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1.. _sphere:
2
3Sphere
4=======================================================
5
6Spheres with uniform scattering length density
7
8=========== ================================= ============ =============
9Parameter   Description                       Units        Default value
10=========== ================================= ============ =============
11scale       Source intensity                  None                     1
12background  Source background                 |cm^-1|                  0
13sld         Layer scattering length density   |1e-6Ang^-2|             1
14solvent_sld Solvent scattering length density |1e-6Ang^-2|             6
15radius      Sphere radius                     |Ang|                   50
16=========== ================================= ============ =============
17
18The returned value is scaled to units of |cm^-1|.
19
20
21For information about polarised and magnetic scattering, click here_.
22
23.. _here: polar_mag_help.html
24
25Definition
26----------
27
28The 1D scattering intensity is calculated in the following way (Guinier, 1955)
29
30.. math::
31
32    I(Q) = \frac{\text{scale}}{V} \cdot \left[ \
33        3V(\Delta\rho) \cdot \frac{\sin(QR) - QR\cos(QR))}{(QR)^3} \
34        \right]^2 + \text{background}
35
36where *scale* is a volume fraction, $V$ is the volume of the scatterer,
37$R$ is the radius of the sphere, *background* is the background level and
38*sld* and *solvent_sld* are the scattering length densities (SLDs) of the
39scatterer and the solvent respectively.
40
41Note that if your data is in absolute scale, the *scale* should represent
42the volume fraction (which is unitless) if you have a good fit. If not,
43it should represent the volume fraction times a factor (by which your data
44might need to be rescaled).
45
46The 2D scattering intensity is the same as above, regardless of the
47orientation of $\vec q$.
48
49Our model uses the form factor calculations as defined in the IGOR
50package provided by the NIST Center for Neutron Research (Kline, 2006).
51
52Validation
53----------
54
55Validation of our code was done by comparing the output of the 1D model
56to the output of the software provided by the NIST (Kline, 2006).
57Figure :num:`figure #sphere-comparison` shows a comparison of the output
58of our model and the output of the NIST software.
59
60.. _sphere-comparison:
61
62.. figure:: img/sphere_comparison.jpg
63
64    Comparison of the DANSE scattering intensity for a sphere with the
65    output of the NIST SANS analysis software. The parameters were set to:
66    *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and
67    *background* = 0.01 |cm^-1|.
68
69
70Reference
71---------
72
73A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
74John Wiley and Sons, New York, (1955)
75
76*2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.*
77
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