Computer simulation of radioactive decay y now you should have worked your way through the introduction to Maple, as well as the introduction to data analysis using Excel Now we will explore radioactive decay with the aid of Excel 1 Theoretical nalysis You are not required to hand in your work for this section, but do make the effort to understand it, since it will form the basis of the second section 1 1 ackground When a radioactive substance decays, the rate of decay is proportional to the amount of substance present at that time That is: dn = k N dt where N(t) is the amount of, in numbers of nuclei, present at time t The proportionality constant, k, is called the decay constant of the substance The decay constant and the half-life T are related by: ln 2 k = T The solution to the differential equation can be found by separation of variables, and is: N where o is the number of nuclei present at t = 0 = exp( k t) (1) o In many cases, the first radioactive element decays into another radioactive element, which then decays with its own decay constant, ie, C The rate at which the number of nuclei of the second element changes with time is given by: dn dn = k N dt dt where k is the decay constant of element This differential equation has the solution: k N = o exp( k t) + o [ exp( k t) exp( k t) ] k k (2) Where o is the number of nuclei of substance present at time t = 0 Radioactive decay - 1
12 asic plotting Let s play around with equations (1) and (2) Start Excel and enter the following into the spreadsheet: C D 1 0 = 100 T_ = 40 2 0 = 0 T_ = 30 3 4 Time N_ N_ ln(n_) 5 0 6 1 205 200 To put in the numbers 0200 in column, type 0 into cell 5, and [Enter] Single-click cell 5 with the mouse, then go to the Edit menu, and select Fill and Series In the dialog box that comes up, choose columns in the Series in field, and specify the stop value to be 200 (make sure that Type is linear, and that Step is 1 ) The entries in cells 4 to D4 are titles for their columns In cell 5, enter the formula =$$1*EXP(- LN(2)/$D$1*5) To carry over the formula to the rest of the column, click on the bottom right corner of cell 5 when the mouse pointer turns into a black + sign, and drag the mouse straight down the column until cell 205 The cells calculate their values via the formula using the value in the column in the current row In cell C5, enter the formula =$$2*EXP(-LN(2)/$D$2*$5)+$$1* $D$2/($D$1-$D$2)*(EXP(-LN(2)/$D$1*$5)-EXP(- LN(2)/$D$2*$5)), and then carry over the formula to the rest of column C to cell C205 in the same way as for column In cell D5, enter the formula =ln(5), and carry over this formula to cell D205 13 Validity of operations In cell D2, change the value from 30 to 40 to equalize the values of T and T The spreadsheet automatically updates all references to cell D2 What happens? Notice that column fills with #DIV/0! Why does this happen? 14 Plotting the Curves Return the value of 30 to cell D2 Now highlight the data in the Time, N and N columns (4:C205) When doing so, include the cells with the Time, N_ and N_ labels, as these will serve as labels in the plot that Excel creates With the data highlighted, click the Chart Wizard button on the toolbar: this is a button with three colour bars (blue, yellow, red from left to right) in what appears to be small bar chart dialog box will come up, asking you to select a type of chart: choose XY (Scatter) and click Next> The range of data being plotted and a preview of the chart is shown: if it is Radioactive decay - 2
not correct, change the data range t this point, you can also change the name of the series of data being plotted by clicking on the Series tab and entering a new series name: this new name will appear in the chart legend The series you have should currently be called N_ and N_ in the series list, in which case no changes are necessary If this is not the case, however, make it so Once you are satisfied with the data range for plotting and the series names, click Next> You now have the opportunity to enter axis labels and a chart title, modifying the properties of the axes, gridlines, legend, data labels and data table For a chart title, enter mount of radioactive substances and as a function of time For the x-axis title, enter Time (s), and for the y-axis title, enter N_(t), N_(t) Make other adjustments as you wish, and click Finish when done Have a look at the behaviour of the curves plotted Observe that the amount of substance decays away exponentially, as expected, but the amount of substance rises before decaying Depending on the half-lives, there may be more of substance than of substance at large times Solve equation (1) to determine the expected value of t when N (t) = 36 Now look down columns and in your spreadsheet to determine the time value in column in the same row as when column has an entry very close to 36 Compare these two values of t Do they match? Now look again at your spreadsheet, and find the time value in column when the value in column is 50 You should find that this time value is 40, equal to the half life of substance Why should this be the case? 15 Logarithmic plot You have a third column of data that has not been plotted yet Highlight the data contained in cells D4:D205, and click on the Chart Wizard button Go through the steps of making an XY (Scatter) chart Note that you need to set the x values range as 5:205 under the Series tab in step 2 The data series in the legend should be called ln(n_) Give the chart a title of Natural Logarithm of N_ versus time You should observe that the data are linear as a function of time: this is confirmation that the N data are exponential in time What does the slope of the graph tell you? 2 Data nalysis In this section, you will be analyzing a more realistic data set which contains some degree of randomness You are required to hand in your work for this section Have a look at the list of expectations for the lab write-up following this section The data you are to work with is in the file decaytxt, which can be found on the course webpage (wwwphysicsuoguelphca/~omeara/phys2470) First download this file from the website by right clicking on the link and selecting Save Target s Now open this file in Excel, as you did with the fallballtxt file in the previous spreadsheet activity This file contains three columns: time, amount of substance and the amount of substance No units are given for the amounts of substance; you may Radioactive decay - 3
assume that time is in seconds We will import this data into Excel to plot, and observe that the data does indeed look like the theoretical curves First, however, we should consider the uncertainty inherent in the data 21 Poisson statistics s you will learn in the error analysis section of PHYS*2440, events which occur at random, but at a definite average rate, are governed by Poisson statistics We will not go into detail here, but will discuss only the result concerning uncertainties In a counting experiment such as radioactive decay, each data point is simply the number of events per unit time The uncertainty on this value can be shown to be the square root of the value (see Taylor, p249, 2 nd ed) Thus, for example, if we counted 50 decays in one second, the uncertainty on this value would be 50 7 Plot, but do not hand in, a graph of the data for substance with y error bars The sizes of the error bars can be calculated in your spreadsheet after you have imported the data Then make an XY (Scatter) plot in the usual way When the plot is complete, click on the special icon CHRT tools, choose LYOUT, then Error ars, followed by More Options You now get a dialog box with a tab called Y Error ars: click on this tab, and then select Custom For the + error bar, click on the red arrow there, then highlight the column of error bar values you just calculated in the spreadsheet, and finally click on the red button of the dialog box Do the same for the - error bar, or just copy the data range showing in the + error bar field into the field for the - error bar When done, click Ok on the dialog box, and y error bars appear in the chart If we had x error bars to worry about here, we would have done the same for the x errors Note that the facility for placing both x and y error bars is only available with XY (Scatter) charts 22 nalysis You are to find the decay constant k for substance, and its associated error You can do this by fitting a line to the natural logarithm of the decay data ecause the uncertainties are not uniform, you should use weighted linear regression See the appendix in your lab manual for the weighted linear regression formulae These are easily implemented using a spreadsheet Note that when you are doing weighted linear regression, your x values are time and your y values are ln(n ) You will need to calculate the latter in your spreadsheet Further, the weighted linear regression formulae require the uncertainty on each of the y values This uncertainty is not N, since your y values are ln(n ) You need to use error propagation to calculate the uncertainty on ln(n ) That is, given that we know N and that σ(n)= N, what is σ ( ln( N ))? (s a hint, you will find the beginning discussion in section 6 of the Maple tutorial you went through earlier to be helpful) What is the meaning of the slope and intercept you obtain from the weighted linear regression? Plot the data for ln(n), with error bars, and the fitted line on the same graph To accomplish this, you should generate data points from your calculated slope and intercept of the fit line, and plot this together with the original data Plot the fitted values as an XY scatter chart with no markers for the individual data points (do this by right- Radioactive decay - 4
clicking on your fit points, and select Chart Type choose the bottom right-hand box in the sub-type section for XY scatter charts) re most of the data within two standard deviations (ie, two error bar lengths) of the fitted line? Hand in this graph along with your discussion Radioactive decay - 5
Informal Lab Report Expectations Maximum report length: 5 pages on 85 by 11 sheets, smallest font = 12 pt (regardless of whether you type or hand write) Please follow the organizational format given below Section 1 State the equation for radioactive decay of a single substance Explain why we expect the natural logarithm of the data to be linear in time; state the equation for the natural logarithm of the data vs time Section 2 Results and nalysis Give error propagation calculation for σ (ln( N )) Show weighted linear regression calculations: Give sample of all columns of spreadsheet calculations (it s not necessary to give the entire spreadsheet, just a few rows to sample them) Include also the summary calculations: the sum of each column,,,, σ and σ If space permits, it will be useful to hand in two versions of the spreadsheet: one showing values and the other showing formulae To show formulae in Excel, go to the Tools menu and select Options In the Options dialog box, click on the View tab and select Formulas under Window Options Deselect this to show the values State slope and intercept with errors, and proper significant digits Include plot of data: Show all data points, with error bars, plotted together with the fitted line Include axis labels and units on the axes, a meaningful legend, and a meaningful title summarizing the graph content and quoting the fit coefficients and with their uncertainties Section 3 Conclusion Do the data behave as expected? (ie is ln(n ) vs t linear?) Does the line of best fit actually fit the data well? How well? What is the meaning of the slope and the intercept of the fit line? Is there systematic error anywhere? Radioactive decay - 6