Question

Please answer the following questions regarding hydroxyapatite. Please show all work.

a.)    How many atoms are in the unit cell for Hap?

b.)    What are the unit cell parameters? Please use the attached diagram to determine.

c.)    Calculate the unit cell volume for HAp.

d.)    Calculate the density for HAp.

e.)    Calculate the packing factor for HAp.

f.)     Derive an equation for calculating the specific surface area for hydroxyapatite particles consisting of unit cell dimension particles. Calculate the value for the specific surface area of these particles in meters squared per gram (m^2/g). What is interesting about the magnitude of this number?

g.)    How would you differentiate between a polycrystalline, a single crystalline, and a nanocrystalline sample of HAp?

Answer 1

a) It is well known that Hydroxylapatite crystallizes in the hexagonal crystal system (hcp)

general diagram of hcp is

It is clear from the above diagram that hcp has three layers ( 7+3+7, ABA type layer) and it can be analysed as below

• 3 atoms in the B layer are not shared by any other prism. They fully count as 3

• 2 atoms in the centre of the top and bottom A layer are each shared by any other prism (above and below). Each of them counts 1/2. So there's another 2×1/2 =1 atom.

• besides, Top and bottom A layers each have 6 atoms. all the atoms are shared by three neighbors in same layer and with another hexagonal prism. It may be count as 2×6×1/3×1/2 =2 atoms.

So, total count is 6.

b)

c.

it is well known that unit cell has 17 spheres with radius “r” (assumed) and edge length of unit cell should be “2r.”

Volume = Base Area x height

Base Area = 6 x √3/4 x ( 2r )²
Height of Unit Cell = 4r x √(2/3)
Total Volume = 6√3 r² x 4r √(2/3)
                    = 24 x √2 x r³ unit

d) density = mass of the unit cell / volume of the unit cell

e)  Packing Fraction = Volume Occupied by Spheres / Volume of Unit Cell

Volume of the spheres = 6 x 4/3 x π x r³ = 8 π r³ unit
Packing Fraction = ( 8 π r³ )/(24√2 r³)
                         = π /3√2
   = 0.74
   = 74%