Form Factor:Sphere

This page provides the equations for calculating the form factor of a sphere (including derivations).

Equations

For spheres of radius R (volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{sphere}=4\pi R^3/3} ):

Form Factor Amplitude

$F_{sphere}(q)=\left\{{\begin{array}{c l}3\Delta \rho V_{sphere}{\frac {\sin(qR)-qR\cos(qR)}{(qR)^{3}}}&\mathrm {when} \,\,q\neq 0\\\Delta \rho V_{sphere}&\mathrm {when} \,\,q=0\\\end{array}}\right.$

Isotropic Form Factor Intensity

$P_{sphere}(q)=\left\{{\begin{array}{c l}36\pi \Delta \rho ^{2}V_{sphere}^{2}{\frac {(\sin(qR)-qR\cos(qR))^{2}}{(qR)^{6}}}&\mathrm {when} \,\,q\neq 0\\4\pi \Delta \rho ^{2}V_{sphere}^{2}&\mathrm {when} \,\,q=0\\\end{array}}\right.$

Sources

NCNR

From NCNR SANS Models documentation:

$P(q)={\frac {\rm {scale}}{V}}\left[{\frac {3V(\Delta \rho )(\sin(qr)-qr\cos(qr))}{(qr)^{3}}}\right]^{2}+{\rm {background}}$

• Parameters:
  1. ${\rm {scale}}$ : Intensity scaling
  2. $r$ : sphere radius (Å)
  3. $\Delta \rho$ : scattering contrast (Å⁻²), $\Delta \rho =SLD_{core}-SLD_{solvent}$
  4. ${\rm {background}}$ : incoherent background (cm⁻¹)

Pedersen

From Pedersen review, Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6

$F(q,r)={\frac {3\left[\sin(qr)-qr\cos(qr)\right]}{(qr)^{3}}}$

• Parameters:
  1. $r$ : sphere radius (Å)

IsGISAXS

From IsGISAXS, Born form factors:

$F(\mathbf {q} ,r)=4\pi r^{3}{\frac {\sin(qr)-qr\cos(qr)}{(qr)^{3}}}\exp {(iq_{z}r)}$

$V={\frac {4}{3}}\pi r^{3},S=\pi r^{2}$

• Parameters:
  1. $r$ : sphere radius (Å)

Code

    def sphere(self, q, r, scale=1.0, contrast=0.1, background=0.0):
        
        V = (4/3)*numpy.pi*(r**3)

        return (scale/V)*(( 3*V*contrast*(sin(q*r)-q*r*cos(q*r) )/( (q*r)**3 ) )**2) + background
        

Derivations

Form Factor

For a sphere of radius R, the volume is:

$V_{sphere}={\frac {4}{3}}\pi R^{3}$

We can use a spherical coordinates, where $\theta$ denotes the angle with respect to the $+q_{z}$ axis, and $\phi$ is the in-plane angle (i.e. with respect to the $+x$ axis):

${\begin{alignedat}{2}&\mathbf {r} =(x,y,z)=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta )\\&\mathbf {q} =(q_{x},q_{y},q_{z})\\&q=|\mathbf {q} |^{2}={\sqrt {q_{x}^{2}+q_{y}^{2}+q_{z}^{2}}}\end{alignedat}}$

Where the form factor is:

${\begin{alignedat}{2}F_{sphere}(\mathbf {q} )&=\int \limits _{V}e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} \mathbf {r} \\&=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi }\int _{r=0}^{R}e^{i\mathbf {q} \cdot \mathbf {r} }r^{2}\mathrm {d} r\sin \theta \mathrm {d} \theta \mathrm {d} \phi \\\end{alignedat}}$

We take advantage of spherical symmetry. E.g. we can rotate any q onto a particular axis, such as $q_{z}$. So that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}=(0,0,q_z) }

${\begin{alignedat}{2}\mathbf {q} \cdot \mathbf {r} &=q_{x}x+q_{y}y+q_{z}z\\&=q_{z}z\\&=qr\cos \theta \end{alignedat}}$

And so:

${\begin{alignedat}{2}F_{sphere}(q)&=\int _{0}^{2\pi }\mathrm {d} \phi \int _{0}^{\pi }\int _{0}^{R}e^{iqr\cos \theta }r^{2}\mathrm {d} r\sin \theta \mathrm {d} \theta \\&=[2\pi ]\int _{0}^{\pi }\int _{0}^{R}(\cos(qr\cos \theta )+i\sin(qr\cos \theta ))r^{2}\mathrm {d} r\sin \theta \mathrm {d} \theta \end{alignedat}}$

A simple variable substitution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = q r \cos\theta}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}u = - q r \sin\theta \mathrm{d}\theta}

Yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} F_{sphere}(q) & = 2\pi \int_{0}^{R} r^2 \left[\int_{0}^{\pi}( \cos(u) + i \sin(u) )\frac{-\mathrm{d}u}{q r} \right] \mathrm{d}r \\ & = 2\pi \int_{0}^{R} r^2 \frac{-1}{qr} \left[ \sin(u) - i \cos(u) \right]_{\theta=0}^{\pi} \mathrm{d}r \\ & = 2\pi \int_{0}^{R} \frac{-r}{q} \left[ \sin(q r \cos\theta) - i \cos(q r \cos\theta) \right]_{\theta=0}^{\pi} \mathrm{d}r \\ & = 2\pi \int_{0}^{R} \frac{-r}{q} \left[ \sin(- q r ) - i \cos(- q r) - \sin(q r) + i \cos(q r) \right] \mathrm{d}r \\ & = 2\pi \int_{0}^{R} \frac{r}{q} \left[ 2 \sin(q r ) \right] \mathrm{d}r \\ & = \frac{4\pi}{q} \int_{0}^{R} r \sin(q r ) \mathrm{d}r \\ \end{alignat} }

Using the fact that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x\sin ax\;dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C\,\!}

We integrate:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} F_{sphere}(q) & = \frac{4\pi}{q} \left[ \frac{\sin(qr)}{q^2} - \frac{r\cos(qr)}{q} \right]_{r=0}^R \\ & = \frac{4\pi}{q} \left[ \frac{\sin(qR)}{q^2} - \frac{R\cos(qR)}{q} - \frac{\sin(0)}{q^2} + \frac{0\cos(q0)}{q} \right] \\ & = \frac{4\pi}{1} \left[ \frac{\sin(qR)}{q^3} - \frac{R\cos(qR)}{q^2} - 0 + 0 \right] \\ & = 4\pi \left[ \frac{\sin(qR)}{q^3} - \frac{R\cos(qR)}{q^2} \right] \\ & = 4\pi R^3 \left[ \frac{\sin(qR)}{q^3R^3} - \frac{qR\cos(qR)}{q^3R^3} \right] \\ & = \frac{3\times4\pi R^3}{3} \left[ \frac{\sin(qR)-qRcos(qR)}{q^3R^3} \right] \\ & = 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \end{alignat} }

Form Factor at q=0

At very small q:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \lim_{q\to0}F_{sphere}(q) & = \frac{3 V_{sphere}}{R^3} \lim_{q\to0} \frac{ \sin(qR)- qR \cos(qR) }{ q^3 } \\ & = 4\pi \lim_{q\to0} \frac{ \sin(qR)- qR \cos(qR) }{ q^3 } \\ & = 4\pi \lim_{q\to0} \frac{ 1 }{q^3}\left[ qR-\frac{(qR)^3}{3!}+... \right]- \frac{R}{q^2} \left[ 1-\frac{(qR)^2}{2!}+... \right] \\ & = 4\pi \lim_{q\to0} \frac{R}{q^2}-\frac{R^3}{3!}+\frac{O((qR)^5)}{q^3} -\frac{R}{q^2}+\frac{R^3}{2!}+\frac{R\times O((qR)^4)}{q^2} \\ & = 4\pi \lim_{q\to0} R^3\left( \frac{1}{2}-\frac{1}{6}\right) + O(q^2) \\ & = 4\pi \lim_{q\to0} \frac{R^3}{3} + O(q^2) \\ & = \frac{4\pi R^3}{3} \\ & = V_{sphere} \\ \end{alignat} }

Isotropic Form Factor Intensity

To average over all possible orientations, we use:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} P(q) & = \int\limits_{S} | F(\mathbf{q}) |^2 \mathrm{d}\mathbf{s} \\ & = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \end{alignat} }

For a sphere:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} P_{sphere}(q) & = \int_{0}^{2\pi}\int_{0}^{\pi} | F_{sphere}(q) |^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\ & = \int_{0}^{2\pi}\int_{0}^{\pi} \left| 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \end{alignat} }

Note that the spherical symmetry guarantees that the integrand does not depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} P_{sphere}(q) & = \left( 3 V_{sphere} \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right)^2 \int_{0}^{2\pi}\int_{0}^{\pi} \sin\theta\mathrm{d}\theta\mathrm{d}\phi \\ & = 3^2 V_{sphere}^2 \left( \frac{ \sin(qR)-qR \cos(qR) }{ (qR)^3 } \right)^2 \left[\int_{0}^{2\pi}\mathrm{d}\phi\right]\left[\int_{0}^{\pi} \sin\theta\mathrm{d}\theta\right] \\ & = 9 V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \left[ 2\pi \right]\left[ 2 \right] \\ & = 36 \pi V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \\ \end{alignat} }

Isotropic Form Factor Intensity at q=0

At q=0, we expect:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{sphere}\left(0\right) = 4 \pi V_{sphere}^2 }

Isotropic Form Factor Intensity at large q

Note that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} P_{sphere}(q) & = 36 \pi V_{sphere}^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ (qR)^6 } \\ & = 36 \pi \left( \frac{4 \pi R^3}{3} \right)^2 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ q^6 R^6 } \\ & = 64 \pi^3 \frac{ (\sin(qR)-qR \cos(qR))^2 }{ q^6 } \\ \end{alignat} }

For large q, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q R} term dominates the numerator:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \lim_{q \rightarrow \infty} P_{sphere}(q) & = \lim_{q \rightarrow \infty} 64 \pi^3 \frac{ (\sin(qR) - qR \cos(qR))^2 }{ q^6 } \\ & = \lim_{q \rightarrow \infty} 64 \pi^3 \frac{ q^2 R^2 \cos^2(qR) }{ q^6 } \\ & = 64 \pi^3 R^2 \lim_{q \rightarrow \infty} \frac{ \cos^2(qR) }{ q^4 } \\ \end{alignat} }

The oscillation of the numerator is overwhelmed by the decay of the denominator:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \lim_{q \rightarrow \infty} P_{sphere}(q) & \approx \frac{ 64 \pi^3 R^2 }{ q^4 } \\ \end{alignat} }