Reciprocal lattice volume

In order to decide whether a reflection is partial or not for a given crystal orientation the sample reflection rocking width A or a spherical reciprocal lattice volume of radius E can be compared with a unit Ewald sphere these are given by ... 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

The true unit cell is not necessarily the smallest unit that will account for all the reciprocal lattice points it is also necessary that the cell chosen should conform to the crystal symmetry. The reflections of crystals with face-centred or body-centred lattices can be accounted for by unit cells which have only a fraction of the volume of the true unit cell, but the smallest unit cells for such crystals are rejected in favour of the smallest that conforms to the crystal symmetry. The... 

You can see from Table 8.1 that 98% of the reflections available out to 2.7 A [those lying within a sphere of radius 1/(2.7 A) centered at the origin of the reciprocal lattice] were measured, and on the average, each reflection was measured four times. Additional reflections were measured out to 2.4 A. The number of available reflections increases with the third power of the radius of the sampled region in the reciprocal lattice (because the volume of a sphere of radius r is proportional to r3), so a seemingly small increase in resolution from 2.7 to 2.4 A requires 40% more data. [Compare (1/2.4)3 with (1/2.7)3.] For a rough calculation of the number of available reflections at specified resolution, see annotations of the 4/92 paper. 

The N q vectors allowed by the boundary conditions just fill the first Brillouin zone (BZ) of volume equal to vb, the volume of the primitive unit cell of the reciprocal lattice. Because of this dense, uniform distribution of q vectors it is possible to treat q as a continuous variable and thus replace... 

From the six equivalent points 100 there are thus produced six planes which enclose a cube of side 1 a and volume (1/tf)3. The density of the states is also 1 per (reciprocal) unit volume. This cube thus encloses (1/tf)3 states. A crystal with dimensions 1X1X1 cm with a simple cubic lattice contains (1 a)z atoms so that there is also 1 state available per atom in this zone, or there is space for 2 electrons per atom. If, for example, the reflection from the octahedral planes 111 is, as in... 

The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The Wigner-Seitz cell of the FCC lattice is the rhombic dodecahedron in Figure A. b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high symmetry points are shown in Table 4.4. 

In Eq. [30], erfc(a ) denotes the complementary error function, N is number of atoms in the simulation box of volume V, and and are charges of atoms i and j, respectively. K = 27tH, K = IKI, and H stands for a vector of the reciprocal lattice defined for the simulation box and are parameters controlling convergence of the direct and reciprocal sums. 

Figure 5.6. Schematic representations of the fractions of the volume of the sphere (r = 1/X) in the reciprocal space in which the list of hkl triplets should be generated in six powder Laue classes to ensure that all symmetrically independent points in the reciprocal lattice have been included in the calculation of Bragg angles using a proper form of Eq. 5.2. The monoclinic crystal system is shown in the alternative setting, i.e. with the unique two-fold axis parallel to c instead of the standard setting, where the two-fold axis is parallel to b. ...

We note that the benchmarks listed above must be applied altogether. For example, it is nearly always possible to choose the highest symmetry crystal system (i.e. cubic) and a large unit cell to dubiously assign index triplets to all observed Bragg peaks and obtain acceptable e. This happens because the density of points in the reciprocal lattice is proportional to the volume of the... 

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