Module Powder Sample

# VITESS Module S(Q)-sample

The module S(Q)-sample module describes the coherent and incoherent elastic scattering of a S(Q)-sample. The coordinate system of an incoming neutron is defined by the preceding module as well as the main flight path of the neutrons. The main flight direction of the neutrons defines the +x-axis. The y-axis is defined as the horizontal axis and the z-axis as the vertical one. The module S(Q)-sample uses the angles q and f. q is defined as the angle between a vector r and the +x-axis and covers the range from [0..p]. f is the angle between the projection of the vector r in the yz-plane and the +y-axis and has the range of [0..2p[.

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 [q-Dq, q+Dq], [f-Df, f+Df] by the input values for q, Dq, f, Df. The direction of the trajectory i after the sample is given by (qi, fi ). For both, coherent scattering and incoherent scattering, both angles qi and fi 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 = Df/p * Dq

(see below). With the equation for the differential cross section (Squires, Thermal Neutron Scattering (5.29)):

ds/dW = Nscoh/(4p) S(Q)

it is possible to calculate the cross section for scattering into a solid angle W. (N is the number of nuclei in the scattering system.) For a discrete number of orientations (qi, fi ):

scoh(W) = ds/dW (fmax-fmin)(qmax-qmin) 1/Ni S sin qi

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

scoh(W) = Nscoh/(4p) S(Q) (fmax-fmin)(qmax-qmin) 1/Ni S sin qi
= Gcoh N scoh S(Q) 1/Ni S sin qi

The probability of scattering neutrons into this solid angle is:

Pcoh = Iout,coh / Iin = scoh(W) /Asmpl

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

Pcoh = Gcoh 1/Asmpl *Vsmpl/v0 *scoh S(Q) 1/NiS sin qi
= Gcoh Li mcohS(Q) 1/Ni S sin qi
=: Gcoh pcoh*1/Ni S sin qi

S(Q) denotes the structure factor as a function of the momentum transfer Q. Vsmpl the sample volume, and v0 the volume per nuclei. mcoh = scoh/v0 denotes the macroscopic coherent scattering cross section and Li 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 S(Q) can be taken from the structure factor file or be calculated in the function "CalcStructureFactor" in "sample_s_q.c". As default, a function using the Percus-Yevick hard sphere model with special parameters is called.

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*sinc/Asmpl = Vsmpl/v0*sinc/Asmpl = Li*sinc/v0 = Li*minc

sinc is the incoherent scattering cross section of the unit cell; minc= sinc/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 * (mtot + mabs*l/(1.798 Ang))}

Ln is the total neutron flight path in the sample (surface -> point of scattering -> surface). mtot = stot/v0 denotes the total macroscopic scattering cross section to be interpreted as wavelength independent, mabs = sabs/v0 is the macroscopic absorption cross section (to be provided for l= 1.798 Ang).

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

P = S (Gcoh pcohsin qi + Ginc pinc sin qj) * At / Ni

where sin qi considers the q-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 compositions of the sample. | -S

Theta, dTheta,
Phi, dPhi
[deg]

These parameters describe the solid angle covered by the detector. The direction (q,f) points to the middle of the covered area, which extends from [q-Dq, q+Dq] and [f-Df, f+Df].
q is defined as the angle between +x-axis (main flight direction of the neutrons) and the Vector R to be described. f 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 Dq, q, Df, f, you must specify all of them.!!!
The default is a coverage of 4p.

-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. | -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
D # S(Q) data from file
SiO_300.str # file containing the structure factor data for the unit cell
0.0082 0.0824 0.0139 # Scattering cross sections: incoherent, total and absorption

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 | data from file ('D') or from an analytic function ('A')
structure factor file | This file (*.str) contains two columns, the first one for the momentum transfer values Q [Ang-1] and the second column contains the appropriate values for S
incoherent scattering, coherent scattering, absorption [cm-1, cm-1, cm-1Ang-1] | macroscopic cross sections

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