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| 5 | </head> |
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| 7 | <body lang=EN-US> |
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| 8 | |
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| 9 | <div class=WordSection1> |
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| 10 | |
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| 11 | <p class=MsoNormal><h3><span style='font-family:"Times New Roman","serif"'>Polydisperisty |
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| 12 | and Angular Distributions</span></h3></p> |
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| 13 | |
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| 14 | <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Calculates |
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| 15 | the form factor for a polydisperse and/or angular population of particles with |
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| 16 | uniform scattering length density. The resultant form factor is normalized by |
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| 17 | the average particle volume such that P(q) = scale*<F*F>/Vol + bkg, where |
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| 18 | F is the scattering amplitude and the < > denote an average over the size |
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| 19 | distribution. Users should use PD (polydispersity: this definition is different from the typical definition in polymer science) |
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| 20 | for a size distribution and Sigma for an |
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| 21 | angular distribution (see below).</span></p> |
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| 22 | <p> Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for |
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| 23 | more than one paramters or increasing Npts values might need extensive patience to complete the computation. Also |
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| 24 | note that even though it is time consuming, it is safer to have larger values of Npts and Nsigmas.</p> |
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| 25 | |
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| 26 | <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span |
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| 27 | style='font-family:"Times New Roman","serif"'>The following five distribution |
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| 28 | functions are provided;</span></p> |
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| 29 | <ul> |
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| 30 | <li><a href="#Rectangular">Rectangular distribution</a></li> |
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| 31 | <li><a href="#Array">Array distribution</a></li> |
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| 32 | <li><a href="#Gaussian">Gaussian distribution</a></li> |
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| 33 | <li><a href="#Lognormal">Lognormal distribution</a></li> |
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| 34 | <li><a href="#Schulz">Schulz distribution</a></li> |
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| 35 | </ul> |
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| 36 | <p> </p> |
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| 37 | <p><a name="Rectangular"><h4>Rectangular distribution</a></h4></p> |
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| 38 | <p><img src="./img/pd_image001.png"></p> |
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| 39 | <p> </p> |
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| 40 | <p>The x<sub>mean</sub> is the mean |
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| 41 | of the distribution, w is the half-width, and Norm is a normalization factor |
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| 42 | which is determined during the numerical calculation. Note that the Sigma and |
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| 43 | the half width <i>w</i> are different.</p> |
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| 44 | <p>The standard deviation is </p> |
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| 45 | <p><img src="./img/pd_image002.png"></p> |
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| 46 | <p> </p> |
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| 47 | <p>The PD (polydispersity) is </p> |
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| 48 | <p><img src="./img/pd_image003.png"></p> |
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| 49 | <p> </p> |
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| 50 | <p><img width=511 height=270 |
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| 51 | id="Picture 1" src="./img/pd_image004.jpg" alt=flat.gif></p> |
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| 52 | <p> </p> |
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| 53 | <p> </p> |
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| 54 | <p><a name="Array"><h4>Array distribution</h4></a></p> |
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| 55 | |
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| 56 | <p>This distribution is to be given |
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| 57 | by users as a txt file where the array should be defined by two columns in the |
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| 58 | order of x and f(x) values. The f(x) will be normalized by SansView during the |
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| 59 | computation.</p> |
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| 60 | |
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| 61 | <p>Example of an array in the file;</p> |
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| 62 | |
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| 63 | <p>30 0.1</p> |
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| 64 | |
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| 65 | <p>32 0.3</p> |
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| 66 | |
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| 67 | <p>35 0.4</p> |
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| 68 | |
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| 69 | <p>36 0.5</p> |
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| 70 | |
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| 71 | <p>37 0.6</p> |
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| 72 | |
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| 73 | <p>39 0.7</p> |
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| 74 | |
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| 75 | <p>41 0.9</p> |
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| 76 | |
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| 77 | <p'> </p> |
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| 78 | |
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| 79 | <p>We use only these array values in |
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| 80 | the computation, therefore the mean value given in the control panel, for |
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| 81 | example radius = 60, will be ignored.</p> |
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| 82 | <p> </p> |
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| 83 | |
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| 84 | |
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| 85 | <p><a name="Gaussian"><h4>Gaussian distribution</h4></a></p> |
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| 86 | <p> </p> |
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| 87 | |
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| 88 | <p><img src="./img/pd_image005.png"></p> |
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| 89 | |
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| 90 | <p> </p> |
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| 91 | |
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| 92 | <p>The x<sub>mean</sub> is the mean |
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| 93 | of the distribution and Norm is a normalization factor which is determined |
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| 94 | during the numerical calculation.</p> |
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| 95 | |
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| 96 | <p> </p> |
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| 97 | |
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| 98 | <p>The PD (polydispersity) is </p> |
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| 99 | |
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| 100 | <p><img src="./img/pd_image003.png"></p> |
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| 101 | |
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| 102 | <p> </p> |
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| 103 | |
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| 104 | <p><img width=518 height=275 |
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| 105 | id="Picture 2" src="./img/pd_image006.jpg" alt=gauss.gif></p> |
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| 106 | |
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| 107 | <p> </p> |
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| 108 | |
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| 109 | <p><a name="Lognormal"><h4>Lognormal distribution</h4></a></p> |
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| 110 | |
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| 111 | <p> </p> |
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| 112 | |
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| 113 | <p><img src="./img/pd_image007.png"></p> |
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| 114 | |
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| 115 | <p> </p> |
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| 116 | |
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| 117 | <p>The μ = ln(x<sub>med</sub>), x<sub>med</sub> |
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| 118 | is the median value of the distribution, and Norm is a normalization factor |
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| 119 | which will be determined during the numerical calculation. The median value is |
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| 120 | the value given in the size parameter in the control panel, for example, |
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| 121 | radius = 60.</p> |
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| 122 | |
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| 123 | <p > </p> |
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| 124 | |
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| 125 | <p>The PD (polydispersity) is given |
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| 126 | by σ,</p> |
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| 127 | |
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| 128 | <p><img src="./img/pd_image008.png"></p> |
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| 129 | |
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| 130 | <p> </p> |
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| 131 | |
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| 132 | <p>For the angular distribution,</p> |
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| 133 | |
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| 134 | <p><img src="./img/pd_image009.png"></p> |
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| 135 | |
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| 136 | <p> </p> |
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| 137 | |
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| 138 | <p>The mean value is given by x<sub>mean</sub> |
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| 139 | =exp(μ+p<sup>2</sup>/2).</p> |
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| 140 | |
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| 141 | <p>The peak value is given by x<sub>peak</sub>=exp(μ-p<sup>2</sup>).</span></p> |
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| 142 | |
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| 143 | <p> </p> |
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| 144 | |
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| 145 | <p><img width=450 height=239 |
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| 146 | id="Picture 7" src="./img/pd_image010.jpg" alt=lognormal.gif></p> |
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| 147 | |
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| 148 | <p> </p> |
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| 149 | |
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| 150 | <p>This distribution function |
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| 151 | spreads more and the peak shifts to the left as the p increases, requiring |
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| 152 | higher values of Nsigmas and Npts.</p> |
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| 153 | |
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| 154 | <p> </p> |
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| 155 | |
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| 156 | <p><a name="Schulz"><h4>Schulz distribution</h4></a></p> |
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| 157 | |
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| 158 | <p> </p> |
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| 159 | |
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| 160 | <p><img src="./img/pd_image011.png"></p> |
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| 161 | |
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| 162 | <p> </p> |
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| 163 | |
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| 164 | <p>The x<sub>mean</sub> is the mean |
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| 165 | of the distribution and Norm is a normalization factor which is determined |
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| 166 | during the numerical calculation. </p> |
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| 167 | |
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| 168 | <p>The z = 1/p<sup>2</sup> 1.</p> |
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| 169 | <p> </p> |
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| 170 | |
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| 171 | <p>The PD (polydispersity) is </p> |
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| 172 | <p'><img src="./img/pd_image012.png"></p> |
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| 173 | <p>Note that the higher PD (polydispersity) |
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| 174 | might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and radisus = 60 A, |
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| 175 | Npts >= 160, and Nsigmas >= 15 at least.</p> |
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| 176 | <p/> |
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| 177 | <p><img width=438 height=232 |
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| 178 | id="Picture 4" src="./img/pd_image013.jpg" alt=schulz.gif></p> |
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| 179 | |
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| 180 | </div> |
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| 181 | <br> |
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| 182 | </body> |
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