Core-Chain-Chain (CCC) Model

This form factor describes scattering from spherical cores (nanoparticle, micellar, etc.) that have chains coming off normal from their surface. In the case of
the Core-Chain-Chain (CCC) Model, these chains have two different regions of conformation, size, and scattering length density. The transition from region 1 (near
the nanoparticle core, CPB) to region 2 (SDPB) is given by the parameter $r_{c}$, which is the distance from the center of the nanoparticle to the junction point. See
fig. 1 of reference 1 for a schematic of the geometry.

The scattering intensity is the sum of 8 terms, and an incoherent background term $B$. This model returns the following scattering intensity:
$$
I(q) = \frac{scale}{(V_{core} + N_{c}V_{c})} \times \left[ P_{core}(q) + N_{c}P_{CPB}(q) + N_{c}P_{SDPB}(q) + 2N_{c}F_{core}(q)j_{0}(qr_{core})F_{CPB}(q) + 2N_{c}F_{core}(q)j_{0}(qr_{c})F_{SDPB}(q) + N_{c}(N_{c} - 1)F_{CPB}(q)^{2}j_{0}(qr_{core})^2 + N_{c}(N_{c} - 1)F_{SDPB}(q)^{2}j_{0}(qr_{c})^2 + N_{c}^{2}F_{CPB}(q)j_{0}(qr_{core})j_{0}(qr_{c})F_{SDPB}(q) \right] + B
$$

where $N_{c}$ is the number of chains grafted to the nanoparticle, $r_{core}$ is the nanoparticle radius, and $r_{c}$ is the position of the junction between the CPB and
SDPB regions. The terms $`P_{core}(q)$, $P_{CPB}(q)$, and $P_{SDPB}(q)$ are the form factors of the spherical core, polymer in the CPB region, and polymer in the SDPB
region, respectively. $F_{core}(q)$, $F_{CPB}(q)$, and $F_{SDPB}(q)$ are the form factor amplitudes of the respective regions. For the nanoparticle core, these terms are related as:

$$
P_{core}(q) = \left| F_{core}(q) \right|^{2} = \left| V_{core}(\rho_{core} - \rho_{solv})\frac{3j_{1}(qr_{core})}{qr_{core}} \right|^{2}
$$
where $\rho_{core}$ is the scattering length density of the nanoparticle core, $\rho_{solv}$ is the scattering length density of the solvent/matrix, $V_{core}$ is the
volume of the nanoparticle core, and $j_{1}(.)$ is a spherical Bessel function.

Scattering from the CPB and SDPB regions is described by the form factor amplitudes and form factors of these regions. For a given region $i$, these terms read:

$$
P_{i}(q,N_{i}) = V_{i}^{2}(\rho_{i} - \rho_{solv})^{2} \left[ \frac{1}{\nu_{i} U_{i}^{1/2\nu_{i}}}\gamma \left( \frac{1}{2\nu_{i}}, U_{i}\right) - \frac{1}{\nu_{i} U_{i}^{1/\nu_{i}}}\gamma \left(\frac{1}{\nu_{i}}, U_{i} \right) \right]
$$
and
$$
F_{i}(q,N_{i}) = V_{i}(\rho_{i} - \rho_{solv})\frac{1}{2\nu_{i} U_{i}^{1/2\nu_{i}}}\gamma \left( \frac{1}{2\nu_{i}}, U_{i}\right)
$$

where $N_{i}$ is the degree of polymerization of the portion of polymer in region $i$, $V_{i}$ is the volume of polymer in region $i$, $\nu_{i}$ is the excluded volume parameter of the polymer in region $i$, $\rho_{i}$ is the scattering length density of polymer in region $i$, $U_{i} = q^{2} b^{2} N_{i}^{2\nu_{i}}/6$, $b$ is the polymer's Kuhn length, and $\gamma$ is the lower incomplete gamma function.

Example Data: