Changeset 094b9eb in sasview

Ignore:
Timestamp:
Apr 6, 2017 5:42:22 PM (3 years ago)
Branches:
master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, magnetic_scatt, release-4.2.2, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
Children:
727c05f
Parents:
Message:

docs: update invariant to use latex for math markup

Location:
src/sas/sasgui/perspectives/invariant/media
Files:
4 deleted
1 edited

Legend:

Unmodified
 r6aad2e8 ----------- The scattering, or Porod, invariant (Q*\) is a model-independent quantity that The scattering, or Porod, invariant ($Q^*$) 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 wave transfer (Q) multiplied by the scattering cross section over the full range of Q from zero to infinity, that is 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 .. image:: image001.png .. math:: where *g = q* for pinhole geometry (SAS) and *g = q*\ :sub:v (the slit height) for slit geometry (USAS). Q^* = \int_0^\infty q^2I(q)\,dq 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 in the case of pinhole geometry. For slit geometry the invariant is given by .. math:: Q^* = \Delta q_v \int_0^\infty qI(q)\,dq where $\Delta q_v$ 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 difficulty with using Q*\  arises from the fact that experimental data is never measured over the range 0 =< *Q* =< infinity. At best, combining USAS and WAS data might cover the range 1e-5 =< *Q* =< 10 1/\ |Ang| . Thus it is usually necessary to extrapolate the experimental data to low and high *Q*. For this 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 High-*Q* region (>= *Qmax* in data) High-\ $Q$ region (>= *Qmax* in data) *  The power law function *C*/*Q*\ :sup:4 is used where the constant *C* (= 2.\ |pi|\ .(\ |bigdelta|\ |rho|\ ).\ *Sv*\ ) is to be found by fitting part of data within the range *Q*\ :sub:N-m to *Q*\ :sub:N (where m < N). *  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 $m < N$). Low-*Q* region (<= *Qmin* in data) Low-\ $Q$ region (<= *Qmin* in data) *  The Guinier function *I0.exp(-Rg*\ :sup:2\ *Q*\ :sup:2\ */3)* where *I0* and *Rg* are obtained by fitting as for the high-*Q* region above. *  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. Alternatively a power law can be used. 2) Load some data with the *Data Explorer*. 3) Select a dataset and use the *Send To* button on the *Data Explorer* to load 3) Select a dataset and use the *Send To* button on the *Data Explorer* to load the dataset into the *Invariant* panel. 4) 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 4) 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. 5) Adjust the extrapolation range as necessary. In most cases the default 5) Adjust the extrapolation range as necessary. In most cases the default values will suffice. 6) Click the *Compute* button. 7) To include a lower and/or higher Q range, check the relevant *Enable 7) To include a lower and/or higher $Q$ range, check the relevant *Enable Extrapolate* check boxes. 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 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. 8) If the value of Q*\  calculated with the extrapolated regions is invalid, a 8) If the value of $Q^*$  calculated with the extrapolated regions is invalid, a red warning will appear at the top of the *Invariant* panel. The details of the calculation are available by clicking the *Details* The details of the calculation are available by clicking the *Details* button in the middle of the panel. ^^^^^^^^^^^^^^^ The volume fraction |phi| is related to Q*\  by The volume fraction $\phi$ is related to $Q^*$  by .. image:: image002.png .. math:: where |bigdelta|\ |rho| is the SLD contrast. \phi(1 - \phi) = \frac{Q^*}{2\pi^2(\Delta\rho)^2} \equiv A .. image:: image003.png where $\Delta\rho$ is the SLD contrast. .. math:: \phi = \frac{1 \pm \sqrt{1 - 4A}}{2} .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ ^^^^^^^^^^^^^^^^^^^^^ The specific surface area *Sv* is related to Q*\  by The specific surface area $S_v$ is related to $Q^*$  by .. image:: image004.png .. math:: where *Cp* is the Porod constant. 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. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ