# FUNDAMENTALS OF CRYSTALLOGRAPHY GIACOVAZZO PDF

1 Symmetry in crystals. 1. Carmelo Giacovazzo. The crystalline State and isometric. Operations. 1. Symmetry elements. 3. Axes of rotational symmetry. 3. Axes of. Download as PDF, TXT or read online from Scribd Library of Congress Cataloging in Publication Data Fundamentals of crystallography/C. Giacovazzo. Get this from a library! Fundamentals of crystallography. [Carmelo Giacovazzo; International Union of Crystallography.;] -- Crystallography is a.

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attending the School, the first 12 pages of Fundamental of Crystallography, by Oxford resourceone.info 1 Symmetry in crystals l. Carmelo Giacovazzo. The crystalline state and isometric operations. 1. Symmetry elements. 3. Axes of rotational symmetry. Carmelo Giacovazzo, Hugo Luis Monaco, Gilberto Artioli, Davide Viterbo, Marco Rigorous description of the fundamentals of crystallography; Guide to.

If all the properties of the space remain unchanged after a given operation has been carried out, the operation will be a symmetry operation. Symmetry elements are points, axes, or planes with respect to. In the following these elements will be considered in more detail, while the description of translation operators will be treated in subsequent sections.

Axes of rotational symmetry If all the properties of the space remain unchanged after a rotation of 2nIn around an axis, this will be called a symmetry axis of order n ; its written symbol is n.

We will be mainly interested cf. Axis 1 is trivial, since, after a rotation of " around whatever direction the space properties will always remain the same. The graphic symbols for the 2, 3, 4, 6 axes called two-, three-, four-, sixfold axes are shown in Table 1. In the first column of Fig. There is no graphic symbol for the 1 axis. Note that a 4 axis is at the same time a 2 axis, and a 6 axis is at the same time a 2 and a 3 axis. Carmelo Giacovazzo Table 1. Graphical symbols for symmetry elements: a axes normal to the pfane of projection; b axes 2 and 2, ,parallel to the plane of projection; c axes parallel or inclined to the plane of projection; d symmetry pfanes normar to the plane of projection; e symmetry planes parallel to the plane of projection Fig.

Arrangements of symmetry-equivalent objects as an effect of rotation, inversion, and screw axes. Symmetry in crystals Axes of rototranslation or screw axes A rototranslational symmetry axis will have an order n and a translational component t, if all the properties of the space remain unchanged after a 2nln rotation around the axis and the translation by t along the axis. Axes of inversion An inversion axis of order n is present when all the properties of the space remain unchanged after performing the product of a 2nln rotation around the axis by an inversion with respect to a point located on the same axis.

The written symbol is fi read 'minus n' or 'bar n'. As we shall see on p. According to international notation, if an object is represented by a circle, its enantiomorph is depicted by a circle with a comma inside.

When the two enantiomorphous objects fall one on top of the other in the projection plane of the picture, they are represented by a single circle divided into two halves, one of which contains a comma.

We may note that: 1 the direction of the i axis is irrelevant, since the operation coincides with an inversion with respect to a point; 2 the 2 axis is equivalent to a reflection plane perpendicular to it; the properties of the half-space on one side of the plane are identical to those of the other half-space after the reflection operation.

The written symbol of this plane is m; 3 the 3 axis is equivalent to the product of a threefold rotation by an inversion: i. Axes of rotoreflection A rotoreflection axis of order n is present when all the properties of the space do not change after performing the product of a 2nln rotation around an axis by a reflection with respect to a plane normal to it. The written symbol of this axis is fi. The effects on the space of the 1, 2, 3, 4, 6 axes coincide with those caused by an inversion axis generally of a different order.

From now on we will no longer consider the ii axes but their equivalent inversion axes. Carmelo Giacovazzo Reflection planes with translational component glide planes 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 the plane by a translation parallel to the plane.

Symmetry operations relating objects referred by direct congruence are called proper we will also refer to proper symmetry axes while those relating objects referred by opposite congruence are called improper we will also refer to improper axes. Lattices Translational periodicity in crystals can be conveniently studied by considering the geometry of the repetion rather than the properties of the motif which is repeated.

If the motif is periodically repeated at intervals a, b, and c along three non-coplanar directions, the repetition geometry can be fully described by a periodic sequence of points, separated by intervals a, b, c along the same three directions.

This collection of points will be called a lattice. We will speak of line, plane, and space lattices, depending on whether the periodicity is observed in one direction, in a plane, or in a three-dimensional space. An example is illustrated in Fig. If we replace the molecule with a point positioned at its centre of gravity, we obtain the lattice of Fig.

Note that, if instead of placing the lattice point at the centre of gravity, we locate it on the oxygen atom or on any other point of the motif, the lattice does not change.

Therefore the position of the lattice with respect to the motif is completely arbitrary. If any lattice point is chosen as the origin of the lattice, the position of any other point in Fig. The vectors a and b define a parallelogram which is called the unit cell: a and b are the basis vectors of the cell.

The choice of the vectors a and b is rather arbitrary. In Fig. Nevertheless we are allowed to choose different types of unit cells, such as those shown in Fig. We will indicate by a. As p varies over all integer values. When along the same axis a proper axis and an inversion axis are simultaneously present. The set of crystals having the same point group is called crystal class and its symbol is that of the point group.

Classes coinciding with other classes already quoted in the table are enclosed in brackets. Often point group and crystal class are used as synonyms.

Let us suppose that there are two proper axes I. The I. The graphic symbols for glide planes are given in Table 1. The 13 independent combinations of this type are described in Table We will consider here only those combinations of operators which do not imply translations.

We may conclude that if one of the three symmetry operators is an inversion axis also another must be an inversion one. The only allowed combinations are n In the two tables the combinations coinciding with previously considered ones are closed within brackets.

For each combination of symmetry axes the minimum angles between axes are given. For these combinations the smallest angles between the axes are listed in Table 1. Note that the combination is also consistent with a tetrahedral symmetry and with a cubic and octahedral symmetry. Then the objects in P and in Q will be directly congruent. P and Q are therefore directly congruent and this implies the existence of another proper operator which repeats the object in P directly in R.

The Fig. Therefore the third operator relating R to P will be an inversion axis. Arrangement of equivalent objects around two intersecting symmetry axes. For each angle the types of symmetry axes are quoted in parentheses Combination of symmetry axes 2 2 2 2 3 3 2 2 2 2 3 2 cu ded 90 90 90 90 54 35 22 2 3 2 4 2 6 44'08" 2 3 15'52" 2 3 B ded 90 2 2 90 2 3 90 2 4 90 2 6 54 44'08" 2 3 45 2 4 Y ded 0 Fig.

Arrangement of proper symmetry axes for six point groups. In Table 1. Suppose now that in Fig. It is very important to understand how the symmetry of the physical properties of a crystal relates to its point group this subject is more extensively described in Chapter 9. If an even-order axis and a? In any case the symmetry of the crystal will belong to one of them.

Crystallographic point groups with more than one axis. The conclusions reached so far do not exclude the possibility of crystallizing molecules with a molecular symmetry different from that of the 32 point groups for instance with a 5 axis.

## Fundamentals of crystallography

To help the reader. In Tables 1. Symmetry in crystals Table 1. It may be noted that crystals with inversion symmetry operators have an equal number of 'left' and 'right' moieties.

If two of the three axes are symmetry equivalent. On the other hand. Of basic relevance to this is a postulate. Haiiy Miller indices can be used as form symbols. This is in general a three-axis ellipsoid: The origin within the crystal is usually chosen so that faces hkl and hit are parallel faces an opposite sides of the crystal.

The indices of well-developed faces on natural crystals tend to have small values of h. From a morphological point of view. Sodium chloride grows as cubic crystals from neutral aqueous solution and as octahedral from active solutions in the latter case cations and anions play a different energetic role.

Symmetry in crystals 2. We shall now see how it is possible to guess about the point group of a crystal through some of its physical properties: Steno and D. Rome' de l'Ile. The morphology of a crystal tends to conform to its point group symmetry. The morphology of different samples of the same compound can show different types of face.

Thus a pedion is a single face. The orientation of the faces is more important than their extension. The set of symmetryequivalent faces constitutes a form: Guglielmini A crystal form is named according to the number of its faces and to their nature. For example. The variation of the refractive index of the crystal with the vibration direction of a plane-polarized light wave is represented by the optical indicatrix see p. Such faces correspond to lattice planes with a high density of lattice points per unit area.

In crystal classes belonging to tetragonal. The orientations can be represented by the set of unit vectors normal to them. In Fig. This set will tend to assume the point-group symmetry of the given crystal 1 15 b Fig. This property. But at the same temperature crystals will all have constant dihedral angles between corresponding faces J. An important extension of this law is obtained if space group symmetry see p.

Because of non-linear optical susceptibility. Ferroelectric crystals show a permanent dipole moment which can be changed by application of an electric field. In these groups not all directions are polar: The morphological analysis of a crystalline sample may be used to get some.

Thus they can only belong to one of the ten polar classes. It then follows that a polar direction can only exist in the 21 non-centrosymmetric point groups the only exception is the class. Etch figures produced on the crystal faces by chemical attack reveal the face symmetry one of the following 10 two-dimensional point groups. A point group is said to be polar if a polar direction.

A polar axis is a rational direction which is not symmetry equivalent to its opposite direction. Along this direction a permanent electric dipole may be measured. Electrical charges of opposite signs-may appear at the two hands of a polar axis of a crystal subject to compression.

The symmetry of a crystal containing only one enantiomer of an optically active molecule must belong to one of the 11 point groups which do not contain inversion axes. The number of crystallographic point groups in one dimension is 2: This phenomenon is described by a third-rank tensor. The total number of point groups in the plane is Point groups in one and two dimensions The derivation of the crystallographic point groups in a two-dimensional space is much easier than in three dimensions.

The ten polar classes are: Nevertheless when these effects are not detectable. In fact the reflection with respect to a plane is substituted by a reflection with respect to a line the same letter m will also indicate this operation. Some simple crystal forms: This is called the cubic system.

In particular this happens when the measured quantities do not depend on the atomic positions. If we assume these as reference axes. Symmetry in crystals 1 17 The Laue classes In agreement with Neumann's principle.

It is therefore convenient to group together the symmetry classes with common features in such a way that crystals belonging to these classes can be described by unit cells of the same type. The presence of the threefold axes ensures that these directions are symmetry equivalent. These classes belong to the orthorhombic system.

For the crystals with only one threefold or sixfold axis [3. The seven crystal systems If the crystal periodicity is only compatible with rotation or inversion axes of order 1. Point groups 1 and i have no symmetry axes and therefore no constraint axes for the unit cell. Point groups differing only by the presence of an inversion centre will not be differentiated by these experiments.

In turn. For the seven groups with only one fourfold axis [4. Crystals with four threefold axes [ Classes These point groups are collected together in the trigonal and hexagonal systems. Classes 1 and are said to belong to the triclinic system. These crystals belong to the tetragonal system. Crystals with symmetry 2. If we assume that this axis coincides with the b axis of the unit cell. When these groups are collected in classes they form the 11 Laue classes listed in Table 1.

Groups 2. The five plane lattices and the corresponding two-dimensional point groups. These are called Bravais lattices. This plane lattice has an oblique primitive cell. The cell is primitive and compatible with the point groups 4 and 4mm. If the row indicated by m in Fig.

A unit cell with a rhombus shape and angles of 60" and " also called hexagonal may be chosen. Each of these primitive cells defines a lattice type.

In particular we will consider as different two lattice types which can not be described by the same unit-cell type. Note that the unit cell is primitive and compatible with the point groups m and 2mm.

A centred rectangular cell can also be selected. In this section we shall describe the five possible plane lattices and fourteen possible space lattices based both on primitive and non-primitive cells. This can be seen by choosing the rectangular centred cell defined by the unit vectors a' and b'. Plane lattices An oblique cell see Fig. This cell is primitive and has point group 2. There are also other types of lattices. Also the lattice illustrated in Fig.

This orthogonal cell is more convenient because a simpler coordinate system is allowed. It is worth noting that the two lattices shown in Figs. In fact a cell which is at the same time A and B. A cell which is at the same time body and face centred can always be reduced to a conventional centred cell.

C P P Point group of the net 2 2mm 4mm 6mm Lattice parameters a. The conventional types of unit cell Symbol Type Positions of additional lattice points Number of lattice points per cell 1 2 2 2 2 2 4 3 Fig. Reduction of an I. It is worth noting that the positions of the additional lattice points in Table 1. Let us now examine the different types of three-dimensional lattices grouped in the appropriate crystal systems.

Their fairly limited number can be explained by the following or similar observations: A cell with two centred faces must be of type F. An F cell with axes a. Monoclinic lattices The conventional monoclinic cell has the twofold axis parallel to b. P and I. We will now show that there is a lattice with a C monoclinic cell which is not amenable to a lattice having a P monoclinic cell. A C cell is always amenable to another tetragonal P cell. Therefore a lattice with a B-type monoclinic cell can always be reduced to a lattice with a P monoclinic cell.

A B-centred monoclinic cell with unit vectors a. The latter is then amenable to a tetragonal I cell. In fact. Monoclinic lattices: It can be easily verified that because of the fourfold symmetry an A cell will always be at the same time a B cell and therefore an F cell. Thus only two different tetragonal lattices. With arguments similar to those used for monoclinic lattices.

Therefore a lattice with an I monoclinic cell may always be described by an A monoclinic cell. It can then be concluded that there are two distinct monoclinic lattices. Since c' lies on the a. An I cell with axes a. Rhombohedra1 lattice.

Centred cells are easily amenable to the conventional P trigonal cell. Three further triple hexagonal cells. Hexagonal lattices In the conventional hexagonal cell the sixfold axis is parallel to c. Six triple rhombohedral cells with basis vectors a. Because of the presence of a threefold axis some lattices can exist which may be described via a P cell of rhombohedral shape.

P is the only type of hexagonal Bravais lattice.. There are three cubic lattices.

Symmetry in crystals 1 21 cell is also an F cell. Trigonal lattices As for the hexagonal cell. CH defined according to[61 These hexagonal cells are said to be in obverse setting. The basis of the. Such lattices may also be described by three triple hexagonal cells with basis vectors UH. The hexagonal cells in obverse setting have centring points see again Fig.

C for a different type of cell. In the first case the nodes lying on the different planes normal to the threefold axis will lie exactly one on top of the other. If the third axis say d on the a.

In conclusion. Since a decrease or increase of the three coordinates by the same amount j does not change the point P.

## Fundamentals of crystallography

In conclusion the direction [mnw] may be represented in the new notation as [uviw]. The family of planes hkl see Fig. A detailed description of the metric properties of crystal lattices will be given in Chapter 2.

From the same figure it can be seen that the negative side of d is divided in h k parts. Each triple rhomobohedral cell will have centring points at O. There are seven three-dimensional lattice point groups. Following pp. A last remark concerns the point symmetry of a lattice. On the contrary. Intersections of the set of crystallographic planes h k l with the three. For instance 1 2 -3 5. The four-index symbol is useful to display the symmetry.

The 14 Bravais lattices are illustrated in Fig. If we introduce the third axis d in the plane. CH by choosing: In two dmensions four holohedries exist: The Schoenflies approach was most practical and is described briefly in the following. As a consequence of the presence of symmetry elements. By combining the 32 point groups with the 14 Bravais lattices i. They were first derived at the end of the last century by the mathematicians Fedorov and Schoenflies and are listed in Table 1.

In Fedorov's mathematical treatment each space group is represented by a set of three equations: The total number of crystallographic space groups is Appendix l. On pp. The others may be obtained by introducing a further variation: The 14 three-dimensional Bravais lattices.

## Fundamentals of Crystallography_Giacovazzo 2000

We will call the smallest part of the unit cell which will generate the whole cell when applying to it the symmetry. Space groups and enantiomorphous pairs that are uniquely determinable from the symmetry of the diffraction pattern and from systematic absences see p.

The asymmetric unit is not usually uniquely defined and can be chosen with some degree of freedom. It is nevertheless obvious that when rotation or inversion axes are present.

C2 Pm. The three-dimensional space groups arranged by crystal systems and point groups. Symmetry in crystals 1 25 According to the international Hermann-Mauguin notation. For monoclinic groups: The space groups include 11 enantiomorphous pairs: P4J2 P P41 P Biological molecules are enantiomorphous and will then crystallize in space groups with no inversion centres or mirror planes.

Only two space groups exist: PI and PI. Because of the tetragonal symmetry. For trigonal and hexagonal groups: For triclinic groups: Thus Pca2. Two settings are used: P P For cubic groups: P6i P The point group to which the space group belongs is easily obtained from the space-group symbol by omitting the lattice symbol and by replacing.

Such a set is organized according to the following rules: The combination of the Bravais lattices with symmetry elements with no translational components yields the 73 so-called symmorphic space groups. For tetragonal groups: Examples are: For orthorhombic groups: We note that: While in the short symbols symmetry planes are suppressed as much as possible.

In order of decreasing frequency we have: The frequency of the different space groups is not uniform. The standard compilation of the plane and of the three-dimensional space groups is contained in volume A of the International Tables for Crystallography.

Two types of space group diagrams as orthogonal projections along a cell axis are given: At the second line: At the first line: Indeed most of the inorganic compounds considered by Mighell and Rodgers crystallize in space groups with orthorhombic or higher symmetry.

Organic compounds tend to crystallize in the space groups that permit close packing of triaxial ellipsoids. For instance.. The same results cannot be applied to inorganic compounds. Mighell and Rodgers [9] examined 21 organic compounds of known crystal structure. Close to the graphical symbols of a symmetry plane or axis parallel to the projection plane the 'height' h as a fraction of the shortest lattice translation normal to the projection plane is printed.

For each space groups the Tables include see Figs Short and full symbols differ only for the monoclinic space groups and for space groups with point group mmm..

In Figs. The block positions called also Wyckoff positions contains the general position a set of symmetrically equivalent points. Wyckoff letter a code scheme starting with a at the bottom position and continuing upwards in alphabetical order.

The condition may be general it is obeyed irrespective of which Wyckoff positions are occupied by atoms see Chapter 3. For example..

For non-centrosymmetric space groups the origin is chosen at a point of highest symmetry e. Information about maximal subgroups and minimal supergroups see Appendix l.

Information is given about: The symbol adoptedr9]for describing the site symmetry displays the same sequence of symmetry directions as the space group symbol. Some or all the symmetry elements are traced as indicated in the space-group symbol. This is often a trivial task. To each Wyckoff position a reflection condition. The origin of the cell for centrosymmetric space groups is usually chosen on an inversion centre. The first three block columns give information about multiplicity number of equivalent points per unit cell.

E selected to generate all symmetrical equivalent points described in block 'Positions'. A dot marks those directions which do not contribute any element to the site symmetry. In order to obtain space group diagrams the reader should perform the following operations: E is given. Three orthogonal projections for each space group are listed: Symmetry of special projections. A second description is given if points of high site symmetry not coincident with the inversion centre occur.

Symmetry in crystals 1 27 4. Asymmetric unit Symmetry operations Fig. Representation of the group Pbcn as inlnternational Tables for Crystallography.

Positions Mulliplicity. Wyckoff letter. Site Spmetry 1 0 0. Some space group diagrams are collected in Fig. Two simple crystal structures are shown in Figs 1. Representation of the group P4. In the symbol g stays for a glide plane. Once conveniently located. If P'. There are two line groups: New symmetry elements are then placed in the unit cell so producing the second type of diagram. The plane and line groups There are 17 plane groups.

Any space group in projection will conform to one of these plane groups. A periodic decoration of the plane according to the 17 plane groups is shown in Fig.

The first type of diagram is so obtained. O hhl: Positions Mulliplicily. Wyckofl kllcr. As we shall see in Chapter 2 its elements may be 0. T is the matrix of the translational component of the operation. Some space group diagrams.

A list of all the rotation matrices needed to conventionally describe the space groups are given in Appendix 1. The R matrix is the rotational component proper or improper of the symmetry operation. Acta Cryst. Gaetani Manfredotti. Guastini Viterbo Glide planes are emphasized by the shading. TI to a point at the end of a vector r. If we then apply to r' the symmetry operator C2.

Nov Loreto and M. Because of 1. In the P6. When we take them into account. If only two generators are sufficient. Any transformation which will keep the distances unchanged will be called an isometry or. TI which will bring r ' back to r. For this reason we will say that the 6. A The isometric transformations It is convenient to consider a Cartesian basis el. All symmetry operators of a group can be generated from at most three generators.

Symmetry in crystals Since the symmetry operators form a mathematical group. Once we have defined the R and T matrices corresponding to an anti-clockwise rototranslation of 60" around z. The theory of symmetry groups will be outlined in Appendix 1.

In fact the set of operations which will transfer a point r in a given cell into its equivalent points in any cell are: So far we have deliberately excluded from our considerations the translation operations defined by the Bravais lattice type.

When all the operators of the group can be generated from only one operator indicated as the generator of the group we will say that the group is cyclic.

Each of the 12 different operators of the group may be obtained as C'. The list of the generators of all point groups is given in Appendix 1.

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It will be a linear transformation. Since the determinant of the product of two matrices is equal to the product of the two determinants.

We will refer to direct or opposite movements and to direct or opposite congruence relating an object and its transform.

A movement. Since det R. The two independent equations will define a line. Therefore one of the three equations represented by 1. R indicates the transpose of the matrix R and I is the identity matrix. If in eqn 1. When the translation is parallel to the rotation axis the movement will be indicated as rototranslation. Another example is the anti-clockwise rotation around the z axis of an angle 8.

An example of direct movement is the transformation undergone by the points of a rigid body when it is moved. We note that X and X' are the matrices of the components of the vectors r and r' respectively. Opposite movements An opposite movement can be obtained from a direct one by changing the sign to one or three rows of the R matrix.

Let us suppose that an isometric transformation relates the three non-collinear points A. If 1 is perpendicular to a. We will now show that any direct movement can be carried out by means of a translation or a rotation or a rototranslation.

Changing the signs of all three rows of the R matrix implies the substitution of the vector x'. We may conclude that each direct movement. Monaco, G. Artioli, D. Viterbo, G. Ferraris, G. Gilli, G. Zanotti, and M.

Catti This book offers a comprehensive account of the wide range of crystallography in many branches of science. The fundamentals, the most frequently used proce- dures and experimental techniques are all described in a detailed way.

A number of appendices are devoted to more specialist aspects.A centred rectangular cell can also be selected. A number of appendices are devoted to more specialist aspects.

For each angle the types of symmetry axes are quoted in parentheses Combination of symmetry axes 2 2 2 2 3 3 2 2 2 2 3 2 cu ded 90 90 90 90 54 35 22 2 3 2 4 2 6 44'08" 2 3 15'52" 2 3 B ded 90 2 2 90 2 3 90 2 4 90 2 6 54 44'08" 2 3 45 2 4 Y ded 0 Fig. In conclusion. C Scattering of X-rays by gases, liquids, and amorphous solids 3. The symmetry of a crystal containing only one enantiomer of an optically active molecule must belong to one of the 11 point groups which do not contain inversion axes.

There are three cubic lattices. An example in two dimensions is given in Fig. Symmetry in crystals 1 9 indicate that the planes of the family divide a in h parts.

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