Figure 1: Test Setup
Properly referenced Goro Sato's work
Added a note about validity of eq. 2 (Rev. B)
The purpose of this experiment was to test a method for finding the μτ product for electrons and for holes using two spectra taken at different bias voltages. The use of two bias voltages for finding the μτ product for electrons was proposed to me by Goro Sato of Tokyo University, who used a similar, though more sophisticated, technique to determine μτ products for CZT detectors.1 The analysis requires:
When I first began, all three of these data sets weren't yet available on any test sandwiches, so I took measurements on a single-detector setup (similar to the setup used by Goro Sato) using the Canberra system in Bert's lab.
At first, I made measurements using a simple setup involving only a source, a detector, and the eV Products brass detector holder. I found that there were scattering features in the spectrum that weren't consistent with a simple model of just the holder and source. In order to control these features so that I could properly model them with grmcflight, I determined to shield the entire setup.
The gain was adjusted so that a 150 keV range covered approximately 4000 channels. The LLD was adjusted to just eliminate the noise spike at the low end of the spectrum. Two spectra were taken, each for 1 hour (live time): one at 300 V bias voltage and one at 100 V bias voltage. Then a pulser sweep was recorded at 7 positions spanning the 4000 channel range.
When the bias voltage was set to 100 V, the spectrum shifted to lower channels.
The Hecht Relation describes the charge trapping effects of a detector. It gives the charge collection efficiency as a function of depth of interaction:
(1)Figure 3 shows charge collection efficiency vs. depth of interaction for some typical values of Vb, (μτ)e, and (μτ)h:
Because of this effect, photons that interact deeper within the detector are "measured" as having deposited less energy. The overall effect is that photons of a given energy produce a peak having a tail that extends to lower energies.
Charge collected is related to channel by:
(2)Note that this equation only holds if the induced charge Q is proportional to the induced voltage pulse height Vinduced, and only if voltage pulse height V is linear with channel Ch.
For each pulser peak, I recorded the dial setting and measured the voltage height on an oscilloscope. Next I found the the channel positions of each pulser peak by fitting it to a gaussian. I fit a line to the voltage heights vs. the channel positions and thus obtained the electronics gain and offset.
Photons interacting near the front surface of the detector (z=0) are the most numerous and result in the highest efficiencies, so they produce counts in the main, high-channel part of the peak. Setting z=0 in equation 1 produces:
.(3)Note that this equation is independent of (μτ)h.
If the position of a peak at one bias voltage and the position of the same peak at another bias voltage are determined, these values can be substituted into equation 3, along with the known values of A, O, and D, producing two equations with two unknowns: (μτ)e and ChMax.
The positions of the 122 keV peak for the two spectra were:
| Bias Voltage | Peak Position |
|---|---|
| 100 V | 2602 ± 3 |
| 300 V | 2730 ± 3 |
Solving equations 3 numerically for (μτ)e gives:
(μτ)e = (2.72 ± 0.13) x 10-3 cm2/V
I wrote a routine, in IDL, which takes 6 parameters:
and returns the spectrum that would result from a stream of incoming photons with a given energy and intensity, assuming uniform illumination and neglecting Compton scattering.
I have used this function in the past to fit spectra and try to obtain the best-fit (μτ)e and (μτ)h, but since (μτ)e, gain, and offset all affect the position of a peak, and gain and offset cannot be determined from a single spectrum, there were many possible pairs of (μτ)e and (μτ)h that, together with suitable values of the other 4 parameters, all fit a single peak equally well (without necessarily fitting the other lines in the spectrum well). Fitting an entire spectrum was problematic because the function does not include effects due to Compton scattering. Having an independent measure of (μτ)e helps.
Gain and offset had previously been determined in voltage units. To obtain them in energy units required a little algebra, involving ChMax found from the numerical solution of equation 3.
With (μτ)e known and fixed, I selected a 200 channel region surrounding the 122 keV peak of the 300 V spectrum (channels 2400 through 2800) and fit my IDL function to it. The fitting procedure I used did not converge to a local mininum very well (for some reason). Therefore, I used many different starting values for the parameters and recorded the set of best fit parameters and Χ2 for each fit.
There is a cluster of values at (2.93 ± 0.13) x 10-5, but the values of (μτ)h with the minimum Χ2 are at (3.33± 0.17) x 10-5.
I Recognized that including more of the peak tail in the fit might lead to a more trustworthy value of (μτ)h, so I selected a 300 and 400 channel region and repeated the process. The results were quite different:
| Channel Range | (μτ)h |
|---|---|
| 2600 - 2800 | 3.33±0.17 |
| 2500 - 2800 | 2.65±0.06 |
| 2400 - 2800 | 2.27±0.08 |
The plots of Χ2/n vs. (μτ)h for all 3 channel ranges are shown in Figure 4:
I plotted the real data verses the simulated spectra corresponding to best fit parameters at the center and extremes of each error bar. For example, for the channel range of 2600 to 2800, I used best fit parameters for which (μτ)h converged to 3.50, 3.67, and 3.84 x 10-5. These plots are shown in Figures 5 thru 7:
Looking at the spectra, it is apparent that the parameters within each error bar produce spectra that are nearly identical. The plots of all 9 best fit spectra next to the real data show that the there are small differences between them (see Figures 8 and 9).
By taking a pulser sweep and spectra at 2 different bias voltages, I was able to obtain a value for (μτ)e that I feel confident with. But in trying to obtain a reliable value for (μτ)h, I found that the techniques I used led to some serious uncertainty.
An interesting note is that when both lines (122 keV and 136 keV) were used in the fit, the plot of Χ2/n vs. (μτ)h doesn't have a clear minimum trough as it did when only the 122 keV line was used in the fit.
In the case of (μτ)e, a deviation of 0.1 x 10-3 cm2/V changes the the peak position by 0.2% or less (depending on the initial values of (μτ)e and (μτ)h), the peak height by 0.9% or less, and the tail height by 0.9% or less (I somewhat arbitrarily defined the tail height to be the height of the tail at a point 250 channels (10.75 keV) below the peak).
In the case of (μτ)h, a deviation of 0.1 x 10-5 cm2/V changes the the peak position by 0.03% or less, the peak height by 2.0% or less, and the tail height by 0.02% or less. (0.03% was about the size of my error due to the channel spacing.)
1See Goro Sato, Tadayuki Takahashi, Masahiko Sugiho, Manabu Kouda, Shin Watanabe, Yuu Okada, Takefumi Mitani, and Kazuhiro Nakazawa, Characterization of CdTe/CdZnTe detectors, submitted to IEEE Transactions on Nuclear Science 2001