Earlier this week, I collected data to determine dead time according to the ratio method of Heinrich et al. (1966) using the Ti Kα and Kβ lines. As in previous cases, I used uncoated and recently polished Ti metal for the measurements and measured at the Kα or Kβ peak on three spectrometers simultaneously using LiF, LiFL, and LiFH.

When working with Ti using this method in conjunction with a sealed Xe counter, it is important to note that, while the energy of Ti Kα lies below that of the Xe L

_{3} absorption edge, the energy of Ti Kβ falls above it, and so electronic gain, anode bias, and baseline voltage must all be considered especially carefully. When measuring Ti Kβ, the baseline needs to be set so that it always fully excludes the Xe escape peak when working at current ranging in my case from 5 nA to 540 nA in measurement set 1 and from 5 nA to 700 nA in measurement set 2. During the first measurement set, I counted simultaneously at the Ti Kα peak position on channel 2/LIFL and at the Kβ position on channels 3/LiF and 5/LiFH. During the second measurement set, I counted simultaneously at the Ti Kβ peak position on channel 2/LIFL and at the Kα position on channels 3/LiF and 5/LiFH. At

*I*_{PCD} = 700 nA (

*V* = 15 kV), the measured Ti Kα count rate on channel 3/LiF was ~60 kcps; on channel 5/LiFH, it was ~227 kcps.

The obtained values for

*τ* are as follows:

Ti Kα/Kβ ratio method [μs]:

channel 2/LiFL: 1.43, 1.46

channel 3/LIF: 1.37

channel 5/LIFH: 1.42

Current-based method, Ti Kα [μs]:

channel 2/LiFL: 1.25

channel 3/LiF: 1.03

channel 5/LiFH: 1.20

Current-based method, Ti Kβ [μs]:

channel 2/LiFL: 0.81

channel 3/LiF: -0.06

channel 5/LiFH: 1.04

Note that 1) the Ti dead time values calculated using the ratio method are essentially the same as those calculated for Cu and Fe and 2) use of the current-based method once again causes underestimation of the dead time, with channel 3/LiF showing the greatest departure. Combining the results of the ratio method for Cu, Fe, and Ti, for channel 2 (six calculations total), the range of calculated dead time values is 1.43-1.50 μs, with an average of 1.46 μs. The observed variation likely can be ascribed to counting error. Dead time values for channel 3 (three calculations) are 1.45, 1.41, and 1.37 μs (Cu, Fe, and Ti, respectively), giving an average of 1.41 μs. For channel 5 (three calculations), I’ve obtained 1.42, 1.38, and 1.41 μs, producing an average of 1.40 μs.

As an aside, assuming that linear behavior is maintained on a plot of

*N’*/

*N* versus

*N’* up to 80 kcps and assuming that

*τ* = 1.45 μs, then

*N’*/

*N* = 0.884 (exactly) at that measured count rate, and so true count rate (

*N*) is 90498 s

^{-1}. This represents a roughly 13% correction relative to the measured count rate. Once again, I prefer to work with dead time corrections smaller than this, and so I’m happy to remain within the linear correction range.

Returning to the current-based method, an anomaly appears in all plots of

*N’*/

*I* versus

*N’* collected in both Ti measurement sets. For instance, in the plot of the data for Ti Kα on channel 2/LiFL (below), the data take a trajectory in the form of an arc, such that initial data (lowest current) plot below the regression line, then subsequently well above it (around 35 kcps), and then, in the high end of the linear range (up to ~85 kcps), most fall slightly below it. Above 85 kcps, significantly non-linear behavior truly is present, though assessing its onset on the plot is virtually impossible. Whatever the nature/source of the problem (certainly related to the picoammeter), the anomaly appears to vary over time in magnitude and form. Compare, for instance with my plots for Cu (anomaly pronounced but different) and Fe (anomaly much less pronounced). Regardless, since it is present in each set of simultaneous measurements, the error cancels when calculating ratios. Note that, had I chosen to use only apparent count rates falling below 35 kcps, my calculated dead time would have been 1.01 μs rather than 1.25 μs.

The same pattern is easily visible on channel 5/LiFH. In calculation of dead time, had I considered only count rates below 35 kcps, I would have obtained 0.98 μs rather than 1.20 μs.

The following plot gives a more accurate depiction of the magnitude to which departure from linearity affects the dead time correction at high count rates. I’ve plotted measured Ti Kβ count rate on channel 2/LiFL divided by the measured Ti Kα count rate on channel 5/LiFH versus the measured Ti Kβ count rate on channel 2/LiFL. The Ti Kβ count rate on LiFL reaches only 33 kcps (at 700 nA), and so all obvious non-linearity should be due to Ti Kα on LiFH. Above 85 kcps (for Ti Kα), all ratio values fall above the regression line, though only very slightly up to 113 kcps (

*I*_{PCD} = 260 nA), and so I’ll take 85 kcps as my upper limit on channel 5; corresponding plots for Cu and Fe support this limit, which appears to be applicable to the other Xe counters as well.

My likely next step, when I get a chance, will be to determine dead time via the ratio method for Ti using my three PET crystals. I’ll then move on to the spectrometers with gas-flow counters.