Crystallography Comprehensive Exam Notes

Preface – Why Learn Crystallography?

  • Rapid growth of crystal‐structure results demands that chemists read crystallographic literature.
  • Goal: arm non-specialists with the “language” of crystallography while minimising mathematics.
  • Four core chapters build vocabulary (lattices, symmetry, crystal systems, space groups).
  • Fourier‐series treatment (Chs. 5–6) shows where diffraction data come from, using only elementary calculus.
  • Final chapter presents simple structures; understanding (not memorisation) stressed.
  • Exercises of graded difficulty + full solutions enable self-study.

Chapter 1 – Crystals and Lattices

1-1 Definition of a Crystal

  • Periodic 3-D arrangement of atoms/molecules/ions.
  • Polycrystalline vs single crystals; faces (morphology) are not essential to definition.
  • Example discussions: benzene solidifies via molecular ordering; NaCl lattice balances electrostatic + size effects.

1-2 Lattice Points & Nets

  • Pick one point; all identical (equivalent) points form lattice.
  • 1 D row → 2 D net → 3 D lattice (space lattice).

1-3 Unit Cells

  • Parallelepiped defined by vectors a, b, c.
  • Six lattice parameters: a,b,c,α,β,γa,b,c,\alpha,\beta,\gamma.
  • Knowing atomic arrangement in one cell defines whole crystal.

1-4 Fractional Coordinates

  • Point x,y,zx,y,z reached by moving xa,  yb,  zcxa,\; yb,\; zc from origin.
  • Coordinates differing by integers denote equivalent points.

1-5 Useful Formulae

  • Cell volume V=abc1−cos⁡2α−cos⁡2β−cos⁡2γ+2cos⁡αcos⁡βcos⁡γV = abc\sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}.
  • Distance between two points r12r_{12} given by generalized metric form (Eq. 1-2).

1-6 Primitive vs Centered Cells

  • Primitive (P): lattice points only at corners.
  • Centered cells (A,B,C,I,F,R) contain extra lattice points; # points = volume ratio relative to P cell.

Chapter 2 – Symmetry

2-1 to 2-3 Basics

  • Symmetry operation: movement leaving object indistinguishable.
  • Elements: rotation axes C<em>nC<em>n, mirror planes mm, inversion centre ii, improper axes S</em>nS</em>n.

2-4 Rotation Axes

  • CnC_n ⇒ rotation 360∘/n360^\circ/n.
  • Examples: C<em>2C<em>2 in H$2$O, C<em>3C<em>3 in CHCl$3$.

2-5 Mirror Planes

  • $
    \sigmah$ (horizontal), $\sigmav$ (vertical), $\sigma_d$ (diagonal).
  • Illustrated with PCl$_5$, staggered ethane, etc.

2-6 Identity

  • E=C1E=C_1 always present.

2-7 Centre of Symmetry

  • Point through which x,y,z→−x,−y,−zx,y,z \rightarrow -x,-y,-z.
  • Example trans-CHClBr-CHClBr.

2-8 Improper Axes

  • Schoenflies SnS_n = rotation + reflection; Hermann-Mauguin nˉ\bar n = rotation + inversion.
  • Key equalities: S<em>1=mS<em>1=m, S</em>2=iS</em>2=i, odd nn ⇒ S<em>nS<em>n implies C</em>n+mC</em>n+m.

2-9-2-15 Point-Group Theory

  • Point symmetry elements leave ≥1 fixed point ⇒ 32 crystallographic point groups.
  • Group properties: closure, associativity, identity, inverse; multiplication tables (e.g. C${2v}$, C${3v}$).
  • Nomenclature: Schoenflies vs Hermann-Mauguin; mapping rules.
  • Determination algorithm (linear → cubic steps).
  • Combinatorial limits: only n=1,2,3,4,6n=1,2,3,4,6 allowed for rotations in crystals (Fig. 3-3 proof).

Chapter 3 – Crystal Systems & Geometry

3-1 to 3-3 Seven Systems

  • Restrictions on a,b,c,α,β,γa,b,c,\alpha,\beta,\gamma derived from symmetry.
  • Systems: Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal, Cubic.

3-4 Limitation Proof (n ≤ 6)

  • Lattice translations + rotations ⇒ only 1,2,3,4,6-fold possible.

3-5 Hermann-Mauguin Review

  • 32 point-group symbols organised by system (Table 3-1).

3-6 Bravais Lattices

  • 14 distinct 3-D lattices (P, I, F, A, B, C, R variants) listed (Table 3-2).

3-7 Trigonal vs Hexagonal Lattices

  • Rhombohedral (primitive; a=b=c,α=β=γ≠90∘a=b=c,\alpha=\beta=\gamma\neq 90^\circ) vs triply-primitive hexagonal description; relation c<em>hex=3 a</em>rh 1−cos⁡α /sin⁡αc<em>{hex}=\sqrt{3}\,a</em>{rh}\,\sqrt{1-\cos\alpha}\,/\sin\alpha.

3-8 – 3-10 Crystal Planes

  • Miller indices (hkl)(hkl) = reciprocals of intercepts.
  • Law of rational indices.
  • Interplanar spacing orthorhombic: 1d2=h2a2+k2b2+l2c2\dfrac{1}{d^2}=\dfrac{h^2}{a^2}+\dfrac{k^2}{b^2}+\dfrac{l^2}{c^2} (general form Eq. 3-3).

Chapter 4 – Space Groups & Equivalent Positions

4-1 Translational Symmetry

  • Screw axes n<em>pn<em>p = C</em>nC</em>n + translation p/np/n along axis.
  • Glide planes (a,b,c,n,d) = mirror + translation.

4-2–4-4 230 Space Groups

  • Combine 32 point groups with 14 lattices + translational elements ⇒ 230 unique groups (Appendix I).

4-5 Diffraction Symmetry vs Physical Properties

  • Macroscopic tensor properties dictated by point group (Laue group symmetry from diffraction).

4-6–4-10 Equivalent & Special Positions

  • Derivation illustrated with P4$_2$/m (8k general, 4j, 4i, … 2a).
  • Choice of origin can reduce multiplicity; e.g. Bmab Cl$_2$ origin shift to centre.
  • Worked examples: PdS (P4$2$/m), HgBr$2$ (Bm2$1$b), $$‐quartz (P3$12$1), Cl$_2$, etc.

Chapter 5 – X-Ray Diffraction

5-1–5-3 Historical Basis

  • Von Laue (1912) proved wave nature of X rays; diffraction by CuSO$_4$ crystal.

5-4 Laue Equations

\begin{aligned}
a(\cos\alpha0-\cos\alpha)&=h\lambda\ b(\cos\beta0-\cos\beta)&=k\lambda\
c(\cos\gamma_0-\cos\gamma)&=l\lambda
\end{aligned}

5-5 Rotating Crystal Method

  • Cones of reflections; formula a=hλsin⁡tan⁡−1(y/r)a = \dfrac{h\lambda}{\sin\tan^{-1}(y/r)} gives aa from film (Eq. 5-3).

5-6 Bragg’s Law (modern form)

2dsin⁡θ=λ2d\sin\theta = \lambda

  • Generalisation: absorb order nn into $hkl$ → use non-coprime indices.

5-7–5-10 Data-Collection Cameras

  • Weissenberg: isolate layer lines via moving screen + film translation.
  • Precession (Buerger): undistorted reciprocal-lattice photographs; complement Weissenberg.

5-11–5-13 Intensities & Fourier Theory

  • Electron density ρ(xyz)\rho(xyz) periodic ⇒ Fourier series (Eq 5-13).
  • Structure factor F(hkl)=∑<em>jf</em>j e2πi(hx<em>j+ky</em>j+lzj)F(hkl)=\sum<em>j f</em>j\,e^{2\pi i(hx<em>j+ky</em>j+lz_j)}.
  • Measured intensity I(hkl)∝∣F∣2I(hkl)\propto |F|^2 – inability to measure phases → phase problem.

5-14–5-19 Advanced Factors

  • Thermal vibration: multiply by e−Bjsin⁡2θ/λ2e^{-B_j\sin^2\theta/\lambda^2}.
  • Centrosymmetric crystals: FF real; simplifies phase issue.

5-20 Friedel’s Law

  • ∣F(hkl)∣=∣F(hˉkˉlˉ)∣|F(hkl)| = |F(\bar h\bar k\bar l)| except near anomalous dispersion.

5-21 Laue Groups

  • 11 centrosymmetric categories dictate diffraction symmetry.

5-22–5-24 Systematic Absences

  • Centering conditions (Table 5-2), glide, and screw extinctions identify space group.
  • NaCl example: only $h,k,l$ all even or all odd give reflections.

Chapter 6 – Determination of Atomic Positions

6-1 Why Direct Solution Fails

  • Unknown phases prevent direct inversion of ∣F∣2|F|^2 to structure.

6-2 Patterson Function

P(u,v,w)=∑<em>hkl∣F</em>hkl∣2cos⁡2π(hu+kv+lw)P(u,v,w)=\sum<em>{hkl}|F</em>{hkl}|^2\cos2\pi(hu+kv+lw)

  • Peaks at inter-atomic vectors; always centrosymmetric.
  • Useful when few atoms or presence of heavy atoms.

6-3-6-4 Heavy-Atom & Isomorphous Replacement

  • Locate heavy atom via Patterson; use intensity differences among derivatives to fix phases (e.g. proteins haemoglobin/myoglobin).

6-5 Superposition / Minimum Function (Buerger)

  • Overlay shifted Patterson maps; intersection peaks reveal atom sites.

6-6 Inequality Relations (Harker–Kasper)

  • Cauchy-derived bounds link ∣U(hkl)∣|U(hkl)|; mostly for strong reflections.

6-7 Sayre–Cochran–Zachariasen (SCZ)

  • Probable phase relation for centrosymmetric: S(h+k)=S(h)S(k)S(h+k) = S(h)S(k).

6-8 Hauptman–Karle Direct Methods

  • Probabilistic phase determination yielding large-structure solutions (e.g. p,p'-dimethoxybenzophenone).

6-9 From Phases to Map

  • Compute ρ(xyz)\rho(xyz) via Fourier; locate peaks → atom list.

6-10 Refinement

  • Least-squares adjust coordinates & thermal parameters; residual R=∑∣∣F<em>o∣−∣F</em>c∣∣∑∣Fo∣R = \frac{\sum||F<em>o|-|F</em>c||}{\sum|F_o|}; good structures R<0.10.

Chapter 7 – Archetypal Structures

7-1–7-3 Close Packing

  • 2-D hexagonal packing ⇒ 12-coordination in 3-D.
  • Cubic close packed (ccp, fcc, ABCABC…) η=0.7405\eta=0.7405.
  • Hexagonal close packed (hcp, ABAB…) c/a=1.633c/a=1.633 ideal.

7-4 Body-Centred Cubic

  • 8 + 6 coordination; η=0.6802\eta=0.6802.

7-5 Diamond & 7-6 Graphite

  • Diamond: tetrahedral sp3sp^3 net (Fd3m).
  • Graphite: layered sp2sp^2 networks (P6_3/mmc); weak inter-layer bonding.

7-7 Other Elements

  • Mention polymorphism (Fe, Mn, Hg) & complex B allotropes.

7-8–7-10 Ionic Structures

  • NaCl (Fm3m): 6-coordination; radius ratio r-/r+<2.414.
  • CsCl (Pm3m): 8-coordination; requires nearly equal radii.
  • CaF$_2$ (fluorite, Fm3m): cations fcc, anions in tetrahedral holes.

7-11 Rutile (TiO$_2$)

  • Ti in octahedral O$6$ cages; tetragonal P42/mnm.

7-12 Zinc Blende vs 7-13 Wurtzite (ZnS/ZnO)

  • Zinc blende: ccp →\rightarrow tetrahedral sp3sp^3; F4_3m.
  • Wurtzite (zincite): hcp analogue; P6_3mc.

7-14 Tables & Data Sources

  • Pearson, Wyckoff, Structure Reports, Crystal Data for comprehensive structural libraries.

Key Formulae Recap

  • Cell volume: V=abc1−cos⁡2α−cos⁡2β−cos⁡2γ+2cos⁡αcos⁡βcos⁡γV = abc\sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}
  • Interplanar spacing (general): d^{-2}=\frac{1}{V^2}\Big[h^2b^2c^2\sin^2\alpha+k^2a^2c^2\sin^2\beta+l^2a^2b^2\sin^2\gamma \
    +2hlab^2c(\cos\alpha\cos\gamma-\cos\beta)+2hkabc^2(\cos\alpha\cos\beta-\cos\gamma)+2kla^2bc(\cos\beta\cos\gamma-\cos\alpha)\Big]
  • Structure factor: F<em>hkl=∑</em>jf<em>je2πi(hx</em>j+ky<em>j+lz</em>j)F<em>{hkl}=\sum</em>j f<em>j e^{2\pi i(hx</em>j+ky<em>j+lz</em>j)}
  • Patterson: P(u,v,w)=∑<em>hkl∣F</em>hkl∣2cos⁡2π(hu+kv+lw)P(u,v,w)=\sum<em>{hkl}|F</em>{hkl}|^2\cos2\pi(hu+kv+lw)
  • Bragg: 2dsin⁡θ=λ2d\sin\theta=\lambda.