Difference between revisions of "Form Factor:Superball"

From GISAXS
Jump to: navigation, search
Line 1: Line 1:
 
A '''superball''' is a general mathematical shape that can be used to describe '''rounded cubes'''. In fact, it is a general parametrization that can describe, via a parameter <math>p</math>:
 
A '''superball''' is a general mathematical shape that can be used to describe '''rounded cubes'''. In fact, it is a general parametrization that can describe, via a parameter <math>p</math>:
 +
* Empty space (<math>p=0.0</math>)
 
* Concave octahedra (<math>p<0.5</math>)
 
* Concave octahedra (<math>p<0.5</math>)
 
* [[Form Factor:Octahedron|Octahedra]] (<math>p=0.5</math>)
 
* [[Form Factor:Octahedron|Octahedra]] (<math>p=0.5</math>)

Revision as of 11:53, 14 June 2014

A superball is a general mathematical shape that can be used to describe rounded cubes. In fact, it is a general parametrization that can describe, via a parameter 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} :

  • Empty space (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=0.0} )
  • Concave octahedra (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<0.5} )
  • Octahedra (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=0.5} )
  • Convex octahedra (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 0.5<p<1} )
  • Spheres (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=1} )
  • Rounded cubes (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>1} )
  • Cubes (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 \to \infty} )


Superball examples.png

The general equation is parametrized by the size, 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 R} , and the curvature 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} :

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} \left | \frac{x}{R} \right | ^{2p} + \left | \frac{y}{R} \right | ^{2p} + \left | \frac{z}{R} \right | ^{2p} & \le 1 \\ | x | ^{2p} + | y | ^{2p} + | z | ^{2p} & \le |R|^{2p} \\ \end{alignat} }

Obviously for 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=1} , we recover the equation for a sphere. In the limit of large 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} , we obtain a cube.

Volume

The normalized volume for a superball is:

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 \frac{ V_{\mathrm{sb}} }{R^3} = \frac{2}{2p} \mathrm{B}\left( \frac{1}{p} , \frac{2p+1}{2p} \right) \mathrm{B}\left( \frac{1}{2p} , \frac{p+1}{p} \right) }

Where 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{B}\left( x,y \right) = \Gamma(x)\Gamma(y)/\Gamma(x+y)} and 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 \Gamma(x)} is the usual Euler gamma function.

Superball volume.png

Equations

The form factor for a superball is likely not analytic. However, it can be computed numerically.

References

Mathematical descriptions of superballs

Application to nanoscience

Retrieved from "http://gisaxs.com/index.php?title=Form_Factor:Superball&oldid=644"