Radioisotopes, Poisson Statistics and Half Lives
Abstract: Using a Geiger-Muller detector with a measured dead52
104
time of 284(26)ms, the half lives of the radioisotopes V and Rh
have been determined. From linear least-squares fits to the dead52
time and background corrected decay data, the half life of V was
105
determined to be 3.62(12)mins, and the two half-lives of Rh
were determined to be 4.3(3)mins and 45.7(1.8)s. The literature
values are 3.743mins, 4.43mins and 42.3s, respectively. Decay
data were also collected from the unknown sample “element X”
and its two half-lives were determined to be 186(3)s and 30.7(5)s.
0.6
Residual in ln (countrate)
ln (corrected countrate)
4
3.5
3
2.5
2
1.5
The decay data from element X were collected for 20 minutes,
and the raw counts corrected for the background and detector
deadtime. The time dependence of the corrected count rate is
shown in Fig. 4(a), and the log of the count rate in Fig. 4(b).
-1
100
200
300
400
500
600
700
800
0
900
100
200
300
400
500
600
700
800
time (s)
time (s)
Figure 1: (Left) Natural log of the corrected decay rate data for V52. The dashed line through the data point is
the best-fitting straight line from the least-squares fit. (Right) The residuals from the least-squares fit.
(a)
Corrected countrate vs time for Rh105
4
(b)
3
Ln (Corrected countrate) vs time for Rh105
(c)
5
(d)
Least-squares fit to high-t Rh105 data
2
From a non-weighted least-squares fit to the high-t (>300s) data
using LINEST (see Fig. 5(a)), the decay constant of the longer
lived element X isotope was determined to be 186(22)seconds.
The contribution of this longer-lived state to the low-t data was
determined and subtracted from the measured count rate. The
subsequent least-squares fit to the low-t (<120s) data is shown in
Fig. 5(b), from which the half-life of the shorter lived isotope
was determined to be 31(1)s.
Least-squares fit to low-t Rh105 data
4.5
1
50
40
30
20
2
0
-2
ln(Corrected Countrate/s)
60
0
-1
-2
-3
4
3.5
3
2.5
-4
The element X decay curve was also fitted using non-linear
fitting techniques. Using the SOLVER function in Excel, the full
decay curve was modeled as a sum of two exponential decays
-4
10
2
-5
-6
0
0
500
1000
0
1500
time (s)
500
1000
1.5
-6
300
1500
800
time (s)
time (s)
0
1300
50
time (s)
100
150
Figure 2: (a) The corrected count rate from Rh105, and (b) the natural log of the countrate. Graph (c) shows
the high-time data, and the dashed line through the data points is the best-fitting straight line from leastsquares fit. Graph (d) shows the least-squares fit to the low-time data after the contribution from the longerlived isotope has been subtracted.
Corrected Countrate/s
Corrected countrate/s
50
40
400
6
350
5
300
4
ln( Corrected Countrate/s)
Corrected countrate vs time in Rh104
showing the sum of the two decays
250
200
150
100
30
Rh104 (1+)
50
Rb104 (5+)
20
N = N1e -l t + N 2e -l t
1
2
with initial values for the decay constants taken from the linear
fits above. The best fitting decay constants were determined by
2
minimising c , and their uncertainties were determined by
2
2
finding the values where c increased by 1. Plots of c versus the
two decay constants in the vicinity of their best fitting values are
shown in Figure 6. The half lives were determined to be 124(10)s
and 24(3)s. The final fit to the total decay curve is shown in
Figure 7.
3
2
1
0
-1
0
-2
250
500
750
time (s)
1000
0
250
500
750
time (s)
1000
10
Figure 4: (a) The corrected count rate from element X, and
(b) the natural log of the countrate.
0
500
time (s)
1000
400
Corrected Counrate vs time in Element X
1500
350
6
4
98
98
ChiSquares vs Decay Constant 2
ChiSquares vs Decay Constant 1
y = 4518085.2x 2 - 50779.2x + 239.7
1
0
97.8
5
97.6
97.6
4.5
4
3.5
-1
-2
300
97.8
3
550
800
time (s)
1050
0
30
60
time (s)
90
120
Chi-Squared
2
97.4
97.4
97.2
97.2
97
97
96.8
0.005
0.0054
0.0058
Isotope 1 Decay Const (s^-1)
2
0.0062
300
y = 84811.0x 2 - 4905.5x + 168.0
5.5
Chi-Squared
ln( Corrected Countrate/s)
3
Figure 3: Corrected decay data for Rh105,
showing the contributions from the Rh105(1+)
and Rh105(5+) states. The calculated
contributions are determined from the
measured half-lives.
Corrected countrate (s^-1)
0
ln( Corrected Countrate/s)
From a non-weighted least-squares fit to the high-t (>300s) data
using LINEST (see Fig. 2(c)), the decay constant of the longer
lived Rh105(5+) state was determined to be 4.3(3)mins. The
contribution of this longer-lived state to the low-t data was
determined and subtracted from the measured countrate. The
subsequent least-squares fit to the low-t (<150s) data is shown in
Fig. 2(c), from which the half-life of the Rh105(1+) state was
determined to be 45.7(1.8)s.
0
-0.4
-0.8
0
0
Half-life of Rh105: The decay data from the Rh105 sample were
collected for 25minutes, and the raw counts corrected for the
background and detector deadtime. The time dependence of the
corrected count rate is shown in Fig. 2(a), and the log of the count
rate in Fig. 2(b).
0.2
-0.2
1
60
Half-life of V52: The decay data for the vanadium sample were
collected for 14 minutes, and the raw counts corrected for the
background and the dead-time of the detector. The resulting
corrected and linearised decay data are shown in Figure 1. A nonweighted least-squares fit to the data using LINEST gave a decay
constant of 0.00322(11)s-1, and hence a half-life of 3.62(12)mins.
0.4
-0.6
ln(Corrected Countrate/s)
Before measuring decay-data, the dead time of the Geiger-Muller
detector was determined. The background radiation was first
determined by counting for 20 consecutive 60s intervals and was
found to be 24.4(1.1)counts/min. Using a split calibration source
204
of Tl , the count rates for the two individual half sources were
221.7(1.4)counts/s and 215.1(1.3) counts/s, respectively, while
the count rate for the combined sources was 409.7(1.8) counts/s.
The resulting deadtime was determined to be 284(26)ms.
Residual versus time for Vanadium52
0.8
4.5
ln(Corrected Countrate/s)
where l is the "decay constant".
ln (corrected count rate) versus time for Vanadium52
5
Corrected Countrate/s
Radioactive decay is a random process governed by the laws of
quantum physics. If a sample contains a large number N0 of
unstable nuclei at time t=0, then the number remaining at some
later time t is given by:
N = N 0 e - lt
Half-life of Element X: Finally, the half-lives of the unknown
element X were determined. A trial run suggested that, as in Rh,
there were two half lives, and that after 20mins, the count rate
dropped to background.
1
5.5
96.8
0.025
250
200
Isotope 1
150
Isotope 2
100
50
0
0.027
0.029
0.031
Isotope 2 Decay Const (s^-1)
0.033
0
200
400
600
800
1000
1200
time (s)
Figure 5: (a) The high-time data for element X. The dashed Figure 6: Plots of c vesus the two decay constants in element Figure 7: Non-linear fit to the decay
line through the data points is the best-fitting straight line X. The lines through the data points are second-order data from element X showing the
from least-squares fit. (B) The least-squares fit to the low- polynomial fits, the equations of which are given on the graphs. contributions from the two isotopes.
time data after the contribution from the longer-lived isotope
has been subtracted.