1 | .. _barbell: |
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2 | |
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3 | Barbell |
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4 | ======================================================= |
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5 | |
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6 | Cylinder with spherical end caps |
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7 | |
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8 | =========== ================================= ============ ============= |
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9 | Parameter Description Units Default value |
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10 | =========== ================================= ============ ============= |
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11 | scale Source intensity None 1 |
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12 | background Source background |cm^-1| 0 |
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13 | sld Barbell scattering length density |1e-6Ang^-2| 4 |
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14 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 1 |
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15 | bell_radius Spherical bell radius |Ang| 40 |
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16 | radius Cylindrical bar radius |Ang| 20 |
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17 | length Cylinder bar length |Ang| 400 |
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18 | theta In plane angle degree 60 |
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19 | phi Out of plane angle degree 60 |
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20 | =========== ================================= ============ ============= |
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21 | |
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22 | The returned value is scaled to units of |cm^-1|. |
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23 | |
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24 | |
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25 | |
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26 | Calculates the scattering from a barbell-shaped cylinder (This model simply |
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27 | becomes the DumBellModel when the length of the cylinder, *L*, is set to zero). |
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28 | That is, a sphereocylinder with spherical end caps that have a radius larger |
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29 | than that of the cylinder and the center of the end cap radius lies outside |
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30 | of the cylinder. All dimensions of the BarBell are considered to be |
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31 | monodisperse. See the diagram for the details of the geometry and restrictions |
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32 | on parameter values. |
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33 | |
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34 | Definition |
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35 | ---------- |
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36 | |
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37 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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38 | |
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39 | The barbell geometry is defined as |
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40 | |
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41 | .. image:: img/barbell_geometry.jpg |
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42 | |
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43 | where *r* is the radius of the cylinder. All other parameters are as defined |
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44 | in the diagram. |
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45 | |
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46 | Since the end cap radius |
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47 | *R* >= *r* and by definition for this geometry *h* < 0, *h* is then |
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48 | defined by *r* and *R* as |
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49 | |
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50 | *h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) |
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51 | |
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52 | The scattered intensity *I(q)* is calculated as |
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53 | |
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54 | .. math:: |
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55 | |
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56 | I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right> |
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57 | |
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58 | where the amplitude *A(q)* is given as |
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59 | |
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60 | .. math:: |
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61 | |
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62 | A(Q) =&\ \pi r^2L |
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63 | {\sin\left(\tfrac12 QL\cos\theta\right) |
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64 | \over \tfrac12 QL\cos\theta} |
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65 | {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\ |
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66 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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67 | \cos\left[ Q\cos\theta |
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68 | \left(Rt + h + {\tfrac12} L\right)\right] |
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69 | \times (1-t^2) |
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70 | {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right] |
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71 | \over QR\sin\theta \left(1-t^2\right)^{1/2}} |
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72 | |
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73 | The < > brackets denote an average of the structure over all orientations. |
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74 | <*A* :sup:`2`\ *(q)*> is then the form factor, *P(q)*. The scale factor is |
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75 | equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast |
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76 | is the difference of scattering length densities of the cylinder and the |
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77 | surrounding solvent. |
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78 | |
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79 | The volume of the barbell is |
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80 | |
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81 | .. math:: |
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82 | |
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83 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
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84 | |
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85 | |
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86 | and its radius-of-gyration is |
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87 | |
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88 | .. math:: |
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89 | |
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90 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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91 | + R^4\left(6h+\tfrac32 L\right) |
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92 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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93 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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94 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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95 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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96 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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97 | |
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98 | **The requirement that** *R* >= *r* **is not enforced in the model!** It is |
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99 | up to you to restrict this during analysis. |
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100 | |
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101 | This example dataset is produced by running the Macro PlotBarbell(), |
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102 | using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 |Ang^-1|, |
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103 | *sld* = 4e-6 |Ang^-2| and the default model values. |
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104 | |
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105 | .. image:: img/barbell_1d.jpg |
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106 | |
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107 | *Figure. 1D plot using the default values (w/256 data point).* |
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108 | |
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109 | For 2D data: The 2D scattering intensity is calculated similar to the 2D |
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110 | cylinder model. For example, for |theta| = 45 deg and |phi| = 0 deg with |
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111 | default values for other parameters |
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112 | |
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113 | .. image:: img/barbell_2d.jpg |
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114 | |
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115 | *Figure. 2D plot (w/(256X265) data points).* |
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116 | |
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117 | .. image:: img/orientation.jpg |
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118 | |
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119 | Figure. Definition of the angles for oriented 2D barbells. |
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120 | |
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121 | .. image:: img/orientation2.jpg |
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122 | |
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123 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
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124 | |
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125 | REFERENCE |
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126 | --------- |
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127 | |
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128 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
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129 | |
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130 | H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
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131 | |
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132 | |
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