Analytical SLD profiles

Oscillatory profile

Use refnx.reflect.FunctionalForm to describe an oscillating SLD profile that can be used to describe the surface of a liquid metal. See “X-ray reflectivity study of temperature-dependent surface layering in liquid Hg”, DiMasi, Tostmann, Ocko, Pershan, Deutsch, Phys Rev B, 58. 10.1103/PhysRevB.58.R13419 for further details

import numpy as np
import matplotlib.pyplot as plt
from refnx.analysis import Parameter
from refnx.reflect import SLD, Slab, FunctionalForm, ReflectModel

FunctionalForm requires a callable of signature profile(z, extent, left_sld, right_sld, **kwds). kwds is used to supply parameters describing the shape of the profile. left_sld, right_sld provide the SLDs of the structure to the left and right of the Component. extent is the total width of the Component. z is an array provided to profile, a list of distances at which the function needs to return a (possibly complex) SLD. profile needs to return a tuple (sld, vfsolv), where sld is an array of the same shape as z, and vfsolv is the volume fraction of solvent at each point in z. If sld already incorporates a solvent contribution, then return (sld, None).

For further details see FunctionalForm.

def rho(z, extent, left_sld, right_sld, d=2.72, sigma_t=1.0, sigma_bar=0.46, offset=5):
    # d, sigma_t, sigma_bar are parameters that describe the shape of the profile
    def term(n):
        sigma_n = n * sigma_bar**2 + sigma_t**2
        prefactor = d / sigma_n / np.sqrt(2 * np.pi)
        return prefactor * np.exp(-0.5 * ((z - offset - n * d) / sigma_n) ** 2)

    _rho = np.zeros_like(z)
    for i in range(0, 20):
        _rho += term(i)
    return left_sld + _rho * (right_sld - left_sld), None

air = SLD(0)
d2o = SLD(6.36)
sigma_bar = Parameter(0.46, "sigma_bar")
d = Parameter(2.72, "d")
sigma_t = Parameter(1.0, "sigma_t")
offset = Parameter(
    5
)  # don't allow me to vary. It's used because rho needs to be evaluated at negative distances of z.

f = FunctionalForm(
    20,
    rho,
    microslab_max_thickness=0.1,
    sigma_bar=sigma_bar,
    d=d,
    sigma_t=sigma_t,
    offset=offset,
)

s = air | f | d2o(0, 0)

s.plot();