Changeset 7f42aad in sasview for src/sans/models/media/model_functions.rst
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src/sans/models/media/model_functions.rst (modified) (163 diffs)
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src/sans/models/media/model_functions.rst
r6386cd8 r7f42aad 389 389 NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 390 390 391 .. image:: img/image005. JPG391 .. image:: img/image005.jpg 392 392 393 393 Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. … … 436 436 ============== ======== ============= 437 437 438 .. image:: img/image009. JPG438 .. image:: img/image009.jpg 439 439 440 440 *Figure. 1D plot using the default values above (w/200 data point).* … … 504 504 ============== ======== ============= 505 505 506 .. image:: img/image012. JPG506 .. image:: img/image012.jpg 507 507 508 508 *Figure. 1D plot using the default values (w/200 data point).* … … 525 525 The structure is: 526 526 527 .. image:: img/raspberry_pic. JPG527 .. image:: img/raspberry_pic.jpg 528 528 529 529 where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the … … 561 561 ============== ======== ============= 562 562 563 .. image:: img/raspberry_plot. JPG563 .. image:: img/raspberry_plot.jpg 564 564 565 565 *Figure. 1D plot using the values of /2000 data points.* … … 625 625 NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 626 626 627 .. image:: img/image014. JPG627 .. image:: img/image014.jpg 628 628 629 629 Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS … … 686 686 *qmax* = 0.7 -1 and the above default values. 687 687 688 .. image:: img/image015. JPG688 .. image:: img/image015.jpg 689 689 690 690 *Figure: 1D plot using the default values (w/200 data point).* … … 692 692 The scattering length density profile for the default sld values (w/ 4 shells). 693 693 694 .. image:: img/image016. JPG694 .. image:: img/image016.jpg 695 695 696 696 *Figure: SLD profile against the radius of the sphere for default SLDs.* … … 721 721 The *I* :sub:`0` is calculated in the following way (King, 2002) 722 722 723 .. image:: img/secondmeq1. JPG723 .. image:: img/secondmeq1.jpg 724 724 725 725 where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the … … 748 748 ============== ======== ============= 749 749 750 .. image:: img/secongm_fig1. JPG750 .. image:: img/secongm_fig1.jpg 751 751 752 752 REFERENCE … … 764 764 solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). 765 765 766 .. image:: img/image020. JPG766 .. image:: img/image020.jpg 767 767 768 768 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as … … 791 791 is the number of shells. 792 792 793 .. image:: img/image021. JPG793 .. image:: img/image021.jpg 794 794 795 795 *Figure. 1D plot using the default values (w/200 data point).* … … 818 818 The 1D scattering intensity is calculated in the following way 819 819 820 .. image:: img/image022. GIF821 822 .. image:: img/image023. GIF820 .. image:: img/image022.gif 821 822 .. image:: img/image023.gif 823 823 824 824 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 825 825 826 .. image:: img/image024. GIF826 .. image:: img/image024.gif 827 827 828 828 so that 829 829 830 .. image:: img/image025. GIF831 832 .. image:: img/image026. GIF833 834 .. image:: img/image027. GIF830 .. image:: img/image025.gif 831 832 .. image:: img/image026.gif 833 834 .. image:: img/image027.gif 835 835 836 836 Here we assumed that the SLDs of the core and solvent are constant against *r*. … … 838 838 Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by 839 839 840 .. image:: img/image028. GIF840 .. image:: img/image028.gif 841 841 842 842 An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and … … 846 846 For \| *A* \| > 0, 847 847 848 .. image:: img/image029. GIF848 .. image:: img/image029.gif 849 849 850 850 For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, … … 852 852 so this case is equivalent to 853 853 854 .. image:: img/image030. GIF855 856 .. image:: img/image031. GIF857 858 .. image:: img/image032. GIF859 860 .. image:: img/image033. GIF854 .. image:: img/image030.gif 855 856 .. image:: img/image031.gif 857 858 .. image:: img/image032.gif 859 860 .. image:: img/image033.gif 861 861 862 862 For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is … … 864 864 factor contributed by the shells is 865 865 866 .. image:: img/image034. GIF867 868 .. image:: img/image035. GIF866 .. image:: img/image034.gif 867 868 .. image:: img/image035.gif 869 869 870 870 In the equation 871 871 872 .. image:: img/image036. GIF872 .. image:: img/image036.gif 873 873 874 874 Finally, the form factor can be calculated by 875 875 876 .. image:: img/image037. GIF876 .. image:: img/image037.gif 877 877 878 878 where 879 879 880 .. image:: img/image038. GIF880 .. image:: img/image038.gif 881 881 882 882 and 883 883 884 .. image:: img/image039. GIF884 .. image:: img/image039.gif 885 885 886 886 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 887 887 defined as 888 888 889 .. image:: img/image040. GIF889 .. image:: img/image040.gif 890 890 891 891 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. … … 909 909 NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. 910 910 911 .. image:: img/image041. JPG911 .. image:: img/image041.jpg 912 912 913 913 *Figure. 1D plot using the default values (w/400 point).* 914 914 915 .. image:: img/image042. JPG915 .. image:: img/image042.jpg 916 916 917 917 *Figure. SLD profile from the default values.* … … 945 945 and a shell thickness, *t*. 946 946 947 .. image:: img/image018. JPG947 .. image:: img/image018.jpg 948 948 949 949 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is … … 970 970 NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. 971 971 972 .. image:: img/image019. JPG972 .. image:: img/image019.jpg 973 973 974 974 *Figure. 1D plot using the default values (w/200 data point).* … … 1001 1001 The 1D scattering intensity is calculated in the following way: 1002 1002 1003 .. image:: img/image022. GIF1004 1005 .. image:: img/image043. GIF1003 .. image:: img/image022.gif 1004 1005 .. image:: img/image043.gif 1006 1006 1007 1007 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 1008 1008 1009 .. image:: img/image024. GIF1009 .. image:: img/image024.gif 1010 1010 1011 1011 so that 1012 1012 1013 .. image:: img/image044. GIF1014 1015 .. image:: img/image045. GIF1016 1017 .. image:: img/image046. GIF1018 1019 .. image:: img/image047. GIF1020 1021 .. image:: img/image048. GIF1022 1023 .. image:: img/image027. GIF1013 .. image:: img/image044.gif 1014 1015 .. image:: img/image045.gif 1016 1017 .. image:: img/image046.gif 1018 1019 .. image:: img/image047.gif 1020 1021 .. image:: img/image048.gif 1022 1023 .. image:: img/image027.gif 1024 1024 1025 1025 Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between … … 1028 1028 1) Exp 1029 1029 1030 .. image:: img/image049. GIF1030 .. image:: img/image049.gif 1031 1031 1032 1032 2) Power-Law 1033 1033 1034 .. image:: img/image050. GIF1034 .. image:: img/image050.gif 1035 1035 1036 1036 3) Erf 1037 1037 1038 .. image:: img/image051. GIF1038 .. image:: img/image051.gif 1039 1039 1040 1040 The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is … … 1044 1044 to the form factor *P(q)* 1045 1045 1046 .. image:: img/image052. GIF1047 1048 .. image:: img/image053. GIF1049 1050 .. image:: img/image054. GIF1046 .. image:: img/image052.gif 1047 1048 .. image:: img/image053.gif 1049 1050 .. image:: img/image054.gif 1051 1051 1052 1052 where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. … … 1054 1054 In the equation 1055 1055 1056 .. image:: img/image055. GIF1056 .. image:: img/image055.gif 1057 1057 1058 1058 Finally, the form factor can be calculated by 1059 1059 1060 .. image:: img/image037. GIF1060 .. image:: img/image037.gif 1061 1061 1062 1062 where 1063 1063 1064 .. image:: img/image038. GIF1064 .. image:: img/image038.gif 1065 1065 1066 1066 and 1067 1067 1068 .. image:: img/image056. GIF1068 .. image:: img/image056.gif 1069 1069 1070 1070 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 1071 1071 defined as 1072 1072 1073 .. image:: img/image040. GIF1073 .. image:: img/image040.gif 1074 1074 1075 1075 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. … … 1098 1098 NB: *rad_core0* represents the core radius (*R1*). 1099 1099 1100 .. image:: img/image057. JPG1100 .. image:: img/image057.jpg 1101 1101 1102 1102 *Figure. 1D plot using the default values (w/400 point).* 1103 1103 1104 .. image:: img/image058. JPG1104 .. image:: img/image058.jpg 1105 1105 1106 1106 *Figure. SLD profile from the default values.* … … 1249 1249 and |phi|. Those angles are defined in Figure 1. 1250 1250 1251 .. image:: img/image061. JPG1251 .. image:: img/image061.jpg 1252 1252 1253 1253 *Figure 1. Definition of the angles for oriented cylinders.* 1254 1254 1255 .. image:: img/image062. JPG1255 .. image:: img/image062.jpg 1256 1256 1257 1257 *Figure 2. Examples of the angles for oriented pp against the detector plane.* … … 1286 1286 NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. 1287 1287 1288 .. image:: img/image065. JPG1288 .. image:: img/image065.jpg 1289 1289 1290 1290 *Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis* … … 1301 1301 distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. 1302 1302 1303 .. image:: img/image066. JPG1303 .. image:: img/image066.jpg 1304 1304 1305 1305 *Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the* … … 1351 1351 ============== ======== ============= 1352 1352 1353 .. image:: img/image074. JPG1353 .. image:: img/image074.jpg 1354 1354 1355 1355 *Figure. 1D plot using the default values (w/1000 data point).* … … 1358 1358 (Kline, 2006). 1359 1359 1360 .. image:: img/image061. JPG1360 .. image:: img/image061.jpg 1361 1361 1362 1362 *Figure. Definition of the angles for the oriented HollowCylinderModel.* 1363 1363 1364 .. image:: img/image062. JPG1364 .. image:: img/image062.jpg 1365 1365 1366 1366 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1388 1388 The Capped Cylinder geometry is defined as 1389 1389 1390 .. image:: img/image112. JPG1390 .. image:: img/image112.jpg 1391 1391 1392 1392 where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius … … 1397 1397 The scattered intensity *I(q)* is calculated as 1398 1398 1399 .. image:: img/image113. JPG1399 .. image:: img/image113.jpg 1400 1400 1401 1401 where the amplitude *A(q)* is given as 1402 1402 1403 .. image:: img/image114. JPG1403 .. image:: img/image114.jpg 1404 1404 1405 1405 The < > brackets denote an average of the structure over all orientations. <\ *A*\ :sup:`2`\ *(q)*> is then the form … … 1409 1409 The volume of the Capped Cylinder is (with *h* as a positive value here) 1410 1410 1411 .. image:: img/image115. JPG1411 .. image:: img/image115.jpg 1412 1412 1413 1413 and its radius-of-gyration 1414 1414 1415 .. image:: img/image116. JPG1415 .. image:: img/image116.jpg 1416 1416 1417 1417 **The requirement that** *R* >= *r* **is not enforced in the model! It is up to you to restrict this during analysis.** … … 1432 1432 ============== ======== ============= 1433 1433 1434 .. image:: img/image117. JPG1434 .. image:: img/image117.jpg 1435 1435 1436 1436 *Figure. 1D plot using the default values (w/256 data point).* … … 1439 1439 |theta| = 45 deg and |phi| =0 deg with default values for other parameters 1440 1440 1441 .. image:: img/image118. JPG1441 .. image:: img/image118.jpg 1442 1442 1443 1443 *Figure. 2D plot (w/(256X265) data points).* 1444 1444 1445 .. image:: img/image061. JPG1445 .. image:: img/image061.jpg 1446 1446 1447 1447 *Figure. Definition of the angles for oriented 2D cylinders.* … … 1485 1485 the outer shell is given by *L+2t*. *J1* is the first order Bessel function. 1486 1486 1487 .. image:: img/image069. JPG1487 .. image:: img/image069.jpg 1488 1488 1489 1489 To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two … … 1520 1520 NIST (Kline, 2006). Figure 1 shows a comparison of the 1D output of our model and the output of the NIST software. 1521 1521 1522 .. image:: img/image070. JPG1522 .. image:: img/image070.jpg 1523 1523 1524 1524 *Figure 1: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS* … … 1531 1531 2D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a cross-check. 1532 1532 1533 .. image:: img/image071. JPG1533 .. image:: img/image071.jpg 1534 1534 1535 1535 *Figure 2: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and* … … 1538 1538 *Solvent_sld* = 1e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. 1539 1539 1540 .. image:: img/image061. JPG1540 .. image:: img/image061.jpg 1541 1541 1542 1542 *Figure. Definition of the angles for oriented core-shell cylinders.* 1543 1543 1544 .. image:: img/image062. JPG1544 .. image:: img/image062.jpg 1545 1545 1546 1546 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1591 1591 All angle parameters are valid and given only for 2D calculation; ie, an oriented system. 1592 1592 1593 .. image:: img/image101. JPG1593 .. image:: img/image101.jpg 1594 1594 1595 1595 *Figure. Definition of angles for 2D* 1596 1596 1597 .. image:: img/image062. JPG1597 .. image:: img/image062.jpg 1598 1598 1599 1599 *Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* … … 1614 1614 ============== ======== ============= 1615 1615 1616 .. image:: img/image102. JPG1616 .. image:: img/image102.jpg 1617 1617 1618 1618 *Figure. 1D plot using the default values (w/1000 data point).* … … 1625 1625 and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). 1626 1626 1627 .. image:: img/image103. GIF1627 .. image:: img/image103.gif 1628 1628 1629 1629 *Figure. Comparison between 1D and averaged 2D.* … … 1632 1632 the results of the averaging by varying the number of angular bins. 1633 1633 1634 .. image:: img/image104. GIF1634 .. image:: img/image104.gif 1635 1635 1636 1636 *Figure. The intensities averaged from 2D over different numbers of bins and angles.* … … 1660 1660 *2.1.19.1. Definition* 1661 1661 1662 .. image:: img/image075. JPG1662 .. image:: img/image075.jpg 1663 1663 1664 1664 The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff … … 1683 1683 ============== ======== ============= 1684 1684 1685 .. image:: img/image076. JPG1685 .. image:: img/image076.jpg 1686 1686 1687 1687 *Figure. 1D plot using the default values (w/1000 data point).* … … 1738 1738 - The scattering function is negative for a range of parameter values and q-values that are experimentally accessible. A correction function has been added to give the proper behavior. 1739 1739 1740 .. image:: img/image077. JPG1740 .. image:: img/image077.jpg 1741 1741 1742 1742 The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff … … 1780 1780 ============== ======== ============= 1781 1781 1782 .. image:: img/image078. JPG1782 .. image:: img/image078.jpg 1783 1783 1784 1784 *Figure. 1D plot using the default values (w/200 data points).* … … 1807 1807 and SLDs. 1808 1808 1809 .. image:: img/image240. PNG1809 .. image:: img/image240.png 1810 1810 1811 1811 *(Graphic from DOI: 10.1039/C0NP00002G)* … … 1839 1839 *Figure. 1D plot using the default values (w/200 data point).* 1840 1840 1841 .. image:: img/image061. JPG1841 .. image:: img/image061.jpg 1842 1842 1843 1843 *Figure. Definition of the angles for the oriented CoreShellBicelleModel.* 1844 1844 1845 .. image:: img/image062. JPG1845 .. image:: img/image062.jpg 1846 1846 1847 1847 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1869 1869 The barbell geometry is defined as 1870 1870 1871 .. image:: img/image105. JPG1871 .. image:: img/image105.jpg 1872 1872 1873 1873 where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. … … 1892 1892 The volume of the barbell is 1893 1893 1894 .. image:: img/image108. JPG1894 .. image:: img/image108.jpg 1895 1895 1896 1896 1897 1897 and its radius-of-gyration is 1898 1898 1899 .. image:: img/image109. JPG1899 .. image:: img/image109.jpg 1900 1900 1901 1901 **The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. … … 1916 1916 ============== ======== ============= 1917 1917 1918 .. image:: img/image110. JPG1918 .. image:: img/image110.jpg 1919 1919 1920 1920 *Figure. 1D plot using the default values (w/256 data point).* … … 1923 1923 |theta| = 45 deg and |phi| = 0 deg with default values for other parameters 1924 1924 1925 .. image:: img/image111. JPG1925 .. image:: img/image111.jpg 1926 1926 1927 1927 *Figure. 2D plot (w/(256X265) data points).* 1928 1928 1929 .. image:: img/image061. JPG1929 .. image:: img/image061.jpg 1930 1930 1931 1931 *Figure. Examples of the angles for oriented pp against the detector plane.* 1932 1932 1933 .. image:: img/image062. JPG1933 .. image:: img/image062.jpg 1934 1934 1935 1935 Figure. Definition of the angles for oriented 2D barbells. … … 1963 1963 *2.1.23.1 Definition* 1964 1964 1965 .. image:: img/image079. GIF1965 .. image:: img/image079.gif 1966 1966 1967 1967 The scattering intensity *I(q)* is … … 2007 2007 ============== ======== ============= 2008 2008 2009 .. image:: img/image085. JPG2009 .. image:: img/image085.jpg 2010 2010 2011 2011 *Figure. 1D plot using the default values (w/1000 data point).* 2012 2012 2013 .. image:: img/image086. JPG2013 .. image:: img/image086.jpg 2014 2014 2015 2015 *Figure. Examples of the angles for oriented stackeddisks against the detector plane.* 2016 2016 2017 .. image:: img/image062. JPG2017 .. image:: img/image062.jpg 2018 2018 2019 2019 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 2038 2038 This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). 2039 2039 2040 .. image:: img/image241. PNG2040 .. image:: img/image241.png 2041 2041 2042 2042 *(Graphic from Matt Henderson, matt@matthen.com)* … … 2128 2128 above. 2129 2129 2130 .. image:: img/image121. JPG2130 .. image:: img/image121.jpg 2131 2131 2132 2132 The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering 2133 2133 kernel and the 1D scattering intensity use the c-library from NIST. 2134 2134 2135 .. image:: img/image122. JPG2135 .. image:: img/image122.jpg 2136 2136 2137 2137 *Figure. The angles for oriented ellipsoid.* … … 2143 2143 software. 2144 2144 2145 .. image:: img/image123. JPG2145 .. image:: img/image123.jpg 2146 2146 2147 2147 *Figure 1: Comparison of the SasView scattering intensity for an ellipsoid with the output of the NIST SANS analysis* … … 2154 2154 cross-check. 2155 2155 2156 .. image:: img/image124. JPG2156 .. image:: img/image124.jpg 2157 2157 2158 2158 *Figure 2: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the* … … 2186 2186 all orientations for 1D. 2187 2187 2188 .. image:: img/image125. GIF2188 .. image:: img/image125.gif 2189 2189 2190 2190 The returned value is in units of |cm^-1|, on absolute scale. … … 2220 2220 ============== ======== ============= 2221 2221 2222 .. image:: img/image127. JPG2222 .. image:: img/image127.jpg 2223 2223 2224 2224 *Figure. 1D plot using the default values (w/1000 data point).* 2225 2225 2226 .. image:: img/image122. JPG2226 .. image:: img/image122.jpg 2227 2227 2228 2228 *Figure. The angles for oriented CoreShellEllipsoid.* … … 2313 2313 where the volume *V* = (4/3)\ |pi| (*Ra* *Rb* *Rc*), and the averaging < > is applied over all orientations for 1D. 2314 2314 2315 .. image:: img/image128. JPG2315 .. image:: img/image128.jpg 2316 2316 2317 2317 The returned value is in units of |cm^-1|, on absolute scale. … … 2349 2349 ============== ======== ============= 2350 2350 2351 .. image:: img/image130. JPG2351 .. image:: img/image130.jpg 2352 2352 2353 2353 *Figure. 1D plot using the default values (w/1000 data point).* … … 2360 2360 angles of |theta|, |phi|, and |psi| respectively). 2361 2361 2362 .. image:: img/image131. GIF2362 .. image:: img/image131.gif 2363 2363 2364 2364 *Figure. Comparison between 1D and averaged 2D.* 2365 2365 2366 .. image:: img/image132. JPG2366 .. image:: img/image132.jpg 2367 2367 2368 2368 *Figure. The angles for oriented ellipsoid.* … … 2399 2399 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2400 2400 2401 .. image:: img/image040. GIF2401 .. image:: img/image040.gif 2402 2402 2403 2403 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, … … 2414 2414 ============== ======== ============= 2415 2415 2416 .. image:: img/image135. JPG2416 .. image:: img/image135.jpg 2417 2417 2418 2418 *Figure. 1D plot using the default values (w/1000 data point).* … … 2444 2444 The form factor is 2445 2445 2446 .. image:: img/image137. JPG2446 .. image:: img/image137.jpg 2447 2447 2448 2448 where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), … … 2451 2451 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2452 2452 2453 .. image:: img/image040. GIF2453 .. image:: img/image040.gif 2454 2454 2455 2455 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, … … 2468 2468 ============== ======== ============= 2469 2469 2470 .. image:: img/image138. JPG2470 .. image:: img/image138.jpg 2471 2471 2472 2472 *Figure. 1D plot using the default values (w/1000 data point).* … … 2519 2519 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2520 2520 2521 .. image:: img/image040. GIF2521 .. image:: img/image040.gif 2522 2522 2523 2523 The returned value is in units of |cm^-1|, on absolute scale. … … 2535 2535 ============== ======== ============= 2536 2536 2537 .. image:: img/image142. JPG2537 .. image:: img/image142.jpg 2538 2538 2539 2539 *Figure. 1D plot using the default values (w/6000 data point).* … … 2587 2587 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2588 2588 2589 .. image:: img/image040. GIF2589 .. image:: img/image040.gif 2590 2590 2591 2591 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, … … 2607 2607 ============== ======== ============= 2608 2608 2609 .. image:: img/image144. JPG2609 .. image:: img/image144.jpg 2610 2610 2611 2611 *Figure. 1D plot using the default values (w/6000 data point).* … … 2634 2634 The scattering intensity *I(q)* is calculated as 2635 2635 2636 .. image:: img/image145. JPG2636 .. image:: img/image145.jpg 2637 2637 2638 2638 The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* 2639 2639 2640 .. image:: img/image146. JPG2640 .. image:: img/image146.jpg 2641 2641 2642 2642 Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume … … 2647 2647 Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values 2648 2648 2649 .. image:: img/image147. JPG2649 .. image:: img/image147.jpg 2650 2650 2651 2651 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 2652 2652 2653 .. image:: img/image040. GIF2653 .. image:: img/image040.gif 2654 2654 2655 2655 The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. … … 2668 2668 ============== ======== ============= 2669 2669 2670 .. image:: img/image148. JPG2670 .. image:: img/image148.jpg 2671 2671 2672 2672 *Figure. 1D plot using the default values above (w/20000 data point).* … … 2695 2695 The scattering intensity *I(q)* is calculated as 2696 2696 2697 .. image:: img/image149. JPG2697 .. image:: img/image149.jpg 2698 2698 2699 2699 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2707 2707 and nearest neighbor separation *D* is 2708 2708 2709 .. image:: img/image150. JPG2709 .. image:: img/image150.jpg 2710 2710 2711 2711 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2712 2712 2713 .. image:: img/image151. JPG2713 .. image:: img/image151.jpg 2714 2714 2715 2715 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2717 2717 The simple cubic lattice is 2718 2718 2719 .. image:: img/image152. JPG2719 .. image:: img/image152.jpg 2720 2720 2721 2721 For a crystal, diffraction peaks appear at reduced *q*\ -values given by 2722 2722 2723 .. image:: img/image153. JPG2723 .. image:: img/image153.jpg 2724 2724 2725 2725 where for a simple cubic lattice any *h*\ , *k*\ , *l* are allowed and none are forbidden. Thus the peak positions 2726 2726 correspond to (just the first 5) 2727 2727 2728 .. image:: img/image154. JPG2728 .. image:: img/image154.jpg 2729 2729 2730 2730 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2748 2748 default values. 2749 2749 2750 .. image:: img/image155. JPG2750 .. image:: img/image155.jpg 2751 2751 2752 2752 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2756 2756 computation. 2757 2757 2758 .. image:: img/image156. JPG2759 2760 .. image:: img/image157. JPG2758 .. image:: img/image156.jpg 2759 2760 .. image:: img/image157.jpg 2761 2761 2762 2762 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2786 2786 The scattering intensity *I(q)* is calculated as 2787 2787 2788 .. image:: img/image158. JPG2788 .. image:: img/image158.jpg 2789 2789 2790 2790 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2798 2798 *R* and nearest neighbor separation *D* is 2799 2799 2800 .. image:: img/image159. JPG2800 .. image:: img/image159.jpg 2801 2801 2802 2802 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2803 2803 2804 .. image:: img/image160. JPG2804 .. image:: img/image160.jpg 2805 2805 2806 2806 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2808 2808 The face-centered cubic lattice is 2809 2809 2810 .. image:: img/image161. JPG2810 .. image:: img/image161.jpg 2811 2811 2812 2812 For a crystal, diffraction peaks appear at reduced q-values given by 2813 2813 2814 .. image:: img/image162. JPG2814 .. image:: img/image162.jpg 2815 2815 2816 2816 where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all even are allowed and reflections where 2817 2817 *h*\ , *k*\ , *l* are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5) 2818 2818 2819 .. image:: img/image163. JPG2819 .. image:: img/image163.jpg 2820 2820 2821 2821 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2839 2839 default values. 2840 2840 2841 .. image:: img/image164. JPG2841 .. image:: img/image164.jpg 2842 2842 2843 2843 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2847 2847 computation. 2848 2848 2849 .. image:: img/image165. GIF2850 2851 .. image:: img/image166. JPG2849 .. image:: img/image165.gif 2850 2851 .. image:: img/image166.jpg 2852 2852 2853 2853 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2877 2877 The scattering intensity *I(q)* is calculated as 2878 2878 2879 .. image:: img/image167. JPG2879 .. image:: img/image167.jpg 2880 2880 2881 2881 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2889 2889 *R* and nearest neighbor separation *D* is 2890 2890 2891 .. image:: img/image159. JPG2891 .. image:: img/image159.jpg 2892 2892 2893 2893 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2894 2894 2895 .. image:: img/image160. JPG2895 .. image:: img/image160.jpg 2896 2896 2897 2897 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2899 2899 The body-centered cubic lattice is 2900 2900 2901 .. image:: img/image168. JPG2901 .. image:: img/image168.jpg 2902 2902 2903 2903 For a crystal, diffraction peaks appear at reduced q-values given by 2904 2904 2905 .. image:: img/image162. JPG2905 .. image:: img/image162.jpg 2906 2906 2907 2907 where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and 2908 2908 reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) 2909 2909 2910 .. image:: img/image169. JPG2910 .. image:: img/image169.jpg 2911 2911 2912 2912 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2930 2930 default values. 2931 2931 2932 .. image:: img/image170. JPG2932 .. image:: img/image170.jpg 2933 2933 2934 2934 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2938 2938 computation. 2939 2939 2940 .. image:: img/image165. GIF2941 2942 .. image:: img/image171. JPG2940 .. image:: img/image165.gif 2941 2942 .. image:: img/image171.jpg 2943 2943 2944 2944 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2967 2967 For information about polarised and magnetic scattering, click here_. 2968 2968 2969 .. image:: img/image087. JPG2969 .. image:: img/image087.jpg 2970 2970 2971 2971 *2.1.37.1. Definition* … … 2991 2991 parallel to the *x*-axis of the detector. 2992 2992 2993 .. image:: img/image090. JPG2993 .. image:: img/image090.jpg 2994 2994 2995 2995 *Figure. Definition of angles for 2D*. 2996 2996 2997 .. image:: img/image091. JPG2997 .. image:: img/image091.jpg 2998 2998 2999 2999 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 3010 3010 ============== ======== ============= 3011 3011 3012 .. image:: img/image092. JPG3012 .. image:: img/image092.jpg 3013 3013 3014 3014 *Figure. 1D plot using the default values (w/1000 data point).* … … 3021 3021 angles of |theta|, |phi|, and |psi| respectively). 3022 3022 3023 .. image:: img/image093. GIF3023 .. image:: img/image093.gif 3024 3024 3025 3025 *Figure. Comparison between 1D and averaged 2D.* … … 3055 3055 dimensions *A*, *B*, *C* such that *A* < *B* < *C*. 3056 3056 3057 .. image:: img/image087. JPG3057 .. image:: img/image087.jpg 3058 3058 3059 3059 There are rectangular "slabs" of thickness *tA* that add to the *A* dimension (on the *BC* faces). There are similar 3060 3060 slabs on the *AC* (= *tB*) and *AB* (= *tC*) faces. The projection in the *AB* plane is then 3061 3061 3062 .. image:: img/image094. JPG3062 .. image:: img/image094.jpg 3063 3063 3064 3064 The volume of the solid is … … 3096 3096 parallel to the *x*-axis of the detector. 3097 3097 3098 .. image:: img/image090. JPG3098 .. image:: img/image090.jpg 3099 3099 3100 3100 *Figure. Definition of angles for 2D*. 3101 3101 3102 .. image:: img/image091. JPG3102 .. image:: img/image091.jpg 3103 3103 3104 3104 *Figure. Examples of the angles for oriented cspp against the detector plane.* … … 3125 3125 ============== ======== ============= 3126 3126 3127 .. image:: img/image096. JPG3127 .. image:: img/image096.jpg 3128 3128 3129 3129 *Figure. 1D plot using the default values (w/256 data points).* 3130 3130 3131 .. image:: img/image097. JPG3131 .. image:: img/image097.jpg 3132 3132 3133 3133 *Figure. 2D plot using the default values (w/(256X265) data points).* … … 3392 3392 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3393 3393 3394 .. image:: img/image040. GIF3394 .. image:: img/image040.gif 3395 3395 3396 3396 ============== ======== ============= … … 3402 3402 ============== ======== ============= 3403 3403 3404 .. image:: img/image173. JPG3404 .. image:: img/image173.jpg 3405 3405 3406 3406 *Figure. 1D plot using the default values (w/200 data point).* … … 3429 3429 The scattering intensity *I(q)* is calculated as 3430 3430 3431 .. image:: img/image174. JPG3431 .. image:: img/image174.jpg 3432 3432 3433 3433 Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. … … 3435 3435 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3436 3436 3437 .. image:: img/image040. GIF3437 .. image:: img/image040.gif 3438 3438 3439 3439 ================== ======== ============= … … 3449 3449 ================== ======== ============= 3450 3450 3451 .. image:: img/image175. JPG3451 .. image:: img/image175.jpg 3452 3452 3453 3453 *Figure. 1D plot using the default values (w/200 data point).* … … 3473 3473 The scattering intensity *I(q)* is calculated as 3474 3474 3475 .. image:: img/image176. JPG3475 .. image:: img/image176.jpg 3476 3476 3477 3477 The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function … … 3484 3484 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3485 3485 3486 .. image:: img/image040. GIF3486 .. image:: img/image040.gif 3487 3487 3488 3488 ==================== ======== ============= … … 3497 3497 ==================== ======== ============= 3498 3498 3499 .. image:: img/image177. JPG3499 .. image:: img/image177.jpg 3500 3500 3501 3501 *Figure. 1D plot using the default values (w/500 data points).* … … 3524 3524 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3525 3525 3526 .. image:: img/image040. GIF3526 .. image:: img/image040.gif 3527 3527 3528 3528 ============== ======== ============= … … 3534 3534 ============== ======== ============= 3535 3535 3536 .. image:: img/image179. JPG3536 .. image:: img/image179.jpg 3537 3537 3538 3538 * Figure. 1D plot using the default values (w/200 data point).* … … 3563 3563 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3564 3564 3565 .. image:: img/image040. GIF3565 .. image:: img/image040.gif 3566 3566 3567 3567 ============== ======== ============= … … 3573 3573 ============== ======== ============= 3574 3574 3575 .. image:: img/image181. JPG3575 .. image:: img/image181.jpg 3576 3576 3577 3577 * Figure. 1D plot using the default values (w/200 data point).* … … 3606 3606 ============== ======== ============= 3607 3607 3608 .. image:: img/image183. JPG3608 .. image:: img/image183.jpg 3609 3609 3610 3610 *Figure. 1D plot using the default values (w/200 data point).* … … 3629 3629 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3630 3630 3631 .. image:: img/image040. GIF3631 .. image:: img/image040.gif 3632 3632 3633 3633 ============== ======== ============= … … 3640 3640 ============== ======== ============= 3641 3641 3642 .. image:: img/image185. JPG3642 .. image:: img/image185.jpg 3643 3643 3644 3644 *Figure. 1D plot using the default values (w/200 data point).* … … 3673 3673 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3674 3674 3675 .. image:: img/image040. GIF3675 .. image:: img/image040.gif 3676 3676 3677 3677 ============== ======== ============= … … 3687 3687 ============== ======== ============= 3688 3688 3689 .. image:: img/image187. JPG3689 .. image:: img/image187.jpg 3690 3690 3691 3691 *Figure. 1D plot using the default values (w/200 data point).* … … 3705 3705 *2.2.9.1. Definition* 3706 3706 3707 .. image:: img/mass_fractal_eq1. JPG3707 .. image:: img/mass_fractal_eq1.jpg 3708 3708 3709 3709 where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, … … 3724 3724 ============== ======== ============= 3725 3725 3726 .. image:: img/mass_fractal_fig1. JPG3726 .. image:: img/mass_fractal_fig1.jpg 3727 3727 3728 3728 *Figure. 1D plot using default values.* … … 3764 3764 ============== ======== ============= 3765 3765 3766 .. image:: img/surface_fractal_fig1. JPG3766 .. image:: img/surface_fractal_fig1.jpg 3767 3767 3768 3768 *Figure. 1D plot using default values.* … … 3812 3812 ============== ======== ============= 3813 3813 3814 .. image:: img/masssurface_fractal_fig1. JPG3814 .. image:: img/masssurface_fractal_fig1.jpg 3815 3815 3816 3816 *Figure. 1D plot using default values.* … … 3838 3838 *2.2.12.1. Definition* 3839 3839 3840 .. image:: img/fractcore_eq1. GIF3840 .. image:: img/fractcore_eq1.gif 3841 3841 3842 3842 The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 … … 3854 3854 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3855 3855 3856 .. image:: img/image040. GIF3856 .. image:: img/image040.gif 3857 3857 3858 3858 ============== ======== ============= … … 3870 3870 ============== ======== ============= 3871 3871 3872 .. image:: img/image188. JPG3872 .. image:: img/image188.jpg 3873 3873 3874 3874 *Figure. 1D plot using the default values (w/500 data points).* … … 3895 3895 The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) 3896 3896 3897 .. image:: img/image189. JPG3897 .. image:: img/image189.jpg 3898 3898 3899 3899 |bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" … … 3907 3907 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3908 3908 3909 .. image:: img/image040. GIF3909 .. image:: img/image040.gif 3910 3910 3911 3911 =================================== ======== ============= … … 3919 3919 =================================== ======== ============= 3920 3920 3921 .. image:: img/image190. JPG3921 .. image:: img/image190.jpg 3922 3922 3923 3923 *Figure. 1D plot using the default values (w/500 data points).* … … 3947 3947 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3948 3948 3949 .. image:: img/image040. GIF3949 .. image:: img/image040.gif 3950 3950 3951 3951 ============== ======== ============= … … 3988 3988 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3989 3989 3990 .. image:: img/image040. GIF3990 .. image:: img/image040.gif 3991 3991 3992 3992 ============== ======== ============= … … 4017 4017 The following functional form is used 4018 4018 4019 .. image:: img/image193. JPG4019 .. image:: img/image193.jpg 4020 4020 4021 4021 This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D … … 4026 4026 Enforcing the continuity of the Guinier and Porod functions and their derivatives yields 4027 4027 4028 .. image:: img/image194. JPG4028 .. image:: img/image194.jpg 4029 4029 4030 4030 and 4031 4031 4032 .. image:: img/image195. JPG4032 .. image:: img/image195.jpg 4033 4033 4034 4034 Note that … … 4054 4054 ============================== ======== ============= 4055 4055 4056 .. image:: img/image196. JPG4056 .. image:: img/image196.jpg 4057 4057 4058 4058 *Figure. 1D plot using the default values (w/500 data points).* … … 4082 4082 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4083 4083 4084 .. image:: img/image040. GIF4084 .. image:: img/image040.gif 4085 4085 4086 4086 ============== ======== ============= … … 4110 4110 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4111 4111 4112 .. image:: img/image040. GIF4112 .. image:: img/image040.gif 4113 4113 4114 4114 ============== ======== ============= … … 4121 4121 ============== ======== ============= 4122 4122 4123 .. image:: img/image199. JPG4123 .. image:: img/image199.jpg 4124 4124 4125 4125 *Figure. 1D plot using the default values (w/500 data points).* … … 4143 4143 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4144 4144 4145 .. image:: img/image040. GIF4145 .. image:: img/image040.gif 4146 4146 4147 4147 ============== ======== ============= … … 4154 4154 ============== ======== ============= 4155 4155 4156 .. image:: img/image201. JPG4156 .. image:: img/image201.jpg 4157 4157 4158 4158 *Figure. 1D plot using the default values (w/500 data points).* … … 4190 4190 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4191 4191 4192 .. image:: img/image040. GIF4192 .. image:: img/image040.gif 4193 4193 4194 4194 This example dataset is produced using 200 data points, using 200 data points, … … 4204 4204 ============== ======== ============= 4205 4205 4206 .. image:: img/image205. JPG4206 .. image:: img/image205.jpg 4207 4207 4208 4208 *Figure. 1D plot using the default values (w/200 data point).* … … 4230 4230 The form factor was originally presented in the following integral form (Benoit, 1957) 4231 4231 4232 .. image:: img/image206. JPG4232 .. image:: img/image206.jpg 4233 4233 4234 4234 where |nu| is the excluded volume parameter (which is related to the Porod exponent *m* as |nu| = 1 / *m*), *a* is the … … 4236 4236 into an almost analytical form as follows (Hammouda, 1993) 4237 4237 4238 .. image:: img/image207. JPG4238 .. image:: img/image207.jpg 4239 4239 4240 4240 where |gamma|\ *(x,U)* is the incomplete gamma function 4241 4241 4242 .. image:: img/image208. JPG4242 .. image:: img/image208.jpg 4243 4243 4244 4244 and the variable *U* is given in terms of the scattering vector *Q* as 4245 4245 4246 .. image:: img/image209. JPG4246 .. image:: img/image209.jpg 4247 4247 4248 4248 The square of the radius-of-gyration is defined as 4249 4249 4250 .. image:: img/image210. JPG4250 .. image:: img/image210.jpg 4251 4251 4252 4252 Note that this model applies only in the mass fractal range (ie, 5/3 <= *m* <= 3) and **does not** apply to surface … … 4256 4256 A low-*Q* expansion yields the Guinier form and a high-*Q* expansion yields the Porod form which is given by 4257 4257 4258 .. image:: img/image211. JPG4258 .. image:: img/image211.jpg 4259 4259 4260 4260 Here |biggamma|\ *(x)* = |gamma|\ *(x,inf)* is the gamma function. … … 4262 4262 The asymptotic limit is dominated by the first term 4263 4263 4264 .. image:: img/image212. JPG4264 .. image:: img/image212.jpg 4265 4265 4266 4266 The special case when |nu| = 0.5 (or *m* = 1/|nu| = 2) corresponds to Gaussian chains for which the form factor is given 4267 4267 by the familiar Debye_ function. 4268 4268 4269 .. image:: img/image213. JPG4269 .. image:: img/image213.jpg 4270 4270 4271 4271 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4272 4272 4273 .. image:: img/image040. GIF4273 .. image:: img/image040.gif 4274 4274 4275 4275 This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.2 |Ang^-1| and the default … … 4285 4285 =================== ======== ============= 4286 4286 4287 .. image:: img/image214. JPG4287 .. image:: img/image214.jpg 4288 4288 4289 4289 *Figure. 1D plot using the default values (w/500 data points).* … … 4369 4369 ======================= ======== ============= 4370 4370 4371 .. image:: img/image215. JPG4371 .. image:: img/image215.jpg 4372 4372 4373 4373 *Figure. 1D plot using the default values (w/500 data points).* … … 4399 4399 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4400 4400 4401 .. image:: img/image040. GIF4401 .. image:: img/image040.gif 4402 4402 4403 4403 =============================== ======== ============= … … 4413 4413 =============================== ======== ============= 4414 4414 4415 .. image:: img/image217. JPG4415 .. image:: img/image217.jpg 4416 4416 4417 4417 *Figure. 1D plot using the default values (w/500 data points).* … … 4435 4435 The scattering intensity *I(q)* is calculated as 4436 4436 4437 .. image:: img/image218. JPG4437 .. image:: img/image218.jpg 4438 4438 4439 4439 where *qc* is the location of the crossover from one slope to the other. The scaling *coef_A* sets the overall … … 4445 4445 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4446 4446 4447 .. image:: img/image040. GIF4447 .. image:: img/image040.gif 4448 4448 4449 4449 ============== ======== ============= … … 4457 4457 ============== ======== ============= 4458 4458 4459 .. image:: img/image219. JPG4459 .. image:: img/image219.jpg 4460 4460 4461 4461 *Figure. 1D plot using the default values (w/500 data points).* … … 4486 4486 The empirical fit function is 4487 4487 4488 .. image:: img/image220. JPG4488 .. image:: img/image220.jpg 4489 4489 4490 4490 For each level, the four parameters *Gi*, *Rg,i*, *Bi* and *Pi* must be chosen. … … 4497 4497 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4498 4498 4499 .. image:: img/image040. GIF4499 .. image:: img/image040.gif 4500 4500 4501 4501 ============== ======== ============= … … 4514 4514 ============== ======== ============= 4515 4515 4516 .. image:: img/image221. JPG4516 .. image:: img/image221.jpg 4517 4517 4518 4518 *Figure. 1D plot using the default values (w/500 data points).* … … 4567 4567 The scattered intensity *I(q)* is calculated as 4568 4568 4569 .. image:: img/image233. GIF4569 .. image:: img/image233.gif 4570 4570 4571 4571 where 4572 4572 4573 .. image:: img/image234. GIF4573 .. image:: img/image234.gif 4574 4574 4575 4575 Note that the first term reduces to the Ornstein-Zernicke equation when *D* = 2; ie, when the Flory exponent is 0.5 … … 4587 4587 ============================ ======== ============= 4588 4588 4589 .. image:: img/image235. GIF4589 .. image:: img/image235.gif 4590 4590 4591 4591 *Figure. 1D plot using the default values (w/300 data points).* … … 4609 4609 For a star with *f* arms: 4610 4610 4611 .. image:: img/star1. PNG4611 .. image:: img/star1.png 4612 4612 4613 4613 where 4614 4614 4615 .. image:: img/star2. PNG4615 .. image:: img/star2.png 4616 4616 4617 4617 and 4618 4618 4619 .. image:: img/star3. PNG4619 .. image:: img/star3.png 4620 4620 4621 4621 is the square of the ensemble average radius-of-gyration of an arm. … … 4646 4646 Also see ReflectivityIIModel_. 4647 4647 4648 .. image:: img/image231. BMP4648 .. image:: img/image231.bmp 4649 4649 4650 4650 *Figure. Comparison (using the SLD profile below) with the NIST web calculation (circles)* 4651 4651 http://www.ncnr.nist.gov/resources/reflcalc.html 4652 4652 4653 .. image:: img/image232. GIF4653 .. image:: img/image232.gif 4654 4654 4655 4655 *Figure. SLD profile used for the calculation (above).* … … 4675 4675 1) Erf 4676 4676 4677 .. image:: img/image051. GIF4677 .. image:: img/image051.gif 4678 4678 4679 4679 2) Power-Law 4680 4680 4681 .. image:: img/image050. GIF4681 .. image:: img/image050.gif 4682 4682 4683 4683 3) Exp 4684 4684 4685 .. image:: img/image049. GIF4685 .. image:: img/image049.gif 4686 4686 4687 4687 The constant *A* in the expressions above (but the parameter *nu* in the model!) is an input. … … 4713 4713 For a 2D plot, the wave transfer is defined as 4714 4714 4715 .. image:: img/image040. GIF4715 .. image:: img/image040.gif 4716 4716 4717 4717 ============== ======== ============= … … 4722 4722 ============== ======== ============= 4723 4723 4724 .. image:: img/image224. JPG4724 .. image:: img/image224.jpg 4725 4725 4726 4726 *Figure. 1D plot using the default values (in linear scale).* … … 4754 4754 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4755 4755 4756 .. image:: img/image040. GIF4756 .. image:: img/image040.gif 4757 4757 4758 4758 ============== ========= ============= … … 4765 4765 ============== ========= ============= 4766 4766 4767 .. image:: img/image226. JPG4767 .. image:: img/image226.jpg 4768 4768 4769 4769 *Figure. 1D plot using the default values (in linear scale).* … … 4806 4806 ============== ======== ============= 4807 4807 4808 .. image:: img/image227. JPG4808 .. image:: img/image227.jpg 4809 4809 4810 4810 *Figure. 1D plot using the default values (in linear scale).* … … 4852 4852 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4853 4853 4854 .. image:: img/image040. GIF4854 .. image:: img/image040.gif 4855 4855 4856 4856 ============== ======== ============= … … 4863 4863 ============== ======== ============= 4864 4864 4865 .. image:: img/image230. JPG4865 .. image:: img/image230.jpg 4866 4866 4867 4867 *Figure. 1D plot using the default values (in linear scale).*
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