Solid State
Shairing of number of atoms per unit cell
| Type of Unit | Number of atoms at corners | Number of atoms on faces | Number of atoms in center | Total |
|---|---|---|---|---|
| Simple Cubic | $8 \times 1 / 8=1$ | 0 | 0 | 1 |
| Body Centred Cubic | $8 \times 1 / 8=1$ | 0 | 1 | 2 |
| Face Centred Cubic | $8 \times 1 / 8=1$ | $6 \times 1 / 2=3$ | 0 | 4 |
| End Centred Cubic | $8 \times 1 / 8=1$ | $2 \times 1 / 2=1$ | 0 | 2 |
Classification of Crystal into Seven System
ANALYSIS OF CUBICAL SYSTEM
| Property | SC | BCC | FCC |
|---|---|---|---|
| (i) atomic radius(r) a = edge length | $ \frac{a}{2} $ | $\frac{\sqrt{3}}{4}a$ | $ \frac{a}{2\sqrt{2}} $ |
| (ii) No. of atoms per unit cell(Z) | 1 | 2 | 4 |
| (iii) C.No. | 6 | 8 | 12 |
| (iv) Packing efficiency | 52% | 68% | 74% |
| (v) No. voids $\newline$ (a) octahedral(Z) $\newline$ (b) tetrahedral(2Z) | $\newline$ $\newline$ $\quad$ __ $\newline$ $\quad$ __ | $\newline$ $\newline$ $\quad$ __ $\newline$ $\quad$ __ | $\newline$ $\newline$ $\quad$ 4 $\newline$ $\quad$ 8 |
NEIGHBOUR HOOD OF A PARTICLE :
(I) Simple Cubic (SC) Structure:
| Type of neighbour | Distance | no. of neighbour |
|---|---|---|
| nearest | a | 6 (shared by 4 cubes) |
| $(next)^1$ | $ a\sqrt{2} $ | 12 (shared by 2 cubes) |
| $ (next)^2 $ | $ a\sqrt{3}$ | 8 (unshared) |
(II) Body Centered Cubic (BCC) Structure :
| Type of neighbour | Distance | no. of neighbour |
|---|---|---|
| nearest | $ 2 r =a \frac{\sqrt{3}}{2} $ | 8 |
| $(next)^1$ | a | 6 |
| $ (next)^2 $ | $ a\sqrt{2} $ | 12 |
(III) Face Centered Cubic (FCC) Structure :
| Type of neighbour | Distance | no. of neighbour |
|---|---|---|
| nearest | $ \frac{a}{\sqrt{2}}$ | $ 12 = \big(\frac{3 \times 8}{2}\big) $ |
| $(next)^1$ | a | $ 6 = \big(\frac{3 \times 8}{4}\big) $ |
| $ (next)^2 $ | $ a\sqrt{\frac{3}{2}} $ | 24 |
DENSITY OF LATTICE MATTER (d) $=\frac{Z}{N_{A}}\left(\frac{M}{a^{3}}\right)$
where $\quad N_{A}=$ Avogadro’s No. $M=$ atomic mass or molecular mass.
IONIC CRYSTALS
C.No. $\quad$ Limiting radius ratio $\left(\frac{r_{+}}{r_{-}}\right)$
$3 \quad \quad 0.155-0.225$ (Triangular)
$4 \quad \quad 0.225-0.414$ (Tetrahedral)
$6 \quad \quad 0.414-0.732$ (Octahedral)
$8 \quad\quad 0.732-0.999$ (Cubic).
EXAMPLES OF A IONIC CRYSTAL
(a) Rock Salt (NaCl) Coordination number $(6: 6)$
(b) CsCl C.No. (8 : 8)
Edge length of unit cell :-
$a_{\mathrm{sc}}=\frac{2}{\sqrt{3}}\left(r_{+}+r_{-}\right)$
(c) Zinc Blende (ZnS) C.No. (4 : 4)
$$ a_{\mathrm{fcc}}=\frac{4}{\sqrt{3}}\left(r_{\mathrm{Zn}^{2+}}+\mathrm{r}_{\mathrm{s}^{2-}}\right) $$
(d) Fluorite structure $\left(\mathrm{CaF}_{2}\right)$ C.No. $(8: 4)$
$$ a_{\mathrm{fcc}}=\frac{4}{\sqrt{3}}\left(r_{\mathrm{Ca}^{2+}}+\mathrm{r}_{\mathrm{F}^{-}}\right) $$
BETA
