[c5b7d07] | 1 | # parallelepiped model |
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| 2 | # Note: model title and parameter table are inserted automatically |
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| 3 | r""" |
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| 4 | The form factor is normalized by the particle volume. |
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| 5 | |
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| 6 | For information about polarised and magnetic scattering, click here_. |
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| 7 | |
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| 8 | Definition |
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| 9 | ---------- |
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| 10 | |
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| 11 | This model provides the form factor, *P(q)*, for a rectangular parallelepiped |
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| 12 | (below) where the form factor is normalized by the volume of the |
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| 13 | parallelepiped. If you need to apply polydispersity, see also the |
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| 14 | RectangularPrismModel_. |
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| 15 | |
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| 16 | The calculated form factor is: |
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| 17 | |
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| 18 | .. math:: |
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| 19 | |
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| 20 | P(Q) = {\text{scale} \over V} F^2(Q) + \text{background} |
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| 21 | |
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| 22 | where the volume *V* = *A B C* and the averaging < > is applied over all |
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| 23 | orientations for 1D. |
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| 24 | |
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| 25 | .. image:: img/parallelepiped.jpg |
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| 26 | |
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| 27 | *Figure. Parallelepiped with the corresponding Definition of sides. |
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| 28 | |
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| 29 | The edge of the solid must satisfy the condition that** *A* < *B* < *C*. |
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| 30 | Then, assuming *a* = *A* / *B* < 1, *b* = *B* / *B* = 1, and |
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| 31 | *c* = *C* / *B* > 1, the form factor is |
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| 32 | |
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| 33 | .. math:: |
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| 34 | |
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| 35 | P(q) = \frac{\textstyle{scale}}{V}\int_0^1 \phi(\mu \sqrt{1-\sigma^2},a) |
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| 36 | [S(\mu c \sigma/2)]^2 d\sigma |
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| 37 | |
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| 38 | with |
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| 39 | |
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| 40 | .. math:: |
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| 41 | |
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| 42 | \phi(\mu,a) = \int_0^1 \{S[\frac{\mu}{2}\cos(\frac{\pi}{2}u)] |
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| 43 | S[\frac{\mu a}{2}\sin(\frac{\pi}{2}u)]\}^2 du |
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| 44 | |
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| 45 | S(x) = \frac{\sin x}{x} |
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| 46 | |
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| 47 | \mu = qB |
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| 48 | |
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| 49 | and the contrast is defined as |
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| 50 | |
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| 51 | .. math:: |
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| 52 | |
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| 53 | \Delta\rho = \rho_{\textstyle p} - \rho_{\textstyle solvent} |
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| 54 | |
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| 55 | The scattering intensity per unit volume is returned in units of |cm^-1|; |
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| 56 | ie, *I(q)* = |phi| *P(q)*\ . |
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| 57 | |
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| 58 | NB: The 2nd virial coefficient of the parallelpiped is calculated based on |
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| 59 | the averaged effective radius (= sqrt(*short_a* \* *short_b* / |pi|)) and |
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| 60 | length(= *long_c*) values, and used as the effective radius for |
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| 61 | *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
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| 62 | |
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| 63 | To provide easy access to the orientation of the parallelepiped, we define |
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| 64 | three angles |theta|, |phi| and |bigpsi|. The definition of |theta| and |phi| |
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| 65 | is the same as for the cylinder model (see also figures below). |
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| 66 | The angle |bigpsi| is the rotational angle around the *long_c* axis against |
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| 67 | the *q* plane. For example, |bigpsi| = 0 when the *short_b* axis is parallel |
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| 68 | to the *x*-axis of the detector. |
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| 69 | |
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| 70 | |
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| 71 | .. _parallelepiped-orientation: |
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| 72 | |
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| 73 | .. figure:: img/orientation.jpg |
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| 74 | |
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| 75 | Definition of the angles for oriented parallelepipeds. |
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| 76 | |
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| 77 | .. figure:: img/orientation2.jpg |
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| 78 | |
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| 79 | Examples of the angles for oriented parallelepipeds against the detector plane. |
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| 80 | |
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| 81 | |
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| 82 | Validation |
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| 83 | ---------- |
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| 84 | |
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| 85 | Validation of the code was done by comparing the output of the 1D calculation |
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| 86 | to the angular average of the output of a 2D calculation over all possible |
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| 87 | angles. The Figure below shows the comparison where the solid dot refers to |
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| 88 | averaged 2D while the line represents the result of the 1D calculation (for |
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| 89 | the averaging, 76, 180, 76 points are taken for the angles of |theta|, |phi|, |
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| 90 | and |psi| respectively). |
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| 91 | |
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| 92 | .. _parallelepiped-compare: |
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| 93 | |
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| 94 | .. figure:: img/parallelepiped_compare.jpg |
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| 95 | |
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| 96 | *Figure. Comparison between 1D and averaged 2D.* |
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| 97 | |
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| 98 | This model reimplements the form factor calculations implemented in a c-library |
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| 99 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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| 100 | |
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| 101 | """ |
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| 102 | |
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| 103 | from numpy import pi, inf |
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| 104 | |
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| 105 | name = "parallelepiped" |
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| 106 | title = "Rectangular parallelepiped with uniform scattering length density." |
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| 107 | description = """ |
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| 108 | P(q)= scale/V*integral from 0 to 1 of ... |
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| 109 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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| 110 | |
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| 111 | phi(mu,a) = integral from 0 to 1 of .. |
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| 112 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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| 113 | S(x) = sin(x)/x |
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| 114 | mu = q*B |
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| 115 | """ |
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| 116 | |
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| 117 | parameters = [ |
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| 118 | # [ "name", "units", default, [lower, upper], "type", |
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| 119 | # "description" ], |
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| 120 | [ "sld", "6e-6/Ang^2", 4, [-inf,inf], "", |
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| 121 | "Parallelepiped scattering length density" ], |
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| 122 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "", |
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| 123 | "Solvent scattering length density" ], |
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| 124 | [ "a_side", "Ang", 35, [0, inf], "volume", |
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| 125 | "Shorter side of the parallelepiped" ], |
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| 126 | [ "b_side", "Ang", 75, [0, inf], "volume", |
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| 127 | "Second side of the parallelepiped" ], |
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| 128 | [ "c_side", "Ang", 400, [0, inf], "volume", |
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| 129 | "Larger side of the parallelepiped" ], |
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| 130 | [ "theta", "degrees", 60, [-inf, inf], "orientation", |
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| 131 | "In plane angle" ], |
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| 132 | [ "phi", "degrees", 60, [-inf, inf], "orientation", |
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| 133 | "Out of plane angle" ], |
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| 134 | [ "psi", "degrees", 60, [-inf, inf], "orientation", |
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| 135 | "Rotation angle around its own c axis against q plane" ], |
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| 136 | ] |
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| 137 | |
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| 138 | source = [ "lib/J1.c", "lib/gauss76.c", "parallelepiped.c" ] |
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| 139 | |
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| 140 | def ER(a_side, b_side, c_side): |
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| 141 | |
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| 142 | # surface average radius (rough approximation) |
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| 143 | surf_rad = sqrt(a_side * b_side / pi) |
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| 144 | |
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| 145 | # DiamCyl recoded here (to check and possibly put in a library?) |
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| 146 | a = surf_rad |
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| 147 | b = 0.5 * c_side |
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| 148 | t1 = a * a * b |
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| 149 | t2 = 1.0 + (b/a)*(1.0+a/b/2.0)*(1.0+pi*a/b/2.0) |
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| 150 | ddd = 3.0 * t1 * t2 |
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| 151 | |
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| 152 | return 0.5 * (ddd)**(1./3.) |
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| 153 | |
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| 154 | # parameters for demo |
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| 155 | demo = dict( |
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| 156 | scale=1, background=0, |
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| 157 | sld=6.3e-6, solvent_sld=1.0e-6, |
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| 158 | a_side=35, b_side=75, c_side=400, |
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| 159 | theta=45, phi=30, psi=15, |
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| 160 | a_side_pd=0.1, a_side_pd_n=10, |
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| 161 | b_side_pd=0.1, b_side_pd_n=1, |
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| 162 | c_side_pd=0.1, c_side_pd_n=10, |
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| 163 | theta_pd=10, theta_pd_n=1, |
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| 164 | phi_pd=10, phi_pd_n=1, |
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| 165 | psi_pd=10, psi_pd_n=10, |
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| 166 | ) |
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| 167 | |
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| 168 | # For testing against the old sasview models, include the converted parameter |
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| 169 | # names and the target sasview model name. |
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| 170 | oldname='ParallelepipedModel' |
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| 171 | oldpars=dict(theta='parallel_theta', phi='parallel_phi', psi='parallel_psi', |
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| 172 | a_side='short_a', b_side='short_b', c_side='long_c', |
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| 173 | sld='sldPipe', solvent_sld='sldSolv') |
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| 174 | |
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