Changeset 62dc889 in sasmodels
 Timestamp:
 Mar 28, 2019 2:08:22 PM (11 months ago)
 Branches:
 master, core_shell_microgels, magnetic_model, ticket1257vesicleproduct, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
 Children:
 adb84f6, 5e1875c
 Parents:
 6607260
 File:

 1 edited
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sasmodels/models/onion.py
r0507e09 r62dc889 6 6 solvent. We currently provide up to 9 shells with this model. 7 7 8 NB: *radius* represents the core radius $r_0$ and 9 *thickness[k]* represents the thickness of the shell, $r_{k+1}  r_k$. 8 .. note:: 9 10 *radius* represents the core radius $r_0$ and *thickness[k]* represents 11 the thickness of the shell, $r_{k+1}  r_k$. 10 12 11 13 Definition … … 56 58 j_1(x) = \frac{\sin(x)}{x^2}  \frac{\cos(x)}{x} 57 59 58 and the volume is $V(r) = \frac{4\pi}{3}r^3$. The volume of the particle 59 is determined by the radius of the outer shell, so $V_\text{particle} = V(r_N)$. 60 61 Now lets consider the SLD of a shell defined by 60 and the volume is $V(r) = \frac{4\pi}{3}r^3$. 61 62 The volume of the particle is determined by the radius of the outer 63 shell, so $V_\text{particle} = V(r_N)$. 64 65 Now consider the SLD of a shell defined by 62 66 63 67 .. math:: … … 74 78 thickness of the $k^\text{th}$ shell in the equation above, respectively. 75 79 76 For $A > 0$, 80 .. figure:: img/onion_geometry.png 81 82 Example of an onion model profile. 83 84 85 **Exponential SLD profiles** ($A > 0$ or $A < 0$): 77 86 78 87 .. math:: … … 87 96  3CV(r_{\text{shell}1}) \frac{j_1(\beta_\text{in})}{\beta_\text{in}} 88 97 89 for 98 where 90 99 91 100 .. math:: … … 95 104 B&=\frac{\rho_\text{out}  \rho_\text{in}}{e^A1} 96 105 & C &= \frac{\rho_\text{in}e^A  \rho_\text{out}}{e^A1} \\ 106 97 107 \alpha_\text{in} &= A\frac{r_{\text{shell}1}}{\Delta t_\text{shell}} 98 108 & \alpha_\text{out} &= A\frac{r_\text{shell}}{\Delta t_\text{shell}} \\ 109 99 110 \beta_\text{in} &= qr_{\text{shell}1} 100 111 & \beta_\text{out} &= qr_\text{shell} \\ 101 112 \end{align*} 102 113 103 where $h$ is 114 and 104 115 105 116 .. math:: 106 117 107 h(x,y) = \frac{x \sin(y)  y\cos(y)}{(x^2+y^2)y}118 h(x,y) = \frac{x \sin(y)  y\cos(y)}{(x^2+y^2)y} 108 119  \frac{(x^2y^2)\sin(y)  2xy\cos(y)}{(x^2+y^2)^2y} 109 120 110 121 111 For $A \sim 0$, e.g., $A = 0.0001$, this function converges to that of the 112 linear SLD profile with 113 $\rho_\text{shell}(r) \approx A(rr_{\text{shell}1})/\Delta t_\text{shell})+B$, 114 so this case is equivalent to 122 123 **Linear SLD profile** ($A \sim 0$): 124 125 For small $A$, say, $A = 0.0001$, the function converges to that of of a linear 126 SLD profile with 127 128 $\rho_\text{shell}(r) \approx A(rr_{\text{shell}1})/\Delta t_\text{shell})+B$, 129 130 which is equivalent to 115 131 116 132 .. math:: … … 140 156 \end{align*} 141 157 142 For $A = 0$, the exponential function has no dependence on the radius (so that 158 159 **Constant SLD** ($A = 0$): 160 161 When $A = 0$ the exponential function has no dependence on the radius (meaning 143 162 $\rho_\text{out}$ is ignored in this case) and becomes flat. We set the constant 144 163 to $\rho_\text{in}$ for convenience, and thus the form factor contributed by … … 153 172 \frac{j_1(qr_\text{in})}{qr_\text{in}} 154 173 155 .. figure:: img/onion_geometry.png156 157 Example of an onion model profile.158 159 174 The 2D scattering intensity is the same as $P(q)$ above, regardless of the 160 175 orientation of the $q$ vector which is defined as … … 184 199 * **Author:** 185 200 * **Last Modified by:** 186 * **Last Reviewed by:** 201 * **Last Reviewed by:** Steve King **Date:** March 28, 2019 187 202 * **Source added by :** Steve King **Date:** March 25, 2019 188 203 """ … … 284 299 285 300 description = """\ 286 Form factor of mu tishells normalized by the volume. Here each shell is301 Form factor of multishells normalized by the volume. Here each shell is 287 302 described by an exponential function; 288 303 … … 297 312 II) For the exact point of A_shell == 0, 298 313 f(r) = sld_in ,i.e., it crosses over flat function 299 Note that the 'sld_out' bec aomes NULL in this case.314 Note that the 'sld_out' becomes NULL in this case. 300 315 301 316 background:background, … … 312 327 # TODO: n is a volume parameter that is not polydisperse 313 328 329 # NOTE: Joachim Wuttke has suggested an alternative parameterisation 330 # in Ticket #1107 331 314 332 # pylint: disable=badwhitespace, linetoolong 315 333 # ["name", "units", default, [lower, upper], "type","description"], … … 318 336 ["radius_core", "Ang", 200., [0, inf], "volume", "Radius of the core"], 319 337 ["sld_solvent", "1e6/Ang^2", 6.4, [inf, inf], "sld", "Solvent scattering length density"], 320 ["n_shells", "", 1, [0, 10], "volume", "number of shells "],338 ["n_shells", "", 1, [0, 10], "volume", "number of shells (must be integer)"], 321 339 ["sld_in[n_shells]", "1e6/Ang^2", 1.7, [inf, inf], "sld", "scattering length density at the inner radius of shell k"], 322 340 ["sld_out[n_shells]", "1e6/Ang^2", 2.0, [inf, inf], "sld", "scattering length density at the outer radius of shell k"],
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