AMERICAN JOURNAL OF PHYSICS VOLUME 38 NUMBER 10 OCTOBER 1970 A Simplified Muon Lifetime Experiment for the Instructional Laboratory Department y P, R. E.HALL, D. A. LIND, AND R. A. RIsTINEN hysics and Astrophysics, University of Colorado, Boulder, Colorado 80802 Received 27 April 1970) An experiment has been developed for the measurement of the half-life of cosmic by using a single scintillator detector. The associated electronies are much simp nventional cosmic ray muon lifetime measuring systems. It is this simplifie lakes the experiment well suited for the instructional laboratory. I INTRODUCTION II. APPARATUS AND ITS OPERATION Lifetime measurements of cosmic ray muons are The simplified system is shown in Fig. 2. Its commonly performed with some type of multiple operation is initiated when a muon traverses a detector telescope. 1 a typical arrangement is portion of the plastic scintillator and comes to shown in Fig. 1. With this system counters 1, 2, rest. The energy lost in the stopping process and 3 have their output signals routed such that ionizes the plastie and gives rise to a pulse of the time-to-amplitude converter(TAC)is started light which is transformed into an electrical pulse only by muons that pass through l and 2 but not 3. and amplified by the photomultiplier tube. The Because of the geometry of this arrangement, a signal for the discriminator is brought directly tart signal to the TAC indicates that a charged off the anode of the photomultiplier tube by a particle has stopped in the absorber. If the stopped 50-02 cable and is proportional in amplitude to the particle was a muon, the subsequent decay into intensity of the light pulse. As a result, if the in an electron plus a neutrino pair sends a stop pulse cident muon traverses enough of the scintillator to the TAC, and an output pulse representing plastic, the electrical pulse delivered to the dis- the start-stop time is recorded by a pulse height criminator will cause it to trigger. The triggered analyzer(PHA). Since the half-life of a free discriminator sends out a 4-nsec wide pulse which muon i8 1.53 usee, corresponding to a mean life is then equally divided between two 50-4 cables of 2.21 usec, the chance of a decay occurring per which drive the start and stop inputs of the TAC. interval of time decreases by half after each 1.53 The cable to the start input of the TAc is ap- usec of time. Therefore, the TAC pulses stored proximately 8 ft longer than the cable to the stoy in successive channels of the PHA will represent input. In its normal state, the TAC is ready to an exponential decay curve. By plotting the accept only a start pulse. It ignores the stop pulse, successive channel accumulations on semilog- and the start pulse starts a timing run. The delay rithmic graph paper, a straight line of negative of the start pulse relative to the stop pulse by the slope is produced. The straight line nature of the 8 ft of additional cable assures that the stop input graph indicates that a radioactive decay phenom- is no longer activated when the pulse gets to the enon is indeed being observed, and by fitting a start input. Failure to include this delay time straight line to the semilog graph of the data would mean that the TAC would be unable to points, one may easily extract the half-life of the accept the start pulse. It should be noted that muon the 8-ft cable extension subtracts approximately While this experimental setup for measuring 8 nsec from the time to which each PHA channel the cosmic ray muon half-life is logical in arrange- would normally correspond. This does not change ment and has worked well for a number of ex- the shape of the spectrum stored in the PHA, perimenters with various modifications, it is rela- and the half-life obtained from the slope of the tively complex and expensive. In this paper we spectrum will be unchanged discuss the development of a simplified apparatus A muon that started the Tac timing run ma which yields essentially the same lifetime measure- be stopped in the scintillator and decay at, some ments as the multiple detector arrangement just later time. The neutrinos from the decay leave described the scintillator without any further effect. The
SIMPLIFIED MUON LIFETIME EXPERIMENT lectron leaves an ionization trail in the scintilla. Muon flux tor, and as a consequence, the decay of the muon Mu ti- Channel causes another input signal to go to the discrimi- nator. This pulse at the stop input of the TAC causes it to measure the time since the last start scintil pulse and to produce a bipolar output pulse of 2 sec width with an amplitude linearly propor- turn recorded by the PhA. The read out process Amplitude takes much longer than the propagation of the decay electron signal from the stop to the start inputs, and as a result, the corresponding start pulse will be ignored etup used for this experiment. The The only piece of equipment needed to per- follows: photomultiplier tube: RCA 7264; discriminator form the experiment which is not available as Ortec model 417; TAC: Ortec model 437; Multichannel standard unit is the seintillator-photomultiplier analyzer: Nuclear Data Corp. model ND]30A tube assembly. because the natural cosmic ray flux is not very intense, approximately 1x10-2 tape. Finally, an aluminum frame was constructed muons/cmsr sec at sea level, and about twice to support and protect this assembly and hold this value at the 5000-ft elevation of the Uni- the photomultiplier tube base versity of Colorado, the need for a scintillator After the components of the system have been with a large horizontal area is indicated. Also, connected, the PM tube high voltage, the dis since the energy spectrum of the muon flux criminator level, and the TAC time range must reaches a maximum in the neighborhood of 1 Bev, be adjusted. The high voltage was selected by the scintillator should be as thick as possible. setting the discriminator to its lowest threshold, These considerations and the availability of a about -0.18 V and then increasing the voltage large piece of scintillator plastic resulted in a applied to the PM tube until a reasonable trigger 68X24X23 cm rectangular block being cut for rate was obtained for the discriminator. At he apparatus. An RCA 7264 photomultiplier -2010 V, a trigger rate of 150/ sec was obtained tube was optically coupled to one of the small To obtain a clear decay curve, the discriminato ends of the scintillator block. The entire assembly setting must be raised. A discriminator setting was then wrapped with aluminum foil and black of -0.8 v was found to give the highest possible TAC output rate and yet provide data repre- senting a smooth decay curve Coincidence III. RESULTS Circuits A long experimental run was made for a period Time to of 695 h with a TAC range of 20 usec and a dis- Conolruder criminator setting of -0.8 V to see where the decay curve merges into the background which is due to discriminator pulses randomly occurring Absorb within 20 usec of each other. This same run was che muon. For the 695-h experimental run,a pulser capable of producing pulse pairs was used along with an oscilloscope to establish a time to Muti- PHA channel gain of 0.112+0.003 usec/ch Analyzer After 695 h, the spectrum of Fig. 3 resulted. Both FiG.I.A typical muon lifetime experimental setup. the muon decay curve and the background are Details of its operation are discussed in the text. clearly evident. Channels 160-180 were average
l198 HALL LIND AND RISTINEN Evaluation gives (104)2(1.12×107sec)(25×10°sec/ serch 30.2 background counts/ch This agrees with the observed background of 31±2/ch Since a 695-h run is almost 29 days long, a much shorter run would be necessitated by the schedules of most instructional laboratory courses A 24-h run was made with a pha channel width of 0.0329+0.0009 usec/ch. to determine the quality of the data to be obtained in this time Figure 5 represents the results of this run. The background is less than one count/channel and FIG 3. Data from a 695-h run before subtracting back- has been ignored. The short time span of the spec ground. The time width of each channel is 0.112=0.003 trum also reduces the significance of any back- ground. A least squares fit of a straight line to these data gives a half-life of 1.64+0. 15 usec to obtain a background of 31=+2 counts. Sub- For a run of this short duration a more accurate time base determination is not needed as the traction of this background gives the time spec- largest error associated with the half-life is due squares fit of a straight line to these data gives to statistical variations in the data a half-life of 1.46_+0.04 usec.(It should be noted It is clear that aside from improving the time ase calibration, the greatest improvement in the that the accepted half-life of the free muon performance of the experimental setup would fact be under the accepted value because about occur if the anal yzing rate to discriminator trigger 45% of the muon flux is u- and 55% u+ 2 The w- rate ratio could be improved. For the 695-h run, particles have a reduced half-life relative to u+ articles in matter since the u particles are cap- tured into atomic orbitals which allow intel actions with the nuclei of the matter. Thus, the observed half-life might be expected to be some- what less than that of free muons, with the exact reduction in lifetime determined by the unknown probability with which a nuclear capture of a muon results in a discriminator pulse. In this work, we have chosen to approximate what may in fact be a complex decay curve by a straight line exponential decay. The background that was subtracted from the 695-h spectrum results from the rather high dis- criminator trigger rate of Since the analyzing rate was only 0.01/sec over-all, almost I of these discriminator pulses were random. CHANNEL NUMath Theoretically, the background per channel of the 695-h spectrum should be FIG. 4. Data from a 695-h run after subtracting 31 counts of background from each channel. The time width of ach channel is 0. 112- -0.003 usec. A computer fit of a Background =(discriminator rate)(time straight line to the data points has been drawn in. Its width of channel)(analyzing time slope represents a half-life of 1.46 usec
SIMPLIFIED MUON LIFETIME EXPERIMENT 1199 it was approximately 1X10-3. It is low for the following reasons. First of all, the muon flux peaks in intensity somewhat above 1 GeV. This means that a large number of muons can easily pass through the scintillator which can only slow a muon down by approximately 2 Mev/emX 38 cm=76 Mev. In order to trigger the discrim inator with a-0.8 V setting, approximately Mev of energy must be deposited in the scintil lator. Consequently, almost all muons passing hrough the counter contribute to the trigger rate without contributing to the analyzing rate Also, those muons which stop in the scintillator FIG. 6. Momentum distribution of electrons or positrons may decay without providing a stop pulse to the produced by muon decay. Taken from Ref. 2 TAC if less than 10 Mev of energy is deposited in the scintillator in the decay process. The mo- plastic, which means that electrons of more than mentum of electrons or positrons produced by 10 Mev of energy can fail to trigger the discrimi- muon decay has a continuous distribution and nator if they are formed within 5 em of the side is described by Fig. 6. Since the neutrinos do not of the scintillator. Another very small contribu- interact with the scintilator, we need only be tion to the trigger rate and not the analysis rate concerned with the electrons. A small number of arises from those muons which enter the scintilla- the decay electrons produced in the scintillator tor with less than 10 Mev of energy. Only a decay will not have sufficient energy to trigger the dis- pulse can then be observed, and these particles criminator. Also, some of those with more than therefore contribute to the random rate 10 Mev of energy may escape the scintillator This information would indicate that the aver- before they deposit 10 MeV of energy. The depo- age muon encountering the scintillator will pro- sition of 10 Mev requires approximately 5 cm of duce one discriminator pulse and not two. In general, one would expect that the ratio of ana- lyzing rate to discriminator trigger rate would be increased if the scintillator were made larger. It would be interesting to examine the cor- respondence between the rate at which muons encounter the scintillator and the discriminator agger rate. The muon flux will be taken as 2X10- muons/cm sr see. Because the scintillator is oriented with its longest axis vertical and the muon flux is most intense near the vertical, the usable solid angle will be taken as 2 sr. The avail- able detector area is 23 X24 cm=552 cm2. TI discriminator trigger rate should be of the orde 2×10-2(552cm2)(28r) To obtain discrim FiG. 5. Data from a 24-h run. The time width of each of this value, the discriminator level must be channel is 0.0329: +0.0009 usec. A computer fit of a straight line to the data points has been drawn in. Its lowered to about -0 4 v. It is also at discrimina- tor levels below -0.4 to -05v that the time
1200 HALL, LIND AND RISTINEN spectrum becomes badly nonlinear. This would ACKNOWLEDGMENTS indicate that the discriminator is probably trig- The assistance of Mr Harry Clark in machining gering due to meaningless noise from the photo- the scintillator plastic and the help of several multiplier tube at these low settin other staff members in the development of this IV SUMMARY apparatus is sincerely appreciated This experiment performs well and is simple *Equipment provided in part by a enough to be used in an undergraduate physics lab. With some care given to least squares methods (Academic, New York, 1967) of data analysis and accurate time calibration, it B. Rossi, High Energy Particles(Prentice-Hall, is capable of producing research quality data. Englewood Cliffs, N.J., 1952) AMERICANJOURNAL OF PHYNICS VOLUME 38, NUMBER 10 Visual appearance of a Moving Vertical Line RAMESH BHANDARI Physics Department, Panjab University, Chandigarh, India (Received 2 March 1970; revision received 19 May 1970) a vertical line moving with velocity 1, when seen, assumes the shape of a hyperbola or para- bola accordingly as v c, which actually is not possible, the line takes the form of an ellipse or parts of an ellipse, or even becomes imaginary, depending upon the length of the line, the magnitude of the velocity, and the distance of the viewer from the line. If, however, a line moving with v<c recedes from the viewer, it seems less curved until at in finity it straightens up to look vertical again In this paper, I have dealt in detail with the of all points lying on the line, the light rays from of ng vertical line and the which, emitted at different instants(At), arrive subsequent changes that take place in it with the at P simultaneously. It is similar to the standard passage of time equation! of a hyperbola in the az plane, drawn Consider a line coincident with the 2 axis of a with the origin shifted to (rl, y ) To the viewer frame moving with velocity v as shown in Fig. 1. therefore, the line looks like a hyperbola with The light ray I from end 0 reaches the viewer's focus F=[y/(1+B),0] and directrix D given by eye fixed in the laboratory frame at(O, y, 0) after a time interval y/c. The ray II from A,on x=-8y/(1+β); the other hand arrives at P at the same time as y here is a constant. The eccentricity e is a funetion after being emitted at an earlier instant At. of v and is equal to 1/ From the figure, it is clear that DEPENDENCE OF APPEARANCE ON D c△+y=[(v△)2+y2+22]/ Atv=0, e=0o. F tends to(, 0) and the di- Multiplying both sides by v/c and squaring the rectrix coincides with the z axis. The hyperbol resulting expression, we arrive at the equation consequently, straightens up coinciding with the x2(1-B2)+2X6y-B22=0 2 axis. However, when the line moves with a finite velocity v<c, it assumes the shape of a hyperbola where X=vAt. This can be further put in the form and at v=c transforms into a parabola given by [(X+B2y)/(8?2y)2]-[Z/(y)2]=1.(3) Equation(3)is in fact the equation of the locus It is interesting to note that with further increase
