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A band of diffraction contrast where local bending brings a crystal structure into a particular Bragg diffraction condition. Its position changes with specimen tilt.

The orientation of a crystal structure relative to a specimen or instrument coordinate frame. It determines which Bragg diffraction conditions are excited.

A technical author whose X-ray system manual explains how instrument geometry turns a detected intensity into a diffraction measurement. Coupled crystal and detector rotation maintains the reflection geometry while the angle is scanned. The description links a mechanical arrangement with the angles needed to analyze Bragg diffraction.^[ref]

A neutron interferometer coherently splits and recombines a neutron matter wave, converting a difference of quantum phase between paths into an output intensity change. A sample in one path can therefore be studied through its phase shift, even when its effect on transmitted intensity is small.

A perfect-crystal implementation uses several diffracting silicon blades machined from one crystal with a common base. Bragg diffraction splits the incident wave into paths, redirects them and recombines them in a geometry analogous to a Mach-Zehnder interferometer. The common base helps preserve relative alignment; thermal and vibration isolation protect phase stability. The institutional apparatus description includes the blade geometry and explains the importance of environmental isolation.^[ref]

Recombination measures a relative phase rather than an individual particle trajectory. Electric, magnetic, gravitational or material interactions can all contribute, so a quantitative result requires controls separating the desired phase from drifts and other path differences. The Colella–Overhauser–Werner experiment is a gravitational example.

Static fluctuations in which chemical element occupies a site produce scattering beyond coherent Bragg diffraction from the mean crystal structure. For independent binary occupations with probabilities $p$ and $1-p$, the variance of the site's atomic scattering factor is $p(1-p)|f_A-f_B{|}^2$. Writing each site amplitude as its mean plus a zero-mean fluctuation shows that the mean squared total amplitude is the coherent contribution plus a term proportional to the number of sites and this variance. Consequently a forbidden reflection of the average crystal structure can still sit on a diffuse background. Correlated occupations add nontrivial scattering vector dependence.

For the conventional body-centred cubic lattice, the crystal basis structure factor of identical atoms isIt vanishes for odd $h+k+l$ and equals $2f_{\mathrm{W}}$ for even $h+k+l$. Thus the first powder diffraction ring is the $\{110\}$ family, with interplanar spacing $d_{110}=a/\sqrt{2}$.

The measured deflection is the angle between incident and scattered rays, $2\theta =40.26^{\circ }$, so the Bragg angle is $\theta =20.13^{\circ }$. Using first-order Bragg diffraction, $2d\sin \theta =\lambda$, givesThe mass density follows from two atoms per conventional unit cell, with molar mass $M=0.1838\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ and Avogadro constant $N_A$:Using $40.26^{\circ }$ as the Bragg angle would confuse $\theta$ with $2\theta$ and give an incorrect lattice constant and mass density.

Replace every coloured site in the drawing by an identical site carrying $\bar{f}$. The face-centering lattice translations now preserve the average crystal structure, whose Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors isA coherent Bragg diffraction peak requires $\mathbf{G}\cdot {\mathbf{a}}_j\in 2\pi \mathbb{Z}$ for all three primitive lattice translation vectors. Thus $k+l$, $h+l$, and $h+k$ must all be even, equivalent to all three Miller indices having the same integer parity. This is the face-centred cubic selection rule: $(110)$ is forbidden and $(111)$ is allowed. The restoration of these lattice translations explains the disappearance of the ordered superlattice reflections.

The stoichiometry fixes the site probabilities to $1/4$ gold and $3/4$ copper. Every site in the average crystal structure therefore has the same atomic scattering factorThe average structure factor isAt $(110)$ the bracket vanishes, while at $(111)$ it is four. HenceThe $(111)$ powder diffraction multiplicity is unchanged, so its coherent Bragg diffraction wave intensity equals the ordered value under the stated equal-lattice constant approximation.

The zero refers to the coherent Bragg diffraction peak of the average crystal structure. A random alloy can still give occupational diffuse scattering at the same scattering vector: the average of the squared amplitude includes fluctuations as well as the square of the average amplitude. Such background is not a surviving $(110)$ Bragg diffraction peak.

For a cubic crystal, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and first-order bragg diffraction requires $2d_{hkl}\sin \theta =\lambda$. The face-centred lattice sum isIt is four if $h,k,l$ are all even or all odd, and zero for mixed parity. With the NaCl basis displaced by $(\frac{1}{2},0,0)$, the allowed amplitudes are $F_{hkl}=4[f_{\mathrm{Cl}}+(-1{)}^{h+k+l}{f}_{\mathrm{Na}}]$, up to the arbitrary choice of origin/species. Both allowed parity classes survive because the two ionic scattering factors are not identical. Intensities depend on $|F{|}^2$, geometry, absorption and scattering vector.

There are three separate tests: nonzero structure factor, accessible Bragg angle, and accessible plane orientation. In a symmetric reflection scan the scattering vector is normal to the specimen face, selecting the surface-parallel family and its allowed orders. Other families need a different mounting, independent orientation control or transmission/another facet. The original diagrams distinguish these geometric requirements; angles in the reciprocal-space panel are between plane normals, not Bragg angles.

Choose an objective aperture around a precipitate superlattice reflection, such as $100$, which is absent from the disordered aluminium matrix. The selected beam then makes precipitates bright in dark-field transmission electron microscopy, while the matrix gives little intensity in that beam. Tilt the grain to place the chosen reflection close to its Bragg diffraction condition and preferably approach a two-beam geometry. This reduces excitation error and can increase precipitate brightness; the exact maximum also depends on foil thickness, strain and dynamical scattering. Selecting a shared fundamental reflection would give much poorer chemical-order selectivity.

Use conventional bright-field transmission electron microscopy, with nearly parallel illumination, a suitably small objective aperture and low magnification covering the whole cross-section. More strongly scattering material sends more intensity outside the selected central beam and therefore looks darker. Tilt crystalline layers away from strong Bragg diffraction conditions to reduce diffraction contrast, and use a near-focus image so that electron phase contrast does not dominate.

The nominally uniform $150\mathrm{nm}$ thickness helps separate composition from thickness variations, but does not make the contrast purely compositional: density, atomic number and multiple scattering still matter. Choose an accelerating voltage sufficient to transmit this thickness, commonly $200$–$300\mathrm{kV}$ for a robust inorganic film, subject to its damage limit. High-angle annular dark-field imaging in scanning transmission electron microscopy is a useful alternative for approximately monotonic mass-thickness contrast, but is a different acquisition mode and still requires thickness and detector-angle control.

Diffraction contrast arises because elastic scattering transfers transmitted electron intensity between the direct beam and Bragg diffraction beams. A local change in crystal orientation, strain or thickness changes that transfer. An objective aperture selecting the direct beam makes strongly diffracting regions relatively dark in bright-field transmission electron microscopy; selecting a diffracted beam reverses the selection in dark-field transmission electron microscopy. This is a change in scattering conditions, not necessarily a change in chemical composition.

Three characteristic effects are bend contours, thickness fringes and dislocation contrast. Bend contours follow positions where a gently curved crystal meets a particular Bragg diffraction condition. Thickness fringes arise from thickness-dependent exchange of intensity between coupled transmitted and diffracted waves. A dislocation distorts nearby planes, changing their diffraction condition and producing localized contrast; changing the diffraction vector can suppress particular dislocations. Moiré fringes from overlapping lattices provide another acceptable example, but their coarse spacing is a beat spacing rather than an atomic spacing.

Use a sufficiently high SEM acceleration voltage to transmit through the particle/support combination and provide useful diffracted intensity; operation near the high-voltage end of the SEM range is a reasonable starting point. The supplied wavelength is used directly for the camera-length calculation, rather than assigning a precise voltage from an approximate wavelength relation.

Choose the smallest beam current consistent with usable counting statistics, because the stated source-size contribution obeysHigher gun brightness permits more current at the same small probe. A relatively short working distance can give a larger convergence semiangle and reduce this term. In the specified fixed lens-to-detector geometry, shortening working distance also increases the specimen-to-detector camera length, reducing the detector's collection angles.

The objective aperture should be optimized, not made arbitrarily large: increasing $\alpha$ reduces the source-size and diffraction contributions, but spherical/chromatic aberrations and angular broadening eventually worsen the probe or diffraction selectivity. A suitable aperture balances those effects while keeping the chosen diffraction rings within the annulus. Adjust specimen height to meet that angular condition, focus and align at the operating voltage, and control drift and contamination.

Low convergence favors angular discrimination of diffraction, whereas high convergence can favor probe size. The optimum therefore depends on whether spatial detail or sharply selected Bragg diffraction contrast is the limiting requirement.

For a face-centered cubic lattice, allowed reflections have indices all odd or all even. The first two are $\{111\}$ and $\{200\}$, withFor first-order Bragg diffraction, $2d\sin \theta =\lambda$. The detector intercepts the beam at the full scattering angle $2\theta$, givingThe inner detector radius supplies the minimum length for each ring:Thus both rings reach the annulus when $L\gtrsim 90.2\mathrm{mm}$. At this minimum the 200 ring is at roughly $3.12\mathrm{mm}$, well inside the $10\mathrm{mm}$ outer radius. Both remain within the detector for approximatelyEquality at the inner edge gives vanishing acceptance for an ideal infinitely narrow ring, so practical operation requires a small margin. The calculation concerns ring positions; individual nanoparticle orientation and the finite probe convergence also affect observed diffraction contrast. Using the Bragg angle instead of the full scattering angle would introduce an erroneous factor of about two.