1 | .. _parallelepiped: |
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2 | |
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3 | Parallelepiped |
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4 | ======================================================= |
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5 | |
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6 | Rectangular parallelepiped with uniform scattering length density. |
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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 Parallelepiped 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 | a_side Shorter side of the parallelepiped |Ang| 35 |
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16 | b_side Second side of the parallelepiped |Ang| 75 |
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17 | c_side Larger side of the parallelepiped |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 | psi Rotation angle around its own c axis against q plane degree 60 |
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21 | =========== ==================================================== ============ ============= |
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22 | |
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23 | The returned value is scaled to units of |cm^-1|. |
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24 | |
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25 | |
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26 | The form factor is normalized by the particle volume. |
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27 | |
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28 | For information about polarised and magnetic scattering, click here_. |
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29 | |
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30 | Definition |
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31 | ---------- |
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32 | |
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33 | This model provides the form factor, *P(q)*, for a rectangular parallelepiped |
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34 | (below) where the form factor is normalized by the volume of the |
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35 | parallelepiped. If you need to apply polydispersity, see also the |
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36 | RectangularPrismModel_. |
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37 | |
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38 | The calculated form factor is: |
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39 | |
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40 | .. math:: |
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41 | |
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42 | P(Q) = {\text{scale} \over V} F^2(Q) + \text{background} |
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43 | |
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44 | where the volume *V* = *A B C* and the averaging < > is applied over all |
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45 | orientations for 1D. |
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46 | |
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47 | .. image:: img/parallelepiped.jpg |
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48 | |
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49 | *Figure. Parallelepiped with the corresponding Definition of sides. |
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50 | |
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51 | The edge of the solid must satisfy the condition that** *A* < *B* < *C*. |
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52 | Then, assuming *a* = *A* / *B* < 1, *b* = *B* / *B* = 1, and |
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53 | *c* = *C* / *B* > 1, the form factor is |
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54 | |
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55 | .. math:: |
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56 | |
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57 | P(q) = \frac{\textstyle{scale}}{V}\int_0^1 \phi(\mu \sqrt{1-\sigma^2},a) |
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58 | [S(\mu c \sigma/2)]^2 d\sigma |
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59 | |
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60 | with |
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61 | |
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62 | .. math:: |
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63 | |
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64 | \phi(\mu,a) = \int_0^1 \{S[\frac{\mu}{2}\cos(\frac{\pi}{2}u)] |
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65 | S[\frac{\mu a}{2}\sin(\frac{\pi}{2}u)]\}^2 du |
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66 | |
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67 | S(x) = \frac{\sin x}{x} |
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68 | |
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69 | \mu = qB |
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70 | |
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71 | and the contrast is defined as |
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72 | |
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73 | .. math:: |
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74 | |
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75 | \Delta\rho = \rho_{\textstyle p} - \rho_{\textstyle solvent} |
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76 | |
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77 | The scattering intensity per unit volume is returned in units of |cm^-1|; |
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78 | ie, *I(q)* = |phi| *P(q)*\ . |
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79 | |
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80 | NB: The 2nd virial coefficient of the parallelpiped is calculated based on |
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81 | the averaged effective radius (= sqrt(*short_a* \* *short_b* / |pi|)) and |
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82 | length(= *long_c*) values, and used as the effective radius for |
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83 | *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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84 | |
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85 | To provide easy access to the orientation of the parallelepiped, we define |
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86 | three angles |theta|, |phi| and |bigpsi|. The definition of |theta| and |phi| |
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87 | is the same as for the cylinder model (see also figures below). |
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88 | The angle |bigpsi| is the rotational angle around the *long_c* axis against |
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89 | the *q* plane. For example, |bigpsi| = 0 when the *short_b* axis is parallel |
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90 | to the *x*-axis of the detector. |
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91 | |
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92 | |
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93 | .. _parallelepiped-orientation: |
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94 | |
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95 | .. figure:: img/orientation.jpg |
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96 | |
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97 | Definition of the angles for oriented parallelepipeds. |
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98 | |
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99 | .. figure:: img/orientation2.jpg |
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100 | |
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101 | Examples of the angles for oriented parallelepipeds against the detector plane. |
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102 | |
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103 | |
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104 | Validation |
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105 | ---------- |
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106 | |
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107 | Validation of the code was done by comparing the output of the 1D calculation |
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108 | to the angular average of the output of a 2D calculation over all possible |
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109 | angles. The Figure below shows the comparison where the solid dot refers to |
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110 | averaged 2D while the line represents the result of the 1D calculation (for |
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111 | the averaging, 76, 180, 76 points are taken for the angles of |theta|, |phi|, |
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112 | and |psi| respectively). |
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113 | |
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114 | .. _parallelepiped-compare: |
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115 | |
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116 | .. figure:: img/parallelepiped_compare.jpg |
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117 | |
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118 | *Figure. Comparison between 1D and averaged 2D.* |
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119 | |
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120 | This model reimplements the form factor calculations implemented in a c-library |
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121 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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122 | |
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123 | |
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