IN26 · Size hidden near the beam

Small-Angle X-ray Scattering

Can a circular scattering halo determine a particle’s size and shape? Small-angle scattering constrains particle size through the angular variation of scattered intensity. Randomly oriented spheres, rods and clusters can all produce circular halos. Distinguishing them needs enough angular coverage and a corrected background. A beamstop that hides the central region can remove the information needed to determine size.

Move the detector and beamstop, then compare spheres, rods and clusters. Save sample, empty-cell and dark photons to see which shapes the scattering can distinguish.

Drag the gold detector carriage. Save counts after a change; moving the detector also needs a fresh geometry calibration.

Particle familyParticle gyration radius

9\,\mathrm{nm}

Sample–detector distance

850\,\mathrm{mm}

Beamstop radius

0.65\,\mathrm{mm}

Large spheres: number fraction

0

A full simulated record and independent geometry standards are loaded.

Counting and material assumptions

Particle volume fraction

0.001

Exposure / seconds

2

Random seed

260199

A one-millimetre path, unit detector efficiency and electron-density contrast of one hundred electrons per cubic nanometre are assumed. Rods have a fixed aspect ratio; clusters contain four non-overlapping spheres. All orientations are random. The large-particle control applies to spheres only.

Model assumptions

Synthetic dilute particles scatter once in transmission. Random orientations give isotropic patterns for all three families. Known ring standards calibrate a perpendicular detector; a second ring tests the result independently. Background, solid angle and saved transmission are corrected before restricted fitting. The shape and concentration results depend on declared contrast, thickness and candidate families. A finite angular range cannot uniquely reconstruct arbitrary particles or size distributions. Interparticle structure, multiple scattering and instrumental blur are omitted. Particle size is enlarged in the apparatus; beam width and photon motion are schematic.

Retained simulated counts

Subtract the cell before choosing a shape

Saved record

Conditional fitted gyration radius

9.00\,\mathrm{nm}

Conditional volume fraction

0.0996\,\%

Saved sample transmission

73.3\,\%

First usable momentum transfer

0.0619\,\mathrm{nm}^{-1}

Sample photons

Empty-cell photons

Dark photons

Each image has its own logarithmic count scale; black marks the saved beamstop/support exclusion. Masked raw counts are retained in both exports.

Calibration, model comparison and provenance

Confidence intervals and shape comparisons are conditional on the three declared families, known contrast and thickness. Counting covariance includes shared transmitted-monitor and monitor-dark noise. Pixel-dark noise uses its shared subtraction coefficient. Geometry, wavelength, contrast, thickness and detector-efficiency systematic uncertainties are not propagated. The small-angle absorption correction treats cell and particles as equally attenuated.

Physical limits and schematic scale

Dilute, homogeneous, non-interacting particles scatter once. Interparticle structure and multiple scattering are omitted; intra-cluster interference is included. A round halo is compatible with every randomly oriented family. A finite set of models cannot prove a unique structure or recover an arbitrary size distribution. The beam is monochromatic and unpolarized; refraction, divergence, detector blur, dead time and flat-field errors are omitted. The physical detector uses the same pixel rays and beamstop shadow as the count model. Particles are enlarged six thousand times within the sample view; one representative orientation stands for an isotropic ensemble. Beam width, ray brightness and photon speed are schematic. The two-dimensional fallback compresses axial distance and labels this projection.

Start with the essentials

Focus question: When do the recorded angles distinguish size and shape?
One-sentence intuition: The radial intensity profile carries information that a circular outline loses. Background and missing angles determine how much can be inferred.

Core mathematical model

Angle selects a spatial scale

q=\frac{4\pi}{\lambda}\sin\frac{\theta}{2}

The scattering angle is between incident and outgoing propagation directions. Wavelength and calibrated detector geometry determine momentum transfer.

The small-angle size limit

I(q)\simeq I(0)\exp\!\left(-\frac{q^2R_g^2}{3}\right)

The Guinier approximation applies at sufficiently small momentum transfer. Gyration radius measures the contrast-weighted spread of scattering matter around its centre. Uniform contrast is assumed here.

Scattering combines particle form and correlations

I(q)\propto N(\Delta\rho_e)^2V^2P(q)S(q)

The form factor describes one particle, including coherent subunits within a cluster. The structure factor describes correlations among different particles; it is fixed to one in this dilute model. Multiple scattering requires a different model.

Common difficulties

A circular image versus a spherical particle

Typical misconceptionA circular halo proves that every particle is a sphere.

Better mental modelRandom orientations erase directional information. Rods and clusters can also give circular patterns.

A good curve versus a unique structure

Typical misconceptionA close fitted curve identifies the only possible particle.

Better mental modelOther shapes, distributions or interparticle correlations can resemble the same finite range. The inference here compares only three specified independent-particle families.

Run the experiment

01
Follow the receiving plane
Open Beamline, then Detector + beamstop. Drag the gold carriage.
What to observe: Moving the same detector changes the angle covered by each pixel. The saved record stays unchanged until another acquisition.
02
Calibrate the new position
Acquire geometry standards after moving the detector.
What to observe: The first raw ring sets centre and distance. A separately acquired ring checks the result and is excluded from the fit.
03
Compare circular patterns
Choose the sphere, rod and four-sphere presets.
What to observe: All three random ensembles have circular halos. Their radial profiles distinguish them when the accepted range is sufficient.
04
Include the empty cell
Compare the sample, empty-cell and dark images below the apparatus.
What to observe: Different exposure, flux and transmission need measured normalization. Shared monitor and dark counts correlate the corrected intensities.
05
Remove the centre
Choose The missing centre.
What to observe: A candidate may follow the visible tail while several sizes remain possible. The low-angle coverage check withholds a size conclusion.
06
Mix two sizes
Choose A few large particles.
What to observe: A larger particle scatters coherently from more electrons. Number fraction and scattering fraction differ; one-size candidates can fail a two-size sample.

Small-Angle X-ray Scattering

Subtract the cell before choosing a shape

A halo contains more than its outline

Core mathematical model

Angle selects a spatial scale

The small-angle size limit

Scattering combines particle form and correlations

Common difficulties

A circular image versus a spherical particle

A good curve versus a unique structure

Run the experiment

Follow the receiving plane

Calibrate the new position

Compare circular patterns

Include the empty cell

Remove the centre

Mix two sizes

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