[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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| 8 | Polydispersity Distributions |
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| 9 | ---------------------------- |
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| 10 | |
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| 11 | With some models in sasmodels we can calculate the average form factor for a |
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| 12 | population of particles that exhibit size and/or orientational |
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| 13 | polydispersity. The resultant form factor is normalized by the average |
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| 14 | particle volume such that |
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| 15 | |
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| 16 | .. math:: |
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| 17 | |
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| 18 | P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} |
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| 19 | |
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| 20 | where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an |
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| 21 | average over the size distribution. |
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| 22 | |
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| 23 | Each distribution is characterized by its center $\bar x$, its width $\sigma$, |
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| 24 | the number of sigmas $N_\sigma$ to include from the tails, and the number of |
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| 25 | points used to compute the average. The center of the distribution is set by the |
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| 26 | value of the model parameter. Volume parameters have polydispersity *PD* |
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| 27 | (not to be confused with a molecular weight distributions in polymer science) |
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| 28 | leading to a size distribution of width $\text{PD} = \sigma / \bar x$, but |
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| 29 | orientation parameters use an angular distributions of width $\sigma$. |
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| 30 | $N_\sigma$ determines how far into the tails to evaluate the distribution, with |
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| 31 | larger values of $N_\sigma$ required for heavier tailed distributions. |
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| 32 | The scattering in general falls rapidly with $qr$ so the usual assumption |
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| 33 | that $G(r - 3\sigma_r)$ is tiny and therefore $f(r - 3\sigma_r)G(r - 3\sigma_r)$ |
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| 34 | will not contribute much to the average may not hold when particles are large. |
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| 35 | This, too, will require increasing $N_\sigma$. |
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| 36 | |
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| 37 | Users should note that the averaging computation is very intensive. Applying |
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| 38 | polydispersion to multiple parameters at the same time or increasing the |
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| 39 | number of points in the distribution will require patience! However, the |
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| 40 | calculations are generally more robust with more data points or more angles. |
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| 41 | |
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[a5a12ca] | 42 | The following six distribution functions are provided: |
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[990d8df] | 43 | |
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| 44 | * *Rectangular Distribution* |
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| 45 | * *Gaussian Distribution* |
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| 46 | * *Lognormal Distribution* |
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| 47 | * *Schulz Distribution* |
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| 48 | * *Array Distribution* |
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[a5a12ca] | 49 | * *Boltzmann Distribution* |
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[990d8df] | 50 | |
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| 51 | These are all implemented as *number-average* distributions. |
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| 52 | |
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| 53 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 54 | |
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| 55 | Rectangular Distribution |
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| 56 | ^^^^^^^^^^^^^^^^^^^^^^^^ |
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| 57 | |
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| 58 | The Rectangular Distribution is defined as |
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| 59 | |
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| 60 | .. math:: |
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| 61 | |
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| 62 | f(x) = \frac{1}{\text{Norm}} |
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| 63 | \begin{cases} |
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| 64 | 1 & \text{for } |x - \bar x| \leq w \\ |
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| 65 | 0 & \text{for } |x - \bar x| > w |
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| 66 | \end{cases} |
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| 67 | |
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| 68 | where $\bar x$ is the mean of the distribution, $w$ is the half-width, and |
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| 69 | *Norm* is a normalization factor which is determined during the numerical |
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| 70 | calculation. |
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| 71 | |
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| 72 | Note that the standard deviation and the half width $w$ are different! |
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| 73 | |
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| 74 | The standard deviation is |
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| 75 | |
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| 76 | .. math:: \sigma = w / \sqrt{3} |
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| 77 | |
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| 78 | whilst the polydispersity is |
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| 79 | |
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| 80 | .. math:: \text{PD} = \sigma / \bar x |
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| 81 | |
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| 82 | .. figure:: pd_rectangular.jpg |
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| 83 | |
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| 84 | Rectangular distribution. |
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| 85 | |
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| 86 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 87 | |
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| 88 | Gaussian Distribution |
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| 89 | ^^^^^^^^^^^^^^^^^^^^^ |
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| 90 | |
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| 91 | The Gaussian Distribution is defined as |
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| 92 | |
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| 93 | .. math:: |
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| 94 | |
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| 95 | f(x) = \frac{1}{\text{Norm}} |
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| 96 | \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) |
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| 97 | |
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[1f058ea] | 98 | where $\bar x$ is the mean of the distribution and *Norm* is a normalization |
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| 99 | factor which is determined during the numerical calculation. |
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[990d8df] | 100 | |
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| 101 | The polydispersity is |
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| 102 | |
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| 103 | .. math:: \text{PD} = \sigma / \bar x |
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| 104 | |
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| 105 | .. figure:: pd_gaussian.jpg |
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| 106 | |
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| 107 | Normal distribution. |
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| 108 | |
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| 109 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 110 | |
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| 111 | Lognormal Distribution |
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| 112 | ^^^^^^^^^^^^^^^^^^^^^^ |
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| 113 | |
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| 114 | The Lognormal Distribution is defined as |
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| 115 | |
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| 116 | .. math:: |
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| 117 | |
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| 118 | f(x) = \frac{1}{\text{Norm}} |
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| 119 | \frac{1}{xp}\exp\left(-\frac{(\ln(x) - \mu)^2}{2p^2}\right) |
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| 120 | |
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| 121 | where $\mu=\ln(x_\text{med})$ when $x_\text{med}$ is the median value of the |
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| 122 | distribution, and *Norm* is a normalization factor which will be determined |
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| 123 | during the numerical calculation. |
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| 124 | |
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[1f058ea] | 125 | The median value for the distribution will be the value given for the |
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| 126 | respective size parameter, for example, *radius=60*. |
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[990d8df] | 127 | |
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| 128 | The polydispersity is given by $\sigma$ |
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| 129 | |
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| 130 | .. math:: \text{PD} = p |
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| 131 | |
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| 132 | For the angular distribution |
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| 133 | |
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| 134 | .. math:: p = \sigma / x_\text{med} |
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| 135 | |
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| 136 | The mean value is given by $\bar x = \exp(\mu+ p^2/2)$. The peak value |
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| 137 | is given by $\max x = \exp(\mu - p^2)$. |
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| 138 | |
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| 139 | .. figure:: pd_lognormal.jpg |
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| 140 | |
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| 141 | Lognormal distribution. |
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| 142 | |
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| 143 | This distribution function spreads more, and the peak shifts to the left, as |
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| 144 | $p$ increases, so it requires higher values of $N_\sigma$ and more points |
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| 145 | in the distribution. |
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| 146 | |
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| 147 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 148 | |
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| 149 | Schulz Distribution |
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| 150 | ^^^^^^^^^^^^^^^^^^^ |
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| 151 | |
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| 152 | The Schulz distribution is defined as |
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| 153 | |
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| 154 | .. math:: |
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| 155 | |
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| 156 | f(x) = \frac{1}{\text{Norm}} |
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| 157 | (z+1)^{z+1}(x/\bar x)^z\frac{\exp[-(z+1)x/\bar x]}{\bar x\Gamma(z+1)} |
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| 158 | |
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| 159 | where $\bar x$ is the mean of the distribution and *Norm* is a normalization |
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| 160 | factor which is determined during the numerical calculation, and $z$ is a |
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| 161 | measure of the width of the distribution such that |
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| 162 | |
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| 163 | .. math:: z = (1-p^2) / p^2 |
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| 164 | |
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| 165 | The polydispersity is |
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| 166 | |
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| 167 | .. math:: p = \sigma / \bar x |
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| 168 | |
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| 169 | Note that larger values of PD might need larger number of points and $N_\sigma$. |
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| 170 | For example, at PD=0.7 and radius=60 |Ang|, Npts>=160 and Nsigmas>=15 at least. |
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| 171 | |
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| 172 | .. figure:: pd_schulz.jpg |
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| 173 | |
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| 174 | Schulz distribution. |
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| 175 | |
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| 176 | For further information on the Schulz distribution see: |
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| 177 | M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461. |
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| 178 | |
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| 179 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 180 | |
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| 181 | Array Distribution |
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| 182 | ^^^^^^^^^^^^^^^^^^ |
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| 183 | |
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[a5a12ca] | 184 | This user-definable distribution should be given as a simple ASCII text |
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[990d8df] | 185 | file where the array is defined by two columns of numbers: $x$ and $f(x)$. |
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| 186 | The $f(x)$ will be normalized to 1 during the computation. |
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| 187 | |
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| 188 | Example of what an array distribution file should look like: |
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| 189 | |
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| 190 | ==== ===== |
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| 191 | 30 0.1 |
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| 192 | 32 0.3 |
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| 193 | 35 0.4 |
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| 194 | 36 0.5 |
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| 195 | 37 0.6 |
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| 196 | 39 0.7 |
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| 197 | 41 0.9 |
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| 198 | ==== ===== |
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| 199 | |
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| 200 | Only these array values are used computation, therefore the parameter value |
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| 201 | given for the model will have no affect, and will be ignored when computing |
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| 202 | the average. This means that any parameter with an array distribution will |
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[a5a12ca] | 203 | not be fitable. |
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| 204 | |
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| 205 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 206 | |
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| 207 | Boltzmann Distribution |
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| 208 | ^^^^^^^^^^^^^^^^^^^^^^ |
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| 209 | |
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| 210 | The Boltzmann Distribution is defined as |
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| 211 | |
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| 212 | .. math:: |
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| 213 | |
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| 214 | f(x) = \frac{1}{\text{Norm}} |
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| 215 | \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) |
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| 216 | |
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| 217 | where $\bar x$ is the mean of the distribution and *Norm* is a normalization |
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| 218 | factor which is determined during the numerical calculation. |
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| 219 | The width is defined as |
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| 220 | |
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| 221 | .. math:: \sigma=\frac{k T}{E} |
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| 222 | |
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| 223 | which is the inverse Boltzmann factor, |
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| 224 | where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a |
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| 225 | characteristic energy per particle. |
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| 226 | |
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| 227 | .. figure:: pd_boltzmann.jpg |
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| 228 | |
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| 229 | Boltzmann distribution. |
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[990d8df] | 230 | |
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| 231 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 232 | |
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| 233 | Note about DLS polydispersity |
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| 234 | ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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| 235 | |
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| 236 | Many commercial Dynamic Light Scattering (DLS) instruments produce a size |
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[1f058ea] | 237 | polydispersity parameter, sometimes even given the symbol $p$\ ! This |
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[990d8df] | 238 | parameter is defined as the relative standard deviation coefficient of |
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| 239 | variation of the size distribution and is NOT the same as the polydispersity |
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| 240 | parameters in the Lognormal and Schulz distributions above (though they all |
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| 241 | related) except when the DLS polydispersity parameter is <0.13. |
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| 242 | |
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| 243 | For more information see: |
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| 244 | S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 |
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| 245 | |
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| 246 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 247 | |
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| 248 | *Document History* |
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| 249 | |
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| 250 | | 2015-05-01 Steve King |
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| 251 | | 2017-05-08 Paul Kienzle |
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