Chemistry and Maths #2: Crystals

Mathematics is essential to the study of crystals and their lattice structure.

A unit cell is the smallest group of particles that contains the repeating pattern of the structure. Therefore, the unit cell completely defines the structure of the lattice, which is built from repetitive translation of the unit cell along an axis. These types of lattices are called Bravais Lattices; all lattice points are equivalent.

Classification

Classifying crystals is down to the symmetry of their lattice structure, as it is their defining property. By ‘symmetry’ I mean that under certain ‘operations’ the crystal remains unchanged.

7 lattice systems:

TRICLINIC

Triclinic

MONOCLINIC

Simple:         Base-centred:

Monoclinic, simple    Monoclinic, centered

ORTHORHOMBIC:

Simple:           Base-centred:       Body-centred:      Face-centred:

Orthorhombic, simple      Orthorhombic, base-centered             Orthorhombic, body-centered             Orthorhombic, face-centered

RHOMBOHEDRAL:

Rhombohedral

TETRAGONAL:

Simple:                Body-centred:

Tetragonal, simple          Tetragonal, body-centered

HEXAGONAL:

Hexagonal

CUBIC:

Simple:         Body-Centred:       Face-Centred:

Cubic, simple     Cubic, body-centered              Cubic, face-centered

Note that circles in the diagrams represent the atoms in the lattice.

To give a few examples, Zinc and Magnesium have a hexagonal lattice structure, whilst Iron and Chromium have a body-centred cubic structure.

Atomic Packing Fraction

The Atomic packing fraction gives the efficiency with which the available space is being filled by atoms. It is defined as:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mi»P«/mi»«mi»F«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»V«/mi»«mi»o«/mi»«mi»l«/mi»«mi»u«/mi»«mi»m«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mi»t«/mi»«mi»o«/mi»«mi»m«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mi»i«/mi»«mi»n«/mi»«mo»§nbsp;«/mo»«mi»u«/mi»«mi»n«/mi»«mi»i«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»e«/mi»«mi»l«/mi»«mi»l«/mi»«/mrow»«mrow»«mi»V«/mi»«mi»o«/mi»«mi»l«/mi»«mi»u«/mi»«mi»m«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»u«/mi»«mi»n«/mi»«mi»i«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»e«/mi»«mi»l«/mi»«mi»l«/mi»«/mrow»«/mfrac»«/math»

Let’s look at two examples.

Cubic

The volume of atoms in a unit cell«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»*«/mo»«mn»4«/mn»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/mrow»«mn»3«/mn»«/mfrac»«/math».

Looking at the diagram to the right, the volume of unit cell=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»a«/mi»«mn»3«/mn»«/msup»«mo»=«/mo»«mo»(«/mo»«mn»2«/mn»«mi»r«/mi»«msup»«mo»)«/mo»«mn»3«/mn»«/msup»«mo»=«/mo»«mn»8«/mn»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/math»

Hence, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mi»P«/mi»«mi»F«/mi»«mo»=«/mo»«mfrac»«mrow»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/mrow»«mrow»«mn»8«/mn»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mn»52«/mn»«mo»%«/mo»«/math».

Face-centred cube

Considering a cube of length a and atoms of radius r are placed at the corners as well as at the face centre. Length of face diagonal √2a=4r.

Volume of unit cell«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mrow»«mn»4«/mn»«mo»*«/mo»«mn»4«/mn»«/mrow»«mn»3«/mn»«/mfrac»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«mo»=«/mo»«mfrac»«mn»16«/mn»«mn»3«/mn»«/mfrac»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/math».

In a face centred cube, each face has one atom along with 8 corner atoms. The atoms at the faces are equally shared by two unit cells and the corner atoms by 8 unit cells. So the number of atoms per unit cell is=(1/8)*8(corner atoms)+(1/2)*6(atoms at face)=4.

Volume of a unite cell«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«msup»«mi»a«/mi»«mn»3«/mn»«/msup»«mo»=«/mo»«mo»(«/mo»«mfrac»«mrow»«mn»4«/mn»«mi»r«/mi»«/mrow»«msqrt»«mn»2«/mn»«/msqrt»«/mfrac»«msup»«mo»)«/mo»«mn»3«/mn»«/msup»«/math»

Thus,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mi»P«/mi»«mi»F«/mi»«mo»=«/mo»«mfrac»«mrow»«mfrac»«mn»16«/mn»«mn»3«/mn»«/mfrac»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«/mrow»«mrow»«mo»(«/mo»«mfrac»«mrow»«mn»4«/mn»«mi»r«/mi»«/mrow»«msqrt»«mn»2«/mn»«/msqrt»«/mfrac»«msup»«mo»)«/mo»«mn»3«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»§#960;«/mi»«mrow»«mn»3«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«/mfrac»«mo»=«/mo»«mn»74«/mn»«mo»%«/mo»«/math»

 

Check out my post on Wednesday about statistical thermodynamics! M x

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4 comments

  1. I am looking quite forward to a post about thermodynamics. It’s maybe not my first academic love but it’s the first field that really captured my imagination in eye-opening ways.

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