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In the book Condensed Matter Physics by Marder I have read that an FCC lattice can be obtained by expanding a bcc lattice along one axis by a factor of $\sqrt{2}$. How can I get that mathematically?

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Yes, that's correct. You can see a useful picture of this in a question posted on Chemistry StackExchange. (The question itself is not the same as yours). That picture shows two unit cells of the fcc lattice. But in between them, rotated by $45^\circ$, is the unit cell of the bcc lattice, stretched along the $z$ axis. So it is properly described as a body-centred tetragonal lattice. You can see that, if the fcc cell dimensions are taken to be of unit length, the body-centred tetragonal cell has dimensions $\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}} \times 1$. So scaling the fcc $z$ axis by a factor $\frac{1}{\sqrt{2}}$ will produce a regular bcc lattice; or equivalently elongating a bcc lattice along $z$ by a factor $\sqrt{2}$ will convert it into fcc.

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