PHYS272_Lab11

=Physics for Scientists and Engineers - Lab 11 PreLab =  Assignment: Read Chapter 7 of the book Experimentation: An Introduction to Measurement Theory and Experiment Design. Chapter 7 details how to write a clear scientific report. Examine one of the lab writeups for this semester and answer the following questions. This assignment is due at the start of the laboratory period.

= = =Physics for Scientists and Engineers – Lab 11 =
 * 1) Does the title fulfil the criteria given in section 7.2? If yes, explain your answer. If no, write an appropriate alternative title that does fulfil the criteria.
 * 2) Does the introduction satisfy the criteria given in section 7.4? If yes, explain your answer. If no, explain how the introduction can be improved.
 * 3) Does the procedure satisfy the criteria given in section 7.5? If yes, explain your answer. If no, explain how the procedure section can be improved.

Objective:

 * To test the exponential law of decay of a radioactive source, and to measure the half-lives and the decay constants of neutron activated silver and indium.

Equipment:

 * GM tube in stand (geiger counter)
 * Indium (In-115 95.7%, In-113 4.3%)
 * Silver (Ag-107 51.83%, Ag-109 48.17%)
 * 100 six sided dice
 * wire basket to hold dice
 * Pasco Signal Interface II, Science Workshop and Graphical Analysis software

Physical Principles:
 Many nuclear species are unstable and make transitions to other species by absorbing or emitting particles and photons of high energy, gamma rays. The number of nuclei dN that decay in a time interval dt is directly proportional to the number N of nuclei present at that instant and time interval dt is given by

math (1) \ \ \ \ \ \ d \,N \ = \ -\,\lambda \,N \,d \, t math

so that the decay rate dN/dt is given by

math \displaystyle (2) \ \ \ \ \ \ \frac {d \,N}{d\, t} \ = \ -\,\lambda \,N math

where λ is the decay constant. The value of λ differs for each radioactive nuclear species. Equation (2) is satisfied by the exponential decay relation

math (3) \ \ \ \ \ \ N \ = \ N_0\,e^{-\lambda \, t} math

where the half life τ is given by

math \tau \ =\ \ln(2)\,/\,\lambda \ = \ 0.693\,/\, \lambda. math

This follows from the definition of half life which implies

math N\ =\ N_0 \ 2^{-\, t \, / \, \tau}. math

After a time of four half lives N is reduced to 1/2 4 = 1/16 or .0625 of its initial value.

Nuclear detectors such as the GM tube give a measure of the number of particles emitted per second by a piece of radioactive matter. The number of particles detected each second, C, is proportional to the total number of nuclei N present. Substituting C for N in equation (2) gives,

math (4) \ \ \ \ \ \ \ \ C\ = \ C_0 e^{-\, \lambda\, t} math

where C 0 is the initial count rate at t = 0. When two radio-active species are present the count rate is given by

math (5)\ \ \ \ \ \ \ C \ = \ C_{01}\, e^{-\, \lambda _1 \, t} \ +\ C_{02}\, e^{-\, \lambda _2 \, t} math

If the half life of one species is much shorter than the other after about four half lives of the shorter lived species the remaining activity will be from the long lived species.

Neutrons that have low kinetic energies are readily absorbed by many nuclei thus transmuting them to an isotope with a mass number that is one unit larger. Neutrons are provided by a plutonium beryllium source that is located in a cylindrical tank filled with water. Energetic neutrons are slowed by collisions with the hydrogen nuclei in the water. The two naturally occurring isotopes of silver (atomic number 47) with mass numbers of 107 and 109 each absorb a neutron and become radioactive silver with mass numbers of 108 and 110. Similarly indium nuclei (mass number of 115, atomic number 49) each absorb a neutron to become radioactive indium (mass number 116).

1. Determination of the Half-life of Indium 116
 Plug the PASCO GM tube into the digital interface of the Science Workshop. Open the PASCO Capstone software, and indicate that a Geiger counter is connected. Open a table for Geiger counter data. Set the sample rate to 30 seconds. With no source near take a background count for 30 seconds and record this as the back ground count, BC. Ask the lab instructor to bring your indium sample and place it beneath the opening in the geiger counter. Open a graph of counts/time for data from the counter. Set stop time to 60 minutes and begin counting on the sample. Once you begin counting do not disturb the apparatus.

Data Studio allows curve fitting to be done while the data is being collected. After about 3 to 5 minutes, fit a linear graph to the data. Show, by replacing the e -λt in Eq. (4) by the series expansion (1 - λt + …) for small values of λt, that λ = the slope divided by the intercept of the linear graph. Determine λ from this graph. After more than 5 minutes, the small value approximation for λt no longer holds, and the data must be fit to a decaying exponential.

//While you are waiting for the 60 minutes of data collection to be completed, proceed to part 3, the dice simulation.//

When the hour of counting is completed, fit the data to an exponential fit of

math C=C_0e^{-\lambda t} - BC, math

where BC is the measured background count value and C 0 and λ are fitted variables. Given the fitted value of λ, calculate the half-life τ and compare to the accepted value of 54.0 minutes. Save the graph to include in your lab report.

2. Determination of the Half-lives of Ag 108 and Ag 110
 Change the Sampling Options to 5 seconds with a stop time of 420 s. As quickly as possible, bring the radioactive silver from the howitzer (within a few seconds) and place it as close as possible to the opening of the geiger counter.

Select the set of data points greater than 150 seconds, and fit to an exponential curve. The value of λ from this data corresponds to the half-life of the longer lived component, Ag-108, τ = 2.4 minutes.

Select the set of data points from 0 to 100 seconds, and fit to an exponential curve. The value of λ from this data corresponds to the half-life of the shorter lived component, Ag-110, τ = 24 seconds.

In order to improve accuracy in the half-life determination, fit the entire set of data to a “user-defined” double exponential curve (Eq 5, above). Type the values of λ 1 and C 01 for the longer-lived component into the function, and refit for the λ 2 and C 02 for the shorter-lived component. Compare these results to those from the previous fit. **Include copies of your three fits in your lab report.**

3. Simulation of Radioactive Decay Using Dice.
 Beginning with 100 dice, roll all of them and remove the ones that come up with a single dot. Record the number of dice that remain and roll them again. Remove the ones that come up with a single dot and again record the number remaining. Continue in this fashion until only a few dice are left. Open Graphical Analysis and make a graph of N vs. Trial number.
 * 1) Estimate the half life by determining how many trials it takes to have approximately 50 dice left (one half life), how many trials it takes to have approximately 25 dice left (two half lives), and how many trials it takes to have approximately 12 or 13 dice left (three half lives).
 * 2) Get a better determination of half-life by fitting an exponential function A 0 e -λt, where A 0 is the initial number of dice (100). Determine the half-life τ from the decay constant λ. **Include a copy of your graph in your lab report.**

Compare the measured half-life with the theoretical value. The theoretical value of the half-life in this situation where the probability that a given dice decays on a given throw is 1/6th. This is predicted to be

math \tau = 6 \ln(2) = 4.16 \ \mbox{trials} math

The experimental data from all groups will be compiled on the blackboard. Add all of the data together and repeat the half-life determination by fitting an exponential function to the combined data (part B). Compare the combined measured half-life with the theoretical value.