-                      r417c03f
+                      rabcac7a
 -----------
 The scattering, or Porod, invariant ($Q^*$) is a model-independent quantity that
+The scattering, or Porod, invariant (:math:`Q^*`) is a model-independent quantity that
 can be easily calculated from scattering data.
 For two phase systems, the scattering invariant is defined as the integral of
+the square of the wavevector transfer ($Q$) multiplied by the scattering cross section
+over the full range of $Q$ from zero to infinity, that is
+the square of the wavevector transfer (:math:`Q`) multiplied by the scattering cross section
+over the full range of :math:`Q` from zero to infinity, that is
 .. math::
     Q^* = \int_0^\infty q^2I(q)\,dq
+    Q^* = \int_0^ \infty q^2 I(q)\,dq
+in the case of pinhole geometry. For slit geometry the invariant is given by
+in the case of pinhole geometry (SAS).
+For slit geometry (USAS) the invariant is given by
 .. math::
     Q^* = \Delta q_v \int_0^\infty qI(q)\,dq
+    Q^* =  \int_0^\infty \Delta q_v \, qI(q)\,dq
 where $\Delta q_v$ is the slit height.
+where :math:`\Delta q_v` is the slit height.
 The worth of $Q^*$  is that it can be used to determine the volume fraction and
+the specific area of a sample. Whilst these quantities are useful in their own
 right they can also be used in further analysis.
+The worth of :math:`Q^*` is that it can be used to determine the volume fraction
+and the specific area of a sample. Whilst these quantities are useful in their
+own right, they can also be used in further analysis.
+The difficulty with using $Q^*$  arises from the fact that experimental data is
+never measured over the range $0 \le Q \le \infty$. At best, combining USAS and
+WAS data might cover the range $10^{-5} \le Q \le 10$ 1/\ |Ang| . Thus it is usually
+necessary to extrapolate the experimental data to low and high $Q$. For this
+The difficulty with using :math:`Q^*`  arises from the fact that experimental
+data is never measured over the range :math:`0 \le Q \le \infty`. At best,
+combining USAS and WAS data might cover the range
+:math:`10^{-5} \le Q \le 10`|Ang|:math:`^{-1}`. Thus it is usually
+necessary to extrapolate the experimental data to low and high :math:`Q`.
+For this
 High-\ $Q$ region (>= *Qmax* in data)
+High-\ :math:`Q` region (>= *Qmax* in data)
+*  The power law function $C/Q^4$ is used where the constant
+   $C = 2 \pi \Delta\rho S_v$ is to be found by fitting part of data
+   within the range $Q_{N-m}$ to $Q_N$ (where $m < N$).
+*  The power law function :math:`C/Q^4` is used where the constant
+   :math:`C = 2 \pi \Delta\rho\, S_v` with
+   :math:`\Delta\rho`, the scattering length density (SLD) contrast and
+   :math:`S_v`, the specific surface area. The value of :math:`C` is to be found
+   by fitting part of data within the range :math:`Q_{N-m}` to :math:`Q_N`
+   (where :math:`m < N`), .
 Low-\ $Q$ region (<= *Qmin* in data)
+Low-\ :math:`Q` region (<= *Qmin* in data)
+*  The Guinier function $I_0 exp(-R_g^2 Q^2/3)$ where $I_0$
+   and $R_g$ are obtained by fitting as for the high-\ $Q$ region above.
+*  The Guinier function :math:`I(Q)=I(0) \exp (-R_g^2 Q^2/3)` where :math:`R_g`
+   is the radius of gyration. The values of :math:`I(0)` and :math:`R_g` are
+   obtained by fitting as for the high-\ :math:`Q` region above.
    Alternatively a power law can be used.
 …
    the dataset into the *Invariant* panel.
+) Use the *Customised Input* boxes on the *Invariant* panel to subtract
+   any background, specify the contrast (i.e. difference in SLDs - this must be
+   specified for the eventual value of $Q^*$  to be on an absolute scale), or to
+   rescale the data.
+.. image:: image_invariant_load_data.png
+) Adjust the extrapolation range as necessary. In most cases the default
+   values will suffice.
+) Use the *Customised Input* box on the *Options* tab to subtract any
+   background, specify the contrast (i.e. difference in SLDs - this must be
+   specified for the eventual value of :math:`Q^*` to be on an absolute scale),
+   or to rescale the data.
+) Click the *Compute* button.
+) Adjust the extrapolation range in the *Options* tab as necessary. In most
+   cases the default values will suffice.
+) To include a lower and/or higher $Q$ range, check the relevant *Enable
+) Click the *Calculate* button.
+) To include a lower and/or higher :math:`Q` range, check the relevant *Enable
    Extrapolate* check boxes.
+   .. figure:: image_invariant_option_tab.png
+       ..
+       *Option tab of the Invariant panel.*
    If power law extrapolations are chosen, the exponent can be either held
    fixed or fitted. The number of points, Npts, to be used for the basis of the
    extrapolation can also be specified.
+   extrapolation can also be specified in the related *Power* box(es).
+) If the value of $Q^*$  calculated with the extrapolated regions is invalid, a
    red warning will appear at the top of the *Invariant* panel.
+   .. figure:: image_invariant_outplot_plot.png
+       :width: 300pt
+   The details of the calculation are available by clicking the *Details*
+   button in the middle of the panel.
+       ..
+.. image:: image005.png
+       *Output plot generated after calculations.*
+) If the value of :math:`Q^*` calculated with the extrapolated regions is
+   invalid, the related box will be highlighted in red.
+   The details of the calculation are available by clicking the *Status*
+   button at the bottom of the panel.
+   .. image:: image_invariant_details.png
+      :width: 300pt
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 …
 ^^^^^^^^^^^^^^^
 The volume fraction $\phi$ is related to $Q^*$  by
+The volume fraction :math:`\phi` is related to :math:`Q^*`  by
 .. math::
 …
     \phi(1 - \phi) = \frac{Q^*}{2\pi^2(\Delta\rho)^2} \equiv A
 where $\Delta\rho$ is the SLD contrast.
+where :math:`\Delta\rho` is the SLD contrast.
 .. math::
 …
 ^^^^^^^^^^^^^^^^^^^^^
 The specific surface area $S_v$ is related to $Q^*$  by
+The specific surface area :math:`S_v` is related to :math:`Q^*`  by
 .. math::
 …
     S_v = \frac{2\pi\phi(1-\phi)C_p}{Q^*} = \frac{2\pi A C_p}{Q^*}
 where $C_p$ is the Porod constant.
+where :math:`C_p` is the Porod constant.
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

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Changeset abcac7a in sasview

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