Module SANS Sample

# VITESS Module SANS Sample

The module sample_sans describes small angle scattering at monodispers hard particles. The coordinate system of an incoming neutron is defined by the preceding module. The direction of the beamline defines the +x-axis. The y-axis is defined as the horizontal axis to the left and the z-axis as the vertical one upwards. The module SANS sample uses the angles θ and φ. θ is defined as the angle between a vector a vector R defining the flight direction of the neutron and the +x-axis and covers a range of [0..π]. φ is the angle between the projection of the vector r in the yz-plane and the +y-axis and has a range of [0..2π[.

A neutron is written to the output by this module, if the neutron arrives at the sample surface after scattering. The coordinate system has still the same orientation, but the origin has moved to the center of the sample. All sample modules consider the divergence of each neutron without any approximation to obtain the true direction of the flightpath after the scattering process.

Description of the scattering:
First of all it is determined if the neutron intersects the sample. If the neutron does not intersect the sample, it is discarded. Otherwise the neutron is scattered along its path through the sample at a certain distance Ls from its entrance, which is determined by a Monte Carlo choice.
The scattering is restricted to an angular range of [θ-Δθ, θ+Δθ] by the input values for θ, Δθ, φ and Δφ. The direction of the trajectory i after the sample is given by (θi, φi). For both, coherent scattering and incoherent scattering, both angles θi and φi are determined by a Monte Carlo choice in these ranges. Therefore, the number of trajectories per solid angle in this range depends on the choice of the range. To get the real flux into this solid angle, this has to be considered by the factors

Ginc = Gcoh = Δφ/π * Δθ

(see below). The differential cross section is given by the square of the scattering amplitude (see e.g. Feigin, Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering (2.2), or P. Fratzl in 'PSI Proceedings 97-01, Cold Neutrons - High Resolution'):

dσ/dΩ = |A(Q)|2 = | ∫ ρ(r) exp(iQ*r) dV)|2

Integration yields for N identical particles of volume Vp and scattering length density ρp:

= N rp2 Vp2 F(Q)

with the Form Factor F(Q) = |S(Q)|2 = | ∫ exp(iQ*r)dV / Vp|2. Considering also the scattering length density ρs of the solution:

dσ/dΩ = N(ρp-ρs)2 Vp2 F(Q) = (ρp-ρs)2 fp VpVsmplF(Q)

where fp is the volume fraction of the particles and Vsmpl the sample volume. Now it is possible to calculate the cross section for scattering into a solid angle Ω. For a discrete number of orientations (θi,φi):

σcoh(Ω) = dσ/dΩ (φmax-φmin)(θmax-θmin) 1/Ni ∑ sin θi

(The meaning of Ni is explained below). Here, we get:

σcoh(Ω) = (ρp-ρs)2 fp Vp Vsample F(Q) (φmax-φmin)(θmax-θmin) 1/Ni ∑ sin θi

= Gcoh 4π (ρp-ρs)2 fp Vp Vsmpl F(Q) 1/Ni ∑ sin θi

The probability of scattering neutrons into this solid angle is:

Pcoh = Iout,coh / Iin = σcoh(Ω) / Asmpl< /p>

Iin denotes the incoming neutron current, Iout,coh the current of the neutrons scattered into the solid angle Ω. Asmpl is the area of the sample perpendicular to the beam direction. With the preceding equation:

Pcoh = Gcoh 4π (ρp-ρs)2 fp Vp Vsmpl/Asmpl F(Q) 1/Ni ∑ sin θi
= Gcoh 4π (ρp-ρs)2 fp Vp Li F(Q) ∑ sin θi
=: Gcoh pcoh 1/Ni ∑ sin θi
for the real samples. For the isotropic scatterer the form factor F(Q) and the product (ρp-ρs)2 fp Vp are set to 1.

Li denotes the total length of the flightpath of the trajectory under consideration through the sample. Ni is the number of trajectories times the number of repetitions, the sum is over these Ni orientations. (In practice, the influence of the number of trajectories on the probability is already taken into account in the module 'Source', but the number Nr of repetitions must be considered here.) pcoh is the probability that a neutron is scattered at all.
The values for F(Q) can be calculated from size and shape for particles of a simple geometry, e.g. for spheres

Fspheres(Q) =

Apart from spheres, the form factors of ellipsoids, cylinders and parallelepipeds are introduced in the program. The formula are taken from the literature mentioned above. Additionally, isotropicscttering can be chosen for testing normalization purposes. Apart from the spheres, the orientation of the particle relatively to the neutron flight direction is determined by a Monte Carlo choice before the treatment of the scattering event.

In addition to the coherent scattering, the incoherent scattering can be treated. This gives an isotropic distribution of the incoherently scattered neutrons. This scattering probability is given by

pinc = Iout,inc / Iin = N σinc/ Asmpl = Vsmpl/v0 * σinc/Asmpl = Li σinc/v0 = Li μinc

σinc is the incoherent scattering cross section of the unit cell; μinc = σinc/v is the macroscopic incoherent scattering cross section, which can be given as an input in the sample file.

For all trajectories, an attenuation At is considered as follows:

At = exp{-Ln (μtot + μabs λ/1.798 Å)}

Ln is the total neutron flight path in the sample (surface -> point of scattering -> surface). μtot = σtot/v0 denotes the total macroscopic scattering cross section to be interpreted as wavelength independent, μabs = σabs/v0is the macroscopic absorption cross section (to be provided for λ= 1.798 Å).

For each incoming trajectory, Nr - only coherent scattering - or 2*Nr - with incoherent scattering - trajectories are generated. Summing over these trajectories, the probability for the scattering of a trajectory is then composed by:

P = ∑ (Gcoh pcohsin θi + Ginc pinc sin θi) * At / Nr

where sin θi considers the θ-dependence of the isotropic distribution of the scattering orientations.

### Module parameters

Parameter Unit | Description | Command option
sample file | The sample file describes the geometry and properties of the sample. | -S

θ, Δθ
φ, Δφ
[deg]

These parameters describe the solid angle covered by the detector. The direction (q,f) points to the middle of the covered area in th range [θ-Δθ, θ+Δθ] and [φ-Δφ, φ+Δφ].
θ is defined as the angle between +x-axis (beam line) and the Vector R of the flight direction of the neutron. φ is the angle between the +y-axis and the projection of R to the yz-plane. x,y and z form a right-handed system.
!!!Note: If you specify any parameter of θ, Δθ φ, Δφ you must specify all of them.!!!
The default is a coverage of 4π.

-D, -d
-P, -p

repetitions | 'repetitions' specifies the number of data sets (trajectories) generated for each scattered trajectory. A larger number of repetitions enriches the population on the detector and gives therefore better statistics in the spectrum, but the number of density of events in the parameter space is not increases, e.g. the number of different wavelength values remains the same. Therefore, an increase of the number of trajectories created by the source module is favorable. | -A
incoherent scattering | yes: neutrons are additionally scattered incoherently no: incoherent scattering is omitted | -I

### Sample file parameters

first an example (the order written to the sample file differs from the sequence chosen in the GUI):

2.0 0.0 0.0 # sample position relative to the coordinate system defined by the preceding module is: 2 cm along the x-axis
cub # sample has the geometry of a cuboid
0.1 1.0 1.0 # thickness, height and width of the cuboid are 0.1 x 1 x 1 cm
1.0 0.0 0.0 # orientation of the cuboid: thickness is in x-direction, height in z-direction (vector need not be normalised)
S 500.0 # sperical particles of radius 500 Å
10000000000.0 30000000000.0 0.03 # scattering length densities: particles 1E10 1/cm², solution 3E10 1/cm², 3 % vol. frac. of particles
0.0 0.5 0.05 # Scattering cross sections: incoherent=0, total=0.5 cm-1 and absorption=0.05 cm-1Å-1

Anything after the # character is interpreted as a comment.

Parameter Unit | Description
x,y,z [cm] | position of the sample centre relative to the coordinate system defined by the preceding module
sample geometry | cylinder or sphere or cuboid
thickness or radius [cm] | thickness of cuboid, or radius of sphere, or radius of cylinder
height [cm] | height of cuboid, or height of cylinder
width [cm] | width of cuboid

x,y,z direction

vector components describing the orientation of the sample (it is not necessary to give a normalized vector).
cylinder: the vector is always perpendicular to the top of the cylinder (standard cyl. position (001), height along the z-axis).
cuboid: Standard is the (1,0,0) direction, i.e. the sample has a thickness in x-direction, a width in y-direction and height in z-direction. By giving a different vector the whole sample is rotated in this direction, i.e. the planes separated by 'thickness' remain perpendicular to this vector.
sphere: no values needed.

scattering objects | geometry of the particles in solution: spheres, ellipsoids, parallelepipeds, cylinders additionally, isotropically scattering objects can be chosen for testing normalization purposes

radius 1 or length
radius 2 or thickness
radius 3 or height
[Å]

sizes of the particles in solution
spheres: the radius of the spheres must be given in radius 1, the other parameters are not needed
ellipsoids: the three radii of the ellipsoid must be given
parallelepipeds: length, thickness and height of the parallelepiped must be given
cylinders: the first two parameters are radius 1 and radius 2 of the cylinder, the third is the length
isotropic: no parameter needed

scat. len. dens. particl. [1/cm²] | scattering length density of the particles
scat. len. dens. solution [1/cm²] | scattering length density of the solution
vol. fraction of particles | volume fraction of the particles in solution (for 5 %: 0.05)
incoherent scattering total scattering absorption [cm-1, cm-1, cm-1Å-1] | macroscopic cross sections

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