+++
title = "Yoneda peak"
weight = 30
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## Yoneda peak

The Yoneda peak is an enhancement of diffuse scattering
that occurs when the exit angle $\alpha_\text{f}$ equals
the critical angle $\alpha_\text{c}$ for total external
reflection. In [off-specular reflectometry](/ref/sim/class/offspec)
and [GISAS](/ref/sim/class/scattering), it appears as a prominent
ridge across the full width of the pattern at
$\alpha_\text{f} = \alpha_\text{c}$.

### Physical origin

For x-rays and neutrons the refractive index of matter is
slightly less than unity, so total external reflection occurs
for $\alpha_\text{i} < \alpha_\text{c}$. The Fresnel
transmission amplitude

$$T(\alpha) = \bigl|1 + r(\alpha)\bigr|,$$

where $r(\alpha)$ is the Fresnel reflection coefficient, reaches
its maximum value $T(\alpha_\text{c}) = 2$ exactly at the
critical angle, and decays to unity well above it.

In the [distorted-wave Born approximation](/ref/sim/setup/bg)
(DWBA) the differential diffuse-scattering cross-section is
proportional to
$|T(\alpha_\text{i})|^2\,|T(\alpha_\text{f})|^2$.
At the Yoneda condition $\alpha_\text{f} = \alpha_\text{c}$
the exit-side factor reaches its maximum of 4 — a fourfold
enhancement over the large-angle background — while
$T(\alpha_\text{i})$ acts as a fixed global prefactor.

### Observation in off-specular scattering

In an off-specular experiment one scans $\alpha_\text{i}$
at fixed detector angle, or vice versa. Because the cross-section
contains the product $|T(\alpha_\text{i})|^2 |T(\alpha_\text{f})|^2$,
the scan shows an intensity maximum — the **Yoneda wing** — whenever
$\alpha_\text{i}$ or $\alpha_\text{f}$ passes through $\alpha_\text{c}$.

When the sample is a thin film on a substrate with two critical angles
$\alpha_\text{c,film}$ and $\alpha_\text{c,substrate}$, the pattern
shows two Yoneda ridges. The region of elevated scattering between them
is the **Yoneda band**. The streak positions directly yield the
scattering densities of the film and the substrate.

Because the Yoneda-peak position depends only on $\alpha_\text{c}$,
which is set by the scattering density of the surface layer and the
radiation wavelength, it provides an absolute angular reference in
every off-specular or GISAS measurement. It is routinely used to
refine the incident angle, the sample-detector distance, and the
beam-center coordinates.

### BornAgain simulation

The BornAgain example below demonstrates how to set up an
off-specular simulation in which the Yoneda peak is the dominant
feature.

**Sample:** vacuum | mixed layer M | substrate S.

Layer M contains a dilute 2D assembly of short cylinders
(radius $R = 5\,\text{nm}$, height $H = 3\,\text{nm}$)
embedded in vacuum. The particle volume fraction $f = 0.006$
gives an effective refractive-index decrement

$$\delta_\text{eff} = f\,\delta_\text{P},$$

chosen to place the V→M critical angle at $\alpha_\text{c} = 2°$.

**Suppression of the M/S interface reflection:** the substrate
refractive index is set equal to $\delta_\text{eff}$, so the
substrate is index-matched to the effective medium. This removes
the Fabry–Pérot resonance at the M/S interface and leaves the
V/M Yoneda ridge as the sole dominant feature.

**Scan range:** $\alpha_\text{i}$ and $\alpha_\text{f}$ both
run from $1.6°$ to $3.4°$, bracketing $\alpha_\text{c} = 2°$.
The Yoneda peak appears as a bright ridge along the diagonal
$\alpha_\text{f} = 2°$ and along $\alpha_\text{i} = 2°$.

{{< figscg src="/img/auto/offspec/YonedaBA.png"
    caption="Off-specular simulation showing the Yoneda peak at the critical angle α_c = 2°. Scan range 1.6°–3.4° in both α_i and α_f."
    class="center" width="500px" >}}

{{< show-ex file="offspec/YonedaBA.py" >}}
