basics of crystallography

Isometric correspondences

Correspondences between two objects that are congruent

Two objects are congruent if: each point of one object corresponds to a point of the other, and distance between two points of one object is equal to the distance between the corresponding two points of the other object

direct

Corresponding angles have same signs

opposite

Corresponding angles have opposite signs

1

2Other types of correspondences

In conformal correspondences (or mapping), for example, only angles are preserved, not distances....

If the isometric congruence is direct, one object can be brought to coincide with the other by a movement, which can be:

(1) A translation(2) A rotation around an axis(3) A rototranslation or srew

movement (combination or rotation around one axis+translation along the axial direction)

3

(1) An inversion with respect to a point (inversion)(2) A reflection with respect to a plane (reflection)(3) The product of a rotation around one axis by an inversion with respect to a

point of the axis (rotoinversion)(4) The product of a reflection by a translation parallel to the reflection plane

(the plane is called glide plane)(5) The product of a rotation by a reflection with respect to a plane

perpendicular to the axis (rotoreflection)

For opposite congruence, the object will be said to be enantiomorphic with respect to the other. The two obects will be brought to coincidence by the following operations:

4

Symmetry operations and symmetry elements

If the isometric operations not only bring to coincidence a couple of congruent objects, but act on the entire space, and all the properties of the space are unchanged after an operation, then the operation is a symmetry operation.

Symmetry elements are points, axes or planes with respect to which a symmetry operation is performed.

5

Symmetry Elements

Translation

6

LatticesTranslational periodicity can be studied by considering the geometry of repetition of a motif (for example a molecule).

2D crystal....

Corresponding lattice with some examples of primitive cells

Corresponding lattice with some examples of multiple cells

Crystal = Lattice + Motif7

+

Lattice

Motif

8

Crystal

=

9

Lattices

Once a lattice point is chosen as origin of the lattice, any other lattice point is defined by:

,

Otherwise the cell is multiple (or centered). In the latter case, u and v are no longer restricted to integer values

u, v: positive or negative integers, : basis vectors of the cell (they define the parallelogram called unit cell)

Choice of basis vectors is arbitrary

If they define cells containing one lattice point each, the cell is primitive

10

Lattices

The same holds for linear and space lattices. For a space lattice we have:,,

u, v,w: integer (for primitive cells) or rational (for multiple cells) numbers, , : basis vectors of the cell (they define the parallelepiped, called unit cel)

Volume of the unit cell:

Lattice points are always characterized by rational numbers!

Direction specified by , , are the X, Y, Z crystallographic axes, and the angles between them are indicated by , and

11

The six lattice parameters a, b, c, , ,

The cell of the lattice

lattice

crystal

+ Motif

12

Crystallographic directions

Two lattice points define a lattice row

A lattice row defines a crystallographic direction

For example, the two lattice vectors ,, and ,, define two different lattice points, but only one direction

,

,

Direction is uniquely defined by a vector with no common factor among the indices:

For example ,, can be uniquely defined by ,,

Lattice rows and planes

13

Rotation

symmetry axes (1, 2, 3, 4, or 6) rotation of 360, 180, 120, 90, or 60 around a rotation axis

2-fold

3-fold

4-fold

6-fold

If all the properties of space remain unchanged after a rotation of 2 around an axis, this will be called a symmetry axis of order n

14

Arrangemements of symmetry-equivalent objects as an effect of a rotation

15

Symmetry restrictions for rotations due to lattice periodicity

Suppose that we have an n axis of symmetry in a crystal. Because of periodicity, we will have an n axis at each lattice point. Let T be the period vector passing through the origin of the lattice.

We will have lattice points at:,, ,

Then also must be a lattice vector. Since this is parallel to , we have:

In scalar form: 2cos 2 integer

But the equation above is verified only for n=1, 2, 3, 4, 6. A 5 axis is not allowed!

It is simple to see that a 5 axis is not allowed: it is impossible to pave a plane with pentagons.

16

Axes of rototranslation or screw axes (rotation + translation)

A rototranslational symmetry will have an order n and a translational component t, if all the properties of space are unchanged after a 2 rotation around the axis and the translation t along that axis.

17

Effects of screw axes on the surrounding space

18


19


20

Right-handed helix Left-handed helix 21


22

Restrictions imposed by periodicity to the translational components t of a screw axis

The axis lies along a row with period T. Rotational components would be n=1, 2, 3, 4, 6. If we apply the traslational component n times, we have a displacement equal to nt.

Because of the periodicity of the lattice, we must have with integer p.

Therefore

Example: for a screw axis of order 4, the allowed translational components are(0/4)T, (1/4)T, (2/4)T, (3/4)T, (4/4)T, (5/4)T

- p can be restricted to 0 - Also, for p=0 we have the simple n-fold axis with no translation.

Therefore, we will have only (1/4)T, (2/4)T, (3/4)T 23

Arrangemements of symmetry-equivalent objects as an effect of a screw axis

24

Arrangemements of symmetry-equivalent objects as an effect of a screw axis

25

Axes of inversion

An axis of inversion of order n is present when all the properties are unchanged after performin the product of a 2 rotation around an axis followed by inversion to a point located on the same axis. Symbol is

Simplest case: axis of inversion of order 1, indicated as 1:No rotation, only inversion with respect to a center

Enantiomorph:Circle with a comma inside

26

Simplest case: axis of inversion of order 2, indicated as 2rotation by followed by inversion

This is equivalent to a reflection plane perpendicular to the 2axis, which is indicated by m

Effect of a 3 axis:This is equivalent to the product of a threefold rotation by an inversion: 3 31

27

The 4 axis is also a 2 axis

The 6 axis is equivalent to the product of a threefold rotation by a reflection to a plane perpendicular to it:

6 3 m

28

Symmetry elements: mirror plane and inversion center

The handedness is changed.

29

Axes of rotoreflection

An axis of rotoreflection of order n is present when all the properties are unchanged after performin the product of a 2 rotation around an axis followed by reflection with respect to a plane normal to it. Symbol is

An example is the axis 3 3 6

In reality, the effect of these axes is the same of that of the inversion axes:

1 2 2 1

3 6

4 46 3 30

Arrangemements of symmetry-equivalent objects as an effect of various symmetry operations

31

Glide reflection (mirror plane + translation)

Glide Plane

A glide plane operator is present if the properties of the half space on one side of the plane are identical to those of the other half-space after the product of a reflection with respect to that plane by a translation parallel to the plane.

32

Graphical Symbols for symmetry elements elements

34

Crystallographic planes

Three lattice points define a crystallographic plane

Suppose the plane interesects the crystallographic axes X, Y, Z at the lattice points (p, 0, 0), (0, q, 0) and (0, 0, r). Let m be the minimum common multiple of p, q, r.

Equation of the plane is:

1

Introducing the fractional coordinates:

,

,

The equation of the plane becomes:

1

(p, 0, 0)(0, q, 0)

(0, 0, r)

a b

c

35

And the intercepts of this plane on the axes are

;

;

(

, 0, 0)

(0,

, 0)

(0, 0,

)

We now multiply both sides by m (the least common multiple):

We then define:

;

;

The equation of the plane becomes:

36

(

, 0, 0)

(0,

, 0)

(0, 0,

)

The equation of the plane below is:

1.1

Then the equation of the plane below,

with intercepts

, 00, 0,

, 0, 0,0,

is:

11.2

(0, 0,

)

(0,

, 0)

(

, 0, 0)

Eq. (1.1) refers then to a plane, the distance of which, from the origin, is m times that of plane (1.2)

37

The plane closest to the origin is:

2 3 6 1

Now let us consider that m can vary from - to + . Then the expression above defines a set of identical and equally spaced crystallographic planes. The three indexes h, k, l define the family and are its Miller indices. The plane closest to the origin cuts the axes at 1/h, 1/k, 1/l.

Let us make an example with numbers. See figure of the right:Intercepts are (3,0,0), (0,2,0) and (0,0,1).

Least common multiple is 6, the equation of the plane is:

6

Therefore: 2 3 6 6

The set of lattice planes is therefore (236)

38

Other examples

Crystallogrpahic planes parallel to one of the axes X, Y, Z are defined by(0kl), (h0l), (hk0)

Crystallogrpahic planes parallel to faces A, B, C are defined by(h00), (0k0), (00l)

39

Another numerical example for a plane

9

6

15 1

Least common multiple is 90. Then: 10 15 6 90

However, the first plane with integer intersections on the three axes will be the 30th. This is because the least common multiple of 10, 15 and 6 is 30.

3

2

5 1

Indeed, if we divide p, q, and r in ** by their common integer factor (3), we have:

From which we get: 10+15+6=30

Conclusion: a family of crystallographic planes is uniquely defined by three indices h, k and l having the largest common integer factor equal to unity

40

Family of planes is (10 15 6)

Miller Indices

Rules for determining Miller Indices:

1. Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.2. Divide by any common integer factor (if bigger than 1)3. Take the reciprocals4. Clear fractions

An example of the (111) plane (h=1, k=1, l=1) is shown on the right.

41

Another example:

42

Crystal lattice planes

43

44

Planes with different Miller indices in cubic crystals

Point groups and symmetry classes

In a crystal, more symmetry axes, proper and/or improper, with or without translational componenets, might coexist

Let us consider only combinations of operators that do not imply translations. These are called point groups (operators form a group and leave one point fixed).

The number of crystallographic point groups for three-dimensional crystals is 32.

Proper axis Improper axis Proper and improper axis

1 1 1 1 1

2 2 2 2 2

3 3 31 3 3 3

4 4 4 4 4

6 6 3 6 6 6

5 + 5 +3 =13

Single-axis crystallographic point groups

45

Point groups in which more than one symmetry axis pass by a common point

If P is congruent with Q and Q is congruent with R, then P is congruent with R and there must be a third l3 axis bringing P into R.

Only combinations of axes allowed: n22 (n=2, 3, 4, 5, 6), 233, 432, 532

In crystals: 222, 322, 422, 622, 233, 432

Arrangement of proper symmetry axes for six point groups

46

Crystallographic point groups with more than one axis

Crystallographic point groups with more than one axis, each axis being proper and improper at the same time

47

Molecular examples of some point groups

48

The seven crystal systems

The crystal periodicity is only compatible with rotation or inversion axes of order 1, 2, 3, 4, 6

The presence of these axes will impose restrictions on the geometry of the lattice

It is convenient therefore to group together the point groups with common features into systems, such that crystals belonging to these systems can be described by unit cells of the same type.

49

These are known as crystals systems

The 32 point groups and their grouping into crystal sytems

Laue classes: classes including point groups differing from each other only by the presenc of an inversion center.

50

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

51

The symmetry elements of the lattice are contained in the lattice point group

+ Motif.

52

As a reminder....This is the lattice

=

53

....This is the crystal.

The underlying lattice in general has a higher symmetry than that of the crystal!

Neumanns principle

The symmetry elements of any physical property must include the symmetry elements of the crystal point group

Some physical experiments, for example diffraction, show the symmetry one would obtain by adding one inversion center. Therefore they reveal the Laue class of the crystal

54

1. Triclinic system

Point groups (i.e. classes) 1 and 1

No symmetry axes, therefore no constraint axes for unit cell.

Ratios a:b:c and angles ,, can assume any value

Lattice point group: 1

55

2. Monoclinic system

Point groups (i.e. classes) 2, m and 2 m

These groups have a n=2 axis.

We assume that this coincides with b axis

a and c can be chosen on the plane normal to b

Ratios a:b:c unrestricted

Angles = =90 and unrestricted

56Lattice point group: 2 m

3. Orthorombic system

Point groups (i.e. classes) 222, mm2 and mmm

These groups have have three mutually orthogonal twofold rotation or inversion axes

We assume these as reference axes

Ratios a:b:c unrestricted

Angles = = =90

57

Lattice point group: mmm

4. Tetragonal system

Point groups (i.e. classes): 4, 4, 4 m , 422, 4mm, 42m, 4 mmm

These groups have only one fourfold axis.

The c axis is chosen for the direction of this fourfold axis

a and b axes are symmetry equivalent, on the plane normal to c

Ratios a:b:c = 1:1:c

Angles = = =90

58

Lattice point group: 4 mmm

5, 5. Trigonal and hexagonal systems

Point groups (i.e. classes):3, 3,32, 3m, 3m

6, 6,6 m , 622, 6mm, 62m, 6 mmm

These groups have only one threefold or sixfold axis.

The c axis is chosen for the direction of the threefold or sixfold axis

a and b axes are symmetry equivalent, on the plane normal to c

Ratios a:b:c = 1:1:c

Angles = =90, =120

59Lattice point group:6 mmm

7. Cubic system

Point groups (i.e. classes):

23,m3,43m,m3m

These groups have four threefold axes, distruted as the diagonals of the cube.

a, b and c axes are chosen as coinciding with the cube edges

Ratios a:b:c = 1:1:1

Angles = = =90

60

Lattice point group: m3m

basics of crystallography

Documents