Changes in / [727c05f:02b01877] in sasview
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src/sas/sasgui/perspectives/invariant/media/invariant_help.rst
r094b9eb r6aad2e8 10 10 ----------- 11 11 12 The scattering, or Porod, invariant ( $Q^*$) is a model-independent quantity that12 The scattering, or Porod, invariant (Q*\) is a model-independent quantity that 13 13 can be easily calculated from scattering data. 14 14 15 For two phase systems, the scattering invariant is defined as the integral of 16 the square of the wave vector transfer ($Q$) multiplied by the scattering cross section17 over the full range of $Q$from zero to infinity, that is15 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 18 18 19 .. math::19 .. image:: image001.png 20 20 21 Q^* = \int_0^\infty q^2I(q)\,dq 21 where *g = q* for pinhole geometry (SAS) and *g = q*\ :sub:`v` (the slit height) for 22 slit geometry (USAS). 22 23 23 in 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 29 where $\Delta q_v$ is the slit height. 30 31 The worth of $Q^*$ is that it can be used to determine the volume fraction and 32 the specific area of a sample. Whilst these quantities are useful in their own 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 33 26 right they can also be used in further analysis. 34 27 35 The difficulty with using $Q^*$ arises from the fact that experimental data is36 never measured over the range $0 \le Q \le \infty$. At best, combining USAS and37 WAS data might cover the range $10^{-5} \le Q \le 10$ 1/\ |Ang| . Thus it is usually38 necessary to extrapolate the experimental data to low and high $Q$. For this28 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 39 32 40 High- \ $Q$region (>= *Qmax* in data)33 High-*Q* region (>= *Qmax* in data) 41 34 42 * The power law function $C/Q^4$ is used where the constant43 $C = 2 \pi \Delta\rho S_v$ is to be found by fitting part of data44 within the range $Q_{N-m}$ to $Q_N$ (where $m < N$).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). 45 38 46 Low- \ $Q$region (<= *Qmin* in data)39 Low-*Q* region (<= *Qmin* in data) 47 40 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.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. 50 43 Alternatively a power law can be used. 51 44 … … 59 52 2) Load some data with the *Data Explorer*. 60 53 61 3) Select a dataset and use the *Send To* button on the *Data Explorer* to load 54 3) Select a dataset and use the *Send To* button on the *Data Explorer* to load 62 55 the dataset into the *Invariant* panel. 63 56 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 to57 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 67 60 rescale the data. 68 61 69 5) Adjust the extrapolation range as necessary. In most cases the default 62 5) Adjust the extrapolation range as necessary. In most cases the default 70 63 values will suffice. 71 64 72 65 6) Click the *Compute* button. 73 66 74 7) To include a lower and/or higher $Q$ range, check the relevant *Enable67 7) To include a lower and/or higher Q range, check the relevant *Enable 75 68 Extrapolate* check boxes. 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 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 79 72 extrapolation can also be specified. 80 73 81 8) If the value of $Q^*$ calculated with the extrapolated regions is invalid, a74 8) If the value of Q*\ calculated with the extrapolated regions is invalid, a 82 75 red warning will appear at the top of the *Invariant* panel. 83 76 84 The details of the calculation are available by clicking the *Details* 77 The details of the calculation are available by clicking the *Details* 85 78 button in the middle of the panel. 86 79 … … 95 88 ^^^^^^^^^^^^^^^ 96 89 97 The volume fraction $\phi$ is related to $Q^*$by90 The volume fraction |phi| is related to Q*\ by 98 91 99 .. math::92 .. image:: image002.png 100 93 101 \phi(1 - \phi) = \frac{Q^*}{2\pi^2(\Delta\rho)^2} \equiv A 94 where |bigdelta|\ |rho| is the SLD contrast. 102 95 103 where $\Delta\rho$ is the SLD contrast. 104 105 .. math:: 106 107 \phi = \frac{1 \pm \sqrt{1 - 4A}}{2} 96 .. image:: image003.png 108 97 109 98 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 112 101 ^^^^^^^^^^^^^^^^^^^^^ 113 102 114 The specific surface area $S_v$ is related to $Q^*$by103 The specific surface area *Sv* is related to Q*\ by 115 104 116 .. math::105 .. image:: image004.png 117 106 118 S_v = \frac{2\pi\phi(1-\phi)C_p}{Q^*} = \frac{2\pi A C_p}{Q^*} 119 120 where $C_p$ is the Porod constant. 107 where *Cp* is the Porod constant. 121 108 122 109 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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