Changeset 77cfcf0 in sasview


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Timestamp:
Apr 5, 2014 10:56:46 AM (10 years ago)
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Updated by SMK

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  • src/sans/models/media/model_functions.rst

    r38d4102 r77cfcf0  
    168168- CoreShellCylinderModel_ 
    169169- EllipticalCylinderModel_ 
    170 - FlexibleCylinderModel 
    171 - FlexCylEllipXModel 
    172 - CoreShellBicelleModel 
    173 - BarBellModel 
    174 - StackedDisksModel 
    175 - PringleModel 
     170- FlexibleCylinderModel_ 
     171- FlexCylEllipXModel_ 
     172- CoreShellBicelleModel_ 
     173- BarBellModel_ 
     174- StackedDisksModel_ 
     175- PringleModel_ 
    176176 
    177177Ellipsoid-based 
     
    180180- EllipsoidModel 
    181181- CoreShellEllipsoidModel 
     182- CoreShellEllipsoidXTModel 
    182183- TriaxialEllipsoidModel 
    183184 
     
    615616for a diagram and documentation. 
    616617 
    617 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     618The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    618619 
    619620Be careful! The SLDs and scale can be highly correlated. Hold as many of these parameters fixed as possible. 
     
    13431344*2.1.16.1. Definition* 
    13441345 
    1345 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     1346The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    13461347 
    13471348The Capped Cylinder geometry is defined as 
     
    17521753**2.1.21 CoreShellBicelleModel** 
    17531754 
    1754 This model provides the form factor for a circular cylinder with a 
    1755 core-shell scattering length density profile. The form factor is 
    1756 normalized by the particle volume. This model is a more general case 
    1757 of core-shell cylinder model (seeabove and reference below) in that 
    1758 the parameters of the shell are separated into a face-shell and a rim- 
    1759 shell so that users can set different values of the thicknesses and 
    1760 slds. 
    1761  
    1762  
    1763  
    1764 The returned value is scaled to units of |cm^-1| and the parameters of 
    1765 the core-shell cylinder model are the following: 
     1755This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The 
     1756form factor is normalized by the particle volume. 
     1757 
     1758This model is a more general case of core-shell cylinder model (see above and reference below) in that the parameters 
     1759of the shell are separated into a face-shell and a rim-shell so that users can set different values of the thicknesses 
     1760and SLDs. 
     1761 
     1762.. image:: img/image240.PNG 
     1763 
     1764*(Graphic from DOI: 10.1039/C0NP00002G)* 
     1765 
     1766The returned value is scaled to units of |cm^-1| and the parameters of the CoreShellBicelleModel are the following 
    17661767 
    17671768==============  ========  ============= 
     
    17821783==============  ========  ============= 
    17831784 
    1784 The output of the 1D scattering intensity function for randomly 
    1785 oriented cylinders is then given by the equation above. 
    1786  
    1787 The *axis_theta* and axis *_phi* parameters are not used for the 1D 
    1788 output. Our implementation of the scattering kernel and the 1D 
    1789 scattering intensity use the c-library from NIST. 
    1790  
    1791  
    1792  
    1793  
     1785The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. 
     1786 
     1787The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering kernel 
     1788and the 1D scattering intensity use the c-library from NIST. 
     1789 
     1790.. image:: img/cscylbicelle_pic.jpg 
    17941791 
    17951792*Figure. 1D plot using the default values (w/200 data point).* 
    17961793 
    1797  
    1798  
    1799 Figure. Definition of the angles for the oriented Core-Shell Cylinder 
    1800 Bicelle Model. 
    1801  
    1802  
    1803  
    1804 Figure. Examples of the angles for oriented pp against the detector 
    1805 plane. 
     1794.. image:: img/image061.JPG 
     1795 
     1796*Figure. Definition of the angles for the oriented CoreShellBicelleModel.* 
     1797 
     1798.. image:: img/image062.JPG 
     1799 
     1800*Figure. Examples of the angles for oriented pp against the detector plane.* 
    18061801 
    18071802REFERENCE 
    1808 Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle 
    1809 X-Ray and Neutron Scattering", Plenum Press, New York, (1987). 
     1803L. A. Feigin and D. I. Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, 
     1804New York, (1987) 
    18101805 
    18111806 
     
    18151810**2.1.22. BarBellModel** 
    18161811 
    1817 Calculates the scattering from a barbell-shaped cylinder (This model 
    1818 simply becomes the DumBellModel when the length of the cylinder, L, is 
    1819 set to zero). That is, a sphereocylinder with spherical end caps that 
    1820 have a radius larger than that of the cylinder and the center of the 
    1821 end cap radius lies outside of the cylinder All dimensions of the 
    1822 barbell are considered to be monodisperse. See the diagram for the 
    1823 details of the geometry and restrictions on parameter values. 
    1824  
    1825 *1.1. Definition* 
    1826  
    1827 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     1812Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of 
     1813the cylinder, *L*, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than 
     1814that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell 
     1815are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. 
     1816 
     1817*2.1.22.1. Definition* 
     1818 
     1819The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    18281820 
    18291821The barbell geometry is defined as 
    18301822 
    1831  
    1832  
    1833 r is the radius of the cylinder. All other parameters are as defined 
    1834 in the diagram. Since the end cap radius R >= r and by definition for 
    1835 this geometry h > 0, h is then defined by r and R as 
    1836  
    1837 h = sqrt(R^2 - r^2). 
    1838  
    1839 The scattering intensity I(q) is calculated as: 
    1840  
    1841  
    1842  
    1843 where the amplitude A(q) is given as: 
    1844  
    1845  
    1846  
    1847  
    1848  
    1849  
    1850  
    1851 The < > brackets denote an average of the structure over all 
    1852 orientations. <A^2(q)> is then the form factor, P(q). The scale factor 
    1853 is equivalent to the volume fraction of cylinders, each of volume, V. 
    1854 Contrast is the difference of scattering length densities of the 
    1855 cylinder and the surrounding solvent. 
    1856  
    1857 The volume of the barbell is: 
    1858  
    1859  
    1860  
    1861 and its radius of gyration: 
    1862  
    1863  
    1864  
    1865 The necessary conditions of R >= r is not enforced in the model. It is 
    1866 up to you to restrict this during analysis. 
    1867  
    1868 REFERENCES 
    1869  
    1870 H. Kaya, J. Appl. Cryst. (2004) 37, 223-230. 
    1871  
    1872 H. Kaya and N-R deSouza, J. Appl. Cryst. (2004) 37, 508-509. (addenda 
    1873 and errata) 
    1874  
    1875 TEST DATASET 
    1876  
    1877 This example dataset is produced by running the Macro PlotBarbell(), 
    1878 using 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 -1 and the above 
    1879 default values. 
     1823.. image:: img/image105.JPG 
     1824 
     1825where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. 
     1826 
     1827Since the end cap radius 
     1828*R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as 
     1829 
     1830*h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) 
     1831 
     1832The scattered intensity *I(q)* is calculated as 
     1833 
     1834.. image:: img/image106.PNG 
     1835 
     1836where the amplitude *A(q)* is given as 
     1837 
     1838.. image:: img/image107.PNG 
     1839 
     1840The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form 
     1841factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is 
     1842the difference of scattering length densities of the cylinder and the surrounding solvent. 
     1843 
     1844The volume of the barbell is 
     1845 
     1846.. image:: img/image108.JPG 
     1847 
     1848 
     1849and its radius of gyration is 
     1850 
     1851.. image:: img/image109.JPG 
     1852 
     1853**The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. 
     1854 
     1855This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, *qmin* = 0.001 |Ang^-1|, 
     1856*qmax* = 0.7 |Ang^-1| and the following default values 
    18801857 
    18811858==============  ========  ============= 
     
    18911868==============  ========  ============= 
    18921869 
    1893  
     1870.. image:: img/image110.JPG 
    18941871 
    18951872*Figure. 1D plot using the default values (w/256 data point).* 
    18961873 
    1897 For 2D data: The 2D scattering intensity is calculated similar to the 
    1898 2D cylinder model. At the theta = 45 deg and phi =0 deg with default 
    1899 values for other parameters, 
    1900  
    1901  
     1874For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for 
     1875|theta| = 45 deg and |phi| = 0 deg with default values for other parameters 
     1876 
     1877.. image:: img/image111.JPG 
    19021878 
    19031879*Figure. 2D plot (w/(256X265) data points).* 
    19041880 
    1905  
    1906  
    1907  
    1908  
    1909 Figure. Examples of the angles for oriented pp against the detector 
    1910 plane. 
     1881.. image:: img/image061.JPG 
     1882 
     1883*Figure. Examples of the angles for oriented pp against the detector plane.* 
     1884 
     1885.. image:: img/image062.JPG 
    19111886 
    19121887Figure. Definition of the angles for oriented 2D barbells. 
    19131888 
     1889REFERENCE 
     1890H. Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 
     1891H. Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) 
     1892 
    19141893 
    19151894 
     
    19181897**2.1.23. StackedDisksModel** 
    19191898 
    1920 This model provides the form factor, *P(q)*, for stacked discs 
    1921 (tactoids) with a core/layer structure where the form factor is 
    1922 normalized by the volume of the cylinder. Assuming the next neighbor 
    1923 distance (d-spacing) in a stack of parallel discs obeys a Gaussian 
    1924 distribution, a structure factor S(q) proposed by Kratky and Porod in 
    1925 1949 is used in this function. Note that the resolution smearing 
    1926 calculation uses 76 Gauss quadrature points to properly smear the 
    1927 model since the function is HIGHLY oscillatory, especially around the 
    1928 q-values that correspond to the repeat distance of the layers. 
    1929  
    1930 The 2D scattering intensity is the same as 1D, regardless of the 
    1931 orientation of the *q* vector which is defined as . 
    1932  
    1933  
    1934  
    1935  
    1936  
    1937  
    1938  
    1939 The returned value is in units of [|cm^-1| |sr^-1|, on absolute scale. 
    1940  
    1941 The scattering intensity I(q) is: 
    1942  
    1943  
    1944  
    1945 where the contrast, 
    1946  
    1947  
    1948  
    1949 N is the number of discs per unit volume, ais the angle between the 
    1950 axis of the disc and q, and Vt and Vc are the total volume and the 
    1951 core volume of a single disc, respectively. 
    1952  
    1953  
    1954  
    1955  
    1956  
    1957  
    1958  
    1959 where d = thickness of the layer (layer_thick), 2h= core thickness 
    1960 (core_thick), and R = radius of the disc (radius). 
    1961  
    1962  
    1963  
    1964 where n = the total number of the disc stacked (n_stacking), D=the 
    1965 next neighbor center to cent distance (d-spacing), and sD= the 
    1966 Gaussian standard deviation of the d-spacing (sigma_d). 
    1967  
    1968 To provide easy access to the orientation of the stackeddisks, we 
    1969 define the axis of the cylinder using two angles and . Similarly to 
    1970 the case of the cylinder, those angles are defined on Figure 2 of 
    1971 CylinderModel. 
    1972  
    1973 For P*S: The 2nd virial coefficient of the solid cylinder is calculate 
    1974 based on the (radius) and length = n_stacking*(core_thick 
    1975 +2*layer_thick) values, and used as the effective radius toward S(Q) 
    1976 when P(Q)*S(Q) is applied. 
     1899This model provides the form factor, *P(q)*, for stacked discs (tactoids) with a core/layer structure where the form 
     1900factor is normalized by the volume of the cylinder. Assuming the next neighbor distance (d-spacing) in a stack of 
     1901parallel discs obeys a Gaussian distribution, a structure factor *S(q)* proposed by Kratky and Porod in 1949 is used 
     1902in this function. 
     1903 
     1904Note that the resolution smearing calculation uses 76 Gauss quadrature points to properly smear the model since the 
     1905function is HIGHLY oscillatory, especially around the *q*-values that correspond to the repeat distance of the layers. 
     1906 
     1907The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
     1908 
     1909.. image:: img/image008.PNG 
     1910 
     1911The returned value is in units of |cm^-1| |sr^-1|, on absolute scale. 
     1912 
     1913*2.1.23.1 Definition* 
     1914 
     1915.. image:: img/image079.GIF 
     1916 
     1917The scattering intensity I(q) is 
     1918 
     1919.. image:: img/image081.PNG 
     1920 
     1921where the contrast 
     1922 
     1923.. image:: img/image082.PNG 
     1924 
     1925and *N* is the number of discs per unit volume, |alpha| is the angle between the axis of the disc and *q*, and *Vt* 
     1926and *Vc* are the total volume and the core volume of a single disc, respectively. 
     1927 
     1928.. image:: img/image083.PNG 
     1929 
     1930where *d* = thickness of the layer (*layer_thick*), 2\ *h* = core thickness (*core_thick*), and *R* = radius of the 
     1931disc (*radius*). 
     1932 
     1933.. image:: img/image084.PNG 
     1934 
     1935where *n* = the total number of the disc stacked (*n_stacking*), *D* = the next neighbor center-to-center distance 
     1936(*d-spacing*), and |sigma|\ D= the Gaussian standard deviation of the d-spacing (*sigma_d*). 
     1937 
     1938To provide easy access to the orientation of the stacked disks, we define the axis of the cylinder using two angles 
     1939|theta| and |phi|. These angles are defined on Figure 2 of CylinderModel. 
     1940 
     1941NB: The 2nd virial coefficient of the cylinder is calculated based on the *radius* and *length* = *n_stacking* \* 
     1942(*core_thick* + 2 \* *layer_thick*) values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 
    19771943 
    19781944==============  ========  ============= 
     
    19911957==============  ========  ============= 
    19921958 
    1993  
     1959.. image:: img/image085.JPG 
    19941960 
    19951961*Figure. 1D plot using the default values (w/1000 data point).* 
    19961962 
    1997  
    1998  
    1999 Figure. Examples of the angles for oriented stackeddisks against the 
    2000 detector plane. 
    2001  
    2002  
    2003  
    2004 Figure. Examples of the angles for oriented pp against the detector 
    2005 plane. 
    2006  
    2007 Our model uses the form factor calculations implemented in a c-library 
    2008 provided by the NIST Center for Neutron Research (Kline, 2006): 
     1963.. image:: img/image086.JPG 
     1964 
     1965*Figure. Examples of the angles for oriented stackeddisks against the detector plane.* 
     1966 
     1967.. image:: img/image062.JPG 
     1968 
     1969*Figure. Examples of the angles for oriented pp against the detector plane.* 
     1970 
     1971Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research 
     1972(Kline, 2006) 
    20091973 
    20101974REFERENCE 
    2011  
    2012 Guinier, A. and Fournet, G., "Small-Angle Scattering of X-Rays", John 
    2013 Wiley and Sons, New York, 1955. 
    2014  
    2015 Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949. 
    2016  
    2017 Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", 
    2018 Clarendon, Oxford, 1994. 
     1975A. Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, 1955 
     1976O. Kratky and G. Porod, *J. Colloid Science*, 4, (1949) 35 
     1977J. S. Higgins and H. C. Benoit, *Polymers and Neutron Scattering*, Clarendon, Oxford, 1994 
    20191978 
    20201979 
     
    20241983**2.1.24. PringleModel** 
    20251984 
    2026 This model provides the form factor, *P(q)*, for a 'pringle' or 
    2027 'saddle-shaped' object (a hyperbolic paraboloid). 
    2028  
    2029  
     1985This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). 
     1986 
     1987.. image:: img/image241.PNG 
     1988 
     1989*(Graphic from Matt Henderson, matt@matthen.com)* 
    20301990 
    20311991The returned value is in units of |cm^-1|, on absolute scale. 
    20321992 
    2033 The form factor calculated is: 
    2034  
    2035  
     1993The form factor calculated is 
     1994 
     1995.. image:: img/pringle_eqn_1.jpg 
    20361996 
    20371997where 
    20381998 
    2039  
    2040  
    2041  
    2042  
    2043 The parameters of the model and a plot comparing the pringle model 
    2044 with the equivalent cylinder are shown below. 
     1999.. image:: img/pringle_eqn_2.jpg 
     2000 
     2001The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below. 
    20452002 
    20462003==============  ========  ============= 
     
    20572014==============  ========  ============= 
    20582015 
    2059  
     2016.. image:: img/pringle-vs-cylinder.png 
    20602017 
    20612018*Figure. 1D plot using the default values (w/150 data point).* 
    20622019 
    20632020REFERENCE 
    2064  
    20652021S. Alexandru Rautu, Private Communication. 
    20662022 
     
    22482204 
    22492205 
     2206.. _CoreShellEllipsoidXTModel: 
     2207 
     2208**2.1.27. CoreShellEllipsoidXTModel** 
     2209 
     2210An alternative version of *P(q)* for the core-shell ellipsoid (see CoreShellEllipsoidModel), having as parameters the 
     2211core axial ratio *X* and a shell thickness, which are more often what we would like to determine. 
     2212 
     2213This model is also better behaved when polydispersity is applied than the four independent radii in 
     2214CoreShellEllipsoidModel. 
     2215 
     2216*2.1.27.1 Definition* 
     2217 
     2218.. image:: img/image125.gif 
     2219 
     2220The geometric parameters of this model are 
     2221 
     2222  *equat_core* = equatorial core radius = *Rminor_core* 
     2223  *X_core* = *polar_core* / *equat_core* = *Rmajor_core* / *Rminor_core* 
     2224  *T_shell* = *equat_outer* - *equat_core* = *Rminor_outer* - *Rminor_core* 
     2225  *XpolarShell* = *Tpolar_shell* / *T_shell* = (*Rmajor_outer* - *Rmajor_core*)/(*Rminor_outer* - *Rminor_core*) 
     2226 
     2227In terms of the original radii 
     2228 
     2229  *polar_core* = *equat_core* \* *X_core* 
     2230  *equat_shell* = *equat_core* + *T_shell* 
     2231  *polar_shell* = *equat_core* \* *X_core* + *T_shell* \* *XpolarShell* 
     2232 
     2233  (where we note that "shell" perhaps confusingly, relates to the outer radius) 
     2234 
     2235When *X_core* < 1 the core is oblate; when *X_core* > 1  it is prolate. *X_core* = 1 is a spherical core. 
     2236 
     2237For a fixed shell thickness *XpolarShell* = 1, to scale the shell thickness pro-rata with the radius 
     2238*XpolarShell* = *X_core*. 
     2239 
     2240When including an *S(q)*, the radius in *S(q)* is calculated to be that of a sphere with the same 2nd virial 
     2241coefficient of the **outer** surface of the ellipsoid. This may have some undesirable effects if the aspect ratio of 
     2242the ellipsoid is large (ie, if *X* << 1 or *X* >> 1), when the *S(q)* - which assumes spheres - will not in any case 
     2243be valid. 
     2244 
     2245If SANS data are in absolute units, and the SLDs are correct, then *scale* should be the total volume fraction of the 
     2246"outer particle". When *S(q)* is introduced this moves to the *S(q)* volume fraction, and *scale* should then be 1.0, 
     2247or contain some other units conversion factor (for example, if you have SAXS data). 
     2248 
     2249==============  ========  ============= 
     2250Parameter name  Units     Default value 
     2251==============  ========  ============= 
     2252background      |cm^-1|   0.001 
     2253equat_core      |Ang|     20 
     2254scale           None      0.05 
     2255sld_core        |Ang^-2|  2.0e-6 
     2256sld_shell       |Ang^-2|  1.0e-6 
     2257sld_solv        |Ang^-2|  6.3e-6 
     2258T_shell         |Ang|     30 
     2259X_core          None      3.0 
     2260XpolarShell     None      1.0 
     2261==============  ========  ============= 
     2262 
     2263REFERENCE 
     2264R. K. Heenan, Private communication 
     2265 
     2266 
     2267 
    22502268.. _TriaxialEllipsoidalModel: 
    22512269 
    2252 **2.1.27. TriaxialEllipsoidModel*** 
     2270**2.1.28. TriaxialEllipsoidModel** 
    22532271 
    22542272This model provides the form factor, *P(q)*, for an ellipsoid (below) 
     
    23322350.. _LamellarModel: 
    23332351 
    2334 **2.1.28. LamellarModel** 
     2352**2.1.29. LamellarModel** 
    23352353 
    23362354This model provides the scattering intensity, I( *q*), for a lyotropic 
     
    23852403.. _LamellarFFHGModel: 
    23862404 
    2387 **2.1.29. LamellarFFHGModel** 
     2405**2.1.30. LamellarFFHGModel** 
    23882406 
    23892407This model provides the scattering intensity, I( *q*), for a lyotropic 
     
    24422460.. _LamellarPSModel: 
    24432461 
    2444 **2.1.30. LamellarPSModel** 
     2462**2.1.31. LamellarPSModel** 
    24452463 
    24462464This model provides the scattering intensity ( *form factor* \* 
     
    25122530.. _LamellarPSHGModel: 
    25132531 
    2514 **2.1.31. LamellarPSHGModel** 
     2532**2.1.32. LamellarPSHGModel** 
    25152533 
    25162534This model provides the scattering intensity ( *form factor* \* 
     
    25932611.. _LamellarPCrystalModel: 
    25942612 
    2595 **2.1.32. LamellarPCrystalModel** 
     2613**2.1.33. LamellarPCrystalModel** 
    25962614 
    25972615Lamella ParaCrystal Model: Calculates the scattering from a stack of 
     
    26612679.. _SCCrystalModel: 
    26622680 
    2663 **2.1.33. SCCrystalModel** 
     2681**2.1.34. SCCrystalModel** 
    26642682 
    26652683Calculates the scattering from a simple cubic lattice with 
     
    26692687characterized by a Gaussian distribution. 
    26702688 
    2671 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     2689The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    26722690 
    26732691The scattering intensity I(q) is calculated as 
     
    27692787.. _FCCrystalModel: 
    27702788 
    2771 **2.1.34. FCCrystalModel** 
     2789**2.1.35. FCCrystalModel** 
    27722790 
    27732791Calculates the scattering from a face-centered cubic lattice with 
     
    27772795characterized by a Gaussian distribution. 
    27782796 
    2779 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     2797The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    27802798 
    27812799The scattering intensity I(q) is calculated as: 
     
    28682886.. _BCCrystalModel: 
    28692887 
    2870 **2.1.35. BCCrystalModel** 
     2888**2.1.36. BCCrystalModel** 
    28712889 
    28722890Calculates the scattering from a body-centered cubic lattice with 
     
    28752893Paracrystalline distortion is assumed to be isotropic and 
    28762894characterized by a Gaussian distribution.The returned value is scaled 
    2877 to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     2895to units of |cm^-1|\ |sr^-1|, absolute scale. 
    28782896 
    28792897The scattering intensity I(q) is calculated as: 
     
    29772995.. _ParallelepipedModel: 
    29782996 
    2979 **2.1.36. ParallelepipedModel** 
     2997**2.1.37. ParallelepipedModel** 
    29802998 
    29812999This model provides the form factor, *P(q)*, for a rectangular 
     
    30673085.. _CSParallelepipedModel: 
    30683086 
    3069 **2.1.37. CSParallelepipedModel** 
     3087**2.1.38. CSParallelepipedModel** 
    30703088 
    30713089Calculates the form factor for a rectangular solid with a core-shell 
     
    31473165 
    31483166This example dataset is produced by running the Macro 
    3149 Plot_CSParallelepiped(), using 100 data points, *qmin* = 0.001 -1, *qmax* 
     3167Plot_CSParallelepiped(), using 100 data points, *qmin* = 0.001 |Ang^-1|, *qmax* 
    31503168= 0.7 -1 and the below default values. 
    31513169 
     
    32543272structures). 
    32553273 
    3256 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     3274The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    32573275 
    32583276The scattering intensity I(q) is calculated by: 
     
    33273345a low-Q signal and a high-Q signal 
    33283346 
    3329 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     3347The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    33303348 
    33313349The scattering intensity I(q) is calculated by: 
     
    39603978provided. 
    39613979 
    3962 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     3980The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    39633981 
    39643982See each of these individual models for full documentation. 
     
    40414059structures. 
    40424060 
    4043 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4061The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    40444062 
    40454063The scattering intensity I(q) is calculated as (eqn 5 from the 
     
    42404258 
    42414259The returned value is P(Q) as written in equation (1), plus the 
    4242 incoherent background term. The result is in the units of [|cm^-1|\ |sr^-1|, 
     4260incoherent background term. The result is in the units of |cm^-1|\ |sr^-1|, 
    42434261absolute scale. 
    42444262 
     
    44894507molecular weight distribution. 
    44904508 
    4491 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4509The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    44924510 
    44934511 
     
    45124530 
    45134531This example dataset is produced by running the Poly_GaussCoil, using 
    4514 200 data points, *qmin* = 0.001 -1, *qmax* = 0.7 -1 and the default values 
     4532200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 -1 and the default values 
    45154533below. 
    45164534 
     
    45664584Calculates the scattering from polymers with excluded volume effects. 
    45674585 
    4568 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4586The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    45694587 
    45704588The returned value is P(Q) as written in equation (2), plus the 
    4571 incoherent background term. The result is in the units of [|cm^-1|\ |sr^-1|, 
     4589incoherent background term. The result is in the units of |cm^-1|\ |sr^-1|, 
    45724590absolute scale. 
    45734591 
     
    48244842a two Lorentzian functions. 
    48254843 
    4826 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4844The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    48274845 
    48284846The scattering intensity I(q) is calculated by: 
     
    49024920two power laws. 
    49034921 
    4904 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4922The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    49054923 
    49064924 
     
    49634981*3.25. UnifiedPower(Law and)Rg(Model)* 
    49644982 
    4965 The returned value is scaled to units of [|cm^-1|\ |sr^-1|, absolute scale. 
     4983The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. 
    49664984 
    49674985Note that the level 0 is an extra function that is the inverse 
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