2 1 Carmelo Giacovazzo, basic crystal periodicity is modulated by periodic distortions incom. mensurated with the basic periods i e in incommensurately modulated. structures IMS It has however been shown p 171 and Appendix. 3 E that IMSs are periodic in a suitable 3 d dimensional space. 2 Some polymers only show a bi dimensional order and most fibrous. materials are ordered only along the fiber axis, 3 Some organic crystals when conveniently heated assume a state. intermediate between solid and liquid which is called the mesomorphic. or liquid crystal state, These examples indicate that periodicity can be observed to a lesser or. greater extent in crystals depending on their nature and on the thermo. dynamic conditions of their formation It is therefore useful to introduce the. concept of a real crystal to stress the differences from an ideal crystal with. perfect periodicity Although non ideality may sometimes be a problem. more often it is the cause of favourable properties which are widely used in. materials science and in solid state physics, In this chapter the symmetry rules determining the formation of an ideal. crystalline state are considered the reader will find a deeper account in. some papers devoted to the subject or some exhaustive or in the. theoretical sections of the International Tables for Cryst llography. In order to understand the periodic and ordered nature of crystals it is. necessary to know the operations by which the repetition of the basic. molecular motif is obtained An important step is achieved by answering the. following question given two identical objects placed in random positions. and orientations which operations should be performed to superpose one. object onto the other, The well known coexistence of enantiomeric molecules demands a. second question given two enantiomorphous the term enantiomeric will. only be used for molecules objects which are the operations required to. superpose the two objects, An exhaustive answer to the two questions is given by the theory of. isometric transformations the basic concepts of which are described in. Appendix 1 A while here only its most useful results will be considered. Two objects are said to be congruent if to each point of one object. corresponds a point of the other and if the distance between two points of. one object is equal to the distance between the corresponding points of the. other As a consequence the corresponding angles will also be equal in. absolute value In mathematics such a correspondence is called isometric. The congruence may either be direct or opposite according to whether. the corresponding angles have the same or opposite signs If the congruence. is direct one object can be brought to coincide with the other by a. convenient movement during which it behaves as a rigid body The. movement may be, 1 a translation when all points of the object undergo an equal. displacement in the same direction, 2 a rotation around an axis all points on the axis will not change their. Symmetry in crystals 3, 3 a rototranslation or screw movement which may be considered as the. combination product of a rotation around the axis and a transl ation. along the axial direction the order of the two operations may be. If the congruence is opposite then one object will be said to be. enantiomorphous with respect to the other The two objects may be brought. to coincidence by the following operations, 1 a symmetry operation with respect to a point known as inversion. 2 a symmetry operation with respect to a plane known as reflection. 3 the product of a rotation around an axis by an inversion with respect to. a point on the axis the operation is called rotoinversion. 4 the product of a reflection by a translation parallel to the reflection. plane the plane is then called a glide plane, 5 the product of a rotation by a reflection with respect to a plane. perpendicular to the axis the operation is called rotoreflection. Symmetry elements, Suppose that the isometric operations described in the preceding section. not only bring to coincidence a couple of congruent objects but act on the. entire space 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. which symmetry operations are performed, In the following these elements will be considered in more detail while. the description of translation operators will be treated in subsequent. 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 p 9 in the axes 1 2 3 4 6. Axis 1 is trivial since after a rotation of 360 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 1 In the first column of Fig 1 1 their effects on the space are. illustrated In keeping with international notation an object is represented. by a circle with a or sign next to it indicating whether it is above or. below the page plane 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. 4 1 Carmelo Giacovazzo, Table 1 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 1 1 Arrangements of symmetry equivalent,objects as an effect of rotation inversion and. screw axes,Symmetry in crystals 1 5,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 On p. 10 we will see that in crystals only screw axes of order 1 2 3 4 6 can exist. with appropriate translational components,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 9. we will be mainly interested in 1 2 3 4 6 axes and their graphic symbols. are given in Table 1 1 while their effects on the space are represented in the. second column of Fig 1 1 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 To. each half the appropriate or sign is assigned,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 e 3 31,4 the 4 axis is also a 2 axis, 5 the 6 axis is equivalent to the product of a threefold rotation by a. reflection with respect to a plane normal to it this will be indicated by. 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 In particular i m 2 1 3 6 4 4 6 3 From now on we will. no longer consider the ii axes but their equivalent inversion axes. 6 1 Carmelo Giacovazzo, Reflection planes with translational component glide. 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 On p 11 we shall see which are the glide planes found in crystals. 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, Translational periodicity in crystals can be conveniently studied by con. sidering 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. Cl Cl Cl Cl Cl Cl Cl, c along three non coplanar directions the repetition geometry can be fully. o H H H H H H H described by a periodic sequence of points separated by intervals a b c. C1 Cl Cl Cl C1 7 1 Cl 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 1 2 a where. HOCl is a geometrical motif repeated at intervals a and b If we replace the. molecule with a point positioned at its centre of gravity we obtain the. lattice of Fig 1 2 b 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 1 2 b is uniquely defined by the vector. where u and v are positive or negative integers 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 1 2 b. four possible choices are shown they are all characterized by the property. that each lattice point satisfies relation 1 1 with integer u and v. Nevertheless we are allowed to choose different types of unit cells such. as those shown in Fig 1 2 c having double or triple area with respect to. those selected in Fig 1 2 b In this case each lattice point will still satisfy. 1 1 but u and v are no longer restricted to integer values For instance the. point P is related to the origin 0 and to the basis vectors a and b through. 4 v 112 112, The different types of unit cells are better characterized by determining. the number of lattice points belonging to them taking into account that the. i a R of a graphical motif as an points on sides and on corners are only partially shared by the given cell. example of a two dimensional crystal b The cells shown in Fig 1 2 b contain only one lattice point since the. corresponding lattice with some examples Of four points at the corners of each cell belong to it for only 114 These cells. primitive cells c corresponding lattice with, some examples of multiple cells are called primitive The cells in Fig 1 2 c contain either two or three. Symmetry in crystals 1 7, points and are called multiple or centred cells Several kinds of multiple. cells are possible i e double cells triple cells etc depending on whether. they contain two three etc lattice points, The above considerations can be easily extended to linear and space. lattices For the latter in particular given an origin 0 and three basis II. vectors a b and c each node is uniquely defined by the vector. ua ub W C 1 2, The three basis vectors define a parallelepiped called again a unit cell a. Fig Notation for a unit cell, The directions specified by the vectors a b and c are the X Y Z. crystallographic axes respectively while the angles between them are. indicated by a 0 and y with a opposing a opposing b and y opposing. c cf Fig 1 3 The volume of the unit cell is given by. where the symbol indicates the scalar product and the symbol A the. vector product The orientation of the three crystallographic axes is usually. chosen in such a way that an observer located along the positive direction of. c sees a moving towards b by an anti clockwise rotation The faces of the. unit cell facing a b and c are indicated by A B C respectively If the. chosen cell is primitive then the values of u u w in 1 2 are bound to be. integer for all the lattice points If the cell is multiple then u u w will have. rational values To characterize the cell we must recall that a lattice point at. vertex belongs to it only for 1 8th a point on a edge for 114 and one on a. face for 112,The rational properties of lattices, Since a lattice point can always be characterized by rational numbers the. lattice properties related to them are called rational Directions defined by. CARMELO GIACOVAZZO The crystalline state and isometric operations Matter is usually classified into three states gaseous liquid and solid Gases are composed of almost isolated particles except for occasional collisions they tend to occupy all the available volume which is subject to variation following changes in pressure In liquids the attraction between nearest neighbour particles is

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