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The unit cell

 

Since the crystal lattice is made up of a regular arrangement which repeats in three dimensions, we can save ourselves a great deal of work by considering the simple repeating unit rather than the entire crystal lattice. The basic repeating unit is known as the unit cell. Crystalline solids often have flat, well-defined faces that make definite angles with their neighbors and break cleanly when struck. These faces lie along well-defined directions in the unit cell.

The unit cell is the smallest, most symmetrical repeating unit that, when translated in three dimensions, will generate the entire crystal lattice.

 

It is possible to have a number of different choices for the unit cell. By convention, the unit cell that reflects the highest symmetry of the lattice is the one that is chosen. A unit cell may be thought of as being like a brick which is used to build a building (a crystal). Many bricks are stacked together to create the entire structure. Because the unit cell must translate in three dimensions, there are certain geometrical constraints placed upon its shape. The main criterion is that the opposite faces of the unit cell must be parallel. Because of this restriction there are only six parameters that we need to define in order to define the shape of the unit cell. These include three edge lengths a, b, and c and three angles α size 12{α} {} , β size 12{β} {} and γ size 12{γ} {} . Once these are defined all other distances and angles in the unit cell are set. As a result of symmetry, some of these angles and edge lengths may be the same. There are only seven different shapes for unit cells possible. These are given in the chart below.

Unit Cell Type Restrictions on Unit Cell Parameters Highest Type of Symmetry Element Required
Triclinic a is not equal to b is not equal to c; α size 12{α} {} is not equal to β size 12{β} {} is not equal to γ size 12{γ} {} . no symmetry is required, an inversioncenter may be present
Monoclinic a is not equal to b is not equal to c α size 12{α} {} = γ size 12{γ} {}  90 ° size 12{"90"°} {} β size 12{β} {} is not equal to 90 ° size 12{"90"°} {} . highest symmetry element allowed is aC2 axis or a mirror plane
Orthorhombic a is not equal to b is not equal to c α size 12{α} {} = β size 12{β} {} = γ size 12{γ} {}  90 ° size 12{"90"°} {} has three mutually perpendicularmirror planes and/or C2 axes
Tetragonal a =b is not equal to c α size 12{α} {} = β size 12{β} {} = γ size 12{γ} {}  90 ° size 12{"90"°} {} has one C4 axis
Cubic a =b =c α size 12{α} {} = β size 12{β} {} = γ size 12{γ} {}  90 ° size 12{"90"°} {} has C3 and C4 axes
Hexagonal, Trigonal a =b is not equal to c α size 12{α} {} = β size 12{β} {} = 90 ° size 12{"90"°} {}  γ size 12{γ} {}  120 ° size 12{"120"°} {} C6 axis (hexagonal); C3 axis (trigonal)
Rhombohedral* a =b =c α size 12{α} {} = β size 12{β} {} = γ size 12{γ} {} is not equal to 90 ° size 12{"90"°} {} C3 axis (trigonal) 

*There is some discussion about whether the rhombohedral unit cell is a different group or is really a subset of the trigonal/hexagonal types of unit cell.

Stoichiometry

You will be asked to count the number of atoms in each unit cell in order to determine the stoichiometry (atom-to-atom ratio) or empirical formula of the compound. However, it is important to remember that solid state structures are extended, that is, they extend out in all directions such that the atoms that lie on the corners, faces, or edges of a unit cell will be shared with other unit cells, and therefore will only contribute a fraction of that boundary atom. As you build crystal lattices in these exercises you will note that eight unit cells come together at a corner. Thus, an atom which lies exactly at the corner of a unit cell will be shared by eight unit cells which means that only⅛of the atom contributes to the stoichiometry of any particular unit cell. Likewise, if an atom is on an edge, only¼of the atom will be in a unit cell because four unit cells share an edge. An atom on a face will only contribute½to each unit cell since the face is shared between two unit cells.

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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Ramkumar Reply
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Sravani Reply
what is system testing?
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Source:  OpenStax, Honors chemistry lab fall. OpenStax CNX. Nov 15, 2007 Download for free at http://cnx.org/content/col10456/1.16
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