Changeset 7f42aad in sasview for src


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Timestamp:
Dec 14, 2014 7:31:17 AM (10 years ago)
Author:
Peter Parker
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:
f8063bf
Parents:
d51acbc
Message:

Refs #221 - Change instances of image file paths in the rst file that have incorrect case. These worked fine on Windows, but Linux is case sensitive when it comes to file paths, so these need to be changed. I whipped up a quick python script to parse the problemtic files from the jenkins docs build log, and then generate a new rst with correct case. A better fix would have been to standardise our image naming conventions, but life is too short to work out how to tell SVN to deal with that in a quick way.

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

    r6386cd8 r7f42aad  
    389389NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 
    390390 
    391 .. image:: img/image005.JPG 
     391.. image:: img/image005.jpg 
    392392 
    393393Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. 
    … …  
    436436==============  ========  ============= 
    437437 
    438 .. image:: img/image009.JPG 
     438.. image:: img/image009.jpg 
    439439 
    440440*Figure. 1D plot using the default values above (w/200 data point).* 
    … …  
    504504==============  ========  ============= 
    505505 
    506 .. image:: img/image012.JPG 
     506.. image:: img/image012.jpg 
    507507 
    508508*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    525525The structure is: 
    526526 
    527 .. image:: img/raspberry_pic.JPG 
     527.. image:: img/raspberry_pic.jpg 
    528528 
    529529where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the 
    … …  
    561561==============  ========  ============= 
    562562 
    563 .. image:: img/raspberry_plot.JPG 
     563.. image:: img/raspberry_plot.jpg 
    564564 
    565565*Figure. 1D plot using the values of /2000 data points.* 
    … …  
    625625NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 
    626626 
    627 .. image:: img/image014.JPG 
     627.. image:: img/image014.jpg 
    628628 
    629629Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS 
    … …  
    686686*qmax* = 0.7 -1 and the above default values. 
    687687 
    688 .. image:: img/image015.JPG 
     688.. image:: img/image015.jpg 
    689689 
    690690*Figure: 1D plot using the default values (w/200 data point).* 
    … …  
    692692The scattering length density profile for the default sld values (w/ 4 shells). 
    693693 
    694 .. image:: img/image016.JPG 
     694.. image:: img/image016.jpg 
    695695 
    696696*Figure: SLD profile against the radius of the sphere for default SLDs.* 
    … …  
    721721The *I* :sub:`0` is calculated in the following way (King, 2002) 
    722722 
    723 .. image:: img/secondmeq1.JPG 
     723.. image:: img/secondmeq1.jpg 
    724724 
    725725where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the 
    … …  
    748748==============  ========  ============= 
    749749 
    750 .. image:: img/secongm_fig1.JPG 
     750.. image:: img/secongm_fig1.jpg 
    751751 
    752752REFERENCE 
    … …  
    764764solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). 
    765765 
    766 .. image:: img/image020.JPG 
     766.. image:: img/image020.jpg 
    767767 
    768768The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    … …  
    791791is the number of shells. 
    792792 
    793 .. image:: img/image021.JPG 
     793.. image:: img/image021.jpg 
    794794 
    795795*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    818818The 1D scattering intensity is calculated in the following way 
    819819 
    820 .. image:: img/image022.GIF 
    821  
    822 .. image:: img/image023.GIF 
     820.. image:: img/image022.gif 
     821 
     822.. image:: img/image023.gif 
    823823 
    824824where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 
    825825 
    826 .. image:: img/image024.GIF 
     826.. image:: img/image024.gif 
    827827 
    828828so that 
    829829 
    830 .. image:: img/image025.GIF 
    831  
    832 .. image:: img/image026.GIF 
    833  
    834 .. image:: img/image027.GIF 
     830.. image:: img/image025.gif 
     831 
     832.. image:: img/image026.gif 
     833 
     834.. image:: img/image027.gif 
    835835 
    836836Here we assumed that the SLDs of the core and solvent are constant against *r*. 
    … …  
    838838Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by 
    839839 
    840 .. image:: img/image028.GIF 
     840.. image:: img/image028.gif 
    841841 
    842842An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and 
    … …  
    846846For \| *A* \| > 0, 
    847847 
    848 .. image:: img/image029.GIF 
     848.. image:: img/image029.gif 
    849849 
    850850For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, 
    … …  
    852852so this case is equivalent to 
    853853 
    854 .. image:: img/image030.GIF 
    855  
    856 .. image:: img/image031.GIF 
    857  
    858 .. image:: img/image032.GIF 
    859  
    860 .. image:: img/image033.GIF 
     854.. image:: img/image030.gif 
     855 
     856.. image:: img/image031.gif 
     857 
     858.. image:: img/image032.gif 
     859 
     860.. image:: img/image033.gif 
    861861 
    862862For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is 
    … …  
    864864factor contributed by the shells is 
    865865 
    866 .. image:: img/image034.GIF 
    867  
    868 .. image:: img/image035.GIF 
     866.. image:: img/image034.gif 
     867 
     868.. image:: img/image035.gif 
    869869 
    870870In the equation 
    871871 
    872 .. image:: img/image036.GIF 
     872.. image:: img/image036.gif 
    873873 
    874874Finally, the form factor can be calculated by 
    875875 
    876 .. image:: img/image037.GIF 
     876.. image:: img/image037.gif 
    877877 
    878878where 
    879879 
    880 .. image:: img/image038.GIF 
     880.. image:: img/image038.gif 
    881881 
    882882and 
    883883 
    884 .. image:: img/image039.GIF 
     884.. image:: img/image039.gif 
    885885 
    886886The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    887887defined as 
    888888 
    889 .. image:: img/image040.GIF 
     889.. image:: img/image040.gif 
    890890 
    891891NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 
    … …  
    909909NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. 
    910910 
    911 .. image:: img/image041.JPG 
     911.. image:: img/image041.jpg 
    912912 
    913913*Figure. 1D plot using the default values (w/400 point).* 
    914914 
    915 .. image:: img/image042.JPG 
     915.. image:: img/image042.jpg 
    916916 
    917917*Figure. SLD profile from the default values.* 
    … …  
    945945and a shell thickness, *t*. 
    946946 
    947 .. image:: img/image018.JPG 
     947.. image:: img/image018.jpg 
    948948 
    949949The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    … …  
    970970NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. 
    971971 
    972 .. image:: img/image019.JPG 
     972.. image:: img/image019.jpg 
    973973 
    974974*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    10011001The 1D scattering intensity is calculated in the following way: 
    10021002 
    1003 .. image:: img/image022.GIF 
    1004  
    1005 .. image:: img/image043.GIF 
     1003.. image:: img/image022.gif 
     1004 
     1005.. image:: img/image043.gif 
    10061006 
    10071007where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 
    10081008 
    1009 .. image:: img/image024.GIF 
     1009.. image:: img/image024.gif 
    10101010 
    10111011so that 
    10121012 
    1013 .. image:: img/image044.GIF 
    1014  
    1015 .. image:: img/image045.GIF 
    1016  
    1017 .. image:: img/image046.GIF 
    1018  
    1019 .. image:: img/image047.GIF 
    1020  
    1021 .. image:: img/image048.GIF 
    1022  
    1023 .. image:: img/image027.GIF 
     1013.. image:: img/image044.gif 
     1014 
     1015.. image:: img/image045.gif 
     1016 
     1017.. image:: img/image046.gif 
     1018 
     1019.. image:: img/image047.gif 
     1020 
     1021.. image:: img/image048.gif 
     1022 
     1023.. image:: img/image027.gif 
    10241024 
    10251025Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between 
    … …  
    102810281) Exp 
    10291029 
    1030 .. image:: img/image049.GIF 
     1030.. image:: img/image049.gif 
    10311031 
    103210322) Power-Law 
    10331033 
    1034 .. image:: img/image050.GIF 
     1034.. image:: img/image050.gif 
    10351035 
    103610363) Erf 
    10371037 
    1038 .. image:: img/image051.GIF 
     1038.. image:: img/image051.gif 
    10391039 
    10401040The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is 
    … …  
    10441044to the form factor *P(q)* 
    10451045 
    1046 .. image:: img/image052.GIF 
    1047  
    1048 .. image:: img/image053.GIF 
    1049  
    1050 .. image:: img/image054.GIF 
     1046.. image:: img/image052.gif 
     1047 
     1048.. image:: img/image053.gif 
     1049 
     1050.. image:: img/image054.gif 
    10511051 
    10521052where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. 
    … …  
    10541054In the equation 
    10551055 
    1056 .. image:: img/image055.GIF 
     1056.. image:: img/image055.gif 
    10571057 
    10581058Finally, the form factor can be calculated by 
    10591059 
    1060 .. image:: img/image037.GIF 
     1060.. image:: img/image037.gif 
    10611061 
    10621062where 
    10631063 
    1064 .. image:: img/image038.GIF 
     1064.. image:: img/image038.gif 
    10651065 
    10661066and 
    10671067 
    1068 .. image:: img/image056.GIF 
     1068.. image:: img/image056.gif 
    10691069 
    10701070The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 
    10711071defined as 
    10721072 
    1073 .. image:: img/image040.GIF 
     1073.. image:: img/image040.gif 
    10741074 
    10751075NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. 
    … …  
    10981098NB: *rad_core0* represents the core radius (*R1*). 
    10991099 
    1100 .. image:: img/image057.JPG 
     1100.. image:: img/image057.jpg 
    11011101 
    11021102*Figure. 1D plot using the default values (w/400 point).* 
    11031103 
    1104 .. image:: img/image058.JPG 
     1104.. image:: img/image058.jpg 
    11051105 
    11061106*Figure. SLD profile from the default values.* 
    … …  
    12491249and |phi|. Those angles are defined in Figure 1. 
    12501250 
    1251 .. image:: img/image061.JPG 
     1251.. image:: img/image061.jpg 
    12521252 
    12531253*Figure 1. Definition of the angles for oriented cylinders.* 
    12541254 
    1255 .. image:: img/image062.JPG 
     1255.. image:: img/image062.jpg 
    12561256 
    12571257*Figure 2. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    12861286NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. 
    12871287 
    1288 .. image:: img/image065.JPG 
     1288.. image:: img/image065.jpg 
    12891289 
    12901290*Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis* 
    … …  
    13011301distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. 
    13021302 
    1303 .. image:: img/image066.JPG 
     1303.. image:: img/image066.jpg 
    13041304 
    13051305*Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the* 
    … …  
    13511351==============  ========  ============= 
    13521352 
    1353 .. image:: img/image074.JPG 
     1353.. image:: img/image074.jpg 
    13541354 
    13551355*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    13581358(Kline, 2006). 
    13591359 
    1360 .. image:: img/image061.JPG 
     1360.. image:: img/image061.jpg 
    13611361 
    13621362*Figure. Definition of the angles for the oriented HollowCylinderModel.* 
    13631363 
    1364 .. image:: img/image062.JPG 
     1364.. image:: img/image062.jpg 
    13651365 
    13661366*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    13881388The Capped Cylinder geometry is defined as 
    13891389 
    1390 .. image:: img/image112.JPG 
     1390.. image:: img/image112.jpg 
    13911391 
    13921392where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius 
    … …  
    13971397The scattered intensity *I(q)* is calculated as 
    13981398 
    1399 .. image:: img/image113.JPG 
     1399.. image:: img/image113.jpg 
    14001400 
    14011401where the amplitude *A(q)* is given as 
    14021402 
    1403 .. image:: img/image114.JPG 
     1403.. image:: img/image114.jpg 
    14041404 
    14051405The < > brackets denote an average of the structure over all orientations. <\ *A*\ :sup:`2`\ *(q)*> is then the form 
    … …  
    14091409The volume of the Capped Cylinder is (with *h* as a positive value here) 
    14101410 
    1411 .. image:: img/image115.JPG 
     1411.. image:: img/image115.jpg 
    14121412 
    14131413and its radius-of-gyration 
    14141414 
    1415 .. image:: img/image116.JPG 
     1415.. image:: img/image116.jpg 
    14161416 
    14171417**The requirement that** *R* >= *r* **is not enforced in the model! It is up to you to restrict this during analysis.** 
    … …  
    14321432==============  ========  ============= 
    14331433 
    1434 .. image:: img/image117.JPG 
     1434.. image:: img/image117.jpg 
    14351435 
    14361436*Figure. 1D plot using the default values (w/256 data point).* 
    … …  
    14391439|theta| = 45 deg and |phi| =0 deg with default values for other parameters 
    14401440 
    1441 .. image:: img/image118.JPG 
     1441.. image:: img/image118.jpg 
    14421442 
    14431443*Figure. 2D plot (w/(256X265) data points).* 
    14441444 
    1445 .. image:: img/image061.JPG 
     1445.. image:: img/image061.jpg 
    14461446 
    14471447*Figure. Definition of the angles for oriented 2D cylinders.* 
    … …  
    14851485the outer shell is given by *L+2t*. *J1* is the first order Bessel function. 
    14861486 
    1487 .. image:: img/image069.JPG 
     1487.. image:: img/image069.jpg 
    14881488 
    14891489To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two 
    … …  
    15201520NIST (Kline, 2006). Figure 1 shows a comparison of the 1D output of our model and the output of the NIST software. 
    15211521 
    1522 .. image:: img/image070.JPG 
     1522.. image:: img/image070.jpg 
    15231523 
    15241524*Figure 1: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS* 
    … …  
    153115312D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a cross-check. 
    15321532 
    1533 .. image:: img/image071.JPG 
     1533.. image:: img/image071.jpg 
    15341534 
    15351535*Figure 2: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and* 
    … …  
    15381538*Solvent_sld* = 1e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. 
    15391539 
    1540 .. image:: img/image061.JPG 
     1540.. image:: img/image061.jpg 
    15411541 
    15421542*Figure. Definition of the angles for oriented core-shell cylinders.* 
    15431543 
    1544 .. image:: img/image062.JPG 
     1544.. image:: img/image062.jpg 
    15451545 
    15461546*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    15911591All angle parameters are valid and given only for 2D calculation; ie, an oriented system. 
    15921592 
    1593 .. image:: img/image101.JPG 
     1593.. image:: img/image101.jpg 
    15941594 
    15951595*Figure. Definition of angles for 2D* 
    15961596 
    1597 .. image:: img/image062.JPG 
     1597.. image:: img/image062.jpg 
    15981598 
    15991599*Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* 
    … …  
    16141614==============  ========  ============= 
    16151615 
    1616 .. image:: img/image102.JPG 
     1616.. image:: img/image102.jpg 
    16171617 
    16181618*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    16251625and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). 
    16261626 
    1627 .. image:: img/image103.GIF 
     1627.. image:: img/image103.gif 
    16281628 
    16291629*Figure. Comparison between 1D and averaged 2D.* 
    … …  
    16321632the results of the averaging by varying the number of angular bins. 
    16331633 
    1634 .. image:: img/image104.GIF 
     1634.. image:: img/image104.gif 
    16351635 
    16361636*Figure. The intensities averaged from 2D over different numbers of bins and angles.* 
    … …  
    16601660*2.1.19.1. Definition* 
    16611661 
    1662 .. image:: img/image075.JPG 
     1662.. image:: img/image075.jpg 
    16631663 
    16641664The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff 
    … …  
    16831683==============  ========  ============= 
    16841684 
    1685 .. image:: img/image076.JPG 
     1685.. image:: img/image076.jpg 
    16861686 
    16871687*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    17381738- 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. 
    17391739 
    1740 .. image:: img/image077.JPG 
     1740.. image:: img/image077.jpg 
    17411741 
    17421742The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff 
    … …  
    17801780==============  ========  ============= 
    17811781 
    1782 .. image:: img/image078.JPG 
     1782.. image:: img/image078.jpg 
    17831783 
    17841784*Figure. 1D plot using the default values (w/200 data points).* 
    … …  
    18071807and SLDs. 
    18081808 
    1809 .. image:: img/image240.PNG 
     1809.. image:: img/image240.png 
    18101810 
    18111811*(Graphic from DOI: 10.1039/C0NP00002G)* 
    … …  
    18391839*Figure. 1D plot using the default values (w/200 data point).* 
    18401840 
    1841 .. image:: img/image061.JPG 
     1841.. image:: img/image061.jpg 
    18421842 
    18431843*Figure. Definition of the angles for the oriented CoreShellBicelleModel.* 
    18441844 
    1845 .. image:: img/image062.JPG 
     1845.. image:: img/image062.jpg 
    18461846 
    18471847*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    18691869The barbell geometry is defined as 
    18701870 
    1871 .. image:: img/image105.JPG 
     1871.. image:: img/image105.jpg 
    18721872 
    18731873where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. 
    … …  
    18921892The volume of the barbell is 
    18931893 
    1894 .. image:: img/image108.JPG 
     1894.. image:: img/image108.jpg 
    18951895 
    18961896 
    18971897and its radius-of-gyration is 
    18981898 
    1899 .. image:: img/image109.JPG 
     1899.. image:: img/image109.jpg 
    19001900 
    19011901**The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. 
    … …  
    19161916==============  ========  ============= 
    19171917 
    1918 .. image:: img/image110.JPG 
     1918.. image:: img/image110.jpg 
    19191919 
    19201920*Figure. 1D plot using the default values (w/256 data point).* 
    … …  
    19231923|theta| = 45 deg and |phi| = 0 deg with default values for other parameters 
    19241924 
    1925 .. image:: img/image111.JPG 
     1925.. image:: img/image111.jpg 
    19261926 
    19271927*Figure. 2D plot (w/(256X265) data points).* 
    19281928 
    1929 .. image:: img/image061.JPG 
     1929.. image:: img/image061.jpg 
    19301930 
    19311931*Figure. Examples of the angles for oriented pp against the detector plane.* 
    19321932 
    1933 .. image:: img/image062.JPG 
     1933.. image:: img/image062.jpg 
    19341934 
    19351935Figure. Definition of the angles for oriented 2D barbells. 
    … …  
    19631963*2.1.23.1 Definition* 
    19641964 
    1965 .. image:: img/image079.GIF 
     1965.. image:: img/image079.gif 
    19661966 
    19671967The scattering intensity *I(q)* is 
    … …  
    20072007==============  ========  ============= 
    20082008 
    2009 .. image:: img/image085.JPG 
     2009.. image:: img/image085.jpg 
    20102010 
    20112011*Figure. 1D plot using the default values (w/1000 data point).* 
    20122012 
    2013 .. image:: img/image086.JPG 
     2013.. image:: img/image086.jpg 
    20142014 
    20152015*Figure. Examples of the angles for oriented stackeddisks against the detector plane.* 
    20162016 
    2017 .. image:: img/image062.JPG 
     2017.. image:: img/image062.jpg 
    20182018 
    20192019*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    20382038This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). 
    20392039 
    2040 .. image:: img/image241.PNG 
     2040.. image:: img/image241.png 
    20412041 
    20422042*(Graphic from Matt Henderson, matt@matthen.com)* 
    … …  
    21282128above. 
    21292129 
    2130 .. image:: img/image121.JPG 
     2130.. image:: img/image121.jpg 
    21312131 
    21322132The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering 
    21332133kernel and the 1D scattering intensity use the c-library from NIST. 
    21342134 
    2135 .. image:: img/image122.JPG 
     2135.. image:: img/image122.jpg 
    21362136 
    21372137*Figure. The angles for oriented ellipsoid.* 
    … …  
    21432143software. 
    21442144 
    2145 .. image:: img/image123.JPG 
     2145.. image:: img/image123.jpg 
    21462146 
    21472147*Figure 1: Comparison of the SasView scattering intensity for an ellipsoid with the output of the NIST SANS analysis* 
    … …  
    21542154cross-check. 
    21552155 
    2156 .. image:: img/image124.JPG 
     2156.. image:: img/image124.jpg 
    21572157 
    21582158*Figure 2: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the* 
    … …  
    21862186all orientations for 1D. 
    21872187 
    2188 .. image:: img/image125.GIF 
     2188.. image:: img/image125.gif 
    21892189 
    21902190The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    22202220==============  ========  ============= 
    22212221 
    2222 .. image:: img/image127.JPG 
     2222.. image:: img/image127.jpg 
    22232223 
    22242224*Figure. 1D plot using the default values (w/1000 data point).* 
    22252225 
    2226 .. image:: img/image122.JPG 
     2226.. image:: img/image122.jpg 
    22272227 
    22282228*Figure. The angles for oriented CoreShellEllipsoid.* 
    … …  
    23132313where the volume *V* = (4/3)\ |pi| (*Ra* *Rb* *Rc*), and the averaging < > is applied over all orientations for 1D. 
    23142314 
    2315 .. image:: img/image128.JPG 
     2315.. image:: img/image128.jpg 
    23162316 
    23172317The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    23492349==============  ========  ============= 
    23502350 
    2351 .. image:: img/image130.JPG 
     2351.. image:: img/image130.jpg 
    23522352 
    23532353*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    23602360angles of |theta|, |phi|, and |psi| respectively). 
    23612361 
    2362 .. image:: img/image131.GIF 
     2362.. image:: img/image131.gif 
    23632363 
    23642364*Figure. Comparison between 1D and averaged 2D.* 
    23652365 
    2366 .. image:: img/image132.JPG 
     2366.. image:: img/image132.jpg 
    23672367 
    23682368*Figure. The angles for oriented ellipsoid.* 
    … …  
    23992399The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    24002400 
    2401 .. image:: img/image040.GIF 
     2401.. image:: img/image040.gif 
    24022402 
    24032403The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, 
    … …  
    24142414==============  ========  ============= 
    24152415 
    2416 .. image:: img/image135.JPG 
     2416.. image:: img/image135.jpg 
    24172417 
    24182418*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    24442444The form factor is 
    24452445 
    2446 .. image:: img/image137.JPG 
     2446.. image:: img/image137.jpg 
    24472447 
    24482448where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), 
    … …  
    24512451The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    24522452 
    2453 .. image:: img/image040.GIF 
     2453.. image:: img/image040.gif 
    24542454 
    24552455The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 
    … …  
    24682468==============  ========  ============= 
    24692469 
    2470 .. image:: img/image138.JPG 
     2470.. image:: img/image138.jpg 
    24712471 
    24722472*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    25192519The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    25202520 
    2521 .. image:: img/image040.GIF 
     2521.. image:: img/image040.gif 
    25222522 
    25232523The returned value is in units of |cm^-1|, on absolute scale. 
    … …  
    25352535==============  ========  ============= 
    25362536 
    2537 .. image:: img/image142.JPG 
     2537.. image:: img/image142.jpg 
    25382538 
    25392539*Figure. 1D plot using the default values (w/6000 data point).* 
    … …  
    25872587The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    25882588 
    2589 .. image:: img/image040.GIF 
     2589.. image:: img/image040.gif 
    25902590 
    25912591The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, 
    … …  
    26072607==============  ========  ============= 
    26082608 
    2609 .. image:: img/image144.JPG 
     2609.. image:: img/image144.jpg 
    26102610 
    26112611*Figure. 1D plot using the default values (w/6000 data point).* 
    … …  
    26342634The scattering intensity *I(q)* is calculated as 
    26352635 
    2636 .. image:: img/image145.JPG 
     2636.. image:: img/image145.jpg 
    26372637 
    26382638The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* 
    26392639 
    2640 .. image:: img/image146.JPG 
     2640.. image:: img/image146.jpg 
    26412641 
    26422642Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume 
    … …  
    26472647Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values 
    26482648 
    2649 .. image:: img/image147.JPG 
     2649.. image:: img/image147.jpg 
    26502650 
    26512651The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 
    26522652 
    2653 .. image:: img/image040.GIF 
     2653.. image:: img/image040.gif 
    26542654 
    26552655The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. 
    … …  
    26682668==============  ========  ============= 
    26692669 
    2670 .. image:: img/image148.JPG 
     2670.. image:: img/image148.jpg 
    26712671 
    26722672*Figure. 1D plot using the default values above (w/20000 data point).* 
    … …  
    26952695The scattering intensity *I(q)* is calculated as 
    26962696 
    2697 .. image:: img/image149.JPG 
     2697.. image:: img/image149.jpg 
    26982698 
    26992699where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    27072707and nearest neighbor separation *D* is 
    27082708 
    2709 .. image:: img/image150.JPG 
     2709.. image:: img/image150.jpg 
    27102710 
    27112711The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    27122712 
    2713 .. image:: img/image151.JPG 
     2713.. image:: img/image151.jpg 
    27142714 
    27152715where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    27172717The simple cubic lattice is 
    27182718 
    2719 .. image:: img/image152.JPG 
     2719.. image:: img/image152.jpg 
    27202720 
    27212721For a crystal, diffraction peaks appear at reduced *q*\ -values given by 
    27222722 
    2723 .. image:: img/image153.JPG 
     2723.. image:: img/image153.jpg 
    27242724 
    27252725where for a simple cubic lattice any *h*\ , *k*\ , *l* are allowed and none are forbidden. Thus the peak positions 
    27262726correspond to (just the first 5) 
    27272727 
    2728 .. image:: img/image154.JPG 
     2728.. image:: img/image154.jpg 
    27292729 
    27302730**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    27482748default values. 
    27492749 
    2750 .. image:: img/image155.JPG 
     2750.. image:: img/image155.jpg 
    27512751 
    27522752*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    27562756computation. 
    27572757 
    2758 .. image:: img/image156.JPG 
    2759  
    2760 .. image:: img/image157.JPG 
     2758.. image:: img/image156.jpg 
     2759 
     2760.. image:: img/image157.jpg 
    27612761 
    27622762*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    27862786The scattering intensity *I(q)* is calculated as 
    27872787 
    2788 .. image:: img/image158.JPG 
     2788.. image:: img/image158.jpg 
    27892789 
    27902790where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    27982798*R* and nearest neighbor separation *D* is 
    27992799 
    2800 .. image:: img/image159.JPG 
     2800.. image:: img/image159.jpg 
    28012801 
    28022802The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    28032803 
    2804 .. image:: img/image160.JPG 
     2804.. image:: img/image160.jpg 
    28052805 
    28062806where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    28082808The face-centered cubic lattice is 
    28092809 
    2810 .. image:: img/image161.JPG 
     2810.. image:: img/image161.jpg 
    28112811 
    28122812For a crystal, diffraction peaks appear at reduced q-values given by 
    28132813 
    2814 .. image:: img/image162.JPG 
     2814.. image:: img/image162.jpg 
    28152815 
    28162816where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all even are allowed and reflections where 
    28172817*h*\ , *k*\ , *l* are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5) 
    28182818 
    2819 .. image:: img/image163.JPG 
     2819.. image:: img/image163.jpg 
    28202820 
    28212821**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    28392839default values. 
    28402840 
    2841 .. image:: img/image164.JPG 
     2841.. image:: img/image164.jpg 
    28422842 
    28432843*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    28472847computation. 
    28482848 
    2849 .. image:: img/image165.GIF 
    2850  
    2851 .. image:: img/image166.JPG 
     2849.. image:: img/image165.gif 
     2850 
     2851.. image:: img/image166.jpg 
    28522852 
    28532853*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    28772877The scattering intensity *I(q)* is calculated as 
    28782878 
    2879 .. image:: img/image167.JPG 
     2879.. image:: img/image167.jpg 
    28802880 
    28812881where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume 
    … …  
    28892889*R* and nearest neighbor separation *D* is 
    28902890 
    2891 .. image:: img/image159.JPG 
     2891.. image:: img/image159.jpg 
    28922892 
    28932893The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 
    28942894 
    2895 .. image:: img/image160.JPG 
     2895.. image:: img/image160.jpg 
    28962896 
    28972897where *g* is a fractional distortion based on the nearest neighbor distance. 
    … …  
    28992899The body-centered cubic lattice is 
    29002900 
    2901 .. image:: img/image168.JPG 
     2901.. image:: img/image168.jpg 
    29022902 
    29032903For a crystal, diffraction peaks appear at reduced q-values given by 
    29042904 
    2905 .. image:: img/image162.JPG 
     2905.. image:: img/image162.jpg 
    29062906 
    29072907where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and 
    29082908reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) 
    29092909 
    2910 .. image:: img/image169.JPG 
     2910.. image:: img/image169.jpg 
    29112911 
    29122912**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** 
    … …  
    29302930default values. 
    29312931 
    2932 .. image:: img/image170.JPG 
     2932.. image:: img/image170.jpg 
    29332933 
    29342934*Figure. 1D plot in the linear scale using the default values (w/200 data point).* 
    … …  
    29382938computation. 
    29392939 
    2940 .. image:: img/image165.GIF 
    2941  
    2942 .. image:: img/image171.JPG 
     2940.. image:: img/image165.gif 
     2941 
     2942.. image:: img/image171.jpg 
    29432943 
    29442944*Figure. 2D plot using the default values (w/200X200 pixels).* 
    … …  
    29672967For information about polarised and magnetic scattering, click here_. 
    29682968 
    2969 .. image:: img/image087.JPG 
     2969.. image:: img/image087.jpg 
    29702970 
    29712971*2.1.37.1. Definition* 
    … …  
    29912991parallel to the *x*-axis of the detector. 
    29922992 
    2993 .. image:: img/image090.JPG 
     2993.. image:: img/image090.jpg 
    29942994 
    29952995*Figure. Definition of angles for 2D*. 
    29962996 
    2997 .. image:: img/image091.JPG 
     2997.. image:: img/image091.jpg 
    29982998 
    29992999*Figure. Examples of the angles for oriented pp against the detector plane.* 
    … …  
    30103010==============  ========  ============= 
    30113011 
    3012 .. image:: img/image092.JPG 
     3012.. image:: img/image092.jpg 
    30133013 
    30143014*Figure. 1D plot using the default values (w/1000 data point).* 
    … …  
    30213021angles of |theta|, |phi|, and |psi| respectively). 
    30223022 
    3023 .. image:: img/image093.GIF 
     3023.. image:: img/image093.gif 
    30243024 
    30253025*Figure. Comparison between 1D and averaged 2D.* 
    … …  
    30553055dimensions *A*, *B*, *C* such that *A* < *B* < *C*. 
    30563056 
    3057 .. image:: img/image087.JPG 
     3057.. image:: img/image087.jpg 
    30583058 
    30593059There are rectangular "slabs" of thickness *tA* that add to the *A* dimension (on the *BC* faces). There are similar 
    30603060slabs on the *AC* (= *tB*) and *AB* (= *tC*) faces. The projection in the *AB* plane is then 
    30613061 
    3062 .. image:: img/image094.JPG 
     3062.. image:: img/image094.jpg 
    30633063 
    30643064The volume of the solid is 
    … …  
    30963096parallel to the *x*-axis of the detector. 
    30973097 
    3098 .. image:: img/image090.JPG 
     3098.. image:: img/image090.jpg 
    30993099 
    31003100*Figure. Definition of angles for 2D*. 
    31013101 
    3102 .. image:: img/image091.JPG 
     3102.. image:: img/image091.jpg 
    31033103 
    31043104*Figure. Examples of the angles for oriented cspp against the detector plane.* 
    … …  
    31253125==============  ========  ============= 
    31263126 
    3127 .. image:: img/image096.JPG 
     3127.. image:: img/image096.jpg 
    31283128 
    31293129*Figure. 1D plot using the default values (w/256 data points).* 
    31303130 
    3131 .. image:: img/image097.JPG 
     3131.. image:: img/image097.jpg 
    31323132 
    31333133*Figure. 2D plot using the default values (w/(256X265) data points).* 
    … …  
    33923392For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    33933393 
    3394 .. image:: img/image040.GIF 
     3394.. image:: img/image040.gif 
    33953395 
    33963396==============  ========  ============= 
    … …  
    34023402==============  ========  ============= 
    34033403 
    3404 .. image:: img/image173.JPG 
     3404.. image:: img/image173.jpg 
    34053405 
    34063406*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    34293429The scattering intensity *I(q)* is calculated as 
    34303430 
    3431 .. image:: img/image174.JPG 
     3431.. image:: img/image174.jpg 
    34323432 
    34333433Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. 
    … …  
    34353435For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    34363436 
    3437 .. image:: img/image040.GIF 
     3437.. image:: img/image040.gif 
    34383438 
    34393439==================  ========  ============= 
    … …  
    34493449==================  ========  ============= 
    34503450 
    3451 .. image:: img/image175.JPG 
     3451.. image:: img/image175.jpg 
    34523452 
    34533453*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    34733473The scattering intensity *I(q)* is calculated as 
    34743474 
    3475 .. image:: img/image176.JPG 
     3475.. image:: img/image176.jpg 
    34763476 
    34773477The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function 
    … …  
    34843484For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    34853485 
    3486 .. image:: img/image040.GIF 
     3486.. image:: img/image040.gif 
    34873487 
    34883488====================  ========  ============= 
    … …  
    34973497====================  ========  ============= 
    34983498 
    3499 .. image:: img/image177.JPG 
     3499.. image:: img/image177.jpg 
    35003500 
    35013501*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    35243524For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    35253525 
    3526 .. image:: img/image040.GIF 
     3526.. image:: img/image040.gif 
    35273527 
    35283528==============  ========  ============= 
    … …  
    35343534==============  ========  ============= 
    35353535 
    3536 .. image:: img/image179.JPG 
     3536.. image:: img/image179.jpg 
    35373537 
    35383538* Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    35633563For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    35643564 
    3565 .. image:: img/image040.GIF 
     3565.. image:: img/image040.gif 
    35663566 
    35673567==============  ========  ============= 
    … …  
    35733573==============  ========  ============= 
    35743574 
    3575 .. image:: img/image181.JPG 
     3575.. image:: img/image181.jpg 
    35763576 
    35773577* Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36063606==============  ========  ============= 
    36073607 
    3608 .. image:: img/image183.JPG 
     3608.. image:: img/image183.jpg 
    36093609 
    36103610*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36293629For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    36303630 
    3631 .. image:: img/image040.GIF 
     3631.. image:: img/image040.gif 
    36323632 
    36333633==============  ========  ============= 
    … …  
    36403640==============  ========  ============= 
    36413641 
    3642 .. image:: img/image185.JPG 
     3642.. image:: img/image185.jpg 
    36433643 
    36443644*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    36733673For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    36743674 
    3675 .. image:: img/image040.GIF 
     3675.. image:: img/image040.gif 
    36763676 
    36773677==============  ========  ============= 
    … …  
    36873687==============  ========  ============= 
    36883688 
    3689 .. image:: img/image187.JPG 
     3689.. image:: img/image187.jpg 
    36903690 
    36913691*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    37053705*2.2.9.1. Definition* 
    37063706 
    3707 .. image:: img/mass_fractal_eq1.JPG 
     3707.. image:: img/mass_fractal_eq1.jpg 
    37083708 
    37093709where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, 
    … …  
    37243724==============  ========  ============= 
    37253725 
    3726 .. image:: img/mass_fractal_fig1.JPG 
     3726.. image:: img/mass_fractal_fig1.jpg 
    37273727 
    37283728*Figure. 1D plot using default values.* 
    … …  
    37643764==============  ========  ============= 
    37653765 
    3766 .. image:: img/surface_fractal_fig1.JPG 
     3766.. image:: img/surface_fractal_fig1.jpg 
    37673767 
    37683768*Figure. 1D plot using default values.* 
    … …  
    38123812==============  ========  ============= 
    38133813 
    3814 .. image:: img/masssurface_fractal_fig1.JPG 
     3814.. image:: img/masssurface_fractal_fig1.jpg 
    38153815 
    38163816*Figure. 1D plot using default values.* 
    … …  
    38383838*2.2.12.1. Definition* 
    38393839 
    3840 .. image:: img/fractcore_eq1.GIF 
     3840.. image:: img/fractcore_eq1.gif 
    38413841 
    38423842The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 
    … …  
    38543854For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    38553855 
    3856 .. image:: img/image040.GIF 
     3856.. image:: img/image040.gif 
    38573857 
    38583858==============  ========  ============= 
    … …  
    38703870==============  ========  ============= 
    38713871 
    3872 .. image:: img/image188.JPG 
     3872.. image:: img/image188.jpg 
    38733873 
    38743874*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    38953895The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) 
    38963896 
    3897 .. image:: img/image189.JPG 
     3897.. image:: img/image189.jpg 
    38983898 
    38993899|bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" 
    … …  
    39073907For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39083908 
    3909 .. image:: img/image040.GIF 
     3909.. image:: img/image040.gif 
    39103910 
    39113911===================================  ========  ============= 
    … …  
    39193919===================================  ========  ============= 
    39203920 
    3921 .. image:: img/image190.JPG 
     3921.. image:: img/image190.jpg 
    39223922 
    39233923*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    39473947For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39483948 
    3949 .. image:: img/image040.GIF 
     3949.. image:: img/image040.gif 
    39503950 
    39513951==============  ========  ============= 
    … …  
    39883988For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    39893989 
    3990 .. image:: img/image040.GIF 
     3990.. image:: img/image040.gif 
    39913991 
    39923992==============  ========  ============= 
    … …  
    40174017The following functional form is used 
    40184018 
    4019 .. image:: img/image193.JPG 
     4019.. image:: img/image193.jpg 
    40204020 
    40214021This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D 
    … …  
    40264026Enforcing the continuity of the Guinier and Porod functions and their derivatives yields 
    40274027 
    4028 .. image:: img/image194.JPG 
     4028.. image:: img/image194.jpg 
    40294029 
    40304030and 
    40314031 
    4032 .. image:: img/image195.JPG 
     4032.. image:: img/image195.jpg 
    40334033 
    40344034Note that 
    … …  
    40544054==============================  ========  ============= 
    40554055 
    4056 .. image:: img/image196.JPG 
     4056.. image:: img/image196.jpg 
    40574057 
    40584058*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    40824082For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    40834083 
    4084 .. image:: img/image040.GIF 
     4084.. image:: img/image040.gif 
    40854085 
    40864086==============  ========  ============= 
    … …  
    41104110For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    41114111 
    4112 .. image:: img/image040.GIF 
     4112.. image:: img/image040.gif 
    41134113 
    41144114==============  ========  ============= 
    … …  
    41214121==============  ========  ============= 
    41224122 
    4123 .. image:: img/image199.JPG 
     4123.. image:: img/image199.jpg 
    41244124 
    41254125*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    41434143For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    41444144 
    4145 .. image:: img/image040.GIF 
     4145.. image:: img/image040.gif 
    41464146 
    41474147==============  ========  ============= 
    … …  
    41544154==============  ========  ============= 
    41554155 
    4156 .. image:: img/image201.JPG 
     4156.. image:: img/image201.jpg 
    41574157 
    41584158*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    41904190For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    41914191 
    4192 .. image:: img/image040.GIF 
     4192.. image:: img/image040.gif 
    41934193 
    41944194This example dataset is produced using 200 data points, using 200 data points, 
    … …  
    42044204==============  ========  ============= 
    42054205 
    4206 .. image:: img/image205.JPG 
     4206.. image:: img/image205.jpg 
    42074207 
    42084208*Figure. 1D plot using the default values (w/200 data point).* 
    … …  
    42304230The form factor  was originally presented in the following integral form (Benoit, 1957) 
    42314231 
    4232 .. image:: img/image206.JPG 
     4232.. image:: img/image206.jpg 
    42334233 
    42344234where |nu| is the excluded volume parameter (which is related to the Porod exponent *m* as |nu| = 1 / *m*), *a* is the 
    … …  
    42364236into an almost analytical form as follows (Hammouda, 1993) 
    42374237 
    4238 .. image:: img/image207.JPG 
     4238.. image:: img/image207.jpg 
    42394239 
    42404240where |gamma|\ *(x,U)* is the incomplete gamma function 
    42414241 
    4242 .. image:: img/image208.JPG 
     4242.. image:: img/image208.jpg 
    42434243 
    42444244and the variable *U* is given in terms of the scattering vector *Q* as 
    42454245 
    4246 .. image:: img/image209.JPG 
     4246.. image:: img/image209.jpg 
    42474247 
    42484248The square of the radius-of-gyration is defined as 
    42494249 
    4250 .. image:: img/image210.JPG 
     4250.. image:: img/image210.jpg 
    42514251 
    42524252Note that this model applies only in the mass fractal range (ie, 5/3 <= *m* <= 3) and **does not** apply to surface 
    … …  
    42564256A low-*Q* expansion yields the Guinier form and a high-*Q* expansion yields the Porod form which is given by 
    42574257 
    4258 .. image:: img/image211.JPG 
     4258.. image:: img/image211.jpg 
    42594259 
    42604260Here |biggamma|\ *(x)* = |gamma|\ *(x,inf)* is the gamma function. 
    … …  
    42624262The asymptotic limit is dominated by the first term 
    42634263 
    4264 .. image:: img/image212.JPG 
     4264.. image:: img/image212.jpg 
    42654265 
    42664266The special case when |nu| = 0.5 (or *m* = 1/|nu| = 2) corresponds to Gaussian chains for which the form factor is given 
    42674267by the familiar Debye_ function. 
    42684268 
    4269 .. image:: img/image213.JPG 
     4269.. image:: img/image213.jpg 
    42704270 
    42714271For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    42724272 
    4273 .. image:: img/image040.GIF 
     4273.. image:: img/image040.gif 
    42744274 
    42754275This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.2 |Ang^-1| and the default 
    … …  
    42854285===================  ========  ============= 
    42864286 
    4287 .. image:: img/image214.JPG 
     4287.. image:: img/image214.jpg 
    42884288 
    42894289*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    43694369=======================  ========  ============= 
    43704370 
    4371 .. image:: img/image215.JPG 
     4371.. image:: img/image215.jpg 
    43724372 
    43734373*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    43994399For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    44004400 
    4401 .. image:: img/image040.GIF 
     4401.. image:: img/image040.gif 
    44024402 
    44034403===============================  ========  ============= 
    … …  
    44134413===============================  ========  ============= 
    44144414 
    4415 .. image:: img/image217.JPG 
     4415.. image:: img/image217.jpg 
    44164416 
    44174417*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    44354435The scattering intensity *I(q)* is calculated as 
    44364436 
    4437 .. image:: img/image218.JPG 
     4437.. image:: img/image218.jpg 
    44384438 
    44394439where *qc* is the location of the crossover from one slope to the other. The scaling *coef_A* sets the overall 
    … …  
    44454445For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    44464446 
    4447 .. image:: img/image040.GIF 
     4447.. image:: img/image040.gif 
    44484448 
    44494449==============  ========  ============= 
    … …  
    44574457==============  ========  ============= 
    44584458 
    4459 .. image:: img/image219.JPG 
     4459.. image:: img/image219.jpg 
    44604460 
    44614461*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    44864486The empirical fit function is  
    44874487 
    4488 .. image:: img/image220.JPG 
     4488.. image:: img/image220.jpg 
    44894489 
    44904490For each level, the four parameters *Gi*, *Rg,i*, *Bi* and *Pi* must be chosen.  
    … …  
    44974497For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    44984498 
    4499 .. image:: img/image040.GIF 
     4499.. image:: img/image040.gif 
    45004500 
    45014501==============  ========  ============= 
    … …  
    45144514==============  ========  ============= 
    45154515 
    4516 .. image:: img/image221.JPG 
     4516.. image:: img/image221.jpg 
    45174517 
    45184518*Figure. 1D plot using the default values (w/500 data points).* 
    … …  
    45674567The scattered intensity *I(q)* is calculated as 
    45684568 
    4569 .. image:: img/image233.GIF 
     4569.. image:: img/image233.gif 
    45704570 
    45714571where 
    45724572 
    4573 .. image:: img/image234.GIF 
     4573.. image:: img/image234.gif 
    45744574 
    45754575Note that the first term reduces to the Ornstein-Zernicke equation when *D* = 2; ie, when the Flory exponent is 0.5 
    … …  
    45874587============================  ========  ============= 
    45884588 
    4589 .. image:: img/image235.GIF 
     4589.. image:: img/image235.gif 
    45904590 
    45914591*Figure. 1D plot using the default values (w/300 data points).* 
    … …  
    46094609For a star with *f* arms: 
    46104610 
    4611 .. image:: img/star1.PNG 
     4611.. image:: img/star1.png 
    46124612 
    46134613where 
    46144614 
    4615 .. image:: img/star2.PNG 
     4615.. image:: img/star2.png 
    46164616 
    46174617and 
    46184618 
    4619 .. image:: img/star3.PNG 
     4619.. image:: img/star3.png 
    46204620 
    46214621is the square of the ensemble average radius-of-gyration of an arm. 
    … …  
    46464646Also see ReflectivityIIModel_. 
    46474647 
    4648 .. image:: img/image231.BMP 
     4648.. image:: img/image231.bmp 
    46494649 
    46504650*Figure. Comparison (using the SLD profile below) with the NIST web calculation (circles)* 
    46514651http://www.ncnr.nist.gov/resources/reflcalc.html 
    46524652 
    4653 .. image:: img/image232.GIF 
     4653.. image:: img/image232.gif 
    46544654 
    46554655*Figure. SLD profile used for the calculation (above).* 
    … …  
    467546751) Erf 
    46764676 
    4677 .. image:: img/image051.GIF 
     4677.. image:: img/image051.gif 
    46784678 
    467946792) Power-Law 
    46804680 
    4681 .. image:: img/image050.GIF 
     4681.. image:: img/image050.gif 
    46824682 
    468346833) Exp 
    46844684 
    4685 .. image:: img/image049.GIF 
     4685.. image:: img/image049.gif 
    46864686 
    46874687The constant *A* in the expressions above (but the parameter *nu* in the model!) is an input. 
    … …  
    47134713For a 2D plot, the wave transfer is defined as 
    47144714 
    4715 .. image:: img/image040.GIF 
     4715.. image:: img/image040.gif 
    47164716 
    47174717==============  ========  ============= 
    … …  
    47224722==============  ========  ============= 
    47234723 
    4724 .. image:: img/image224.JPG 
     4724.. image:: img/image224.jpg 
    47254725 
    47264726*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    47544754For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    47554755 
    4756 .. image:: img/image040.GIF 
     4756.. image:: img/image040.gif 
    47574757 
    47584758==============  =========  ============= 
    … …  
    47654765==============  =========  ============= 
    47664766 
    4767 .. image:: img/image226.JPG 
     4767.. image:: img/image226.jpg 
    47684768 
    47694769*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    48064806==============  ========  ============= 
    48074807 
    4808 .. image:: img/image227.JPG 
     4808.. image:: img/image227.jpg 
    48094809 
    48104810*Figure. 1D plot using the default values (in linear scale).* 
    … …  
    48524852For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 
    48534853 
    4854 .. image:: img/image040.GIF 
     4854.. image:: img/image040.gif 
    48554855 
    48564856==============  ========  ============= 
    … …  
    48634863==============  ========  ============= 
    48644864 
    4865 .. image:: img/image230.JPG 
     4865.. image:: img/image230.jpg 
    48664866 
    48674867*Figure. 1D plot using the default values (in linear scale).* 
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