🌟 Lecture Summary — X-ray Generation, Diffraction & Resolution

This lecture explains how X-rays are produced, how they scatter from protein crystals, and the core theory (Bragg’s law, lattice planes, reciprocal space) behind protein crystallography.

⚡ How X-rays are produced (X-ray tubes)

To perform X-ray diffraction experiments, we first need an X-ray source.

🔧 Sealed X-ray tube mechanism

• A cathode emits electrons.
• The tube is under vacuum, and a high voltage accelerates electrons toward a metal anode.
• When electrons hit the anode:
  • 🔥 Most energy → heat
  • 🌈 Small fraction → X-rays

Two types of X-ray radiation are generated:

🌊 1. Continuous radiation (Bremsstrahlung)

• Produced when charged particles decelerate.
• Gives a smooth spectrum with a sharp cutoff at short wavelengths.
• Minimum wavelength depends on accelerating voltage (e.g., ~0.4 nm at 35 kV).

🎯 2. Characteristic radiation (sharp peaks)

• Due to electronic transitions in the anode atoms.
• Example:
  • Cu Kα wavelength = 1.5418 Å → very commonly used in protein crystallography.

💥 What happens when X-rays hit a crystal?

X-rays interact with electrons inside the crystal and are scattered.

Two types of scattering:

✅ Elastic (coherent / Thomson scattering)

• No energy loss → same wavelength.
• Produces useful diffraction spots.

❌ Inelastic scattering

• Energy lost → lower wavelength.
• Adds noise to diffraction pattern.

👉 Therefore crystallographers focus on elastic scattering only.

🔬 Experimental diffraction setup

Typical crystallography experiment includes:

• 💡 Incoming X-ray beam
• 🧊 Protein crystal mounted in a loop
• 🔄 Goniometer (rotates crystal: φ, χ, ω angles)
• ❄️ Nitrogen cryostream (~100 K) → reduces radiation damage
• 🛑 Beam stop → blocks strong direct beam
• 📟 Detector → records scattered X-rays

Only ~1–2% of X-rays are scattered — signals are very weak.

🧊 Cryoprotection (very important!)

Crystals are flash-frozen to prevent damage.

Common cryoprotectants:

• Alcohols (glycerol, ethylene glycol)
• PEG 400
• Sugars (trehalose, sucrose)
• DMSO
• Oils

Crystals are tiny (loop size ~0.05–1 mm) and embedded in a thin cryosolution film.

🌊 Diffraction basics — Huygens’ principle & interference

Every point on a wavefront generates a secondary wave.

When waves from multiple scatterers meet:

➕ Constructive interference

• Peaks align with peaks → strong signal
• Amplitude increases (e.g., from e → 2e).

➖ Destructive interference

• Peaks meet troughs → signal cancels.

👉 Diffraction pattern depends on distance between scatterers → structural information can be deduced.

🧱 Crystal lattice & Miller indices (hkl)

Inside a crystal:

• Unit cell repeats in 3D.
• We can define sets of equally spaced lattice planes.

These planes are labeled using Miller indices (h, k, l):

• h → divisions along a-axis
• k → divisions along b-axis
• l → divisions along c-axis

Example:

• If axes are divided into 2, 3, and 4 parts → planes have index (2 3 4).

Each reflection on the detector corresponds to a specific (hkl) plane.

📸 Detectors & measuring intensities

Modern detectors:

• CMOS (new)
• CCD (older)
• Image plates / film (very old)

Each diffraction spot:

• Contains many pixels.
• Spot intensity is fitted to a Gaussian peak.
• Integrated intensity ∝ square of wave amplitude → important for structure calculation.

❄️ Artifacts in diffraction images

Common experimental issues:

🔵 Ice rings

• Caused by tiny frozen solvent crystals.
• Produce continuous rings instead of discrete spots.

🧵 Fiber diffraction

• From mounting loop material.

🛑 Beamstop shadow

• Blocks central region.

These can often be tolerated if not too strong.

📐 Diffraction condition — Bragg’s Law

Constructive interference occurs when:

nlambda = 2d sin heta

Where:

• λ = X-ray wavelength
• d = distance between lattice planes
• θ = scattering angle

Key consequences:

🔎 Smaller d → Larger θ → Higher resolution

• Small plane spacing samples finer structural details.

Typical protein crystal resolutions:

• ~3 Å (moderate)
• 1.8–2 Å (good)
• ~1 Å (exceptional)

🔬 Why crystals are needed (signal amplification)

Single molecule → scattering too weak.

Crystal lattice:

• Many identical molecules.
• Scattering adds constructively.
• Crystal acts as a molecular signal amplifier.

🧠 Phase problem (important future topic)

Diffraction gives:

• Intensities (amplitudes)

But phases are missing.

Determining phases is essential to: ➡ reconstruct the electron density map ➡ build the protein structure.

This will be discussed later in the course.

🌐 Real space vs Reciprocal space

Crystallography uses two coordinate systems:

🧱 Real space

• Physical crystal lattice (unit cell a, b, c)

✨ Reciprocal space

• Mathematical lattice where:
  • Each point = one reflection (hkl)
  • Reciprocal axes (a*, b*, c*) are perpendicular to real-space planes.

Important relationships:

• Reciprocal spacing ∝ 1/d
• Real cell volume × reciprocal cell volume = 1.

⭐ Big Picture Takeaways

✅ X-rays are produced via electron acceleration → bremsstrahlung + characteristic lines ✅ Elastic scattering from electrons gives diffraction spots ✅ Crystals amplify weak molecular scattering ✅ Diffraction depends on interference from lattice planes ✅ Bragg’s law links resolution ↔ scattering angle ↔ plane spacing ✅ Miller indices label reflections ✅ Reciprocal space is the natural framework for analyzing diffraction