Introduction to Crystallography: Lecture 11 — Structure Solutions

Main ideas & lessons (Structure Solution in Crystallography)

1) After indexing, you still don’t know the structure

• Indexing gives the unit cell, but not the atomic arrangement.
• The next major task is structure solution: determining how atoms are positioned inside the unit cell.

2) Before solving: estimate what’s in the unit cell (for correct intensity scaling)

Crystallographers ask: “What’s in your unit cell?” because:

• The total scattering strength depends on the number of electrons in the unit cell.
• Measured intensities require correct scaling, since experimental intensities can include losses (e.g., absorption).

Elemental analysis may be requested:

• Proteins: elements are largely known from sequence information.
• Small molecules: often known from NMR/IR (organic chemistry context).
• Inorganic solids: can be more uncertain; may require methods like:
  • Atomic absorption
  • ICP
  • XPS
  • XRF
  • (depending on the sample)

3) Stoichiometry is not enough—need Z

• Stoichiometry gives ratios of elements.
• Programs also need Z = number of formula units per unit cell.
• Consistency rule:
  • Example concept: for Al₂O₃
    • If stoichiometry is Al₂O₃ and Z = 6, then the unit cell contains:
      • 12 Al and 18 O
    • If stoichiometry is expressed differently (e.g., fractional), Z must be chosen so totals match the formula definition.

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Methodology: how to determine Z (detailed constraints)

Use density constraints

• If a proposed structure/unit-cell content implies a physically impossible density, reject it.
• Measure density:
  • Organics: use crystal floating methods with solvents of different densities; infer density at flotation equilibrium.
  • If crystals float in a single solvent, density is approximated from that solvent condition.
• Displacement method (works for some solids):
  • Weigh powder in a volumetric flask, fill to a known mark.
  • Subtract missing mass to infer displaced volume:
    • Example logic: if a flask holds 1 g water at the mark but only 0.95 g is present, crystals displaced 0.05 mL.
  • Convert mass / occupied volume → density.
• Use density to select among candidate Z values so the chosen Z gives realistic density.

Estimate using volume-per-atom / packing rules (especially for inorganic solids)

• Rule of thumb: ~10–20 ų per atom (depends on ionic vs covalent character).
• Ionic compounds:
  • Often consider primarily the cation sublattice packing (anions fill interstitial spaces).
  • Example rule: ~20 ų per metal cation.
  • Supports compositions like nitrites / oxynitrides, where charge balance may change the effective anion count without drastically changing packing.
• Covalent compounds:
  • No clear sublattices → consider all atoms using volume-per-atom for the whole formula.

Use symmetry-based likelihood constraints

Once lattice symmetry is known, some Z values are unlikely/invalid:

• Hexagonal cells: pair naturally with Z values consistent with 6-fold relationships; Z = 5 is unlikely.
• Tetragonal: tends to favor Z consistent with 4-fold symmetry (e.g., Z = 4 more plausible than 3).
• Trigonal: expects compatibility with its symmetry.

Multiplicity check from space-group tables:

• If a space group has general position multiplicity = 8, then for one atomic type:
  • the number of equivalent positions contributing to the unit cell must be compatible with 8
  • e.g., 8, or values obtainable as 8 divided by integers consistent with special positions.
• Therefore, Z choices must permit atomic site counts realizable by allowed special/general position multiplicities.

Crystallographers often double-check suspicious Z values (especially odd ones) because they may violate symmetry/multiplicity constraints.

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Core crystallography concept: the phase problem (why structure solution is hard)

1) Goal

• Determine the electron density distribution in the unit cell: (\rho(x,y,z))

2) Fourier relationship

• In principle, electron density can be obtained by Fourier synthesis from structure factors (F_{hkl}).

3) What’s measured vs missing

• Experiment measures intensities (I) versus angle.
• Intensity relates to the magnitude of structure factors:
  • (|F_{hkl}| \propto \sqrt{I_{hkl}})
• Phase is lost because only intensity (magnitude squared) is measured.

4) Phase problem

• Without correct phases, assigning an interpretable structure is not feasible.
• The fundamental challenge is: recover phase information.

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Methodology: constraints used to recover phases (conceptual rules)

• Constraint 1: electron density must be positive
  • Electron density corresponds to electron counts, so it cannot be negative anywhere.
• Constraint 2: electron density concentrates in “atomic balls”
  • Except for hydrogen, atoms have more than one electron → electron density should appear as localized regions consistent with atomic shape.
• Constraint 3: atoms cannot overlap
  • Atoms require a minimum separation (no physically impossible overlap).

Why these constraints help:

• Electron density is a Fourier transform of structure factors.
• Every electron-density location influences all measured intensities.
• You can’t isolate one region independently; the whole unit-cell density must remain consistent with the data.

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Scenarios for phase recovery: centrosymmetric vs non-centrosymmetric

1) Centrosymmetric structures (easier)

• Phases can only be 0 or 180 degrees.
• This avoids discontinuities and leverages cosine-like behavior from inversion symmetry.
• Ambiguity still grows exponentially with reflections, but phase search is far simpler.

2) Non-centrosymmetric structures (harder)

• Phases can take any value from 0 to 360 degrees.
• In practice, you often don’t need perfect precision:
  • using 90° increments may be sufficient.
• Still, brute-force search space remains enormous → requires smarter methods than random trial.

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Friedel’s law (reciprocal-space symmetry)

• Statement: X-ray diffraction intensities are centrosymmetric in reciprocal space: [ I_{hkl} = I_{\bar{h}\bar{k}\bar{l}} ]

• Reason (as explained):

  • Structure factor magnitude depends on (F^2 = a^2 + b^2).
  • Swapping (hkl \to \bar{h}\bar{k}\bar{l}) flips sign of phase-dependent components, but the squared magnitude stays the same.
• Practical implication:
  • Summations can be reduced (e.g., summing over half the reflections and doubling the contribution).

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Outcome: electron density maps (what you look at)

Electron density maps show:

• Contours of equal electron density (like geology contour lines, but height = electron density)
• Optionally slices through planes in the unit cell

Interpretation rule-of-thumb:

• Higher peaks → heavier atoms (more electrons).

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Patterson method (no phase needed) — main takeaway

1) Alternative approach

• Fourier synthesis for electron density needs phases.
• Patterson functions avoid phase recovery by using a different transform.

2) Who introduced it

• Paterson (1934) introduced the Patterson function using intensities only.

3) What Patterson computes

• The Patterson map is based on the convolution of electron density with itself:
  • depends on interatomic vectors/distances, not directly on atom positions.
• Peaks correspond to interatomic distances ((u,v,w)).

4) Key properties explained

• The Patterson map is always centrosymmetric:
  • interatomic distances are direction-independent (A→B and B→A produce opposite vectors with the same magnitude).
• It includes:
  • a strong origin peak (self-vectors where distance = 0)
  • additional peaks at vectors matching interatomic separations

5) Interpretation logic

• You don’t directly obtain atomic coordinates, but you can infer interatomic relationships.
• With known space-group symmetry and equivalences, Patterson peaks can help deduce possible atom positions.

6) Purpose and relevance

• Patterson remains useful in modern crystallography, especially in small-molecule crystallography.
• It is valuable when phases are unknown because it uses only intensities.

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Speakers / sources featured

• The lecture lecturer / course instructor (unnamed; speaks throughout)
• Paterson (1934) — credited with introducing the Patterson function
• Friedel — Friedel’s law mentioned (full first name not provided in subtitles)