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b. If the radius of each atom in this cell is r, what is the equation that describes the volume of the cube generated in terms of r? (Note that the face of the cube is defined by the position of the rods and does not include the whole sphere.)



c. Draw the z-diagram for the unit cell layers.



d. To how many different cells does a corner atom belong? What is the fractional contribution of a single corner atom to a particular unit cell?


e. How many corner spheres does a single unit cell possess?


f. What is the net number of atoms in a unit cell? (Number of atoms multiplied by the fraction they contribute)


g. Pick an interior sphere in the extended array. What is the coordination number (CN) of that atom? In other words, how many spheres are touching it? .


h. What is the formula for the volume of a sphere with radius r?


i. Calculate the packing efficiency of a simple cubic unit cell (the % volume or space occupied by atomic material in the unit cell). Hint: Consider the net number of atoms per simple cubic unit cell (question g) the volume of one sphere (question i), and the volume of the cube (question b).




B. body-centered cubic (bcc) structure

a. Draw the z diagrams for the layers.



 b. Fill out the table below for a BCC unit cell

Atom type Number Fractional Contribution Total Contribution Coordination Number

c. What is the total number of atoms in the unit cell?


d. Look at the stacking of the layers. How are they arranged if we call the layers a, b, c, etc.?


e. If the radius of each atom in this cell is r, what is the formula for the volume of the cube generated in terms of the radius of the atom? (See diagrams below.)





f. Calculate the packing efficiency of the bcc cell: Find the volume occupied by the net number of spheres per unit cell if the radius of each sphere is r; then calculate the volume of the cube using r of the sphere and the Pythagoras theorem ( a 2 + b 2 = c 2 size 12{a rSup { size 8{2} } +b rSup { size 8{2} } =c rSup { size 8{2} } } {} ) to find the diagonal of the cube.







C. the face centered cubic (fcc) unit cell

A. fill out the following table for a fcc unit cell.

Atom type Number Fractional Contribution Total Contribution Coordination Number

b. What is the total number of atoms in the unit cell?

c. Using a similar procedure to that applied in Part B above; calculate the packing efficiency of the face-centered cubic unit cell.





  • Close-Packing

a. Compare the hexagonal and cubic close-packed structures.



b. Locate the interior sphere in the layer of seven next to the new top layer. For this interior sphere, determine the following:

Number of touching spheres: hexagonal close-packed (hcp) cubic close-packed (ccp)
on layer below    
on the same layer    
on layer above    
TOTAL CN of the interior sphere    

c. Sphere packing that has this number (write below) of adjacent and touching nearest neighbors is referred to as close-packed. Non-close-packed structures will have lower coordination numbers.


d. Are the two unit cells the identical? 


e. If they are the same, how are they related? If they are different, what makes them different? 


f. Is the face-centered cubic unit cell aba or abc layering? Draw a z-diagram.


 III.Interstitial sites and coordination number (CN)

a. If the spheres are assumed to be ions, which of the spheres is most likely to be the anion and which the cation, the colorless spheres or the colored spheres? Why?


b. Consider interstitial sites created by spheres of the same size. Rank the interstitial sites, as identified by their coordination numbers, in order of increasing size (for example, which is biggest, the site with coordination number 4, 6 or 8?).


 c. Using basic principles of geometry and assuming that the colorless spheres are the same anion with radius r A in all three cases, calculate in terms of rA the maximum radius, rC, of the cation that will fit inside a hole of CN 4, CN 6 and CN 8. Do this by calculating the ratio of the radius of to cation to the radius of the anion: r C / r A size 12{r rSub { size 8{C} } /r rSub { size 8{A} } } {} .



d. What terms are used to describe the shapes (coordination) of the interstitial sites of CN 4, CN 6 and CN 8?

CN 4: ________________

CN 6: _______________

CN 8: ________________


Iv.ionic solids

A. Cesium Chloride

1. Fill the table below for Cesium Chloride

  Colorless spheres Green spheres
Type of cubic structure    
Atom represented    



2. Using the simplest unit cell described by the colorless spheres, how many net colorless and net green spheres are contained within that unit cell?


3. Do the same for a unit cell bounded by green spheres as you did for the colorless spheres in question 4.


4. What is the ion-to-ion ratio of cesium to chloride in the simplest unit cell which contains both? (Does it make sense? Do the charges agree?)


B. Calcium Fluoride

 1. Draw the z diagrams for the layers (include both colorless and green spheres).



2. Fill the table below for Calcium Fluoride

  Colorless spheres Green spheres
Type of cubic structure    
Atom represented    



3. What is the formula for fluorite (calcium fluoride)?


C. Lithium Nitride

1. Draw the z diagrams for the atom layers which you have constructed.



2. What is the stoichiometric ratio of green to blue spheres?


3. Now consider that one sphere represents lithium and the other nitrogen. What is the formula?

4. How does this formula correspond to what might be predicted by the Periodic Table?


D. Zinc Blende and Wurtzite

Fill in the table below:

  Zinc Blende Wurtzite
Stoichiometric ratio of colorless to pink spheres    
Formula unit (one sphere represents and the other the sulfide ion)    
Compare to predicted from periodic table    
Type of unit cell    


Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Gen chem lab. OpenStax CNX. Oct 12, 2009 Download for free at http://cnx.org/content/col10452/1.51
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