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+
+<p class=MsoNormal><h3><span style='font-family:"Times New Roman","serif"'>Polydisperisty
+and Angular Distributions</span></h3></p>
+
+<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Calculates
+the form factor for a polydisperse and/or angular population of particles with
+uniform scattering length density. The resultant form factor is normalized by
+the average particle volume such that P(q) = scale*&lt;F*F&gt;/Vol + bkg, where
+F is the scattering amplitude and the &lt; &gt; denote an average over the size
+distribution.  Users should use PD (polydispersity: this definition is different from the typical definition in polymer science) 
+for a size distribution and Sigma for an
+angular distribution (see below).</span></p>
+<p> Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for 
+more than one paramters or increasing Npts values might need extensive patience to complete the computation. Also
+note that even though it is time consuming, it is safer to have larger values of Npts and Nsigmas.</p>
+
+<p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span
+style='font-family:"Times New Roman","serif"'>The following five distribution
+functions are provided;</span></p>
+<ul>
+<li><a href="#Rectangular">Rectangular distribution</a></li>
+<li><a href="#Array">Array distribution</a></li>
+<li><a href="#Gaussian">Gaussian distribution</a></li>
+<li><a href="#Lognormal">Lognormal distribution</a></li>
+<li><a href="#Schulz">Schulz distribution</a></li>
+</ul>
+<p>&nbsp;</p>
+<p><a name="Rectangular"><h4>Rectangular distribution</a></h4></p>
+<p><img src="img/pd_image001.png"></p>
+<p>&nbsp;</p>
+<p>The x<sub>mean</sub> is the mean
+of the distribution, w is the half-width, and Norm is a normalization factor
+which is determined during the numerical calculation. Note that the Sigma and
+the half width <i>w</i> are different.</p>
+<p>The standard deviation is </p>
+<p><img src="img/pd_image002.png"></p>
+<p>&nbsp;</p>
+<p>The PD (polydispersity) is </p>
+<p><img src="img/pd_image003.png"></p>
+<p>&nbsp;</p>
+<p><img width=511 height=270
+id="Picture 1" src="img/pd_image004.jpg" alt=flat.gif></p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p><a name="Array"><h4>Array distribution</h4></a></p>
+
+<p>This distribution is to be given
+by users as a txt file where the array should be defined by two columns in the
+order of x and f(x) values. The f(x) will be normalized by SasView during the
+computation.</p>
+
+<p>Example of an array in the file;</p>
+
+<p>30        0.1</p>
+
+<p>32        0.3</p>
+
+<p>35        0.4</p>
+
+<p>36        0.5</p>
+
+<p>37        0.6</p>
+
+<p>39        0.7</p>
+
+<p>41        0.9</p>
+
+<p'>&nbsp;</p>
+
+<p>We use only these array values in
+the computation, therefore the mean value given in the control panel, for
+example radius = 60, will be ignored.</p>
+<p>&nbsp;</p>
+
+
+<p><a name="Gaussian"><h4>Gaussian distribution</h4></a></p>
+<p>&nbsp;</p>
+
+<p><img src="img/pd_image005.png"></p>
+
+<p>&nbsp;</p>
+
+<p>The x<sub>mean</sub> is the mean
+of the distribution and Norm is a normalization factor which is determined
+during the numerical calculation.</p>
+
+<p>&nbsp;</p>
+
+<p>The PD (polydispersity) is </p>
+
+<p><img src="img/pd_image003.png"></p>
+
+<p>&nbsp;</p>
+
+<p><img width=518 height=275
+id="Picture 2" src="img/pd_image006.jpg" alt=gauss.gif></p>
+
+<p>&nbsp;</p>
+
+<p><a name="Lognormal"><h4>Lognormal distribution</h4></a></p>
+
+<p>&nbsp;</p>
+
+<p><img src="img/pd_image007.png"></p>
+
+<p>&nbsp;</p>
+
+<p>The &#956; = ln(x<sub>med</sub>),  x<sub>med</sub>
+is the median value of the distribution, and Norm is a normalization factor
+which will be determined during the numerical calculation. The median value is
+the value given in the size parameter in the control panel, for example,
+radius = 60.</p>
+
+<p >&nbsp;</p>
+
+<p>The PD (polydispersity) is given
+by &#963;,</p>
+
+<p><img src="img/pd_image008.png"></p>
+
+<p>&nbsp;</p>
+
+<p>For the angular distribution,</p>
+
+<p><img src="img/pd_image009.png"></p>
+
+<p>&nbsp;</p>
+
+<p>The mean value is given by x<sub>mean</sub>
+=exp(&#956;+p<sup>2</sup>/2).</p>
+
+<p>The peak value is given by x<sub>peak</sub>=exp(&#956;-p<sup>2</sup>).</span></p>
+
+<p>&nbsp;</p>
+
+<p><img width=450 height=239
+id="Picture 7" src="img/pd_image010.jpg" alt=lognormal.gif></p>
+
+<p>&nbsp;</p>
+
+<p>This distribution function
+spreads more and the peak shifts to the left as the p increases, requiring
+higher values of Nsigmas and Npts.</p>
+
+<p>&nbsp;</p>
+
+<p><a name="Schulz"><h4>Schulz distribution</h4></a></p>
+
+<p>&nbsp;</p>
+
+<p><img src="img/pd_image011.png"></p>
+
+<p>&nbsp;</p>
+
+<p>The x<sub>mean</sub> is the mean
+of the distribution and Norm is a normalization factor which is determined
+during the numerical calculation. </p>
+
+<p>The z = 1/p<sup>2</sup>  1.</p>
+<p>&nbsp;</p>
+
+<p>The PD (polydispersity) is </p>
+<p'><img src="img/pd_image012.png"></p>
+<p>Note that the higher PD (polydispersity)
+ might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and  radisus = 60 A, 
+ Npts >= 160, and Nsigmas >= 15 at least.</p>
+ <p/>
+<p><img width=438 height=232
+id="Picture 4" src="img/pd_image013.jpg" alt=schulz.gif></p>
+
+</div>
+<br>
+</body>
