Space lattices rotation inversion

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

The reciprocal lattice is spatially linked to the crystal because of the way the lattice points are defined, so if we rotate the crystal, the reciprocal lattice rotates with it. So now when you think of a crystal, and imagine the many identical unit cells stretching out in all directions (real space), imagine also a lattice of points in reciprocal space, points whose lattice spacing is inversely proportional to the interplanar spacings within the crystal. 

The Hermann- Mauguin notation is generally used by crystallographers to describe the space group. Tables exist to convert this notation to the Schoen-flies notation. The first symbol is a capital letter and indicates whether the lattice is primitive. The next symbol refers to the principal axis, whether it is rotation, inversion, or screw, e.g.,... 

Figure 3.7 demonstrates this rotation inversion symmetry operation in space lattices. 

Note As 1, that is, onefold rotation and inversion is same as centre of symmetry and 2, that is, twofold rotations and inversion is also same as the mirror plane, they are not included as separate symmetry elements in space lattices. 

Figure 2 View looking down on the real-space mesh (a) and the corresponding view of the reciprocal-space mesh (b) for a crystal plane with a nonrectangular lattice. The reciprocal-space mesh resembles the real-space mesh, but rotated 90°. Note that the magnitude of the reciprocal lattice vectors is inversely related to the spacing of atomic rows.

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

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. 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

Both are body-centered Bravais lattices and for both the site symmetry of the origin is identical with the short space group symbol. The body-center position is of the lowest multiplicity (two-fold) and highest symmetry, and thus is considered as the origin in the lA/mmm space group. However, in the tetragonal lattice, a = b c. Hence, the body center position is not an inversion center. It possesses four-fold rotational symmetry (the axis is parallel to c) with a perpendicular mirror plane and two additional perpendicular mirror planes that contain the rotation axis. 

The symmetry elements of the point groups—proper rotation, reflection, inversion, and improper rotation—all still apply. For example, the lattice described by Fig. 13.2 has the same point group operations of the octahedral group—several C3 and Q rotation axes, various mirror planes, improper rotation axes, and inversion. These can also be combined with the translation operators. We call the groups that contain the point group and translation operations the space groups and use them to classify different crystal structures. 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

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