Plotting on a Log Scale¶

Scattering data spans many orders of magnitude in both \(q\) and \(I(q)\). Plotting on a linear scale hides almost everything interesting. This page shows why log-log plots are the standard for scattering data and walks through building a publication-ready figure with Matplotlib.

Why linear scale fails¶

Add a code cell to your notebook and plot \(P(q)\) on a linear scale first:

import matplotlib.pyplot as plt

R = 50.0
P = sphere_form_factor(q, R)

fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(q, P, color="steelblue")
ax.set_xlabel(r"$q$ (Å$^{-1}$)")
ax.set_ylabel(r"$P(q)$")
ax.set_title("Linear scale")
plt.tight_layout()
plt.show()

The curve appears to drop immediately to zero and stay there. All of the oscillation structure — the features that carry information about the sphere radius — is invisible because the intensity at large \(q\) is millions of times smaller than at small \(q\).

Switching to log-log¶

Add another code cell and use logarithmic axes:

fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(q, P, color="steelblue")
ax.set_xscale("log")  # (1)
ax.set_yscale("log")
ax.set_xlabel(r"$q$ (Å$^{-1}$)")
ax.set_ylabel(r"$P(q)$")
ax.set_title("Log-log scale")
plt.tight_layout()
plt.show()

1. ax.set_xscale("log") and ax.set_yscale("log") switch both axes to logarithmic spacing. This is a one-line change from the linear version above.

Now you can see the full structure of the curve: the flat plateau at low \(q\), the oscillations at intermediate \(q\), and the power-law decay at large \(q\).

Side-by-side comparison¶

Seeing both scales together makes the difference concrete. Add this cell:

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

for ax in axes:
    ax.plot(q, P, color="steelblue")
    ax.set_xlabel(r"$q$ (Å$^{-1}$)")
    ax.set_ylabel(r"$P(q)$")

axes[0].set_title("Linear scale")
axes[1].set_xscale("log")
axes[1].set_yscale("log")
axes[1].set_title("Log-log scale")

plt.tight_layout()
plt.show()

Building a clean scattering plot¶

For figures you will use in presentations or papers, a few small additions make a significant difference. Add this cell as the template you will use for all remaining plots in this tutorial:

def plot_form_factor(q, P, label=None, ax=None, color="steelblue"):
    """Plot a form factor on log-log axes with standard formatting."""
    if ax is None:
        fig, ax = plt.subplots(figsize=(7, 5))

    ax.plot(q, P, color=color, label=label, linewidth=1.5)
    ax.set_xscale("log")
    ax.set_yscale("log")
    ax.set_xlabel(r"$q$ (Å$^{-1}$)", fontsize=13)
    ax.set_ylabel(r"$P(q)$", fontsize=13)
    ax.tick_params(which="both", direction="in", top=True, right=True)  # (1)

    if label is not None:
        ax.legend(fontsize=11)

    return ax

1. which="both" adds tick marks for both major and minor gridlines. direction="in" points the ticks inward, which is the convention in most scientific journals.

Test the function:

fig, ax = plt.subplots(figsize=(7, 5))
plot_form_factor(q, P, label=f"R = {R:.0f} Å", ax=ax)
plt.tight_layout()
plt.show()

fig, ax = plt.subplots(figsize=(7, 5))
plot_form_factor(q, P, label=f"R = {R:.0f} Å", ax=ax)

# Overlay a q^-4 reference line
q_ref = q[q > 0.1]           # only the high-q region
scale = P[q > 0.1][0]        # match the amplitude at the start of the region
ax.plot(q_ref, scale * (q_ref / q_ref[0])**(-4),
        color="tomato", linestyle="--", label=r"$q^{-4}$")
ax.legend()
plt.tight_layout()
plt.show()

$ git add chapter_02_sphere.ipynb
$ git commit -m "add plot_form_factor log-log plotting helper"

What's next: Effect of Sphere Size — varying the radius and observing how the scattering curve changes.