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.

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:

  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:

  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#

  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#

  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:

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:

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:

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:

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#

TOTAL_EVENTS = EVENTS+OVER+UNDER

FALCONX#

TRIGGERS     = PULSES_ACCEPTED + PULSES_REJECTED
TOTAL_EVENTS = PULSES_ACCEPTED