Changeset 34e0c32 in sasview for src/sas/models/media/olddocs


Ignore:
Timestamp:
Mar 18, 2016 7:13:07 PM (9 years ago)
Author:
smk78
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:
354524d, 3689302
Parents:
b7a2ebfd
Message:

Image links changed to \img\olddocs

File:
1 edited

Legend:

Unmodified
Added
Removed

  • src/sas/models/media/olddocs/model_functions.rst

    r6f17afa r34e0c32  
    102102the particle 
    103103 
    104 .. image:: img/olddocs/image001.PNG 
     104.. image:: ..\img\olddocs\image001.PNG 
    105105 
    106106with 
    107107 
    108 .. image:: img/olddocs/image002.PNG 
     108.. image:: ..\img\olddocs\image002.PNG 
    109109 
    110110where |P0|\ *(q)* is the un-normalized form factor, |rho|\ *(r)* is the scattering length density at a given 
    … …  
    114114by the particle volume fraction 
    115115 
    116 .. image:: img/olddocs/image003.PNG 
     116.. image:: ..\img\olddocs\image003.PNG 
    117117 
    118118Our so-called 1D scattering intensity functions provide *P(q)* for the case where the scatterer is randomly oriented. In 
    … …  
    337337The 1D scattering intensity is calculated in the following way (Guinier, 1955) 
    338338 
    339 .. image:: img/olddocs/image004.PNG 
     339.. image:: ..\img\olddocs\image004.PNG 
    340340 
    341341where *scale* is a volume fraction, *V* is the volume of the scatterer, *r* is the radius of the sphere, *bkg* is 
    … …  
    372372NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 
    373373 
    374 .. image:: img/olddocs/image005.jpg 
     374.. image:: ..\img\olddocs\image005.jpg 
    375375 
    376376Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. 
    … …  
    391391solution 
    392392 
    393 .. image:: img/olddocs/image006.PNG 
     393.. image:: ..\img\olddocs\image006.PNG 
    394394 
    395395where *Sij* are the partial structure factors and *fi* are the scattering amplitudes of the particles. The subscript 1 
    … …  
    397397where *n* = the number density) is internally calculated based on 
    398398 
    399 .. image:: img/olddocs/image007.PNG 
     399.. image:: ..\img\olddocs\image007.PNG 
    400400 
    401401The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    402402 
    403 .. image:: img/olddocs/image008.PNG 
     403.. image:: ..\img\olddocs\image008.PNG 
    404404 
    405405The parameters of the BinaryHSModel are the following (in the names, *l* (or *ls*\ ) stands for larger spheres 
    … …  
    419419==============  ========  ============= 
    420420 
    421 .. image:: img/olddocs/image009.jpg 
     421.. image:: ..\img\olddocs\image009.jpg 
    422422 
    423423*Figure. 1D plot using the default values above (w/200 data point).* 
    … …  
    445445The scattering intensity *I(q)* is calculated as: 
    446446 
    447 .. image:: img/olddocs/image010.PNG 
     447.. image:: ..\img\olddocs\image010.PNG 
    448448 
    449449where the amplitude *A(q)* is given as the typical sphere scattering convoluted with a Gaussian to get a gradual 
    450450drop-off in the scattering length density 
    451451 
    452 .. image:: img/olddocs/image011.PNG 
     452.. image:: ..\img\olddocs\image011.PNG 
    453453 
    454454Here |A2|\ *(q)* is the form factor, *P(q)*. The scale is equivalent to the volume fraction of spheres, each of 
    … …  
    471471For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    472472 
    473 .. image:: img/olddocs/image008.PNG 
     473.. image:: ..\img\olddocs\image008.PNG 
    474474 
    475475This example dataset is produced by running the FuzzySphereModel, using 200 data points, *qmin* = 0.001 -1, 
    … …  
    487487==============  ========  ============= 
    488488 
    489 .. image:: img/olddocs/image012.jpg 
     489.. image:: ..\img\olddocs\image012.jpg 
    490490 
    491491*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    508508The structure is: 
    509509 
    510 .. image:: img/olddocs/raspberry_pic.jpg 
     510.. image:: ..\img\olddocs\raspberry_pic.jpg 
    511511 
    512512where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the 
    … …  
    523523For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    524524 
    525 .. image:: img/olddocs/image008.PNG 
     525.. image:: ..\img\olddocs\image008.PNG 
    526526 
    527527This example dataset is produced by running the RaspBerryModel, using 2000 data points, *qmin* = 0.0001 |Ang^-1|, 
    … …  
    544544==============  ========  ============= 
    545545 
    546 .. image:: img/olddocs/raspberry_plot.jpg 
     546.. image:: ..\img\olddocs\raspberry_plot.jpg 
    547547 
    548548*Figure. 1D plot using the values of /2000 data points.* 
    … …  
    568568The 1D scattering intensity is calculated in the following way (Guinier, 1955) 
    569569 
    570 .. image:: img/olddocs/image013.PNG 
     570.. image:: ..\img\olddocs\image013.PNG 
    571571 
    572572where *scale* is a scale factor, *Vs* is the volume of the outer shell, *Vc* is the volume of the core, *rs* is the 
    … …  
    608608NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 
    609609 
    610 .. image:: img/olddocs/image014.jpg 
     610.. image:: ..\img\olddocs\image014.jpg 
    611611 
    612612Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS 
    … …  
    669669*qmax* = 0.7 -1 and the above default values. 
    670670 
    671 .. image:: img/olddocs/image015.jpg 
     671.. image:: ..\img\olddocs\image015.jpg 
    672672 
    673673*Figure: 1D plot using the default values (w/200 data point).* 
    … …  
    675675The scattering length density profile for the default sld values (w/ 4 shells). 
    676676 
    677 .. image:: img/olddocs/image016.jpg 
     677.. image:: ..\img\olddocs\image016.jpg 
    678678 
    679679*Figure: SLD profile against the radius of the sphere for default SLDs.* 
    … …  
    704704The *I* :sub:`0` is calculated in the following way (King, 2002) 
    705705 
    706 .. image:: img/olddocs/secondmeq1.jpg 
     706.. image:: ..\img\olddocs\secondmeq1.jpg 
    707707 
    708708where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the 
    … …  
    731731==============  ========  ============= 
    732732 
    733 .. image:: img/olddocs/secongm_fig1.jpg 
     733.. image:: ..\img\olddocs\secongm_fig1.jpg 
    734734 
    735735REFERENCE 
    … …  
    747747solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). 
    748748 
    749 .. image:: img/olddocs/image020.jpg 
     749.. image:: ..\img\olddocs\image020.jpg 
    750750 
    751751The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    752752 
    753 .. image:: img/olddocs/image008.PNG 
     753.. image:: ..\img\olddocs\image008.PNG 
    754754 
    755755NB: The outer most radius (= *core_radius* + *n_pairs* \* *s_thickness* + (*n_pairs* - 1) \* *w_thickness*) is used 
    … …  
    774774is the number of shells. 
    775775 
    776 .. image:: img/olddocs/image021.jpg 
     776.. image:: ..\img\olddocs\image021.jpg 
    777777 
    778778*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    801801The 1D scattering intensity is calculated in the following way 
    802802 
    803 .. image:: img/olddocs/image022.gif 
    804  
    805 .. image:: img/olddocs/image023.gif 
     803.. image:: ..\img\olddocs\image022.gif 
     804 
     805.. image:: ..\img\olddocs\image023.gif 
    806806 
    807807where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 
    808808 
    809 .. image:: img/olddocs/image024.gif 
     809.. image:: ..\img\olddocs\image024.gif 
    810810 
    811811so that 
    812812 
    813 .. image:: img/olddocs/image025.gif 
    814  
    815 .. image:: img/olddocs/image026.gif 
    816  
    817 .. image:: img/olddocs/image027.gif 
     813.. image:: ..\img\olddocs\image025.gif 
     814 
     815.. image:: ..\img\olddocs\image026.gif 
     816 
     817.. image:: ..\img\olddocs\image027.gif 
    818818 
    819819Here we assumed that the SLDs of the core and solvent are constant against *r*. 
    … …  
    821821Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by 
    822822 
    823 .. image:: img/olddocs/image028.gif 
     823.. image:: ..\img\olddocs\image028.gif 
    824824 
    825825An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and 
    … …  
    829829For \| *A* \| > 0, 
    830830 
    831 .. image:: img/olddocs/image029.gif 
     831.. image:: ..\img\olddocs\image029.gif 
    832832 
    833833For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, 
    … …  
    835835so this case is equivalent to 
    836836 
    837 .. image:: img/olddocs/image030.gif 
    838  
    839 .. image:: img/olddocs/image031.gif 
    840  
    841 .. image:: img/olddocs/image032.gif 
    842  
    843 .. image:: img/olddocs/image033.gif 
     837.. image:: ..\img\olddocs\image030.gif 
     838 
     839.. image:: ..\img\olddocs\image031.gif 
     840 
     841.. image:: ..\img\olddocs\image032.gif 
     842 
     843.. image:: ..\img\olddocs\image033.gif 
    844844 
    845845For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is 
    … …  
    847847factor contributed by the shells is 
    848848 
    849 .. image:: img/olddocs/image034.gif 
    850  
    851 .. image:: img/olddocs/image035.gif 
     849.. image:: ..\img\olddocs\image034.gif 
     850 
     851.. image:: ..\img\olddocs\image035.gif 
    852852 
    853853In the equation 
    854854 
    855 .. image:: img/olddocs/image036.gif 
     855.. image:: ..\img\olddocs\image036.gif 
    856856 
    857857Finally, the form factor can be calculated by 
    858858 
    859 .. image:: img/olddocs/image037.gif 
     859.. image:: ..\img\olddocs\image037.gif 
    860860 
    861861where 
    862862 
    863 .. image:: img/olddocs/image038.gif 
     863.. image:: ..\img\olddocs\image038.gif 
    864864 
    865865and 
    866866 
    867 .. image:: img/olddocs/image039.gif 
     867.. image:: ..\img\olddocs\image039.gif 
    868868 
    869869The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    870870defined as 
    871871 
    872 .. image:: img/olddocs/image040.gif 
     872.. image:: ..\img\olddocs\image040.gif 
    873873 
    874874NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 
    … …  
    892892NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. 
    893893 
    894 .. image:: img/olddocs/image041.jpg 
     894.. image:: ..\img\olddocs\image041.jpg 
    895895 
    896896*Figure. 1D plot using the default values (w/400 point).* 
    897897 
    898 .. image:: img/olddocs/image042.jpg 
     898.. image:: ..\img\olddocs\image042.jpg 
    899899 
    900900*Figure. SLD profile from the default values.* 
    … …  
    918918The 1D scattering intensity is calculated in the following way (Guinier, 1955) 
    919919 
    920 .. image:: img/olddocs/image017.PNG 
     920.. image:: ..\img\olddocs\image017.PNG 
    921921 
    922922where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total 
    … …  
    928928and a shell thickness, *t*. 
    929929 
    930 .. image:: img/olddocs/image018.jpg 
     930.. image:: ..\img\olddocs\image018.jpg 
    931931 
    932932The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    933933defined as 
    934934 
    935 .. image:: img/olddocs/image008.PNG 
     935.. image:: ..\img\olddocs\image008.PNG 
    936936 
    937937NB: The outer most radius (= *radius* + *thickness*) is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* 
    … …  
    953953NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. 
    954954 
    955 .. image:: img/olddocs/image019.jpg 
     955.. image:: ..\img\olddocs\image019.jpg 
    956956 
    957957*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    984984The 1D scattering intensity is calculated in the following way: 
    985985 
    986 .. image:: img/olddocs/image022.gif 
    987  
    988 .. image:: img/olddocs/image043.gif 
     986.. image:: ..\img\olddocs\image022.gif 
     987 
     988.. image:: ..\img\olddocs\image043.gif 
    989989 
    990990where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 
    991991 
    992 .. image:: img/olddocs/image024.gif 
     992.. image:: ..\img\olddocs\image024.gif 
    993993 
    994994so that 
    995995 
    996 .. image:: img/olddocs/image044.gif 
    997  
    998 .. image:: img/olddocs/image045.gif 
    999  
    1000 .. image:: img/olddocs/image046.gif 
    1001  
    1002 .. image:: img/olddocs/image047.gif 
    1003  
    1004 .. image:: img/olddocs/image048.gif 
    1005  
    1006 .. image:: img/olddocs/image027.gif 
     996.. image:: ..\img\olddocs\image044.gif 
     997 
     998.. image:: ..\img\olddocs\image045.gif 
     999 
     1000.. image:: ..\img\olddocs\image046.gif 
     1001 
     1002.. image:: ..\img\olddocs\image047.gif 
     1003 
     1004.. image:: ..\img\olddocs\image048.gif 
     1005 
     1006.. image:: ..\img\olddocs\image027.gif 
    10071007 
    10081008Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between 
    … …  
    101110111) Exp 
    10121012 
    1013 .. image:: img/olddocs/image049.gif 
     1013.. image:: ..\img\olddocs\image049.gif 
    10141014 
    101510152) Power-Law 
    10161016 
    1017 .. image:: img/olddocs/image050.gif 
     1017.. image:: ..\img\olddocs\image050.gif 
    10181018 
    101910193) Erf 
    10201020 
    1021 .. image:: img/olddocs/image051.gif 
     1021.. image:: ..\img\olddocs\image051.gif 
    10221022 
    10231023The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is 
    … …  
    10271027to the form factor *P(q)* 
    10281028 
    1029 .. image:: img/olddocs/image052.gif 
    1030  
    1031 .. image:: img/olddocs/image053.gif 
    1032  
    1033 .. image:: img/olddocs/image054.gif 
     1029.. image:: ..\img\olddocs\image052.gif 
     1030 
     1031.. image:: ..\img\olddocs\image053.gif 
     1032 
     1033.. image:: ..\img\olddocs\image054.gif 
    10341034 
    10351035where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. 
    … …  
    10371037In the equation 
    10381038 
    1039 .. image:: img/olddocs/image055.gif 
     1039.. image:: ..\img\olddocs\image055.gif 
    10401040 
    10411041Finally, the form factor can be calculated by 
    10421042 
    1043 .. image:: img/olddocs/image037.gif 
     1043.. image:: ..\img\olddocs\image037.gif 
    10441044 
    10451045where 
    10461046 
    1047 .. image:: img/olddocs/image038.gif 
     1047.. image:: ..\img\olddocs\image038.gif 
    10481048 
    10491049and 
    10501050 
    1051 .. image:: img/olddocs/image056.gif 
     1051.. image:: ..\img\olddocs\image056.gif 
    10521052 
    10531053The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    10541054defined as 
    10551055 
    1056 .. image:: img/olddocs/image040.gif 
     1056.. image:: ..\img\olddocs\image040.gif 
    10571057 
    10581058NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 
    … …  
    10811081NB: *rad_core0* represents the core radius (*R1*). 
    10821082 
    1083 .. image:: img/olddocs/image057.jpg 
     1083.. image:: ..\img\olddocs\image057.jpg 
    10841084 
    10851085*Figure. 1D plot using the default values (w/400 point).* 
    10861086 
    1087 .. image:: img/olddocs/image058.jpg 
     1087.. image:: ..\img\olddocs\image058.jpg 
    10881088 
    10891089*Figure. SLD profile from the default values.* 
    … …  
    11041104of each string is assumed to be negligible. 
    11051105 
    1106 .. image:: img/olddocs/linearpearls.jpg 
     1106.. image:: ..\img\olddocs\linearpearls.jpg 
    11071107 
    11081108*2.1.12.1. Definition* 
    … …  
    11101110The output of the scattering intensity function for the LinearPearlsModel is given by (Dobrynin, 1996) 
    11111111 
    1112 .. image:: img/olddocs/linearpearl_eq1.gif 
     1112.. image:: ..\img\olddocs\linearpearl_eq1.gif 
    11131113 
    11141114where the mass *m*\ :sub:`p` is (SLD\ :sub:`pearl` - SLD\ :sub:`solvent`) \* (volume of *N* pearls). V is the total 
    … …  
    11331133NB: *num_pearls* must be an integer. 
    11341134 
    1135 .. image:: img/olddocs/linearpearl_plot.jpg 
     1135.. image:: ..\img\olddocs\linearpearl_plot.jpg 
    11361136 
    11371137REFERENCE 
    … …  
    11501150distance. 
    11511151 
    1152 .. image:: img/olddocs/pearl_fig.jpg 
     1152.. image:: ..\img\olddocs\pearl_fig.jpg 
    11531153 
    11541154*2.1.13.1. Definition* 
    … …  
    11561156The output of the scattering intensity function for the PearlNecklaceModel is given by (Schweins, 2004) 
    11571157 
    1158 .. image:: img/olddocs/pearl_eq1.gif 
     1158.. image:: ..\img\olddocs\pearl_eq1.gif 
    11591159 
    11601160where 
    11611161 
    1162 .. image:: img/olddocs/pearl_eq2.gif 
    1163  
    1164 .. image:: img/olddocs/pearl_eq3.gif 
    1165  
    1166 .. image:: img/olddocs/pearl_eq4.gif 
    1167  
    1168 .. image:: img/olddocs/pearl_eq5.gif 
    1169  
    1170 .. image:: img/olddocs/pearl_eq6.gif 
     1162.. image:: ..\img\olddocs\pearl_eq2.gif 
     1163 
     1164.. image:: ..\img\olddocs\pearl_eq3.gif 
     1165 
     1166.. image:: ..\img\olddocs\pearl_eq4.gif 
     1167 
     1168.. image:: ..\img\olddocs\pearl_eq5.gif 
     1169 
     1170.. image:: ..\img\olddocs\pearl_eq6.gif 
    11711171 
    11721172and 
    11731173 
    1174 .. image:: img/olddocs/pearl_eq7.gif 
     1174.. image:: ..\img\olddocs\pearl_eq7.gif 
    11751175 
    11761176where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* (volume of the *N* pearls/rods). *V* is the 
    … …  
    11981198NB: *num_pearls* must be an integer. 
    11991199 
    1200 .. image:: img/olddocs/pearl_plot.jpg 
     1200.. image:: ..\img\olddocs\pearl_plot.jpg 
    12011201 
    12021202REFERENCE 
    … …  
    12191219The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955) 
    12201220 
    1221 .. image:: img/olddocs/image059.PNG 
     1221.. image:: ..\img\olddocs\image059.PNG 
    12221222 
    12231223where 
    12241224 
    1225 .. image:: img/olddocs/image060.PNG 
     1225.. image:: ..\img\olddocs\image060.PNG 
    12261226 
    12271227and |alpha| is the angle between the axis of the cylinder and the *q*-vector, *V* is the volume of the cylinder, 
    … …  
    12321232and |phi|. Those angles are defined in Figure 1. 
    12331233 
    1234 .. image:: img/olddocs/image061.jpg 
     1234.. image:: ..\img\olddocs\image061.jpg 
    12351235 
    12361236*Figure 1. Definition of the angles for oriented cylinders.* 
    12371237 
    1238 .. image:: img/olddocs/image062.jpg 
     1238.. image:: ..\img\olddocs\image062.jpg 
    12391239 
    12401240*Figure 2. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    12591259The output of the 1D scattering intensity function for randomly oriented cylinders is then given by 
    12601260 
    1261 .. image:: img/olddocs/image063.PNG 
     1261.. image:: ..\img\olddocs\image063.PNG 
    12621262 
    12631263The *cyl_theta* and *cyl_phi* parameter are not used for the 1D output. Our implementation of the scattering kernel 
    … …  
    12691269NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. 
    12701270 
    1271 .. image:: img/olddocs/image065.jpg 
     1271.. image:: ..\img\olddocs\image065.jpg 
    12721272 
    12731273*Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis* 
    … …  
    12771277In general, averaging over a distribution of orientations is done by evaluating the following 
    12781278 
    1279 .. image:: img/olddocs/image064.PNG 
     1279.. image:: ..\img\olddocs\image064.PNG 
    12801280 
    12811281where *p(*\ |theta|,\ |phi|\ *)* is the probability distribution for the orientation and |P0|\ *(q,*\ |alpha|\ *)* is 
    … …  
    12841284distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. 
    12851285 
    1286 .. image:: img/olddocs/image066.jpg 
     1286.. image:: ..\img\olddocs\image066.jpg 
    12871287 
    12881288*Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the* 
    … …  
    13091309The 1D scattering intensity is calculated in the following way (Guinier, 1955) 
    13101310 
    1311 .. image:: img/olddocs/image072.PNG 
     1311.. image:: ..\img\olddocs\image072.PNG 
    13121312 
    13131313where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x* - *x* cos *x*)/ *x*\ :sup:`2`. 
    … …  
    13341334==============  ========  ============= 
    13351335 
    1336 .. image:: img/olddocs/image074.jpg 
     1336.. image:: ..\img\olddocs\image074.jpg 
    13371337 
    13381338*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    13411341(Kline, 2006). 
    13421342 
    1343 .. image:: img/olddocs/image061.jpg 
     1343.. image:: ..\img\olddocs\image061.jpg 
    13441344 
    13451345*Figure. Definition of the angles for the oriented HollowCylinderModel.* 
    13461346 
    1347 .. image:: img/olddocs/image062.jpg 
     1347.. image:: ..\img\olddocs\image062.jpg 
    13481348 
    13491349*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    13711371The Capped Cylinder geometry is defined as 
    13721372 
    1373 .. image:: img/olddocs/image112.jpg 
     1373.. image:: ..\img\olddocs\image112.jpg 
    13741374 
    13751375where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius 
    … …  
    13801380The scattered intensity *I(q)* is calculated as 
    13811381 
    1382 .. image:: img/olddocs/image113.jpg 
     1382.. image:: ..\img\olddocs\image113.jpg 
    13831383 
    13841384where the amplitude *A(q)* is given as 
    13851385 
    1386 .. image:: img/olddocs/image114.jpg 
     1386.. image:: ..\img\olddocs\image114.jpg 
    13871387 
    13881388The < > brackets denote an average of the structure over all orientations. <\ *A*\ :sup:`2`\ *(q)*> is then the form 
    … …  
    13921392The volume of the Capped Cylinder is (with *h* as a positive value here) 
    13931393 
    1394 .. image:: img/olddocs/image115.jpg 
     1394.. image:: ..\img\olddocs\image115.jpg 
    13951395 
    13961396and its radius-of-gyration 
    13971397 
    1398 .. image:: img/olddocs/image116.jpg 
     1398.. image:: ..\img\olddocs\image116.jpg 
    13991399 
    14001400**The requirement that** *R* >= *r* **is not enforced in the model! It is up to you to restrict this during analysis.** 
    … …  
    14151415==============  ========  ============= 
    14161416 
    1417 .. image:: img/olddocs/image117.jpg 
     1417.. image:: ..\img\olddocs\image117.jpg 
    14181418 
    14191419*Figure. 1D plot using the default values (w/256 data point).* 
    … …  
    14221422|theta| = 45 deg and |phi| =0 deg with default values for other parameters 
    14231423 
    1424 .. image:: img/olddocs/image118.jpg 
     1424.. image:: ..\img\olddocs\image118.jpg 
    14251425 
    14261426*Figure. 2D plot (w/(256X265) data points).* 
    14271427 
    1428 .. image:: img/olddocs/image061.jpg 
     1428.. image:: ..\img\olddocs\image061.jpg 
    14291429 
    14301430*Figure. Definition of the angles for oriented 2D cylinders.* 
    14311431 
    1432 .. image:: img/olddocs/image062.jpg 
     1432.. image:: ..\img\olddocs\image062.jpg 
    14331433 
    14341434*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    14531453The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006) 
    14541454 
    1455 .. image:: img/olddocs/image067.PNG 
     1455.. image:: ..\img\olddocs\image067.PNG 
    14561456 
    14571457where 
    14581458 
    1459 .. image:: img/olddocs/image068.PNG 
    1460  
    1461 .. image:: img/olddocs/image239.PNG 
     1459.. image:: ..\img\olddocs\image068.PNG 
     1460 
     1461.. image:: ..\img\olddocs\image239.PNG 
    14621462 
    14631463and |alpha| is the angle between the axis of the cylinder and the *q*\ -vector, *Vs* is the volume of the outer shell 
    … …  
    14681468the outer shell is given by *L+2t*. *J1* is the first order Bessel function. 
    14691469 
    1470 .. image:: img/olddocs/image069.jpg 
     1470.. image:: ..\img\olddocs\image069.jpg 
    14711471 
    14721472To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two 
    … …  
    15031503NIST (Kline, 2006). Figure 1 shows a comparison of the 1D output of our model and the output of the NIST software. 
    15041504 
    1505 .. image:: img/olddocs/image070.jpg 
     1505.. image:: ..\img\olddocs\image070.jpg 
    15061506 
    15071507*Figure 1: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS* 
    … …  
    151415142D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a cross-check. 
    15151515 
    1516 .. image:: img/olddocs/image071.jpg 
     1516.. image:: ..\img\olddocs\image071.jpg 
    15171517 
    15181518*Figure 2: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and* 
    … …  
    15211521*Solvent_sld* = 1e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. 
    15221522 
    1523 .. image:: img/olddocs/image061.jpg 
     1523.. image:: ..\img\olddocs\image061.jpg 
    15241524 
    15251525*Figure. Definition of the angles for oriented core-shell cylinders.* 
    15261526 
    1527 .. image:: img/olddocs/image062.jpg 
     1527.. image:: ..\img\olddocs\image062.jpg 
    15281528 
    15291529*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    15451545to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. 
    15461546 
    1547 .. image:: img/olddocs/image098.gif 
     1547.. image:: ..\img\olddocs\image098.gif 
    15481548 
    15491549*Figure.* *a* = *r_minor* and |nu|\ :sub:`n` = *r_ratio* (i.e., *r_major* / *r_minor*). 
    … …  
    15511551The function calculated is 
    15521552 
    1553 .. image:: img/olddocs/image099.PNG 
     1553.. image:: ..\img\olddocs\image099.PNG 
    15541554 
    15551555with the functions 
    15561556 
    1557 .. image:: img/olddocs/image100.PNG 
     1557.. image:: ..\img\olddocs\image100.PNG 
    15581558 
    15591559and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector *q*\ . 
    … …  
    15741574All angle parameters are valid and given only for 2D calculation; ie, an oriented system. 
    15751575 
    1576 .. image:: img/olddocs/image101.jpg 
     1576.. image:: ..\img\olddocs\image101.jpg 
    15771577 
    15781578*Figure. Definition of angles for 2D* 
    15791579 
    1580 .. image:: img/olddocs/image062.jpg 
     1580.. image:: ..\img\olddocs\image062.jpg 
    15811581 
    15821582*Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* 
    … …  
    15971597==============  ========  ============= 
    15981598 
    1599 .. image:: img/olddocs/image102.jpg 
     1599.. image:: ..\img\olddocs\image102.jpg 
    16001600 
    16011601*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    16081608and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). 
    16091609 
    1610 .. image:: img/olddocs/image103.gif 
     1610.. image:: ..\img\olddocs\image103.gif 
    16111611 
    16121612*Figure. Comparison between 1D and averaged 2D.* 
    … …  
    16151615the results of the averaging by varying the number of angular bins. 
    16161616 
    1617 .. image:: img/olddocs/image104.gif 
     1617.. image:: ..\img\olddocs\image104.gif 
    16181618 
    16191619*Figure. The intensities averaged from 2D over different numbers of bins and angles.* 
    … …  
    16391639The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    16401640 
    1641 .. image:: img/olddocs/image040.gif 
     1641.. image:: ..\img\olddocs\image040.gif 
    16421642 
    16431643*2.1.19.1. Definition* 
    16441644 
    1645 .. image:: img/olddocs/image075.jpg 
     1645.. image:: ..\img\olddocs\image075.jpg 
    16461646 
    16471647The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff 
    … …  
    16661666==============  ========  ============= 
    16671667 
    1668 .. image:: img/olddocs/image076.jpg 
     1668.. image:: ..\img\olddocs\image076.jpg 
    16691669 
    16701670*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    17211721- 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. 
    17221722 
    1723 .. image:: img/olddocs/image077.jpg 
     1723.. image:: ..\img\olddocs\image077.jpg 
    17241724 
    17251725The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff 
    … …  
    17451745For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    17461746 
    1747 .. image:: img/olddocs/image008.PNG 
     1747.. image:: ..\img\olddocs\image008.PNG 
    17481748 
    17491749This example dataset is produced by running the Macro FlexCylEllipXModel, using 200 data points, *qmin* = 0.001 |Ang^-1|, 
    … …  
    17631763==============  ========  ============= 
    17641764 
    1765 .. image:: img/olddocs/image078.jpg 
     1765.. image:: ..\img\olddocs\image078.jpg 
    17661766 
    17671767*Figure. 1D plot using the default values (w/200 data points).* 
    … …  
    17901790and SLDs. 
    17911791 
    1792 .. image:: img/olddocs/image240.png 
     1792.. image:: ..\img\olddocs\image240.png 
    17931793 
    17941794*(Graphic from DOI: 10.1039/C0NP00002G)* 
    … …  
    18181818and the 1D scattering intensity use the c-library from NIST. 
    18191819 
    1820 .. image:: img/olddocs/cscylbicelle_pic.jpg 
     1820.. image:: ..\img\olddocs\cscylbicelle_pic.jpg 
    18211821 
    18221822*Figure. 1D plot using the default values (w/200 data point).* 
    18231823 
    1824 .. image:: img/olddocs/image061.jpg 
     1824.. image:: ..\img\olddocs\image061.jpg 
    18251825 
    18261826*Figure. Definition of the angles for the oriented CoreShellBicelleModel.* 
    18271827 
    1828 .. image:: img/olddocs/image062.jpg 
     1828.. image:: ..\img\olddocs\image062.jpg 
    18291829 
    18301830*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    18521852The barbell geometry is defined as 
    18531853 
    1854 .. image:: img/olddocs/image105.jpg 
     1854.. image:: ..\img\olddocs\image105.jpg 
    18551855 
    18561856where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. 
    … …  
    18631863The scattered intensity *I(q)* is calculated as 
    18641864 
    1865 .. image:: img/olddocs/image106.PNG 
     1865.. image:: ..\img\olddocs\image106.PNG 
    18661866 
    18671867where the amplitude *A(q)* is given as 
    18681868 
    1869 .. image:: img/olddocs/image107.PNG 
     1869.. image:: ..\img\olddocs\image107.PNG 
    18701870 
    18711871The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form 
    … …  
    18751875The volume of the barbell is 
    18761876 
    1877 .. image:: img/olddocs/image108.jpg 
     1877.. image:: ..\img\olddocs\image108.jpg 
    18781878 
    18791879 
    18801880and its radius-of-gyration is 
    18811881 
    1882 .. image:: img/olddocs/image109.jpg 
     1882.. image:: ..\img\olddocs\image109.jpg 
    18831883 
    18841884**The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. 
    … …  
    18991899==============  ========  ============= 
    19001900 
    1901 .. image:: img/olddocs/image110.jpg 
     1901.. image:: ..\img\olddocs\image110.jpg 
    19021902 
    19031903*Figure. 1D plot using the default values (w/256 data point).* 
    … …  
    19061906|theta| = 45 deg and |phi| = 0 deg with default values for other parameters 
    19071907 
    1908 .. image:: img/olddocs/image111.jpg 
     1908.. image:: ..\img\olddocs\image111.jpg 
    19091909 
    19101910*Figure. 2D plot (w/(256X265) data points).* 
    19111911 
    1912 .. image:: img/olddocs/image061.jpg 
     1912.. image:: ..\img\olddocs\image061.jpg 
    19131913 
    19141914*Figure. Examples of the angles for oriented pp against the detector plane.* 
    19151915 
    1916 .. image:: img/olddocs/image062.jpg 
     1916.. image:: ..\img\olddocs\image062.jpg 
    19171917 
    19181918Figure. Definition of the angles for oriented 2D barbells. 
    … …  
    19401940The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    19411941 
    1942 .. image:: img/olddocs/image008.PNG 
     1942.. image:: ..\img\olddocs\image008.PNG 
    19431943 
    19441944The returned value is in units of |cm^-1| |sr^-1|, on absolute scale. 
    … …  
    19461946*2.1.23.1 Definition* 
    19471947 
    1948 .. image:: img/olddocs/image079.gif 
     1948.. image:: ..\img\olddocs\image079.gif 
    19491949 
    19501950The scattering intensity *I(q)* is 
    19511951 
    1952 .. image:: img/olddocs/image081.PNG 
     1952.. image:: ..\img\olddocs\image081.PNG 
    19531953 
    19541954where the contrast 
    19551955 
    1956 .. image:: img/olddocs/image082.PNG 
     1956.. image:: ..\img\olddocs\image082.PNG 
    19571957 
    19581958and *N* is the number of discs per unit volume, |alpha| is the angle between the axis of the disc and *q*, and *Vt* 
    19591959and *Vc* are the total volume and the core volume of a single disc, respectively. 
    19601960 
    1961 .. image:: img/olddocs/image083.PNG 
     1961.. image:: ..\img\olddocs\image083.PNG 
    19621962 
    19631963where *d* = thickness of the layer (*layer_thick*), 2\ *h* = core thickness (*core_thick*), and *R* = radius of the 
    19641964disc (*radius*). 
    19651965 
    1966 .. image:: img/olddocs/image084.PNG 
     1966.. image:: ..\img\olddocs\image084.PNG 
    19671967 
    19681968where *n* = the total number of the disc stacked (*n_stacking*), *D* = the next neighbor center-to-center distance 
    … …  
    19901990==============  ========  ============= 
    19911991 
    1992 .. image:: img/olddocs/image085.jpg 
     1992.. image:: ..\img\olddocs\image085.jpg 
    19931993 
    19941994*Figure. 1D plot using the default values (w/1000 data point).* 
    19951995 
    1996 .. image:: img/olddocs/image086.jpg 
     1996.. image:: ..\img\olddocs\image086.jpg 
    19971997 
    19981998*Figure. Examples of the angles for oriented stackeddisks against the detector plane.* 
    19991999 
    2000 .. image:: img/olddocs/image062.jpg 
     2000.. image:: ..\img\olddocs\image062.jpg 
    20012001 
    20022002*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    20212021This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). 
    20222022 
    2023 .. image:: img/olddocs/image241.png 
     2023.. image:: ..\img\olddocs\image241.png 
    20242024 
    20252025*(Graphic from Matt Henderson, matt@matthen.com)* 
    … …  
    20292029The form factor calculated is 
    20302030 
    2031 .. image:: img/olddocs/pringle_eqn_1.jpg 
     2031.. image:: ..\img\olddocs\pringle_eqn_1.jpg 
    20322032 
    20332033where 
    20342034 
    2035 .. image:: img/olddocs/pringle_eqn_2.jpg 
     2035.. image:: ..\img\olddocs\pringle_eqn_2.jpg 
    20362036 
    20372037The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below. 
    … …  
    20502050==============  ========  ============= 
    20512051 
    2052 .. image:: img/olddocs/pringle-vs-cylinder.png 
     2052.. image:: ..\img\olddocs\pringle-vs-cylinder.png 
    20532053 
    20542054*Figure. 1D plot using the default values (w/150 data point).* 
    … …  
    20712071The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987) 
    20722072 
    2073 .. image:: img/olddocs/image059.PNG 
     2073.. image:: ..\img\olddocs\image059.PNG 
    20742074 
    20752075where 
    20762076 
    2077 .. image:: img/olddocs/image119.PNG 
     2077.. image:: ..\img\olddocs\image119.PNG 
    20782078 
    20792079and 
    20802080 
    2081 .. image:: img/olddocs/image120.PNG 
     2081.. image:: ..\img\olddocs\image120.PNG 
    20822082 
    20832083|alpha| is the angle between the axis of the ellipsoid and the *q*\ -vector, *V* is the volume of the ellipsoid, *Ra* 
    … …  
    21112111above. 
    21122112 
    2113 .. image:: img/olddocs/image121.jpg 
     2113.. image:: ..\img\olddocs\image121.jpg 
    21142114 
    21152115The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering 
    21162116kernel and the 1D scattering intensity use the c-library from NIST. 
    21172117 
    2118 .. image:: img/olddocs/image122.jpg 
     2118.. image:: ..\img\olddocs\image122.jpg 
    21192119 
    21202120*Figure. The angles for oriented ellipsoid.* 
    … …  
    21262126software. 
    21272127 
    2128 .. image:: img/olddocs/image123.jpg 
     2128.. image:: ..\img\olddocs\image123.jpg 
    21292129 
    21302130*Figure 1: Comparison of the SasView scattering intensity for an ellipsoid with the output of the NIST SANS analysis* 
    … …  
    21372137cross-check. 
    21382138 
    2139 .. image:: img/olddocs/image124.jpg 
     2139.. image:: ..\img\olddocs\image124.jpg 
    21402140 
    21412141*Figure 2: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the* 
    … …  
    21692169all orientations for 1D. 
    21702170 
    2171 .. image:: img/olddocs/image125.gif 
     2171.. image:: ..\img\olddocs\image125.gif 
    21722172 
    21732173The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    21772177The form factor calculated is 
    21782178 
    2179 .. image:: img/olddocs/image126.PNG 
     2179.. image:: ..\img\olddocs\image126.PNG 
    21802180 
    21812181To provide easy access to the orientation of the core-shell ellipsoid, we define the axis of the solid ellipsoid using 
    … …  
    22032203==============  ========  ============= 
    22042204 
    2205 .. image:: img/olddocs/image127.jpg 
     2205.. image:: ..\img\olddocs\image127.jpg 
    22062206 
    22072207*Figure. 1D plot using the default values (w/1000 data point).* 
    22082208 
    2209 .. image:: img/olddocs/image122.jpg 
     2209.. image:: ..\img\olddocs\image122.jpg 
    22102210 
    22112211*Figure. The angles for oriented CoreShellEllipsoid.* 
    … …  
    22342234*2.1.27.1. Definition* 
    22352235 
    2236 .. image:: img/olddocs/image125.gif 
     2236.. image:: ..\img\olddocs\image125.gif 
    22372237 
    22382238The geometric parameters of this model are 
    … …  
    23012301where the volume *V* = (4/3)\ |pi| (*Ra* *Rb* *Rc*), and the averaging < > is applied over all orientations for 1D. 
    23022302 
    2303 .. image:: img/olddocs/image128.jpg 
     2303.. image:: ..\img\olddocs\image128.jpg 
    23042304 
    23052305The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    23092309The form factor calculated is 
    23102310 
    2311 .. image:: img/olddocs/image129.PNG 
     2311.. image:: ..\img\olddocs\image129.PNG 
    23122312 
    23132313To provide easy access to the orientation of the triaxial ellipsoid, we define the axis of the cylinder using the 
    … …  
    23372337==============  ========  ============= 
    23382338 
    2339 .. image:: img/olddocs/image130.jpg 
     2339.. image:: ..\img\olddocs\image130.jpg 
    23402340 
    23412341*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    23482348angles of |theta|, |phi|, and |psi| respectively). 
    23492349 
    2350 .. image:: img/olddocs/image131.gif 
     2350.. image:: ..\img\olddocs\image131.gif 
    23512351 
    23522352*Figure. Comparison between 1D and averaged 2D.* 
    23532353 
    2354 .. image:: img/olddocs/image132.jpg 
     2354.. image:: ..\img\olddocs\image132.jpg 
    23552355 
    23562356*Figure. The angles for oriented ellipsoid.* 
    … …  
    23772377The scattering intensity *I(q)* is 
    23782378 
    2379 .. image:: img/olddocs/image133.PNG 
     2379.. image:: ..\img\olddocs\image133.PNG 
    23802380 
    23812381The form factor is 
    23822382 
    2383 .. image:: img/olddocs/image134.PNG 
     2383.. image:: ..\img\olddocs\image134.PNG 
    23842384 
    23852385where |delta| = bilayer thickness. 
    … …  
    23872387The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    23882388 
    2389 .. image:: img/olddocs/image040.gif 
     2389.. image:: ..\img\olddocs\image040.gif 
    23902390 
    23912391The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, 
    … …  
    24022402==============  ========  ============= 
    24032403 
    2404 .. image:: img/olddocs/image135.jpg 
     2404.. image:: ..\img\olddocs\image135.jpg 
    24052405 
    24062406*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    24282428The scattering intensity *I(q)* is 
    24292429 
    2430 .. image:: img/olddocs/image136.PNG 
     2430.. image:: ..\img\olddocs\image136.PNG 
    24312431 
    24322432The form factor is 
    24332433 
    2434 .. image:: img/olddocs/image137.jpg 
     2434.. image:: ..\img\olddocs\image137.jpg 
    24352435 
    24362436where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), 
    … …  
    24392439The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    24402440 
    2441 .. image:: img/olddocs/image040.gif 
     2441.. image:: ..\img\olddocs\image040.gif 
    24422442 
    24432443The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 
    … …  
    24562456==============  ========  ============= 
    24572457 
    2458 .. image:: img/olddocs/image138.jpg 
     2458.. image:: ..\img\olddocs\image138.jpg 
    24592459 
    24602460*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    24842484The scattering intensity *I(q)* is 
    24852485 
    2486 .. image:: img/olddocs/image139.PNG 
     2486.. image:: ..\img\olddocs\image139.PNG 
    24872487 
    24882488The form factor is 
    24892489 
    2490 .. image:: img/olddocs/image134.PNG 
     2490.. image:: ..\img\olddocs\image134.PNG 
    24912491 
    24922492and the structure factor is 
    24932493 
    2494 .. image:: img/olddocs/image140.PNG 
     2494.. image:: ..\img\olddocs\image140.PNG 
    24952495 
    24962496where 
    24972497 
    2498 .. image:: img/olddocs/image141.PNG 
     2498.. image:: ..\img\olddocs\image141.PNG 
    24992499 
    25002500Here *d* = (repeat) spacing, |delta| = bilayer thickness, the contrast |drho| = SLD(headgroup) - SLD(solvent), 
    … …  
    25072507The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    25082508 
    2509 .. image:: img/olddocs/image040.gif 
     2509.. image:: ..\img\olddocs\image040.gif 
    25102510 
    25112511The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    25232523==============  ========  ============= 
    25242524 
    2525 .. image:: img/olddocs/image142.jpg 
     2525.. image:: ..\img\olddocs\image142.jpg 
    25262526 
    25272527*Figure. 1D plot using the default values (w/6000 data point).* 
    … …  
    25502550The scattering intensity *I(q)* is 
    25512551 
    2552 .. image:: img/olddocs/image139.PNG 
     2552.. image:: ..\img\olddocs\image139.PNG 
    25532553 
    25542554The form factor is 
    25552555 
    2556 .. image:: img/olddocs/image143.PNG 
     2556.. image:: ..\img\olddocs\image143.PNG 
    25572557 
    25582558The structure factor is 
    25592559 
    2560 .. image:: img/olddocs/image140.PNG 
     2560.. image:: ..\img\olddocs\image140.PNG 
    25612561 
    25622562where 
    25632563 
    2564 .. image:: img/olddocs/image141.PNG 
     2564.. image:: ..\img\olddocs\image141.PNG 
    25652565 
    25662566where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), 
    … …  
    25752575The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    25762576 
    2577 .. image:: img/olddocs/image040.gif 
     2577.. image:: ..\img\olddocs\image040.gif 
    25782578 
    25792579The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 
    … …  
    25952595==============  ========  ============= 
    25962596 
    2597 .. image:: img/olddocs/image144.jpg 
     2597.. image:: ..\img\olddocs\image144.jpg 
    25982598 
    25992599*Figure. 1D plot using the default values (w/6000 data point).* 
    … …  
    26222622The scattering intensity *I(q)* is calculated as 
    26232623 
    2624 .. image:: img/olddocs/image145.jpg 
     2624.. image:: ..\img\olddocs\image145.jpg 
    26252625 
    26262626The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* 
    26272627 
    2628 .. image:: img/olddocs/image146.jpg 
     2628.. image:: ..\img\olddocs\image146.jpg 
    26292629 
    26302630Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume 
    … …  
    26352635Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values 
    26362636 
    2637 .. image:: img/olddocs/image147.jpg 
     2637.. image:: ..\img\olddocs\image147.jpg 
    26382638 
    26392639The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    26402640 
    2641 .. image:: img/olddocs/image040.gif 
     2641.. image:: ..\img\olddocs\image040.gif 
    26422642 
    26432643The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. 
    … …  
    26562656==============  ========  ============= 
    26572657 
    2658 .. image:: img/olddocs/image148.jpg 
     2658.. image:: ..\img\olddocs\image148.jpg 
    26592659 
    26602660*Figure. 1D plot using the default values above (w/20000 data point).* 
    … …  
    26832683The scattering intensity *I(q)* is calculated as 
    26842684 
    2685 .. image:: img/olddocs/image149.jpg 
     2685.. image:: ..\img\olddocs\image149.jpg 
    26862686 
    26872687where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    26952695and nearest neighbor separation *D* is 
    26962696 
    2697 .. image:: img/olddocs/image150.jpg 
     2697.. image:: ..\img\olddocs\image150.jpg 
    26982698 
    26992699The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    27002700 
    2701 .. image:: img/olddocs/image151.jpg 
     2701.. image:: ..\img\olddocs\image151.jpg 
    27022702 
    27032703where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    27052705The simple cubic lattice is 
    27062706 
    2707 .. image:: img/olddocs/image152.jpg 
     2707.. image:: ..\img\olddocs\image152.jpg 
    27082708 
    27092709For a crystal, diffraction peaks appear at reduced *q*\ -values given by 
    27102710 
    2711 .. image:: img/olddocs/image153.jpg 
     2711.. image:: ..\img\olddocs\image153.jpg 
    27122712 
    27132713where for a simple cubic lattice any *h*\ , *k*\ , *l* are allowed and none are forbidden. Thus the peak positions 
    27142714correspond to (just the first 5) 
    27152715 
    2716 .. image:: img/olddocs/image154.jpg 
     2716.. image:: ..\img\olddocs\image154.jpg 
    27172717 
    27182718**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    27362736default values. 
    27372737 
    2738 .. image:: img/olddocs/image155.jpg 
     2738.. image:: ..\img\olddocs\image155.jpg 
    27392739 
    27402740*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    27442744computation. 
    27452745 
    2746 .. image:: img/olddocs/image156.jpg 
    2747  
    2748 .. image:: img/olddocs/image157.jpg 
     2746.. image:: ..\img\olddocs\image156.jpg 
     2747 
     2748.. image:: ..\img\olddocs\image157.jpg 
    27492749 
    27502750*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    27742774The scattering intensity *I(q)* is calculated as 
    27752775 
    2776 .. image:: img/olddocs/image158.jpg 
     2776.. image:: ..\img\olddocs\image158.jpg 
    27772777 
    27782778where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    27862786*R* and nearest neighbor separation *D* is 
    27872787 
    2788 .. image:: img/olddocs/image159.jpg 
     2788.. image:: ..\img\olddocs\image159.jpg 
    27892789 
    27902790The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    27912791 
    2792 .. image:: img/olddocs/image160.jpg 
     2792.. image:: ..\img\olddocs\image160.jpg 
    27932793 
    27942794where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    27962796The face-centered cubic lattice is 
    27972797 
    2798 .. image:: img/olddocs/image161.jpg 
     2798.. image:: ..\img\olddocs\image161.jpg 
    27992799 
    28002800For a crystal, diffraction peaks appear at reduced q-values given by 
    28012801 
    2802 .. image:: img/olddocs/image162.jpg 
     2802.. image:: ..\img\olddocs\image162.jpg 
    28032803 
    28042804where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all even are allowed and reflections where 
    28052805*h*\ , *k*\ , *l* are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5) 
    28062806 
    2807 .. image:: img/olddocs/image163.jpg 
     2807.. image:: ..\img\olddocs\image163.jpg 
    28082808 
    28092809**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    28272827default values. 
    28282828 
    2829 .. image:: img/olddocs/image164.jpg 
     2829.. image:: ..\img\olddocs\image164.jpg 
    28302830 
    28312831*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    28352835computation. 
    28362836 
    2837 .. image:: img/olddocs/image165.gif 
    2838  
    2839 .. image:: img/olddocs/image166.jpg 
     2837.. image:: ..\img\olddocs\image165.gif 
     2838 
     2839.. image:: ..\img\olddocs\image166.jpg 
    28402840 
    28412841*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    28652865The scattering intensity *I(q)* is calculated as 
    28662866 
    2867 .. image:: img/olddocs/image167.jpg 
     2867.. image:: ..\img\olddocs\image167.jpg 
    28682868 
    28692869where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    28772877*R* and nearest neighbor separation *D* is 
    28782878 
    2879 .. image:: img/olddocs/image159.jpg 
     2879.. image:: ..\img\olddocs\image159.jpg 
    28802880 
    28812881The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    28822882 
    2883 .. image:: img/olddocs/image160.jpg 
     2883.. image:: ..\img\olddocs\image160.jpg 
    28842884 
    28852885where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    28872887The body-centered cubic lattice is 
    28882888 
    2889 .. image:: img/olddocs/image168.jpg 
     2889.. image:: ..\img\olddocs\image168.jpg 
    28902890 
    28912891For a crystal, diffraction peaks appear at reduced q-values given by 
    28922892 
    2893 .. image:: img/olddocs/image162.jpg 
     2893.. image:: ..\img\olddocs\image162.jpg 
    28942894 
    28952895where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and 
    28962896reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) 
    28972897 
    2898 .. image:: img/olddocs/image169.jpg 
     2898.. image:: ..\img\olddocs\image169.jpg 
    28992899 
    29002900**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    29182918default values. 
    29192919 
    2920 .. image:: img/olddocs/image170.jpg 
     2920.. image:: ..\img\olddocs\image170.jpg 
    29212921 
    29222922*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    29262926computation. 
    29272927 
    2928 .. image:: img/olddocs/image165.gif 
    2929  
    2930 .. image:: img/olddocs/image171.jpg 
     2928.. image:: ..\img\olddocs\image165.gif 
     2929 
     2930.. image:: ..\img\olddocs\image171.jpg 
    29312931 
    29322932*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    29552955For information about polarised and magnetic scattering, click here_. 
    29562956 
    2957 .. image:: img/olddocs/image087.jpg 
     2957.. image:: ..\img\olddocs\image087.jpg 
    29582958 
    29592959*2.1.37.1. Definition* 
    … …  
    29622962*b* = *B* / *B* = 1, and *c* = *C* / *B* > 1, the form factor is 
    29632963 
    2964 .. image:: img/olddocs/image088.PNG 
     2964.. image:: ..\img\olddocs\image088.PNG 
    29652965 
    29662966and the contrast is defined as 
    29672967 
    2968 .. image:: img/olddocs/image089.PNG 
     2968.. image:: ..\img\olddocs\image089.PNG 
    29692969 
    29702970The scattering intensity per unit volume is returned in units of |cm^-1|; ie, *I(q)* = |phi| *P(q)*\ . 
    … …  
    29792979parallel to the *x*-axis of the detector. 
    29802980 
    2981 .. image:: img/olddocs/image090.jpg 
     2981.. image:: ..\img\olddocs\image090.jpg 
    29822982 
    29832983*Figure. Definition of angles for 2D*. 
    29842984 
    2985 .. image:: img/olddocs/image091.jpg 
     2985.. image:: ..\img\olddocs\image091.jpg 
    29862986 
    29872987*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    29982998==============  ========  ============= 
    29992999 
    3000 .. image:: img/olddocs/image092.jpg 
     3000.. image:: ..\img\olddocs\image092.jpg 
    30013001 
    30023002*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    30093009angles of |theta|, |phi|, and |psi| respectively). 
    30103010 
    3011 .. image:: img/olddocs/image093.gif 
     3011.. image:: ..\img\olddocs\image093.gif 
    30123012 
    30133013*Figure. Comparison between 1D and averaged 2D.* 
    … …  
    30433043dimensions *A*, *B*, *C* such that *A* < *B* < *C*. 
    30443044 
    3045 .. image:: img/olddocs/image087.jpg 
     3045.. image:: ..\img\olddocs\image087.jpg 
    30463046 
    30473047There are rectangular "slabs" of thickness *tA* that add to the *A* dimension (on the *BC* faces). There are similar 
    30483048slabs on the *AC* (= *tB*) and *AB* (= *tC*) faces. The projection in the *AB* plane is then 
    30493049 
    3050 .. image:: img/olddocs/image094.jpg 
     3050.. image:: ..\img\olddocs\image094.jpg 
    30513051 
    30523052The volume of the solid is 
    30533053 
    3054 .. image:: img/olddocs/image095.PNG 
     3054.. image:: ..\img\olddocs\image095.PNG 
    30553055 
    30563056**meaning that there are "gaps" at the corners of the solid.** 
    … …  
    30843084parallel to the *x*-axis of the detector. 
    30853085 
    3086 .. image:: img/olddocs/image090.jpg 
     3086.. image:: ..\img\olddocs\image090.jpg 
    30873087 
    30883088*Figure. Definition of angles for 2D*. 
    30893089 
    3090 .. image:: img/olddocs/image091.jpg 
     3090.. image:: ..\img\olddocs\image091.jpg 
    30913091 
    30923092*Figure. Examples of the angles for oriented cspp against the detector plane.* 
    … …  
    31133113==============  ========  ============= 
    31143114 
    3115 .. image:: img/olddocs/image096.jpg 
     3115.. image:: ..\img\olddocs\image096.jpg 
    31163116 
    31173117*Figure. 1D plot using the default values (w/256 data points).* 
    31183118 
    3119 .. image:: img/olddocs/image097.jpg 
     3119.. image:: ..\img\olddocs\image097.jpg 
    31203120 
    31213121*Figure. 2D plot using the default values (w/(256X265) data points).* 
    … …  
    33983398calculation. **NB: No size polydispersity is included in this model, use the** Poly_GaussCoil_ **Model instead** 
    33993399 
    3400 .. image:: img/olddocs/image172.PNG 
     3400.. image:: ..\img\olddocs\image172.PNG 
    34013401 
    34023402For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    34033403 
    3404 .. image:: img/olddocs/image040.gif 
     3404.. image:: ..\img\olddocs\image040.gif 
    34053405 
    34063406==============  ========  ============= 
    … …  
    34123412==============  ========  ============= 
    34133413 
    3414 .. image:: img/olddocs/image173.jpg 
     3414.. image:: ..\img\olddocs\image173.jpg 
    34153415 
    34163416*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    34393439The scattering intensity *I(q)* is calculated as 
    34403440 
    3441 .. image:: img/olddocs/image174.jpg 
     3441.. image:: ..\img\olddocs\image174.jpg 
    34423442 
    34433443Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. 
    … …  
    34453445For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    34463446 
    3447 .. image:: img/olddocs/image040.gif 
     3447.. image:: ..\img\olddocs\image040.gif 
    34483448 
    34493449==================  ========  ============= 
    … …  
    34593459==================  ========  ============= 
    34603460 
    3461 .. image:: img/olddocs/image175.jpg 
     3461.. image:: ..\img\olddocs\image175.jpg 
    34623462 
    34633463*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    34833483The scattering intensity *I(q)* is calculated as 
    34843484 
    3485 .. image:: img/olddocs/image176.jpg 
     3485.. image:: ..\img\olddocs\image176.jpg 
    34863486 
    34873487The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function 
    … …  
    34943494For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    34953495 
    3496 .. image:: img/olddocs/image040.gif 
     3496.. image:: ..\img\olddocs\image040.gif 
    34973497 
    34983498====================  ========  ============= 
    … …  
    35073507====================  ========  ============= 
    35083508 
    3509 .. image:: img/olddocs/image177.jpg 
     3509.. image:: ..\img\olddocs\image177.jpg 
    35103510 
    35113511*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    35283528The Ornstein-Zernicke model is defined by 
    35293529 
    3530 .. image:: img/olddocs/image178.PNG 
     3530.. image:: ..\img\olddocs\image178.PNG 
    35313531 
    35323532The parameter *L* is the screening length. 
    … …  
    35343534For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    35353535 
    3536 .. image:: img/olddocs/image040.gif 
     3536.. image:: ..\img\olddocs\image040.gif 
    35373537 
    35383538==============  ========  ============= 
    … …  
    35443544==============  ========  ============= 
    35453545 
    3546 .. image:: img/olddocs/image179.jpg 
     3546.. image:: ..\img\olddocs\image179.jpg 
    35473547 
    35483548* Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    35673567*2.2.5.1. Definition* 
    35683568 
    3569 .. image:: img/olddocs/image180_corrected.PNG 
     3569.. image:: ..\img\olddocs\image180_corrected.PNG 
    35703570 
    35713571The parameter *L* is the correlation length. 
    … …  
    35733573For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    35743574 
    3575 .. image:: img/olddocs/image040.gif 
     3575.. image:: ..\img\olddocs\image040.gif 
    35763576 
    35773577==============  ========  ============= 
    … …  
    35833583==============  ========  ============= 
    35843584 
    3585 .. image:: img/olddocs/image181.jpg 
     3585.. image:: ..\img\olddocs\image181.jpg 
    35863586 
    35873587* Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36043604This model describes a simple power law with background. 
    36053605 
    3606 .. image:: img/olddocs/image182.PNG 
     3606.. image:: ..\img\olddocs\image182.PNG 
    36073607 
    36083608Note the minus sign in front of the exponent. The parameter *m* should therefore be entered as a **positive** number. 
    … …  
    36163616==============  ========  ============= 
    36173617 
    3618 .. image:: img/olddocs/image183.jpg 
     3618.. image:: ..\img\olddocs\image183.jpg 
    36193619 
    36203620*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36353635*2.2.7.1. Definition* 
    36363636 
    3637 .. image:: img/olddocs/image184.PNG 
     3637.. image:: ..\img\olddocs\image184.PNG 
    36383638 
    36393639For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    36403640 
    3641 .. image:: img/olddocs/image040.gif 
     3641.. image:: ..\img\olddocs\image040.gif 
    36423642 
    36433643==============  ========  ============= 
    … …  
    36503650==============  ========  ============= 
    36513651 
    3652 .. image:: img/olddocs/image185.jpg 
     3652.. image:: ..\img\olddocs\image185.jpg 
    36533653 
    36543654*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36733673*2.2.8.1. Definition* 
    36743674 
    3675 .. image:: img/olddocs/image186.PNG 
     3675.. image:: ..\img\olddocs\image186.PNG 
    36763676 
    36773677The *scale* parameter is the volume fraction of the building blocks, *R0* is the radius of the building block, *Df* is 
    … …  
    36833683For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    36843684 
    3685 .. image:: img/olddocs/image040.gif 
     3685.. image:: ..\img\olddocs\image040.gif 
    36863686 
    36873687==============  ========  ============= 
    … …  
    36973697==============  ========  ============= 
    36983698 
    3699 .. image:: img/olddocs/image187.jpg 
     3699.. image:: ..\img\olddocs\image187.jpg 
    37003700 
    37013701*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    37153715*2.2.9.1. Definition* 
    37163716 
    3717 .. image:: img/olddocs/mass_fractal_eq1.jpg 
     3717.. image:: ..\img\olddocs\mass_fractal_eq1.jpg 
    37183718 
    37193719where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, 
    … …  
    37343734==============  ========  ============= 
    37353735 
    3736 .. image:: img/olddocs/mass_fractal_fig1.jpg 
     3736.. image:: ..\img\olddocs\mass_fractal_fig1.jpg 
    37373737 
    37383738*Figure. 1D plot using default values.* 
    … …  
    37553755*2.2.10.1. Definition* 
    37563756 
    3757 .. image:: img/olddocs/surface_fractal_eq1.gif 
     3757.. image:: ..\img\olddocs\surface_fractal_eq1.gif 
    37583758 
    37593759where *R* is the radius of the building block, *Ds* is the **surface** fractal dimension, |zeta| is the cut-off length, 
    … …  
    37743774==============  ========  ============= 
    37753775 
    3776 .. image:: img/olddocs/surface_fractal_fig1.jpg 
     3776.. image:: ..\img\olddocs\surface_fractal_fig1.jpg 
    37773777 
    37783778*Figure. 1D plot using default values.* 
    … …  
    38023802The scattered intensity *I(q)* is  calculated using a modified Ornstein-Zernicke equation 
    38033803 
    3804 .. image:: img/olddocs/masssurface_fractal_eq1.jpg 
     3804.. image:: ..\img\olddocs\masssurface_fractal_eq1.jpg 
    38053805 
    38063806where *Rg* is the size of the cluster, *rg* is the size of the primary particle, *Ds* is the surface fractal dimension, 
    … …  
    38223822==============  ========  ============= 
    38233823 
    3824 .. image:: img/olddocs/masssurface_fractal_fig1.jpg 
     3824.. image:: ..\img\olddocs\masssurface_fractal_fig1.jpg 
    38253825 
    38263826*Figure. 1D plot using default values.* 
    … …  
    38483848*2.2.12.1. Definition* 
    38493849 
    3850 .. image:: img/olddocs/fractcore_eq1.gif 
     3850.. image:: ..\img\olddocs\fractcore_eq1.gif 
    38513851 
    38523852The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 
    38533853 
    3854 .. image:: img/olddocs/image013.PNG 
     3854.. image:: ..\img\olddocs\image013.PNG 
    38553855 
    38563856while the fractal structure factor S(q) is 
    38573857 
    3858 .. image:: img/olddocs/fractcore_eq3.gif 
     3858.. image:: ..\img\olddocs\fractcore_eq3.gif 
    38593859 
    38603860where *Df* = frac_dim, |xi| = cor_length, *rc* = (core) radius, and *scale* = volume fraction. 
    … …  
    38643864For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    38653865 
    3866 .. image:: img/olddocs/image040.gif 
     3866.. image:: ..\img\olddocs\image040.gif 
    38673867 
    38683868==============  ========  ============= 
    … …  
    38803880==============  ========  ============= 
    38813881 
    3882 .. image:: img/olddocs/image188.jpg 
     3882.. image:: ..\img\olddocs\image188.jpg 
    38833883 
    38843884*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    39053905The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) 
    39063906 
    3907 .. image:: img/olddocs/image189.jpg 
     3907.. image:: ..\img\olddocs\image189.jpg 
    39083908 
    39093909|bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" 
    … …  
    39173917For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39183918 
    3919 .. image:: img/olddocs/image040.gif 
     3919.. image:: ..\img\olddocs\image040.gif 
    39203920 
    39213921===================================  ========  ============= 
    … …  
    39293929===================================  ========  ============= 
    39303930 
    3931 .. image:: img/olddocs/image190.jpg 
     3931.. image:: ..\img\olddocs\image190.jpg 
    39323932 
    39333933*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    39493949*2.2.14.1. Definition* 
    39503950 
    3951 .. image:: img/olddocs/image191.PNG 
     3951.. image:: ..\img\olddocs\image191.PNG 
    39523952 
    39533953where *K* is the contrast factor for the polymer, *Lb* is the Bjerrum length, *h* is the virial parameter, *b* is the 
    … …  
    39573957For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39583958 
    3959 .. image:: img/olddocs/image040.gif 
     3959.. image:: ..\img\olddocs\image040.gif 
    39603960 
    39613961==============  ========  ============= 
    … …  
    39923992This model fits the Guinier function 
    39933993 
    3994 .. image:: img/olddocs/image192.PNG 
     3994.. image:: ..\img\olddocs\image192.PNG 
    39953995 
    39963996to the data directly without any need for linearisation (*cf*. Ln *I(q)* vs *q*\ :sup:`2`). 
    … …  
    39983998For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39993999 
    4000 .. image:: img/olddocs/image040.gif 
     4000.. image:: ..\img\olddocs\image040.gif 
    40014001 
    40024002==============  ========  ============= 
    … …  
    40274027The following functional form is used 
    40284028 
    4029 .. image:: img/olddocs/image193.jpg 
     4029.. image:: ..\img\olddocs\image193.jpg 
    40304030 
    40314031This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D 
    … …  
    40364036Enforcing the continuity of the Guinier and Porod functions and their derivatives yields 
    40374037 
    4038 .. image:: img/olddocs/image194.jpg 
     4038.. image:: ..\img\olddocs\image194.jpg 
    40394039 
    40404040and 
    40414041 
    4042 .. image:: img/olddocs/image195.jpg 
     4042.. image:: ..\img\olddocs\image195.jpg 
    40434043 
    40444044Note that 
    … …  
    40524052For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    40534053 
    4054 .. image:: img/olddocs/image008.PNG 
     4054.. image:: ..\img\olddocs\image008.PNG 
    40554055 
    40564056==============================  ========  ============= 
    … …  
    40644064==============================  ========  ============= 
    40654065 
    4066 .. image:: img/olddocs/image196.jpg 
     4066.. image:: ..\img\olddocs\image196.jpg 
    40674067 
    40684068*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    40834083This model fits the Porod function 
    40844084 
    4085 .. image:: img/olddocs/image197_corrected.PNG 
     4085.. image:: ..\img\olddocs\image197_corrected.PNG 
    40864086 
    40874087to the data directly without any need for linearisation (*cf*. Log *I(q)* vs Log *q*). 
    … …  
    40924092For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    40934093 
    4094 .. image:: img/olddocs/image040.gif 
     4094.. image:: ..\img\olddocs\image040.gif 
    40954095 
    40964096==============  ========  ============= 
    … …  
    41134113This model describes a Gaussian shaped peak on a flat background 
    41144114 
    4115 .. image:: img/olddocs/image198.PNG 
     4115.. image:: ..\img\olddocs\image198.PNG 
    41164116 
    41174117with the peak having height of *I0* centered at *q0* and having a standard deviation of *B*.  The FWHM (full-width 
    … …  
    41204120For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    41214121 
    4122 .. image:: img/olddocs/image040.gif 
     4122.. image:: ..\img\olddocs\image040.gif 
    41234123 
    41244124==============  ========  ============= 
    … …  
    41314131==============  ========  ============= 
    41324132 
    4133 .. image:: img/olddocs/image199.jpg 
     4133.. image:: ..\img\olddocs\image199.jpg 
    41344134 
    41354135*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    41474147This model describes a Lorentzian shaped peak on a flat background 
    41484148 
    4149 .. image:: img/olddocs/image200.PNG 
     4149.. image:: ..\img\olddocs\image200.PNG 
    41504150 
    41514151with the peak having height of *I0* centered at *q0* and having a HWHM (half-width half-maximum) of B.  
    … …  
    41534153For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    41544154 
    4155 .. image:: img/olddocs/image040.gif 
     4155.. image:: ..\img\olddocs\image040.gif 
    41564156 
    41574157==============  ========  ============= 
    … …  
    41644164==============  ========  ============= 
    41654165 
    4166 .. image:: img/olddocs/image201.jpg 
     4166.. image:: ..\img\olddocs\image201.jpg 
    41674167 
    41684168*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    41884188The scattering intensity *I(q)* is calculated as 
    41894189 
    4190 .. image:: img/olddocs/image202.PNG 
     4190.. image:: ..\img\olddocs\image202.PNG 
    41914191 
    41924192where the dimensionless chain dimension is 
    41934193 
    4194 .. image:: img/olddocs/image203.PNG 
     4194.. image:: ..\img\olddocs\image203.PNG 
    41954195 
    41964196and the polydispersity is 
    41974197 
    4198 .. image:: img/olddocs/image204.PNG 
     4198.. image:: ..\img\olddocs\image204.PNG 
    41994199 
    42004200For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    42014201 
    4202 .. image:: img/olddocs/image040.gif 
     4202.. image:: ..\img\olddocs\image040.gif 
    42034203 
    42044204This example dataset is produced using 200 data points, using 200 data points, 
    … …  
    42144214==============  ========  ============= 
    42154215 
    4216 .. image:: img/olddocs/image205.jpg 
     4216.. image:: ..\img\olddocs\image205.jpg 
    42174217 
    42184218*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    42404240The form factor  was originally presented in the following integral form (Benoit, 1957) 
    42414241 
    4242 .. image:: img/olddocs/image206.jpg 
     4242.. image:: ..\img\olddocs\image206.jpg 
    42434243 
    42444244where |nu| is the excluded volume parameter (which is related to the Porod exponent *m* as |nu| = 1 / *m*), *a* is the 
    … …  
    42464246into an almost analytical form as follows (Hammouda, 1993) 
    42474247 
    4248 .. image:: img/olddocs/image207.jpg 
     4248.. image:: ..\img\olddocs\image207.jpg 
    42494249 
    42504250where |gamma|\ *(x,U)* is the incomplete gamma function 
    42514251 
    4252 .. image:: img/olddocs/image208.jpg 
     4252.. image:: ..\img\olddocs\image208.jpg 
    42534253 
    42544254and the variable *U* is given in terms of the scattering vector *Q* as 
    42554255 
    4256 .. image:: img/olddocs/image209.jpg 
     4256.. image:: ..\img\olddocs\image209.jpg 
    42574257 
    42584258The square of the radius-of-gyration is defined as 
    42594259 
    4260 .. image:: img/olddocs/image210.jpg 
     4260.. image:: ..\img\olddocs\image210.jpg 
    42614261 
    42624262Note that this model applies only in the mass fractal range (ie, 5/3 <= *m* <= 3) and **does not** apply to surface 
    … …  
    42664266A low-*Q* expansion yields the Guinier form and a high-*Q* expansion yields the Porod form which is given by 
    42674267 
    4268 .. image:: img/olddocs/image211.jpg 
     4268.. image:: ..\img\olddocs\image211.jpg 
    42694269 
    42704270Here |biggamma|\ *(x)* = |gamma|\ *(x,inf)* is the gamma function. 
    … …  
    42724272The asymptotic limit is dominated by the first term 
    42734273 
    4274 .. image:: img/olddocs/image212.jpg 
     4274.. image:: ..\img\olddocs\image212.jpg 
    42754275 
    42764276The special case when |nu| = 0.5 (or *m* = 1/|nu| = 2) corresponds to Gaussian chains for which the form factor is given 
    42774277by the familiar Debye_ function. 
    42784278 
    4279 .. image:: img/olddocs/image213.jpg 
     4279.. image:: ..\img\olddocs\image213.jpg 
    42804280 
    42814281For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    42824282 
    4283 .. image:: img/olddocs/image040.gif 
     4283.. image:: ..\img\olddocs\image040.gif 
    42844284 
    42854285This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.2 |Ang^-1| and the default 
    … …  
    42954295===================  ========  ============= 
    42964296 
    4297 .. image:: img/olddocs/image214.jpg 
     4297.. image:: ..\img\olddocs\image214.jpg 
    42984298 
    42994299*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    43794379=======================  ========  ============= 
    43804380 
    4381 .. image:: img/olddocs/image215.jpg 
     4381.. image:: ..\img\olddocs\image215.jpg 
    43824382 
    43834383*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    44014401The scattering intensity *I(q)* is calculated as 
    44024402 
    4403 .. image:: img/olddocs/image216.jpg  
     4403.. image:: ..\img\olddocs\image216.jpg  
    44044404 
    44054405where *A* = Lorentzian scale factor #1, *C* = Lorentzian scale #2, |xi|\ :sub:`1` and |xi|\ :sub:`2` are the 
    … …  
    44094409For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    44104410 
    4411 .. image:: img/olddocs/image040.gif 
     4411.. image:: ..\img\olddocs\image040.gif 
    44124412 
    44134413===============================  ========  ============= 
    … …  
    44234423===============================  ========  ============= 
    44244424 
    4425 .. image:: img/olddocs/image217.jpg 
     4425.. image:: ..\img\olddocs\image217.jpg 
    44264426 
    44274427*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    44454445The scattering intensity *I(q)* is calculated as 
    44464446 
    4447 .. image:: img/olddocs/image218.jpg 
     4447.. image:: ..\img\olddocs\image218.jpg 
    44484448 
    44494449where *qc* is the location of the crossover from one slope to the other. The scaling *coef_A* sets the overall 
    … …  
    44554455For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    44564456 
    4457 .. image:: img/olddocs/image040.gif 
     4457.. image:: ..\img\olddocs\image040.gif 
    44584458 
    44594459==============  ========  ============= 
    … …  
    44674467==============  ========  ============= 
    44684468 
    4469 .. image:: img/olddocs/image219.jpg 
     4469.. image:: ..\img\olddocs\image219.jpg 
    44704470 
    44714471*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    44964496The empirical fit function is  
    44974497 
    4498 .. image:: img/olddocs/image220.jpg 
     4498.. image:: ..\img\olddocs\image220.jpg 
    44994499 
    45004500For each level, the four parameters *Gi*, *Rg,i*, *Bi* and *Pi* must be chosen.  
    … …  
    45074507For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    45084508 
    4509 .. image:: img/olddocs/image040.gif 
     4509.. image:: ..\img\olddocs\image040.gif 
    45104510 
    45114511==============  ========  ============= 
    … …  
    45244524==============  ========  ============= 
    45254525 
    4526 .. image:: img/olddocs/image221.jpg 
     4526.. image:: ..\img\olddocs\image221.jpg 
    45274527 
    45284528*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    45424542This calculates the simple linear function 
    45434543 
    4544 .. image:: img/olddocs/image222.PNG 
     4544.. image:: ..\img\olddocs\image222.PNG 
    45454545 
    45464546**NB: For 2D plots,** *I(q)* = *I(qx)*\ *\ *I(qy)*, **which is a different definition to other shape independent models.** 
    … …  
    45774577The scattered intensity *I(q)* is calculated as 
    45784578 
    4579 .. image:: img/olddocs/image233.gif 
     4579.. image:: ..\img\olddocs\image233.gif 
    45804580 
    45814581where 
    45824582 
    4583 .. image:: img/olddocs/image234.gif 
     4583.. image:: ..\img\olddocs\image234.gif 
    45844584 
    45854585Note that the first term reduces to the Ornstein-Zernicke equation when *D* = 2; ie, when the Flory exponent is 0.5 
    … …  
    45974597============================  ========  ============= 
    45984598 
    4599 .. image:: img/olddocs/image235.gif 
     4599.. image:: ..\img\olddocs\image235.gif 
    46004600 
    46014601*Figure. 1D plot using the default values (w/300 data points).* 
    … …  
    46194619For a star with *f* arms: 
    46204620 
    4621 .. image:: img/olddocs/star1.png 
     4621.. image:: ..\img\olddocs\star1.png 
    46224622 
    46234623where 
    46244624 
    4625 .. image:: img/olddocs/star2.png 
     4625.. image:: ..\img\olddocs\star2.png 
    46264626 
    46274627and 
    46284628 
    4629 .. image:: img/olddocs/star3.png 
     4629.. image:: ..\img\olddocs\star3.png 
    46304630 
    46314631is the square of the ensemble average radius-of-gyration of an arm. 
    … …  
    46564656Also see ReflectivityIIModel_. 
    46574657 
    4658 .. image:: img/olddocs/image231.bmp 
     4658.. image:: ..\img\olddocs\image231.bmp 
    46594659 
    46604660*Figure. Comparison (using the SLD profile below) with the NIST web calculation (circles)* 
    46614661http://www.ncnr.nist.gov/resources/reflcalc.html 
    46624662 
    4663 .. image:: img/olddocs/image232.gif 
     4663.. image:: ..\img\olddocs\image232.gif 
    46644664 
    46654665*Figure. SLD profile used for the calculation (above).* 
    … …  
    468546851) Erf 
    46864686 
    4687 .. image:: img/olddocs/image051.gif 
     4687.. image:: ..\img\olddocs\image051.gif 
    46884688 
    468946892) Power-Law 
    46904690 
    4691 .. image:: img/olddocs/image050.gif 
     4691.. image:: ..\img\olddocs\image050.gif 
    46924692 
    469346933) Exp 
    46944694 
    4695 .. image:: img/olddocs/image049.gif 
     4695.. image:: ..\img\olddocs\image049.gif 
    46964696 
    46974697The constant *A* in the expressions above (but the parameter *nu* in the model!) is an input. 
    … …  
    47174717The calculation uses the Percus-Yevick closure where the interparticle potential is 
    47184718 
    4719 .. image:: img/olddocs/image223.PNG 
     4719.. image:: ..\img\olddocs\image223.PNG 
    47204720 
    47214721where *r* is the distance from the center of the sphere of a radius *R*. 
    … …  
    47234723For a 2D plot, the wave transfer is defined as 
    47244724 
    4725 .. image:: img/olddocs/image040.gif 
     4725.. image:: ..\img\olddocs\image040.gif 
    47264726 
    47274727==============  ========  ============= 
    … …  
    47324732==============  ========  ============= 
    47334733 
    4734 .. image:: img/olddocs/image224.jpg 
     4734.. image:: ..\img\olddocs\image224.jpg 
    47354735 
    47364736*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    47584758The interaction potential is: 
    47594759 
    4760 .. image:: img/olddocs/image225.PNG 
     4760.. image:: ..\img\olddocs\image225.PNG 
    47614761 
    47624762where *r* is the distance from the center of the sphere of a radius *R*. 
    … …  
    47644764For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    47654765 
    4766 .. image:: img/olddocs/image040.gif 
     4766.. image:: ..\img\olddocs\image040.gif 
    47674767 
    47684768==============  =========  ============= 
    … …  
    47754775==============  =========  ============= 
    47764776 
    4777 .. image:: img/olddocs/image226.jpg 
     4777.. image:: ..\img\olddocs\image226.jpg 
    47784778 
    47794779*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    48034803For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    48044804 
    4805 .. image:: img/olddocs/image040.gif 
     4805.. image:: ..\img\olddocs\image040.gif 
    48064806 
    48074807==============  ========  ============= 
    … …  
    48164816==============  ========  ============= 
    48174817 
    4818 .. image:: img/olddocs/image227.jpg 
     4818.. image:: ..\img\olddocs\image227.jpg 
    48194819 
    48204820*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    48424842that smaller |tau| means stronger attraction. 
    48434843 
    4844 .. image:: img/olddocs/image228.PNG 
     4844.. image:: ..\img\olddocs\image228.PNG 
    48454845 
    48464846where the interaction potential is 
    48474847 
    4848 .. image:: img/olddocs/image229.PNG 
     4848.. image:: ..\img\olddocs\image229.PNG 
    48494849 
    48504850The Percus-Yevick (PY) closure was used for this calculation, and is an adequate closure for an attractive interparticle 
    … …  
    48624862For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    48634863 
    4864 .. image:: img/olddocs/image040.gif 
     4864.. image:: ..\img\olddocs\image040.gif 
    48654865 
    48664866==============  ========  ============= 
    … …  
    48734873==============  ========  ============= 
    48744874 
    4875 .. image:: img/olddocs/image230.jpg 
     4875.. image:: ..\img\olddocs\image230.jpg 
    48764876 
    48774877*Figure. 1D plot using the default values (in linear scale).* 
Note: See TracChangeset for help on using the changeset viewer.