.. _guinier:

guinier
=======================================================



========== =============================== ======= =============
Parameter  Description                     Units   Default value
========== =============================== ======= =============
scale      Scale factor or Volume fraction None                1
background Source background               |cm^-1|         0.001
rg         Radius of Gyration              |Ang|              60
========== =============================== ======= =============

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


**Definition**

This model fits the Guinier function

.. math::

    I(q) = \text{scale} \cdot \exp{\left[ \frac{-Q^2 R_g^2 }{3} \right]}
            + \text{background}

to the data directly without any need for linearization (*cf*. the usual
plot of $\ln I(q)$ vs $q^2$\ ). Note that you may have to restrict the data
range to include small q only, where the Guinier approximation actually
applies. See also the guinier_porod model.

For 2D data the scattering intensity is calculated in the same way as 1D,
where the $q$ vector is defined as

.. math:: q = \sqrt{q_x^2 + q_y^2}

In scattering, the radius of gyration $R_g$ quantifies the object's
distribution of SLD (not mass density, as in mechanics) from the object's
SLD centre of mass. It is defined by

.. math:: R_g^2 = \frac{\sum_i\rho_i\left(r_i-r_0\right)^2}{\sum_i\rho_i}

where $r_0$ denotes the object's SLD centre of mass and $\rho_i$ is the SLD at
a point $i$.

Notice that $R_g^2$ may be negative (since SLD can be negative), which happens
when a form factor $P(Q)$ is increasing with $Q$ rather than decreasing. This
can occur for core/shell particles, hollow particles, or for composite
particles with domains of different SLDs in a solvent with an SLD close to the
average match point. (Alternatively, this might be regarded as there being an
internal inter-domain "structure factor" within a single particle which gives
rise to a peak in the scattering).

To specify a negative value of $R_g^2$ in SasView, simply give $R_g$ a negative
value ($R_g^2$ will be evaluated as $R_g |R_g|$). Note that the physical radius
of gyration, of the exterior of the particle, will still be large and positive.
It is only the apparent size from the small $Q$ data that will give a small or
negative value of $R_g^2$.


.. figure:: img/guinier_autogenfig.png

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


**Source**

:download:`guinier.py <src/guinier.py>`

**References**

#. A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*,
   John Wiley & Sons, New York (1955)

**Authorship and Verification**

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