

ORIGINAL ARTICLE 

Year : 2022  Volume
: 45
 Issue : 1  Page : 2832 


Estimation of Fano factor for oversquare HPGe detector
M Manohari, V Sugumar, R Mathiyarasu, B Venkatraman
Indira Gandhi Centre for Atomic Research, Safety, Quality and Resource Management Group, Kalpakkam, Tamil Nadu, India
Date of Submission  17Jan2022 
Date of Decision  04Apr2022 
Date of Acceptance  04Apr2022 
Date of Web Publication  28Jun2022 
Correspondence Address: M Manohari Room No. 102, HASL Building, HPS/HISD/SQ&RMG, IGCAR, Kalpakkam, Tamil Nadu India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/rpe.rpe_1_22
Fano factor is a quantity used to measure the departure of the observed variance in the number of charge carriers produced from that predicted using poison statistics. Fano factor varies with the detector type. In case of proportional counters and semiconductor detectors, it is substantially less than unity, whereas, for scintillator detectors, it would be unity. Even in case of semiconductor detectors, it varies with the type and the shape of the detector. Another source of fluctuations that gives the overall resolution of the detector is preamplifier noise. Oversquare coaxial HPGe detectors are a recent development in the field. Literature on the Fano factor of oversquare large volume coaxial detector is not available. In this work, the Fano factor and the electronic noise of an oversquare HPGe coaxial detector are estimated after optimizing the shaping parameters. The Fano factor for the oversquare HPGe detector is estimated as 0.1291 keV, which agrees with the ideal value of 0.13 by 99.98%. The preamplifier noise was found to be 0.048 keV which is 35% lesser than the reported value.
Keywords: Baseline restorer, electronic noise, Fano factor, FWHM, oversquare coaxial HPGe detector, resolution, shaping time
How to cite this article: Manohari M, Sugumar V, Mathiyarasu R, Venkatraman B. Estimation of Fano factor for oversquare HPGe detector. Radiat Prot Environ 2022;45:2832 
How to cite this URL: Manohari M, Sugumar V, Mathiyarasu R, Venkatraman B. Estimation of Fano factor for oversquare HPGe detector. Radiat Prot Environ [serial online] 2022 [cited 2022 Nov 27];45:2832. Available from: https://www.rpe.org.in/text.asp?2022/45/1/28/348723 
Introduction   
HPGe detectors are used in various fields such as fundamental research, nuclear material safety, environmental radioactivity measurements, and human safety due to their high resolution. The main factor affecting the resolution of the detector is the variance in the electronhole pair production. This variance depends on the Fano factor. Theoretically, the Fano factor is a characteristic of the detector material. Papp^{[1]} has reported Fano factor values of 0.059 and 0.067 for Ge and Si respectively using Xrays of energies less than 60 keV. The literature reports are mainly for the Ge (Li) detectors. Fano factor as small as 0.058 has been reported for a Ge (Li) having a volume of 40 cm^{3}.^{[1]} Only a few literatures are found on the Fano factor of HPGe detector. These few literatures report that for the same material, Fano factor changes with the configuration of the detectors.^{[2]} A fano factor of 0.123 for a coaxial HPGe detector of 115 cm3[3] and 0.115 for a planar detector of 1 mm thickness^{[5]} has been reported in literature. The oversquare detector has a diameter greater than the thickness. There is no literature on the estimation of Fano factors for this configuration. The authors wanted to know what would be the variation in the Fano factor for this configuration. The volume of the detector is 170 cm^{3,} and Pehl and Goulding^{[4]} have reported that the Fano factor increases with the detector size.
In our laboratory, the estimation of Fano factor has not been carried out till now. Hence, an attempt has been made to estimate the Fano factor of an oversquare HPGe detector. There are various methods for the estimation of Fano factor. These methods include theoretical, numerical as well as experimental ones. The theoretical ones are Alkhazov's approach and Klein's approach.^{[5]} These two methods are based on the actual number of pairs created in each stage and the average number of ion pairs produced theoretically. They have derived the Fano factor using band gap energy (E_{g}), ionization energy (ε), ionization efficiency (Y), phonon loss, relative phonon loss (K), optical phonon energy, and photon generation ratio. The numerical one uses the probability distributions for ion pair creation and phonon creation. The first moment and the second moment give the average pair creation energy and Fano factor, respectively.^{[6]} Samedov^{[7]} has generated a probability generating function to describe the conversion of photon energy into the output signal and, using this expressed the amplitude and the variance in terms of a power series of the inverse bias voltage. He has determined the Fano factor using the coefficients of this power series. The experimental ones are from the intercept of dependence curve of FWHM (Full width at Half maximum) on bias voltage and from the slope of energy versus FWHM curve.^{[2],[3]} In this paper, the Fano factor is estimated using the slope of energy versus FWHM curve. The electronic noise is also estimated for the oversquare HPGe coaxial detector. The paper also compares the estimated values with the literaturereported values.
The overall resolution of a semiconductor detector (fluctuations in the energy measured by the system) is affected by the uncertainty on the energy of the gammaray (intrinsic width) (Ri), uncertainty in the production of electronhole pairs in the detector (R_{p}), uncertainty in the collection of charge carriers (R_{C}), the uncertainty introduced by the electronic noise in processing the pulse (R_{E}). The overall resolution is given by Eq. (1).
The uncertainty in the gammaray energy (intrinsic width) is inversely proportional to the halflife of the metastable states and is negligible compared to the overall resolution because of the lesser halflife of the metastable states of the gammaemitting radionuclides.^{[8]}
Uncertainty in electronhole pair production: It takes on an average of 2.96 eV(ε)^{[9]} to produce an ion pair in germanium crystal. The uncertainty in terms of FWHM in keV is given as:
F is the Fano factor given by Eq. (3).
Uncertainty in charge collection: Usually, the statistical fluctuation in charge collection is negligible to that of production as long as the operating voltage is kept at the bias voltage.^{[8]} Hence, the uncertainty in charge collection was assumed to be zero.
Now the overall resolution is given by Eq. (5).
Electronic noise: The uncertainties in the baseline of the electrical signals are called as electronic noise. The electronic noise is independent of pulse height but dependent on the shaping time of the amplifier. The electronic noise can be estimated using a pulse generator or from the plot of FWHM^{2} as a function of energy. The generator peak will not have production or collection uncertainties, and so the width of the peak in the spectrum is the electronic noise.
In measurement, the FWHM is the overall R.
Substituting in Eq. (4) we get:
Substituting the value of Rp from Eq. (4), we get,
Eq. 7 is of the form similar to linear equation y = mx + c; where y = FWHM^{2} (keV^{2}), x = E (keV), m(slope) = (2.35^{2} ∈ F). Hence, the Fano factor can be estimated from the slope of the energy versus FWHM curve using Eq. (8).
The intercept of the FWHM calibration on the yaxis represents the electronic noise alone.
C and Δ C, the noise and its uncertainty can be estimated using Eq. (9).
Materials and Methods   
The experiment was carried out using Bruker make oversquare coaxial HPGe detector with a relative efficiency of 45%. Oversquare detectors have a better resolution at lower energies. They have better efficiency at low energies for extended samples. The detector of dimension 8.5 cm × 3.03 cm has Be entrance window and is housed inside a graded shield room.^{[10]} The resolution and P/C ratio at 1332 keV are 1.9 keV and 93:1 respectively. The detector preamplifier has a low noise JFET. It has a transistor resistive feedback. The detector is attached to the BOSON Digital signal processor, and the spectrum is acquired using SPECTRALINE MCA software.^{[11]} BOSON uses Gaussian pulse shaping with CRRC circuit. The shaping time and the baseline restorer of the digital signal processor are optimized to give the best resolution. The variation of FWHM for various combinations of shaping time and baseline restorer was studied using 137Cs source of 3.7 ± 0.15 kBq activity kept at a distance of 25 cm. To study the variation of FWHM with energy 15.1 kBq ^{152}Eu point source standard (Ecker and Zeigler) emitting gamma photons in the range of 100–1500 keV,^{[12]} kept at a distance of 29 cm was used. This measurement setup produces a count rate equal to that encountered when a person has an activity equal to one recording level.^{[13]} The energies used are 121.8, 344.3, 778.9, 1112.1, and 1408 keV. [Figure 1] shows the measurement geometry of the oversquare detector. The dead time was <5%. The measurement time (50,000 s) was fixed to have the counting statistics error <1% in all energy regions. The software fits the FWHM values using a linear fit. FWHM values of these energies were noted down. The measurements were repeated 11 times to check the reproducibility of the system. OriginPro was used to fit the FWHM2 values linearly as a function of energy in a straight line using leastsquare fit and obtain the values of m, Δm, C and ΔC.
Results and Discussion   
Optimization of shaping parameters
Shaping time: To have the best resolution, the photopeak should be Gaussian. However, in practical measurement, even for a detector with the best resolution, the photopeak has a lowenergy tailing due to the incomplete charge collection, and the highenergy side has a contribution from the overlapping of noise with the signal. Tailing at the lower end is also due to the escape of the electrons produced due to the Compton scattering at the periphery of the detector. The peak shape varies with the pulse shaping parameters.^{[3]} The BOSON digital processor has options to change the shaping time and the baseline restorer. The processor gives a Gaussian output. The spectra of 661.62 keV were recorded for various shaping times and baseline restorer (BLR). [Figure 2] shows the change in the peak shape as a function of shaping time. [Figure 3] gives the variation of FWHM with the shaping time and the baseline restorer (BLR). The resolution becomes poorer (FWHM increases) as the shaping time decreases due to the ballistic deficit, which depends on the radial distribution of the original ion pairs. Hence, shaping time that is larger than the rise time of the pulses should be used to minimize the effect of the ballistic deficit on the energy resolution. Trapping of electrons can cause a deterioration in the energy resolution due to the variation in the amount of charge lost per pulse. Shaping time greater than the detrapping time will lower the FWHM. The shaping time of 6 μs, which gives the lowest FWHM and near Gaussian shape, was chosen for this study. The variation in FWHM with baseline restorer at this shaping time shows that for BLR greater than 20 ms, the FWHM remains constant at a minimum. Hence, any value above 20 ms can be used. [Figure 4] gives a sample spectrum of ^{137}Cs, and [Figure 5] the high energy portion of ^{152}Eu.  Figure 3: FWHM of 661.62 photopeak as a function of BLR for a different time constant
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Both the figures demonstrate the quality of the detector along with the digital signal processor.
The incomplete charge collection tailing and the electronic noise are small. [Table 1] lists the FWHM values for different energies. [Figure 6] shows a linear fit for one experimental set. Linear fit shows that the ballistic contribution to FWHM is nil.^{[3]} [Table 2] shows the Fano factor and the electronic noise obtained using origin from the straightline graph of (FWHM)^{ 2} versus E.  Table 1: Measured FMHM values of oversquare detector for different energies of 152^{Eu}
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 Table 2: Calculated Fano factor and preamplifier noise for 11 datasets of oversquare detector
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χ^{2} values are close to zero, indicating that FWHM^{2} and energy perfectly fit linearly. From [Table 2], it can be seen that F and R_{E} value ranges from 0.127–0.132 to 0.047–0.049 keV for the oversquare detector. The average value of the Fano factor is 0.1291, which agrees within 0.8% of the most quoted value of 0.13.^{[14]} The average value of noise is 0.048 keV.
The electronic noise is due to the leakage current in the detector as well as the preamplifier, preamplifier shot noise, and the acoustic pickup. The leakage current noise is produced by the current flowing through the reversebiased diode (I_{d}) and the equivalent thermal noise of any preamplifier input resistance. Electronic noise is also produced in the amplifier and the digital signal processor. The noise is 35% lesser than the earlier value reported by Samat and Priharti. This reduction is perhaps due to the improvement in the electronic component of the semiconductor detector.
Conclusion   
The Fano factor of an oversquare HPGe detector is estimated as 0.1291, which agrees within 0.8% lower than the theoretical value. The electronic noise was also low compared to the literature reported value.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6]
[Table 1], [Table 2]
