Extended Czjzek distribution¶
The Extended Czjzek distribution models random variations of a second-rank traceless symmetric tensors about a non-zero tensor. See Extended Czjzek distribution and references within for a brief description of the model.
Extended Czjzek distribution of symmetric shielding tensors¶
To generate an extended Czjzek distribution, use the
ExtCzjzekDistribution
class as follows.
from mrsimulator.models import ExtCzjzekDistribution
shielding_tensor = {"zeta": 80, "eta": 0.4}
shielding_model = ExtCzjzekDistribution(shielding_tensor, eps=0.1)
The ExtCzjzekDistribution class accepts two arguments. The first argument is the
dominant tensor about which the perturbation applies, and the second parameter, eps
,
is the perturbation fraction. The minimum value of the eps
parameter is 0, which means
the distribution is a delta function at the dominant tensor parameters. As the value of
eps
increases, the distribution gets broader; at values greater than 1, the extended
Czjzek distribution approaches a Czjzek distribution. In the above example, we create an
extended Czjzek distribution about a second-rank traceless symmetric shielding tensor
described by anisotropy of 80 ppm and an asymmetry parameter of 0.4. The perturbation
fraction is 0.1.
As before, you may either draw random samples from this distribution or generate a
probability distribution function. Let’s first draw points from this distribution, using
the rvs()
method of the instance.
zeta_dist, eta_dist = shielding_model.rvs(size=50000)
In the above example, we draw size=50000 random points of the distribution. The output
zeta_dist
and eta_dist
hold the tensor parameter coordinates of the points, defined
in the Haeberlen convention.
The scatter plot of these coordinates is shown below.
import matplotlib.pyplot as plt
plt.scatter(zeta_dist, eta_dist, s=4, alpha=0.01)
plt.xlabel("$\zeta$ / ppm")
plt.ylabel("$\eta$")
plt.xlim(60, 100)
plt.ylim(0, 1)
plt.tight_layout()
plt.show()
Extended Czjzek distribution of symmetric quadrupolar tensors¶
The extended Czjzek distribution of symmetric quadrupolar tensors follows a similar
setup as the extended Czjzek distribution of symmetric shielding tensors, shown above.
In the following example, we generate the probability distribution
function using the pdf()
method.
import numpy as np
Cq_range = np.arange(100) * 0.04 + 2 # pre-defined Cq range in MHz.
eta_range = np.arange(21) / 20 # pre-defined eta range.
quad_tensor = {"Cq": 3.5, "eta": 0.23} # Cq assumed in MHz
model_quad = ExtCzjzekDistribution(quad_tensor, eps=0.2)
Cq, eta, amp = model_quad.pdf(pos=[Cq_range, eta_range])
As with the case with Czjzek distribution, to generate a probability distribution of the
extended Czjzek distribution, we need to define a grid system over which the distribution
probabilities will be evaluated. We do so by defining the range of coordinates along the
two dimensions. In the above example, Cq_range
and eta_range
are the
range of \(\text{Cq}\) and \(\eta_q\) coordinates, which is then given as the
argument to the pdf()
method. The output
Cq
, eta
, and amp
hold the two coordinates and amplitude, respectively.
The plot of the extended Czjzek probability distribution is shown below.
import matplotlib.pyplot as plt
plt.contourf(Cq, eta, amp, levels=10)
plt.xlabel("$C_q$ / MHz")
plt.ylabel("$\eta$")
plt.tight_layout()
plt.show()
Note
The pdf
method of the instance generates the probability distribution function
by first drawing random points from the distribution and then binning it
onto a pre-defined grid.
Mini-gallery using the extended Czjzek distributions¶
Extended Czjzek distribution (Shielding and Quadrupolar)
Simulating site disorder (crystalline)