source: sasview/src/sans/models/media/model_functions_v0.rest @ 38d4102

**Model Functions**


+ **Introduction**
+ **Shapes**:

    + Sphere based: > <a href="#SphereModel">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: > <a href="#CylinderModel">CylinderModel (Magnetic
      2D Model)< a>, CoreShellCylinderModel, CoreShellBicelleModel,
      HollowCylinderModel, FlexibleCylinderModel, FlexCylEllipXModel,
      StackedDisksModel, EllipticalCylinderModel, BarBellModel,
      CappedCylinderModel, PringleModel
    + Parallelpipeds: > <a href="#ParallelepipedModel">ParallelepipedModel
      (Magnetic 2D Model)< a>, CSParallelepipedModel
    + Ellipsoids: > <a href="#EllipsoidModel">EllipsoidModel< a>,
      CoreShellEllipsoidModel, TriaxialEllipsoidModel
    + Lamellar: > <a href="#LamellarModel">LamellarModel< a>,
      LamellarFFHGModel, LamellarPSModel, LamellarPSHGModel
    + Paracrystals: > <a
      href="#LamellarPCrystalModel">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*<f^2>/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*<f^2>/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*<f^2>/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*<f^2>/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. <A^2(q)> 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. <A^2(q)> 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*<f^2>/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 A<B

By this definition, assuming

a = A/B<1; b=B/B=1; c=C/B>1, 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*<f^2>/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*<f^2>/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*<f^2>/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<mass_dim<6.



Parameter name

Units

Default value

scale

None

1

radius



10.0

mass_dim

None

1.9

co_length



100.0

background



0.0



**Figure. 1D plot**





References:

D. Mildner, and P. Hall, J. Phys. D.: Appl. Phys., 19, 1535-1545
(1986), Equation(9).

2013/09/09 - Description reviewed by King, S. and Parker, P.





**3.10.** ** ** ** SurfaceFractalModel**

Calculates the scattering based on the Mildner reference (below).










The R is the radius of the building block, Ds is the surface 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 surface fractal dimension is valid for 1<surface_dim<3. Also
it is valid in a limited q range (see the reference for details).



Parameter name

Units

Default value

scale

None

1

radius



10.0

surface_dim

None

2.0

co_length



500.0

background



0.0



**Figure. 1D plot**





References:

D. Mildner, and P. Hall, J. Phys. D.: Appl. Phys., 19, 1535-1545
(1986), Equation(13).





**3.11.** ** ** **MassSurfaceFractal**

A number of natural and commercial processes form high-surface area
materials as a result of the vapour-phase aggregation of primary
particles. Examples of such materials include soots, aerosols, and
fume or pyrogenic silicas. These are all characterised by cluster mass
distributions (sometimes also cluster size distributions) and internal
surfaces that are fractal in nature. The scattering from such
materials displays two distinct breaks in log-log representation,
corresponding to the radius-of-gyration of the primary particles, rg,
and the radius-of-gyration of the clusters (aggregates), Rg. Between
these boundaries the scattering follows a power law related to the
mass fractal dimension, Dm, whilst above the high-Q boundary the
scattering follows a power law related to the surface fractal
dimension of the primary particles, Ds.

The scattered intensity I(Q) is then calculated using a modified
Ornstein-Zernicke equation:










The Rg is for the cluster, rg is for the primary, Ds is the surface
fractal dimension, Dm is the mass fractal dimension, *solvent* is the
scattering length density of the solvent, and *p* is the scattering
length density of particles.

Note: The surface and mass fractal dimensions are valid for
0<surface_dim<6, 0<mass_dim<6, and (surface_mass+mass_dim)<6.



Parameter name

Units

Default value

scale

None

1

primary_rg



4000.0
cluster_rg 86.7
surface_dim

None

2.3
mass_dim None 1.8
background



0.0



**Figure. 1D plot**





References:

P. Schmidt, J Appl. Cryst., 24, 414-435 (1991), Equation(19).

Hurd, Schaefer, Martin, Phys. Rev. A, 35, 2361-2364 (1987),
Equation(2).





**3.12.** ** ** ** FractalCoreShell(Model)**

Calculates the scattering from a fractal structure with a primary
building block of core-shell spheres.




The formfactor P(q) is CoreShellModel with bkg = 0,
,

while the fractal structure factor S(q);



where Df = frac_dim, = cor_length, rc = (core) radius, and scale =
volfraction.
The fractal structure is as documented in the fractal model. This
model could find use for aggregates of coated particles, or aggregates
of vesicles.The polydispersity computation of radius and thickness is
provided.

The returned value is scaled to units of [cm-1sr-1], absolute scale.

See each of these individual models for full documentation.

For 2D plot, the wave transfer is defined as .



Parameter name

Units

Default value

volfraction

0.05

frac_dim

2

thickness



5.0

raidus

20.0

cor_length



100.0

core_sld

-2

3.5e-6

shell_sld

-2

1e-6

solvent_sld

-2

6.35e-6

background

cm-1

0.0



**Figure. 1D plot using the default values (w/500 data points).**





References:

See the PolyCore and Fractal documentation. ** **

**3.13.** ** ** ** GaussLorentzGel(Model)**

Calculates the scattering from a gel structure, typically a physical
network. It is modeled as a sum of a low-q exponential decay plus a
lorentzian at higher q-values. It is generally applicable to gel
structures.

The returned value is scaled to units of [cm-1sr-1], absolute scale.

The scattering intensity I(q) is calculated as (eqn 5 from the
reference):





Uppercase Zeta is the static correlations in the gel, which can be
attributed to the "frozen-in" crosslinks of some gels. Lowercase zeta
is the dynamic correlation length, which can be attributed to the
fluctuating polymer chain between crosslinks. IG(0) and IL(0) are the
scaling factors for each of these structures. Your physical system may
be different, so think about the interpretation of these parameters.

Note that the peaked structure at higher q values (from Figure 2 of
the reference below) is not reproduced by the model. Peaks can be
introduced into the model by summing this model with the PeakGauss
Model function.

For 2D plot, the wave transfer is defined as .



Parameter name

Units

Default value

dyn_colength(=Dynamic correlation length)



20.0

scale_g(=Gauss scale factor)

100

scale_l(=Lorentzian scale factor)

50

stat_colength(=Static correlation Z)



100.0

background

cm-1

0.0



**Figure. 1D plot using the default values (w/500 data points).**





REFERENCE:

G. Evmenenko, E. Theunissen, K. Mortensen, H. Reynaers, Polymer 42
(2001) 2907-2913.

**3.14.** ** ** ** BEPolyelectrolyte Model**

Calculates the structure factor of a polyelectrolyte solution with the
RPA expression derived by Borue and Erukhimovich. The value returned
is in cm-1.







K is a contrast factor of the polymer, Lb is the Bjerrum length, h is
the virial parameter, b is the monomer length, Cs is the concentration
of monovalent salt, is the ionization degree, Ca is the polymer molar
concentration, and background is the incoherent background.

For 2D plot, the wave transfer is defined as .

Parameter name

Units

Default value

K

Barns = 10-24 cm2

10

Lb



7.1

h

-3

12

b



10

Cs

Mol/L

0

alpha

None

0.05

Ca

Mol/L

0.7

background

cm-1

0.0

References:

Borue, V. Y., Erukhimovich, I. Y. Macromolecules 21, 3240 (1988).

Joanny, J.-F., Leibler, L. Journal de Physique 51, 545 (1990).

Moussaid, A., Schosseler, F., Munch, J.-P., Candau, S. J. Journal de
Physique II France

3, 573 (1993).

Raphal, E., Joanny, J.-F., Europhysics Letters 11, 179 (1990).

****

**3.15.** ** ** **Guinier (Model)**

A Guinier analysis is done by linearizing the data at low q by
plotting it as log(I) versus Q2. The Guinier radius Rg can be obtained
by fitting the following model:





For 2D plot, the wave transfer is defined as .

****

Parameter name

Units

Default value

scale

cm-1

1.0

Rg



0.1

****

**3.16.** ** ** **GuinierPorod (Model)**

Calculates the scattering for a generalized Guinier/power law object.
This is an empirical model that can be used to determine the size and
dimensionality of scattering objects.

The returned value is P(Q) as written in equation (1), plus the
incoherent background term. The result is in the units of [cm-1sr-1],
absolute scale.

A Guinier-Porod empirical model can be used to fit SAS data from
asymmetric objects such as rods or platelets. It also applies to
intermediate shapes between spheres and rod or between rods and
platelets. The following functional form is used:

(1)



This is based on the generalized Guinier law for such elongated
objects [2]. For 3D globular objects (such as spheres), s = 0 and one
recovers the standard Guinier formula. For 2D symmetry (such as for
rods) s = 1 and for 1D symmetry (such as for lamellae or platelets) s
= 2. A dimensionality parameter 3-s is defined, and is 3 for spherical
objects, 2 for rods, and 1 for plates.

Enforcing the continuity of the Guinier and Porod functions and their
derivatives yields:



and





Note that the radius of gyration for a sphere of radius R is given by
Rg = R sqrt(3/5) ,

that for the cross section of an randomly oriented cylinder of radius
R is given by Rg = R / sqrt(2).

The cross section of a randomly oriented lamella of thickness T is
given by Rg = T / sqrt(12).

The intensity given by Eq. 1 is the calculated result, and is plotted
below for the default parameter values.

REFERENCE

[1] Guinier, A.; Fournet, G. "Small-Angle Scattering of X-Rays", John
Wiley and Sons, New York, (1955).

[2] Glatter, O.; Kratky, O., Small-Angle X-Ray Scattering, Academic
Press (1982). Check out Chapter 4 on Data Treatment, pages 155-156.

For 2D plot, the wave transfer is defined as .

****

Parameter name

Units

Default value

Scale(=Guinier scale, G)

cm-1

1.0

rg



100

dim(=Dimensional Variable, s)

1

m(=Porod exponent)

3

background

0.1

****

** **

**Figure. 1D plot using the default values (w/500 data points).**

****

****

**3.17.** ** ** **PorodModel**

A Porod analysis is done by linearizing the data at high q by plotting
it as log(I) versus log(Q). In the high q region we can fit the
following model:





C is the scale factor and Sv is the specific surface area of the
sample and is the contrast factor.

The background term is added for data analysis.

For 2D plot, the wave transfer is defined as .

****

Parameter name

Units

Default value

scale

-4

0.1

background

cm-1

0

**3.18.** ** ** **PeakGaussModel**

Model describes a Gaussian shaped peak including a flat background,







with the peak having height of I0 centered at qpk having standard
deviation of B. The fwhm is 2.354*B.

Parameters I0, B, qpk, and BGD can all be adjusted during fitting.

REFERENCE: None

For 2D plot, the wave transfer is defined as .

****

Parameter name

Units

Default value

scale

cm-1

100

q0



0.05

B

0.005

background

1

****

****

** **

**Figure. 1D plot using the default values (w/500 data points).**

****

**3.19.** ** ** **PeakLorentzModel**

Model describes a Lorentzian shaped peak including a flat background,







with the peak having height of I0 centered at qpk having a hwhm (half-
width-half-maximum) of B.

The parameters I0, B, qpk, and BGD can all be adjusted during fitting.

REFERENCE: None

For 2D plot, the wave transfer is defined as .

****

Parameter name

Units

Default value

scale

cm-1

100

q0



0.05

B

0.005

background

1

****

****

** **

**Figure. 1D plot using the default values (w/500 data points).**

**3.20. Poly_GaussCoil (Model)**

Polydisperse Gaussian Coil: Calculate an empirical functional form for
scattering from a polydisperse polymer chain ina good solvent. The
polymer is polydisperse with a Schulz-Zimm polydispersity of the
molecular weight distribution.

The returned value is scaled to units of [cm-1sr-1], absolute scale.



where the dimensionless chain dimension is:



and the polydispersion is

.

The scattering intensity I(q) is calculated as:

The polydispersion in rg is provided.



For 2D plot, the wave transfer is defined as .

TEST DATASET

This example dataset is produced by running the Poly_GaussCoil, using
200 data points, qmin = 0.001 -1, qmax = 0.7 -1 and the default values
below.

Parameter name

Units

Default value

Scale

None

1.0

rg



60.0

poly_m

Mw/Mn

2

background

cm-1

0.001





**Figure. 1D plot using the default values (w/200 data point).**



Reference:

Glatter & Kratky - pg.404.

J.S. Higgins, and H.C. Benoit, Polymers and Neutron Scattering, Oxford
Science

Publications (1996).

**3.21. PolymerExclVolume (Model)**

Calculates the scattering from polymers with excluded volume effects.

The returned value is scaled to units of [cm-1sr-1], absolute scale.

The returned value is P(Q) as written in equation (2), plus the
incoherent background term. The result is in the units of [cm-1sr-1],
absolute scale.

A model describing polymer chain conformations with excluded volume
was introduced to describe the conformation of polymer chains, and has
been used as a template for describing mass fractals. The form factor
for that model (Benoit, 1957) was originally presented in the
following integral form:

(1)

Here n is the excluded volume parameter which is related to the Porod
exponent m as n = 1/m, a is the polymer chain statistical segment
length and n is the degree of polymerization. This integral was later
put into an almost analytical form (Hammouda, 1993) as follows:

(2)

Here, g(x,U) is the incomplete gamma function which is a built-in
function in computer libraries.



The variable U is given in terms of the scattering variable Q as:



The radius of gyration squared has been defined as:



Note that this model describing polymer chains with excluded volume
applies only in the mass fractal range ( 5/3 <= m <= 3) and does not
apply to surface fractals ( 3 < m <= 4). It does not reproduce the
rigid rod limit (m = 1) because it assumes chain flexibility from the
outset. It may cover a portion of the semiflexible chain range ( 1 < m
< 5/3).

The low-Q expansion yields the Guinier form and the high-Q expansion
yields the Porod form which is given by:



Here G(x) = g(x,inf) is the gamma function. The asymptotic limit is
dominated by the first term:



The special case when n = 0.5 (or m = 1/n = 2) corresponds to Gaussian
chains for which the form factor is given by the familiar Debye
function.



The form factor given by Eq. 2 is the calculated result, and is
plotted below for the default parameter values.

REFERENCE

Benoit, H., Comptes Rendus (1957). 245, 2244-2247.

Hammouda, B., SANS from Homogeneous Polymer Mixtures A Unified
Overview, Advances in Polym. Sci. (1993), 106, 87-133.

For 2D plot, the wave transfer is defined as .

TEST DATASET

This example dataset is produced, using 200 data points, qmin = 0.001
-1, qmax = 0.2 -1 and the default values below.

Parameter name

Units

Default value

Scale

None

1.0

rg



60.0

m(=Porod exponent)

3

background

cm-1

0.0





**Figure. 1D plot using the default values (w/500 data points).**



**3.22.** ** ** ** RPA10Model**

Calculates the macroscopic scattering intensity (units of cm^-1) for a
multicomponent homogeneous mixture of polymers using the Random Phase
Approximation. This general formalism contains 10 specific cases:

Case 0: C/D Binary mixture of homopolymers

Case 1: C-D Diblock copolymer

Case 2: B/C/D Ternary mixture of homopolymers

Case 3: C/C-D Mixture of a homopolymer B and a diblock copolymer C-D

Case 4: B-C-D Triblock copolymer

Case 5: A/B/C/D Quaternary mixture of homopolymers

Case 6: A/B/C-D Mixture of two homopolymers A/B and a diblock C-D

Case 7: A/B-C-D Mixture of a homopolymer A and a triblock B-C-D

Case 8: A-B/C-D Mixture of two diblock copolymers A-B and C-D

Case 9: A-B-C-D Four-block copolymer

Note: the case numbers are different from the IGOR/NIST SANS package.

****

Only one case can be used at any one time. Plotting a different case
will overwrite the original parameter waves.

The returned value is scaled to units of [cm-1].

Component D is assumed to be the "background" component (all contrasts
are calculated with respect to component D).

Scattering contrast for a C/D blend= {SLD (component C) - SLD
(component D)}2

Depending on what case is used, the number of fitting parameters
varies. These represent the segment lengths (ba, bb, etc) and the Chi
parameters (Kab, Kac, etc). The last one of these is a scaling factor
to be held constant equal to unity.

The input parameters are the degree of polymerization, the volume
fractions for each component the specific volumes and the neutron
scattering length densities.

This RPA (mean field) formalism applies only when the multicomponent
polymer mixture is in the homogeneous mixed-phase region.

REFERENCE

A.Z. Akcasu, R. Klein and B. Hammouda, Macromolecules 26, 4136 (1993)



Fitting parameters for Case0 Model

Parameter name

Units

Default value

background

cm-1

0.0

scale

1

bc(=Seg. Length bc)

5

bd(=Seg. Length bd)

5

Kcd(Chi Param. Kcd)

-0.0004

****

****

Fixed parameters for Case0 Model

Parameter name

Units

Default value

Lc(= Scatter. Length_c)

1e-12

Ld(= Scatter. Length_d)

0

Nc(=Deg.Polym.c)

1000

Nd(=Deg.Polym.d)

1000

Phic(=Vol. fraction of c)

0.25

Phid(=Vol. fraction of d)

0.25

vc(=Spec. vol. of c)

100

vd(=Spec. vol. of d)

100

****

****



**Figure. 1D plot using the default values (w/500 data points).**

****

**3.23.** ** ** **TwoLorentzian(Model)**

Calculate an empirical functional form for SANS data characterized by
a two Lorentzian functions.

The returned value is scaled to units of [cm-1sr-1], absolute scale.

The scattering intensity I(q) is calculated by:





A = Lorentzian scale #1

C = Lorentzian scale #2

where scale is the peak height centered at q0, and B refers to the
standard deviation of the function.

The background term is added for data analysis.

For 2D plot, the wave transfer is defined as .

**Default input parameter values**

Parameter name

Units

Default value

scale_1(=A)

10

scale_2(=C)

1

1ength_1 (=Correlation length1)



100

1ength_2(=Correlation length2)



10

exponent_1(=n)

3

exponent_2(=m)

2

Background(=B)

cm-1

0.1







**Figure. 1D plot using the default values (w/500 data points).**



**REFERENCE: None**

**3.24.** ** ** **TwoPowerLaw(Model)**

Calculate an empirical functional form for SANS data characterized by
two power laws.

The returned value is scaled to units of [cm-1sr-1], absolute scale.



The scattering intensity I(q) is calculated by:





qc is the location of the crossover from one slope to the other. The
scaling A, sets the overall intensity of the lower Q power law region.
The scaling of the second power law region is scaled to match the
first. Be sure to enter the power law exponents as positive values.

For 2D plot, the wave transfer is defined as .

**Default input parameter values**

Parameter name

Units

Default value

coef_A

1.0

qc

-1

0.04

power_1(=m1)

4

power_2(=m2)

4

background

cm-1

0.0







**Figure. 1D plot using the default values (w/500 data points).**



**3.25.** ** ** **UnifiedPower(Law and)Rg(Model)**

The returned value is scaled to units of [cm-1sr-1], absolute scale.

Note that the level 0 is an extra function that is the inverse
function; I (q) = scale/q + background.

Otherwise, program incorporates the empirical multiple level unified
Exponential/Power-law fit method developed by G. Beaucage. Four
functions are included so that One, Two, Three, or Four levels can be
used.

The empirical expressions are able to reasonably approximate the
scattering from many different types of particles, including fractal
clusters, random coils (Debye equation), ellipsoidal particles, etc.
The empirical fit function is





For each level, the four parameters Gi, Rg,i, Bi and Pi must be
chosen.

For example, to approximate the scattering from random coils (Debye
equation), set Rg,i as the Guinier radius, Pi = 2, and Bi = 2 Gi /
Rg,i

See the listed references for further information on choosing the
parameters.



For 2D plot, the wave transfer is defined as .

**Default input parameter values**

Parameter name

Units

Default value

scale

1.0

Rg2



21

power2

2

G2

cm-1sr-1

3

B2

cm-1sr-1

0.0006

Rg1



15.8

power1

4

G1

cm-1sr-1

400

B1

cm-1sr-1

4.5e-006

background

cm-1

0.0







**Figure. 1D plot using the default values (w/500 data points).**



REFERENCES

G. Beaucage (1995). J. Appl. Cryst., vol. 28, p717-728.

G. Beaucage (1996). J. Appl. Cryst., vol. 29, p134-146.

**3.26.** ** ** ** LineModel**

This is a linear function that calculates:





where A and B are the coefficients of the first and second order
terms.

**Note:** For 2D plot, I(q) = I(qx)*I(qy) which is defined differently
from other shape independent models.

Parameter name

Units

Default value

A

cm-1

1.0

B



1.0



**3.27.** ** ** **ReflectivityModel**

This model calculates the reflectivity and uses the Parrett algorithm.
Up to nine film layers are supported between Bottom(substrate) and
Medium(Superstrate where the neutron enters the first top film). Each
layers are composed of [ of the interface(from the previous layer or
substrate) + flat portion + of the interface(to the next layer or
medium)]. Only two simple interfacial functions are selectable, error
function and linear function. The each interfacial thickness is
equivalent to (- 2.5 sigma to +2.5 sigma for the error function,
sigma=roughness).

Note: This model was contributed by an interested user.



**Figure. Comparison (using the SLD profile below) with NISTweb
calculation (circles):
http://www.ncnr.nist.gov/resources/reflcalc.html.**



**Figure. SLD profile used for the calculation(above).**

**3.28.** ** ** **ReflectivityIIModel**

Same as the ReflectivityModel except that the it is more customizable.
More interfacial functions are supplied. The number of points
(npts_inter) for each interface can be choosen. The constant (A below
but 'nu' as a parameter name of the model) for exp, erf, or power-law
is an input. The SLD at the interface between layers, *rinter_i*, is
calculated with a function chosen by a user, where the functions are:

1) Erf;



2) Power-Law;







3) Exp;





Note: This model was implemented by an interested user.

**3.29.** ** ** **GelFitModel**

Unlike a concentrated polymer solution, the fine-scale polymer
distribution in a gel involves at least two characteristic length
scales, a shorter correlation length (a1) to describe the rapid
fluctuations in the position of the polymer chains that ensure
thermodynamic equilibrium, and a longer distance (denoted here as a2)
needed to account for the static accumulations of polymer pinned down
by junction points or clusters of such points. The letter is derived
from a simple Guinier function.

The scattered intensity I(Q) is then calculated as:



Where:







Note the first term reduces to the Ornstein-Zernicke equation when
D=2; ie, when the Flory exponent is 0.5 (theta conditions). In gels
with significant hydrogen bonding D has been reported to be ~2.6 to
2.8.

Note: This model was implemented by an interested user.

**Default input parameter values**

Parameter name

Units

Default value

Background

cm-1

0.01

Guinier scale

cm-1

1.7

Lorentzian scale

cm-1

3.5

Radius of gyration



104

Fractal exponent

2

Correlation length



16







**Figure. 1D plot using the default values (w/300 data points,
qmin=0.001, and qmax=0.3).**



REFERENCES

Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C. Han, J. Chem. Phys.
1992, 97 (9), 6829-6841.

Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R. Rennie, Erik
Geissler, Macromolecules 1991, 24, 543-548.



**3.30.** ** ** ** Star Polymer with Gaussian Statistics **

For a star with *f* arms:







where is the ensemble average radius of gyration squared of an arm.



References:

H. Benoit, J. Polymer Science., 11, 596-599 (1953)





**4.** ** ** **Customized Models **

****

Customized model functions can be redefined or added by users (See
SansView tutorial for details).

**4.1.** **** **testmodel**

****

This function, as an example of a user defined function, calculates
the intensity = A + Bcos(2q) + Csin(2q).

**4.2.** **** **testmodel_2 **

This function, as an example of a user defined function, calculates
the intensity = scale * sin(f)/f, where f = A + Bq + Cq2 + Dq3 + Eq4 +
Fq5.

**4.3.** ** ** **sum_p1_p2 **

This function, as an example of a user defined function, calculates
the intensity = scale_factor * (CylinderModel + PolymerExclVolume
model). To make your own sum(P1+P2) model, select 'Easy Custom Sum'
from the Fitting menu, or modify and compile the file named
'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. It
works only for single functional models.

**4.4.** **** **sum_Ap1_1_Ap2 **

This function, as an example of a user defined function, calculates
the intensity = (scale_factor * CylinderModel + (1-scale_factor) *
PolymerExclVolume model). To make your own A*p1+(1-A)*p2 model, modify
and compile the file named 'sum_Ap1_1_Ap2.py' from 'Edit Custom Model'
in the 'Fitting' menu. It works only for single functional models.

**4.5.** ** ** **polynomial5 **

This function, as an example of a user defined function, calculates
the intensity = A + Bq + Cq2 + Dq3 + Eq4 + Fq5. This model can be
modified and compiled from 'Edit Custom Model' in the 'Fitting' menu.

**4.6.** **** **sph_bessel_jn **

This function, as an example of a user defined function, calculates
the intensity = C*sph_jn(Ax+B)+D where the sph_jn is spherical Bessel
function of the order n. This model can be modified and compiled from
'Edit Custom Model' in the 'Fitting' menu.

**5.** ** ** **Structure Factors**



The information in this section is originated from NIST SANS IgorPro
package.

**5.1.** ** ** **HardSphere Structure **

This calculates the interparticle structure factor for monodisperse
spherical particles interacting through hard sphere (excluded volume)
interactions. The calculation uses the Percus-Yevick closure where the
interparticle potential is:





where r is the distance from the center of the sphere of a radius R.

For 2D plot, the wave transfer is defined as .

Parameter name

Units

Default value

effect_radius



50.0

volfraction

0.2



**Figure. 1D plot using the default values (in linear scale).**

References:

Percus, J. K.; Yevick, J. Phys. Rev. 110, 1. (1958).

**5.2.** ** ** **SquareWell Structure **

This calculates the interparticle structure factor for a squar well
fluid spherical particles The mean spherical approximation (MSA)
closure was used for this calculation, and is not the most appropriate
closure for an attractive interparticle potential. This solution has
been compared to Monte Carlo simulations for a square well fluid,
showing this calculation to be limited in applicability to well depths
e < 1.5 kT and volume fractions f < 0.08.

Positive well depths correspond to an attractive potential well.
Negative well depths correspond to a potential "shoulder", which may
or may not be physically reasonable.

The well width (l) is defined as multiples of the particle diameter
(2*R)

The interaction potential is:





where r is the distance from the center of the sphere of a radius R.

For 2D plot, the wave transfer is defined as .

Parameter name

Units

Default value

effect_radius



50.0

volfraction

0.04

welldepth

kT

1.5

wellwidth

diameters

1.2



**Figure. 1D plot using the default values (in linear scale).**

References:

Sharma, R. V.; Sharma, K. C. Physica, 89A, 213. (1977).



**5.3.** ** ** **HayterMSA Structure **

This calculates the Structure factor (the Fourier transform of the
pair correlation function g(r)) for a system of charged, spheroidal
objects in a dielectric medium. When combined with an appropriate form
factor (such as sphere, core+shell, ellipsoid etc), this allows for
inclusion of the interparticle interference effects due to screened
coulomb repulsion between charged particles. This routine only works
for charged particles. If the charge is set to zero the routine will
self destruct. For non-charged particles use a hard sphere potential.

The salt concentration is used to compute the ionic strength of the
solution which in turn is used to compute the Debye screening length.
At present there is no provision for entering the ionic strength
directly nor for use of any multivalent salts. The counterions are
also assumed to be monovalent.

For 2D plot, the wave transfer is defined as .

Parameter name

Units

Default value

effect_radius



20.8

charge

19

volfraction

0.2

temperature

K

318

salt conc

M

0

dielectconst

71.1



**Figure. 1D plot using the default values (in linear scale).**

References:

JP Hansen and JB Hayter, Molecular Physics 46, 651-656 (1982).

JB Hayter and J Penfold, Molecular Physics 42, 109-118 (1981).

**5.4.** ** ** **StickyHS Structure **

This calculates the interparticle structure factor for a hard sphere
fluid with a narrow attractive well. A perturbative solution of the
Percus-Yevick closure is used. The strength of the attractive well is
described in terms of "stickiness" as defined below. The returned
value is a dimensionless structure factor, S(q).

The perturb (perturbation parameter), epsilon, should be held between
0.01 and 0.1. It is best to hold the perturbation parameter fixed and
let the "stickiness" vary to adjust the interaction strength. The
stickiness, tau, is defined in the equation below and is a function of
both the perturbation parameter and the interaction strength. Tau and
epsilon are defined in terms of the hard sphere diameter (sigma = 2R),
the width of the square well, delta (same units as R), and the depth
of the well, uo, in units of kT. From the definition, it is clear that
smaller tau mean stronger attraction.







where the interaction potential is





The Percus-Yevick (PY) closure was used for this calculation, and is
an adequate closure for an attractive interparticle potential. This
solution has been compared to Monte Carlo simulations for a square
well fluid, with good agreement.

The true particle volume fraction, f, is not equal to h, which appears
in most of the reference. The two are related in equation (24) of the
reference. The reference also describes the relationship between this
perturbation solution and the original sticky hard sphere (or adhesive
sphere) model by Baxter.

NOTES: The calculation can go haywire for certain combinations of the
input parameters, producing unphysical solutions - in this case errors
are reported to the command window and the S(q) is set to -1 (it will
disappear on a log-log plot). Use tight bounds to keep the parameters
to values that you know are physical (test them) and keep nudging them
until the optimization does not hit the constraints.

For 2D plot, the wave transfer is defined as .

Parameter name

Units

Default value

effect_radius



50

perturb

0.05

volfraction

0.1

stickiness

K

0.2



**Figure. 1D plot using the default values (in linear scale).**

References:

Menon, S. V. G., Manohar, C. and K. Srinivas Rao J. Chem. Phys.,
95(12), 9186-9190 (1991).

**References**

Feigin, L. A, and D. I. Svergun (1987) "Structure Analysis by Small-
Angle X-Ray and Neutron Scattering", Plenum Press, New York.

Guinier, A. and G. Fournet (1955) "Small-Angle Scattering of X-Rays",
John Wiley and Sons, New York.

Kline, S. R. (2006) *J Appl. Cryst.* **39**(6), 895.

Hansen, S., (1990) * J. Appl. Cryst. *23, 344-346.

Henderson, S.J. (1996) *Biophys. J. *70, 1618-1627

Stckel, P., May, R., Strell, I., Cejka, Z., Hoppe, W., Heumann, H.,
Zillig, W. and Crespi, H. (1980) *Eur. J. Biochem. *112, 411-417.

McAlister, B.C. and Grady, B.P., (1998) J. Appl. Cryst. 31, 594-599.

Porod, G. (1982) in Small Angle X-ray Scattering, editors Glatter, O.
and Kratky, O., Academic Press.

*Also, see the references at the end of the each model function
descriptions.


.. _Polarization/Magnetic Scattering: polar_mag_help.html