4. Direct and
reciprocal lattices
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Fragment of a distribution of a set of objects that produce a direct lattice in 2 dimensions. As an example, one of the infinite sets of motifs (small tiles) that produce the repetitive and periodic distribution is shown inside the yellow squares. The dimensions of the yellow square represent the translations of the direct lattice.  Fragment of a mosaic in The Alhambra, showing a 2dimensional periodic pattern. These periodic translations can be discovered in the mosaic and produce a twodimensional direct lattice  Periodic stacking of balls, producing a 3dimensional network (direct lattice). The motif being repeated in the three directions of space is the contents of the small box with blue edges. 
Elementary
cell (or unit cell) defined by the 3 noncoplanar translations
known as reticular axes (cell axes or lattice axes)

Formation of an octahedral crystal by stacking unit cells in the 3 directions of space 
In general, inside the unit cell there is a minimum set of atoms (ions or molecules) which are repeated inside the cell due to the symmetry elements of the crystal structure. This minimum set of atoms (ions or molecules) which generate the whole contents of the unit cell (after applying the symmetry elements to them) is known as the asymmetric unit.
A structural motif or asymmetric unit 
The structural
motif shown in
the left figure is repeated by a symmetry element (symmetry operation),
in this case a screw axis

The repetiion of
the assymetric unit generates the whole contents of the unit cell

The repetition of unit cells (due to the periodic translations) generates the crystal 
The lattice, which is a purely mathematical concept, can be selected in various ways in the same real periodic distribution. However, only one of these lattices "fits" best with the symmetry of the periodic distribution of the motifs.
Periodic distribution of one motif that contains two objects (a triangle and a circle) 
Unit
cells corresponding to possible direct lattices that can be drawn over
the periodic distribution shown above. Only one of the unit cells (the
red one) is more appropriate because it fits much better with the
symmetry
of the distribution

The red
cell on the left figure (a centered lattice) fits better with
the
symmetry of the distribution, and can be decomposed in two identical
lattices, one for each object of the motif.

As is shown in the figures above, although especially in the right one, any lattice that describes the repetition of the motif (triangle + circle) can be decomposed into two identical equivalent lattices (one for each object of the motif). Thus, the concept of lattice is independent of the complexity of the motif, so that we can use only one lattice, since it represents all the remaining equivalent ones.
Once we have chosen a representative lattice, appropriate to the symmetry of the structure, any reticular point (or lattice node) can be described by a vector that is a linear combination (with integer numbers) of the direct reticular axes: R = m a + n b + p c, where m, n and p are integers. Nonreticular points can be reached using the nearest R vector, and adding to it the corresponding fractions of the reticular axes to reach it:
r = R + r' = (m a + n b + p c) + (x a + y b + z c) Position vector for any nonreticular point of a direct lattice 
where x, y, z represent the corresponding dimensionless fractions of axes X/a, Y/b, Z/c, and X, Y, Z the corresponding lengths.

Position vector for a nonreticular point (black circle) 
From a geometric point of view, on a lattice we can consider some reticular lines and reticular planes which are those passing through the reticular points (or reticular nodes).
Just as we did with the lattices (choosing only one of them from all the equivalent ones), we do the same with the reticular lines and planes. A reticular line or a reticular plane can be used as a representative of the entire family of parallel lines or parallel planes.




The plane drawn on the left side of the figure above cuts the a axis in 2 equal parts, the b axis in 2 parts and the c axis in 1 part. Hence, the numerical triplet identifying the plane will be (221). The plane drawn on the right side of the figure cuts the a axis into 2 parts, is parallel to the b axis and cuts the c axis in 1 part. Therefore, the numerical triplet will be (201). 
A unique plane, as the one drawn in the top right figure, defined by the numerical triplet known as Miller indices, represents and describes the whole family of parallel planes passing through every element of the motif. Thus, in a crystal structure, there will be as many plane families as possible numerical triplets exist with the condition that these numbers are primes, one to each other (not having a common divisor). The Miller indices are generically represented by the triplet of letters hkl.
If there are common divisors among the Miller indices, the numerical triplet would represent a single family of planes only. For example, the family with indices (330), which are not strictly reticular, can be regarded as the representative of 3 families of indices (110) with a geometric outofphase distance (among the families) of 1/3 of the original (see the figures below).


Thus, the concept of Miller indices, previously restricted to numerical triplets (being prime numbers), can now be generalized to any triplet of integers. In this way, every family of planes, will "cover" the whole crystal. And therefore, for every point of the crystal we can draw an infinite number of plane families with infinite orientations.
Through
a point in the crystal (in the example in the center of the cell) we
can draw an infinite number of plane families with an infinite number
of
orientations. In this case only 3 families and 3 orientations are
shown.

Of course, interplanar spacings can be directly calculated from the Miller indices (hkl) and the values of the reticular parameters (unit cell axes). The table below shows that these relations can be simplified for the corresponding metric of the different lattices.

Any plane can also be characterized by a vector (σ_{hkl}) perpendicular to it. Therefore, the projection of the position vector of any point (belonging to the plane), over that perpendicular line is constant and independent of the point. It is the distance of the plane to the origin, ie, the spacing (d_{hkl} ).

Any plane can be represented by a vector perpendicular to it. 
Consider the family of planes hkl with the interplanar distance d_{hkl}. From the set of vectors normal to the planes' family, we take the one (σ_{hkl}) with length 1/d_{hkl}. The scalar product between this vector and the position vector (d'_{hkl} ) of a point belonging to a plane from the family is an integer (n), and this integer gives us the order of that plane in the hkl family. That is:
(σ_{hkl}) . (d'_{hkl}) = (1/d_{hkl}) . (n.d_{hkl}) = n (see left figure below)
Thus, σ_{hkl} represents the whole family of hkl planes having an interplanar spacing given by d_{hkl}. In particular, for the first plane we get: σ_{hkl} d_{hkl} = 1.


If from this normal vector σ_{hkl} of length 1/d_{hkl}, we take another vector, n times (integer) longer (n.σ_{hkl}), the above mentioned product (σ_{hkl} d_{hkl} = 1) would imply that the new vector (n.σ_{hkl}) will correspond to a family of planes of indices nh,nk,nl having an interplanar spacing n times smaller. In other words, for instance, the lengths of the following interplanar spacings will bear the relation: d_{100} = 2.(d_{200})= 3.(d_{ 300})..., so that σ_{100} = (1/2).σ_{200} = (1/3).σ_{300 }... and similarly for other hkl planes.
Therefore, it appears that the moduli (lengths) of the perpendicular vectors (σ_{hkl}) are reciprocal to the interplanar spacings. The end points of these vectors (blue arrows in figure below) also produce a periodic lattice that, due to this reciprocal property, is known as the reciprocal lattice of the original direct lattice. The reciprocal points obtained in this way (green points in figure below) are identified with the same numerical triplets hkl (Miller indices) which represent the corresponding plane family.

Animated example showing how to obtain the reciprocal points from a direct lattice 
It should now be clear that the direct lattice, and its reticular planes, are directly associated (linked) with the reciprocal lattice. Moreover, in this reciprocal lattice we can also define a unit cell (reciprocal unit cell) whose periodic translations will be determined by three reciprocal axes that form reciprocal angles among them.
If the unit cell axes and angles of the direct cell are known by the letters a, b, c, α, β, γ, the corresponding parameters for the reciprocal cell are written with the same symbols, adding an asterisk: a*, b*, c*, α*, β*, γ*. It should also be clear that these reciprocal axes (a*, b*, c*) will correspond to the vectors σ_{100}, σ_{010} and σ_{001}, respectively, so that any reciprocal vector can be expressed as a linear combination of these three reciprocal vectors:
Position vector of any reciprocal point 
Relationships between direct (black) and
reciprocal (green)
cells . Both cells, and their corresponding lattices, are
stuck together.

Two lattices (direct with blue points, reciprocal in green) showing their geometrical relations. For clarity it is assumed that the third direct axis (c), not shown, is perpendicular to the screen. According to what has been said before, the corresponding third reciprocal axis (c*) will also be perpendicular to the screen. 
The geometric relationships between the direct and reciprocal cells imply the following welldefined metric relations between them:
V
= (a x
b) . c
= a. b. c (1  cos^{2}α 
cos^{2}β 
cos^{2}γ
+
2 cos α
cos β cos γ)^{1/2}

Note that, in accordance with the definitions given above, the length of a* is the inverse of the interplanar spacing d_{100} (a* = 1/d_{100}), and that b* = 1/d_{010}, and that c* = 1/d_{001}. Therefore, the following scalar products (dot products) can be written: a.a* = 1, a.b* = 0 and similarly with the other pairs of axes.
In
addition to this, we recommend using the Java applet offered by Nicolas Schoeni y Gervais Chapuis of the
Ecole Polytechnique
Fédéral de Lausanne (Suiza) to understand the
relation between direct and reciprocal lattices and how to build the
latter from a direct lattice. In case of problems using this applet,
please follow the indications
shown in this link.
See also the pages on reciprocal space offered by the University of Cambridge through this link.
And although we are revealing aspects corresponding to the next chapter (see the last paragraph of this page), the reader should also look at the video made by www.PhysicsReimagined.com, showing the geometric relationships between direct and reciprocal lattices, displayed below as an animated gif:


