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efficiency of packing in face-centred cubic cell &end -centred cubic cell?

PACKING FRACTIONS

Both of the above patterns of packing (i.e. hcp & ccp) though different in form are equally efficient. They occupy the maximum possible space which is about 74% of the available volume. Hence they are called closest packing.

In addition to the above two types of arrangements a third type of arrangement found in metals is body centred cubic (bcc) in which space occupied is about 68%.

CALCULATION OF THE SPACE OCCUPIED
1965_cubic unit cell.JPG

 
 
In simple cubic unit cell   

 Let ‘a’ be the edge length of the unit cell and r be the radius of sphere.

As sphere are touching each other

Therefore a = 2r

No. of spheres per unit cell = 1/8 × 8 = 1

Volume of the sphere = 4/3 ?r3

Volume of the cube = a3= (2r)3 = 8r3

? Fraction of the space occupied = 1/3?r3 / 8r3 = 0.524

? % occupied = 52.4 %

In face centred unit cell

1129_face centred unit cell.JPG

Let ‘r’ be the radius of sphere and ‘a’ be the edge length of the cube

As there are 4 sphere in fcc unit cell

? Volume of four spheres = 4 (4/3 ?r3)

In fcc, the corner spheres are in touch with the face centred sphere. Therefore, face diagonal AD is equal to four times the radius of sphere

AD = 4r

But from the right angled triangle ACD

AD = ?AC2 + DC2 = ?a2 + a2= ?2a

 ?2a = 4r or a = 4/?2 r

? volume of cube = (4/?2 r)3 = 64 / 2?2 r3

percentage of space occupied by sphere
= volume of sphere / volume of cube × 100 = 16/3 ?r3 / 64 /2?2 r3  × 100 = 74%


In body centred cubic unit cell

          1457_body centred cubic unit cell.JPG

Let ‘r’ be the radius of sphere and ‘a’ be the edge length of the cube

As the sphere at the centre touches the sphere at the corner. Therefore body diagonal

AD = 4r

Face diagonal AC = ?AB2 + BC2 = ?a2 + a2 = ?2a

In right angled triangle ACD =

AD = ?AC2 + CD2 = ?2a2 + a2 = ?3a

 ?3a = 4r
a = 4r  / ?3

? Volume of the unit cell = a3 = (4r / ?3)3 = 64r3 / 3?3

No. of spheres in bcc = 2

? volume of 2 spheres = 2 × 4/3?r3

 ? percentage of space occupied by spheres

= volume of sphere / volume of cube × 100 = 8/3 ?r3 × 100 / 64r3 / 3?3 = 8/3 × 22/7 × 3?3/64 = 68%


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