Introduction to Crystallography: Lecture 11 — Structure Solutions 2

Main ideas / lessons from the lecture (Structure solutions and phase-finding)

1) Patterson maps basics (continuing from last lecture)

• A Patterson map analyzes interatomic vectors in a crystal.
• The lecture emphasizes that you usually don’t plot measured intensities directly. Instead you often use:
  • E-values rather than raw intensities/F-values.
• Why E-values? They are “cleaner” because normalization:
  • removes the angle-dependent drop-off from atomic scattering factors (as a function of (\sin\theta/\lambda))
  • removes or partly accounts for thermal motion effects
• Patterson maps can be plotted using different transformations of normalized structure factors:
  • (E^2)-type maps (conceptually similar to (F^2) intensities), where the origin peak remains.
  • (E^2-1)-type maps, which subtract out the origin peak.
    • The origin peak is large because self-vectors (atom to itself) dominate.
    • Using (E^2-1) makes other peaks easier to interpret.
• Caution: If you sharpen too aggressively, you can introduce spurious peaks due to truncation/finite-resolution effects.

2) Harker lines and planes (space-group symmetry signatures)

• Strong Patterson peaks often lie on Harker lines or Harker planes.
• These come from space-group symmetry and help infer the space group.
• Examples discussed:
  • Mirror plane / glide plane (2D symmetry element) → produce Harker lines
    • Example: a monoclinic space group with two equivalent positions related by a mirror plane.
    • The equivalent atoms create vectors with fixed components in Patterson coordinates, producing peaks along a line.
  • Rotation axis / screw axis (1D symmetry element) → produce Harker planes
    • Equivalent positions involve a relation that leaves two variables changing.
    • Peaks therefore trace a plane in Patterson space.

3) Distinguishing space groups without systematic absences

• The lecture contrasts:
  • using systematic absences vs.
  • using Patterson symmetry (alignment of peaks as lines/planes).
• Some space groups may not have distinct systematic absences, but can still be distinguished by Patterson peak geometry.

4) Why Patterson symmetry uses ((u,v,w)) rather than ((x,y,z))

• Patterson maps don’t use real atomic coordinates directly.
• Patterson coordinates ((u,v,w)) represent interatomic vector components derived from ((x,y,z)).
• Therefore, “lines/planes” in Patterson correspond to fixed vector relations implied by symmetry.

5) How Patterson maps help locate heavy atoms (initial phases)

• Key use: locate the heaviest atom(s).
• Reason: in the Patterson, peak heights scale roughly with products like (Z \cdot Z’), so heavy atoms dominate.
• Once heavy-atom positions are found:
  • you estimate phases (approximate at first),
  • compute a map (e.g., an electron density map) to find additional atoms.
• The process is iterative:
  • update phases and continue locating more structure.

6) Map types used after initial heavy-atom placement

• Difference / Fourier map workflow
  • Start with initial atoms → compute (F_{\text{calc}}).
  • Use Fourier difference maps to find missing atoms.
• Map types:
  • (F_{\text{obs}}-F_{\text{calc}})-type maps (difference maps):
    • use measured amplitudes with phases from the current model
    • show where atoms are missing via peaks (and sometimes “holes” for wrong atoms)
  • (|F_{\text{calc}}|)-only maps (Fourier synthesis from the model):
    • mostly reproduce what’s already in the model, so are less useful alone
• The lecture emphasizes difference maps as the main tool:
  • incorrect/missing atoms produce positive peaks or negative density depending on subtraction.

7) Resolution and interpretability of electron density maps

• You can’t achieve full Fourier completeness:
  • only finite measured data → omission of high-resolution terms
  • causes smeared detail and truncation artifacts (“ripples”)
• Interpretive rules:
  • small negative density is unavoidable
  • very large negative density (e.g., “too negative”) often signals a wrong model/assignment
• Resolution guidance (conceptual):
  • ~6 Å: blobs; interpretation is unreliable without strong prior knowledge
  • ~2.5 Å: partial interpretation if connectivity is known
  • ~1.5 Å: side-chain orientations start to be meaningful
  • ~0.8 Å: clear densities; light atoms (N/O/F) can often be distinguished
• Hydrogen visibility:
  • X-ray diffraction struggles with H because it has ~1 electron (or less effective density depending on bonding/polarization)
  • neutrons can see hydrogen better
  • otherwise use chemical restraints to place H and refine

8) Worked example for an actual Patterson structure solution (heavy atom strategy)

• A textbook-style example is discussed:
  • heavy atom candidate: mercury (Hg) because it’s much heavier than others (e.g., K).
• High-level steps:
  1. Use crystal data + density to estimate the number of formula units (Z) → infer expected counts of K and Hg per unit cell.
  2. Use reflection conditions (systematic absences) to narrow space-group candidates.
  3. Use International Tables-style information:
     • choose special positions with correct multiplicity so you don’t “overfill” the unit cell.
  4. Apply bonding/geometry constraints:
     • eliminate Hg sites that would place Hg atoms unrealistically close given unit-cell dimensions.
  5. Check Patterson maps:
     • locate the Patterson vector peaks predicted by candidate positions.
     • measure peak positions to solve for heavy atom coordinates in Patterson space.
  6. Apply symmetry operators to generate symmetrically related atoms.
  7. Compute specific interatomic vectors and correlate them to Patterson peaks.

9) Alternatives to Patterson (initial phases when heavy atoms exist or not)

A) Isomorphous replacement (multiple heavy-atom derivatives)

• Assumption: derivative crystals are identical except for the heavy atom identity/position.
• Notation/idea:
  • native structure factor: (F_P)
  • derivative structure factors: (F_1), (F_2) (two heavy-atom substitutions)
  • in vector sense:
    • (F_1 = F_M + F_H) (native contribution + heavy-atom contribution)
• Use differences:
  • (F_1 - F_2) cancels the common “rest of structure,” leaving mainly the heavy-atom difference.
• Phase determination:
  • if heavy atom dominates, Patterson yields initial heavy-atom phase/magnitude information
  • in proteins, heavy atoms often don’t dominate completely, so further refinement is needed
• Ambiguity handling:
  • with one derivative, you often get two possible phase choices
  • with more derivatives, intersections narrow to the correct phase (or a small range)

B) Anomalous scattering (use wavelength dependence near absorption edge)

• Uses wavelength dependence: [ f = f_0 + \Delta f’ + i\Delta f’’ ]

• Near an absorption edge:

  • (\Delta f’‘) becomes significant
  • Friedel pairs differ:
    • (F(hkl)\neq F(\bar{h}\bar{k}\bar{l})) in amplitude/phase
• Consequence:
  • Friedel’s law breaks down
  • phase information becomes accessible even without isomorphous derivatives
• Often done at synchrotrons with tunable X-ray wavelength.

C) Direct methods (when there is little scattering contrast / no heavy atoms)

• Used especially for small-molecule organic crystallography.
• Core assumptions in the direct-methods program:
  • electron density is non-negative
  • electron density is concentrated at atoms
  • uses normalized structure factors (E-values) and their statistics
• Statistical tests:
  • determine whether the structure is centrosymmetric or non-centrosymmetric
  • uses expected distributions for quantities like ( \langle E^2 \rangle) and relationships involving (E^2-1)
  • distinguishability is “gray but computable,” not a strict single threshold
• Origin:
  • direct methods still require a consistent origin (origin choice can be arbitrary but must be consistent).

10) Triplet relationships in centrosymmetric structures (phase probability)

• Direct methods assign phases probabilistically via phase relationships.

• For a centrosymmetric structure:

  • triplets relate three reflections such that the third reflection is the vector difference of the first two: [ (h_3k_3l_3)=(h_1-h_2,\;k_1-k_2,\;l_1-l_2) ]

• Key probability idea:

  • the product of the three phase signs is likely to be positive.
• Once signs/phases are assigned for strong reflections:
  • they become a basis to extend assignments to additional reflections.
• The lecture references:
  • early triplet work (Signer/Sayre-like relationship)
  • later improvements (Hauptman–Karle) using more triplets.

---

Detailed bullet-point methodology / instruction-style content

Patterson / E-map processing guidance

• Compute normalized structure factors:
  • use E-values instead of raw intensities/F-values
• Choose Patterson map formulation:
  • consider (E^2) if you want the origin peak included
  • prefer (E^2-1) if the origin peak obscures other vectors
• Interpret Harker features:
  • locate Harker lines/planes
    • lines → typically mirror/glide constraints (2D symmetry element)
    • planes → typically screw/rotation constraints (1D symmetry element)
• Apply caution:
  • avoid over-sharpening to prevent spurious peaks from truncation.

Heavy-atom placement and phase initiation (Patterson-centered workflow)

• Identify heavy atoms:
  • choose the strongest Patterson peaks, treating them as heavy-atom vector relationships.
• Use symmetry to expand candidate sites:
  • generate symmetry-equivalent heavy-atom positions from the space group operators.
• Compute approximate phases from a heavy-atom model:
  • use vector relations ((u,v,w)\leftrightarrow(x,y,z)) to estimate phase information.
• Produce electron density maps:
  • go from approximate phases → electron density map
  • refine by locating additional peaks and iterating the phase/model update.

Difference/Fourier map workflow to find remaining atoms

• Start with a current model:
  • compute (F_{\text{calc}}) and compare to measured (F_{\text{obs}}).
• Compute map types:
  • model-only ((F_{\text{calc}})) maps are less informative for finding missing atoms
  • difference maps (F_{\text{obs}}-F_{\text{calc}}) highlight:
    • missing atoms → peaks
    • wrong atoms → holes/negative density
• Use resolution limits:
  • accept small negative density as an artifact
  • interpret peaks by sign/magnitude consistency with expected electron density.

Resolution-based interpretability rule-of-thumb (as taught)

• If resolution is:
  • poor (e.g., ~6 Å): blobs; atomic assignment needs strong prior knowledge
  • moderate (~2–2.5 Å): molecules may be placed if connectivity is known
  • good (~1.5 Å): side-chain orientations begin to be meaningful
  • very good (~0.8 Å): light atoms (N/O/F, etc.) can often be distinguished
• Hydrogen:
  • typically not visible directly by X-ray; use neutron data or restraints/chemical knowledge.

Example-based Patterson solution steps (Hg/K case, summarized)

• Estimate (Z) from unit-cell data and density → infer atom counts.
• Use reflection conditions to narrow space group candidates.
• Use International Tables multiplicities:
  • restrict to special positions consistent with expected atom counts (avoid impossible overfilling).
• Eliminate candidate sites using chemistry/geometry:
  • rule out Hg sites that force unreasonably short Hg–Hg distances.
• Predict Patterson vector peaks from candidate symmetry positions:
  • compare predicted peaks to observed Patterson map features.
• Measure Patterson peak positions to solve for heavy atom coordinates:
  • use fractional coordinates and interatomic vector relations.
• Once heavy atom coordinates are known:
  • generate symmetrically equivalent heavy atoms
  • compute interatomic vectors and verify consistency with Patterson peaks.

Isomorphous replacement / phase refinement (conceptual vector-diagram procedure)

• Prepare at least one derivative crystal differing by a heavy atom.
• Measure (F_{\text{obs}}) amplitudes for native and derivatives.
• Locate heavy atoms (often via Patterson).
• Construct vector diagram:
  • (F_{PH} = F_P + F_H)
  • infer (F_P) given (F_H) and measured (|F_{PH}|)
  • typically yields two possible phase choices (circle intersections).
• Reduce ambiguity:
  • add more derivatives so intersections converge to a unique phase (or narrow a phase range).

Anomalous scattering phase approach

• Use tunable X-ray wavelength near absorption edge:
  • exploit (f = f_0 + \Delta f’ + i\Delta f’‘).
• Measure Friedel pairs separately:
  • (F(hkl)) and (F(\bar{h}\bar{k}\bar{l})) differ → phase constraints.
• Often needs synchrotron tunability.

Direct methods / triplets approach (centrosymmetric case)

• Use normalized structure factor statistics (E-values) to infer centrosymmetry.
• Determine an origin consistently.
• For centrosymmetric structures:
  • set phase sign assignments using triplet relationships
  • probability rule: product of phase signs in a triplet is likely positive.
• Use strong reflections as seeds:
  • extend sign assignments using more triplets.

---

Speakers / sources featured

Speaker

• Unspecified lecturer/instructor (no name given in the subtitles)

Historical scientists referenced

• Patterson (Patterson method/maps)
• Harker (Harker lines/planes)
• Sayre (triplet relationship; sign probability concept)
• Hauptman and Karle (augmented triplet/direct methods; Nobel Prize referenced)

General reference sources

• International Tables for Crystallography (space group diagrams and atomic positions)