AcademyDiffraction
Academy
X-Ray Diffraction
Level 1 - Physics topic page in Diffraction.
Principle
X-ray diffraction probes crystal structure because x-ray wavelengths are comparable with atomic plane spacings. Reflections from adjacent crystal planes interfere constructively only at Bragg angles.
Notation
\(d\)
spacing between crystal planes
\(\mathrm{m}\)
\(\theta\)
Bragg angle measured from the planes
\(\mathrm{rad}\)
\(\lambda\)
x-ray wavelength
\(\mathrm{m}\)
\(m\)
diffraction order
1
\(\Delta r\)
path difference between reflections
\(\mathrm{m}\)
Method
Derivation 1: Path difference
Rays reflecting from adjacent crystal planes travel an extra distance down to the lower plane and back out.
Extra path
\[\Delta r=2d\sin\theta\]
Constructive condition
\[\Delta r=m\lambda\]
Derivation 2: Bragg law
Combining the path difference with constructive interference gives Bragg's law.
Bragg law
\[2d\sin\theta=m\lambda\]
Plane spacing
\[d=\frac{m\lambda}{2\sin\theta}\]
Derivation 3: Order limits
Only orders with \(\sin\theta\le1\) are possible.
Allowed order
\[m\lambda\le2d\]
Maximum order
\[m_{\max}=\left\lfloor\frac{2d}{\lambda}\right\rfloor\]
Rules
Bragg law
\[2d\sin\theta=m\lambda\]
Plane spacing
\[d=\frac{m\lambda}{2\sin\theta}\]
Allowed orders
\[m\lambda\le2d\]
Examples
Question
X-rays of wavelength
\[0.154\,\mathrm{nm}\]
reflect from planes spaced \[0.200\,\mathrm{nm}\]
Find the first-order Bragg angle.Answer
\[\sin\theta=\frac{0.154}{2(0.200)}=0.385\]
so \[\theta=22.6^\circ\]
Checks
- The Bragg angle is measured relative to the crystal planes.
- The x-ray wavelength must be comparable with the plane spacing.
- Higher order means a larger angle for the same \(d\) and \(\lambda\).
- No order exists if the required sine is greater than 1.