Case 1: ∆t << Td

Cascaded Dead Times:

As explained in the introduction to the Fast Timing Discriminators section, the dead time experienced in the counting chain is typically composed of two cascaded components, T_e and T_ne. T_e is the extending dead time caused by the duration of the analog signal from the detector at the noise threshold of the timing (or counting) discriminator. It is an extending dead time because a second analog pulse occuring during a preceeding pulse extends the dead time by one pulse width from the arrival time of the second pulse. The non-extending dead time, T_ne, can be caused by the pulse width of the discriminator output driver, or it can be a longer dead time contributed by a circuit in the MCS or time digitizer. Sufficient accuracy^1,2 will be achieved if one chooses the longer of these two dead times to represent T_ne. A second pulse occurring during Tne is ignored and does not affect the dead time. It is convenient to define the approximate dead time in the system as

T_d ≈ T_e + U(T_ne–T_e) (T_ne–T_e)           (1)

where U(T_ne–T_e) is a unit step function defined by

U(T_ne–T_e) = 1 for T_ne > T_e
                                = 0 for T_ne ≤ T_e           (2)

Under that definition, the equations for Case 1 are valid if the quantization interval, Δt, is insignificant compared to T_d. This is the practical situation encountered in the Model 9308 picosecond TIME ANALYZER. For the Model 9308, T_d ≈ 45 ns and the maximum size of the bin width, Δt, is 2.5 ns.

Presume a time digitizer that has summed the repetitive spectra from n start triggers. The counts in the ith bin of the resulting spectrum (after suffering dead time losses) are defined to be q_i, and the time, t, is related to the bin number by

t = i Δt           (3)

By analogy to equation (3) it is convenient to define the quantized dead times, t_e, t_ne, and t_d, by equations (4).

T_e = t_e Δt           (4a)
T_ne = t_ne Δt         (4b)
T_d = t_d Δt           (4c)

Note that i, t_e, t_ne and t_d are all rounded to integer values.

The number of counts that would have been recorded in bin i if the dead time were zero is defined to be Q_i. The distorted spectrum recorded in the measurement is represented by q_i, whereas Q_i is the undistorted spectrum that is sought.

When the counting rates are low enough to yield single-ion or single-photon counting, one can apply statistical sampling theory. Poisson Statistics can also be applied directly, provided the dead time losses are negligible³.

In equation (5), q_i /n is the probability of recording an event in the ith bin during a single pass through the time span. It is composed of three probabilities⁴, as described in the right hand side of the equation.

Cascaded Dead Time Equation:

    (5)

The first term, Q_i /n is the probability that an event will arrive at the detector at a time suitable to be categorized in bin i. In order to be recognized as distinct from previous analog pulses there must be no pulses arriving at the detector in the time interval t_e preceeding the pulse for bin i. That probability is given by the exponential term in equation (5). If a pulse had been counted by triggering the non-extending dead time in the time interval t_ne preceeding the pulse for bin i, the pulse for bin i would be lost. Consequently, the term in the square brackets is the probability that no pulses are recorded in the preceeding time interval t_ne. Note that the sum stops at j = i – t_e –1 because the exponential term already guarantees no pulses occured in the time interval from j = i – t_e to i –1.

Equation (5) can be rearranged to get the formula for computing the corrected spectrum, Q_i, from the distorted spectrum, q_i.

    (6)

One applies the correction algorithm by starting at bin i = 0, while presuming that q_i, q_j, and Q_j are all zero for negative values of i and j. For each i, the value Q_i is calculated from equation (6) using the recorded values q_i and q_j along with the Q_j values calculated for the previous values of i. This correction calculation is applied bin by bin until the maximum bin in the spectrum has been treated. At that point the list of Q_i values is the corrected spectrum.

If there truly were no counts to be detected for negative values of i, then the Q_i data near i = 0 will represent the true spectrum before dead time losses. If the detector was actually responding to events for negative values of i, then Q_i will be underestimated until the bin number exceeds several times t_d. Frequently, this shortcoming can be eliminated by inserting a coaxial cable delay of the appropriate length between the detector and the stop input on the time digitizer.

Single, Extending Dead Time:

For a system where the detector pulses are longer than any other dead times in the system (t_e > t_ne), equation (5) simplifies to the equation for a single, extending dead time⁴.

i –1
q_i = Q_i exp {–Σ Q_j /n }           (7)
       j = i – t_e

Single, Non-extending Dead Time:

The other extreme is a system in which detector pulse widths are negligible compared to the non-extending dead time in the MCS or time digitizer. In that case, equation (6) simplifies to^4,5

          (8)

Accuracy of the Dead Time Correction:

It can be demonstrated by substituting known values into equations (6), (7) and (8) that all three equations yield predictions of Q_i / q_i that are within 1% of each other provided Q_i / q_i < 1.15 (i.e., a dead time correction < 15%), and provided td is substituted for the single dead times in equations (7) and (8). This allows a simpler correction algorithm to be implemented using equation (8). In fact, the algorithm using equation (8) can start at the maximum value of i and proceed towards i = 0, while replacing q_i with Q_i in the data file. This is the procedure used in the Model 9308 software.

Without the dead time correction algorithm, one would have to limit the counting rate to achieve <1% dead time losses in order to limit the spectrum distortion to <1%. By applying the dead time correction algorithm, one can typically operate at a factor of 10 higher dead time loss, while still achieving <1% spectrum distortion. This implies a factor of 10 higher data rates. However, the accuracy of the correction is limited by the factors discussed next.

If the time spectrum is constant across all bins, it is easy to show that a 10% error in the assumed value for the dead time in equations (6), (7), or (8) will lead to < 1% error in the corrected counts if Q_i / q_i < 1.10. A more serious case is a narrow peak centered at bin i, and preceeded by an intense, narrow peak centered at bin j = i – t_d. A small error in the presumed value for t_d can result in either including or excluding the peak at bin j in the dead time corrections. This can make a large difference in the dead time correction applied to the peak at bin i. An additional issue is the error in rounding off the presumed dead time to the nearest integer value. This leads to round-off errors at the two limits of the sums in the equations. That effect can be restricted to an error <1% if one ensures that q_j /n < 0.005 for all j. Clearly, it is important to use an accurately measured dead time in the correction formula.

By applying basic probability theory, it can be shown that the statistical variance in the recorded counts q_i is given by⁴

σ_qi² = q_i (1– q_i /n)           (9)
           ≈ q_i             for q_i /n << 1

Moreover, for the sum of the recorded counts in any number of bins, such as

i –1
         M = Σ q_j           (10)
     j = i – t

the statistical variance in the sum is

 i –1
σ_M² = M = Σ q_j           (11)
      j = i – t

A straight forward propagation-of-errors computation predicts the statistical variance in the corrected counts Q_i calculated from equation (8) to be^4,5

σ_qi² = Q_i k_i           (12)

where k_i defines the magnification factor arising from the variances in the q_i and q_j in equation (8), i.e.,

k_i = (Q_i / q_i) + (Q_i / q_i)² [(Q_i / q_i) –1] (q_i /n)           (13)

The effect of k_i is small for dead time corrections Q_i / q_i < 1.15, but the second term of k_i escalates rapidly for higher dead time corrections.<sup>5

1</sup>Jörg W. Müller, Nucl. Instr. Meth. 112, (1973), 47–57, Fig. 3.
²D.R. Beaman, et al., J. of Physics E: Sci. Instr. (1972), 5, 767–776.
³Ron Jenkins, R.W. Gould, and Dale Gedcke, Quantitative X-Ray Spectrometry, Marcel Dekker, New York, (first edition), 1981, Chap. 4.
⁴D.A. Gedcke, Development notes and private communication, Nov.Dec. 1996.
⁵P.B. Coates, Rev. Sci. Instrum. 63 (3), March 1992, 2084.