``HRIXS``: The Heisenberg Resonant Inelastic X-Ray Scattering Spectrometer
==========================================================================

Technical movement
------------------

The movements of the HRIXS are controlled by a PLC. On the PLC level we assure
that the spectrometer cannot be moved outside of its safe working conditions,
as defined by the instruction manual.

Two of those parameters are trivial, the grating postion *G* (motor ``GTZ``)
and the detector position *D* (motor ``DTZ``) simply have a minimum and a
maximum value.

.. figure:: hrixs_geometry.jpg

   Geometry of the HRIXS, from Figure 35 (page 79) in the instruction manual


The height *Y* of the detector should be kept such that the angle δ of the
detector is below 15°. So the detector height should be calculated from
*Y* = (*D* - *G*) tan δ. The height is adjusted using the two motors ``DTY1``
and ``DTY2``, which always must move in parallel.

It is also important that the detector pitch follows this angle δ, but it
needs to be adjustable by a small angle ɣ, -5° < ɣ < 5° (ɣ is shown
exaggerated in the figure). So the detector pitch θ (motor ``DRX``)
is θ = δ + ɣ.

Summarizing, we have four parameters, *D*, *G*, δ and ɣ, from which the motor
positions are calculated as follows:

- ``GTZ`` = *G*
- ``DTZ`` = *D*
- ``DTY1`` = ``DTY2`` = (*D* - *G*) tan δ
- ``DRX`` = δ + Ɣ

Solution using interlocks
-------------------------

Unfortunately, it is not as easy as desired to implement coupled motions as
described above on the PLC level, so also this will be done in Karabo. Instead,
the PLC will only have interlocks which stop all four motors mentioned above
if we leave the safe range of parameters.

Solving the above equations leads to the following inequalities:

- ``DRX`` > arctan(``DTY`` / (``DTZ`` - ``GTZ``)) - 5°
- ``DRX`` < arctan(``DTY`` / (``DTZ`` - ``GTZ``)) + 5°
- ``DTY`` < (``DTZ`` - ``GTZ``) tan 15°

These must be fulfilled at all times, otherwise all mentioned motors must stop.
In order to get out of an interlocked condition one can move the motors on the
left side of the inequalities such that those are again fulfilled.

At the same time, the distance between grating chamber (GTZ) and detector
chamber (DTZ) should remain between 2190 mm and 3242 mm. So

- ``DTZ`` - ``GTZ`` < 3242 mm
- ``DTZ`` - ``GTZ`` < 2190 mm

Physical parameters
-------------------

The HRIXS uses a variable line spacing (VLS) grating to disperse the incoming
X-Ray beam onto a detector. Two different gratings are available, with a nominal
line spacing of *n*\ :sub:`0` = 1000 lines / mm or 
*n*\ :sub:`0` = 3000 lines / mm. Calculations of such a grating can be found in
`Svitozar Serkez' Thesis <https://bib-pubdb1.desy.de/record/275967/files/Thesis%20PDF.pdf>`_
on page 37.

We need to calculate the parameters for the technical movement as described
above from the photon enery *E* and the diffraction order *m*.

The grating equation applies, with entrance angle α and exit angle β we have

.. math:: 

   \sin α - \sin β = m n_0 λ
  
where *m* is the diffraction order and λ is the wavelength.
(Obviously *E* λ = *hc* with Planck's constant *h* and the speed of light *c*)

The VLS grating focusses the beam onto the detector following 

.. math::

   \frac{\cos α + \cos β}{R} - \frac{\cos^2 α}{r_1} - \frac{\cos^2 β}{r_2} = m λ n_1
   
where *n*\ :sub:`1` represents the linear chirp of the grating spacing,
*R* is the radius of curvature of the grating, and *r*\ :sub:`1` and
*r*\ :sub:`2` are the lenght of the entrance and exit arm of the spectrometer.

The coma abberation is minimized if

.. math::

   \left(\frac{\cos^2 α}{r_1}-\frac{\cos α}{R}\right)\frac{\sin α}{r_1}
   + \left(\frac{\cos^2 β}{r_2}-\frac{\cos β}{R}\right)\frac{\sin β}{r_2}
   = \frac{2}{3} m λ n_2
   
where *n*\ :sub:`2` represents the linear and quadratic chirp of the grating
spacing.

By now we have four physical parameters (α, β, *r*\ :sub:`1`, *r*\ :sub:`2`)
but only three equations. As a fourth equation we could use the position
of the sagittal focus, as proposed by Svitozar. But usually this is not a
critical parameter, so instead we leave one variable free (*r*\ :sub:`1`),
and use this variable to optimize resolution vs. transmission: if the grating
is close to the source, we improve transmission, if it is further away, we
improve the resolution.

The technical parameters can be calculated as follows: *D* = *r*\ :sub:`1`,
α + β + δ = 180°, and *G* - *D* = *r*\ :sub:`2` cos δ.