Fundamentals. Crystal patterns and crystal structures. Lattices, their symmetry and related basic concepts

Didactic material for the MaThCryst schools

Massimo Nespolo, Université de Lorraine, France [email protected] Ideal vs. real crystal, perfect vs. imperfect crystal

Ideal crystal: Perfect periodicity, no static (vacancies, dislocations, chemical heterogeneities, even the surface!) or dynamic (phonons) defects. Real crystal: A crystal whose structure differs from that of an ideal crystal for the presence of static or dynamic defects.

Perfect crystal: A real crystal whose structure contains only equilibrium defects.

Imperfect crystal: A real crystal whose structure contains also non-equilibrium defects (dislocations, chemical heterogeneities...).

What follows describes the structure of an ideal crystals (something that does not exist!), whereas a real crystal is rarely in thermodynamic equilibrium. 2 / 129 Massimo Nespolo, Université de Lorraine What do you get from a (conventional) diffraction experiment? Time and space averaged structure! “Time-averaged” because the time span of a diffraction experiment is much larger that the time of an atomic vibration.

The instantaneous position of an atom is replaced by the envelop (most often an ellipsoid) that describes the volume spanned by the atom during its vibration.

“Space-averaged” because a conventional diffraction experiment gives the average of the atomic position in the whole crystal volume, which corresponds to “the” position of the atom only if perfect periodicity is respected.

Importance of studying the “ideal” crystal Non-conventional experiments (time-resolved crystallography, nanocrystallography etc.) allow to go beyond the ideal crystal model, but also some information that are often neglected in a conventional experiment (diffuse scattering) can give precious insights on the real structure of the crystal under investigation. 3 / 129 Massimo Nespolo, Université de Lorraine Space and time-averaged structure

4 / 129 Massimo Nespolo, Université de Lorraine Transformations

Affine transformation

An affine transformation is a deformation that sends corners to corners, parallel lines to parallel lines, mid-points of edges to mid-point of edges but does not preserves distances or angles.

Euclidean transformation or isometry

An isometry is a special case of affine transformation which is not a deformation: the object on which it acts can change its orientation and position in space. 1 2 4 1 Symmetry operation 4 3 3 2

A symmetry operation is a special case of isometry: the object on which it acts can change its orientation and position in space only in such a way that its configuration (position, orientation) after the action cannot be distinguished from its configuration before the action. 5 / 129 Massimo Nespolo, Université de Lorraine Symmetry operations of a crystal pattern

●A crystal pattern is an idealized crystal structure which makes abstraction of the defects (including the surface!) and of the atomic nature of the structure. ●A crystal pattern is therefore infinite and perfect. ●A crystal pattern has both translational symmetry and point symmetry; these are described by its space group. ●We have to define the concepts of group and its declinations in crystallography.

6 / 129 Massimo Nespolo, Université de Lorraine Elements and operations

● geometric element：the point, line or plane left invariant by the symmetry operation. ● symmetry element: the geometric element defined above together with the set of operations (called element set) that leave in invariant. ● symmetry operation： an isometry that leave invariant the object to which it is applied. geometric element symmetry element

element set symmetry operation

symmetry operation

symmetry operation

The operation that share a given geometric element differ by a lattice vector. The one characterized by the shortest vector is called definingdefining operationoperation. 7 / 129 Massimo Nespolo, Université de Lorraine Crystal structure/pattern vs. crystal lattice

Crystal

Molecule or coordination polyhedron

Crystal pattern (crystal structure)

Unit cell

Translations Lattice nodes

8 / 129 Massimo Nespolo, Université de Lorraine Example of crystal pattern which is not a crystal structure

9 / 129 Massimo Nespolo, Université de Lorraine The minimal unit you need to describe a crystal pattern

Asymmetric unit*

fi

Unit cell *In mathematics, it is called “fundamental region”

10 / 129 Massimo Nespolo, Université de Lorraine The concepts of chirality and handedness (the left-right difference)

Symmetry operations are classified into first kind (keep the handedness) and second kind (change the handedness). If the object on which the operation is applied is non-chiral, the effect of the operation on the handedness is not visible but the nature of the operation (first or second kind) is not affected! The determinant of the matrix representation of a symmetry operation is +1 (first kind) or -1 (second kind) Chirality: property of an object not being superimposable to its mirror image by a first-kind operation. Handedness: one of the two configurations (left or right) of chiral object. Also known as chirality sense.

11 / 129 Massimo Nespolo, Université de Lorraine Handedness of the object and nature of the operation

12 / 129 Massimo Nespolo, Université de Lorraine Crystal structure, fractional atomic coordinates, crystallographic orbits, Wyckoff positions

Crystal structure： atomic distribution in space that complies with the order and periodicity of the crystal

Fractional atomic coordinates： atomic coordinates x,y,z within a unit cell with respect to the basis vectors a,b,c. a,b,c (bold)： basis vectors a,b,c (italics)： reference axes and cell parameters x,y,z (italics)： fractional atomic coordinates

Don’t forget the translations! crystallographic orbit： the infinite set of atoms obtained by applying all the symmetry operations of the space group to a given atom in the unit cell.

Wyckoff positions：classification of the crystallographic orbits on the basis of the symmetry of the atomic positions (site-symmetry group) （N to 1 mapping）*.

*More about this follows.

13 / 129 Massimo Nespolo, Université de Lorraine Global vs. local domain of action

Substructure S2 Substructure S1

Fi,00 Structure F i,10  Substructure S0

Substructure S3

 Substructure S4

Fpp local operations: act only on a specific substructure

Fqp partial operations: relate only a specific pair of substructures Global (total) operations：that subset of local and partial operations that actually act on the whole structure

https://doi.org/10.2465/gkk1952.14.Special2_215https://doi.org/10.2465/gkk1952.14.Special2_215

14 / 129 Massimo Nespolo, Université de Lorraine Binary operations acting on a set

A binary operation ○

An “internal law of composition” S ○ S → S (closure property)

We've got a “magma” A set S

Note: An “external law of composition” acts on two disjoint sets S ○ S' → S

S S' 15 / 129 Massimo Nespolo, Université de Lorraine Possible properties associated to a magma

Associativity: a ○ (b ○ c) = (a ○ b) ○ c Presence of identity (neutral element): left identity e ○ a = a right identity a ○ e = a Presence of singular element: left singular s ○ a = s right singular a ○ s = s Presence of the inverse (symmetric) element of each element: left inverse: a-1 ○ a = e right inverse: a ○ a-1 = e Commutativity: a ○ b = b ○ a

Presence of a second binary operation: a  S  b  S = c  S

16 / 129 Massimo Nespolo, Université de Lorraine Algebraic structures

globality associativity identity inverse element Magma    

Semigroup    

Monoid    

Group    

Semicategory    

Category    

Groupoid    

17 / 129 Massimo Nespolo, Université de Lorraine The notion of symmetry group

A symmetry group (G,○) is a set G whose elements are symmetry operations having the following features: ● the combination ○ (successive application) of two symmetry operations

gi and gj of the set G is still a symmetry operation gk of the set G

(closure property): giG ○ gjG → gkG (gigj = gk)

● the binary operation is associative: gi(gjgk) = (gigj)gk

● the set G includes the identity (left and right identical): egi = gie ● for each element of the set G (each symmetry operation) the inverse -1 -1 element (inverse symmetry operation) is in the set G: gi gi = gigi = e

In the following we will normally speak of a group G: it is a shortened expression for group (G,○) where ○ is the “successive application” of symmetry operations g G. Rigorously speaking, G is not a group but just a set!

18 / 129 Massimo Nespolo, Université de Lorraine Abelian and cyclic groups

Commutativity is NOT included in the group properties (closure, associativity, presence of identity and of the inverse of each element).

A group which includes the commutativity property is called Abelian.

A special case of Abelian groups: cyclic groups A cyclic group is a special Abelian group in which all elements of the group are generated from a single element (the generator).

G = {g, g2, g3, …. , gn = e}

19 / 129 Massimo Nespolo, Université de Lorraine The notion of “order”

Order of a group element: the smallest n such that gn = e.

If n = 2, then g-1 = g and the element is known as an involution.

Order of a group: the number of elements of the group (finite or infinite)

If n =  the group is infinite

20 / 129 Massimo Nespolo, Université de Lorraine Homomorphism

Let G and H be two sets, and * and # two binary operations acting on G and H respectively. The mapping f : G → H that satisfies the relation f(u * v) = f(u) # f(v) is called a homomorphism.

● If G contains the neutral element 1 , then H too contains a neutral element 1 and the G H f(u) = u' homomorphism f maps them:1 →1 G H f(v) = v' ● If G contains the inverse of each element g, then H too contains the inverse of each element h, and the homomorphism f maps the respective elements: f(u-1) = u'-1 = f(u)-1. f(w) = w' w = u*v w' = u'#v' f(u * v) = f(w) = w' u = u' # v' = f(u) # f(v) u' * v w # v' Domain w' Codomain

f (G, *) (H, #) If H  G, the homomorphism takes the name of endomorphismendomorphism 21 / 129 Massimo Nespolo, Université de Lorraine Kernel and image

The kernel of the homomorphism f : G → H is the subset of elements of G that is mapped by f on to neutral element of H. ＝ ker f {gG: f(g) = 1H}

The image of the homomorphism f : G → H is the set of elements of G that are mapped by f on H. The image may coincide with the codomain or be a subset of it.

Domain Codomain Kernel 1 of f H image of f

f (G,*) (H, #)

22 / 129 Massimo Nespolo, Université de Lorraine Isomorphism

A 1:1 mapping f : G → H is called an isomorphism. For an isomorphism Ker(f) = 1. We are especially interested in isomorphic groups, which have the same structure but differ only for the labelling of the elements.

1G = 1H Ker(f)

f (G,*) (H, #)

23 / 129 Massimo Nespolo, Université de Lorraine Dimensions of the space and periodicity of the pattern

n n-dimensional space, m-dimensional periodicity Gm m = 0:point groups；n = m：space groups；0 < m < n: subperiodic groups n m No. of types of groups Name 1 0 2 1-dimensional point groups 1 2 Line groups： 1-dimensional space groups 2 0 10 2-dimensional point groups 1 7 Frieze groups 2 17 Plane groups, wallpaper groups: 2-dimensional space groups 3 0 32 3-dimensional point-groupspoint-groups 1 75 RodRod groupsgroups 2 80 Layer groups 3 230 (3-dimensional) Space groups

24 / 129 Massimo Nespolo, Université de Lorraine Symmetry group of the crystal

Space group：shows the symmetry of the crystal structure and is obtained as intersection of eigensymmetries that build up the structure.

G = iEi

Translation group：the group containing only the translations of the crystal structure。It is a normal subgroup of the space group G.

T G t T, g G : g t g -1 = t  → j i i j i j

Point group：shows the morphological symmetry of the crystal as well as the symmetry of its physical properties. It is isomorphic to the factor group of the space group and its translation subgroup. P  G/T Some of the definitions will follow

25 / 129 Massimo Nespolo, Université de Lorraine Bravais lattices

● Bravais lattices are sets of (zero-dimensional) points (nodes).

● The lattice nodes are (quite obviously…) different from atoms. Warning: In the literature, this difference is often overlooked!

● The space group of a Bravais lattice is a Bravais group (B).

● The subgroup T of B which contains, apart from the identity, only the translations of B, is the translation subgroup.

● The subgroup H of B obtained by removing all the translations from B is isomorphic to the point group P of the lattice. B ={(W,w)} Projection Endomorphism Isomorphism T＝{(I,w)｝ H = {(W,0)} P＝{W} The minimal point group of a Bravais lattice is built on｛I,-I}.

W = nn matrix, I = nn identity matrix、w = n1 matrix, 0 = n1 zero matrix

26 / 129 Massimo Nespolo, Université de Lorraine Bravais lattices

t t

origin -I I -I I -I I etc…. The minimal point group of a Bravais lattice is built on｛I,-I}. 1-dimensional space：x → x: -I = reflection (or inversion) 2-dimensional space：xy → xy: -I = rotation 3-dimensional space：xyz → xyz: -I = inversion etc…. 27 / 129 Massimo Nespolo, Université de Lorraine -I in En

1  x  x    y   y  1      1  z  z        ...... ...      1  w  w 

Odd-dimensional space：-1

n Second kind operation det( -In) = (-1) Even-dimensional space： +1 First kind operation

TheThe inversioninversion doesdoes notnot existexist inin even-dimensionaleven-dimensional spacesspaces

28 / 129 Massimo Nespolo, Université de Lorraine One-dimensional lattices

Symmetry operations

Identity Translations Reflections

How many 1D lattices are there? Infinite! etc. etc. How many types of 1D lattice are there? One

I = 1, -I = m  point group of a Bravais lattice: m = {1,m}

29 / 129 Massimo Nespolo, Université de Lorraine The world in two dimensions

E2 : the two-dimensional Euclidean space

30 / 129 Massimo Nespolo, Université de Lorraine Symmetry operations in E2

Operations that leave invariant all the space (2D): the identity Operations that leave invariant one direction of the space (1D): reflections Operations that leave invariant one point of the space (0D): rotations Operations that do not leave invariant any point of the space: translations

The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 2 here)

Two independent directions in E2  two axes (a, b) and one interaxial angle (g)

I = 1, -I = 2  Minimal point group of a Bravais lattice: 2 = {1,2}

31 / 129 Massimo Nespolo, Université de Lorraine Symmetry elements in E2

First kind operations Second kind operation

Graphic Hermann-Mauguin Meaning Graphic Hermann-Mauguin Meaning symbol symbol symbol symbol

2 2-fold rotation point m Reflection line (mirror)

3 3-fold rotation point

4 4-fold rotation point

6 6-fold rotation point

Operations obtained as combination with a translation are introduced later The orientation in space of a reflection line is always expressed with respect to the lattice direction to which it is perpendicular 32 / 129 Massimo Nespolo, Université de Lorraine Lattice node coordinates uv, lattice direction indices [uv]

00 01 02 03 b 10 11 12 13 a axis? [10]

20 21 22 23 b axis? [01] 30 31 32 33 [11] 40 41 42 43 [13]

[42] direction ? [21]

a

33 / 129 Massimo Nespolo, Université de Lorraine Choice of the unit cell

t(1,0), t(0,1) t(1,0), t(0,1), t(⅔,⅓), t(⅓,⅔) t(1,0), t(0,1), t(½,½) t(1,0), t(0,1), t(¾,¼), t(½,½), t(¼,¾) 34 / 129 Massimo Nespolo, Université de Lorraine Change of reference

1 5 1 1 1  1 1  1 2  3 3      5  4 2  0 1  0 1  2 2  3 3  Determ.: 1 Determ.: 1 Determ.: 1.5 Determ.: ⅔

1 2 0  2 0  3 3 1   1  2 2  3 1  1 1  2 1  1 1  1  2 2  3 1  Determ.: 2 Determ.: ½ Determ.: 1.5 Determ.: ⅔

1 1 1 1  2 2      1 1 7 5 1 5 1 1  2 2  4 4  3 3  1 1  1 7  Determ.: 2 Determ.: ½ 4 4  3 3  Determ.: ¾ Determ.: 4/3 2 0  1 0    2 1 1 1    2 1  Determ.: 2 Determ.: ½

3 1 1 1 5 1 2 2 2    2 2  4 4  3 3  3 1  1 1 4 2 1 1  2 2          1 1  3 5  3 1 1 1 3 2 2 4 4 1 2  3 1    1 1    4 2      1 1  2 2  Determ.: 4 Determ.: ¼ Determ.: 2 Determ.: ½ Determ.: 3 Determ.: ⅓ Determ.: 2 Determ.: ½

1 1   1 1 3 1 1 2 1 2 2 2 2  1 1  4 4  1 1  1 2  2 1      3 3 3 1 3 1              2 2  2 2 1 1 1 1 2   1 3  4 4  0 2  0 2  1 2  3 3  Determ.: 1 Determ.: 1 Determ.: 4 Determ.: ½ Determ.: 2 Determ.: ½ Determ.: 3 Determ.: ⅓

35 / 129 Massimo Nespolo, Université de Lorraine Change of the unit cell (cont.) t(1,0), t(0,1) : primitive (p) unit cell t(1,0), t(0,1), t(½,½) : centred (c) unit cell

Cartesian (orthonormal) cell: unsuitable (in general)

Conventional unit cell 1. Edges of the cell are parallel to the symmetry directions of the lattice (if any); 2. If more than one unit cell satisfies the above condition, the smallest one is the conventional cell.

Reduced cell (Lagrange-Gauss reduction) 1. Basis vectors correspond to the shortest lattice translation vectors; 2. ||a|| ≤ ||b||, ||b|| ≤ ||b+qa||, q any integer. For q = 1, this condition means that the sides of the unit cell are not longer than its diagonals.

36 / 129 Massimo Nespolo, Université de Lorraine Lattice node coordinates uv, lattice direction indices [uv]

00 01 02 03 04 [01] 1 1 1 3 1 5 1 7 2 2 2 2 2 2 2 2

10 11 12 13 14

3 1 3 3 3 5 3 7 2 2 2 2 2 2 2 2 ½[13] 20 21 22 23 24

5 1 5 3 5 5 2 2 2 2 2 2

30 31 32 33 34

7 1 7 3 7 5 2 2 2 2 2 2

40 41 42 43 44

½[11] [10] https://doi.org/10.1107/S1600576717010548 37 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell fi

38 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

39 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

40 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

41 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

no symmetry direction for this lattice point group of the lattice: 2 = {I, -I} no symmetry restriction on the cell parameters: a; b; g

O

a b

42 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

43 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

two symmetry directions for this lattice, that are taken as axes a and b

fi

point group of the lattice: 2 m m

[10][01]

symmetry restriction on the cell parameters: a; b; g = 90º

O

bp bc ac

ap Conventional unit cell Primitive unit cell

44 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

45 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

two symmetry directions for this lattice, that are taken as axes a and b

fi

point group of the lattice: 2 m m

[10][01] symmetry restriction on the cell parameters: a; b; g = 90º

O b ac=p c=p

Conventional (primitive) unit cell

46 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

47 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a1 and a2.

48 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

b,a2 fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

a,a1 This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a1 and a2. 49 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

b,a2 fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

a,a1 This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a1 and a2. 50 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

b,a2 fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

a,a 1 Point group of the lattice symmetry: 4 m m

10 11

51 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

b,a2 fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

a,a 1 Point group of the lattice symmetry: 4 m m

10 11 p cell (conventional)

52 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

b,a2 fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

a,a1 Restrictions on cell parameters: a = b; g = 90°

53 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

This lattice has six symmetry directions. Two among the three corresponding to the shortest period are taken as axes a1 et a2.

54 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

55 / 129 Massimo Nespolo, Université de Lorraine Exercise: Find the lattice symmetry and thenthen choose a suitable unit cell

fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi

b,a Point group of the lattice： 2 6 m m p cell (conventional)

a,a1 h cell (triple) 10 11 Restrictions on the cell parameters: a = b; g = 120°

56 / 129 Massimo Nespolo, Université de Lorraine Direction indices [uv] in the hexagonal lattice of E2

[01] a2

a1

[12] [11]

[10] [21] [11]

57 / 129 Massimo Nespolo, Université de Lorraine Why not 5, then?

58 / 129 Massimo Nespolo, Université de Lorraine Crystal patterns in 2D

59 / 129 Massimo Nespolo, Université de Lorraine The concept of holohedry

Lattice

« Object » (content of the unit cell)

Pattern

Symmetry of the lattice : SL = 4mm

Symmetry of the object : So = 4mm

Symmetry of the pattern : Sp = 4mm

Sp = SL : holohedry

60 / 129 Massimo Nespolo, Université de Lorraine The concept of holohedry

Symmetry of the lattice: SL= 4mm

Symmetry of the object: So = 3m

Symmetry of the pattern : Sp = m = 4mm  3m

Sp < SL : merohedry

61 / 129 Massimo Nespolo, Université de Lorraine The concept of holohedry

Symmetry of the lattice: SL= 4mm

Symmetry of the object: So = 3m

Symmetry of the pattern : Sp = 1 = 4mm  3m

SP < SL: merohedry

62 / 129 Massimo Nespolo, Université de Lorraine The concept of holohedry

The crystallographic restriction (n = 1, 2, 3, 4, 6) applies to the lattice and to the pattern, but not to the content of the unit cell!

Symmetry of the lattice : SL = 4mm

Symmetry of the object : So = 5m

Symmetry of the pattern : Sp = m = 4mm  5m

Sp < SL : merohedry

63 / 129 Massimo Nespolo, Université de Lorraine TheThe notionnotion ofof subgroupsubgroup

64 / 129 Massimo Nespolo, Université de Lorraine Exercise: find the elements, the operations and the symmetry group of a square in E2

Geometric elements: one point (centre of the square), four lines ([10], [01], [11], [11]) 3 2 Symmetry elements: one fourfold rotation point: 4

four reflection lines: m[10], m[01], m[11], m[11] Symmetry operations: Four rotations (41, 42 = 2, 43 = 4-1, 44 =1) m[10] Four reflections (m[10], m[01], m[11], m[11])

Check that these operations do form a group

● Isometries are always associative 4 1 ● The identity belong to the set (it corresponds to 44 and m2) ● The inverse of each operation belongs to the set (1-1=1; m-1 m m [11] [11] = m; 2-1 = 2, 4-1 = 43) m[01] ● To check the closure, let us build the multiplication (Cayley) table.

a2 (b)

a1 (a)

65 / 129 Massimo Nespolo, Université de Lorraine Subgroups

From the group (G,○) we select a subset of elements forming a subset H. If the (H,○) is a group under the H G same binary operation ○ as (G,○), then (H,○) is a subgroup of (G,○).

1 3 1 2 4 4 m[10] m[01] m[11] m[11] 3 2 1 1 3 11 1 2 4 4 m[10] m[01] m[11] m[11] 3 1 2 2 1 4 4 m[01] m[10] m[11] m[11] 1 41 3 4 1 1 4 2 1 m[11] m[11] m[01] m[10] m 3 43 1 [10] 4 1 1 4 1 2 m[11] m[11] m[10] m[01] m 3 1 m[10] [10] m[01] m[11] m[11] 1 2 4 4 m 1 3 m[01] [01] m[10] m[11] m[11] 2 1 4 4 m 1 3 m[11] [11] m[11] m[10] m[01] 4 4 1 2 m m m m m 43 41 2 1 0 1 [11] [1 1 ] [11] [01] [10]

m[11] m[11] Multiplication table of the group （G,○）

m[01] Multiplication table of the subgroup (H,○) 66 / 129 Massimo Nespolo, Université de Lorraine Order and index of a subgroup group G, order |G| H  G group H, order |H|

|H| is a divisor of |G| (Lagrange’s theorem)

The ratio iG(H) = |G|/|H| is called the indexindex ofof HH inin GG

： iG(H) = 2 hemihedry ： iG(H) = 4 tetartohedry

iG(H) = 8：ogdohedry（3-dimensional space）

67 / 129 Massimo Nespolo, Université de Lorraine Cosets

By decomposing (G,○) with respect to (H,○) we get i = |G|/|H| cosets. Each coset has the same number of C1=HH C2G elements |H| as the subgroup, which is called the length of the coset.

1 3 1 2 4 4 m[10] m[01] m[11] m[11] 3 2 1 2 41 43 m m m m 11 1 [10] [01] [11] [11] 3 1 2 2 1 4 4 m[01] m[10] m[11] m[11] 1 41 3 4 1 1 4 2 1 m[11] m[11] m[01] m[10] m 3 43 1 [10] 4 1 1 4 1 2 m[11] m[11] m[10] m[01] m 3 1 m[10] [10] m[01] m[11] m[11] 1 2 4 4 m 1 3 m[01] [01] m[10] m[11] m[11] 2 1 4 4 m 1 3 m[11] [11] m[11] m[10] m[01] 4 4 1 2 m m m m m 43 41 2 1 0 1 [11] [1 1 ] [11] [01] [10] Group st m[11] m[11] Subgroup = 1 coset nd m[01] 2 coset 68 / 129 Massimo Nespolo, Université de Lorraine Cosets

C3 By decomposing (G,○) with respect to (H,○) we get i = |G|/|H| cosets. Each coset has the C2C1=HH G same number of elements |H| as the subgroup, which is called the length of the coset. C4

1 3 1 2 4 4 m[10] m[01] m[11] m[11] 3 2 1 2 41 43 m m m m 11 1 [10] [01] [11] [11] 3 1 2 2 1 4 4 m[01] m[10] m[11] m[11] 41 41 3 1 1 4 2 1 m[11] m[11] m[01] m[10] 3 3 1 m[10] 4 4 1 1 4 1 2 m[11] m[11] m[10] m[01] m m 3 1 [10] [10] m[01] m[11] m[11] 1 2 4 4 m m 1 3 [01] [01] m[10] m[11] m[11] 2 1 4 4 m m 1 3 [11] [11] m[11] m[10] m[01] 4 4 1 2 m m 3 1 0 1 [11] [1 1 ] m[11] m[01] m[10] 4 4 2 1 Group Subgroup = 1st coset m m[11] [11] nd rd th m[01] 2 , 3 , 4 coset 69 / 129 Massimo Nespolo, Université de Lorraine Desymmetrization of the square

Order Index

1 1 3 8 4mm = {1, 4 , 2, 4 , m[10], m[01], m[11], m[11]}

1 3 4 2 {1, 4 , 2, 4 } = 4 {1, 2, m[10], m[01], } = 2mm.{1, 2, m[11], m[11]} = 2.mm

2 4 {m } = .m. {1, m } = ..m {1, m } = ..m {1, 2} = 2 {1, m[10]} = .m. [01] [11] [11]

1 8 {1} = 1

70 / 129 Massimo Nespolo, Université de Lorraine Desymmetrization of the square

Order Index 8 4mm 1

4 2mm. 4 2.mm 2

2 .m[10]. .m[01]. 2 ..m[11] ..m[11] 4

1 1 8

71 / 129 Massimo Nespolo, Université de Lorraine Cosets Conjugacy classes Conjugate subgroups Normal or invariant subgroups

72 / 129 Massimo Nespolo, Université de Lorraine Left and right cosets

G = HiCi =HigiH, gi(H,Cj

1 2 3 -1 G = 4mm = {1, 4 , 4 = 2, 4 = 4 , m[10], m[01], m[11], m[11]} H = 2mm. 1 3 4mm = {2mm.}g{2mm.} ={1, 2, m[10], m[01]}{4 , 4 , m[11], m[11]} H = 2.mm 1 3 4mm = {2.mm}g{2.mm} ={1, 2, m[11], m[11]}{4 , 4 , m[10], m[01]} H = 4 1 3 4mm = {4}g{4} ={1, 4 , 2, 4 }{m[10], m[01], m[11], m[11]} Etc. etc.

G = HiCi =HiHgi, gi(H,Cj

Usually, giH  Hgi

If giH = Hgi, i, then H is called a normalnormal (invariant)(invariant) subgroupsubgroup. 73 / 129 Massimo Nespolo, Université de Lorraine Decomposition of 4mm with respect to 2 and m[10]

1 4mm = {1,2}  4 {1,2}  m[10]{1,2}  m[11]{1,2} =

1 3 {1,2}  {4 ,4 }  {m[10],m[01]}  {m[11],m[11]}

1 Normal 4mm = {1,2}  {1,2}4  {1,2}m[10]  {1,2}m[11] = = subgroup

1 3 {1,2}  {4 ,4 }  {m[10],m[01]}  {m[11],m[11]}

1 3 4mm = {1,m[10]}  4 {1,m[10]}  4 {1,m[10]}  2{1,m[10]} =

1 3 {1,m[10]}  {4 ,m[11]}  {4 ,m[11]}  {2,m[01]} 4mm = {1,m }  {1,m }41  {1,m }43  {1,m }2 = [10] [10] [10] [10] ¹

1 3 {1,m[10]}  {4 ,m[11]}  {4 ,m[11]}  {2,m[01]}

74 / 129 Massimo Nespolo, Université de Lorraine Conjugate and normal subgroups

H  G

-1 ghig = hj; hiH, gG : conjugate elements Conjugation is a similarity transformation: “do the same thing somewhere else”

ghi = hjg; hiH, gG

If hi = hj, gG, hi is a self-conjugateself-conjugate elementelement

-1 The set of conjugate elements form a class igihgi = Cl(h) conjugacy class Cl(h) = { h' ∈∈ G |  g  G : h' = ghg-1 } Next consider not one element h but the whole subgroup H

-1 ・・・ igiHgi = {H, H', H'' } are conjugate subgroups of G ・・・ （ ）、 If H' = H'' = = H giH = Hgi, gG H is a normal or invariant subgroup of G: H  G. 75 / 129 Massimo Nespolo, Université de Lorraine Conjugacy classes of 4mm

-1 igihgi = Cl(h): conjugacy class

-1  g g -1 = h = 1 igi1gi = i i i {1} h = 2 -1 igi2gi = {2}

1 1 -1 {41, 43} h = 4 igi4 gi = h = m -1 {m , m } [10] igim[10]gi = [10] [01]

 g m g -1 = {m , m } h = m[11] i i [11] i [11] [11]

1 3 4mm = {1}, {2}, {4 , 4 }, {m[10], m[01]}, {m[11], m[11]}

76 / 129 Massimo Nespolo, Université de Lorraine When we apply an isometry・・・ The object is transformed directly(O→ O') The symmetry group of the object is transformed by conjugation

hO = O, hH h'O' = O', h'H' gO = O', gH,H'

H' = gHg-1

77 / 129 Massimo Nespolo, Université de Lorraine Effects of the operations in a coset

H = 2mm. = {1, 2, m , m } -1 [10] [01] gHg = H  H = 2mm.  G = 4mm Normal subgroup

1 3 {4 , 4 , m[11], m[11]}

b a gHg-1 = 2mm. 78 / 129 Massimo Nespolo, Université de Lorraine Effects of the operations in a coset

H = 2.mm = {1, 2, m , m } [11] [11] -1 gHg = H  H = 2.mm  G = 4mm Normal subgroup 1 3 {4 , 4 , m[10], m[01]}

b a gHg-1 = 2.mm

79 / 129 Massimo Nespolo, Université de Lorraine Effects of the operations in a coset

3 H = 4 = {1, 4, 2, 4 } -1 gHg = H  H = 4  G = 4mm Normal subgroup

{m[10], m[01], m[11], m[11]}

b a

gHg-1 = 4

80 / 129 Massimo Nespolo, Université de Lorraine Effects of the operations in a coset

-1 H = {1, m[10]} gHg = {1, m[10]} g{1,m }g-1 = {1,m } [01] [10] {2, m }  .m. G = 4mm [01] onjugate subgroups

3 1 1 {4 , m[11]} {4 , m[11]} {4 , m[11]}

b {2, m[01]}

a -1 -1 gHg = {1, m[01]} gHg = {1, m[01]} 81 / 129 Massimo Nespolo, Université de Lorraine Desymmetrization of the square

Order Index

8 4mm 1

4 2mm. 4 2.mm 2

2 .m[10]. .m[01]. 2 ..m[11] ..m[11] 4

1 1 8

82 / 129 Massimo Nespolo, Université de Lorraine Extra-simpleExtra-simple exerciseexercise Show that a subgroup of index 2 is always normal

83 / 129 Massimo Nespolo, Université de Lorraine Holohedries and merohedries in the two-dimensional space

84 / 129 Massimo Nespolo, Université de Lorraine Holohedry 2 : {1,2} Conjugacy classes {1}, {2} Subgroup 1 : {1}

Holohedry 2mm : {1,2,m[10],m[01]}

Conjugacy classes {1}, {2}, {m[10]}, {m[01]}

Subgroups 2 : {1,2}, m : {1,m[10]}, {1,m[01]}, 1 : {1}

1 2 3 -1 Holohedry 4mm : {1, 4 , 4 = 2, 4 = 4 , m[10], m[01], m[11], m[11]} 1 3 Conjugacy classes {1}, {2}, {4 , 4 }, {m[10], m[01]}, {m[11], m[11]} 1 2 3 Subgroups 4 : {1, 4 , 4 , 4 }, 2mm : {1, 2, m[10], m[01]}, {1, 2, m[11], m[11]},

2 : {1, 2}, m : {1,m[10]}, {1,m[01]}, {1,m[11]}, {1,m[11]}, 1 : {1} 1 2 3 4 -1 5 -1 Holohedry 6mm : {1, 6 , 6 = 3, 6 = 2, 6 = 3 , 6 = 6 , m[10], m[01], m[11], m[11], m[12], m[21]} Conjugacy classes: 1 2 1 5 {1}, {2}, {3 , 3 }, {6 , 6 }, {m[10], m[01], m[11], }, {m[11], m[12], m[21]} 1 2 3 4 5 1 2 1 Subgroups 6 : {1, 6 , 6 , 6 , 6 , 6 }, 3m : {1, 3 , 3 , m[10], m[01], m[11]}, {1, 3 , 2 -1 3 , m[11], m[12], m[21]}, 3 : {1, 3, 3 }, 2mm : {1, 2, m[10], m[12]}, {1, 2, m[01],

m [21]}, {1, 2, m[11], m[11]}, 2 : {1,2}, m : {1,m[10]}, {1,m[01]}, {1,m[11]}, {1,m[11]},

{1,m[12]}, {1,m[21]}, 1 : {1} Tree of group-subgroups in E2

6mm 6mm 4mm 4mm

6 3m 3m1 31m 2.mm 2mm. 4 2mm

2mm 3 2 m

m m [010] [100] 1

Tree of maximal group-subgroups (in bold the holohedries)

Normal subgroup Conjugate subgroups

86 / 129 Massimo Nespolo, Université de Lorraine Crystal families

Point-group types that satisfy the following criteria belong to the same crystal family: 1.they correspond to the same holohedry; 2.they are in group-subgroup relation; 3.the types of Bravais lattice on which they act have the same number of free parameters.

6mm Hexagonal family：1 free parameter (a) Tetragonal family：1 4mm free parameter (a)

6 3m 4 2mm

3 2 m Orthorhombic family：2 free parameters (a,b) Monoclinic family： 3 free parameters (a,b,g) 1

87 / 129 Massimo Nespolo, Université de Lorraine Conventional cell parameters and symmetry directions in the four crystal families of E2

monoclinic orthorhombic tetragonal hexagonal point group 2 point group 2mm point group 4mm point group 6mm (minimal point group) No restriction a = b ; g = 90º a = b ; g = 120º No restriction on a, b ; 10 ([10] and [01]) 10 ([10], [01] and [11]) on a, b, g g = 90º 11 ([11] and [11]) 11 ([21], [12] and [11]) No symmetry direction [10] and [01] g g b g b a2

a a a2 a 1 a g mp op 1 a b hp p p tp

g b c a Crystal families: monoclinic, c orthorhombic, tetragonal, hexagonal Type of lattice*: primitive, centred

oc

*Lattice whose conventional unit cell is primitive or centred Kettle S.F.A, Norrby L.J., J. Chem. Ed. 70(12), 1993, 959-963

88 / 129 Massimo Nespolo, Université de Lorraine Crystal systems

Point-group types that act on the same types of Bravais lattices belong to the same crystal system. Type of group mp op oc tp hp Crystal system 2, 1      monoclinic 2mm, m     orthorhombic 4mm, 4  tetragonal 6mm, 6, 3m, 3  hexagonal No. of free parameters 3 2 2 1 1

89 / 129 Massimo Nespolo, Université de Lorraine Lattice systems

Point-group types that correspond to the same lattice symmetry belong to the same lattice system.

6mm, 6, 3m, 3 : hexagonal lattice system 4mm, 4 : tetragonal lattice system 2mm, m : orthorhombic lattice system 2, 1 : monoclinic lattice system

90 / 129 Massimo Nespolo, Université de Lorraine The world in three dimensions

E3 : the three-dimensional Euclidean space

91 / 129 Massimo Nespolo, Université de Lorraine Symmetry operations in E3

Operations that leave invariant all the space (3D): the identity Operations that leave invariant a plane (2D): the reflections Operations that leave invariant one direction of the space (1D): the rotations Operations that leave invariant one point of the space (0D): the roto-inversions Operations that do not leave invariant any point of the space : the translations

The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 3 here)

Three independent directions in E3  three axes (a, b, c) and three interaxial angles (a, b, g)

I = 1, -I = 1  Minimal point group of a Bravais lattice: 1 = {1,1}

92 / 129 Massimo Nespolo, Université de Lorraine Inversion centre vs. inversion point

Rotoinversion: n On a mono-dimensional element (axis) a zero-dimensional element (point) exists: ● if n is odd, the inversion operation exists as an independent operation and the corresponding element is called an inversion centrecentre; ● if n is even, the inversion operation does not exist as an independent operation and the corresponding element is an inversion pointpoint.

3 (order 6) 4 (order 4)

inversion centre inversion point

93 / 129 Massimo Nespolo, Université de Lorraine Labelling of axes and angles in E3

c

b a

g b

a

94 / 129 Massimo Nespolo, Université de Lorraine Graphic symbols for symmetry elements n-fold rotation axis n-fold rotoinversion axis n-fold rotation axis and mirror plane perpendicular to it 2 2-fold rotation axis 1 one-fold rotoinversion axis (inversion centre) 2/m 3 three-fold rotoinversion axis 3 3-fold rotation axis 4/m

4 four-fold rotoinversion axis 6/m 4 4-fold rotation axis 6 (3/m) six-fold rotoinversion axis 6 6-fold rotation axis

two-fold rotoinversion axis is a mirror plane

First-kind operations Second-kind operations Operations including translations are introduced later The orientation of a mirror plane is indicated by the vector normal to it

95 / 129 Massimo Nespolo, Université de Lorraine The orientation of a mirror plane is indicated by the vector normal to it

Only one

Infinite!

96 / 129 Massimo Nespolo, Université de Lorraine An isometry of the second kind, fII, can always be

expressed as 1fI

x   x'  x   x'  x   x'          Associativity     f y y ' f11 y y ' fII 1 1 y y ' II     II            z   z '  z   z '  z   z ' 

x   x'  x   x'      Commutativity     fI 1 y y ' 1fI y y '     1      z   z '    z   z '  1  1    1 

If fII is a symmetry operation of a given object, and if that object is not

centrosymmetric, fI is not a symmetry operation of the object. 97 / 129 Massimo Nespolo, Université de Lorraine Equivalence of 2 and m

I b d

II II

q

98 / 129 Massimo Nespolo, Université de Lorraine Combination of 2 and 1 gives m

I b d

II II II

p q

I Applies to even-fold rotations as well, because they all “contain” a twofold rotation

2 3 4 = 2; 6 = 2

99 / 129 Massimo Nespolo, Université de Lorraine Choice of the unit cell

t(1,0,0), t(0,1,0), t(0,0,1) : Primitive cell (P) t(1,0,0), t(0,1,0), t(0,0,1), t(0,½,½): A t(1,0,0), t(0,1,0), t(0,0,1), t(½,0,½): B t(1,0,0), t(0,1,0), t(0,0,1), t(½,½,0): C t(1,0,0), t(0,1,0), t(0,0,1), t(½,½,½): I t(1,0,0), t(0,1,0), t(0,0,1), t(½,½,0), t(½,0,½), t(0,½,½): F t(1,0,0), t(0,1,0), t(0,0,1), t(⅔,⅓,0), t(⅓,⅔,0): H t(1,0,0), t(0,1,0), t(0,0,1), t(⅔,⅓,⅓), t(⅓,⅔,⅔): R t(1,0,0), t(0,1,0), t(0,0,1), t(⅓,⅓,⅓), t(⅔,⅔,⅔): D Conventional unit cell 1. the basis vectors a, b, c form a right-handed reference; 2. edges of the cell are parallel to the symmetry directions of the lattice (if any); 3. if more than one unit cell satisfies the above condition, the smallest one is the conventional cell. Reduced cell 1. the basis vectors a, b, c form a right-handed reference; 2. its faces are two-dimensional reduced unit cells and its edges are not longer than its diagonals; concretely: a. basis vectors correspond to the shortest lattice translation vectors; b. the angles between pairs of basis vectors are either all acute (type I reduced cell) or non-acute (type II reduced cell). 100 / 129 Massimo Nespolo, Université de Lorraine Types of unit cells that bring a letter in E3

c c c c

b b b b

a P a C a B a A

S

c c c c c

b b b b b I F R H D a a a a a

Seldom used 101 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the triclinic (aanorthic) family

g b a + a third direction inclined on the plane 2D 3D mp

2 1  1 = 1 triclinic crystal family (anorthic)

● One symmetry element: the inversion centre ● No symmetry direction ● The conventional unit cell is not defined – no reason to choose a priori a centred cell ● Point group of the lattice: 1 aP ● No restriction on a, b, c, a, b, g

102 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the mmonoclinic family

g b a + a third direction perpendicular to the plane

mp 2D 3D

2 2  1 = 2/m

monoclinic crystal family

● One symmetry direction (usually taken as the b axis). ● The conventional unit cell has two right angles (a and g) ● Point group of the lattice: 2/m ● Two independent types of unit cell respect these conditions mP mS

103 / 129 Massimo Nespolo, Université de Lorraine A lattice of type mP is equivalent to mB

c c c B B P

b a B a = (a -c )/2 ; c = (a +c )/2 a B P B B P B B B b  b B P a P

c B  c a = (a -c )/2 P P B B

a B b  b B P a P

104 / 129 Massimo Nespolo, Université de Lorraine A lattice of type mP is NOT equivalent to mC

c C b P

b C b a C a C P a C

c C b P

b C b a C a C P a C

105 / 129 Massimo Nespolo, Université de Lorraine Lattices of type mC, mA, mI and mF are all equivalent

cC c mA mC A b  b A C a = -c b A C a C c = a C a A C A mC mF c = -a +2c mC mI F C C a = a -c I C C c c I F

a  a b  b F C I A a b  b I F C

106 / 129 Massimo Nespolo, Université de Lorraine Three monoclinic settings

b-unique c-unique a-unique

b unrestricted g unrestricted a unrestricted by symmetry by symmetry by symmetry mB = mP mC = mP mA = mP mA = mC = mI = mF mA = mB = mI = mF mB = mC = mI = mF

The symbol of a monoclinic space group will change depending on which setting and what type of unit cell you choose, leading to up to 21 possible symbols for a monoclinic space group (the “monoclinic monster”).

http://dx.doi.org/10.1107/S2053273316009293

107 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the oorthorhombic family

g g b b a a

op oc

+ a third direction perpendicular to the plane orthorhombic crystal family 2D 3D

2mm 2mm  1 = 2/m2/m2/m

oP oI oS oF

● Three symmetry directions (axes a, b, c) ● The conventional unit cell has three right angles (a, b, g) ● Point group of the lattice: 2/m 2/m 2/m ● Four types of cell respect these conditions ● The unit cell with one pair of faces centred can be equivalently described as A, B or C by a permutation of the (collective symbol : S)

108 / 129 Massimo Nespolo, Université de Lorraine Three possible setting for the oS type of lattice in E3

c b a

b c a c a b oC oB oA oS

109 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the ttetragonal family

g b 2D 3D a 4mm 4mm  1

tp tP tI = 4/m2/m2/m + a third direction perpendicular to the plane tetragonal crystal family ● Five symmetry directions (c, a&b, the two diagonals in the a-b plane) ● Point group of the lattice: 4/m 2/m 2/m ● The conventional cell has three right angles and two identical edges ● Two independent types of unit cell respect these conditions: tP (equivalent to tC) and tI (equivalent to tF) ½ ½

½ ½ ½ tC tI

tP ½ tF ½

110 / 129 Massimo Nespolo, Université de Lorraine Why unit cells of type tA et tB cannot exist?

0 1/2 0 1/2 0

4-fold axis!

1/2 1/2

0 0 0

a2 a 1 0 1/2 0 1/2 0

0 1/2 1/2 0 0 111 / 129 Massimo Nespolo, Université de Lorraine Why unit cells of type tA et tB cannot exist?

0 1/2 0 1/2 0

4-fold axis! 1/2 1/2 1/2

1/2 1/2

0 0 0

1/2 1/2 1/2

a2 a 1 0 1/2 0 1/2 0

1/2 1/2 1/2

0 1/2 1/2 0 0 112 / 129 Massimo Nespolo, Université de Lorraine Why unit cells of type tA et tB cannot exist?

0 1/2 0 1/2 0

4-fold axis! 1/2 1/2 1/2

1/2 1/2

0 0 0

1/2 1/2 1/2

a2 a 1 0 1/2 0 1/2 0

1/2 1/2 1/2

0 1/2 1/2 0 0 113 / 129 Massimo Nespolo, Université de Lorraine Why unit cells of type tA et tB cannot exist?

0 1/2 0 1/2 0

4-fold axis! 1/2 1/2 0 1/2 0

1/2 1/2

0 0 0

1/2 0 1/2 0 1/2

a2 a 1 0 1/2 0 1/2 0

1/2 tF! 1/2 0 1/2 0

0 1/2 1/2 0 0 114 / 129 Massimo Nespolo, Université de Lorraine Why unit cells of type tA et tB cannot exist?

0 1/2 0 1/2 0

4-fold axis! 1/2 1/2 0 1/2 0

1/2 1/2

0 0 0

1/2 0 1/2 0 1/2

a2 a 1 0 1/2 0 1/2 0

1/2 tF! tI! 1/2 0 1/2 0

0 1/2 1/2 0 0 115 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the ccubic family

g a 2 + a third direction perpendicular to the plane AND c = a = b

a 1 cubic crystal family

tp

cP cI cF

116 / 129 Massimo Nespolo, Université de Lorraine Three-fold rotoinversion along the 111 direction

First and fourth plane Observer from the observer Second plane from the observer

Third plane from the observer

117 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the ccubic family

g a 2 + a third direction perpendicular to the plane AND c = a = b

a1

tp cP cI cF cubic crystal family

2D 3D

4mm 4mm  3 = 4/m32/m

● Thirteen symmetry directions (the 3 axes; the 4 body diagonals ; the six face diagonals) ● Point group of the lattice: 4/m 3 2/m ● The conventional unit cell has three right angles and three identical edges ● Three types of unit cell respect these conditions : cP, cI et cF

118 / 129 Massimo Nespolo, Université de Lorraine Crystal families and types of lattices in E3: the hexagonal family

+ a third direction perpendicular to the plane a2

a1 g

hp

2D 3D C 6mm 6mm  1 = 6/m2/m2/m

hexagonal crystal family

+ a new type of unit cell!

A1 A2 119 / 129 Massimo Nespolo, Université de Lorraine Peculiarity of the hhexagonal crystal family in E3

Symmetry directions

C

a 3 a2

a1 A2

a3

a2

a1 A1

A1, A2, C hexagonal axes parallel to the symmetry directions A 1 A2 a1, a2, a3 rhombohedral axes NOT parallel to the symmetry directions Two types of lattice with different symmetry in the hexagonal crystal family 120 / 129 Massimo Nespolo, Université de Lorraine Symmetry difference between hP and hR types of lattice

[210] 6/m2/m2/m [110] H [100] H A H 3 32/m [101] [110] R R [110] H [120] H

a a 2 [010] 3 [010] H H A [011] 2 [011] R R a 1 [120] H

z = 0 [110] A H 1 [100] [210] [110] z = 1/3 H H H [101] z = 2/3 [110] R R 121 / 129 Massimo Nespolo, Université de Lorraine Cubic 7 holohedries → 7 lattice systems but 6 crystal families

hexagonal Tetragonal rhombohedral r e d r o orthorhombic p u o r g t monoclinic n i o P

triclinic Hexagonal crystal family: 2 free parameters (a,c or a,a)

Normal subgroup Conjugate subgroups 122 / 129 Massimo Nespolo, Université de Lorraine Lattice systems: classification based on the symmetry of the lattices

aP mP mS tP tI

1 triclinic 2/m monoclinic 4/m 2/m 2/m tetragonal (4/mmm)

oP oI oS oF

2/m 2/m 2/m (mmm) orthorhombic

cP cI cF hP hR 6/m 2/m 2/m 3 2/m 4/m 3 2/m cubic (6/mmm) hexagonal (3m) rhombohedral (m3m) 123 / 129 Massimo Nespolo, Université de Lorraine Crystal systems: morphological (macroscopic) and physical symmetry

124 / 129 Massimo Nespolo, Université de Lorraine Crystal systems: morphological (macroscopic) and physical symmetry

A A 2 3´A 1 2 or m or 2/m 2 1 or 1 Three twofold elements

triclinic monoclinic orthorhombic

A A3 6 A4

tetragonal trigonal hexagonal

4´A3 A = n or n cubic n 125 / 129 Massimo Nespolo, Université de Lorraine Crystal families, crystal systems, lattice systems and types of Bravais lattices in E3

7 crystal systems 7 lattice systems 14 types of Bravais 6 crystal families conventional unit cell (morphological symmetry) (lattice symmetry) lattices(**)

a = anortic* no restriction on triclinic triclinic aP (triclinic, asynmetric, tetartoprismac...) a ; b ; c, a , b , g m = monoclinic mP ( mB ) no restriction on a ; b ; c ; b. (clinorhombic, monosymmetric, binary, a = g = 90º monoclinic monoclinic hemiprismatic, monoclinohedral, ...) mS ( mC, mA, mI, mF ) oP o = orthorhombic no restriction on a ; b ; c. oS ( oC, oA, oB ) (rhombic, trimetric, terbinary, a = b = g = 90º orthorhombic orthorhombic prismatic, anisometric,...) oI oF t = tetragonal a = b ; a = b = g = 90º tetragonal tP ( tC ) (quadratic, dimetric, no restriction on c tetragonal monodimetric, quaternary...) tI ( tF ) rhombohedral trigonal (ternary...) (***) hR h = hexagonal a = b ; a = b = 90º, g = 120º (senaiey, monotrimetric...) no restriction on c hexagonal hP hexagonal

cP c = cubic a = b = c (isometric, monometric, cI a = b = g = 90º cubic cubic triquaternary, regular, tesseral, tessural...) cF

(*) Synonyms within parentheses. (**) S = one pair of faces centred. Within parentheses the types of lattices that are equivalent (axial setting change – see the monoclinic example). (***) Crystals of the trigonal crystal system may have a rhombohedral or hexagonal lattice

126 / 129 Massimo Nespolo, Université de Lorraine Symmetry directions of the lattices in the three-dimensional space (directions in the same box are equivalents by symmetry )

Lattice system Symmetry restrictions on the First symmetry Second symmetry Third symmetry parameters of the conventional cell direction direction direction

triclinic No restriction on any parameter

monoclinic a = g = 90º (b-unique) No restriction on a; b; c; b [010]

a = b = g = 90º orthorhombic No restriction on a; b; c [100] [010] [001]

a = b ; a = b = g = 90º [100] [010]  100 tetragonal No restriction on c [001] [110] [110]  110

rhombohedral a = b = c [110] [011] [101] axes a = b = g [111]  110 rhombohedral a = b; hexagonal [100] [010] [110] a = b = 90º; g = 120º [001]  100 axes No restriction on c a = b; [100] [010] [110] [110] [120] [210] hexagonal a = b = 90º; g = 120º [001]  100  110 No restriction on c [110] [110] [011] a = b = c [001] [100] [010] [111] [111] [111] [111] cubic [011] [101] [101] a = b = g = 90º  001  111  110

127 / 129 Massimo Nespolo, Université de Lorraine Symmetry restrictions on cell parameters (conventional cell)

Restrictions incorrectly given in many textbooks Triclinic family: a  b  c none a  b  g  90°

Monoclinic family a  b  c; For the crystal structure, a = g = 90º a = g = 90°; b  90° too restrictive

Orthorhombic family a  b  c; a = b = g = 90º a = b = g = 90° For the crystal lattice (geometric concept), Tetragonal family insufficient a = b a = b  c; a = b = g = 90º a = b = g = 90°

Hexagonal family Hexagonal axes: a = b; a = b = 90º ; g = 120º a = b  c; a = b = 90°; g = 120° Rhombohedral axes: a = b = c; a = b = g a = b = c; a = b = g  90°

Cubic family a = b = c; a = b = c a = b = g = 90° a = b = g = 90º

128 / 129 Massimo Nespolo, Université de Lorraine Why insufficient for the crystal lattice?

A simple example: mP if b = 90° the lattice is oP (obviously)

Let us suppose cosb = -a/2c

c

oB!

a

129 / 129 Massimo Nespolo, Université de Lorraine