Crystal structure: introduction to lattice properties.

This is a 1670 worded article.


Lecture — I, II. 

All articles in this series can be found here

This article is purported to serve as an introduction to a solid state physics course for the 3 year degree physics honors class. It was delivered to the same class on 25 July 2017. Expect some refinement, add-ons, content expansion etc, in the web version. 


Topics covered. 

Lattice translation vectors. 

Lattice with basis.

Unit, primitive and non-primitive cells. 

Bravais and non-Bravais lattice.

Wigner-Seitz cell.


Lattice and crystals

A lattice is a three-dimensional periodic array of identical building blocks. The building blocks are atoms or groups of atoms. The crystals usually come with imperfection of structure and impurities.

The periodicity of crystals is well established by the experimental studies of X-ray, neutron and electron diffraction patterns.

A solid is a crystal if the positions of the atoms in it are exactly periodic. Here is a diagram that represents this property ideally.

Solid State 1

i. Distance between two nearest neighbors is ‘a’ along x-axis and ‘b’ along y-axis, where x–, and y– axes are not necessarily orthogonal.

ii. A perfect crystal maintains the periodicity for -\infty < x < \infty and -\infty < y < \infty. The points A, B, C are equivalents. That means for an observer at A, the environment at A is exactly the same as it is for an observer at B or C.

This is expressed by saying crystals have translational symmetry. e.g. if the crystal is translated by a vector \vec{R} — joining two atoms, the appearance of the crystal remains unchanged.

The atoms have no restrictions as to which location they preside over, as long as that position can be occupied by any  atom, it can be taken over by any other given atom, and all others would relent.

Imperfection in crystals

There are no perfect crystals though, defined the above way. All crystals have some degree of imperfections. There are 3 basic examples of imperfections.

i. Atoms near the surface have a different environment than atoms deep inside the crystal.

ii. Due to thermal vibrations, equilibrium position of atoms are distorted, which depends on temperature T.

iii. Atoms always contain foreign elements known as impurities.

The effect of imperfections can be neglected in very ideal crystals. Imperfections lead to interesting physical properties of crystals. E.g. Resistivity of metals is a result of thermal vibrations of atoms. — We will discuss this at a later time, in this course.

When atoms are replaced by geometrical points, geometrical patterns depicting the periodicity of the crystals are obtained. They do not have any physical contents. Such geometrical patterns are known as “Lattice” or “crystal Lattice”.

Bravais and non-Bravais lattices

There are two classes of lattices, Bravais and non-Bravais lattices.

Bravais Lattice: In a Bravais lattice all lattice points are equivalent, hence all atoms of the crystal are of the same kind.

Non-Bravais Lattice: In a non-Bravais lattice some of the lattice points are not equivalent.

This is easily understood by the following diagram.

solid-state 2

In the diagram the points A, B, C represent those points which they did in the first diagram — when we defined lattice and crystals. They are shown here in black-writing. The points A', B', C' shown in blue writing represent the 3 red points drawn on the lattice.

equivalent: Points A, B, C are equivalent among themselves.

equivalent: Also points A', B', C' are equivalent among themselves.

not equivalent: But point A and point A' are not equivalent.

This can be seen by considering the following property which we already discussed a little earlier, in this article. If we translate the crystal by a vector \vec{R} connecting the points from A to A', the crystal changes, even when A and A' represent atoms of the same kind.

The above is an example of “lattice with a basis”, another name for a non-Bravais type of lattice. A and A' are known as a basis, as also any other set of non-equivalent points. — This last sentence needs some review to establish the correctness of the statement made.

A non-Bravais lattice is an interpenetration of two Bravais lattices, one: lattice represented by A, B, C and the other: lattice represented by A', B', C'.

Basis vectors

Let us consider the lattice shown in the next figure.

solid-state 3

Let its origin coincide at point A. Then position vector of any lattice point is given as: \boxed{\vec{R}_n = n_1 \vec{a} + n_2 \vec{b}}.

Here \vec{a}, \vec{b} are vectors as shown in the diagram. n_1, n_2 are a pair of integers depending upon the position of the lattice point in the lattice. Check two points D and B in the diagram and see that these integers are given by the following values. D: (n_1, n_2)= (0, 2) and B: (n_1, n_2)= (1, 0). Try finding these integer pair values for other points in the diagram.


Notes 


i. basis vectors\vec{a} and \vec{b} are non-collinear and are known as the basis vectors of the lattice.

ii. general representation: All lattice points can be expressed by the above boxed equation.

iii. lattice vectors: All vectors expressed by the above equation are known as lattice vectors, i.e. lattice vectors is the set of vectors \vec{R}_n.

iv. translation symmetry: Lattice is invariant under group of translation expressed by the above equation. Thus the lattice has translational symmetry under all displacements specified by lattice vectors: \vec{R}_n.

v. choice of basis vectors: Choice of basis vectors is not unique. E.g. \vec{a} and \vec{b}\hspace{2pt}' = \vec{a} + \vec{b} is another set of basis vectors.

Unit cells

i. unit cell: the parallelogram represented by the basis vectors \vec{a} and \vec{b} is called a unit cell of the lattice. When the cell is translated, the whole area of the lattice can be covered — only once.

ii. definition of the unit cell: unit cell is the parallelogram with the smallest area that spans the lattice.

iii. choice of unit cell: choice of unit cell is not unique. This follows from the fact that choice of basis vectors is not unique. E.g. \vec{a} and \vec{b}\hspace{2pt}' represent another parallelogram, hence another unit cell.

iv. relation between different unit cells: All unit cells have the same area.

S_{ab} = |\vec{a} \times \vec{b}|

S_{ab'} = |\vec{a} \times \vec{b}\hspace{2pt}'|=|\vec{a}\times (\vec{a}+\vec{b})|=|\vec{a}\times \vec{b}|=S_{ab} because \vec{a} \times \vec{a}=0

v. lattice points per cell: Only one lattice point belongs to a cell. These points are shared by as many cells as there are points in the corner of the cell.

Primitive and non-primitive cells

A non-primitive cell is one where any lattice point can not be expressed as an integer multiple of its basis vectors. Also there would be additional lattice points associated with the cell. Is this same as saying a lattice with basis, as we mentioned earlier? — needs review.

A primitive cell on the other hand is one unit cell where any lattice point can be obtained by integer multiples of its basis vectors. There would be only one lattice point per such cells. Note that when we said above — under the heading unit cells, that there would be only one point per cell, we meant primitive cells.

Here is a diagram that represent these two types of cells. S_2 represents a non-primitive cell and S_1 represents a primitive type of cell.

solid-state 4


Notes


i. difference between primitive and non-primitive type of unit cells: area of a non-primitive cell is an integral multiple of area of a primitive cell.

ii. difference between non-primitive and non-Bravais lattice: non-primitive cells are unrelated to non-Bravais type of lattice. Non-primitive cells have an arbitrary choice of basis vectors whereas non-Bravais lattices have non-equivalent lattice points.

Three dimensional lattice

A lattice transformation — translation here, defined by the following vector equation represents a 3-dimensional lattice.

\boxed{\vec{r}\hspace{2pt}' = \vec{r} + n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3}

i. volume of the 3-D lattice unit cell: The cells of the parallelepiped constructed by primitive basis vectors \vec{a}_i (i = 1, 2, 3)  has minimum volume.  This volume is given by: V_c = \vec{a}_1.(\vec{a}_2 \times \vec{a}_3).

ii. nature of the basis vectors of 3-D lattice: \vec{a}_i are non-co-planar.

iii. primitive and non-primitive cells in 3-D lattice: All primitive cells have equal volume. Non-primitive cells have additional lattice points.

iv. non-Bravais type in 3-D lattice: Non-Bravais lattices are possible.

v. position of center of j-th atom: \boxed{\vec{r}_j = x_j \vec{a}_1 + y_j \vec{a}_2 + z_j \vec{a}_3, \hspace{10pt}0\leq x_j, y_j, z_j \leq1}.

Here are two diagrams that show case the lattice properties in case of 3-D lattices.

solid-state 5

solid-state 6

Wigner-Seitz cell

All primitive cells do not have all the possible symmetries of a Bravais lattice. Wigner-Seitz cell is one example of a primitive cell which possesses all possible symmetries of a Bravais lattice. This is because in constructing a W-S cell one does not refer to any particular choice of primitive or basis vectors.

Ways of constructing a W-S cell: A particular lattice point is chosen. It is then connected to all neighboring closest points. At the mid-point of these joining lines planes or new lines are drawn so that they bisect the joining line, perpendicularly.

The resulting minimal volume or area is the W-S cell. Here are two diagrams that show the W-S cell and the way to construct it.

Coordinate of any point inside a cell is given by the same expression that we wrote for position of the center of the j-th atom \boxed{\vec{r}_j = x_j \vec{a}_1 + y_j \vec{a}_2 + z_j \vec{a}_3, \hspace{10pt}0\leq x_j, y_j, z_j \leq1}.



Categories: basic physics, Courses I developed, Lectures & Presentations, Physics, Solid Sate Physics, Teaching

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  1. Fundamental types of crystal lattices and their symmetry operations. – "Invariance" from M Dash Foundation

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