Mirror lines

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

If we introduce a longitudinal reflection operation (i.e., the line of translation is also a mirror line) we get class 2. 

Class 4 is obtained by introducing a transverse mirror line. Not only is this line reproduced by the translation operation, but a second set of transverse mirror lines is created. This is similar to what occurred in class 3. The second set is not equivalent to the first, but is brought into existence by it. 

Class 5 arises by a natural extension of what has been done to make classes 2, 3, and 4. If both types of mirror line are introduced simultaneously, we obtain this class. Just as in point groups, the intersection between two mirrors generates a C2 axis and this class therefore also contains two sets of C2 axes. We could also have generated class 5 by introducing the C2 axes and only one type of mirror line the other type would then have arisen as the product of these. The relationships are exactly as in the case of point group C2r. 

In Figure 11.4 we also give the symbols used to specify the symmetries of these lattices. This type of notation will be fully explained in Section 11.4, but we can point out here that a rotation axis of order n is represented simply by the number h and mirror lines (or planes) by m. In addition, p and c specify primitive and centered lattices, respectively. 

Perusal of Figs 1 2 shows why the early shock sensitivity studies that measured only were inadequate, since they provide no information as to the point on the line of slope Poi i about which the mirror line should be drawn. Thus the shock state in the expl remains unknown even if Ut is determined. Knowing Uj, U(, the initial densities of both media and either Pi or Uj, one can obtain approx values of P2, u2 by reflecting a line of slope poi Uj thru a point determined by either Pj or ut on the Rayleigh line of the incident shock. The reason that this procedure gives only approximate values is that P-u plots are curved and not linear... 

Figure 1. Chiral figures in the plane (m = mirror line). Top enantiomorphous scalene triangles. Bottom enantiomorphous oriented circles.

Finally, the hexagonal primitive (hp) lattice, (Figure 3.5i), has a hexad rotation axis at each lattice point. This generates diads and triads as shown. In addition, there are six mirror lines through each lattice point. In other parts of the unit cell, two mirror lines intersect at diads and three mirror lines intersect at triads, (Figure 3.5j). The lattice point symmetry is described by the symbol 6mm. 

Figure 3.8 The plane groups pm and cm (a) the plane lattice op (b) the pattern formed by adding the motif of point group m to the lattice in (a), representative of plane group pm (b) the plane lattice oc (d) the pattern formed by adding the motif of plane group m to the lattice in (c), representative of plane group cm. Mirror lines are heavy, and glide lines in (d) are heavy dashed...

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...

The location of the symmetry elements within the unit cells of the plane groups is illustrated in Figure 3.10. Heavy lines represent mirror lines and heavy dashed lines represent glide lines. The unit cell has a light outline. 

There are parallels between the two-and three-dimensional cases. Naturally, mirror lines in two dimensions become mirror planes, and glide lines in two dimensions become glide planes. The glide translation vector, t, is constrained to be equal to half of the relevant lattice vector, T, for the same reason that the two-dimensional glide vector is half of a lattice translation (Chapter 3). 

Fig. 5.11. TVo-dimensional energy surface for the inner-sphere reorganization in an electron selfexchange reaction. Reorganization of reactants and products is assumed to follow harmonic potentials symbolized by contour lines. The two surfaces intersect along the diagonal mirror line. The point of lowest energy on the intersection parabola is the transition state it is marked with a cross...

Fig. 5.13. Result of principal component analysis on the metallacyclopentene fragment of (s-c(s-/7 -butadiene) metallocene complexes (Figure 5.12). The coordinate PRINl measures the dihedral angle between the CCCC and the CMC planes, PRIN2 measures the bite distance CH2. .. CHj. The point at PRINl = 0 corresponds to a planar fragment. The distribution of data points is roughly semicircular and symmetrical with respect to a vertical mirror line. The length of the arc from this line to any point is defined as the distance Xq between ground-state and transition state...

In Fig. 2.24, m represents a mirror line. Let T be a primitive translation and T the mirror image of T. (A translation T is primitive if 1/2T is not a translation.) T + T and T — T are perpendicular and define a rectangular cell. If T + T and T — T are primitive translations, the rectangular cell is centered because there is a lattice point in the middle. We would thus choose the diamond (T,T) as the primitive cell. In contrast, if 1/2(T + T ) and 1/2(T — T ) are translations, we then obtain a rectangular primitive cell. These two rectangular planar lattices, primitive, on the one hand and centered or diamond, on the other, are representative of two types of lattice that it is important to differentiate. It is impossible to find a primitive diamond-shaped cell for the first, or a primitive rectangular cell for the second. These considerations lead us to an operational definition for the Bravais lattices (or classes). A Bravais lattice is characterized by ... 

METHOD. The group is generated by a glide line g parallel to an edge and a mirror line m parallel to a diagonal of a square. Draw several square unit cells... 

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