nuclear decay and counting statistics: measure the half-life of a … · 2013-10-08 · nuclear...

Nuclear Decay and Counting Statistics:

Measure the Half-life of a Radioactive Nuclide with a Geiger-Müller Counter

Joshua Webster

Partners: Billy Day & Josh Kendrick

PHY 3802L

10/4/2013

Webster Lab 1: Geiger-Müller Counter

1

Abstract:

The purpose of this lab was to observe the nuclear decay of particles from known sources,

perform statistical calculations, and to finally observe the nuclear decay of an unknown

radioactive isotope in order to determine its decay constant, mean-life, and half-life within a

range of uncertainty. The results of this experiment conclude that the unknown radioisotope has

a decay constant of 4.77 ± 0.78 milliseconds-1

, a mean-life of 209.64 ± 34.19 seconds, and a half-

life of 145.31 ± 34.19 seconds.


2

Table of Contents Abstract: ........................................................................................................................................................ 1

Introduction ................................................................................................................................................... 4

Background ................................................................................................................................................... 5

Part 1: ........................................................................................................................................................ 5

Part 2: ........................................................................................................................................................ 5

Part 3: ........................................................................................................................................................ 8

Part 4: ........................................................................................................................................................ 9

Part 5: ........................................................................................................................................................ 9

Experimental Techniques ............................................................................................................................ 12

Diagrams and Images .............................................................................................................................. 12

Part 1: ...................................................................................................................................................... 13

Part 2: ...................................................................................................................................................... 13

Part 3: ...................................................................................................................................................... 13

Part 4: ...................................................................................................................................................... 13

Part 5: ...................................................................................................................................................... 14

Data ............................................................................................................................................................. 15

Part 1: ...................................................................................................................................................... 15

Part 2: ...................................................................................................................................................... 19

Part 3: ...................................................................................................................................................... 19

Part 4: ...................................................................................................................................................... 20

Part 5: ...................................................................................................................................................... 20

Analysis ...................................................................................................................................................... 25

Part 1: ...................................................................................................................................................... 25

Part 2: ...................................................................................................................................................... 25

Part 3: ...................................................................................................................................................... 27

Part 4: ...................................................................................................................................................... 28

Part 5: ...................................................................................................................................................... 28

Discussion ................................................................................................................................................... 32

Part 1: ...................................................................................................................................................... 32

Part 2: ...................................................................................................................................................... 32

Part 3: ...................................................................................................................................................... 32

Part 4: ...................................................................................................................................................... 32


3

Part 5: ...................................................................................................................................................... 32

Questions and Answers: .......................................................................................................................... 33

Conclusion .................................................................................................................................................. 35

Part 1: ...................................................................................................................................................... 35

Part 2: ...................................................................................................................................................... 35

Part 3: ...................................................................................................................................................... 35

Part 4: ...................................................................................................................................................... 35

Part 5: ...................................................................................................................................................... 35

Appendix ..................................................................................................................................................... 36

References ................................................................................................................................................... 38


4

Introduction

Radioactive is the word used when describing nuclear isotope that emits radiation in the

form of particles (alpha and beta), and/or electromagnetic waves (gamma rays). The reason for

the emission of these particles (and radioactivity in general) is due to the instability of an isotope.

Radioactive sources can be naturally occurring or man-made, with half-lives varying from less

than a second to billions of years. The disintegration of an atom or its radioactive decay can be

measured using a device popularly known as a Geiger counter. In the case of the experiments

performed in this report a Geiger-Müller counter was utilized.

The Geiger-Müller (GM) counter consists of a tube (cathode) with a wire running through

it (anode), and is filled with easily ionized gas (in our case halogen). The GM counter was

connected to a counting device that also supplied the voltage and to an oscilloscope. The

oscilloscope was used to visually observe the pulses during detection. The shape of the pulse

corresponds to the cascade of electrons during an “avalanche”. This “avalanche” occurs because

electrons traveling to the anode of the Geiger-Müller tube ionize other atoms, which in turn,

causes a voltage drop that the counter recognizes. This voltage drop is then represented on the

oscilloscope as a pulse, signifying the presence of a detected particle. The avalanche is stopped

by the halogen gas inside the Geiger-Müller tube that acts as a barrier, forcing positive ions to

transfer energy to the gas molecules. Some of the halogen gas molecules are broken down in

place of ionization. Any of these molecules that are accelerated toward the cathode quickly

disassociate and can no longer produce a signal. Put simply, the Geiger-Müller counter acts as a

detector, which sends a signal to the “counting device” which records the number of counts

during a set time interval. An oscilloscope can also be connected to the setup to display a graph

of voltage versus time, and indicate the detection of particles by a “blip” in the graph.

The purposes of the experiments conducted in this report were to achieve a general

knowledge of the procedures involved in detecting radioactive particles and to understand the

statistical nature of radioactive decay. The final experiment conducted was to determine the

decay constant of an unknown radioisotope. The ability to determine a radioisotope’s decay rate

is an important scientific concept that allows us to have a greater understanding for the world we

live in and the universe that surrounds us. The sections of the report that follow will be: The

Background, consisting of theoretical expectations and mathematical derivations; Experimental

Techniques, providing insight into the equipment and procedures used during experimentation

and related diagrams; Data, listing all the data gathered from experiments; Analysis, where

calculations of values will be shown in detail, as well as discussion of uncertainties or other

errors involved; Discussion, in which results will be compared to expected values; Conclusion,

summarizing the results.


5

Background

Part 1:

The first part of the experiment consisted of constructing a Geiger-Müller plateau graph

for our specific device. This will allow us to determine the optimum operating voltage. The

graph should begin from a zero value in counts to an abrupt jump in counts that begins to level

off, or plateau, before it eventually rises exponentially (the discharge region). After taking the

first measurements for the plateau, the slope of the plateau can be calculated. The value that

results from this calculation determines the Geiger-Müller devices reliability, and represents

percent per 100 V. If the value for the slope is greater than 10%, the accuracy of the GM tube is

compromised, and it should no longer be used. The equation for the slope is as follows,

( ) ( )

( )( ) ( )

For the equation above: N1 and N2 are the low end and high end counts respectively, and V1 and

V2 are the lowest and highest voltages.

To understand this equation better it breaks down to,

( )

( )

For the equation above: ∆counts represents the change in counts (N2-N1), ∆voltage represents the

change in voltage (V2-V1), and total counts represents the quantity (N2+N1). This equation is

similar to the standard slope formula, however, since the graph that we are finding the slope for

is not a straight line, we must use the change in counts divided by the change in voltage (instead

of ∆y/∆x), and then divide that by the sum of the high end and low end counts (to give you a

fractional percentage). The 2 multiplier term coupled with a 100 multiplier term normalizes the

equation to (% per) 100V, because in the experiment the voltage was reduced to about 200V

from the starting voltage. There is also a 100 multiplier to convert the fractional percentage to a

percent.

Part 2:

The next section of the experiment deals with finding the dead-time of the Geiger-Muller

counter. This is the amount of time that must pass after one decay is counted until a subsequent

decay can be counted. This is calculated using a split source. The first count is recorded, then

another split source is used and counts are recorded. Finally the sources are combined and the

combined counts recorded. The equation for dead-time, τ:

( )


6

In the above equation, N1 is the count rate of the first half source, N2 is the count rate of the

second half source, Nc is the combined source count rate, and T is the time during counting.

Adapting equation 2 for the dead-time uncertainty, we first take the natural log of both sides,

then take the derivative to obtain dτ, and using the fact that σN = √N (or dN = √N) we square both

sides.

( ) ( ) ( ) ( ) ( )

(

) (

) (

)

(

)

(

)

(

)

(

)

√ (

)

(

)

(

)

( )

Now calculating the dead-time uncertainty, we use Poisson uncertainty for the number of

counts. Equation 2.2 is for calculating the mean. Equation 2.3 is for calculating the standard

deviation. The formulae are as follows:

∑

( )

For the equation above: xi, the counts, are summed over the number of counts, N.

√ √ ( )

The standard deviation, σ, for a Poisson distribution is simply the square root of the mean value

found before.

The dead-time can be used to make corrections to the observed count rate. Finding and equation

to account for the true counting rate:

For a dead-time, τ, we need to define the following variables:

real-time or count time

true time or time detector can record counts


7

total counts

measured count rate, true count rate,

Taking the ratio of counts, we find:

⁄

⁄

We need an equation relating the dead-time. We can approximate the true time:

From the above equation: Cτ is the total time detector dead-time. Continuing, we divide the

equation for above by :

Finally, we relate the L.H.S. of the above equation to the R.H.S. of the equation for the ratio of

counts, and solve for R.

( )

For the equation above: is the true count rate, is the observed count rate, and τ is the dead-

time.

For the uncertainty in the true count rate we need to differentiate the above equation 3 with

respect to r, but since ⁄ is very small:

( ) ( )

Once the count rates are corrected, they should satisfy the following equation (within a margin of

error), where the c’s denote the corrected values:

( ) ( ) ( ) ( )


8

Part 3:

During the next section we conducted another counting experiment, and determined the

mean, standard deviation for a Poisson distribution, individual deviations, and the standard

deviation (square root of the sample variance). Rewriting eq. 2.2 from part 2 algebraically to find

the mean value:

( )

( )

For the equation above: is the mean, x1-xN are the counts, and N is the number of counts. The

true mean can only be determined by averaging an infinite number of measurements, but for our

case (a finite set of measurements) the sample mean is the best approximation.

For a Poisson distribution, the standard deviation is the square root of the mean (also

known as the R.M.S. or Root-Mean-Square). This R.M.S. value represents the magnitude of the

deviations of individual measurements, as shown in equation 2.3 above.

Also in the case of this section of the experiment, the best approximation for the standard

deviation is the unbiased sample variance. The equation for the standard deviation is:

√∑ ( )

( )

Which can expressed algebraically as,

√( ) ( ) ( )

( )

For the equation above: σx represents the standard deviation, is the arithmetic mean (as

calculated before in part 2), xi is the count, and N is the number of counts. This equation shows

that the standard deviation is the sums of the arithmetic mean minus the count squared and

divided by the number of counts minus one, all square rooted.

When only a single observation can be made, the mean can be approximated to be the

single measurement, x, in place of the true mean (m or as we have in this report). The

uncertainty then, is just σx = √x.

For a gross counting rate standard deviation is:

√

√

√

( )


9

For the equation above: σRg is the standard deviation of the gross counting rate, x is the counts,

Rg is the gross counting rate, and t is the time of the counting.

Part 4:

In this section, we conducted measurements of the background radiation over a period of

5 minutes with no source in the GM counter. This background radiation measurement plays an

integral part in the accuracy of all counts taken by a specific GM counter in a specific location.

The background readings are to be subtracted from all counts.

Part 5:

This part of the experiment deals with finding a decay constant of a rapidly decaying

unknown source. The decay constant will be determined by using the equations outlined in this

section.

The activity of a radioisotope, or rather the number of disintegrations per unit time, is determined

by the equation below.

( ) ( ) ( ) ( )

For the equation above: A(t) is the activity at time t, t2 and t1 are the beginning and end times (in

seconds) of the two activities taken into consideration, and λ is the decay constant (specific to a

certain radioisotope, representing the probability per unit time for a radioactive decay to occur).

Taking the natural logarithm of both sides of equation 8 we find:

[ ( )

( )] ( )

which can also be represented as,

[ ( )] ( ) [ ( )]

This is the equation that will be plotted later in this report, and from which the decay constant

can be easily identified. It shows that the decay constant, –λ, represents the slope of the equation.

Solving eq. 8 for λ, we find:

[ ( ) ( )

]

( )

Uncertainty in the decay constant can be found using equation A.3 from the appendix. Note: The

uncertainty in the time is small so it can be left out of the error propagation.


10

√(

)

(

)

(

)

( )

√(

( ))

( ) (

( ))

( )

√(

( )( ))

( ) (

( )( ))

( ) ( )

To find the weighted average decay constant we use the formula:

∑

∑

( )

The variables in this equation are defined for our purposes as (the decay constant) and

(σi is the uncertainty in the decay constant).

The mean life of the isotope can be found using the decay constant in the following equation,

where τ is the mean life:

( )

We can find the uncertainty in the mean-life by taking the ln of equation 9 and differentiating:

( )

The half-life of a radioisotope is the interval of time in which the activity decreases to one-half

its value in the beginning of the time interval.

( )

( )

( )

To calculate the uncertainty in the half-life we must find the derivative of equation 10. It is easy

to do if we take the natural log of both sides, and we can also negate the ln(2) term, because it is

small:


11

( ) (

)

( )

The relationship below determines the error bars for Graph 2 in Part 5 of the Data section.

σR is the uncertainty in the corrected counts, and R is the corrected count.

( ) ( )


12

Experimental Techniques

Diagrams and Images

Diagram 1: This diagram depicts the device setup for the series of experiments performed in this report.

The only difference between this setup and our setup is that our scaler, timer, high voltage supply (HV),

and voltmeter were all in one device called the counter. Our setup also included an oscilloscope and GM

tube.

Image 1: The picture below was taken in the lab, and shows our exact laboratory setup. The

digital oscilloscope (Tektronix TDS 2002B) is on the left, SPECTECH ST360 counter (high

voltage supply, timer, and scaler) is in the middle, and GM tube is on the right. What might not

be easily visible is the makeshift cardboard shelf that we had to construct for our GM tube, as it

was missing its originally included glass shelf.


13

Part 1:

A radioactive source was placed into the GM tube. The radioactive source used was

Cesium-137. The plateau curve for the specific GM tube used was found during this part of the

experiment. The voltage was raised on the counter/HV until counts were detected. The minimum

voltage required to detect counts was 760 V (starting voltage). The counter was set to a counting

time of 30 seconds, and the voltage was set to the starting voltage. The second shelf in the GM

tube was used, as it provided at least 1000 counts once the voltage was raised 50 V above the

starting voltage. Every 30 seconds of counting, the voltage and counts were recorded, and the

timer and counts were reset. The voltage was raised after each 30 second count time by 40 V. As

the voltage was increased, the count rate rose. The pulse height in the oscilloscope was around

3.6 V as depicted in Image 2 in the data section of this report. The highest voltage reached was

1200 V, and no exponential increase in counts was detected at this voltage. The data was then

plotted, and the slope of the plateau was determined. The optimum operating voltage was

determined by this experiment to be 1000 V.

Part 2:

A radioactive split source was used to determine the dead-time of the GM tube. The split

source used was Titanium-204. One half of the source was placed on the GM tube tray, and the

tray was placed on the second shelf under the GM tube. The HV was set to the determined

operating voltage of 1000 V. As a trial, the counter-timer was set to 60 seconds, and a count rate

higher than 20,000 was detected (which is desirable). An order of magnitude estimate on the

dead-time was made by observing the shortest time interval between two pulses on the

oscilloscope. The counter-timer was then set to 200 seconds, and the counts were recorded as N1.

Then, very carefully, the other half of the source was placed directly next to the piece already

counted. The timer was reset to 200 seconds, the count number was reset to 0, and the counting

for the combined pieces began. The count was then recorded as Nc. Finally, the first half counted

was then removed, leaving the final half to be counted. The timer was reset to 200 seconds, and

the count number was reset to 0. The count of the remaining half was recorded as N2.

Part 3:

Using Titanium-204, we adjusted the GM tube shelf height to produce about 2000 counts

per minute. The count-timer was set to 30 seconds, and the HV was set to the operating voltage

of 1000 V. A total of 10 separate counting intervals were made, and deemed “Set A”. Then,

another set of 10 separate counting intervals were made and deemed “Set B”. Statistical

calculations were then made using the equations found in the background section of this report.

Part 4:

Measurements were conducted in the section of the experiment to determine the

background radiation count. All radioactive sources were removed from the GM tube tray, the

tray was put back in place under the GM tube, the count-timer was set to 300 seconds (5

minutes), the HV was set to 1000 V, and the background radiation count was recorded.


14

Part 5:

An unknown radioactive sample was obtained from the instructor. The count-timer was

set to 30 seconds, HV supply was set to the operating voltage of 1000 V, the source was placed

under the GM tube on the second shelf, and a total of 20 separate counting intervals were

recorded with as little time between each counting interval as possible. Once the 30 seconds for a

specific counting interval was over, the counter-timer was immediately reset to begin counting

the next 30 second interval. The decay constant was then determined using the equations found

in the background section of this report, as well as by the slope of the logarithmic graph found in

the data section.


15

Data

Part 1:

Image 2: The picture below was taken in the lab, and shows the pulse shape as seen through the

oscilloscope for the Cesium-137 source used in part 1. This graph plots the voltage (2.00 V

increments) on the vertical axis, and time (500 μs increments) on the horizontal axis. Using the

graph, the voltage of the pulse is estimated to be at 3.6 V, and the pulse duration around 750

microseconds. Note: Channel 1 (CH1) is the channel that was being used. This image was taken

with a camera and digitally enhanced to improve contrast.


16

Table 1: The voltage and counts represented in this table are the data collected from this part of

the experiment using Cesium-137. They comprise the points on the Geiger plateau graphs

(graphs 1 & 2 below). The voltage uncertainty is an estimated uncertainty based on the accuracy

of the device. The uncertainty for the counts was calculated using standard deviation for the

individual counts.

Plateau Points

Time (s) Voltage (V) Counts Count Uncertainty Voltage Uncertainty

30 800 11839 108.81 0.5

30 840 27434 165.63 0.5

30 880 32567 180.46 0.5

30 920 36325 190.59 0.5

30 960 40821 202.04 0.5

30 1000 43263 208.00 0.5

30 1040 46373 215.34 0.5

30 1080 47155 217.15 0.5

30 1120 49152 221.70 0.5

30 1160 49601 222.71 0.5

30 1200 50955 225.73 0.5


17

Graph 1: This graph plots the plateau points given in Table 1 with counts as the vertical axis and

voltage in volts as the horizontal axis. The slope of the plateau part of this graph needs to be less

than 10% per 100 V for the GM tube to be considered reliable. This graph shows the fast jump in

counts after the first detection interval, and the plateau area which is consistent with theoretical

expectations. The vertical error bars were calculated using the uncertainty for a single

measurement, the horizontal error bars shown are ± 0.5 volts due to uncertainty in the accuracy

of the device. The manual for the HV supply (ST360) does not state the accuracy of the internal

voltmeter.

0

10000

20000

30000

40000

50000

60000

700 800 900 1000 1100 1200

Co

un

ts

Voltage (V)

Geiger Plateau Graph

Counts vs. Voltage


18

Graph 2: This is another graph for the Geiger plateau which was created with the program

LineFit using the same data, but fitting a line to the points considered to be the “plateau”

(starting at 960 V). Error bars are included. The equation of the fit line shown is y = 41.5x +

1.90.


19

Part 2:

The data shown in the section will be used to calculate the dead-time. The dead-time

uncertainty will also be calculated, as well as corrections to the observed count rates.

Table 2: This table lists the counts for the split source of Titanium-204. The first count, N1, is

for the first half. The second count, Nc, is for the two combined split sources. The third count,

N2, is for the second half. All counts were taken for a time of 200 seconds at the operating

voltage of 1000 V. The count rate is simply the counts for the trial divided by the time, and the

units are counts per second.

Split Source Titanium-204

Trial Time (s) Counts Count Rate

N1 200 70570 352.85

Nc 200 126998 634.99

N2 200 70903 354.52

Part 3:

In this section there were ten counts recorded for two sets (Set A and Set B), giving 20

separate data points as shown in Table 3 below. The source used was Titanium-204. The data

from the table is later used for statistical analysis.

Table 3: This table shows the data recorded after counting for 30 seconds a total of 10 times for

two separate sets (A and B). It also shows the individual count deviations from the mean, and

then shows that they sum to a value near zero.

Trial

Time

(s)

Set A

Counts

Set B

Counts

Set A Individual

Devs.

Set B Individual

Devs.

1 30 1117 1077 29 1

2 30 1061 1058 -27 -18

3 30 1080 1061 -8 -15

4 30 1053 1045 -35 -31

5 30 1044 1096 -44 20

6 30 1082 1038 -6 -38

7 30 1079 1063 -9 -13

8 30 1120 1140 32 64

9 30 1120 1101 32 25

10 30 1132 1077 44 1

Sums

8 -4


20

Part 4:

A background radiation count was determined during this section of the experiment. The

voltage used was the operating voltage of 1000 V. The background count data in the table below

is used to adjust the count numbers in other parts of the experiment.

Table 4: This table shows that after a 300 second counting interval and background count of 707

was detected.

Background Count Time (s)

707 300

Part 5:

A rapidly decaying unknown source was obtained. The counts in Table 5 were recorded

at 30 second time intervals at the operating voltage of 1000 V. The data in Table 5 is used to

determine the decay constant and associated uncertainty. Image 3 displays the pulses on the

oscilloscope during the counting.

Image 3: This image shows the pulses on the oscilloscope during this part of the lab. The large

amount of pulses displayed in short time intervals represents a high count rate. The operating

channel of the oscilloscope is channel one (CH1). Voltage (in 1 V increments) is along the

vertical axis, and time (1.00 ms increments) is along the horizontal axis. This image was taken

with a camera and digitally enhanced to improve contrast.


21

Table 5: Shown in this table are the counts recorded during 30 second counting time intervals.

Also shown are the corrected values accounting for the background count, and the dead-time.

The Corrected Counts column uses the dead-time equation and multiplies the result by the 30

second time interval. The Ln(Corrected Count Rate) column is for use in plotting the decay as a

linear function. The uncertainty for the count rate, σ(corrected count rate), was calculated using

equation 3.1, and the error for the log function was calculated using equation 11 in the

Background section. The time uncertainty was due to human error and was determined to be a

maximum of 2 seconds.

Unknown Source Data

Trial

Elapsed

Time (s) Counts

Count

Rate

Corrected

Count Rate

Corrected

Counts

Ln(Corrected

Count Rate)

σ(corrected

count rate) σln

σ(time)

(s)

1 30 11955 398.50 448.069 15442 6.10 25.50 0.06 2

2 60 10359 345.30 381.264 12854 5.94 22.94 0.06 2

3 90 9221 307.37 335.007 11128 5.81 21.12 0.06 2

4 120 8085 269.50 289.931 9493 5.67 19.31 0.07 2

5 150 7013 233.77 248.365 8027 5.51 17.59 0.07 2

6 180 6226 207.53 218.432 6994 5.39 16.31 0.07 2

7 210 5431 181.03 188.681 5987 5.24 14.98 0.08 2

8 240 4813 160.43 165.885 5227 5.11 13.93 0.08 2

9 270 4152 138.40 141.815 4436 4.95 12.77 0.09 2

10 300 3545 118.17 119.992 3729 4.79 11.65 0.10 2

11 330 3138 104.60 105.507 3265 4.66 10.88 0.10 2

12 360 2757 91.90 92.053 2837 4.52 10.12 0.11 2

13 390 2451 81.70 81.321 2498 4.40 9.48 0.12 2

14 420 2097 69.90 68.986 2112 4.23 8.71 0.13 2

15 450 1895 63.17 61.985 1893 4.13 8.25 0.13 2

16 480 1713 57.10 55.702 1698 4.02 7.81 0.14 2

17 510 1411 47.03 45.325 1378 3.81 7.05 0.16 2

18 540 1268 42.27 40.433 1227 3.70 6.66 0.16 2

19 570 1113 37.10 35.146 1065 3.56 6.22 0.18 2

20 600 980 32.67 30.621 927 3.42 5.83 0.19 2


22

Graph 3: The graph below plots corrected counts vs. time for the unknown radioactive isotope.

For the vertical axis the error bars were calculated using equation 3.1 modified as described in

the appendix (A.5), and for the horizontal axis the error bars are due to human error and were

estimated to be ±2 seconds.

0.000

50.000

100.000

150.000

200.000

250.000

300.000

350.000

400.000

450.000

500.000

0 60 120 180 240 300 360 420 480 540 600 660

Co

un

ts

Time (s)

Unknown Source Counts vs. Time Graph

Counts vs. Time Plot


23

Graph 4: This graph depicts the logarithmic of the counts of the unknown radioactive source vs.

time. Plotting logarithmically allows us to see the decay as a linear function. Using the natural

log of the count rates makes the slope of the fit line a good estimate of the decay constant (the

decay constant equals negative slope of the natural logarithmic function). There are vertical and

horizontal error bars, but the horizontal are very small (due to human error). The vertical error

bars are calculated using equation A.6 in the Appendix, and the values are shown in Table 5.

This linear fit accounts for x and y error, and the equation of the line is ( )

.


24

Table 6: This table represents the data analysis of the individual decay constants and

uncertainties associated. It also shows the values for the average decay constant and the average,

standard deviation, and shows the values used to calculate the weighted average. This weighted

average of the decay constant is another means of finding the decay constant value.

Trial

Elapsed Time

(s)

Decay

Const. λ(s-1

)

Uncertainty

σi (s-1

)

Dev. From

Mean λi/σi2

Weight

(1/σi2)

1 30 n/a n/a n/a n/a n/a

2 60 5.38E-03 2.76E-03 4.55E-07 7.06E+02 1.31E+05

3 90 4.31E-03 2.90E-03 1.57E-07 5.11E+02 1.19E+05

4 120 4.82E-03 3.06E-03 1.20E-08 5.15E+02 1.07E+05

5 150 5.16E-03 3.24E-03 2.03E-07 4.91E+02 9.52E+04

6 180 4.28E-03 3.43E-03 1.82E-07 3.64E+02 8.50E+04

7 210 4.88E-03 3.63E-03 2.99E-08 3.70E+02 7.58E+04

8 240 4.29E-03 3.85E-03 1.72E-07 2.89E+02 6.74E+04

9 270 5.23E-03 4.10E-03 2.68E-07 3.10E+02 5.94E+04

10 300 5.57E-03 4.41E-03 7.44E-07 2.86E+02 5.13E+04

11 330 4.29E-03 4.72E-03 1.76E-07 1.92E+02 4.49E+04

12 360 4.55E-03 5.02E-03 2.57E-08 1.80E+02 3.96E+04

13 390 4.13E-03 5.34E-03 3.31E-07 1.45E+02 3.51E+04

14 420 5.48E-03 5.73E-03 6.02E-07 1.67E+02 3.05E+04

15 450 3.57E-03 6.11E-03 1.30E-06 9.54E+01 2.68E+04

16 480 3.56E-03 6.44E-03 1.31E-06 8.58E+01 2.41E+04

17 510 6.87E-03 6.98E-03 4.68E-06 1.41E+02 2.05E+04

18 540 3.81E-03 7.55E-03 8.10E-07 6.67E+01 1.75E+04

19 570 4.67E-03 8.06E-03 1.29E-09 7.18E+01 1.54E+04

20 600 4.59E-03 8.66E-03 1.30E-08 6.12E+01 1.33E+04

AVG Decay

Const. (λ)

Std. Dev. Sum

Weighted

AVG λ

4.71E-03 7.77E-04 5.05E+03 4.77E-03


25

Analysis

Part 1:

Equation 1 is used to find the slope of the Geiger plateau found in Graph 1 using the data in Table

1. A slope of less than 10% signifies a GM tubes reliability. A GM tube with a slope greater than 10%

should not be used. A sample calculation is shown below.

( ) ( )

( )( ) ( )

( ) ( )

( )( )

Possible errors in this value can be attributed to the counts and dead-time, although dead-time

isn’t necessary to account for in this part of the experiment. Also, another error could also possibly be

associated with the HV. Since there was no separate voltmeter attached to the device, it could not be

verified that the HV supply was delivering every volt it stated it was. Furthermore, if the power outlet that

the HV supply was connected to had a drop in voltage due to environmental effects (improper wiring of

electrical outlets, or a bad circuit breaker) this could possibly affect the voltage delivered to the GM tube

resulting in systematic errors. It is not believed that this occurred during the experiment, but it also could

not be verified due to not having an available voltmeter to test/observe the voltage delivered by the outlet.

Doing this experiment in Part 1 and the other parts of this lab, we are working on the assumption that the

HV supply is able to maintain a constant voltage, and there are no internal faults in the device itself.

While these environmental errors are systematic (affecting all parts of this experiment) I will not re-iterate

those aforementioned.

Part 2:

This section of the analysis uses values from Table 2. Equation 2 is used to find the dead-time, τ,

of the GM tube. Equation 2.1 is also used to find the uncertainty associated with the dead-time, στ.

( )

( )( ) ( ) (

)

Now calculating the dead-time uncertainty using Poisson uncertainty for the numbers of counts,

we will use equation 2.1.


26

√ (

)

(

)

(

)

( )

To make things a bit easier, calculate the common fractional term ahead of time:

( )√ (

)

(

)

(

)

( )√( ) ( ) ( )

Therefore,

To make a dead-time correction to count N1 using equation 3, the count rates need to be used

(shown in Table 2). The value then obtained will be the corrected count rate. This value then needs to be

multiplied by the count time, 200 seconds, to obtain the value for the corrected count.

( )

( )( )

For the equation below: N1c represents the corrected count value for N1.

(

) ( )

To calculate the error propagation we can use equation 3.1, noting that σr is just the √r:

( ) ( )


27

( ( )( )

Using the same steps as before we obtain the values, and . These

values can now be checked by ensuring that they satisfy equation 4.

( ) ( ) ( ) ( )

This result is acceptable because it is within the uncertainty of counts. Although it is

insignificant, it should also be stated that the counter (ST360) itself has a resolving time (dead-

time). From the ST360 manual, “The resolving time of the ST360 RADIATION COUNTER is

very short and is not a significant factor compared to that of the GM tube.”1

Part 3:

Using the data from Table 3, equation 5 is used to determine the arithmetic average, and

equation 2.3 is used to determine the standard deviation of the mean. A sample set of

calculations for Set A of the average and standard deviation is shown below.

( )

( )

For Set A:

(

)

√ √ ( )

√

Now using equation 6.1, we find the standard deviation by taking the square root of the sample

variance.

1 (Spectrum Techniques, 2002)


28

√( ) ( ) ( )

( )

For Set A:

√( ) ( ) ( ) ( )

7/10 counts in Set A fall within

For Set B: , , and .

7/10 counts in Set B fall within

The two mean values are fairly close with a difference of 12 between the two sets. The

values for sigma from set to set are very close. The sums of the individual deviations sum to

nearly zero as shown in Table 3.

Part 4:

The background radiation was measured in this part of the experiment. The background

radiation value recorded (707 counts in 300 seconds) seems a bit high. Our GM tube was located

near the box containing other radioactive sources, so this may have been the cause of the high

background count. Also, the GM tube was missing its glass shelf, so a shelf had to be constructed

out of cardboard. This cardboard could have added a minor effect to the background radiation.

Calculating the uncertainty associated with this single measurement:

√

Part 5:

Table 5 shows the corrected count values after doing the dead-time calculation in Part 3

and accounting for the background radiation. Accounting for the background radiation and dead-

time is an important step when accuracy is required; such is the case when determining the decay

constant of an unknown radioactive isotope. Though it is basic algebra, for thoroughness, sample

calculations to obtain one of the corrected counts is shown below.

Using Trial 10, which has 3545 counts, the count rate will be calculated, the background

radiation will be taken into account, and the dead-time correction will be applied. To calculate

the count rate, simply divide the counts by the time interval they were detected in.


29

Accounting for background radiation we need to use the value obtained in Part 4, which

was 707 counts in 300 seconds. Since the counts were detected over a 300 second time interval,

we divide 707 counts by 300 to obtain the count rate.

The background count rate must then be subtracted from the count rate of the sample to obtain

the background corrected count rate for Trial 10.

Now the count rate needs to account for dead-time, so we apply equation 3.

( )

( )( )

To find the corrected count we simply multiply this (background and dead-time) corrected count

rate by the 30 second time interval.

(

) ( )

Using equation 3.1 we can account for the propagation of errors:

( ) ( )

( ( )( )

To determine the decay constant we use eq. 8.1. We can then find the mean-life using eq.

9. With the mean-life determined, we simply plug that value into equation 10 and solve for the

half-life. Finding the decay constant for the first two activity values in Table 5:

[ ( ) ( )

]

( )


30

[ ]

Uncertainty in the decay constant can now be found using equation 8.2, noting that the

uncertainties in the activities at time 2 and time 1 were previously calculated and shown in Table

5:

√(

( )( ))

( ) (

( )( ))

( ) ( )

√(

( ))

( ) (

( ))

( )

Using the weighted average for the decay constant we found using the values tabulated in

Table 6, we can find the mean-life by inserting this value into equation 9. The mean life can also

be calculated using each individual decay constant or the average decay constant, but in theory

the more reliable value would be the weighted average decay constant, so we shall use it here.

( )

The uncertainty in the mean-life can be determined by using equation 9.1:

( )

( )( )

We can now solve the final equation (10) to find the half-life of the unknown radioactive isotope:

( )

( )

( )


31

( ) ( )

( )

The uncertainty in the half-life is found using equation 10.1. The value used for σλ is the standard

deviation of the individual values, as calculated in Table 6.

( )( )

The decay constant can also be estimated by taking the negative value of the slope of the

logarithmic graph, which was determined to be 4.71 ± 0.14 ms-1

. The error on this value is given

by the program LineFit as part of the equation of the line. This value is fairly close to the value

arrived at by using the weighted average decay constant.

Human errors also come in to effect with Part 5. The errors can be attributed to the fact

that a human cannot press the button on the count-timer (in order to reset the counts and begin

counting again) at the exact moment the prior 30 second interval has finished. This error shows

up in the counts, which in turn affects the count rate. These effects are more pronounced when

viewing the graph, resulting in a slightly imperfect linear graph (when in theory it should be

perfectly linear).


32

Discussion

Part 1:

The results for the Geiger plateau are considered reliable. The slope was determined to be

9.20% and the accepted value for reliability is any value under 10%.

Part 2:

The value obtained for the dead-time of the Geiger-Müller tube, 292.29 ± 9.29 μs, seems

to be an acceptable value. A reference to known or acceptable dead-times was not found, so a

comparison could not be made. After comparing this result with colleagues, it was found to be an

accepted value.

Part 3:

The statistical calculations performed in Part 3 produce anticipated results. The two mean

values are fairly close with a difference of 12 between the two sets. The values for sigma from

set to set are very close. The sums of the individual deviations sum to nearly zero as shown in

Table 3.

Part 4:

At first glance 707 counts seems like an above average background radiation count. Upon

further inspection, it was realized that our GM tube was located near the box that contained all of

the radioactive sources. If repeated, it would be preferable to keep the GM away from any

coincidental radioactive sources. A smaller background count could result in fewer errors due to

a fluctuating background count. Furthermore, repeated background counts allowing for the

statistical treatment of the background would result in an even greater accuracy.

Part 5:

The values obtained for the decay constant, mean-life, half-life, and associated

uncertainties are listed in a table in the conclusion for easy comparisons. The half-life seems to

be fairly short. It is definitely within the broadly acceptable values of half-lives.


33

Questions and Answers:

What are physical phenomena that are initiated by the passage of an ionizing particle in the GM

tube?

The Townsend avalanche (or gas multiplication) is a physical phenomenon initiated by

the passage of an ionizing particle in the GM tube. This “avalanche” occurs because electrons

traveling to the anode of the Geiger-Müller counter ionize other atoms, which in turn, causes a

voltage drop that the counter recognizes. This voltage drop is then represented on the

oscilloscope as a pulse, signifying the presence of a detected particle. The avalanche is stopped

by the halogen gas inside the Geiger-Müller counter that acts as a barrier forcing positive ions to

transfer energy to the gas molecules. Some of the halogen gas molecules are broken down in

place of ionization. Any of these molecules that are accelerated toward the cathode quickly

disassociate and can no longer produce a signal.

Coincidence is another physical phenomenon. A decaying particle or a ray can produce

an ion pair by ionization inside the GM tube. Electrons arrive at the anode faster than the positive

ions arrive at the cathode. While the positive ions are still traveling to the cathode there is a short

time interval in which no other radiation can be detected. The other “coincidental” radiation

produces another Townsend avalanche that effectively adds to the one already present. The GM

tube then sees this as one large avalanche.

What makes it possible for the GM tube to also count X-rays and γ-rays (photons)?

An absorber can be used to shield the tube from incident X-rays. Taking into account

counts recorded for a source with no absorber present, and counts with an absorber present the

X-rays can be differentiated from the γ-rays.

What is the size (in V) of the smallest signal from the GM tube that is big enough to trigger the

counter circuit?

From the oscilloscope it seemed that 1 V was large enough to trigger the counter. I am

not entirely certain; it could be that less than 1 V could trigger the counter. The starting voltage

on the counter however, was around 760 Volts to produce any counts.

What is the smallest time between two successive GM tube output signals that you can see on the

oscilloscope?

From close examination of the pulses on the oscilloscope, the smallest time interval

between two pulses was recorded to be around 50 μs. Unfortunately an image was not taken of

this. It can be seen in Image 3 that each block has a time interval of 1.00 ms. Some of the pulses

in the image seem to overlap. From this we can determine that the time between two successive

pulses must be much less than 1.00 ms.


34

Given the dead-time determined by you, what is the counting rate at which the error due to the

finite dead-time is 1% of the rate?

as calculated in the Appendix section (A.3)

When taking the counts for the decay constant measurement, why is it important that the time

between two successive measurement periods be minimized? If you had appreciable delays

between successive counting periods (without correcting for this), would that cause you to over-

or underestimate the half-life? (Explain)

When trying to determine the decay constant of a radioactive isotope it is important to

minimize the time between counts. This is because the isotope is rapidly decaying, and any

added time between two counts produces an error if it is not accounted for and corrected. This

added time would result in a lower count, because the count is continually decreasing. Without

correction, the original total time (t2-t1) would still be used. Solving the equations outlined in part

5 using a lower count rate along with the same time interval would result in the decay constant

being underestimated.


35

Conclusion

Part 1:

The slope was determined to be 9.20% per 100V, which is an acceptable value. It isn’t a

great value, but it is under the 10% per 100V which is necessary for accuracy.

Part 2:

The dead-time of the device was determined to be 289 ± 9.29 μs. Possibly more advanced

or specialized equipment would improve this dead-time.

Part 3:

For Set A, the mean was determined to be 1088, the standard deviation was 32.98, the

root-mean-square deviation was 31.44. For Set B, the mean was determined to be 1076, the

standard deviation was 32.80, and the root-mean-square deviation was 30.27.

Part 4:

Background radiation was determined to be 707 ± 26.6 counts over a period of 300

seconds. A longer background count time and/or more individual background counts would yield

more accurate results for use in Part 5.

Part 5:

The values obtained for the decay constant, mean-life, half-life, and associated

uncertainties are listed in the table below.

Decay Constant

λ (s-1

)

Mean-life

τ (s)

Half-life

T1/2 (s) σλ (s-1

) στ (s) σT1/2 (s)

Average 4.71E-03 212.31 147.17 7.78E-04 35.07 16.85

Weighted Average 4.77E-03 209.64 145.31 7.78E-04 34.19 16.43

From Graph 4.72E-03 211.86 146.85 1.32E-04 5.93 2.85

Equipment that could record counts at set intervals and reset automatically would greatly

improve the results in this part of the experiment.


36

Appendix

A.2 Graph: This is another graph of the natural logarithm of count rates. Made in Excel, it

shows a slightly different perspective. Horizontal and vertical error bars are shown. Vertical

errors calculated by the standard deviation for each individual count. The plot points are fit with

a line. This linear fit does not take the error bars into account like the Graph 4 does. The equation

of the line is also shown, with the coefficient of determination. We can see that without taking

the error bars into account the value of the slope changes slightly.

A.3 Propagation of errors equation:

For an equation, f, that is dependent on the variables x, y, …, n the equation below is used for

finding the uncertainty in f that has propagated from the other variables uncertainties (σ).

√(

)

(

)

(

)

( )

y = -4.691E-03x + 6.227E+00 R² = 9.996E-01

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

0 100 200 300 400 500 600 700

Log(

cou

nt)

Time (s)

Logarithm of Counts Graph

Series1

Linear (Series1)


37

A.4 Rate at which the error due to dead time effects (i.e. the difference between raw and

true counting rate) is 1%.

Using equation 3:

( )

We want to satisfy the following relation:

Solving for :

( )

Now plug this value back into equation 3:

( )

( ) ( )

We know τ, so we can solve for R:

A.5 Calculating the error bars for Graph 3:

In order to calculate the error bars for Graph 3, we must use a modified version of equation 3.1

that accounts for the uncertainty in the count being the Poisson distribution for an individual

count. That is, σr is just √r.

( ) ( )

A.6 Calculating error bars for Graph 4 (logarithmic decay graph):

To calculate the errors associated with the logarithmic values the following equation is used:

( )

Using the first trial values from Table 5:

( )


38

References

Lindberg, V. (2000, July 1). Uncertainties and Error Propagation. Retrieved September 19,

2013, from rit.edu: http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart2.html

Spectrum Techniques. (2002, June). Instrument Manuals. Retrieved September 19, 2013, from

SPECTECH: http://spectrumtechniques.com/manuals/ST360manual.pdf

Spectrum Techniques. (n.d.). Lab Manuals. Retrieved September 19, 2013, from SPECTECH:

http://spectrumtechniques.com/Lab_Manuals/Studentmanual.pdf

Stuve, E. M. (2004, December). Estimating and Plotting Logarithmic Error Bars. Retrieved

September 19, 2013, from University of Washington:

http://faculty.washington.edu/stuve/uwess/log_error.pdf

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