---
title: X-ray and Neutron Diffraction
module: Crystal Structure
moduleNumber: 3
lessonNumber: 4
order: 304
summary: >
  A crystal diffracts radiation whose wavelength matches its atomic spacing. This
  lesson derives the Bragg condition, the equivalent Laue condition 2k dot G equals
  G squared, and the Ewald-sphere construction, then computes the geometric
  structure factor that produces systematic absences for bcc and fcc, the atomic
  form factor, and the powder method. It closes on why neutrons and electrons
  complement X-rays.
topics: [Crystal Structure]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 6 — Determination of Crystal Structures by X-ray Diffraction"
  - book: Kittel
    ref: "Ch. 2 — Wave Diffraction and the Reciprocal Lattice"
  - book: Hook & Hall
    ref: "Ch. 11 — Experimental methods; diffraction"
---

The spacing of atoms in a crystal is a few tenths of a nanometer, matching the
wavelength of X-rays, thermal neutrons, and fast electrons. A wave of that
wavelength scattering off the periodic array interferes constructively only in
sharply defined directions, and the pattern of those directions reveals the
lattice while their intensities reveal the basis. The geometry of the allowed
directions is fixed entirely by the
[reciprocal lattice](/condensed-matter/crystal-structure/reciprocal-lattice-and-brillouin-zones):
diffraction occurs when the scattering vector equals a reciprocal-lattice vector.

## The Bragg condition

Bragg treated the crystal as a stack of parallel atomic planes, spacing $d$, each
reflecting a small fraction of the incident wave specularly. Rays reflected from
adjacent planes travel path lengths differing by $2d\sin\theta$, where $\theta$ is
the glancing angle measured from the plane. Constructive interference requires this
to be a whole number of wavelengths.

$$
n\lambda = 2d\sin\theta.
$$

$$
% caption: Bragg reflection. Waves reflecting from adjacent planes a distance d
% apart differ in path by 2 d sin theta; when this equals an integer number of
% wavelengths, the reflections add in phase.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % two lattice planes
  \draw[black, thick] (0,0) -- (6,0);
  \draw[black, thick] (0,-1.4) -- (6,-1.4);
  \foreach \x in {0.6,1.6,2.6,3.6,4.6,5.4}{
    \fill[black] (\x,0) circle (1.6pt);
    \fill[black] (\x,-1.4) circle (1.6pt);
  }
  \node[black, anchor=west] at (6.05,0) {upper plane};
  \node[black, anchor=west] at (6.05,-1.4) {lower plane};
  % incident and reflected rays on upper plane
  \draw[acc, very thick, ->] (0.4,1.5) -- (2.6,0);
  \draw[acc, very thick, ->] (2.6,0) -- (4.8,1.5);
  % rays to lower plane
  \draw[acc, thick, ->] (1.1,1.5) -- (3.3,-1.4);
  \draw[acc, thick, ->] (3.3,-1.4) -- (5.5,1.5);
  % spacing d
  \draw[black, <->] (0.15,0) -- (0.15,-1.4);
  \node[black, anchor=east] at (0.1,-0.7) {$d$};
  \node[acc, anchor=south] at (0.5,1.5) {incident};
  \node[acc, anchor=south] at (4.7,1.5) {outgoing};
\end{tikzpicture}
$$

The Bragg picture is heuristic — atoms are points, not mirror planes — but the
condition it gives is exact. Its limitation is that it says nothing about
intensity; that requires the structure factor below.

## The Laue condition and its equivalence

Von Laue treated every atom as a scatterer and summed the outgoing amplitudes. An
incident plane wave with wavevector $\vec k$ scatters into $\vec k'$; the path
difference between waves scattered from lattice points separated by $\vec R$ is
$\vec R \cdot (\hat k - \hat k')$, so all scatterers add in phase when

$$
(\vec k' - \vec k)\cdot \vec R = 2\pi\,m \quad\text{for all } \vec R.
$$

By the definition of the reciprocal lattice this holds exactly when the scattering
vector $\Delta\vec k = \vec k' - \vec k$ is a reciprocal-lattice vector $\vec G$.
For elastic scattering $\lvert\vec k'\rvert = \lvert\vec k\rvert$, so writing $\vec
k' = \vec k + \vec G$ and squaring gives the standard Laue form,

$$
2\,\vec k \cdot \vec G = G^2.
$$

This is the Bragg condition in disguise. Take $\vec G$ to be the shortest
reciprocal vector normal to a plane family, $\lvert\vec G\rvert = 2\pi/d$. The
component of $\vec k$ along $\vec G$ is $k\sin\theta$, so $2k\sin\theta = G =
2\pi/d$, and with $k = 2\pi/\lambda$ this is $2d\sin\theta = \lambda$. Higher
orders $n$ correspond to reciprocal vectors $n$ times as long.

## The Ewald sphere

The Laue condition has a geometric solution. Place the tail of $\vec k$ so its head
lands on a reciprocal-lattice point (the origin). Sweep out the sphere of radius
$k$ centered at the tail. A reciprocal point $\vec G$ lies on this **Ewald sphere**
exactly when $\vec k' = \vec k + \vec G$ has length $k$, i.e. when the Laue
condition holds. Diffraction occurs for every reciprocal point the sphere passes
through.

$$
% caption: The Ewald sphere. The incident wavevector k_i ends on the origin of
% the reciprocal lattice; the sphere of radius k centered on its tail intersects
% a reciprocal point G, and the vector to that point is the diffracted
% wavevector k_f. Only points on the sphere diffract.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % reciprocal lattice points
  \foreach \i in {-2,-1,0,1,2}{
    \foreach \j in {-2,-1,0,1,2}{
      \fill[black] (\i,\j) circle (1.6pt);
    }
  }
  % Ewald circle centered at C=(-2,0), radius 2, passes through origin (0,0)
  \draw[acc, thick] (-2,0) circle (2);
  \fill[acc] (-2,0) circle (1.8pt);
  \node[acc, anchor=north] at (-2,-0.1) {center};
  % incident k from center to origin
  \draw[acc, very thick, ->] (-2,0) -- (0,0) node[midway, above] {$k_i$};
  \node[anchor=north west] at (0.05,-0.05) {origin};
  % a reciprocal point on the circle: (-2+2cos, 2 sin). take angle 60deg -> (-1, 1.732)
  \draw[acc, very thick, ->] (-2,0) -- (-1,1.732) node[midway, left] {$k_f$};
  % G = k' - k = from origin to that point
  \draw[black, thick, ->] (0,0) -- (-1,1.732) node[midway, right] {$G$};
\end{tikzpicture}
$$

The construction shows why a stationary single crystal in monochromatic radiation
usually gives no reflections: the sphere passes through the origin but rarely
through a second point. Rotating the crystal, using a range of wavelengths (Laue
method), or using a powder (all orientations) brings reciprocal points onto the
sphere.

## The geometric structure factor

The Bragg and Laue conditions locate the allowed reflections but not their
strength. When the basis has more than one atom, the waves scattered by the atoms
within a cell can interfere and cancel. Summing over the basis positions $\vec
d_j$ with atomic scattering amplitudes $f_j$, the amplitude of the reflection
$\vec G = (hkl)$ is the **structure factor**

$$
S_{\vec G} = \sum_j f_j\, e^{-i\,\vec G \cdot \vec d_j}
= \sum_j f_j\, e^{-2\pi i (h u_j + k v_j + l w_j)},
$$

where $\vec d_j = u_j\vec a_1 + v_j\vec a_2 + w_j\vec a_3$. The measured intensity
is $I \propto \lvert S_{\vec G}\rvert^2$. When $S_{\vec G} = 0$ the reflection is
**systematically absent**, even though the Bragg condition is satisfied.

> **Worked example — bcc extinctions.** The bcc structure is a simple-cubic
> lattice with basis $\vec d_1 = (0,0,0)$ and $\vec d_2 =
> (\tfrac12,\tfrac12,\tfrac12)$, both the same atom $f$. Then
> $$
> S_{hkl} = f\left[1 + e^{-i\pi(h+k+l)}\right] =
> \begin{cases} 2f, & h+k+l \text{ even},\\ 0, & h+k+l \text{ odd}.\end{cases}
> $$
> The $(100)$, $(111)$, $(210)$ reflections vanish; $(110)$, $(200)$, $(211)$
> survive.

> **Worked example — fcc extinctions.** The fcc basis is $(0,0,0)$,
> $(\tfrac12,\tfrac12,0)$, $(\tfrac12,0,\tfrac12)$, $(0,\tfrac12,\tfrac12)$. Then
> $$
> S_{hkl} = f\left[1 + e^{-i\pi(h+k)} + e^{-i\pi(h+l)} + e^{-i\pi(k+l)}\right] =
> \begin{cases} 4f, & h,k,l \text{ all even or all odd},\\ 0, & \text{mixed
> parity}.\end{cases}
> $$
> So $(111)$ and $(200)$ appear but $(100)$, $(110)$, $(210)$ are extinct.

$$
% caption: Systematic absences in reciprocal space. For bcc (left) only points
% with h plus k plus l even survive (filled); for fcc (right) only all-even or
% all-odd indices survive. Open circles are reflections extinguished by the
% structure factor.
\begin{tikzpicture}[scale=0.95, >=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- bcc: h+k+l even survives (2D slice l=0: h+k even) ----
  \begin{scope}
    \foreach \i in {0,1,2,3}{
      \foreach \j in {0,1,2,3}{
        \pgfmathtruncatemacro{\idxsum}{\i+\j}
        \pgfmathtruncatemacro{\parbcc}{mod(\idxsum,2)}
        \ifnum\parbcc=0
          \fill[acc] (\i,\j) circle (2.4pt);
        \else
          \draw[black] (\i,\j) circle (2.4pt);
        \fi
      }
    }
    \node[anchor=north] at (1.5,-0.5) {bcc: h+k even};
  \end{scope}
  % ---- fcc: all even or all odd (2D slice: h,k same parity) ----
  \begin{scope}[xshift=5.5cm]
    \foreach \i in {0,1,2,3}{
      \foreach \j in {0,1,2,3}{
        \pgfmathtruncatemacro{\parih}{mod(\i,2)}
        \pgfmathtruncatemacro{\parjk}{mod(\j,2)}
        \ifnum\parih=\parjk
          \fill[acc] (\i,\j) circle (2.4pt);
        \else
          \draw[black] (\i,\j) circle (2.4pt);
        \fi
      }
    }
    \node[anchor=north] at (1.5,-0.5) {fcc: same parity};
  \end{scope}
\end{tikzpicture}
$$

## The atomic form factor

The amplitude $f_j$ is not a constant. Each atom scatters from its whole electron
cloud $n(\vec r)$, and the contributions from different parts of the cloud
interfere. The **atomic form factor** is the Fourier transform of the electron
density,

$$
f(\vec G) = \int n(\vec r)\, e^{-i\,\vec G \cdot \vec r}\,\d^3 r.
$$

At $\vec G = 0$ (forward scattering) the phases align and $f(0) = Z$, the number of
electrons. As $\lvert\vec G\rvert$ grows — equivalently as $\sin\theta/\lambda$
grows — the cloud's finite size dephases the sum and $f$ falls off. Light atoms
with few electrons scatter X-rays weakly, and hydrogen is nearly invisible, which
motivates neutron diffraction below.

$$
% caption: The atomic form factor falls from f equals Z in the forward direction
% as the scattering angle grows, because the finite electron cloud dephases the
% scattered waves at large G. Heavier atoms (more electrons) start higher.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.2,0) node[right, black] {scattering angle};
  \draw[->, black] (0,0) -- (0,3.4) node[above, black] {$f$};
  % heavier atom curve (starts at 3, decays)
  \draw[acc, very thick, domain=0:5.8, samples=80, variable=\x]
    plot ({\x},{3.0*exp(-\x*\x/9)});
  \node[acc, anchor=west] at (1.2,2.15) {heavier atom};
  % lighter atom curve (starts at 1.6)
  \draw[black, thick, domain=0:5.8, samples=80, variable=\x]
    plot ({\x},{1.6*exp(-\x*\x/9)});
  \node[black, anchor=west] at (2.2,0.95) {lighter atom};
  \draw[black, dashed] (0,3.0) -- (0.3,3.0);
  \node[black, anchor=east] at (-0.05,3.0) {$Z$};
\end{tikzpicture}
$$

## Methods and probes

Two experimental geometries dominate. In the **single-crystal** method one
oriented crystal is rotated to bring successive reciprocal points onto the Ewald
sphere, mapping the full three-dimensional reciprocal lattice. In the **powder
(Debye-Scherrer)** method a mass of randomly oriented crystallites presents every
plane orientation at once, so each family $(hkl)$ with $S_{\vec G}\neq 0$ diffracts
into a cone of half-angle $2\theta$, recorded as a ring. The ring radii give the
$d$-spacings and identify the lattice without a single crystal.

$$
% caption: A powder pattern. Randomly oriented crystallites turn each allowed
% reflection into a cone, recorded as a ring; the ring radii encode the
% d-spacings d = lambda over 2 sin theta of the surviving reflections.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % incident beam
  \draw[very thick, ->] (-3.2,0) -- (-1.4,0) node[midway, above] {beam};
  \fill[black] (-1.4,0) circle (2pt);
  \node[black, anchor=north] at (-1.4,-0.1) {sample};
  % rings on a detector plane (concentric)
  \begin{scope}[xshift=2.4cm]
    \draw[black] (0,0) circle (1.7);
    \draw[acc, thick] (0,0) circle (1.2);
    \draw[acc, thick] (0,0) circle (0.7);
    \fill[black] (0,0) circle (1.4pt);
    \node[black, anchor=north] at (0,-1.95) {detector rings};
  \end{scope}
\end{tikzpicture}
$$

The scattering amplitude depends on the probe. **X-rays** scatter from electrons,
so $f \propto Z$ and light atoms are hard to locate. **Neutrons** scatter from
nuclei with amplitudes that vary irregularly with isotope rather than with $Z$, so
they see hydrogen and distinguish neighboring elements; and because the neutron
carries a magnetic moment, it also scatters from ordered electron spins, making
neutron diffraction the primary probe of magnetic structure. **Electrons**
interact strongly through the Coulomb potential, giving intense scattering from
thin samples and surfaces, at the cost of multiple-scattering complications. The
three probes are complementary: X-rays for the heavy-atom skeleton, neutrons for
light atoms and magnetism, electrons for thin films and surfaces.

The diffraction pattern is the experimental face of the reciprocal lattice. Its
peak positions fix the Bravais lattice and cell dimensions, and its intensities,
through $\lvert S_{\vec G}\rvert^2$ and the form factor, fix the basis — the two
ingredients that define the
[crystal structure](/condensed-matter/crystal-structure/bravais-lattices-and-crystal-systems).