Unit cell direct-lattice

We also need to define how large the unit-cell is in terms of both the length of its sides and its volume. We do so by defining the unit-cell directions in terms of its "lattice unit-vectors". That is, we define it in terms of the x, y, z directions of the unit cell with specific vectors having directions corresponding to ... 

Note think of the atom in the middle of the bcc unit cell — at lattice position (1/2,1/2,1/2). Since there are no atoms on the unit cell faces in a bcc array, there are no atoms that he directly along the x, y, and z axes emanating from this central atom. 

It then seems that the SbF salt has reached the compactness observed in the two other salts at 4 K. This needs to be discussed more in detail, since these contacts point in the direction of the diagonal (c - b) of the unit cell this lattice direction does not contract upon cooling. This is illustrated in Table 3 the greatest variation of (c - b) is observed for (TMTTF)2PF5, while no signifcant change is noticed for the other salts. This stiffness speaks for strong coulombic... 

X-ray diffraction measurements show that polyethylene crystallizes in a body-centered orthorhombic unit cell with lattice parameters a = 0.7417 nm, /> = 0.4945 nm, and c = 0.2547 nm at 25 °C. The a, b, and c axes are orthogonal. The chain axes run in the c direction and there are two repeating units per unit cell (one running up the center and one-quarter in each of the four comers). Calculate the crystal density of polyethylene. 

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

So it is essential to relate the LEED pattern to the surface structure itself As mentioned earlier, the diffraction pattern does not indicate relative atomic positions within the structural unit cell, but only the size and shape of that unit cell. However, since experiments are mostly perfonned on surfaces of materials with a known crystallographic bulk structure, it is often a good starting point to assume an ideally tenuinated bulk lattice the actual surface structure will often be related to that ideal structure in a simple maimer, e.g. tluough the creation of a superlattice that is directly related to the bulk lattice. 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

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.

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

For a two-dimensional array of equally spaced holes the difftaction pattern is a two-dimensional array of spots. The intensity between the spots is very small. The crystal is a three-dimensional lattice of unit cells. The third dimension of the x-ray diffraction pattern is obtained by rotating the crystal about some direction different from the incident beam. For each small angle of rotation, a two-dimensional difftaction pattern is obtained. 

The crystallographic requirement for tire formation of G-P zones is that the material within the zones shall have an epitaxial relationship with the maUix, and tlrus the eventual precipitate should have a similar unit cell size in one direction as tha maUix. In dre Al-Cu system, the f.c.c. structure of aluminium has a lattice parameter of 0.4014 nm, and the tetragonal CuAl2 compound has lattice parameters a — 0.4872 and b — 0.6063 nm respectively. 

Fig. 8.12. The structure of 0.8% carbon martensite. During the transformation, the carbon atoms put themselves into the interstitial sites shown. To moke room for them the lattice stretches along one cube direction (and contracts slightly along the other two). This produces what is called a face-centred tetragonal unit cell. Note that only a small proportion of the labelled sites actually contain a carbon atom.

The translational direction of the lattice (c), which is the direction that the crystallite axis is not, because of the triclinic crystallographic system, is perpendicular to the plane of the unit cell base (ab). 

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