EXERCISES (PRACTICAL SESSION II):

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1 EXERCISES (PRACTICAL SESSION II): Exercise 1 Generate all the atomic positions for a structure with symmetry Cmcm and atoms located on 4a and 4c sites (y=0.2). Which is the total number of atoms in the cell? Hints: Use the WYCKPOS tool and choose the space group. Exercise 2 Which are the cartesian coordinates in units of 2π/a of the special X and M k-points for the space group Fm3m?. Click on KVEC tool and choose the space group. A table with the k-vector types appear. Hint: CDML refers to a primitive reciprocal basis. Go to for a definition of the reciprocal lattice vectors for a fcc structure. Exercise 3 Which are the cartesian coordinates in units of 2π/a of the special S and R k-points for a structure with the space group Cmcm and lattice parameters a=4.5 Å, b=6 Å and c= 7 Å?. Click on KVEC tool and choose the space group. A table with the k-vector types appear. Hint: The conventional reciprocal lattice is orthorhombic base centered with lattice parameters 4π/a, 4π/b y 2π/c. A. Basics on subgroup-groups relationships Relations between crystal structures imply relations between their space groups, which can often be expressed by group-subgroup relations. A set {H i } of symmetry operations of a space group G is called a subgroup H of G if {H i } obeys the group conditions. The subgroup H is called a proper subgroup of G if there are symmetry operations of G not contained in H. A subgroup H of a space group G is called a maximal subgroup of G if there is no proper supergroup M of G, such that H < M < G. The symmetry reduction in the subgroups can occur in three different ways: (i) by reducing the order of the point group, i.e. by eliminating all symmetry operations of some kind, (ii) by loss of translations, (iii) by combination of (i) and (ii). Subgroups of the first kind (i) are called t subgroups because the set of translations is retained. In case (ii), the point group is unchanged. These groups are called k subgroups. In the general case (iii), the subgroup has lost translations and belongs to a crystal class of lower order. Fortunately, the Theorem of Hermann states that the maximal subgroups os a space group G are of type (i) or (ii). Go to and read the definition of the transformation index. Any non maximal subgroup H of a given space group G may be obtained via a chain of maximal subgroups s Z i : G > Z 1 >... > Z n > H. The index of H is given by the multiplication of the intermediate indexes. A1. Transformation matrix A point X in a crystal is defined with respect to the basis vectors a, b, c and the origin O by the coordinates (x,y,z) of the position vector r. The same point is given with respect to a new coordinate system ( a, b, c and origin O ) by the position vector, r = x a +y b +z c. (1) The transformation of the coordinate system consists of two parts, the (3x3) matrix P and the the column matrix p. (i) The matrix P implies a change of orientation, lenght or both of the

2 basis vectors a, b, c, ( a b c ) = ( a b c ) P = ( a b c ) P 11 P 12 P 13 P 21 P 22 P 23 P 31 P 32 P 33. (2) (ii) A shift of origin is defined by the shift vector, p = p 1 a+p 2 b+p3 c, (3) The new basis vectors are fixed at the origin O which has the coordinates p 1,p 2,p 3 in the old coordinate system. The transformation of the components of a position vector r (coordinates of a point X in direct space x,y,z is given by, x y z = Q where Q is the inverse matrix of P Q = P 1 and q = P 1 p. x y z + q. (4) For more information, go to Exercise 4 Consider the B1 structure of NaCl, Fm 3m (B1), Z=4; a I = 4.84 Å; Na 0, 0, 0 (4a m 3m), Cl 1, 1, 1 (4b m 3m); Using SUBGROUPGRAPH find the lowest index relating the space groups F m 3m (No. 225) and R 3m (No. 166). Trough the associated matrix transformation, determine the lattice parameters and atomic positions in the R 3m subgroup. Hint: WYCKSPLIT can be used to check that your atomic positions are OK. A2. Structural analysis of the phase transition S1 S2 The conventional S1 and S2 cells can be represented in the basis of a common subgroup thanks to the previous equations. It is useful to analyze the transformation mechanism S1 S2. It is also reasonable to think that those transformations involving smaller lattice strains and atomic displacements should be favoured. (a) Lattice strain Consider an homogenous deformation of the crystal lattice that changes the parameters from the initial values (a 0.b 0,c 0,α 0,β 0,γ 0 ) to the end values (a.b,c,α,β,γ). The components of a cartesian basis with orthonormal vectors with respect to the basis of the lattices without/with deformation are given by, ( i, j, k) = ( a 0, b 0, c 0 )R 0 = ( a, b, c)r A reference crystallographic system can be orthonormalized in infinite ways. For instance, R 0 = 1/a 0 cosγ 0 /(a 0 sinγ 0 ) a 0cosβ 0 0 1/(b 0 sinγ 0 ) b 0cosα c 0,

3 withananalogousexpressionforr,andwherea = bcsinα/v,b = acsinβ/v,c = absinγ/v, cosα = (cosβcosγ cosα)/(sinβsinγ) y cosβ = (cosαcosγ cosβ)/(sinαsinγ), V being the unit cell volume, V = a. b c. Thefractionalcoordinatesx = (x,y,z) t ofageneralpointonthespacedonotchangewhenahomogenous deformation happens. Therefore, if X 0 y X (X 0 = (X 0,Y 0,Z 0 ) t and X = (X,Y,Z) t ) are the cartesian coordinates of that point before and after the deformation (X 0 = R 1 0 x, X = R 1 x), the equality condition of the coordinates with respect to the undeformed and deformed lattice is, RX = R 0 X 0 The strain tensor e is defined as, X X 0 = ex 0, X = (I+e)X 0 = DX 0, and using RX = R 0 X 0, where I is the identity matrix. e = XX 1 0 I = R 1 R 0 I = D I, In general, the e tensor consist of a antisymmetric component (rigid rotation) and a symmetry component (ǫ = 1 2 (e+et )), that corresponds to the physically relevant part of the deformation ǫ. The lineal Lagrangian strain tensor, is adequate for small strains, whereas, η = 1 2 (e+et +ee t ), the finite Lagrangian strain tensor is adequate for finite strains of any magnitude order. η can be written as, η = 1 2 (DDt I) and, using the relation G = (R 1 ) t R 1 (it comes from ( i, j, k) t ( i, j, k) = I = R t GR), where G is the metric matrix of the deformed lattice, η = 1 2 Rt 0(G G 0 )R 0, where G 0 is the metric matrix of the undeformed lattice. The strain tensor η depends only on the metrics of the deformed lattice, not on its orientation with respect to the undeformed lattice. In short, η is a transformation to a cartesian basis of the change in the metric tensor G G 0 induced by the deformation of the crystal lattice. The η tensor is symmetric and can be diagonalized. A convenient parameter for a quantitative evaluation of the degree of deformation of a lattice is given by,

4 S = 1 η1 2 +η2 2 +η where η 1, η 2 and η 3 are the eigenvalues of the finite Lagrangian strain tensor. (b) Atomic mappings. The strain tensor, η, relates to a change of the metric tensor G G 0 that corresponds to a homogeneous deformation with fixed atomic positions. If the atomic coordinates change, a internal deformation emerges. The internal deformation (relaxation of the atomic positions) happens generally to minimize the energy of the deformed lattice. The total deformation includes both effects. The changes of the interatomic distances only due to the internal deformation can be used to establish the atom-atom relation in the two structures. It seems reasonable to choose the atomic correspondence (i and j atoms) that gives the smallest distances between the same type atoms in both structures. The distances are calculated using the cell parameters in the S1 structures (atomic displacements δ S1 (i j) or δ 0 (i j)): where δ 2 0 = V t 0V 0 = (X 0 X 0 ) t (X 0 X 0 ) = (x x) t G 0 (x x), V 0 = X 0 X 0 = R 0 1 (x x) represents the pure internal deformation in the cartesian reference system whereas x x refers to the lattice. The coordinates of the i atom can be referred to, (i) the lattice basis before (x(i)) and after (x (i)) the internal deformation; and (ii) the cartesian basis before (X 0 (i)) and after (X 0(i)) the internal deformation. Exercise 5 Consider the transformation B3 B1 for BeO. F 43m (B3), Z=4; a I = Å; Be 1 4, 1 4, 1 4 (4c 43m), O 0, 0, 0 (4a 43m); Fm 3m (B1), Z=4; a II = Å; Be 1, 1, 1 (4b m 3m), O 0, 0, 0 (4a m 3m); a) Find the transformation matrices relating the B3 and B1 structures with the intermediate R3m structure. Hint: you can use the utilities MAXSUB, SUBGROUPGRAPH and WYCKS- PLIT. b) Calculate the lattice parameters and atomic positions in the R3m structure for B3 and B1. Do they agree with the data shown on the slide? You can use the SETSTRU tool in structure utilites to transform the hexagonal setting to a rombohedral one. Exercise 6 Propose an alternative B3 B1 mechanism involving an body centered intermediate structure (Z=2). Hint: you can use the COMMONSUB tool. a) Calculate the lattice parameters and atomic positions in this intermediate structure. b) Which order paramater would you choose? Exercise 7 Consider the Buerger s mechanism for the B1 B2 transformation in NaCl. Fm 3m (B1), Z=4; a I = 4.84 Å; Na 0, 0, 0 (4a m 3m), Cl 1, 1, 1 (4b m 3m);

5 Pm 3m (B2), Z=1; a II = 2.98 Å; Na 0, 0, 0 (1a m 3m), Cl 1, 1, 1 (1b m 3m); The mechanism involves an intermediate state with space group R 3m (166) and Z = 1 (rombohedral axis). Consider the cells of the B1 and B2 structures represented in the basis of the intermediate subgroup R 3m. a) Calculate the lattice parameters and the positions of atoms. b) Which order parameter would you choose? c) Can you propose an alternative mechanism involving a intermediate structure with symmetry P mmm and Z=2? Exercise 8 How many paths do you find for the B1 B2 transformation in NaCl, with maximum lattice strain (S tot =0.5) and maximum displacement tol =2.5Å(maximum k-index equal to 2)? a) Can you identify the Buerger s mechanism? b) If you had to choose another one, which one would you choose? c) Can you find more paths if you relax the thresholds? Exercise 9 Consider the two mechanisms described above for the B3 B1 transformation. Which one would you select? Exercise 10 Look for three different experimental structures for Sn using the COD Database. Go to Search in Accesing COD Data (left bar) and read the hints and tips. Save the.cif files and visualize the different structures using VESTA. Which are the IC, distances and coordination polyhedra? Exercise 11 Visualize the Buerger s mechanism for NaCl using VESTA. Which are the changes of IC, coordination polyhedra and distances along the transition? Exercise 12 The β and γ phases of C 3 N 4 belong to the hexagonal P6 3 /m and the cubic Fd 3m space groups, respectively. The primitive hexagonal cell contains 14 atoms. All C atoms are located at 6h (x C,y C, 1 4 ) sites. The nitrogen atoms are located at 6h (x N,y N, 1 4 ) sites) and 2d (1 3,2 3,3 4 ) sites. The conventional cell of the γ phase (Fd 3m) contains 56 atoms. All N atoms at 32e (u,u,u) sites are equivalent and four-fold coordinated. C atoms located at 8a ( 1 8,1 8,1 ) sites occupy the center 8 of regular tetrahedra, whereas C atoms located at 16d ( 1 2,1 2,1 ) sites are the centers of distorted 2 octahedra. Determine the irreducible representations of the Γ-point vibrational modes for both structure. Which are the numbers of expected Raman and Infrared peaks? Hint: You can use the SAM tool. References S. Bhagavantam. Crystal Symmetry and Physical Properties. Academic Press, London (1966). C. Giacovazzo, H.L. Monaco, D. Viterbo, F. Scordari, et al., Fundamentals of Crystallography, 1st Edition, Oxford Universtiy Press, 1992, Chapt. 2 (C. Giacovazzo), Chapt. 9 (M. Catti). M. Catti. Acta Cryst. A41, (1985); A45, (1989). César Capillas López. Métodos de la cristalografía computacional en el análisis de transiciones de fase estructurales. Tesis Doctoral, Universidad del País Vasco, Mayo 2006.

6 M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov & H. Wondratschek. Bilbao Crystallographic Server I: Databases and crystallographic computing programs. Zeitschrift fuer Kristallographie 221, 1, (2006). M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato & H. Wondratschek. Bilbao Crystallographic Server II: Representations of crystallographic point groups and space groups. Acta Cryst. A62, (2006). M. A. Blanco, J. M. Recio, A. Costales, & R. Pandey. Phys. Rev. B 62, R (2000). M. Catti, Phys. Rev. Lett. 87, (2001). J. M. Buerger. Phase Transformations in Solids NewYork: Willey (1951).

INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY AND WORKSHOP ON STRUCTURAL PHASE TRANSITIONS. 30 August - 4 September 2017

INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY AND WORKSHOP ON STRUCTURAL PHASE TRANSITIONS 30 August - 4 September 2017 ROURKELA INTERNATIONAL CRYSTALLOGRAPHY SCHOOL BILBAO CRYSTALLOGRAPHIC SERVER