Presets
Playback
Source and slit
Energy selects the Bragg angle; the slit sets beam size and flux.
Optics
Detune the second crystal; the mirror angle decides whether harmonics survive.
Sample and detector
Distance, exposure, crystallite size and thermal motion shape the rings.
Layers
Model checks
The science
Bragg's law and the monochromator
A crystal reflects X-rays only when the path difference between adjacent lattice planes is a whole number of wavelengths: n lambda = 2 d sin(theta), with lambda[A] = 12.398 / E[keV] and, for a cubic crystal, d = a / sqrt(h^2 + k^2 + l^2). A two-crystal silicon monochromator in the fixed-exit geometry turns the broad synchrotron spectrum into a narrow band: tilting the pair to the Bragg angle theta selects the energy. Because n lambda fits too, the same angle also passes 3E, 5E ... (harmonics) unless something removes them.
Energy resolution and rocking curve
Differentiating Bragg's law gives dE/E = dtheta / tan(theta). The intrinsic (Darwin) width of each reflection sets the best possible resolution: typical literature values for silicon are about 1.3e-4 for (111), 5.6e-5 for (220), 2.7e-5 for (311) and 8e-6 for (333). The rocking-curve width in arcseconds is (dE/E) tan(theta); detuning the second crystal by that much halves the transmitted flux. The page models the curve as a Gaussian, a simplification of the true Darwin shape.
Source and mirror
The bending-magnet spectrum is approximated by S(y) = 1.8 y^(1/3) exp(-y) with y = E / Ec. A grazing-incidence mirror reflects X-rays only below the critical angle, approximately theta_c[mrad] = 32 / E[keV] for a heavy-metal coating. The Fresnel reflectivity without absorption is R = (x - sqrt(x^2 - 1))^4 with x = theta / theta_c, and R = 1 for x < 1. Set the mirror angle between theta_c(3E) and theta_c(E): the fundamental is reflected, the harmonic is thrown away.
Slits, sample and detector
The slit passes the fraction erf(s / (2 sqrt2 sigma)) of a 300 micrometre FWHM Gaussian beam. The powder sample gives Debye rings at 2 theta = 2 asin(lambda / 2d); on a flat detector at distance D the ring radius is R = D tan(2 theta). Ring intensity is the sum of |F|^2 over all equivalent reflections (so multiplicity comes out automatically) times the Lorentz-polarisation factor (1 + cos^2 2theta) / (sin^2 theta cos theta). The structure factor F = sum f_j exp(2 pi i (h x + k y + l z)) is summed over the real atom positions, so the systematic absences (fcc: mixed parity, bcc: h+k+l odd) appear on their own. Ring width combines beam size / D, the mono bandwidth and the Scherrer term 0.9 lambda / (L cos theta).
Honest simplifications
The page is an educational model of a generic bending-magnet beamline, not any real facility. Form factors are an illustrative Gaussian in sin(theta)/lambda, the Debye-Waller factor is a single isotropic B, absorption, fluorescence, air scatter and detector response are only suggested by a smooth background and Poisson noise, the second crystal is assumed perfectly matched, and the mirror is a flat reflector with fixed geometry. Mirror angle deflection is exaggerated in the drawing. The 3D page is a separate museum-style exhibit of a beamline; this page adds the physics.
What to try
- Move the energy slider and watch the silicon crystals tilt and the Debye rings shrink or grow (R = D tan 2 theta).
- Raise the mirror angle above the harmonic critical angle: the magenta third-harmonic beam is reflected less than the fundamental. Then pick the "Mirror off" preset.
- Detune the monochromator by the rocking-curve width and see the flux halve.
- Shrink the crystallite size to 8 nm: the rings broaden (Scherrer).
- Compare Si, NaCl, Au, Fe and LaB6: missing rings are the systematic absences.