Determining an Upper Limit

An upper limit for the number of counts R in a counting experiment, assuming a

Poission distribution may be calculated using

R≤n√B (6.8)

where n is the confidence level at which the result is expressed (n=1 (67%), n=1.64

(90%) ,n=2 (99%), etc.) and B is number of counts in a given energy bin.

Peak Window Optimization

Our goal is to choose a region-of-interest (ROI) of optimal width when determining an

upper limit. We will focus on the case where the signal is Gaussian and the background

is flat. Although we chose this shape since it pertains toγ-ray spectroscopy using HPGe

detectors, it may be extended for any signal and background shape.

Varying the width will change two things: (1)B, the amount of background in the

ROI will depend on the width, (2) the fraction of the potential Gaussian signal shape

contained in the ROI. Following [Ell11], we construct a figure of merit (FoM) based on

the parts of the upper limit calculation that will be affected by the width, given as

F oM =

√ B ROI

, (6.9)

where B is the background in the ROI and ROI is the efficiency factor due to the

fraction of the line shape contained in the peak. For a Gaussian signal (assume the

Gaussian is centered at x= 0 ie. µ=0) on a flat background, the FoM becomes

F oM = √ x Rx 0 e −x02_/2σ2 dx0. (6.10)

Integrating the gaussian, this becomes F oM = √ x √ π·σ·erf(₂x_σ) (6.11) We now attempt to findσ such that our FoM is a minimum.

∂ ∂x( √ x √ π·σ·erf(x 2σ) ) = √ π·σ·erf(₂x_σ)−2xe−x2/4σ2 2π·σ2√_x·erf(x 2σ) 2 (6.12)

Setting this equation to zero and solving for x numerically, we find that our FoM has

a minimum value at 1.4σ. We therefore conclude that the optimum width for an ROI

for a gaussian signal on a flat background is ±1.4σ about the mean, or a total window width of 2.8σ.

6.4.3

Detector Efficiencies

The full-energy peak (FEP) detection efficiency is defined as the ratio of the number

of events detected in theγ-ray peak to the number of events emitted from the source.

In general, this depends on many factors, including the crystal, cryostat, shielding and

source geometries. There are several methods to determine the FEP detection efficiency:

(1) Use analytical calculations [Wan95]. This technique is limited to simple geometries

and requires complex calculations. (2) A physical model of the sample can be created

using known standards [Smi09]. This process is complicated and time consuming and

is of limited accuracy for complex geometries. (3) Perform a point source calibration

at several points near the detector. The efficiency curve generated is then corrected for

absorption by a sample matrix [Sae04]. (4) Perform a detailed Monte-Carlo simulation

for each sample [Fin11]. A distribution of γ rays are tracked from the emission at

the source to absorption in the detector active region. By using a pure Monte-Carlo

is accounted for and there are no limitations on source or detector configurations.

However, Monte-Carlo simulations are limited by the knowledge of the response and

geometry of the detector and targets, and the micro-physics models used.

For this experiment, the γ-ray detection efficiency (γ) was measured using 17

γ rays from 152_Eu, 60_{Co and} 137_{Cs point sources each placed in the center of the}

array. For each γ ray, the detection efficiency was calculated using the number of

events in the photopeak, known source activity, γ-ray branching ratios and measure-

ment live time. These measured efficiencies were fit to derive an efficiency curve for each

detector. A simulation of the efficiency was also done using MaGe [Bos11]; a Monte-

Carlo framework developed by the _Majorana and GERDA collaborations based on GEANT4 [Ago03, All06]. The detector geometries were adapted from the simulation of

the GEANIE detectors done using MCNP by Ref.[McN00a]. Simulated mono-energetic

γ rays were tracked from a point source in 10 keV increments from 10 to 4000 keV.

The efficiency was calculated for eachγ-ray energy using

γ =

Npeak

Nsim

(6.13)

where Npeak is the number of events in the peak and Nsim is the number of events

simulated. Enough events were generated for each γ-ray energy so that statistical

uncertainties were less than 1%. An example efficiency curve constructed from the

simulated data and source data are shown in Fig. 6.9. The data were fit to a power

law above 200 keV. At lower energies, the efficiency peaks and then sharply decreases

due to absorption in the detector endcap and cryostat. The simulated efficiency curve

was consistent with the fit to the experimental data to within 6% from 200 – 3200 keV,

Energy (keV)

200 400 600 800 1000 1200 1400 1600 1800 2000

Efficiency

0.000 0.002 0.004 0.006 0.008 0.010 0.012 Data Simulation

Figure 6.9: An example efficiency curve (detector GeQ). The black circles are data taken with the 152Eu point source. Error bars on the data points are not visible on the scale of this graph. The black curve is a power law fit to the data for energies above 200 keV. The red curve is the simulation.

Energy (keV)

2 10 103

Efficiency

-4 10 -3 10 -2 10 Point Source

Gas Cell - 0 degrees Gas Cell - 90 degrees

Figure 6.10: Simulated efficiency curves for GeQ using a point source and argon gas cell. The black curve is the efficiency for a point source in the center of the GEANIE array. The red and blue curves are the efficiencies for the gas cell oriented at 0 degrees and 90 degrees with respect to the beam axis, respectively.

The effects ofγ-ray attenuation and extended source geometry were investigated us-

ing additional Monte-Carlo simulations. Mono-energeticγ rays were generated isotrop-

ically in a simulated gas cell in 10 keV increments from 10 to 4000 keV. The gas cell

was oriented at 0 degrees and at 90 degrees with respect to the beam axis, representing

the most extreme placements of the gas cell relative to any detector in the array. A

comparison of the gas cell filled with 2.75 atm argon and point source simulations is

shown in Fig. 6.10. While the effect of absorption in the gas cell is significant at γ-ray

energies of about 10−50 keV, the correction due to γ-ray attenuation in the gas target and aluminum cell and extended source geometry was negligible for the measuredγ-ray

Beam Profile Effects

A measurement of the neutron beam profile and correction for beam profile effects was

discussed in [McN00b]. The γ rays emitted from a target are distributed according to

the neutron beam profile, which was approximately Gaussian with a 1.27 cm diameter.

Since there is more geometric phase space for neutrons further away from the beam

center (the area of a ring about the center of width dr is rdr, i.e. is proportional to

the distance from the center of the beam profile), the efficiency can be sensitive to the

effects of the beam profile. Any part of the neutron beam which passes through the

fission chamber is counted, but are not equally likely to result in an observed γ ray.

To determine how the efficiency decreases for detecting γ rays as the source moves

away from the center, Ref.[McN00b] performed a series of MCNP calculations for rings

of source material 0.1 inches in thickness. For example, the first calculation was for

source material evenly distributed from 0 < r < 0.1 in, the next calculation was for

source material evenly distributed from 0.1< r <0.2 in, and so on. It was determined

from the flux-weighted beam profile that the majority of the neutrons in the beam were

located approximately 5 mm from the center of the beam spot. At this location, the

correction factor was determined to be about 10% with a minor dependence on γ-ray

energy.

We performed a similar simulation using 1000 keVγ rays for rings of source material

1 mm in thickness usingMaGe. The efficiency was calculated relative to a point source.

The results of the simulation are shown in Fig. 6.11. The relative efficiency is constant,

with less than 2% variation to a ring radius of about 14 mm at which point we expected

thatγ rays would be significantly attenuated by the lead collimators and BGO shields.

Ring radius (mm)

0 2 4 6 8 10 12 14 16 18 20

Relative Efficiency

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04

Figure 6.11: The efficiency of GeQ for 1000 keVγ rays distributed in 2-mm thick rings relative to a point source.

ADC channel

0 2000 4000 6000 8000 10000 12000 14000 16000

counts/channel

1 10 2 10 3 10

Figure 6.12: A sample ADC spectrum for the 238_{U fission chamber summed over all} neutron energies from 1 to 100 MeV. The peak on the left is from α-particle detection and the peak on the right is from fission fragments. A cut was placed at the dashed line to excludeα-particle events from the neutron flux analysis.

In document 5037.pdf (Page 182-189)