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)	[math] R_{Measure} (Hz) [/math]	Solid Angle (sterradians)	[math] R_{Theory} [/math]	[math] \frac{R_{Measured}}{R_{Theory}} [/math]
DetA_DeadTime_90cm	90	10036 +/- 100.180	365.632	27.448 +/- 0.274	0.00668	116.211	0.2362 +/- 0.0024
DetA_DeadTime_85cm	85	7824 +/- 88.453	276.940	28.252 +/- 0.319	0.00749	130.304	0.2168 +/- 0.0024
DetA_DeadTime_80cm	80	5040 +/- 70.993	168.989	29.824 +/- 0.420	0.00845	147.005	0.2029 +/- 0.0029
DetA_DeadTime_75cm	75	5020 +/- 70.196	165.028	30.419 +/- 0.425	0.00962	167.359	0.1818 +/- 0.0025
DetA_DeadTime_70cm	70	5037 +/- 70.972	157.465	31.988 +/- 0.451	0.01104	192.063	0.1665 +/- 0.0023
DetA_DeadTime_65cm	65	5051 +/- 71.070	143.480	35.204 +/- 0.495	0.01281	222.856	0.1580 +/- 0.0022
DetA_DeadTime_60cm	60	5035 +/- 70.958	135.545	37.146 +/- 0.524	0.01503	261.477	0.1421 +/- 0.0020
DetA_DeadTime_55cm	55	5036 +/- 70.965	124.963	40.300 +/- 0.568	0.01789	311.232	0.1295 +/- 0.0018
DetA_DeadTime_50cm	50	6955 +/- 83.397	163.809	42.458 +/- 0.509	0.02164	376.471	0.1128 +/- 0.0014
DetA_DeadTime_45cm	45	5079 +/- 71.267	99.950	50.815 +/- 0.713	0.02671	464.673	0.1094 +/- 0.0015
DetA_DeadTime_40cm	40	5048 +/- 71.049	91.293	55.294 +/- 0.781	0.03381	588.193	0.0940 +/- 0.0013
DetA_DeadTime_35cm	35	5262 +/- 72.540	79.486	66.200 +/- 0.913	0.04417	768.425	0.0862 +/- 0.0012
DetA_DeadTime_30cm	30	5078 +/- 71.601	61.545	82.509 +/- 1.163	0.06011	1045.734	0.0789 +/- 0.0011
DetA_DeadTime_25cm	25	5848 +/- 76.472	53.951	108.395 +/- 1.417	0.08657	1506.058	0.0720 +/- 0.0009
DetA_DeadTime_20cm	20	5380 +/- 73.348	35.239	152.672 +/- 2.081	0.13527	2353.292	0.0649 +/- 0.0013
DetA_DeadTime_15cm	15	6078 +/- 77.962	24.858	244.509 +/- 3.136	0.24047	4183.457	0.0584 +/- 0.0007
DetA_DeadTime_10cm	10	5380 +/- 73.348	11.665	461.209 +/- 6.288	0.54106	9412.821	0.0490 +/- 0.0007
DetA_DeadTime_30cm_LargerSource	30	32989 +/- 181.629	40.822	808.118 +/- 4.4493	0.06012	12583.5759	0.0642 +/- 0.0004
DetA_DeadTime_25cm_LargerSource	25	11424 +/- 106.883	9.895	1154.522 +/- 10.802	0.08657	18119.7633	0.0637 +/- 0.0006
DetA_DeadTime_20cm_LargerSource	20	60557 +/- 246.083	37.121	1631.341 +/- 6.629	0.13527	28313.0458	0.0576 +/- 0.0002
DetA_DeadTime_15cm_LargerSource	15	57329 +/- 239.434	22.402	2559.101 +/- 10.688	0.24047	50332.2106	0.0508 +/- 0.0002
DetA_DeadTime_10cm_LargerSource	10	48392 +/- 219.982	10.354	4673.749 +/- 21.246	0.54106	113247.997	0.0413 +/- 0.0002