1 | .. _core-shell-cylinder: |
---|
2 | |
---|
3 | Core shell cylinder |
---|
4 | ======================================================= |
---|
5 | |
---|
6 | Right circular cylinder with a core-shell scattering length density profile. |
---|
7 | |
---|
8 | =========== ======================================== ============ ============= |
---|
9 | Parameter Description Units Default value |
---|
10 | =========== ======================================== ============ ============= |
---|
11 | scale Source intensity None 1 |
---|
12 | background Source background |cm^-1| 0 |
---|
13 | core_sld Cylinder core scattering length density |1e-6Ang^-2| 4 |
---|
14 | shell_sld Cylinder shell scattering length density |1e-6Ang^-2| 4 |
---|
15 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 1 |
---|
16 | radius Cylinder core radius |Ang| 20 |
---|
17 | thickness Cylinder shell thickness |Ang| 20 |
---|
18 | length Cylinder length |Ang| 400 |
---|
19 | theta In plane angle degree 60 |
---|
20 | phi Out of plane angle degree 60 |
---|
21 | =========== ======================================== ============ ============= |
---|
22 | |
---|
23 | The returned value is scaled to units of |cm^-1|. |
---|
24 | |
---|
25 | |
---|
26 | The form factor is normalized by the particle volume. |
---|
27 | |
---|
28 | Definition |
---|
29 | ---------- |
---|
30 | |
---|
31 | The output of the 2D scattering intensity function for oriented core-shell |
---|
32 | cylinders is given by (Kline, 2006) |
---|
33 | |
---|
34 | .. math:: |
---|
35 | |
---|
36 | P(Q,\alpha) = {\text{scale} \over V_s} F^2(Q) + \text{background} |
---|
37 | |
---|
38 | where |
---|
39 | |
---|
40 | .. math:: |
---|
41 | |
---|
42 | F(Q) = &\ (\rho_c - \rho_s) V_c |
---|
43 | {\sin \left( Q \tfrac12 L\cos\alpha \right) |
---|
44 | \over Q \tfrac12 L\cos\alpha } |
---|
45 | {2 J_1 \left( QR\sin\alpha \right) |
---|
46 | \over QR\sin\alpha } \\ |
---|
47 | &\ + (\rho_s - \rho_\text{solv}) V_s |
---|
48 | {\sin \left( Q \left(\tfrac12 L+T\right) \cos\alpha \right) |
---|
49 | \over Q \left(\tfrac12 L +T \right) \cos\alpha } |
---|
50 | { 2 J_1 \left( Q(R+T)\sin\alpha \right) |
---|
51 | \over Q(R+T)\sin\alpha } |
---|
52 | |
---|
53 | and |
---|
54 | |
---|
55 | .. math:: |
---|
56 | |
---|
57 | V_s = \pi (R + T)^2 (L + 2T) |
---|
58 | |
---|
59 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, |
---|
60 | $V_s$ is the volume of the outer shell (i.e. the total volume, including |
---|
61 | the shell), $V_c$ is the volume of the core, $L$ is the length of the core, |
---|
62 | $R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$ |
---|
63 | is the scattering length density of the core, $\rho_s$ is the scattering |
---|
64 | length density of the shell, $\rho_\text{solv}$ is the scattering length |
---|
65 | density of the solvent, and *background* is the background level. The outer |
---|
66 | radius of the shell is given by $R+T$ and the total length of the outer |
---|
67 | shell is given by $L+2T$. $J1$ is the first order Bessel function. |
---|
68 | |
---|
69 | .. _core-shell-cylinder-geometry: |
---|
70 | |
---|
71 | .. figure:: img/core_shell_cylinder_geometry.jpg |
---|
72 | |
---|
73 | Core shell cylinder schematic. |
---|
74 | |
---|
75 | To provide easy access to the orientation of the core-shell cylinder, we |
---|
76 | define the axis of the cylinder using two angles $\theta$ and $\phi$. As |
---|
77 | for the case of the cylinder, those angles are defined in |
---|
78 | :num:`figure #cylinder-orientation`. |
---|
79 | |
---|
80 | NB: The 2nd virial coefficient of the cylinder is calculated based on |
---|
81 | the radius and 2 length values, and used as the effective radius for |
---|
82 | $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
---|
83 | |
---|
84 | The $\theta$ and $\phi$ parameters are not used for the 1D output. Our |
---|
85 | implementation of the scattering kernel and the 1D scattering intensity |
---|
86 | use the c-library from NIST. |
---|
87 | |
---|
88 | Validation |
---|
89 | ---------- |
---|
90 | |
---|
91 | Validation of our code was done by comparing the output of the 1D model to |
---|
92 | the output of the software provided by the NIST (Kline, 2006). |
---|
93 | :num:`Figure #core-shell-cylinder-1d` shows a comparison |
---|
94 | of the 1D output of our model and the output of the NIST software. |
---|
95 | |
---|
96 | .. _core-shell-cylinder-1d: |
---|
97 | |
---|
98 | .. figure:: img/core_shell_cylinder_1d.jpg |
---|
99 | |
---|
100 | Comparison of the SasView scattering intensity for a core-shell cylinder |
---|
101 | with the output of the NIST SANS analysis software. The parameters were |
---|
102 | set to: *scale* = 1.0 |Ang|, *radius* = 20 |Ang|, *thickness* = 10 |Ang|, |
---|
103 | *length* =400 |Ang|, *core_sld* =1e-6 |Ang^-2|, *shell_sld* = 4e-6 |Ang^-2|, |
---|
104 | *solvent_sld* = 1e-6 |Ang^-2|, and *background* = 0.01 |cm^-1|. |
---|
105 | |
---|
106 | Averaging over a distribution of orientation is done by evaluating the |
---|
107 | equation above. Since we have no other software to compare the |
---|
108 | implementation of the intensity for fully oriented cylinders, we can |
---|
109 | compare the result of averaging our 2D output using a uniform |
---|
110 | distribution $p(\theta,\phi) = 1.0$. |
---|
111 | :num:`Figure #core-shell-cylinder-2d` shows the result |
---|
112 | of such a cross-check. |
---|
113 | |
---|
114 | .. _core-shell-cylinder-2d: |
---|
115 | |
---|
116 | .. figure:: img/core_shell_cylinder_2d.jpg |
---|
117 | |
---|
118 | Comparison of the intensity for uniformly distributed core-shell |
---|
119 | cylinders calculated from our 2D model and the intensity from the |
---|
120 | NIST SANS analysis software. The parameters used were: *scale* = 1.0, |
---|
121 | *radius* = 20 |Ang|, *thickness* = 10 |Ang|, *length* = 400 |Ang|, |
---|
122 | *core_sld* = 1e-6 |Ang^-2|, *shell_sld* = 4e-6 |Ang^-2|, |
---|
123 | *solvent_sld* = 1e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
---|
124 | |
---|
125 | 2013/11/26 - Description reviewed by Heenan, R. |
---|
126 | |
---|