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1.. invariant_help.rst
2
3.. This is a port of the original SasView html help file to ReSTructured text
4.. by S King, ISIS, during SasView CodeCamp-III in Feb 2015.
5
6.. |Ang| unicode:: U+212B
7.. |pi| unicode:: U+03C0
8.. |bigdelta| unicode:: U+0394
9.. |rho| unicode:: U+03C1
10.. |phi| unicode:: U+03C6
11
12Invariant Calculation Perspective
13=================================
14
15Description
16-----------
17
18The scattering, or Porod, invariant (Q*\) is a model-independent quantity that
19can be easily calculated from scattering data.
20
21For two phase systems, the scattering invariant is defined as the integral of
22the square of the wave transfer (Q) multiplied by the scattering cross section
23over the full range of Q from zero to infinity, that is
24
25.. image:: image001.gif
26
27where *g = Q* for pinhole geometry (SAS) and *g = Qv* (the slit height) for 
28slit geometry (USAS).
29
30The worth of Q*\  is that it can be used to determine the volume fraction and
31the specific area of a sample. Whilst these quantities are useful in their own
32right they can also be used in further analysis.
33
34The difficulty with using Q*\  arises from the fact that experimental data is
35never measured over the range 0 =< *Q* =< infinity. At best, combining USAS and
36WAS data might cover the range 1e-5 =< *Q* =< 10 1/\ |Ang| . Thus it is usually
37necessary to extrapolate the experimental data to low and high *Q*. For this
38
39High-*Q* region (>= *Qmax* in data)
40
41*  The power law function *C*/*Q*\ :sup:`4` is used where the constant
42   *C* (= 2.\ |pi|\ .(\ |bigdelta|\ |rho|\ ).\ *Sv*\ ) is to be found by fitting part of data
43   within the range *Q*\ :sub:`N-m` to *Q*\ :sub:`N` (where m < N).
44
45Low-*Q* region (<= *Qmin* in data)
46
47*  The Guinier function *I0.exp(-Rg*\ :sup:`2`\ *Q*\ :sup:`2`\ */3)* where *I0*
48   and *Rg* are obtained by fitting as for the high-*Q* region above.
49   Alternatively a power law can be used.
50
51.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
52
53Using the perspective
54---------------------
55
561) Select *Invariant* from the *Analysis* menu on the SasView toolbar.
57
582) Load some data with the *Data Explorer*.
59
603) Select a dataset and use the *Send To* button on the *Data Explorer* to load
61   the dataset into the *Invariant* perspective.
62
634) Use the *Customised Input* boxes on the *Invariant* perspective to subtract
64   any background, specify the contrast (i.e. difference in SLDs - this must be
65   specified for the eventual value of Q*\  to be on an absolute scale), or to
66   rescale the data.
67
685) Adjust the extrapolation range as necessary. In most cases the default
69   values will suffice.
70
716) Click the *Compute* button.
72
737) To include a lower and/or higher Q range, check the relevant *Enable
74   Extrapolate* check boxes.
75   
76   If power law extrapolations are chosen, the exponent can be either held
77   fixed or fitted. The number of points, Npts, to be used for the basis of the
78   extrapolation can also be specified.
79
808) If the value of Q*\  calculated with the extrapolated regions is invalid, a
81   red warning will appear at the top of the *Invariant* perspective panel.
82
83   The details of the calculation are available by clicking the *Details*
84   button in the middle of the panel.
85
86.. image:: image005.gif
87
88.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
89
90Parameters
91----------
92
93Volume Fraction
94^^^^^^^^^^^^^^^
95
96The volume fraction |phi| is related to Q*\  by
97
98.. image:: image002.gif
99
100where |bigdelta|\ |rho| is the SLD contrast.
101
102.. image:: image003.gif
103
104.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
105
106Specific Surface Area
107^^^^^^^^^^^^^^^^^^^^^
108
109The specific surface area *Sv* is related to Q*\  by
110
111.. image:: image004.gif
112
113where *Cp* is the Porod constant.
114
115.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
116
117Reference
118---------
119
120O. Glatter and O. Kratky
121Chapter 2 in *Small Angle X-Ray Scattering*
122Academic Press, New York, 1982
123
124http://physchem.kfunigraz.ac.at/sm/
125
126.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
127
128.. note::  This help document was last changed by Steve King, 01May2015
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