Deadtime
========

When detecting events from a random process, a pulse is generated for each
event which has a finite time width τ. Either pulses are rejected
during this time period (non-extendable dead time) or overlapping pulses are
counted as one pulse (extendable dead time).

Deadtime fraction
-----------------

``DT``
    The fraction of counts or time lost due to pulse processing.

``1-DT``
    The fraction of counts or time left after pulse processing.

.. parsed-literal::

    COUNTS_MEASURED = COUNTS_REAL * (1-DT)
    LIVE_TIME = REAL_TIME * (1-DT)

``LIVE_TIME``
    The effective time of ``COUNTS_MEASURED``.

``REAL_TIME``
    The effective time of ``COUNTS_REAL``.

Deadtime correction
-------------------

Counts that would have been measured if there was no pulse loss:

.. parsed-literal::

      COUNTS_REAL = COUNTS_MEASURED * DT_COR
    ⇔ DT_COR = 1 / (1-DT) = REAL_TIME / LIVE_TIME

Counts that would have been measured if there was no pulse loss
and the exposure time was not REAL_TIME but NORM_TIME:

.. parsed-literal::

      COUNTS_NORM = COUNTS_MEASURED * LT_COR
    ⇔ LT_COR = NORM_TIME / LIVE_TIME

The latter is referred to as a *live time correction*.
So deadtime correction is a special case of live time correction
with ``NORM_TIME=REAL_TIME``. Both cannot be applied at the same time: only one or the other.
The goal is to correct measured counts to a value that would have
been measured in a specific time with a perfect detector (one
that does not loose counts and therefore does not loose time).

Counting pulses
---------------

There are three types of "counts":

    * ``PHOTONS``: the real number of pulses.
    * ``TRIGGERS``: the number of pulses being detected.
    * ``TOTAL_EVENTS``: the number of pulses being accepted.

Pulses are lost whenever counting something:

    * ``PHOTONS`` → ``TRIGGERS``: pulses are lost due to overlap and electronics overhead, e.g. resets in a dual-channel signal processor.
    * ``TRIGGERS`` → ``TOTAL_EVENTS``: pulses are rejected if they do not fulfill certain criteria.

Loss of pulses: PHOTONS ➡ TRIGGERS
++++++++++++++++++++++++++++++++++

.. parsed-literal::

      DT_TRIGGERS   = (PHOTONS-TRIGGERS)/PHOTONS         # lost fraction in counts
                    = (REAL_TIME-LT_TRIGGERS)/REAL_TIME  # lost fraction in time
    ⇔ 1-DT_TRIGGERS = TRIGGERS/PHOTONS                   # remaining fraction in counts
                    = LT_TRIGGERS/REAL_TIME              # remaining fraction in time

``LT_TRIGGERS``
    Time not occupied by detected pulses and electronics overhead (e.g. resets).

Loss of pulses: PHOTONS ➡ TOTAL_EVENTS
++++++++++++++++++++++++++++++++++++++

.. parsed-literal::

      DT_EVENTS   = (PHOTONS-TOTAL_EVENTS)/PHOTONS    # lost fraction in counts
                  = (REAL_TIME-LT_EVENTS)/REAL_TIME   # lost fraction in time
    ⇔ 1-DT_EVENTS = TOTAL_EVENTS/PHOTONS              # remaining fraction in counts
                  = LT_EVENTS/REAL_TIME               # remaining fraction in time

      LT_EVENTS   = REAL_TIME.TOTAL_EVENTS/PHOTONS
                  = LT_TRIGGERS.TOTAL_EVENTS/TRIGGERS

``LT_EVENTS`` (called "energy live time" by XIA)
    Time remaining after loss due to pulse processing (pulse rejection and electronics overhead).

Correcting MCA counts
---------------------

Correct measured counts to counts of a perfect detector with exposure time ``REAL_TIME``:

.. parsed-literal::

    MCA_REAL = MCA * DT_COR

    DT_COR = 1/(1-DT_EVENTS)
           = REAL_TIME/LT_EVENTS
           = PHOTONS/TOTAL_EVENTS

Correct measured counts to counts of a perfect detector with exposure time ``NORM_TIME``:

.. parsed-literal::

    MCA_NORM = MCA * LT_COR

    LT_COR = NORM_TIME/LT_EVENTS
    LT_EVENTS = LT_TRIGGERS.TOTAL_EVENTS/TRIGGERS

XIA definitions
---------------

XIA defines the following quantities:

.. parsed-literal::

    OCR       = TOTAL_EVENTS/REAL_TIME
    ICR       = TRIGGERS/LT_TRIGGERS
    LT_ENERGY = LT_EVENTS

As a result we can express ``LT_EVENTS`` and ``DT_COR`` as:

.. parsed-literal::

    LT_EVENTS = LT_TRIGGERS.TOTAL_EVENTS/TRIGGERS
              = TOTAL_EVENTS/ICR
              = REAL_TIME.OCR/ICR

    DT_COR = REAL_TIME/LT_EVENTS
           = REAL_TIME.ICR/TOTAL_EVENTS
           = ICR/OCR

You can use any of these expressions depending on which quantities are provided with the MCA data.
Most data processing tools can handle only deadtime correction with ``LT_EVENTS``.

XMAP/MERCURY/SATURN
+++++++++++++++++++

.. parsed-literal::

    TOTAL_EVENTS = EVENTS+OVER+UNDER

FALCONX
+++++++

.. parsed-literal::

    TRIGGERS     = PULSES_ACCEPTED + PULSES_REJECTED
    TOTAL_EVENTS = PULSES_ACCEPTED