[990d8df] | 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 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 7 | |
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[eda8b30] | 8 | .. _polydispersityhelp: |
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| 9 | |
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[990d8df] | 10 | Polydispersity Distributions |
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| 11 | ---------------------------- |
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| 12 | |
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[eda8b30] | 13 | With some models in sasmodels we can calculate the average intensity for a |
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[990d8df] | 14 | population of particles that exhibit size and/or orientational |
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[eda8b30] | 15 | polydispersity. The resultant intensity is normalized by the average |
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[990d8df] | 16 | particle volume such that |
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| 17 | |
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| 18 | .. math:: |
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| 19 | |
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| 20 | P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} |
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| 21 | |
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| 22 | where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an |
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| 23 | average over the size distribution. |
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| 24 | |
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[5026e05] | 25 | Each distribution is characterized by a center value $\bar x$ or $x_\text{med}$, |
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| 26 | a width parameter $\sigma$ (note this is *not necessarily* the standard deviation, so read |
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| 27 | the description carefully), the number of sigmas $N_\sigma$ to include from the |
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| 28 | tails of the distribution, and the number of points used to compute the average. |
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| 29 | The center of the distribution is set by the value of the model parameter. |
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| 30 | |
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| 31 | Volume parameters have polydispersity *PD* (not to be confused with a molecular |
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| 32 | weight distributions in polymer science), but orientation parameters use angular |
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| 33 | distributions of width $\sigma$. |
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| 34 | |
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[990d8df] | 35 | $N_\sigma$ determines how far into the tails to evaluate the distribution, with |
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| 36 | larger values of $N_\sigma$ required for heavier tailed distributions. |
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| 37 | The scattering in general falls rapidly with $qr$ so the usual assumption |
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| 38 | that $G(r - 3\sigma_r)$ is tiny and therefore $f(r - 3\sigma_r)G(r - 3\sigma_r)$ |
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| 39 | will not contribute much to the average may not hold when particles are large. |
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| 40 | This, too, will require increasing $N_\sigma$. |
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| 41 | |
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| 42 | Users should note that the averaging computation is very intensive. Applying |
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| 43 | polydispersion to multiple parameters at the same time or increasing the |
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| 44 | number of points in the distribution will require patience! However, the |
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| 45 | calculations are generally more robust with more data points or more angles. |
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| 46 | |
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[22279a4] | 47 | The following distribution functions are provided: |
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[990d8df] | 48 | |
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[75e4319] | 49 | * *Uniform Distribution* |
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[5026e05] | 50 | * *Rectangular Distribution* |
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[990d8df] | 51 | * *Gaussian Distribution* |
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[5026e05] | 52 | * *Boltzmann Distribution* |
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[990d8df] | 53 | * *Lognormal Distribution* |
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| 54 | * *Schulz Distribution* |
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| 55 | * *Array Distribution* |
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| 56 | |
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| 57 | These are all implemented as *number-average* distributions. |
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| 58 | |
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[5026e05] | 59 | Additional distributions are under consideration. |
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[990d8df] | 60 | |
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[5026e05] | 61 | Suggested Applications |
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| 62 | ^^^^^^^^^^^^^^^^^^^^^^ |
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[990d8df] | 63 | |
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[5026e05] | 64 | If applying polydispersion to parameters describing particle sizes, use |
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| 65 | the Lognormal or Schulz distributions. |
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[990d8df] | 66 | |
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[5026e05] | 67 | If applying polydispersion to parameters describing interfacial thicknesses |
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| 68 | or angular orientations, use the Gaussian or Boltzmann distributions. |
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[990d8df] | 69 | |
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[5026e05] | 70 | The array distribution allows a user-defined distribution to be applied. |
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[990d8df] | 71 | |
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[5026e05] | 72 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[990d8df] | 73 | |
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[5026e05] | 74 | Uniform Distribution |
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| 75 | ^^^^^^^^^^^^^^^^^^^^ |
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[990d8df] | 76 | |
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[5026e05] | 77 | The Uniform Distribution is defined as |
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[990d8df] | 78 | |
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[5026e05] | 79 | .. math:: |
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[990d8df] | 80 | |
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[5026e05] | 81 | f(x) = \frac{1}{\text{Norm}} |
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| 82 | \begin{cases} |
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| 83 | 1 & \text{for } |x - \bar x| \leq \sigma \\ |
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| 84 | 0 & \text{for } |x - \bar x| > \sigma |
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| 85 | \end{cases} |
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[990d8df] | 86 | |
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[5026e05] | 87 | where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, |
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| 88 | $\sigma$ is the half-width, and *Norm* is a normalization factor which is |
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| 89 | determined during the numerical calculation. |
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[990d8df] | 90 | |
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[5026e05] | 91 | The polydispersity in sasmodels is given by |
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[990d8df] | 92 | |
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[5026e05] | 93 | .. math:: \text{PD} = \sigma / \bar x |
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[92d330fd] | 94 | |
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[5026e05] | 95 | .. figure:: pd_uniform.jpg |
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[3d58247] | 96 | |
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[5026e05] | 97 | Uniform distribution. |
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[990d8df] | 98 | |
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[5026e05] | 99 | The value $N_\sigma$ is ignored for this distribution. |
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| 100 | |
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| 101 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 102 | |
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| 103 | Rectangular Distribution |
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[75e4319] | 104 | ^^^^^^^^^^^^^^^^^^^^^^^^ |
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| 105 | |
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[5026e05] | 106 | The Rectangular Distribution is defined as |
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[75e4319] | 107 | |
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| 108 | .. math:: |
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| 109 | |
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| 110 | f(x) = \frac{1}{\text{Norm}} |
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| 111 | \begin{cases} |
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[5026e05] | 112 | 1 & \text{for } |x - \bar x| \leq w \\ |
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| 113 | 0 & \text{for } |x - \bar x| > w |
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[75e4319] | 114 | \end{cases} |
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| 115 | |
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[5026e05] | 116 | where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, |
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| 117 | $w$ is the half-width, and *Norm* is a normalization factor which is determined |
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| 118 | during the numerical calculation. |
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[75e4319] | 119 | |
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[5026e05] | 120 | Note that the standard deviation and the half width $w$ are different! |
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[75e4319] | 121 | |
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[5026e05] | 122 | The standard deviation is |
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[75e4319] | 123 | |
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[5026e05] | 124 | .. math:: \sigma = w / \sqrt{3} |
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[75e4319] | 125 | |
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[5026e05] | 126 | whilst the polydispersity in sasmodels is given by |
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[92d330fd] | 127 | |
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[5026e05] | 128 | .. math:: \text{PD} = \sigma / \bar x |
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| 129 | |
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| 130 | .. figure:: pd_rectangular.jpg |
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| 131 | |
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| 132 | Rectangular distribution. |
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| 133 | |
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| 134 | .. note:: The Rectangular Distribution is deprecated in favour of the Uniform Distribution |
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| 135 | above and is described here for backwards compatibility with earlier versions of SasView only. |
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[75e4319] | 136 | |
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[990d8df] | 137 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 138 | |
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| 139 | Gaussian Distribution |
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| 140 | ^^^^^^^^^^^^^^^^^^^^^ |
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| 141 | |
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| 142 | The Gaussian Distribution is defined as |
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| 143 | |
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[5026e05] | 144 | .. math:: |
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| 145 | |
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| 146 | f(x) = \frac{1}{\text{Norm}} |
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| 147 | \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) |
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[990d8df] | 148 | |
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[5026e05] | 149 | where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution |
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| 150 | and *Norm* is a normalization factor which is determined during the numerical calculation. |
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[990d8df] | 151 | |
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[5026e05] | 152 | The polydispersity in sasmodels is given by |
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[990d8df] | 153 | |
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[5026e05] | 154 | .. math:: \text{PD} = \sigma / \bar x |
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| 155 | |
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| 156 | .. figure:: pd_gaussian.jpg |
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| 157 | |
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| 158 | Normal distribution. |
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| 159 | |
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| 160 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 161 | |
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| 162 | Boltzmann Distribution |
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| 163 | ^^^^^^^^^^^^^^^^^^^^^^ |
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| 164 | |
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| 165 | The Boltzmann Distribution is defined as |
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[990d8df] | 166 | |
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[5026e05] | 167 | .. math:: |
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[990d8df] | 168 | |
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[5026e05] | 169 | f(x) = \frac{1}{\text{Norm}} |
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| 170 | \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) |
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[990d8df] | 171 | |
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[5026e05] | 172 | where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution |
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| 173 | and *Norm* is a normalization factor which is determined during the numerical calculation. |
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| 174 | |
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| 175 | The width is defined as |
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| 176 | |
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| 177 | .. math:: \sigma=\frac{k T}{E} |
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| 178 | |
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| 179 | which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, |
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| 180 | $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. |
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| 181 | |
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| 182 | .. figure:: pd_boltzmann.jpg |
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| 183 | |
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| 184 | Boltzmann distribution. |
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[990d8df] | 185 | |
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| 186 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 187 | |
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| 188 | Lognormal Distribution |
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| 189 | ^^^^^^^^^^^^^^^^^^^^^^ |
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| 190 | |
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[5026e05] | 191 | The Lognormal Distribution describes a function of $x$ where $\ln (x)$ has a normal distribution. |
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| 192 | The result is a distribution that is skewed towards larger values of $x$. |
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| 193 | |
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[990d8df] | 194 | The Lognormal Distribution is defined as |
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| 195 | |
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[5026e05] | 196 | .. math:: |
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[990d8df] | 197 | |
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[5026e05] | 198 | f(x) = \frac{1}{\text{Norm}} |
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| 199 | \frac{1}{x\sigma}\exp\left(-\frac{1}{2}\bigg(\frac{\ln(x) - \mu}{\sigma}\bigg)^2\right) |
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| 200 | |
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| 201 | where *Norm* is a normalization factor which will be determined during the numerical calculation, |
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| 202 | $\mu=\ln(x_\text{med})$ and $x_\text{med}$ is the *median* value of the *lognormal* distribution, |
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| 203 | but $\sigma$ is a parameter describing the width of the underlying *normal* distribution. |
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[990d8df] | 204 | |
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[5026e05] | 205 | $x_\text{med}$ will be the value given for the respective size parameter in sasmodels, for |
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| 206 | example, *radius=60*. |
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[990d8df] | 207 | |
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[5026e05] | 208 | The polydispersity in sasmodels is given by |
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[990d8df] | 209 | |
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[5026e05] | 210 | .. math:: \text{PD} = p = \sigma / x_\text{med} |
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[990d8df] | 211 | |
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[5026e05] | 212 | The mean value of the distribution is given by $\bar x = \exp(\mu+ p^2/2)$ and the peak value |
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| 213 | by $\max x = \exp(\mu - p^2)$. |
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[990d8df] | 214 | |
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[5026e05] | 215 | The variance (the square of the standard deviation) of the *lognormal* distribution is given by |
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[990d8df] | 216 | |
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[5026e05] | 217 | .. math:: |
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[990d8df] | 218 | |
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[5026e05] | 219 | \nu = [\exp({\sigma}^2) - 1] \exp({2\mu + \sigma^2}) |
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[990d8df] | 220 | |
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[5026e05] | 221 | Note that larger values of PD might need a larger number of points and $N_\sigma$. |
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| 222 | |
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| 223 | .. figure:: pd_lognormal.jpg |
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[990d8df] | 224 | |
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[5026e05] | 225 | Lognormal distribution. |
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[990d8df] | 226 | |
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[5026e05] | 227 | For further information on the Lognormal distribution see: |
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| 228 | http://en.wikipedia.org/wiki/Log-normal_distribution and |
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| 229 | http://mathworld.wolfram.com/LogNormalDistribution.html |
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[990d8df] | 230 | |
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| 231 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 232 | |
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| 233 | Schulz Distribution |
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| 234 | ^^^^^^^^^^^^^^^^^^^ |
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| 235 | |
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[5026e05] | 236 | The Schulz (sometimes written Schultz) distribution is similar to the Lognormal distribution, |
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| 237 | in that it is also skewed towards larger values of $x$, but which has computational advantages |
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| 238 | over the Lognormal distribution. |
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| 239 | |
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[990d8df] | 240 | The Schulz distribution is defined as |
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| 241 | |
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[5026e05] | 242 | .. math:: |
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[990d8df] | 243 | |
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[5026e05] | 244 | f(x) = \frac{1}{\text{Norm}} |
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| 245 | (z+1)^{z+1}(x/\bar x)^z\frac{\exp[-(z+1)x/\bar x]}{\bar x\Gamma(z+1)} |
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[990d8df] | 246 | |
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[5026e05] | 247 | where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, |
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| 248 | *Norm* is a normalization factor which is determined during the numerical calculation, |
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| 249 | and $z$ is a measure of the width of the distribution such that |
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[990d8df] | 250 | |
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[5026e05] | 251 | .. math:: z = (1-p^2) / p^2 |
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[990d8df] | 252 | |
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[5026e05] | 253 | where $p$ is the polydispersity in sasmodels given by |
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[990d8df] | 254 | |
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[5026e05] | 255 | .. math:: PD = p = \sigma / \bar x |
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[990d8df] | 256 | |
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[5026e05] | 257 | and $\sigma$ is the RMS deviation from $\bar x$. |
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| 258 | |
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| 259 | Note that larger values of PD might need a larger number of points and $N_\sigma$. |
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| 260 | For example, for PD=0.7 with radius=60 |Ang|, at least Npts>=160 and Nsigmas>=15 are required. |
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[990d8df] | 261 | |
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[5026e05] | 262 | .. figure:: pd_schulz.jpg |
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[990d8df] | 263 | |
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[5026e05] | 264 | Schulz distribution. |
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[990d8df] | 265 | |
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| 266 | For further information on the Schulz distribution see: |
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[5026e05] | 267 | M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461 and |
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| 268 | M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 |
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[990d8df] | 269 | |
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| 270 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 271 | |
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| 272 | Array Distribution |
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| 273 | ^^^^^^^^^^^^^^^^^^ |
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| 274 | |
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[a5a12ca] | 275 | This user-definable distribution should be given as a simple ASCII text |
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[990d8df] | 276 | file where the array is defined by two columns of numbers: $x$ and $f(x)$. |
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| 277 | The $f(x)$ will be normalized to 1 during the computation. |
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| 278 | |
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| 279 | Example of what an array distribution file should look like: |
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| 280 | |
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| 281 | ==== ===== |
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| 282 | 30 0.1 |
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| 283 | 32 0.3 |
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| 284 | 35 0.4 |
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| 285 | 36 0.5 |
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| 286 | 37 0.6 |
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| 287 | 39 0.7 |
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| 288 | 41 0.9 |
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| 289 | ==== ===== |
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| 290 | |
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| 291 | Only these array values are used computation, therefore the parameter value |
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| 292 | given for the model will have no affect, and will be ignored when computing |
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| 293 | the average. This means that any parameter with an array distribution will |
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[a5a12ca] | 294 | not be fitable. |
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| 295 | |
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| 296 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 297 | |
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[990d8df] | 298 | Note about DLS polydispersity |
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| 299 | ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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| 300 | |
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| 301 | Many commercial Dynamic Light Scattering (DLS) instruments produce a size |
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[1f058ea] | 302 | polydispersity parameter, sometimes even given the symbol $p$\ ! This |
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[990d8df] | 303 | parameter is defined as the relative standard deviation coefficient of |
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| 304 | variation of the size distribution and is NOT the same as the polydispersity |
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| 305 | parameters in the Lognormal and Schulz distributions above (though they all |
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| 306 | related) except when the DLS polydispersity parameter is <0.13. |
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| 307 | |
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[5026e05] | 308 | .. math:: |
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| 309 | |
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| 310 | p_{DLS} = \sqrt(\nu / \bar x^2) |
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| 311 | |
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| 312 | where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. |
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| 313 | |
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[990d8df] | 314 | For more information see: |
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| 315 | S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 |
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| 316 | |
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| 317 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 318 | |
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| 319 | *Document History* |
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| 320 | |
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| 321 | | 2015-05-01 Steve King |
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| 322 | | 2017-05-08 Paul Kienzle |
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[5026e05] | 323 | | 2018-03-20 Steve King |
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