💎 Solid State & Crystal Lattices

Solid state chemistry analyzes the geometric periodicity and packing symmetry of crystal lattices. Competitive examinations emphasize void coordinates & radius ratios, coordination geometry transitions, stoichiometric and non-stoichiometric defects (F-centers), and Bragg's Law of X-ray diffraction.

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1. 🧊 Unit Cell Geometry & Packing Metrics

1.1 The 3 Cubic Unit Cells & HCP

Lattice Type	Atoms per Unit Cell ($Z$)	Radius-Edge Length Relation ($r-a$)	Coordination Number (CN)	Packing Efficiency (PE)	Nearest Neighbor Distance ($d$)
Simple Cubic (SC)	$8\times \frac{1}{8}=1$	$a=2r$	$6$	$\frac{\pi }{6}\approx 52.36\mathrm{\%}$	$d=a$
Body-Centered Cubic (BCC)	$8\times \frac{1}{8}+1=2$	$4r=\sqrt{3}a$	$8$	$\frac{\sqrt{3}\pi }{8}\approx 68.02\mathrm{\%}$	$d=\frac{\sqrt{3}a}{2}$
Face-Centered Cubic (FCC/CCP)	$8\times \frac{1}{8}+6\times \frac{1}{2}=4$	$4r=\sqrt{2}a$	$12$	$\frac{\sqrt{2}\pi }{6}\approx 74.05\mathrm{\%}$	$d=\frac{a}{\sqrt{2}}$
Hexagonal Close Packing (HCP)	$12\times \frac{1}{6}+2\times \frac{1}{2}+3=6$	$a=2r,c=4r\sqrt{\frac{2}{3}}=1.633a$	$12$	$\frac{\pi }{3\sqrt{2}}\approx 74.05\mathrm{\%}$	$d=a$

$$\text{Density of Crystal: }\rho \mathbf{=}\frac{\mathbf{Z}\times \mathbf{M}}{{\mathbf{N}}_{\mathbf{A}}\times {\mathbf{a}}^{\mathbf{3}}}$$

where $M$ is molar mass in $\text{g/mol}$, $a$ is edge length in $\text{cm}$, and $N_A=6.023\times {10}^{23}$.

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2. 🕳️ Voids in Close Packings & Limiting Radius Ratios

In any close packed lattice (FCC/CCP or HCP) containing $N$ atoms ($Z$ atoms per unit cell):

• Number of Octahedral Voids (OV) = $N=Z$
  • Location in FCC: $1$ at body center $+12\times \frac{1}{4}$ at edge centers $=4$.
• Number of Tetrahedral Voids (TV) = $2N=2Z$
  • Location in FCC: $2$ along each of the $4$ body diagonals at distance $\frac{\sqrt{3}a}{4}$ from each corner $=8$.

2.1 Limiting Radius Ratio ($r_{+}/r_{-}$) Rules

Radius Ratio Range ($r_{+}/r_{-}$)	Coordination Number	Geometry	Void Type	Example Structure
$<0.155$	$2$	Linear	Linear Void	$Be{F}_2$ vapor
$0.155-0.225$	$3$	Trigonal Planar	Triangular Void	$B_2{O}_3$
$0.225-0.414$	$4$	Tetrahedral	Tetrahedral Void	$ZnS$ (Zinc Blende)
$0.414-0.732$	$6$	Octahedral	Octahedral Void	$NaCl$ (Rock Salt)
$0.732-1.000$	$8$	Cubic	Cubic Void	$CsCl$

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3. 🏛️ Archetypal Ionic Crystal Structures

1. $NaCl$ (Rock Salt Structure, $6:6$ Coordination):
   • $C{l}^{-}$ forms FCC lattice ($Z=4$).
   • $N{a}^{+}$ occupies all octahedral voids ($4$).
   • Formula units per cell $=4$. Distance: $r_{N{a}^{+}}+r_{C{l}^{-}}=a/2$.
2. $CsCl$ (Cubic Structure, $8:8$ Coordination):
   • $C{l}^{-}$ forms Simple Cubic lattice.
   • $C{s}^{+}$ occupies the single body-center cubic void.
   • Formula units per cell $=1$. Distance: $r_{C{s}^{+}}+r_{C{l}^{-}}=\frac{\sqrt{3}a}{2}$.
3. $ZnS$ (Zinc Blende Structure, $4:4$ Coordination):
   • $S^{2-}$ forms FCC lattice ($Z=4$).
   • $Z{n}^{2+}$ occupies alternate tetrahedral voids ($4$).
4. $Ca{F}_2$ (Fluorite Structure, $8:4$ Coordination):
   • $C{a}^{2+}$ forms FCC lattice ($Z=4$).
   • $F^{-}$ occupies all tetrahedral voids ($8$).
5. $N{a}_{2}O$ (Antifluorite Structure, $4:8$ Coordination):
   • $O^{2-}$ forms FCC lattice ($Z=4$).
   • $N{a}^{+}$ occupies all tetrahedral voids ($8$).

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4. ⚠️ Point Defects & X-Ray Diffraction

4.1 Stoichiometric Defects

• Schottky Defect: Equal numbers of cations and anions missing from lattice sites.
  • Decreases crystal density. Highly favored in ionic solids with high coordination numbers and similar ionic radii ($NaCl,KCl,CsCl,AgBr$).
• Frenkel Defect: A cation is dislocated from its lattice site into an interstitial void.
  • Density remains unchanged. Highly favored in solids with large cation-anion size difference ($AgCl,AgBr,ZnS$).
  • Note: $AgBr$ exhibits both Schottky and Frenkel defects.

4.2 Non-Stoichiometric Defects & F-Centres

• Metal Excess Defect (Anion Vacancy): Heating $NaCl$ crystal in sodium vapor creates $C{l}^{-}$ vacancies occupied by trapped electrons (F-Centres / Farbenzentren).
  • Imparts yellow color to $NaCl$, pink to $LiCl$, violet to $KCl$, and induces paramagnetism and semiconductor conductivity.

4.3 Bragg's Law of Diffraction

$$n\lambda =2d\sin \theta$$

For a cubic crystal with Miller indices $(h,k,l)$:

$$d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}$$