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 li:
scoh,d(cone) = Vsmpl*li3/(4v0²) * |Fd|2 / sin(0.5*qsc,i) (Squires (3.103))
=> pcoh,d = Iout,coh / Iin = scoh,d(cone) / Asmpl = Li*li3/(4*v0²) * |Fd|2 / sin(0.5*qsc,i)
with qsc,i = 2*asin(li/(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.) qsc,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 (qi, fi
). The angle qi is determined by qsc,i ; fi
is given by a Monte Carlo choice between f - Df
and f + Df.
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/v0*sinc= 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.
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. For coherent scattering, qi is determined by the Bragg condition . If it is out of the range [q-Dq, q+Dq], the trajectory is discarded. Therefore, no correction for this angle is necessary. For incoherent scattering, qi is determined by a Monte Carlo choice in the range [q-Dq, q+Dq]. Therefore, the intensity in this range depends on the choice of the range. This is corrected by a factor Dq. A limitation of the f-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 = Df/p * Dq
Gcoh = Df/p .
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 one incoming trajectory, Ncoh+Ninc trajectories are generated. Summing over these trajectories, the total probability for the scattering of a trajectory is then composed by:
P = (Gcoh pcoh,d + Ginc pinc sin qj) * At / Nr
where Nr denotes the number of repetitions. sin qi considers the q-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.
| Parameter Unit |
Description | Command option |
| sample file | The sample file describes the geometry and properties 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
Al_300.str
# file containing the structure factor data for the unit cell
0.000494 0.09057 0.01392 # macroscopic cross sections: incoherent, total
and absorption
66.38
# unit cell volume in cubic Angstrom
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-1Ang-1] |
macroscopic cross sections |
| unit cell volume [Ang³] |
volume of the unit cell |
Last modified: Tuesday, 03-Jul-2007 16:14:08 CEST