LB Det A Dead Time

Due to the difficulty in getting the activity ratios to match the expected values, I decided to investigate the dead time of the detector in use (Detector A). Below is a table of values used to generate the plot. This will also give the reader a good idea of what the dead times are for certain count rates.

Count Rate (Hz)	% Dead
129.47 +/- 7.84	0.24 +/- 0.10
154.33 +/- 10.36	0.29 +/- 0.11
194.83 +/- 8.36	0.41 +/- 0.11
253.69 +/- 14.05	0.51 +/- 0.18
257.11 +/- 11.24	0.53 +/- 0.14
338.8 +/- 11.68	0.65 +/- 0.15
477.61 +/- 15.06	0.84 +/- 0.16
619.3 +/- 10.56	1.34 +/- 0.18
680.83 +/- 17.5	1.33 +/- 0.23
807.37 +/- 15.85	1.55 +/- 0.18
889.6 +/- 16.22	1.65 +/- 0.34
1051.33 +/- 26.74	1.92 +/- 0.26
1213.93 +/- 23.62	2.26 +/- 0.33
1389.45 +/- 24.75	2.53 +/- 0.31
1628.47 +/- 17.17	3.06 +/- 0.30
2084.6 +/- 33.77	3.95 +/- 0.42
2576.5 +/- 31.53	5.06 +/- 0.39
3058.20 +/- 35.62	5.87 +/- 0.54
3362.13 +/- 23.12	6.59 +/- 0.43
4067.2 +/- 45.37	8.18 +/- 0.66
4564.37 +/- 61.94	9.09 +/- 0.60
5511.6 +/- 64.60	10.86 +/- 0.85

Below is a plot of the data

Improved Measurement

Since the initial measurement of the dead time had a high error associated with the fluctuating of the count rate, a different measurement was made to try and minimized the error. This measurement was taken by using Ba-133 sources on detector A at the IAC. The source was placed at some position in front of the detector, then data was collected. For the next run the source was moved closer to the detector. A larger source was used as well in this manner to try and reach the higher dead times. The measurement gives the number of counts seen in some solid angle. I will compare this with theory by using the fact that the solid angle can be written as

[math] \Omega = \frac{Area}{r^2} [/math]

We can also calculate the theoretical activity of the source. Using this we can find a theoretical rate of photons incident in the solid angle, or

[math] R_{Theory} = A_{Theory} \times \Omega [/math]

Once the theoretical rate has been found, take a ratio between the measured rate and the theoretical rate to get the percent dead time, or

[math] Percent Dead = \frac{A_{Measured}}{A_{Theory}} [/math]

The source information was as follows:

The smaller source used was a 0.9671 uCi source with a reference date of January 1st, 2007 and serial number 1219-33-2

The larger source used was a 10.54 uCi source with a reference date of July 1, 2008 and serial number 129791.

Theoretical Activities

To find the theoretical activities of the sources, we must find the activity on the date that the measurement was taken. The half life of Ba-133 is 10.51 years, or 3836.15 days, which gives a decay constant of

[math] \lambda = \frac{\ln{2}}{t_{\frac{1}{2}}} = \frac{\ln{2}}{3836.15} = 1.80688237 \times 10^{-4} Days^{-1} [/math]

For the smaller source, the number of days between the date printed on the source and the date of measurement, which was December 5, 2017 is 3991 days. By using the decay equation, the activity on the date of the measurement is

[math] A_0 \times e^{-\lambda \times t} = 0.9671 \times e^{-\lambda \times 3991} = 0.4702 \mu Ci \rightarrow 17397 Hz [/math]

For the larger source, the number of days between the date printed on the source and the date of the measurement, which was December 5, 2017 is 3444 days. By using the decay equation, the activity of the source on the day of measurement is

[math] A_0 \times e^{-\lambda \times t} = 10.54 \times e^{-\lambda \times 3444} = 5.6570 \mu Ci \rightarrow 209307.65 Hz [/math]

Below is a table of information about the measurements that were made along with the theoretical rate within the solid angle. The Area of the detector face is 54.1061 [math] cm^2 [/math]

Run	Distance from Detector (cm)	Number of Histogram Entries	Runtime (seconds)	R_{Measure} (Hz)	Solid Angle (sterradians)	[math] R_{Theory} [/math]	[math] \frac{R_{Measured}}{R_{Theory}} [/math]
DetA_DeadTime_90cm	90	10036	365.632
DetA_DeadTime_85cm	85	7824	276.940
DetA_DeadTime_80cm	80	5040	168.989
DetA_DeadTime_75cm	75	5020	165.028
DetA_DeadTime_70cm	70	5037	157.465
DetA_DeadTime_65cm	65	5051	143.480
DetA_DeadTime_60cm	60	5035	135.545
DetA_DeadTime_55cm	55	5036	124.963
DetA_DeadTime_50cm	50	6955	163.809
DetA_DeadTime_45cm	45	5079	99.950
DetA_DeadTime_40cm	40	5048	91.293
DetA_DeadTime_35cm	35	5262	79.486
DetA_DeadTime_30cm	30	5078	61.545
DetA_DeadTime_25cm	25	5848	53.951
DetA_DeadTime_20cm	20	5380	35.239
DetA_DeadTime_15cm	15	6078	24.858
DetA_DeadTime_10cm	10	5380	11.665
DetA_DeadTime_30cm_LargerSource	30	32989	40.822
DetA_DeadTime_25cm_LargerSource	25	11424	9.895
DetA_DeadTime_20cm_LargerSource	20	60557	37.121
DetA_DeadTime_15cm_LargerSource	15	57329	22.402
DetA_DeadTime_10cm_LargerSource	10	48392	10.354