crystalline structure, planes, directions and point ... and...crystalline structure, planes,...

Crystalline Structure, Planes, Directions and Point Symmetry Operations

1. The 07 crystal Systems2. The 14 Bravais Lattices3. The 32 Crystallographic Point Groups 4. The 230 Space Groups5. Wyckoff Positions6. Stereographic Projection

C.S. Barrett and T.B. Massalski, The Structure of Metals, 3rd Edition, Elsevier, 1980. cap.1

B.D. Cullity and S.R. Stock, Elements of X-Ray Diffraction, 3rd Edition, Prentice Hall, 2001.cap.2

Y. Waseda , E. Matsubara, K Shinoda, X-Ray Diffraction CrystallographyIntroduction, Examples and Solved Problems. Cap.6

G. Burns and A.M. Glazer, Space Groups for Solid State Scientists, 2nd Edition, Academic Press, 1990. caps 2, 3, 4

Crystalline Structure, Planes, Directions and Point Symmetry Operations

• 7 Crystal Systems– Based on the number of self consistent combinations of rotation axis in 3D -

defines basic P units cells

• 14 Bravais Lattices – Arrangements of lattice points consistent with the above combinations of

rotation axes i.e. some unit cells can also be F,I or C centred

• 32 Point Groups– Combinations of symmetry elements acting through a point - each belongs

to a crystal group

• 11 Laue Groups – As above but where the centre of symmetry cannot be identified

• 230 Space Groups– Point group symmetry plus translational symmetry e.g. screw axes and

glide planes

1. The 07 crystal Systems

Lattice:

A lattice is an infinite array of points in space, in which each point has identical surroundings to all others.

The simplest way of generating such an array is by invoking the property of translational invariance, which is the most fundamental feature of normal crystals.

This property can be conveniently expressed by the primitive translation vectortn = n1a + n2b + n3c

where ni is any integer (negative, zero or positive) and a, b, c are vectorschosen so that a and b are not collinear and c is not coplanar with the ab-plane:a, b and c start from the same origin and serve as axes of reference.

A crystal may be represented as a lattice, that is, a three-dimensional array of points (lattice points), each of which has identical surroundings.


Points of the space lattice of a crystal are positions occupied by: single atom (in many common metals and elements)group or basis of atoms

Perfect crystal: each group is identical in composition and orientation with every other.

Fundamental translation vectors serve as references axes for the crystal axis.

Frequently “lattice” is loosely used as synonym for crystal structure (incorrect).

Lattice should be used to refer to the scheme of repetition in the crystal, and not the actual arrangement of atoms in a crystal.

Atomic arrangement is the structure of the crystal.

Only 14 lattices is possible but a much larger number of different structure can be found.


If we choose a particular point as an origin (marked O) and consider the others with respect to it, then Eqn [2.1] gives us the vectors to these points.

tn = 2a + 3b + 5c

Figure 2.2 A lattice in projection


Figure 2.3 Some examples of primitive (P) and non-primitive (NP) unit cells.

Completing the parallelepiped formed by the vectors a, b and c, we enclose a volume in space that, when repeated fills all space and generates the lattice.

primitive (P) unit cell : contains only one lattice point

non-primitive (NP) or multiply primitive unit cells: contains more than one lattice point

Note that the volume (area in projection) of each primitive unit cell is the same irrespective of the choice of axes used.


Convention:

For unit cell, we use the customary right-handed set of axes, labeled a, b and c with interaxial angles α, β and γ.

Conventional labels for axes and angles.


In defining a lattice with three non-coplanar lattice vectors, units cells of variousshapes can result, depending on the length and orientation of the vectors.

For example, if the vectors a , b , c are of equal length and at right angles to one another, or α = β = γ and , the unit cell is cubic.

Giving special values to the axial lengths and angles, produces unit cells of various shapes and therefore various kinds of point lattices, since the points of the lattice are located at the primitive unit cell corners.

Thus, only seven different kinds of cells are necessary to include all the possible point lattices.

These correspond to the seven crystal systems into which all crystals can be classified.


Study: Draw a cubic unit cell with lattice points on all corners. Add a lattice point to the center of each unit-cell face. Show, by shifting theorigin of the unit cell, that there are four lattice points in such a cell.

2. The 14 Bravais Lattices

Seven different point lattices can be obtained simply by putting points at the corners of the unit cells of the seven crystal systems.

However, there are other arrangements of points which fulfill the requirements of a point lattice, namely, that each lattice point have identical surroundings.

Bravais (1848) demonstrated that there are fourteen possible point lattices and no more!

Bravais lattice and point lattice are synonymous.

For example, if a point is placed at the center of each cell of a cubic point lattice, the new array of points also forms a point lattice.

Similarly, another point lattice can be based on a cubic unit cell having lattice points at each corner and in the center of each face.

Some unit cells are simple, or primitive, cells (symbol P or R), and some are non-primitive cells (any other symbol):

A lattice point in the interior of a cell “belongs” to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight.

F - face-centered cell

I - body-centered cell

A, B, and C: face-centered cells, centered on one pair of opposite faces A, B, or C.

(The A face is the face defined by the b and c axes, etc.)

R - used especially for the rhombohedral system.



Is it possible to have more than 14 Bravais lattices?

Why not, for example, a base-centered tetragonal lattice?

The full lines delineate a base-centered lattice, centered onthe C face, but the same array of lattice points can be referred to the simple tetragonal cell shown by dashed lines, so that the base-centered arrangement of points is not a new lattice.

However, the base-centered cell is a perfectly good unit cell and may be used rather than the simple cell.

The lattice points in a nonprimitive unit cell can be extended through space byrepeated applications of the unit-cell vectors a, b, c just like those of a primitive cell.


The lattice points associated with a unit cell can be translated one by one or as a group.

In either case, equivalent lattice points in adjacent unit cells are separated by one of the vectors a, b, c, wherever these points happen to be located in the cell (Fig. 12).

2. The 14 Bravais Lattices – indices de Miller

There are in all four macroscopic symmetry operations or elements:reflectionrotationinversion rotation-inversion

n -fold rotational symmetry about an axis : 360°/n brings it into self-coincidence.

Cube: 4-fold axis normal to each face3-fold axis along each body diagonal2-fold axes joining the center of opposite edges.

3. The 32 Crystallographic Point Groups: Point Symmetry Operations

Points A1, A2 , A3, and A4 are related by the four-fold rotation axis (Fig. 8(b)) while points A1 and A4 are also related by the two-fold axis inclined with respect to the four-fold axis.

In general, rotation axes may be 1-, 2-, 3-, 4- or 6-fold. Multiple 1- fold axes are present in all objects, and these are normally not shown while a 5-fold axis or one of higher degree than 6 are impossible, in the sense that unit cells having such symmetry cannot be made to fill space without leaving gaps.


Study: demonstrate that 5 and 7-fold rotations cannot occur in a conventional lattice (pg33 Burns).

A body has an inversion center if corresponding points of the body are locatedat equal distances from the center on a line drawn through the center.

A body having an inversion center will come into coincidence with itself if every point in the body is inverted, or “reflected” in the inversion center.

A cube has inversion centers at the intersection of body diagonals [Fig. 8(c)].

Finally, a body may have a rotation inversion axis, either 1-, 2-, 3-, 4- or 6-fold.

If it has an n -fold rotation-inversion axis, it can be brought into coincidence with itself by a rotation of 360°/n about the axis followed by an inversion in a center lying on the axis.



Figure 1.7 The point symmetry proper (pure) rotation operations and the symbols used to designate them. Hands are used to show the results of the operations. The symbols used to designate the symmetry operations are in the International (Schoenflies) notation.

Consider next all of the positions and orientations an object or motif must takedue to the operation of various symmetry elements (Fig. 9).

The motif must appear even more frequently if, for example as in Fig. 9(g) and (h), two symmetry operators operate through the same point.

The combined operation of a two-fold axis lying within a mirror plane “produces” a second mirror plane, perpendicular to the first mirror and also containing the two-fold axis i.e., horizontal in Fig. 9(g).


When a four-fold axis lies within a single mirror plane as shown in Fig. 9(h) symmetry requires a total of eight identical motifs (in various orientations) and four mirror planes to be present.

The different symmetry operations acting through a point are termed point groups.

In two-dimensions there are ten point groups which can be included in lattices.

In three-dimensions, the number of point groups increases to thirty-two: unlike in two-dimensions, inversion centers are no longer equivalent to two-fold axes, and combinations such as mirrors perpendicular to rotation axes are possible.


2D point groups


The possession of a certain minimum set of symmetry elements is a fundamental property of each crystal system, and one system is distinguished from another just as much by its symmetry elements as by the values of its axial lengths and angles. In fact, these are interdependent .For example: A 4-fold rotation axes normal to the faces of a cubic cell requires that the cell edges be equal in length and at to one another.

A tetragonal cell has only one 4-fold axis, and this symmetry requires that only two cell edges be equal, namely, the two that are at right angles to the rotation axis.

Minimum number of symmetry elements possessed by each crystal system


If a symmetry operation leaves a locus, such as a point, a line, or a planeunchanged (i.e., same atomic position), this locus is referred to as the symmetryelement.

For any operation excluding lattice translation for space group, the symmetryoperation belongs to one of four cases:

inversion (I) expressed by a change from (x; y; z) to (x; y; z);

rotation-inversion (¯n );

reflection (m) a mirror plane: expressed by a change from (x; y; z ) to (x; y; z );

rotation (n), a rotation axis; expressed by a change (360/n) about an axis.

In crystals, only one-, two-, three-, four-, and six-fold rotation axes can be accepted.


Three important aspects:

(1) Since the rotation-inversion operation of ¯1 is a rotation of 360° followed by inversion through a point on the one-fold rotation-inversion axis, it is identical to inversion (i), simply called center of symmetry or inversion center.

(2) The rotation-inversion operation of ¯2 is represented by rotation through an angle of 180° followed by inversion to take one point into an equivalent one. However, these two points are also related to one another by reflection in a plane normal to the axis, so that the rotation-inversion of ¯2 is identical to a mirror reflection (m).


(3) Successive applications of the rotation-inversion operation of ¯3 alter a point into altogether six equivalent positions.

This variation can be reproduced by combining operations with a three-fold rotation axis and inversion (I). Similarly, the rotation-inversion operation of ¯6 is also represented by combining operations with a three-fold rotation axis and a two-fold rotation-inversion axis, or mirror plane perpendicular to the axis.


These three points suggest that the rotation-inversion operations except for ¯4 result in no new operation, so that ¯1, ¯2, ¯3, and ¯6 are not included in the independent symmetry operations.

In conclusion, the independent symmetry operations for the symmetryof the three-dimensional atomic arrangement are eight:

Inversion (i), Reflection (m), Rotation (1, 2, 3, 4, 6), Rotation-inversion of ¯4.

This means that the whole periodic array observed in crystals can be covered by repeating the parallel translation (translational operation) of the structure derived from these eight kinds of symmetry element.

In other words, there are seven crystal systems for classification, which consist of 14 kinds of Bravais lattices and a crystal is classified into 32 point groups on the basis of eight kinds of symmetry element.


3. The 32 Crystallographic Point Groups

Study: organize a full description of the notations of all the point groups listed in Table 6.1. See slide 47 or Barret pg.27.

3. The 32 Crystallographic Point Groups

Until here it was shown the symmetry of the external faces of crystals which permits a classification of all crystals into 32 crystal classes, which are divided among the 7 systems.

The symmetry of each of these classes is described by macroscopic symmetry elements grouped at a point, termed a point group or a group of equivalent points that can be written in the form x, y, z; -x, -y, -z, etc. The operation of all the symmetry elements of the point group upon any one of the equivalent points will produce all the others.

Another classification of crystals specifies the total symmetry of the arrangements of atoms in a crystal!

Space group: is an array of symmetry elements in three dimensions on a space lattice. Thus space group is a group of symmetry in space (as point group is a group of symmetry at a point).

Each element of symmetry has a specific location in a unit cell as well as a specific direction with respect to the axes of the cell, and each unit cell in the crystal has an identical array of symmetry elements within it.

4. The 230 Space Groups

In the point groups the operation of a symmetry element located at the origin on a points x, y, z will always produce equivalent points that are equidistant from the origin.

In space groups it is possible to have symmetry elements in which a translation is involved, and equivalent points will consequently be at different distances from the origin.

Thus, it is necessary to consider another symmetry elements, which are possible in a space group but not possible in a point group.

Two compound symmetry operations must be included:

rotation and translation (screw axis)

reflection and translation (glide plane).

4. The 230 Space Groups: screw rotation and glide reflection

The symmetry operation “screw rotation” consists of a rotation of 360°/n wheren is 2, 3, 4, and 6 and a translation by a vector parallel to the axis.

The screw axis is expressed by nm and its operation is to translate by (m/n)times the length of a unit lattice vector along the direction of a rotation axis everyone operation with respect to n-axis of rotation.

Although the direction of rotation itself is not so important in the screw rotation, the definition is illustrated in Fig. 6.4 using a right-handed axial system as an example.

This case: rotation around a c -axis from the a -axis toward the b –axis by an angle Φ followed by a positive translation along a c -axis, called the motion of a right-handed screw.

4. The 230 Space Groups: screw rotation

All of possible screw rotation axes:

4. The 230 Space Groups: screw rotation

The compound symmetry operation of a glide reflection consists of a reflectionand a translation by the vector qg parallel to the plane of reflection.

For convenience, Fig. 6.5 shows a comparison of the operation of a glide plane with that of a mirror plane on a point lying off the planes.

4. The 230 Space Groups: glide reflection

There are five kinds of glide reflection planes in all.(pg 223 Waseda)

The atomic distribution in crystals is characterized by its periodicity in a regularthree-dimensional lattice and it is classified into 32 point groups using eight symmetry elements.

In addition, the periodicity in regular three dimensional lattice can be analyzed by the concept of symmetry in space groups.

Using: 08 kinds of symmetry elements: reflection (m)rotation (1, 2, 3, 4 and 6)inversion (i)rotation-inversion (¯4)

11 screw axes 05 glide planes

all the possible geometric arrangement of atoms in three dimensional lattice space can be classified into 230 in all, called “space groups.”



Thus, the number of geometric arrangements with periodicity is limited in three-dimensional lattice space.

Or, any crystal can be described only by one of the 230 space groups.

In addition, the real crystal structures are not evenly distributed over these 230 space groups. It is rather unevenly distributed, so that there are many space groups that do not represent any real crystal structure.

4. The 230 Space Groups: notation

The first alphabetical capital letter is to show the lattice symbol of the Bravaislattices (P, F, I, A, B, C, and R ).

Next three characters indicate symmetry elements related to the particular orientation in crystal systems as summarized in Table 6.4 .

Cmm2 (orthorhombic) C - space lattice is base-centered (C ) mm2 – inform the symmetry elements with respect to directions of [100], [010], and [001], respectively. That is, this orthorhombic space lattice shows:

mirror plane m, perpendicular to both a – and b -axes twofold rotation axis along the c- axis.

Another example, P121 /c1P - space lattice is primitive (simple) and it has the 21 screw axis parallel to b -axis and the c –glide plane which is perpendicular to the 21 screw axis.

Since the description of space groups is generally used in a simplified form as much as possible, so-called short space group symbol, for example: P121/c1 → P21/c

F 4/m‾32/m → Fm‾3m

4. The 230 Space Groups: notation

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list


How the structure of real crystal are related to:point latticescrystal systemssymmetry elements?

Atoms of a crystal are set in space in some fixed relation to the points of a Bravais lattice.

It follows from this that the atoms of a crystal will be arranged periodically in three dimensions and that this arrangement of atoms will exhibit many of the properties of a Bravais lattice, in particular many of its symmetry elements.

The features associated with each lattice point are termed the basis of the lattice,and this applies to one- and two-dimensional lattices as well as three-dimensionalcrystal structures.

4. The 230 Space Groups – crystal structure

Three different bases for a one-dimensional lattice; the vertical dashed lines mark the end of the unit cells.

The basis for lattice (b) is a single dot-dash, with the dot to the right of the dash, The basis for lattice (c) is a dash-dot dot-dash combinationThe basis for lattice (d) is a dash-dot dash-dot combination.


The symmetry in Fig. 16(c) can be represented by mirrors (solid vertical lines in the figure) or by 2-fold rotation axes perpendicular to the page.

The mirror at “O” (or the 2-fold axis at “O” in the alternate version) acts throughout the entire onedimensional space: the features at A and B appear at A’ and B’.


Study: Summarize the discussion concerning HCP structure, which that two atoms of the same kind are “associated with” each point of a Bravais lattice, found in Cullity, pg 54).

Study: Summarize Question 6.2 Waseda:A crystal structure is known to be characterized by points in an infinite three-dimensional regular array. Explain the geometry of crystals using symmetry elements. Keep in mind the viewpoint of symmetry given in a crystal structure.


Study: Explain the affirmations below for the cubic, tetragonal, hexagonal and trigonal crystal systems


5. Wickoff Positions

6. Stereographic Projection

C.S. Barrett and T.B. Massalski – Cap.1

G. Burns and A.M. Glazer -

crystalline structure, planes, directions and point ... and...crystalline structure, planes,...

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