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Calculate the efficiency of packing in case of a metal crystal for (iii) face centred cubic (with the assumptions that atoms are touching each other).

1.10     Calculate the efficiency of packing in case of a metal crystal for

                  (iii) face-centred cubic (with the assumptions that atoms are touching each other).

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In fcc, we know that we have a total of 4 efficient atoms present.

Also, radius and edge length of the sphere can be related by the following relation :         

                                               r = \frac{a}{2\sqrt{2}}

The above relation can be found out by equating diagonal of 1 face to 4r. (2r from the atom at face centre and r from each atom at a corner.)

Thus packing efficiency becomes:- 

                        Packing\ efficiency = \frac{Volume\ occupied\ by\ atoms}{Total\ volume\ of\ unit\ cell}\times100\%

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

                                                                     = 74\%

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