Question

An element crystallises both in fcc and bcc lattice if the density of the element in the two forms is same ratio of unit cell length of fcc to that of bcc lattice is

Answer 1

Answer:

Ratio of unit cell length is

$\frac{a}{b} = \sqrt[3]{2}$

EXPLAINATION

Let a, b be the unit cell length in fcc amd bcc crystal respectively and m be the atomic mass of the element.

ᴅᴇɴsɪᴛʏ ɪɴ ғᴄᴄ ᴄʀʏsᴛᴀʟ

Effective No. of atom in FCC = 4

$\rho = \frac{mass \: of \: unit \: cell}{volume \: of \: unit \: cell}$

$\rho \: = \frac{4m}{ {a}^{3} } ...........(1)$

ᴅᴇɴsɪᴛʏ ɪɴ ʙᴄᴄ ᴄʀʏsᴛᴀʟ

Effective No. of atom in BCC = 2

$\rho = \frac{2m}{ {b}^{3} } ........(2)$

ʀᴀᴛɪᴏ ᴏғ ᴜɴɪᴛ ᴄᴇʟʟ ʟᴇɴɢᴛʜ

Since the density is same, from (1) and (2)

$\frac{4m}{ {a}^{3} } = \frac{2m}{ {b}^{3} }$

$2 = {( \frac{a}{b} )}^{3}$

$\frac{a}{b} = \sqrt[3]{2}$