# 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 bornagain as ba
from bornagain import deg, angstrom, nm
from matplotlib import pyplot as plt

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, -2.0*deg, 2.0*deg,
200, 0.0*deg, 2.0*deg)
simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg)
simulation.getOptions().setMonteCarloIntegration(integration_flag, 50)
simulation.setTerminalProgressMonitor()
return simulation

def run_simulation():
"""
Run simulation and plot results 4 times: for small and large cylinders,
with and without integration
"""

fig = plt.figure(figsize=(12.80, 10.24))

# conditions to define cylinders scale factor and integration flag
conditions = [
{'title': "Small cylinders, analytical calculations",
'scale': 1,   'integration': False},

{'title': "Small cylinders, Monte-Carlo integration",
'scale': 1,   'integration': True},

{'title': "Large cylinders, analytical calculations",
'scale': 100, 'integration': False},

{'title': "Large cylinders, Monte-Carlo integration",
'scale': 100, 'integration': True}
]

# run simulation 4 times and plot results
for i_plot, condition in enumerate(conditions):
scale = condition['scale']
integration_flag = condition['integration']

default_cylinder_height*scale)
simulation = get_simulation(integration_flag)
simulation.setSample(sample)
simulation.runSimulation()
result = simulation.getIntensityData()

# plotting results
plt.subplot(2, 2, i_plot+1)