deerlab.dd_shellsphere#

dd_shellsphere = <deerlab.model.Model object>#

Particles uniformly distributed on a sphere and on an outer spherical shell.

Parameters:

rarray_like: Distance axis, in nanometers.
radiusscalar: Inner shell radius.
thicknessscalar: Inner shell thickness.

Returns:

Pndarray: Distance distribution.

Notes

Parameter List

Name	Lower	Upper	Type	Frozen	Unit	Description
`radius`	0.1	20	nonlin	No	nm	Inner shell radius
`thickness`	0.1	20	nonlin	No	nm	Inner shell thickness

Model

$P(r) = \frac{3}{16R_1^3(R_2^3 - R_1^3)}\begin{cases} 12r^3R_1^2 - r^5 \quad \text{for} \quad 0\leq r < \min(2R_1,R_2 - R_1) \\ 8r^2(R_2^3 - R_1^3) - 3r(R_2^2 - R_1^2)^2 - 6r^3(R_2 - R_1)(R_2 + R_1) \quad \text{for} \quad R_2-R_1 \leq r < 2R_1 \\ 16r^2R_1^3 \quad \text{for} \quad 2R_1\leq r < R_2 - R_1 \\ r^5 - 6r^3(R_2^2 + R_1^2) + 8r^2(R_2^3 + R_1^3) - 3r(R_2^2 - R1_2)^2 \quad \text{for} \quad \max(R_2-R_1,2R_1) \leq r < R_1+R_2 \\ 0 \quad \text{for} \quad \text{otherwise} \end{cases}$

with

$R_1 = R$

$R_2 = R + w_1$

where $R$ is the inner sphere’s radius, and $w$ is the outer shell’s thickness.

References

[1]

D.R. Kattnig, D. Hinderberger, Analytical distance distributions in systems of spherical symmetry with applications to double electron-electron resonance, JMR, 230, 50-63, 2013

Examples

Example of the model evaluated at the start values of the parameters:

(Source code, png, hires.png, pdf)