The Hollow Sphere

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The Hollow Sphere

On the scattering length density page in Section 2, we asked: what happens when the core SLD equals the matrix SLD? We said the question would come back. It has.

Work through this page in your notebook — the result is more interesting than you might expect.

Setting up the experiment

Set the core SLD equal to the matrix SLD and compute the form factor:

R_core     = 45.0    # core radius in Å
t_shell    = 10.0    # shell thickness in Å
rho_shell  = 14.0e-6 # Å^-2
rho_matrix =  9.47e-6  # water

# The key change: core SLD = matrix SLD
rho_core   = rho_matrix

P_hollow = multi_shell_form_factor(
    q,
    radii=[R_core, R_core + t_shell],
    slds=[rho_core, rho_shell],
    rho_matrix=rho_matrix,
)

Now plot it alongside the equivalent solid sphere (same outer radius, same shell SLD, but a solid core with rho_core = rho_shell):

P_solid_sphere = sphere_amplitude(q, R_core + t_shell)**2

rho_core_solid = rho_shell  # filled core
P_filled = multi_shell_form_factor(
    q,
    radii=[R_core, R_core + t_shell],
    slds=[rho_core_solid, rho_shell],
    rho_matrix=rho_matrix,
)

fig, ax = plt.subplots(figsize=(8, 5))
plot_form_factor(q, P_solid_sphere, label="Solid sphere", ax=ax, color="steelblue")
plot_form_factor(q, P_filled,       label="Filled shell (core = shell SLD)", ax=ax, color="seagreen")
plot_form_factor(q, P_hollow,       label="Hollow sphere (core = matrix SLD)", ax=ax, color="tomato")
ax.set_title("Hollow sphere vs filled alternatives")
plt.tight_layout()
plt.show()

Look carefully at the three curves. Then read on.

What is happening physically

Key Concept: The Hollow Sphere

When rho_core = rho_matrix, the core is indistinguishable from the surrounding medium. The particle that the X-rays "see" is not a filled object — it is purely the shell. The core volume contributes no contrast; it scatters identically to the matrix and therefore produces no signal.

What remains is the scattering from a thin spherical shell: a hollow sphere.

The hollow sphere curve has a distinctly different character from a solid sphere of the same outer radius:

  • The first peak shifts to lower \(q\) compared to the solid sphere. The dominant length scale is now the vesicle radius \(R\), not the core radius.
  • The oscillation spacing at high \(q\) is controlled by the shell thickness \(t\), not the overall radius. Thinner shells push these features to larger \(q\).
  • The curve falls more steeply at intermediate \(q\) than a solid sphere.

Exploring shell thickness

Compare hollow spheres with the same outer radius but different shell thicknesses:

R_outer = 55.0   # fixed outer radius in Å
thicknesses = [5.0, 10.0, 20.0, 35.0]
colors = ["steelblue", "tomato", "seagreen", "darkorchid"]

fig, ax = plt.subplots(figsize=(8, 5))

for t, color in zip(thicknesses, colors):
    R_core = R_outer - t
    if R_core <= 0:
        continue
    P = multi_shell_form_factor(
        q,
        radii=[R_core, R_outer],
        slds=[rho_matrix, rho_shell],   # core = matrix
        rho_matrix=rho_matrix,
    )
    plot_form_factor(q, P, label=f"t = {t:.0f} Å", ax=ax, color=color)

ax.set_title(f"Hollow sphere: fixed R = {R_outer} Å, varying shell thickness")
plt.tight_layout()
plt.show()

As the shell becomes thinner, the high-\(q\) oscillations shift outward. As it approaches the full radius (a solid sphere), the curve converges toward the solid sphere limit.

Try It Yourself

  1. Fix the outer radius at 55 Å and the shell thickness at 10 Å. Increase rho_shell from just above rho_matrix to well above it. How does the overall intensity (proportional to \(F_0^2\)) change? Why?
  2. What happens to the curve if you make the shell SLD lower than the matrix SLD? Is that physically possible? (Hint: look at the SLD table from Section 2.)
  3. Confirm numerically that setting rho_core = rho_shell in the hollow sphere model gives back the solid sphere form factor.
Solution

Part 1: Increasing rho_shell increases the contrast \(\Delta\rho = \rho_{\text{shell}} - \rho_{\text{matrix}}\). The normalization factor \(F_0\) grows, meaning the unnormalized intensity \(\propto F_0^2\) increases quadratically with contrast. The shape of \(P(q)\) does not change — it is the prefactor that scales.

Part 2: Yes — a shell SLD below the matrix SLD is physically possible. Lipid hydrocarbon chains have SLD \(\approx 7.0 \times 10^{-6}\) Å\(^{-2}\), which is lower than water (\(9.47 \times 10^{-6}\) Å\(^{-2}\)). The contrast is still non-zero; the sign of \(\Delta\rho\) flips, but \((\Delta\rho)^2\) is unchanged, so the form factor shape is identical. Only the overall sign of the amplitude changes.

Part 3: 

R_outer = 55.0
t = 10.0
R_core = R_outer - t

# Hollow model with rho_core = rho_shell (filled)
rho_fill = 14.0e-6
P_filled_test = multi_shell_form_factor(
    q,
    radii=[R_core, R_outer],
    slds=[rho_fill, rho_fill],
    rho_matrix=rho_matrix,
)
P_solid_test = sphere_amplitude(q, R_outer)**2

print(f"Max difference: {np.max(np.abs(P_filled_test - P_solid_test)):.2e}")

Both curves should be identical (difference at floating-point precision). Setting rho_core = rho_shell zeroes the first contrast term, leaving only the outermost boundary — a solid sphere of radius R_outer with \(\Delta\rho = \rho_{\text{shell}} - \rho_m\).

Git checkpoint

$ git add chapter_04_shells.ipynb
$ git commit -m "explore hollow sphere geometry and vesicle approximation"

What's next: Connection to Vesicles — connecting the hollow sphere model to the physical picture of a lipid bilayer membrane.

Multiple Shells Connection to Vesicles