statistical nature of radioactive decay

8/18/2019 Statistical nature of radioactive decay

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Statistic descrtiption of radioactive decays

Mostarac Deniz; 1349358

April 18, 2016

1 Condition for the validity of the Poisson dis-

tribution

The purpose of the first experiment is to understand how can one effectivelymodel the statistical nature of radioactive decay. We have performed a countof radiation using a Geiger Muller counter. The quantity that we measured wasbackground radiation. From preforming the measurement in equal time periodssome number of times we were able to assume a general behaviour. By calculat-ing the mean value and the standard deviation of our experimental results, wegot useful values which can then be compared to the values one would obtainwhen assuming a model.

There is a sensible model for this type of distribution. Here is how we esti-mated what model would make sense. Since nuclear radiation can be describedas random events, with no necessary correlation between one another, and thenature of the reaction is binary, one can conclude that the binomial distributionwould be a good starting point.

P ( p, k) =

n

k

pkq n−k

where the total number of steps, p denotes the probability for the event to oc-cur with a frequency k and q denotes the probability of a non-event with thefrequency n-k.

Under the assumption the probability for an event to happen is small, the onlysensible choice for obtaining a non 0 Poisson distribution would be that k needsto be small. We take the mean value of counts to be the probability for an event

to occur times the number of events, and the probability for a non-reaction to

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be q=1-p.

q n−

k = (1 − p)n−k

= (1 − p)1

p p(n−k)

lim p→0

(1 − p)1

p p(n−k)

≈ eµ

We can also consider that since n >> k:

n!

k!(n − k)! ≈ nk

With all of theese aproximations in mind one gets the following form, which is

coincidentally the expression for a Poisson distribution:

P (k) = µke−µ

k!

Since all of above used assumptions are sensible considerations regarding thegeneral knowledge about radioactive decay, it follows that the Poisson distribu-tion should theoretically be a good model for this reaction. Now, in order toexperimentally verify the validity of the model for describing radioactive decay,one can compare the experimentally calculated values with the ones assumedby the model.

We performed 50 count measurements in time intervals of 5 seconds, at a con-

stant device operating voltage of 450V. We have calculated the frequency of counts in our measured range and subsequently plotted that information alongwith the uncertainty in our calculation due the counting statistics. Then weused the Poisson distribution model to estimate the frequencies, along with pre-forming the error propagation that goes with such a model. All of the resultingdata is plotted on the Figure 1.The formula used for calculating the experimental standard deviation, which isthe square root of the experimental variance, is:

σ =

Σ|xi − x̄|2

N − 1

The formula for the standard deviation assumed by the model is:

σ =√

µ

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where we calculated the uncertanty in the scatter using the folowing expression:

∆σ = 1 2(n − 1)

The uncertanty in the mean value was calculated using the formula:

µ = σ√

n

Basically we have proved that out assumption of the Poisson distribution alongwith being theoretically a sensible choice, proves to be a good choice experi-mentally! I have previously stated that by comparing the calculated scatter of experimental data and the one assumed by the model one can see how closely

the model represents the real life results. In addition to the one can considerthe following. One can calculate the error in the mean experimentally. Sinceout model assumes the scatter the be the square root of the mean, it is sensibleto assume the error in the scatter to be of the order of the square of the un-certainty in the mean. And since the uncertainty in the mean decreases as theno of counts increases this describes the behaviour that we actually observe andthe uncertainties in the scatter calculated both ways match almost perfectly.This behaviour can be seen on Figure 1, along with the data beeing stated inthe Table 1 and Table 2.

Table 1: Frequency Distributions

In order to further our understanding of theoretical modelling of the statisticaldescription of radioactive decay we performed an additional experiment. The

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Table 2: Mean analysis

Figure 1: Plot of Frequency distributions

setup used was more adequate for accurate measurements and used specialisedsoftware for analysing and interpreting the data. The point of the experimentwas to demonstrate that in certain cases, aside for the Poisson distribution, theGaussian distribution is also a sound theoretical description. This statement isvalid in the cases that we have a large expectation value µ :

limµ→∞P (k) = µke−µ

k! ≈

1

√ 2πµe

−(x − µ)22µ

Firstly, we adjusted the time resolution of the detector, which resulted withhigher number of counts of events in spite of the fact that the number of mea-surements was always kept at 256(device pre-set). With the increase numberof counts, our mean value also increased. We preformed 4 measurements, in-

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creasing the time resolution of the device with each measurement. We imple-mented booth the Poisson distribution and the Gaussian distribution as theo-

retical models in order to determine theoretically the frequency distributions.As mathematically stated above, we observed that as the mean value of ourmeasurements increased, the Poisson predicted frequency distributions come into a better agreement with the Gaussian distribution.

The data we accumulated experimentally will not be stated here due to thesheer amount of it. What I will present is the graphical depictions the 4 differ-ent time resolutions and the resulting superposition of the Poisson and Gaussianpredicted frequency distributions along with the experimental frequencies.

One other important fact that we should mention is that addition of Poissondistributions is a Poisson distribution. The same goes for the Gaussian distri-bution. I state this because the data shows that it doesn’t matter if we performone measurement under same conditions several times, or just perform one mea-surement with a larger amount of data. In either case, the scatter turns out tobe exactly same.


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2 Statistical uncertanty of the net counting rate

The purpose of this experiment was to understand how do counting statisticswork, and how do they influence out ability to make a valid conclusion abouta measurement, especially in the case when we do not have an isolated sourceand are dealing with the superposition of the background radiation and the ra-diation of the source.

We have used a non-isolated Geiger counter to measure the count rates of aparticular source. The main focus of this experiment was tackling the problemof background radiation. Firstly what we needed to estimate is what would bethe sensible time intervals in measuring the background radiation and the source

radiation.The relation used for estimating the measurment time intervals was:ttot

tb=

rtot

rb

We concluded that it was optimal to use 100s for total count measurements, and70s for background count measurements. After performing the measurements

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for both background radiation and then the total radiation with the source, onecan estimate the ratio of the rates. After deciding what the sensible choice for

measurement times is, we perform the measurements several times with theirrespective time intervals and proceed to estimate their individual rates, simplyby subtraction.

From the counting statistics one can discern the error for each respective mea-surement and also the error in the mean calculated value for the cases of back-ground radiation, total count rate and form that evaluated source rate. Sincethe rates are functions of measured number of counts the necessary standard de-viations are obrained by standard error propagarion of measurments. In generalthere is no way to get a smaller uncertainty in the measurements that what isproposed by the counting statistics. What this means that the lower thresholdfor out measurement uncertainties can only be reproduced by counting statis-tics were all other errors estimated by the models can tend towards that limitbut not exceed it as demonstrated by the calculations performed. The obtaineddata alond with uncertanty calculations which are indicative of the statementsi have given can be seein in Tables 3, 4, and 5.

From that one can immediately see that our setup was not optimal for mea-suring the source radiation since the background radiation rate is larger. Thisalso leads to the observed the fact that in this case out experimentally calcu-lated standard deviations cannot be closely reproduced by the assumed modeldefinition of standard deviation.

For this experiment the error calculations, whether by model assumption orfor experimental data analysis, were of paramount importance. That’s why I

find it useful to explain the formulas I used for the analysis. It is important tounderstand how error propagations work, and how to obtain the relevant formsfor the functions one is trying to estimate the errors. Quantities that are of in-terest usually tend to be determined for various measurements, which are proneto random error. If the quantities are independent, and the desired quantity x isa function of x = x(a,b,c,...) then the independent contributions to the globalerror add in the following fashion:

δ xa = |∂x∂a

|δ awhere the contributions add in quadrature in the following fashion:

δ x = δ 2xa + δ 2xb + δ

2xc + ...

A prticullarly important case of this for this experiment is when x(a) = ka,where k is a constant:

δ x =

δ 2xa = kδ a

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Table 3: Background data

Table 4: Background+ Source data

Table 5: Source data

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Figure 6: Plot of Rates

3 Experimental verification of the interval den-

sity distribution

The goal of this experiment is to understand and observe how can one, under

the assumption of a Poisson distribution, obtain the frequency distribution of the time intervals between individual detector pulses (interval density distribu-tion).The Experimental setup that enabled us to achieve these measurements con-sisted of a switching unit, which sets the successive pulses alternately to theSTART and STOP channels. The time-to-amplitude converter (TAC) measuredthe time interval between two signals (START-STOP). The measured time dif-ference appears at the output an amplitude-proportional signal. By sorting theTAC output signals according to their amplitude via a multichannel analyser,we obtained the spectrum of the time intervals.The Poisson process is disturbed by any kind of event that does not occurpurely coincidentally. A common experimental practice is to use a pulse pro-cessing electronic module of the reducer. It produces a significant change in the

distribution of the pulse intervals. A dual reducer passes every second pulse of a pulse sequence, and so on for 4 and 8 fold reduction. This type of behaviouris exactly what we observed experimentally as we can see on Figures 8, 9, 10

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and 11. The formula used modelling the interval density distributions is:

I N (t)dt = (rt)N −1

e−rt

(N − 1)! rdt

We tried to describe the obtained data by fitting a form of the above statedfunction for various N until we get a fit. In order to be able to attempt that,it was necessity to get the data it form which has physical sense, because theformula in itself has no purpose in the context of number of count per channel of the device. It was necessary to obtain a relationship between counts per channeland counts per unit time. In that form it was possible to fit the data with aparticular form of the above stated function.

We used the data described by Figure 12, with the assumption that an equalamount of time passed between the peaks, to get a correlation of channels and

time. We observed 8 peaks, with some anomalous occurrences (we neglectedthose on principle). Just to clarify the logic, it the device was counting for adetermined amount of time, and we know the number of operational channels,even thou the ”filling” of the respective channels with signals is a random andprobabilistic process it is reasonable to assume that on average it would take anequal time to fill up each of the channels with the signals (basically total timeover the no of channels). Conversely it can be taught of as if the machine isdoing the following. It is opening each channel for a particular amount of time,after which it opens a new channel while closing the first one, and so on. Inprinciple it is the same logic which justifies the method we used to determine arelationship between the channels and time.(Table 6)

Table 6: Recalibration data

After assuming equidistant time intervals between the signals we proceeded toplot the number of counts in the respective channels against the time intervalsin order to obtain a linear dependence between the two. Obtaining the gradient

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and the y intercept of the linear fit gave us the parameters with which we cancorrelate the channels with a time interval(Figure 7).

Figure 7: Calibration plot

At this point we plotted the obtained, rescaled data for each of the cases of no, 2, 4, and 8 fold reduction, and attempted to fit a form of the above statedfunction(with the exception of fitting a function for 8 fold reduction).

With this we have experimentally demonstrated the formalism works in thesecases. An important quantity that we can also deduce form the function fittingis the actual rate of counts. We performed an initial count measurement in orderto understand what rate of counts to expect. The initially assessed count rateper second was 40 000. The rates obtained from the function fitting should besimilar to the expected rate in all cases (no, 2, and 4 fold reduction). We haveobtained data that verify our statement, which can be seen on Figures 8, 9, and

10. One should have in mind that the units on the figures, when transformed toseconds reproduce quite closely the expected rate of 40000 counts per second.

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As a final note, I would like to stress the fact that figure has the number of counts plotted against time, while the rest of figures wave their respective rates

plotted against time. I choose to do this just to demonstrate the fact that inthe context of the actual values we obtain form the function fitting, nothingchanges. The obtained rates form the function fitting is still as expected. Thiswas however not obvious to me, so I choose to leave one graph different in thatrespect.

Figure 8: No reduction

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Figure 9: 2 fold reduction


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Figure 12: Visualisation of the calibration data

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statistical nature of radioactive decay

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