.. algorithm::

.. summary::

.. relatedalgorithms::

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

This algorithm Corrects the time of flight (TOF) of an indirect geometry
instrument by substracting a time offset :math:`t_0` linearly dependent
on the wavelength of the neutron when emitted through the moderator.
This algorithm is suitable to data reduction of indirect instruments
featuring a neutron flux with a narrow distribution of wavelengths. A
empirical formula for the correction, stored in the instrument
definition file, is taken as linear on the initial neutron wavelength
:math:`\lambda_i`: :math:`t_0 = a * \lambda_i + b`. Gradient :math:`a` is
in units of microsec/Angstrom and Intercept :math:`b` is in units
of microsec. Below is the example XML code included in BASIS beamline
parameters file.

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

    <!-- Moderator Tzero/LambdaZero Parameters  -->
    <parameter name="Moderator.TimeZero.Gradient">
        <value val="11.967"/>
    </parameter>
    <parameter name="Moderator.TimeZero.Intercept">
        <value val="-5.0"/>
    </parameter>

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

-  :math:`t_0`: emission time from the moderator
-  :math:`t_i`: time from moderator to sample
-  :math:`t_f`: time from sample to detector

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

For an indirect geometry instrument, :math:`\lambda_i` is not known but
the final energy, :math:`E_f`, selected by the analyzers is known. For
this geometry:

-  :math:`t_f = L_f/v_f`, with :math:`L_f`: distance from sample to
   detector, :math:`v_f`: final velocity derived from :math:`E_f`
-  :math:`t_i = L_i/v_i`, with :math:`L_i`: distance from moderator to
   sample, :math:`v_i`: initial velocity unknown
-  :math:`t_0 = a'/v_i+b'`, with :math:`a'` and :math:`b'` constants derived from the
   aforementioned empirical formula
   :math:`a' = a \cdot 3.956 \cdot 10^{-3}` with :math:`a'` in units of meters

and :math:`b' = b` with :math:`b'` in units of microseconds.

Putting all together:
:math:`TOF' = \frac{L_i}{L_i+a'} \cdot (TOF-t_f-b') + t_f`, with
[TOF']=microsec

If the detector is a monitor, then we can treat it as both sample and
detector. Thus, we use the previous formula inserting the time from
sample to detector :math:`t_f = 0` and with the initial fligh path
:math:`L_i` as the distance from source to monitor.

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