Delta Lorentz¶

Model of Delta function and Lorentzian with intensities given by the Debye-Waller factor:

\( I = K \exp \left( \frac{-\langle u^2 \rangle Q^2}{3} \right)[A_0 \delta(E) + A_1 L(E, \Gamma)] \),

where \(K\) is the scale factor, \(\langle u^2 \rangle\) is the mean square displacement, \(Q\) is the scattering vector, \(A_0\) and \(A_1\) are the relative amplitudes of the delta function and Lorentzian, respectively, with the constraint that \(A_0+A_1=1\), and \(L(E, \Gamma)\) is the Lorentzian function with width \(\Gamma\). \(A_0\), \(A_1\) and the width of the Lorentzian can be the same at all \(Q\) or be allowed to vary with \(Q\).

In [1]:

import matplotlib.pyplot as plt
import numpy as np

import easydynamics as edyn

%matplotlib widget

import matplotlib.pyplot as plt import numpy as np import easydynamics as edyn %matplotlib widget

In [2]:

Q = np.linspace(0.5, 2, 7)
energy = np.linspace(-2, 2, 501)
scale = 1.0
mean_u_squared = 0.5
A_0 = 0.01
lorentzian_width = 0.2

diffusion_model = edyn.DeltaLorentz(
    scale=scale,
    mean_u_squared=mean_u_squared,
    A_0=A_0,
    lorentzian_width=lorentzian_width,
    allow_Q_variation={'A_0': True, 'lorentzian_width': True},
    Q=Q,
    lorentzian_name='Lorentzian',
    delta_name='Delta function',
)

Q = np.linspace(0.5, 2, 7) energy = np.linspace(-2, 2, 501) scale = 1.0 mean_u_squared = 0.5 A_0 = 0.01 lorentzian_width = 0.2 diffusion_model = edyn.DeltaLorentz( scale=scale, mean_u_squared=mean_u_squared, A_0=A_0, lorentzian_width=lorentzian_width, allow_Q_variation={'A_0': True, 'lorentzian_width': True}, Q=Q, lorentzian_name='Lorentzian', delta_name='Delta function', )

Both A_0 and lorentzian_width are here allowed to vary with Q. We here change a few of them just to show how this impacts the model. The # ruff: ignore[private-member-access] comment is because we are accessing private members of the model (ones beginning with _), which is generally discouraged. Because of these changes, in the figure below, the delta function at Q=1.25 Å^-1 is much larger than the other ones, and the Lorentzian at Q=1.75 Å^-1 is much narrower and taller than the other ones.

In [3]:

diffusion_model._A_0_list[3].value = 0.02  # ruff: ignore[private-member-access]
diffusion_model._lorentzian_width_list[5].value = 0.1  # ruff: ignore[private-member-access]

diffusion_model._A_0_list[3].value = 0.02 # ruff: ignore[private-member-access] diffusion_model._lorentzian_width_list[5].value = 0.1 # ruff: ignore[private-member-access]

In [4]:

component_collections = diffusion_model.get_component_collections()
cmap = plt.cm.jet
nQ = len(component_collections)
plt.figure()
for Q_index in range(len(component_collections)):
    color = cmap(Q_index / (nQ - 1))
    y = component_collections[Q_index].evaluate(energy)
    plt.plot(energy, y, label=f'Q={Q[Q_index]} Å^-1', color=color)

plt.legend()
plt.show()
plt.xlabel('Energy (meV)')
plt.ylabel('Intensity (arb. units)')
plt.title('Delta-Lorentz Model')

component_collections = diffusion_model.get_component_collections() cmap = plt.cm.jet nQ = len(component_collections) plt.figure() for Q_index in range(len(component_collections)): color = cmap(Q_index / (nQ - 1)) y = component_collections[Q_index].evaluate(energy) plt.plot(energy, y, label=f'Q={Q[Q_index]} Å^-1', color=color) plt.legend() plt.show() plt.xlabel('Energy (meV)') plt.ylabel('Intensity (arb. units)') plt.title('Delta-Lorentz Model')

Out[4]:

Text(0.5, 1.0, 'Delta-Lorentz Model')

In [5]:

diffusion_model.get_all_variables()

diffusion_model.get_all_variables()

Out[5]:

[<Parameter 'scale': 1.0000 meV, bounds=[0.0:inf]>,
 <Parameter 'mean_u_squared': 0.5000 Å^2, bounds=[0.0:inf]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0200, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'A_0': 0.0100, bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9800 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'DeltaLorentz A_1': 0.9900 (dependent), bounds=[0.0:1.0]>,
 <Parameter 'Lorentzian area': 0.9496 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0096 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.9014 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0091 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.8380 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0085 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.7553 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0154 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.6804 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0069 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.5942 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.1000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0060 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian area': 0.5083 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Lorentzian center': 0.0000 meV (fixed), bounds=[-inf:inf]>,
 <Parameter 'Lorentzian width': 0.2000 meV, bounds=[1e-10:inf]>,
 <Parameter 'Delta function area': 0.0051 meV (dependent), bounds=[0.0:inf]>,
 <Parameter 'Delta function center': 0.0000 meV (fixed), bounds=[-inf:inf]>]