and N(t) ~ exp(-t/ ),

Transcription

1 Muon Lifetime Experiment Introduction Charged and neutral particles with energies in excess of ev from Galactic and extra Galactic sources impinge on the earth. Here we speak of the earth as the terrestrial mass and its captive atmosphere. These cosmic rays are mostly hadrons which undergo strong interactions and therefore interact in the earth s atmosphere producing showers of charged and neutral particles that in turn interact resulting in many charged mesons with short lifetimes that eventually decay yielding many high energy muons. At the earth s surface we are bombarded by muons of positive ( ) and negative ( ) charge at the rate of about one particle per square centimeter per minute. Muons belong to a class of particles called leptons. They are spin ½ particles with a rest mass m0 = ± MeV or about 200 times the mass of an electron, which is also a lepton as is the much heavier. The muon lifetime is measured to be = ( ± ) x10-6 s. These charged members of the lepton family do not interact strongly. They have only electromagnetic and weak interactions and that is why we observe them at the earth s surface. The stopped muon will decay into an electron or positron depending on the initial charge of the muon and two neutrinos. For example and with a distribution of decay times given by N(t) ~ exp(-t/ ), The bar over the neutrino symbol indicates it is an antineutrino. Where N(t) is the number of decays between t and t+dt. The measurement of the lifetime is complicated by the fact that negative muons can be captured by a nucleus via the following reaction:, with considerable energy carried off by the neutrino. In this case the nucleus is left in an excited state, and the muon absorption is followed by gamma and/or beta decay from the nucleus, which can be sensed by the detector. This additional channel for negative muons has the consequence that the measured lifetime can be significantly shorter than the measured lifetime. The effect of nuclear absorption is strongly dependent on the atomic number Z. For higher elements, the absorption probability is roughly proportional to. This rule may be explained by the mechanism of absorption, where the trapped muon first enters a K shell of the atom, and then the overlap of the muon wave function with the protons in 1

2 nucleus governs the absorption probability. Positive muons cannot enter into an electronlike orbital state around a nucleus, and so they do not get absorbed. Since we do not have a magnetic field to bend positive and negative particles in opposite directions to identify the charge we cannot tell if a given muon stopping in our detector is a positive or negative muon. The ratio of positive to negative cosmic muons striking the earth s surface is N( Exercise 1. Given their very short life time and their origin far from the earth s surface how do you explain that we observe so many muons at the surface of the earth? (Hint, most of the muons are produced with ~ 1). To illustrate your answer, calculate the life time of 10 GeV muon as observed in the earth s reference or rest frame. Discuss the result you obtain in your lab report. Experimental Method There are several different arrangements of scintillation detectors that can be used to measure the muon lifetime with cosmic ray muons. In the laboratory you will find a muon telescope that consists of the three scintillator counters (A,B,C) and a slab of absorber arranged as shown in figure 1. Figure 1: Schematic view of experiment. The procedure is to use a NIM logic unit to define an incoming muon that stops in the absorber. We demand a coincidence of signals from the A and B scintillation counters and no signal from scintillation counter C (A AND B AND NOT C) followed by a subsequent decay electron that traverses scintillation counter C within a time interval of 10 microseconds. 2

3 The scintillation counter pulses are sent to discriminators and the discriminator outputs go to a logic unit that forms the logical signal ABC that is the "start" signal for the Time to Amplitude Converter (TAC). The electron/positron emitted when the muon decays traverses scintillation counter C with a delay that depends on the lifetime of the muon. This signal is used to stop the TAC by taking a separate output from the C counter discriminator and connecting it to the TAC stop input. To summarize: the clock (TAC) is started by ABC and stopped by the signal from C when the electron or positron passes through C. Using coincidences reduces the sensitivity to noise and the use of a C-veto enables you to get a cleaner sample of stopping muons and decay electrons. A B Absorber C Figure 2: Photograph of the muon lifetime paddle array MUON1. The absorber is simply another slab of scintillator material. NOTE: The use of the veto from scintillation counter C selects stopping muons and prevents us from triggering on radiation associated with the muons that appear within a few nanoseconds after the muon stops. Thus a veto pulse must occur in the logic unit a little before the AB coincidence and should extend beyond the duration of the AB overlap time about 10 ns or so. To accomplish this, the discriminator pulse width for C is set wider than the pulse widths for A and B, and delays are added to the A and B channels to insure that a veto from C arrives earlier at the logic unit. 3

4 Setting up the Counters and Electronics The apparatus and the position of the scintillator detectors are shown in Figure 2. High voltages and thresholds need to be set to count cosmic-ray muons and decay electrons efficiently while avoiding phototube noise and low-energy background radiation to the extent possible. In this apparatus between 1900 and 2200 volts (depending on the detector) has been shown to work well for the high voltage bias. The PMTs in this experiment are 12 dynode units that need a significantly higher voltage than the 10 dynode types used elsewhere in the lab. A first task will be to choose the discriminator levels and adjust the relative timing of the signals by using appropriate cable lengths between the PMT signal outputs and the discriminator inputs. Important: The PMTs used in the muon lifetime apparatus require NEGATIVE voltages applied to the bases (most PMTs used in the other experiments require positive voltages). Using three long lengths of 50 ohm cable) that are the same length (about 2m) connect the signal outputs of the PMTs to three inputs of the 4 channel digital oscilloscope, Tektronix MSO2024. Don t forget to terminate the scope inputs with 50 ohm terminators. Set the recommended voltage for the given PMT, and observe the outputs of each scintillating detector to endure all are working. Decide how to set the scope controls to be able to observe pulses of about 50 ns width with amplitudes of a few hundred millivolts. The TA or instructor can demonstrate some features of this particular scope that are different from the old analog scopes. Now place a 60 Co source in the middle of the rectangular region of the top paddle (A). Set the oscilloscope to trigger on the signal from paddle A. Use normal triggering (not auto ), negative slope, and adjust the trigger level to be at the (negative) voltage of around 100 mv. One feature of the MSO2024 scope is that it shows the rate of triggering events as a frequency displayed on the small information boxes at the bottom of the display. One box gives the volts/div of each active channel, another box the sec/div of the horizontal axis (also called the timebase ), and one box that gives the trigger source, trigger slope, the trigger level and the trigger rate. Note that the trigger rate increases as the trigger level approaches 0 volts. The trigger level should be set so that the trigger frequency is a few 10s of khz (i.e., khz). Slowly vary the trigger level from close to 0 volts to larger negative voltages and note at which point the trigger rate falls rapidly. This will occur at the signal amplitude that is close to the Compton edge, and is proportional to the energy of the 60 Co gamma rays. Record this value. Based on the response to the 60 Co source, estimate the pulse heights expected from passing muons and from the electrons emitted by a decaying muon that is at rest. To estimate the pulse height from a passing muon, take into account the range and de/dx or energy loss per unit length of the muon traversing the scintillator. A relativistic particle ( vc 1) will lose approximately 1.5 MeV /[gm/cm 2 ] as it traverses a mass-thickness x of material, where is the density of the material in units of gm/cm 3 and x the thickness. 4

5 Material properties of the plastic scintillator are available in the lab or online and you can measure the thickness of the paddles with a ruler. Question: For the decaying muon, would you expect the electron to have a unique energy? Note that the muon decays into three particles: an electron or positron and two neutrinos. Discuss this in your report in detail, including calculations, explaining clearly any assumptions that you make. Disconnect the cables from the scope inputs and connect each one to a discriminator input (so that each PMT output feeds directly into a discriminator). Connect cables to the discriminator outputs, and observe the output of each discriminator on the scope using the same lengths of cable for each channel. Set the discriminator output widths to 40 ns. You may need to adjust the discriminator threshold levels in order to see good pulses. Using the test point on the discriminator set the discriminator threshold for paddle A to be a bit under the level that found for the Compton edge of the 60 Co above. For example, if you found that the Compton edge corresponded to a trigger level of 110 mv, set the discriminator threshold between 100 mv and 90 mv. (Remember that the test point voltage is 10 times the actual discriminator level.) You will also need to adjust the scope to trigger on the square pulse produced by the discriminator (typically a negative voltage of a few hundred mv). Make a survey of the counting sensitivity of paddle A by mapping the trigger rate at various locations of the 60 Co source placed on top of the scintillator paddle. Measure and/or estimate the active area of the plastic scintillator in paddle A. Note in particular the boundary between the active scintillator and the Lucite light pipe that connects the scintillator to the PMT. Between 9 and 12 separate regions on the paddle should be sufficient. For your report. Make a table and map of the scintillator paddles showing the counting rates at different locations. Comment on any variations in counting rate you observe and discuss why there might be such variations. Estimate the effective area of the paddle from the measured area and variation in counting rate. Setting discriminator levels for muon counting After mapping your top paddle, reconnect the PMT outputs directly to the oscilloscope channels (terminated, of course). Remove any radioactive sources and put them away in storage; you will not need them anymore. The signals you see will be from cosmic rays, background radiation and PMT noise. Your goal in this step is to find discriminator levels that minimize the counts coming from background radiation and noise. Based on your calculation of the energy loss of a passing cosmic ray muon and the discriminator level you chose for the survey of paddle A, estimate the discriminator level you would use for picking out muons above the noise. Use this value as the trigger level on the oscilloscope channel connected to paddle A. If your estimate is good, you should see two things: (1) the trigger rate will be in the neighborhood of 100 Hz (i.e., between about 50 Hz and about 300 Hz); (2) many simultaneous pulses: when you see a pulse from paddle A, you also see pulses from 5

6 paddles B and C. These simultaneous pulses are from muons passing through all three paddles, and picking these out is the goal of setting the discriminator levels. If you do not see the above two features, adjust the trigger level up or down until you do. Note and record the trigger level that works. This will be your discriminator level for that channel. Also note and record the trigger rate at that setting. You will want to compare that rate with the rate you get from the discriminator output later. Repeat the above process by changing the trigger source to the other two paddles: select the trigger source to be that paddle of interest, and adjust the level until you see mostly simultaneous pulses and a rate of around 100 Hz. Record the final levels and their rates. (Note: one feature of the MSO2024 scope is that the trigger level associated with a particular source is saved if you switch the trigger source between two inputs, the trigger level changes accordingly. This is especially convenient for comparing the signals between channels.) Now, connect the PMT outputs to the discriminator inputs, and connect the discriminator outputs to the scope, as you had done earlier. Then use the discriminator level test points and the digital multimeter to dial in the discriminator levels so that they match the trigger levels that you recorded for each PMT. Adjust the trigger levels on the scope so that each channel will trigger on the square pulses produced by the discriminator. Then compare the trigger rate that you get when triggering the scope from the discriminator output with the rate you recorded from triggering on the PMT directly, for each channel. They should be about the same. If they are not, make fine adjustments to each discriminator threshold until they are. It is not uncommon for there to be some differences; and the test points are only accurate to about 20% of the actual level. Record the final levels you are using and compare them with the previously used values, which should be posted on the paddle array. If your values are a lot different, you should check with the TA or instructor to make sure there is not a problem. Detector Timing As noted earlier, cable should be added to the outputs from paddles A and B to delay their signals relative to C in order to make sure that a simultaneous pulse in C does not start the TAC clock. To establish accurate timing between the counters first look at all three discriminator outputs on the scope. Fine tune the widths of the pulses from paddles A and B so that they are as close to the same as possible, at 30 ns. Then note whether one channel appears to start consistently later than the other. If you see this, estimate how much delay there is between them, and add an appropriate length of cable to the output of one or the other to bring them into sync. The pulse from the discriminator from paddle C needs to be set considerably wider than the pulses from paddles A and B. Trigger the scope on the signal from paddle C and adjust the width to about 70 ns. 6

7 Finally add cable lengths of 16 ns to the outputs of paddles A and B, and look at the outputs of all three discriminators on the scope. You should see the pulse from paddle C appear before the pulses from A and B by about ns and last past the end of these pulses by about the same amount. Wiring the Logic Use the LeCroy 365AL logic unit to create a pulse when you get the condition A AND B AND NOT C. There are two ways to get the NOT condition: (1) by using the veto input on the logic channels or (2) by using the complement output from the discriminator. Either way will work, but the veto method uses two-fold coincidence and the complement method other uses three-fold coincidence. The output of this logic will be the START pulse for the TAC. The STOP pulse to the TAC comes simply from paddle C. The output from the TAC will feed into the input of the pulse height analyzer. Note: be careful in hooking up the TAC a common mistake is to misread the labels on the front panel. Before you start collecting data, you need to calibrate the TAC and pulse height analyzer. Calibrating the TAC and pulse height analyzer The data set in this experiment is the frequency distribution of time intervals, ti, between the ABC start pulse and the follow-on pulse in C from the decay electron. A standard procedure is to convert a given t to a voltage pulse V( t)= t, where α is a constant of proportionality, and record the distribution of V( t) with a pulse height analyzer. Thus we need a device that converts a time interval to a voltage pulse and does so linearly. The TAC, (time to amplitude converter), which is used for this is designed to convert times between a start and a stop signal to a voltage that is proportional to the time interval. To be able to use the recorded data it is necessary to determine the constant by calibrating the time scale (channels per microsecond) using a calibrated pulse generator. This is accomplished using the EG&G model 9650 Digital Delay Generator that generates a pulse on one output (start pulse) followed by a second pulse on a different output after a preset time interval. First, determine out how the pulse generator works. Connect a BNC cable to the T0 output and a second BNC cable of identical length to the A output and connect the other ends of the cable to the two oscilloscope input channels. These pulses are fast NIM signals and need to be properly terminated. Triggering on the T0 output, note how the position (t) of the leading edge of the A output channel changes as the settings on the A readout are changed. (You should see something like A ns. To change the delay, use the + or buttons together with the cursor switch.) Second, connect the T0 output pulse to the TAC Start input and the A output pulse to the TAC Stop input. Setting the oscilloscope to trigger on a positive-slope trigger, observe the TAC output pulse. Note how the square positive pulse from the TAC changes its height (voltage) as you change the A-output delay. The range selectors are used to 7

8 select the time delay and the output voltage depends linearly on the delay setting. The maximum output voltage of the TAC is 10 volts. With the range selector set at 100 ns and the multiplier set to 100, a 10 microsecond difference between the start and stop pulses would make a TAC output pules equal to 10 volts. Now, connect the TAC output to the pulse height analyzer and take some calibration spectra. The pulse generator has very stable output pulses, which can be verified by looking carefully at the pulse height spectrum. Confirm this by noting the distribution of channels corresponding to a fixed time delay. Take measurements while changing the A delay to cover the range of times you need. It is possible to (and advisable) to get a complete calibration spectrum in one data set. Record the value of each time delay so you can make a calibration curve (channel number vs. Δt). Data Taking As soon as you have the calibration data, make the appropriate connections from the logic unit and discriminator to the TAC and begin the experiment (collecting data). It is helpful to connect some additional counter/scalars to verify that your trigger is working well. Using the Ortec model 872 Quad Counter/Timer or a pair of smaller counter/timers you can keep track of the following: Real time The Ortek 872 has an internal clock that you can look at on Channel 1. Using a 0.1 second time base, you can record the overall counting time. If using one of the other counters, you will need to take the real time from the MCA readout. Valid conversions On the back of the TAC there is a TTL-level output that goes high whenever the TAC receives a start pulse and a subsequent stop pulse within the time window set by the range switches. If using the Quad counter, you will need to convert these pulses to negative fast NIM with a TTL-NIM converter first. When using one of the other counters, plug the TAC s Valid Conversions output into the Positive counter input. Valid starts Keep track of valid A+B+NOT-C (aka ABCʹ) pulses by running an additional line from the logic unit output to a counter input. C singles rate In order to estimate the background counting rate, keep track of the singles rate from scintillation counter C by using another counter input. Start pulse height analysis and reset the counter simultaneously. Then the total numbers of counts should track with the valid conversions rate. After starting data collection, watch the analyzer and counter/timer unit to see if everything is working as expected. After a little while you should see the valid conversions tending towards about one count every 12 seconds with a few more counts beginning to pile up near the shorter time interval end of the PHA scale. The main data collection should occur over a 5 6 day period; before you leave be sure that everything is in order and that the MCA is collecting data and will not stop on its own, etc. Post signs that data taking is in progress (do not touch apparatus!) so that no 8

9 one will stop your experiment, erase the data, take cables etc. Each group needs to have someone stop by each day to verify that the experiment is working properly. At the end of the week one or more designated members of the group need to turn off the triggers (stop the experiment) and save the data stored in the multi channel analyzer. When you have recorded the data, leave the apparatus in the same state it was when you began. Items to record Make a rough sketch of the geometry of the set-up showing relevant dimensions and a diagram of the electronic logic. High voltages and thresholds need to be recorded. You should have enough information so that you can easily set up the experiment again if it were necessary and give a good description of apparatus in your lab report. Record the rate of valid conversions, the rate of the ABCʹ coincidences and the singles rate of the C counter. These numbers are needed to estimate the rate of accidental triggers. Discuss in lab report: Does the rate of valid starts agree with what you would expect, given the known flux of muons at sea level? The integral intensity of vertical muons with energy above 1 GeV at sea level is approximately 70 sr 1 cm 2 min 1, which gives about one muon per cm 2 per minute in a horizontal detector. Data analysis An important part of this experiment is the data analysis. You are expected to do some of the analysis on PCs or MACs and to present graphs and results in your lab report along with comments about sources of systematic and statistical uncertainty. Begin the analysis by making a quick fit of the data set using the analysis dialog box on the LabVIEW Norland Interface. This will fit the data to an exponential plus a background term and report the reduced chi-square for the overall fit as well as the uncertainties for each of the parameters. However, the results are quoted in terms of channel numbers. Note: take care to exclude the first few (dead) channels in your fit. This plot is also very useful for estimating the background that can be compared with the background you obtain when you fit the data to the expected time distribution given below. If this step is omitted there is no way to check if the results of your fit are reasonable. For a more thorough analysis, sum successive bins (a spreadsheet works well for this) to improve the statistical error of each bin as well as making it more convenient to plot the results. Approximately fifteen to twenty bins makes a nice plot. Take care to choose the appropriate time to associate with each summed bin. From this plot you can make a fit to the expected time distribution, where B is the background term, N(0)is a normalization parameter and τ is the mean lifetime. This can be done using any of the well known analysis packages that exist such 9

10 as Mathematica, Maple or KaleidaGraph. (KaleidaGraph is available on the lab computers.) Be sure to include the error on each bin in your fit. Fit your data to the above function and report the results as τ στ and B ± σb, that is, the values and standard deviations, obtained from your fit procedure. In addition, you should to report the χ 2 and discuss the goodness of your fit. To discuss clearly in your report 1. Is it better to fit the background value or to estimate it from a graph and leave that estimate fixed in your fit (try both)? 2. Is the background approximately consistent with expectations from accidental coincidences? To answer this it is necessary to calculate the accidental start pulse rate and the expected time distribution of this background source. (Note: the background used in the fit can be fixed in the Norland Interface analysis set-up dialog.) 3. Are there any sources of background that do not have a uniform time distribution? 4. In discussing the sources of uncertainty the following effects need to be considered: calibration, pulse height variation, random background, muon absorber (stopping material), electron energy spectrum, negative muons, muon travel time from the top of atmosphere, muon stopping time, cable delays, phototube transit times, etc. Which of these, if any, cause a systematic shift in the results and how big an effect is it? Revised October 2015 Prepared by H. Lubatti and D. Pengra 10