X-ray reflectivity, or XRR, measures how a surface reflects X-rays as the measurement geometry changes. The resulting curve can constrain a thin film’s thickness and electron-density structure, including the transitions between layers.
The gold standard follows the measured intensities into a clearly stated depth model. A fitted layer drawing is an interpretation of the curve, with resolution and assumptions, rather than a direct cross-sectional photograph.
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Read the angle convention, estimate thickness from fringes and recognise what a reflectivity fit still needs to establish.
Full chapter index · Practice and transfer · Education root · How X Works
Full chapter index
Define the geometry · Chapters 1–2
Work the thickness · Chapters 3–4
Test the measurement · Chapters 5–7
Compare and transfer · Chapters 8–10
Choose specialist reading · Chapters 11–12
A thin film may have a surface, one or more internal interfaces and a supporting substrate. XRR investigates an electron-density profile normal to those interfaces through the angular dependence of reflected intensity.
Useful questions include whether a deposition produced the expected thickness or whether a layered model explains a measured change. The quantity of interest should determine the required measurement range and supporting evidence.
Draw the proposed layers with approximate thicknesses and label the unknowns before fitting. This makes it easier to see whether a parameter was actually constrained by the measurement or simply fixed from prior knowledge.
CHAPTER 2 OF 12 · Define the geometry
2. Specular reflection has a defined angle convention
In a specular geometry, the incident and reflected angles to the sample surface are equal. If each is θ, the angle between the incident beam direction continued forward and the reflected beam is 2θ.
Check the axis before inserting an angle into a formula. A scan plotted against 2θ must not be interpreted as though it directly reports θ. Also distinguish grazing angle measured from the surface from an angle measured from its normal.
At small grazing angles, X-rays can show strong external reflection. Above the relevant critical region, the reflected intensity declines with a shape influenced by the interface structure and experimental conditions.
For specular reflection in the simple external geometry, the normal momentum-transfer magnitude is qz = 4πsinθ/λ. Use the same length unit for wavelength and the inverse unit for qz.
With hypothetical λ = 0.154 nm and θ = 0.50°, qz ≈ 4πsin0.50°/0.154 ≈ 0.712 nm⁻¹. If the scan axis displays 2θ = 0.50°, then θ = 0.25° and qz ≈ 0.356 nm⁻¹.
The factor-of-two angle error nearly doubles qz at these small angles. Label the original angle and the converted coordinate in the working so a later thickness estimate remains traceable.
CHAPTER 4 OF 12 · Work the thickness
4. Worked example: estimate thickness from fringes
For a simple film with suitable contrast and resolved high-q Kiessig fringes, a useful first estimate is t ≈ 2π/Δqz. This approximation uses the spacing of successive equivalent fringes away from the critical region.
Let Δqz = 0.314 nm⁻¹. Then t ≈ 2π/0.314 ≈ 20.0 nm. If the spacing doubles to 0.628 nm⁻¹ in the same model, the estimated thickness halves to about 10.0 nm.
A thicker simple film produces more closely spaced fringes. Do not apply this shortcut indiscriminately to multiple overlapping periods, unresolved fringes or the nonuniform oscillations close to a critical edge. Those cases require a fuller model.
CHAPTER 5 OF 12 · Test the measurement
5. Normalise intensities before comparing curves
Reflectivity is a ratio tied to the incident intensity and the instrument geometry. Source changes, detector response and background affect a measured curve, particularly where the reflected signal becomes small.
In an invented simplified case, take reflected and background rates of 100 and 10 counts/s, with a compatible incident monitor rate of 100,000 counts/s. Assuming the monitor conversion and other factors are already handled, the corrected ratio is (100 − 10)/100,000 = 0.0009.
This arithmetic does not replace a real instrument correction. State how monitor rates, acquisition durations and detector settings are converted into comparable intensities. A fall in raw counts alone need not be a change in the specimen.
CHAPTER 6 OF 12 · Test the measurement
6. The beam footprint changes with grazing angle
For a beam of vertical height h incident at angle θ to a flat surface, an approximate footprint length is h/sinθ. At small angles, a narrow beam can illuminate a surprisingly long strip.
Take h = 0.10 mm and θ = 0.50°. The footprint is about 11.46 mm. A sample only 10 mm long would not intercept that entire ideal footprint. At 0.25°, the footprint grows to about 22.92 mm.
Consequently, angle-dependent illumination can modify the intensity comparison. Check sample dimensions, alignment and the instrument’s footprint treatment before assigning every low-angle change to a density profile.
CHAPTER 7 OF 12 · Test the measurement
7. A good fit still needs an identifiable model
The reflectivity curve depends on the assumed electron-density profile, interface widths and experimental resolution. Different parameter combinations can sometimes describe the available data similarly. A small residual alone does not establish a unique layer structure.
Keep the measured range, background and fixed parameters with the result. Compare plausible alternative models and identify what independent information constrains them.
An interface width can represent roughness, a gradual density transition or contributions from both. Report what the model parameter means rather than treating it automatically as one microscopic surface property.
| Feature or result | Possible information | Question to retain |
|---|---|---|
| Resolved fringe spacing | Simple-film thickness scale | Is one period and the approximation appropriate? |
| Critical-region behaviour | Electron-density contrast | Were angle and illumination checked? |
| Fitted interface width | A broadened density transition | Could roughness and grading be distinguished? |
| Small fit residual | Agreement within the chosen model | Are alternative models also compatible? |
CHAPTER 8 OF 12 · Compare and transfer
8. Uncertainty follows the observed fringe spacing
Using t ≈ 2π/Δqz, consider an estimated spacing between 0.300 and 0.330 nm⁻¹. The corresponding thickness estimates are about 20.94 and 19.04 nm. The inverse relation means a larger spacing produces a smaller thickness.
This range only explores sensitivity to the spacing in the simplified model. It omits angle calibration, instrumental resolution and assumptions about the film.
A clear report distinguishes that limited sensitivity from a complete uncertainty estimate. If the fringes are poorly resolved, printing more decimal places in the fitted thickness does not supply missing information.
CHAPTER 9 OF 12 · Compare and transfer
9. Choose complementary methods by the missing quantity
XRR addresses the depth-dependent electron-density structure of a surface region. Diffraction can add information about crystalline ordering. Spectroscopic ellipsometry supplies a different optical response, also interpreted through a film model.
These methods can constrain a shared specimen from different directions, but their parameters are not automatically interchangeable. A thickness from one model should be compared with a thickness defined consistently in the other.
For a learner, write the measured observable for each method before comparing the output diagrams. This keeps agreement between two fitted thicknesses from being mistaken for two direct images of the same boundary.
First, estimate thickness for Δqz = 0.0250 Å⁻¹. Second, calculate the ideal footprint of a 0.05 mm beam at 0.25°. Third, explain whether equal fitted thicknesses establish identical films.
Checks: t ≈ 2π/0.0250 ≈ 251 Å, or 25.1 nm. The footprint is 0.05/sin0.25° ≈ 11.46 mm. Equal thicknesses alone do not establish equal density profiles, interfaces or composition.
Now change a single-layer specimen into a multilayer stack. Explain why selecting just one visually convenient oscillation period can become ambiguous. The transfer test is recognising where the simple estimate no longer answers the full structural question.
Follow the X-ray measurement
- X-ray Optics — connect grazing geometry with mirrors and other optical components.
- X-ray Diffraction — compare thin-film depth information with structural ordering.
Add another experimental view
- Spectroscopic ellipsometry and thin-film evidence — compare an optical model of the same broad class of specimens.
- Synchrotron light and beamlines — place a reflectivity experiment within a tunable source and beamline.
Use the geometry route when the difficulty is the incident angle or q coordinate. Use the complementary-method route when the question concerns which part of a layer model another measurement could test.
- Diamond Light Source: X-ray reflectivity — specular geometry and surface-property interpretation.
- NIST Special Publication 960-13 — X-ray reflectivity models, Kiessig fringes and experimental considerations.
- ISO 16413 overview — the scope of thickness, density and interface-width evaluation by X-ray reflectometry.
The numerical examples are invented for teaching. They do not constitute an instrument protocol or certify a deposited film. Practical measurements require the appropriate experimental facility and its procedures.
The learning checkpoint is a structural claim supported by a visible chain: measured curve, coordinate convention, fitted model and unresolved alternatives. Continue through the Education root to practise that chain in a different measurement problem.