Consider the bcc unit cells of the solid 1 an | vedclass

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(A) In a $bcc$ unit cell,the body diagonal is $\sqrt{3}a$,where $a$ is the edge length.Along the body diagonal,the atoms are in contact: $2r_A + 2r_B

Vedclass · Accepted Answer

$90$

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(A) In a $bcc$ unit cell,the body diagonal is $\sqrt{3}a$,where $a$ is the edge length.Along the body diagonal,the atoms are in contact: $2r_A + 2r_B = \sqrt{3}a$.Given $r_B = 2r_A$,so $2r_A + 2(2r_A) = \sqrt{3}a$,which gives $6r_A = \sqrt{3}a$,or $a = 2\sqrt{3}r_A$.The packing efficiency is given by the ratio of the volume of atoms to the volume of the unit cell.Volume of atoms = $8 	imes (\frac{1}{8}) 	imes \frac{4}{3}\pi r_A^3 + 1 	imes \frac{4}{3}\pi r_B^3 = \frac{4}{3}\pi r_A^3 + \frac{4}{3}\pi (2r_A)^3 = \frac{4}{3}\pi r_A^3 (1 + 8) = 12\pi r_A^3$.Volume of unit cell = $a^3 = (2\sqrt{3}r_A)^3 = 8 	imes 3\sqrt{3} r_A^3 = 24\sqrt{3} r_A^3$.Packing efficiency = $\frac{12\pi r_A^3}{24\sqrt{3} r_A^3} 	imes 100 = \frac{\pi}{2\sqrt{3}} 	imes 100 \approx \frac{3.14159}{3.464} 	imes 100 \approx 90.6\%$.Since the packing efficiency depends only on the ratio of radii and the structure type,it is independent of the absolute edge length. Thus,the packing efficiency in solid $2$ is approximately $90\%$.

Consider the $bcc$ unit cells of the solid $1$ and $2$ with the position of atoms as shown below. The radius of atom $B$ is twice that of atom $A$. The unit cell edge length is $50\%$ more in solid $2$ than in $1$. What is the approximate packing efficiency in solid $2$? $........... \%$

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