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28 | <link rel="up" title="2.1.6. Sphere Functions" href="../ref/models/shape-sphere.html" /> |
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32 | <body> |
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33 | <div class="header"><h1 class="heading"><a href="../index.html"> |
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34 | <span>Home</span></a></h1> |
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35 | <h2 class="heading"><span>2.1.6.2. Sphere</span></h2> |
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36 | </div> |
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37 | <div class="topnav"> |
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38 | |
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39 | <p> |
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40 | «  <a href="sphere.html">2.1.6.1. Sphere</a> |
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41 |   ::   |
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42 | <a class="uplink" href="../index.html">Contents</a> |
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43 |   ::   |
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44 | <a href="../ref/models/shape-independent.html">2.2. Shape-Independent Functions</a>  Â» |
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45 | </p> |
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46 | |
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47 | </div> |
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48 | <div class="content"> |
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49 | |
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50 | |
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51 | <div class="section" id="sphere"> |
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52 | <span id="id1"></span><h1>2.1.6.2. Sphere<a class="headerlink" href="#sphere" title="Permalink to this headline">¶</a></h1> |
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53 | <p>Spheres with uniform scattering length density</p> |
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54 | <table border="1" class="docutils"> |
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55 | <colgroup> |
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56 | <col width="16%" /> |
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57 | <col width="48%" /> |
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58 | <col width="17%" /> |
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59 | <col width="19%" /> |
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60 | </colgroup> |
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61 | <thead valign="bottom"> |
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62 | <tr class="row-odd"><th class="head">Parameter</th> |
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63 | <th class="head">Description</th> |
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64 | <th class="head">Units</th> |
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65 | <th class="head">Default value</th> |
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66 | </tr> |
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67 | </thead> |
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68 | <tbody valign="top"> |
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69 | <tr class="row-even"><td>scale</td> |
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70 | <td>Source intensity</td> |
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71 | <td>None</td> |
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72 | <td>1</td> |
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73 | </tr> |
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74 | <tr class="row-odd"><td>background</td> |
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75 | <td>Source background</td> |
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76 | <td>cm<sup>-1</sup></td> |
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77 | <td>0</td> |
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78 | </tr> |
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79 | <tr class="row-even"><td>sld</td> |
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80 | <td>Layer scattering length density</td> |
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81 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
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82 | <td>1</td> |
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83 | </tr> |
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84 | <tr class="row-odd"><td>solvent_sld</td> |
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85 | <td>Solvent scattering length density</td> |
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86 | <td>10<sup>-6</sup>â«<sup>-2</sup></td> |
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87 | <td>6</td> |
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88 | </tr> |
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89 | <tr class="row-even"><td>radius</td> |
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90 | <td>Sphere radius</td> |
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91 | <td>â«</td> |
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92 | <td>50</td> |
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93 | </tr> |
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94 | </tbody> |
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95 | </table> |
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96 | <p>The returned value is scaled to units of cm<sup>-1</sup>.</p> |
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97 | <p>For information about polarised and magnetic scattering, click <a class="reference external" href="polar_mag_help.html">here</a>.</p> |
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98 | <div class="section" id="definition"> |
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99 | <h2>Definition<a class="headerlink" href="#definition" title="Permalink to this headline">¶</a></h2> |
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100 | <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955)</p> |
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101 | <div class="math"> |
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102 | \[I(Q) = \frac{\text{scale}}{V} \cdot \left[ \ |
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103 | 3V(\Delta\rho) \cdot \frac{\sin(QR) - QR\cos(QR))}{(QR)^3} \ |
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104 | \right]^2 + \text{background}\]</div> |
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105 | <p>where <em>scale</em> is a volume fraction, <span class="math">\(V\)</span> is the volume of the scatterer, |
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106 | <span class="math">\(R\)</span> is the radius of the sphere, <em>background</em> is the background level and |
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107 | <em>sld</em> and <em>solvent_sld</em> are the scattering length densities (SLDs) of the |
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108 | scatterer and the solvent respectively.</p> |
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109 | <p>Note that if your data is in absolute scale, the <em>scale</em> should represent |
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110 | the volume fraction (which is unitless) if you have a good fit. If not, |
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111 | it should represent the volume fraction times a factor (by which your data |
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112 | might need to be rescaled).</p> |
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113 | <p>The 2D scattering intensity is the same as above, regardless of the |
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114 | orientation of <span class="math">\(\vec q\)</span>.</p> |
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115 | <p>Our model uses the form factor calculations as defined in the IGOR |
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116 | package provided by the NIST Center for Neutron Research (Kline, 2006).</p> |
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117 | </div> |
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118 | <div class="section" id="validation"> |
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119 | <h2>Validation<a class="headerlink" href="#validation" title="Permalink to this headline">¶</a></h2> |
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120 | <p>Validation of our code was done by comparing the output of the 1D model |
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121 | to the output of the software provided by the NIST (Kline, 2006). |
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122 | Figure <a class="pageref" href="#sphere-comparison">figure 1</a> shows a comparison of the output |
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123 | of our model and the output of the NIST software.</p> |
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124 | <div class="figure" id="sphere-comparison"> |
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125 | <img alt="../_images/sphere_comparison.jpg" src="../_images/sphere_comparison.jpg" /> |
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126 | <p class="caption">Figure 1: Comparison of the DANSE scattering intensity for a sphere with the |
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127 | output of the NIST SANS analysis software. The parameters were set to: |
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128 | <em>scale</em> = 1.0, <em>radius</em> = 60 â«, <em>contrast</em> = 1e-6 â«<sup>-2</sup>, and |
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129 | <em>background</em> = 0.01 cm<sup>-1</sup>.</p> |
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130 | </div> |
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131 | </div> |
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132 | <div class="section" id="reference"> |
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133 | <h2>Reference<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2> |
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134 | <p>A Guinier and G. Fournet, <em>Small-Angle Scattering of X-Rays</em>, |
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135 | John Wiley and Sons, New York, (1955)</p> |
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136 | <p><em>2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.</em></p> |
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137 | </div> |
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138 | </div> |
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139 | |
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142 | <div class="bottomnav"> |
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143 | |
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144 | <p> |
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145 | «  <a href="sphere.html">2.1.6.1. Sphere</a> |
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146 |   ::   |
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147 | <a class="uplink" href="../index.html">Contents</a> |
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148 |   ::   |
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149 | <a href="../ref/models/shape-independent.html">2.2. Shape-Independent Functions</a>  Â» |
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150 | </p> |
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