# Large Particle Form Factor

This example demonstrates, that for large particles (~1000nm) the contribution to the scattered intensity from the form factor oscillates rapidly within one detector bin and analytical calculations (performed for the bin center) give completely a wrong intensity pattern. In this case Monte-Carlo integrations over detector bin should be used.

The simulation generates four plots using different sizes of the particles, (radius=10 nm, height=20nm) or (radius=1 μm, height=2 μm), and different calculation methods: analytical calculations or Monte-Carlo integration. The other parameters are identical:

• The sample is made of a monodisperse distribution of cylinders, deposited randomly on a substrate.
• There is no interference between the scattered waves.
• The wavelength is equal to 1 Å.
• The incident angles are αi = 0.2° and Φi = 0°.
Real-space model:
Intensity Image:
Python Script:
"""
Large cylinders in DWBA.

This example demonstrates that for large particles (~1000nm) the formfactor
oscillates rapidly within one detector bin and analytical calculations
(performed for the bin center) give completely wrong intensity pattern.
In this case Monte-Carlo integration over detector bin should be used.
"""
import numpy, sys
import bornagain as ba
from bornagain import deg, angstrom, nm

phi_min, phi_max = -2.0, 2.0
alpha_min, alpha_max = 0.0, 2.0
default_cylinder_height = 20*nm

"""
Returns a sample with cylindrical particles on a substrate.
"""
# defining materials
m_ambience = ba.HomogeneousMaterial("Air", 0.0, 0.0)
m_substrate = ba.HomogeneousMaterial("Substrate", 6e-6, 2e-8)
m_particle = ba.HomogeneousMaterial("Particle", 6e-4, 2e-8)

# collection of particles
cylinder = ba.Particle(m_particle, cylinder_ff)
particle_layout = ba.ParticleLayout()

air_layer = ba.Layer(m_ambience)
substrate_layer = ba.Layer(m_substrate)

multi_layer = ba.MultiLayer()
return multi_layer

def get_simulation(integration_flag):
"""
Returns a GISAXS simulation with defined beam and detector.
If integration_flag=True, the simulation will integrate over detector bins.
"""
simulation = ba.GISASSimulation()
simulation.setDetectorParameters(
200, phi_min*deg, phi_max*deg, 200, alpha_min*deg, alpha_max*deg)
simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg)
simulation.getOptions().setMonteCarloIntegration(integration_flag, 50)

return simulation

def simulate(condi):
"""
Runs simulation and returns result.
"""
scale = condi['scale']
integration_flag = condi['integration']
default_cylinder_height*scale)
simulation = get_simulation(integration_flag)
simulation.setSample(sample)
simulation.setTerminalProgressMonitor()
simulation.runSimulation()
return simulation.getIntensityData()

def plot(result, nframe, title):
plt.subplot(2, 2, nframe+1)
im = plt.imshow(
result.getArray(),
norm=matplotlib.colors.LogNorm(1.0, result.getMaximum()),
extent=[result.getXmin()/deg, result.getXmax()/deg,
result.getYmin()/deg, result.getYmax()/deg],
aspect='auto')
cb = plt.colorbar(im)
cb.set_label(r'Intensity (arb. u.)', size=16)
plt.xlabel(r'$\phi_f (^{\circ})$', fontsize=16)
plt.ylabel(r'$\alpha_f (^{\circ})$', fontsize=16)
plt.text(0.0, 2.1, title, horizontalalignment='center',
verticalalignment='center',  fontsize=13)

if __name__ == '__main__':
"""
Runs one simulation for each condition, and plots results on a single canvas.
Conditions are small and large cylinders, with and without integration.
"""
arg = ba.getFilenameOrPlotflag()

# conditions to define cylinders scale factor and Monte-Carlo integration flag
conditions = [
{'name': "SmallAn",
'title': "Small cylinders, analytical calculations", 'scale': 1,
'integration': False, 'max': 1e+08},
{'name': "SmallMC",
'title': "Small cylinders, Monte-Carlo integration", 'scale': 1,
'integration': True,  'max': 1e+08},
{'name': "LargeAn",
'title': "Large cylinders, analytical calculations", 'scale': 100,
'integration': False, 'max': 1e+12},
{'name': "LargeMC",
'title': "Large cylinders, Monte-Carlo integration", 'scale': 100,
'integration': True,  'max': 1e+12}
]

if arg == "-p":
import matplotlib
from matplotlib import pyplot as plt
from matplotlib import rc
plt.figure(figsize=(12.80, 10.24))
for nplot in range(len(conditions)):
condi = conditions[nplot]
title = condi['title']
print("Generating intensity map for " + title)
intensities = simulate(condi)
plot(intensities, nplot, title)
plt.show()
else:
for condi in conditions:
intensities = simulate(condi)
fname = "%s.%s.int" % (arg, condi['name'])
ba.IntensityDataIOFactory.writeIntensityData(intensities, fname)
print("Stored intensity map in " + fname)