Early in the history of modern science it was suggested that the regular external form of crystals implied an internal regularity of their constituents.
Two objects are said congruent if to each point of one object corresponds a point of the other and if the distance between two points of an object is equal to the distance between corresponding points of the other.
The corresponding angles will also be equal in absolute value. The congruence may be either direct or opposite, according to whether the corresponding angles have the same or opposite signs.
For direct congruents, the objects can be superimposed, one object being brought to coincide with the other by a movement that may be :
- a translation following a direction
- a rotation around an axis
- a rototranslation (screw movement), a combination of a translation and a rotation.
For opposite congruents, one object is said to be enantiomorphous with respect to the other (images in a mirror). The two objects may be brought to coincidence by the following operations :
- an inversion (versus a point)
- a reflection through a plane
- a rotoinversion, a combination of a rotation and an inversion
- a reflection combined with a translation parallel to the reflection plane (glide plane)
- a rotoreflection, a combination of a reflection and a rotation
A few symmetry elements (with their symbols) and their effect (symmetry operation) is given in Fig. 4.
Fig. 4. Some common symmetry elements (with their symbols) and their effect. x and o are used to represent the relative position of the objects (eg x = above the plane and o under the plane). For opposite congruents, a different color is given.
.2. Finite objects : point groups
On a finite object (eg a molecule) different symmetry elements can be combined (with the exception of translations, nonexistent in finite objects) (Fig. 5). The combination of those operators defines a point group (in the mathematical sense).
A series of rules govern the combination of the symmetry operators:
- all elements join at (at least) one point (otherwise they would propagate to infinity)
- X = 2 fold a
xis _|_ m -->> -1
- X + m -->> X _|_ m
- X in m -->> x planes m (angle between planes =180/x). E.g. 2mm, 4mm, 6mm, but 3m
- If only one X > 2 + 2 fold axis -->> 2 _|_ X (noted X2).
- 2 _|_ X -->> x 2 fold axis (angle between 2 fold axis = 180/x). E.g. 222, 422, 622, but 3m
- If more than one X > 2 -->> only possibilities 23, 432, 532
Fig. 5. Combination of symmetry elements. x and o are used to represent the relative position of the objects (eg x = abo ve the plane and o under the plane). For opposite congruents, a different color is given.
The different combinations of symmetry elements that can be applied on finite 1D, 2D, or 3D objects are summarized in Table 1. Note that planar molecules are usually considred as 2D objects.
Table 1. Point groups in 1D, 2D, or 3D objects.
- crystal = regular repetition in the three-dimensional space of an object made of atoms, ions, molecules or groups of molecules, extending over a distance corresponding to thousands of molecular dimensions.
- asymmetric unit = ind
ividual entities from which the crystal is build
- lattice = pattern formed by points representing the location of the
asymmetric unit. Formally, it is a 3D, infinite array
of points, each of which is surrounded in an identical way by its neighbours.
- unit cell = fundamental unit from which the entire crystal may be constructed by purely translational displacements (like bricks in a wall).
operations (point group) lattice translation
Bravais lat
tices :
There is only a limited number of possibilities to cover the space by a periodic assembly of regular parallelograms. In practice, a plane (2D space) can be filled by regular parallelograms in only 5 ways (Table 2 and Fig. 6 a) while there are only 14 distinct crystal lattices (Bravais lattices) in three dimensions (Table 2 and Fig. 6b). Those parallelograms (forming the unit cells) are classified into crystal systems and possess a symmetry (Table 2).
Table 2. The
crystal systems in two and three dimensions
2D
oblique p a not = b, a not = 90° 2
rectangle p a b, a = 90° 2mm
rectangle c a b, a =
90° 2mm
square p a = b, a = 90° 4mm
hexagone p a = b, a = 120° 6mm
3D
triclinic P a not= b not= c, alpha not= beta not= gamma not= 90° -1
monoclini
c P, C a not= b not= c, beta not= 90°, alpha = gamma = 90° 2/m
orthorhombic P, C, I, F a not= b not= c, alpha = beta = gamma = 90° mmm
trigonal R a not= b not= c, alpha = beta = gamma not= 90° -3m
tetragonal P, I a = b c
, alpha = beta = gamma = 90° 4/mm
m
hexagonal
a = b c, alpha = beta =
90°, gamma = 120° 6/mmm
cubic P, I, F a = b = c, alpha = beta = gamma = 90° m3m
Depending on the position of the lattice points, one finds primitive cells denoted P (each cell has one lattice point
at its origin) or non-primitive cells (additional lattice points appear at the center of the cell or on pairs of faces) : body-centered (I), face-centered (F) or side-centered (A, B, C) unit cells. In 2D, the primitive cells are denoted p and there is only non.primitive type
of cell: centered (c).
Fig. 5. Combination of symmetry elements. x and o are used to represent the relative position of the objects (eg x = abo ve the plane and o under the plane). For opposite congruents, a different color is given.
The different combinations of symmetry elements that can be applied on finite 1D, 2D, or 3D objects are summarized in Table 1. Note that planar molecules are usually considred as 2D objects.
| 1D | 1 and -1 | |||||||||||
| 2D | X | 1 | 2 | 3 | 4 | 5 | 6 | |||||
| Xm | m | 2m m | 3m | 4mm | 5m | 6mm | ||||||
| 3D | X | 1 | 2 | 3 | 4 | 5 | 6 | |||||
| -X | -1 | -2 = m | -3 = 3-1 | -4 | -5 = 5-1 | -6 | ||||||
| Xm | m | 2mm | 3m | 4mm | 5m | 6mm | ||||||
| X/m | m | 2/m | 3/m | 4/m | 5/m | 6/m | ||||||
| X2 | 2 | 222 | 32 | 422 | 52 | 622 | ||||||
| -Xm | -1m = 2/m | = 2mm | -3m | -42m | -5m | -62m | ||||||
| X/mm | 2mm | mmm | = -62m | 4/mmm | 5/mm | 6/mmm | ||||||
| + 7 groups | 23, m3, 432, -43m, m3m, 532, -53m | |||||||||||
.3. Infinite objects : lattices, unit cells, and space groups
Definitions :
- asymmetric unit = ind
ividual entities from which the crystal is build
- lattice = pattern formed by points representing the location of the
asymmetric unit. Formally, it is a 3D, infinite array
of points, each of which is surrounded in an identical way by its neighbours.
- unit cell = fundamental unit from which the entire crystal may be constructed by purely translational displacements (like bricks in a wall).
operations (point group) lattice translation
Bravais lat
tices :
There is only a limited number of possibilities to cover the space by a periodic assembly of regular parallelograms. In practice, a plane (2D space) can be filled by regular parallelograms in only 5 ways (Table 2 and Fig. 6 a) while there are only 14 distinct crystal lattices (Bravais lattices) in three dimensions (Table 2 and Fig. 6b). Those parallelograms (forming the unit cells) are classified into crystal systems and possess a symmetry (Table 2).
Table 2. The
crystal systems in two and three dimensions
2D
oblique p a not = b, a not = 90° 2
rectangle p a b, a = 90° 2mm
rectangle c a b, a =
90° 2mm
square p a = b, a = 90° 4mm
hexagone p a = b, a = 120° 6mm
3D
triclinic P a not= b not= c, alpha not= beta not= gamma not= 90° -1
monoclini
c P, C a not= b not= c, beta not= 90°, alpha = gamma = 90° 2/m
orthorhombic P, C, I, F a not= b not= c, alpha = beta = gamma = 90° mmm
trigonal R a not= b not= c, alpha = beta = gamma not= 90° -3m
tetragonal P, I a = b c
, alpha = beta = gamma = 90° 4/mm
m
hexagonal
a = b c, alpha = beta =
90°, gamma = 120° 6/mmm
cubic P, I, F a = b = c, alpha = beta = gamma = 90° m3m
Depending on the position of the lattice points, one finds primitive cells denoted P (each cell has one lattice point
at its origin) or non-primitive cells (additional lattice points appear at the center of the cell or on pairs of faces) : body-centered (I), face-centered (F) or side-centered (A, B, C) unit cells. In 2D, the primitive cells are denoted p and there is only non.primitive type
of cell: centered (c).
Space groups :
In the case of periodic, infinite systems, a translation movment is allowed. In particular, axis can become helicoidal (screw) axis and mirror planes (m) may become glide planes.
The combi
nation of all available symmetry operations (point groups plus glides and screws) with the Bravais translations leads to exactly 17 combinations (so-called space groups) in 2D objects and 230 space grousp in 3D objects. Only the 2D space groups and some common 3D space groups are discussed here. A detailed descriptions of those space groups is available in the International Tables of Crystallography that list those by symbol and number, together with symmetry operators, origins, reflection conditions, and space group projection diagrams. Note that crys
tals that contain only one enantiomer (one mirror-image of a chiral molecule) as is the case in most of natural macromolecules (eg proteins) are restricted to non-mirror and non-inversion symmetry operations. This effectively reduc
es the number of possible groups to the 65 chiral space groups.
Lattice planes and Miller indices :
The spacing of the lattice points in a crystal is an important quantitative aspect of its structure. This concept is crucial for understanding diffraction techniques.For a given lattice (a 2D example for simplicity), there are different sets of planes, and one needs to be able to label them (Fig. 7).
- Each plane can be distinguished by the distance at which in intersects the a and b axis
(1a, 1b) (1/2a, 1/3b) (-1a, 1b) (inf a, 1b)
- For si
mplicity, let's express the distances as multiples of the lenghts of the unit cell
(1, 1) (1/2, 1/3) (-1, 1) (inf, 1)
- the general 3D notation
(all planes intersect the z axis at infinity) becomes
(1, 1,inf) (1/2, 1/3, inf) (-1, 1, inf) (inf, 1, inf)
- The Miller indices are the reciprocals of the intersection distances :
(110) (230) (-110) (010)
The (hkl) familly of planes devides the a, b, and c axis into h, k, and l equal parts respectively. The smaller the value of h in (hkl), the more nearly parralel the plane is to the a-axis. When h = 0, the plane intersects th
e a-axis at inf, so the (0kl) planes a
re parallel to the a-axis. Similarly, the (h0l) planes are parallel to b and the (hk0) planes parallel to c. The Miller (hkl) notation (and the concept of reciprocal space) will prove very useful in interpreting diffraction data.
Separation of planes :
The Miller indices are very useful for expressing the separation of planes, dhkl (the separation of planes in the direct space where the lattice is defined by the a, b, c axis).In the squ
are lattice shown in Fig. 8. the separation of the (hk0) planes is given by :
1/d2hkl = (h2 + k2) / a2 , or dhkl = a / (h2 + k2)1/2
Fig. 8. Calculation of the separation of planes (hkl) in terms of the Miller indices for a square lattice.
Generalization for a cubic lattice gives : 1/d2hkl = (h2 + k2+ l2) / a2 , or dhkl = a / (h2 + k2 + l2)1/2
The corresponding expression in an orthorhonbic lattice is : 1/d2hkl = h2 / a2 + k2 / b2 + l2 / c2
Reciprocal lattice :
The crystal lattice defined previously (point 1.3) provides information on the 3D symmetry and periodicity of the internal structure of the crystal. However, it provides no information on the detailed arrangement of atoms within a unit cell. In 1921, Ewald introduced the concept of reciprocal lattice, which is mathematically constructed from the crystal (direct) lattice. The concept of reciprocity has already appeared intuitively in previous chapters : in the interference experiment of Young, the spacing of the resulting spots is inversely related to the positions of the slits; the Miller indices correspond to reciprocals of the intersection distances.
The reciprocal lattice is very useful in analyses of X-ray diffraction patterns. It is the reciprocal lattice points that provide sampling regions in the diffraction pattern of the contents of one unit cell.
The reciprocal lattice is constructed on the a*, b*, c* axis derived from the (direct) crystal lattice axis so that :
a* . b =
a* . c = b* . a = b* . c = c* . a = c* . b = 0 and
a* . a = b* . b = c* . c = 1
This suggests that a* is normal to the (b,c) plane, b* is normal to the (a,c) plane, and c* is normal to the (a,b) plane (Fig. 9). The reciprocal axis are thus the scalar product of axis from the direct lattice :
a* = [b x c ] / v, b* = [c x a] / v, c* = [a x b] / v where v is the volume of the direct lattice.
Fig. 9. Definition of reciprocal vectors based on vectors of the (direct) crystal lattice.
Fuente: http://perso.fundp.ac.be/~jwouters/DRX/diffraction.html#chap1
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