1. Twin-symmetry analysis
Crystallography Online: International School on the Use and Applications of the Bilbao Crystallographic Server
Twin-symmetry analysis
A simple way to provide you a nice headache
Massimo Nespolo
CRM2 UMR-CNRS 7036
Nancy-Université, France

2. What is a twin?
No space group
With space group
A twin is a heterogeneous edifice built by homogeneous crystals (individuals) of the same phase in different orientations, related by an operation (the twin operation) that does not belong to the point group of the individual.
Georges Friedel, 1904

3. Twin operation, twin element, twin law
Twin operation: the movement mapping the orientation of one individual onto the orientation of another individual.
Twin element: the geometrical element in direct lattice (plane, axis, centre) about which the twin operation is performed.
Correspondingly, twins are classified as reflection twins, rotation twins and inversion twins
Twin law: the set of equivalent twin operations, obtained by coset decomposition.

4. Example
Crystal belonging to the geometric crystal class 2 (b-unique) twinned by 120º about [001]hex
{1, 2[010]}  {3+[001], 2[110]}  {3-[001], 2[100]}
Two twin laws: the two cosets {3+[001], 2[110]} and {3-[001], 2[100]}
Four twin operations – the four operations in the two cosets
Three twin elements: [001], [110] and [100]
Symmetry of the twin expressed by a trichromatic point group (twin point group): (3(3)2(2,1))(3)

5. Mapping of individuals in twins
Reflection in
(100)
Rotation about
[001]
Reflection in
{031}
(cyclic twin)
Reflection in
{1122}

6. Notation General G : space group H : subgroup of G
 : isomorphism, isomorphic
Specific to twins
H : type of point group of the individual
H* : intersection group of the point groups of the individuals in their respective orientations
K : chromatic point group obtained by extension of H* through the twin operation(s). Also, the achromatic point group isomorphic to it.

7. Two orientational domains (variants)
A bit of terminology
Two individuals
Two orientational domains (variants)
with N domains

8. Transformation twins
Twins occurring as the consequence of a phase transition implying a reduction of point symmetry.
But a phase transition may involve more than just a point-symmetry reduction and produce more than just twins!

9. Transformation twins
A phase transition that is connected to a symmetry reduction will result in new phases that consist of:
twin domains
if the formed phase belongs to a crystal class with reduced symmetry
antiphase domains (translational domains)
if translation symmetry is lost
The total number of domains depends on the number of nucleation sites
(Ulrich Müller, 2008, MaThCryst school)

10. Transformation twins G H Parent phase Daughter phase Twin domains
translationengleiche subgroup
klassengleiche subgroup H
Daughter phase
Twin domains
Antiphase domains
G is a supergroup of H
(as Monsieur de la Palisse would have said)
Number of domains: N = O(G)/O(H)

11. H is a translationengleiche subgroup of G of order n:
tn-subgroup, t-index n
H and G belong to the same geometric crystal class
k-index of H in G is 1

12. Example: Swiss twin in the transformation ba-quartz
b-quartz
G = P6222 or P6422
573ºC a1 a2 a3 a1 a2 a3
a-quartz
H = P3221 or P3121
t = 2[001] ^

13. Define the twin law and the twin operations

14. And do not forget the transformation matrix!

15. Different choice of the origin

16. Do it yourself!

17. Results.... subgroup H Ignore the SNoT 2[001],6-[001] 6+[001],2[110]
2[120],2[210]

18. Twin point group of the Swiss twina1 a2 a3
H = 321
H* = 321
2[001],6-[001],6+[001]
2[110]
2[120]
2[210]
K = 6'22'

19. b-quartz vs. Swiss twin of a-quartz
Hexagonal
prism
Hexagonal
bipyramid
Two
rhombohedra
Two
trigonal bipyramids
Hexagonal
bipyramid
Hexagonal
trapezohedron
Two trigonal
trapezohedra

20. Other types of twins
Growth twins: micro or macro-crystals attach with precise crystallographic orientation.
Mechanical twins: atomic planes slip under an oriented pressure and a part of the crystal take a twin orientation with respect to the other.
No parent phase group (twin group) known a priori

21. Growth twin: Brazil twin in quartz^ a1 a2 a3
H = 321
H* = 321
K = 3'2/m'

22. Is K always a supergroup of H?
No!
despite several wrong statements in this direction...

23. Growth twin: Japan twin in quartz: {1122}c a3 c a1 a2
t = m ^
The intersection group is H * = 1
K = m'
Reflection in
{11-22}

24. Therefore.... K is always a supergroup of H*
K can be a supergroup or a subgroup of H, be of the same type as H, or even be not directly related to it

25. Melilite, (Ca,Na)2(Al,Mg,Fe)(Si,Al)2O7
An example of K = H
Melilite, (Ca,Na)2(Al,Mg,Fe)(Si,Al)2O7
P421m twinned on (120)
H = 42m
H* = 4
K = 42m

26. What about the lattice? Yes, it does!
Swiss and Brazil twins of quartz: the whole lattice is in common
Japan twin of quartz: half of the lattice is in common
Melilite twin: one fifth of the lattice is in common
Does the lattice play a role in twinning?
Yes, it does!

27. Reticular theory of twinning - why?
Twinning is governed by the structural match at the interface of the individuals
To study this structural match means to investigate twins case by case
The reticular theory makes abstraction of the structure and concentrates on the lattice
This approach is reasonable, although approximate, because the lattice represents the periodicity of the structure
A good lattice match is a necessary, although not sufficient, condition for a good structural match

28. Twinning by merohedry All nodes are restored by the twin operation:
we say that the twin index is n = 1

29. Twinning by pseudo-merohedry
All nodes are quasi-restored by the twin operation:
we say that the twin index is n = 1

30. Definition of obliquityw

31. Twinning by reticular merohedry (K  H) or reticular polyholohedry (K = H)
One node out of three is restored by the twin operation:
we say that the twin index is n = 3

32. Twinning by reticular pseudo-merohedry or reticular pseudo-polyholohedry
One node out of three is quasi-restored by the twin operation:
we say that the twin index is n = 3

33. Occurrence probability of twins
The lower are the twin index and the obliquity, the higher is the probability that the twin occurs.
Friedel's empirical criteria: n  6, w  6º.
Exceptions are known: high index twins occurs, low index twins never or seldom found, corresponding twins ( ex. albite (010) and pericline [010] twins in triclinic pseudo-monoclinic plagioclases) have different occurrence frequency.

34. Zero-obliquity (reticular) pseudo-merohedry
b  c c
90º b d
twin misfit

35. Computation of the obliquityw
[hkl]*
[uvw]
L*(hkl)
L(uvw)
(hkl)
L*(hkl)L(uvw)cosw = hkl|a*b*c*abc|uvw = |hu+kv+lw|
w = cos-1|hu+kv+lw|/L*(hkl)L(uvw)

36. How to find the direction [uvw] quasi-perpendicular to (hkl)?
Easy! Find the irrational expression of [hkl]* in direct space
[hkl]* a b c a* b* c*
[uvw]
How?

37. Easy!
Find u,v,w (in general non-integer) satisfying:
and of course...

38. Exercise
Celestine, SrSO4, Pbnm a = 8.359Å, b = 5.352Å, c = 6.866Å, Twinned on (210)
Find the directions quasi-perpendicular to (210) and CHOOSE ONE! u v
v/u 1 2 3
1.5 4
1.333 5
1.25

39. Calculate the obliquity
w = cos-1|hu+kv+lw|/L*(hkl)L(uvw)
uvw w
110
5.36º
120
14.03º
230
5.86º
340
2.50º
450
0.69º

40. bI
[120]
[110]
[230]
[340]
uvw w
110
5.36º
230
5.86º
340
2.50º
450
0.69º
[450] aI

41. Summary
uvw w n
110
5.36º 3
230
5.86º 7
340
2.50º 5
450
0.69º 13

42. What is the twin point group of SrSO4 (210)?
H = 2/m 2/m 2/m
H* = 2/m
K = 2'/m' 2/m 2'/m'
K  H !
within the approximation of the obliquity
In twinning, the pseudo-symmetry is often more important than the true symmetry

43. Cell parameters of the twin lattice
A matter of basis transformation....
But check the determinant!
[u1,hklv1,hklw1,hkl] and [u2,hklv2,hklw2,hkl] are contained in (hkl)
(choose the shortest!)
[uvw] is the direction quasi-perpendicular to (hkl)

44. Directions [uvw] contained in a plane (hkl)
A plane of the family (hkl) which passes through the origin is hx+ky+lz = 0.
A direction [uvw] passes through the origin and the node uvw.
The direction [uvw] is contained in the plane (hkl) if hu+kv+lw = 0.
Calculate the cell parameters of the (210) twin in celestine.

45. Cell parameters of the (210) twin in celestine
[u1,hklv1,hklw1,hkl] = [001]
[u2,hklv2,hklw2,hkl] [120]
[uvw]=[340]
|P| = 10 > 0
N.B. n = 5 but |P| = 10. Why?

46. Cell parameters of the (210) twin in celestine
aT = cI = Å
bT = Å
cT = Å

47. Twin lattice and pseudo-symmetry of (210) twin in celestine
K = 2'/m' 2/m 2'/m'
within the approximation of the obliquity aI bI
[120]
[340] bT
aT Å; bT = Å:
cT = Å; aT = 92.50º cT
mA, y-oA (easily transformed to mC, y-oC)
In twinning, the pseudo-symmetry is often more important than the true symmetry

48. TLS vs. TLQS
Twin Lattice Symmetry: the twin operation restores exactly a certain subset of the lattice of the individual (zero obliquity and, zero twin misfit)
Twin Lattice Quasi Symmetry: the twin operation restores approximately a certain subset of the lattice of the individual (non-zero obliquity or non-zero twin misfit)

49. i-TLS vs. e-TLS
intrinsic Twin Lattice Symmetry: when the perpendicularity (hkl)/[uvw] does not depend on the metric
Lattice system
lattice plane
lattice direction
triclinic
---
monoclinic (b-unique)
(010)
[010]
orthorhombic
(100) (010) (001)
[100]
[001]
tetragonal
(001)
(hk)
[hk]
rhombohedral and hexagonal (hexagonal axes)
(0001)
(hki)
[2h+k,h+2k,0]
Cubic
(hkl)
[hkl]
Rotation about
111

50. example: orthorhombic crystal with primitive lattice, pair (121)/[561]
i-TLS vs. e-TLS
extrinsic Twin Lattice Symmetry: when the perpendicularity (hkl)/[uvw] does depend on the metric
example: orthorhombic crystal with primitive lattice, pair (121)/[561] a b c
w(º)
type of twinning
4.00
5.000
8.00
2.85
TLQS
9.00
1.81
5.200
0.36
5.165
8.95
e-TLS