Reflection, across a plane

Translation, i.e.- from one plane to another Rotation about an axis of the crystal Reflection across a plane Inversion through a point... 

A C axis is often called a proper rotational axis and the rotation about it a proper rotation. An improper rotation may be visualized as occurring in two steps rotation by 360E/ followed by reflection across a plane perpendicular to the rotational axis. Neither the axis of rotation nor the mirror plane need be true symmetry elements that can stand alone. For example, we have seen that SiF4 has C3 axes but no C4 axis. Nevertheless, it has three S4 axes, one through each pair of opposite faces of the cube below ... 

Glide plane A glide plane involves reflection across a plane combined with a translation. For an axial glide plane, denoted a, b or c, the translational component is or f, respectively. For a c-glide plane, if the mirror plane is perpendicular... 

The relationship between the two parts of a twinned crystal is described by the symmetry operation which will bring one part into coincidence with the other or with an extension of the other. Two main kinds of twinning are distinguished, depending on whether the symmetry operation is 180° rotation about an axis, called the twin axis, or reflection across a plane, called the twin plane. The plane on which the two parts of a twinned crystal are united is called the composition plane. In the case of a reflection twin, the composition plane may or may not coincide with the twin plane. 

Deformation twins are found in both BCC and HCP lattices and are all that their name implies, since, in both cases, the cause of twinning is deformation. In each case, the orientation relationship between parent crystal and twin is that of reflection across a plane. 

Alternating symmetry axis (S ) An axis about which a rotation by an angle of 360/n, followed by a reflection across a plane perpendicular to the axis results in an entity that is indistinguishable from (superimposable on) the original. Also called a rotation-reflection axis. See also symmetry axis. 

Reflection across a plane Plane of symmetry or mirror m... 

Glide plane, which combines a translation in the plane and a reflection across a plane. 

This two-fold procedure—calculate similarity among the samples, then spread the samples across a plane in a way that reflects that similarity—is exactly the task for which the SOM is designed. 

The reflected ray may also be constructed by considering the mirror image of the source as reflected across the plane of the wall. In figure 3.2, the image source thus constructed is denoted AThis technique of reflecting sources across wall surfaces is called the source image method. The method allows a source with reflective boundaries to be modeled as multiple sources with no boundaries. 

HREM is particularly suitable for determining the nature and extent of twinning in faujasite crystals which are prone (8) to twinning by reflection across 111 planes as shown in Fig. 2. Multiple twinning (not necessarily on the same member of the family of 111 planes) is common and in Fig. 3 we show a crystal ( 110] zone-axis orientation) exhibiting the consequences of simultaneous twinning on (111) and (111) planes. In such a situation two further twin planes inclined to 010 are theoretically possible (i.e. (111) and (111)), but these twinned individual would be projected along a <1JM > type zone-axis. 

For alternant conjugated systems (those having no odd-membered rings), the Id values are reflected across a mirror plane placed horizontally, halfway between ip2 and 3, and also across a mirror plane placed vertically, halfway between C-2 and C-3. It is only necessary therefore to calculate four of the 16 numbers in Fig. 1.31, and deduce the rest from the symmetry. 

Another type of symmetry operation is reflection across a mirror plane (Figure 4.4). Reflection across mirror planes converts an object into its mirror image. Reflection symmetry occurs when a plane can be constructed such that an object or molecule on one side of the plane is... 

FIGURE 4,12. A glide plane causes a reflection across a mirror plane plus a translation of half the unit cell edge (translation = a/2) A left-handed molecule is converted to a right-handed molecule by this symmetry operation. Two such operations give the original molecule translated to the next unit cell (translation = a). 

The simplest symmetry elements are the centre, plane, and axes of symmetry. A cube, for example, is symmetrical about its body-centre, that is, every point (xyz) on its surface is matched by a point (xyz). It is said to possess a centre of symmetry or to be centrosymmetrical a tetrahedron does not possess this type of symmetry. Reflection of one-half of an object across a plane of symmetry (regarded as a mirror, hence the alternative name mirror plane) reproduces the other half. It can easily be checked that a cube has no fewer than nine planes of symmetry. The presence of an -fold axis of symmetry implies that the appearance of an object is the same after rotation through 3607 l a cube has six 2-fold, four 3-fold, and three 4-fold axes of symmetry. We postpone further discussion of the symmetry of finite solid bodies because we shall adopt a more general approach to the symmetry of repeating patterns which will eventually bring us back to a consideration of the symmetry of finite groups of points. 

Figure 3.9 The glide operation (a) reflection across a mirror line (b) translation parallel to the mirror plane by a vector t, which is constrained to be equal to T/2, where T is the lattice repeat vector parallel to the glide line. The lattice unit cell is shaded...

In contrast, improper rotations are not physically possible. An S involves a rotation of (360/ )°, combined with a reflection across a mirror plane that is perpendicular to the rotation axis (see examples on the next page). Note that Si is equivalent to just a mirror reflection (denoted with a a), while Sz is equivalent to a center of inversion (denoted with an i). The Cj operation also exists. It leaves an object completely unmoved and is also termed the identity operation, sometimes symbolized as . An internal cr plane that includes a C2 axis is designated a (Ty, while a a plane perpendicular to a C2 axis is designated Oh-... 

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