4. Direct and reciprocal lattices
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Let's start with a summary of several concepts seen in previous chapters...


Any repetitive and periodic distribution of an object (or motif) can be characterized, or described, by the translations that repeat the object periodically. This set of translations generate what we call a direct lattice.
 
Fragment of a repetitive distribution of objects that produce a direct lattice in 2 dimensions Fragment of a mosaic in The Alhambra, showing a 2-dimensional periodic pattern. These periodic translations can be discovered in the mosaic and produce a two-dimensional direct lattice

The translations that describe the periodicity in crystals can be expressed as a linear combination of three basic translations, not coplanar, ie independent, known as reticular or lattice axes (or unit cell axes). These axes define a parallelogram (in 2 dimensions), or a parallelepiped (in 3 dimensions) known as a unit cell(or elementary cell). This elementary area (in 2-dimensional cases), or elementary volume (in 3-dimensional cases), which holds the minimum set of the periodic distribution, generates (by translations)  the full distribution which, in our 3-dimensional case, we call crystal.






Elementary cell (or unit cell) defined by the 3 non-coplanar 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. Non-reticular 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 non-reticular 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 non-reticular point (black circle)


Let's now see some new concepts on direct lattices...

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.

Of the two families of equivalent lattices shown (red and blue) we can choose only one of them, on the understanding that it represents the remaining equivalent ones. Note that the distance between the planes drawn on each lattice  (interplanar spacing) is the same for the blue or red families. However, the family of red planes is separated from the family of blue planes by a distance that depends on the separation between the objects which produced the lattice. This distance between the planes of different families can be called the geometric out-of-phase distance.

Family of reticular planes cutting the vertical axis of the cell in 2 parts and the horizontal axis in 1 part. These planes are parallel to the third reticular axis (not shown in the figure).
Family of reticular planes cutting the vertical axis of the cell in 3 parts and the horizontal axis in 1 part. These planes are parallel to the third reticular axis  (not shown in the figure).
The number of parts in which a family of planes cut the cell axes can be associated with a triplet of numbers that identify that family of planes. In the three previous figures, the number of cuts, and therefore the numerical triplets would be (110), (210) and (310), respectively, according to the the vertical, horizontal and perpendicular-to-the-figure axes. In this figure, the numerical triplets for the planes drawn are (022), that is, the family of planes does not cut the a axis, but cuts the b and c axes in 2 identical parts, respectively.



The plane drawn on the left side of the figure cuts the a  axis in 2 equal parts, cuts 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 out-of-phase distance (among the families) of 1/3 of the original (see the figures below).

Three families of reticular planes, with indices (110) in three equivalent lattices, showing an out-of-phase distance between them of 1/3 of the interplanar spacing in each family.

The same set of planes of the figure on the left drawn over one of the equivalent lattices. Therefore its Miller indices are (330) and its interplanar spacing is 1/3 of the interplanar spacing of the (110) family.

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.

Formula to calculate the interplanar spacings (dhkl) for a family of planes with Miller indices hkl in a unit cell of parameters a, b, cαβ, γ. Vertical bars (for the triclinic case) mean the function "determinant".  In the trigonal case a=b=c=Aα=β=γ.  In all cases, obviously, the calculated interplanar spacing also represents the distance between the cell origin and the nearest plane of the family.
 
The interested reader should also have a look into the chapter on lattice planes and Miller indices offered by the University of Cambridge.


And now some more concepts on lattices: the so called reciprocal lattice ...

Any plane can also be characterized by a vector (σhkl) 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 (dhkl ).

Any plane can be represented by a vector perpendicular to it.

Consider the family of planes hkl with the interplanar distance dhkl. From the set of vectors normal  to the planes' family, we take the one (σhkl) with length 1/dhkl. 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:

(σhkl) . (d'hkl)  =  (1/dhkl) . (n.dhkl)  =  n    (see left figure below)

n will be 0 for the plane passing through the origin, 1 for the first plane, 2 for the second, etc.

Thus, σhkl represents the whole family of hkl planes having an interplanar spacing given by dhkl. In particular, for the first plane we get:  |σhkl| dhkl = 1.


If we define 1/dhkl, as the length of the vector σhkl, the product of this vector, times the dhkl spacing of the planes family is the unit. 

If we take a vector 2 times longer than σhkl , the interplanar spacing of the corresponding new family of planes would be a half.

If from this normal vector σhkl of length 1/dhkl, we take another vector, n times (integer) longer (n.σhkl), the above mentioned product (|σhkl| dhkl = 1) would imply that the new vector (n.σhkl) 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: d100 = 2.(d200)= 3.(d 300)..., so that  σ100 = (1/2).σ200(1/3).σ300 ...  and similarly for other hkl planes.

Therefore, it appears that the moduli (lengths) of the perpendicular vectors (σhkl) are reciprocal to the interplanar spacings. The end points of these vectors 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, are identified with the same numerical triplets hkl (Miller indices) which represent the corresponding plane family.

Geometrical construction of some points of a reciprocal lattice (green points) from a direct lattice. To simplify, we assume that the third axis of the direct lattice (c) is pependicular to the screen. The red lines represent the reticular planes (perpendicular to the screen) and whose Miller indices are shown in blue. As an example: the reciprocal point with indices (3,1,0) will be located on a vector perpendicular to the plane (3,1,0) and its distance to the origin O is inversely proportional to the spacing of that family of planes.

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, cαβγ, 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 σ100σ010 and σ001, respectively, so that any reciprocal vector can be expressed as a linear combination of these three reciprocal vectors:
σhkl = h a* + k b* + l c*

Position vector of any reciprocal point 

 
Relationships between direct and reciprocal cells . Both cells, and their corresponding lattices, are stuck together.

Two lattices (direct in blue, 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 well-defined metric relations between them:

Metrical relations among the parameters defining the direct and reciprocal cells. V represents the volume of the direct cell and the symbol  x means the cross product between two vectors.  The same type of equations can be written by changing the asterisks to the right side of the equations.  The volume of the direct cell can be calculated as:
V = (a x b) . c = a. b. c (1 - cos2α - cos2β - cos2γ + 2 cos α cos β + 2 cos α  cos γ + 2 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 d100  (|a*| = 1/d100), and that |b*| = 1/d010, and that |c*| = 1/d001. 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 we recommend to use 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 latticeIn case of problems using this applet, please follow the indications shown in this link.

Similarly, it is very educational to visit the pages that on reciprocal space are offered by the University of Cambridge through this link.


The reader is probably asking himself why we need this new concept (the reciprocal lattice). Well, there are reasons which justify it. One of them is that a family of planes can be represented by just one point, which obviously simplifies things. And another important reason is that this new lattice offer us a very simple geometric model that can interpret the diffraction phenomena in crystals. But this will be described in another chapter. Continue!

To the next suggested chapter: Scattering and diffraction
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