combinations of two sets of nine integers, corresponding to the components of the Q and Q 2 matrices, are considered. Two limiting conditions: - a reasonable limit on the maximum length of the primitive lattice basis vectors of T - a reasonable limit on the total strain involved in the T T 2 transformation, which can be calculated from the Q Q - 2 matrix. Once the superlattice T is defined, its symmetry point group P has to be found; Let P and P 2 be the point groups of T and T 2, respectively: then P = P P 2. P is found simply by selecting the point group operators of G and G 2 which, in the reference frame of T, are represented by matrices with integer components. The point group P' of H must be a subgroup of P: P' P. P' and H are found by selecting, within the symmetry operators of G and G 2, only those which are compatible with P and T. M. Catti Lekeitio 29 9

10 2) Program TRANPATH of the Bilbao Crystallographic Package A separate search is performed for the subgroups of G and G 2, and the common subgroup types shared by both symmetry groups are determined (COMMONSUBS module), within the constraint of a maximum value of the i k index: i k, i k, i k,2 i k. For a given common subgroup type H, the lists of all subgroups H p (p=,...m) G of the first branch, and of all subgroups H q 2 (q=,...n) G 2 of the second branch are obtained. The indexes p and q label different classes of conjugated subgroups; conjugated subgroups of the same class are completely equivalent and then they are represented by a single member of the class. Every H p or H 2 q subgroup is associated to a transformation matrix Q relating the basis vectors of G to those of the subgroup, according to (a,b,c) H = (a,b,c) G Q. This matrix is by no means unique, of course, because different basis can be chosen to represent the same lattice. M. Catti Lekeitio 29

11 Each pair (H p,h q 2 ) defines an independent possible transformation path relating G and G 2 with common subgroup type H. Every path is checked for compatibility of the Wyckoff position splittings in the two G H p and G 2 H q 2 branches (WYCKSPLIT module). The WP's occupied by a given atom in G and G 2 must give rise to the same WP for that atom in H. The lattice strain in the H reference frame is computed for the G G 2 transformation, and its value is compared to a threshold given in input to TRANPATH. The coordinates of all independent atoms are computed in the H reference frame for the two G and G 2 end structures. The corresponding atomic shifts are compared to a threshold value given in input to TRANPATH. M. Catti Lekeitio 29

12 B3/B reconstructive phase transition (cf. Catti, PRL 2 and PRB 22) zincblende (G = F43m) to rocksalt (G 2 = Fm3m) structure in ZnS and SiC under pressure Two examples of maximal common subgroups, giving rise to well-studied transition mechanisms: H = R3m, Imm2 F43m (B3) Z=4 M ¼, ¼, ¼ (4c, 43m); X,, (4a, 43m) a I Fm3m (B) Z=4 M,, (4b, m3m); X,, (4a, m3m) a II Intermediate states: I - H = R3m Z= M x, x, x (3a, 3m); X,, (3a, 3m) order parameter: x(m) (¼ ) M. Catti Lekeitio 29 2

13 3 F43m R3m Q = Q - = Fm3m R3m Q 2 = Q = F43m Fm3m Q Q 2 - = R3m (B3): a R = a I / 2, α R = 6 ; R3m (B): a R = a II / 2, α R = 6 II - H = Imm2 Z=2 M,, z (¼ ) (2b, mm2); X,, (2a, mm2) Order parameter: z(m) F43m Imm2 Q = - Q - = - M. Catti Lekeitio 29

14 4 Fm3m Imm2 Q 2 = - Q 2 - = - F43m Fm3m Q Q 2 - = - - Imm2 (B3): a = b=a I / 2, c = a I ; Imm2 (B): b = c =a II / 2, a= a II M. Catti Lekeitio 29

15 Imm2 and R3m mechanisms of the B3/B high-pressure phase transition Imm2 R3m 5

16 Imm2 pathway of the B3/B phase transition of ZnS and SiC B3(F43m) Imm2 B(Fm3m) M. Catti Lekeitio 29 6

17 2.4 R3m p= 2..6 R3m p=92 Imm2 p= H (ev).2 Imm2 p=3.8.4 Imm2 p=6 Imm2 p=92 GPa z(c) Enthalpy of the intermediate state of SiC along the B3-B transformation path vs. order parameter ξ at several p values for two different pathways: Imm2 (closed symbols) and R3m (dashed lines) M. Catti Lekeitio 29 7

18 8 Example: rocksalt (G = Fm3m) to CsCl-type (G 2 = Pm3m) structure in NaCl under pressure Fm3m (B) Z=4 M,, (4a, m3m); X,, (4b, m3m) a I Pm3m (B2) Z= M,, (a, m3m); X,, (b, m3m) a II Intermediate states: I - H = Pmmn Z=2 M ¼, ¼, z (¼ ) (2a, mm2); X ¼, ¾, z+ (¾ ) (2b, mm2) Fm3m Pmmn Q = - Q - = - Pm3m Pmmn Q 2 = Q 2 - = - Fm3m Pm3m Q Q 2 - = ¼ -¼ -¼ ¼ (Q Q 2 - ) - = - - M. Catti Lekeitio 29

19 9 Pmmn (B): a = a I, b= c = a I / 2; Pmmn (B2): a = b = a II 2, c = a II II - H = R3m Z= M,, (3a, 3m); X,, (3b, 3m) Fm3m R3m Q = Q - = Pm3m R3m Q 2 = Q 2 - = Fm3m Pm3m Q Q 2 - = Q = (Q Q 2 - ) - = Q - = R3m (B): a R = a I / 2, α R = 6 ; R3m (B2): a R = a II, α R = 9 M. Catti Lekeitio 29

20 2 II2 - H = P2 /m Z=2 M x (¼), ¼, z () (2e, m); X x 2 (¾), ¼, z 2 () (2e, m) Fm3m P2 /m Q = - Q - = - - Pm3m P2 /m, R3m P2 /m Q 2 = - Q 2 - = - Fm3m Pm3m Q Q 2 - = (Q Q 2 - ) - = P2 /m (B): a = a I (3/2), b = c = a I / 2, β = arcos(/ 3) = ; P2 /m (B2): a = b = a II 2, c = a II, β = 9 ; P2 /m (R3m): a = a R [2(+cosα R )] /2, b = a R [2(-cosα R )] /2, c = a R, β = arcos[cosα R /cos(α R /2)] M. Catti Lekeitio 29

21 2 II3 - R3m Z=4 M,, (3m); M 2 x (), x (), z () (3m) X x 2 (), x 2 (),x 2 () (3m); X 2 x 3 (), x 3 (), z 2 () (3m) Fm3m R3m Q = Q - = Pm3m R3m, R3m R3m Q 2 = - - Q 2 - = Fm3m Pm3m Q Q 2 - = (Q Q 2 - ) - = R3m (B): a R = a I, α R = 9 ; R3m (B2): a R = a II 3, α R = arcos(-/3)=9.47 ; R3m (R3m): a R = a R (3-2cosα R ) /2, α R = arcos[(2cosα R -)/(3-2cosα R )] M. Catti Lekeitio 29

22 R3m pathway of the B/B2 phase transition B(Fm3m) R3m B2(Pm3m) M. Catti Lekeitio 29 22

23 Pmmn pathway of the B/B2 phase transition B(Fm3m) Pmmn B2(Pm3m) M. Catti Lekeitio 29 23

24 P2 /m pathway of the B/B2 phase transition B P2 /m B2 24 M. Catti Lekeitio 29

25 NaCl p=26 GPa R-3m.6 Pmmn H (ev) P2/m ξ Enthalpy of the intermediate state of NaCl along the B-B2 transformation path vs. order parameter ξ for three different pathways: rhombohedral R3m, monoclinic P2 /m orthorhombic Pmmn M. Catti Lekeitio 29 25

26 Intermediate metastable Cmcm phase along the P2 /m pathway: TlI-like structure with both Na and Cl in seven-fold coordination M. Catti Lekeitio 29 26

27 4.4 NaCl GPa Na-Cl distance (Å) ξ Na-Cl8 (full diamonds) and Na-Cl7 (full triangles) interatomic distances versus the order parameter ξ along the P2 /m pathway; open diamonds indicate the corresponding Na-Cl distance along the Pmmn path. M. Catti Lekeitio 29 27

28 .2 NaCl 5 GPa GPa H (ev) GPa ξ Enthalpy of the intermediate state of NaCl along the B-B2 transformation path vs. order parameter ξ at three p values for two different pathways: P2 /m (closed symbols) and Pmmn (open symbols) M. Catti Lekeitio 29 28

29 Multiple reconstructive phase transition of AgI under pressure (cf. Catti, PRB 25) zincblende (G = F43m) to anti-litharge (G 2 =P4/nmm) to rocksalt (G 3 = Fm3m) structure F43m Z=4 Ag (4a),, ; I (4c) ¼, ¼, ¼; a I P4/nmm Z=2 Ag (2a),,; I (2c),, z; origin a III, c III Ag (2a) ¼, -¼, z; I (2c) ¼, ¼, z; origin 2 Fm3m Z=4 Ag (4a),, ; I (4b),, a II Transformation pathway within the non-maximal common subgroup Pm (derived from maximal common subgroup Pmm2): Intermediate state: Pm Z=2 Ag (a),,; Ag2 (b) x(ag2),, z(ag2); I (b) x(i),, z(i); I2 (a) x(i2),, z(i2) Order parameter : z(ag2) ( ) M. Catti Lekeitio 29 29

30 3 F43m Pm Q = - Q - = - P4/nmm Pm Q 2 = F43m P4/nmm Q Q 2 - = Q = - Fm3m Pm Q 3 = - Q 3 - = - P4/nmm Fm3m Q 2 Q 3 - = Q 3 - = - Pm (zincblende): a = b = a I / 2, c = a I ; Pm (anti-litharge): a = b = a III, c = c III Pm (rocksalt): a = c = a II / 2, b = a II M. Catti Lekeitio 29

31 Z I I I2 I Ag2 I I2 X Ag (a) Y (b) (c) (d) (f) (e) Pm monoclinic mechanisms for the zincblende (a) to anti-litharge (d), and anti-litharge (d) to rocksalt (f) phase transformations of AgI. The pseudo-orthorhombic Bmm2 intermediate state (c) is present in both pathways. M. Catti Lekeitio 29 3

32 .5..5 H/eV F-43m P4/nmm Fm-3m p/gpa Theoretical enthalpy differences H - H(F43m) plotted vs. pressure for the AgI phases P4/nmm (antilitharge, circles), Fm3m (rocksalt, squares), and Bmm2 (metastable phase, diamonds). Vertical lines bound the predicted pressure stability fields M. Catti Lekeitio 29 32

33 H (ev) AgI Pm.5 GPa F-43m - P4/nmm Fm-3m - P4/nmm H (ev) AgI Pm.64 GPa Fm-3m - P4/nmm F-43m - P4/nmm z(ag2) z(ag2) Theoretical enthalpy difference H(z(Ag2)) - H(zincblende) for the monoclinic Pm intermediate state of AgI along the zincblende to anti-litharge (open circles), and anti-litharge to rocksalt (full triangles) phase transitions. Zincblende/anti-litharge (left) and anti-litharge/rocksalt (right) equilibrium pressures. M. Catti Lekeitio 29 33

34 AgI Pm.5 GPa F-43m - P4/nmm Fm-3m - P4/nmm V (Å 3 ) z(ag2) Theoretical molecular volume for the monoclinic Pm intermediate state of AgI along the zincblende to anti-litharge (open circles), and anti-litharge to rocksalt (full triangles) phase transitions at p =.5 GPa. M. Catti Lekeitio 29 34

35 Cell constants and atomic fractional coordinates of the Pm monoclinic intermediate state of AgI along the F43m (zinc-blende) to P4/nmm (anti-litharge) phase transformation, optimized for fixed z(ag2) order parameter at the equilibrium pressure.5 GPa. Coordinates constrained by symmetry: x(ag)=y(ag)=z(ag)=y(i2)=; y(ag2)=y(i)=/2. The enthalpy values per formula unit, referred to that of the F43m phase, are also given. z(ag2) a/å b/å c/å β/deg x(ag2) x(i) z(i) x(i2) z(i2) H/eV M. Catti Lekeitio 29 35

36 Cell constants and atomic fractional coordinates of the Pm monoclinic intermediate state of AgI along the P4/nmm (anti-litharge) to Fm3m (rock-salt) phase transformation, optimized for fixed z(ag2) order parameter at the equilibrium pressure.64 GPa. The enthalpy values per formula unit, referred to that of the P4/nmm phase, are also given. z(ag2) a/å b/å c/å β/deg x(ag2) x(i) z(i) x(i2) z(i2) H/eV M. Catti Lekeitio 29 36

37 References Capillas C, PhD Thesis, Bilbao (26) Catti, M Phys. Rev. Lett (2) Catti M, Phys. Rev. B (22) Catti M, Phys. Rev. B 68 (R) (23) Catti M, J. Phys.: Condens. Matter (24) Catti M, Phys. Rev. B (25) Stokes H T and Hatch D M, Phys. Rev. B (22) Toledano P and Dmitriev V, 'Reconstructive Phase Transitions' Singapore: World Scientific (996) Toledano J C and Toledano P, 'The Landau Theory of Phase Transitions' Singapore: World Scientific (987) 37 M. Catti Lekeitio 29