## Sawtooth ripple

A ripple with an asymmetric saw-tooth profile that is uniform in $x$-direction.

#### Constructor

SawtoothRippleBox(L, W, H, d)
SawtoothRippleGauss(L, W, H, d)
SawtoothRippleLorentz(L, W, H, d)


Parameters:

• L, length
• W, width
• H, height
• d, asymmetry

Constraint:

$|d| \le W/2$

#### Usage

As for any other Form factor.

#### Implementation

Class SawtoothRipple inherits from the interface class IFormfactor .

Form factor is computed as

$$F(\mathbf{q}) = f_\parallel(q_x) f_\bot(q_y,q_z),$$

where $$f_\bot(q_y,q_z) = Hi\frac{\text{e}^{-i q_y d}}{q_y} \left[ \text{e}^{i \alpha_{-}/2} \text{sinc}\left( \frac{\alpha_{+}}{2} \right) - \text{e}^{i \alpha_{+}/2} \text{sinc}\left( \frac{\alpha_{-}}{2} \right) \right],$$

with the notation $$\alpha_{+} = H q_z + \frac{q_y W}{2} + q_y d, \quad \alpha_{-} = H q_z - \frac{q_y W}{2} + q_y d.$$

Corresponding factor $f_\parallel(q_x)$ is chosen according to longitudinal profile.

Volume has been validated against $$V=\dfrac{LWH}{2}.$$

#### Scattering

Scattering by uncorrelated, oriented sawtooth ripples for horizontal incidence. Rotation around $z$ axis:

Generated by Examples/ff/SawtoothRippleBox.py .

#### History

“SawtoothRippleBox” replicates “Ripple2” from FitGISAXS [Babonneau 2013].