Case 2: T_d << Δt, with Counts Essentially Constant Across Δt

This solution is applicable to multichannel scalers. Typically the dead time T_d is < 20 ns. If the dwell time is large compared to 20 ns (for example: 2000 ns) and the counting rate varies insignificantly during the dwell time of a bin, then one can transform equation (6) into the form that applies for approximately constant counting rate.⁴ The result is

Cascaded Dead Times:

          (14)

Note that Q_i appears on both sides of the equation. So equation (14) must be solved by iteration. If one substitutes the instantaneous counting rates defined by equation (15) in equation (14), the constant counting rate formula in the Fast Timing Discriminator section are generated.

          (15a)

          (15b)

In some applications the detector pulse width will exceed the non-extending dead time (T_e > T_ne)leading to

Single, Extending Dead Time:

          (16)

If the detector pulse width is negligible compared to the non-extending dead time, equation (14) becomes

Single, Non-extending Dead Time:

          (17a)

or

          (17b)

As in Case 1, it is convenient and adequate to use equation (17a) when the dead time losses are less than 15%, provided T_ne is replaced with T_d. Most of the caveats in Case 1 concerning accuracy apply here as well. The one exception is the statistical variance in the recorded counts q_i which is given by³

σ_qi² = q_i (q_i / Q_i)²            (18)

for the case of non-extending dead time. The statistical variance in the Q_i calculated from equation (17a) is given by³

σ_Qi² = Q_i (Q_i / q_i)           (19)

Note that large dead time corrections magnify σ_Qi (the standard deviation in the corrected counts) by the square root of the correction factor, Q_i / q_i. Equations (18) and (19) are valid if Δt >> Td and Δt is large compared to the mean spacing between pulses. The expressions for the σ_qi and σ_Qi corresponding to extending dead time are approximately the same as equations (18) and (19) for Q_i / q_i < 1.1, but diverge wildly from the non-extending case for large correction factors³. This provides an incentive for limiting dead time losses to < 10%.