Evanescent wave intensity

From the BornAgain documentation one can see, that for the $q > q_c$ there are two solutions for the Schrödinger equation: reflected wave and transmitted wave. However, for the case $q < q_c$ there is also a real solution, that shows that even when the potential barier is higher than the neutron energy normal to the surface, it can still penetrate to a characteristic depth. This evanescent wave travels along the surface with wave vector $k_{||}$ and after a very short time it is ejected out of the bulk in the specular direction [1].

Evanescent wave intensity is calculated in BornAgain as

$$I_{ew}(z) = \left|\Psi(z)\right|^2 = \left|R\cdot e^{ik_zz} + T\cdot e^{-ik_zz}\right|^2$$

Simulate neutron penetration depth for the resonator [2]

BornAgain uses the following conventions for the DepthProbe geometry description. The surface of the sample will be assigned to $z=0$, while the direction of z-axis will be from the bulk of the sample out to the ambient. Thus, $z<0$ corresponds to the region from the substrate to the surface of the sample, and $z>0$ corresponds to the ambient media.

For the sample under consideration, Si block is considered as an ambient medium and D$_2$O as a substrate.

GUI

Start a new project Welcome view->New project
Go to the Instrument view and add an DepthProbe instrument.
Set the instrument parameters as follows.
Switch to the Sample view. Create a sample made of 6 layers (from bottom to top)
- D$_2$O substrate, SLD-based material with SLD=$6.34\cdot 10^{-6} + i\cdot 1.13\cdot 10^{-13}$
- TiO$_2$, thickness 3 nm, SLD-based material with SLD=$2.63\cdot 10^{-6} + i\cdot 5.4\cdot 10^{-10}$
- Ti, thickness 10 nm, SLD-based material with SLD=$2.8\cdot 10^{-6} + i\cdot 5.75\cdot 10^{-10}$
- Pt, thickness 32 nm, SLD-based material with SLD=$6.36\cdot 10^{-6} + i\cdot 1.9\cdot 10^{-9}$
- Ti, thickness 13 nm
- Si ambient layer, SLD-based material with SLD=$2.07\cdot 10^{-6} + i\cdot 2.38\cdot 10^{-11}$
Switch to Simulation view. Click Run simulation.

Exercise:

Go back to the Instrument view and add a Gaussian wavelength distribution with $\frac{\Delta\lambda}{\lambda}=10$%.

If got stucked, see the solution

Python API

Since Export to Python script does not work for Depth probe in BornAgain 1.14.0, see the prepared script below.

Exercise

Modify script below to simulate resonator with 3 Ti/Pt double layers. Hint: use for statement, take care about indentations. How number of double layers influences the simulation result?
Check for beam divergence parameters for MARIA instrument. Add Gaussian divergency for incident angle and the wavelength. How does the beam divergency influences the simulation result? What influences more: angular divergence or wavelength divergence?



In [3]:

    
%matplotlib inline



In [4]:

    
# %load depthprobe_ex.py
import numpy as np
import bornagain as ba
from bornagain import deg, angstrom, nm

# layer thicknesses in angstroms
t_Ti = 130.0 * angstrom
t_Pt = 320.0 * angstrom
t_Ti_top = 100.0 * angstrom
t_TiO2 = 30.0 * angstrom

#  beam data
ai_min = 0.0 * deg  # minimum incident angle
ai_max = 1.0 * deg  # maximum incident angle
n_ai_bins = 500    # number of bins in incident angle axis
beam_sample_ratio = 0.01  # beam-to-sample size ratio
wl = 10 * angstrom  # wavelength in angstroms

# angular beam divergence from https://mlz-garching.de/maria
d_ang = np.degrees(3.0e-03)*deg  # spread width for incident angle
n_points = 50  # number of points to convolve over
n_sig = 3  # number of sigmas to convolve over

# wavelength divergence from https://mlz-garching.de/maria
d_wl = 0.1*wl  # spread width for the wavelength
n_points_wl = 50
n_sig_wl = 2

#  depth position span
z_min = -100 * nm  # 300 nm to the sample and substrate
z_max = 100 * nm   # 100 nm to the ambient layer
n_z_bins = 500


def get_sample():
    """
    Constructs a sample with one resonating Ti/Pt layer
    """

    # define materials
    m_Si = ba.MaterialBySLD("Si", 2.07e-06, 2.38e-11)
    m_Ti = ba.MaterialBySLD("Ti", 2.8e-06, 5.75e-10)
    m_Pt = ba.MaterialBySLD("Pt", 6.36e-06, 1.9e-09)
    m_TiO2 = ba.MaterialBySLD("TiO2", 2.63e-06, 5.4e-10)
    m_D2O = ba.MaterialBySLD("D2O", 6.34e-06, 1.13e-13)

    # create layers
    l_Si = ba.Layer(m_Si)
    l_Ti = ba.Layer(m_Ti, 130.0 * angstrom)
    l_Pt = ba.Layer(m_Pt, 320.0 * angstrom)
    l_Ti_top = ba.Layer(m_Ti, 100.0 * angstrom)
    l_TiO2 = ba.Layer(m_TiO2, 30.0 * angstrom)
    l_D2O = ba.Layer(m_D2O)

    # construct sample
    sample = ba.MultiLayer()
    sample.addLayer(l_Si)

    # put your code here (1 line), take care of correct indents
    sample.addLayer(l_Ti)
    sample.addLayer(l_Pt)

    sample.addLayer(l_Ti_top)
    sample.addLayer(l_TiO2)
    sample.addLayer(l_D2O)

    return sample


def get_simulation():
    """
    Returns a depth-probe simulation.
    """
    footprint = ba.FootprintFactorSquare(beam_sample_ratio)
    simulation = ba.DepthProbeSimulation()
    simulation.setBeamParameters(wl, n_ai_bins, ai_min, ai_max, footprint)
    simulation.setZSpan(n_z_bins, z_min, z_max)

    fwhm2sigma = 2*np.sqrt(2*np.log(2))

    # add angular beam divergence
    # put your code here (2 lines)

    # add wavelength divergence
    # put your code here (2 lines)

    return simulation


def run_simulation():
    """
    Runs simulation and returns its result.
    """
    sample = get_sample()
    simulation = get_simulation()
    simulation.setSample(sample)
    simulation.runSimulation()
    return simulation.result()


if __name__ == '__main__':
    result = run_simulation()
    ba.plot_simulation_result(result)

Solution

Run the line below to see the solution



In [ ]:

    
%load depthprobe.py

References

[1] R. Cubitt, G. Fragneto, Chapter 2.8.3 - Neutron Reflection:: Principles and Examples of Applications, Editor(s): Roy Pike, Pierre Sabatier, Scattering, Academic Press, 2002, pp. 1198-1208

[2] H. Frielinghaus,et. al., NIMA, 871 (2017), pp. 72-76



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