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Crystallographic restriction theorem

The crystallographic restriction theorem generally states that the rotational symmetries of a crystal are limited.

We have seen from a previous section that a lattice is formed by repeating lattice points that have the same environment. It is this fundamental geometric property of a lattice that restricts the number of rotational symmetries of a space lattice and hence the types of unit cells.

To illustrate the crystallographic restriction theorem, let’s consider a row of lattice points (indicated by the line connecting blue dots in figure I of the below diagram) that are equally spaced by the basis vector  a.

Assuming that the lattice has a rotational symmetry given by the angle \alpha, a rotation about the lattice point X by +\alpha, around an axis perpendicular to the plane of this page, maps the blue lattice point X- to the pink lattice point Y. Similarly, a rotation about the same lattice point X by –\alpha maps the blue lattice point X+ to the green lattice point Z.

Since the rotations are symmetry operations, Y and Z are also lattice points. As mentioned in an earlier article, the lattice points in a two-dimensional lattice are described by the position vector r = ua + vb where u, v ∈ \mathbb{Z}. Therefore, YZ must be a lattice vector separated by an integer multiple of a, i.e. YZ = ua + 0b or YZ = ua, where u ∈ \mathbb{Z} . With reference to figure II of the above diagram,

cos\alpha =\frac{u\textbf{\textit{a}/2}}{\textbf{\textit{a}}}=\frac{u}{2}\; \; \; \; \; \; \; (4)

Since -1 ≤ cos\alpha ≤ 1,

-1\leq \frac{u}{2}\leq 1\; \; \; \; \Rightarrow \; \; \; \;-2\leq u\leq 2\; \; \; \;u\in \mathbb{Z}\; \; \; \; \; \; \; (5)

Substituting eq5 in eq4, the possible values of \alpha are 180o, 120o, 90o, 60o, 0o (or 360o). As we have assumed that a rotation of the lattice by the angle \alpha transforms the lattice into a state that is indistinguishable from the starting state, the possible values of \alpha correspond to 2-, 3-, 4-, 6- and 1-fold symmetry of the two-dimensional lattice.

For a space lattice in three dimensions, consider a three-dimensional lattice vector r = ua + vb + wc that is neither parallel nor perpendicular to the axis of rotation (see diagram below). A rotation by \alpha maps r onto another lattice vector r’ that has the same magnitude as r. Since both r and r’ are lattice vectors, their difference r’ – r is also a lattice vector, which lies on a plane that is perpendicular to the axis of rotation.

By varying the integers u, v and w, we can extend m to a row of points mno that are spaced \vert \boldsymbol{\mathit{r'}}-\boldsymbol{\mathit{r}}\vert=a apart. This reduces a three-dimensional lattice proof of the crystallographic restriction theorem to a two-dimensional one, which is already given above.

We have proven the crystallographic restriction theorem with trigonometry. The proof can also be made with linear algebra (see this article).

 

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Bragg’s law (crystallography)

Bragg’s law states that the incidence and scattered waves in x-ray diffraction make the same angle with a lattice plane.

X-ray is a form of electromagnetic radiation, which has an alternating electric field component. When a beam of X-rays encounters a charged particle, such as an electron in an atom, its rapidly alternating electric field causes the electron to oscillate and form a dipole. Since an accelerated charge emits electromagnetic radiation, the accelerated electron (sinusoidal change of velocity with time) reemits X-ray at the same frequency as the incident radiation. This phenomenon is known as scattering.

X-ray diffraction is a technique that relies on the interference of X-rays, which are scattered by atoms in a crystal, for analysing properties of the crystal. As the overall electron density of an atom is approximately spherical, X-ray scattered by the atom is consequently spherical (see diagram below). This is analogous to the atoms being the source of spherical wavelets, as stated in Huygens’ principle. The scattered X-rays therefore interfere with one another to form a distinct pattern.

In 1913, William Bragg and his son Lawrence Bragg considered the specular waves scattered by atoms in a crystal, i.e., incident and scattered waves that make the same angle with a lattice plane (see diagram below). They proposed that scattered specular X-ray waves, from atoms separated by an interplanar distance dhkl, interfere constructively when the difference between their respective total path lengths, which is AB + BC, is an integer multiple of the wavelength of the X-ray.

AB=BC=d_{hkl}sin\theta \; \; \; \; \; \; \Rightarrow \; \; \; \; \; \; AB+BC=2d_{hkl}sin\theta

So, according to Bragg’s law:

2d_{hkl}sin\theta =n\lambda \; \; \; \; \; \; \; (10)

where n ∈ \mathbb{Z} and n is called the order of diffraction.

Since both the order of diffraction and the common factor between planes are integers, we can substitute eq5 where dhkl = ndnh,nk,nl in eq10:

2d_{nh,nk,nl}sin\theta =\lambda \; \; \; \; \; \; \; (11)

Combining eq7 where \frac{1}{{d_{nh,nk,nl}}^{2}}=\frac{(nh)^2+(nk)^2+(nl)^2}{a^2}  for a primitive cubic unit cell and eq11, we have:

sin\theta =\frac{\lambda \sqrt{(nh)^2+(nk)^2+(nl)^2}}{2a}\; \; \; \; \; \; \; (12)

where n is now the common factor between planes.

This eliminates the need to describe a signal in terms of the order of diffraction. In other words, each signal now refers to the first order diffraction from the (nh nk nl) plane. Eq12 is often written simply as:

sin\theta =\frac{\lambda \sqrt{h^2+k^2+l^2}}{2a}\; \; \; \; \; \; \; (13)

because it is understood that h = nh, k = nk and l = nl.

The significance of eq12 and eq13 is that a set of lattice planes, with a specific Miller index (hkl) in a solid composed of a particular unit cell dimension, scatters X-rays that interfere constructively at a specific specular angle. Therefore, eq12 and similar equations for other unit cell types allow us to determine unit cell dimensions, if the angle of diffraction and the Miller index (hkl) are known. This is accomplished experimentally via X-ray diffraction techniques, one of which is powder X-ray diffraction.

 

Question

What if we consider a scattered X-ray vector in a different direction with respect to the one used in Bragg’s law, e.g. one that points below the lattice plane?

Answer

You get a different constructive interference (diffraction) formula known as the Laue equations, i.e., you have a different mathematical approach using different criteria to describe the same diffraction pattern.

 

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Miller indices (crystallography)

The Miller index is a system developed by British mineralogist William Miller in 1839 for describing the orientation of lattice planes. Since a normal vector to a plane describes the orientation or direction of the plane, the Miller index of a plane must have parameters consistent with the normal vector. Recall that the scalar equation of a plane is given by eq2 of the previous section:

\frac{x}{D/A}+\frac{y}{D/B}+\frac{z}{D/C}=1

In the case of a space lattice, let’s rewrite it as:

\frac{x}{a/h}+\frac{y}{b/k}+\frac{z}{c/l}=1

where a/h, b/k and c/l are the intercepts of the plane with the x-axis, y-axis and z-axis respectively and h, k, l are by definition, integers.

In the diagram above, the plane intercepts the x-axis, y-axis and z-axis at a, b/2 and ∞c respectively. Hence,

\frac{a}{h}=a\; \; \; \Rightarrow \; \; \; h=1

\frac{b}{k}=\frac{b}{2}\; \; \; \Rightarrow \; \; \; k=2

\frac{c}{l}=\infty\: c\; \; \; \Rightarrow \; \; \; l=0

The variables h, k, l are called the Miller indices and are used to describe the orientation of planes via the notation (hkl).

The plane, in this case, is (120). If we draw a position vector that is perpendicular to the plane (depicted as the purple arrow in the diagram), we can describe the vector with the direction [120], which is consistent with the Miller index of the plane. Furthermore, the Miller index of a plane may contain negative numbers, e.g. (1 -2 0) which is denoted by \left ( 1\bar{2}0 \right ).

Consider a crystal with monoclinic unit cells. Due to the periodic arrangement of lattice points, we can find many different planes intersecting the points. The diagram below depicts parallel planes with the middle one having the Miller index of (001) and the top plane of (00½). As whole numbers are preferred in the (hkl) notation, (00½) is rewritten as (001) by multiplying each of the three numbers by the smallest integer that will give whole numbers.

If we shift the origin from the far left lattice point of the bottom layer to the far left lattice point of the middle layer, the top plane becomes (001) and the bottom plane becomes (00\bar{1}). Furthermore, the parallel planes are equivalent by a rotation of the lattice and form a family of planes. In general, we denote the set of all planes that are equivalent to a particular reference plane (hkl) by the symmetry of the lattice as {hkl}. Therefore, we describe this family of planes as {001}. This example shows that the Miller notation does not require axes to be orthogonal to each other.

 

Question

Can we denote the family of planes in the above diagram as \left \{ 00\bar{1} \right \}?

Answer

Yes. However, the convention is to select the plane that is closest to the origin that has positive indices as the reference plane for the family. Hence, the notation {001} is preferred.

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Bravais lattice 2D (crystallography)

A Bravais lattice has lattice points with the same environment such that the lattice has at least one of the rotational symmetries predicted by the crystallographic restriction theorem. In other words, lattices are Bravais lattices only if they meet the symmetry and ‘same environment’ criteria. Such lattices have patterns depicted in figures I to V in the diagram below.

Notice that a lattice with one-fold rotational symmetry is not included among the two-dimensional Bravais lattices. The reason is as follows:

The purpose of defining two-dimensional Bravais lattices is to use them as a foundation for constructing three-dimensional Bravais lattices and their associated unit cells, which are ultimately of interest to chemists. Since there are many possible lattices with one-fold rotational symmetry in two dimensions, it is more practical to utilise the five specified two-dimensional lattices to develop all possible unit cells, including those that can replicate to form lattices with one-fold rotational symmetry (see the next few articles for details).

 

Question

Why is the honeycomb lattice (figure VI) not a Bravais lattice?

Answer

Even though the honeycomb lattice satisfies the rotational symmetry criterion, not all lattice points have the same environment. An easy way to determine whether all lattice points have the same environment is to see if their nearest neighbours are in the same directions. With reference to the diagram above, lattice point 1 of figure VI has nearest neighbours in the northeast, northwest and south directions, while lattice point 2 has nearest neighbours in the southeast, southwest and north directions. The honeycomb lattice fails the ‘same environment’ criterion and therefore is not a Bravais lattice. Nonetheless, we can demarcate a unit cell for the lattice by combining two lattice points to establish a single basis (shaded pink). This method of pairing lattice points to form unit cells is commonly used in ionic crystals to determine their lattice types.

 

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Proof for distance between lattice layers

 

Question 1

Show that the triple hexagonal lattice (Vc) becomes a cubic P lattice when the perpendicular distance between two layer of lattices, h, is(IaI√6)/6.

Answer 1

The diagram below shows two layers of the triple hexagonal lattice Vc.

Focusing on just one segment of the space lattice (IVb), IaI refers to the separation between two same-layer lattice points of Vc and is also the surface diagonal of IVb. The body diagonal AB of IVb is determined using Pythagoras’ theorem as follows:

Each side of the cubic P unit cell is IaI/√2 and

(AB)^{2}=(BC)^{2}+(AC)^{2}

AB=\sqrt{\left | \textbf{\textit{a}}\right |^{2}+(\frac{\left |\textbf{\textit{a}} \right |}{\sqrt{2}})^{2}}\; =\left | \textbf{\textit{a}}\right |\sqrt{\frac{3}{2}}

Furthermore, AB is thrice the perpendicular distance between two layers of the triple hexagonal lattice Vc (see below diagram, which is IVb rotated for a better view). 

Since AB = 3h,

h=\frac{\left |\textbf{\textit{a}} \right |}{3}\sqrt{\frac{3}{2}}=\frac{\left | \textbf{\textit{a}}\right |\sqrt{6}}{6}

 

 

Question 2

Show that the triple hexagonal lattice (Vc) becomes a cubic F lattice when the perpendicular distance between two layer of lattices, h, is(IaI√6)/3.

Answer 2

Similar to the cubic P unit cell, IaI refers to the separation between two same-layer lattice points of Vc in the cubic F unit cell, e.g. the length of BC. Using Pythagoras’ theorem,

(CE)^{2}=(CH)^{2}+(HE)^{2}

By the symmetry of the cubic F unit cell, BC = CE = IaI and CH = HE. Therefore, CH = IaI/√2 and hence, AI = 2IaI/√2. Furthermore,

(AJ)^{2}=(AI)^{2}+(IJ)^{2}

Since (IJ)2 = (KI)2 + (KJ)2 = (AI)2 + (AI)2 = 2(AI)2 = 4IaI2,

AJ=\sqrt{\left ( \frac{2\left | \textbf{\textit{a}}\right |}{\sqrt{2}} \right )^{2}+4\left | \textbf{\textit{a}}\right |^{2}}=\left | \textbf{\textit{a}}\right |\sqrt{6}

Just like the cubic P unit cell, the body diagonal AJ of the cubic F unit cell (IVe) is thrice the perpendicular distance between two layers of Vc. Therefore,

h=\frac{\left | \textbf{\textit{a}}\right |\sqrt{6}}{3}

 

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Fractional coordinates (crystallography)

Fractional coordinates are numbers (between 0 and 1) that indicate the positions of lattice points in a unit cell.

The edges of a unit cell, which may be primitive or non-primitive, are defined by the basis vectors a, b and c, and any lattice point within the unit cell is described by the position vector

\textbf{\textit{r}}=x'\textbf{\textit{a}}+y'\textbf{\textit{b}}+z'\textbf{\textit{c}}

where x’, y’, z‘ are fractions with 0 ≤ x’ ≤ 1, 0 ≤ y’ ≤ 1, 0 ≤ z’ ≤ 1.

Therefore, (x’, y’, z’) are called fractional coordinates (see diagram below).

With the different combinations of the positions of lattice points and angles between primitive translation vectors, one may think that there are possibly an infinite number of types of unit cell. However, the number is finite as explained in the next section.

 

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Unit cell and lattice (crystallography)

A unit cell is a parallelepiped that serves as the simplest repeating unit of a crystal. A unit cell must fill all the space of the crystal when replicated. As a result, spherical unit cells do not exist.

 

 

At each corner of the unit cell is a blue point known as a lattice point (see above diagram). The environment of any lattice point is equivalent to the environment of any other lattice point in the crystal. An infinite three-dimensional array of lattice points forms a three-dimension lattice called a space lattice. A unit cell that only has lattice points at its corners is called a primitive unit cell. Unit cells are sometimes chosen in the non-primitive form for convenience.

The edges of a unit cell are, by convention, chosen to be right-handed (a × b is the direction of c). The angles α, β and γ are between b and c, c and a, and a and b, respectively.

 

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X-ray crystallography: overview

X-ray crystallography is a technique that uses x-rays to determine three-dimensional structures of crystals. A single crystal of a chemical species is irradiated with a monochromatic beam of x-ray, which diffracts as it passes through the crystal. The diffracted radiation is collected by an image sensor and the data is analysed by a computer to give relative positions of atoms, bond lengths and angles, symmetry of the crystal and dimensions of the unit cell.

To fully comprehend the details of how x-ray crystallography works, we need to understand a few more concepts.

 

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Lattice vectors (crystallography)

In crystallography, it is convenient to explain certain concepts using vectors and matrices. For example, a lattice point in a two-dimensional lattice can be described by the position vector, r, in the form:

\textbf{\textit{r}}=u\textbf{\textit{a}}+v\textbf{\textit{b}}

where u and v are integers; a and b are primitive translation vectors or basis vectors.

The red lattice point in the above diagram is represented by r = a + 2b. Basis vectors in two dimensions are usually chosen to be the shortest vectors linking two pairs of neighbouring lattice points. They can be selected anywhere in the lattice as long as they originate from the same lattice point. In three dimensions, we have:

\textbf{\textit{r}}=u\textbf{\textit{a}}+v\textbf{\textit{b}}+w\textbf{\textit{c}}

 

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X-ray crystallography: triclinic and monoclinic

As mentioned in previous sections, a unit cell is a parallelepiped that is the simplest repeating unit of a three-dimensional Bravais lattice, which is obtained by replicating one of the five two-dimensional lattices and stacking the replicated lattices above one another. The first two types of unit cells that form Bravais lattices are the triclinic and monoclinic unit cells.

There are three ways to stack the layers of the two-dimensional lattice depicted in figure I (where IaI ≠ IbI and the angle between the basis vectors is not 90o) such that the space lattice maintains a 2-fold rotational symmetry.

The first way is to stack successive layers that are directly above one another (Ia), resulting in a primitive monoclinic unit cell(Ib), with IaI ≠ IbI ≠ IcI and α = β = 90o, γ≠ 90o.

 

The second way is to stack a second layer such that its lattice points are in between the lattice points of the first layer (Ic). A third layer is then stacked directly above the first layer, giving a non-primitive unit cell known as a base-centred monoclinic unit cell (Id), with IaI ≠ IbI ≠ IcI and α = β = 90o, γ≠ 90o.

The third way is the stack the second layer with its lattice points above the middle of parallelograms formed by the lattice points of the first layer (Ie). A third layer is then stacked directly above the first layer. With a different choice of basis vectors (If), we again obtain a based-centred monoclinic unit cell (Id).

If the second layer is stacked in a way that 2-fold rotational symmetry is no longer preserved in the space lattice (Ig), we have a triclinic unit cell (Ih) with IaI ≠ IbI ≠ IcI and α ≠ β ≠ γ.

All monoclinic units cells are described by the parameters of IaI ≠ IbI ≠ IcI and α = β = 90o, γ≠ 90o. Furthermore, all monoclinic unit cells have just one 2-fold rotational axis of symmetry. Although it is possible to outline a primitive unit cell for the based-centred monoclinic lattice, the primitive unit cell has two angles that are not equal to 900 and is therefore less symmetrical and not monoclinic.

A triclinic unit cell, on the other hand, has no essential symmetry as it only has a 1-fold rotational axis of symmetry, which is trivial. We can therefore construct different three-dimensional Bravais lattices, each with 1-fold rotational symmetry, by replicating triclinic unit cells of different dimensions. This is the reason why a lattice with one-fold rotational symmetry is not included as one of the two-dimensional Bravais lattices (see earlier article).

 

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