Symmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES

Transcription

1 Symmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES

2 Acknowledgements: Bilbao: M. Aroyo, D. Orobengoa, J.M. Igartua Grenoble: J. Rodriguez-Carvajal Provo, Utah: H.T. Stokes, B. J. Campbell Sydney: B.J. Kennedy, C.J. Howard, K.S. Knight (Chilton, ISIS)

3 What is a mode decomposition? Fourier decomposition: = mode decomposition: =

4 Modes in the dynamics of solids: ħω j (k) Mohr et al. PRB 2007 Energy as a function of the normal (dynamic) coordinates: E = Eo 1/2 Σ ω 2 j(k) Q j dyn (k) 2 ω 2 j (k) >0 Zero mean value of mode coordinates : < Q i dyn >=0 Symmetry of vibrational modes: irreducible representations (group theory)

5 Why symmetry-adapted modes? to first-order, irrep modes only couple with modes of the same symmetry atomic displacements: resultant atomic forces: modes of different symmetry are uncoupled to first order

6 Modes in the statics of low-symmetry distorted phases: The natural language to describe a symmetry break/phase transition (Landau Theory) primary distortion mode order parameter Unstable collective degree of freedom: F. Energy T>Tc Q T<Tc 2 E = Eo 1/2 κ(t) Q κ(t)<0 T <Tc

7 Collective modes is the natural language to describe the structure of (displacive) distorted phases: Hierarchy of modes: Von Neumann principle: all modes compatible with the symmetry will be present in the total distortion. But not all with the same weight!: primary mode(s): unstable it at the origin of the distortion secondary modes: induced by the presence of the primary one(s)

8 Hierarchy of modes κ 1 <0 κ 2 >0 Example of a (free) energy map with primary (Q 1 ) and secondary (Q 2 ) distortion modes: Q 2 E = Eo ½ κ 1 Q 1 κ 1 <0 κ 2 >0 2 2 ½ κ 2 Q 2 3 γ Q 1 Q 2 Q 1 Anharmonic allowed coupling Equivalent ferroic stable structures

9 AMPLIMODES Symmetry Modes Analysis

10 Modes in the statics of low-symmetry distorted phases: Distorted Structure = High-symmetry Struct frozen modes distortion mode = Amplitude x polarization vector Description of a mode : e 2 e 1 e 3 e 4 u(atoms) = Q e amplitude polarization vector e = ( e 1,e 2,e 3,e 4 ) normalization: e 1 2 e 2 2 e e 4 2 =1 (within a unit cell) AMPLIMODES calculates the amplitudes and polarization vectors of all distortion modes with different symmetries (irreps) frozen in a distorted structure.

11 We can compare the amplitudes of different frozen distortion modes: Q 2 e 2 Q 1 e 1 Q 1 and Q 2 have the same dimensions and their values can be compared

12 Symmetry-mode decomposition A second simple example: the triclinic structure of NbS 3 P2 1 /m Z=2 P-1 (a,2b,c;000) Z=4 c b a parent (average) struct parent P2 1 /m struct. in the P-1 setting: real struct 24 free positional parameters Nb 1 2i Nb 1_2 2i S 1 2i S 1_2 2i S 2 2i S 2_2 2i S 3 2i S 3_2 2i

13 Symmetry-mode decomposition of NbS 3 structure: b Z1 distortion c a GM2 distortion GM1 (a,b,c) k-vector irrep direction isotropy subgroup dim. amplitude(å) (a,2b,c) GM2 (0,0,0) GM1 (a) P2 1 /m (a,b,c;0,0,0) (5) (0,0,0) GM2 (a) P-1 (a,b,c;0,0,0) (3) (0,1/2,0) Z1 (0,a) P-1 (a,2b,c;0,0,0) (4) Z1 max atom. displ. : 0.18 Å

14 Symmetry-mode decomposition of NbS 3 structure. Quantitative description: GM2 distortion (dim 4) Amplitude: 0.036(3) Å 4 allowed orthogonal symmetry-modes (normalized) polarization vector: Atom δx δy δz Nb Nb1_ S S1_ S S2_ S S3_ positional parameters as a 4 dim vector in terms of normalized symmetry modes (3 indep. parameters): Nb1 1 S1 1 S2 1 S

15 Symmetry-mode decomposition of NbS 3 structure Z1 distortion (dim 12) Amplitude: 0.520(4) Å 12 allowed orthogonal symmetry-modes (normalized) polarization vector: Atom δx δy δz Nb Nb1_ S S1_ S S2_ positional parameters S S3_ or as a 12 dim vector in terms of normalized symmetry modes (11 indep. parameters): Nb1 1 Nb1 2 Nb1 3 S1 1 S1 2 S1 3 S2 1 S2 2 S2 3 S3 1 S3 2 S

16 Example of input of AMPLIMODES: Amm2 phase of BaTiO 3 Pm-3m b Amm2 c 4 parameters Transf.

17 The orthorhombic Amm2 structure of BaTiO 3 ( Kwei et al. (1993) neutron-powder 190 K ) b c max. atomic displ. : 0.13 Ǻ Mode decomposition of distortion: Q GM4- x Q GM5- x GM4- mode GM5- mode polar ferroelectric mode Q GM4- >> Q GM5-

18 Example of output of AMPLIMODES:

19 Example of output of AMPLIMODES: amplitude of the GM4- distortion basis for this 4dim vector polarization vector in two forms crystallographic form

20 The orthorhombic Amm2 structure of BaTiO 3 ( Kwei et al. (1993) neutron-powder 190 K ) b Perovskite in Amm2 setting c δx δy δz Ba Ti O O polarization vector GM4- polarization vector GM5- δx δy δz δx δy δz Q GM4- Ba Ti O O Q GM5- Ba Ti O O Q GM4- = Å Q GM5- = Å

21 Leucite KAlSi 2 O 6 I4 1 /a Palmer et. (Amer. Miner. 82 (1997) 16 max. atomic displ. : 1.04Ǻ Ia-3d Γ 3 Γ 4 I4 1 /acd I4 1 /a DISTORTION AMPLITUDES VS. TEMPERATURE: Q Γ 3 Γ 4 I4 1 /a Induced effect : 2 Q Γ3 = α Q Γ4 T (K)

22 Polarization vectors in Leucite I4 1 /a Palmer et. (Amer. Miner. 82 (1997) 16 Ia-3d Γ 3 KAlSi 2 O 6 Γ 4 I4 1 /ac I4 1 /a Q Γ 3 Γ 4 I4 1 /a dot product e(4k).e(t) Γ 4 Γ 1 Γ 3 Induced effect : 2 Q Γ3 = α Q Γ4 T (K)

23 Sequence of transitions in SrZrO 3 (Howard et al & data from B. Kennedy) 20 C Pnma Pm3m P4/mmm M 2 P4/mbm M 3 P4/mbm Cmmm Pbam X 5 Cmcm R 4, R 5 Imma two primary modes necessary Q M2 = Å Q M3 = Å Q X5 = Å Q R4 = Å Q R5 = Å Pnma Pnma - Each primary mode is a different instability mechanism - each primary mode condenses in general at different temperatures : two phase transitions Expected transition sequence: T > Pnma Imma Pm3m ( M 3, R 4 ) R 4

24 Sequence of transitions in SrZrO 3 M 3 R 4 X 5 R Å 0.79 Å Å 0.07 Å M Å (data provided by B. Kennedy) Temperature variation: Amplitudes: They deform the octahedra. X 5 has significant amplitude modes polarization vectors:

25 Another distorted perovskite. NaNbO 3 X3- M5- R4,R5 T4 T2 DT5 k-vector irrep direction isotropy subgroup dim. amplitude (Å) (0,1/4,0) DT5 (0,0,0,0,0,0,0,0,a,0,0,-a) Cmcm (a-b,ab,4c;0,0,1/2) (1/2,1/2,1/2) R4 (0,a,a) Imma (ab,2c, a-b,;0,0,0) (1/2,1/2,1/2) R5 (0,a,-a) Imma (ab,2c, a-b,;0,0,0) (0,1/2,0) X3- (0,a,0) P4/mmm( a,b,2c;0,0,1/2) (1/2,1/2,0) M5- (0,0,0,-a,0,0) Pmma (ab,c,a-b;0,1/2,0) (1/2,1/2,1/4) T2 (a,a,0,0,0,0) I4/mcm (a-b,ab,4c;1/2,1/2,1/2) (1/2,1/2,1/4) T4 (a,-a,0,0,0,0) I4/mcm (a-b,ab,4c;0,0,1/2) max atom. displ. : 0.40 Å

26 Predominant irrep distortions in NaNbO 3 primary distortions (RUMs) secondary distortions R4 T4 DT Å 1.07 Å 0.55 Å q 1 = (1/2,1/2,1/2) q 2 = (1/2,1/2,1/4) q 3 = q 1 -q 2 =(0,0,1/4)

27 Mode decomposition of NaNbO 3 primary wavevectors: q 1 = (1/2,1/2,1/2) q 2 = (1/2,1/2,1/4) k-vector irrep isotropy subgroup dim. amplitude (Å) q 1 -q 2, q 1 q 2 q 1 3q 1 2q 2 2q 2 -q 1 3q 2 q 2 (0,0,1/4) DT5 Cmcm (a-b,ab,4c;0,0,1/2) (1/2,1/2,1/2) R4 Imma (ab,2c, a-b,;0,0,0) (1/2,1/2,1/2) R5 Imma (ab,2c, a-b,;0,0,0) (0,0,1/2) X3- P4/mmm( a,b,2c;0,0,1/2) (1/2,1/2,0) M5- Pmma (ab,c,a-b;0,1/2,0) (1/2,1/2,1/4) T2 I4/mcm (a-b,ab,4c;1/2,1/2,1/2) (1/2,1/2,1/4) T4 I4/mcm (a-b,ab,4c;0,0,1/2)

28 Mode decomposition vs. ab-initio calculations SrAl 2 O 4 (Larsson et al. 2008) P P2 1 P two different displacive instabilities: Γ 4 P312 C222 1 Γ 5 (b) M 3-1q Γ 6 Γ 6 C2 M 3-1q P P M 2-1q E(mRy) M 2-1q P2 1

29 SrAl 2 O 4 Comparison of mode decomposition of experimental and ab-initio structures Amplitudes and dot products of polarization vectors : irrep M 2-1q Γ 6 M 3-1q Γ 5 Γ 4 dim Amp. prod. Amp. prod. Amp. prod. Amp. prod. Amp. prod. Exp. Struct ab-initio

30 A detailed review with more examples: "Mode crystallography of distorted structure", Acta Cryst. A, to be published J.M. Perez-Mato, D. Orobengoa and M.I. Aroyo preprint available at the Bilbao crystallographic server (

31 Symmetry-mode coordinates in the structure refinement, instead of the individual atomic coordinates: AMPLIMODES FullProf (Juan Rodriguez-Carvajal) (for powder or single crystal diffraction refinement) One expects: a natural hierarchy of parameters less correlations with atomic (thermal) displacement parameters better control of the refinement

32 Example of PCR file for FullProf corresponding to the compound LaMnO 3! Polarisation Vectors of Symmetry Modes for each atom V_MODES 12 Symbols of the Irreducible representations! Nm Atm Irrep Vx Vy Vz Coeff 1 O1 R O2 R La R O1 R Polarisation. vectors components 7 O2 M ! Amplitudes of Symmetry Modes A_MODES Q1_R Q2_R Q3_R Q4_X Q5_X Q6_M Q7_M ! > Profile Parameters for Pattern # 1 Keyword, # of modes, output for FST Names of amplitudes, values and refinement codes (allowing constraints)! Scale Shape1 Bov Str1 Str2 Str3 Strain-Model E

33 Example of direct refinement of mode amplitudes with FullProf: Symmetry Mode amplitudes have very different amplitudes and their standard deviations may differ in orders of magnitude => Amplitudes of symmetry modes ==> Name Value Sigma Q1_GM Q2_GM Q3_GM Q4_GM Q5_GM Q6_GM Q7_GM Q8_X Q9_X Q10_X Q11_X Q12_X dominant irrep distortions with much smaller standard deviations

34 Other programs: