Changeset abcac7a in sasview
 Timestamp:
 Jun 4, 2018 2:22:47 AM (3 months ago)
 Branches:
 ESS_GUI, ESS_GUI_beta_approx, ESS_GUI_iss957, ESS_GUI_iss959, ESS_GUI_iss971, ESS_GUI_iss976
 Children:
 397037e
 Parents:
 4ac377c
 Location:
 src/sas/qtgui/Perspectives/Invariant/media
 Files:

 4 added
 1 edited
Legend:
 Unmodified
 Added
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src/sas/qtgui/Perspectives/Invariant/media/invariant_help.rst
r417c03f rabcac7a 10 10  11 11 12 The scattering, or Porod, invariant ( $Q^*$) is a modelindependent quantity that12 The scattering, or Porod, invariant (:math:`Q^*`) is a modelindependent quantity that 13 13 can be easily calculated from scattering data. 14 14 15 15 For two phase systems, the scattering invariant is defined as the integral of 16 the square of the wavevector transfer ($Q$) multiplied by the scattering cross section 17 over the full range of $Q$ from zero to infinity, that is 16 the square of the wavevector transfer (:math:`Q`) multiplied by the scattering cross section 17 over the full range of :math:`Q` from zero to infinity, that is 18 18 19 19 20 .. math:: 20 21 21 Q^* = \int_0^ \infty q^2I(q)\,dq22 Q^* = \int_0^ \infty q^2 I(q)\,dq 22 23 23 in the case of pinhole geometry. For slit geometry the invariant is given by 24 25 in the case of pinhole geometry (SAS). 26 27 28 For slit geometry (USAS) the invariant is given by 24 29 25 30 .. math:: 26 31 27 Q^* = \Delta q_v \int_0^\inftyqI(q)\,dq32 Q^* = \int_0^\infty \Delta q_v \, qI(q)\,dq 28 33 29 where $\Delta q_v$is the slit height.34 where :math:`\Delta q_v` is the slit height. 30 35 31 The worth of $Q^*$ is that it can be used to determine the volume fraction and32 the specific area of a sample. Whilst these quantities are useful in their own 33 rightthey can also be used in further analysis.36 The worth of :math:`Q^*` is that it can be used to determine the volume fraction 37 and the specific area of a sample. Whilst these quantities are useful in their 38 own right, they can also be used in further analysis. 34 39 35 The difficulty with using $Q^*$ arises from the fact that experimental data is 36 never measured over the range $0 \le Q \le \infty$. At best, combining USAS and 37 WAS data might cover the range $10^{5} \le Q \le 10$ 1/\ Ang . Thus it is usually 38 necessary to extrapolate the experimental data to low and high $Q$. For this 40 The difficulty with using :math:`Q^*` arises from the fact that experimental 41 data is never measured over the range :math:`0 \le Q \le \infty`. At best, 42 combining USAS and WAS data might cover the range 43 :math:`10^{5} \le Q \le 10`Ang:math:`^{1}`. Thus it is usually 44 necessary to extrapolate the experimental data to low and high :math:`Q`. 45 For this 39 46 40 High\ $Q$region (>= *Qmax* in data)47 High\ :math:`Q` region (>= *Qmax* in data) 41 48 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 $Q_{Nm}$ to $Q_N$ (where $m < N$). 49 * The power law function :math:`C/Q^4` is used where the constant 50 :math:`C = 2 \pi \Delta\rho\, S_v` with 51 :math:`\Delta\rho`, the scattering length density (SLD) contrast and 52 :math:`S_v`, the specific surface area. The value of :math:`C` is to be found 53 by fitting part of data within the range :math:`Q_{Nm}` to :math:`Q_N` 54 (where :math:`m < N`), . 45 55 46 Low\ $Q$region (<= *Qmin* in data)56 Low\ :math:`Q` region (<= *Qmin* in data) 47 57 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. 58 * The Guinier function :math:`I(Q)=I(0) \exp (R_g^2 Q^2/3)` where :math:`R_g` 59 is the radius of gyration. The values of :math:`I(0)` and :math:`R_g` are 60 obtained by fitting as for the high\ :math:`Q` region above. 50 61 Alternatively a power law can be used. 51 62 … … 62 73 the dataset into the *Invariant* panel. 63 74 64 4) 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 67 rescale the data. 75 .. image:: image_invariant_load_data.png 68 76 69 5) Adjust the extrapolation range as necessary. In most cases the default 70 values will suffice. 77 4) Use the *Customised Input* box on the *Options* tab to subtract any 78 background, specify the contrast (i.e. difference in SLDs  this must be 79 specified for the eventual value of :math:`Q^*` to be on an absolute scale), 80 or to rescale the data. 71 81 72 6) Click the *Compute* button. 82 5) Adjust the extrapolation range in the *Options* tab as necessary. In most 83 cases the default values will suffice. 73 84 74 7) To include a lower and/or higher $Q$ range, check the relevant *Enable 85 86 6) Click the *Calculate* button. 87 88 7) To include a lower and/or higher :math:`Q` range, check the relevant *Enable 75 89 Extrapolate* check boxes. 90 91 .. figure:: image_invariant_option_tab.png 92 93 .. 94 95 *Option tab of the Invariant panel.* 96 76 97 77 98 If power law extrapolations are chosen, the exponent can be either held 78 99 fixed or fitted. The number of points, Npts, to be used for the basis of the 79 extrapolation can also be specified .100 extrapolation can also be specified in the related *Power* box(es). 80 101 81 8) If the value of $Q^*$ calculated with the extrapolated regions is invalid, a 82 red warning will appear at the top of the *Invariant* panel.102 .. figure:: image_invariant_outplot_plot.png 103 :width: 300pt 83 104 84 The details of the calculation are available by clicking the *Details* 85 button in the middle of the panel. 105 .. 86 106 87 .. image:: image005.png 107 *Output plot generated after calculations.* 108 109 8) If the value of :math:`Q^*` calculated with the extrapolated regions is 110 invalid, the related box will be highlighted in red. 111 112 The details of the calculation are available by clicking the *Status* 113 button at the bottom of the panel. 114 115 116 .. image:: image_invariant_details.png 117 :width: 300pt 118 88 119 89 120 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 95 126 ^^^^^^^^^^^^^^^ 96 127 97 The volume fraction $\phi$ is related to $Q^*$by128 The volume fraction :math:`\phi` is related to :math:`Q^*` by 98 129 99 130 .. math:: … … 101 132 \phi(1  \phi) = \frac{Q^*}{2\pi^2(\Delta\rho)^2} \equiv A 102 133 103 where $\Delta\rho$is the SLD contrast.134 where :math:`\Delta\rho` is the SLD contrast. 104 135 105 136 .. math:: … … 112 143 ^^^^^^^^^^^^^^^^^^^^^ 113 144 114 The specific surface area $S_v$ is related to $Q^*$by145 The specific surface area :math:`S_v` is related to :math:`Q^*` by 115 146 116 147 .. math:: … … 118 149 S_v = \frac{2\pi\phi(1\phi)C_p}{Q^*} = \frac{2\pi A C_p}{Q^*} 119 150 120 where $C_p$is the Porod constant.151 where :math:`C_p` is the Porod constant. 121 152 122 153 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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