.. algorithm::

.. summary::

.. relatedalgorithms::

.. properties::

Description
-----------

Corrects the time of flight (TOF) by a time offset that is dependent
on the energy of the neutron after passing through the moderator. A
heuristic formula for the correction is stored in the instrument
definition file. Below is shown the entry in the instrument file for the
VISION beamline:

.. rstcheck: ignore-next-code-block
.. code-block:: xml

  <!-- formula for t0 calculation. See http://muparser.sourceforge.net/mup_features.html#idDef2 for available operators -->
  <parameter name="t0_formula" type="string">
    <value val="(incidentEnergy < 34.7332) ? 37.011296*incidentEnergy^(-0.052874) : (incidentEnergy < 88.7556) ? 124.267307*incidentEnergy^(-0.394282) : (incidentEnergy < 252.471) ? 963.775145*incidentEnergy^(-0.850919) : (incidentEnergy < 420.145) ? 33.225834*incidentEnergy^(-0.242105) : (incidentEnergy < 100000.0) ? 120.569231*incidentEnergy^(-0.455477) : 0.0" />
  </parameter>

The recorded :math:`TOF = t_0 + t_i + t_f` with

- :math:`t_0`: emission time from the moderator
- :math:`t_1`: time from moderator to sample or monitor
- :math:`t_2`: time from sample to detector

This algorithm will replace :math:`TOF` with :math:`TOF^* = TOF-t_0 = t_i+t_f`

For a direct geometry instrument, the incident energy :math:`E_1` is
the same for all neutrons. Hence, the moderator emission time is the
same for all neutrons. For an indirect geometry instrument, :math:`E_1`
is different for each neutron and is not known. However, the final
energy :math:`E_2` selected by the analyzers is known.

- :math:`t_0 = func(E_1)` , a function of the incident energy
- :math:`t_1 = L_1/v_1` with :math:`L_1` the distance from moderator to
  sample, and :math:`v_1` the initial unknown velocity (:math:`E_1=1/2*m*v_1^2`)
- :math:`t_2 = L_2/v_2` with :math:`L_2` the distance from sample to
  detector, and :math:`v_2` is the final fixed velocity (:math:`E_2=1/2*m*v_2^2`)

.. note::

   We obtain :math:`TOF^*` as an iterative process,
   taking into account the fact that the correction :math:`t_0` is much
   smaller than :math:`t_i+t_f`. Thus
   :math:`TOF-t_0^{(n)} = L_1/v_1^{(n)} + L_2/v_2` , :math:`n=0, 1, 2,..`
   Set :math:`t_0^{(0)}=0` and obtain :math:`v_1^{(0)}` from the previous
   formula. From :math:`v_1^{(0)}` we obtain :math:`E_1^{(0)}`
   Set :math:`t_0^{(1)}=func( E_1^{(0)} )` and repeat the steps until
   :math:`|t_0^{(n+1)} - t_0^{(n+1)}| < tolTOF`. With
   :math:`tolTOF=0.1 microsecond`, only one iteration is needed for convergence.

Here's the result of applying ModeratorTzero to both the event list and
the histogrammed data of a run in the VISION beamline. The
transformation of either events or histograms shifts the curve to
smaller TOF's. The transformed curves are not supposed to be identical,
but similar and differentiated from the original curve.

.. figure:: /images/ModeratorTzero_Fig.1.jpeg
   :width:  400px
   :alt:    Summed Histogram

.. figure:: /images/ModeratorTzero_Fig.2.jpeg
   :width:  400px
   :alt:    Elastic Line

.. figure:: /images/ModeratorTzero_Fig.3.jpeg
   :width:  400px
   :alt:    Inelastic Peaks

For indirect instruments featuring an incoming neutron flux having a
sufficiently narrow distribution of energies, a linear relationship
between :math:`t_0` and the wavelength of the incoming neutron can be
established. This relation allows for coding of an algorithm with faster
execution times. For indirect instruments that comply with these
conditions, use of :ref:`algm-ModeratorTzeroLinear` is
preferred.

.. categories::

.. sourcelink::