Module Powder Sample

# VITESS Module Powder sample

The module sample_powder describes the coherent and incoherent elastic scattering of a powder sample. 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 powder sample module uses the angles θ and φ. θ is defined as the angle between 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 to the yz-plane and the +y-axis and has a range of [0..2 π[.

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 probability pcoh (see e.g. Squires (3.103)) of the scattering process results from summing up the single contributions Pcoh, d for all d-spacings which satisfy the Bragg equation for a neutron with wavelength λi:

σcoh,d(cone) = Vsmpl*λi3/(4v0²) * |Fd|2 / sin(0.5*θsc,i) (Squires (3.103))

=> pcoh,d = Iout,coh /Iin = σcoh,d(cone) / Asmpl = Li*λi3/(4*v0²) * |Fd|2 / sin(0.5*θsc,i)

with θsc,i = 2*asin(λi/(2*d))

Fd denotes the structure factor; it is the sum over all reciprocal lattice vectors with the same d-spacing d. (d and |Fd|2 are taken from the structure factor file.) θsc,i is the scattering angle of the i-th trajectory, Vsmpl the sample volume, and v0 the unit cell volume. Iin denotes the incoming neutron current, Iout,coh the current of the neutrons scattered to the cone. Asmpl is the area of the sample perpendicular to the beam direction. Li denotes the total length of the flightpath of the trajectory under consideration through the sample. The values for |Fd|2 contain the Debye-Waller factor.
The direction of the trajectory i after the sample is given by (θi, φi). The angle θi is determined by &thetasc,i ; φi is given by a Monte Carlo choice between &phi - Δφ and &phi + Δφ.

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

pinc = Iout,inc / Iin = Nσinc/Asmpl = Vsmpl/v0 σinc/Asmpl = Li/v0*σinc = 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.

The scattering is restricted to an angular range of [θ-Δθ, θ+Δθ],[φ-Δφ, φ+Δφ] by the input values θ, Δθ, φ and Δφ. For coherent scattering, θi is determined by the Bragg condition . If it is out of the range [θ-Δθ, θ+Δθ], the trajectory is discarded. Therefore, no correction for this angle is necessary. For incoherent scattering, θi is determined by a Monte Carlo choice in the range [θ-Δθ, θ+Δθ]. Therefore, the intensity in this range depends on the choice of the range. This is corrected by a factor Δθ. A limitation of the φ-range must be corrected for both kinds of scattering, because the value is chosen by a Monte Carlo choice in the given range (see above). So, the restricted solid angle is taken into account by using the factors

Ginc = Δφ/π * Δθ
Gcoh = Δφ/π .

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/v0 is the macroscopic absorption cross section (to be provided for λ = 1.798 Å).

For one incoming trajectory, Nr(Ncoh + 1) trajectories are generated, where Nr denotes the number of repetitions. Summing over these trajectories, the total probability for the scattering of a trajectory is then given by:

P = (Gcoh pcoh,d + Ginc pinc sin θi) At / Nr

sin θi considers the θ-dependence of the isotropic distribution of the scattering orientations. Due to considering all accessible Bragg peaks, Ncoh is often greater than 1, even if Nr = 1. So don't worry about obtaining more output than input trajectories.

### 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 into which scattering occurs. The direction (θ,φ) points to the middle of the covered range, which extends from [θ-Δθ, θ+Δθ] and [φ-Δφ, φ+Δφ].
θ is the scattering angle defined as the angle between +x-axis (main flight direction of the neutrons) and the vector R describing the flight direction after scattering. φ is the direction on the cone defined as the angle between the +y-axis and the projection of R to the yz-plane. (x,y and z form a right-handed system.)
The default is Theta=DelTheta=90°, Phi=DelPhi=180° corresponding to a coverage of 4π.
Example: 20° to 160° scattering only to the right: θ=90°, Δθ=70°, φ=180° Δφ=90°

-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 at 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):

100.0 0.0 0.0 # sample position relative to the coordinate system defined by the preceding module
cyl # sample geometry (cyl, bal or cub)
2.0 10.0 # radius and height of the cylinder in cm
1.0 0.0 0.0 # orientation of the cylinder; need not be normalised
Al_300.str # file containing the structure factor data for the unit cell. If the format is neither .str nor .laz, the meaning of individual columns needs to be specified (see below).
0.000494 0.09057 0.01392 # macroscopic cross sections: incoherent, total and absorption
66.38 # unit cell volume in cubic Angstrom
6 13 0 17 0 1 # Structure file format, if the file is not given as .str or .laz. The last number is the scale factor that is needed to express the squared structure factor in barn, if it is given in a different unit in the used structure factor file.

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.

structure factor file | This file (*.str) contains two columns, the first one for the d-spacings and the second column contains the appropriate values for |Fd|2.
incoherent scattering total scattering absorption [cm-1, cm-1, cm-1Å-1] | macroscopic cross sections
unit cell volume [Å³] | volume of the unit cell
Column numbers for individual parameters of the look-up table | In case the structure factor file has neither the .str (VITESS) nor .laz format, the meaning of individual columns needs to be specified so that the file can be used as structure file.
Scale factor | In case the structure factor file has neither the .str (VITESS) nor .laz format, this scale factor is needed, if the (squared) structure factor is not given in barn^0.5 (barn).

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