Plane lattices rotational

We found that without any exception in all of our simulations Bain s lattice correspondence actually applies, i.e. one set of (110) planes of the bcc structure corresponds to a set of fee (111) planes, while the bcc [001] direction lying in these planes is transformed into the [110] direction of the fee phase. Moreover, these directions are exactly parallel to each other. This would correspond to a Nishiyama-Wassermann orientational relationship if the (110) and (111) planes would also be parallel to each other. But this is not the case. These planes are rotated around [001] by an angle between 0 and 9 during the transformation. This angle differs between the simulations in a non-systematic way. 

X-ray radiation wavelength—that is, 1/ X. When the crystal is rotated, the reciprocal lattice rotates with it and different points within the lattice are brought to diffraction. The diffracted beams are called reflections because each of them can be regarded as a reflection of the primary X-ray beam against planes in the crystal. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

Fig. 80. Formation of reciprocal lattice rotation diagram, set of planes is the corresponding reciprocal lattice point V, the distance of which from the reciprocal lattice origin X is inversely proportional to...

Fig. 90. Monoclinic reciprocal lattice rotated round normal to a b plane (c axis of real cell). Above general view. Right real cell, same orientation. Below view (on smaller scale) looking straight down c axis. 

In considering the hexagonal lattice, our attention is strongly drawn to the question of the symmetry of lattices. It is a question that must eventually be addressed in more detail for this as well as the other four plane lattices and we shall do so shortly. However, we shall first deal with a geometrical aspect of plane lattices that hinges on just one of their possible symmetry properties, namely, rotational symmetry. When we have done this it will be clear why the five lattices just described are the only ones possible. We shall understand why it is that we need not look for some special value of y that would allow for fivefold or sevenfold, eightfold, and so on rotational symmetry. 

The geometrical features of shear deformation are shown in Fig. 24.5. Here, the shear is on the K plane in the direction of d. The initial unit sphere is deformed into an ellipsoid and the Ki plane is an invariant plane. The K2 plane is rotated by the shear into the K 2 position and remains undistorted. A reasonable slip system to assume for the lattice-invariant shear deformation is slip in a (111) direction on a 112 plane in the b.c.t. lattice, which corresponds to slip in a (110) direction on a 110 plane in the f.c.c. lattice. 

When the point-group symmetries are combined with the plane lattices, 17 two-dimensional space groups can be produced. In such treatment, severe limitations are imposed on the possible point groups that may be combined with lattices to produce space groups. Some symmetry elements, such as the fivefold rotation axis, are not compatible with translational symmetry and from this, forbidden symmetries follow in classical crystallography. This and the lifting of such limitations in modem crystallography will be examined in Chapter 9. 

All crystal planes having indices hk ) are represented by points lying on a plane (called the / = 1 layer ) in the reciprocal lattice, normal to hi,. When the reciprocal lattice rotates, this plane cuts the reflection sphere in the small circle shown, and any points on the / = 1 layer which touch the sphere surface must... 

As described in Chapter 1, a crystal is defined by the fact that the whole structure can be built by the regular stacking of a unit cell that is translated but neither rotated nor reflected. The same is true for two-dimensional crystals or patterns. This imposes a limitation upon the combinations of symmetry elements that are compatible with the use of unit cells to build up a two-dimensional pattern or a three-dimensional crystal. To understand this, consider the rotational symmetry of the five unique plane lattices, described in the previous chapter. 

Suppose that a rotational axis of value n is normal to a plane lattice. It is convenient, (but not... 

Figure 22. Overlap of two hp lattices rotated about an axis normal to the plane and passing through the origin by the angle cp of the compound tessellation 3, 6 [13 3, 6 ]. One node out of 13 is restored. Three hexagonal meshes containing each 13 nodes are also shown.

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