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**Relations between crystal structures**

Relations between crystal structures. Bärnighausen trees of crystal families. Computer tools on BCS for the study of crystal-structure relationships. Yuri E.Kitaev

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**PLAN OF THE LECTURE Space groups, structure types, structures**

Symmetry relationships between structure types a) ascending Bärnighausen tree: group-supergroup tree b) descending Bärnighausen tree: group-subgroup tree c) non-characteristic orbits d) aristotypes, hettotypes, “dead-ends” Construction of Bärnighausen trees for two main cases a) symmetry relationships between different phases b) symmetry relationships between the structure types derived from the parent structure by various substitutions Exercises

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**Space groups, structure types, structures**

Space group No Fm-3m NaCl structure type CaF2 structure type Na – 4a, Cl – 4b Ca – 4a, F- 8c

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**CaF2 (fluorite) structure type**

MeO2 (Me=Rb; Zr, Hf; Sn; Po; Si; Ce, Pr, Tb, Te; Th, Pa, U, Np, Pu, Am, Bk, Cf ) MeF2 (Me= Ca, Sr, Ba, Ra; Ti; Cd, Hg; Pb;Sm, Eu) MeCl2 (Me= Sr, Ba) MeH2 (Me= Sc,Y; Ti, Zr; V, Nb, Ta; Cr; La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu; Np, Pu, Am)

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**MoSi2 structure type Space group G139 D4h17(I4/mmm)**

BaO2, KO2, CsO2, CaC2, NdC2, SrC2, SrN2 A – 2a (000) B – 4e (00z) (00-z) + (½½½)

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**Ascending Bärnighausen tree**

Choice of the parent structure Construction of an ascending Bärnighausen tree “Dead ends” Predictions: Are high-symmetry (high-temperature) phases possible? ? G139 2a, 4e

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**Index of a group-subgroup relation**

i = ik ∙ it ik is the k-index (klassengleich index) = cell multiplication in the case of primitive cells it is the t-index (translationgleich index) = ratio of the orders of the point groups G and H

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**Minimal supergroups (of index 2, 3 and 4) of group 139 (I4/mmm) MINSUP**

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**Minimal supergroups (of index 2) isomorphic to the group 123 (P4/mmm) of the group 139 (I4/mmm)**

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**Wyckoff Positions Splitting for group - subgroup pair P4/mmm(123)>I4/mmm(139)**

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**Ascending Bärnighausen tree for G139 (2a,4e)**

The structure type at ambient conditions was chosen as the parent structure type The ascending Bärnighausen tree is constructed using MINSUP The G139 (2a,4e) structure type is the “dead end” of the ascending Bärnighausen tree, i.e. the archetype structure Displacive-type transitions from the G139 (2a,4e) structure type into the high-symmetry (high-temperature) phases are forbidden G123 G221 G225 G139 (index 3,5,7,9) G139 2a, 4e

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**Descending Bärnighausen tree for G139(2a,4e)**

Choice of the aristotype Construction of an descending Bärnighausen tree Structure types with non-characteristic orbits Possible paths into the low-symmetry structure types Predictions of intermediate structure types G139 2a, 4e ? ? ? ?

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**Maximal subgroups of group 139 (I4/mmm) MAXSUB**

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**Wyckoff Positions Splitting for group - subgroup pair I4/mmm(139)>I4/m(87)**

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**Transition into the structure with non-characteristic orbits**

Structures G139 (2a,4e) and G87 (2a, 4e) are indistinguishable: Atoms occupy the same points in space

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**Descending Bärnighausen tree for G139 (2a,4e)**

The parent structure of the ascending tree was taken as the aristotype for the descending Bärnighausen tree Descending Bärnighausen tree is constructed using MAXSUB Structure types with non-characteristic orbits have been found G69, G71 – lattice strain G87,G97,G119, G121,G126, G128,G131,G134,G136,G137 - Structure types with occupied non-characteristic orbits G139 2a, 4e G69 4a,8i G71 2a,4i G107 2a, 2a+2a G123 1a+1d, 2g+2h G129 2c,2c+2c G139 2a+4e, 4e+4e+4e etc G87 2a,4e G97 2a,4e G119 2a,4e G121 2a,4e G126 2a,4e G128 2a,4e G131 2c,4i G134 2a,4g G136 2a,4e G137 2a,4c

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Symmetry relationships between the parent structure type and the structure types derived by various substitutions of atoms GaAs parent structure (GaAs)m(AlAs)n [hkl] derivative structures Space group G216 Td2 (F-43m) Ga : 4a (000) As : 4c (¼¼ ¼)

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**Maximal subgroups of group 216 (F-43m) MAXSUB**

We choose cell multiplication along the [001] direction: tetragonal G119 group

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**G216 → G119 Ga : 4a → 2a As : 4c → 2c Lattice strain No WP splitting**

[ 1/2 1/2 0 ] [0] [ -1/2 1/2 0 ] [0] [ ] [0] Ga : 4a → 2a As : 4c → 2c Lattice strain No WP splitting

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**Maximal subgroups of group 119 (I-4m2)**

The tools of BCS allow one to obtain results by different ways. One can obtain directly the WP splitting G216 → G115 using WYCKSPLIT, the knowledge of the TRANSFORM MATRIX being needed

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**Maximal subgroup(s) of type 115 (P-4m2) of index 2 for Space Group 119 (I-4m2)**

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**Wyckoff Positions Splitting for group - subgroup pair I-4m2(119)>P-4m2(115) (class a)**

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**Wyckoff Positions Splitting for group - subgroup pair I-4m2(119)>P-4m2(115) (class b)**

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**Possible derivative structures**

1a1 - Ga 1c1 - Al 2g1- As Ga – Al (1x1) G216 Ga: 4a As: 4c G119 Ga: 2a As: 2c G115 Ga:1a+1c As: 2g

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**EXERCISES EXERCISE 1. Anatase structure type:**

Space group G141 (I41/amd) Tetragonal system Ti: 4a O: 8e a) Find the minimal supergroups of the anatase space group and show that it is the archetype of the tree. b) Find maximal subgroups of the anatase space group, occupied Wyckoff position splittings, structure types with non-characteristic orbits. c) Find possible paths to the cottunite structure type G62 (Pnma) (4c, 4c+4c).

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**Anatase structure Space group G141 D2h19( I41/amd) Ti: 4a (0 0 0),**

(0 ½ ¼) O: 8e (0 0 z), (0 ½ z+¼), (½ 0 –z+¾), (½ ½ -z + ½), + (½ ½ ½)

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**Minimal supergroups (of index 2, 3 and 4) of group 141 (I41/amd) [origin choice 2] MINSUP**

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Minimal supergroups (of index 2) isomorphic to the group 134 (P42/nnm) [origin choice 2] of the group 141 (I41/amd) [origin choice 2]

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**Wyckoff Positions Splitting for group - subgroup pair P42/nnm(134)>I41/amd(141)**

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**Ascending Bärnighausen tree for the anatase structure type**

(index 3,5,7,9) G134 G141 4a, 8e The G141(4a, 8e) structure type is the “dead end” of the tree

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**Maximal subgroups of group 141 (I41/amd) MAXSUB**

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**Descending Bärnighausen tree for the anatase structure type**

4a, 8e G70 8a,16g G74 4e,4e+4e G88 4a, 8c G98 4b, 8c G109 4a,4a+4a G119 2b+2d,4e+4f G122 4b,8c G141 4a+8e,8e+8e+8e etc G70 – lattice strain G88, G98, G122 – structure types with occupied non-characteristic orbits

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**Group-Subgroup Lattice and Chains of Maximal Subgroups SUBGROUPGRAPH**

I41/amd 4a, 8e The transition from anatase G141 into cottunite G62 G74 Imma 4e, 4e+4e G62 Pnma 4c, 4c+4c

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**Exercises Exercise 2. Wurtzite structure type G186 (P63mc) (2b, 2b)**

a) For the wurtzite parent structure, find possible (GaN)m(AlN)n superlattice families specified by one of the maximal subgroups in the Bärnighausen tree. b) Determine the superlattice growth direction, i.e. the direction of the unit cell multiplication. c) Find the possible combinations of occupations of the splitted Wyckoff positions.

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Symmetry relationships between the parent wurtzite structure type and the structure types derived by various substitutions of atoms Ga – 2b (1/3 2/3 z1) (2/3 1/3 z1+1/2) N – 2b (1/3 2/3 z2) (2/3 1/3 z2+1/2)

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**Maximal subgroups of group 186 (P63mc)**

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**Wyckoff Positions Splitting for group - subgroup pair P63mc(186)>P63mc(186)**

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**Ga-Ga-Ga-Ga-Al-Ga-Ga-Ga-Ga-Al (4x1) **

Possible structures for the derivative structure type with the subgroup G186 (index k=5, t=1) for space group G186 Ga – 2b → 2b1+2b2 +2b3+2b4+2b5 N – 2b → 2b6+2b7 +2b8+2b9+2b10 Ga-Ga-Ga-Ga-Al-Ga-Ga-Ga-Ga-Al (4x1) Ga-Ga-Ga-Al-Al-Ga-Ga-Ga-Al-Al (3x2) Ga-Al-Ga-Al-Al-Ga-Al-Ga-Al-Al (1x1x1x2) etc n=2, k=5 N=(nk – 2)/k N=6 combinations

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**Symmetry relationship tree for the wurzite structure type**

G186 Ga:2b N: 2b [k=5, t=1] G186 Ga1:2b Ga2:2b Ga3:2b Ga4:2b Al:2b N1:2b N2:2b N3:2b N4:2b N5:2b [k=3, t=1] [k=1, t=2] G186 Ga1:2b Ga2:2b Al:2b N1:2b N2:2b N3:2b G156 Ga:1b Al:1c N1:1b N2:1c 1 type 2 types 6 types

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**Acknowledgements The author acknowledges the support of IKERBASQUE**

Basque Foundation for Science.

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