# linear_pearls

Linear pearls model of scattering from spherical pearls.

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | None | 1
background | Source background | cm-1 | 0.001
radius | Radius of the pearls | Å | 80
edge_sep | Length of the string segment - surface to surface | Å | 350
num_pearls | Number of the pearls | None | 3
sld | SLD of the pearl spheres | 10-6Å-2 | 1
sld_solvent | SLD of the solvent | 10-6Å-2 | 6.3

The returned value is scaled to units of cm-1 sr-1, absolute scale.

This model provides the form factor for \(N\) spherical pearls of radius \(R\) linearly joined by short strings (or segment length or edge separation) \(l\) \((= A - 2R)\). \(A\) is the center-to-center pearl separation distance. The thickness of each string is assumed to be negligible.

Definition

The output of the scattering intensity function for the linear_pearls model is given by (Dobrynin, 1996)

\[P(Q) = \frac{\text{scale}}{V}\left[ m_{p}^2 \left(N+2\sum_{n-1}^{N-1}(N-n)\frac{\sin(qnl)}{qnl}\right) \left( 3\frac{\sin(qR)-qR\cos(qR)}{(qr)^3}\right)^2\right]\]

where the mass \(m_p\) is \((SLD_{pearl}-SLD_{solvent})*(volume\ of\ N\ pearls)\). V is the total volume.

The 2D scattering intensity is the same as P(q) above, regardless of the orientation of the q vector.

Fig. 82 1D plot corresponding to the default parameters of the model.

Source

`linear_pearls.py` \(\ \star\ \) `linear_pearls.c` \(\ \star\ \) `sas_3j1x_x.c`

References

1. A V Dobrynin, M Rubinstein and S P Obukhov, Macromol., 29 (1996) 2974-2979

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