Changeset 1c03e14 in sasview
- Timestamp:
- Apr 5, 2014 7:16:12 AM (10 years ago)
- Branches:
- master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
- Children:
- b50b057
- Parents:
- beda555
- Location:
- src/sans/models/media
- Files:
-
- 1 edited
- 1 moved
-
model_functions.rst (modified) (45 diffs)
-
model_functions_v0.rest (moved) (moved from src/sans/models/media/model_functions_v0.rst)
Legend:
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src/sans/models/media/model_functions.rst
rc5442fb r1c03e14 1 .. model_functions.rst1 .. model_functions.rst 2 2 3 3 .. This is a port of the original SasView model_functions.html to ReSTructured text … … 162 162 -------------- 163 163 164 - CylinderModel (including magnetic 2D version)165 - HollowCylinderModel 164 - CylinderModel_ (including magnetic 2D version) 165 - HollowCylinderModel_ 166 166 - CappedCylinderModel 167 167 - CoreShellCylinderModel … … 244 244 ------------------------------ 245 245 246 - HardSphereStructure 247 - SquareWellStructure 248 - HayterMSAStructure 249 - StickyHSStructure 246 - HardSphereStructure_ 247 - SquareWellStructure_ 248 - HayterMSAStructure_ 249 - StickyHSStructure_ 250 250 251 251 .. _Customised: … … 267 267 268 268 269 .. _R EFERENCE269 .. _References: 270 270 271 271 3. References … … 914 914 915 915 .. image:: img/image008.PNG 916 917 For P*S: toward S(Q) when P(Q)*S(Q) is applied.918 916 919 917 NB: The outer most radius (= *radius* + *thickness*) is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* … … 1188 1186 **2.1.14. CylinderModel** 1189 1187 1190 This model provides the form factor for a right circular cylinder with 1191 uniform scattering length density. The form factor is normalized by 1192 the particle volume. 1188 This model provides the form factor for a right circular cylinder with uniform scattering length density. The form 1189 factor is normalized by the particle volume. 1193 1190 1194 1191 For information about polarised and magnetic scattering, click here_. 1195 1192 1196 *1.1. Definition* 1197 1198 The output of the 2D scattering intensity function for oriented 1199 cylinders is given by (Guinier, 1955) 1200 1201 1202 1203 1204 1205 where is the angle between the axis of the cylinder and the q-vector, 1206 V is the volume of the cylinder, L is the length of the cylinder, r is 1207 the radius of the cylinder, and * (contrast) is the scattering length 1208 density difference between the scatterer and the solvent. J1 is the 1209 first order Bessel function. 1210 1211 To provide easy access to the orientation of the cylinder, we define 1212 the axis of the cylinder using two angles theta and phi. Those angles 1213 are defined on Figure 2. 1214 1215 1216 1217 Figure 2. Definition of the angles for oriented cylinders. 1218 1219 1220 1221 Figure. Examples of the angles for oriented pp against the detector 1222 plane. 1223 1224 For P*S: The 2nd virial coefficient of the cylinder is calculate based 1225 on the radius and length values, and used as the effective radius 1226 toward S(Q) when P(Q)*S(Q) is applied. 1227 1228 The returned value is scaled to units of |cm^-1| and the parameters of 1229 the cylinder model are the following: 1230 1231 Parameter name 1232 1233 Units 1234 1235 Default value 1236 1237 scale 1238 1239 None 1240 1241 1.0 1242 1243 radius 1244 1245 1246 1247 20.0 1248 1249 length 1250 1251 1252 1253 400.0 1254 1255 contrast 1256 1257 -2 1258 1259 3.0e-6 1260 1261 background 1262 1263 |cm^-1| 1264 1265 0.0 1266 1267 cyl_theta 1268 1269 degree 1270 1271 60 1272 1273 cyl_phi 1274 1275 degree 1276 1277 60 1278 1279 The output of the 1D scattering intensity function for randomly 1280 oriented cylinders is then given by: 1281 1282 1283 1284 The *cyl_theta* and *cyl_phi* parameter are not used for the 1D 1285 output. Our implementation of the scattering kernel and the 1D 1286 scattering intensity use the c-library from NIST. 1287 1288 *2.1. Validation of the cylinder model* 1289 1290 Validation of our code was done by comparing the output of the 1D 1291 model to the output of the software provided by the NIST (Kline, 1292 2006). Figure 3 shows a comparison of the 1D output of our model and 1293 the output of the NIST software. 1294 1295 In general, averaging over a distribution of orientations is done by 1296 evaluating the following: 1297 1298 1299 1300 where *p(,* *)* is the probability distribution for the orientation 1301 and *P0(q,* *)* is the scattering intensity for the fully oriented 1302 system. Since we have no other software to compare the implementation 1303 of the intensity for fully oriented cylinders, we can compare the 1304 result of averaging our 2D output using a uniform distribution *p(,* 1305 *)* = 1.0. Figure 4 shows the result of such a cross-check. 1306 1307 1308 1309 1310 1311 Figure 3: Comparison of the SasView scattering intensity for a cylinder 1312 with the output of the NIST SANS analysis software. The parameters 1313 were set to: Scale=1.0, Radius=20 , Length=400 , Contrast=3e-6 -2, and 1314 Background=0.01 cm -1. 1315 1316 1317 1318 1319 1320 1321 1322 Figure 4: Comparison of the intensity for uniformly distributed 1323 cylinders calculated from our 2D model and the intensity from the NIST 1324 SANS analysis software. The parameters used were: Scale=1.0, Radius=20 1325 , Length=400 , Contrast=3e-6 -2, and Background=0.0 cm -1. 1193 *2.1.14.1. Definition* 1194 1195 The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955) 1196 1197 .. image:: img/image059.PNG 1198 1199 where 1200 1201 .. image:: img/image060.PNG 1202 1203 and |alpha| is the angle between the axis of the cylinder and the *q*-vector, *V* is the volume of the cylinder, 1204 *L* is the length of the cylinder, *r* is the radius of the cylinder, and |bigdelta|\ |rho| (contrast) is the 1205 scattering length density difference between the scatterer and the solvent. *J1* is the first order Bessel function. 1206 1207 To provide easy access to the orientation of the cylinder, we define the axis of the cylinder using two angles |theta| 1208 and |phi|. Those angles are defined in Figure 1. 1209 1210 .. image:: img/image061.JPG 1211 1212 *Figure 1. Definition of the angles for oriented cylinders.* 1213 1214 .. image:: img/image062.JPG 1215 1216 *Figure 2. Examples of the angles for oriented pp against the detector plane.* 1217 1218 NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and length values, and used as the 1219 effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 1220 1221 The returned value is scaled to units of |cm^-1| and the parameters of the CylinderModel are the following: 1222 1223 ============== ======== ============= 1224 Parameter name Units Default value 1225 ============== ======== ============= 1226 scale None 1.0 1227 radius |Ang| 20.0 1228 length |Ang| 400.0 1229 contrast |Ang^-2| 3.0e-6 1230 background |cm^-1| 0.0 1231 cyl_theta degree 60 1232 cyl_phi degree 60 1233 ============== ======== ============= 1234 1235 The output of the 1D scattering intensity function for randomly oriented cylinders is then given by 1236 1237 .. image:: img/image063.PNG 1238 1239 The *cyl_theta* and *cyl_phi* parameter are not used for the 1D output. Our implementation of the scattering kernel 1240 and the 1D scattering intensity use the c-library from NIST. 1241 1242 *2.1.14.1. Validation of the CylinderModel* 1243 1244 Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the 1245 NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. 1246 1247 .. image:: img/image065.JPG 1248 1249 Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis 1250 software. The parameters were set to: *Scale* = 1.0, *Radius* = 20 |Ang|, *Length* = 400 |Ang|, 1251 *Contrast* = 3e-6 |Ang^-2|, and *Background* = 0.01 |cm^-1|. 1252 1253 In general, averaging over a distribution of orientations is done by evaluating the following 1254 1255 .. image:: img/image064.PNG 1256 1257 where *p(*\ |theta|,\ |phi|\ *)* is the probability distribution for the orientation and |P0|\ *(q,*\ |alpha|\ *)* is 1258 the scattering intensity for the fully oriented system. Since we have no other software to compare the implementation 1259 of the intensity for fully oriented cylinders, we can compare the result of averaging our 2D output using a uniform 1260 distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. 1261 1262 .. image:: img/image066.JPG 1263 1264 Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the intensity 1265 from the NIST SANS analysis software. The parameters used were: *Scale* = 1.0, *Radius* = 20 |Ang|, *Length* = 400 |Ang|, 1266 *Contrast* = 3e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. 1326 1267 1327 1268 … … 1331 1272 **2.1.15. HollowCylinderModel** 1332 1273 1333 This model provides the form factor, P( *q*), for a monodisperse 1334 hollow right angle circular cylinder (tube) where the form factor is 1335 normalized by the volume of the tube: 1336 1337 P(q) = scale*<f^2>/Vshell+background where the averaging < > id 1338 applied only for the 1D calculation. The inside and outside of the 1339 hollow cylinder have the same SLD. 1274 This model provides the form factor, *P(q)*, for a monodisperse hollow right angle circular cylinder (tube) where the 1275 form factor is normalized by the volume of the tube 1276 1277 *P(q)* = *scale* \* *<F*\ :sup:`2`\ *>* / *V*\ :sub:`shell` + *background* 1278 1279 where the averaging < > is applied only for the 1D calculation. 1280 1281 The inside and outside of the hollow cylinder are assumed have the same SLD. 1340 1282 1341 1283 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 1342 1284 1343 1344 1345 1346 1347 where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x - *x* cos *x*)/ *x*\ :sup:`2`. 1348 1349 1350 1351 To provide easy access to the orientation of the core-shell cylinder, 1352 we define the axis of the cylinder using two angles and . Similarly to 1353 the case of the cylinder, those angles are defined on Figure 2 in 1354 Cylinder Model. 1355 1356 For P*S: The 2nd virial coefficient of the solid cylinder is calculate 1357 based on the (radius) and 2(length) values, and used as the effective 1358 radius toward S(Q) when P(Q)*S(Q) is applied. 1359 1360 In the parameters, the contrast represents SLD (shell) - SLD (solvent) 1361 and the radius = Rhell while core_radius = Rcore. 1362 1363 1364 1365 Parameter name 1366 1367 Units 1368 1369 Default value 1370 1371 scale 1372 1373 None 1374 1375 1.0 1376 1377 radius 1378 1379 1380 1381 30 1382 1383 length 1384 1385 1386 1387 400 1388 1389 core_radius 1390 1391 1392 1393 20 1394 1395 sldCyl 1396 1397 -2 1398 1399 6.3e-6 1400 1401 sldSolv 1402 1403 -2 1404 1405 5e-06 1406 1407 background 1408 1409 |cm^-1| 1410 1411 0.01 1412 1413 1285 .. image:: img/image072.PNG 1286 1287 where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x* - *x* cos *x*)/ *x*\ :sup:`2`. 1288 1289 To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two 1290 angles |theta| and |phi|\ . As for the case of the cylinder, those angles are defined in Figure 2 of the CylinderModel. 1291 1292 NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and 2 length values, and used as the 1293 effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 1294 1295 In the parameters, the contrast represents SLD :sub:`shell` - SLD :sub:`solvent` and the *radius* = *R*\ :sub:`shell` 1296 while *core_radius* = *R*\ :sub:`core`. 1297 1298 ============== ======== ============= 1299 Parameter name Units Default value 1300 ============== ======== ============= 1301 scale None 1.0 1302 radius |Ang| 30 1303 length |Ang| 400 1304 core_radius |Ang| 20 1305 sldCyl |Ang^-2| 6.3e-6 1306 sldSolv |Ang^-2| 5e-06 1307 background |cm^-1| 0.01 1308 ============== ======== ============= 1309 1310 .. image:: img/image074.JPG 1414 1311 1415 1312 *Figure. 1D plot using the default values (w/1000 data point).* 1416 1313 1417 Our model uses the form factor calculations implemented in a c-library 1418 provided by the NIST Center for Neutron Research (Kline, 2006). 1419 1314 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 1315 (Kline, 2006). 1316 1317 .. image:: img/image061.JPG 1420 1318 1421 1319 … … 1501 1399 -1 and the above default values. 1502 1400 1503 Parameter name 1504 1505 Units 1506 1507 Default value 1508 1509 scale 1510 1511 None 1512 1513 1.0 1514 1515 len_cyl 1516 1517 1518 1519 400.0 1520 1521 rad_cap 1522 1523 1524 1525 40.0 1526 1527 rad_cyl 1528 1529 1530 1531 20.0 1532 1533 sld_capcyl 1534 1535 -2 1536 1537 1.0e-006 1538 1539 sld_solv 1540 1541 -2 1542 1543 6.3e-006 1544 1545 background 1546 1547 0 1401 ============== ======== ============= 1402 Parameter name Units Default value 1403 ============== ======== ============= 1404 scale None 1.0 1405 len_cyl |Ang| 400.0 1406 rad_cap |Ang| 40.0 1407 rad_cyl |Ang| 20.0 1408 sld_capcyl |Ang^-2| 1.0e-006 1409 sld_solv |Ang^-2| 6.3e-006 1410 background |cm^-1| 0 1411 ============== ======== ============= 1412 1548 1413 1549 1414 … … 1614 1479 the core-shell cylinder model are the following: 1615 1480 1616 Parameter name 1617 1618 Units 1619 1620 Default value 1621 1622 scale 1623 1624 None 1625 1626 1.0 1627 1628 radius 1629 1630 1631 1632 20.0 1633 1634 thickness 1635 1636 1637 1638 10.0 1639 1640 length 1641 1642 1643 1644 400.0 1645 1646 core_sld 1647 1648 -2 1649 1650 1e-6 1651 1652 shell_sld 1653 1654 -2 1655 1656 4e-6 1657 1658 solvent_sld 1659 1660 -2 1661 1662 1e-6 1663 1664 background 1665 1666 |cm^-1| 1667 1668 0.0 1669 1670 axis_theta 1671 1672 degree 1673 1674 90 1675 1676 axis_phi 1677 1678 degree 1679 1680 0.0 1481 ============== ======== ============= 1482 Parameter name Units Default value 1483 ============== ======== ============= 1484 scale None 1.0 1485 radius |Ang| 20.0 1486 thickness |Ang| 10.0 1487 length |Ang| 400.0 1488 core_sld |Ang^-2| 1e-6 1489 shell_sld |Ang^-2| 4e-6 1490 solvent_sld |Ang^-2| 1e-6 1491 background |cm^-1| 0.0 1492 axis_theta degree 90 1493 axis_phi degree 0.0 1494 ============== ======== ============= 1681 1495 1682 1496 The output of the 1D scattering intensity function for randomly … … 1709 1523 parameters were set to: Scale=1.0, Radius=20 , Thickness=10 , 1710 1524 Length=400 , Core_sld=1e-6 -2, Shell_sld=4e-6 -2, Solvent_sld=1e-6 -2, 1711 and Background=0.01 cm -1.1525 and Background=0.01 |cm^-1|. 1712 1526 1713 1527 … … 1721 1535 the NIST SANS analysis software. The parameters used were: Scale=1.0, 1722 1536 Radius=20 , Thickness=10 , Length=400 , Core_sld=1e-6 -2, 1723 Shell_sld=4e-6 -2, Solvent_sld=1e-6 -2, and Background=0.0 cm -1.1537 Shell_sld=4e-6 -2, Solvent_sld=1e-6 -2, and Background=0.0 |cm^-1|. 1724 1538 1725 1539 … … 1802 1616 P(Q)*S(Q) is applied. 1803 1617 1804 Parameter name 1805 1806 Units 1807 1808 Default value 1809 1810 scale 1811 1812 None 1813 1814 1.0 1815 1816 r_minor 1817 1818 1819 1820 20.0 1821 1822 r_ratio 1823 1824 1825 1826 1.5 1827 1828 length 1829 1830 1831 1832 400.0 1833 1834 sldCyl 1835 1836 -2 1837 1838 4e-6 1839 1840 sldSolv 1841 1842 -2 1843 1844 1e-006 1845 1846 background 1847 1848 0 1618 ============== ======== ============= 1619 Parameter name Units Default value 1620 ============== ======== ============= 1621 scale None 1.0 1622 r_minor |Ang| 20.0 1623 r_ratio |Ang| 1.5 1624 length |Ang| 400.0 1625 sldCyl |Ang^-2| 4e-06 1626 sldSolv |Ang^-2| 1e-06 1627 background |cm^-1| 0 1628 ============== ======== ============= 1849 1629 1850 1630 … … 1886 1666 **2.1.19. FlexibleCylinderModel** 1887 1667 1888 This model provides the form factor, P( *q*), for a flexible cylinder1668 This model provides the form factor, *P(q)*, for a flexible cylinder 1889 1669 where the form factor is normalized by the volume of the cylinder: 1890 1670 Inter-cylinder interactions are NOT included. P(q) = … … 1904 1684 respectively. 1905 1685 1906 1907 1908 1909 1910 Parameter name 1911 1912 Units 1913 1914 Default value 1915 1916 scale 1917 1918 None 1919 1920 1.0 1921 1922 radius 1923 1924 1925 1926 20 1927 1928 length 1929 1930 1931 1932 1000 1933 1934 sldCyl 1935 1936 -2 1937 1938 1e-06 1939 1940 sldSolv 1941 1942 -2 1943 1944 6.3e-06 1945 1946 background 1947 1948 |cm^-1| 1949 1950 0.01 1951 1952 kuhn_length 1953 1954 1955 1956 100 1686 ============== ======== ============= 1687 Parameter name Units Default value 1688 ============== ======== ============= 1689 scale None 1.0 1690 radius |Ang| 20 1691 length |Ang| 1000 1692 sldCyl |Ang^-2| 1e-06 1693 sldSolv |Ang^-2| 6.3e-06 1694 background |cm^-1| 0.01 1695 kuhn_length |Ang| 100 1696 ============== ======== ============= 1957 1697 1958 1698 … … 2066 1806 -1 and the default values below. 2067 1807 2068 Parameter name 2069 2070 Units 2071 2072 Default value 2073 2074 axis_ratio 2075 2076 1.5 2077 2078 background 2079 2080 |cm^-1| 2081 2082 0.0001 2083 2084 Kuhn_length 2085 2086 2087 2088 100 2089 2090 (Contour) length 2091 2092 2093 2094 1e+3 2095 2096 radius 2097 2098 2099 2100 20.0 2101 2102 scale 2103 2104 1.0 2105 2106 sldCyl 2107 2108 -2 2109 2110 1e-6 2111 2112 sldSolv 2113 2114 -2 2115 2116 6.3e-6 1808 ============== ======== ============= 1809 Parameter name Units Default value 1810 ============== ======== ============= 1811 axis_ratio None 1.5 1812 background |cm^-1| 0.0001 1813 Kuhn_length |Ang| 100 1814 Contour length |Ang| 1e+3 1815 radius |Ang| 20.0 1816 scale None 1.0 1817 sldCyl |Ang^-2| 1e-6 1818 sldSolv |Ang^-2| 6.3e-6 1819 ============== ======== ============= 2117 1820 2118 1821 … … 2139 1842 the core-shell cylinder model are the following: 2140 1843 2141 Parameter name 2142 2143 Units 2144 2145 Default value 2146 2147 scale 2148 2149 None 2150 2151 1.0 2152 2153 radius 2154 2155 2156 2157 20.0 2158 2159 rim_thick 2160 2161 2162 2163 10.0 2164 face_thick 10.0 2165 length 2166 2167 2168 2169 400.0 2170 2171 core_sld 2172 2173 -2 2174 2175 1e-6 2176 2177 rim_sld 2178 2179 -2 2180 2181 4e-6 2182 face_sld -2 4e-6 2183 solvent_sld 2184 2185 -2 2186 2187 1e-6 2188 2189 background 2190 2191 |cm^-1| 2192 2193 0.0 2194 2195 axis_theta 2196 2197 degree 2198 2199 90 2200 2201 axis_phi 2202 2203 degree 2204 2205 0.0 1844 ============== ======== ============= 1845 Parameter name Units Default value 1846 ============== ======== ============= 1847 scale None 1.0 1848 radius |Ang| 20.0 1849 rim_thick |Ang| 10.0 1850 face_thick |Ang| 10.0 1851 length |Ang| 400.0 1852 core_sld |Ang^-2| 1e-6 1853 rim_sld |Ang^-2| 4e-6 1854 face_sld |Ang^-2| 4e-6 1855 solvent_sld |Ang^-2| 1e-6 1856 background |cm^-1| 0.0 1857 axis_theta degree 90 1858 axis_phi degree 0.0 1859 ============== ======== ============= 2206 1860 2207 1861 The output of the 1D scattering intensity function for randomly … … 2302 1956 default values. 2303 1957 2304 Parameter name 2305 2306 Units 2307 2308 Default value 2309 2310 scale 2311 2312 None 2313 2314 1.0 2315 2316 len_bar 2317 2318 2319 2320 400.0 2321 2322 rad_bar 2323 2324 2325 2326 20.0 2327 2328 rad_bell 2329 2330 2331 2332 40.0 2333 2334 sld_barbell 2335 2336 -2 2337 2338 1.0e-006 2339 2340 sld_solv 2341 2342 -2 2343 2344 6.3e-006 2345 2346 background 2347 2348 0 1958 ============== ======== ============= 1959 Parameter name Units Default value 1960 ============== ======== ============= 1961 scale None 1.0 1962 len_bar |Ang| 400.0 1963 rad_bar |Ang| 20.0 1964 rad_bell |Ang| 40.0 1965 sld_barbell |Ang^-2| 1.0e-006 1966 sld_solv |Ang^-2| 6.3e-006 1967 background |cm^-1| 0 1968 ============== ======== ============= 2349 1969 2350 1970 … … 2375 1995 **2.1.23. StackedDisksModel** 2376 1996 2377 This model provides the form factor, P( *q*), for stacked discs1997 This model provides the form factor, *P(q)*, for stacked discs 2378 1998 (tactoids) with a core/layer structure where the form factor is 2379 1999 normalized by the volume of the cylinder. Assuming the next neighbor … … 2433 2053 when P(Q)*S(Q) is applied. 2434 2054 2435 Parameter name 2436 2437 Units 2438 2439 Default value 2440 2441 background 2442 2443 |cm^-1| 2444 2445 0.001 2446 2447 core_sld 2448 2449 -2 2450 2451 4e-006 2452 2453 core_thick 2454 2455 2456 2457 10 2458 2459 layer_sld 2460 2461 -2 2462 2463 0 2464 2465 layer_thick 2466 2467 2468 2469 15 2470 2471 n_stacking 2472 2473 1 2474 2475 radius 2476 2477 2478 2479 3e+003 2480 2481 scale 2482 2483 0.01 2484 2485 sigma_d 2486 2487 0 2488 2489 solvent_sld 2490 2491 -2 2492 2493 5e-006 2055 ============== ======== ============= 2056 Parameter name Units Default value 2057 ============== ======== ============= 2058 background |cm^-1| 0.001 2059 core_sld |Ang^-2| 4e-006 2060 core_thick |Ang| 10 2061 layer_sld |Ang^-2| 0 2062 layer_thick |Ang| 15 2063 n_stacking None 1 2064 radius |Ang| 3e+03 2065 scale None 0.01 2066 sigma_d |Ang| 0 2067 solvent_sld |Ang^-2| 5e-06 2068 ============== ======== ============= 2494 2069 2495 2070 … … 2526 2101 **2.1.24. PringleModel** 2527 2102 2528 This model provides the form factor, P( *q*), for a 'pringle' or2103 This model provides the form factor, *P(q)*, for a 'pringle' or 2529 2104 'saddle-shaped' object (a hyperbolic paraboloid). 2530 2105 … … 2546 2121 with the equivalent cylinder are shown below. 2547 2122 2548 Parameter name 2549 2550 Units 2551 2552 Default value 2553 2554 background 2555 2556 |cm^-1| 2557 2558 0.0 2559 2560 alpha 2561 2562 2563 2564 0.001 2565 2566 beta 2567 2568 2569 2570 0.02 2571 2572 radius 2573 2574 60 2575 2576 scale 2577 2578 2579 2580 1 2581 2582 sld_pringle 2583 2584 -2 2585 2586 1e-006 2587 2588 sld_solvent 2589 2590 -2 2591 2592 6.3e-006 2593 2594 thickness 2595 2596 2597 2598 10 2123 ============== ======== ============= 2124 Parameter name Units Default value 2125 ============== ======== ============= 2126 background |cm^-1| 0.0 2127 alpha None 0.001 2128 beta None 0.02 2129 radius |Ang| 60 2130 scale None 1 2131 sld_pringle |Ang^-2| 1e-06 2132 sld_solvent |Ang^-2| 6.3e-06 2133 thickness |Ang| 10 2134 ============== ======== ============= 2599 2135 2600 2136 … … 2645 2181 the ellipsoid model are the following: 2646 2182 2647 Parameter name 2648 2649 Units 2650 2651 Default value 2652 2653 scale 2654 2655 None 2656 2657 1.0 2658 2659 radius_a (polar) 2660 2661 2662 2663 20.0 2664 2665 radius_b (equatorial) 2666 2667 2668 2669 400.0 2670 2671 sldEll 2672 2673 -2 2674 2675 4.0e-6 2676 2677 sldSolv 2678 2679 -2 2680 2681 1.0e-6 2682 2683 background 2684 2685 |cm^-1| 2686 2687 0.0 2688 2689 axis_theta 2690 2691 degree 2692 2693 90 2694 2695 axis_phi 2696 2697 degree 2698 2699 0.0 2183 ================ ======== ============= 2184 Parameter name Units Default value 2185 ================ ======== ============= 2186 scale None 1.0 2187 radius_a (polar) |Ang| 20.0 2188 radius_b (equat) |Ang| 400.0 2189 sldEll |Ang^-2| 4.0e-6 2190 sldSolv |Ang^-2| 1.0e-6 2191 background |cm^-1| 0.0 2192 axis_theta degree 90 2193 axis_phi degree 0.0 2194 ================ ======== ============= 2700 2195 2701 2196 … … 2744 2239 parameters were set to: Scale=1.0, Radius_a=20 , Radius_b=400 , 2745 2240 2746 Contrast=3e-6 -2, and Background=0.01 cm -1.2241 Contrast=3e-6 -2, and Background=0.01 |cm^-1|. 2747 2242 2748 2243 … … 2762 2257 **2.1.26. CoreShellEllipsoidModel** 2763 2258 2764 This model provides the form factor, P( *q*), for a core shell2259 This model provides the form factor, *P(q)*, for a core shell 2765 2260 ellipsoid (below) where the form factor is normalized by the volume of 2766 2261 the cylinder. P(q) = scale*<f^2>/V+background where the volume V= … … 2795 2290 P(Q)*S(Q) is applied. 2796 2291 2797 2798 2799 Parameter name 2800 2801 Units 2802 2803 Default value 2804 2805 background 2806 2807 |cm^-1| 2808 2809 0.001 2810 2811 equat_core 2812 2813 2814 2815 200 2816 2817 equat_shell 2818 2819 2820 2821 250 2822 2823 sld_solvent 2824 2825 -2 2826 2827 6e-006 2828 2829 ploar_shell 2830 2831 2832 2833 30 2834 2835 ploar_core 2836 2837 2838 2839 20 2840 2841 scale 2842 2843 1 2844 2845 sld_core 2846 2847 -2 2848 2849 2e-006 2850 2851 sld_shell 2852 2853 -2 2854 2855 1e-006 2292 ============== ======== ============= 2293 Parameter name Units Default value 2294 ============== ======== ============= 2295 background |cm^-1| 0.001 2296 equat_core |Ang| 200 2297 equat_shell |Ang| 250 2298 sld_solvent |Ang^-2| 6e-06 2299 ploar_shell |Ang| 30 2300 ploar_core |Ang| 20 2301 scale None 1 2302 sld_core |Ang^-2| 2e-06 2303 sld_shell |Ang^-2| 1e-06 2304 ============== ======== ============= 2856 2305 2857 2306 … … 2880 2329 **2.1.27. TriaxialEllipsoidModel*** 2881 2330 2882 This model provides the form factor, P( *q*), for an ellipsoid (below)2331 This model provides the form factor, *P(q)*, for an ellipsoid (below) 2883 2332 where all three axes are of different lengths, i.e., Ra =< Rb =< Rc 2884 2333 (Note that users should maintains this inequality for the all … … 2915 2364 radius toward S(Q) when P(Q)*S(Q) is applied. 2916 2365 2917 2918 2919 2920 2921 Parameter name 2922 2923 Units 2924 2925 Default value 2926 2927 background 2928 2929 |cm^-1| 2930 2931 0.0 2932 2933 semi_axisA 2934 2935 2936 2937 35 2938 2939 semi_axisB 2940 2941 2942 2943 100 2944 2945 semi_axisC 2946 2947 2948 2949 400 2950 2951 scale 2952 2953 1 2954 2955 sldEll 2956 2957 -2 2958 2959 1.0e-006 2960 2961 sldSolv 2962 2963 -2 2964 2965 6.3e-006 2366 ============== ======== ============= 2367 Parameter name Units Default value 2368 ============== ======== ============= 2369 background |cm^-1| 0.0 2370 semi_axisA |Ang| 35 2371 semi_axisB |Ang| 100 2372 semi_axisC |Ang| 400 2373 scale None 1 2374 sldEll |Ang^-2| 1.0e-06 2375 sldSolv |Ang^-2| 6.3e-06 2376 ============== ======== ============= 2966 2377 2967 2378 … … 3024 2435 and bi_thick = the thickness of the bilayer. 3025 2436 3026 3027 3028 Parameter name 3029 3030 Units 3031 3032 Default value 3033 3034 background 3035 3036 |cm^-1| 3037 3038 0.0 3039 3040 sld_bi 3041 3042 -2 3043 3044 1e-006 3045 3046 bi_thick 3047 3048 3049 3050 50 3051 3052 sld_sol 3053 3054 -2 3055 3056 6e-006 3057 3058 scale 3059 3060 1 2437 ============== ======== ============= 2438 Parameter name Units Default value 2439 ============== ======== ============= 2440 background |cm^-1| 0.0 2441 sld_bi |Ang^-2| 1e-06 2442 bi_thick |Ang| 50 2443 sld_sol |Ang^-2| 6e-06 2444 scale None 1 2445 ============== ======== ============= 3061 2446 3062 2447 … … 3105 2490 the head group. 3106 2491 3107 3108 3109 Parameter name 3110 3111 Units 3112 3113 Default value 3114 3115 background 3116 3117 |cm^-1| 3118 3119 0.0 3120 3121 sld_head 3122 3123 -2 3124 3125 3e-006 3126 3127 scale 3128 3129 1 3130 3131 sld_solvent 3132 3133 -2 3134 3135 6e-006 3136 3137 h_thickness 3138 3139 3140 3141 10 3142 3143 t_length 3144 3145 3146 3147 15 3148 3149 sld_tail 3150 3151 -2 3152 3153 0 2492 ============== ======== ============= 2493 Parameter name Units Default value 2494 ============== ======== ============= 2495 background |cm^-1| 0.0 2496 sld_head |Ang^-2| 3e-06 2497 scale None 1 2498 sld_solvent |Ang^-2| 6e-06 2499 h_thickness |Ang| 10 2500 t_length |Ang| 15 2501 sld_tail |Ang^-2| 0 2502 ============== ======== ============= 3154 2503 3155 2504 … … 3211 2560 The returned value is in units of |cm^-1|, on absolute scale. 3212 2561 3213 3214 3215 Parameter name 3216 3217 Units 3218 3219 Default value 3220 3221 background 3222 3223 |cm^-1| 3224 3225 0.0 3226 3227 contrast 3228 3229 -2 3230 3231 5e-006 3232 3233 scale 3234 3235 1 3236 3237 delta 3238 3239 3240 3241 30 3242 3243 n_plates 3244 3245 20 3246 3247 spacing 3248 3249 3250 3251 400 3252 3253 caille 3254 3255 -2 3256 3257 0.1 2562 ============== ======== ============= 2563 Parameter name Units Default value 2564 ============== ======== ============= 2565 background |cm^-1| 0.0 2566 contrast |Ang^-2| 5e-06 2567 scale None 1 2568 delta |Ang| 30 2569 n_plates None 20 2570 spacing |Ang| 400 2571 caille |Ang^-2| 0.1 2572 ============== ======== ============= 3258 2573 3259 2574 … … 3322 2637 head group, and sld_solvent = SLD of the solvent. 3323 2638 3324 3325 3326 Parameter name 3327 3328 Units 3329 3330 Default value 3331 3332 background 3333 3334 |cm^-1| 3335 3336 0.001 3337 3338 sld_head 3339 3340 -2 3341 3342 2e-006 3343 3344 scale 3345 3346 1 3347 3348 sld_solvent 3349 3350 -2 3351 3352 6e-006 3353 3354 deltaH 3355 3356 3357 3358 2 3359 3360 deltaT 3361 3362 3363 3364 10 3365 3366 sld_tail 3367 3368 -2 3369 3370 0 3371 3372 n_plates 3373 3374 30 3375 3376 spacing 3377 3378 3379 3380 40 3381 3382 caille 3383 3384 -2 3385 3386 0.001 2639 ============== ======== ============= 2640 Parameter name Units Default value 2641 ============== ======== ============= 2642 background |cm^-1| 0.001 2643 sld_head |Ang^-2| 2e-06 2644 scale None 1 2645 sld_solvent |Ang^-2| 6e-06 2646 deltaH |Ang| 2 2647 deltaT |Ang| 10 2648 sld_tail |Ang^-2| 0 2649 n_plates None 30 2650 spacing |Ang| 40 2651 caille |Ang^-2| 0.001 2652 ============== ======== ============= 2653 3387 2654 3388 2655 … … 3439 2706 pd_spacing= polydispersity of spacing): 3440 2707 3441 Parameter name 3442 3443 Units 3444 3445 Default value 3446 3447 background 3448 3449 |cm^-1| 3450 3451 0 3452 3453 scale 3454 3455 1 3456 3457 Nlayers 3458 3459 20 3460 3461 pd_spacing 3462 3463 0.2 3464 3465 sld_layer 3466 3467 -2 3468 3469 1e-6 3470 3471 sld_solvent 3472 3473 -2 3474 3475 6.34e-6 3476 3477 spacing 3478 3479 3480 3481 250 3482 3483 thickness 3484 3485 3486 3487 33 2708 ============== ======== ============= 2709 Parameter name Units Default value 2710 ============== ======== ============= 2711 background |cm^-1| 0 2712 scale None 1 2713 Nlayers None 20 2714 pd_spacing None 0.2 2715 sld_layer |Ang^-2| 1e-6 2716 sld_solvent |Ang^-2| 6.34e-6 2717 spacing |Ang| 250 2718 thickness |Ang| 33 2719 ============== ======== ============= 3488 2720 3489 2721 … … 3569 2801 (Corrections to FCC and BCC lattice structure calculation) 3570 2802 3571 3572 3573 Parameter name 3574 3575 Units 3576 3577 Default value 3578 3579 background 3580 3581 |cm^-1| 3582 3583 0 3584 3585 dnn 3586 3587 3588 3589 220 3590 3591 scale 3592 3593 1 3594 3595 sldSolv 3596 3597 -2 3598 3599 6.3e-006 3600 3601 radius 3602 3603 3604 3605 40 3606 3607 sld_Sph 3608 3609 -2 3610 3611 3e-006 3612 3613 d_factor 3614 3615 0.06 2803 ============== ======== ============= 2804 Parameter name Units Default value 2805 ============== ======== ============= 2806 background |cm^-1| 0 2807 dnn |Ang| 220 2808 scale None 1 2809 sldSolv |Ang^-2| 6.3e-06 2810 radius |Ang| 40 2811 sld_Sph |Ang^-2| 3e-06 2812 d_factor None 0.06 2813 ============== ======== ============= 3616 2814 3617 2815 TEST DATASET … … 3713 2911 (Corrections to FCC and BCC lattice structure calculation) 3714 2912 3715 3716 3717 3718 3719 Parameter name 3720 3721 Units 3722 3723 Default value 3724 3725 background 3726 3727 |cm^-1| 3728 3729 0 3730 3731 dnn 3732 3733 3734 3735 220 3736 3737 scale 3738 3739 1 3740 3741 sldSolv 3742 3743 -2 3744 3745 6.3e-006 3746 3747 radius 3748 3749 3750 3751 40 3752 3753 sld_Sph 3754 3755 -2 3756 3757 3e-006 3758 3759 d_factor 3760 3761 0.06 2913 ============== ======== ============= 2914 Parameter name Units Default value 2915 ============== ======== ============= 2916 background |cm^-1| 0 2917 dnn |Ang| 220 2918 scale None 1 2919 sldSolv |Ang^-2| 6.3e-06 2920 radius |Ang| 40 2921 sld_Sph |Ang^-2| 3e-06 2922 d_factor None 0.06 2923 ============== ======== ============= 3762 2924 3763 2925 TEST DATASET … … 3847 3009 (Corrections to FCC and BCC lattice structure calculation) 3848 3010 3849 3850 3851 3852 3853 Parameter name 3854 3855 Units 3856 3857 Default value 3858 3859 background 3860 3861 |cm^-1| 3862 3863 0 3864 3865 dnn 3866 3867 3868 3869 220 3870 3871 scale 3872 3873 1 3874 3875 sldSolv 3876 3877 -2 3878 3879 6.3e-006 3880 3881 radius 3882 3883 3884 3885 40 3886 3887 sld_Sph 3888 3889 -2 3890 3891 3e-006 3892 3893 d_factor 3894 3895 0.06 3011 ============== ======== ============= 3012 Parameter name Units Default value 3013 ============== ======== ============= 3014 background |cm^-1| 0 3015 dnn |Ang| 220 3016 scale None 1 3017 sldSolv |Ang^-2| 6.3e-006 3018 radius |Ang| 40 3019 sld_Sph |Ang^-2| 3e-006 3020 d_factor None 0.06 3021 ============== ======== ============= 3896 3022 3897 3023 TEST DATASET … … 3930 3056 **2.1.36. ParallelepipedModel** 3931 3057 3932 This model provides the form factor, P( *q*), for a rectangular3058 This model provides the form factor, *P(q)*, for a rectangular 3933 3059 cylinder (below) where the form factor is normalized by the volume of 3934 3060 the cylinder. P(q) = scale*<f^2>/V+background where the volume V= ABC … … 3977 3103 plane. 3978 3104 3979 Parameter name 3980 3981 Units 3982 3983 Default value 3984 3985 background 3986 3987 |cm^-1| 3988 3989 0.0 3990 3991 contrast 3992 3993 -2 3994 3995 5e-006 3996 3997 long_c 3998 3999 4000 4001 400 4002 4003 short_a 4004 4005 -2 4006 4007 35 4008 4009 short_b 4010 4011 4012 4013 75 4014 4015 scale 4016 4017 1 3105 ============== ======== ============= 3106 Parameter name Units Default value 3107 ============== ======== ============= 3108 background |cm^-1| 0.0 3109 contrast |Ang^-2| 5e-06 3110 long_c |Ang| 400 3111 short_a |Ang^-2| 35 3112 short_b |Ang| 75 3113 scale None 1 3114 ============== ======== ============= 4018 3115 4019 3116 … … 4130 3227 = 0.7 -1 and the below default values. 4131 3228 4132 Parameter name 4133 4134 Units 4135 4136 Default value 4137 4138 background 4139 4140 |cm^-1| 4141 4142 0.06 4143 4144 sld_pcore 4145 4146 -2 4147 4148 1e-006 4149 4150 sld_rimA 4151 4152 -2 4153 4154 2e-006 4155 4156 sld_rimB 4157 4158 -2 4159 4160 4e-006 4161 4162 sld_rimC 4163 4164 -2 4165 4166 2e-006 4167 4168 sld_solv 4169 4170 -2 4171 4172 6e-006 4173 4174 rimA 4175 4176 4177 4178 10 4179 4180 rimB 4181 4182 4183 4184 10 4185 4186 rimC 4187 4188 4189 4190 10 4191 4192 longC 4193 4194 4195 4196 400 4197 4198 shortA 4199 4200 4201 4202 35 4203 4204 midB 4205 4206 4207 4208 75 4209 4210 scale 4211 4212 1 3229 ============== ======== ============= 3230 Parameter name Units Default value 3231 ============== ======== ============= 3232 background |cm^-1| 0.06 3233 sld_pcore |Ang^-2| 1e-06 3234 sld_rimA |Ang^-2| 2e-06 3235 sld_rimB |Ang^-2| 4e-06 3236 sld_rimC |Ang^-2| 2e-06 3237 sld_solv |Ang^-2| 6e-06 3238 rimA |Ang| 10 3239 rimB |Ang| 10 3240 rimC |Ang| 10 3241 longC |Ang| 400 3242 shortA |Ang| 35 3243 midB |Ang| 75 3244 scale None 1 3245 ============== ======== ============= 4213 3246 4214 3247 … … 6325 5358 The information in this section is originated from NIST SANS IgorPro package. 6326 5359 5360 .. _HardSphereStructure: 5361 6327 5362 **2.3.1. HardSphereStructure Factor** 6328 5363 … … 6332 5367 The calculation uses the Percus-Yevick closure where the interparticle potential is 6333 5368 6334 6335 6336 6337 6338 where r is the distance from the center of the sphere of a radius R. 5369 .. image:: img/image223.PNG 5370 5371 where *r* is the distance from the center of the sphere of a radius *R*. 6339 5372 6340 5373 For a 2D plot, the wave transfer is defined as 6341 5374 6342 Parameter name 6343 6344 Units 6345 6346 Default value 6347 6348 effect_radius 6349 6350 6351 6352 50.0 6353 6354 volfraction 6355 6356 0.2 6357 6358 5375 .. image:: img/image040.GIF 5376 5377 ============== ======== ============= 5378 Parameter name Units Default value 5379 ============== ======== ============= 5380 effect_radius |Ang| 50.0 5381 volfraction None 0.2 5382 ============== ======== ============= 5383 5384 .. image:: img/image224.JPG 6359 5385 6360 5386 *Figure. 1D plot using the default values (in linear scale).* … … 6364 5390 6365 5391 5392 5393 .. _SquareWellStructure: 6366 5394 6367 5395 **2.3.2. SquareWellStructure Factor** … … 6370 5398 approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive 6371 5399 interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing 6372 this calculation to be limited in applicability to well depths e < 1.5 kT and volume fractions f< 0.08.5400 this calculation to be limited in applicability to well depths |epsilon| < 1.5 kT and volume fractions |phi| < 0.08. 6373 5401 6374 5402 Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential 6375 5403 "shoulder", which may or may not be physically reasonable. 6376 5404 6377 The well width ( l) is defined as multiples of the particle diameter (2*R)5405 The well width (*l*\ ) is defined as multiples of the particle diameter (2\*\ *R*\ ) 6378 5406 6379 5407 The interaction potential is: 6380 5408 6381 6382 6383 6384 6385 where r is the distance from the center of the sphere of a radius R. 6386 6387 For 2D plot, the wave transfer is defined as . 6388 6389 Parameter name 6390 6391 Units 6392 6393 Default value 6394 6395 effect_radius 6396 6397 6398 6399 50.0 6400 6401 volfraction 6402 6403 0.04 6404 6405 welldepth 6406 6407 kT 6408 6409 1.5 6410 6411 wellwidth 6412 6413 diameters 6414 6415 1.2 6416 6417 5409 .. image:: img/image225.PNG 5410 5411 where *r* is the distance from the center of the sphere of a radius *R*. 5412 5413 For 2D plot, the wave transfer is defined as 5414 5415 .. image:: img/image040.GIF 5416 5417 ============== ========= ============= 5418 Parameter name Units Default value 5419 ============== ========= ============= 5420 effect_radius |Ang| 50.0 5421 volfraction None 0.04 5422 welldepth kT 1.5 5423 wellwidth diameters 1.2 5424 ============== ========= ============= 5425 5426 .. image:: img/image226.JPG 6418 5427 6419 5428 *Figure. 1D plot using the default values (in linear scale).* … … 6424 5433 6425 5434 5435 .. _HayterMSAStructure: 5436 6426 5437 **2.3.3. HayterMSAStructure Factor** 6427 5438 6428 This calculates the Structure factor (the Fourier transform of the pair correlation function g(r)) for a system of5439 This calculates the structure factor (the Fourier transform of the pair correlation function *g(r)*) for a system of 6429 5440 charged, spheroidal objects in a dielectric medium. When combined with an appropriate form factor (such as sphere, 6430 core+shell, ellipsoid etc), this allows for inclusion of the interparticle interference effects due to screened coulomb 6431 repulsion between charged particles. This routine only works for charged particles. If the charge is set to zero the 6432 routine will self destruct. For non-charged particles use a hard sphere potential. 5441 core+shell, ellipsoid, etc), this allows for inclusion of the interparticle interference effects due to screened coulomb 5442 repulsion between charged particles. 5443 5444 **This routine only works for charged particles**. If the charge is set to zero the routine will self-destruct! 5445 For non-charged particles use a hard sphere potential. 6433 5446 6434 5447 The salt concentration is used to compute the ionic strength of the solution which in turn is used to compute the Debye … … 6436 5449 multivalent salts. The counterions are also assumed to be monovalent. 6437 5450 6438 For 2D plot, the wave transfer is defined as . 6439 6440 Parameter name 6441 6442 Units 6443 6444 Default value 6445 6446 effect_radius 6447 6448 6449 6450 20.8 6451 6452 charge 6453 6454 19 6455 6456 volfraction 6457 6458 0.2 6459 6460 temperature 6461 6462 K 6463 6464 318 6465 6466 salt conc 6467 6468 M 6469 6470 0 6471 6472 dielectconst 6473 6474 71.1 6475 6476 5451 For 2D plot, the wave transfer is defined as 5452 5453 .. image:: img/image040.gif 5454 5455 ============== ======== ============= 5456 Parameter name Units Default value 5457 ============== ======== ============= 5458 effect_radius |Ang| 20.8 5459 charge *e* 19 5460 volfraction None 0.2 5461 temperature K 318 5462 salt conc M 0 5463 dielectconst None 71.1 5464 ============== ======== ============= 5465 5466 .. image:: img/image227.JPG 6477 5467 6478 5468 *Figure. 1D plot using the default values (in linear scale).* … … 6483 5473 6484 5474 5475 .. _StickyHSStructure: 6485 5476 6486 5477 **2.3.4. StickyHSStructure Factor** 6487 5478 6488 This calculates the interparticle structure factor for a hard sphere 6489 fluid with a narrow attractive well. A perturbative solution of the 6490 Percus-Yevick closure is used. The strength of the attractive well is 6491 described in terms of "stickiness" as defined below. The returned 6492 value is a dimensionless structure factor, S(q). 6493 6494 The perturb (perturbation parameter), epsilon, should be held between 6495 0.01 and 0.1. It is best to hold the perturbation parameter fixed and 6496 let the "stickiness" vary to adjust the interaction strength. The 6497 stickiness, tau, is defined in the equation below and is a function of 6498 both the perturbation parameter and the interaction strength. Tau and 6499 epsilon are defined in terms of the hard sphere diameter (sigma = 2R), 6500 the width of the square well, delta (same units as R), and the depth 6501 of the well, uo, in units of kT. From the definition, it is clear that 6502 smaller tau mean stronger attraction. 6503 6504 6505 6506 6507 6508 5479 This calculates the interparticle structure factor for a hard sphere fluid with a narrow attractive well. A perturbative 5480 solution of the Percus-Yevick closure is used. The strength of the attractive well is described in terms of "stickiness" 5481 as defined below. The returned value is a dimensionless structure factor, *S(q)*. 5482 5483 The perturb (perturbation parameter), |epsilon|, should be held between 0.01 and 0.1. It is best to hold the 5484 perturbation parameter fixed and let the "stickiness" vary to adjust the interaction strength. The stickiness, |tau|, 5485 is defined in the equation below and is a function of both the perturbation parameter and the interaction strength. 5486 |tau| and |epsilon| are defined in terms of the hard sphere diameter (|sigma| = 2\*\ *R*\ ), the width of the square 5487 well, |bigdelta| (same units as *R*), and the depth of the well, *Uo*, in units of kT. From the definition, it is clear 5488 that smaller |tau| means stronger attraction. 5489 5490 .. image:: img/image228.PNG 6509 5491 6510 5492 where the interaction potential is 6511 5493 6512 6513 6514 6515 6516 The Percus-Yevick (PY) closure was used for this calculation, and is 6517 an adequate closure for an attractive interparticle potential. This 6518 solution has been compared to Monte Carlo simulations for a square 6519 well fluid, with good agreement. 6520 6521 The true particle volume fraction, f, is not equal to h, which appears 6522 in most of the reference. The two are related in equation (24) of the 6523 reference. The reference also describes the relationship between this 6524 perturbation solution and the original sticky hard sphere (or adhesive 6525 sphere) model by Baxter. 6526 6527 NOTES: The calculation can go haywire for certain combinations of the 6528 input parameters, producing unphysical solutions - in this case errors 6529 are reported to the command window and the S(q) is set to -1 (it will 6530 disappear on a log-log plot). Use tight bounds to keep the parameters 6531 to values that you know are physical (test them) and keep nudging them 5494 .. image:: img/image229.PNG 5495 5496 The Percus-Yevick (PY) closure was used for this calculation, and is an adequate closure for an attractive interparticle 5497 potential. This solution has been compared to Monte Carlo simulations for a square well fluid, with good agreement. 5498 5499 The true particle volume fraction, |phi|, is not equal to *h*, which appears in most of the reference. The two are 5500 related in equation (24) of the reference. The reference also describes the relationship between this perturbation 5501 solution and the original sticky hard sphere (or adhesive sphere) model by Baxter. 5502 5503 NB: The calculation can go haywire for certain combinations of the input parameters, producing unphysical solutions - in 5504 this case errors are reported to the command window and the *S(q)* is set to -1 (so it will disappear on a log-log 5505 plot). Use tight bounds to keep the parameters to values that you know are physical (test them) and keep nudging them 6532 5506 until the optimization does not hit the constraints. 6533 5507 6534 For 2D plot, the wave transfer is defined as . 6535 6536 Parameter name 6537 6538 Units 6539 6540 Default value 6541 6542 effect_radius 6543 6544 6545 6546 50 6547 6548 perturb 6549 6550 0.05 6551 6552 volfraction 6553 6554 0.1 6555 6556 stickiness 6557 6558 K 6559 6560 0.2 6561 6562 5508 For 2D plot, the wave transfer is defined as 5509 5510 .. image:: img/image040.GIF 5511 5512 ============== ======== ============= 5513 Parameter name Units Default value 5514 ============== ======== ============= 5515 effect_radius |Ang| 50 5516 perturb None 0.05 5517 volfraction None 0.1 5518 stickiness K 0.2 5519 ============== ======== ============= 5520 5521 .. image:: img/image230.JPG 6563 5522 6564 5523 *Figure. 1D plot using the default values (in linear scale).*
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