Category: Advanced Chemistry

Posted on

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.

next article: interplanar distance
Previous article: equation of a plane
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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.

 

next article: Bravais lattice (3d)
Previous article: crystallographic restriction theorem
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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}

 

Next article: equation of a plane
Previous article: cubic
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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.

 

next article: crystallographic restriction theorem
Previous article: lattice vectors
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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.

 

next article: lattice vectors
Previous article: overview
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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.

 

next article: Unit cell and lattice
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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}}

 

next article: fractional coordinates
Previous article: unit cell and lattice
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

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 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).

 

next article: orthorhombic
Previous article: bravais lattice (3d)
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

Bravais lattice 3D (crystallography)

A three-dimensional Bravais lattice is obtained by replicating any of the five two-dimensional lattices and stacking the replicated lattices above one another. The space lattices formed have lattice points that are generated by a set of translation operations described by the three-dimensional position vector r = ua + vb + wc.

There are exactly fourteen types of three-dimensional Bravais lattices that are grouped into seven lattice systems:

Lattice system

Primitive

Base-centred

Body-centred

Face-centred
Triclinic \bigstar
Monoclinic \bigstar \bigstar
Orthorhombic \bigstar \bigstar \bigstar \bigstar
Tetragonal \bigstar \bigstar
Rhombohedral \bigstar
Hexagonal \bigstar
Cubic \bigstar \bigstar \bigstar

The unit cells of these lattices form the building blocks of all crystalline solids. We shall derive them in the following sections.

 

next article: triclinic and monoclinic
Previous article: BRAVAIS LATTICE (2D)
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Posted on

X-ray crystallography: orthorhombic

Space lattices that are composed of orthorhombic unit cells are formed by stacking two-dimensional lattices depicted in figure II (where IaI ≠ IbI and the angle between the basis vectors is not 90o) in three different ways 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 (IIa), resulting in the primitive orthorhombic unit cell (IIb), with IaI ≠ IbI ≠ IcI and α = β = γ = 90o.

The second way is to stack a second layer such that its lattice points are in the middle of rectangles formed by the lattice points of the first layer (IIc). A third layer is then stacked directly above the first layer, giving a non-primitive unit cell known as body-centred orthorhombic (IId) with IaI ≠ IbI ≠ IcI and α = β = γ = 90o. 

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

The base-centred orthorhombic unit cell (IIIb) happens to be the same unit cell obtained by stacking the two-dimensional lattices of figure III (see diagram below) directly above one another (IIIa) to give figure IIIb’ (see above diagram).

We have used the non-primitive unit cell (IIIa) of the two-dimensional lattice of figure III as the reference cell to form IIIb’ instead of the primitive rhombic unit cell. This is because the resultant three-dimensional unit cell that is formed by IIIa is easier to visualize and reveals the higher symmetry of the orthorhombic C lattice than the resultant unit cell formed by the primitive rhombic unit cell.

The other way to generate a lattice (using figure III) with a 2-fold rotational symmetry is to stagger the second layer as shown in IIIc, with a third layer directly above the first layer. This gives a non-primitive unit cell called face-centred orthorhombic (IIId), with IaI ≠ IbI ≠ IcI and α = β = γ = 90o. All orthorhombic unit cells are described by the same parameters of IaI ≠ IbI ≠ IcI and α = β = γ = 90o. All orthorhombic unit cells also have three perpendicular two-fold rotational axes.

 

next article: tetragonal
Previous article: triclinic and monoclinic
Content page of X-ray crystallography
Content page of advanced chemistry
Main content page
Mono Quiz