The scattering invariant (Q*)
The scattering invariant (Q*) is a model-independent quantity that can be easily calculated from
scattering data. For two phase systems, the scattering invariant, Q*, is defined as the integral of
the square of the wave transfer (q) multiplied by the scattering cross section over the full range of q.
Q* is given by the following equation,
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This model independent quantity (Q*) is calculated from the scattering data that can be used to determine the volume fraction and the specific area of the sample under consideration. These quantities are useful in their own right and can be used in further analysis. With this scattering invariant module users will also be able to determine the consistency of those properties between data. There is no real data defined from zero to infinity, there usually have limited range. Q* is not really computed from zero to infinity. Our maximum q range is 1e-5 ~ 10 (1/A). The lower and/or higher q range than data given can be extrapolated by fitting some data nearby.
The scattering invariant is computed as follow:
• I(q) = I(q) w/o background : If the data includes a background, user sets the value to subtract the background for the Q* computation.
• Reset I(q) = I(q)* scaling factor , deltaI(q) = deltaI(q)*scaling factor : If non-zero scaling factor is given, it will be considered.
• Invariant:
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where, g =q for the pinhole geometry and g = qv(the slit height) for the slit geometry which can be given in data or as a value.
• Higher q-region (>= qmax in data):
Power law (w/o background term) function = C/q4 will be used
where the constant C(=2pi(delta(rho))Sv) is to be found by fitting part of data with the range of qN-m to qN (m<N).
• Lower q-region (<= qmin in data):
Guinier function = I0exp(-Rg2q2/3) where I0 and Rg are obtained by fitting,
similarly to the high q region above.
Power law can also be used.
The volume fraction:
where delta(rho) is the SLD contrast of which value is given by users.
Thus,
where 0 =< A =<1/4 in order for these values to be physically valid.
The specific surface area :
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where A and Q* are obtained from previous sections, and the Porod constant Cp is given by users.
Definitions:
Q : the magnitude of neutron (or X-ray) momentum transfer vector.
I(Q): the scattering intensity as a function of the momentum transfer Q.
Invariant total is the sum of the invariant calculated from data’s q range and the invariant resulting from extrapolation at low q range and at high q range if considered.
Porod's chapter (2) from O. Glatter and O. Kratky, "Small Angle X-Ray Scattering", Academic Press, New York, 1982
Otto's web page: http://physchem.kfunigraz.ac.at/sm/
i) Loading data to the panel: Open the data file from File in the menu bar. Select loaded data from a plot panel by highlighting that it until its color turns yellow. Then right click on that the data and selects the option Compute Invariant. The application automatically computes the invariant value if the data loaded is valid.
ii) To subtract a background or/and to rescale the data, type the values in Customized Input box.
iii) If you want to calculate the volume fraction and the specific surface area, type the optional inputs in the customized input box, and then press ‘Compute’ button.
iv) The invariant can also be calculated including the outside of the data Q range: To include the lower Q and/or the higher Q range, check in the enable extrapolation check box in ‘Extrapolation ‘box. If the power low is chosen, the power (exponent) can be either held or fitted by checking the corresponding radio button. The Npts that is being used for the extrapolation can be specified.
v) If the invariant calculated from the extrapolated region is too large, it will be warn in red at the top of the panel, which means that your data is not proper to calculate the invariant.
vi) The details of the calculation is available by clicking the ‘Details’ button in the middle of the panel.
