[58f41fe] | 1 | #barbell model |
---|
| 2 | # Note: model title and parameter table are inserted automatically |
---|
| 3 | r""" |
---|
[eb69cce] | 4 | Calculates the scattering from a barbell-shaped cylinder. Like |
---|
| 5 | :ref:`capped-cylinder`, this is a sphereocylinder with spherical end |
---|
| 6 | caps that have a radius larger than that of the cylinder, but with the center |
---|
| 7 | of the end cap radius lying outside of the cylinder. See the diagram for |
---|
| 8 | the details of the geometry and restrictions on parameter values. |
---|
[58f41fe] | 9 | |
---|
| 10 | Definition |
---|
| 11 | ---------- |
---|
| 12 | |
---|
[eb69cce] | 13 | .. figure:: img/barbell_geometry.jpg |
---|
[58f41fe] | 14 | |
---|
[eb69cce] | 15 | Barbell geometry, where $r$ is *radius*, $R$ is *bell_radius* and |
---|
| 16 | $L$ is *length*. Since the end cap radius $R \geq r$ and by definition |
---|
| 17 | for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
---|
| 18 | $h = - \sqrt{R^2 - r^2}$ |
---|
[58f41fe] | 19 | |
---|
[eb69cce] | 20 | The scattered intensity $I(q)$ is calculated as |
---|
[58f41fe] | 21 | |
---|
| 22 | .. math:: |
---|
| 23 | |
---|
[eb69cce] | 24 | I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q)\right> |
---|
[58f41fe] | 25 | |
---|
[eb69cce] | 26 | where the amplitude $A(q)$ is given as |
---|
[58f41fe] | 27 | |
---|
| 28 | .. math:: |
---|
| 29 | |
---|
[eb69cce] | 30 | A(q) =&\ \pi r^2L |
---|
| 31 | \frac{\sin\left(\tfrac12 qL\cos\theta\right)} |
---|
| 32 | {\tfrac12 qL\cos\theta} |
---|
| 33 | \frac{2 J_1(qr\sin\theta)}{qr\sin\theta} \\ |
---|
[58f41fe] | 34 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
---|
[eb69cce] | 35 | \cos\left[ q\cos\theta |
---|
[58f41fe] | 36 | \left(Rt + h + {\tfrac12} L\right)\right] |
---|
| 37 | \times (1-t^2) |
---|
[eb69cce] | 38 | \frac{J_1\left[qR\sin\theta \left(1-t^2\right)^{1/2}\right]} |
---|
| 39 | {qR\sin\theta \left(1-t^2\right)^{1/2}} |
---|
[58f41fe] | 40 | |
---|
[eb69cce] | 41 | The $\left<\ldots\right>$ brackets denote an average of the structure over |
---|
| 42 | all orientations. $\left<A^2(q)\right>$ is then the form factor, $P(q)$. |
---|
| 43 | The scale factor is equivalent to the volume fraction of cylinders, each of |
---|
| 44 | volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length |
---|
| 45 | densities of the cylinder and the surrounding solvent. |
---|
[58f41fe] | 46 | |
---|
| 47 | The volume of the barbell is |
---|
| 48 | |
---|
| 49 | .. math:: |
---|
| 50 | |
---|
| 51 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
---|
| 52 | |
---|
| 53 | |
---|
[eb69cce] | 54 | and its radius of gyration is |
---|
[58f41fe] | 55 | |
---|
| 56 | .. math:: |
---|
| 57 | |
---|
| 58 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
---|
| 59 | + R^4\left(6h+\tfrac32 L\right) |
---|
| 60 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
---|
| 61 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
---|
| 62 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
---|
| 63 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
---|
| 64 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
---|
| 65 | |
---|
[eb69cce] | 66 | .. note:: |
---|
| 67 | The requirement that $R \geq r$ is not enforced in the model! It is |
---|
| 68 | up to you to restrict this during analysis. |
---|
[58f41fe] | 69 | |
---|
[2f0c07d] | 70 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
---|
[58f41fe] | 71 | |
---|
[2f0c07d] | 72 | .. figure:: img/cylinder_angle_definition.jpg |
---|
[58f41fe] | 73 | |
---|
[eb69cce] | 74 | Definition of the angles for oriented 2D barbells. |
---|
[58f41fe] | 75 | |
---|
[2f0c07d] | 76 | .. figure:: img/cylinder_angle_projection.jpg |
---|
[58f41fe] | 77 | |
---|
[eb69cce] | 78 | Examples of the angles for oriented pp against the detector plane. |
---|
[58f41fe] | 79 | |
---|
[eb69cce] | 80 | References |
---|
| 81 | ---------- |
---|
[58f41fe] | 82 | |
---|
| 83 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
---|
| 84 | |
---|
| 85 | H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
---|
| 86 | """ |
---|
[3c56da87] | 87 | from numpy import inf |
---|
[58f41fe] | 88 | |
---|
| 89 | name = "barbell" |
---|
| 90 | title = "Cylinder with spherical end caps" |
---|
| 91 | description = """ |
---|
[dcdf29d] | 92 | Calculates the scattering from a barbell-shaped cylinder. |
---|
| 93 | That is a sphereocylinder with spherical end caps that have a radius larger |
---|
| 94 | than that of the cylinder and the center of the end cap radius lies outside |
---|
| 95 | of the cylinder. |
---|
| 96 | Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when |
---|
| 97 | rad_bar = rad_bell, it is a spherocylinder. |
---|
| 98 | It must be that rad_bar <(=) rad_bell. |
---|
[58f41fe] | 99 | """ |
---|
[a5d0d00] | 100 | category = "shape:cylinder" |
---|
[dcdf29d] | 101 | # pylint: disable=bad-whitespace, line-too-long |
---|
[5ef0633] | 102 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
[42356c8] | 103 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Barbell scattering length density"], |
---|
| 104 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
---|
[dcdf29d] | 105 | ["bell_radius", "Ang", 40, [0, inf], "volume", "Spherical bell radius"], |
---|
| 106 | ["radius", "Ang", 20, [0, inf], "volume", "Cylindrical bar radius"], |
---|
| 107 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder bar length"], |
---|
| 108 | ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"], |
---|
| 109 | ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"], |
---|
[5ef0633] | 110 | ] |
---|
[dcdf29d] | 111 | # pylint: enable=bad-whitespace, line-too-long |
---|
[58f41fe] | 112 | |
---|
[26141cb] | 113 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"] |
---|
[58f41fe] | 114 | |
---|
| 115 | # parameters for demo |
---|
[5ef0633] | 116 | demo = dict(scale=1, background=0, |
---|
[02a0920] | 117 | sld=6, sld_solvent=1, |
---|
[5ef0633] | 118 | bell_radius=40, radius=20, length=400, |
---|
| 119 | theta=60, phi=60, |
---|
| 120 | radius_pd=.2, radius_pd_n=5, |
---|
| 121 | length_pd=.2, length_pd_n=5, |
---|
| 122 | theta_pd=15, theta_pd_n=0, |
---|
| 123 | phi_pd=15, phi_pd_n=0, |
---|
| 124 | ) |
---|