Changeset 318b5bbb in sasview for sansmodels/src/sans/models


Ignore:
Timestamp:
Dec 18, 2012 10:55:24 AM (12 years ago)
Author:
Jae Cho <jhjcho@…>
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:
6550b64
Parents:
0203ade
Message:

Added polarization and magnetic stuffs

Location:
sansmodels/src/sans/models
Files:
14 added
51 edited

Legend:

Unmodified
Added
Removed

  • sansmodels/src/sans/models/BaseComponent.py

    r7fdb332 r318b5bbb  
    3939        ## parameters with orientation 
    4040        self.orientation_params = [] 
     41        ## magnetic parameters 
     42        self.magnetic_params = [] 
    4143        ## store dispersity reference 
    4244        self._persistency_dict = {} 

  • sansmodels/src/sans/models/CoreMultiShellModel.py

    r279e371 r318b5bbb  
    1313    def __init__(self, multfactor=1): 
    1414        BaseComponent.__init__(self) 
     15        """ 
     16        :param n_shells: number of shells in the model, assumes 1<= n_shells <=4. 
     17        """ 
    1518 
    1619        ## Setting  model name model description 
     20        self.description="" 
    1721        model = CoreFourShellModel() 
    1822        self.model = model 
    1923        self.name = "CoreMultiShellModel" 
    20         self.description = "" 
     24        self.description="" 
    2125        self.n_shells = multfactor 
    2226        ## Define parameters 
    … …  
    3943        #list of parameter that can be fitted 
    4044        self._set_fixed_params()   
     45        self.orientation_params = [] 
     46        self.magnetic_params = [] 
    4147         
    4248        ## functional multiplicity info of the model 
    … …  
    4551        self.multiplicity_info = [max_nshells, "No. of Shells:", [], ['Radius']] 
    4652 
    47         ## parameters with orientation: 
    48         for item in self.model.orientation_params: 
    49             self.orientation_params.append(item) 
     53        ## parameters with orientation: can be removed since there is no orientational params 
     54        self._set_orientation_params() 
     55                 
    5056         
    5157    def _clone(self, obj): 
    … …  
    6268        return obj 
    6369     
     70     
    6471    def _set_dispersion(self): 
    6572        """ 
    … …  
    7077        for name , value in self.model.dispersion.iteritems():       
    7178            nshell = 0 
    72             if name.split('_')[0] == 'thick': 
     79            if name.split('_').count('thick') > 0: 
    7380                while nshell < self.n_shells: 
    7481                    nshell += 1 
    75                     if name.split('_')[1] == 'shell%s' % str(nshell): 
     82                    if name.split('_')[-1] == 'shell%s' % str(nshell): 
    7683                        self.dispersion[name] = value 
    7784                    else:  
    … …  
    7986            else: 
    8087                self.dispersion[name] = value 
    81                                
     88                 
     89    def _set_orientation_params(self): 
     90        """ 
     91        model orientation and magnetic parameters, same params for this model 
     92        """  
     93        ##set dispersion from model  
     94        for param in self.model.orientation_params:      
     95            nshell = 0 
     96            if param.split('_')[-1].count('shell') < 1: 
     97                #print "param", param, param.split('_')[-1].count('shell') 
     98                self.orientation_params.append(param) 
     99                self.magnetic_params.append(param) 
     100                continue 
     101            while nshell < self.n_shells: 
     102                nshell += 1 
     103                if param.split('_')[-1] == 'shell%s' % str(nshell): 
     104                    self.orientation_params.append(param) 
     105                    self.magnetic_params.append(param) 
     106                    continue                            
    82107 
    83108    def _set_params(self): 
    … …  
    89114        for name , value in self.model.params.iteritems(): 
    90115            nshell = 0 
    91             if name.split('_')[0] == 'thick' or name.split('_')[0] == 'sld': 
    92                 if name.split('_')[1] == 'solv' \ 
    93                     or name.split('_')[1] == 'core0': 
    94                     self.params[name] = value 
     116            if name.split('_').count('thick') > 0 or \ 
     117                        name.split('_').count('sld') > 0 or \ 
     118                        name[0] == 'M': 
     119                if name.split('_')[-1] == 'solv' or \ 
     120                        name.split('_')[-1] == 'core0': 
     121                    self.params[name]= value 
    95122                    continue 
    96123                while nshell < self.n_shells: 
    97124                    nshell += 1 
    98                     if name.split('_')[1] == 'shell%s' % str(nshell): 
    99                         self.params[name] = value 
     125                    if name.split('_')[-1] == 'shell%s' % str(nshell): 
     126                        self.params[name]= value 
    100127                        continue 
    101128            else: 
    102                 self.params[name] = value 
    103              
     129                self.params[name]= value 
    104130             
    105131        # set constrained values for the original model params 
    … …  
    113139        for name ,detail in self.model.details.iteritems(): 
    114140            if name in self.params.iterkeys(): 
    115                 self.details[name] = detail 
     141                self.details[name]= detail 
    116142             
    117143     
    … …  
    123149        for key in self.model.params.iterkeys(): 
    124150            if key not in self.params.keys(): 
    125                 if  key.split('_')[0] == 'thick': 
     151                if  key.split('_').count('thick') > 0: 
    126152                    self.model.setParam(key, 0) 
    127153                    continue 
    128154 
    129155                for nshell in range(self.n_shells,max_nshells): 
    130                     if key.split('_')[0] == 'sld' and \ 
    131                         key.split('_')[1] == 'shell%s' % str(nshell+1): 
     156                     if key.split('_').count('sld') > 0 and \ 
     157                            key.split('_')[-1] == 'shell%s' % str(nshell+1): 
    132158                        try: 
    133                             value = self.model.params['sld_solv'] 
    134                             self.model.setParam(key, value) 
    135                         except: 
    136                             message = "CoreMultiShellModel evaluation problem" 
    137                             raise RuntimeError, message 
     159                            if key[0] != 'M': 
     160                                value = self.model.params['sld_solv'] 
     161                                self.model.setParam(key, value) 
     162                            else: 
     163                                self.model.setParam(key, 0.0) 
     164                        except: pass 
     165                      
    138166 
    139167    def getProfile(self): 
    … …  
    187215        ## setParam to model  
    188216        if name == 'sld_solv': 
    189             # the sld_*** model.params not in params  
    190             # must set to value of sld_solv 
     217            # the sld_*** model.params not in params must set to value of sld_solv 
    191218            for key in self.model.params.iterkeys(): 
    192                 if key not in self.params.keys()and key.split('_')[0] == 'sld': 
    193                     self.model.setParam(key, value) 
     219                if key not in self.params.keys(): 
     220                    if key.split('_')[0] == 'sld': 
     221                        self.model.setParam(key, value) 
     222                    elif key.split('_')[1] == 'sld': 
     223                        # mag params 
     224                        self.model.setParam(key, 0.0) 
    194225        self.model.setParam( name, value) 
    195226 
    … …  
    213244                return 
    214245         
    215         raise ValueError, "Model does not contain parameter %s" % name 
     246        #raise ValueError, "Model does not contain parameter %s" % name 
    216247              
    217248    
    … …  
    250281    ## Now (May27,10) directly uses the model eval function  
    251282    ## instead of the for-loop in Base Component. 
    252     def evalDistribution(self, x): 
     283    def evalDistribution(self, x = []): 
    253284        """  
    254285        Evaluate the model in cartesian coordinates 
    … …  
    259290        # set effective radius and scaling factor before run 
    260291        return self.model.evalDistribution(x) 
    261  
     292     
     293    def calculate_ER(self): 
     294        """  
     295        Calculate the effective radius for P(q)*S(q) 
     296         
     297        :return: the value of the effective radius 
     298         
     299        """        
     300        return self.model.calculate_ER()  
     301     
     302    def calculate_VR(self): 
     303        """  
     304        Calculate the volf ratio for P(q)*S(q) 
     305         
     306        :return: the value of the volf ratio 
     307         
     308        """        
     309        return self.model.calculate_VR() 
     310     
    262311    def set_dispersion(self, parameter, dispersion): 
    263312        """ 

  • sansmodels/src/sans/models/MultiplicationModel.py

    rae4c139 r318b5bbb  
    3636        self.p_model = p_model 
    3737        self.s_model = s_model         
    38         
     38        self.magnetic_params = [] 
    3939        ## dispersion 
    4040        self._set_dispersion() 
    … …  
    5353        for item in self.p_model.orientation_params: 
    5454            self.orientation_params.append(item) 
    55              
     55        for item in self.p_model.magnetic_params:   
     56            self.magnetic_params.append(item)  
    5657        for item in self.s_model.orientation_params: 
    5758            if not item in self.orientation_params: 

  • sansmodels/src/sans/models/media/model_functions.html

    r657e52c r318b5bbb  
    77<ul style="margin-top: 0in;" type="disc"> 
    88<li style="line-height: 115%;"><a href="#Introduction"><strong>Introduction</strong></a></li> 
    9 <li style="line-height: 115%;"><a href="#Shapes"><strong>Shapes</strong></a>: <a href="#SphereModel">SphereModel</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel</a>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 
    10 <li style="line-height: 115%;"><a href="#Shape-Independent"><strong>Shape-Independent</strong></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DAB_Model</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li> 
     9<li style="line-height: 115%;"><a href="#Shapes"><strong>Shapes</strong></a>: <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 
     10<li style="line-height: 115%;"><a href="#Shape-Independent"><strong>Shape-Independent</strong></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DABModel</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li> 
    1111<li style="line-height: 115%;"><a href="#Model"><strong>Customized Models</strong></a>: <a href="#testmodel">testmodel</a>, <a href="#testmodel_2">testmodel_2</a>, <a href="#sum_p1_p2">sum_p1_p2</a>, <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, <a href="#polynomial5">polynomial5</a>, <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li> 
    1212<li style="line-height: 115%;"><a href="#Structure_Factors"><strong>Structure Factors</strong></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a href="#SquareWellStructure">SquareWellStructure</a>, <a href="#HayterMSAStructure">HayterMSAStructure</a>, <a href="#StickyHSStructure">StickyHSStructure</a>.</li> 
    … …  
    1717<p>&nbsp;Many of our models use the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research and thus some content and figures in this document are originated from or shared with the NIST Igor analysis package.</p> 
    1818<p style="margin-left: 0.25in; text-indent: -0.25in;"><strong><span style="font-size: 16pt;">2.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp; </span></strong><a name="Shapes"></a><strong><span style="font-size: 16pt;">Shapes (Scattering Intensity Models)</span></strong></p> 
    19 <p>This software provides form factors for various particle shapes. After giving a mathematical definition of each model, we draw the list of parameters available to the user. Validation plots for each model are also presented. Instructions on how to use the software is available with the source code, available from SVN:</p> 
    20 <p style="text-indent: 0.25in;"><em>https://sansviewproject.svn.sourceforge.net/svnroot/sansviewproject/</em></p> 
     19<p>This software provides form factors for various particle shapes. After giving a mathematical definition of each model, we draw the list of parameters available to the user. Validation plots for each model are also presented. Instructions on how to use the software is available with the source code.</p> 
     20 
    2121<p>To easily compare to the scattering intensity measured in experiments, we normalize the form factors by the volume of the particle:</p> 
    2222<p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image001.PNG" alt="" /></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    … …  
    2727<p>Our so-called 1D scattering intensity functions provide <em>P(q) </em>for the case where the scatterer is randomly oriented. In that case, the scattering intensity only depends on the length of q. The intensity measured on the plane of the SANS detector will have an azimuthal symmetry around <em>q</em>=0.</p> 
    2828<p>Our so-called 2D scattering intensity functions provide <em>P(q, </em><em><span style="font-family: 'Arial','sans-serif';">&phi;</span>)</em> for an oriented system as a function of a q-vector in the plane of the detector. We define the angle <span style="font-family: 'Arial','sans-serif';">&phi;</span> as the angle between the q vector and the horizontal (x) axis of the plane of the detector.</p> 
    29 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.1.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="SphereModel"></a><strong><span style="font-size: 14pt;">Sphere Model</span></strong></p> 
     29<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.1.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="SphereModel"></a><strong><span style="font-size: 14pt;">Sphere Model (Magnetic 2D Model)</span></strong></p> 
    3030<p>This model provides the form factor, P(q), for a monodisperse spherical particle with uniform scattering length density. The form factor is normalized by the particle volume as described below.</p> 
     31For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
    3132<p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</strong></p> 
    3233<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
    … …  
    242243<p>where the amplitude A(q) is given as the typical sphere scattering convoluted with a Gaussian to get a gradual drop-off in the scattering length density:</p> 
    243244<p style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="img/image011.PNG" alt="" /></span></p> 
    244 <br> 
    245245<p>Here A2(q) is the form factor, P(q). The &lsquo;scale&rsquo; is equivalent to the volume fraction of spheres, each of volume, V. Contrast (<em><span style="font-family: 'Arial','sans-serif';">&Delta;</span>&rho;</em> ) is the difference of scattering length densities of the sphere and the surrounding solvent.</p> 
    246246<p>The poly-dispersion in radius and in fuzziness is provided.</p> 
    … …  
    453453<p style="text-align: center;" align="center">&nbsp;</p> 
    454454<p>&nbsp;</p> 
    455 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.5.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CoreShellModel"></a><strong><span style="font-size: 14pt;">Core Shell (Sphere) Model</span></strong></p> 
     455<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.5.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CoreShellModel"></a><strong><span style="font-size: 14pt;">Core Shell (Sphere) Model (Magnetic 2D Model)</span></strong></p> 
    456456<p>This model provides the form factor, P(<em>q</em>), for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume.</p> 
     457For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
    457458<p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</strong></p> 
    458459<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
    … …  
    686687<p>&nbsp;</p> 
    687688<p style="text-align: center; page-break-after: avoid;" align="center"><img style="width: 526px; height: 333px;" src="img/secongm_fig1.jpg" alt="core_scondmoment_1D_validation" /></p> 
    688 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.7.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CoreMultiShellModel"></a><strong><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model</span></strong></p> 
     689<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.7.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CoreMultiShellModel"></a><strong><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model (Magnetic 2D Model)</span></strong></p> 
    689690<p>This model provides the scattering from spherical core with from 1 up to 4 shell structures. It&nbsp;has a core of a specified radius, with four shells. The SLDs of the core and each shell are individually specified.&nbsp;</p> 
     691For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
    690692<p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</strong></p> 
    691693<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
    … …  
    17071709<p><a name="PearlNecklaceModel"></a>R. Schweins and K. Huber, &lsquo;Particle Scattering Factor of Pearl Necklace Chains&rsquo;, Macromol. Symp., 211, 25-42, 2004.</p> 
    17081710<p><a name="PearlNecklaceModel"></a>&nbsp;</p> 
    1709 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a><strong><span style="font-size: 14pt;">2.14.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CylinderModel"></a><strong><span style="font-size: 14pt;">Cylinder Model</span></strong></p> 
     1711<p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a><strong><span style="font-size: 14pt;">2.14.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CylinderModel"></a><strong><span style="font-size: 14pt;">Cylinder Model (Magnetic 2D Model)</span></strong></p> 
    17101712<p>This model provides the form factor for a right circular cylinder with uniform scattering length density. The form factor is normalized by the particle volume.</p> 
     1713For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
    17111714<p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</strong></p> 
    17121715<p>The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955):</p> 
    … …  
    17151718<p>where <span style="font-family: 'Arial','sans-serif';">&alpha;</span> is the angle between the axis of the cylinder and the q-vector, V is the volume of the cylinder, L is the length of the cylinder, r is the radius of the cylinder, and <em><span style="font-family: 'Arial','sans-serif';">&Delta;</span>&rho;</em> (contrast) is the scattering length density difference between the scatterer and the solvent. J1 is the first order Bessel function.</p> 
    17161719<p>To provide easy access to the orientation of the cylinder, we define the axis of the cylinder using two angles theta and phi. Those angles are defined on Figure 2.</p> 
    1717 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" alt="cylinderangles.gif" width="478" height="258" /></p> 
    1718 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2a. Definition of the angles for oriented cylinders.</p> 
    1719 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="cylinderangles2.gif" width="464" height="313" /></p> 
    1720 <p style="text-align: center;" align="center">Figure 2b. Examples of the angles for oriented cylinders against the detector plane.</p> 
     1720<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     1721<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2. Definition of the angles for oriented cylinders.</p> 
     1722 
     1723<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     1724<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     1725 
     1726 
    17211727<p>For P*S: The 2nd virial coefficient of the cylinder is calculate based on the radius and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 
    17221728<p>The returned value is scaled to units of [cm-1] and the parameters of the cylinder model are the following:</p> 
    … …  
    19811987<p style="text-align: center; page-break-after: avoid;" align="center">&nbsp;</p> 
    19821988<p><a name="_Ref173307204"></a>Figure 9: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius=20 &Aring;, Thickness=10 &Aring;, Length=400 &Aring;, Core_sld=1e-6 &Aring; -2, Shell_sld=4e-6 &Aring; -2, Solvent_sld=1e-6 &Aring; -2, and Background=0.0 cm -1.</p> 
     1989 
     1990<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     1991<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented core-shell cylinders.</p> 
     1992<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     1993<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     1994 
    19831995<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.16.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CoreShellBicelleModel"></a><strong><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></strong></p> 
    19841996<p>This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The form factor is normalized by the particle volume. This model is a more general case of <a href="#CoreShellCylinderModel">core-shell cylinder model </a> (see&nbsp;above and reference below) in that the parameters of the shell are separated into a face-shell and a rim-shell so that users can set different values of the thicknesses and slds.&nbsp;</p> 
    … …  
    21272139<p style="text-align: center;" align="center"><img id="cscylbicelle" style="width: 512px; height: 377px;" src="img/cscylbicelle_pic.jpg" alt="" /></p> 
    21282140<p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/200 data point).</strong></p> 
     2141 
     2142 
     2143<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     2144<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for the  
     2145oriented Core-Shell Cylinder Bicelle Model.</p> 
     2146<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     2147<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     2148 
     2149 
    21292150<p>&nbsp;REFERENCE<br /> Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> 
    21302151<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.17.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="HollowCylinderModel"></a><strong><span style="font-size: 14pt;">HollowCylinderModel</span></strong></p> 
    … …  
    22372258<p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 
    22382259<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
     2260 
     2261<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     2262<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for the  
     2263oriented HollowCylinderModel.</p> 
     2264<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     2265<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     2266 
    22392267<p>REFERENCE</p> 
    22402268<p>Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> 
    … …  
    23412369<p style="text-align: center;" align="center"><img id="Picture 228" src="img/image076.jpg" alt="" width="465" height="345" /></p> 
    23422370<p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 
     2371 
    23432372<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    23442373<p>From the reference, "Method 3 With Excluded Volume" is used. The model is a parametrization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous limit.&nbsp; See equations (13,26-27) in the original reference for the details.</p> 
    … …  
    26162645<p style="text-align: center;" align="center"><img src="img/image085.jpg" alt="" width="451" height="334" /></p> 
    26172646<p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 
    2618 <p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg" alt="" width="377" height="215" /></p> 
     2647<p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg" /></p> 
    26192648<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented stackeddisks against the detector plane.</p> 
     2649 
     2650<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     2651<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     2652 
     2653 
    26202654<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    26212655<p>REFERENCE</p> 
    … …  
    26232657<p>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949.</p> 
    26242658<p>Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", Clarendon, Oxford, 1994.</p> 
    2625 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.21.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="ParallelepipedModel"></a><strong><span style="font-size: 14pt;">ParallelepipedModel</span></strong></p> 
     2659<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.21.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="ParallelepipedModel"></a><strong><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></strong></p> 
    26262660<p>This model provides the form factor, P(<em>q</em>), for a rectangular cylinder (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*&lt;f^2&gt;/V+background where the volume V= ABC and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p> 
     2661For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
    26272662<p><span style="font-size: 14pt;">&nbsp;</span></p> 
    26282663<p style="text-align: center;" align="center"><img src="img/image087.jpg" alt="" width="326" height="247" /></p> 
    … …  
    26362671<p>For P*S: The 2nd virial coefficient of the solid cylinder is calculate based on the averaged effective radius (= sqrt(short_a*short_b/pi)) and length( = long_c) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 
    26372672<p>To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using two angles &theta; , <span style="font-family: 'Arial','sans-serif';">&phi; </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, &theta; &nbsp;and <span style="font-family: 'Arial','sans-serif';">&phi;,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 
    2638 <p style="text-align: center;" align="center"><img src="img/image090.jpg" alt="" width="352" height="264" /></p> 
     2673<p style="text-align: center;" align="center"><img src="img/image090.jpg"/></p> 
    26392674<p style="text-align: center;" align="center"><strong>Figure. Definition of angles for 2D</strong>.</p> 
    26402675<p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p> 
    … …  
    27492784<p>For P*S: The 2nd virial coefficient of this CSPP is calculate based on the averaged effective radius (= sqrt((short_a+2*rim_a)*(short_b+2*rim_b)/pi)) and length( = long_c+2*rim_c) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 
    27502785<p>To provide easy access to the orientation of the CSparallelepiped, we define the axis of the cylinder using two angles &theta; , <span style="font-family: 'Arial','sans-serif';">&phi; </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, &theta; &nbsp;and <span style="font-family: 'Arial','sans-serif';">&phi;,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 
    2751 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" alt="" width="352" height="264" /></p> 
     2786<p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" /></p> 
    27522787<p style="text-align: center;" align="center"><strong>Figure. Definition of angles for 2D</strong>.</p> 
    27532788<p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p> 
    … …  
    29182953<p style="text-align: center;" align="center"><img id="Picture 34" src="img/image097.jpg" alt="" width="451" height="339" /></p> 
    29192954<p style="text-align: center;" align="center"><strong>Figure. 2D plot using the default values (w/(256X265) data points).</strong></p> 
     2955 
    29202956<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    29212957<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; REFERENCE</p> 
    … …  
    29402976<p>To provide easy access to the orientation of the elliptical, we define the axis of the cylinder using two angles &theta; , <span style="font-family: 'Arial','sans-serif';">&phi; </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, &theta; &nbsp;and <span style="font-family: 'Arial','sans-serif';">&phi;,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the r_minor axis is parallel to the x-axis of the detector.</p> 
    29412977<p>All angle parameters are valid and given only for 2D calculation (Oriented system).</p> 
    2942 <p style="text-align: center;" align="center"><img id="Picture 105" src="img/image101.jpg" alt="" width="370" height="277" /></p> 
     2978<p style="text-align: center;" align="center"><img id="Picture 105" src="img/image101.jpg" /></p> 
    29432979<p style="text-align: center;" align="center"><strong>Figure. Definition of angels for 2D</strong>.</p> 
    2944 <p style="text-align: center;" align="center"><img id="Picture 114" src="img/image091.jpg" alt="" width="379" height="256" /></p> 
     2980<p style="text-align: center;" align="center"><img id="Picture 114" src="img/image062.jpg" alt="" width="379" height="256" /></p> 
    29452981<p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented elliptical cylinders </span></p> 
    29462982<p style="text-align: center;" align="center"><span style="font-size: 12pt;">against the detector plane.</span></p> 
    … …  
    31733209<p style="text-align: center;" align="center"><img id="Picture 66" src="img/image111.jpg" alt="" width="425" height="346" /></p> 
    31743210<p style="text-align: center;" align="center"><strong>Figure. 2D plot (w/(256X265) data points).</strong></p> 
     3211 
     3212<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     3213<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     3214<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     3215 
     3216 
     3217<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented 2D barbells.</p> 
     3218 
     3219 
    31753220<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.25.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="CappedCylinderModel"></a><strong><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></strong></p> 
    31763221<p>Calculates the scattering from a cylinder with spherical section end-caps(This model simply becomes the ConvexLensModel when the length of the cylinder L = 0. &nbsp;That is, a sphereocylinder with end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies within the cylinder. See the diagram for the details of the geometry and restrictions on parameter values.</p> 
    … …  
    32953340<p style="text-align: center;" align="center"><img id="Picture 71" src="img/image118.jpg" alt="" width="402" height="334" /></p> 
    32963341<p style="text-align: center;" align="center"><strong>Figure. 2D plot (w/(256X265) data points).</strong></p> 
     3342<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     3343<p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented 2D cylinders.</p> 
     3344<p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 
     3345<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     3346 
     3347 
     3348 
    32973349<p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.26.</span></strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="EllipsoidModel"></a><strong><span style="font-size: 14pt;">Ellipsoid Model</span></strong></p> 
    32983350<p>This model provides the form factor for an ellipsoid (ellipsoid of revolution) with uniform scattering length density. The form factor is normalized by the particle volume.</p> 
    … …  
    34143466<p>The output of the 1D scattering intensity function for randomly oriented ellipsoids is then given by the equation above.</p> 
    34153467<p>The <em>axis_theta</em> and axis<em>_phi</em> parameters are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.</p> 
    3416 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="396" height="297" /></p> 
    3417 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented ellipsoid </span></p> 
    3418 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">against the detector plane</span>.</p> 
     3468<p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 
     3469<p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. The angles for oriented ellipsoid </span></p> 
     3470 
    34193471<p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>2.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the ellipsoid model</strong></p> 
    34203472<p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 5 shows a comparison of the 1D output of our model and the output of the NIST software.</p> 
    … …  
    35583610<p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 
    35593611<p style="text-align: center;" align="center"><strong>&nbsp;</strong></p> 
    3560 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="396" height="297" /></p> 
    3561 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented coreshellellipsoid against the detector plane where a =polar axis.</p> 
     3612<p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 
     3613<p style="text-align: center;" align="center">Figure. The angles for oriented coreshellellipsoid .</p> 
    35623614<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    35633615<p>REFERENCE</p> 
    … …  
    36743726<p style="text-align: center;" align="center"><img src="img/image131.gif" alt="" width="438" height="272" /></p> 
    36753727<p style="text-align: center;" align="center"><strong>Figure. Comparison between 1D and averaged 2D.</strong></p> 
    3676 <p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="396" height="297" /></p> 
    3677 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented ellipsoid against the detector plane.</p> 
     3728<p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="379" height="256" /></p> 
     3729<p style="text-align: center;" align="center">Figure. The angles for oriented ellipsoid.</p> 
    36783730<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    36793731<p>REFERENCE</p> 
    … …  
    43684420<p style="text-align: center;" align="center"><strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p> 
    43694421<p>&nbsp;The 2D (Anisotropic model) is based on the reference (above) which I(q) &nbsp;is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 
    4370 <p style="text-align: center;" align="center"><img id="Object 23" src="img/image156.gif" alt="" width="304" height="321" /></p> 
     4422<p style="text-align: center;" align="center"><img id="Object 23" src="img/image156.jpg" /></p> 
    43714423<p style="text-align: center;" align="center">&nbsp;</p> 
    43724424<p>&nbsp;</p> 
    … …  
    44934545<p style="text-align: center;" align="center"><strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p> 
    44944546<p>&nbsp;The 2D (Anisotropic model) is based on the reference (above) in which I(q)&nbsp; is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 
    4495 <p style="text-align: center;" align="center"><img src="img/image165.gif" alt="" width="304" height="321" /></p> 
     4547<p style="text-align: center;" align="center"><img src="img/image165.gif" /></p> 
    44964548<p style="text-align: center;" align="center">&nbsp;</p> 
    44974549<p>&nbsp;</p> 
    … …  
    46174669<p style="text-align: center;" align="center"><strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p> 
    46184670<p>&nbsp;The 2D (Anisotropic model) is based on the reference (1987) in which I(q) &nbsp;is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 
    4619 <p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" alt="" width="304" height="321" /></p> 
     4671<p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" /></p> 
    46204672<p style="text-align: center;" align="center">&nbsp;</p> 
    46214673<p>&nbsp;</p> 

  • sansmodels/src/sans/models/media/pd_help.html

    r17574ae r318b5bbb  
    1 <html> 
    2  
    31<head> 
    42<meta http-equiv=Content-Type content="text/html; charset=windows-1252"> 
    … …  
    2927style='font-family:"Times New Roman","serif"'>The following five distribution 
    3028functions are provided;</span></p> 
    31 <p>&nbsp;</p> 
    3229<ul> 
    3330<li><a href="#Rectangular">Rectangular distribution</a></li> 
    … …  
    3835</ul> 
    3936<p>&nbsp;</p> 
    40 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    41 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     37<p><a name="Rectangular"><h4>Rectangular distribution</a></h4></p> 
     38<p><img src="./img/pd_image001.png"></p> 
     39<p>&nbsp;</p> 
     40<p>The x<sub>mean</sub> is the mean 
     41of the distribution, w is the half-width, and Norm is a normalization factor 
     42which is determined during the numerical calculation. Note that the Sigma and 
     43the half width <i>w</i> are different.</p> 
     44<p>The standard deviation is </p> 
     45<p><img src="./img/pd_image002.png"></p> 
     46<p>&nbsp;</p> 
     47<p>The PD (polydispersity) is </p> 
     48<p><img src="./img/pd_image003.png"></p> 
     49<p>&nbsp;</p> 
     50<p><img width=511 height=270 
     51id="Picture 1" src="./img/pd_image004.jpg" alt=flat.gif></p> 
     52<p>&nbsp;</p> 
     53<p>&nbsp;</p> 
     54<p><a name="Array"><h4>Array distribution</h4></a></p> 
    4255 
    43 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    44 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     56<p>This distribution is to be given 
     57by users as a txt file where the array should be defined by two columns in the 
     58order of x and f(x) values. The f(x) will be normalized by SansView during the 
     59computation.</p> 
    4560 
    46 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    47 style='font-family:"Times New Roman","serif"'><a name="Rectangular"><h4>Rectangular distribution</a></h4></span></p> 
     61<p>Example of an array in the file;</p> 
    4862 
    49 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    50 style='font-family:"Times New Roman","serif";position:relative;top:22.0pt'><img 
    51 width=248 height=67 src="./img/pd_image001.png"></span></p> 
     63<p>30        0.1</p> 
     64 
     65<p>32        0.3</p> 
     66 
     67<p>35        0.4</p> 
     68 
     69<p>36        0.5</p> 
     70 
     71<p>37        0.6</p> 
     72 
     73<p>39        0.7</p> 
     74 
     75<p>41        0.9</p> 
     76 
     77<p'>&nbsp;</p> 
     78 
     79<p>We use only these array values in 
     80the computation, therefore the mean value given in the control panel, for 
     81example ‘radius = 60’, will be ignored.</p> 
     82<p>&nbsp;</p> 
     83 
     84 
     85<p><a name="Gaussian"><h4>Gaussian distribution</h4></a></p> 
     86<p>&nbsp;</p> 
     87 
     88<p><img src="./img/pd_image005.png"></p> 
    5289 
    5390<p>&nbsp;</p> 
    54 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    55 style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean 
    56 of the distribution, w is the half-width, and Norm is a normalization factor 
    57 which is determined during the numerical calculation. Note that the Sigma and 
    58 the half width <i>w</i> are different.</span></p> 
    5991 
    60 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    61 style='font-family:"Times New Roman","serif"'>The standard deviation is </span></p> 
     92<p>The x<sub>mean</sub> is the mean 
     93of the distribution and Norm is a normalization factor which is determined 
     94during the numerical calculation.</p> 
    6295 
    63 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    64 style='font-family:"Times New Roman","serif";position:relative;top:4.0pt'><img 
    65 width=72 height=24 src="./img/pd_image002.png"></span><span 
    66 style='font-family:"Times New Roman","serif"'>. </span></p> 
     96<p>&nbsp;</p> 
    6797 
    68 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    69 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     98<p>The PD (polydispersity) is </p> 
    7099 
    71 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    72 style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p> 
     100<p><img src="./img/pd_image003.png"></p> 
    73101 
    74 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    75 style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img 
    76 width=93 height=24 src="./img/pd_image003.png"></span><span 
    77 style='font-family:"Times New Roman","serif"'>.</span></p> 
     102<p>&nbsp;</p> 
    78103 
    79 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    80 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     104<p><img width=518 height=275 
     105id="Picture 2" src="./img/pd_image006.jpg" alt=gauss.gif></p> 
    81106 
    82 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    83 style='font-family:"Times New Roman","serif"'><img width=511 height=270 
    84 id="Picture 1" src="./img/pd_image004.jpg" alt=flat.gif></span></p> 
     107<p>&nbsp;</p> 
    85108 
    86 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    87 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     109<p><a name="Lognormal"><h4>Lognormal distribution</h4></a></p> 
    88110 
    89 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    90 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     111<p>&nbsp;</p> 
    91112 
    92 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    93 style='font-family:"Times New Roman","serif"'><a name="Array"><h4>Array distribution</h4></a></span></p> 
     113<p><img src="./img/pd_image007.png"></p> 
    94114 
    95 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    96 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     115<p>&nbsp;</p> 
    97116 
    98  
    99 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    100 style='font-family:"Times New Roman","serif"'>This distribution is to be given 
    101 by users as a txt file where the array should be defined by two columns in the 
    102 order of x and f(x) values. The f(x) will be normalized by SansView during the 
    103 computation.</span></p> 
    104  
    105 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    106 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    107  
    108 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    109 style='font-family:"Times New Roman","serif"'>Example of an array in the file;</span></p> 
    110  
    111 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    112 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    113  
    114 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    115 style='font-family:"Times New Roman","serif"'>30        0.1</span></p> 
    116  
    117 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    118 style='font-family:"Times New Roman","serif"'>32        0.3</span></p> 
    119  
    120 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    121 style='font-family:"Times New Roman","serif"'>35        0.4</span></p> 
    122  
    123 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    124 style='font-family:"Times New Roman","serif"'>36        0.5</span></p> 
    125  
    126 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    127 style='font-family:"Times New Roman","serif"'>37        0.6</span></p> 
    128  
    129 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    130 style='font-family:"Times New Roman","serif"'>39        0.7</span></p> 
    131  
    132 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    133 style='font-family:"Times New Roman","serif"'>41        0.9</span></p> 
    134  
    135 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    136 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    137  
    138 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    139 style='font-family:"Times New Roman","serif"'>We use only these array values in 
    140 the computation, therefore the mean value given in the control panel, for 
    141 example ‘radius = 60’, will be ignored.</span></p> 
    142  
    143 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    144 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    145  
    146 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    147 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    148  
    149 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    150 style='font-family:"Times New Roman","serif"'><a name="Gaussian"><h4>Gaussian distribution</h4></a></span></p> 
    151  
    152 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    153 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    154  
    155 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    156 style='font-family:"Times New Roman","serif";position:relative;top:12.0pt'><img 
    157 width=212 height=44 src="./img/pd_image005.png"></span></p> 
    158  
    159 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    160 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    161  
    162 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    163 style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean 
    164 of the distribution and Norm is a normalization factor which is determined 
    165 during the numerical calculation.</span></p> 
    166  
    167 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    168 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    169  
    170 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    171 style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p> 
    172  
    173 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    174 style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img 
    175 width=93 height=24 src="./img/pd_image003.png"></span><span 
    176 style='font-family:"Times New Roman","serif"'>.</span></p> 
    177  
    178 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    179 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    180  
    181 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    182 style='font-family:"Times New Roman","serif"'><img width=518 height=275 
    183 id="Picture 2" src="./img/pd_image006.jpg" alt=gauss.gif></span></p> 
    184  
    185 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    186 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    187  
    188 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    189 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    190  
    191 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    192 style='font-family:"Times New Roman","serif"'><a name="Lognormal"><h4>Lognormal distribution</h4></a></span></p> 
    193  
    194 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    195 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    196  
    197 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    198 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    199  
    200 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    201 style='font-family:"Times New Roman","serif";position:relative;top:14.0pt'><img 
    202 width=236 height=47 src="./img/pd_image007.png"></span></p> 
    203  
    204 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    205 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    206  
    207 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    208 style='font-family:"Times New Roman","serif"'>The mu = ln(x<sub>med</sub>),  x<sub>med</sub> 
     117<p>The &#956; = ln(x<sub>med</sub>),  x<sub>med</sub> 
    209118is the median value of the distribution, and Norm is a normalization factor 
    210119which will be determined during the numerical calculation. The median value is 
    211120the value given in the size parameter in the control panel, for example, 
    212 “radius = 60”.</span></p> 
     121“radius = 60”.</p> 
    213122 
    214 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    215 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     123<p >&nbsp;</p> 
    216124 
    217 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    218 style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is given 
    219 by sigma,</span></p> 
     125<p>The PD (polydispersity) is given 
     126by &#963;,</p> 
    220127 
    221 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    222 style='font-family:"Times New Roman","serif";position:relative;top:5.0pt'><img 
    223 width=55 height=21 src="./img/pd_image008.png"></span><span 
    224 style='font-family:"Times New Roman","serif"'>.</span></p> 
     128<p><img src="./img/pd_image008.png"></p> 
    225129 
    226 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    227 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     130<p>&nbsp;</p> 
    228131 
    229 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    230 style='font-family:"Times New Roman","serif"'>For the angular distribution,</span></p> 
     132<p>For the angular distribution,</p> 
    231133 
    232 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    233 style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img 
    234 width=76 height=24 src="./img/pd_image009.png"></span></p> 
     134<p><img src="./img/pd_image009.png"></p> 
    235135 
    236 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    237 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     136<p>&nbsp;</p> 
    238137 
    239 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    240 style='font-family:"Times New Roman","serif"'>The mean value is given by x<sub>mean</sub> 
    241 =exp(mu+p^2/2).</span></p> 
     138<p>The mean value is given by x<sub>mean</sub> 
     139=exp(&#956;+p<sup>2</sup>/2).</p> 
    242140 
    243 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    244 style='font-family:"Times New Roman","serif"'>The peak value is given by x<sub>peak</sub>=exp(mu-p^2).</span></p> 
     141<p>The peak value is given by x<sub>peak</sub>=exp(&#956;-p<sup>2</sup>).</span></p> 
    245142 
    246 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    247 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     143<p>&nbsp;</p> 
    248144 
    249 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    250 style='font-family:"Times New Roman","serif"'><img width=450 height=239 
    251 id="Picture 7" src="./img/pd_image010.jpg" alt=lognormal.gif></span></p> 
     145<p><img width=450 height=239 
     146id="Picture 7" src="./img/pd_image010.jpg" alt=lognormal.gif></p> 
    252147 
    253 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    254 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     148<p>&nbsp;</p> 
    255149 
    256 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    257 style='font-family:"Times New Roman","serif"'>This distribution function 
     150<p>This distribution function 
    258151spreads more and the peak shifts to the left as the p increases, requiring 
    259 higher values of Nsigmas and Npts.</span></p> 
     152higher values of Nsigmas and Npts.</p> 
    260153 
    261 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    262 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     154<p>&nbsp;</p> 
    263155 
    264 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    265 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     156<p><a name="Schulz"><h4>Schulz distribution</h4></a></p> 
    266157 
    267 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    268 style='font-family:"Times New Roman","serif"'><a name="Schulz"><h4>Schulz distribution</h4></a></span></p> 
     158<p>&nbsp;</p> 
    269159 
    270 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    271 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     160<p><img src="./img/pd_image011.png"></p> 
    272161 
    273 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    274 style='font-family:"Times New Roman","serif";position:relative;top:15.0pt'><img 
    275 width=347 height=45 src="./img/pd_image011.png"></span></p> 
     162<p>&nbsp;</p> 
    276163 
    277 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    278 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
     164<p>The x<sub>mean</sub> is the mean 
     165of the distribution and Norm is a normalization factor which is determined 
     166during the numerical calculation. </p> 
    279167 
    280 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    281 style='font-family:"Times New Roman","serif"'>The x<sub>mean</sub> is the mean 
    282 of the distribution and Norm is a normalization factor which is determined 
    283 during the numerical calculation. </span></p> 
     168<p>The z = 1/p<sup>2</sup> – 1.</p> 
     169<p>&nbsp;</p> 
    284170 
    285 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    286 style='font-family:"Times New Roman","serif"'>The z = 1/p^2 – 1.</span></p> 
    287  
    288 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    289 style='font-family:"Times New Roman","serif"'>&nbsp;</span></p> 
    290  
    291 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    292 style='font-family:"Times New Roman","serif"'>The PD (polydispersity) is </span></p> 
    293  
    294 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    295 style='font-family:"Times New Roman","serif";position:relative;top:6.0pt'><img 
    296 width=80 height=24 src="./img/pd_image012.png"></span><span 
    297 style='font-family:"Times New Roman","serif"'>.</span></p> 
    298 <p/> 
    299 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    300 style='font-family:"Times New Roman","serif"'>Note that the higher PD (polydispersity) 
     171<p>The PD (polydispersity) is </p> 
     172<p'><img src="./img/pd_image012.png"></p> 
     173<p>Note that the higher PD (polydispersity) 
    301174 might need higher values of Npts and Nsigmas. For example, at PD = 0.7 and  radisus = 60 A,  
    302  Npts >= 160, and Nsigmas >= 15 at least.</span></p> 
     175 Npts >= 160, and Nsigmas >= 15 at least.</p> 
    303176 <p/> 
    304 <p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span 
    305 style='font-family:"Times New Roman","serif"'><img width=438 height=232 
    306 id="Picture 4" src="./img/pd_image013.jpg" alt=schulz.gif></span></p> 
     177<p><img width=438 height=232 
     178id="Picture 4" src="./img/pd_image013.jpg" alt=schulz.gif></p> 
    307179 
    308180</div> 
    309  
     181<br> 
    310182</body> 
    311  
    312 </html> 

  • sansmodels/src/sans/models/media/smear_computation.html

    r17574ae r318b5bbb  
    1 <html> 
    21 
    32<head> 
    … …  
    3029smeared scattering intensity for SANS is defined by</span></p> 
    3130 
    32 <p class=MsoNormal><img width=349 height=49 
     31<p class=MsoNormal><img  
    3332src="./img/sm_image002.gif" align=left hspace=12></p> 
    3433 
    3534<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>            
    36                                  1)</span><br clear=all> 
     35                                 ---- 1)</span><br clear=all> 
    3736<span style='font-family:"Times New Roman","serif"'>where Norm = <span 
    38 style='position:relative;top:15.0pt'><img width=137 height=49 
     37style='position:relative;top:15.0pt'><img  
    3938src="./img/sm_image003.gif"></span>.</span></p> 
    40  
     39<br> 
    4140<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>The 
    42 functions <span style='position:relative;top:6.0pt'><img width=43 height=25 
    43 src="./img/sm_image004.gif"></span>and <span style='position: 
    44 relative;top:6.0pt'><img width=43 height=25 
    45 src="./img/sm_image005.gif"></span>refer to the slit width weighting 
     41functions <span style='position:relative;top:6.0pt'><img  
     42src="./img/sm_image004.gif"></span> and <span style='position: 
     43relative;top:6.0pt'><img  
     44src="./img/sm_image005.gif"></span> refer to the slit width weighting 
    4645function and the slit height weighting determined at the q point, respectively. 
    4746 Here, we assumes that the weighting function is described by a rectangular 
    4847function, i.e.,</span></p> 
    4948 
    50 <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=134 
    51 height=26 src="./img/sm_image006.gif">                                                                                                        
     49<p class=MsoNormal><span style='position:relative;top:7.0pt'><img  
     50 src="./img/sm_image006.gif">                                                                                                        
    5251  </span><span style='font-family:"Times New Roman","serif";position:relative; 
    53 top:7.0pt'>2)</span></p> 
     52top:7.0pt'> ---- 2)</span></p> 
    5453 
    5554<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>and </span></p> 
    5655 
    57 <p class=MsoNormal><span style='position:relative;top:7.0pt'><img width=136 
    58 height=26 src="./img/sm_image007.gif"></span>,                                                                               
    59                          <span style='font-family:"Times New Roman","serif"'>3)</span></p> 
    60  
    61 <p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>so that </span><span 
    62 style='position:relative;top:6.0pt'><img width=58 height=23 
    63 src="./img/sm_image008.gif"></span> <span style='position:relative; 
    64 top:16.0pt'><img width=76 height=51 src="./img/sm_image009.gif"></span> <span 
    65 style='font-family:"Times New Roman","serif"'>for</span>  <span 
    66 style='position:relative;top:3.0pt'><img width=40 height=15 
    67 src="./img/sm_image010.gif"></span> <span style='font-family: 
    68 "Times New Roman","serif"'>and <i>u</i>. The </span><span style='position:relative; 
    69 top:6.0pt'><img width=28 height=24 src="./img/sm_image011.gif"></span> <span 
    70 style='font-family:"Times New Roman","serif"'>and </span><span 
    71 style='position:relative;top:6.0pt'><img width=28 height=24 
    72 src="./img/sm_image012.gif"> </span><span style='font-family: 
    73 "Times New Roman","serif"'>stand for the slit height (FWHM/2) and the slit 
     56<p class=MsoNormal><span style='position:relative;top:7.0pt'><img  
     57 src="./img/sm_image007.gif"></span>,                                                                               
     58                         <span style='font-family:"Times New Roman","serif"'> ---- 3)</span></p> 
     59 
     60<p>so that <img  
     61src="./img/sm_image008.gif"> <img src="./img/sm_image009.gif">  for <img  
     62src="./img/sm_image010.gif"> and <i>u</i>. The <img src="./img/sm_image011.gif">  
     63 and <img src="./img/sm_image012.gif">  stand for the slit height (FWHM/2) and the slit 
    7464width (FWHM/2) in the q space. Now the integral of Eq. (1) is simplified to</span></p> 
    7565 
    76 <p class=MsoNormal><img width=283 height=52 
     66<p class=MsoNormal><img  
    7767src="./img/sm_image013.gif" align=left hspace=12><span 
    7868style='font-family:"Times New Roman","serif"'>                                                   
    79          4)</span></p> 
     69          ---- 4)</span></p> 
    8070 
    8171<p class=MsoNormal><span style='font-family:"Times New Roman","serif"; 
    … …  
    8979style='font-family:"Times New Roman","serif"'>1)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    9080</span></span><span style='font-family:"Times New Roman","serif"'>For </span><span 
    91 style='position:relative;top:6.0pt'><img width=28 height=24 
    92 src="./img/sm_image014.gif"></span>= 0  <span style='font-family: 
     81style='position:relative;top:6.0pt'><img  
     82src="./img/sm_image012.gif"></span>= 0  <span style='font-family: 
    9383"Times New Roman","serif"'>and </span><span style='position:relative; 
    94 top:6.0pt'><img width=28 height=24 src="./img/sm_image015.gif"></span> = 
     84top:6.0pt'><img src="./img/sm_image011.gif"></span> = 
    9585<span style='font-family:"Times New Roman","serif"'>constant:</span></p> 
    9686 
    97 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
     87<p> 
    9888<img src="./img/sm_image016.gif"></p> 
    9989 
    100 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    101 style='font-family:"Times New Roman","serif"'>For discrete q values, at the q 
     90<p> For discrete q values, at the q 
    10291values from the data points and at the q values extended up to  q<sub>N</sub>= 
    103 q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img width=28 
    104 height=24 src="./img/sm_image011.gif"></span><span 
    105 style='font-family:"Times New Roman","serif"'>, the smeared intensity can be 
    106 calculated approximately,</span></p> 
    107  
    108 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><img  
     92q<sub>i</sub> +  <img src="./img/sm_image011.gif"> , the smeared intensity can be 
     93calculated approximately, </p> 
     94 
     95<p><img  
    10996src="./img/sm_image017.gif">.                                                            
    110 <span style='font-family:"Times New Roman","serif"'>5)</span></p> 
    111  
    112 <p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    113 style='position:relative;top:7.0pt'><img width=23 height=25 
     97 ---- 5)</p> 
     98 
     99<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
     100style='position:relative;top:7.0pt'><img  
    114101src="./img/sm_image018.gif"></span> <span style='font-family: 
    115102"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    123110style='font-family:"Times New Roman","serif"'>2)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    124111</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span 
    125 style='position:relative;top:6.0pt'><img width=28 height=24 
    126 src="./img/sm_image014.gif"></span>= <span style='font-family: 
     112style='position:relative;top:6.0pt'><img  
     113src="./img/sm_image012.gif"></span>= <span style='font-family: 
    127114"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and 
    128 </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
    129 src="./img/sm_image015.gif"></span> = <span style='font-family: 
     115</span><span style='position:relative;top:6.0pt'><img  
     116src="./img/sm_image011.gif"></span> = <span style='font-family: 
    130117"Times New Roman","serif"'>0:</span></p> 
    131118 
    … …  
    135122<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
    136123<img src="./img/sm_image019.gif">                                                                                         
    137 <span style='font-family:"Times New Roman","serif"'>6)</span></p> 
     124<span style='font-family:"Times New Roman","serif"'> ---- 6)</span></p> 
    138125 
    139126<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    140127style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub> 
    141 - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
     128- </span><span style='position:relative;top:6.0pt'><img  
    142129src="./img/sm_image012.gif"></span><span style='font-family: 
    143130"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> 
    144131= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img 
    145 width=28 height=24 src="./img/sm_image012.gif"></span>.  <span 
    146 style='position:relative;top:7.0pt'><img width=23 height=25 
     132 src="./img/sm_image012.gif"></span>.  <span 
     133style='position:relative;top:7.0pt'><img  
    147134src="./img/sm_image018.gif"></span> <span style='font-family: 
    148135"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    155142style='font-family:"Times New Roman","serif"'>3)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
    156143</span></span><span style='font-family:"Times New Roman","serif"'>For  </span><span 
    157 style='position:relative;top:6.0pt'><img width=28 height=24 
    158 src="./img/sm_image014.gif"></span>= <span style='font-family: 
     144style='position:relative;top:6.0pt'><img  
     145src="./img/sm_image011.gif"></span>= <span style='font-family: 
    159146"Times New Roman","serif"'>constant </span> <span style='font-family:"Times New Roman","serif"'>and 
    160 </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
    161 src="./img/sm_image015.gif"></span> = <span style='font-family: 
     147</span><span style='position:relative;top:6.0pt'><img  
     148src="./img/sm_image011.gif"></span> = <span style='font-family: 
    162149"Times New Roman","serif"'>constant:</span></p> 
    163150 
    … …  
    177164<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'> 
    178165<img src="./img/sm_image020.gif">    <span style='font-family: 
    179 "Times New Roman","serif"'>(7)</span></p> 
     166"Times New Roman","serif"'> ---- (7)</span></p> 
    180167 
    181168<p class=MsoListParagraphCxSpMiddle style='margin-left:.25in'><span 
    182169style='font-family:"Times New Roman","serif"'>for  q<sub>p</sub> = q<sub>i</sub> 
    183 - </span><span style='position:relative;top:6.0pt'><img width=28 height=24 
     170- </span><span style='position:relative;top:6.0pt'><img  
    184171src="./img/sm_image012.gif"></span><span style='font-family: 
    185172"Times New Roman","serif"'> and</span> <span style='font-family:"Times New Roman","serif"'>q<sub>N</sub> 
    186173= q<sub>i</sub> + </span><span style='position:relative;top:6.0pt'><img 
    187 width=28 height=24 src="./img/sm_image012.gif"></span>.  <span 
    188 style='position:relative;top:7.0pt'><img width=23 height=25 
     174 src="./img/sm_image012.gif"></span>.  <span 
     175style='position:relative;top:7.0pt'><img  
    189176src="./img/sm_image018.gif"></span> <span style='font-family: 
    190177"Times New Roman","serif"'>= 0 for <i>I<sub>s</sub></i> in</span> <i><span 
    … …  
    207194 
    208195<p class=MsoNormal><img src="./img/sm_image021.gif"><span 
    209 style='font-family:"Times New Roman","serif"'>                                                         (8)</span></p> 
     196style='font-family:"Times New Roman","serif"'>   ---- (8)</span></p> 
    210197 
    211198<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>For all 
    … …  
    228215 
    229216<p class=MsoNormal><img src="./img/sm_image022.gif"><span 
    230 style='font-family:"Times New Roman","serif"'>                   (9)</span></p> 
     217style='font-family:"Times New Roman","serif"'>     ---- (9)</span></p> 
    231218 
    232219<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>In Eq 
    233 (9), x<sub>0</sub> = qcos</span><span style='font-family:Symbol'>(theta)</span><span 
    234 style='font-family:"Times New Roman","serif"'> and y<sub>0</sub>=qsin</span><span 
    235 style='font-family:Symbol'>(theta)</span><span style='font-family:"Times New Roman","serif"'> 
    236 , and the primed axes are in the coordinate rotated by an angle </span><span 
    237 style='font-family:Symbol'>theta</span><span style='font-family:"Times New Roman","serif"'> 
    238 around z-axis (below) so that x’<sub>0</sub> =  x<sub>0</sub>cos</span><span 
    239 style='font-family:Symbol'>(theta) + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
    240 sin</span><span style='font-family:Symbol'>(theta)  </span><span style='font-family: 
    241 "Times New Roman","serif"'>and y’<sub>0</sub> =  -x<sub>0</sub>sin</span><span 
    242 style='font-family:Symbol'>(theta) + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
    243 cos</span><span style='font-family:Symbol'>(theta) .</span><span style='font-family: 
     220(9), x<sub>0</sub> = qcos&#952;</span><span 
     221style='font-family:"Times New Roman","serif"'> and y<sub>0</sub> = qsin&#952;</span><span style='font-family:"Times New Roman","serif"'> 
     222, and the primed axes are in the coordinate rotated by an angle &#952;</span><span style='font-family:"Times New Roman","serif"'> 
     223around z-axis (below) so that x’<sub>0</sub> =  x<sub>0</sub>cos&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
     224sin&#952;  </span><span style='font-family: 
     225"Times New Roman","serif"'>and y’<sub>0</sub> =  -x<sub>0</sub>sin&#952; + </span><span style='font-family:"Times New Roman","serif"'>y<sub>0</sub> 
     226cos&#952;.</span><span style='font-family: 
    244227"Times New Roman","serif"'> Note that the rotation angle is zero for x-y 
    245 symmetric elliptical Gaussian distribution</span><span style='font-family:Symbol'>. 
    246 </span><span style='font-family:"Times New Roman","serif"'>The  A is a 
     228symmetric elliptical Gaussian distribution</span>. 
     229<span style='font-family:"Times New Roman","serif"'>The  A is a 
    247230normalization factor.</span></p> 
    248231 
    249232<p class=MsoNormal align=center style='text-align:center'><span 
    250 style='font-family:"Times New Roman","serif"'><img width=439 height=376 
     233style='font-family:"Times New Roman","serif"'><img  
    251234id="Object 1" src="./img/sm_image023.gif"></span></p> 
    252235 
    … …  
    254237 
    255238<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Now we 
    256 consider a numerical integration where each bins in </span><span 
    257 style='font-family:Symbol'>THETA</span><span style='font-family:"Times New Roman","serif"'> 
     239consider a numerical integration where each bins in </span> &#920; </span><span style='font-family:"Times New Roman","serif"'> 
    258240and R are <b>evenly </b>(this is to simplify the equation below) distributed by 
    259 </span><span style='font-family:Symbol'>Delta_THETA </span><span style='font-family: 
    260 "Times New Roman","serif"'>and </span><span style='font-family:Symbol'>Delta</span><span 
     241</span>&#916;&#920; </span><span style='font-family: 
     242"Times New Roman","serif"'>and </span> &#916;</span><span 
    261243style='font-family:"Times New Roman","serif"'>R, respectively, and it is 
    262244assumed that I(x’, y’) is constant within the bins which in turn becomes</span></p> 
    … …  
    265247 
    266248<p class=MsoNormal>                                                                                                                                                                                <span 
    267 style='font-family:"Times New Roman","serif"'>(10)</span></p> 
     249style='font-family:"Times New Roman","serif"'> ---- (10)</span></p> 
    268250 
    269251<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>Since we 
    270252have found the weighting factor on each bin points, it is convenient to 
    271 transform x’-y’ back to x-y coordinate (rotating it by -</span><span 
    272 style='font-family:Symbol'>(theta)</span><span style='font-family:"Times New Roman","serif"'> 
     253transform x’-y’ back to x-y coordinate (rotating it by -&#952;</span><span style='font-family:"Times New Roman","serif"'> 
    273254around z axis).  Then, for the polar symmetric smear,</span></p> 
    274255 
    275256<p class=MsoNormal><img src="./img/sm_image025.gif"><span 
    276 style='position:relative;top:35.0pt'>                                                         </span>(11)</p> 
     257style='position:relative;top:35.0pt'>      </span> ---- (11)</p> 
    277258 
    278259<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> 
    279260 
    280 <p class=MsoNormal><img src="./img/sm_image026.gif"></p> 
     261<p class=MsoNormal><img src="./img/sm_image026.gif">,</p> 
    281262 
    282263<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>while 
    … …  
    284265 
    285266<p class=MsoNormal><img src="./img/sm_image027.gif"><span 
    286 style='font-family:"Times New Roman","serif"'>                                                                                          (12)</span></p> 
     267style='font-family:"Times New Roman","serif"'>  ---- (12)</span></p> 
    287268 
    288269<p class=MsoNormal><span style='font-family:"Times New Roman","serif"'>where,</span></p> 
    … …  
    301282</body> 
    302283 
    303 </html> 
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