Laue and Bragg Diffraction Conditions

Derivation

Laue and Bragg Diffraction Conditions

D-248 Home PU-303 Threads waves · matter · symmetry Depends on Reciprocal Lattice from the Bravais Lattice

Statement

For elastic scattering of X-rays from a crystal, the amplitude summed over all unit cells is appreciable only when the momentum transfer \( \vec{q} = \vec{k}' - \vec{k} \) equals a reciprocal-lattice vector \( \vec{G} \) (the Laue condition). This condition is algebraically equivalent to Bragg's law \( 2d_{hkl}\sin\theta = n\lambda \), and the intensity of each allowed reflection is set by the geometric structure factor \( F(\vec{G}) = \sum_{\alpha} f_\alpha\, e^{-i\vec{G}\cdot\vec{d}_\alpha} \), whose zeros are the systematic absences.

Why it matters

X-ray diffraction is the primary experimental route to crystal structure: peak positions encode the reciprocal lattice (hence the Bravais lattice and cell dimensions), while peak intensities, through the structure factor, encode the arrangement of atoms in the basis. Every solved protein and every phase diagram of a solid rests on this correspondence.

The Laue formulation makes the physics geometric — the Ewald sphere intersecting reciprocal-lattice points — and unifies X-ray, neutron, and electron diffraction under one momentum-conservation statement. Systematic absences are the fingerprint that distinguishes SC, BCC and FCC, and later reveals glide planes and screw axes.

Assumptions

Elastic (kinematic) scattering:if photons lose energy (Compton, thermal diffuse), then \( |\vec{k}'|\neq|\vec{k}| \) and the sharp Laue condition smears into a continuous background rather than delta-function peaks. Single scattering (first Born approximation):if a photon scatters more than once before leaving the crystal (dynamical diffraction in large perfect crystals), then intensities are no longer \( \propto|F|^2 \) and forbidden reflections can reappear via multiple-scattering paths. Infinite, perfectly periodic lattice:if the crystal is finite or disordered, then the lattice sum broadens each peak (finite-size / Scherrer broadening) and defects add diffuse scattering between the Bragg spots. Rigid lattice at \(T=0\):if atoms vibrate thermally, then each amplitude is multiplied by a Debye–Waller factor \( e^{-W} \), reducing high-angle intensities without shifting peak positions. Scattering from point-like or spherical charge densities:if the atomic charge distribution has non-trivial extent, the per-atom amplitude becomes the \(\vec{q}\)-dependent atomic form factor \( f_\alpha(\vec q) \) rather than a constant.

Derivation

\[ \psi_{\text{sc}}(\vec{R}_{\text{det}}) \;\propto\; \sum_{j} n_j\, e^{i\vec{k}\cdot\vec{r}_j}\, e^{-i\vec{k}'\cdot\vec{r}_j} \;=\; \sum_{j} n_j\, e^{-i(\vec{k}'-\vec{k})\cdot\vec{r}_j} \]

Each scatterer at \(\vec r_j\) re-radiates a spherical wave; in the far field the path-length phase relative to the origin is \( (\vec k-\vec k')\cdot\vec r_j \). Define the momentum transfer \( \vec q\equiv\vec k'-\vec k \). A

\[ \vec{r}_j = \vec{R}_n + \vec{d}_\alpha, \qquad \psi_{\text{sc}} \;\propto\; \underbrace{\left(\sum_{n} e^{-i\vec{q}\cdot\vec{R}_n}\right)}_{\text{lattice sum } S(\vec q)} \underbrace{\left(\sum_{\alpha} f_\alpha\, e^{-i\vec{q}\cdot\vec{d}_\alpha}\right)}_{\text{structure factor } F(\vec q)} \]

Split every atomic position into a Bravais lattice vector \(\vec R_n\) plus a basis offset \(\vec d_\alpha\). The exponential factorises because \( e^{-i\vec q\cdot(\vec R_n+\vec d_\alpha)}=e^{-i\vec q\cdot\vec R_n}e^{-i\vec q\cdot\vec d_\alpha} \); the atomic weight becomes the form factor \(f_\alpha\). B

\[ \left| S(\vec q) \right| = \left| \sum_{n=0}^{N-1} e^{-i\vec{q}\cdot\vec{R}_n} \right| \;\longrightarrow\; N \quad\text{iff}\quad \vec{q}\cdot\vec{R}_n = 2\pi m \;\; (m\in\mathbb{Z})\ \ \forall\, \vec{R}_n \]

A geometric sum of \(N\) unit-modulus phasors is \(O(1)\) unless every term has the same phase, in which case it is \(N\). Constructive addition of all \(\sim10^{23}\) cells therefore demands \( \vec q\cdot\vec R_n\in2\pi\mathbb Z \) for all lattice translations. B

\[ e^{-i\vec{q}\cdot\vec{R}_n}=1\ \ \forall n \;\;\Longleftrightarrow\;\; \boxed{\ \vec{q} = \vec{G}_{hkl} \equiv h\vec{b}_1 + k\vec{b}_2 + l\vec{b}_3\ } \]

By the defining relation \( \vec b_i\cdot\vec a_j = 2\pi\delta_{ij} \) of the reciprocal lattice (prior result), the set of vectors satisfying \( \vec G\cdot\vec R_n\in2\pi\mathbb Z \) for all \(\vec R_n\) is exactly the reciprocal lattice. This is the Laue condition. B

\[ \vec{k}' = \vec{k} + \vec{G}, \qquad |\vec{k}'|^2 = |\vec{k}|^2 \;\Rightarrow\; |\vec{k}|^2 + 2\,\vec{k}\cdot\vec{G} + |\vec{G}|^2 = |\vec{k}|^2 \]

Impose elasticity \( |\vec k'|=|\vec k| \) (energy conservation) on the Laue condition and expand the square. This is the Ewald-sphere statement: allowed \(\vec G\) are those with tip on a sphere of radius \(k\). C

\[ 2\,\vec{k}\cdot\vec{G} + |\vec{G}|^2 = 0 \;\Longrightarrow\; -2\,\vec{k}\cdot(-\vec G) = |\vec{G}|^2 \;\Longrightarrow\; 2k\sin\theta = |\vec{G}| \]

Cancel \(|\vec k|^2\) and rearrange. Since \(\vec G\) is normal to the \((hkl)\) planes and \(\theta\) is the glancing angle between \(\vec k\) and those planes, \( \vec k\cdot(-\vec G)=k|\vec G|\sin\theta \). C

\[ |\vec{G}_{hkl}| = \frac{2\pi n}{d_{hkl}}, \quad k=\frac{2\pi}{\lambda} \;\Longrightarrow\; 2\left(\frac{2\pi}{\lambda}\right)\sin\theta = \frac{2\pi n}{d_{hkl}} \;\Longrightarrow\; \boxed{\,2 d_{hkl}\sin\theta = n\lambda\,} \]

The shortest reciprocal vector normal to a plane family has magnitude \(2\pi/d_{hkl}\); an \(n\)-th order reflection uses \(\vec G=n\vec G_{hkl}\). Substituting \(k=2\pi/\lambda\) and simplifying recovers Bragg's law, proving the two formulations equivalent. A

\[ I(\vec{G}) \;\propto\; |S(\vec G)|^2\,|F(\vec G)|^2 = N^2\left|\sum_{\alpha} f_\alpha\, e^{-i\vec{G}\cdot\vec{d}_\alpha}\right|^2, \qquad F(\vec G)=0 \Rightarrow \text{systematic absence} \]

At \(\vec q=\vec G\) the lattice sum saturates at \(N\); the measured intensity is then governed entirely by the structure factor from step 2. Whenever the basis phases interfere destructively, \(F(\vec G)=0\) and a Laue-allowed reflection is extinguished. B

Result

\[ \vec{q}=\vec{k}'-\vec{k}=\vec{G}_{hkl} \;\;\Longleftrightarrow\;\; 2d_{hkl}\sin\theta=n\lambda, \qquad I \propto \Big|\sum_\alpha f_\alpha\, e^{-i\vec G\cdot\vec d_\alpha}\Big|^2 \]

Reading. A crystal diffracts only when the change in the photon's wavevector lands exactly on a reciprocal-lattice point. Equivalently, the classical Bragg picture of mirror-like reflection from lattice planes at the special angle \(\theta\) is nothing but this momentum-matching condition rewritten in real space. Which reflections actually appear — and how bright they are — is decided by the structure factor, the phased sum of atomic form factors over the basis; when that sum cancels, the reflection is systematically absent regardless of geometry.

Units check. \(\vec q\) and \(\vec G\) are wavevectors with units \(\text{m}^{-1}\) (or \(\text{nm}^{-1}\)); \(k=2\pi/\lambda\) matches. In Bragg's law \(2d\sin\theta=n\lambda\), \(\sin\theta\) is dimensionless so both sides are lengths (\(\text{nm}\)). The form factors \(f_\alpha\) are in units of the Thomson (single-electron) amplitude, so \(F\) is dimensionless and \(|F|^2\) a relative intensity.

Limiting cases

Single-atom basis (\(\vec d_1=0\)): \(F=f_1\) for every \(\vec G\); no systematic absences, so all reciprocal-lattice points reflect (e.g. simple cubic).
Long wavelength \(\lambda>2d_{\max}\): \(\sin\theta=n\lambda/2d>1\) has no solution — no Bragg peaks at all; this is why visible light cannot image atomic planes.
Forward scattering \(\vec q\to 0\) (the \(\vec G=0\) point): \(F(0)=\sum_\alpha Z_\alpha\), the total electron count — the unscattered beam.
Small-angle limit \(\theta\to0\): \(|\vec q|=(4\pi/\lambda)\sin\theta\to 0\), probing large real-space distances \(\sim2\pi/|\vec q|\) (mesoscopic order, SAXS).
High symmetry basis: BCC keeps \(h+k+l\) even, FCC keeps \(h,k,l\) all-even or all-odd — the surviving subset shrinks as the basis fills the cell.

Breaks when

Dynamical (multiple) scattering in thick, near-perfect crystals: intensities no longer follow \(|F|^2\), Pendellösung oscillations appear, and forbidden reflections (e.g. Si 222) gain intensity through Umweganregung — kinematic theory fails.
Inelastic channels dominate: Compton scattering and thermal diffuse scattering violate \(|\vec k'|=|\vec k|\), so the sharp Laue delta-functions dissolve into broad backgrounds; the derivation assumed pure elastic scattering.
Loss of long-range order: amorphous solids, liquids, or heavily disordered alloys have no reciprocal lattice — one gets broad structure-factor humps \(S(q)\), not Bragg peaks. Quasicrystals diffract sharply but need a higher-dimensional reciprocal lattice.
Finite / nanoscale crystals: with only \(N\) cells the lattice sum has width \(\sim2\pi/(Na)\), broadening peaks (Scherrer) so the exact \(\vec q=\vec G\) equality becomes an approximation.

Failure modes

Measuring \(\theta\) from the wrong reference: Bragg's \(\theta\) is the glancing angle from the plane, not from the plane normal, and the detector sits at \(2\theta\) from the incident beam — mixing these gives a factor-of-two error.
Confusing "\(n\)-th order" with a new plane family: the \(n\)-th order reflection of \((hkl)\) is the first order of \((nh,nk,nl)\); double-counting inflates the peak list.
Forgetting the elastic constraint: writing \(\vec q=\vec G\) alone (without \(|\vec k'|=|\vec k|\)) and expecting every reciprocal point to reflect at fixed \(\lambda\) — most lie off the Ewald sphere.
Dropping the phase in \(F\): summing \(|f_\alpha|\) instead of the complex \(f_\alpha e^{-i\vec G\cdot\vec d_\alpha}\) hides all systematic absences.
Using \(d=a/n\) blindly: \(d_{hkl}=a/\sqrt{h^2+k^2+l^2}\) only for the cubic system; applying it to tetragonal or hexagonal cells is wrong.
Assuming absent \(\Rightarrow\) atoms missing: a zero of \(F\) is destructive interference of present atoms, not a vacancy.

Discussion

The deepest lesson is that a diffraction pattern is a map of the reciprocal lattice, sampled wherever a reciprocal point pierces the Ewald sphere of radius \(k=2\pi/\lambda\). Rotating the crystal sweeps points through the sphere; powder methods average over all orientations so every allowed \(|\vec G|\) appears as a ring. The Laue condition is thus momentum conservation for the crystal as a whole: the lattice can only absorb momentum in quanta \(\hbar\vec G\), the crystalline analogue of Umklapp processes in phonon and electron transport.

Position and intensity carry complementary information. Peak positions fix the reciprocal-lattice geometry — the metric of the Bravais lattice — and are basis-independent. Intensities, through \(F(\vec G)=\sum_\alpha f_\alpha e^{-i\vec G\cdot\vec d_\alpha}\), encode the basis: what sits where inside the cell. This split is exactly why solving a structure is really a phase problem: detectors record \(|F|^2\), discarding the phase of \(F\), and recovering the electron density \(\rho(\vec r)=\frac{1}{V}\sum_{\vec G}F(\vec G)e^{i\vec G\cdot\vec r}\) requires that lost phase back.

The structure factor also unifies the empirical extinction rules with symmetry. A body-centring translation \((\tfrac12,\tfrac12,\tfrac12)\) forces \(F\propto1+(-1)^{h+k+l}\); a screw axis or glide plane imposes its own conditional absences (e.g. a \(2_1\) axis kills odd \(00l\)). Reading which reflections vanish is therefore a direct readout of the space-group symmetry operations — the reciprocal-space shadow of the crystal's point and translational symmetry, and the reason systematic absences are tabulated in the International Tables.

Common misconceptions. Bragg "reflection" is not specular reflection off physical mirrors: there are no reflecting sheets, only atoms whose scattered wavelets add in phase along the direction fixed by \(\vec q=\vec G\). And the Laue and Bragg conditions are not two competing theories — they are the same equation in reciprocal and real space, related by \(|\vec G_{hkl}|=2\pi/d_{hkl}\).

Worked examples

Example 1 — Bragg angle of the aluminium (111) reflection (FCC).

\[ d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}} \;\Rightarrow\; d_{111}=\frac{a}{\sqrt{3}} \]

Cubic-system plane spacing; \((111)\) is Laue-allowed for FCC since \(h,k,l\) are all odd. A

\[ d_{111}=\frac{0.405\ \text{nm}}{\sqrt{3}} = 0.2338\ \text{nm} \]

Insert the aluminium lattice parameter \(a=0.405\ \text{nm}\). A

\[ \sin\theta=\frac{n\lambda}{2 d_{111}} = \frac{(1)(0.15406\ \text{nm})}{2(0.2338\ \text{nm})} = 0.3295 \]

First order \(n=1\), Cu K\(\alpha\) radiation \(\lambda=0.15406\ \text{nm}\), via Bragg's law. A

\[ \theta=\arcsin(0.3295)=19.24^\circ, \qquad 2\theta=38.5^\circ \]

The diffractometer records the peak at the scattering angle \(2\theta\). A

\[ 2\theta_{111}\approx 38.5^\circ \]

Reading. Matches the measured Al (111) powder line at \(2\theta\approx38.5^\circ\) for Cu K\(\alpha\), confirming both the FCC allowed-reflection rule and the cell size.

Units check. \(\sin\theta\) is a ratio of nm/nm — dimensionless — as required.

Example 2 — Structure factor and first peak of BCC iron.

\[ F(hkl)=f\left(1+e^{-i\pi(h+k+l)}\right)=f\left(1+(-1)^{h+k+l}\right) \]

BCC basis \(\vec d_1=(0,0,0),\ \vec d_2=(\tfrac12,\tfrac12,\tfrac12)\); using \(\vec G\cdot\vec d_2=\pi(h+k+l)\) and one atomic species (\(f_1=f_2=f\)). B

\[ (100):\ h+k+l=1\ \text{odd} \Rightarrow F=0 \ \ (\text{absent}); \qquad (110):\ h+k+l=2\ \text{even} \Rightarrow F=2f \ \ (\text{present}) \]

Systematic absence rule for BCC: reflections survive only when \(h+k+l\) is even. The lowest-angle allowed peak is therefore \((110)\), not \((100)\). B

\[ d_{110}=\frac{a}{\sqrt{2}}=\frac{0.2866\ \text{nm}}{\sqrt{2}}=0.2027\ \text{nm} \]

Cubic spacing with \(\alpha\)-Fe lattice parameter \(a=0.2866\ \text{nm}\). A

\[ \sin\theta=\frac{\lambda}{2 d_{110}}=\frac{0.15406}{2(0.2027)}=0.380 \;\Rightarrow\; \theta=22.35^\circ,\ \ 2\theta=44.7^\circ \]

First-order Bragg reflection with Cu K\(\alpha\). A

\[ (100)\ \text{absent}, \qquad (110)\ \text{present at}\ 2\theta\approx44.7^\circ \]

Reading. The missing \((100)\) line and a strong first peak at \(2\theta\approx44.7^\circ\) are the classic signature of a body-centred cubic metal — exactly what is measured for \(\alpha\)-iron with Cu K\(\alpha\).

Units check. \(d\) in nm, \(\lambda\) in nm, ratio dimensionless; angles in degrees.

Problems

Momentum transfer. Monochromatic X-rays of \(\lambda=0.100\ \text{nm}\) scatter through \(2\theta=60^\circ\). Find \(|\vec q|\).
Solution
\(|\vec q|=|\vec k'-\vec k|=2k\sin\theta=\dfrac{4\pi}{\lambda}\sin\theta\). With \(\theta=30^\circ\), \(\sin\theta=0.5\): \(|\vec q|=\dfrac{4\pi(0.5)}{0.100\ \text{nm}}=62.8\ \text{nm}^{-1}\). Real-space scale probed \(\sim2\pi/|\vec q|=0.10\ \text{nm}\).
Simple cubic Bragg angle. A simple-cubic crystal has \(a=0.300\ \text{nm}\). Using \(\lambda=0.154\ \text{nm}\), find \(2\theta\) for the first-order \((200)\) reflection.
Solution
\(d_{200}=a/\sqrt{2^2+0+0}=0.300/2=0.150\ \text{nm}\). \(\sin\theta=\lambda/2d=0.154/0.300=0.513\Rightarrow\theta=30.9^\circ\), so \(2\theta=61.7^\circ\). (SC has no systematic absences, so \((200)\) is allowed.)
Laue-to-Bragg algebra. Starting from \(\vec k'=\vec k+\vec G\) with \(|\vec k'|=|\vec k|\), derive \(2k\sin\theta=|\vec G|\) and identify \(\theta\).
Solution
Square: \(|\vec k|^2=|\vec k|^2+2\vec k\cdot\vec G+|\vec G|^2\Rightarrow 2\vec k\cdot\vec G=-|\vec G|^2\). Since \(\vec G\parallel\) plane normal and \(\theta\) is the glancing angle, \(\vec k\cdot\vec G=-k|\vec G|\sin\theta\) (incident beam pointing into the planes). Thus \(2k|\vec G|\sin\theta=|\vec G|^2\Rightarrow 2k\sin\theta=|\vec G|\). With \(|\vec G|=2\pi/d\), \(k=2\pi/\lambda\): \(2d\sin\theta=\lambda\). \(\theta\) is the angle between the incident beam and the reflecting planes (half the detector angle \(2\theta\)).
FCC allowed reflections. For an FCC crystal, determine which of \((111),(200),(210),(220)\) are allowed and which are systematically absent.
Solution
FCC basis gives \(F=f\big[1+e^{-i\pi(h+k)}+e^{-i\pi(h+l)}+e^{-i\pi(k+l)}\big]=4f\) when \(h,k,l\) are all even or all odd, and \(0\) otherwise (mixed parity). \((111)\): all odd \(\Rightarrow\) allowed \((F=4f)\). \((200)\): all even \(\Rightarrow\) allowed. \((210)\): mixed \(\Rightarrow\) absent \((F=0)\). \((220)\): all even \(\Rightarrow\) allowed. So the first FCC lines are \(111,200,220,\dots\)
Highest observable order. With Cu K\(\alpha\) (\(\lambda=0.15406\ \text{nm}\)) and a plane family of spacing \(d=0.204\ \text{nm}\), how many Bragg orders \(n\) can be observed, and at what \(2\theta\) does the last one occur?
Solution
Require \(\sin\theta=n\lambda/2d\le1\Rightarrow n\le 2d/\lambda=2(0.204)/0.15406=2.65\), so \(n=1,2\) are observable (\(n=3\) needs \(\sin\theta=1.19>1\)). For \(n=2\): \(\sin\theta=2(0.15406)/(2\cdot0.204)=0.755\Rightarrow\theta=49.0^\circ\), \(2\theta=98.1^\circ\). The last observable order is \(n=2\) at \(2\theta\approx98^\circ\).