[da53353] | 1 | .. pd_help.rst |
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| 2 | |
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| 3 | .. This is a port of the original SasView html help file to ReSTructured text |
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| 4 | .. by S King, ISIS, during SasView CodeCamp-III in Feb 2015. |
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| 5 | |
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| 6 | .. |inlineimage004| image:: sm_image004.gif |
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| 7 | .. |inlineimage005| image:: sm_image005.gif |
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| 8 | .. |inlineimage008| image:: sm_image008.gif |
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| 9 | .. |inlineimage009| image:: sm_image009.gif |
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| 10 | .. |inlineimage010| image:: sm_image010.gif |
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| 11 | .. |inlineimage011| image:: sm_image011.gif |
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| 12 | .. |inlineimage012| image:: sm_image012.gif |
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| 13 | .. |inlineimage018| image:: sm_image018.gif |
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| 14 | .. |inlineimage019| image:: sm_image019.gif |
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| 15 | |
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| 16 | |
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| 17 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 18 | |
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| 19 | Polydispersity Distributions |
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| 20 | ---------------------------- |
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| 21 | |
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[f256d9b] | 22 | With some models SasView can calculate the average form factor for a population |
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| 23 | of particles that exhibit size and/or orientational polydispersity. The resultant |
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| 24 | form factor is normalized by the average particle volume such that |
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[da53353] | 25 | |
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[f256d9b] | 26 | *P(q) = scale* * \ <F*\F> / *V + bkg* |
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[da53353] | 27 | |
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[f256d9b] | 28 | where F is the scattering amplitude and the \<\> denote an average over the size |
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| 29 | distribution. |
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[da53353] | 30 | |
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[f256d9b] | 31 | Users should note that this computation is very intensive. Applying polydispersion |
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| 32 | to multiple parameters at the same time, or increasing the number of *Npts* values |
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| 33 | in the fit, will require patience! However, the calculations are generally more |
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| 34 | robust with more data points or more angles. |
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[da53353] | 35 | |
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[f256d9b] | 36 | SasView uses the term *PD* for a size distribution (and not to be confused with a |
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| 37 | molecular weight distributions in polymer science) and the term *Sigma* for an |
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| 38 | angular distribution. |
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| 39 | |
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| 40 | The following five distribution functions are provided: |
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[da53353] | 41 | |
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[a0637de] | 42 | * *Rectangular Distribution* |
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| 43 | * *Gaussian Distribution* |
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| 44 | * *Lognormal Distribution* |
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| 45 | * *Schulz Distribution* |
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[f256d9b] | 46 | * *Array Distribution* |
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[da53353] | 47 | |
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[a910c788] | 48 | These are all implemented in SasView as *number-average* distributions. |
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| 49 | |
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[a0637de] | 50 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[da53353] | 51 | |
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| 52 | Rectangular Distribution |
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[892a2cc] | 53 | ^^^^^^^^^^^^^^^^^^^^^^^^ |
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[da53353] | 54 | |
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[f256d9b] | 55 | The Rectangular Distribution is defined as |
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| 56 | |
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[da53353] | 57 | .. image:: pd_image001.png |
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| 58 | |
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[f256d9b] | 59 | where *xmean* is the mean of the distribution, *w* is the half-width, and *Norm* is a |
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| 60 | normalization factor which is determined during the numerical calculation. |
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| 61 | |
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| 62 | Note that the standard deviation and the half width *w* are different! |
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[da53353] | 63 | |
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| 64 | The standard deviation is |
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| 65 | |
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| 66 | .. image:: pd_image002.png |
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| 67 | |
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[f256d9b] | 68 | whilst the polydispersity is |
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[da53353] | 69 | |
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| 70 | .. image:: pd_image003.png |
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| 71 | |
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| 72 | .. image:: pd_image004.jpg |
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| 73 | |
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[a0637de] | 74 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[da53353] | 75 | |
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| 76 | Gaussian Distribution |
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[892a2cc] | 77 | ^^^^^^^^^^^^^^^^^^^^^ |
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[da53353] | 78 | |
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[f256d9b] | 79 | The Gaussian Distribution is defined as |
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| 80 | |
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[da53353] | 81 | .. image:: pd_image005.png |
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| 82 | |
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[f256d9b] | 83 | where *xmean* is the mean of the distribution and *Norm* is a normalization factor |
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[da53353] | 84 | which is determined during the numerical calculation. |
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| 85 | |
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[f256d9b] | 86 | The polydispersity is |
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[da53353] | 87 | |
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| 88 | .. image:: pd_image003.png |
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| 89 | |
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[f256d9b] | 90 | |
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[da53353] | 91 | .. image:: pd_image006.jpg |
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| 92 | |
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[a0637de] | 93 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[da53353] | 94 | |
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| 95 | Lognormal Distribution |
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[892a2cc] | 96 | ^^^^^^^^^^^^^^^^^^^^^^ |
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[da53353] | 97 | |
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[f256d9b] | 98 | The Lognormal Distribution is defined as |
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| 99 | |
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[da53353] | 100 | .. image:: pd_image007.png |
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| 101 | |
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[f256d9b] | 102 | where |mu|\ =ln(*xmed*), *xmed* is the median value of the distribution, and |
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| 103 | *Norm* is a normalization factor which will be determined during the numerical |
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| 104 | calculation. |
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| 105 | |
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| 106 | The median value for the distribution will be the value given for the respective |
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[b64b87c] | 107 | size parameter in the *FitPage*, for example, radius = 60. |
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[da53353] | 108 | |
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[f256d9b] | 109 | The polydispersity is given by |sigma| |
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[da53353] | 110 | |
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| 111 | .. image:: pd_image008.png |
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| 112 | |
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| 113 | For the angular distribution |
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| 114 | |
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| 115 | .. image:: pd_image009.png |
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| 116 | |
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[f256d9b] | 117 | The mean value is given by *xmean*\ =exp(|mu|\ +p\ :sup:`2`\ /2). The peak value |
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| 118 | is given by *xpeak*\ =exp(|mu|-p\ :sup:`2`\ ). |
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[da53353] | 119 | |
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| 120 | .. image:: pd_image010.jpg |
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| 121 | |
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[f256d9b] | 122 | This distribution function spreads more, and the peak shifts to the left, as *p* |
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| 123 | increases, requiring higher values of Nsigmas and Npts. |
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[da53353] | 124 | |
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[a0637de] | 125 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[da53353] | 126 | |
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| 127 | Schulz Distribution |
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[892a2cc] | 128 | ^^^^^^^^^^^^^^^^^^^ |
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[da53353] | 129 | |
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[f256d9b] | 130 | The Schulz distribution is defined as |
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| 131 | |
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[da53353] | 132 | .. image:: pd_image011.png |
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| 133 | |
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[f256d9b] | 134 | where *xmean* is the mean of the distribution and *Norm* is a normalization factor |
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| 135 | which is determined during the numerical calculation, and *z* is a measure of the |
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| 136 | width of the distribution such that |
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[da53353] | 137 | |
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[f256d9b] | 138 | z = (1-p\ :sup:`2`\ ) / p\ :sup:`2` |
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[da53353] | 139 | |
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[f256d9b] | 140 | The polydispersity is |
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[da53353] | 141 | |
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| 142 | .. image:: pd_image012.png |
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| 143 | |
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[f256d9b] | 144 | Note that larger values of PD might need larger values of Npts and Nsigmas. |
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| 145 | For example, at PD=0.7 and radius=60 |Ang|, Npts>=160 and Nsigmas>=15 at least. |
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[da53353] | 146 | |
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| 147 | .. image:: pd_image013.jpg |
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| 148 | |
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[f256d9b] | 149 | For further information on the Schulz distribution see: |
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| 150 | M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461. |
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| 151 | |
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[da53353] | 152 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[f256d9b] | 153 | |
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| 154 | Array Distribution |
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| 155 | ^^^^^^^^^^^^^^^^^^ |
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| 156 | |
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| 157 | This user-definable distribution should be given as as a simple ASCII text file |
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| 158 | where the array is defined by two columns of numbers: *x* and *f(x)*. The *f(x)* |
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| 159 | will be normalized by SasView during the computation. |
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| 160 | |
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| 161 | Example of what an array distribution file should look like: |
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| 162 | |
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| 163 | ==== ===== |
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| 164 | 30 0.1 |
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| 165 | 32 0.3 |
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| 166 | 35 0.4 |
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| 167 | 36 0.5 |
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| 168 | 37 0.6 |
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| 169 | 39 0.7 |
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| 170 | 41 0.9 |
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| 171 | ==== ===== |
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| 172 | |
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| 173 | SasView only uses these array values during the computation, therefore any mean |
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[b64b87c] | 174 | value of the parameter represented by *x* present in the *FitPage* |
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[f256d9b] | 175 | will be ignored. |
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| 176 | |
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| 177 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 178 | |
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| 179 | Note about DLS polydispersity |
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| 180 | ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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| 181 | |
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| 182 | Many commercial Dynamic Light Scattering (DLS) instruments produce a size |
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| 183 | polydispersity parameter, sometimes even given the symbol *p*! This parameter is |
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| 184 | defined as the relative standard deviation coefficient of variation of the size |
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| 185 | distribution and is NOT the same as the polydispersity parameters in the Lognormal |
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| 186 | and Schulz distributions above (though they all related) except when the DLS |
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| 187 | polydispersity parameter is <0.13. |
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| 188 | |
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| 189 | For more information see: |
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| 190 | S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 |
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| 191 | |
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| 192 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 193 | |
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| 194 | .. note:: This help document was last changed by Steve King, 01May2015 |
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