Changeset 34e0c32 in sasview for src


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Mar 18, 2016 5:13:07 PM (8 years ago)
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smk78
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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
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Image links changed to \img\olddocs

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  • 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).* 
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