1.17 Which of the following lattices has the highest packing efficiency

(i) simple cubic

(ii) body-centred cubic and

(iii) hexagonal close-packed lattice?

Answers (1)

(i) Simple cubic:-  In a simple cubic lattice the atoms are located only on the corners of the cube.

                              Thus, the edge length or side of the cube ‘a’, and the radius of each particle, r are related as a = 2r

                              Volume of cubic unit cell = (2r)^3 = 8r^3

                            And  Volume of 1 atom : 

                                                                =\frac{4}{3}\Pi r^3

                              Packing\ efficiency = \frac{Volume\ of\ one\ atom}{Volume\ of\ cubic\ unit\ cell}\times100 \%  

                                                                          = \frac{\frac{4}{3}\Pi r^3}{8\Pi r^3}\times100 \% = \frac{\Pi }{6}\times100\%

                                                                          = 52.4\%

(ii) Body centred cubic:-  In body centred cubic, we have atoms at all corners and at body centre.

                                        Clearly, the atom at the centre will be in touch with the other two atoms diagonally arranged.

                                         b = \sqrt{2}a ;                      and           c = \sqrt{3}a           

      Also, the length of body diagonal is equal to 4r.

                                            \sqrt{3}a = 4r

                                                a = \frac{4r}{ \sqrt{3}}

The volume of the cube              :             

                                                       = a^3 = \left ( \frac{4r}{ \sqrt{3}} \right )^3                                                                                                                                         

In BCC, a total number of atoms is 2. 

                                               

                                                 Packing\ efficiency = \frac{Volume\ of\ one\ atom}{Volume\ of\ cubic\ unit\ cell}\times100 \%

                                                                                             = \frac{2\times(\frac{4}{3}\Pi r^3)}{(\frac{4}{\sqrt{3}}r)^3}\times100 \%   

                                                                                              = 68\%

(iii) Hexagonal close-packed:-    We know that both types of (hcp and ccp) are equally efficient. We also know that the packing efficiency of ccp is 74 percent.

(i) Simple cubic = 52.4%

(ii) Body centred cubic=68%

(iii) Hexagonal close-packed=74%

Thus among all, packing efficiency of hcp is the highest.

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