## Radius of gyration ### Definition of $R_g$ This quantity is **defined for scattering**, the mass density is replaced with scattering length density. Recall the original form: $R_g^2 = \frac{1}{M} \int_V \mathrm{d}^3\mathbf{r}\, m(\mathbf{r}) (\mathbf{r} - \mathbf{R})^2$ Here $M$ is total mass, $M = \int_V \mathrm{d}^3r\, m(\mathbf{r})$; $\mathbf{R} = \frac{1}{M} \int_V \mathrm{d}^3r\, m(r) \mathbf{r}$ is center of mass; $m(\mathbf{r})$ is mass density. Then $R_g^2 = \frac{1}{2M^2} \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r'\, m(\mathbf{r}) m(\mathbf{r}') (\mathbf{r} - \mathbf{r}')^2$ For scattering, we change mass density to SLD, $\rho(\mathbf{r})$. And retrieve the term $R_{g}$ from $F(\mathbf{q})$ and $P(\mathbf{q})$. This radius of gyration can be obtained from the scattering by [[Form factor]] at $qR_{g}\lesssim 1 $ namely for relatively small $q$, around the particle size. >[!Notice] >Although in the original definition, this $R_{g}$ is defined for scattering, but the quantity is the same for radius of gyration defined for mass (or geometry), if the particles have homogeneous SLD (which will be cancelled in the end). For spheres with homogeneous composition, radius of gyration is $\sqrt{\frac{3}{5}} R$. ### $R_g$ and form factor Recall the expression of $F(\mathbf{q})$ (for one particle), $F(\mathbf{q}) = \int_{V} \mathrm{d}\mathbf{r}_{p} \Delta \rho_{p} (\mathbf{r}_{p}) e^{-i \mathbf{q} \cdot \mathbf{r}_{p}}$ $F(\mathbf{q}) = \iint_{V} \mathrm{d}\mathbf{r}_{p} \mathrm{d}\mathbf{r}_{p}' \Delta \rho_{p} (\mathbf{r}_{p})\, \Delta \rho_{p} (\mathbf{r}_{p}) e^{-i \mathbf{q} \cdot (\mathbf{r}_{p}-\mathbf{r}_{p}')}$ Assume **a very dilute suspension** with anisotropic particles in **all orientations**. $\begin{aligned} \langle F^2(\mathbf{q}) \rangle &= \frac{1}{4\pi} \int_0^{2\pi} \mathrm{d}\phi \int_0^{\pi} \mathrm{d}\theta \sin \theta \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r' \rho(\mathbf{r}) \rho(\mathbf{r}') e^{-iq|\mathbf{r} - \mathbf{r}'| \cos \theta} \\ &= \frac{1}{2} \int_{-1}^{1} \mathrm{d}\xi \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r' \rho(\mathbf{r}) \rho(\mathbf{r}') e^{-iq|\mathbf{r} - \mathbf{r}'| \xi} \\ &= \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r' \rho(\mathbf{r}) \rho(\mathbf{r}') \frac{\sin(q|\mathbf{r} - \mathbf{r}'|)}{q|\mathbf{r} - \mathbf{r}'|} \end{aligned} $ In above derivation, $\xi=\cos \theta$. $\theta$ is the angle between $\mathbf{q}$ and $|\mathbf{r} - \mathbf{r}'|$. Take the first two terms in Taylor expansion **near zero**, $\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \frac{x^4}{120} + \ldots $ we get an approximation for $\langle F^2(\mathbf{q}) \rangle$, $\langle F^2(q) \rangle \approx \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r' \rho(\mathbf{r}) \rho(\mathbf{r}') - \frac{q^2}{6} \int_V \mathrm{d}^3r \int_V \mathrm{d}^3r' \rho(\mathbf{r}) \rho(\mathbf{r}') |\mathbf{r} - \mathbf{r}'|^2 $ The last integration contains our expression for $R_{g}$. If we define total SLD in a particle as $M$, then $M = \int_V \mathrm{d}^3r\, \rho(r)=F(0)$, and above expression becomes $\langle F^2(q) \rangle \approx M^2 - \frac{q^2}{6} 2M^2 R_g^2$ And recall [[Differential scattering cross-section in SAS#Form and structure factor for identical particles]], form factor is $P(q)=\frac{\langle F^2(q) \rangle}{F(0)^2} = \frac{\langle F^2(q) \rangle}{M^2} = 1 - \frac{q^2 R_g^2}{3}. $ This estimation is valid when $qR_{g}$ is small, typical range can be $qR_{g}< 0.1$ or $qR_{g}< 0.3$. >[!Note] >Remember the condition to estimate $R_{g}$ is checking $P(q)$ and $q$ plot and apply $1 - \frac{q^2 R_g^2}{3}$ estimation in small $q$ range, the doing a linear fit for $\ln P$ vs $q^{2}$. The transition between flat $P$ (Guinier) to oscillatory (Porod) region indicates the proper range to extract this quantity. >After extraction of $R_{g}$, one may confirm $qR_{g}\lesssim 1$ region and do further analysis. ### Special case: negative SLD and complex $R_g$ The fact that scattering length density could be negative makes it possible that the $R_{g}$, unlike the simple case for mass density, having strange result, like complex value. Consider a core-shell system with the core radius $R_{c}$ and shell (entire particle) radius $R_{s}$. The scattering length density is defined as $\rho_{c}$ and $\rho_{s}$. And the term "integrated SLD", which is mass in original expression, is kept as $M$. Therefore, we have For core: $M_c = \frac{4}{3} \pi R_c^3 \rho_c$ For shell: $M_s = \frac{4}{3} \pi (R_s^3 - R_c^3) \rho_s$ And combined: $M = M_c + M_s = \frac{4}{3} \pi \left( R_c^3 \rho_c + (R_s^3 - R_c^3) \rho_s \right)$ Recall the definition of $R_{g}$, and for this core-shell system, $R_g^2 = \frac{1}{M} \left( \int_{\text{core}} r^2 \rho_c \, \mathrm{d}V + \int_{\text{shell}} r^2 \rho_s \, \mathrm{d}V \right)$ Integrate for the core: $\begin{aligned} \int_{\text{core}} r^2 \rho_c \, \mathrm{d}V &= \rho_c \int_0^{R_c} 4 \pi r^4 \, \mathrm{d}r \\ &= \rho_c \cdot 4 \pi \left[ \frac{r^5}{5} \right]_0^{R_c} \\ &= \rho_c \cdot 4 \pi \cdot \frac{R_c^5}{5} \\ &= \frac{4 \pi \rho_c R_c^5}{5} \end{aligned}$ And the shell: $\begin{aligned} \int_{\text{shell}} r^2 \rho_s \, \mathrm{d}V &= \rho_s \int_{R_c}^{R_s} 4 \pi r^4 \, \mathrm{d}r \\ &= \rho_s \cdot 4 \pi \left[ \frac{r^5}{5} \right]_{R_c}^{R_s} \\ &= \rho_s \cdot 4 \pi \left( \frac{R_s^5}{5} - \frac{R_c^5}{5} \right) \\ &= \frac{4 \pi \rho_s}{5} (R_s^5 - R_c^5) \end{aligned}$Gives the overall result as: $\begin{aligned} \int_V r^2 \rho \, \mathrm{d}V &= \frac{4 \pi \rho_c R_c^5}{5} + \frac{4 \pi \rho_s}{5} (R_s^5 - R_c^5) \end{aligned}$ Insert $M$ to the expression, we have $\begin{aligned} R_g^2 &= \frac{1}{M} \left( \frac{4 \pi \rho_c R_c^5}{5} + \frac{4 \pi \rho_s}{5} (R_s^5 - R_c^5) \right) \\ &= \frac{3}{4 \pi (R_c^3 \rho_c + (R_s^3 - R_c^3) \rho_s)} \left( \frac{4 \pi \rho_c R_c^5}{5} + \frac{4 \pi \rho_s}{5} (R_s^5 - R_c^5) \right) \\ &= \frac{3}{R_c^3 \rho_c + (R_s^3 - R_c^3) \rho_s} \cdot \frac{1}{5} \left( \rho_c R_c^5 + \rho_s (R_s^5 - R_c^5) \right) \\ &= \frac{3}{5} \cdot \frac{\rho_c R_c^5 + \rho_s (R_s^5 - R_c^5)}{\rho_c R_c^3 + \rho_s (R_s^3 - R_c^3)} \end{aligned}$ Therefore, the radius of gyration for core-shell system would be $R_g = \sqrt{ \frac{3}{5} \cdot \frac{\rho_c R_c^5 + \rho_s (R_s^5 - R_c^5)}{\rho_c R_c^3 + \rho_s (R_s^3 - R_c^3)} }$ One may easily notice, due to $\rho_{s}$ or $\rho_{c}$ could be negative, we cannot ensure this $R_{g}$ is real! An example is given below. >[!Example] >Use the system we have in [[Form factor#Form factor for core-shell particles]]. Neglect $\sigma_{p}$. Use $R =R_{s}= 100$ nm, $R_c = 80$ nm, $\rho_{\text{shell}} = 1$ a.u., $\rho_{\text{core}} = -1$ a.u., then calculate $R_{g}$. >$R_g = \sqrt{ \frac{3}{5} \cdot \frac{\rho_c R_c^5 + \rho_s (R_s^5 - R_c^5)}{\rho_c R_c^3 + \rho_s (R_s^3 - R_c^3)} }$ >This gives >$R_g^2 = \frac{3}{5} \cdot \frac{100^5 - 2 \cdot 80^5}{100^3 - 2 \cdot 80^3}$ >The denominator is $100^3 - 2 \cdot 80^3 = -2.4 \times 10^4$. This is a negative value! And the numerator is $3.4464 \times 10^9$, which is positive. This means we have $R_{g}^{2}$ a negative value, and $R_{g}$ being imaginary. >Under this condition, it could be hard to find a proper physical interpretation for $R_{g}$, but $R_{g}^{2}$ is still valid, and can be seen in the Guinier region inside the form factor plot (i.e., $1 - \frac{q^2 R_g^2}{3}$ trend). >![[Drawing 2024-05-20 19.37.41.excalidraw.svg]] > >In the plot, the $b_{core}=-1$ case shows a increasing $P$ with $q$ getting larger. This is because $- \frac{q^2 R_g^2}{3}$ is positive in this case. > >Another thing worth to note is, under this condition, we have a form factor $P$ larger than 1.