Introduction to Crystallography: Lecture 7 — Reciprocal Space

Main Ideas / Concepts Taught

Administrative Notes + Homework Guidance

• Updated web address for the Asher playing group / symmetry resources (the previous address was changed).
• Encouraged practice using symmetry and drawing tools.
• Homework clarifications:
  • How to tell whether a problem is 2D or 3D:
    • Plane groups → 2D
    • Space groups → 3D
  • If symmetry elements are “missing”:
    • Find only what’s missing
    • Don’t add extra elements
    • Don’t modify the provided drawing

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What Diffraction Experiments Actually Measure

• Bragg-law “concepts vs reality”:
  • The Bragg-law picture (reflection from lattice planes) is a useful conceptual model, not a literal atomic-scale description.
  • In reality, X-rays scatter from electron clouds, not from “planes.”
  • Because crystals are periodic, scattering from the repeating electron density produces interference.
• Neutron diffraction:
  • Neutrons primarily scatter from nuclei.
  • They can also interact with magnetic moments/spins.
• What is measured:
  • Diffraction experiments measure intensity vs. scattering angle (and related detector geometry).
  • Intensity connects to electron density through a Fourier transform:
    • it’s not a simple linear mapping,
    • but the relationship is structured and invertible in theory.

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Direct Space vs. Reciprocal Space

• Direct space: the crystal’s periodic electron density (set by the lattice/unit-cell content and its repetition).
• Reciprocal space: where diffraction data “live.”
• The diffraction pattern follows a Fourier relationship between:
  • electron density in direct space
  • scattering amplitude/intensity in reciprocal space
• Crystallographers often work with the derived reciprocal lattice.

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Reciprocal Lattice Construction (Graphical Method)

• The reciprocal lattice has the same symmetry as the real lattice.
• Construction steps described:
  1. Draw (hkl) plane families in real space.
  2. Choose an origin in the reciprocal-space drawing.
  3. For each (hkl) plane family, draw the normal to those planes.
  4. Mark equal-spaced points along the normals.
  5. Spacing between points is denoted ( d^*_{hkl} ) (reciprocal spacing).

• Key quantitative relationships:

  • Magnitude relation: [ d^* = \frac{1}{d} ]

  • Direction comes from the plane normal.

  • Star notation indicates reciprocal quantities (e.g., (a^), (b^), (c^), (d^_{hkl})).
  • Finding reciprocal lattice constants:
  • Use plane families like (100) and (010) to obtain (a^), (b^), etc.
  • Special note (non-orthogonal lattices):
  • If real-space angles are not (90^\circ), reciprocal axes are not simply parallel to real-space axes.
  • This will be tested in homework.

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Numbering / Reciprocal-Grid Intuition

• Reciprocal lattice points can be labeled like a grid:
  • origin = (000)
  • nearby points correspond to indices such as (100), (010), (001), etc.
• The “counting rule” reflects the inverse-spacing nature of reciprocal space.

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Mathematical Definition of Reciprocal Lattice Constants (Vector Form)

• Unlike the graphical normal method, the vector definition gives precise formulas.

• Reciprocal lattice “constants” are vectors: [ \mathbf{a}^ = \frac{\mathbf{b}\times\mathbf{c}}{V_c},\quad \mathbf{b}^ = \frac{\mathbf{c}\times\mathbf{a}}{V_c},\quad \mathbf{c}^* = \frac{\mathbf{a}\times\mathbf{b}}{V_c} ]

• Unit cell volume: [ V_c = \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) ]

• Dot/cross product rules emphasized:

  • Cross product → a vector (direction matters via the right-hand rule).
  • Dot product → a scalar (order doesn’t matter because cosine symmetry applies).

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Where Reciprocal Space Comes From (Conceptual Scattering Derivation)

• Reciprocal space is motivated by considering scattering from two atoms separated by a vector ( \mathbf{R} ).

• Physical reasoning steps:

  • Define incoming/outgoing wave directions:
    • incoming: ( \mathbf{s}_0 )
    • scattered: ( \mathbf{s} )

  • The key factor is path difference, which yields a phase difference: [ \text{phase shift} \propto \frac{2\pi}{\lambda}(\text{path difference}) ]

  • Define the scattering vector: [ \mathbf{S} = \mathbf{s}-\mathbf{s}_0 ]

  • The scattering amplitude relates to electron density via a Fourier transform:

    • amplitude (A(\mathbf{S})) is the Fourier transform of electron density (\rho(\mathbf{r})).
  • Intensity vs. amplitude:
  • Experiments measure intensity, not complex amplitude.
  • Intensity equals amplitude times its complex conjugate, canceling imaginary components and yielding real-valued results.

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How Periodicity Produces Discrete Spots / Reciprocal-Lattice Planes

• For a periodic crystal, electron density can be built from repeating the unit cell.
• Periodic sums and exponential properties imply:
  • constructive interference occurs only when phase conditions align.
• Reciprocal-space interpretation:
  • Along each crystallographic direction, solutions form sets of equally spaced parallel planes.
  • Considering three non-parallel directions simultaneously, the intersection of three plane families yields discrete points.
  • Thus, a 3D periodic crystal produces reciprocal-lattice “spots” (diffraction peaks).

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Physical Consequence for Real Crystals

• If periodicity is limited (e.g., thin platelets / finite repeats in one direction):
  • Instead of sharp spots in reciprocal space, you observe extended features:
    • streaks, rods, or smeared intensity
• Reciprocal-space understanding helps interpret these patterns.

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Methodologies / “Instruction-Like” Processes

A) How to Handle Symmetry Homework Correctly

• Determine dimension from the prompt:
  • Plane group → 2D
  • Space group → 3D
• If symmetry elements are missing:
  • Add only the missing elements
  • Do not add extra symmetry
  • Do not modify the given drawing/layout
• Avoid changing the reference system/drawing in ways that would produce impossible symmetry outcomes (e.g., wrong-handedness or non-existent space groups).

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B) Graphical Construction of Reciprocal Lattice / Reciprocal Spacings

• Draw the relevant real-space (hkl) plane families.
• Pick an origin in reciprocal space.
• For each (hkl) set:
  • draw the normal
  • mark equally spaced points along it
  • interpret the spacing as (d^*_{hkl})
• Use magnitude intuition:
  • ( d^ ) follows ( d^ = 1/d )
• Determine reciprocal constants:
  • use (100), (010), etc., to derive (a^), (b^), etc.
• For non-(90^\circ) real lattices:
  • expect reciprocal directions to not align with real-space axes in a simple way.

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C) Vector/Mathematical Definition of Reciprocal Lattice Constants

• Compute the unit cell volume:
  • ( V_c = \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) )
• Compute reciprocal vectors:
  • ( \mathbf{a}^* = \dfrac{\mathbf{b}\times\mathbf{c}}{V_c} )
  • ( \mathbf{b}^* = \dfrac{\mathbf{c}\times\mathbf{a}}{V_c} )
  • ( \mathbf{c}^* = \dfrac{\mathbf{a}\times\mathbf{b}}{V_c} )
• Apply correct product rules:
  • cross product order/direction matters (right-hand rule)
  • dot product order does not matter

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D) Conceptual Derivation Workflow (Scattering → Reciprocal-Space Condition)

1. Start with scattering from atom pairs separated by (\mathbf{R}).
2. Define wave directions:
   • incoming ( \mathbf{s}_0 )
   • outgoing ( \mathbf{s} )
3. Compute the path difference and convert it to phase difference:
   • multiply by (2\pi/\lambda)
4. Define the scattering vector:
   • ( \mathbf{S} = \mathbf{s}-\mathbf{s}_0 )
5. Relate scattering amplitude to electron density:
   • amplitude is a Fourier transform of (\rho(\mathbf{r}))
6. Use periodicity:
   • reduce full-space sums to sums over unit-cell translations
   • constructive interference only at reciprocal-lattice-consistent conditions
7. Interpret results:
   • plane families in each direction
   • discrete reciprocal-lattice “spots” from intersections of three families

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Speakers / Sources Featured

• Primary speaker (lecturer): “everybody / good morning” instructor (name not provided in subtitles).