**Model Functions** + **Introduction** + **Shapes**: + Sphere based: > SphereModel (Magnetic 2D Model)< a>, BinaryHSModel, FuzzySphereModel, RaspBerryModel, CoreShellModel (Magnetic 2D Model), Core2ndMomentModel, CoreMultiShellModel (Magnetic 2D Model), VesicleModel, MultiShellModel, OnionExpShellModel, SphericalSLDModel, LinearPearlsModel, PearlNecklaceModel + Cylinder based: > CylinderModel (Magnetic 2D Model)< a>, CoreShellCylinderModel, CoreShellBicelleModel, HollowCylinderModel, FlexibleCylinderModel, FlexCylEllipXModel, StackedDisksModel, EllipticalCylinderModel, BarBellModel, CappedCylinderModel, PringleModel + Parallelpipeds: > ParallelepipedModel (Magnetic 2D Model)< a>, CSParallelepipedModel + Ellipsoids: > EllipsoidModel< a>, CoreShellEllipsoidModel, TriaxialEllipsoidModel + Lamellar: > LamellarModel< a>, LamellarFFHGModel, LamellarPSModel, LamellarPSHGModel + Paracrystals: > LamellarPCrystalModel< a>, SCCrystalModel, FCCrystalModel, BCCrystalModel + **Shape-Independent**: AbsolutePower_Law, BEPolyelectrolyte, BroadPeakModel, CorrLength, DABModel, Debye, FractalModel, FractalCoreShell, GaussLorentzGel, Guinier, GuinierPorod, Lorentz, MassFractalModel, MassSurfaceFractal, PeakGaussModel, PeakLorentzModel, Poly_GaussCoil, PolyExclVolume, PorodModel, RPA10Model, StarPolymer, SurfaceFractalModel, Teubner Strey, TwoLorentzian, TwoPowerLaw, UnifiedPowerRg, LineModel, ReflectivityModel, ReflectivityIIModel, GelFitModel. + **Customized Models**: testmodel, testmodel_2, sum_p1_p2, sum_Ap1_1_Ap2, polynomial5, sph_bessel_jn. + **Structure Factors**: HardSphereStructure, SquareWellStructure, HayterMSAStructure, StickyHSStructure. + **References** **1.** ** ** **Introduction ** 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. **2.** ** ** **Shapes (Scattering Intensity Models)** 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. To easily compare to the scattering intensity measured in experiments, we normalize the form factors by the volume of the particle: with where *P*0 *( **q**)* is the un-normalized form factor, *( **r**)* is the scattering length density at a given point in space and the integration is done over the volume *V* of the scatterer. For systems without inter-particle interference, the form factors we provide can be related to the scattering intensity by the particle volume fraction: . Our so-called 1D scattering intensity functions provide *P(q) *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 *q*=0. Our so-called 2D scattering intensity functions provide *P(q, * *)* for an oriented system as a function of a q-vector in the plane of the detector. We define the angle as the angle between the q vector and the horizontal (x) axis of the plane of the detector. **2.1.** ** ** **Sphere Model (Magnetic 2D Model)** 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. For magnetic scattering, please see the '`Polarization/Magnetic Scattering`_' in Fitting Help. **1.1.** ** Definition** The 1D scattering intensity is calculated in the following way (Guinier, 1955): where scale is a volume fraction, V is the volume of the scatterer, r is the radius of the sphere, bkg is the background level and sldXXX is the scattering length density (SLD) of the scatterer or the solvent.<\p> Note that if your data is in absolute scale, the 'scale' should represent the volume fraction (unitless) if you have a good fit. If not, it should represent the volume fraction * a factor (by which your data might need to be rescaled). The 2D scattering intensity is the same as above, regardless of the orientation of the q vector. The returned value is scaled to units of [cm-1] and the parameters of the sphere model are the following: Parameter name Units Default value scale None 1 radius 60 sldSph -2 2.0e-6 sldSolv -2 1.0e-6 background cm-1 0 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). **2.1.** ** Validation of the sphere model** 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 1 shows a comparison of the output of our model and the output of the NIST software. Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=60 , Contrast=1e-6 -2, and Background=0.01 cm -1. 2013/09/09 and 2014/01/06 - Description reviewed by King, S. and Parker, P. **2.2.** ** ** **BinaryHSModel** This model (binary hard sphere model) provides the scattering intensity, for binary mixture of spheres including hard sphere interaction between those particles. Using Percus-Yevick closure, the calculation is an exact multi-component solution: where Sij are the partial structure factors and fi are the scattering amplitudes of the particles. And the subscript 1 is for the smaller particle and 2 is for the larger. The number fraction of the larger particle, ( *x* = n2/(n1+n2), n = the number density) is internally calculated based on: . The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as . The parameters of the binary hard sphere are the following (in the names, l (or ls) stands for larger spheres while s (or ss) for the smaller spheres): Parameter name Units Default value background cm-1 0.001 l_radius 100.0 ss_sld -2 0.0 ls_sld -2 3e-6 solvent_sld -2 6e-6 s_radius 25.0 vol_frac_ls 0.1 vol_frac_ss 0.2 **Figure. 1D plot using the default values above (w/200 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). See the reference for details. REFERENCE N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156 (1967) 685-692. [Errata found in Phys. Rev. 166 (1968) 934.] **2.3.** ** ** **FuzzySphereModel** ****This model is to calculate the scattering from spherical particles with a "fuzzy" interface. **1.1.** ** Definition** The 1D scattering intensity is calculated in the following way (Guinier, 1955): The returned value is scaled to units of [cm-1 sr-1], absolute scale. The scattering intensity I(q) is calculated as: 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: Here A2(q) is the form factor, P(q). The scale is equivalent to the volume fraction of spheres, each of volume, V. Contrast ( ** ) is the difference of scattering length densities of the sphere and the surrounding solvent. The poly-dispersion in radius and in fuzziness is provided. (direct from the reference) The "fuzziness" of the interface is defined by the parameter (sigma)fuzzy. The particle radius R represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core density. The (sigma)fuzzy is the width of the smeared particle surface: i.e., the standard deviation from the average height of the fuzzy interface. The inner regions of the microgel that display a higher density are described by the radial box profile extending to a radius of approximately Rbox ~ R - 2(sigma). the profile approaches zero as Rsans ~ R + 2(sigma). For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . REFERENCE M. Stieger, J. S. Pedersen, P. Lindner, W. Richtering, Langmuir 20 (2004) 7283-7292. TEST DATASET This example dataset is produced by running the FuzzySphereModel, using 200 data points, qmin = 0.001 -1, qmax = 0.7 A-1 and the default values: Parameter name Units Default value scale None 1.0 radius 60 fuzziness 10 sldSolv -2 3e-6 sldSph -2 1e-6 background cm-1 0.001 **Figure. 1D plot using the default values (w/200 data point).** **2.4.** **RaspBerryModel** Calculates the form factor, P(q), for a "Raspberry-like" structure where there are smaller spheres at the surface of a larger sphere, such as the structure of a Pickering emulsion. **1.1.** ** Definition** The structure is: Ro = the radius of thelarge sphere Rp = the radius of the smaller sphere on the surface delta = the fractional penetration depth surface coverage = fractional coverage of the large sphere surface (0.9 max) The large and small spheres have their own SLD, as well as the solvent. The surface coverage term is a fractional coverage (maximum of approximately 0.9 for hexagonally packed spheres on a surface). Since not all of the small spheres are necessarily attached to the surface, the excess free (small) spheres scattering is also included in the calculation. The function calculated follows equations (8)-(12) of the reference below, and the equations are not reproduced here. The returned value is scaled to units of [cm-1]. No interparticle scattering is included in this model. For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . REFERENCE Kjersta Larson-Smith, Andrew Jackson, and Danilo C Pozzo, "Small angle scattering model for Pickering emulsions and raspberry particles." Journal of Colloid and Interface Science (2010) vol. 343 (1) pp. 36-41. TEST DATASET This example dataset is produced by running the RaspBerryModel, using 2000 data points, qmin = 0.0001 -1, qmax = 0.2 A-1 and the default values, where Ssph/Lsph stands for Smaller/Large sphere andsurfrac_Ssph for the surface fraction of the smaller spheres. Parameter name Units Default value delta_Ssph 0 radius_Lsph 5000 radius_Ssph 100 sld_Lsph -2 -4e-07 sld_Ssph -2 3.5e-6 sld_solv -2 6.3e-6 surfrac_Ssph 0.4 volf_Lsph 0.05 volf_Lsph 0.005 background cm-1 0 **Figure. 1D plot using the values of /2000 data points.** **2.5.** ** ** **Core Shell (Sphere) Model (Magnetic 2D Model)** This model provides the form factor, P( *q*), for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume. For magnetic scattering, please see the '`Polarization/Magnetic Scattering`_' in Fitting Help. **1.1.** ** Definition** The 1D scattering intensity is calculated in the following way (Guinier, 1955): where *scale* is a scale factor, *Vs* is the volume of the outer shell, *Vc* is the volume of the core, *rs* is the radius of the shell, *rc* is the radius of the core, *c* is the scattering length density of the core, *s* is the scattering length density of the shell, solv is the scattering length density of the solvent, and *bkg* is the background level. The 2D scattering intensity is the same as P(q) above, regardless of the orientation of the q vector. For P*S: The outer most radius (= radius + thickness) is used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the core-shell sphere model are the following: Here, radius = the radius of the core and thickness = the thickness of the shell. Parameter name Units Default value scale None 1.0 (core) radius 60 thickness 10 core_sld -2 1e-6 shell_sld -2 2e-6 solvent_sld -2 3e-6 background cm-1 0.001 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). REFERENCE Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955). **2.1.** ** Validation of the core-shell sphere model** 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 1 shows a comparison of the output of our model and the output of the NIST software. Figure 7: Comparison of the DANSE scattering intensity for a core- shell sphere with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=60 , Contrast=1e-6 -2, and Background=0.001 cm -1. **2.6.** ** ** **Core2ndMomentModel** This model describes the scattering from a layer of surfactant or polymer adsorbed on spherical particles under the conditions that (i) theparticles (cores) are contrast-matched to the dispersion medium, (ii) S(Q)~1 (ie, the particle volume fraction is dilute), (iii) the particle radius is >> layer thickness (ie, the interface is locally flat), and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is absent or has been corrected for. Unlike a core-shell model, this model does not assume any form for the density distribution of the adsorbed species normal to the interface (cf, a core-shell model which assumes the density distribution to be a homogeneous step-function). For comparison, if the thickness of a (core-shell like) step function distribution is t, the second moment, sigma = sqrt((t^2)/12). Thesigma is the second moment about the mean of the density distribution (ie, the distance of the centre-of-mass of the distribution from the interface). **1.1.** ** Definition** The I0 is calculated in the following way (King, 2002): where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer,solv is the sld of the solvent/medium and cores, phi_cores is the volume fraction of the core paraticles, and Gamma and delta arethe adsorbed amount and the bulk density of the polymers respectively. The sigma is the second moment of the thickness distribution. Note that all parameters except the 'sigma' are correlated for fitting so that fittingthose with more than one parameters will be generally failed. And note that unlike other shape models, no volume normalization was applied to this model. The returned value is scaled to units of [cm-1] and the parameters are the following: Parameter name Units Default value scale None 1.0 density_poly g/cm2 0.7 radius_core 500 ads_amount mg/m2 1.9 second_moment 23.0 volf_cores None 0.14 sld_poly -2 1.5e-6 sld_solv -2 6.3e-6 background cm-1 0.0 REFERENCE S. King, P. Griffiths, J. Hone, and T. Cosgrove, "SANS from Adsorbed Polymer Lyaers", Macromol. Symp. 190, 33-42 (2002). **2.7.** ** ** **CoreMultiShell(Sphere)Model (Magnetic 2D Model)** This model provides the scattering from spherical core with from 1 up to 4 shell structures. Ithas a core of a specified radius, with four shells. The SLDs of the core and each shell are individually specified. For magnetic scattering, please see the '`Polarization/Magnetic Scattering`_' in Fitting Help. **1.1.** ** Definition** The returned value is scaled to units of [cm-1sr-1], absolute scale. This model is a trivial extension of the CoreShell function to a larger number of shells. See the CoreShell function for a diagram and documentation. Be careful that the SLDs and scale can be highly correlated. Hold as many of these fixed as possible. The 2D scattering intensity is the same as P(q) of 1D, regardless of the orientation of the q vector. For P*S: The outer most radius (= radius + 4 thicknesses) is used as the effective radius toward S(Q) if P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the CoreFourshell sphere model are the following: Here, rad_core = the radius of the core, thick_shelli = the thickness of the shell i and sld_shelli = the SLD of the shell i. And the sld_core and the sld_solv are the SLD of the core and the solvent, respectively. Parameter name Units Default value scale None 1.0 rad_core 60 sld_core -2 6.4e-6 sld_shell1 -2 1e-6 sld_shell2 -2 2e-6 sld_shell3 -2 3e-6 sld_shell4 -2 4e-6 sld_solv -2 6.4e-6 thick_shell1 10 thick_shell2 10 thick_shell3 10 thick_shell4 10 background cm-1 0.001 Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). REFERENCE See the CoreShell documentation. TEST DATASET This example dataset is produced by running the CoreMultiShellModel using 200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the above default values. **Figure: 1D plot using the default values (w/200 data point).** The scattering length density profile for the default sld values (w/ 4 shells). **Figure: SLD profile against the radius of the sphere for default SLDs.** **2.8.** ** ** **VesicleModel** This model provides the form factor, P( *q*), for an unilamellar vesicle. The form factor is normalized by the volume of the shell. The 1D scattering intensity is calculated in the following way (Guinier, 1955): where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total volume, *R1* is the radius of the core, *r2* is the outer radius of the shell, *1* is the scattering length density of the core and the solvent, *2* is the scattering length density of the shell, and *bkg* is the background level. And *J1* = (sin *x *- *x*cos *x*)/ *x*2. The functional form is identical to a "typical" core-shell structure, except that the scattering is normalized by the volume that is contributing to the scattering, namely the volume of the shell alone. Also, the vesicle is best defined in terms of a core radius (= R1) and a shell thickness, t. The 2D scattering intensity is the same as *P*( *q*) above, regardless of the orientation of the *q* vector which is defined as . For P*S: The outer most radius (= radius + thickness) is used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the vesicle model are the following: In the parameters, the radius represents the core radius (R1) and the thickness (R2 R1) is the shell thickness. Parameter name Units Default value scale None 1.0 radius 100 thickness 30 core_sld -2 6.3e-6 shell_sld -2 0 background cm-1 0.0 **Figure. 1D plot using the default values (w/200 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). REFERENCE Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955). **2.9.** ** ** **MultiShellModel** This model provides the form factor, P( *q*), for a multi-lamellar vesicle with N shells where the core is filled with solvent and the shells are interleaved with layers of solvent. For N = 1, this return to the vesicle model (above). The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as . For P*S: The outer most radius (= core_radius + n_pairs * s_thickness + (n_pairs -1) * w_thickness) is used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the multi-shell model are the following: In the parameters, the s_thickness is the shell thickness while the w_thickness is the solvent thickness, and the n_pair is the number of shells. Parameter name Units Default value scale None 1.0 core_radius 60.0 n_pairs None 2.0 core_sld -2 6.3e-6 shell_sld -2 0.0 background cm-1 0.0 s_thickness 10 w_thickness 10 **Figure. 1D plot using the default values (w/200 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). REFERENCE Cabane, B., Small Angle Scattering Methods, Surfactant Solutions: New Methods of Investigation, Ch.2, Surfactant Science Series Vol. 22, Ed. R. Zana, M. Dekker, New York, 1987. **2.10.** ** ** **OnionExpShellModel** This model provides the form factor, *P*( *q*), for a multi-shell sphere where the scattering length density (SLD) of the each shell is described by an exponential (linear, or flat-top) function. The form factor is normalized by the volume of the sphere where the SLD is not identical to the SLD of the solvent. We currently provide up to 9 shells with this model. The 1D scattering intensity is calculated in the following way: where, for a spherically symmetric particle with a particle density *r*( *r*) [L.A.Feigin and D.I.Svergun, Structure Analysis by Small- Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987], so that Here we assumed that the SLDs of the core and solvent are constant against *r*. Now lets consider the SLD of a shell, *rshelli*, defineded by An example of a possible SLD profile is shown below where sld_in_shelli ( *rin* ) and thick_shelli ( *Dtshelli* ) stand for the SLD of the inner side of the ith shell and the thickness of the ith shell in the equation above, respectively. For |A|>0, For A **~ **0 (eg., A = - 0.0001), this function converges to that of the linear SLD profile (ie, *rshelli*( *r*) = *A *****( *r* - *rshelli-1*) / *Dtshelli*) + *B *****), so this case it is equivalent to For A = 0, the exponential function has no dependence on the radius (so that sld_out_shell# ( *rout*) is ignored this case) and becomes flat. We set the constant to *rin* for convenience, and thus the form factor contributed by the shells is In the equation, Finally, the form factor can be calculated by where The 2D scattering intensity is the same as *P*( *q*) above, regardless of the orientation of the *q* vector which is defined as . For P*S: The outer most radius is used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of this model are the following: In the parameters, the rad_core represents the core radius (R1) and the thick_shell1 (R2 R1) is the thickness of the shell1, etc. Note: Only No. of shells = 1 is given below. Parameter name Units Default value A_shell1 None 1 scale None 1.0 rad_core 200 thick_shell1 50 sld_core -2 1.0e-06 sld_in_shell1 -2 1.7e-06 sld_out_shell1 -2 2.0e-06 sld_solv -2 6.4e-06 background cm-1 0.0 **Figure. 1D plot using the default values (w/400 point).** **Figure. SLD profile from the default values.** REFERENCE L.A.Feigin and D.I.Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987 **2.11.** ** ** **SphericalSLDModel** Similarly to the OnionExpShellModel, this model provides the form factor, *P*( *q*), for a multi-shell sphere, where the interface between the each neighboring shells can be described by one of the functions including error, power-law, and exponential functions. This model is to calculate the scattering intensity by building a continuous custom SLD profile against the radius of the particle. The SLD profile is composed of a flat core, a flat solvent, a number (up to 9 shells) of flat shells, and the interfacial layers between the adjacent flat shells (or core, and solvent) (See below). Unlike OnionExpShellModel (using an analytical integration), the interfacial layers are sub-divided and numerically integrated assuming each sub- layers are described by a line function. The number of the sub-layer can be given by users by setting the integer values of npts_inter# in GUI. The form factor is normalized by the total volume of the sphere. The 1D scattering intensity is calculated in the following way: where, for a spherically symmetric particle with a particle density *r*( *r*) [L.A.Feigin and D.I.Svergun, Structure Analysis by Small- Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987], so that Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between shells, *rinter_i*, is calculated with a function chosen by an user, where the functions are: 1) Exp; 2) Power-Law; 3) Erf; Then the functions are normalized so that it varies between 0 and 1 and they are constrained such that the SLD is continuous at the boundaries of the interface as well as each sub-layers and thus the B and C are determined. Once the *rinter_i* is found at the boundary of the sub-layer of the interface, we can find its contribution to the form factor P(q); where we assume that rho_inter_i (r) can be approximately linear within a sub-layer j. In the equation, Finally, the form factor can be calculated by where The 2D scattering intensity is the same as *P*( *q*) above, regardless of the orientation of the *q* vector which is defined as . For P*S: The outer most radius is used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of this model are the following: In the parameters, the rad_core0 represents the core radius (R1). Note: Only No. of shells = 1 is given below. Parameter name Units Default value background cm-1 0.0 npts_inter 35 scale 1 sld_solv -2 1e-006 func_inter1 Erf nu_inter 2.5 thick_inter1 50 sld_flat1 -2 4e-006 thick_flat1 100 func_inter0 Erf nu_inter0 2.5 rad_core0 50 sld_ core0 -2 2.07e-06 thick_ core0 50 **Figure. 1D plot using the default values (w/400 point).** **Figure. SLD profile from the default values.** REFERENCE L.A.Feigin and D.I.Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, 1987 **2.12.** **LinearPearlsModel** This model provides the form factor for pearls linearly joined by short strings: N pearls (homogeneous spheres), the radius R and the string segment length (or edge separation) l (= A- 2R)). The A is the center to center pearl separation distance. The thickness of each string is assumed to be negligable. **1.1.** ** Definition** The output of the scattering intensity function for the linearpearls model is given by (Dobrynin, 1996): where the mass mp is (sld(of a pearl) sld(of solvent)) * (volume of the N pearls), V is the total volume. The 2D scattering intensity is the same as P(q) above, regardless of the orientation of the q vector. The returned value is scaled to units of [cm-1] and the parameters are the following: Parameter name Units Default value scale None 1.0 radius 80.0 edge_separation 350.0 num_pearls (integer) 3 sld_pearl -2 1e-6 sld_solv -2 6.3e-6 background cm-1 0.0 REFERENCE A. V. Dobrynin, M. Rubinstein and S. P. Obukhov, Macromol. 29, 2974-2979, 1996. **2.13.** ** ** **PearlNecklaceModel** This model provides the form factor for a pearl necklace composed of two elements: N pearls (homogeneous spheres) freely jointed by M rods (like strings) (with a total mass Mw = M *mr + N * ms, the radius R and the string segment length (or edge separation) l (= A- 2R)). The A is the center to center pearl separation distance. **1.1.** ** Definition** The output of the scattering intensity function for the pearlnecklace model is given by (Schweins, 2004): where , , , , , and . where the mass mi is (sld(of i) sld(of solvent)) * (volume of the N pearls/rods), V is the total volume of the necklace. The 2D scattering intensity is the same as P(q) above, regardless of the orientation of the q vector. The returned value is scaled to units of [cm-1] and the parameters are the following: Parameter name Units Default value scale None 1.0 radius 80.0 edge_separation 350.0 num_pearls (integer) 3 sld_pearl -2 1e-6 sld_solv -2 6.3e-6 sld_string -2 1e-6 thick_string (=rod diameter) 2.5 background cm-1 0.0 REFERENCE R. Schweins and K. Huber, Particle Scattering Factor of Pearl Necklace Chains, Macromol. Symp., 211, 25-42, 2004. **2.14.** ** ** **Cylinder Model (Magnetic 2D Model)** 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. For magnetic scattering, please see the '`Polarization/Magnetic Scattering`_' in Fitting Help. **1.1.** ** Definition** The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955): where 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 ** (contrast) is the scattering length density difference between the scatterer and the solvent. J1 is the first order Bessel function. 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. Figure 2. Definition of the angles for oriented cylinders. Figure. Examples of the angles for oriented pp against the detector plane. 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. The returned value is scaled to units of [cm-1] and the parameters of the cylinder model are the following: Parameter name Units Default value scale None 1.0 radius 20.0 length 400.0 contrast -2 3.0e-6 background cm-1 0.0 cyl_theta degree 60 cyl_phi degree 60 The output of the 1D scattering intensity function for randomly oriented cylinders is then given by: The *cyl_theta* and *cyl_phi* parameter are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST. **2.1.** ** Validation of the cylinder model** 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 3 shows a comparison of the 1D output of our model and the output of the NIST software. In general, averaging over a distribution of orientations is done by evaluating the following: where *p(,* *)* is the probability distribution for the orientation and *P0(q,* *)* is the scattering intensity for the fully oriented system. Since we have no other software to compare the implementation of the intensity for fully oriented cylinders, we can compare the result of averaging our 2D output using a uniform distribution *p(,* *)* = 1.0. Figure 4 shows the result of such a cross-check. Figure 3: Comparison of the DANSE scattering intensity for a cylinder with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 , Length=400 , Contrast=3e-6 -2, and Background=0.01 cm -1. Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius=20 , Length=400 , Contrast=3e-6 -2, and Background=0.0 cm -1. **2.15.** ** ** **Core-Shell Cylinder Model** 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. **1.1.** ** Definition** The output of the 2D scattering intensity function for oriented core- shell cylinders is given by (Kline, 2006): where is the angle between the axis of the cylinder and the q-vector, *Vs* is the volume of the outer shell (i.e. the total volume, including the shell), *Vc* is the volume of the core, *L* is the length of the core, *r* is the radius of the core, *t* is the thickness of the shell, *c* is the scattering length density of the core, *s* is the scattering length density of the shell, solv is the scattering length density of the solvent, and *bkg* is the background level. The outer radius of the shell is given by *r+t* and the total length of the outer shell is given by *L+2t*. J1 is the first order Bessel function. To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two angles and . Similarly to the case of the cylinder, those angles are defined on Figure 2 in Cylinder Model. For P*S: The 2nd virial coefficient of the solid cylinder is calculate based on the (radius+thickness) and 2(length +thickness) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the core-shell cylinder model are the following: Parameter name Units Default value scale None 1.0 radius 20.0 thickness 10.0 length 400.0 core_sld -2 1e-6 shell_sld -2 4e-6 solvent_sld -2 1e-6 background cm-1 0.0 axis_theta degree 90 axis_phi degree 0.0 The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. The *axis_theta* and axis *_phi* 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. **2.1.** ** Validation of the core-shell cylinder model** 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 8 shows a comparison of the 1D output of our model and the output of the NIST software. Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software to compare the implementation of the intensity for fully oriented core-shell cylinders, we can compare the result of averaging our 2D output using a uniform distribution *p(,* *)* = 1.0. Figure 9 shows the result of such a cross-check. Figure 8: Comparison of the DANSE scattering intensity for a core- shell cylinder with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=20 , Thickness=10 , Length=400 , Core_sld=1e-6 -2, Shell_sld=4e-6 -2, Solvent_sld=1e-6 -2, and Background=0.01 cm -1. 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 , Thickness=10 , Length=400 , Core_sld=1e-6 -2, Shell_sld=4e-6 -2, Solvent_sld=1e-6 -2, and Background=0.0 cm -1. Figure. Definition of the angles for oriented core-shell cylinders. Figure. Examples of the angles for oriented pp against the detector plane. 2013/11/26 - Description reviewed by Heenan, R. **2.16.** ** ** **Core-Shell (Cylinder) Bicelle Model** 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 core-shell cylinder model (seeabove 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. The returned value is scaled to units of [cm-1] and the parameters of the core-shell cylinder model are the following: Parameter name Units Default value scale None 1.0 radius 20.0 rim_thick 10.0 face_thick 10.0 length 400.0 core_sld -2 1e-6 rim_sld -2 4e-6 face_sld -2 4e-6 solvent_sld -2 1e-6 background cm-1 0.0 axis_theta degree 90 axis_phi degree 0.0 The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. The *axis_theta* and axis *_phi* 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. **Figure. 1D plot using the default values (w/200 data point).** Figure. Definition of the angles for the oriented Core-Shell Cylinder Bicelle Model. Figure. Examples of the angles for oriented pp against the detector plane. REFERENCE Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987). **2.17.** ** ** **HollowCylinderModel** This model provides the form factor, P( *q*), for a monodisperse hollow right angle circular cylinder (tube) where the form factor is normalized by the volume of the tube: P(q) = scale*/Vshell+background where the averaging < > id applied only for the 1D calculation. The inside and outside of the hollow cylinder have the same SLD. The 1D scattering intensity is calculated in the following way (Guinier, 1955): where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1* (x)= (sin *x *- *x*cos *x*)/ *x*2. To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two angles and . Similarly to the case of the cylinder, those angles are defined on Figure 2 in Cylinder Model. For P*S: The 2nd virial coefficient of the solid cylinder is calculate based on the (radius) and 2(length) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. In the parameters, the contrast represents SLD (shell) - SLD (solvent) and the radius = Rhell while core_radius = Rcore. Parameter name Units Default value scale None 1.0 radius 30 length 400 core_radius 20 sldCyl -2 6.3e-6 sldSolv -2 5e-06 background cm-1 0.01 **Figure. 1D plot using the default values (w/1000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). Figure. Definition of the angles for the oriented HollowCylinderModel. Figure. Examples of the angles for oriented pp against the detector plane. REFERENCE Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987). **2.18.** ** ** **FlexibleCylinderModel** This model provides the form factor, P( *q*), for a flexible cylinder where the form factor is normalized by the volume of the cylinder: Inter-cylinder interactions are NOT included. P(q) = scale*/V+background where the averaging < > is applied over all orientation for 1D. The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as . The chain of contour length, L, (the total length) can be described a chain of some number of locally stiff segments of length lp. The persistence length,lp, is the length along the cylinder over which the flexible cylinder can be considered a rigid rod. The Kuhn length (b = 2*lp) is also used to describe the stiffness of a chain. The returned value is in units of [cm-1], on absolute scale. In the parameters, the sldCyl and sldSolv represent SLD (chain/cylinder) and SLD (solvent) respectively. Parameter name Units Default value scale None 1.0 radius 20 length 1000 sldCyl -2 1e-06 sldSolv -2 6.3e-06 background cm-1 0.01 kuhn_length 100 **Figure. 1D plot using the default values (w/1000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): 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. See equations (13,26-27) in the original reference for the details. REFERENCE Pedersen, J. S. and P. Schurtenberger (1996). Scattering functions of semiflexible polymers with and without excluded volume effects. Macromolecules 29: 7602-7612. Correction of the formula can be found in: Wei-Ren Chen, Paul D. Butler, and Linda J. Magid, "Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles" Langmuir, August 2006. **2.19.** ** ** **FlexCylEllipXModel** **Flexible Cylinder with Elliptical Cross-Section: **Calculates the form factor for a flexible cylinder with an elliptical cross section and a uniform scattering length density. The non-negligible diameter of the cylinder is included by accounting for excluded volume interactions within the walk of a single cylinder. The form factor is normalized by the particle volume such that P(q) = scale*/Vol + bkg, where < > is an average over all possible orientations of the flexible cylinder. **1.1.** ** Definition** The function calculated is from the reference given below. From that paper, "Method 3 With Excluded Volume" is used. The model is a parameterization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudo- continuous limit. See equations (13, 26-27) in the original reference for the details. NOTE: there are several typos in the original reference that have been corrected by WRC. Details of the corrections are in the reference below. - Equation (13): the term (1-w(QR)) should swap position with w(QR) - Equations (23) and (24) are incorrect. WRC has entered these into Mathematica and solved analytically. The results were converted to code. - Equation (27) should be q0 = max(a3/sqrt(RgSquare),3) instead of max(a3*b/sqrt(RgSquare),3) - 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. The chain of contour length, L, (the total length) can be described a chain of some number of locally stiff segments of length lp. The persistence length, lp, is the length along the cylinder over which the flexible cylinder can be considered a rigid rod. The Kuhn length (b) used in the model is also used to describe the stiffness of a chain, and is simply b = 2*lp. The cross section of the cylinder is elliptical, with minor radius a. The major radius is larger, so of course, the axis ratio (parameter 4) must be greater than one. Simple constraints should be applied during curve fitting to maintain this inequality. The returned value is in units of [cm-1], on absolute scale. The sldCyl = SLD (chain), sldSolv = SLD (solvent). The scale, and the contrast are both multiplicative factors in the model and are perfectly correlated. One or both of these parameters must be held fixed during model fitting. If the scale is set equal to the particle volume fraction, f, the returned value is the scattered intensity per unit volume, I(q) = f*P(q). However, no inter-particle interference effects are included in this calculation. For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . REFERENCE Pedersen, J. S. and P. Schurtenberger (1996). Scattering functions of semiflexible polymers with and without excluded volume effects. Macromolecules 29: 7602-7612. Corrections are in: Wei-Ren Chen, Paul D. Butler, and Linda J. Magid, "Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles" Langmuir, August 2006. TEST DATASET This example dataset is produced by running the Macro FlexCylEllipXModel, using 200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the default values below. Parameter name Units Default value axis_ratio 1.5 background cm-1 0.0001 Kuhn_length 100 (Contour) length 1e+3 radius 20.0 scale 1.0 sldCyl -2 1e-6 sldSolv -2 6.3e-6 **Figure. 1D plot using the default values (w/200 data points).** **2.20.** ** ** **StackedDisksModel ** This model provides the form factor, P( *q*), for stacked discs (tactoids) with a core/layer structure where the form factor is normalized by the volume of the cylinder. Assuming the next neighbor distance (d-spacing) in a stack of parallel discs obeys a Gaussian distribution, a structure factor S(q) proposed by Kratky and Porod in 1949 is used in this function. Note that the resolution smearing calculation uses 76 Gauss quadrature points to properly smear the model since the function is HIGHLY oscillatory, especially around the q-values that correspond to the repeat distance of the layers. The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as . The returned value is in units of [cm-1 sr-1], on absolute scale. The scattering intensity I(q) is: where the contrast, N is the number of discs per unit volume, ais the angle between the axis of the disc and q, and Vt and Vc are the total volume and the core volume of a single disc, respectively. where d = thickness of the layer (layer_thick), 2h= core thickness (core_thick), and R = radius of the disc (radius). where n = the total number of the disc stacked (n_stacking), D=the next neighbor center to cent distance (d-spacing), and sD= the Gaussian standard deviation of the d-spacing (sigma_d). To provide easy access to the orientation of the stackeddisks, we define the axis of the cylinder using two angles and . Similarly to the case of the cylinder, those angles are defined on Figure 2 of CylinderModel. For P*S: The 2nd virial coefficient of the solid cylinder is calculate based on the (radius) and length = n_stacking*(core_thick +2*layer_thick) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. Parameter name Units Default value background cm-1 0.001 core_sld -2 4e-006 core_thick 10 layer_sld -2 0 layer_thick 15 n_stacking 1 radius 3e+003 scale 0.01 sigma_d 0 solvent_sld -2 5e-006 **Figure. 1D plot using the default values (w/1000 data point).** Figure. Examples of the angles for oriented stackeddisks against the detector plane. Figure. Examples of the angles for oriented pp against the detector plane. Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Guinier, A. and Fournet, G., "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, 1955. Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949. Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", Clarendon, Oxford, 1994. **2.21.** ** ** **Elliptical Cylinder Model** This function calculates the scattering from an oriented elliptical cylinder. **For 2D (orientated system):** The angles theta and phi define the orientation of the axis of the cylinder. The angle psi is defined as the orientation of the major axis of the ellipse with respect to the vector Q. A gaussian poydispersity can be added to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. **Figure. a= r_minor and ** **n= r_ratio (i.e., r_major/r_minor).** The function calculated is: with the functions: and the angle psi is defined as the orientation of the major axis of the ellipse with respect to the vector Q. **For 1D (no preferred orientation):** The form factor is averaged over all possible orientation before normalized by the particle volume: P(q) = scale*/V . The returned value is scaled to units of [cm-1]. To provide easy access to the orientation of the elliptical, we define the axis of the cylinder using two angles , andY. Similarly to the case of the cylinder, those angles, and , are defined on Figure 2 of CylinderModel. The angle Y is the rotational angle around its own long_c axis against the q plane. For example, Y = 0 when the r_minor axis is parallel to the x-axis of the detector. All angle parameters are valid and given only for 2D calculation (Oriented system). **Figure. Definition of angels for 2D**. Figure. Examples of the angles for oriented elliptical cylinders against the detector plane. **For P*S**: The 2nd virial coefficient of the solid cylinder is calculate based on the averaged radius (=sqrt(r_minor^2*r_ratio)) and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. Parameter name Units Default value scale None 1.0 r_minor 20.0 r_ratio 1.5 length 400.0 sldCyl -2 4e-6 sldSolv -2 1e-006 background 0 **Figure. 1D plot using the default values (w/1000 data point).** **Validation of the elliptical cylinder 2D model** Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and psi respectively). **Figure. Comparison between 1D and averaged 2D.** **** In the 2D average, more binning in the angle phi is necessary to get the proper result. The following figure shows the results of the averaging by varying the number of bin over angles. **Figure. The intensities averaged from 2D over different number ** **of points of binning of angles.** REFERENCE L. A. Feigin and D. I. Svergun Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987). **2.22.** ** ** **BarBell(/DumBell)Model** Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of the cylinder, L, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies outside of the cylinder All dimensions of the barbell are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. **1.1.** ** Definition** The returned value is scaled to units of [cm-1sr-1], absolute scale. The barbell geometry is defined as: r is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius R >= r and by definition for this geometry h > 0, h is then defined by r and R as: h = sqrt(R^2 - r^2). The scattering intensity I(q) is calculated as: where the amplitude A(q) is given as: The < > brackets denote an average of the structure over all orientations. is then the form factor, P(q). The scale factor is equivalent to the volume fraction of cylinders, each of volume, V. Contrast is the difference of scattering length densities of the cylinder and the surrounding solvent. The volume of the barbell is: and its radius of gyration: The necessary conditions of R >= r is not enforced in the model. It is up to you to restrict this during analysis. REFERENCES H. Kaya, J. Appl. Cryst. (2004) 37, 223-230. H. Kaya and N-R deSouza, J. Appl. Cryst. (2004) 37, 508-509. (addenda and errata) TEST DATASET This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the above default values. Parameter name Units Default value scale None 1.0 len_bar 400.0 rad_bar 20.0 rad_bell 40.0 sld_barbell -2 1.0e-006 sld_solv -2 6.3e-006 background 0 **Figure. 1D plot using the default values (w/256 data point).** For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. At the theta = 45 deg and phi =0 deg with default values for other parameters, **Figure. 2D plot (w/(256X265) data points).** Figure. Examples of the angles for oriented pp against the detector plane. Figure. Definition of the angles for oriented 2D barbells. **2.23.** ** ** **CappedCylinder(/ConvexLens)Model** 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. 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. **1.1.** ** Definition** The returned value is scaled to units of [cm-1sr-1], absolute scale. The Capped Cylinder geometry is defined as: r is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius R >= r and by definition for this geometry h < 0, h is then defined by r and R as: h = -1*sqrt(R^2 - r^2). The scattering intensity I(q) is calculated as: where the amplitude A(q) is given as: The < > brackets denote an average of the structure over all orientations. is then the form factor, P(q). The scale factor is equivalent to the volume fraction of cylinders, each of volume, V. Contrast is the difference of scattering length densities of the cylinder and the surrounding solvent. The volume of the Capped Cylinder is: (with h as a positive value here) and its radius of gyration: The necessary conditions of R >= r is not enforced in the model. It is up to you to restrict this during analysis. REFERENCES H. Kaya, J. Appl. Cryst. (2004) 37, 223-230. H. Kaya and N-R deSouza, J. Appl. Cryst. (2004) 37, 508-509. (addenda and errata) TEST DATASET This example dataset is produced by running the Macro CappedCylinder(), using 200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the above default values. Parameter name Units Default value scale None 1.0 len_cyl 400.0 rad_cap 40.0 rad_cyl 20.0 sld_capcyl -2 1.0e-006 sld_solv -2 6.3e-006 background 0 **Figure. 1D plot using the default values (w/256 data point).** For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. At the theta = 45 deg and phi =0 deg with default values for other parameters, **Figure. 2D plot (w/(256X265) data points).** Figure. Definition of the angles for oriented 2D cylinders. Figure. Examples of the angles for oriented pp against the detector plane. **2.24.** ** ** **PringleModel** This model provides the form factor, P( *q*), for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). The returned value is in units of [cm-1], on absolute scale. The form factor calculated is: where The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below. Parameter name Units Default value background cm-1 0.0 alpha 0.001 beta 0.02 radius 60 scale 1 sld_pringle -2 1e-006 sld_solvent -2 6.3e-006 thickness 10 **Figure. 1D plot using the default values (w/150 data point).** REFERENCE S. Alexandru Rautu, Private Communication. **2.25.** ** ** **ParallelepipedModel (Magnetic 2D Model) ** This model provides the form factor, P( *q*), for a rectangular cylinder (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*/V+background where the volume V= ABC and the averaging < > is applied over all orientation for 1D. For magnetic scattering, please see the '`Polarization/Magnetic Scattering`_' in Fitting Help. The side of the solid must be satisfied the condition of A1, the form factor, The contrast is defined as The scattering intensity per unit volume is returned in the unit of [cm-1]; I(q) = fP(q). 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. To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using two angles , andY. Similarly to the case of the cylinder, those angles, and , are defined on Figure 2 of CylinderModel. The angle Y is the rotational angle around its own long_c axis against the q plane. For example, Y = 0 when the short_b axis is parallel to the x-axis of the detector. **Figure. Definition of angles for 2D**. Figure. Examples of the angles for oriented pp against the detector plane. Parameter name Units Default value background cm-1 0.0 contrast -2 5e-006 long_c 400 short_a -2 35 short_b 75 scale 1 **Figure. 1D plot using the default values (w/1000 data point).** **Validation of the parallelepiped 2D model** Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for the averaging, 76, 180, 76 points are taken over the angles of theta, phi, and psi respectively). **Figure. Comparison between 1D and averaged 2D.** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211. Equations (1), (13-14). (in German) **2.26.** ** ** **CSParallelepipedModel** Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or "rim" can be different on all three (pairs) of faces. The form factor is normalized by the particle volume such that P(q) = scale*/Vol + bkg, where < > is an average over all possible orientations of the rectangular solid. An instrument resolution smeared version is also provided. The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions ABC such that A < B < C. There are rectangular "slabs" of thickness tA that add to the A dimension (on the BC faces). There are similar slabs on the AC (=tB) and AB (=tC) faces. The projection in the AB plane is then: The volume of the solid is: meaning that there are "gaps" at the corners of the solid. The intensity calculated follows the parallelepiped model, with the core-shell intensity being calculated as the square of the sum of the amplitudes of the core and shell, in the same manner as a core-shell sphere. For the calculation of the form factor to be valid, the sides of the solid MUST be chosen such that A < B < C. If this inequality in not satisfied, the model will not report an error, and the calculation will not be correct. FITTING NOTES: If the scale is set equal to the particle volume fraction, f, the returned value is the scattered intensity per unit volume, I(q) = f*P(q). However, no interparticle interference effects are included in this calculation. There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical. Constraints must be applied during fitting to ensure that the inequality A < B < C is not violated. The calculation will not report an error, but the results will not be correct. The returned value is in units of [cm-1], on absolute scale. 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. To provide easy access to the orientation of the CSparallelepiped, we define the axis of the cylinder using two angles , andY. Similarly to the case of the cylinder, those angles, and , are defined on Figure 2 of CylinderModel. The angle Y is the rotational angle around its own long_c axis against the q plane. For example, Y = 0 when the short_b axis is parallel to the x-axis of the detector. **Figure. Definition of angles for 2D**. Figure. Examples of the angles for oriented cspp against the detector plane. TEST DATASET This example dataset is produced by running the Macro Plot_CSParallelepiped(), using 100 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the below default values. Parameter name Units Default value background cm-1 0.06 sld_pcore -2 1e-006 sld_rimA -2 2e-006 sld_rimB -2 4e-006 sld_rimC -2 2e-006 sld_solv -2 6e-006 rimA 10 rimB 10 rimC 10 longC 400 shortA 35 midB 75 scale 1 **Figure. 1D plot using the default values (w/256 data points).** **** **Figure. 2D plot using the default values (w/(256X265) data points).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE see: Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211. Equations (1), (13-14). (yes, it's in German) **2.27.** ** ** **Ellipsoid Model** 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. **1.1.** ** Definition** The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987): where is the angle between the axis of the ellipsoid and the q-vector, V is the volume of the ellipsoid, Ra is the radius along the rotation axis of the ellipsoid, Rb is the radius perpendicular to the rotation axis of the ellipsoid and ** (contrast) is the scattering length density difference between the scatterer and the solvent. To provide easy access to the orientation of the ellipsoid, we define the rotation axis of the ellipsoid using two angles and . Similarly to the case of the cylinder, those angles are defined on Figure 2. For the ellipsoid, is the angle between the rotation axis and the z-axis. For P*S: The 2nd virial coefficient of the solid ellipsoid is calculate based on the radius_a and radius_b values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. The returned value is scaled to units of [cm-1] and the parameters of the ellipsoid model are the following: Parameter name Units Default value scale None 1.0 radius_a (polar) 20.0 radius_b (equatorial) 400.0 sldEll -2 4.0e-6 sldSolv -2 1.0e-6 background cm-1 0.0 axis_theta degree 90 axis_phi degree 0.0 The output of the 1D scattering intensity function for randomly oriented ellipsoids is then given by the equation above. The *axis_theta* and axis *_phi* 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. Figure. The angles for oriented ellipsoid **2.1.** ** Validation of the ellipsoid model** 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. Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software to compare the implementation of the intensity for fully oriented ellipsoids, we can compare the result of averaging our 2D output using a uniform distribution *p(,* *)* = 1.0. Figure 6 shows the result of such a cross-check. ** **** The discrepancy above q=0.3 -1 is due to the way the form factors are calculated in the c-library provided by NIST. A numerical integration has to be performed to obtain P(q) for randomly oriented particles. The NIST software performs that integration with a 76-point Gaussian quadrature rule, which will become imprecise at high q where the amplitude varies quickly as a function of q. The DANSE result shown has been obtained by summing over 501 equidistant points in . Our result was found to be stable over the range of q shown for a number of points higher than 500. ** ** Figure 5: Comparison of the DANSE scattering intensity for an ellipsoid with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius_a=20 , Radius_b=400 , Contrast=3e-6 -2, and Background=0.01 cm -1. Figure 6: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius_a=20 , Radius_b=400 , Contrast=3e-6 -2, and Background=0.0 cm -1. **2.28.** ** ** **CoreShellEllipsoidModel ** This model provides the form factor, P( *q*), for a core shell ellipsoid (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*/V+background where the volume V= 4pi/3*rmaj*rmin2 and the averaging < > is applied over all orientation for 1D. The returned value is in units of [cm-1], on absolute scale. The form factor calculated is: To provide easy access to the orientation of the coreshell ellipsoid, we define the axis of the solid ellipsoid using two angles , . Similarly to the case of the cylinder, those angles, and , are defined on Figure 2 of CylinderModel. The contrast is defined as SLD(core) SLD(shell) or SLD(shell solvent). In the parameters, equat_core = equatorial core radius, polar_core = polar core radius, equat_shell = rmin (or equatorial outer radius), and polar_shell = = rmaj (or polar outer radius). For P*S: The 2nd virial coefficient of the solid ellipsoid is calculate based on the radius_a (= polar_shell) and radius_b (= equat_shell) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. Parameter name Units Default value background cm-1 0.001 equat_core 200 equat_shell 250 sld_solvent -2 6e-006 ploar_shell 30 ploar_core 20 scale 1 sld_core -2 2e-006 sld_shell -2 1e-006 **Figure. 1D plot using the default values (w/1000 data point).** **** Figure. The angles for oriented coreshellellipsoid . Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461. Berr, S. J. Phys. Chem., 1987, 91, 4760. **2.29.** ** ** **TriaxialEllipsoidModel** This model provides the form factor, P( *q*), for an ellipsoid (below) where all three axes are of different lengths, i.e., Ra =< Rb =< Rc (Note that users should maintains this inequality for the all calculations). P(q) = scale*/V+background where the volume V= 4pi/3*Ra*Rb*Rc, and the averaging < > is applied over all orientation for 1D. The returned value is in units of [cm-1], on absolute scale. The form factor calculated is: To provide easy access to the orientation of the triaxial ellipsoid, we define the axis of the cylinder using the angles , andY. Similarly to the case of the cylinder, those angles, and , are defined on Figure 2 of CylinderModel. The angle Y is the rotational angle around its own semi_axisC axis against the q plane. For example, Y = 0 when the semi_axisA axis is parallel to the x-axis of the detector. The radius of gyration for this system is Rg2 = (Ra2*Rb2*Rc2)/5. The contrast is defined as SLD(ellipsoid) SLD(solvent). In the parameters, semi_axisA = Ra (or minor equatorial radius), semi_axisB = Rb (or major equatorial radius), and semi_axisC = Rc (or polar radius of the ellipsoid). For P*S: The 2nd virial coefficient of the solid ellipsoid is calculate based on the radius_a (=semi_axisC) and radius_b (=sqrt(semi_axisA* semi_axisB)) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied. Parameter name Units Default value background cm-1 0.0 semi_axisA 35 semi_axisB 100 semi_axisC 400 scale 1 sldEll -2 1.0e-006 sldSolv -2 6.3e-006 **Figure. 1D plot using the default values (w/1000 data point).** **Validation of the triaxialellipsoid 2D model** Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and psi respectively). **Figure. Comparison between 1D and averaged 2D.** Figure. The angles for oriented ellipsoid. Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE L. A. Feigin and D. I. Svergun Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, 1987. **2.30.** ** ** **LamellarModel** This model provides the scattering intensity, I( *q*), for a lyotropic lamellar phase where a uniform SLD and random distribution in solution are assumed. The ploydispersion in the bilayer thickness can be applied from the GUI. The scattering intensity I(q) is: The form factor is, where d = bilayer thickness. The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . The returned value is in units of [cm-1], on absolute scale. In the parameters, sld_bi = SLD of the bilayer, sld_sol = SLD of the solvent, and bi_thick = the thickness of the bilayer. Parameter name Units Default value background cm-1 0.0 sld_bi -2 1e-006 bi_thick 50 sld_sol -2 6e-006 scale 1 **Figure. 1D plot using the default values (w/1000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502. also in J. Phys. Chem. B, 105, (2001) 11081-11088. **2.31.** ** ** **LamellarFFHGModel** This model provides the scattering intensity, I( *q*), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region. The scattering intensity I(q) is: The form factor is, where dT = tail length (or t_length), dH = heasd thickness (or h_thickness) , DrH = SLD (headgroup) - SLD(solvent), and DrT = SLD (tail) - SLD(headgroup). The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . The returned value is in units of [cm-1], on absolute scale. In the parameters, sld_tail = SLD of the tail group, and sld_head = SLD of the head group. Parameter name Units Default value background cm-1 0.0 sld_head -2 3e-006 scale 1 sld_solvent -2 6e-006 h_thickness 10 t_length 15 sld_tail -2 0 **Figure. 1D plot using the default values (w/1000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502. also in J. Phys. Chem. B, 105, (2001) 11081-11088. **2.32.** ** ** **LamellarPSModel** This model provides the scattering intensity ( **form factor** ***** **structure factor**), I( *q*), for a lyotropic lamellar phase where a random distribution in solution are assumed. The scattering intensity I(q) is: The form factor is and the structure is where Here d= (repeat) spacing, d = bilayer thickness, the contrast Dr = SLD (headgroup) - SLD(solvent), K=smectic bending elasticity, B=compression modulus, and N = number of lamellar plates (n_plates). Note: When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due to the complication of the model function, users are responsible to make sure that all the assumptions are handled accurately: see the original reference (below) for more details. The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . The returned value is in units of [cm-1], on absolute scale. Parameter name Units Default value background cm-1 0.0 contrast -2 5e-006 scale 1 delta 30 n_plates 20 spacing 400 caille -2 0.1 **Figure. 1D plot using the default values (w/6000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502. also in J. Phys. Chem. B, 105, (2001) 11081-11088. **2.33.** ** ** **LamellarPSHGModel** This model provides the scattering intensity ( **form factor** ***** **structure factor**), I( *q*), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region. The scattering intensity I(q) is: The form factor is, The structure factor is where where dT = tail length (or t_length), dH = heasd thickness (or h_thickness) , DrH = SLD (headgroup) - SLD(solvent), and DrT = SLD (tail) - SLD(headgroup). Here d= (repeat) spacing, K=smectic bending elasticity, B=compression modulus, and N = number of lamellar plates (n_plates). Note: When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect. And due to the complication of the model function, users are responsible to make sure that all the assumptions are handled accurately: see the original reference (below) for more details. The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as . The returned value is in units of [cm-1], on absolute scale. In the parameters, sld_tail = SLD of the tail group, sld_head = SLD of the head group, and sld_solvent = SLD of the solvent. Parameter name Units Default value background cm-1 0.001 sld_head -2 2e-006 scale 1 sld_solvent -2 6e-006 deltaH 2 deltaT 10 sld_tail -2 0 n_plates 30 spacing 40 caille -2 0.001 **Figure. 1D plot using the default values (w/6000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006): REFERENCE Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502. also in J. Phys. Chem. B, 105, (2001) 11081-11088. **2.34.** ** ** **LamellarPCrystalModel** Lamella ParaCrystal Model: Calculates the scattering from a stack of repeating lamellar structures. The stacks of lamellae (infinite in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further characterized by a Gaussian polydispersity. This model can be used for large multilamellar vesicles. The scattering intensity I(q) is calculated as: The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness t. Here, the scale factor is used instead of the mass per area of the bilayer (G). The scale factor is the volume fraction of the material in the bilayer, not the total excluded volume of the paracrystal. ZN(q) describes the interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) from the Bergstrom reference below. Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values: The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as . The parameters of the model are the following (Nlayers= no. of layers, pd_spacing= polydispersity of spacing): Parameter name Units Default value background cm-1 0 scale 1 Nlayers 20 pd_spacing 0.2 sld_layer -2 1e-6 sld_solvent -2 6.34e-6 spacing 250 thickness 33 **Figure. 1D plot using the default values above (w/20000 data point).** Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). See the reference for details. REFERENCE M. Bergstrom, J. S. Pedersen, P. Schurtenberger, S. U. Egelhaaf, J. Phys. Chem. B, 103 (1999) 9888-9897. **2.35.** ** ** **SC(Simple Cubic Para-)CrystalModel** Calculates the scattering from a simple cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. The returned value is scaled to units of [cm-1sr-1], absolute scale. The scattering intensity I(q) is calculated as: where scale is the volume fraction of spheres, Vp is the volume of the primary particle, V(lattice) is a volume correction for the crystal structure, P(q) is the form factor of the sphere (normalized) and Z(q) is the paracrystalline structure factor for a simple cubic structure. Equation (16) of the 1987 reference is used to calculate Z(q), using equations (13)-(15) from the 1987 paper for Z1, Z2, and Z3. The lattice correction (the occupied volume of the lattice) for a simple cubic structure of particles of radius R and nearest neighbor separation D is: The distortion factor (one standard deviation) of the paracrystal is included in the calculation of Z(q): where g is a fractional distortion based on the nearest neighbor distance. The simple cubic lattice is: For a crystal, diffraction peaks appear at reduced q-values givn by: where for a simple cubic lattice any h, k, l are allowed and none are forbidden. Thus the peak positions correspond to (just the first 5): NOTE: The calculation of Z(q) is a double numerical integral that must be carried out with a high density of points to properly capture the sharp peaks of the paracrystalline scattering. So be warned that the calculation is SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This makes a triple integral. Very, very slow. Go get lunch. REFERENCES Hideki Matsuoka et. al. Physical Review B, 36 (1987) 1754-1765. (Original Paper) Hideki Matsuoka et. al. Physical Review B, 41 (1990) 3854 -3856. (Corrections to FCC and BCC lattice structure calculation) Parameter name Units Default value background cm-1 0 dnn 220 scale 1 sldSolv -2 6.3e-006 radius 40 sld_Sph -2 3e-006 d_factor 0.06 TEST DATASET This example dataset is produced using 200 data points, qmin = 0.01 -1, qmax = 0.1 -1 and the above default values. **Figure. 1D plot in the linear scale using the default values (w/200 data point).** The 2D (Anisotropic model) is based on the reference (above) which I(q) 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. ** ** **Figure. 2D plot using the default values (w/200X200 pixels).** **2.36.** ** ** **FC(Face Centered Cubic Para-)CrystalModel** Calculates the scattering from a face-centered cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. The returned value is scaled to units of [cm-1sr-1], absolute scale. The scattering intensity I(q) is calculated as: where scale is the volume fraction of spheres, Vp is the volume of the primary particle, V(lattice) is a volume correction for the crystal structure, P(q) is the form factor of the sphere (normalized) and Z(q) is the paracrystalline structure factor for a face-centered cubic structure. Equation (1) of the 1990 reference is used to calculate Z(q), using equations (23)-(25) from the 1987 paper for Z1, Z2, and Z3. The lattice correction (the occupied volume of the lattice) for a face-centered cubic structure of particles of radius R and nearest neighbor separation D is: The distortion factor (one standard deviation) of the paracrystal is included in the calculation of Z(q): where g is a fractional distortion based on the nearest neighbor distance. The face-centered cubic lattice is: For a crystal, diffraction peaks appear at reduced q-values givn by: where for a face-centered cubic lattice h, k, l all odd or all even are allowed and reflections where h, k, l are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5): NOTE: The calculation of Z(q) is a double numerical integral that must be carried out with a high density of points to properly capture the sharp peaks of the paracrystalline scattering. So be warned that the calculation is SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This makes a triple integral. Very, very slow. Go get lunch. REFERENCES Hideki Matsuoka et. al. Physical Review B, 36 (1987) 1754-1765. (Original Paper) Hideki Matsuoka et. al. Physical Review B, 41 (1990) 3854 -3856. (Corrections to FCC and BCC lattice structure calculation) Parameter name Units Default value background cm-1 0 dnn 220 scale 1 sldSolv -2 6.3e-006 radius 40 sld_Sph -2 3e-006 d_factor 0.06 TEST DATASET This example dataset is produced using 200 data points, qmin = 0.01 -1, qmax = 0.1 -1 and the above default values. **Figure. 1D plot in the linear scale using the default values (w/200 data point).** The 2D (Anisotropic model) is based on the reference (above) in which I(q) 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. **Figure. 2D plot using the default values (w/200X200 pixels).** **2.37.** ** ** **BC(Body Centered Cubic Para-)CrystalModel** Calculates the scattering from a body-centered cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution.The returned value is scaled to units of [cm-1sr-1], absolute scale. The scattering intensity I(q) is calculated as: where scale is the volume fraction of spheres, Vp is the volume of the primary particle, V(lattice) is a volume correction for the crystal structure, P(q) is the form factor of the sphere (normalized) and Z(q) is the paracrystalline structure factor for a body-centered cubic structure. Equation (1) of the 1990 reference is used to calculate Z(q), using equations (29)-(31) from the 1987 paper for Z1, Z2, and Z3. The lattice correction (the occupied volume of the lattice) for a body-centered cubic structure of particles of radius R and nearest neighbor separation D is: The distortion factor (one standard deviation) of the paracrystal is included in the calculation of Z(q): where g is a fractional distortion based on the nearest neighbor distance. The body-centered cubic lattice is: For a crystal, diffraction peaks appear at reduced q-values givn by: where for a body-centered cubic lattice, only reflections where (h+k+l) = even are allowed and reflections where (h+k+l) = odd are forbidden. Thus the peak positions correspond to (just the first 5): NOTE: The calculation of Z(q) is a double numerical integral that must be carried out with a high density of points to properly capture the sharp peaks of the paracrystalline scattering. So be warned that the calculation is SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This makes a triple integral. Very, very slow. Go get lunch. REFERENCES Hideki Matsuoka et. al. Physical Review B, 36 (1987) 1754-1765. (Original Paper) Hideki Matsuoka et. al. Physical Review B, 41 (1990) 3854 -3856. (Corrections to FCC and BCC lattice structure calculation) Parameter name Units Default value background cm-1 0 dnn 220 scale 1 sldSolv -2 6.3e-006 radius 40 sld_Sph -2 3e-006 d_factor 0.06 TEST DATASET This example dataset is produced using 200 data points, qmin = 0.001 -1, qmax = 0.1 -1 and the above default values. **Figure. 1D plot in the linear scale using the default values (w/200 data point).** The 2D (Anisotropic model) is based on the reference (1987) in which I(q) 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. **Figure. 2D plot using the default values (w/200X200 pixels).** **3.** ** ** **Shape-Independent Models ** The following are models used for shape-independent SANS analysis. **3.1.** ** ** **Debye (Model)** The Debye model is a form factor for a linear polymer chain. In addition to the radius of gyration, Rg, a scale factor "scale", and a constant background term are included in the calculation. For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale None 1.0 rg 50.0 background cm-1 0.0 **Figure. 1D plot using the default values (w/200 data point).** Reference: Roe, R.-J., "Methods of X-Ray and Neutron Scattering in Polymer Science", Oxford University Press, New York (2000). **3.2.** ** ** **BroadPeak Model** Calculate an empirical functional form for SANS data characterized by a broad scattering peak. Many SANS spectra are characterized by a broad peak even though they are from amorphous soft materials. The d-spacing corresponding to the broad peak is a characteristic distance between the scattering inhomogeneities (such as in lamellar, cylindrical, or spherical morphologies or for bicontinuous structures). The returned value is scaled to units of [cm-1sr-1], absolute scale. The scattering intensity I(q) is calculated by: Here the peak position is related to the d-spacing as Q0 = 2pi/d0. Soft systems that show a SANS peak include copolymers, polyelectrolytes, multiphase systems, layered structures, etc. For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale_l (= C) 10 scale_p (=A) 1e-05 length_l (=x) 50 q_peak (= Q0) -1 0.1 exponent_p (=n) 2 exponent_l (=m) 3 Background (=B) cm-1 0.1 **Figure. 1D plot using the default values (w/200 data point).** Reference: None. 2013/09/09 - Description reviewed by King, S. and Parker, P. **3.3.** ** ** **CorrLength (CorrelationLengthModel)** Calculate an empirical functional form for SANS data characterized by a low-Q signal and a high-Q signal The returned value is scaled to units of [cm-1sr-1], absolute scale. The scattering intensity I(q) is calculated by: The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function describing scattering from polymer chains (exponent = m). This second term characterizes the polymer/solvent interactions and therefore the thermodynamics. The two multiplicative factors A and C, the incoherent background B and the two exponents n and m are used as fitting parameters. The final parameter (xi) is a correlation length for the polymer chains. Note that when m = 2, this functional form becomes the familiar Lorentzian function. For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale_l (= C) 10 scale_p (=A) 1e-06 length_l (=x) 50 exponent_p (=n) 2 exponent_l (=m) 3 Background (=B) cm-1 0.1 **Figure. 1D plot using the default values (w/500 data points).** REFERENCE B. Hammouda, D.L. Ho and S.R. Kline, Insight into Clustering in Poly(ethylene oxide) Solutions, Macromolecules 37, 6932-6937 (2004). 2013/09/09 - Description reviewed by King, S. and Parker, P. **3.4.** ** ** **(Ornstein-Zernicke) Lorentz (Model)** The Ornstein-Zernicke model is defined by: The parameter L is referred to as the screening length. For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale None 1.0 length 50.0 background cm-1 0.0 ** ** **Figure. 1D plot using the default values (w/200 data point).** **3.5.** ** ** **DAB (Debye-Anderson-Brumberger)_Model** **** Calculates the scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system is characterized by a single length scale, the correlation length, which is a measure of the average spacing between regions of phase 1 and phase 2. The model also assumes smooth interfaces between the phases and hence exhibits Porod behavior (I ~ Q-4) at large Q (Q*correlation length >> 1). The parameter L is referred to as the correlation length. For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale None 1.0 length 50.0 background cm-1 0.0 ** ** **Figure. 1D plot using the default values (w/200 data point).** References: Debye, Anderson, Brumberger, "Scattering by an Inhomogeneous Solid. II. The Correlation Function and its Application", J. Appl. Phys. 28 (6), 679 (1957). Debye, Bueche, "Scattering by an Inhomogeneous Solid", J. Appl. Phys. 20, 518 (1949). 2013/09/09 - Description reviewed by King, S. and Parker, P. **3.6.** ** ** ** Absolute Power_Law ** This model describes a power law with background. Note the minus sign in front of the exponent. Parameter name Units Default value Scale None 1.0 m None 4 Background cm-1 0.0 **Figure. 1D plot using the default values (w/200 data point).** **3.7.** ** ** **Teubner Strey (Model)** This function calculates the scattered intensity of a two-component system using the Teubner-Strey model. **** For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale None 0.1 c1 None -30.0 c2 None 5000.0 background cm-1 0.0 **Figure. 1D plot using the default values (w/200 data point).** References: Teubner, M; Strey, R. J. Chem. Phys., 87, 3195 (1987). Schubert, K-V., Strey, R., Kline, S. R. and E. W. Kaler, J. Chem. Phys., 101, 5343 (1994). **3.8.** ** ** ** FractalModel** Calculates the scattering from fractal-like aggregates built from spherical building blocks following the Texiera reference. The value returned is in cm-1. The scale parameter is the volume fraction of the building blocks, R0 is the radius of the building block, Df is the fractal dimension, is the correlation length, *solvent* is the scattering length density of the solvent, and *block* is the scattering length density of the building blocks. **The polydispersion in radius is provided.** For 2D plot, the wave transfer is defined as . Parameter name Units Default value scale None 0.05 radius 5.0 fractal_dim None 2 corr_length 100.0 block_sld -2 2e-6 solvent_sld -2 6e-6 background cm-1 0.0 **Figure. 1D plot using the default values (w/200 data point).** References: J. Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-785 **** **3.9.** ** ** **MassFractalModel** Calculates the scattering from fractal-like aggregates based on the Mildner reference (below). The R is the radius of the building block, Dm is the mass fractal dimension, is the correlation (or cutt-off) length, *solvent* is the scattering length density of the solvent, and *particle* is the scattering length density of particles. Note: The mass fractal dimension is valid for 1