Didactic material for the MaThCryst schools
Massimo Nespolo, Université de Lorraine, [email protected]
Fundamentals. Crystal patterns and crystal structures.Lattices, their symmetry and related basic concepts
Massimo Nespolo, Université de Lorraine2 / 129
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.
Ideal vs. real crystal, perfect vs. imperfect crystal
Massimo Nespolo, Université de Lorraine3 / 129
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” crystalNon-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.
What do you get from a (conventional) diffraction experiment?
Massimo Nespolo, Université de Lorraine4 / 129
Space and time-averaged structure
Massimo Nespolo, Université de Lorraine5 / 129
Transformations
Affinetransformation
Euclidean transformationor isometry
Symmetryoperation
1 2
34
4 1
23
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.
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.
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.
Massimo Nespolo, Université de Lorraine6 / 129
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.
Massimo Nespolo, Université de Lorraine7 / 129
● 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
element set
symmetry element
symmetry operation
The operation that share a given geometric element differ by a lattice vector. The one characterized by the shortest vector is called defining operationdefining operation.
symmetry operation
symmetry operation
Elements and operations
Massimo Nespolo, Université de Lorraine8 / 129
Crystal structure/pattern vs. crystal lattice
Crystal
Crystal pattern(crystal structure)
Molecule orcoordination polyhedron
Unit cell
Lattice nodes
Translations
Massimo Nespolo, Université de Lorraine9 / 129
Example of crystal pattern which is not a crystal structure
Massimo Nespolo, Université de Lorraine10 / 129
Unit cell
Asymmetric unit*
*In mathematics, it is called “fundamental region”
fi
The minimal unit you need to describe a crystal pattern
Massimo Nespolo, Université de Lorraine11 / 129
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)
ChiralityChirality: property of an object not being superimposable to its mirror image by a first-kind operation. HandednessHandedness: one of the two configurations (left or right) of chiral object. Also known as chirality sensechirality sense.
The concepts of chirality and handedness(the left-right difference)
Massimo Nespolo, Université de Lorraine12 / 129
Handedness of the object and nature of the operation
Massimo Nespolo, Université de Lorraine13 / 129
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
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.
Don’t forget the translations!
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.
Massimo Nespolo, Université de Lorraine14 / 129
Substructure S4
Substructure S1
Substructure S0
Substructure S2
Substructure S3
Fi,00
Fi,10
Global vs. local domain of action
Fpp
local operations: act only on a specific substructure
Global (total) operations:that subset of local and partial operations that actually act on the whole structure
Structure
Fqp
partial operations: relate only a specific pair of substructures
https://doi.org/10.2465/gkk1952.14.Special2_215https://doi.org/10.2465/gkk1952.14.Special2_215
Massimo Nespolo, Université de Lorraine15 / 129
A set S
A binary operation ○
An “internal law of composition” S ○ S → S(closure property)
We've got a “magma”
Note: An “external law of composition” acts on two disjoint sets S ○ S' → S
S S'
Binary operations acting on a set
Massimo Nespolo, Université de Lorraine16 / 129
Associativity: a ○ (b ○ c) = (a ○ b) ○ c
Presence of identity (neutral element): left identity e ○ a = aright identity a ○ e = a
Presence of the inverse (symmetric) element of each element:left inverse: a-1 ○ a = eright inverse: a ○ a-1 = e
Commutativity: a ○ b = b ○ a
Presence of singular element: left singular s ○ a = sright singular a ○ s = s
Presence of a second binary operation: a S b S = c S
Possible properties associated to a magma
Massimo Nespolo, Université de Lorraine17 / 129
Algebraic structures
globality associativity identity inverse element
Magma
Semigroup
Monoid
Group
Semicategory
Category
Groupoid
Massimo Nespolo, Université de Lorraine18 / 129
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 g
j of the set G is still a symmetry operation g
k of the set G
(closure property): giG ○ g
jG → g
kG (g
ig
j = g
k)
● the binary operation is associative: gi(g
jg
k) = (g
ig
j)g
k
● the set G includes the identity (left and right identical): egi = g
ie
● for each element of the set G (each symmetry operation) the inverse element (inverse symmetry operation) is in the set G: g
i-1g
i = g
ig
i-1 = 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!
The notion of symmetry group
Massimo Nespolo, Université de Lorraine19 / 129
A special case of Abelian groups: cyclic groupsA 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}
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.
Abelian and cyclic groups
Massimo Nespolo, Université de Lorraine20 / 129
Order of a group element: the smallest n such that gn = e.
Order of a group: the number of elements of the group (finite or infinite)
If n = the group is infinite
If n = 2, then g-1 = g and the element is known as an involution.
The notion of “order”
Massimo Nespolo, Université de Lorraine21 / 129
(G, *) (H, #)f
Let G and H be two sets, and * and # two binary operations acting on G and H respectively.
*u'
#v' w'
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 1G, then H too contains a neutral element 1
H and the
homomorphism f maps them:1G →1
H● 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.
Domain Codomain
u
wv
f(u) = u'
f(v) = v'
f(w) = w'
w = u*v
w' = u'#v'
f(u * v) = f(w) = w' = u' # v' = f(u) # f(v)
If H G, the homomorphism takes the name of endomorphismendomorphism
Homomorphism
Massimo Nespolo, Université de Lorraine22 / 129
1H
Kernelof f image of f
(G,*) (H, #)f
The kernelkernel of the homomorphism f : G → H is the subset of elements of G that is mapped by f on to neutral element of H.
The imageimage 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.
DomainCodomain
ker f ={gG: f(g) = 1H}
Kernel and image
Massimo Nespolo, Université de Lorraine23 / 129
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.
1H
(G,*) (H, #)f
1G =
Ker(f)
Isomorphism
Massimo Nespolo, Université de Lorraine24 / 129
Dimensions of the space and periodicity of the pattern
nmG n-dimensional space, m-dimensional periodicity
m = 0:point groupspoint groups;n = m:space groupsspace groups;0 < m < n: subperiodic groupssubperiodic groups
n m No. of types of groups Name
1 0 2 1-dimensional point groupspoint groups
1 2 Line groups:1-dimensional space groups1-dimensional space groups
2 0 10 2-dimensional point groupspoint groups
1 7 Frieze groupsFrieze groups
2 17 Plane groups, wallpaper groups:2-dimensional space groups2-dimensional space groups
3 0 32 3-dimensional point-groupspoint-groups
1 75 Rod groupsRod groups
2 80 Layer groupsLayer groups
3 230 (3-dimensional) Space groupsSpace groups
Massimo Nespolo, Université de Lorraine25 / 129
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 = iE
i
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.
Translation group:the group containing only the translations of the crystal structure。It is a normal subgroup of the space group G.
TG → tjT, g
iG : g
itjg
i-1 = t
j
P G/T
Some of the definitions will follow
Massimo Nespolo, Université de Lorraine26 / 129
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)}
H = {(W,0)}
W = nn matrix, I = nn identity matrix、w = n1 matrix, 0 = n1 zero matrix
T={(I,w)} P={W}Isomorphism
EndomorphismProjection
The minimal point group of a Bravais lattice is built on{I,-I}.
Massimo Nespolo, Université de Lorraine27 / 129
The minimal point group of a Bravais lattice is built on{I,-I}.
origin
tt
I-I
I-I
I-I
etc….
1-dimensional space:x → x: -I = reflection (or inversion)
2-dimensional space:xy → xy: -I = rotation
3-dimensional space:xyz → xyz: -I = inversion
etc….
Bravais lattices
Massimo Nespolo, Université de Lorraine28 / 129
1
1
1
...
1
...
x
y
z
w
...
x
y
z
w
det( -In) = (-1)n
Odd-dimensional space:-1
Even-dimensional space: +1
Second kind operation
First kind operation
The inversion does not exist in even-dimensional spacesThe inversion does not exist in even-dimensional spaces
-I in En
Massimo Nespolo, Université de Lorraine29 / 129
Symmetry operations
Translations
Identity
Reflections
How many 1D lattices are there? Infinite!
How many types of 1D lattice are there? One
etc. etc.
I = 1, -I = m point group of a Bravais lattice: m = {1,m}
One-dimensional lattices
Massimo Nespolo, Université de Lorraine30 / 129
The world in two dimensions
E2 : the two-dimensional Euclidean space
Massimo Nespolo, Université de Lorraine31 / 129
Operations that leave invariant all the space (2D): the identityOperations that leave invariant one direction of the space (1D): reflectionsOperations that leave invariant one point of the space (0D): rotationsOperations that do not leave invariant any point of the space: translations
Two independent directions in E2 two axes (a, b) and one interaxial angle (g)
The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 2 here)
I = 1, -I = 2 Minimal point group of a Bravais lattice: 2 = {1,2}
Symmetry operations in E2
Massimo Nespolo, Université de Lorraine32 / 129
2 2-fold rotation point m Reflection line(mirror)
First kind operations Second kind operation
Hermann-Mauguin symbol
Graphicsymbol
Meaning
3 3-fold rotation point
4 4-fold rotation point
6 6-fold rotation point
Hermann-Mauguin symbol
Graphicsymbol
Meaning
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
Symmetry elements in E2
Massimo Nespolo, Université de Lorraine33 / 129
a
00
10
20
30
01 02 03
11 12 13
21 22 23
31 32 33
b
[42] direction ?
[10]
[01]
a axis?
b axis?
[13]
[11]
[21]
40 41 42 43
Lattice node coordinates uv, lattice direction indices [uv]
Massimo Nespolo, Université de Lorraine34 / 129
t(1,0), t(0,1)
t(1,0), t(0,1), t(½,½)
t(1,0), t(0,1), t(⅔,⅓), t(⅓,⅔)
t(1,0), t(0,1), t(¾,¼), t(½,½), t(¼,¾)
Choice of the unit cell
Massimo Nespolo, Université de Lorraine35 / 129
1 1
0 1
Determ.: 1
2 0
1 1
Determ.: 2
1 1
0 1
Determ.: 1
12
12
0
1
Determ.: ½
1 1
1 1
Determ.: 2
1 12 2
1 12 2
Determ.: ½
2 0
1 1
Determ.: 2
12
12
0
1
Determ.: ½
1 1
1 1
Determ.: 2
1 12 2
1 12 2
Determ.: ½
1 12 2
3 12 2
Determ.: 1
1 12 2
3 12 2
Determ.: 1
2 2
1 3
Determ.: 4
3 14 2
1 14 2
Determ.: ¼
1 1
1 3
Determ.: 4
3 14 4
1 14 4
Determ.: ½
1 12 2
3 52 2
Determ.: 2 Determ.: ½
5 14 4
3 14 4
1 1
0 2
Determ.: 2
12
12
1
0
Determ.: ½
3 3
1 2
Determ.: 3
23
13
1
1
Determ.: ⅓
2 1
1 2
Determ.: 3
2 13 3
1 23 3
Determ.: ⅓
12
52
1
2
Determ.: 1.5
5 13 3
4 23 3
Determ.: ⅔
3 32 2
1 12 2
Determ.: 1.5
13
13
1
1
Determ.: ⅔
7 54 4
1 14 4
Determ.: ¾
513 3
713 3
Determ.: 4/3
Change of reference
Massimo Nespolo, Université de Lorraine36 / 129
t(1,0), t(0,1) : primitive (p) unit cell
t(1,0), t(0,1), t(½,½) : centred (c) unit cell
Conventional unit cell1. 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.
Cartesian (orthonormal) cell: unsuitable (in general)
Change of the unit cell (cont.)
Massimo Nespolo, Université de Lorraine37 / 129
312 2
512 2
712 2
1 12 2
00
10
20
30
01 02 03
11 12 13
21 22 23
31 32 33
40 41 42 43
24
44
34
14
04
[10]
[01]
½[11]
½[13]
3 32 2
3 52 2
3 72 2
3 12 2
5 52 2
5 32 2
5 12 2
7 52 2
7 32 2
7 12 2
https://doi.org/10.1107/S1600576717010548
Lattice node coordinates uv, lattice direction indices [uv]
Massimo Nespolo, Université de Lorraine38 / 129
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine39 / 129
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine40 / 129
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine41 / 129
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
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fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fi fi fi fi fi fi fi fi fi fi fi fi fi fifi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine42 / 129
ba
O
no symmetry direction for this latticepoint group of the lattice: 2 = {I, -I}no symmetry restriction on the cell parameters:a; b; g
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine43 / 129
fi
fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine44 / 129
ap
bc
O
bpa
c
Primitive unit cellConventional unit cell
fi
two symmetry directions for this lattice, that are taken as axes a and b
point group of the lattice: 2 m m
[10][01]
symmetry restriction on the cell parameters:a; b; g = 90º
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine45 / 129
fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fifi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
fifi fi fi fi fi fi fi fi fi fi fi fi fi fi fi fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine46 / 129
bc=p
Oa
c=p
Conventional (primitive) unit cell
fi
two symmetry directions for this lattice, that are taken as axes a and b
point group of the lattice: 2 m m
[10][01]
symmetry restriction on the cell parameters:a; b; g = 90º
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine47 / 129
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine48 / 129
This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a
1 and a
2.
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine49 / 129
b,a2
a,a1
This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a
1 and a
2.
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine50 / 129
This lattice has four symmetry directions. Those corresponding to the shortest period are taken as axes a
1 and a
2.
b,a2
a,a1
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine51 / 129
Point group of the lattice symmetry: 4 m m
10 11
b,a2
a,a1
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine52 / 129
Point group of the lattice symmetry: 4 m m
10 11
b,a2
a,a1
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
p cell (conventional)
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine53 / 129
b,a2
a,a1
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fifi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
fi
Restrictions on cell parameters: a = b; g = 90°
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine54 / 129
fi
fi fi fififi
fi fifi fi
fifififififififififi
fi fi fi fi fi
fifififififififififi
fi fi fi fi
fififififi
fi
This lattice has six symmetry directions. Two among the three corresponding to the shortest period are taken as axes a
1 et a
2.
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine55 / 129
fi
fi fi fififi
fi fifi fi
fifififififififififi
fi fi fi fi fi
fifififififififififi
fi fi fi fi
fififififi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine56 / 129
b,a2
a,a1
p cell (conventional)
h cell (triple)
Point group of the lattice:6 m m
10 11
Restrictions on the cell parameters: a = b; g = 120°
fi
fi fi fififi
fi fifi fi
fifififififififififi
fi fi fi fi fi
fifififififififififi
fi fi fi fi
fififififi
fi
Exercise:Find the lattice symmetry and thenthen choose a suitable unit cell
Massimo Nespolo, Université de Lorraine57 / 129
[10] [11]
[01]
[21]
[12][11]
a1
a2
Direction indices [uv] in the hexagonal lattice of E2
Massimo Nespolo, Université de Lorraine58 / 129
Why not 5, then?
Massimo Nespolo, Université de Lorraine59 / 129
Crystal patterns in 2D
Massimo Nespolo, Université de Lorraine60 / 129
Symmetry of the lattice : SL = 4mm
Lattice
« Object » (content of the unit cell)
Pattern
Symmetry of the object : So = 4mm
Symmetry of the pattern : Sp = 4mm
Sp = S
L : holohedry
The concept of holohedry
Massimo Nespolo, Université de Lorraine61 / 129
Symmetry of the lattice: SL= 4mm
Symmetry of the object: So = 3m
Symmetry of the pattern : Sp = m = 4mm 3m
Sp < S
L : merohedry
The concept of holohedry
Massimo Nespolo, Université de Lorraine62 / 129
Symmetry of the lattice: SL= 4mm
Symmetry of the object: So = 3m
Symmetry of the pattern : Sp = 1 = 4mm 3m
SP < S
L: merohedry
The concept of holohedry
Massimo Nespolo, Université de Lorraine63 / 129
Symmetry of the lattice : SL = 4mm
Symmetry of the object : So = 5m
Symmetry of the pattern : Sp = m = 4mm 5m
Sp < S
L : merohedry
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!
The concept of holohedry
Massimo Nespolo, Université de Lorraine64 / 129
The notion of subgroupThe notion of subgroup
Massimo Nespolo, Université de Lorraine65 / 129
m[10]
m[11]
m[01]
m[11]
Symmetry elements:
Symmetry operations:
one fourfold rotation point: 4four reflection lines: m
[10], m
[01], m
[11], m
[11]
Four rotations (41, 42 = 2, 43 = 4-1, 44 =1)Four reflections (m
[10], m
[01], m
[11], m
[11])
Check that these operations do form a group
● Isometries are always associative● 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; 2-1 = 2, 4-1 = 43)● To check the closure, let us build the multiplication
(Cayley) table.
1
2
4
3
a1 (a)
a2 (b)
Geometric elements: one point (centre of the square), four lines ([10], [01], [11], [11])
Exercise: find the elements, the operations and the symmetry group of a square in E2
Massimo Nespolo, Université de Lorraine66 / 129
H GFrom the group (G,○) we select a subset of elements forming a subset H. If the (H,○) is a group under the same binary operation ○ as (G,○), then (H,○) is a subgroup of (G,○).
m[10]
m[11]
m[01]
m[11]
1
2
0
3
Multiplication table of the group (G,○)Multiplication table of the subgroup (H,○)
1 2 41 43 m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
2 41 43 m[10]
m[01]
m[11]
m[11]
1 43 41 m[01]
m[10]
m[11]
m[11]
43 2 1 m[11]
m[11]
m[01]
m[10]
41 1 2 m
[11]m
[11]m
[10]m
[01]
m
[01]m
[11]m
[11]1 2 43 41
m
[10]m
[11]m
[11] 2 1 41 43
m
[11]m
[10]m
[01]41 43 1 2
m
[11]m
[01]m
[10]43 41 2 1
Subgroups
Massimo Nespolo, Université de Lorraine67 / 129
Order and index of a subgroup
group G, order |G|
group H, order |H|H G
|H| is a divisor of |G| (Lagrange’s theorem)
The ratio iG(H) = |G|/|H| is called the index of H in Gindex of H in G
iG(H) = 8:ogdohedry(3-dimensional space)
iG(H) = 4:tetartohedry
iG(H) = 2:hemihedry
Massimo Nespolo, Université de Lorraine68 / 129
m[10]
m[11]
m[01]
m[11]
1
2
0
3
Group Subgroup = 1st coset2nd coset
By decomposing (G,○) with respect to (H,○) we get i = |G|/|H| cosets. Each coset has the same number of elements |H| as the subgroup, which is called the length of the coset.
1 2 41 43 m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
2 41 43 m[10]
m[01]
m[11]
m[11]
1 43 41 m[01]
m[10]
m[11]
m[11]
43 2 1 m[11]
m[11]
m[01]
m[10]
41 1 2 m
[11]m
[11]m
[10]m
[01]
m
[01]m
[11]m
[11]1 2 43 41
m
[10]m
[11]m
[11] 2 1 41 43
m
[11]m
[10]m
[01]41 43 1 2
m
[11]m
[01]m
[10]43 41 2 1
H GC1=H C2
Cosets
Massimo Nespolo, Université de Lorraine69 / 129
m[10]
m[11]
m[01]
m[11]
1
2
0
3
Group Subgroup = 1st coset2nd, 3rd, 4th coset
1 2 41 43 m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
11
241
1
431
m[10]
m[01]
m[11]
m[11]
2 41 43 m[10]
m[01]
m[11]
m[11]
1 43 41 m[01]
m[10]
m[11]
m[11]
43 2 1 m[11]
m[11]
m[01]
m[10]
41 1 2 m
[11]m
[11]m
[10]m
[01]
m
[01]m
[11]m
[11]1 2 43 41
m
[10]m
[11]m
[11] 2 1 41 43
m
[11]m
[10]m
[01]41 43 1 2
m
[11]m
[01]m
[10]43 41 2 1
H GC1=H
C3
C2
C4
By decomposing (G,○) with respect to (H,○) we get i = |G|/|H| cosets. Each coset has the same number of elements |H| as the subgroup, which is called the length of the coset.
Cosets
Massimo Nespolo, Université de Lorraine70 / 129
Order Index
8
4
2
1
1
2
4
8
1 4mm = {1, 41, 2, 43, m[10]
, m[01]
, m[11]
, m[11]
}
{1, 41, 2, 43} = 4 {1, 2, m[10]
, m[01]
, } = 2mm. {1, 2, m[11]
, m[11]
} = 2.mm
{1, 2} = 2 {1, m[10]
} = .m. {m[01]
} = .m. {1, m[11]
} = ..m {1, m[11]
} = ..m
{1} = 1
Desymmetrization of the square
Massimo Nespolo, Université de Lorraine71 / 129
4mm
2mm. 2.mm4
.m[10]
. .m[01]
. 2 ..m[11]
..m[11]
1
Order Index
8
4
2
1
1
2
4
8
1
Desymmetrization of the square
Massimo Nespolo, Université de Lorraine72 / 129
CosetsConjugacy classes
Conjugate subgroupsNormal or invariant subgroups
Massimo Nespolo, Université de Lorraine73 / 129
G = 4mm = {1, 41, 42 = 2, 43 = 4-1, m[10]
, m[01]
, m[11]
, m[11]
}
4mm = {2mm.}g{2mm.} ={1, 2, m[10]
, m[01]
}{41, 43, m[11]
, m[11]
}
4mm = {2.mm}g{2.mm} ={1, 2, m[11]
, m[11]
}{41, 43, m[10]
, m[01]
}
4mm = {4}g{4} ={1, 41, 2, 43}{m[10]
, m[01]
, m[11]
, m[11]
}
G = HiC
i =H
ig
iH, g
i(H,C
j<i)
H = 2mm.
H = 2.mm
H = 4
Etc. etc.
G = HiC
i =H
iHg
i, g
i(H,C
j<i) Right coset
Usually, giH Hg
i
If giH = Hg
i, i, then H is called a normal (invariant) subgroupnormal (invariant) subgroup.
Left coset
Left and right cosets
Massimo Nespolo, Université de Lorraine74 / 129
4mm = {1,2} 41{1,2} m[10]
{1,2} m[11]
{1,2} =
{1,2} {41,43} {m[10]
,m[01]
} {m[11]
,m[11]
}
4mm = {1,2} {1,2}41 {1,2}m[10]
{1,2}m[11]
=
{1,2} {41,43} {m[10]
,m[01]
} {m[11]
,m[11]
}
=
4mm = {1,m[10]
} 41{1,m[10]
} 43{1,m[10]
} 2{1,m[10]
} =
{1,m[10]
} {41,m[11]
} {43,m[11]
} {2,m[01]
}
Normal subgroup
4mm = {1,m[10]
} {1,m[10]
}41 {1,m[10]
}43 {1,m[10]
}2 =
{1,m[10]
} {41,m[11]
} {43,m[11]
} {2,m[01]
}
Decomposition of 4mm with respect to 2 and m[10]
Massimo Nespolo, Université de Lorraine75 / 129
H G
ghig-1 = h
j; h
iH, gG : conjugate elements
ghi = h
jg; h
iH, gG
ig
ihg
i-1 = Cl(h) conjugacy classconjugacy class
ig
iHg
i-1 = {H, H', H''・・・} are conjugate subgroupsconjugate subgroups of G
If H' = H'' = ・・・ = H (giH = Hg
i, gG)、H is a normal or invariant normal or invariant
subgroupsubgroup of G: H H G G.
The set of conjugate elements form a class
Next consider not one element h but the whole subgroup H
If hi = h
j, gG, h
i is a self-conjugate elementself-conjugate element
Cl(h) = { h' G | ∈Cl(h) = { h' G | ∈ g g G : h' = ghg G : h' = ghg-1-1 } }
Conjugation is a similarity transformation: “do the same thing somewhere else”
Conjugate and normal subgroups
Massimo Nespolo, Université de Lorraine76 / 129
ig
ihg
i-1 = Cl(h): conjugacy class
h = 1 ig
i1g
i-1 =
ig
ig
i-1 = {1}
h = 2
h = 41
h = m[10]
h = m[11]
ig
i2g
i-1 =
ig
i1g
i-1 =
{2}
ig
i41g
i-1 = {41, 43}
ig
im
[10]g
i-1 = {m
[10], m
[01]}
ig
im
[11]g
i-1 = {m
[11], m
[11]}
4mm = {1}, {2}, {41, 43}, {m[10]
, m[01]
}, {m[11]
, m[11]
}
Conjugacy classes of 4mm
Massimo Nespolo, Université de Lorraine77 / 129
When we apply an isometry・・・The object is transformed directly(O→ O')
The symmetry group of the object is transformed by conjugation
hO = O, hH gO = O', gH,H'h'O' = O', h'H'
H' = gHg-1
Massimo Nespolo, Université de Lorraine78 / 129
H = 2mm. = {1, 2, m[10]
, m[01]
}
gHg-1 = 2mm.
gHg-1 = H H = 2mm. G = 4mm Normal subgroup
{41, 43, m[11]
, m[11]
}
a
b
Effects of the operations in a coset
Massimo Nespolo, Université de Lorraine79 / 129
H = 2.mm = {1, 2, m[11]
, m[11]
}
gHg-1 = 2.mm
gHg-1 = H H = 2.mm G = 4mmNormal subgroup
{41, 43, m[10]
, m[01]
}
a
b
Effects of the operations in a coset
Massimo Nespolo, Université de Lorraine80 / 129
H = 4 = {1, 4, 2, 43}
gHg-1 = 4
gHg-1 = H H = 4 G = 4mmNormal subgroup
{m[10]
, m[01]
, m[11]
, m[11]
}
a
b
Effects of the operations in a coset
Massimo Nespolo, Université de Lorraine81 / 129
H = {1, m[10]
} gHg-1 = {1, m[10]
}
{2, m[01]
}
a
b
g{1,m[01]
}g-1 = {1,m[10]
}
.m. G = 4mmonjugate subgroups
{41, m[11]
}{43, m
[11]}
{2, m[01]
}
{41, m[11]
}
gHg-1 = {1, m[01]
} gHg-1 = {1, m[01]
}
Effects of the operations in a coset
Massimo Nespolo, Université de Lorraine82 / 129
4mm
2mm. 2.mm4
.m[10]
. .m[01]
. 2 ..m[11]
..m[11]
1
Order Index
8
4
2
1
1
2
4
8
1
Desymmetrization of the square
Massimo Nespolo, Université de Lorraine83 / 129
Extra-simple exerciseExtra-simple exerciseShow that a subgroup of index 2 is always
normal
Massimo Nespolo, Université de Lorraine84 / 129
Holohedries and merohedries in the two-dimensional space
Holohedry 2 : {1,2}
Subgroup 1 : {1}
Holohedry 2mm : {1,2,m[10]
,m[01]
}
Subgroups 2 : {1,2}, m : {1,m[10]
}, {1,m[01]
}, 1 : {1}
Holohedry 4mm : {1, 41, 42 = 2, 43 = 4-1, m[10]
, m[01]
, m[11]
, m[11]
}
Subgroups 4 : {1, 41, 42, 43}, 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}
Holohedry 6mm : {1, 61, 62 = 3, 63 = 2, 64 = 3-1, 65 = 6-1, m[10]
, m[01]
, m[11]
, m[11]
, m
[12], m
[21]}
Subgroups 6 : {1, 61, 62, 63, 64, 65}, 3m : {1, 31, 32, m[10]
, m[01]
, m[11]
}, {1, 31, 32, m
[11], m
[12], m
[21]}, 3 : {1, 3, 3-1}, 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}
Conjugacy classes {1}, {2}
Conjugacy classes {1}, {2}, {m[10]
}, {m[01]
}
Conjugacy classes {1}, {2}, {41, 43}, {m[10]
, m[01]
}, {m[11]
, m[11]
}
Conjugacy classes:{1}, {2}, {31, 32}, {61, 65}, {m
[10], m
[01], m
[11], }, {m
[11], m
[12], m
[21]}
Massimo Nespolo, Université de Lorraine86 / 129
6mm
4mm
6 3m
4 2mm
2
1
m3
4mm
2.mm 2mm.
6mm
3m1 31m
2mm
m[010]
m[100]
Tree of maximal group-subgroups (in bold the holohedries)
Conjugate subgroups
Normal subgroup
Tree of group-subgroups in E2
Massimo Nespolo, Université de Lorraine87 / 129
Orthorhombic family:2 free parameters (a,b)
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
4mm
6 3m
4 2mm
2
1
m3
Hexagonal family:1 free parameter (a)Tetragonal family:1
free parameter (a)
Monoclinic family: 3 free parameters (a,b,g)
Massimo Nespolo, Université de Lorraine88 / 129
mp
tp
op
hp
monoclinicpoint group 2
(minimal point group)No restriction
on a, b, gNo symmetry direction
orthorhombicpoint group 2mm
No restriction on a, b ;g = 90º
[10] and [01]
tetragonalpoint group 4mm
a = b ; g = 90º10 ([10] and [01])11 ([11] and [11])
hexagonalpoint group 6mma = b ; g = 120º
10 ([10], [01] and [11])11 ([21], [12] and [11])
b
a
g b
a
g
a1
a2
g
a1
a2
g
Crystal families: monoclinic, orthorhombic, tetragonal, hexagonalType of lattice*: primitive, centred
*Lattice whose conventional unit cell is primitive or centredKettle S.F.A, Norrby L.J., J. Chem. Ed. 70(12), 1993, 959-963
g bc
ac
bp
ap
oc
Conventional cell parameters and symmetry directions in the four crystal families of E2
Massimo Nespolo, Université de Lorraine89 / 129
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 hpCrystal system
2, 1 monoclinic
2mm, m orthorhombic
4mm, 4 tetragonal
6mm, 6, 3m, 3 hexagonal
No. of free parameters 3 2 2 1 1
Massimo Nespolo, Université de Lorraine90 / 129
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
Massimo Nespolo, Université de Lorraine91 / 129
The world in three dimensions
E3 : the three-dimensional Euclidean space
Massimo Nespolo, Université de Lorraine92 / 129
Operations that leave invariant all the space (3D): the identityOperations that leave invariant a plane (2D): the reflectionsOperations that leave invariant one direction of the space (1D): the rotationsOperations that leave invariant one point of the space (0D): the roto-inversionsOperations that do not leave invariant any point of the space : the translations
Three independent directions in E3 three axes (a, b, c) and three interaxial angles (a, b, g)
The subspace left invariant (if any) by the symmetry operation has dimensions from 0 to N (= 3 here)
I = 1, -I = 1 Minimal point group of a Bravais lattice: 1 = {1,1}
Symmetry operations in E3
Massimo Nespolo, Université de Lorraine93 / 129
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)
inversion centre
4 (order 4)
inversion point
Massimo Nespolo, Université de Lorraine94 / 129
c
b
a
ab
g
Labelling of axes and angles in E3
Massimo Nespolo, Université de Lorraine95 / 129
1 one-fold rotoinversion axis(inversion centre)
2 2-fold rotation axis
3 3-fold rotation axis
4 4-fold rotation axis
6 6-fold rotation axis
2/m
3 three-fold rotoinversion axis 4/m
4 four-fold rotoinversion axis 6/m
6 (3/m) six-fold rotoinversion axis
n-fold rotation axis n-fold rotoinversion axis n-fold rotation axis and mirror plane perpendicular to it
two-fold rotoinversion axis is a mirror plane
First-kind operations Second-kind operations
Operations including translations are introduced laterThe orientation of a mirror plane is indicated by the vector normal to it
Graphic symbols for symmetry elements
Massimo Nespolo, Université de Lorraine96 / 129
Infinite!
Only one
The orientation of a mirror plane is indicated by the vector normal to it
Massimo Nespolo, Université de Lorraine97 / 129
II
'
'
'
x x
f y y
z z
II
'
11 '
'
x x
f y y
z z
II
'
1 1 '
'
x x
f y y
z z
1
1 1
1
I
'
1 '
'
x x
f y y
z z
I
'
1 '
'
x x
f y y
z z
Associativity
Commutativity
If fII is a symmetry operation of a given object, and if that object is not centrosymmetric, f
I is not a symmetry operation of the object.
An isometry of the second kind, fII, can always be
expressed as 1fI
Massimo Nespolo, Université de Lorraine98 / 129
db
q
I
IIII
Equivalence of 2 and m
Massimo Nespolo, Université de Lorraine99 / 129
db
q
I
IIII
p
I
II
Applies to even-fold rotations as well, because they all “contain” a twofold rotation
42 = 2; 63 = 2
Combination of 2 and 1 gives m
Massimo Nespolo, Université de Lorraine100 / 129
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(½,½,0): Ct(1,0,0), t(0,1,0), t(0,0,1), t(½,0,½): Bt(1,0,0), t(0,1,0), t(0,0,1), t(0,½,½): At(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): Ft(1,0,0), t(0,1,0), t(0,0,1), t(⅔,⅓,0), t(⅓,⅔,0): Ht(1,0,0), t(0,1,0), t(0,0,1), t(⅔,⅓,⅓), t(⅓,⅔,⅔): Rt(1,0,0), t(0,1,0), t(0,0,1), t(⅓,⅓,⅓), t(⅔,⅔,⅔): D
Conventional unit cell1. 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 cell1. 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).
Choice of the unit cell
Massimo Nespolo, Université de Lorraine101 / 129
a
b
c
P a
b
c
a
b
c
C B a
b
c
A
S
a
b
c
Ia
b
c
F
Types of unit cells that bring a letter in E3
a
b
c
R a
b
c
H a
b
c
D
Seldom used
Massimo Nespolo, Université de Lorraine102 / 129
mp
b
a
g+ a third direction inclined on the plane
● 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● No restriction on a, b, c, a, b, g
triclinic crystal family (anorthic)
aP
2D 3D
2 1 1 = 1
Crystal families and types of lattices in E3: the triclinic (aanorthic) family
Massimo Nespolo, Université de Lorraine103 / 129
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
mp
b
a
g+ a third direction perpendicular to the plane
2D 3D
2 2 1 = 2/m
Crystal families and types of lattices in E3: the mmonoclinic family
Massimo Nespolo, Université de Lorraine104 / 129
cB
bBa
Ba
P = (a
B-c
B)/2 ; c
P = (a
B+c
B)/2
bP
cP
aP
cB
aB
bB
aP
cB
aB
aP = (a
B-c
B)/2
bB
bP
cP
A lattice of type mP is equivalent to mB
Massimo Nespolo, Université de Lorraine105 / 129
cC
bCa
C
bC
aC
bP
aP
cC
bCa
C
bC
aC
bP
aP
A lattice of type mP is NOT equivalent to mC
Massimo Nespolo, Université de Lorraine106 / 129
aA = -c
C
cA = a
C
mA mC
mC mF
cF = -a
C+2c
CmC mIa
I = a
C-c
Cc
I
aI
bI b
A
cF
bF b
C
aF a
C
cC
bCa
C
aA
cA b
A b
C
Lattices of type mC, mA, mI and mF are all equivalent
Massimo Nespolo, Université de Lorraine107 / 129
b-unique c-unique a-unique
b unrestricted by symmetry
g unrestricted by symmetry
a unrestricted by symmetry
mB = mPmA = mC = mI = mF
mC = mPmA = mB = mI = mF
mA = mPmB = 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
Three monoclinic settings
Massimo Nespolo, Université de Lorraine108 / 129
op
ba
b
a
oc
g g
oP oI oS oF
orthorhombic crystal family
● 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)
+ a third direction perpendicular to the plane
2D 3D
2mm 2mm 1 = 2/m2/m2/m
Crystal families and types of lattices in E3: the oorthorhombic family
Massimo Nespolo, Université de Lorraine109 / 129
a b
c
a
b
c
a
bc
oC oB oA
oS
Three possible setting for the oS type of lattice in E3
Massimo Nespolo, Université de Lorraine110 / 129
tp
a
bg
tP tI
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)
tP
tC
½
tF
½
½
+ a third direction perpendicular to the plane
½
tI
½
½
½
2D 3D
4mm 4mm 1
= 4/m2/m2/m
Crystal families and types of lattices in E3: the ttetragonal family
Massimo Nespolo, Université de Lorraine111 / 129
Why unit cells of type tA et tB cannot exist?
0 0
0 0
0
0
0 0 0
00
1/2
1/2
1/2
1/2
1/2 1/2
1/2 1/20
4-fold axis!
a1
a2
Massimo Nespolo, Université de Lorraine112 / 129
Why unit cells of type tA et tB cannot exist?
0 0
0 0
0
0
0 0 0
00
1/2
1/2
1/2
1/2
1/2 1/2
1/2 1/20
1/21/2 1/2
1/21/2
1/21/21/2
1/2
a1
a2
4-fold axis!
Massimo Nespolo, Université de Lorraine113 / 129
0 0
0 0
0
0
0 0 0
00
1/2
1/2
1/2
1/2
1/2 1/2
1/2 1/20
1/21/2 1/2
1/21/2
1/21/21/2
1/2
a1
a2
4-fold axis!
Why unit cells of type tA et tB cannot exist?
Massimo Nespolo, Université de Lorraine114 / 129
00
00
00
0 0
0 0
0
0
0 0 0
00
1/2
1/2
1/2
1/2
1/2 1/2
1/2 1/20
1/21/2 1/2
1/21/2
1/21/21/2
1/2
a1
a2
tF!
4-fold axis!
Why unit cells of type tA et tB cannot exist?
Massimo Nespolo, Université de Lorraine115 / 129
00
00
00
0 0
0 0
0
0
0 0 0
00
1/2
1/2
1/2
1/2
1/2 1/2
1/2 1/20
1/21/2 1/2
1/21/2
1/21/21/2
1/2
a1
a2
tF! tI!
4-fold axis!
Why unit cells of type tA et tB cannot exist?
Massimo Nespolo, Université de Lorraine116 / 129
tp
a1
a2
g
cP cI cF
cubic crystal family
+ a third direction perpendicular to the plane AND c = a = b
Crystal families and types of lattices in E3: the ccubic family
Massimo Nespolo, Université de Lorraine117 / 129
ObserverFirst and fourth plane from the observer
Second plane from the observer
Third plane from the observer
Three-fold rotoinversion along the 111 direction
Massimo Nespolo, Université de Lorraine118 / 129
tp
a1
a2
g
cP cI cF
cubic crystal family
● 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
+ a third direction perpendicular to the plane AND c = a = b
2D 3D
4mm 4mm 3 = 4/m32/m
Crystal families and types of lattices in E3: the ccubic family
Massimo Nespolo, Université de Lorraine119 / 129
hp
a1
a2
g
hexagonal crystal family
A1
A2
C
+ a new type of unit cell!
+ a third direction perpendicular to the plane
2D 3D
6mm 6mm 1 = 6/m2/m2/m
Crystal families and types of lattices in E3: the hexagonal family
Massimo Nespolo, Université de Lorraine120 / 129
A1 A
2
C
a1
a2
a3
A1
A2
a1
a2
a3
Symmetry directions
A1, A
2, C hexagonal axes parallel to the symmetry directions
a1, a
2, a
3 rhombohedral axes NOT parallel to the symmetry directions
Two types of lattice with different symmetry in the hexagonal crystal family
Peculiarity of the hhexagonal crystal family in E3
Massimo Nespolo, Université de Lorraine121 / 129
[110]R
[101]R
[011]R
[101]R
[011]R
[110]R
a1
a2a
3
A3
A1
A2
[100]H
[010]H
[110]H
[110]H
[010]H
[100]H
[210]H
[120]H
[110]H
[110]H
[210]H
[120]H
z = 0
z = 1/3
z = 2/3
6/m2/m2/m
32/m
Symmetry difference between hP and hR types of lattice
Massimo Nespolo, Université de Lorraine122 / 129
Conjugate subgroupsNormal subgroup
7 holohedries → 7 lattice systems but 6 crystal families
Cubic
Tetragonalhexagonal
rhombohedral
orthorhombic
monoclinic
triclinicHexagonal crystal family: 2 free
parameters (a,c or a,a)
Poi
nt g
roup
ord
er
Massimo Nespolo, Université de Lorraine123 / 129
aPmP mS tP tI
oP oI oS oF
hP hRcP cI cF
1 triclinic 2/m monoclinic 4/m 2/m 2/m(4/mmm)
tetragonal
2/m 2/m 2/m (mmm) orthorhombic
6/m 2/m 2/m(6/mmm) hexagonal rhombohedral
cubic3 2/m(3m)
4/m 3 2/m(m3m)
Lattice systems: classification based on the symmetry of the lattices
Massimo Nespolo, Université de Lorraine124 / 129
Crystal systems:morphological (macroscopic) and physical symmetry
Massimo Nespolo, Université de Lorraine125 / 129
A1
1 or 1
A2
2 or m or 2/m3A
2
Three twofold elements
A4
A3
A6
4A3
triclinic monoclinic
tetragonal
orthorhombic
hexagonaltrigonal
cubicA
n = n or n
Crystal systems:morphological (macroscopic) and physical symmetry
Massimo Nespolo, Université de Lorraine126 / 129
hexagonal
6 crystal families
aP
mP ( mB )
mS ( mC, mA, mI, mF )
oP
oS ( oC, oA, oB )
oI
oF
tP ( tC )
tI ( tF )
hR
hP
cP
cI
cF
no restriction on a ; b ; c, a , b , g
no restriction on a ; b ; c ; b.a = g = 90º
no restriction on a ; b ; c.a = b = g = 90º
a = b ; a = b = g = 90ºno restriction on c
a = b = ca = b = g = 90º
a = b ; a = b = 90º, g = 120º no restriction on c
conventional unit cell
a = anortic*(triclinic, asynmetric, tetartoprismac...)
m = monoclinic(clinorhombic, monosymmetric, binary,hemiprismatic, monoclinohedral, ...)
o = orthorhombic(rhombic, trimetric, terbinary, prismatic, anisometric,...)
t = tetragonal(quadratic, dimetric, monodimetric, quaternary...)
h = hexagonal(senaiey, monotrimetric...)
c = cubic(isometric, monometric, triquaternary, regular, tesseral, tessural...)
7 crystal systems(morphological symmetry)
trigonal (ternary...) (***)
triclinic
monoclinic
orthorhombic
tetragonal
hexagonal
cubic
7 lattice systems(lattice symmetry)
triclinic
monoclinic
orthorhombic
rhombohedral
cubic
tetragonal
14 types of Bravais lattices(**)
(*) 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
Crystal families, crystal systems, lattice systems and types of Bravais lattices in E3
Massimo Nespolo, Université de Lorraine127 / 129
[110] [011] [101] 110
[100] [010] [110] 100
Lattice system First symmetry direction
triclinic
monoclinic(b-unique)
orthorhombic
tetragonal
rhombohedral
hexagonal
cubic
[010]
[100] [010] [001]
[100] [010] 100[001]
[111]
[001]
[001] [100] [010] 001
[110] [110] 110
rhombohedral axes
hexagonal axes [001]
[100] [010] [110] 100
[111] [111] [111] [111] 111
[110] [110] [011][011] [101] [101]
110
[110] [120] [210] 110
No restriction on any parameter
a g = 90ºNo restriction on a; b; c; b
a b g = 90ºNo restriction on a; b; c
a = b ; a b g = 90ºNo restriction on c
a = b = ca b g = 90º
a = b; a b = 90º; g = 120º No restriction on c
a = b = ca b g
Symmetry restrictions on the parameters of the conventional cell
Second symmetry direction
Third symmetry direction
Symmetry directions of the lattices in the three-dimensional space(directions in the same box are equivalents by symmetry )
a = b; a b = 90º; g = 120º No restriction on c
Massimo Nespolo, Université de Lorraine128 / 129
Triclinic family:none
Monoclinic familya = g = 90º
Orthorhombic familya = b = g = 90º
Tetragonal familya = b
a = b = g = 90º
Hexagonal familyHexagonal axes: a = b; a = b = 90º ; g = 120º
Rhombohedral axes: a = b = c; a = b = g
Cubic familya = b = c
a = b = g = 90º
a b c a b g 90°
a b c; a = g = 90°; b 90°
a b c; a = b = g = 90°
a = b c; a = b = g = 90°
a = b c; a = b = 90°; g = 120°a = b = c; a = b = g 90°
a = b = c; a = b = g = 90°
For the crystal structure, too restrictive
For the crystal lattice (geometric concept), insufficient
Restrictions incorrectly given in many textbooks
Symmetry restrictions on cell parameters (conventional cell)
Massimo Nespolo, Université de Lorraine129 / 129
Why insufficient for the crystal lattice?
A simple example: mP
if b = 90° the lattice is oP (obviously)
Let us suppose cosb = -a/2c
a
c
oB!