Transcript Diperiodic symmetry of crystals

Relations between crystal structures.
Bärnighausen trees of crystal families.
Computer tools on BCS for the study of
crystal-structure relationships.
Yuri E.Kitaev
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
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d) aristotypes, hettotypes, “dead-ends”
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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
Space groups, structure types, structures
Space group No 225 Fm-3m
NaCl structure type
CaF2 structure type
Na – 4a, Cl – 4b
Ca – 4a, F- 8c
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)
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)
Ascending Bärnighausen tree


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Choice of the parent structure
Construction of an ascending Bärnighausen tree
“Dead ends”
Predictions: Are high-symmetry (high-temperature) phases possible?
?
G139
2a, 4e
Index of a group-subgroup relation
i = ik ∙ i t
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
Minimal supergroups (of index 2, 3 and 4) of group 139 (I4/mmm)
MINSUP
Minimal supergroups (of index 2) isomorphic to the
group 123 (P4/mmm)
of the group 139 (I4/mmm)
Wyckoff Positions Splitting for group - subgroup pair
P4/mmm(123)>I4/mmm(139)
Ascending Bärnighausen tree for G139 (2a,4e)

The structure type at ambient conditions was chosen as the parent structure type
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The ascending Bärnighausen tree is constructed using MINSUP
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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 (hightemperature) phases are forbidden
G123
G221
G225
G139
2a, 4e
G139
(index 3,5,7,9)
Descending Bärnighausen tree for G139(2a,4e)


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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
?
?
?
?
Maximal subgroups of group 139 (I4/mmm)
MAXSUB
Wyckoff Positions Splitting for group - subgroup pair
I4/mmm(139)>I4/m(87)
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
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
G87
2a,4e
G97
2a,4e
G71
2a,4i
G119
2a,4e
G107
2a, 2a+2a
G121
2a,4e
G126
2a,4e
G123
1a+1d, 2g+2h
G128
2a,4e
G131
2c,4i
G129
2c,2c+2c
G134
2a,4g
G139
2a+4e, 4e+4e+4e
etc
G136
2a,4e
G137
2a,4c
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)
Maximal subgroups of group 216 (F-43m)
MAXSUB
We choose cell multiplication along the [001] direction:
tetragonal G119 group
G216 → G119
[ 1/2 1/2 0 ] [0]
[ -1/2 1/2 0 ] [0]
[0
0 1 ] [0]
Ga : 4a → 2a
As : 4c → 2c
Lattice strain
No WP splitting
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
Maximal subgroup(s) of type 115 (P-4m2) of index 2
for Space Group 119 (I-4m2)
Wyckoff Positions Splitting for group - subgroup pair I4m2(119)>P-4m2(115) (class a)
Wyckoff Positions Splitting for group - subgroup pair I4m2(119)>P-4m2(115) (class b)
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
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).
Anatase structure
Space group G141
D2h19( I41/amd)
Ti:
4a
O:
8e
(0 0 0),
(0 ½ ¼)
(0 0 z),
(0 ½ z+¼),
(½ 0 –z+¾),
(½ ½ -z + ½),
+ (½ ½ ½)
Minimal supergroups (of index 2, 3 and 4) of group 141 (I41/amd)
[origin choice 2]
MINSUP
Minimal supergroups (of index 2) isomorphic to the
group 134 (P42/nnm) [origin choice 2]
of the group 141 (I41/amd) [origin choice 2]
Wyckoff Positions Splitting for group - subgroup pair
P42/nnm(134)>I41/amd(141)
Ascending Bärnighausen tree
for the anatase structure type
G134
G227
G141
(index 3,5,7,9)
G141
4a, 8e
The G141(4a, 8e) structure type is the “dead end” of the tree
Maximal subgroups of group 141 (I41/amd)
MAXSUB
Descending Bärnighausen tree for the anatase
structure type
G141
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
G70 – lattice strain
G88, G98, G122 – structure types with
occupied non-characteristic orbits
G141
4a+8e,8e+8e+8e
etc
Group-Subgroup Lattice and Chains of Maximal Subgroups
SUBGROUPGRAPH
The transition
from anatase G141 into cottunite G62
G141
I41/amd
4a, 8e
G74
Imma
4e, 4e+4e
G62
Pnma
4c, 4c+4c
Exercises
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
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.
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)
Maximal subgroups of group 186 (P63mc)
Wyckoff Positions Splitting for group - subgroup pair
P63mc(186)>P63mc(186)
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
Symmetry relationship tree for the wurzite
structure type
G186
Ga:2b
N: 2b
[k=5, t=1]
[k=1, t=2]
[k=3, t=1]
G156
Ga:1b
Al:1c
N1:1b
N2:1c
G186
Ga1:2b
Ga2:2b
Al:2b
N1:2b
N2:2b
N3:2b
1 type
2 types
G186
Ga1:2b
Ga2:2b
Ga3:2b
Ga4:2b
Al:2b
N1:2b
N2:2b
N3:2b
N4:2b
N5:2b
6 types
Acknowledgements
The author acknowledges the support of
IKERBASQUE
Basque Foundation for Science.