🧊 Lecture Summary — Reciprocal Lattice, Ewald Sphere & Diffraction Geometry

🔁 What is the Reciprocal Lattice (Big Idea)

The reciprocal lattice is a mathematical way to describe X-ray diffraction patterns.

• Each reciprocal lattice point = one diffracted X-ray beam
• Each beam comes from a set of crystal planes with spacing ( d )
• These planes are labeled by Miller indices (hkl)

👉 Key construction rule:

• From the real crystal lattice:
  • Draw planes (like 110, 120, 130…)
  • From the origin draw a vector perpendicular to that plane
  • Length of vector = ( 1/d_ )
  • The endpoint is a reciprocal lattice point with same (hkl)

This means:

Real space planes → become points in reciprocal space

This is the fundamental mapping used in crystallography.

📏 Relationship Between Miller Indices and Distance

As Miller index increases:

• Real space plane spacing decreases
• Therefore:

rac{1}{d} ext{ increases}

So reciprocal lattice vectors become longer.

💡 Example:

• 110 → shorter reciprocal vector
• 120 → longer
• 130 → even longer

Because planes get closer together.

📐 Reciprocal Axes vs Real Axes

There is a reciprocal relationship between unit cell sizes:

Example insight:

• Proteins → very large unit cells
• Therefore:
  • Reciprocal lattice is dense
  • Diffraction spots on detector are close together

Small molecule crystals:

• Small unit cell
• Diffraction spots widely spaced

This is something you actually observe experimentally.

📐 Direction Relationships (Very Important Exam Concept)

Even when angles ≠ 90°:

• ( a^* ) is perpendicular to plane BC
• ( b^* ) is perpendicular to plane AC
• ( c^* ) is perpendicular to plane AB

So reciprocal axes are always:

Normal vectors to real lattice planes

🌐 The Ewald Sphere (Geometrical Condition for Diffraction)

Now comes one of the most important visual models in crystallography.

🎯 Concept

• Incoming X-ray has wavevector length = ( 1/lambda )
• Draw a sphere with radius ( 1/lambda )
• Place crystal at center
• Place reciprocal lattice origin at beam tail

👉 Diffraction happens ONLY when a reciprocal lattice point touches the sphere

Because:

• This means Bragg’s law is satisfied

So:

Reciprocal point on Ewald sphere → diffraction spot recorded.

This is purely a geometrical representation of Bragg’s law.

🔄 Why We Rotate the Crystal

When you rotate the crystal:

• Real lattice rotates
• Reciprocal lattice rotates
• New reciprocal points intersect the Ewald sphere

Therefore:

👉 New reflections appear on detector.

This is how we collect thousands of reflections for structure solution.

🎯 Detector Geometry & Missing Reflections

Not all possible reflections are recorded.

Reasons:

1️⃣ Detector size limitation

A reflection may satisfy Bragg’s law but:

• If diffracted beam misses detector → not recorded

Solution:

• Move detector closer

2️⃣ Resolution sphere (Crystal quality limit)

Crystal disorder limits:

• Maximum observable reciprocal lattice radius
• Therefore limits resolution

Outside this sphere:

• No diffraction (even if geometry allows)

3️⃣ Beam stop shadow

Most X-rays are not diffracted.

To protect detector:

• A beam stop blocks direct beam
• Creates a shadow region with no data

Very common feature in diffraction images.

📸 Precession Photographs (Special Case)

If crystal is perfectly aligned:

• Diffraction pattern becomes highly symmetric
• You can directly see:
  • HK plane
  • Axes directions
  • Reflection spacing

From this:

You can calculate reciprocal lattice spacing → then unit cell dimensions

Because:

A' = rac{A^*}{CF} cdot lambda

This shows:

Unit cell parameters can be determined directly from diffraction geometry.

🧠 Space Group Determination Logic

In real experiments:

• Crystal orientation is random
• We collect many reflections during rotation

Then software:

1. Builds 3D reciprocal lattice
2. Tests all possible space groups
3. Chooses best symmetry fit

Important reasoning:

• Low symmetry (triclinic) can fit anything
• But if high symmetry fits → very likely correct

Because symmetry produces systematic reflection patterns.

⭐ Ultra-Short Take-Home Messages

✅ Reciprocal lattice = diffraction language ✅ Reciprocal vector length = (1/d) ✅ Large protein unit cell → dense diffraction spots ✅ Diffraction occurs when reciprocal point hits Ewald sphere ✅ Crystal rotation = sampling reflections ✅ Detector geometry + crystal order limit data ✅ Diffraction symmetry → helps determine space group