This appendix collects the notation of the book. Each symbol links to the chapter that introduces it.
Units
The book uses the units of X-Ray Calc, so every number matches the screen.
- Every length (thickness, roughness, period, wavelength) is in ångströms, Å. 10 Å = 1 nm.
- Density is in g/cm³.
- Angles are in degrees, and every angle is labeled θ or 2θ.
- Photon energy is in eV. It appears only as an input that the program converts to λ.
- Reflectivity has no unit. Measured intensity is in counts, or in counts/s for an
.xrdmlimport.
Numbers use a decimal point. A range is written with an en-dash and no spaces: 0.012–0.015°. A number keeps the precision of its source.
θ and 2θ
θ is the grazing angle, measured from the surface. Every formula in the book uses θ. 2θ is the scattering angle. It is what the X-Ray Calc chart shows by default, with the 2θ box ticked. Every angle read off the chart, or typed into a range, a limit or a trim, is therefore 2θ. The conversion is θ = (2θ)/2.
The resolution Δθ is always a θ value, whatever the chart shows. On a 2θ axis the program doubles it.
Symbols
Beam and geometry
Table A-1: Beam and geometry.
| Symbol | Meaning | Unit | Introduced in |
|---|---|---|---|
| θ | The grazing angle of incidence, measured from the surface. Every formula uses θ. | ° | Chapter 2 |
| 2θ | The scattering angle. The chart shows it by default. | ° | Chapter 2 |
| λ | The wavelength. On screen: λ(Å). | Å | Chapter 10 |
| E | The photon energy, E = hc/λ, with hc = 12398.6 eV·Å. | eV | Chapter 4 |
| q | The wave-vector transfer in vacuum, q = 4π sin θ / λ. | Å⁻¹ | Chapter 7 |
| q₀, q′ | In the roughness factor: q in vacuum, and q refracted across the interface. | Å⁻¹ | Chapter 7 |
| kj | The wave vector across the layers in medium j. | Å⁻¹ | Chapter 8 |
| Fj, rj | The reflection of the interface under medium j, with its roughness factor; the reflection of everything from that interface down. | none | Chapter 8 |
| Δθ | The angular resolution: the FWHM of the Gaussian blur of the calculated curve. Always a θ value. | ° (θ) | Chapter 8 |
| s, p | The two polarization states. On screen: Polarization. | none | Chapter 8 |
Optical constants
Table A-2: Optical constants.
| Symbol | Meaning | Unit | Introduced in |
|---|---|---|---|
| n | The complex refractive index, n = 1 − δ + iβ. | none | Chapter 4 |
| ε | The dielectric constant, ε = 1 − 2δ + 2iβ. | none | Chapter 8 |
| δ | The refractive decrement, proportional to ρλ²f₁/A. | none | Chapter 4 |
| β | The absorption index, proportional to ρλ²f₂/A. | none | Chapter 4 |
| f₁, f₂ | The atomic scattering factors, real and imaginary, from the Henke tables. | electrons | Chapter 4 |
| A | The molar mass. On screen: A (g/mol). | g/mol | Chapter 4 |
| re | The classical electron radius, 2.82 × 10⁻⁵ Å. | Å | Chapter 4 |
| NA | The Avogadro constant, 6.02 × 10²³ mol⁻¹. | mol⁻¹ | Chapter 4 |
| θt | The angle of the transmitted beam inside a material, measured from the surface. | ° | Chapter 4 |
| θc | The critical angle, θ_c ≈ √(2δ) in radians. | ° | Chapter 4 |
The structure
The structure terms are those of X-Ray Calc.
Table A-3: The structure.
| Symbol | Meaning | Unit | Introduced in |
|---|---|---|---|
| Material | The chemical composition of a layer, such as Mo, SiO2 or B4C. It sets f₁, f₂ and A. It is never fitted. | none | Chapter 1 |
| Layer | One material with three parameters: H, σ and ρ. | none | Chapter 1 |
| Stack | A set of layers, single or repeated N times. | none | Chapter 1 |
| Structure | All the stacks, the substrate included. | none | Chapter 1 |
| Substrate | A stack of one layer of infinite thickness. It is held in a fit. | none | Chapter 12 |
| H | The thickness of a layer. | Å | Chapter 1 |
| σ | The roughness of a layer’s upper interface. The substrate’s σ is the roughness of its surface. | Å | Chapter 7 |
| ρ | The density. ρ = 0 means the bulk density of the material’s table. | g/cm³ | Chapter 4 |
| D | The period of a periodic stack, the sum of the H of its layers. The D box shows the top period. | Å | Chapter 6 |
| N | The number of periods of a stack. | none | Chapter 6 |
| m | The Bragg order index, m = 1, 2, 3, … | none | Chapter 6 |
| k | The period index in a periodic stack, counted from the surface: k = 1 is the top period. | none | Chapter 17 |
| C₀, C₁, … | The coefficients of a polynomial profile: value(k) = C₀ + C₁(k − 1) + C₂(k − 1)² + … C₀ is the value of the top period. | as the parameter | Chapter 17 |
| ΔD | The period drift: the bottom period minus the top one, ΔD = D(N) − D(1). | Å | Chapter 17 |
| θm | The angle of the m-th Bragg order. | ° (θ) | Chapter 6 |
| δ̄ | The refractive decrement averaged over one period, as it enters the Bragg law with refraction. | none | Chapter 6 |
| Dm | The period that the plain Bragg law gives from order m alone, D_m = mλ/(2 sin θ_m). | Å | Chapter 6 |
| Γ | The layer ratio: the thickness of the heavy layer of a two-layer period divided by D. | none | Chapter 6 |
The measured curve and the fit
Table A-4: The measured curve and the fit.
| Symbol | Meaning | Unit | Introduced in |
|---|---|---|---|
| R | The reflectivity, as calculated. | none | Chapter 1 |
| I | The measured intensity. | counts | Chapter 10 |
| θmax, Imax | The angle and the intensity of the measured maximum below the critical edge, the plateau maximum. | °, counts | Chapter 11 |
| background | The median intensity past the last visible order, never less than one count. | counts | Chapter 11 |
| rmin | The floor: calculated values below it are raised to it, in the curve and in χ². On screen: R min. | none | Chapter 14 |
| one count | One detector count on the curve that an .xrdml import scales to 1: 1 / (peak rate × counting time). | none | Chapter 10 |
| λ from the file | For copper with both Kα lines, (Kα1 + r·Kα2)/(1 + r) = 1.541874 Å at r = 0.5; behind a monochromator or hybrid mirror, Kα1 = 1.540598 Å. | Å | Chapter 10 |
| scale | The factor that puts the measured curve on the reflectivity axis. | none | Chapter 14 |
| w | The window of the solved scale: the scale may move by a factor of up to 1 + w from its anchor. On screen: Window. | none | Chapter 14 |
| χ² | The cost function as displayed: the weighted mean squared relative residual of log I, times 1000. | none | Chapter 9 |
| plain χ² | The same sum with both weights set to 1. It is shown, and never minimized. | none | Chapter 9 |
| PW χ² | The peak weighting of χ². | none | Chapter 9 |
| TW χ² | The angle weighting of χ²: none, θ², θ, √θ, 1/θ² or 1/√θ of the chart axis. | none | Chapter 9 |
| Min, Max | The fit limits of one parameter. The parameter is fitted only when Min ≠ Max. | as the parameter | Chapter 15 |
| ΔH, Δσ, Δρ | The fractions that Initialize uses: Min = V(1 − Δ), Max = V(1 + Δ). | none | Chapter 15 |
| p | The polynomial order in Polynomial mode. On screen: Order. | none | Chapter 17 |
| population | The number of particles in the swarm. | none | Chapter 16 |
| iterations | The largest number of iterations of a fit. | none | Chapter 16 |
| seed spread | The sample standard deviation of a fitted parameter over repeated runs. | as the parameter | Chapter 16 |
| seed | The start value of the random numbers of one fit. | none | Chapter 9 |
| SeedR | Start the swarm spread over each parameter’s whole Min–Max range. | none | Chapter 9 |
| Vmax, Ksxr | The largest step, and the scatter around a structure, as fractions of each parameter’s range. | none | Chapter 9 |
| Jmax, SHmax, k1, k2 | The Shake settings: iterations without improvement before a shake; shakes before a return to the best; the lift of the best χ²; the enlargement of Vmax and Ksxr. | none | Chapter 9 |
| Tolerance | The fit stops when the best χ² falls below it. | none | Chapter 9 |
| x, v, b, g, u₁, u₂, c₁, c₂, κ | The particle-swarm step: position, step, best particle of the iteration, best structure so far, random numbers in [0, 1], the two pulls, the constriction factor. | none | Chapter 9 |
The Bragg law with refraction
Every period in the book is read from the orders with this law (Chapter 6: The Bragg law with refraction):
Labels that can mislead
- N means two things on the screen. In the structure panel it is the number of periods. In the calculation settings it is the number of calculated points. The book writes N only for periods.
- Some labels are drawn in the Symbol font. The screen shows θ, Δθ, 2θ, λ(Å), θ1 and θ2. The book quotes the labels as they appear.
- The molar mass in Tools > Edit Henke table… reads A (g/mol) from 3.9.4, and N (a.u.) in earlier versions. It is neither a number of periods nor a number of points.
Abbreviations
Table A-5: Abbreviations.
| Abbreviation | Meaning | First use |
|---|---|---|
| XRR | X-ray reflectivity | Chapter 1 |
| PMM | Periodic multilayer mirror | Chapter 6 |
| LFPSO | Lévy-flight particle swarm optimization, the fitting algorithm of X-Ray Calc | Chapter 9 |
| FWHM | Full width at half maximum | Chapter 8 |
| GUI | Graphical user interface | Chapter 2 |
Sources
- NOTATION.md of this book, §1–3 and §5–7: the units, the angle convention, the symbols and their sources, the on-screen labels and the abbreviations. Each symbol’s own source (the X-Ray Calc Help, the X-Ray Calc source, the fitting procedure, Spiller) is named there and in the chapter linked.
- X-Ray Calc Help: Models → Terminology (material, layer, stack, structure, substrate; from Penkov et al. 2024); Calculation → Algorithm; Fitting → Understanding the Cost Function.
- The X-Ray Calc 3 source: hc = 12398.6 eV·Å (
Shared\Math\math_globals.pas); the resolution doubled on a 2θ axis (Shared\Math\unit_calc.pas); the period index counted from the surface.
Cross-references: Chapter 2: 2θ or θ; Chapter 4: Where δ and β come from; Chapter 6: The Bragg law with refraction; Appendix D: Glossary.