🌟 Diffraction Basics — Full Conceptual Summary

This lecture introduces the fundamental physics behind X-ray diffraction, which is essential for understanding how we determine protein structures.

🦴 Historical Start — First X-ray Image

One of the first X-ray transmission images showed a human hand.

Key ideas:

• X-rays interact with electrons
• Bone scatters more than soft tissue → because bone contains calcium phosphate (many electrons).
• Materials can even be identified by scattering strength (e.g., distinguishing gold vs cubic zirconia).

👉 This shows that scattering intensity depends strongly on electron number.

🌈 Electromagnetic Waves — Why X-rays?

Electromagnetic radiation spans from radio waves → visible light → X-rays → gamma.

Important relationships:

• Shorter wavelength → higher energy
• To see atoms → need wavelength ~ 1 Å (angstrom) → This is why hard X-rays are used in crystallography.

Wave properties:

• Electric field vector ⟂ Magnetic field vector ⟂ Direction of propagation
• The electric field interacts strongly with electrons (magnetic interaction is negligible).

👉 Therefore diffraction theory mainly considers electric field–electron interaction.

⚛️ How X-rays Scatter — Quantum Picture

A common misunderstanding: ❌ Not two beams reflecting from crystal planes.

✔ Correct picture:

• A single photon travels through the crystal
• Usually → nothing happens (~99%)
• Sometimes → photon induces coherent oscillation of electrons
• Photon “disappears” (virtual process) and a new photon appears in a scattered direction.

Diffraction pattern formation:

• Many individual photon scattering events accumulate
• Detectors can count single photons → build pattern gradually.

👉 Diffraction is fundamentally quantum mechanical + probabilistic.

🌊 Interference — Adding Waves

Phase matters!

• If waves are 180° out of phase → destructive interference → zero signal.
• If in phase → constructive interference → strong signal.

This determines where diffraction spots appear.

🧠 Complex Numbers Make Wave Addition Easier

Instead of adding sine waves point-by-point:

We use complex vectors (Argand diagram):

• Vector length = amplitude
• Angle = phase

Resultant wave: F_3 = F_1 + F_2

Intensity: I propto |F|^2

This becomes extremely important later in structure factor calculations.

📐 Wave Function Parameters

General 1D wave:

F(x) = A cos(2pi Hx + alpha)

Where:

• A = amplitude
• H = frequency
• α = phase shift

Changing these gives:

• Higher amplitude → taller peaks
• Higher frequency → more oscillations
• Phase shift → wave moves left/right

🔬 Fourier Series — Key Concept for Crystallography

Jean-Baptiste Fourier showed:

Any periodic function can be described as a sum of simple waves.

Connection to proteins:

• Diffraction waves = simple sine waves
• Electron density of protein = complex target function

Thus:

⭐ Electron density = Fourier sum of diffracted X-rays

This is the mathematical foundation of structure determination.

🎯 Scattering by a Single Atom

Important vectors:

• Incoming wavevector → S₀
• Scattered wavevector → S₁
• Scattering vector → mathbf{S} = mathbf{S_1} - mathbf{S_0}

As scattering angle increases:

• Magnitude of S increases
• Corresponds to higher resolution information

⚡ Thomson Scattering — Intensity Dependence

Key conclusions:

• Scattering mainly from electrons (not nuclei)
• Strongest scattering:
  • Forward direction
  • Backward direction
• Weakest scattering:
  • Perpendicular direction

Also:

• Elastic scattering → energy conserved
• Only direction changes.

🧩 Atomic Scattering Factor

Describes how strongly an atom scatters.

Important trends:

• At zero angle → equals number of electrons
• With increasing angle → scattering intensity falls off

Example:

• Fe (26 electrons) scatters more than:
• S (16) → O (8) → C (6) → H (1)

👉 Heavy atoms give stronger diffraction signal.

🌡️ B-factor (Atomic Displacement Factor)

Atoms are not fixed — they vibrate.

This causes:

• Blurring of electron density
• Loss of high-resolution reflections

B-factor meaning:

• Low B → rigid, well-defined position (protein core)
• High B → flexible, mobile (protein surface)

Typical values:

• 2 – 200 Ų

If B too high:

• Density averages out → atom may be invisible in map.

📉 B-factor vs Resolution

Higher B-factor:

• Faster decay of scattering at high angles
• Limits achievable resolution

Thus: ⭐ Overall protein B-factor correlates with diffraction quality.

🧠 BIG PICTURE — Why This Lecture Matters

This lecture builds the physics + math foundation for:

• Why diffraction patterns form
• How intensity relates to electrons
• How waves interfere
• Why Fourier transforms reconstruct structures
• Why flexible atoms are hard to see
• Why heavy atoms help phasing
• Why resolution depends on motion

👉 These ideas directly lead into:

• Structure factors
• Electron density maps
• Model building
• Refinement