Changeset 094b9eb in sasview


Ignore:
Timestamp:
Apr 6, 2017 5:42:22 PM (3 years ago)
Author:
Paul Kienzle <pkienzle@…>
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:
6aad2e8
Message:

docs: update invariant to use latex for math markup

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

Legend:

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  • src/sas/sasgui/perspectives/invariant/media/invariant_help.rst

    r6aad2e8 r094b9eb  
    1010----------- 
    1111 
    12 The scattering, or Porod, invariant (Q*\) is a model-independent quantity that  
     12The scattering, or Porod, invariant ($Q^*$) is a model-independent quantity that 
    1313can be easily calculated from scattering data. 
    1414 
    15 For two phase systems, the scattering invariant is defined as the integral of  
    16 the square of the wave transfer (Q) multiplied by the scattering cross section  
    17 over the full range of Q from zero to infinity, that is 
     15For two phase systems, the scattering invariant is defined as the integral of 
     16the square of the wavevector transfer ($Q$) multiplied by the scattering cross section 
     17over the full range of $Q$ from zero to infinity, that is 
    1818 
    19 .. image:: image001.png 
     19.. math:: 
    2020 
    21 where *g = q* for pinhole geometry (SAS) and *g = q*\ :sub:`v` (the slit height) for   
    22 slit geometry (USAS). 
     21    Q^* = \int_0^\infty q^2I(q)\,dq 
    2322 
    24 The worth of Q*\  is that it can be used to determine the volume fraction and  
    25 the specific area of a sample. Whilst these quantities are useful in their own  
     23in the case of pinhole geometry. For slit geometry the invariant is given by 
     24 
     25.. math:: 
     26 
     27    Q^* = \Delta q_v \int_0^\infty qI(q)\,dq 
     28 
     29where $\Delta q_v$ is the slit height. 
     30 
     31The worth of $Q^*$  is that it can be used to determine the volume fraction and 
     32the specific area of a sample. Whilst these quantities are useful in their own 
    2633right they can also be used in further analysis. 
    2734 
    28 The difficulty with using Q*\  arises from the fact that experimental data is  
    29 never measured over the range 0 =< *Q* =< infinity. At best, combining USAS and  
    30 WAS data might cover the range 1e-5 =< *Q* =< 10 1/\ |Ang| . Thus it is usually  
    31 necessary to extrapolate the experimental data to low and high *Q*. For this 
     35The difficulty with using $Q^*$  arises from the fact that experimental data is 
     36never measured over the range $0 \le Q \le \infty$. At best, combining USAS and 
     37WAS data might cover the range $10^{-5} \le Q \le 10$ 1/\ |Ang| . Thus it is usually 
     38necessary to extrapolate the experimental data to low and high $Q$. For this 
    3239 
    33 High-*Q* region (>= *Qmax* in data) 
     40High-\ $Q$ region (>= *Qmax* in data) 
    3441 
    35 *  The power law function *C*/*Q*\ :sup:`4` is used where the constant  
    36    *C* (= 2.\ |pi|\ .(\ |bigdelta|\ |rho|\ ).\ *Sv*\ ) is to be found by fitting part of data  
    37    within the range *Q*\ :sub:`N-m` to *Q*\ :sub:`N` (where m < N). 
     42*  The power law function $C/Q^4$ is used where the constant 
     43   $C = 2 \pi \Delta\rho S_v$ is to be found by fitting part of data 
     44   within the range $Q_{N-m}$ to $Q_N$ (where $m < N$). 
    3845 
    39 Low-*Q* region (<= *Qmin* in data) 
     46Low-\ $Q$ region (<= *Qmin* in data) 
    4047 
    41 *  The Guinier function *I0.exp(-Rg*\ :sup:`2`\ *Q*\ :sup:`2`\ */3)* where *I0*  
    42    and *Rg* are obtained by fitting as for the high-*Q* region above.  
     48*  The Guinier function $I_0 exp(-R_g^2 Q^2/3)$ where $I_0$ 
     49   and $R_g$ are obtained by fitting as for the high-\ $Q$ region above. 
    4350   Alternatively a power law can be used. 
    4451 
     
    52592) Load some data with the *Data Explorer*. 
    5360 
    54 3) Select a dataset and use the *Send To* button on the *Data Explorer* to load  
     613) Select a dataset and use the *Send To* button on the *Data Explorer* to load 
    5562   the dataset into the *Invariant* panel. 
    5663 
    57 4) Use the *Customised Input* boxes on the *Invariant* panel to subtract  
    58    any background, specify the contrast (i.e. difference in SLDs - this must be  
    59    specified for the eventual value of Q*\  to be on an absolute scale), or to  
     644) Use the *Customised Input* boxes on the *Invariant* panel to subtract 
     65   any background, specify the contrast (i.e. difference in SLDs - this must be 
     66   specified for the eventual value of $Q^*$  to be on an absolute scale), or to 
    6067   rescale the data. 
    6168 
    62 5) Adjust the extrapolation range as necessary. In most cases the default  
     695) Adjust the extrapolation range as necessary. In most cases the default 
    6370   values will suffice. 
    6471 
    65726) Click the *Compute* button. 
    6673 
    67 7) To include a lower and/or higher Q range, check the relevant *Enable  
     747) To include a lower and/or higher $Q$ range, check the relevant *Enable 
    6875   Extrapolate* check boxes. 
    69     
    70    If power law extrapolations are chosen, the exponent can be either held  
    71    fixed or fitted. The number of points, Npts, to be used for the basis of the  
     76 
     77   If power law extrapolations are chosen, the exponent can be either held 
     78   fixed or fitted. The number of points, Npts, to be used for the basis of the 
    7279   extrapolation can also be specified. 
    7380 
    74 8) If the value of Q*\  calculated with the extrapolated regions is invalid, a  
     818) If the value of $Q^*$  calculated with the extrapolated regions is invalid, a 
    7582   red warning will appear at the top of the *Invariant* panel. 
    7683 
    77    The details of the calculation are available by clicking the *Details*  
     84   The details of the calculation are available by clicking the *Details* 
    7885   button in the middle of the panel. 
    7986 
     
    8895^^^^^^^^^^^^^^^ 
    8996 
    90 The volume fraction |phi| is related to Q*\  by 
     97The volume fraction $\phi$ is related to $Q^*$  by 
    9198 
    92 .. image:: image002.png 
     99.. math:: 
    93100 
    94 where |bigdelta|\ |rho| is the SLD contrast. 
     101    \phi(1 - \phi) = \frac{Q^*}{2\pi^2(\Delta\rho)^2} \equiv A 
    95102 
    96 .. image:: image003.png 
     103where $\Delta\rho$ is the SLD contrast. 
     104 
     105.. math:: 
     106 
     107    \phi = \frac{1 \pm \sqrt{1 - 4A}}{2} 
    97108 
    98109.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    101112^^^^^^^^^^^^^^^^^^^^^ 
    102113 
    103 The specific surface area *Sv* is related to Q*\  by 
     114The specific surface area $S_v$ is related to $Q^*$  by 
    104115 
    105 .. image:: image004.png 
     116.. math:: 
    106117 
    107 where *Cp* is the Porod constant. 
     118    S_v = \frac{2\pi\phi(1-\phi)C_p}{Q^*} = \frac{2\pi A C_p}{Q^*} 
     119 
     120where $C_p$ is the Porod constant. 
    108121 
    109122.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
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