PC1144 Physics IV

Radioactivity

1 Purpose

• Investigate the analogy between the decay of dice “nuclei” and radioactive nuclei.

• Determine experimental and theoretical values of the decay constant λ and the half-lifefor the dice “nuclei”.

• Measure the count rate versus voltage for a Geiger counter to determine its appropriateoperating voltage.

• Investigate how well the observed distributions of counts compares to that predicted bythe normal distribution.

2 Equipment

• 200 dice to simulate radioactive nuclei

• Geiger counter

• 210Po α radiation source

• 90Sr β radiation source

3 Theory

A basic concept of radioactive decay is that the probability of decay for each radioactivenucleus is constant. In other words, there are a predictable number of decays per second eventhough it is not possible to predict which nuclei among the sample will decay. A quantitycalled the decay constant λ characterizes this concept. It is the probability of decay per unittime for one radioactive nucleus. Because λ is constant, it is possible to predict the rate ofdecay for a radioactive sample. The value of the constant λ is different for each radioactivenucleus.

Consider a sample of N radioactive nuclei with a decay constant λ. The rate of decay ofthese nuclei dN/dt is related to λ and N by the equation

dN

dt= −λN (1)

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The negative sign in the equation means that dN/dt is negative because the number of ra-dioactive nuclei is decreasing. The number of radioactive nuclei at t = 0 is designated as N0.At some later time t, the number of radioactive nuclei that are still there N is given by

N = N0 e−λt (2)

Equation (2) states that the number of radioactive nuclei N at some later time t decreasesexponentially from the original number N0 that originally present. The quantity λN is thenumber of decays per second and it is called the activity A of the sample. It can be shownthat it obeys the equation

A = A0 e−λt (3)

Equations (2) and (3) state that both N and A decay exponentially with the same exponentialfactor. For measurements made on real radioactive nuclei, the activity A is usually all thatcan be measured.

The time it takes for N0 to be reduced to N0/2 and the time it takes for A0 to be reducedto A0/2 are the same. It is called the half-life T1/2 of the decay. It is related to λ by

T1/2 =ln(2)

λ=

0.693

λ(4)

Figure 1: Diagram of the essential elements of a Geiger tube.

To study these radioactive processes, we must detect the presence of these particles thatare the product of the decay. We can build devices in many form to accomplish the detection,but there all would have one feature in common. Every practical device that detects radiationallows the particles to interact with matter and then uses that interaction as basis for detection.The particular device we will use is called a Geiger counter. It consists of a tube in which theincident particle interacts and a scaling circuit to count the pulses of electricity produced. Adiagram of a Geiger tube is shown in Figure1.

The Geiger tube is a small metal cylinder with a thin self-supporting wire along the axisof the cylinder. The wire is insulated from the cylinder. The cylindrical wall of the tubeserves as the negative electrode (cathode) and the wire along the axis is the positive electrode(anode). At the entrance end of the tube, there is a thin “window” formed by a very thin piece

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of fragile mica. Inside the counter is a special gas mixture that is ionized by any radiationthat penetrates the “window”.

In operation, a voltage is applied across the electrodes. The particular voltage for eachtube must be determined experimentally. The applied voltage creates a large electric field inthe tube and the field is especially large in the region near the central wire. When radiationpasses through the window and ionizes the gas, the large electric field causes an accelerationof the free electrons. These accelerated electrons cause additional ionizations that creates an“avalanche” effect. The total number of ion-electron pairs created by a single incident particleis of the order of one million.

The electrons are move mobile and drift toward the positive central wire. When theyarrive at the wire, their negative charge causes the voltage of the wire to be lowered andthis sudden drop in voltage creates a pulse that is counted by the electronic circuitry. Eachpulse counted signifies the passage of a particle through the counter. The ions then recombinewith electrons, leaving the gas neutral again and ready for the passage of another particle.The whole process takes a time of the order of 300µmathrms and during that time period ifanother particle goes through the counter, it may not be counted. Thus, one disadvantage ofGeiger counter is this “dead time” during which counts may be missed. This is a negligibleeffect unless the count rate is very high.

The count rate of a Geiger counter is a function of the voltage applied across the electrodes.Therefore, the counter should be operated in a region where the rate at which the count ratechanges with voltage is a minimum. This is accomplished experimentally by measuring thecount rate of some fixed source of radiation as a function of the voltage across applied tothe tube. A graph of the count rate versus the voltage will be made from the data and theoperating point will be chosen to be some voltage where the count rate versus voltage curveis nearly flat as possible. This is referred to as the plateau region.

Figure 2: Typical count rate versus voltage for a Geiger tube.

The counter needs a minimum voltage to produce pulses at all. Both this minimum voltageand the operating voltage are quite variable and depend upon the dimensions of the Geigertube and on the particular gas used in the tube. Therefore, the exact nature of the count rateversus voltage curve depends on the particular tube used. A typical count rate versus voltage

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curve for a Geiger tube is shown in Figure 2.

If all other sources of error are removed from a nuclear counting experiment, there remainsan uncertainty due to the random nature of the nuclear decay process. It is assumed thatthere exists some true mean value of the count which shall be designated as m. However, we,emphasize, do not assume that there is a true value for any individual count Ci. Although m isassumed to exist, it can never be known exactly. Instead, one can approach knowledge of thetrue mean m by a large number of observations. It can be shown that the best approximationto the true mean m is the mean C which is given by

C =1

n

n∑i=1

Ci (5)

where Ci stands for the ith value of the count obtained in n trials. The standard deviationfrom the mean σ and the standard error σC are defined in the usual manner as

σ =

√√√√ 1

n− 1

n∑i=1

(C − Ci

)2and σC =

σ√n

(6)

The way in which the measurements Ci are distributed around the mean C depends uponthe statistical distribution. The binomial distribution is the fundamental law for statistics ofall random events including radiative decay. Calculations are difficult with this distributionand it is often approximated by another integral distribution called the Poisson distribution.For cases of m greater than 20, both the binomial and the Poisson distributions can beapproximated by the Gaussian distribution. It has the advantage that it deals with continuousvariables and thus calculations are much easier with the Gaussian distribution. For mostnuclear counting problems of interest, the Gaussian distribution predicts the same results fornuclear counting that have been assumed for measurements in general. Approximately 68.3%of the measured values of Ci should fall within C±σ and approximately 95.5% of the measuredvalues of Ci should fall within C ± 2σ.

There is one statistical idea valid for nuclear counting experiments that is not true formeasurements in general. For any given single measurement of the count C in a nuclearcounting experiment, an approximation to the standard deviation from the mean σ is givenby

σ ≈√C (7)

For a series of repeated trials of a given count, the most accurate determination is given byC ± σC . If only a single measurement of the count is made, the most accurate statement thatcan be made is given by C ±

√C.

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4 Experimental Procedure

Part I: Simulated Radioactive Decay

P1. Place all 200 of the dice in the box provided. Place the cover over the dice and shakethem vigorously. Remove all dice that come to rest with ONE point face up. Removeonly those that have ONE point that points directly upward. Record in Data Table 1the number of dice that decay (are removed) on the first throw of the dice. Also recordthe number of dice that remain after the ones that decay are removed.

P2. Place the cover on the dice and shake the remaining dice vigorously. Remove the dicethat come to rest with ONE point face up on the second throw. Record in Data Table 1the number of dice removed on the second throw and also record the number of dicethat remain after the ones that decay are removed.

P3. Continue this process of shaking the dice, removing the ones that have ONE point faceup, and recording the number of dice removed and the number of dice left for eachthrow. Continue this procedure for a total of 20 throws of the dice; or until all of thedice have been removed.

P4. Each group should record its data on the whiteboard so that the session results can beplotted as a set of data with better statistics. Sum the total number of dice thrownoriginally for the entire session and sum the number removed at each throw of the dice.Record this session data in Data Table 1.

Part II: Characteristics Curve of the Geiger Counter

P1. All Geiger tubes have a maximum permissible operating voltage above which they aresubject to breakdown. This may permanently damage the tube. Check with yourdemonstrator for the specific maximum voltage of the particular Geiger tube used. DONOT ever exceed this maximum voltage. Before plugging in the power cord of yourinstrument, make sure that the high voltage control is turned to the minimum setting.

P2. Place the 210Po α source as close as possible to the window of the Geiger tube. Oncethe source is positioned, take all the measurements for this procedure without changingthe position of the source relative to the detector.

P3. Turn on the power to the instrument, reset the counter to zero and start the counter ina continuous count mode with the high voltage still set to a minimum. Slowly increasethe high voltage setting until the counts begin to register on the counter. Leave thevoltage at this setting for which the Geiger tube just begins to count. This is called thethreshold voltage.

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P4. Reset the counter to zero, start the counter and let it count for 1 minute. Repeat thisprocedure two more times for a total of three 1-minute counts at this voltage setting.Record the voltage and the three values of the count in Data Table 2.

P5. Raise the high voltage setting by 25 V, repeat step P4 at this new high voltage settingand record the high voltage and the counts for these three trials.

P6. Continue this process up to the maximum permissible voltage of the Geiger tube. Be sureto obtain the proper maximum voltage for your tube from your laboratory demonstrator.

Part III: Counting Statistics

P1. Set the Geiger counter to the proper operating voltage. Place the 90Sr β source about 2or 3 cm from the window of the Geiger tube. Once the source is positioned, take all themeasurements for this procedure without changing the position of the source relative tothe detector.

P2. Reset the counter to zero, start the counter and let it count for 30 seconds. Record thevalue of the count in Data Table 3.

P3. Repeat the count for a total of 50 trials. Make no changes whatsoever in the experimentalarrangement for these 50 trials. Record each count in the Data Table 3.

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5 Data Processing

Part I: Simulated Radioactive Decay

D1. For each of the 20 shakes of the dice (group data), calculate the ratio of the number ofdice removed after a given thrown to the number shaken for that thrown.

Hint: Note carefully that this ratio must be calculated with data from two differentrows in Data Table 1. For example, the number of dice thrown on the fourth throw islisted as the number of dice remaining after the third throw. Thus the ratio is calculatedwith the number removed on each row to the number remaining in the preceding row.

D2. Determine your best experimental value for decay constant λ with the correspondinguncertainty.

D3. Use percentage discrepancy to compare your experimental value for decay constant λwith the theoretical value.

Hint: The percentage discrepancy is defined as

Percentage discrepancy =|Experimental value− Theoretical value|

Theoretical value× 100%

D4. Perform a suitable linear least squares fit to your data (N,#throw) in Data Table 1(session data). Determine the slope and intercept with the corresponding uncertaintiesof the least squares fit to the data.

D5. Plot suitable graph as well as the best fitted line obtained above.

D6. Based on your results from the least squares fit above, determine the experimental valueof half-life T1/2 (number of throws) with the corresponding uncertainty. Use percentagediscrepancy to compare your experimental value of T1/2 with the theoretical value.

Hint: The experimental half-life is the number of throws needed to go from any pointon the line to one-half that value.

Part II: Characteristics Curve of the Geiger Counter

D1. Calculate the mean for the three trials of the count at each voltage.

D2. Plot the mean count rate (counts/min) versus voltage to produce graph like the oneshown in Figure 2.

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Part III: Counting Statistics

D1. Calculate the mean count C, the standard deviation from the mean σ and the standarderror σC for the 50 trials of the count.

D2. For each count Ci, calculate |Ci − C|/σ.

D3. Determine what percentage of the counts Ci are further from C than σ by counting thenumber of times a value of |Ci − C|/σ > 1 occurs. Express this number divided by 50at a percentage.

D4. Count the number of times that |Ci − C|/σ > 2 occurs. Express this number dividedby 50 as a percentage.

D5. Calculate√C.

6 Questions

Q1. For the simulation of a radioactive decay using dice, what are the analogous quantitiesto the real quantities listed below – undecayed nucleus, decayed nucleus, time, decayconstant?

Q2. What quantity can be measured for the simulation laboratory that cannot normally bedirectly measured in a true radioactive decay laboratory?

Q3. Compare the percentage of trials that have |Ci − C|/σ > 1 with that predicted by thenormal distribution. Compare the percentage of trials that have |Ci − C|/σ > 2 withthat predicted by the normal distribution.

Q4. Calculate the percentage difference between√C and σC . Do the results confirm the

expectations of equation (7)?

Hint: The percentage difference is defined as

Percentage difference =|E1 − E2|

(|E1|+ |E2|)/2× 100%

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