tetrapod

Core-shell tetrapod with four cylindrical arms

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

length

Arm length

Å

1000

radius

Arm core radius

Å

50

thickness

Arm shell thickness

Å

10

sld_core

Arm core scattering length density

10-6Å-2

1

sld_shell

Arm shell scattering length density

10-6Å-2

0.5

sld_solvent

Solvent scattering length density

10-6Å-2

0

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Definition

Calculates the scattering from a tetrapod-shaped structure. A tetrapod consists of four cylindrical arms radiating from a central point, oriented along the (1,1,1), (-1,-1,1), (-1,1,-1), and (1,-1,-1) directions.

../../_images/tetrapod.png

Fig. 39 Tetrapod schematic. Each of the four arms is a core–shell cylinder of length \(L\), core radius \(R\) and shell thickness \(t\); the arm cross-section is shown in the inset. The arms radiate from a central junction along tetrahedral directions.

The scattering intensity is calculated as an average over all orientations:

\[I(q) = \frac{(\Delta \rho)^2}{4\pi} \int_0^{\pi} \int_0^{2\pi} \left|\sum_{n=1}^{4} F_n(q, \theta, \varphi)\right|^2 \sin\theta \, d\theta \, d\varphi\]

where \(F_n\) is the core-shell form factor amplitude of the \(n\)-th arm:

\[F_n(q, \theta, \varphi) = \text{sinc}\!\left(\frac{q u_n L}{2}\right) \left[ (\rho_\text{core} - \rho_\text{shell})\, V_\text{core}\, \frac{2 J_1(q \mu_n R)}{q \mu_n R} + (\rho_\text{shell} - \rho_\text{solvent})\, V_\text{outer}\, \frac{2 J_1(q \mu_n (R+t))}{q \mu_n (R+t)} \right]\]

with \(u_n = \hat{q} \cdot \hat{a}_n\), \(\mu_n = \sqrt{1 - u_n^2}\), \(L\) the arm length, \(R\) the arm core radius, \(R + t\) the outer radius (\(t\) = shell thickness), and \(V_\text{core} = \pi R^2 L\), \(V_\text{outer} = \pi (R+t)^2 L\). Expanding the squared modulus into a double sum gives:

\[I(q) = \frac{1}{4\pi} \int \sum_{n=1}^{4}\sum_{m=1}^{4} F_n F_m \cos\!\left(\frac{q(u_n-u_m)L}{2}\right) \sin\theta \, d\theta \, d\varphi\]

The cosine factor is the interference term between the centres of arms \(n\) and \(m\), which are displaced by \(\tfrac{L}{2}\hat{a}_n\) from the junction.

Geometry

The four arms are oriented along tetrahedral directions. With \(A = 109.5 /2\) (the half-angle between arms), the arm unit vectors and the corresponding projections \(u_n\) are

\[u_n = s_n \cos A \cos\theta + \sin A \sin\theta \cos(\varphi - \varphi_n)\]

where \((s_n, \varphi_n) = (+1,\ 0),\ (-1,\ \pi/2),\ (+1,\ \pi),\ (-1,\ 3\pi/2)\) for \(n = 1, 2, 3, 4\) respectively.

Each arm has length \(L\), core radius \(R\), shell thickness \(t\), and hence outer radius \(R + t\).

Note

Each of the four arms is modelled as a complete cylinder of length \(L\) extending from the central junction, so the arms overlap near the origin. This overlap is neglected in two ways, following the treatment used in the reference. First, the particle volume is overestimated (the effect being largest when the arms are short and wide), which affects the calculated intensity through the volume normalisation. Second, the scattering amplitude is the plain sum of the four cylinder amplitudes, so the overlapping region near the origin is counted more than once; because this region is small compared with the arms, the resulting artefacts are expected to appear mainly in the high-\(q\) range. Consequently the model is only valid for long-arm tetrapods, i.e. for arm lengths much larger than the arm width, \(L \gg R + t\).

../../_images/tetrapod_autogenfig.png

Fig. 40 1D plot corresponding to the default parameters of the model.

Source

tetrapod.py \(\ \star\ \) tetrapod.c \(\ \star\ \) gauss76.c \(\ \star\ \) sas_J1.c \(\ \star\ \) polevl.c

References

  1. Seoki Kyoo Seo Korean J. Chem. Eng. 34(2017) 1192-1198 DOI:10.1007/s11814-016-0341-x

Authorship and Verification