This page gives a self-contained derivation of the theory underlying each calculation in SLD Explorer, at a level appropriate for practitioners in neutron scattering. Notation follows the conventions of Sears (1992) and Blundell & Blundell except where noted.
The interaction between a neutron and a nucleus is short-ranged (nuclear force range ~1 fm) compared to thermal neutron wavelengths (1–10 Å). In the low-energy limit the scattering is purely s-wave and the true potential can be replaced by the Fermi pseudo-potential:
where mn is the neutron mass, R is the nucleus position, and b is the bound coherent scattering length. This immediately gives the Born-approximation differential cross section for a single nucleus:
and the total coherent cross section σcoh = 4πbc2.
The scattering length quoted in data tables is the bound value, meaning the nucleus is
assumed infinitely heavy. For a nucleus of mass M the free-nucleus value is
bfree = bbound ⋅ M/(M+mn). For heavy nuclei the
correction is negligible; for hydrogen (M/mn = 1) it halves the effective length.
SLD Explorer uses bound coherent values from periodictable, consistent with
the NIST and ILL databases.
For a real element the scattering length fluctuates between isotopes and between nuclear spin states. The ensemble average decomposes as:
Only the coherent part carries structural information (interference). Incoherent scattering adds an isotropic, Q-independent background that reduces signal-to-noise. Hydrogen has an exceptionally large incoherent cross section (~80 barn) because the two nuclear spin states (I = 0 and I = 1) have scattering lengths of very different magnitude (+6.67 and −47.5 fm, weighted average −3.74 fm). This is the primary motivation for H → D substitution in structural studies.
Unlike X-ray form factors, neutron scattering lengths show no simple trend with atomic number. Values can be positive or negative, and adjacent elements or isotopes can differ enormously. This arises from resonance scattering: compound-nucleus resonances near thermal energies shift the effective scattering length in a sign- and magnitude-dependent way through the Breit–Wigner formula. Notable examples include the sign change from Ti (−3.44 fm) to V (−0.38 fm) to Cr (+3.64 fm), and the enormous contrast between 10B (+−mixed, large absorption) and 11B (+6.65 fm, weak absorption).
The SLD is the sum of scattering lengths per unit volume, averaged over a volume element large compared to interatomic distances but small compared to the neutron coherence volume:
For a homogeneous material with number density N (atoms or formula units per unit volume):
The number density of formula units is related to the macroscopic mass density ρ and formula-unit molar mass M by:
so for a compound with formula units containing atoms of type i with multiplicity ni and individual coherent scattering lengths bi:
where M = Σi ni Ai, with Ai the atomic weight. This is the formula evaluated by the Compound SLD calculator in SLD Explorer.
In small-angle neutron scattering (SANS), the scattered intensity from a two-phase system is proportional to the square of the SLD contrast:
where P(Q) is the particle form factor and S(Q) the structure factor. Maximising |ΔρSLD| between particle and matrix maximises signal. Setting it to zero (contrast matching) suppresses one component entirely, allowing the other to be studied in isolation.
Neutron absorption removes flux from the beam. By the optical theorem, absorption is captured through a complex scattering length:
The imaginary part b'' is related to the total cross section and absorption cross section by:
where k = 2π/λ is the neutron wave vector. Since σabs ∝ 1/v ∝ λ for most nuclei (the 1/v law), b'' is proportional to wavelength:
with the reference wavelength λ0 = 1.798 Å (thermal, 25.3 meV). The Im(SLD) reported by SLD Explorer is evaluated at this reference wavelength. For instruments using cold neutrons (λ ~ 5–10 Å) or hot neutrons (λ ~ 0.5 Å), Im(SLD) should be scaled accordingly. Elements with significant absorption at thermal wavelengths include 10B (~3840 barn), Gd (~49 000 barn), Cd (~2520 barn), and In (~194 barn).
The incoherent cross section also contributes to neutron removal from the coherent beam. SLD Explorer reports an incoherent SLD defined as:
This is a transmission-relevant quantity: it attenuates the beam but carries no structural information. For hydrogenous materials SLDinc dominates the total cross section and determines sample thickness limits in transmission geometry.
A large Im(SLD) means significant absorption. This matters for:
For a compound formula unit An1Bn2..., summing over all atoms in the formula unit:
Isotope enrichment is handled by replacing the natural-abundance average scattering length
with the isotope-specific value. SLD Explorer supports isotope syntax such as
B[11]4C, Li[7]F, D2O, and mixed formulas such as
H[0.9]D[0.1]2O.
When a sample is a physical mixture of two distinct phases A and B that retain their identities (no chemical reaction, no excess volume), the effective SLD is the volume-weighted average:
This is thermodynamically equivalent to ideal volume additivity: Vmix = VA + VB. The model is appropriate for:
SLD Explorer uses this model in Blend Explorer and as one option in Contrast Match.
For solid solutions and alloys where the sample is a single-phase compound at every composition, the formula itself changes with composition. At fractional composition x the effective formula is constructed from the weighted sum of constituent atoms, and ρ and μ are re-estimated at each x:
where ρ0, ρ1 are the endpoint mass densities and μ0, μ1 are the endpoint magnetic moments per formula unit. If endpoint values are not entered, the app estimates ρ from an additive molar-volume rule and μ from stoichiometric sum of elemental moments (see Section 4.5).
Because the SLD is then recomputed from the formula at each x using the formula-unit molar mass at that composition, the SLD vs x curve is generally non-linear even when ρ(x) is linear. This captures the effect of changing formula-unit mass as composition changes.
The number density of atoms in a formula unit (in Å−3):
For base material A and additive C at additive volume fraction x:
When tabulated data for a compound are unavailable, the app uses two standard approximations:
Additive molar volume: each atom contributes its elemental molar volume Mi/ρi:
Stoichiometric magnetic moment:
These are first approximations only. Crystal-field effects, hybridisation, and charge transfer all modify the actual moment in compounds. For quantitative work, measured density and moment should be substituted.
The sensitivity of neutron scattering lengths to nuclear structure means isotope substitution is a uniquely powerful experimental tool, unmatched in X-ray or electron scattering.
H and D have scattering lengths of opposite sign and very different magnitude (bH = −3.741 fm, bD = +6.671 fm). The SLD of H2O and D2O differ by nearly 7 × 10−6 Å−2. By mixing them at fractional deuteration fD:
the solvent SLD can be tuned continuously across the range −0.56 to +6.37 (in 10−6 Å−2), passing through the SLD of most organic materials near fD ≈ 0.08–0.12 (contrast-match point). This enables selective visibility in SANS and reflectometry: a component matched to solvent becomes invisible, and the signal arises purely from the remaining component. This technique is the backbone of structural biology and soft-matter neutron scattering.
10B has a thermal neutron absorption cross section of ~3840 barn and a scattering length b = −0.1 fm (essentially zero coherent scattering). 11B has b = +6.65 fm and an absorption cross section of only ~0.005 barn. Isotopic substitution in boron-containing materials (borides, boron carbide) thus allows independent control of absorption and scattering: 10B-enriched layers are absorbing mirrors or screens; 11B-enriched layers scatter strongly with negligible absorption.
| Nuclide / Material | bc (fm) | σcoh (barn) | σinc (barn) | σabs (barn, λ0) | Bulk SLD (10−6 Å−2) |
|---|---|---|---|---|---|
| H | −3.741 | 1.758 | 80.26 | 0.333 | — |
| D | +6.671 | 5.592 | 2.05 | 0.000519 | — |
| 10B | −0.1 | 0.14 | 3.0 | 3835 | — |
| 11B | +6.65 | 5.56 | 0.21 | 0.005 | — |
| Ti | −3.438 | 1.485 | 2.87 | 6.09 | −1.95 |
| Ni | +10.3 | 13.3 | 5.2 | 4.49 | +9.40 |
| Fe | +9.45 | 11.22 | 0.40 | 2.56 | +8.02 |
| Si | +4.149 | 2.163 | 0.004 | 0.171 | +2.07 |
| H2O | — | — | — | — | −0.56 |
| D2O | — | — | — | — | +6.37 |
Values from Sears (1992) and NIST Neutron Data Booklet (Dianoux & Lander, eds., 2003).
Some element pairs can be combined to produce a zero or near-zero SLD alloy at a specific composition. The classic example is Ti:Zr (bTi = −3.44 fm, bZr = +7.16 fm): at approximately 67 at.% Ti the coherent SLD vanishes, giving a null-scattering matrix useful as a SANS sample container or magnetic matrix. SLD Explorer can identify such null-contrast compositions by inspecting the Composition Scan or SLD Search results.
Neutrons possess a magnetic moment μn = −1.913 μN. They interact with the magnetic field produced by unpaired electron spins and orbital angular momentum in a material. In the dipole approximation, the magnetic scattering amplitude for a single atom carrying moment μ (in Bohr magnetons, μB) is:
where γ = 1.913, re = 2.818 fm is the classical electron radius, and f(Q) is the magnetic form factor. At Q = 0 (the relevant limit for reflectometry), f(0) = 1 and the magnetic scattering length per atom is simply 2.695 fm per μB.
For a material with number density N and ordered magnetic moment μ per formula unit (oriented parallel to the applied field, i.e. along the neutron quantisation axis):
This has the same units as nuclear SLD and can be directly compared to it. For pure iron (μ = 2.22 μB/atom, ρ = 7.874 g cm−3): SLDmag ≈ 5.08 × 10−6 Å−2.
With neutrons polarised parallel (+) or antiparallel (−) to the applied field:
These are the spin-up (non-spin-flip) and spin-down (non-spin-flip) channels. In SLD Explorer the labels spin-up and spin-down refer to these two channels, consistent with the convention used at ILL, ISIS, and SNS beamlines for polarised neutron reflectometry.
The magnetic form factor f(Q) falls off with increasing Q because the electron cloud has a finite spatial extent (~1–3 Å). For 3d transition metals f(Q) is well approximated by a two-Gaussian fit to Hartree–Fock calculations. For 4f rare earths the form factor is much more extended in Q-space. At the Q values relevant to specular reflectometry (Q < 0.3 Å−1), f(Q) ≈ 1 for 3d metals and the approximation SLDmag = N ⋅ 2.695 fm ⋅ μ is essentially exact.
The full magnetic scattering operator includes both spin and orbital contributions. For most 3d transition metals (Fe, Co, Ni) the orbital moment is quenched by the crystal field and the spin contribution dominates. For 4f rare earths (Gd, Dy, Tb, Ho, Er) the orbital contribution is significant and the full Hund's-rule moment must be decomposed. SLD Explorer uses the total effective moment μeff entered by the user, which implicitly includes both contributions for the Q→0 limit.
A neutron propagating through a medium of SLD experiences a coherent optical potential:
This potential shifts the neutron kinetic energy and defines an effective wave vector inside the medium. For a neutron impinging at grazing angle θ with wave vector k = 2π/λ, the z-component of the wave vector inside the medium is:
where kz,0 = k sinθ is the z-component in vacuum.
Total external reflection occurs when kz2 ≤ 0, i.e. when:
For Ni (SLD = 9.40 × 10−6 Å−2) at λ = 4 Å: θc ≈ 0.62°. The critical angle is routinely used to calibrate reflectometer geometry and to verify sample SLD.
At a single sharp interface between media with SLD1 and SLD2, the Fresnel reflection coefficient is:
For Q ≫ Qc the reflectivity falls as Q−4 (Porod law). Interface roughness σ introduces a Debye–Waller-like damping factor exp(−2kz,1kz,2σ2).
For a stratified medium with N layers, the exact specular reflectivity is computed by the Parratt recursion from the substrate upward:
where rj,j+1 is the Fresnel coefficient at the j/(j+1) interface and dj is the thickness of layer j. SLD Explorer does not fit reflectivity curves, but the SLD values computed here are the direct input to Parratt-type codes such as RefNX, Refl1D, GenX, and Motofit.
The reflectometry inverse problem is, in essence, recovering the SLD depth profile ρSLD(z) from measured R(Q). From a forward-design perspective, every materials choice in a multilayer determines a specific ρSLD(z), and the tabs in SLD Explorer are tools for optimising that profile before fabrication.
With a saturating magnetic field applied in-plane and neutrons polarised along the field, the optical potential splits into two independent channels:
The two polarisation channels experience different effective potentials and therefore different critical angles and reflectivities. This is the basis of polarised neutron reflectometry (PNR). The four cross sections measured in a full polarisation analysis are R++, R−− (non-spin-flip, NSF) and R+−, R−+ (spin-flip, SF).
When the magnetisation has a component perpendicular to the neutron polarisation axis (e.g. in-plane rotation, canted moments, domains), spin-flip scattering occurs and R+− ≠ 0. In the collinear geometry (magnetisation || field || polarisation), SF channels vanish and only NSF channels carry information. SLD Explorer operates in the collinear approximation throughout: SLD± = SLDnuc ± SLDmag.
The spin asymmetry SA = (R++ − R−−) / (R++ + R−−) is zero for non-magnetic layers and non-zero for magnetic ones. For a single magnetic layer at Q ≫ Qc:
An ideal neutron polariser transmits or reflects one spin channel completely and suppresses the other. This requires:
This defines the polarising mirror condition: SLDmag > SLDnuc. For Fe (SLDnuc = 8.02, SLDmag = 5.08): spin-up SLD = 13.10, spin-down = 2.94. A Fe/Si bilayer on glass gives very different critical angles for the two spin channels, and is commonly used in Fe/Si supermirror polarisers. The Bilayer Contrast and Element Pairs tabs in SLD Explorer are designed precisely to identify and compare such combinations.
In small-angle scattering from a two-component system (particles in a matrix or solvent), the scattered intensity is:
Setting ρSLD,particle = ρSLD,matrix (contrast match point) makes the particles invisible. In H2O/D2O mixtures this is achieved by tuning fD. For hard materials systems, the contrast match point can instead be reached by varying the composition of one component (e.g. alloying, partial isotope substitution).
The Contrast Match tab finds compositions x, y such that:
For the volume-mixing model:
This is a linear equation in two unknowns. In general there is a family of solutions forming a line in (x,y) space. The interactive matcher line in the plot shows this locus, and the user can drag it to identify physically accessible match points (x,y ∈ [0,1]).
For the formula-based model, the SLDA(x) and SLDB(y) curves are non-linear and the match locus is found numerically. The root-finding is done by sampling both curves on a dense grid and detecting sign changes in SLDA(x) − SLDB(yfixed).
The Composition Scan tab sweeps x from 0 to 1 through a binary formula AxB1−x and plots Re(SLD) vs Im(SLD) for each composition step. This gives an immediate visual overview of:
The plot shows nuclear SLD only and magnetic moment is not included.
Blend Explorer applies the two-phase volume-mixing model to a user-defined material pair, with interactive sliders for the volume fraction and endpoint densities. It is primarily a rapid prototyping tool for estimating the SLD of composite or porous materials.
The Spin Contrast tab maps all element pairs (i,j) in the periodic table onto the plane (|ΔSLD−|, |ΔSLD+|) where:
The x-axis is |ΔSLD−| (spin-down contrast) and the y-axis is |ΔSLD+| (spin-up contrast).
The perpendicular distance from the diagonal quantifies the degree of spin selectivity:
A large dperp combined with a large total contrast |ΔSLD+| + |ΔSLD−| identifies ideal polarising mirror material pairs. The Fe/Si system, for example, sits far above the diagonal with a very large spin-up contrast and modest spin-down contrast, consistent with its use in practical supermirror polarisers.
The custom pair input allows arbitrary compounds (not just elemental pairs) to be mapped onto the same plane, enabling direct comparison with the elemental reference data. The density and magnetic moment must be supplied for each compound; the app then computes SLD± and plots the resulting point with the perpendicular-distance annotation.
periodictable Python package.
periodictable.readthedocs.ioSLD Explorer — Theory reference — 2026