# VITESS New Monochromator-Analyser Module

This module simulates a neutron monochromator or analyser (M/A) as a flat rectangular crystal or a matrix of flat crystal elements (CE-s) forming a focusing geometry.

The new monochromator module is an upgrade of the existing one. The older module is kept for backward-compatibility, however it is strongly recommended to change to the new one since only this module will be maintained. All known functionalities were kept and two were added: The "reflection" and "reflection+transmission" modes, which are explained below. Moreover, the additional simulation run to determine the correct normalisation has become obsolete. The new module now handles the normalisation of the outgoing neutrons properly if the peak reflectivity, i.e. the reflectivity of the monochromator crystal for neutrons of the peak wavelength and negligible divergence, is given. The peak reflectivity is usually known from dedicated measurements. Note that a correct normalisation of the outgoing intensity is only possible, if realistic, non-zero values for the mosaicity and the d-spread are given. If an ideal crystal needs to be simulated, i.e. the mosaicity or d-spread are very close to zero, then it is possible that the outgoing intensity is larger than the incoming one. In this case you can set the reflectivity normalisation to a value below 1. If both the d-spread and the mosaicities are set to 0 then the crystal will only transmit the peak wavelength, resulting in a complete loss of intensity in a simulation.

A monochromator element is characterized by its geometric and crystallographic properties. The corresponding simulation parameters are set either on the main GUI window or in the parameter file. The module can handle flat crystals and focusing geometries. In addition, it is now possible to set up an array of monochromators/analyzers. The geometric parameters of a monochromator element are: position, horizontal and vertical offset angles (conforming to Eulerian rotation), horizontal and vertical 'Bragg angles' with reference to the final coordinate system of the previous module, and size of the CE. The Bragg angles can be rotated w.r.t. the crystal surface to allow for the transmission geometry (offcut orientation). For focusing geometries, additional parameters need to be specified to determine the curved shape of the monochromator element. The crystal structure can be characterized by setting the parameters d-spacing, d-spread (Lorentzian and Gaussian), mosaicity and reflectivity. Detailed parameter definitions can be read from the tables in section B. In the output frame X'Y'Z', all neutron coordinates are referring to the moment when the neutrons are just reflected from the crystal taking the new directions and probability weights.

There are two new modes, "reflection" and "reflection+transmission". In "reflection" regime, the trajectories that are not reflected but are transmitted through the monochromator without any deflection are treated. The "reflection+transmission" regime is necessary, if an array of monochromators needs to be simulated. In this case, trajectories reaching the first or all intermediate monochromators are split in two. One of the trajectories is reflected off and the other one transmitted through the monochromator with corresponding propabilities. After the last monochromator, all trajectories that are passed on to the next module have been reflected by either of the elements. Important: In this regime the global coordinate system is not set by the monochromator module, but needs to be defined in a subsequent frame module.

Fig. CE

A. Options

For flat crystal simulation it is enough to use one single CE. The simulation of a curved focussing surface works with a matrix of CE-s (as they are really constructed), therefore one needs all positions, orientations and size parameters. For convenience, the deviations relative to the corresponding main values given in the Parameter File have to be defined and read from a Focus File properly formatted.

The options are:

1. Crystal_flat: | Simulates a flat rectangular generally (h,k,l) oriented offcut crystal. | -O1
2. Crystal_focus: | Activates the Focus File generation for a) l-focussing, b) spherical c) vertically focussing cylinder or d) double focussing cylinder M/A geometry. | -O2
3. Crystal_focus_dat: | Uses optional Focus File independently generated by separate programs not included in this module. | -O3

Option 2 takes advantage of an internal code of the module, which automatically generates the Focus File for a) l-focussing, b) spherical c) vertical cylinder d) or double focussing M/A geometry. This option is suitable, for example, for setting analysers in a near-backscattering geometry to provide for example constant wavelength selection for neutrons scattered from the center of the sample (Fig. l-focussing).

Fig. l-focussing

One example for vertical focusing monochromator geometry. α is the angle between the tangent of the central element that is perpendicular to the beam in this example and the lowest (the highest) monochromator element (see 'Focus Option').

B. Parameter and file descriptions

Option Crystal_Flat

File | Format | Examples Attached | Command Option
Parameter File | Includes FILE INPUT PARAMETERS. This file can be read or created/modified by the VITESS shell. Values are read from separate rows i.e. 1 value/row for scalar and 3 values/row for vector type variables. | crys.par | -P

- MAIN PARAMETERS

Parameter | Physical Symbol, Description | Range, Examples | Command Option
mosaic fwhm horizontal, vertical [deg] | hY ,hZ Horizontal and vertical fwhm components of the 2-dimensional Gaussian mosaic distribution. If it is set smaller than 0.001, then this minimum value is set automatically. | PG(002): 0.2 0.8 deg General: 0.001 1deg | -m, -M
d-spread [-] | Δd / d Fwhm of the d-spacing distribution function (Lorentzian and Gaussian options) divided by lattice parameter under consideration. It is zero for a perfect crystal. | PG(002): 0.2 - 2´ 10-3 Si(111): 0.1 - 2´ 10-4 | -D
d-distribution | Lorentzian (1), Gaussian (2) | 1, 2 | -d
Reflectivity normalization [-] | R Set this parameter to the (experimentally determined) peak reflectivity of this monochromator depending on the bragg angle. | 1 (default) | -R
repetition rate [-] | If this integer > 1, the neutron is used multiple times for better statistics. | >= 1 | -A

- FILE INPUT PARAMETERS

Parameter | Physical Symbol, Description | Range, Examples
d-spacing [Å] | d Lattice parameter corresponding to a reflection from a (h,k,l) crystal plane | PG(002): 3.332 Å Si(111): 3.135 Å Ge(113): 1.703 Å Cu(220): 1.272 Å
order of reflection [-] | N Conforming to Braggs Law: N l = 2 d sin φBr | 1, 2, ...

main position

[cm]

x, y, z

Generally this position defines the reference point (origin) of the M/A system in the frame provided by the former module(or according to an input data file): the position of each single CE is defined by (x,y,z)+deviation from (x,y,z) the deviation is read from focus data file.

For theFlat Geometry option (x,y,z) is simply the center position of the rectangular CE.

X = distance to CE

Y = Z = 0.0

thickness, width, height of CE [cm] | t, w, h Thickness, width and height give depth, horizontal and vertical dimensions of the rectangular CE. For each single CE the dimensions will be (t,w,h)+deviation from (t,w,h) the last being read from focus data file. | t = 0.0 0.5 cm w = h = 0.5 20 cm
main 'surface offset' angle horizontal, vertical [deg] | f, q Euler angles If f = q = 0.0 (exact backscattering), the normal vector of CE is parallel to the positive X axis direction. A rotation first around the Z axis (f ) and then around the (new)Y axis (q ) gives a proper orientation of the CE. The orientation of each single CE will be (f, q) +deviation from (f, q) the last being read from focus data file. | f = q = 0.0 exact backscattering; f = 0.0, q = 0 ... 90 deg;
main 'Bragg offset' angle horizontal, vertical [deg] | Same as 'main offset angle' above if the (h,k,l) planes are parallel to the crystal plane as it has sense for the focusing geometry options. In case of flat crystal option, the reflection planes can be set non-parallel to the crystal surface. | see one before
output angle horizontal, vertical [deg] | F, Q In case of "user defined output frame", a Frotation about the Z axis and then a Qrotation about the (new)Y axis defines a new reference orientation. According to this and the output frame translation vector (see below) the neutrons are written to the output file. If "standard frame generation" is activated, the output frame is rotated until the new X axis is parallel to the "reflected" original input X axis corresponding to the main offset values. | F = 180 deg, Q= 0.0 exact backscattering; F = 180 deg, Q= - 2q, if qis the main vertical offset angle and j= 0.0
output frame [cm] | x, y, z The position of the output frame origin (O) in the original frame. (x, y, z) represents the translation vector applied to shift the origin of the original (input) frame (O) to the new (output) position. Default setting is: (x, y, z) = (x, y, z) i.e. main position of the M/A system. | x, y, z one point on the reflected beam axis

- Please note that the parameter file format used for the old monochromator module remains unchanged. However, the 'mosaic range' and 'd-range' parameters became obsolete and are not shown anymore, when the parameter file is open for editing.

Option Crystal_Focus

- same as Option Crystal_flat

-FOCUS PARAMETERS

Parameter | Physical Symbol, Description | Range, Examples | Command Option
number of CE horizontal, vertical [-] | The number of columns and rows (nH, nV) of the created CE-matrix. | 2 50 | -H,-V
radius [cm] | Distance from the sample center to the bottom row of the CE-matrix (see figure l-focussing). | 200 cm | -r
radius horiz. [cm] | Radius of focussing in horizontal direction for a double focussing shape. | 200 cm, 0 cm to focus only vertically | -s
angle vertical [deg] | Angular offset α of the bottom row of the CE-matrix relative to the monochromator center (see figure l-focussing). | -3 deg | -a
focusing option: 1 constant lambda 2 spherical 3 vert. cylinder 4 double focussing | Here one can choose the focusing geometry. The 'double focussing' option assumes a flat geometry, in which each element is rotated to give cylindrical focussing with different radii in vert. and hor. direction. The 'vert. cylinder' option really forms a vertically focussing cylinder. | 1, 2, 3, 4 | -g

- description of the focus file (filename must be given as input):

File | Format | Examples Attached | Command Option

focus file

First row includes 2 integer numbers: nH, nV, the number of columns and rows of the created CE-matrix. (H = horizontal, V = vertical.)

Next nH´ nV rows are created by two program loops (internal loop: V) computing 8 values representing deviations from the main values for each single CE . These are interpreted by the module Crystal as follows:

The position of each single CE is (x,y,z)+deviation from (x,y,z) as read from columns 1-3.

The dimensions of each single CE are (t,w,h)+deviation from (t,w,h) as read from columns 4-6.

The orientation of each single CE is (j , q) + deviation from (j , q) as read from columns 7, 8.

lamb_foc.dat,

-G

Option Crystal_focus_dat

- same as Option Crystal_flat

- focus file (as described above) which has to be given externally, focus file name should be give as input.

D. Computing algorithm

For each incoming trajectory the d-spacing d is randomized according to the given d-spread. The randomization takes into account whether the distribution is Gaussian or Lorentzian. If the d-spread is zero, this step is obviously skipped. Then the Bragg-angle φBr of the trajectory is calculated taking into account its wavelength and the current d-spacing value: φBr = sin-1(λ/(2*d)). Now one needs to calculate the probability that the monochromating crystal can reflect this neutron according to the Bragg condition.

In general, the Bragg condition is not fulfilled by the orientation of the CE, see Fig. Cone. The angle φCE between the incoming neutron ki and the CE normal nCE is usually not equal to φCE,Br = 90° - φBr, since the incoming beam can be divergent and the spectrum usually consists of a finite waveband. To assign a proper weight to the outgoing neutron the probability of finding a mosaic piece that fulfills the Bragg condition needs to be calculated. All possible normal orientarions of such mosaic elements describe a Bragg cone around ki with the radius rc = |ki|tan(φCE,Br). Since all monochromators have a finite horizontal and vertical mosaicity, the Bragg cone can be reached starting from nCE by two rotations around the y and z axes. To be in accordance with the given mosaic distributions and to obtain the correct probability, the rotation procedure is done as following: The first (e.g. vertical) rotation (Δφz) is randomised using a Gaussian with the given mosaic spread (FWHM of the corresponding distribution). Obviously, this first rotation fixes Δφy, the angle of the second (e.g. horizontal) rotation, since nM must lie on the Bragg cone. The absolute probability for the reflection of this neutron is then determined by looking up the value of the Gaussian function of the latter (horizontal) mosaicity corresponding to Δφy. The norm of the Gaussian function is such that it is in accordance with the peak reflectivity of the monochromator element. This norm is determined in each simulation during the initialisation of the module and is based on the given Bragg angles, d-spread and mosaicity.

In the output, the new probability weight of a neutron mirrors adequately the given d- and mosaic distributions. The new coordinates and the direction of the neutron ki are computed consecutively by taking into account the exact orientation of the reflecting mosaic piece nM.

Fig. Cone

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Last modified: Dec 03 15:00:00 MET 2013, D. Nekrassov